Filtering N.M.R. Spectra Citation Petersson, George Arthur (1970) Filtering N.M.R. Spectra. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/b9m3-qs59. https://resolver.caltech.edu/CaltechTHESIS:08252015-093052674 Abstract Methods of filtering an n.m.r. spectrum which can improve the resolution by as much as a factor of ten are examined. They include linear filters based upon an information theory approach and non-linear filters based upon a statistical approach. The appropriate filter is determined by the nature of the problem. Once programmed on a digital computer they are both simple to use. These filters are applied to some examples from 13 C and 15 N n.m.r. spectra. Item Type: Thesis (Dissertation (Ph.D.)) Subject Keywords: (Chemistry) Degree Grantor: California Institute of Technology Division: Chemistry and Chemical Engineering Major Option: Chemistry Thesis Availability: Public (worldwide access) Research Advisor(s): McLachlan, Andrew D. (advisor) McKoy, Basil Vincent (advisor) Roberts, John D. (advisor) Thesis Committee: Unknown, Unknown Defense Date: 22 January 1970 Additional Information: Title varies in the 1970 Caltech commencement program: Filtering Nuclear Magnetic Resonance Spectra Funders: Funding Agency Grant Number NIH UNSPECIFIED NSF UNSPECIFIED Caltech UNSPECIFIED Record Number: CaltechTHESIS:08252015-093052674 Persistent URL: https://resolver.caltech.edu/CaltechTHESIS:08252015-093052674 DOI: 10.7907/b9m3-qs59 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 9115 Collection: CaltechTHESIS Deposited By: INVALID USER Deposited On: 25 Aug 2015 17:28 Last Modified: 20 May 2024 22:33 Thesis Files Preview PDF - Final Version See Usage Policy. 42MB Repository Staff Only: item control page

FILTERING N. M . R. SPECTRA Thesis by George A. Petersson In Partial Fulfillment of the Requirements For the Degree of Doctor of Philosophy California Institute of Te.c lmology Pasadena, California January 22, 1970 ... ii To Smiley - Who has shown me the deeper truth which is beauty. iii ACKNOWLEDGMENTS I am deeply indebted to Professor Andrew D. McLachlan, Professor Vincent McKoy, and Professor John D. Roberts for the patience each has shown while he was my research advisor . I am also grateful to my friends--both students and faculty - -for providing an intellectually stimulating and thoroughly enjoyable environment during the past several years. I am particularly grateful to Dr. Robert L. Lichter and Professor John D. Roberts for their help in clarifying some of the more obscure passages in this thesis. Finally, I am indebted to the National Institutes of Health, the National Science Foundation, and the California Institute of Technology for providing financial support for myself and for the research described in this thesis. iv ABSTRACT Methods of filtering an n. m. r. spectrum which can improve the resolution by as much as a factor of ten are examined. They include linear filters based upon an information theory approach and non-linear filters based upon a statistical approach. The appropriate filter is determined by the nature of the problem. Once programmed on a digital computer they are both simple to use. These filters are applied to some examples from 13 C and 15 N n. m. r. spectra. v TABLE OF CONTENTS Page 1. INTRODUCTION 1. 1 Matched Filters 2. THE THEORY OF LINEAR FILTERS 2 3 7 2.1 Line Width 8 2.2 Convolution Integrals and Fourier Transforms 15 2.3 Generalized Harmonic Analysis 18 2.4 Optimum Resolution Enhancement Filters 27 2.5 Numerical Fourier Transforms 32 2.6 RC Filters 43 2.7 Sequences of Filters 46 2.8 The Determination of Signal-to-Noise Ratios 49 3. RESULTS OBTAINED WITH LINEAR FILTERS 53 3. 1 Optimum Line Widths of Weighting Functions 54 3.2 Resolution Enhancement 57 3.3 Line Shapes Resulting from Resolution Enhancement 74 3.4 Signal-to-Noise Ratios 79 3.5 The Effects of Sweep Rate 87 4. NON-LINEAR FILTERS 104 4.1 The Principle of Maximum Likelihood 104 4.2 Least-Squares Filters 111 4.3 Comparison of Linear and Non-Linear Filters 120 vi 5. RESOLUTION OF SATELLITE PEAKS 131 5. 1 The Least-Squares Approach 132 5. 2 Least-Squares Fitting with Non-Linear Parameters 135 5. 3 Filtering the Satellite Peak 141 5. 4 The Pseudo-Linear Approach 149 5. 5 Spectral Analysis Using Satellites 166 ERROR ESTIMATES 170 6. 7. APPLICATIONS 8. 1 73 7. 1 Signal-to- Noise Enhancement 173 7. 2 Resolution Enhancement 1 75 7. 3 Filtering Satellite Peaks 1 78 EXPERIMENTAL 188 APPENDIX A: Using the Program SNARE 190 APPENDIX B: A Listing of the Program SNARE 204 APPENDIX C: A Semiempirical Calculation of Spin-Spin Coupling Constants 217 REFERENCES 226 PROPOSITIONS 231 vii Copyright© by GEORGE ARTHUR PETERSSON 1970 1 FILTERING N. M. R. SPECTRA 2 1. INTRODUCTION A nuclear magnetic resonance spectrum is a plot of the intensity of out-of-phase absorbed radiation as a function of the frequency, JI. We wish to find one or more signals (resonance lines) in a background of noise (random fluctuations in intensity) . A signal is a set of intensities which vary in a well-defined manner given by g(J1), the line-shape function. 1 Noise is essentially all other varia- tions of the intensity with frequency . Since the sweep rate is held constant, the frequency is proportional to the time: JI = at, (1. 1) where a is the sweep rate. We shall find it convenient to consider the signal a function of time s(t). A filtered or transformed spectrum is obtained from the unfiltered spectrum by a process which can be described by an operator H: (1. 2) where fi (t) is the input spectrum and f0 (t) is the output spectrum. Filters may be divided into two general types ; linea r and non-linear. A linear filter satisfies the condition: (1. 3) 3 while a non-linear filter does not. We shall show applications for both types. Linear transformations are more convenient to use since it is not necessary to know the multiplicity of overlapping peaks in advance. Ernst has written an excellent article 2 on filters. We shall partly follow his development of linear filters, but obtain a result for a general line shape instead of restricting ourselves to even functions. First, we shall derive a matched filter from a simple statistical approach. This will give us a feeling for the nature of a filtering process. Later, we shall show that this filter is a special case of a linear resolution enhancement filter. 1. 1 Matched Filters We shall calculate the most probable intensity of absorption at the time t, given the input spectrum, fi (t). The line-shape function, s(t), will typically have the form shown in Figure 1. 1. We shall define the zero of time as the time when s(t) is a maximum. Consider an absorption f(t) with a maximum at the time t 0 , and intensity f(t0 ). The simplest method of measuring f(t0 ) is to simply observe the intensity at t 0 : (1. 1. 1) In fact, however when we record an n. m. r. spectrum we measure f(t 0 ) each time we measure fi (t 1) at any time t 1 : 4 -• ~ 4;I .... Cf) 0 t Figure 1. 1. A typical n . m. r. signal s(t) . (1. 1. 2) where s(t 1 - t 0 ) is the value of s at t 1 - ta . .This is displaced from the maximum, s(O), by the displacement of fi(t 1 ) from fi(t 0 ). Since these measurements are not all of the same accuracy, the most probable value of f(t 0 ) is a weighted average of the values obtained from each of these measurements : (1. 1. 3) 5 where the W(t 1 ) are the statical weights. If the noise is random (i.e., it has a Gaussian distribution of errors), then the proper weights are: (1. 1. 4) wher e a (t 1 ) is the standard deviation of fi (t 1 ). 3 The standard deviation of fi(t 1 ) is simply the inverse of the s i gnal to noise ratio : (1. 1. 5) where this ratio is calculated at fi(t 1 ) . Substitution of Eqs. (1. 1. 4 and 1.1. 5) in Eq. (1.1. 3) gives: -1 00 f 0 (t0 ) = si(O) [ J s i (t 1- t 0 )dt 1 J f 0 0 si(t 1 -t0 ) f/t 1 )dtH -00 (1. 1. 6) -00 for the most probable value of the intensity of a line with a ma ximum at t 0 • If we compute this integral, f 0 (t0 ) for all t 0 , then the output spectrum f0 (t) is a linear transformation of the input spectrum fi (t). This t ransforma tion can be carried out if we know the line shape s/t) of an unfiltered line . Finally, we m a ke the substitution T = t 0 - t 1 in Eq. (1 . 1. 6) and obtain: 00 f0 (t) = s i (O) [ J s ! (T)dT -00 -1 00 J J si(-T) fi(t - T)dT, -00 (1. 1. 7) 6 where we have changed the variable of integration in the first integral. 4 This is the standard form of a matched filter . This particular filter is normalized to be height preserving. A matched filter is the filter which gives the maximum signal-tonoise ratio in f (t). 4 It is also the optimum filter for the determi0 nation of line positions. 5 The output spectrum at a given time f0 (t ') depends upon the input spectrum at all times fi (t ' - T). Thus, it is known much more accurately. Measuring a property from an unfiltered spectrum assumes that we have no knowledge of the correlation between fi(t) and fi(t'). This is not generally true. Both analog and digital filters can easily be constructed. Digital filters can represent any filter arbitrarily accurately and are more easily varied than analog devices . If a filter operates on the spectrum while the spectrometer is in operation, it must be selected in advance and can only use the points measured at an earlier time not those measured at a later time. The most powerful filter is therefore a digital one which is used after the complete spectrum has been determined. 7 2. THE THEORY OF LINEAR FILTERS The intuitive concept of the resolution of a spectrum involves the question of whether adjacent lines appear as one peak with distorted line shape or are resolved into two discrete peaks. If the signal is a rectangular pulse, the width of the pulse is a quantitative measure of the resolution. Two rectangular peaks more than one line width apart appear as separate lines (Fig. 2. la). Two rectangular signals less than one line width apart appear as a single peak of more complex shape (Fig. 2. lb) . This simple situation gives our ... · ... :· a j S, - ·. - b S,+Sa. S, ". s~ ., Figure 2 . 1. The resolution of rectangular peaks. 8 conceptual framework for the discussion of resolution enhancement. We shall now seek to find a suitable definition of the line width of a general line shape. That is, we seek a suitable rectangular model, so that we can derive a linear transformation to improve resolution. Although Fig. 2. lb shows only one peak, it is obvious what the separation between the two peaks is . This is because of the discontinuity in the first derivative of a rectangular pulse. The deriva- tive of the spectrum is two Dirac delta functions separated by the line width. Unfortunately, the noise is greatly increased due to the inherent instability of numerical differentiation. 6 Since both differen- tiation and integration are linear processes, the optimum amount of either is automatically included in the filter we shall derive below. 2. 1 Line Width We define the resolution line width of two peaks as the minimum separation between them which permits their detection as discrete signals. Figure 2.1. 1 shows two peaks of the same line shape with different amplitudes. If s 2 is less than s 1 the peaks are not resolved. If S 2 is greater than S 1 the peaks are resolved. The resolution line width of s 1 is thus a function of the size and line shape of both peaks. The optimum process for improving resolution is therefore non-linear . This is a serious drawback since we frequently will not know the multiplicity of overlapping peaks. We therefore seek a property related to the resolution line width which will lead to a linear process. This property must be a function of the line shape of a single peak. 9 a I ' ' '' ' : ' : TS'+ S I : t J,. : -- -- S, Figure 2. 1. 1. The resolution line width can be a function of relative peak height. So far we have only considered whether or not two overlapping peaks have two discrete maxima. We have not discussed where the maxima occur. In the case of two overlapping Lorentzians (Fig. 2 .1. 2) the maxima will not correspond to the original peak positions . Although they are resolved, their maxima are shifted and a coupling constant measured by the separation of the maxima will be too small. This will occur at any separalion for line shapes which do not go to 10 J f--J•. Figure 2. 1. 2. Two Lorentzians separated by their width at half height. zero at a finite distance from the maxima. At larger separations the distance between maxima approaches the distance between line positions. At smaller separations the ratio of the distance between maxima to the distance between line positions approaches zero. This ratio equals zero at separations smaller than the resolution line width. There is a systematic error in measuring the separation of overlapping peaks. In order to accurately measure line separations 11 we must decrease the resolution line width and make the signal approach zero rapidly. In the case of two triangular peaks (Fig. 2. 1. 3) the resolution line width is equal to the full width at half height. Since this property a b I ..' I ,' ' ,--------, s + s \ I \ I. \ \\ \ \ \ \ \ Figure 2. 1. 3. The resolution of triangular peaks. is easy to measure for an experimental spectrum, it is widely used. This line width depends upon only two points for a general line shape. It is therefore not amenable to optimization procedures. It is also indepehdent of the "tail" of a line shape and minimizing the full width at half height could lead to increased shifts of the maxima. 12 It seems reasonable to define a line width as the width of a rectangle which has the same height as the input signal and some other property the same. The most obvious property is the area. The area of an odd function (e.g., a dispersion signal) is zero. A process which minimizes the area could convert a peak to an odd function (e. g. , by differentiation) without necessarily decreasing the resolution line width. Fully one half the area of a Lorentzian is outside the full width at half height. The area clearly places too much emphasis on the tail. Using the integral of the absolute value of a function would cure the first problem but not the second. Let us cut off a Lorentzian where it reaches half its height in such manner that the sum of two equal peaks separated by their full width at half height is constant between the maxima (Fig. 2. 1. 4). This decreases the area by a factor of 11' / 2. The tail has been removed so well that if two discrete maxima occur, then they correspond to the original peak positions. Unfortunately, the resolution line width has been increased by about 16%. The area places too much emphasis on the tail. A filter based on the area could lead to increased rather than decreased resolution line widths. The first moment is unacceptable since it places even more emphasis on the tail. The second moment may even diverge (e.g., for a Lorentzian line and therefore for an n. m . r . signal). Higher moments offer similar problems . A practical definition of line width is the energy width. 7 Energy here refers to the signal energy and not to the absorbed energy that the signal represents. If the signal is a voltage or current, then the 13 S, + S:a. .. .. Figure 2. 1. 4. Two truncated Lorentzians separated by their full width at half height. square of the signal is the power dissipated. Integration of this power gives the signal energy. The energy width is the width of a rectangle whose height and energy are the same as those of the line . That is: 00 b J s(t) dt / s(0) 2 2 , -00 where s(O) is the height of the peak. (2.1.1 ) 14 Only (7r-2)/211, or about one sixth of the energy width for a Lorentzian arises from the portion outside the full width at half height. Truncating a Lorentzian (as in Fig. 2. 1. 4) decreases the energy width by only a factor of 1T /(6-11'), or about one sixth the improvement calculated by area. A definition of line width based upon the integral of higher powers of the signal will still give a smaller line width for the truncated line than for the Lorentzian. Only (1/2 - 4/ 3 7T), or about 7% of the width based upon s 3(t) arises from the portion of a Lorentzian which is outside the full width at half height. This would not seem to place enough emphasis on the tail of the line. Higher powers of s(t) aggrevate the problem still further. The be~t exponent for s(t) would seem to be about two. The energy width may not be the best definition of line width, but we shall find that it is convenient to optimize and leads to useful filters. When we examine how the resulting filters work we shall find ithat they use the best method known for accurately measuring positions. It will then be clear that only minor improvements could result from modifying our definition of line width. We shall define a process which decreases the energy width b. In orde:.t to be useful the process must retain sufficient signal to noise that we can still find then. m. r. signal after filtering. We therefore must consider what happens to the signal height, the energy width of the signal, and the noise level during the filtering operation. most easily done by considering the Fourier transform. This is 15 A linear time-independent filter can be represented by an 8 integral; the convolution integral: 00 f 0 (t) = f h(T) fi (t - T) dT, (2. 2 . 1) -00 where h(T) is called the weighting function of the filter. problem is to determine the optimum form for h(T). The This can be done by examining the Fourier transform. We shall define F(w) to be the Fourier transform of f(t): 00 f(t) = 2~ J F(w) eiwt dw, (2. 2. 2) -00 where: 00 F(w) = J f(t) e -iwt dt . (2. 2. 3) -00 That is, we shall use lower case for a time function and upper case for the Fourier transform. The angular Fourier frequency w should should not be confused with the frequency of irradiation, v. They have nothing to do with each other. We note from examination of Eq. (2. 2. 3) that if s(t) is even then S(w) is real and even. If s(t) is odd, then S(c.cJ) is imaginary and odd. It follows that for any s (t): 16 S(-w) = s*(w), (2 . 2. 4) where the asterisk denotes the complex conjugate. s(t), S(w) is complex and asymmetric. For asymmetric Furthermore, the inverse Fourier transform of s*(w) gives s(-t) . That is, taking the complex conjugate in frequency space is equivalent to reflecting about t = 0 in time space. Finally, we note from Eq. (2. 2. 2) that wF(w) is the Fourier transform of f' (t) . Multiplying by w in frequency space provides a convenient method of numerical differentiation. This method is considerably more accurate than simple difference techniques . Consider a product of the form of Eq. (2. 2.1): 00 S0 (t) = f h(T) Si (t- T) dT. (2. 2 . 5) -00 The Fourier transform is: 00 So(w) = J -oO 00 J h(T) Si(t- T) dT e-iwt dt' (2 . 2. 6) -00 or : 00 S0 (w) = J -00 00 J h(T) e-iWT si(t - T) e-iw(t-T) dT dt . (2. 2. 7) -00 Reversing the order of integration gives: 17 J®h(r) e -iWTC J ®s;(t- r) e -iw(t- r) d(t- r)} dT. $ 0 ( w) = -00 (2. 2. 8) -00 Therefore : (2. 2. 9) where H( w) (the Fourier transform of the weighting function h(T) ) is called the frequency response function or transfer function of the filter. 9, lO A filter simply takes a product in frequency space. This is the first big advantage we gain by working with Fourier transforms. A filtering process can be expressed very simply. Later, we shall see that the properties of noise are also simplified by working in frequency space. . We quote without proof the Plancherel theorem: 11 00 00 J f (t) dt = 2~ J I F(w) 12 dw. 2 (2. 2. 10) -oo -00 Applying this to Eq. (2. 1. 1) for b, the energy line width, gives : 0() bo = i1T J ls0 (w)1 dw/ s;(o). 2 (2.2.11) -oo Substituting the value of S0 (w) from Eq. (2. 2. 9) leads to : oO bo = 2; J IH(w)l lsi(w)l dw/ s:(o). 2 -00 2 (2. 2. 12) 18 Equations (2. 2. 9) and (2. 2. 12) give the output signal and the output energy line width in terms of the input signal and the filter . The properties of noise are more difficult to describe and will be covered in the next section. Wiener, by developing generalized harmonic analysis, provided the mathematical tools for dealing with random functions. 12 This approach yields a reciprocal relationship between frequency and time. The frequency characteristic is called the power spectrum and the time characteristic the correlation function. We shall use the pro- perties of the power spectrum to develop filters. Random noise has an infinite period. Let the noise function be n(t). We can define a periodic function, nT(t), with period T as follows : It I :::; T / 2' n(t) ; (2. 3. 1) and: Taking the limit as T becomes arbitrarily large we obta in: Limit n (t) T-oo T = n (t) · (2. 3. 2) We shall first determine some properties of the periodic function, nT(t), and later take the limit to obtain the desired properties of the 19 non-periodic function, n(t). Since nT(t) is periodic we can expand it in a Fourier Series: 00 nT(t) = j~O [ aj cos (wjt) + bj sin (wjt)], (2. 3 . 3) where: aj bJ T/ 2 nT(t) cos (wjt) dt, -T/ 2 (2.3.4) T/2 2/ T nT(t) sin (wjt) dt, -T/ 2 (2. 3 . 5) = 2/ T J = J and wj = 2 1Tj / T. We shall also use the complex form: (2 . 3 . 6) where: T/ 2 -iw·t nT (t) e J dt. -T/ 2 = J (2. 3. 7) Suppose nT(t) is a current flowing through a circuit with unit resistance. Then a certain amount of energy, ET, is dissipated in each cycle. From elementary circuit theory: 20 = J T/2 n~ (t) dt. (2. 3. 8) -T/ 2 Substitution of Eq. (2. 3. 3) for nT(t) gives: T / 2 oo = J oo E _E -T/ 2j=Ok=0 [a.cos(w.t)+b.sin(w.t)] J J J J (2.3.9) Rearrangement of this expression leads to: 00 00{a . ak T J ·cos (j e) cos (ke) de 1T .0 6 j=O k=O 3 21T -1T + aj ~ ~ 1T J sin (je) cos (ke) de (2. 3. 10) -11 + bi bk~ /'sin (jB) sin (kB) dB''f\ . -1T Integration simplifies this expression to: 00 = I; ~ [aJ.2 + bJ.2 ) . (2. 3 . 11) j=O We can therefore speak about the component of ET from the frequency w . . The total energy is the sum of these components: ] 21 (2. 3 . 12) where: ET(w.) = T (a.2 + b.2 ), ] 2 ] ] is the energy per cycle from frequency wj" (2. 3. 13) The average power for a cycle, pT, is given by ET/T: oO PT = ~ E(wj) J=O i, (2. 3. 14) or: 00 L; PT = E(wj) j=O 21f ~ wj. (2. 3. 15) A very useful characterization of random functions is the power spectral density: (2. 3. 16) This leads to the representation of the total power as : ao L) WT(w.) ~ w .. J J j=-oO Substituting Eq. (2. 3.13) in Eq. (2. 3.16) we obtain: (2 . 3.17) 22 (2. 3. 18) so that: WT ( w.) J = _!_ 87T (2. 3.19) (a. + i b . )(a. - i b.). J J ] ] We now substitute Eqs. (2. 3. 4) and (2. 3. 5) for aj and bj: WT(wj) ~ i,. {~ {:;:nT(t) [cos (wit) + sin (wit)] dy i T/ } (2.3 . 20) 2 2 x T nT(t)[cos(w.t)-isin(w.t)]dt, { -T / 2 l J J or: 1 = 2 7TT iw·t T/ 2 . ~ { J T / 2nT(t) e -lW· · t } J nT(t) e J dt J dt . { -T/2 -T/ 2 (2. 3. 21) Substitution of Eq. (2. 3. 7) for NT(wj) gives: (2. 3. 22 ) The root-mean-square value of the noise is defined as: (2.3.23) 23 This is simply the square root of the total power per cycle. We therefore obtain from Eq. (2 . 3. 1 7): 00 I; WT(w.) Aw .. J j=-00 (2. 3. 24) J We now examine the result of a filter on nrms· After filtering we have : 00 2 (n rms )T I; WT (w.) Aw., J J = 0 j=-00 (2. 3. 25) 0 or, substituting Eq. (2. 3. 22) for WT (wj) gives: 0 (2. 3 . 26) Since the filter is equivalent to multiplying by H(wj) in frequency space: 00 = 1 ~ T) IH(w.) NT (w.)I j=-00 . 21T ] i ) LI ( 2 Aw .. J (2. 3 . 27) We again use Eq. (2. 3. 22) for WT (w. ) and obtain: i J (2 . 3 . 28) for the mean-square value of the noise after filtering. This is the crucial relationship which shows the effect of a filter on the noise. 24 We shall find the time characteristic useful for developing filters based on probability theory. The autocorrelation function, rT(t), is defined as: rT(t) T/ 2 nT(T) nT(t + T) clT. -T/ 2 = ~ J (2.3.29) This is a measure of the dependence of nT(T) upon nT(t+T) . If nT(T) is independent of nT(t+T) for all t not equal to zero, then rT(t) approaches a Dirac delta function. Substitution of Eq. (2. 3. 6) for nT(t) into Eq. (2. 3. 29) gives: = +J 41T T T/2 -T/ 2 (2.3.30) Rearrangement of this expression leads to: 00 y; ··-' 00 y; .....I NT(wj) NT( wk) 2 7TT j=-00 k=-00 ~wj ~wk { 1 21T T/2 i(w.+w )T } e l k dT . -T/ 2 J (2. 3. 31) The limit as T approaches infinity of the quantity in brackets is the Dirac delta function . 14 25 We now have derived all the desired relationships for the periodic function nT(t). We let T approach infinity and we obtain the following relationships for the non-periodic function, n(t) from Eqs. (2.3.6), (2.3.7), (2.3.22), (2.3.29), (2.3.24), (2.3.28), and (2.3.31), respectively: n(t) ao 1 . t J N(w) e w dw, 211' = 1 - . (2.3.32) -00 ao J n(t) e -iwt dt, N(w) = -ao = Limit IN(w) 12 W(w) T-oo 21TT ' T/ 2 r(t) = Limit _!_ n(T) n(t+T) dT, T-ao T -T/ 2 J (2. 3. 33) (2. 3. 34) (2. 3. 35) 00 n;ms = J W(w) dw, (2.3 . 36) -00 = J co I H(w) 1 2 Wi(w) dw, (2 . 3 . 37) -00 and 00 r(t) = J W(w) eiwt dw. -oo (2.3.38) 26 It also follows from Eqs . (2 . 3 . 32), (2. 3. 33), and (2 . 3 . 38) that: 00 W(w) = i1I J r(t) e-iwt dt . (2. 3 . 39) - 00 The autocorrelation func t ion a nd the power spectral density are r elated through a Fourier transform . This is called the Wiene r Khinchin relation. Finally, we note that white noise is defined as having a power spectral density whic h is frequency independent. That is, W(w)white noise is a constant. All frequencies contribute equally to white noise. Later we shall see that this provides an excellent description of the noise inn. m. r. spectra under normal conditions. The autocorrelation function for white noise is given by: 00 r(t) white noise = W J eiwt dw = 27TW6 (t) , (2 . 3 . 40) so that the data points of a spectrum are completely independent for white noise. They are statistically independent only for ergodic Gaussian noise. We shall consider the general significance of r(t) in a later section. Non-white or colore d noise requires assigning a functional form to W(w) or r(t). two. This could be done by measuring either of the The resulting filters are significantly more complicated than t hose for white noise. 27 This completes our excursion into generalized harmonic analysis. We are now ready to determine H(w) for an optimum linear filter . 2. 4 timum Resolution Enhancement Filters The optimum resolution enhancement filter gives the minimum energy width for a given signal-to-noise ratio. It is simpler to solve for the filter which gives the maximum signal to noise ratio for a given energy width, which is the same thing. The signal-to-noise ratio is defined as the maximum signal voltage, s 0 max divided by the root-mean-square noise voltage. Our definition of the time scale gives the maximum signal voltage as: . Somax = (2.4.1) So (O). Taking the Fourier transform we obtain: 00 · Som ax = 2~ J S0 ( w) dw . (2. 4. 2) -00 Substitution of Eq. (2. 2. 9) for S0 (w) gives: 1 = 211 00 . J H(w) Si(w) dw, (2. 4. 3) -00 for the maximum output signal voltage in terms of the input signal. We maximize the signal-to-noise ratio by holding the signal constant 28 and minimizing the noise. The optimum resolution enhancement filter satisfies the conditions: 00 n~ms = J IH(w)l Wi(w) dw 2 is a minimum; (2.4.4) is a constant; (2.4.5) = 27T J \H(w)l lsi(w)l 2 dw/s~(O) is a constant; (2. 4. 6) -00 oO Somax = i1T J H(w) Si(w) dw -oo . and: 00 b0 2 -oo where the last condition comes from Eq. (2. 2.12). Since the denominator of Eq. (2. 4. 6) appears in Eq. (2. 4. 5), we can restate the problem. We wish to satisfy the following conditions : J lH(w) 1 Wi(w) dw = a minimum 2 (2 . 4. 7) -00 00 J H(w) Si (w) dw (2. 4. 8) -00 00 (2. 4. 9) 29 This problem is readily solved by Lagrange's method of undetermined 13 multipliers. The general method for an extremal problem with k constraints is as follows; Let: X2 J I [ x, y(x), y'(x) ]dx = an extremum, (2. 4. 10) X1 under the k constraints: X2 J Ij[x,y(x),y'(x)]dx = cj j = 1, 2, ... k, (2.4.11) X1 and let: k K = I + ~ A· I., j=l ] ] where the ;\"j are constants. (2.4.12) Then the solution satisfies: aK a aK ay - ax ay' = 0 · (2. 4. 13) When this method is applied to the above problem we obtain: For this problem K does not depend explicitly upon H '(w). of Eq. (2. 4. 13) therefore leads to: Application . 30 (2. 4. 15) I where we have used H(w) I as y(x) in Eq. (2. 4. 13). We now multiply by H*(w)/I H(w) I and obtain: (2. 4. 16) Solving for H*(w) and taking the complex con]ugate gives: . H(w) (2. 4. 17) which is the frequency response function (the Foilrier transform of the weighting function) for the optimum linear resolution enhancement filter. We have held the line width constant by use of the parameter ~· It is clear from its origin in the third term of .Eq. (2. 4. 14) that if A2 = 0 we obtain the filter with the maximum signal-to-noise ratio .regardless of line width, since this eliminates consideration of line width in Eq. (2. 4. 14). If A2 approaches infinity we eliminate the first two terms of Eq. (2. 4.14) and obtain the minimum line width regardless of the signal-to-noise ratio. First, we take the limit as Aa approaches infinity. The limit of the frequency response function as Aa approaches infinity is: 31 Limit H(w) (2. 4. 18) ,\2- 00 We now substitute this limiting form for H(w) in Eq. (2. 2. 9) giving: S0 (w) (2 . 4.19) Taking the Fourier transform of this we obtain: 00 s 0 (t) = J e iwt d w. -,\1 1T ,\ 4 2 (2 . 4 . 20) -00 The integral is easily evaluated 14 giving: (2. 4. 21) The signal has been transformed into a Dirac delta function . This indicates that our definition of line width was realistic, since in the limit it leads to a peak with zero resolution line width and no shift of adjacent peaks. The second situation of interest is the case ,\2 = 0. H(w) = * I 2 Si (w) W i(w), - A1 which is the form for a matched filter. This gives: (2. 4. 22) 4 In addition to giving the maximum signal-to-noise ratio, the maximum of the filtered function 32 is the ~ost probable position of the line. The parameter ~ determines the emphasis of the filter on resolution or sensitivity, while A1 determines the normalization. We now assume that the noise power spectral density is a constant (i. e. , the noise is white or frequency independent) and rearrange Eq. (2. 4 . 17) to: H(w) = c(q) st(w) (2.4.23) 1+q1si(w)l2' where c(q) is the normalization constant and q determines the resolving power of the filter. If we require that: (2. 4. 24) so that the filter is height preserving, then: (2. 4. 25) This frequency response function [ Eqs. (2. 4. 23) and (2 . 4 . 25)] defines a family of linear resolution enhancement filters which depend only on the input line shape. In the next section we consider the implementation of these filters . 2. 5 Numerical Fourier Transforms ~, ~~~ The Fourier transform of the weighting function for the optimum linear resolution enhancement filter has the form given in Eq. (2. 4. 23) . 33 The calculation of this weighting function using the I. B . M. provided subroutine "HARM" has been programmed. This subroutine uses the very efficient numerical F ourier transform algorithm of Cooley and 15 Tukey. The calculation of a Fourier transform at N points using this algorithm requires only N log2 N multiplications rather than the N2 required by a more straightforward approach. The execution time for the complete filtering program is about 10 seconds on an I. B. M. 360/ 75 computer when the H level compiler is used. Both linear and non-linear filters require some lmowledge about the noise function n(t). We obtained the final form of the linear resolution enhancement filters by assuming that the power spectral density of the input noise W i(w) was not a function of w. In order to test the validity of this assumption under the operating conditions of our experiments we recorded the noise from the Varian Model DFS-60 spectrometer. The power spectral density was calculated using the above program. The results are shown in Figure 2. 5.1. The assumption that the noise is white [ W(w) is constant] is clearly quite good. There is no systematic variation of W(w) with w (Fig. 2. 5. 1) . The Fourier transform of an n. m. r. signal (Fig. 2. 5. 2, page 37 ) does vary with w. The only assumption made in deriving Eq. (2. 4. 23) has been verified. We now consider a numerical problem which arises in the use of these filters . The program requires a table of values of the line-shape function, s(t), from which h(t) is calculated. This line shape function is best determined experimentally. We therefore must make the 34 - --·- - - · ··- , I ·-·- , -- --=~'!l_ 0 "'11 Figure 2. 5. 1. The power spectral density of noise from the DFS-60 spectrometer . approximation: (2. 5. 1) where ns (t) is the noise in the peak we are using for the line shape. · This leads to the approximation for the frequency response function: H(w) c(q) [ st(u1) + N 8*(w) ] I (2. 5. 2 ) 1 + q Si(w) + Ns(w) 12 This approximation must be modified to eliminate the domination of high frequency noise. We shall first prove that this is true in principle and then show that it can be important in practice. 35 The line-shape function at t = 0 is given by: oO s/O) = i1T J Si(w) dw, (2. 5 . 3) - oO which implies that : Limit S.(w) I wl-oo l = 0· (2. 5. 4) The noise frequency response function satisfies the limiting equation: (2. 5. 5) where W s(w) is the power spectral density of the noise. noise is white so that W s(w) is independent of w , then independent of w. Since the INs(w) I is also Therefore the limit of the noise spectrum is.: Limit N (w) w-oo s = 11 ''s eiB(w) ~ 0 ' wher 7Js is a constant but e is not. (2 . 5. 6) The limit of the weighting function, Eq. (2.5.2), is therefore: Limit H(w) _ w-o0 c(q) N *(w) 8 l+qlNs(w)l2· We recall that the filtered spectrum is given by : (2. 5. 7) 36 00 fJt) = 2~ J H(w) F /w) eiwt dw, (2.5.8) -oO where Fi(w) is the Fourier transform of the input spectrum. Equations (2. 5. 7) and (2. 5. 8) show that the noise in the line-shape function will dominate the output spectrum. Clearly, there exists an w > 0 such that: 17 (2.5.9) If we use Eq. (2. 5. 2) beyond w we are mainly increasing the noise 17 in the output spectrum. We therefore use w for a cut-off frequency 71 and redefine H(w) to be : c(q, w ) [ s*(w) + N;(w)] 11 1 + q S(w) + Ns (w) 2 I H(w) w I < ~ (2. 5. 10) 0 This is equivalent to replacing the limits of integration by wr/ in Eq. (2.5.8). When performing a numerical Fourier transform, Eq. (2. 5. 8) must be replaced by the approximation : 1 . 21T ~ J H(w) F i(w) e . wt dw, 1 -n where n is determined by the number of data points for fi (t). (2. 5. 11) The 37 above discussion would be purely academic if w 1 were larger than n. 1. Their relative sizes must be determined experimentally. Figure 2. 5. 2 shows a computer drawn plot of the modulus of an experimentally determined line-shape transform IS(w) I· The signal-to-noise ratio for s(t) was approximately 200/1. It is clear -w,, 0 w,, -n Figure 2. 5. 2. The modulus of the line-shape transform, S(w) for an n. m. r. signal. I that in this case w 17 I, is much less than n. A frequency response function which gives high resolution enhancement has been computed from this line shape. The plot of the modulus, IH(w) l, is shown in Figure 2. 5. 3 and the corresponding time function, h(T), is shown in 38 ~ ... -w,, 0 w.., -n Figure 2. 5. 3. The modulus of the frequency response function, H(w) for a high resolution weighting function . I Figure 2. 5. 4. I, It is obviously desirable to make use of the cut-off frequency, wT/. In practice we compute rfs as the average value of \ S/w) + Ns(w) I for the ten highest frequencies available. We then choose w 71 to be the smallest positive value of w for which: (2. 5. 12) 39 • . jY •"l $ :;i''·o._.o_ _ _ =-:------T,,. __.__.___..,.,2.'{"=flE-QUf-llC-Y-111-HZ~._=-----::•. __,,,,,.lllD,,,... . Figure 2. 5. 4. The weighting function h(t) corresponding to Figure 2. 5. 3. This effectively limits the frequency response function to those frequencies for which the signal is greater than the noise. Figure 2. 5. 5 shows the results of using this value of w 71 on IH(w) I· The effect that this has on the weighting function, h(t) is shown :l n Figure 2. 5. 6. We have eliminated the problems caused by high frequency noise. 40 II I i \ 1I ,1 ! I I !I -w,, 0 w,, -n Figure 2. 5. 5. The modulus of the frequency response function H{w) using the cut-off frequency wTJ. I I I I 2. DOCl l.IU'l F Rf. (.llJf tj(.( I M !17 I "·""" I •.ouo ~ Figure 2 . 5. 6. The weighting function h(t) resulting from the frequency response function in Figure 2. 5. 5. 41 . The effects of multiple scans on the above frequency response functions are shown in Figure 2. 5. 7. Comparison with Figures 2 . 5 . 2 and 2 . 5. 3 indicciJes that eliminating the higher frequencies from the single scan data was quite reasonable. The weighting function calcu- lated from the multiple scan data is shown in Figure 2. 5 . 8. The use of the cut-off frequency causes H((v) and h(t) to mor e clos ely resemble the results which could be obta ined by collecting additional data. V./e note that the high frequency wiggles tmvarcls the right in h(t) have been decreased by obtaining b etter dat a . I Figure 2. 5. 7 . l Frequency r es ponse fun ctions obt<d .ne d fro m a m ultiple scan experiment . 42 Figure 2. 5. 8. The weighting function resulting from the high resolution frequency response function shown iI1 Figure 2. 5. 7. The elimination of high _frequency noise is not obviously It would be silly to use data for s i (t) whi ch clid not have a s i g11ificantly higher signal t o noise ratio than the input speclrum fi (t) . The high freque ncy noise i n s i (t) will therefore always b e smsJl. However, t he high freque n cy noise in fi (t) is not necess<:u~ ny srn.a ll, and the filter will tak e t he product of the two . T he mm·e i mportant application of the cut-off criterion will probab ~y be found in the study of the behavior of h(t). In this c::-i.se, we clearly eliminate most of the noise. 43 All weighting functions discussed in the remainder of this thesis have been computed using the above cut-off criterion. 2. 6 RC Filters ~~ These simple analog filters are of particular interest since one is built into most commercially available n. m. r. spectrometers. They consist of one or more pairs of resistors and capacitors (Fig. 2. 6. 1). v, v. Figure 2. 6.1. Single and double section RC filters. From elementary circuit theory 16 we know that: fi (t) = I(t) R + Q(t)/ C, (2.6 . 1) 44 where fi(t) is the input voltage (signal), I(t) is the current, and Q(t) is the charge on the capa citor. Since the current is the rate of change of the charge : f . (t ) 1 Q(t) R + Q(t) / C. (2. 6. 2) The output voltage (sig11al) is given by Q/ C so that: (2. 6. 3) The input and output voltages are connected by a first order linear differential equation. We first put it in standard form: ' \ + RC 1 fo () f o(t, t The general solution is : (2. 6. 4) 17 (2 . 6. 5) where k is an arbitrary constant. Since this filter is a real time device it can not use future data . Therefore, T 1 :5 t. The case of interest also satisfies the initial condition: (2 . 6. 6) The result of using a single section RC filter is therefore : t f 0 (t) -- f e-(T -t)/RC (1 / RC] fi(T 1 - 00 1 ) dT 1 • (2. 6. 7) 45 We now put this in the form of Eq. (2. 2.1) by making the substitution 00 fo(t) ic) J( = e -T/RC fi (t - T) dT. (2.6.8) 0 This gives the weighting function: 1 -T/ RC RC e T > 0 (2. 6. 9) 0 T <0 for a single section RC filter. When two of these filters are placed in sequence the output function is: (2. 6. 10) This can be rearranged to: 00 fo(t) 00 = J j(R~) e-(Ti+T2 ) /RCfi[t-(T1+T2)]dT2dT1. 0 2 (2. ·6.11) 0 We now make the substitution T = T 1 + T2, giving: (2. 6. 12) 46 where the upper limit on r 2 comes from the fact that r 1 can not be negative. Integration over T 2 gives : (2. 6. 13) The weighting function for a double section RC filter is therefore: T > 0 (2. 6. 14) h(T) = 0 ; T <0 This rFsult could also have been obtained by taking the inverse Fourier transform of H~c(w). That is, by working in frequency space. We shall find this approach particularly helpful wheri we consider sequences of filters in the next section. It is sometimes desirable to use a series of successive filters. One example is the double section RC filter. A second is suggested · by the design of an n. m. r. spectrometer. Ann. m. r. spectrometer is equipped with a double section RC filter with variable capacitors. 18 We must decide how to set this device, since all data we obtain will have been filtered by it (or not filtered by it if we decide to remove it). The result of two filters applied in sequence is most easily discussed in frequency space. We therefore take the Fourier 47 transform of the weighting function: (2. 7. 1) or ;c J 00 HRC(w) = e-(iw+ 1/ RC)T dT. (2. 7. 2) 0 Integration yields: 1 1 + iwRC' . (2. 7. 3) for the frequency response function of a single section RC filter. We shaH examine single section RC filters in detail. It will be obvious that multiple section filters behave similarly. We recall from Eq. (2. 2. 9) that the result of a linear filter in frequency space is a simple product: (2.7.4) where SRC(w) is the result of an RC filter on the Fourier transform of the line shape. Similarly: (2. 7. 5) where F RC(w) is the RC filter output signal in frequency space. If 48 we use an RC filter on both our signal and our line shape function, and then construct a matched filter [ Eq. (2:. 4. 22)] based upon this new line shape, we obtain: (2.7.6) Substitution of Eq. (2 . 7. 4) for SRC(w) and Eq. (2. 7. 5) for FRC(w) gives : (2 . 7. 7) or: (2. 7. 8) where Fm (w) is the result of a matched filter on the spectrum. We see that this procedure gives the desired result if and only if: (2. 7. 9) From Eq . (2 . 7. 3) and Eq. (2. 7. 8) we see that: (2 . 7 . 10) We could obtain the results of a matched filter by computing: (2. 7. 11) 49 but this is considerably more time consuming than carrying out the integration in time space: fm(t) = J si(-T) fi(t-T) dT, (2.7.12) -00 which we can do when no RC filter is used. The simplest procedure is clearly to disconnect the RC filter (frequency response control) on the spectrometer. The extension of these results to multiple section filters is obvious. We conclude that the results of an optimum filter are recoverable after the use of any RC filter, but that the most efficient procedure is to disconnect the RC filter on the spectrometer. More generally, this is true of any filter for which the inverse exists ·(i.e., H(w) >= O; for all w ). The RC filter may be disconnected from the Varian Model DFS-60 (or HR-60) spectrometer by using the "scope" setting. The RC filter on an A-60 type spectrometer may be effectively disconnected by setting the filter bandwidth control at 4. At this setting, RC in Eq. (2. 7. 11) can be regarded as zero for most purposes . The most straightforward method of measuring the signal-tonoise ratio is to take the ratio of the peak height to the rms value of the spectrum where the signal is zero. This method works in some cases. In many of the spectra we deal with, the value of the signal 50 is greater than the noise level at all points. A less empirical method is therefore necessary. The output signal-to-noise ratio is given by: (2. 8. 1) 1 00 (S/N) 0 = s 0 (t)max/[ J 1H(w)I2 W i (w) dw 2 J . (2. 8. 2) -00 For white noise this becomes: . (S/ N) 0 = So(t) ~ax [ wi 2 J-2 1 oo J IH(w) I dw 2 . (2. 8. 3) -oo Using the Plancherel theorem 9 gives: (S/N) 0 (2. 8. 4) The improvement in the signal to noise is then: (S/N) 0 = (S/N)i So(t)max [Joo 2 s.(t) h (T) dT 1 max -oo ]-! (2.8.5) J-2 (2. 8. 6) , or approximately: 1 (S/N) 0 fo(t)max[ (S/ N).1 - f. (t) 1 max oo 2 f h (T) dT -oo 51 This is true regardless of whether or not h(r) is based upon the line shape s(t). We can therefore use this relationship to compare optimum filters with other filters assuming only that the noise is white. The above equation can also be derived from a statistical approach. Consider the discrete transformation: (2. 8. 7) The classical statistical equation for the linear propagation of 19 . errors is: (2. 8. 8) Using Eq. (2. 8. 7) for ~0 (tj) and assuming that the noise is random we obtain: ,... 2 v 0 " i h2(-rka., ) 2 =L k 1 (2. 8. 9) or: (2 . 8. 10) The signal-to-noise ratio is given by: (2. 8. 11) 52 The improvement in the signal-to-noise ratio is therefore: (S/N) 0 (S/N)i 8 = o(t)max (: si(t)max i), (2.8.12) 0 or finally: (S/N) 0 So(t)max [ [; h2 T = . (t) .k ( k) (S/N) . s1 max 1 J 1 -2 ' (2.8.13) which is the discrete approximation to the result obtained from the information theory approach, Eq. (2. 8. 5). This relationship, Eq. (2. 8. 6), was used to evaluate the signalto-noise ratio obtained from the filters we later discuss. 53 3. RESULTS OBTAINED WITH LINEAR FILTERS The effects of several linear filters on the types of line shapes resulting from an n. m. r . experiment have been studied. The resolution and signal to noise of spectra can be significantly improved .. with the aid of linear filters . Resolution enhancement requires more detailed knowledge of the line shape than sensitivity enhancement does. Two line shapes have been studied in detail. The first, an experimental n. m. r. signal, was obtained from the proton-decoupled 13 C spectrum of carbon-1 of trans-1, 3-pentadiene using a 0. 2 Hz/ sec. sweep rate. We considered this to be a representative line shape (Fig. 3. 1). The second was a Lorentzian line, the form of an n. rn. r . § 90.0- .. -·--·--· ---tm;-----tcm- · ·-···---~, _.,. --·---itOJ----i(OJ FAECIU€HC't' IN HZ Figure 3. 1. The line shape of a typical n. m . r. signal. 54 signal for infinitely slow pasage. 20 These line shapes are two members of the family of possible line shapes which result from varying the sweep rate. We shall examine the nature of resolution enhancement by careful consideration of these two line shapes . Since a filtering device based upon a Lorentzian line shape is commerically available, 21 we have compared the results obtained by using the correct line shape to filter the above experimental line with those obtained using a Lorentzian line shape. Both these Lorentzian filters and RC filters contain the line width as a variable parameter. It is therefore necessary to optimize this parameter before compari- sons can be made. The line width of the weighting function, h(T) has been varied for several combinations of line-shape function and weighting function. The results of using an RC filter are shown in Fig. 3. 1. 1 for a Lorentzian line shap1= and for the above experimental n. m. r . line shape. The use of a' double section RC filter on a Lorentzian line can give up to 96. 1 % of the signal to noise obtainable with a matched filter. The use of a double section RC filter on the experimental signal gave up to 97. 1% of the signal to noise obtainable with a matched filter. In the case of an n. m. r. line sha pe with "wiggles" the signal to noise peaks more sharply, indicating that setting the time constant of the filter is somewhat mor e critical in this case . 55 l.O -- 0.125 - - - - - - - - - -- - - -- 0.25 0.50 - - -- - - 1.0 •UC/~ Figure 3.1. 1. Signal-to-noise ratios from using a double section RC filter. Curve A results from a Lorentzian line shape and B results from the experimental line shape. vi is the input line width and a is the sweep rate. Whereas the double section RC filter with the optimum time constant broadens a Lorentzian line by 90%, the experimental n. m. r. signal is broadened by only 26%. This is because the optimum time constant for filtering the experimental signal is only 42% of the line width (full width at half height), while for a Lorentzian line it is 50% of the line width. Our results for filtering a Lorentzian agree with thos_e obtained earlier by Ernst. 22 We have also studied filters using Lorentzian weighting functions of varying half width. The results of using this filter ori. both the Lorentzian and the experimental line shapes are shown in 56 Fig. 3. 1. 2. The Lorentzian function achieves a maximum when the line width of the filter function exactly equals that of the signal. This represents a matched filter. The Lorentzian line is broadened by 23 100% and the experimental signal by 53% when the optimum line width weighting function is used. 1.0-- - - -- -· 8 0.7 0.25 0-50 1.0 Figure 3.1. 2. Signal-to-noise ratios from using a Lorentzian weighting function. Curve A is from Lorentzian line shape and B results from an n. m. r. line shape. vi is the input line width and Vf is the filter line width. The Lorentzian weighting function which gives the maximum signal-to-noise for the experimental signal had a line width which was 0. 79 times the line width of t he n. m. r. signal. This is the Lorentzian line width we shall use to evaluate a resolution enhancement filter based upon a Lorentzian line shape but applied to general n . rn. r. signals. 57 3. 2 Resolution Enhancement With the signal-to-noise ratios which are experimentally accessible, one can achieve almost an order of magnitude reduction in the resolution line width. This represents a substa ntial improvement in the capabilities of the instrument. Figure 3. 2. 1 shows an example of the results we have achieved in r esolution enhancement. The spectrum was obtained by adding together two slightly displaced spectra from carbon-1of1, 3-pentadiene. The line shape was also taken from pentadiene . It is clear that a high degree of resolution enhancement is possible. ~L. ' o.o I~ . - ·-----··. -·-iim- - FAEQUF.HC Y IN HZ . '·°"' ..... ... _J Figure 3 . 2.1. Resolution enhancement of an n. m . r. s i gnal. The large peak and its "wiggles" are an unresolved doublet. The two smaller peaks are the same doublet after resolution enhancement (q = 3000). 58 We shall now consider the nature of resolution enhancement in some detail and try to achieve a better intuitive understanding of the process . Resolution enhancement is most easily understood in frequency space. We consider the Fourier transform of the weighting function, h(1), which is the frequency response function, H(w). We recall that this function is given by: (3. 2.1) The degree of resolution enhancement increases with increasing values of q (see section 2. 4). Since c(q) is real and not a function of w, the phase is unchanged from the phase of the frequency response function for a matched filter, Hm(w). However, the amplitude is modified to emphasize higher frequencies. This is clear from the plots of I Hm(w) I and IHr(w) I shown in Fig. 3. 2. 2. To understand the significance of this shift to higher frequencies, we consider the corresponding weighting functions hr(;) and hm (1), shown in Fig. 3. 2. 3. In time space, the emphasis has been shifted from the main peak to the "wiggles" and the weighting function has taken on the appearance of a diffraction pattern. When the frequency response function shifts emphasis to higher frequencies, the effective wavelength of the diffraction pattern h(r) is decreased. Thus, resolution enhancement can be looked upon as using a higher frequency diffraction pattern to fix positions more accurately. 59 ,0 ·- I \ I ': q = 0 ./ \ i q = la4 ' ·'1·. 11 . ., f ' Jv.t.r 'v .'vJ "--·--·----· ------~ . - .. ··r w,, Figure 3. 2. 2. The modulus of the frequency response functions for a matched filter and for a resolution enhancement filter. We note from Fig. 3. 2. 3 that the "wiggles" still line up perfectly. This suggests that unlike sensitivity enhancement, resolution enhancement requires a detailed knowledge of the line shape. We now consider what happens if our knowledge is limited. One method of determining the importance of the "wiggles" in an n. m. r. line is to eliminate them from the line-s hape function and look al the effect on filtering. The above experiment.a l n. m . r. 60 !i ~ .. ~ q = 10 \.1,.1"~ ·-./ -.... ~ , ' ~ . · . • · ·-· · ·--·· q = 0 ~II 1:t----~\ <i ·- l'flllWCY 111 HI Figure 3. 2. 3. .... The weighting function for a matched filter (bottom) and for a resolution enhancement filter (top). The former is the mirror image of the line sha pe. 61 spectrum was filtered with a matched filter based upon the correct line shape and with the corresponding filter based upon a line shape with the "wiggles" removed (Fig. 3. 2. 4). ~§ i9f--- - The results a re shown in ----J lllli! q = 0 9 ~ 'i' ~'0.0---....L.-... -·- - ·····--2.D ·· IAID __ JU _llO _ 'l'Eoua«:Y I H l1Z Figure 3. 2. 4. Figure 3. 2. 5. The weighting function for a matched filter with the "wiggles' 1 removed. The signal-to-noise ratios differ by only 2. 6% and the spectra look quite similar. This is not the case when resolution enhancement filters based upon these two line shapes are used (Fig. 3. 2. 6). When the "wiggles" are removed from the line shape the results of resolution enhancement are not even useful. Thus, we see that while 97% of the signal to noise of an n. m. r . signal comes from 62 ~~ :L__ 0.0 J _ _ ___ ___ u111 _ __ -'-c--- 2.ax> ~---'-=c--- f~~y !~ . t!Z - s.<DJ 11.m:» s.crx Figure 3. 2. 5. The results of filtering an n. m. r. signal with a matched filter which: A includes the "wiggles"; and B does not include the ''wiggles". the broad first maximum, the position of the signal (or resolution) is determined almost entirely by the much narrower wiggles . This raises the question of whether or not we can measure the line shape sufficiently accurately in practice for these techniques to be useful. In an attempt to answer this question we compare the above line shape from pentadiene with a proton-decoupled benzene 13 C spectrum taken under identical conditions (Fig. 3. 2. 7). There is almost no difference between these line shapes ; in particular, the 63 ~' "i §I ~ 1- !~ !r !r q = 3000 z ' "'!!i 0 i I "' ' l ! ! ! ~ii! ~.; ll'§ "i Iii "i !i 9 ~ 'i' ~ 1.0 ..... Figure 3. 2. 6. a- -·"The results of filtering an n. m. r. signal with a resolution enhancement filter which includes the "wiggles" (top), and with one which does not (bottom). 64 Iii ~~ •.•:--~~-->,, ..~ ~~~----c-'-,.~.~~~--.-' •.=~~~~~.~.~,--~~......J • .FREWENCY IN HZ Figure 3. 2. 7. 13 The proton decoupled C spectrum of benzene (top) and the one carbon of trans-1, 3-pentadiene (bottom). "wiggles" line up perfectly. The result of filtering the benzene spectrum with a resolution enhancement filter based upon the line shape from pentadiene (Fig. 3. 2. 8) does not show any of the difficul ties encountered earlier. When we consider the effects of sweep rate we shall see that line shapes generally appear to be sufficiently predictable as to make resolution enhancement based upon the exact line shape practical. Another case of interest is to assume a Lorentzian line shape, since this is the exact line shape for an infinitely slow sweep rate. 21 65 q = 100 §:. r,i ;- ~:. : .:: <;:• l ':. ('.~.t~-_:-- ·--·--·-·--i\)~";':i--- -·- -·-·· ·--·2:~:.1---·----·--·f.::;-··---------··---~~:~::------·- "'--(C ~'J . J.:f;[r.1,,;:::.:~; y P: <~i'. F'i gure 3 . 2. 8. The result of filtering the above benzene spectrum with a resolution enhancement filter which uses the line shape from penta diene. Tha t Lorentzian line width was selected which gave the maxinrn.n1 signa l to noise for a filter which uses the Lorentzian as the weighting function to filter the pentadiene signal (see section 3. l). Weighting functions ba sed upon this Lorentzjan line shape are shovm a longside the equivalent functions based upon the above pentadiene line shap2 in Fig. 3. 2. 9 . Although both are diffraction patterns, they are drastically cliHe:cent. Iviost notably, the Lorentzian pattern is symmetric and remains centered at t = 0. 'What effect the dHforences will have on a filtered sig11al is not obvious . The resElts of using 66 ~~I /\ / \ .\ i II j t \ ! ! ~! ~ q =0 T· I ' \. \ \. \",\JV ..; ./1/'·· _, . ' I i ! .' ii I I ~ ~- 0 t i . I q = 100 1r II -\J v---- ~I 9r ~I ;t !L 11.0 ' 1.- ---'--·--~ :u110 1.cm fAfQUEl«:T IN HZ Figure 3 . 2. 9. A comparison of weighting functions based upon the experimental n . m. r. signal (left) with those based upon the Lorentzian line which gives the maximum signal to noise (right). 67 these two sets of weighting functions to filter n. m. r. spectra are shown in Fig. 3. 2. 10. For low resolution enhancement the filters based upon a Lorentzian line shape give some improvement in the line width, but for high resolution enhancement they give results which are useless. The Lorentzian results are quite similar to those using the correct line shape, but with the "wiggles" removed. We should keep in mind that the sweep rate for the original n. m. r. spectrum was only 0. 2 Hz/ sec. Slower sweep rates than this are often impractical. Under normal conditions of n. m. r. spectroscopy the deviations of the line shape from a Lorentzian are extremely important for resolution enhancement. These deviations are sufficient to make a filter based upon a Lorentzian line shape of questionable value. Some of the difficulties encountered in filtering an n. m. r. signal with a Lorentzian resolution enhancement filter might be eliminated by first filtering the spectrum with an RC filter to remove the "wiggles". It is clear from section 2. 7 that this is not the best method of going about things. If for some reason (e.g., the sweep rate is sufficiently slow, the magnetic field is inhomogeneous, etc. ) the "wiggles" disappear, the Lorentzian assumption is not too bad. Figure 3. 2 . 11 shows the results of using the above Lorentzian resolution enhancement filters on a Lorentzian line. The achievable resolution is not as great as that from fi,_tering an n. m. r. signal containing "wiggles" with the correct filter. This is because the natural diffraction pattern of a Lorentzian • 68 I \. !j· .;r I / - - -- -- / --~-/ I ~~·~c q = 0 !!!~:_ _ __ 'i' i2!! i 'i ---- / r\\ I i ,, / / q = 100 0 ~§ , - - --- - -- -- i I I '"'-___ --· '\ i / I I\ ' I I \ I ~ ('-"·--- \) q = 3 000 I.ODO --',,'-;:_QUE_NC _ Y_ IH_ H_Z ~=...---~•.=-cm· i~0: .. . . .. Figure 3. 2. 10. The results of using the weighting functions based upon the correct n. m . r. line shape (left) and those based upon the Lorentzian line shape (right). . '• .. _:_ 69 ~- i1l~,,--··-~--------i~r:--:;:)------,~~--·- ·- ---1;~;;~----···-7~~'.; Fi':i.: O~f~<CY Figure 3 . 2. 11. / IN tlZ ··- ·-... ~ .. j.::. Resolution enh--1ncement of a Lor entz ian line using the correct 1ine sha pe function. The inner lines represent successively higher resolution e nha ncement. line is not as good as that of an n. m. r . signal containing "wiggles ". Figtffe 3. 2. 11 does indicate however, that these Lor entzian filters cari be quJte effective if the line shape is s irnilar to a Lor e ntz ian. The results of us ing a Lorentz ian res olution enhancement filter on the ex p2rimenhtl n . m. r. signal (F'ig·. 3. 2.10) are quite sirnilar to the mir ror i m a ge of a weighting funcUon b ased upon t he s ame n. rn . r . signal (Fig. 3. 2 . 9) . The similar ity renlains even Ha Lorentzian line is filte red (s e e Figs. 3. 2. 11 and 3. 2 . 9) . under sta nd in freql1 ency space. This sh:n ila rity is easy to The results of using a Lorent z ian 70 resolution enhancement filter are : c(q1) L(w) Si (w) (3 . 2. 2) 2 1 + q 1 L (w) which has the same phase as H*(w) , since L(w) is real and: c(q2 ) Si(w) (3.2.3) 1 + ~1si(w)l2. It is quite reasonable that these two expressions should be almost equal for some value of q/q2 • The experimental n. m. r. signal is more closely related to a Lorentzian line shape than the above discussion indicates . In addition to .Q(t) being the limiting form of si(t) for slow sweep rates, L(w) is I I an excellent approximation to S(w) for any sweep rate. We shall first demonstrate that this statement is true, and then use it to determine some general properties of linear resolution enhancement filters. Consider the normalized Lorentzian function : l(t) = which has a full width at half height equal to 2 b p_. transform of l(t) is given by: (3. 2 . 4) The Fourier 71 0 w Figure 3. 2. 12. The Fourier transform of the experimental n. m. r. line. The curve is an exponential. 72 = L(w) -iwt 1 1T b.f ~00 1 + (t/ b £)2 dt. (3. 2 . 5) Integration gives the simple result: = e -b£1WI. L(w) (3.2 . 6) The modulus of the Fourier transform of the experimental n. m. r. signal also decays exponentially (Fig. 3. 2.12). We can therefore make the approximation: IH(w) c (q ) e I -bs Ir.vi for the frequency response function. as the phase of the modulus, st ( (3.2.7) The phase of H(w) is the same This will depend upon the sweep rate, however , w). IH(.:..v) j, has a form which is independent of sweep rate . The exponent, bs, increases with increasing sweep rate, but the form of IH(w) I does not change. If q is between zero and one, this function has a single maximum at :..v = 0 (Fig. 3. 2. 2). If q is greater than one, IH(w) I will have two maxima at: 1 w q = ± 2b" £n q. s If we define: (3. 2. 8) 73 (3.2.9) then it follows from Eq. (3. 2. 7) that: (3. 2. 10) IH(w) I The effect of changing q is simply to change the position of the maximum, w q· Each branch of IH(w) I is symmetric about this maximum (Fig. 3. 2 . 2). We shall return to the approximation: -bslwl e ' (3. 2. 11) when we discuss line shapes. Diffraction patterns have long been used to measure positions accurately. Since the linear filters we have developed make effective use of this technique, it seems rather unlikely that any modification of our definition of line width would lead to a significant improvement in resolution enhancement. The qualitative conclusions we have reached in this section apply to any n. m. r. signal, whereas the quantitative results are a function of the sweep rate. The two line shapes studied above indicate the nature of the dependence of these results upon the sweep rate . We shall study the details of this dependence later. 74 We concluded in section 2. 1 that in addition to decreasing the resolution line width we wanted to make the signal approach zero rapidly. Figure 3. 3. 1 shows the line shapes of filter e d i:;pcclra for several values of q in Eq. (3. 2.1). These lines obviously have both of the desired properties. The line shapes are similar to those resulting from filtering a Lorentzian with an optimum filter 24 (Fig. 3. 2. 11). In both cases "wiggles" are introduced in the tail of the line shape. These "wiggles" grow with increasing resolution enhancement. ~ 'I gl ~i- i ~lo: .,j ~r- ~L 01 s!1 ~1=------~~~~..;:=-.---~ ::! ? Figure 3. 3. 1. The line shape of the experimental n. m. r. signal after resolution enhancement. The outside line results from a matched filter. The inside lines result from successively higher values of q in Eq. (3. 2.1) . 75 The general features of these line shapes are independent of the sweep rate. The Fourier transform of the line shape resulting from a linear resolution enhancement filter is given by: I c(q) Si (w) 12 = l + qJSi(w)J (3.3.1) 2 ' The output from a rnatched filter (q = 0) is then : (3.3.2) Substituting the approximate form of !Si(w) I given by Eq. (3. 2. 11) we find: (3. 3. 3) The output from a matched filter is therefore : (3. 3. 4) That is, if q equals zero, then the output line shape is approximately Lorentzian. If q is much greater than one, S0 (w) is approximately a step function (Fig. 3.3. 2). Limits (w) q-o0 0 That is, one can make the approximation: ~ { c(q) / q ; 0 ; lwl lwl ::: } (3. 3. 5) 76 r--------:.;;;------..,,------ ---1 1 I I I I I I I I I I I I I I I I I I I I I I I I I I I I 0 w Figure 3. 3. 2. The Fourier transform of a filtered line. Taking the Fourier transform gives: Limit . c(q) s (t) =. -rrq q-o00 l' sin (wq t) t j' (3 . 3.6) ' for the output line shape. As q increases, so does wq. the frequency of the wiggles to increase . This causes The full width at half height of this line shape must be determined numerically. The result is: 3.7910 Limit v q - 00 0 r.uq (3. 3. 7) 77 Substitution of Eq. (3. 2. 8) for ·.uq gives: Limit v q - ~ 0 * 7.5820bs fnq 1. 8955 = _ _ _ vm, (3. 3. 8) .Qn q since the output line width from a matched filter is 4 bs [ Eq. (3. 3. 4) ]. As q approaches infinity, S0 (t) approaches a Dirac delta function according to Eq. (3. 3. 6). If we wish to resolve two overlapping peaks of greatly differing amplitudes, these "wiggles " may prevent us from doing so. A method of avoiding this problem by using nonlinear filters will be discussed later. We shall use linear filters only for lines of comparable magnitude. A quantitative measure of the desirability of a line shape is the ratio of the observed separation to the exact separation for two closely spaced lines (Fig. 3 . 3. 3). Ideally this ratio should always be one. In practice it is zero when the separation is less than the resolution line width and deviates from one for larger separations . The "wiggles" in a resolution enhancement line shape cause this ratio to oscillate about one . The observed separation is not a lower bound to the true separation, however the magnitude of the deviations has · not increased. The resolution line width therefore seems to be an adequate measure of the resolution of these types of line shapes. The situation is somewhat more complicated for other line shapes. Figure 3.3. 4 shows an n.m.r. spectrum of two overlapping lines obtained by addin g together two of the above pentadiene spectra. 78 1.2...------------------~ .2 ' ~ .' ....... . .. ·····--- -····-·-··- ·· · · -- - Figure 3 . 3. 3. The observed separation of a symmetric doublet as a function of the actual separation. Curve A results from a Lorentzian line shape. Curves B and C result from the experimental n. m. r. signal using a matched filter and a resolution enhancement filter respectively. l\A I . / I /\ v,. I ....- o.o -----T.mJ--~, I I \ I}\ . \ lt\ \1 - -,""=.-=-----..,.., ,._= --- - _,,__ FAEOUEtCY IN HZ Figure 3. 3. 4. The n. m. r. spectrum of two overlapping lines before (top) and after (bottom) using a mat ched filter . 79 The raw spectrum is rather difficult to interpret. After filtering this spectrum with a matched filter it is obvious that the spectrum represents a symmetric doublet. The matched filter made the line shape symmetric and easy to interpret. In general, we obtain the Fourier transform of the output line shape by taking the product of the Fourier transform of the input line shape and the frequency response function of the filter : (3. 3 . 9) In the case of an optimum linear resolution enhancement filter based upon the correct line shape this leads to the output function: c(q) Si*(w) Si (w) (3.3.10) I 1 + q S/w) 12 Since S0 (w) is real, the output line-shape function s 0 (t) is symmetric (see Section 2. 2). Only a filter based upon the correct line shape can lead to a symmetric filtered line . The above ;example indicates that using the correct line shape can be important beyond the obvious advantage of a decreased resolution line width. 3. 4 Signal-to-Noise Ratios .........,,....._,..._........ ,....,,....,., .,.....~,-...,.._,.._. ~ Now we consider exactly how much signal to noise we must sacrifice for a given improvement in resolution. The signal-to-noise level was calculated using Eq. (2. 8. 6) and the resolution was 80 measured using the full width at half height since this quantity is easily obtained. The results for several combinations of line shape and filter are shown in Fig. 3. 4. 1. The best resolution improvement clearly results when there are "wiggles" in both the signal and the line shape the filter is based upon. An n. m. r. line with "wiggles" gives considerably better results than a Lorentzian line shape when the correct resolution enhancement filter is used for both (A and D D c c >( .1 0 ~E z ' V) ;S;;: z ~ .01 .001;-'--'--o - -,------ +-- - - . - - -- - , . - - -- - - l - .5 1.5 2.5 Y./~, Figure 3. 4.1. The output signa(fo nol.se -asa-Iunction o:f the line width, where Vi is the input full width at half height and v 0 is the output full width at half height. Curves A and B result from filtering the experimental n. m. r. signal with a resolution enhancement filter based upon the correct and Lorentzian line shapes respectively. Curve C results from filtering this n. m. r. signal with an RC filter. Curve D results from filtering a Lorentzian line with a resolution enhancement filter based upon the correct Lorentzian line shape. 81 in Fig. 3. 4.1). Enhancement of the above experimental n. m. r. signal with "wiggles" is roughly equivalent to that of a Lorentzian line with half the full width at half height. The results of filtering this n. m. r. signal with a Lorentzian-based filter (Bin Fig. 3. 4.1) could not be measured for high resolution enhancement since the resulting line shape (Fig. 3. 2. 10) becomes too ambiguous to interpret. The line width resulting from using an RC filter (C in Fig. 3 . 4. 1) is never less than the original line width. As RC approaches zero the RC weighting function approaches a Dirac delta function (i.e., it becomes equivalent to not using any filter at all). Figure 3. 4. 2. The relative signal-to-noise ratio as a function of the parameter q in Eq. (3. 2. 1). 82 In practice, the resolution enhancement and consequent loss of signal to noise are determined by our choice of the parameter q in the frequency response function LEq. (3. 2. 1)]. The loss in signal to noise is a smooth monotonic function of q (Fig. 3. 4. 2) when the correct line shape is used. The loss in signal to noise varies only slightly between the Lorentzian and the experimental n. m. r. line shapes. That is, it is independent of the sweep rate. Thus, it is possible (and obviously desirable) to know in advance how much signal to noise we shall lose. We sought a simple expression for the loss of signal to noise as a function of q. The following relationship was empirically deter- mined: (3. 4. 1) This equation predicts signal-to-noise losses of up to one thousand to within 30% for all sweep rates of interest. Our program requires only the tolerable signal-to-noise loss (S/N)T as input. It then calculates q from: (3.4.2) When using the program one should keep in mind that a matched filter can give an improvement of up to ten to one over the signal-to-noise ratio of the raw data. 83 The empirical relationship in Eq. (3. 4. 1) is adequate for the useful range of q. A more accurate relationship between q and the signal-to-noise loss can be obtained from the asymptotic behavior of linear resolution enhancement filters. form of lsi(w) We shall use the approximate I given by Eq. (3. 2.11) and the asymptotic form of S0 (w) given by Eq. (3. 3. 5). This leads to the approximation : ~ bslwl ~i~~ IH(w) I ~ q e (3.4. 3) w > wq 0 for the asymptotic form of the frequency response function. From Eq. (2. 8. 3) we see that the loss in signal-to-noise from resolution enhancement is given by : 1 2 (S/ N) 0 (S/ N)m = so(t)max 8 m(t)max 00 2 J IH(w) I dw . (3.4 ~ 4) -00 Integration using the above approximations for gives: ISi (w) I and IH(w) [ 84 l Limit (S/ N)o g -oo (S/ N) m [ c(q)wq/ 7Tq] [ 1/ 2 bs] 2 [ 1/ 2 1T bs] (3.4. 5) Substitution of Eq. (3. 2. 8) for wq gives: Limit (S/ N) 0 q -oo (S/ N)m • .fn q . -1- , (3.4.6) q2 for the asymptotic behavior of resolution enhancement filters. This suggests the . approximation: (S/N) 0 (S/ N)m .. 1 + .fn(q + 1) (3. 4. 7) 1 + q2 which is accurate to within 15% for all q greater than or equal to zero. Unfortunately, we cannot solve Eq. (3.4 . 7) for q as an explicit function of the signal-to-noise loss. We therefore retain Eq. (3. 4. 2) as our basis for selecting q. We have used the full width at half height to compare several filters . A more realistic comparison would use the resolution line 85 width (sec. 2. 1) for two equal lines. The resolution line widths of some typical line shapes are given in Table 3. 4.1. Although they vary TABLE 3. 4.1 Line Shape (R. L. W. )/(F. W. H. H.) Lorentzian 0.864 Experimental N. M. R. 0.664 Exp. Matched Filter 0. 511 Exp. Res. Enhc. (q = 3000) 0.950 by almost a factor of two, they do not effect any of the qualitative conclusions reached on the basis of the full width at half height. The variation of signal-to-noise ratio with line width (Fig. 3 . 4. 1) has been recalculated on the basis of Table 3. 4. 1. The two cases of principal interest are filtering the experimental n. m. r. line and the Lorentzian line with the correct resolution enhancement filters. Figure 3.4. 3 shows the variations in signal to noise with resolution line width for these two cases. The Lorentzian curve (B) is almost unchanged from that in Fig. 3. 4. 1. The curve (A) for an n. m. r. line w'i th "wiggles" shows greater improvement for low resolution enhancement. Even a matched filter can give an improvement in the resolution line width. The curves shown in Fig. 3. 4. 3 are two members of the family of 86 ~ .1 ~E 2: ...... V) ~ 2: ....... ~ .01 .00 1 ;-'~~~--.-~~~---.....---~~~--,.-~~~-+ 0 (IU.W. >o/(1u.w1. I Figure 3. 4. 3. The output signal to noise as a function of the resolution line width. Curve A results from filtering the n. m. r. signal with ' 1wiggles 1 1 using the correct resolution enhancement filter. Curve B results from filtering a Lorentzian signal using the correct resolution enhancement filter. 87 possible curves for resolution enhancement in an n. m. r. experiment with varying sweep rate. Curve B (o: = 0) appears to be the limiting worst case. Significant improvement in resolution can be achieved in any n. m. r. experiment for which we can measure the line shape. We shall now consider the effects of sweep rate in more detail. Changing the sweep rate has dramatic effects upon all aspects of an n. m. r. experiment. To understand the effects involved we consider briefly the physical nature of a typical n. m. r. experiment. If nuclei are placed in a homogeneous magnetic field in the z direction, they tend to align themselves with the field and precess about this direction with the Larmor frequency v0 (Fig. 3. 5.1). A net magnetization ~o develops in the z direction, with the characteristic time constant T 1 • If the magnetic field is turned on at t = 0, then: M "z (3. 5. 1) where M .,...z is the instantaneous magnetization in the z direction. If an rf field, 2 !:! 1 , of frequency v0 is applied in the x direction, it will have components of magnitude H 1 and frequency v0 rotating clockwise and counter-clockwise in the xy plane. The clockwise component is stationary with respect to the nuclei, while the counter-clockwise 88 z / .' I \ ' --------- ---) '' ;; / , ------------ x Figure 3. 5 . 1. Precession of a nuclear magnetic moment µN about a magnetic field in the z direction. 89 component rotates with frequency 2 v0 with respect to the nuclei and has no effect on them . The nuclei will precess about the "stationary" clockwise component. This leads to an induced magnetization in the xy plane, M ....,xy, which trails the clockwise component of the rf field by 90° (Fig. 3. 5. 2) . If !,!1 is turned off, Mxy decays with characteristic time constant T 2 • That is: (3. 5. 2) where !!1 was turned off at t = 0. The torque exerted by !! 1 on ~xy leads to absorption. If the frequency of the rf field, · v rf' is slightly different from v0 , !!1 will slowly rotate with respect to Mxy· If v rf has been increased sufficiently past v0 , then !! 1 no longer affects Mxy and Mxy will rotate at exactly v0 • The rate of change of e (Fig. 3. 5. 2) is then given by : (3. 5; 3) where e is measured in radians. If the frequency (or field) sweep is linear and we passed through resonance at t = 0, then: (3 . 5. 4) where a is the sweep rate in Hz. integrating we obtain: Combining these two equations and 90 --- -- ---, 1' I I I I I I I I Mxy Figure 3. 5. 2. The rf field leads to an induced magnetization in the xy plane. e = 1T at2 + 00 , (3. 5. 5) or: (3.5.6) The torque exerted by H ....., 1 on M .....,xy depends upon the sine of e. We . observe alternating absorption and emission, or "wiggles". The 91 maxima are determined by the condition sin (} = 1. Since e0 is approximately 1T/2: . (} max - eo = 1T-a (v r f - v a) where n is an integer . 2 • =· 2 7Tn ' (3 . 5 . 7) The maxima of the "wiggles" therefore occur when: n = O, 1, 2, . ... (3. 5. 8) The positions of the "wiggles" depend only upon the sweep rate and not on the relaxation times or the magnetogyric ratio of the nuclei (Fig. 3. 5. 3). Therefore, for purposes of resolution enhancement, Figure 3. 5. 3. The approximate positions of the maxima in an n. m. r. experiment. 92 line shapes may be measured by using standard samples. We shall have problems only if either the standard or the unknown is sufficiently broadened to make the "wiggles" disappear. The full width at half height of these "wiggles " is roughly the time it takes for e to change by 2 rr /3 (Fig. 3. 5. 4), so that : "'n =.= v'2(n + 1/3)a - v'2 na ; n =012 · · · ' ' ' . (3. 5. 9) Although this relationship is not accurate for small n, it does give a rough approximation for the full width at half height of the first maximum: V·1 0. 817 ~. (3. 5. 10) I I :H1/2 I H1<t2) -------------~------ - ----I H1(~) I I I I I I • Figure 3. 5. 4. The projection of !! 1 perpendicular to MxyNote that: 8(t2 ) - 8(t 1 ) = 2 rr/ 3. 93 This approximation should be more accurate for large o:, since vrf becomes larger than v0 very quickly. 25 . 20 Ernst and Anderson have solved the Bloch equations irnmerically by the Runge-Kutta technique 26 and found that: Limit j). a-oo 1 0.66~. (3. 5. 11) Our simplified model is really then not too bad in that vi turns out to be proportional to Va , even though with a different proportionality constant than shown in Eq. (3. 5. 11). In order to determine the detailed effects of sweep rate on filtering n. m. r. spectra, we return to a somewhat more pragmatic approach. We have recorded the line shape of the proton signal from chloroform over a wide range of sweep rates. The observed line widths are shown in Fig. 3. 5. 5. The asymptotic form [Eq. (3. 5. 9)] is an excellent approximation at the faster sweep rates, This experimental curve agrees very closely with the theoretical curve obtained by Ernst and Anderson 25 if w~ assume that T = T -;-1 3 sec. 2 The two relaxation times will in general be equal for a non- viscous isotropic sample, and 3 sec is a reasonable relaxation time for protons. 27 The relaxation times of other nuclei with I= i, and whose chemical environments are comparable to those of protons are . b y: 27 given (3. 5 . 12) 94 100-.--- -- - 10 --, - .1-+---~-~--:- _0 01 _m .1 1 10 100 1000 10000 Swe ep Rate (Hz/ sec.) . - ---·---- . ·····-·-- - --· Figure 3. 5. 5. The line width of the proton signal from chloroform as a function of sweep rate. Thus, 1 3 C relaxation times s hould be about 16 times as large as proton relaxation times. This leads to 13 C line widths which would be m uch narrower than the homo geneity of the magnetic field. Fortunately, an inhomogeneous field has approximately the same effect ~-s decreasing the relaxation times. 28 We can rationalize this by considering an inhomogeneous field to be a random field with a long correlation time . 29 If we calibrate our sweep rates in ter ms of line shapes (e.g., the number of "wiggles") rather than Hz, the ratio of T / T 2 will have only minor effects upon these line shapes. In this sense, the conclusions we reach below will be quite general. 95 The signal-to-noise ratio of an unfiltered spectrum varies considerably with the sweep rate. The nature of this variation depends upon how we change the other parameters of our n. m. r. experiment. We shall assume that we are interested in a given region of the spectrum, from v 1 to v 2 • We therefore keep the sweep width constant and vary only the sweep time. We shall also assume that we have a constant amount of total instrument time available, so that the number of scans we record is proportional to the sweep rate. Finally, we assume that we carefully adjust the strength of the rf field for the maximum signal-to-noise ratio at each sweep rate. Ernst and Anderson 25 have determined the theoretical variation of the signal-to-noise level with sweep rate under these conditions (Fig. 3. 5. 6). For sweep rates above or below the region where saturation varies, the signal-to-noise ratio of an unfiltered spectrum approaches a constant value. If we calibrate the sweep rate as discussed above, this curve shows the same general behavior for any ratio of T / T 2 • We shall use this theoretical ~ariation of the input signal-to-noise level since the noise produced ·during amplification is s trongly dependent upon instrument design. The signal-to-noise ratio of a filtered spectrum is of considerably greater interest. The signal-to-noise improvement from a matched filter is proportional to the square root of the number of data points under the line (Sec. 1. 1). The signal-to-noise ratio resulting from the use of a n1a t ched filter is therefore approximately : 96 .9 ~ .8 ~E :z ..... V') .s: z ~ .7 .6 ,5,-r--.001 - . - -- , - --.-.01 .1 - . - -..,....-- ..,....---4 10 1000 10000 100 Sweep Rote (Hz / sec.) Figure 3. 5. 6. The theoretical variation of the input signal-to-noise ratio with the sweep rate . (3. 5. 13) where k 1 is a constant. Although the "wiggles" will dominate the signal-to-'-noise ratio at extremely fast sweep rates, they do not make an important contribution at the sweep rates available with our spectrometers. Recalling that the input signal-to-noise ratio approaches a constant value for large sweep rates and using Eq. (3. 5. 11) for the input line wi.ctth we find that: 97 1 (S/N)m ~ k 2 a 4 . (3. 5. 14) This approximation is valid for moderately large values of a. The general variation with sweep rate of the signal-to-noise ratio resulting from using a matched filter was calculated from Eq. (2. 8. 5) using Fig. 3. 5. 6 for the input signal-to-noise ratio (Fig. 3. 5. 7). We previously observed that lines with "wiggles" were not broadened by matched filters as much as Lorentzian lines were. This is essentially an artifact caused by our restriction of the definition of 100.-----------------, .1 10 Sweep Figure 3 . 5. 7. 100 1000 10000 Rote (Hz/ sec.} The slgnal-to-no:lse ratio--resuTffiig from using a matched filter on the proton signal from CHCl3 • 98 line width to the first maximum. As we increase the sweep rate and more "wiggles" appear, the "average" width of a "wiggle" decreases relative to the width of the first maximum ( Eq. (3. 5. 9)]. Since a matched filter uses the entire line shape, the line width resulting from use of a matched filter depends upon the "average" line width of the input line shape. The ratio of the width of the first maximum to the line width after using a matched filter therefore decreases with increasing sweep rate (Fig. 3. 5. 8). O +-~~~-.--~~-~--,-~~~-"T"""~~~-+ DI 10 Sweep Figure 3 . 5. 8. 100 Rate (Hz/s ec.) The ratio of the full width at half height after using a matched filter to the width of the first maximum before filtering for the proton signal from CHC1 3 as a function of sweep rate. 99 The effects of sweep rate on resolution enhancement filters may be determined by considering the filtering process in frequency space. The Fourier transform of the line shape resulting from a resolution enhancement filter is given by: c(q) Si*(w) Si(w) 1 + qjSi(w)j 2 ., (3. 5. 15) where Si (w) is the Fourier transform of the input line shape. Since filters take simple products in frequency space we can easily break this filter down into a sequence of two filters: (3 . 5. 16) where the first filter is a matched filter. The Fourier transform of the output line shape is therefore : . Sa(w) = . l c(q)/c(O) JSm(w), 2 (3.5.17) l+qls/w)I where Sm (w) is the Fourier transform of the line shape resulting from using a matched filter. The line shape resulting from a matched filter is always symmetric (see Sec. 3. 3) and except for changes in line width shows littlevariationwithsweeprate (Figs. 3.2.11, 3.3.1, and3 . 3.3). The frequency response function in Eq. (3. 5. 17) can also be expressed 100 in terms of Sm (w). The Fourier transform of the output line shape is then: (3. 5. 18) It is clear that the only effect of sweep rate on resolution enhancement filters is through the line width obtained by using a matched filter. If we plot the signal-to-noise ratio as a function of the line width as in section 3 . .4, but normalize the line width to that obtained by the use of a matched filter, then the resulting curve shows no observable trend with changes in the sweep rate (Fig. 3. 5. 9). The asymptotic form of these curves is easily determined by substituting Eq. (3. 3. 8) for !n q in Eq. (3. 4. 6). We find: Limit (S/ N) 0 q - ~ (S/N) m .;. 1. 8955 ( :m) e -1. 8955 vm/ 2 v 0 (3. 5. 19) 0 which is accurate to within 30% for all q greater than or equal to zero. The sweep rate has an important effect on resolution enhancement which does not explicitly involve the filtering process. Both the signal-to-noise ratio and the line width of the input signal vary with t he sweep rate (Figs. 3. 5. 5 and 3. 5. 6). If we plot the signal-to-noise level resulting from use of a matched filter as a function of the line width after using a matched filter, we obtain a curve indicating how the resolution may be enhanced by decreasing the sweep rate . At any sweep rate the resolution may be further enhanced by filtering (Fig. 101 3. 5. 9). The possible results of the combination of these two resolu- tion enhancement processes is indicated in Fig. 3. 5. 10. The curve which is tangent to each of the smaller curves indicates a procedure in which we optimize both the sweep rate and the filter . First, we estimate the line width which will permit observation of a coupling. Then we select the sweep rate (using Fig. 3. 5. 10) such that a >( 0 _e z ....... V) .1 ~ z ........ V) ~01-+'------...----+ .1 Figure 3. 5. 9. .5 - - - -- -------··· The loss in signal-to-noise ratio resulting from resolution ehnancement. The solid line was obtained by filtering a Lorentzian. The points shown were obtained by filtering the above CHCl 3 spectra. v0 is the output full width at half height and Vm is the width resulting from using a matched filter. 102 1 0 0 0 - . - - - - - - - - - - - - -- -- - - - . 100 10 10 100 Figure 3. 5. 10. Resolution enhancement in which we vary both the sweep rate and the filter. The numbers at the points indicate the sweep rate in Hz. This curve was obtained from the above CHC1 3 spectra with the exception of the point a = 0, which was obtained from the asymptotic form. resolution enhancement filter can bring us to the tangent curve at the desired line width. After collecting the data we then filter the spectrum using the proper value of q so that we achieve the desired line width. This procedure gives the maximum signal-to-noise for a given Lne width when linear filters are used for n. m. r . signals. When we select the sweep rate we must keep in mind that it cannot be so fast that the nuclei will not return to equilibrium for the 103 next scan, nor so slow that we cannot see a peak to "tune" on in a single scan. The latter consideration seriously limits the use of extremely slow sweep rates for 13 C n. m. r. In this case, it is critical that we can use the correct line shape for resolution enhance ment (see Sec. 3. 2). Although Fig. 3. 5. 10 was obtained from experiments using a specific sample, the basic form of this curve is quite general. The only important variation is in the scale for the sweep rate . This scale can be adjusted by considering the point labeled 0. 1 to represent the minimum sweep rate for which the first downward "wiggle" goes below zero. 104 4. NON-LINEAR FILTERS These filters are easier to describe than linear filters, but are in general more difficult to apply. The effect of a filter on a single line is given by: (4. 1) where si (t) is the input line shape and s 0 (t) is the output line shape. The effect of this filter on a spectrum composed of two lines is then: (4. 2) where H 1 2 is the non-linear interaction term. This interaction term "' may be comparable to the linear terms . We therefore must know (or assume) the multiplicity of a spectrum before we can apply nonlinear ,:-ilters . Although we could filter a spectrum assuming several different multiplicities and afterwards select the best one, linear filters are clearly simpler to apply in this case. In the discussion which follows we shall assume that the number of lines is known. A very useful group of filters is obtained by a statistical approach. We seek a basis for deciding that some quantity, x, lies in a certain interval. (4 . 1. 1) 105 This decision is clearly valid only if: (4 . 1. 2) where P(X) is the probability that X is true. Since this condition does not uniquely define x 1 and x 2 , any method of measuring x must include additional conditions. There is no rigorous approach to defining the best set of additional conditions. A reasonable approach might be to find the smallest interval which satisfies Eq. (4. 1. 2). This does not appear to be very practical since it does not suggest any method of finding this interval. A very common approach is the principle of maximum likelihood. 30 We select Xo such that P(Xo ~ x ~ x 0 + dx) is a maximum. That is, we use the most probable value of x as an estimate of the true value. We then use the limits to which we can determine Xo for the values of x 1 and x 2 • It is only the interval (x11 x 2 ) that has any statistical significance. Although Xo is the most probable value of x, the interval (x1 , Xz) is not generally the smallest interval satisfying Eq. (4. 1. 2). There is no justification for assuming that the principle of maximum likelihood is the most powerful approach to measuring x. Later, we shall examine an important example for which it is not the most powerful approach. If we wish to determine f, we can do an experiment and obtain a value fi which will differ from f by the error or noise. The probability that the error, fi-f, lies between f' and f' + df is given by: (4. 1. 3) 106 where <j>(f') is the probability density function 31 of the noise. If we measure fa total of NT times, then the definition of the above probability is: I •• P(f ~ fi - f ~ f' + df) = L 1m1t - NT-oo l- N(f' ~f. -f~f' +df) Jl NT . ' --~1~---- where N(X) is the number of measurements that satisfy X. (4 . 1. 4) From Eqs. (4 . 1. 3) and (4. 1. 4) we see that we can measure </>(f') by using the following approximation : i:/J(f' + Af) 2 if we know f. . N(f' ~ fi - f~ f' + Af) v'N(f' ~ f f ~ f' + Af) J. l NT ' ± 1 - ~f (4.1.5) The probability density function of the noise spectrum referred to in section 2. 5 is shown in Fig. 4. 1. 1. The true value of the zero point was assumed to be the average value n. The probability density function of this noise is not distinguishable from a Gaussian. This is the probability density function for the noise measured at different times. In a multiple-scan experiment, the value of the signal at any point in the spectrum is the average of measurements made at many different times (the individual scans). The average over this ensemble of scans is therefore in reality a time average. That is, regardless of the nature of the noise in a single scan, the noise in a multiple-scan experiment is effectively ergodic. We can therefore assume that the noise at each data point of an n. m. r. spectrum has a Gaussian probability density function: 107 75~-- - - - - - - -- -- -- - I 50 25 0-1--a......i.....,,.._~-~-~-~-~=--=-..;"'llo-l............~ -4cr -2cr 0 4cr 2cr n-n Figure 4. 1. 1. The points are the probability density function of noise from the DFS-60 spectrometer. The curve is a Gaussian. (4. 1. 6) where a is the variance of the noise. The probability density function for the N data points of the spectrum is thus : 1 (211') ~Nv' I ~I 32 1 ( t -1( fi - f ) e -2 .fi . . . - .f) ... c , .. .... .... (4. 1. 7) where ! is the vector with components f (tj), fi is the vector with components fi(tj) and l is the transpose of!· The covariance matrix, 108 S2, has components: 00 - J [fi(tj) -f(tj>J [fi(tk) -f(tk)J <P(!i -D tl! . (4.1.8) - 00 This is an ensemble average . Since the noise is effectively ergodic we can substitute a time average: ao cjk = f [fi(tj)-f(tj)][fi(tk)-f(tk)] dtj, (4.1.9) -00 or: 00 J [fi (r) - f(r)] [ fi (tk - tj + r) - f(tk - tj + r)] dT. (4.1 . 10) -ao The last expression is simply the autocorrelation function r(tk - tj) for the noise (Sec. 2. 3). In section 2. 5, it was shown that the noise in an unfiltered n. m. r. spectrum is white. The autocorrelation function is therefore a delta function (Sec. 2. 3). We can check this by calculating this integral using the above experimental noise spectrum. The result (Fig. 4. 1. 2) clearly indicates that the assumption that r(tk - tj) is a delta function is quite good. The components of the covariance matrix are simply: (4.1.11) and the determinant, I ~I, is equal to a2N. lOS :· . . 2- Figure 4.1. 2. The autocorrelation function [ Eq. (2 . 3. 35)] of noise fr om the DFS-60 spectrometer. The t otal pr obability of the spectrum is: p (4 . 1 . 12) 110 which is the product of the probabilities of the individual data points. The data points are therefore independent for white ergodic Gaussian noise. The probability is a maximum when the exponent in Eq. (4. 1. 12) is a minimum. The principle of maximum likelihood leads to a simple least-squares procedure for white Gaussian ergodic noise. We have shown that the noise in an unfiltered n. m. r. spectrum has these properties. The principle of maximum likelihood therefore gives the form of f which minimizes the function: (p = 0 [f.(t.) -f(t.)] 2 , j=l 1 J J (4. 1. 13) as the most probable form, ,.... f 0 , for a multiple-scan n. m. r. spectrum. The situation is more complicated if the noise is not white. lf an RC filter were used, the noise would still be Gaussian 33 and ergodic, but not white. We should then have to minimize the exponent in Eq. (4.1. 7). That is, we must find the minimum of the more general function: t -1( <I>= ( ,t-D <2 !i-D, (4. 1. 14) in order to determine f0 , the most probable ,....f. We shall limit our ,.... discussion to the simpler case of white noise. 111 Let a spectrum be composed of n lines with kj variable parameters for the line shape of line j, where j = 1, 2, · · · n. eters may include position, height, width, etc. These param- There are M variable parameters in all, where: n M = ) ' k •• (_j (4. 2. 1) j=l ] Let ~be a vector of dimension M whose components are the line-shape parameters. Then the spectrum is composed of the sum of the n lines, f(t, ~' and random noise. The function, f(t, ~ may be known either analytically or through a table of numerical values. The most probable value of b,.... is the value which minimizes the function: QO ~ = f [fi(t) - f(t,~]2 dt, (4. 2. 2) -00 where fi (t) is the experimental spectrum. This is in general a non:- linear problem which is ill-conditioned. That is, the bj are strongly coupled, and the problem is difficult to solve numerically. This 'a pproach does have the advantage that there is no systematic error in the result 34 as with linear filters (Sec. 3. 3). If the signal-to-noise ratio is sufficiently large, then the observed separation of two lines exactly equals the calculated separation. The values of the peak positions obtained in~ are the most probabl~ values. If the number and relative size of the lines is known, then the statistical approach 112 is feasible. We shall use two simple filters to illustrate the above remarks. We first examine filtering an input spectrum fi(t) composed of a single line with known line-shape function si (t). The only unknown parame.ters are the position and amplitude of the line. The theoretical spectrum is therefore given by: (4.2.3) where a si (O) is the amplitude and o is the position of the maximum of f(t). Substituting Eq. (4.2. 3) in Eq. (4.2. 2) we obtain: .f> = f 00 2 (4.2.4) [fi(t)-asi(t-0)] dt. -00 Expansion of the integrand gives : ao <I> ao oo = J f;(t) dt - 2af si(t- o) fi(t) dt + a J si(t) dt. 2 (4. 2. 5) -00 The value of a which minimizes <P satisfies the condition: ait> aa 00 00 0 2af si(t) dt - 2 J si(t - o) fi(t) dt, (4 . 2. 6) -00 -00 so that: ao amin = -1 00 l J si(t)dt J J si(t - o) fi (t) dt. -00 -00 (4.2.7) 113 Substituting this value of a in Eq. (4.1. 5) gives: 00 = 00 ·; -1 -2 00 J f;(t)dt - l J st(t)dt J LJ si(t - o) fi(t) dt J -"° (4.2.8) -00 This function is clearly a minimum when the second term is a maximum, but the second term is a maximum when the output from a matched filter [ Eq. (1.1. 6)] is a maximum. The position of a single line obtained by the method of least-squares is therefore identical to that obtained with a matched filter. The amplitude obtained by the method of least squares is given by: 00 f 0 (0 0 ) = amin si(O) = si(O) [ J si(t) dt -00 fJ 1 "°si(t-o0 )fi(t) dt, (4. 2. 9) -00 which is precisely the amplitude obtained by using a matched filter. Since Eq. (4. 2. 9) gives the most probable amplitude of a line with a maximum at 50 , for any 50 , we have simply given an alternate derivation of Eq. (1.1. 6). If the input spectrum is composed of a single line (or several lines which do not overlap), the non-linear interaction term in Eq. (4. 2) vanishes and the method of least-squares is equivalent to using a matched filter. It is of interest to consider the more general case of filtering a single line when the noise is not white. The principle of maximum likelihood then leads to seeking the minimum of <l> in Eq. (4. 1. 14). 114 From Eq. (4. 1. 10), we see that if the noise is colored, but still Gaussian and ergodic, then we must find the minimum of the function: 00 ~ = 00 J J [fi(t )-f(t )][fi(t )-f(t )]r-1(t -t )dt dt 1 1 2 2 2 1 1 2, (4. 2. 10) -00 -00 rather than Eq. (4. 2 . 2). The inverse of the autocorrelation function, r- 1 , is defined by : 00 J r(r) r - (t - T) dr = o(t). 1 (4. 2. 11) -00 where o(t) is the Dirac delta function. 'Jf both sides, Taking the Fourier transform we find: (4. 2. 12 ) where we have used Eqs. (2 . 2. 9) and (2. 3 . 33). Substituting Eq. (2.3 . 39) for R(w) we see that : 1 2rrW(w) • (4. 2. 13) This suggests that these "generalized least-squares" filters [Eq. (4. 2. 10)] are described more concisely in frequency space. We now substitute Eq. (4. 2. 3) for f(t) in Eq. (4. 2. 10). By the same arguments us ed to obtain Eq. (4. 2. 9), we see that we can determine the most probable value of o by finding the maximum of the function : 115 00 si(O) 00 J J fi(t si(t -6) r-\t -t dt dt 1) 00 00 -00 -oo 2 2 1) J J si(t si(t r- (t -t dt dt 1 2 (4. 2. 14) 1 2) 1) 2 1) 1 2 Since the denominator is a constant, independent of o, we can rearrange Eq. (4. 2. 14) to give : 00 00 f 0 (t>) = c J si(t2 -o) J fi(t 1 ) r- 1 (t 2 -t 1 ) dt 1 dt2 • -00 (4. 2. 15) -00 We define the intermediate function f 1 (t2 ) by: 0() f 1 (t 2 ) J fi(t )r- (t -t dt dt 1 2, (4. 2. 16) = c J f 1 (t2 ) si[-(6 - t 2 )] dt 2 • (4. 2. 17) - 1 2 1) 1 -00 so that : 00 f 0 (6) -00 As suggested by the properties of r - i, we now take Fourier transforms of Eqs. (4. 2. 16) and (4. 2. 17) giving: c st(w) = 211W(w) (4 . 2. 18) F' (w) i . This is the general form of a matched filter [Eq. (2. 4. 22)]. For Gaussian ergodic noise, the principle of maximum likelihood leads to a "g;eneraliz.cd least-squares" procedure. In the case of 116 non-overlapping lines, this procedure is equivalent to a matched filter, regardless of the power spectral density of the noise. We now examine filtering an input spectrum composed of two closely spaced lines of equal size, such as we find in either half of an AX spectrum. The theoretical spectrum is given by: f(t,~ = a[si(t-o+J/2) + si(t-o-J/2)], (4. 2. 19) where o is the chemical shift and J is the spin-spin coupling constant. The chemical shift can be measured by using a matched filter and so we assume that it is known. The only unknowns are a and J. Substituting this in Eq. (4. 2. 2) we obtain: 00 00 (4. 2. 20) 00 +a 2 J [si(t- o+J/2) + si(t- o-J/ 2)] dt. 2 -00 The last two integrals can be expressed in terms of the convolution integrals for a matched filter: 00 sm(t) - c(O) J si(-T) si(t -T) dT, (4. 2 . 21) -00 and: 00 fm(t) = c(O) J si(-T) fi(t-T) dT. -oo (4. 2. 22) 117 The function we wish to minimize is then: OQ CI> = fi(t)dt - c~~) [fm(o+ J/2) + fm(o-J/2)] [ 00 (4. 2. 23) It follows that :I> is a minimum only if: a = c(O) f 111 (o + J / 2)+f. (o-J / 2) m 2 [sm(O) + sm(J)] (4. 2. 24) Substituting this value of a in Eq. (4. 2. 23) we find: o0 2 ~m(o+J /2) + fm(o-J / 2)]2 -00 1 2[sm(O) + sm(J)] = J f·(t) dt - - - - - - - - - - (4. 2. 25) The most probable value of J is clearly the one for which the second 2 term on the right is a maximum. We shall call this term g (J). The most probable amplitude of lines with chemical shift o and spin-spin coupling constant J is given by: a m in . s 1.(O) (4.2.26) 2 This function is not necessarily a maximum when g (J) is a maximum. ') Both g-(J) and a(J)min were calculated for the spectrum shown in 118 . Fig. 3. 2 . 1. The most probable value of a, a 0 , is not the maximum value . It is the val ue of a(J)min which corresponds to the maximum of g2(J) (Fig. 4. 2. 1). Rather than study g 2 (J), we shall find it more convenient to determine the maximum of the function: g(J) fm(o +J/2) + fm(o -J/2) o-- - - ----r- - ·- ? -"·-.... .. (4. 2. 27) 12[sm(O) + sm(J) ] ~ - --~-- -------- Jo . . -. . ~ .. ·=-· .... -<~·· 2 Figure 4 . 2. 1. The normalized functions, g (<T) and 1 0· to t'l1a ''I)""'Ctr·L1r:n . CO ~'ti1 a(J,.) l l l lll ' I'I'e"' -'-' l"O' ' , 'l l.. -- ' b , '-- ,; J. ~ - sho',vn i n Figllre 3. 2. 1. 119 This function takes an average of fm(o +J / 2) and fm(o-J / 2). That is, it averages the output of a matched filter about the center (t = o). T his is because we assumed that fi (t) should be symmetric about t = o. If we define the "non-averaged'' function h(t) as: (4. 2. 28) then: g(J) = t[h(o+J/ 2) + h(o-J/ 2)] . From Eqs. (4. 2), (4.2.29) (4. 2. 9), and (4. 2.19) we see that the non-linear interaction term is given by: Least-squares filters are non-linear for the case of overlapping lines. Interaction terms appear in both the position and the amplitude functions. Neither is the sum of two analogous expressions involving the individual lines. If a closely spaced doublet is processed with a matched filter , the maxima in the output spectrum are too close together (Sec . 3. 3.). A least-squares filter multiplies the matched filter output by the function: (4. 2. 31) 120 to give a spectrum with the maxima in the most probable positions . A least-squares filter produces maxima in the most probable positions even if a matched filter gives only one maximum at t = o. Since x(t, o) varies by only a factor of ..ff between t = o and t = ao, the maxima of the output spectrum are not much greater than the output at t = o (Fig. 4. 2. 1). We would not expect the simple expedient of multiplying the matched filter output by x(t, o) to be as powerful a technique as using the linear resolution enhancement filters developed earlier. In the next section we shall make a quantitative comparison between the two . 4. 3 9?~~~~~!~~a~~~~2E;::YE-~~~ We shall examine non-linear filters first. The signal-to-noise ratio of g(J) determines whether or not we can measure J and with what accuracy. From Eq. (4. 2. 28) we see that the signal-to-noise ratio of g(J) is given by : (4.3.1) where (S/N)m is the signal-to-noise ratio resulting from a matched filter. This signal-to-noise level is a measure of the ratio of ( g(J0 ) - g(«>)] to the root-mean-square noise level. This is the signal- to-noise level available to distinguish between two lines separated by J 0 and two lines separated by oo (or no lines at all). We are interested in the signal-to-noise level available to measure J. That is, we wish to distinguish between two lines separated by J 0 and a single line 121 (J = 0). From Fig. 4.2.1 it is clear that the signal-to-noise level available to measure J is given by: = g(Jo) - g(O) (S/N) . (S/N)J g(Jo) (4. 3. 2) g The factor of '12. in Eq. (4. 3. 1) arises from the averaging effect discussed in the previous section. In order to make a valid comparison with linear filters we shall omit this factor and consider the function: (S/N)J - g(J o) - g(O) (S/N) . g(J 0 ) m o (4. 3. 3) This function has been calculated for both a Lorentzian line shape (a= 0) and for the line shape shown in Fig. 3. 1 (a = 0. 2). The results (Fig. 4. 3.1) are almost identical for the two cases, if we normalize the separation, J 0 , to the full width at half height resulting from a matched filter. Just as we previously found for linear filters (Fig. 3. 5. 9), non-linear resolution enhancement filters give results which are independent of the sweep rate if we normalize to the line width from a matched filter. We are now ready to examine linear filters. The output of a linear filter is given by: 00 = f hq(T) fi(t- T) dT -00 (4.3.4) 122 ------ -- -- - - --·--- . -·-· -----------~ ------ -- Figure 4. 3. 1. The signal-to-noise ratio available to measure a spin-spin coupling constant as a function of the coupling, J 0 • The line width resulting from a matched filter is vm, and a is the sweep rate in Hz. where q determines the degree of resolution enhancement (Sec. 3. 4). This function has a signal-to-noise ratio which is approximately given by: (S/N)q : 1 _g_ -2 (S/N)m, l+(l37) (4. 3. 5) 123 where (S/N)m is the signal-to-noise ratio resulting from a matched filter. In order to improve our accuracy, we shall determine (S/N)q numerically (Sec. 2. 8), rather than use Eq. (4. 3. 5) . This is the signal-to-noise ratio available to determine whether or not there are any lines in the spectrum. We are interested in the signal-to-noise level available to measure J. It is clear from Fig. 2. 1. 2 that this is approximately given by: (4.3.6) (S/N)q, J where sq(t) is the output line shape of a single line with a maximum at t = 0. This function is independent of the sweep rate, a, since the 1 line shape resulting from resolution enhancement is independent of 1cr (Sec. 3. 3). We therefore computed (S/N)q,J only for the case a = 0. The results show considerable variation with q (Fig. 4. 3. 2). The "wiggles" in these curves for large q result from the "wiggles" in the line shape (Fig. 3. 3.1). The signal-to-noise ratio, (S/N)q, decreases with increasing q [Eq. (4.3.5)] . The line width, vq, also decreases with increasing q (Fig. 3. 5. 9) . Combining these relations, we obtain the family of curves shown in Fig. 4. 3. 3. We can use Fig. 4. 3. 3 to determine the optimum value of q. The signal-to-noise level available to measure J with this optimized process is given by the envelope (the dashed line in Fig. 4. 3. 3) of the family of curves, (S/N) q, J" 124 1 .1 .01 q" 0 25 45000 .001.-t-----......_~--.,--------1 .l 1 10 Figure 4 . 3. 2. The fraction of the signal-to-noise level available to measure J using linear resolution enhancement filters. The output line width is vq. Finally, we are ready to compare the linear and non-linear filters for the resolution of two identical lines separated by J (Fig. 4. 3 . 4) . We have also included the results obtainable with a matched filter. The resolution enhancement of a filter is essentially any improvement over a matched filter. It seems clear that no filter can resolve two lines separated by .J, if the position of a single line cannot be measured to within ±J. We the r e l'orl) include the signal-to-nois e level available to measure 125 qao q=3000 .1 ~e :z: q= 4eooo ' V) ~ z ' ~ q: 7eQOOO .01 ··· · -·- - -·· ----- - -- ---- Figure 4. 3. 3. The signal-to-noise level available to measure J when linear resolution enhancement filters are used. a peak position of a single line to within ±J in Fig. 4. 3. 4. This is essentially an upper bound on the resolution which should be possible. Linear resolution enhancement filters come very close to this upper bound in the region of primary interest. They are more powerful than least-squares filters for any reasonable loss in the signal-to-noise ratio. Least-squares filters perform well under two conditions. The first is the trivial case of large separation between lines. 126 ----- -- - --·----- --- - -Figure 4. 3. 4. The signal-to-noise level available to measure J by various met~ods. ' ····-·- - - . The least-squares filter is then a matched filter and is identical to ' the linear resolution enhancement filter. The second case is of practical importance. If we wish to resolve two lines of drastically different size, the "wiggles" in the tail of the line shape from linear filters will interfere (Fig. 3. 2. 8). A least-squares filter does not suffer from this problem. Resolving lines of drastically different size is considered in the next section. It is convenient to assign a line width to non-linear filters, in order lo discuss the effects of varying sweep rates on these filters. 127 We define the effective line width as follows: veff.x =([~: J vq)const. (S/N)/(S/N)m' (4.3.7) where J x is the minimum separation which can be measured by method x, J q is the minimum separation which can be measured with linear resolution enhancement filters, and v q is the line width resulting from the optimum linear filter. Suppose we have sufficient signal- to-noise that we can afford to reduce the signal-to-noise ratio to 0. 1 times that available with a matched filter. We can then measure a coupling greater than or equal to 0. 25 v m with linear resolution enhancement filters. Using least-squares filters we can measure a coupling greater than or equal to 0. 55 vm. The linear resolution enhancement filter has an output line width of approximately 0. 1 7 v m when the signal-to-noise ratio is reduced by a factor of 10. The effective line width of the least-squares filter is therefore: vL. S. "' { ( ~: ~~ ) 0. 17 vm} (S/N) = 0. 1 (S/N)m" (4.3.8) We have calculated vL. S. for several values of the available signal-to-noise ratio and obtained a curve similar to tru;it previously obtained for linear resolution enhancement filters (Fig. 3. 5. 9). We then determined the envelope of these curves as the sweep rate is varied, just as we had previously done for linear resolution enhancement filters (Fig. 3. 5. 10) . This envelope (Fig. 4. 3. 5) gives the 128 10 · -- ·-·- ··- - -- - -· - - -·---·-·-·-·-- 100 - - ·- - -- - -··· ·· Figure 4. 3. 5. Resolution enhancement by varioQ.s methods in which the sweep rate is varied (see Figure 3. 5. 10) . maximum signal-to-noise level available as a function of the effective lirie width of a least-squares filter. The upper bound shown in Fig. I 4 ~ 3. 5 was obtained by performing the identical calculations for the upper bound in Fig. 4. 3. 4. The time constant of an RC filter can be used as a resolution enhancement parameter analogous to q (Fig. 3. 4 . 1}, however the ,output line width is never less than the input line width. We can therefore obtain a curve similar to that in Fig. 3. 5. 9 for an RC filter. Although the resulting curve for an RC filter varies with the sweep ' rate since these filters are not based upon the exact line shape, we can 129 still determine the envelope of these curves as the sweep rate is varied. The result is shown in Fig. 4. 3. 5. We have also reproduced the analogous curves for matched filters and for optimum linear resolution enhancement filters in Fig. 4. 3. 5. We can now compare all the important filters that we have discussed so far. First, we should note that our definition of the effective line width [ Eq. (4. 3. 7)] was somewhat arbitrary and therefore the details of the curves for the upper bound and the leastsquares filters are also somewhat arbitrary. They are qualitatively significant however . Least-squares filters are definitely intermediate between optimum linear resolution enhancement filters and matched filters. We should also keep in mind that we have no reason to believe that the upper bound in Fig. 4. 3. 5 is the smallest upper bound. The simple RC filter which is built into most commercially available n. m. r . spectrometers is clearly adequate for all routine work. There are two general circumstances when it should not be used. The first is the obvious case of the resolution of lines separated by less than the "natural line width" [J < vm (o:= 0) / 2]. The "natural line width" is usually determined by the inhomogeneity of the magnetic field (Sec. 3. 5). The second case is that of spectra with inherently low signal-to-noise levels, such as natural abundance 13 C or 15 N spectra. For the sweep rates considered (Fig. 3. 5. 10), optimum linear resolution enhancement filters give an improvement in the signal-to-noise level of about a factor of 2 over RC filters (Fig. 4. 3 . 5). This represents a reduction of the number of scans 130 which must be recorded by a factor of 4. Optimum linear resolution enhancement filters are the most powerful filters we presently have for dealing with lines of comparable size. We now consider the problem of filtering lines of drastically different size. 131 5. RESOLUTION OF SATELLITE PEAKS Let X be a magnetic nucleus (e.g., 13 C) whose natural abundance is small. Let Y be a nucleus which is coupled only to X . If we measure the absorption spectrum of Y, we find a large parent peak at Oy and two small satellite peaks at approximately Oy±Jxy/2. The parent peak results from those molecules which do not contain an X nucleus (e.g., they have a nucleus with spin 1/ 2). 12 C nucleus with spin 0 rather than a 13 C The satellite peaks arise from the small percentage of molecules which contain X. If X and Y have similar resonance frequencies, the satellites will not be symmetric about oy· We shall therefore adopt the following notation for spin-spin coupling .constants: the upfield "shift" shall be labeled J + and the downfield "shift" J J = J + (Fig. 5.1). That is : (5. 1) - J . If J is large, the optimum linear filters discussed earlier give the maximum signal-to- noise available to measure J. We shall examine two approaches for resolving satellite peaks when J is small. The first is a least-squares method. The second is a non-linear method which is closely related to linear resolution enhancement filters. The detection of satellite peaks which overlap the parent peak requires extremely accurate knowledge of the line shape of the parent peak. It is not practical to attempt a separate measurement of the line shape with sufficient accuracy (Figs. 3. 2. 7 and 3. 2. 8) to detect 132 ---)_------)+--- --·---- - - -··---- -------- - - - ----"- Figure 5. 1. The definition of J - and J . + the small satellites. Both approaches we consider therefore · determine the line shape from the spectrum of the sample. The intensity of a satellite peak such as that from 13 C in natural abundance is of the order of 0. 5% of the intensity of the parent peak. To first order we can ignore the satellite and perform a least-squares fit to the parent peak [i.e., we can minimize «I> in Eq. (4. 2. 2) while omitting the satellite from f(t, b)]. ,.... 133 The shape of a resonance signal will depend upon the sweep rate and the relaxation times. The qualitative variation of the line sha pe with the sweep rate, a, was considered in section 3. 5. 35 Jacobsohn and Wangsness have calculated the exact line shapes of n . m . r. absorption signals when a small rf field is applied (Fig. 5. 1. 1). The asymptotic form of these line shapes is identical to the fo r m for adiabatic passage 35 (a = 0) . This form is : Limit g(v) (5. 1. 1) .Q.v-- '.>t!l where iYI is the absolute value of the gyromagnetic ratio, v is the 0 resonance frequency, v is the frequency of the rf field, A.v = v - v0 , lO ,2 0 -.2 -6 -4 -2 0 4 6 • 10 12 14 ,_ ......~: Figure 5.1.1. L ine shapes of n. m. r. absorption signals for varying sweep rates (Jacobsohn and Wangsness 35). 134 2H 1 is the amplitude of the rf field, and T 2 is the total relaxation time (i.e., T 2 takes into account broadening due to inhomogeneity of the strong magnetic field). In an actual experiment it is difficult to avoid mixing in a small amount of dispersion mode. the line width. In fact, this can be helpful, by reducing The proper asymptotic form is therefore: 35 (5.1.2) where d determines the amount of dispersion mode. From Fig. 5. 1. 1 we see that this approximation is fairly good if: AvT2 < -2 . (5. 1. 3) We therefore make the approximation: g(v) ~ a 1 + b(Av) 2 [1-~(Av)]; v ~ v max' (5. 1. 4) where a, b, db, and v 0 are determined by least-squares fitting. The value of v max is chosen so that: . 05 ~ g(vmax)/g(v)max ~ 0. 5. (5. 1. 5) That is, we are able to fit the experimental curve with Eq. (5. 1. 4) up to the point where it reaches 0. 05 to 0. 5 of the peak height. The procedure for the least-squares fit is discussed in the next section. 135 We wish to approximate a set of N experimental points fi(xj), j = 1, 2, ... N by the function f(x, b) . "" The vector bis a set of M " . parameters we shall vary so as to minimize: 4> 2 = '>' L I If.(x.) - f(x ., b) I . j 1 J (5. 2. 1) J " A necessary condition for <I> to be a minimum is that: a<I> dbk = O; k = 1, 2, .... M. (5. 2. 2) If the parameters, ~' are non-linear, this problem cannot in general · be solved directly. The most common approach is the Taylor series method of linear approximation. This is the method currently used by most crystallographers. 36 It consists of taking only the linear term in the Taylor series expansion of the function f: (5. 2. 3) This gives the approximation for <I> : N M ~ {f.(x.) - f(x., b0 ) - L; 1 j =l J l " k =l af(x· b) 0 l' " ahk 2 f\.. ~K } The set of conditions, Eq. (5. 2. 2), then becomes: • (5. 2. 4) 136 i = 1, 2, .... M. ' (5.2.5) Rearranging this, we obtain the set of linear equations: (5. 2. 6) These Aquations can be written more concisely in the matrix form: (5. 2. 7) = ~' where : _ £; ()f(~, ~0) ()f(xj, ~0) j=l abi abk (5. 2 . 8) and: (5.2.9) The solution of these equations is then: §_T = ~ -1 ~. (5. 2. 10) 137 After solving for §_T, we then set ~equal to ~o + ~T' and iterate until ~converges . This set of equations can in principle be solved by Gaussian elimination; however, in practice we find that for the problem of interest the matrix ~ is ill-conditioned and the method This is because the diagonal elements of ~are small compared to the off diagonal elements. 37 does not work. Another common approach is the gradient technique which iteratively moves in the direction of steepest descent on the <I> surface. That is: (5 . 2.11) Substitution of Eq. (5. 2.1) for <I> gives : ok = ~ of(~, b) 2L..J (f.(x.)-f(x· b)][abk"' ]. 1 ] ]' "' (5.2.12) j= 1 Substituting Eq. (4. 5. 9) for gk we find : (5.2.13) so that: ~ = -2~. (5. 2. 14) 138 This method suffers from extremely slow convergence near the minimum. The Taylor series method converges rapidly only near the minimum. Ideally, we should use the gradient method first, and switch to the Taylor series method near the minimum. In order to do this we must find a way to get around the ill-conditioning of ~ · The mos t obvious way to make ~ "better conditioned" is to construct: (5.2.15) and decrease A. as we approach the minimum. Marquardt 38 has shown that: 1) Q.0 minimizes <I> on the "sphere" of radius 11§._0 11; 2) 11 §._0 11 is a 11 00 11 - continuous decreasing function of A., such that as A. - 0; 3) let Y be the angle between §._0 and ~· continuous decreasing function of A such that as A. §._0 rotates toward ~ as A. - between the above methods. oo. oo, Then y is a ~, y - O; i.e., Hence, this method is an interpolation Since 2..T is independent of scale, but o is not, we must rescale A.~ and Q"R to be dimensionless. ;:_g scale is the root-mean-square value of [ ofo(xj)/a~]. A convenient This leads to the definition of the scaled quantities : * A.k J = Ajk n::;_ ../ Akk JJ g.]* = g././"A..' ] J] (5 . 2. 16) (5.2.17) and: (5. 2. 18) 139 where o* for the rth iteration is obtained from the solution of: (A *(r) + A( r) I) o*(r) ,,,-.._ ~,,.... = g,,..,.,*(r) . (5.2.19) The only question remaining is the selection of an appropriate value for A. (r). It is clear that cI>(r+l) must be less than ip(r) or the process will not converge. 38 well is as follows : Let q A procedure which has been found to work = 10 1. If ti> (A.(r-1)/ q) < cI>(r-1) 1 let A. (r) =A (r-l) / q. 2. If cI> (A. (r-1) / q) > :p(r-1), and <Il(A (r-1)) < <Il(r-1)' let \ (r) = A. (r-l). 3. If cI> (A. (r-1) / q) > q,(r-1), and <Il(A.(r-1)) > cI>(r-1), increase \ by successive multiplication by q until for some smallest m, <f>(A (r-l) qm) < <Il(r-l) , let A(r) = A(r-1) qm . 4. If Eq. (4. 5. 19) is ill-conditioned and cannot be solved, increase \ by successive multiplication by q until for some smallest m, we can solve Eq. (5 . 2 . 19), let \(r) = ;\(r-l)qm. The non-linear least-squares procedure involves the calculation of many thousands of quantities af(~, ~/a bk. It is essential that these quantities be calculated in a reasonably efficient manner. Let 140 the function, f, be given by: f(x) = Yo + a [ 1 - DB(x - Xo) ) 1 + b(x - Xo) 2 (5. 2. 20) • We then calculate the necessary quantities according to the following algorithm. Number of Multiplications = b +b 1. B2 = a* DB 2 . ADB 3. DO 12 j = 1, 2, ... N 4. XJ = x(j) - Xo 5. BX2 6. afjlaa 7. f.] = 1. / (1. + B * XJ2) = (1. - DB* XJ) * BX2 = a* (af/aa) + Yo 1 2 (at/aa) * (at/aDB) 1 = 9. DFF 10. at/ab = 11. at/aXo = ADB* BX2 - B2 * DFF 12. at/ay0 2 -a* XJ * BX2 8. at/a DB = 2 = XJ * DFF 1 2 1. 11 Thus, there are a total of approximately llN multiplications required to evaluate these quantities . This leads to an efficient and practical program for filtering spectra. The values of the parameters, b, which ·'"' we obtain have a high degree of uncertainty. This uncertainty results from the ill-conditioning of~' which is caused by the large 141 correlations between the parameters, b. It is irrelevant for our "' present purposes, since the important point is that the deviations between fi and f should be small. This condition can be satisfied.· After performing the least-squares fit of the parent peak, we are ready to search for the satellite peak. We remove the parent peak by subtracting the Lorentzian we have determined by leastsquares fitting from the experimental spectrum: (5. 3. 1) where f/t) is the set of data points. The line shape of the satellite peak is significantly distorted, because the fitting procedure attempts to remove the satellite. The least-squares fit and the resulting distorted line shape are shown in Fig. 5. 3 . 1 where the size of the satellite peak is exaggerated for clarity. This distortion can be eliminated. : The ~pproximate position of the satellite can be determined from f 1 (t) (Fig'. 5. 3. 1) . We then construct: fi {t) l = fi (t) - f s (t)' (5.3.2) where f 8 (t) is the satellite. A new f 1 (t, ~) is then determined by least-squares fitting. to fi 1 (t). We remove the parent peak from fi (t) again: 142 I . Fig;ure 5. 3.1. The line shape of the satellite peak is distorted by removing the parent peak. (5. 3. 3) and obtain a new spectrum of the satellite peak. Additional iterations are possible but not necessary. If f(t, ~) fits the parent peak exactly, then the sum of the squares of the residues, «>, is given by: (5 . 3. 4) . 143 where fs (t) is the satellite peak and ni (t) is the input noise. From Fig. 5. 3. 1 we see that in practice the value of '1> after the first iteration, 'l> 1 is somewhat better described by : (5.3.5) If fs (t) and ni (t) are of the same order of magnitude, we can make the further approximation: t2 q> 1 t2 t J [fs(t)] dt + J [ni(t)] dt, 2 -:- tl 2 (5.3 . 6) tl since the cross terms, fs(t) ni(t) and fs(t+T) ni(t+T) tend to cancel when the noise is white . We can express q> 1 more concisely as: (5. 3. 7) where Es and En are the energy of the satellite peak and the noise respectively [ Eq. (2. 3. 8)]. If we determine the exact position of the satellite peak from the first iteration, then fi (t) in Eq. (5. 3. 2) will 1 contain only the parent peak and noise. The sum of the squares of the residues would then be given by : (5.3 . 8) Combining Eqs . (5. 3. 7) and (5 . 3. 8) we obtain a useful c.o ndition for the existence of a satellite at the position we determined from the 144 first iteration: (5. 3. 9) The only remaining problem is the determination of the position of the satellite peak in the presence of random noise. The least- squares position of the satellite peak can be determined by using a matched filter (Sec. 4. 2) . The parent peak can frequently be used for the line shape function. category. The examples we shall treat are all in this This iterative procedure leads to a least-squares estimate of the satellite and parent peak parameters. We have used what is essentially a perturbation approach. A very high signal-to-noise spectrum of the proton resonance from chloroform was recorded using a Varian Associates Model A-60 spectrometer (Fig. 5. 3. 2). The Lorentzian least-squares function [ Eq. (5. 2. 20)] provides a very accurate description of this experimental spectrum in the region of interest. If fi(t) < 0. 5 fi (o), then !fi(t) - f(t, ~I < 0. 0005 fi(o). A comparison of Figs . 5.1.1 and 5. 3. 2 indicates that the parameters, ~' are not very good estimates of the parameters in Eq. (5.1. 2). In order to determine whether or not the excellent fit we obtain using a Lorentzian is real, we have also performed a least-squares fit of the same data using a Gaussian : f(x) : 1 - DB(x - Xo) = Yo + a L eb(x - Xo )2 J. (5. 3. 10) 145 ~ . . 90~.o:--~~---"".2~.11111=-~~~~-.GllD=-..,-~~~ . t"::"~ ; F1'E.ClJEICY JN HZ Figure 5. 3. 2 . T·he least-squares fit of the proton resonance of CHC1 3 (a= 1 . 0) using a Lorentzian line shape function [Eq.(5.2 .20)]. The result (Fig. 5. 3. 3) is clearly unsatisfactory. In the region of interest, \fi(t) - f(t,!V\ can be greater than 0. 005 fi(o) which is the amplitude of a 13 C satellite. The parameters, b, ,..._ are even less realistic than those obtained using a Lorentzian line shape. The parameter a in Eq. (5. 3. 10) is approximately 7. 5 fi(o) . Recalling the correlation of the Lorentzian parameters, we conclude that the asymptotic form [Eq. (5. 1. 2)] is a valid approximation in the region 146 :[ 0 -· ·--- --·-- .... Figure 5. 3. 3 . The least-squares fit of the proton resonance of CHC1 3 (a = 1. 0) using a Gaussian line shape function [Eq. (5. 3 . 10)]. of interest although the parameters are not well defined. In order to de termine the resolving power of these filters we have artificially added satellite peaks to the spectrum shown in Fig. 5. 3. 2. If we set J _ equal to -1. 90 Hz, the maximum of f 1 (t) falls at -1. 95 Hz (Fig. 5. 3 . 4) . We then set the maximum of fs(t) at -1. 95 Hz [Eq. (5. 3. 2)] and calculate f . (t). A second least-squares fit gives 11 f 2 (t) [Eq. (5 . 3. 3)]. The maximum of f2 (t) falls at -1. 90 Hz and the 147 z Oo -o r- N a. .; - cp1 = O. 197 E s Cf'" 0 V> coo <Io 0 0 ' 8,,,. 0 z ::!8 IN a._ 0 cc D (/) CD a: 0 0 cf>2 = 0. 040 E a Figure 5. 3.4. The functions f 1 (t) (top) and f 2 (t) (bottom) [Eqs. (5. 3.1) and (5. 3. 3)] resulting from the above CHC1 3 spectrum. A matched filter has been used and the scale is such that fi(o) = 100. The error bars represent one standard deviation (Sec. 6). amplitude is within a few per cent of the correct value for the satellite. The condition in Eq. (5. 3 . 9) is approximately satisfied (Fig. 5. 3. 4) and we have successfully resolved our artificial 13 C satellite. By varying where we place this artificial satellite, we find that this filter can resolve a 13 C satellite if the spin-spin coupling is 148 greater than about four times the line width of the parent peak. When using this filter, there is some danger reporting a satellite which is not real. In order to assess this danger we have studied the above chloroform spectrum with no satellite added. The initial filtered spectrum, f 1 (t) shows nothing remotely resembling a satellite (Fig. 5. 3. 5). If however, we assume the existence of a 13 C satellite with J_ equal to -1. 95 Hz, we obtain something strongly resembling a satellite peak at approximately -1. 80 Hz in f 2 (t) (Fig. 5. 3. 5). The sum of the squares of the residues has increased z : Do . -o f-N 1CLo a: . 8 (f) <j> 1 = O. 026 E 5 00 ' erg 0 o. I 0 g . ::!' 0 :z IDo ._N -o a...c:i a: ID en co a: o• CJ <l>z = 0. 329 E s ..- - - -·-·-" - - -- Figure 5. 3. 5. The functions f 1 (t) and f 2 (t) [Eqs. (5. 3.1) and (5. 3. 3)] in the absence of a satellite peak. Note that f 1 (t) (top) is almost entirely within one standard deviation from zero. 149 however, and we therefore conclude that there is no satellite peak at -1. 80 Hz in fi(t). The second output spectrum, f 2 (t), is useful only for the refinement of our estimate of the exact value of J ±. From Fig. 5. 3.4 we see that the signal-to-noise level available to measure a small coupling constant when this least-squares filter can be used is approximately half that available with a matched filter for an infinitely large coupling. A 13 C spin-spin coupling constant greater than four line widths can be measured using this least-squares filter based upon the asymptotic form of the line shape. The loss in signal-to-noise from resolution enhancement is rather modest. Since the "wiggles" in the line shape obscure the satellite at high frequency, we must measure J+ separately by sweeping from high frequency to low frequency . Although this doubles the work we must do, it does provide an important check on the results . 5. 4 ~,.R~~~~~~!:~,Sti" _\Ve shall now consider a class of filters which are conceptually derived from linear resolution enhancement filters, but are actually non-linear. A linear filter requires an independent measurement of the line shape. Since this is not practical for the resolution of satellite peaks, we shall derive some pseudo-linear filters which determine the line shape from the spectrum itself. Consider a spectrum composed of n lines, all of which have the same line shape. If there is no noise, the spectrum is given by : 150 n = .0 a. s. (t - o.), j=l J 1 (5. 4. 1) J where 6. and a. s.(O) are the position and amplitude of the jth line . J J 1 The Fourier transform of the spectrum is then: N . s:.. = s.(w) '(_j' a . e -IW u.J. 1 j=l J (5. 4. 2) If ak is much greater than all the other aj we can make the approxi- mation: (5 . 4. 3) for the Fourier transform of the line shape. The output of a matched filter is given by: (5 . 4. 4) This suggests the construction of the self-filter : (5.4.5) Substituting Eq. (5. 4. 2) for Fi(w), we find that: = 11 11 a . a iw(o. - o - o ) c(O) s.*(w) S· (w) ">_; 6 _L_!. e J k £ . 1 1 j =lf=l ak (5. 4 . 6) 151 Taking the Fourier transform gives: (5. 4. 7) where sm (t) is the line shape resulting from a matched filter. If ak is indeed much greater than all the other aj, then aja .e_lak will be very I small unless j or P.. or both are equal to k . The output of a self-filter is therefore approximately: (5. 4. 8) The output spectrum is symmetric about t = ok. For each line in the input spectrum at t = oj there is a line in the output spectrum at both t = oj and at t = ok + (ok- oj). These two lines are located symmetrically with respect to ok. The simplest case of a self filter is that of a spectrum containing only one line and no noise. The output spectrum [Eq. (5. 4. 7)] is then: (5 . 4 . 9) which is the output from a matched filter. If the input spectrum contains two lines, the output spectrum becomes : 152 (5. 4. 10) where a 1 si (t - 01 ) is the larger of the two input lines. From Eq. (4. 2) we see that the non-linear interaction term is given by: (5. 4. 11) The self-filter is not a linear filter. The first term in Eq. (5. 4. 11) may be neglected when filtering satellite peaks, however, the second term is quite important. If J+ and J_ are unequal, the output spectrum will have four satellite peaks located at o + J +' o - J +' o + J_, and o - J_. If J+ and .T_ are equal, the output spectrum will have two satellite peaks located at o+ J /2 and o-J / 2, and the output amplitudes of the satellites will be twice the input amplitudes. If J+ and J_ are unequal we lose all information about which is larger. Although the sign of oxy is lost, we can still measure the magnitude Ioxy \. These self-matched filters are interesting, but not at all useful. If J is large we can select the portion of the spectrum which contains only the parent peak. Using this as the line shape we can construct a matched filter for the satellites. This procedure permits the determination of the sign of oxy , which can be valuable for the analysis of 13 C spectra. If J is small, J+ and J_ will be nearly equal, and the fact that we cannot determine which is larger with a self-filter 153 becomes unimportant. Unfortunately, a self-matched filter cannot resolve the satellite peaks when J is small. We therefore examine a s elf- resolution enhancement filter, which has the frequency response function: H( w) c(q) sf* ( :.u) = (5 . 4. 12) l+q!Sf(w)i2 , whe re Sf(w) is defined by Eq. (5. 4. 3). We cannot use a very large value of q, because: Limit F (w) q-oo o = Limit H(w) F·(w) q-oo 1 ' (5. 4. 13) or, from Eqs . (5. 4. 12) and (5. 4. 3) : Limit F (w) q-oo o = c(q) Sf*(w) Fi(w) (5.4.14) qi Sf(w) 12 T hat is, for large q, f0 (t) approaches a Dirac delta function of amplitude ak, centered at t = ok. The satellite peaks have been removed. If we use modest values of q (<"'15 ), the satellites will still be present, but with decreased amplitudes. Two 13 C satellite peaks were added to the chloroform spectrum discussed in the previous section (J+ = J_ = 1. 90 Hz). The spectrum was then filtered using a self-resolution enhancement filter [ Eq. (5. 4. 12)] in which the value of q was varied to obtain the minimum interference from the parent peak. The resulting spectrum shows 154 easily visible peaks in the positions of the satellites (Fig. 5. 4. 1). The satellites are nicely resolved and their intensity is still more than 25% of the intensity of satellite peaks filtered with a selfmatched filter~ Unfortunately, the "wiggles" in the tail of the parent peak also look like satellite peaks. A chloroform spectrum with no satellites added has been filtered in the same manner (Fig. 5 . 4. 2). ~ ~ .. ~ a "I' 0 ~ § 0 ~ 0 z ~~ ll:<> I- • 0 ti) 1¥§ 0 ~ 'i' ~ 'i' ~ 'i' ~ 'i'o.o 2.000 - - -- 11 .000 8..Gllll FREC.U:tl::Y lli_HZ - ·-· --- ----·-- -···----·--· .~-- Figure 5. 4. 1. Self-resolution enhancement spectrum [Eq. (5. 4.12)] with q = 12. The peaks at 1. 4 Hz and 5. 2 Hz are 13 C satellites. The amplitude of the parent peaks is reduced by a factor of 100. 155 Figure 5. 4. 2. Self-resolution enhancement in the absence of satellites (see F ig. 5. 4. 1). Comparing the two makes it clear that our interpretation of Fig. 5. 4. 1 was correct. In practice, we will not have Fig. 5. 4. 2 available for comparison and we will not know J in advance. The "wiggles" in the tail of the output line shape destroy the usefulness of these filters. The "wiggles" in the tail of an output line shape are easily understood by considering the limiting form [Eq. (5 . 4.14)] . the Fourier transform of this equation gives: Taking 156 Limitf (t) (5 . 4. 15) q-oo o which is the well known oscillatory form of the Dirac delta function. In order to remove the "wiggles" we must convert the output line shape to some non-oscillating form. We can convert an arbitrary input line shape to any output line shape by setting: (5 . 4 . 16) if S0 (w) exists and S/:.v) is never zero. Ernst 39 points out that such filters exist for the conversion of a Lorentzian line to a Gaussian line of reduced line width. This Lorentzian to Gaussian filter has the frequency response function: H(w) = b 2 ( v0 2 2 w i e - 4£n 2 V:-) (5 . 4 . 1 7) where b describes the line width of the Lorentzian input line shape . The line shape of a typical n . m. r. signal is not Lorentzian and the filter which converts it to a Gaussian : H(w) = does not necessarily exist. (5 . 4 . 18) Even if it does exist, the self-filter : 157 H(w) = (5. 4. 19) leads to a single output peak centered at t = ok. The Gaussian modification of Eq. (5. 4. 12) is given by: -~(vo)2 w2 c(q) e 4 .fn 2 vi st(w) H(w) = I 1 + q Sf(~) 12 (5 . 4. 20) This frequency response function will exist for any value of q, and if q is not too terribly large, it will not remove the satellite peaks. As q approaches infinity, the output line shape from this filter will approa'!h a Gaussian of line width v0 • The parameters vi' v0 , and b must now be determined. We can consider these Gaussian self-filters to be a sequence of two filters . The first is a matched filter and the second has the form of Eq. (5. 4.18) with the denominator equal to Sm(w) . We therefore set vi in Eq. (5. 4. 20) equal to v m' the line width resulting from a matched filter. From Figs. 5. 4. 1 and 5. 4. 2, we see that our problem is not to narrow the output line shape, but only to remove the "wiggles". Since a reduction of the output line width would reduce the signal-to-noise ratio, we set v0 equal to the output line width we would have if b = 0. This will be a function of q. A sufficiently 158 accurate empirical estimate of this dependence is given by: Vq ) 1 . (5. 4. 21) := 1 + 0. 53 £n(q + 1) ( vm which is independent of the sweep rate (see Sec. 3.3). We shall now estimate the proper value of b. The quantity 2 I Sf(w) 1 in Eq. (5. 4. 20) has the line shape which results from a matched filter. Since sm (t) looks very much like a Lorentzian function regardless of the sweep rate (Fig. 3. 3. 1), one could make the approximation: (5. 4. 22) where af and bf may be determined by a least-squares procedure. Substituting our estimates of the parameters vi' v0 , and b in Eq. (5. 4. 20) we obtain: 2 • = c(q) e H(w) - [ 0. 36 b 2 . w . ; f 1 + 0. 53 £n(q+1) ] S *(w) f (5.4.23) 1 + q!Sf(w)l2 for the frequency response function of a Gaussian self-filter. This Gaussian self-filter has been applied to the above chloroform spectrum with satellites. In order to check our estimates of the parameters in H(w), we varied the output line width, v0 • As we predicted above, the optimum value of 110 is approximately v q 159 (Fig. 13. 4. 3). That is, vq is the minimum output l_ine width whi_c h will eliminate the "wiggles". The signal-to- noise ratio of the spectrum without "wiggles n in Fig. 5. 4. 3 (q = 12) is 96% of that resulting from a self-matched filter, however, the size of the satellite peaks has been reduced by a factor of four. This difficulty is easily avoided by using the approximation given by Eq. (5. 4. 22) for the denominator I' I I I . :i :i ~ I :i :i I• :I :i :i I '·· . : I :i : '-. ll /). !! ~~ \\ '..' \~ \"\...;·/ \ ' -~· ,,.. .,- ~t 1.0 - ·- t ---: .75 - - - - .:..:: 'i .5 _ _ __ ._ ...,._ - - - -s'.ro-1- 'o.o ·-----= 2'-::. co::::0 ---~-".:'~."=-=c,:>-c-) FREQUENCY HI t lZ Figu r e 5. 4. 3. Resolut ion enha ncement of the chlor ofor r.o spe ctrum (F i g. 5. 3. 2) with s atellite s adde d at T ---· 1 oru T-Tz 'fhc, Gati <0•c1' n ce]f r ecol 1 u± u u . -..1t 1'0' . enhancen1cnt filt e r [ Eq. (5. 4. 20)] gi ves the best results when 110 is appl' oxima t e ly equa l to v q [ Eq. (5.4.2 3)]. .L -L • •J "·~- -' ,. ..1. .&. \ J 1<.. t. ..J 0 4.A,. J. ).J , . - 1 ! 160 of Eq. (5. 4. 23). The frequency response function of this filter is given by: c(q) e H(w) = -[ 0. 36 b{ w2 1+. 53 ln (q+l) ] sf_*( w) 1 + q af e - bf Iw I (5. 4. 24) where af and bf are determined by a least-squares procedure (Sec. 5. 2. ). The above chloroform spectrum with satellites has been ·filtered using this frequency response function (Fig. 5. 4. 4) . Figure 5. 4. 4. Resolution of the chloroform spectrum using Eq. (5. 4 .. 24). 161 The amplitude of the satellite peaks is independent of q. We now have a very useful pseudo-linear resolution enhancement filter for satellite peaks. The exponential approximation [ Eq. (5. 4. 22)] is valid for large '.V only if all "wiggles" in the input line shape are included in the spectrum. Therefore, these filters should not be used with large values of q for a spectrum such as that shown in Fig. 5. 3. 2. If all "wiggles" are included in the spectrum, we can achieve substantial resolution enhancement (Fig. 5. 4. 5). In this case, large values of q simply reduce the signal-to-noise ratio. They do not introduce any fluctuations which have the appearance of a satellite. In practice, it is difficult to obtain a spectrum of the quality of Fig. 5. 4. 5 when studying {1 3 C, 13 C) coupling constants with natural abundance samples. Several hundred scans are necessary in order to achieve an adequate signal-to-noise ratio. With presently available spectrometers, this introduces a broadening due to the drift of the field (Fig. 5. 4. 6). When large values of q ar~ used in Eq. (5. 4. 24), these spectra tend to produce minima adjacent to the parent peak (Fig. 5. 4. 7). Although this limits the resolution which is possible, there is no problem of producing spurious "satellite" peaks. When the input line shape is determined primarily by field drift, the exponential approximation [Eq. (5. 4. 22)] breaks down for large w. The slow drift of the field limits the resolution, however, we can still obtain very useful spectra when small values of q are used (Fig. 5. 4. 8). 162 ~ - 0 - ______:._____ 0.0 _ _____j__ _ __ __ _ __ _ J _ _ _ __ 0 . 500 1.000 L__ _. - - · ·- ·· I .SOD FREQUENCY IN HZ Figure 5. 4. 5. The proton resonance of CHC1 3 (a= 2. 5) with satellites added (J + = 4. 75 Hz, J _ = -3 . 29 Hz). The frequencies are scaled by a factor of 10. Note that the upfield satellite is easily resolved even though it is completely buried in the "wiggles" before filtering. 163 5CA.!. 'RCTCl'I = 10 1.000 FRfOUENCY IN HZ Figure 5. 4. 6. 1.500 2 .000 The natural abundance 13 C spectrum of carbons two and three from cyclopropyl chloride (925 scans), courtesy of F . J . Weigert. ~ -r <7 § 0 Figure 5. 4. 7. The resolution enhancement of the spectrum in Fig. 5. 4. 6. using Eq. (5. 4. 24) for the filter (q = 1014) . 164 Figure 5. 4. 8 . The resolution enhancement of the spectrum in Fig. 5. 4. 6 using Eq. (5. 4. 24) for the filter (q = 1. 4). The observed value of J (14 . 0 ± 0.1 Hz) compares favorably with the value (13. 9 ± 0. 2 Hz) obtained after 2, 600 scans without resolution enhancement. In principle we could measure an arbitrarily small spin-spin coupling constant using the pseudo-linear filter given in Eq. (5 . 4.. 24). Unfortunately, the signal-to-noise level available to measure J falls off extremely rapidly for couplings smaller than about twic e the full width a t half height resulting from a matched filter (Fig. 5. 4. 9) . This ~s about the limit of the region which is accessible with leastsquares filters (Sec . 5. 3). The asymptotic behavior of these pseudo-linear filters can be determined by the techniques used in section 3. 4 for linear filters . We find that: 165 Limit (S/N)o = ( 1T q - oo (S/N)m ~n 2) (v:10~ 4 1/ 112 ~) e -( l~ 2 l ~~)' (5. 4. 25) The signat-to-noise level falls off much more rapidly than it does with optimum linear resolution enhancement filters [Eq. (3. 5.19)]. 40 The point at which the signal-to-noise level begins to fall off is roughly the same as the point at which "wiggles" begin to appear in the tail of the output line shape. These "wiggles" are not important >( 0 _e z ....... V) :S;: .1 z ........ V) . 01-+----'----.--~---r----"f 1 Figure 5~ 4. 9. 2 4 8 16 The sig11al-to-noise level available to measure the position of a 13 C satellite using the pseudo-linear filter given in Eq. (5. 4. 24). 166 when filtering lines of comparable magnitude since the amplitude of the largest "wiggle" is always less than 12. 8% of the amplitude of the peak [Eq. (5.4 . 15)]. We cannot afford to sacrifice much signal-tonoise when filtering satellite spectra. The importance of removing these "wiggles" by transforming to a Gaussian output function is therefore somewhat questionable, since in practice we can achieve approximately the same resolution without this transformation. If the spin-spin coupling of the satellite peak is greater than eight times the natural line width [vm(a= 0) / 2], we can increase the signal-to-noise level by increasing the sweep rate. The maximum signal-to-noise ratio is achieved when vm is approximately 4J. These filters permit the practical measurement of spin-spin coupling 13 constants as small as 0. 5 Hz for both (1 3 C, H) and (13 C, C) using natural abundance samples. Pseudo-linear and least-squares filters are complementary rather than competitive procedures. Pseudo-linear filters can be used to measure both J +and J _, however they cannot distinguish between the two. Least-squares filters can be used to measure J _ (or J+ if we sweep downfield). By using both pseudo-linear and least-squares filters one can obtain separate measurements of J + and J _ from a single spectrum. If J is large, this can be accomplished by using a linear filter (Sec. 5. 4). The knowledge of J valuable for the analysis of spectra. + and J - can be 167 Natural abundance 13 C spectra are much simpler than equivalent proton spectra. If the protons are decoupled, the 13 C spectrum consists of single lines centered at the resonance frequencies of the various carbons. The chemical shifts can be measured directly, however the assignments of these shifts is not necessarily trivial. 41 Let us assume that the chemical shift of carbon A is known. It is then possible to assign carbon B based upon the measured values of J + and J _ for the (A, B) coupling. The energy levels for an AB system 42 are shown in Table 5. 5.1. Using these values we find that TABLE 5. 5.1 Energy Levels for an AB System Energy Trans itions State + J/4 + (:yHo/2) (2 - aA - aB) 1 4• B2 A2 2 h - J/4 - (1/2)v'J +¥H:(aA-aB) 2 2 2 2 A1 3 -J/4 + (1/2)v'J +¥H:(aA-aB) h B1 4 the transition energies are given by: ~EA 1 = -J/2- ~v'J 2 +~H:(aA-aB) 2 + (~Ifo/2)(2-aA - aB) ; (5 . 5. 1) 168 (5. 5 . 2) ·we now define: J2/~21-t 2(aA - a B )2. ' .L"'() x2 = - (5. 5. 3) and assume that X 2 « 1. The transition energy is then: (5. 5. 4) We can expand the square root in a Taylor's series: ..; 1 2 +X • . 1 2 i 4 1 + 2X - aX + · · · (5. 5. 5) Substituting this in Eq. (5. 5. 4) gives: Using the definitions of J _ and oAB this becomes : (5. 5. 7) Treating Eq. (5 . 5. 2) similarly, we find: (5. 5. 8) 169 Therefore : (5. 5. 9) Finally, we rearrange this expression and obtain: 0 ~ J2 AB . 2 (J + + J _) [ 1 - ~ J2 4 0AB (5. 5. 10) This relationship can be quite useful in the assignment of the resonance line of a carbon which is directly bonded to another carbon whose chemical shift is known. When A and B are directly bonded, JAB and (J + + J ) are both large and can therefore be measured with reasonable - accuracy. 170 6 ERROR ESTIMATES The techniques discussed in this thesis all involve the measure- ment of line positions. It is obviously desirable to have realistic estimates of the errors in these measurements . If we consider the derivation of a matched filter given in Section 1. 1, a suitable method immediately suggests itself. The output spectrum f0 (t) is the weighted average of the values: (6. 1) We can therefore calculate the standard deviation of f0 (t). This serves several very useful purposes . The points of the "base line" should all be within about one standard deviation of a straight line. If a point rises two standard deviations above the base line, the probability is 0. 954 that it is a peak (the filtered noise is Gaussian, since each filtered point is a linear combination of points with 33 Gaussian distributions ) . If a point rises three standard deviations above the base line, the probability is 0. 997 that it is a peak. The standard deviations also provide a method of estimating the error in a measurement of a peak position. value of f 0 (t). Let f(t) be the "true" The upper and lower ends of the error bars (which indicate on e standard deviation) constitute an envelope around f 0 (t) . We shall call the upper curve fo+(t) and the lower curve fo-(t). The probability that any point of f(t) lies within these curves is 0. 683. Therefore, 68. 3% of f(t) lies within this envelope. This is true 171 whether we cons ider f(T i) in F ig. 6. 2 to b e above f 0 (T 1 ) or to the left of f 0 (T 2 ). Therefore, the inter val oft with a proba bility of 0. 683 tha t f(t) takes on a pa rticula r value is the horizontal distance to f0+(t) or £0-(t). Let t 1 and t 2 (Fig. 6. 1) be chosen such that: (6. 2) These points will then s a tisfy: / :;:. P \/J.l 1 <....__ u / ....... t2 ) = . 6"" {j .j ' (6. 3) where 6 is the pos ition of the peak. We ther efore estimate the error in the p eak pos ition to b e ±(t2 - t 1 )/ 2. ' I Figure 6. 1 . The standard deviations of f 0 (t). 172 / •J.n;> Figure 6. 2. The standard deviations of t. It can be shown that a maximum-likelihood estimate is normally distributed. 43 A peak position obtained with a matched filter is therefore normally distributed with standard deviation (t2 - t 1 ) / 2. Peak positions which are measured with resolution enhancement filter s are not necessarily normally distributed, however the output from any linear filter will satisfy Eq. (6 . 3). Since an RC filter is linear, this provides a convenient error estimate for routine work. 173 7. APPLICATIONS The filtering techniques discussed in this thesis are useful for improving the signal-to-noise ratio and the resolution of any spectrum where one or both present a problem. Although Fourier transform spectroscopy promises to provide additional sensitivity enhancement, 44 this new technique is not in competition with linear filtering. On the contrary, Fourier transform spectroscopy will make linear filters even easier to apply. One must simply multiply the spectrum by the frequency response function of the filter before taking the Fourier transform. A set of instructions for the filtering program is included as Appendix A and a complete FORTRAN listing as Appendix B. We shall briefly discuss some experiments which were done with the intention of using this program, since they illustrate the kind of results one can expect. 7.1 Signal-to-Noise Enhancement ~~,,......,.......,,....~ Dr. Robert L. Lichter has found the filtering program developed in this thesis useful for natural- abundance 15 N studies of substituted hydrazines. Although the double section RC filter built into the spectrometer gives near optimum results, the extreme signal-tonoise problem inherent in natural-abundance 15 N work justifies the use of a matched filter. The resonance from nitrogen-2 of 1, 1- dimethyl hydrazine can be observed only after 1300 scans (Fig. 7. 1. 1). 174 The error bars constitute a valuable indication of the statisti cal significance of the peak. In general it would be desirable to obtain a better signal-to-noise ratio for the line shape than that shown in Fig. 7.1. 1, however, the above spectra are quite acceptabl e . ,-- I A ! I .-- B ' : i I~- LI I I L 1- I I ID (L a: E; u·J ! l'.O a: :z ID ,_, z: z: D i- i ,_ I i c r-- i-·- i- i- 11 (L (L ID ID co co a: a: I~ (f) (f) a: er !I LI I a.a I ---- 2a .a I a.a 20 . 0 a.o 20.0 Figure 7. 1. 1. Natur al-abundance 15N spectrum showing the resonance from nitrogen-2 of 1, 1- dimethyl hydrazine, courtesy of Robert L. Lichter. Trace A is the experimental spectrum, B is the line shape (15 N enriched nitric acid) , and C is the filtered spectrum. 175 7. 2 Resolution Enhancement · Dr. Robert L. Lichter has measured the directly bonded C5 N, 13 C} spin-spin coupling in pyridine. Using a 98. 9% tsN labeled ·sample, he found the proton-decoupled 13 C spectrum of carbon-1 to be a singlet when the sweep rate was 1. 0 Hz/sec (Fig. 7. 2. 1). Using a line shape from the cyclopentane solvent (Fig. 7. 2. 2), he was able to resolve this singlet into a doublet with a linear resolution enhancement filter (Fig. 7. 2. 3) . The observed coupling was 0. 45 ± 0. 1 Hz. He then recorded the spectrum with a sweep rate of 0. 1 Hz/ sec and directly observed a coupling of 0. 42 ±0. 05 Hz (Fig. 7. 2. 4). In the above e~mple we could verify the coupling measured with resolution enhancement by direct observation. If the coupling were smaller than the natural line width (or inhomogeneity in the magnetic field), direct observation would not be possible. However, it might still be possible to measure such a coupling using linear resolution enhancement filters. The spectra shown in Figs. 7. 2 . 1 and 7. 2. 2 were recorded with the RC filter connected (frequency response setting = 5 Hz). This improved the signal-to-noise of the unfiltered spectra, and in this case did not interfere with the resolution enhancement. It would be a safer practice to dis connect the RC filter completely. 176 ~- I t___ _ L _ _______l __ _ ____ _ j _ _____ __ . ---- 0.0 2.5 5.0 7 .5 !. _ ___ _ - - J 10'.0 12.S FRE~U_E~~~y IN _!:iL- ----- _ Fig. 7. 2.1. The natural abundance 13 C spectrum from carbon-1 of 15 N labeled pyridine, courtesy of Robert L. Lichter (a= 1. 0 Hz/sec). ~ ~ ~t a: ~ r.n a: i ~ I ;r _J~~- r1 !- I l.~-~___,_____,_____ 0.0 2.5 5.0 7.5 10.0 ___J 12.5 _l_ ______ ff3EQUENCY IN HZ Fig. 7. 2 . 2. The natural abundance 13 C spectrum of cyclopentane, courtesy of Robert L. Lichter (a= 1.0 Hz/sec). 177 ~- ! 1~ ~ I nI 1·' , t-- 11' \ i :z. l I II :&: L I J 'I A c l 1;\ I \ I\ .\ i \/ .\ j:1 \) \V'"I i!\\ ~ rv \,: ~, v J\J · v ,j 1 I ,, A f 1 \; I ·I - ~ I i '\ (\ I I /~ 1 ; )· , 1' . /"\ ./ (\ ._ i 1 1 I v ~- [_ o.o - -' 2 .5 ,,,,_., J ___ ----·--- L. ________.L .. ·-- --- .J 5.0 7 .5 10.0 12.5 FREQUENCY IN HZ Fig. 7. 2. 3. The pyridine spectrum in Fig. 7. 2. 1 has been resolved with a linear resolution enhancement f ilter based upon the line shape in Fig. 7. 2. 2. 1 Hz Fig. 7. 2. 4. The natural-abundance 13 C spectrum from carbon-1 of t 5 N labeled pyridine, courtesy of Robert L . Lichter (a = 0. 1 Hz/ sec) . 178 The pseudo-linear filter given by Eq. (5. 4. 24) permits the observation of 13 C satellites which cannot be observed with conventional filters. We shall use the natural-abundance 13 C spectrum of 2, 3-dimethyl-2-butene to illustrate this point. The proton decoupled spectrum of this molecule consists of two singlets from the vinyl and methyl carbons (Fig. 7. 3. 1). The methyl resonance was recorded with a reduced sweep rate (Fig. 7. 3. 2). This spectrum was then filtered with a double section RC filter, the time constant of which was varied in order to maximize the visibility of the 13 C satellites. Even with the optimum time constant the satellite peaks remain buried in the llOise (Fig. 7. 3. 3). If, however, we use a pseudo-linear filter, the 13 C satellites resulting from the directly bonded coupling are clearly visible (Fig. 7. 3. 4). The pseudo-linear filter has taken full advantage of the symmetry in this "AX" system. For the rather modest resolution required to observe this coupling, the field drift indicated in this spectrum (Fig. 7. 3. 2) did not present any problems . We shall now demonstrate the kind of results which are possible in the absence of significant field drift. The natural-abundance 13 C spectrum of trans-1, 3-pentadiene (Fig. 7. 3. 5) provides examples of applications for several of the filters discussed above. Although the assignment of the methyl resonance is trivial, the assignment of the vinyl carbon resonances is quite difficult. The off-resonance decouplect 45 spectrum (Fig. 7. 3. 6) indicates that the line at -46. 9 ppm res ulls from a carbon directly 179 c, -55.8 0 --. ·------ 47.4 sin PP.M. . ··---·-· · ··- ~ ·Fig. 7. 3. 1. The proton decoupled 13C spectrum of 2, 3-dimethyl-2-butene (a = 100 Hz/sec). z 0 I-CL a: 0 r.n a: ro 0.0 20.0 ij0.0 60.0 80.0 100.0 ______ J:REQUENC!. .. llLHZ Fig. 7. 3. 2. The methyl t 3 c resonance from 2, 3- dimethyl2-butene (a = 4 Hz/ sec). The spinning rate was 30±2 Hz. 180 z 0 ...... 1Cl.. cc 0 (f) ID a: 0.0 20.0 ij0.0 60.0 FREQUENCY IN .Hl . ·- ---- -- -· ·--·· ---- ·- .. 80.0 100.0 Fig. 7. 3. 3. The spectrum shown .in Fig. 7. 3. 2 has been filtered with a double section RC filter and the vertical scale has been atenuated by a factor of 100. z: 0 \A -\; ; .• 11'.L a: 0 UJ aJ a: -~---~ ----__j_ 0.0 20.0 40.0 60 .0 fREQUENCY IN HZ ___ - J __ aa.o _ _ __, lOO.O __ Fig. 7. 3. 4. The spectrum shown in Fig. 7. 3. 2 has been filtered with a pseudo-linear filter [ Eq. (5 . 4. 24)] . •T :::: 43. 9 ± 1 Hz. 181 c, I · 10A . I I ·6l.1 Dloxa ne I .•., I I · 46.CJ 0 ·65.1 ···-- .. 8 -·- ---- -·-- ______ ----· - -- -.- - ·- - - · in P.PM. ... - .. , .. Fig. 7. 3. 5. The noise decoupled natural-abundance 13C spectrum of trans-1, 3-pentadiene. c, · 70 A .... · 46,, 8 in P.P.M. Fig. 7. 3 . 6. The off-resonance decoupled spectrum of the vinyl carbons of trans-1, 3- pentadiene. 182 bonded to two protons, hence this line results from carbon-1. The resonance from carbon-1 was then recorded at a reduced sweep rate (Fig. 7. 3. 7). 13 This high resolution spectrum was used to observe the C satellites. A small portion of the spectrum (Fig. 7. 3. 8) was used to calculate the weighting function for a pseudo-linear resolution enhancement filter. This made the filter pseudo-linear only in the region immediately surrounding the parent peak and linear everywhere else. The asymmetry of the directly bonded (1 3 C, 13 C) spin-spin coupling satellites could therefore be measured (Fig. 7. 3. 9). values of J ly. + and J - The observed were 31. 1±1. 0 Hz and -38. 8 ± 0. 7 Hz, respective- Substituting these values in Eq. (5. 5.10) one calculates that the chemical shift of carbon-2 relative to dioxane is -68. 0 ± 3 ppm. The directly observed chemical shift of carbon-2 relative to dioxane is therefore either -70.4±0.1 ppm or -65.8±0.1 ppm (Fig. 7.3.6), since both of these peaks fall within the calculated interval. The analysis of this spectrum could be completed by scanning the region surrounding these peaks. In addition to the large directly bonded coupling, Fig. 7. 3. 9 also indicates the presence of a smaller coupling (J = 9. 8 ± 0. 6 Hz) . In order to demonstrate that these peaks represent real and reproducible satellites, a slow sweep rate spectrum was recorded (Fig. 7. 3.10). This spectrum was then filtered with a least-squares satellite filter (Sec. 5. 3) . The resulting filtered spectrum (Fig. 7. 3.11) clearly indicates the presence of a long range spin-spin 183 ~fACJflt::IO~ u.soo Fig. 7. 3. 7. l.000 FR~~IJ€NC.Y l . 500 2.cm ~.SC IN MZ The natural-abundance 13 C spectrum from carbon-1 of trans-1, 3-pentadiene (a = 10 Hz/sec). The spinning rate was 15 ± 5 Hz. ~ :r ~- s~ :il.; s ~. 0 z !':!~ ~- ~.:) c "' [11§ 9 ~~ a=o ~ ~~-- ~ ~ ~'? § '? 'i' Fig. 7. 3. 8. Ol ~, .; I The line shape (left) and weighting function (right) used to filter the spectrum in Fig. 7. 3. 7. 184 ~ 0~.•~~~~.~.soo~~~~,~.«Xl ~~~~~, .soo ~~~~~2.~~~~--.J.1110 7 .. . _ . . f!l!_~~.Y IN HZ Fig. 7. 3. 9 . The spectrum in Fig. 7. 3. 7 has been filtered with the weighting function shown in Fig.· 7. 3. 8. ~ 9 ~~ . II • ~~~.,--~~--:2~.~ =-~~~.~.-=-~~~~.oxi,,,,_~~~~ ••~-=--~~-:-=',o.• Ff\E~~J.JN HZ Fig. 7. 3. 10. The natural-abundance tac spectrum from carbon1 of trans-1, 3, -pentadiene (a = 0. 4 Hz/ sec). The spuming rate was 15 ± 5 Hz. 185 1I T ~ 9 0~ . •~~~__,...2.~=-~~~~ • .~=-~~~.~.~=-~~--:l~. ~ =-~~-;::' IO.OO FflElll!fN~U !'__ HZ Fig. 7. 3. 11. The spectrum shown in Fig. 7. 3 . 10 has been filtered with a least- squares satellite filter. coupling of 9. 5 ± 0. 3 Hz . The origin of this satellite is not important for our present interests . The significant point is that the satellite is consistently observed. The directly bonded (1 3 C, 13 C) spin-spin couplings we have measured are consistent with the general trend of the variation of these couplings with the percent S hybridization (Fig. 7. 3.12) . If we base our prediction ·o n the observed coupling in ethylene, the measured values are smaller than the predict ed ones for small percent S and larger for large percent S. This has been rationalized in terms of the contraction of the carbon S orbitals as the multiplicity of the bonding increa ses. 46 186 150 J 100 50 ......~~-.-~~--~~....,.~~---. 20 30 10 0-f"--~~--~~ 0 %S Fig. 7. 3. 12. Directly bonded (13 C, 13C) spin-spin couplings as a function of the product of the S character at the two centers. In order of increasing coupling constant, the e2g)erimental values represent cyclopropane, 47 ethane, 48 tetramethyl ethylene, ethylene, 48 pentadiene, and acetylene. 48 150 a. 100 )( u..i 50 100 50 150 Cale. ---- Fig. 7. 3. 13. Directly bonded {1 3 C, 13 C) spin-spin couplings as a function of the values calculated by extended Hlickel theory. 187 If we compare the experimental spin-spin couplings with those obtained from extended Htickel theory (see Appendix C), the general behavior is unchanged (Fig. 7. 3. 13). It would seem reasonable to use the calculated spin-spin couplings to select more realistic exponents for the 2S orbitals of extended Htickel calculations. These calculations should then give a better description of the effects of hybridization. 188 8. EXPERIMENTAL The 2, 3-dimethyl - 2-butene was purchased from Aldrich Chemical Co., and the trans-1, 3-pentadiene from Matheson Coleman and Bell. These samples were then sealed in 10-mm n. m. r. tubes. The 13 C and 15N spectra were determined with the DFS-6G spectrometer discussed by Weigert and Roberts, 49 and the proton spectra were determined with a Varian Associates A-56/ 60 spectrometer. The spectra were accumulated on a Varian C-1024 computer and later transferred to cards with a Varian C-1001 coupler interface to an I. B. M. 526 printing summary punch. The data were then processed using the program SNARE on an I. B. M . 360/ 75 computer. The spinning rates of the sample tubes were measured with a General Radio Co. model 631-B Strobotac which had been calibrated using the spinning side bands from the A - 56/ 60 spectrometer. Finally, we note that our experimental spectra were really filtered. A completely unfiltered spectrum, fi (t), would have zero signal- to-noise ratio, because the measurement at each frequency must be accomplished in zero time. All n. m. r. spectrometers involve an approximation of the form: f T/ 2 h(T)fi(t-T)dT . (8. 1) -T/ 2 If T is much smaller than the narrowest "wiggle" in the spectrum, t his will have negligible effect on the line shape. 50 A digital s weep 189 spectrometer generally uses a weighting function which is constant. That is, it simply takes the average of the measured signal over a small frequency range, ~11, and uses this for a data point. Continuous sweep spectrometers generally use an RC filter. If we had used it, the recording device on the spectrometer would have introduced additional filtering depending upon the response time of the pen. The exact form of h(T) is unimportant, provided that T is sufficiently small (Sec. 2. 7). We are not faced with the question of whether or not a spectrum should be filtered, but rather how it should be filtered. 190 APPENDIX A USING THE PROGRAM SNARE 191 If one lmows the line shape of an n. m. r. signal, it is possible to distinguish between one or more absorption lines and random noise. Given the line shape of a nuclear magnetic resonance signal as input, the program SNARE will remove most of the random noise in a spectrum or reduce the width of the lines, depending upon the options specified. In these instructions we shall not attempt to discuss the theory of filtering n. m. r. spectra, but limit ourselves to the use of the filtering program SNARE. We strongly recommend that the user of SNARE become moderately acquainted with the theory of filtering so that he may use the program to maximum advantage and avoid incorrect interpretation of the results. We shall begin by describing the types of filtering available with SNARE. Signal-to-Noise Enhancement ~~,....._,.....,.~~ The filter which gives the maximum signal-to-noise ratio for a given spectrum is called a matched filter. SNARE provides a matched filter as one option. The signal-to-noise ratio of a multiple -scan n. m. r. experiment can be further improved by increasing the sweep rate. T he maximum signal-to-noise ratio available for measuring the position of a single line is obtained by using the fastest sweep rate available (i.e., with due consideration for the relaxation times of the nuclei and instr ument limitations). The dependence of the signal-to-noise ratio upon the experimental parameters is approximately: 1 1 S/ N c~ k (sweep rate) .t (sweep width) - 2 . 192 One should therefore use the fastest sweep rate and the smallest sweep width which is practical. However, one should also keep in mind that increasing the sweep rate will increase the line width, although this does not necessarily decrease the accuracy with which the position of a line can be measured. A matched filter requires an independent measurement of the line shape. Resolution Enhancement for Lines of Comparable Magnitude ~~,,......,,..._,--........~"""~~,....~~~~~ The program SNARE includes an optimum linear resolution enhancement filter. Fig. 1. The capabilities of this filter are indicated in These filters "trade off" signal-to-noise for resolution. Decreasing the sweep rate will also trade signal-to-noise for resolution. The optimum procedure is to combine the two as \, __ Figure 1. . Resolution enhancement of an n. m. r . signal. The large peak and its " wiggles " are an unresolved doublet. The two smaller peaks are the same doublet after resolution enhancement (q = 3000). 193 indicated in Fig. 2. The curve which intersects all the points represents the resul~ of varying the sweep rate and using a matched filter. The curves which intersect only one point represent the effect of using a linear resolution enhancement filter and the indicated sweep rate. The point labeled 0. 1 should be interpreted as the minimum sweep rate for which the first downward "wiggle" goes below zero. The numerical value of the sweep rate at which this occurs will depend upon the sample used. The optimum process varies the sweep rate and uses a linear resolution enhancement filter. This gives the maximum signal-to-noise for a given line width and is represented by the envelope of the curves in Fig. 2. To use this optimum process one preselects the sweep rate and resolution enhancement filter such that one obtains the desired line width after resolution enhancement, and the signal-to- noise ratio is a maximum for this line width. We shall illustrate the use of this optimum process with an example. Let us assume that we wish to measure the separation between two closely spaced lines (approximately 0. 5 Hz separation) of a natural abundance 13 C spectrum. We therefore decide that 110 should be 0. 2 Hz in order to provide adequate resolution. We find that the minimum sweep rate for which the first downward "wiggle'' for a sample of CS 2 crosses zero is 0. 05 Hz/ sec, and that the matched filter line width for this sweep rate is O. 27 Hz. We then modify Fig. 2 as indicated in Fig. 2a and find the point on the horizontal scale corresponding to 0. 2 Hz. We see from Fig. 2a that we will 194 tOIU) - ·-··· ·· - ··-- - - - ·-- ···- · · 100 10 1~----.-------,---·~---j 10 .1 Figure 2. 100 Resolution enhancement in which we vary both the sweep rate, a, and tre filter. The output line width is v 0 and the line width resulting from a matched filter is vm. The numbers at the points indicate the sweep rate in Hz. This curve was obtained from several CHC13 spectra with the exception of the point a = 0, which was obtained from the asymptotic form . iooo, - - -- Figure 2a. - - -- The use of Fig. 2 to select the sweep rate for an n. m. r. experiment. 195 obtain the maximum output signal-to-noise by using the following procedure: first collect data using a sweep rate of 0. 5 Hz/ sec, then use a linear resolution enhancement filter to reduce the line width to 0.2. Hz. Several filters are compared in Fig. 3. The optimum linear filter discussed above gives the highest signal-to-noise ratio for a given line width of any filter presenting known. The instrument RC filter is clearly adequate for routine work. The curve representing this filter may be somewhat optimistic, however, because it assumes that the optimum time constant was used at each sweep rate. This is not generally done in practice. If either signal-:-to-noise or resolution is a problem, one should use the optimum linear filter . If one wishes to resolve lines differing in amplitude by more than a factor of five, a pseudo-linear filter, which is described in the next section, should be used. 10 .+-'--~~---·~- IO Figure 3. 100 Resolution enhancement by various methods in which the sweep rate is var ied . 196 A linear resolution enhancement filter requires an independent measurement of the line shape. Pseudo-Linear Filters These filters are similar to linear filters except the spectrum itself is used for the line shape. They produce a symmetric output spectrum regardless of the nature of the input spectrum. If a pair of satellites are not symmetric about the parent peak, the output spectrum will contain four satellites (Fig. 4). If the satellites are It,~_,_ _ /:~~~ nl(;(l,.Vl'.:Y lk /Q Figure 4. The proton resonance of CHCl3 (o = 2. 5) with satellites added 1.J + = 4. 75 Hz, J _ = -3 . 29 Hz) . The frequencies are scaled by a factor of 10. Note that the upfield satellite is easily resolved even though it is completely buried in the "wiggles" before filtering. 197 symmetric in the input spectrum, their intensity will be doubled in the filtered spectrum. Pseudo-linear filters are the most powerful presently known for resolving satellite peaks which are close to the parent peak. They should not be used to resolve lines which differ in amplitude by less than a factor of five. If one wishes to determine whether the closer or farther satellite, of those on one side of the parent peak, is actually to the left of the parent peak, the least...:squares satellite filter described in the next section can be used. The usefulness of these filters is described in the preceding section. This procedure approximates the parent peak by the Lorentzian asymptotic form (Fig. 5). The parameters in the Lorentzian are determined by performing a least-squares fit up to the point where the parent peak reaches "FIT" (a fraction which should be set between 0. 05 and 0. 5) of its maximum value. The resulting least-squares function f(t, b) ,.._ is then subtracted from the input spectrum fi(t), leaving the sat~llite spectrum f 1 (t) (Fig. 6). The position of the satellite can be estimated from f 1 (t). This estimate is then used as data ("SEPRTE") for the second iteration. This second iteration should remove the distortion in f 1 (t) indicated in Fig. 6, and produce a new spectrum, f 2 (t) with a maximum in the correct position (Fig. 7). 198 St;-~~~•.-......~~~,,_._...~~~........f!GWCT 11111! Figure 5. The least-squares fit of the proton resonance of CHC1 3 (o = 1. 0) using a Lorentzian line• shape function . Figure 6 . The line shape of the satellite peak is distorted by removing the parent peak. ,_ - 199 The sum of the squares of the residues from the least-squares fit, ¢, should decrease by about 0. 25 Es (the output parameter Es is a convenient normalization for ¢) during the second iteration (Fig. 7). If ¢ increases, the estimated position of the satellite was incorrect. This procedure should converge after two iterations. The "wiggles" in the line shape must be to the right when the least-squares fit is performed. This is accomplished with the input parameter "NRFLCT" . q, 1 = O. _l 97 E 5 z ~8 I- N CL a: o 0 (/) CD a: 0 0 The functions f,(t) (top) and f 2 (t) (bottom) [Eqs. (5. 3. 1) and (5. 3. 3)] resulting from the above CHCl; spectrum. A matched filter has been used and the scale is such that fi(o) = 100 . The error bars represent one s tandard deviation 200 Data Collection ~"""""""" All spectra must be taken with the instrument RC filter disconnected. This can be accomplished by setting the "function" switch of the DFS-60 spectrometer to "scope" or the "filter bandwidth" of the A-60 spectrometer to "4 cps". If the instrument RC filter is not disconnected, the noise in the spectrum will not be white, and using SNARE to filter spectra will not be valid. If the line shape is determined separately, it is critical that all experimental parameters (sweep time, sweep width, phase, rf field strength, etc.) be as nearly identical to those for the spectrum as is possible. In order to accomplish this one should alternately scan the sample and the line-shape standard. Considerable computing time can be saved by using only the portion of the line shape which differs significantly from zero. The number of data sets to be added to form a spectrum, "NSTITL", can be arbitrarily large. Each data set must be properly '.'rolled over". Any data set in which one of the peaks is truncated is useless. To add m data sets, simply set "NSTITL" equal tom, place a card giving "NPOINT" and "NALIGN" in front of each data set, and place the data sets in the input data deck. The data set with the largest peak farthest to the left must be placed first. Two data sets should not be added unless the same setting of the "gain control" was used for both. 201 The current version of SNARE will cost between $1 and $5 per spectrum depending upon the options specified. The program can select a portion of the input spectrum, FI(T), to use for the line shape, S(T). This will be convenient if the standard line shape sample and the spectrum were scanned simultaneously. To use this option, set "NSIG" equal to zero and set "FIT" slightly larger than (S/ N)- 1 • The program will then find the largest peak in FI(T) and use a segment of FI(T) which includes this peak for the line shape S(T). For absolute control of the segment used for the line shape-duplicate the cards representing this segment and submit them as a separate signal shape. Regardless of how the line shape is measured, the signal-to- noise ratio for the line shape must be better (by at least a factor of two) than that for the signal. The functions F(T), H(T), and FO(T) are the least-squares Lorentzian, the weighting function and the output spectrum, I respectively. The weighting function is calculated from S(T) and is of interest when considering the theory of filtering. A complete list of the input parameters is given on the next two pages. It is assumed that the data has been punched on cards in format (8X, 12¢6), and that there is a maximum of 1024 data points per scan. ·r.-------- __ c_________ c c cr. ··r. ( c c r. c c c c c c c cc-~-~ --- 202 OATA C ARO -c-o·c ~ c-:. .:-"~ : -"·-·" _-_ - " " "-:" - - " ~ --" Slf o o~-( ,,SN•Ro o l A Fl!IJM TH~ ·n~S-b(J. .. . ' "'4S TJTLtl51 t.tUM 8ER llF S P FCT+U TO RF AIHffO 4 <; r. fl'°' Hm I) 1- . i=.«t=Ft F C Tl llN ~-11.:;~ /\ - ~() H 1" N5. TnN1 J ~ ) C C · c;1 <71l 6 I. YIJI J 11-4 I lllk AFFfJ.,() . r. t 1: C C C C C r \ W E fPIFll. ~ 1 C "011 C /l. N <;t4U 0 01t: ~N A ll OO;:! l ' '\, i..• •Q.Of;'.1J. R r; 1-11. TF. R. SNAkfH) 4 C, fl J.'O\ I T I V~ N UHkf.&oe i.-HJ(. l-1 $ N/I N.OC1t,.H Tl) tJ'\ WE H> , <;t..i A4 i.;M,AC T JllN Lrl- THt-; 1 Nl ENS 11 V HF- THF +.' ~~r:r..i T <; NA )( fl04 J SWH-'.f' WID lt-4 JN H/., (.0~..tF: S l-'ON O S C t: FIT I F 11. 5 I TH~ 0rs c PF.AK TO WH1CH F J TT Y'NG ~··~ i;:· _'. PE R F~M F. 0 1114 THE' SN AR. :~~g~.' ~1=~;:;~~~N TH; ;Jz;H~F P~~;.M~:.~=F.NT ~=::005 a llS EO FOR SN•Roos• T~E LINE SHAPF •H F• NSIG=o . YSCALECFll.SI A SCAL ING FA C TOR. FOR T~f FttTfREO SN AROO'if. SPEC TRUM. YSC • tE TIMF S THE !'C A1. F. FnR lH ESN ~ R Ul< I G I NAL Sl'ECTklH4 1 5. THF SG/11.F llF l HF. $ Nlllof. C C i C ' C F'IL,FllEIT SPFtllHii<; ··- · ·· ... I f.. Y 5.CllL ~ .L i; . o . YS CALf Ji;. CA LCU L ATFO T IJ F IT THF. F ll. TfMFI> SPl-!C r Mut~ C C UN l HE P LOT T ING P AJ.'F ~ Ar.J n C (. TO Mrlf<,F TtlAT --%P~~~ r,1,;;~ ;-~~m%~ {t;~i~tr- c r. ''' "''"'''•/·, Gf·IAW00/•-1 I MATC HF IJ ~ltl l;R - Ml 0:. ,.. Alc 0 03 ' ER!COA: 11 A ~S . (,NA ll. O())(, 0 MATCH EO F 11. l fR - t:~RllR \ ,..AIUJf) 3H MARS . ~MAR Oo H . u::. -1 RC F l l l fl.t SNA R0 0 4 f. S ET N~TON EOlJ AL T O SN& AOQ4 A -J r>O• IT H E HALF Wl 0 H1 SNAk0-04J;t AT H ALF HFI GHT r)F THE ~kAA004C N A~RllW ~ S T l I NE I FU ~ HtF. SNAH. 0040 C t S ' •t• "'·" ' ;.i: ~ ~, •." ,,. , ?l.: f, -t, ( J ,. +l IJN F REFlECTJr1N - ll5F. F m1 OO ~"i FJF l.1 1 ~~f(.l~ll f.- k ll"°' l1F t., - 1,c1. V~ t\ " '•'1/ I \ ( (f N/11. I ll NlllSI-: .GT.\ fJPT{l.tfjj,'. ru:'°. 11L •llf f't(I/ <, r.. 4µ 0 11 ·H. HutANU- MF /"1 f JLl J.L. -~f'. f },1~ -'"t N\ 1111\l F'J1Jb L 111 It:'" oi;,r . .. o1n11"3fl l 1~ 1--\ I HI- Ut~S I ~ ~1. tou , 11 ~~.,. ( C r: SNAA ;,.;; ~ <;F.F j)J:. ~., -,_!J. A J() l-~11._. (. C • A•. fJfr.., 11;. TF r., r. S V l- ClMti C c C t; S NA~ OO lF SkAH = - 1 ~ f "111Vf- l' Ak F.1•1 1 P~/\I( \ ' ,",l>t,(t?.\ 11 SI f,NAL SH 11.PF. I f·I <;, IJ~ C TRIJt.; +l S fl,NA L SHM'I- 0.,f ~Ak6 H. '"f..,' '· 'i''~ N ~Fl_i.1' 11 5 ) =- I NO REFLfCTION<;, - •J5.F. FPli 11µ J:.I H . fJ ~"' J.": l"'· \ i.. •t.111ri,;> 1, c c C C 5 NA Roo n SN &!( N\tr. lt 3 l c C: ( SNM{ 0 c C C SNAROO J C S Nfll< \tl ll~ () f'\H S F.PARATE IN S TRUCTI ON~ AS f r! 0 11T /, r.11 •.L !::r: Tf rJ,\ ! <; '·•1>14 1 PSOLIJ 2J • 0 ... SE llDtl - L P.H:A R J:.Jl_ll- W \ i.tll •H ' !')J · '°'E . O L IN F AR ftR t r- tt<;T-\Ct1JllRF- c,; 1: \LT~.. °'" 41.10!<.?t. r. c r. . SNUtOtll~ THE NUMK FR OF f N!)F.PE NUE ·\ IT S P ~ C T RAL ..-Rni... l_FMS HI I-If CllNS l t>F.toil- 1) ., THF. IJ Al /\ C AW I J~ F llM l· ACH PIH)i\L EM ARE" PA."fFA CF.f') l1FQ:F Ft Y 4 f>. AJ , • • •• F T C ~ CAR(l An • • H OSCR.., rt 7.0 A4 1 11 T U: F1 1K THE P i.t 1 NTl:IJ otJTP•JT. <:ARD A l • •• • TITL E(A4 ) TJTLE FOR THI: PUITS . NSWEf P( 1">1. f'llUMk F~ llF P (IJNl S !:fJUIVAU: Nl TCJ THF. S WE-f P WJ UHt. THI S Wf t l GF.N E ~ A L LY RE 107. 4 !"OR SP ECTRA F M. OM THf 4- bO ANO Cil99 FOR S ~ fC T~A 1 • • ••• Pf'SP EC T IJ';I TH F 5.Cfll F I'\ Pl(JNTEft . THE FH ACT l flNJ\ t Nl\TllKllt 11. ~llN I M IJ( f: Hf. lH t: Nl J(.Lf.U5. AES"11iNS I KL~ r 11" r..u: <;A T F.' LllTI-. S Ff'RrF. f J: I J .'H O I S TA •. CE I N HF.RTZ HF. i WE t- roJ lHf \ AHL I.I I tVfl\I( ANI} TH t: J.'/IQfN T l'f/\K C., f.T Tl! t 1 . Jr UNKNttl-'N . CARO A7 • • •• I F PLllTflll1 3 J "'Cl 11 0 Nill Pl.Ill THF.: L FA~T -~Pllflll F S UF HRMI Nt:ll ti JP f:1'1 r l. I A ~ . PLnT THE !.f ASl -<; (} IJAP. F!; F IJNC T Till>/ . I FP LO Jl (? tlf 3 1 = II 1111 NIJT PUil FI P J. l J.' 1.llT FT IT l. 7 1-'LOT ANO I.AKEL f TIT),. A8NIJNC IF 7 . ')) 0 1F l'L 0lf 3 J(1 3 1 s C -C IF+>LOff~l(BI C ('. 0 rm NOT PLOT S(T). I µ LllT SI Tl. ·2 "PL OT - ANO tARF L <;(T) ·. <;N flM O U C. t-' \ N/il(f!Cl"I °'N MH 1n;, ,. C. F'l /llof. ~f>l/\W SNAI( SN/I R•lOh K 1)1'1hM.()ll ht ~l.Jllkll fli'i [1 S l• /\ ROllM:: 5.N/lfHIOh F SNMHIO#,f. SNl\R0fl6H "iN l.l ~OOhl SN /\ RO n7£. S NA~O O?A .. 0 llO NO T .,_ LOT H fTI. l PUil H( TI. 2 PLOT ANO LA BE L H IT J. St-1Mrn o 1 1-1 C C C IFPL OT(5)fl3J .. 0 00 NOT PLOT FJCTJ-FIT) ,. 1 1-'L(Jl Fl(ll-F(TJ. r ----- - - --- - -- · C I FP LOTC61fl 3 ) ""0 DO NO T Pl rtT FOfT),. C C S"'AfWf1<; 1SN l'IWf)ll') F S._,fiMll ll'\ r7 SN /\ R007C S"'A R007 [J St.i6+<007E Sl-IU[00 7F ' 2 ,,CTIT'JmO "CTllc[ - F'J!TICF JTJ ; SN 6 R 00 7G SNUl007f.t 1 PLOT FO ( l I . 2 PLOT ANO LABEL FOtl I. SN AA.0071 ..!_~. . .K'°.!..G..Qf.ll:.l.tli . I!!L!. ~~GTH OF THE g:...All.S__I~• _ ___ SNAR.QQ6Jt C YLNGhHF.7. 2 J THE LENGT H Of TH E Y-AXIS I N INCHES. S "'UWOHA -c--- -·caRo AJK • •-;NPOIN' '15 l-n=rE"""RUfil8"Elf"llr-craTl"'l'tJT'FlrrllrTRT s:;:'t'A:N'"~ . . 'SHTROORll C THERE ARE A · TOTAL OF NS TI TL S C4NS TD Rf SNAR c PR OCE"SEO ·as EACH OF WHtCH" lS PR ECEOEO SNAR C 8V A CAN.O A~K. NPfJ INT WILL BE LESS THAN S,..6flt C ·NSWE F.P IF ANY PART OF THE SPfCTROM lS SN AR' C. ARB JT~ARI LY OH JTTEO OR OMITTE O RE CAU SE OF S NAA ,,,9 -c - ···- --···- - · - - - --·-·--...rorse-;- 'S'PTKC'S·; ·· Err-;-··-· .. ------- - · -· ·-··· --· C NALI GN l( S ) 't C l C .T.:... ·- ·- - ····.. ·· -· · ·- C (', C c <; ' (; C r;. Al. J(; NF: ll. C.,MAl<I Nl\ l, Jt; N ,. L f. - 1 THF MAJC IMA litJl.I. RF l. JN fl) Ill'. f.AAfl S f A l 4 )K • •• AIN..-f11r'flllJM..l • • , •• IMlll POINl\ lfi.ICtl i'llh) , Tl~F F0tl0Wt,..r. r• .l ... llS Alff U~Fll r1 1>11. v "; NSIC; " ... , • f.AHU A"\K• ••• NS.,lllNfll ,.d hH· Nll#Htf.H rn l 1A ll\ "'llfNl~ JN Hff: !l'. ll:toli\l t; Al-!11 '.'ot SNAR THE NUMUER OF P OlN lS HI T ME LEFT TMJ S Sr.A N SN A ~n m4C t~ T0 . 8F. SH IFT ED .. IHF. SCAN w~nsF MA XIHllM 15 SNAROOIHl FARTHl:: S T TU THE LEF T flmST Al: RF /\0 JN .-: JH ~ T. :-iiN/\ROOM f' NALJC; N • 0 NO SHIFT OR. FIRST StllN. $ NAROO "'f: U SI· /I PflS ITIVt: VA l.ll E FOR NA L I GN WHfN l H fMF ~ NAM ·1 ~ - A~1nr.u 1rv - 1n:um1 ~Hn:H PF tiKc; ~"Rm11. n · A ~ <'i~•AH ~;l·Hll'I' • I A 14 11f.•, • • AIN ~l' Hf N l/llH<'l ...... •.IC,fJ/.t SH Al•i t 10' . ) 7. llh) • hAfA llO I Nl 'i (,flHI R:flf t lH; ""'/\Mf10•Ht 'iNAll:1wn1 -, NA IH!ll'lf. l\NAI Hll fJ'IA ~NAJ' llfl~k l\N Akflfl'' ( C: - ... - - - • - - - - - --- - ... - - - - - --- - - - - - - -· - - - - - - - - - -- - - - ---- - ---- - - - - - -- ------- - - - ~ """~ no•.it 1 ' C:AIHI' A ltc • • eA ([N J.>Ul,.,1'111. IK AMI:- NH'l!ATH> K• l tN Sl J TL !\fOMOOCfl(i. t UtnS "o······"""'IP01Nl/1 2 11C. AM.f f(fn11 TFO ~ • • •• NSPF.t'l S NAN ()()q F -r....·" '"~ - ..: "'-= -• ..... __ ..,._ ... .;;..-. ...... . .-._ - - .... ;.: _,.._---.. - - ........ - - - --- ~-.;...;.-=:... .:.:~ ..:i..:. -.:.:;.:~~4:. ..._. .:.·---.:.-sN'" nni;ic; 203 SNARE is currently on a disc library as ,JDR. NMHFLTR. The necessary Job Control Language cards and sample data cards are shown below: II C91,32,GA,,CH•,•GEOAGE A. PETERSSOH • '"SGLEYEl•l JOB PGM•SNA9'E ,lllEG IOH• l '()t(, T JME• 1 OSN-=JOR.NMRFL TR, VOl. =SER,,.C. l TSC.2 , UNI l •SYS04,0I SP•SHR SYSOUT • A ,DCB• ( RECFM:F8SA ,LRECL• l ~3, 8LKS I lE • 1596 J SYSOUT•N,UHI T"'(, SEP•FTObFOO l J EXEC / /STEPLIS 00 // FT06FOO l 00 fl . / /SY SPLTON OD //FT05FOOl 00 l TRANS-l. , 3-PENT ADI ENE ·o 0 6A-F 999 6 1 2 2 2 2 998 .0 I 2 -1 12.50 CAR80N-J 8 zso.o 37 10 SCANS 0~05 0.02 0.01 ·· 10. 00 Data from C. A. T . II .. j' ·--·· These are the only cards necessary to use SNARE. The subroutine structure of SNARE is as follows: ~NARl~ JI' RFLCTI EBPLOT OCTHEX ~ ~ VLABEL RFLCTR CPLOT FILTER . -.:..REMov: \ ,~ Iv1ATCH / . PHICLC ..._ GAMCLC BCALC 1 EQSOLV ~ "' ~ BNDINV RCFLTR LSQFIT ~ RESOLV ~ AGCALC ~ ~HICLt f ~FCALC2 FCALC ""' HARM 204 APPENDIX B A LISTING OF THE PROGRAM SNARE 205 C-------------- ---------------------- ----------------------------------- SNA ROOO A C • S NARE • SNAR C--------------------------------------------------------------- - ----- - -SN•ROO OA C THIS PROGRAM P ERFOR MS A VAR I ETY OF F ILTER !NG UPF.RU I llNS UN SN AR00 0" c NMR SPECTRA. THE UF TAILS AR F. D~SCR !RED I N THE PHO THf S I S OF SN•• n onc c r..A.PETERS SUN, f. ALI FORNI A ll'I STI TIJTE nF TECHNOL<)l;V I 19 701. SN• •nono C THE PROGMAM CA LLS THI· FULLllWINl; PLOTT ING SllAROllTINl· S I S NA MOO O ~ C (. C SNl\ ~ fl OOt- CPLOT t X; Y PLOTTP.+ i; ROllTJ NF. E6PLOTt PLOTS E J(J.1 0 .._ hAM S VLARF.l: LABELS THE AXE S OF l'UJl s . fHESf SUHKOUTINE S AJ< F. NCH INCLUllffJ IN S tH Mfltlllf. S. ~AH n on H THI~ ~ r~ J1K Oflf1I PfHJ Gl-( AM. C ----------------------------------------------------- - ----------- ------\~A ~ fl O J& S NA ~OO l A C t DATA C C C C C C C C C C C C C C C C C C c C C C C C C C C c C C C C C C C C C C C C C C CARO I • ••• • NSPE C TI 15 1 THf NIJ1<8 ER rJF l NllF.PEMl fNT SPECTRAL PR OllLEMS SN AM O() I C: Tri HF. CllN S IDF.REO. T"F. OHA CARDS F OR EACH S"'AR PR O»LEM AR E PREFACEO l< f RE BY AO,A1, . . . . HC . SN •M CARO AO oo oo OSCRPTl 2 0A41 TJTL t FOR THE PRINT ED OUTPUT. SN AROO I F. CARO Al •••• T ITL E IA4) TITLE FOR THE PLOT S . SNAROOIF NS WEEP 11 5 1 NUMBER OF POINTS EQUIVALENT TO TME SWEEP SNA R Wl DTH . Tt<I S Will GEiiER.lLLY BE 1024 FOR SNAR SP EC TRA FROM THE A-60 ANO 999 FOR SPECTRA SNA R FROM THE OFS-60. SNAR NSTITLl l5 1 NUMBER OF SPECTR A TO BE AOOED AS 'A'. SEE SNAROO IH SEP ARATE INSTRUCTION S AS TO DATA COLLECTION . SNAR I PSDLll 2 1 = 0 PSE UOO-LINEAR FILT ER SNAR OOll , NE. O LINEAR OR LEAST- SDUARES FILTER SNAROOZC NSIGC 131 •- I REMOVE P AR E.NT PEAK ,. SNU002 A SNAR0026 0 S IGNAL SHAPE IN S P ECTR~ +I SI GNAL SHAPE SEPARATE. SNAR002C NRFLCTll51 • -I NO REFLECTIONS-USE FOR UPFIEl.0' Sl'ECTMA SNAR0020 FROM A-60 ANQ Ql!.S-60. SN•R002E 0 Two RE FL EC TI ONS-US E FDR OOWl<Fi ELO SNAR002F SNAR002 G SPECTRA FRUM A-60 . +l ONE REFLECTION -USE FOR DOWNFI E U J SN • R<Hl 2H SPECTRA FROM l.>F S-60. SNAR 0 0 2 1 NSTONll51 S I GNAL TO NOI S E .GT.! OPTIMUM RESOLUTI ON SNAR003C ENHANC EMENT FILT ER -Sf. TSN AR NSTON EDllAL TO TE N SNAR003 A TIMES THE LOSS IN SNAR00 3 B SIGNAL TO NOIS E THAT SNAR0 0 3 C vrn i THINK VOIJ CAN S N• ~00 3D AF FORD. SNAR003E MATC.. EO FILTER - NO . SNAR0 03F ERROR BARS. SNAR 0 03G 0 HATCH EO FILTER - EHROR SNAR001'1 BARS. SN AR0031 .LE.-1 RC FILTER SNAR004 & SET NSTON EOIJAL TO SN•R004• -IOO* I THE HALF WI Ol H SNAR OO• R AT HALF HE IGHT OF THF. SNAR Q04C NARROWEST L INE I FOR THESNAH 0040 OP TIMUM TIME CONS TANT SNAR0 04E FOR THE DOUBLE SEC T I ON SNAR 004F RC FILTER . SNAR004G SWEEPI F ll. 5 1 SWEEP WIDTH IN HZ, A POSITIVE NUMBE R WH ICHSNAR004 H . CORRE SPONDS TO NSWEEP. SNAR FITIFll . 51 THE FRAC TION OF TH E INTENSITY OF THE P AREN T SNAROO•I PEAK TO WHICH FITTING IS PERFORMED IN THE SNAR REMOVAL OPERATION. -Olt- THE PARAMETER SNAR WHICH DET ERMINES THE SIZE OF THE SEGMENT SNAR005A USED FOR THE UNE SHAPE HHEll NSI G•O. SNAROOS !I YSCALE CFll. 5 1 A SC ALING FACTOlt FOR THE FILlERED .~ NAR00 5 C ~ C C C C C C SNA~ O O l k C C C C r. C C C C C C t G r. C f. l (. C C l C C. C C $)-IE GTRUM. YSC At E T J "•f:: S Tfo:: SCAL E FIH< l' HE SN A~ OR I GINA L SP ECTRUM I S THE SC ALE UF THE F I LT ERED SPEC TRUM. I F YSCALE .L E. O. YSCAL E I S CALCllLATE D TO F IT THE F I LTERE D SPECTR UM ON TH E PL nT TI NG PAPER ANO THE SC ALE I S PR I NTED . ABNONC CF7 . 5 1 THE F RA CTIONAL NATUR AL AHllNO ANC E OF THE NUCL EUS HESPUN S I BLE FOR THE SATELLIT E. SEPRTE!F l l. 5 1 DIS TANCE 11'1 Hf RTZ BE TWEEN THE SATELL I TE PF. Al< ANO THE PA J.t. ENT Pi:: AK. S ET TO O. UNK NO WN• 0 DO NO T PL O T TH E L E AS T - SO lJA RE~ O ET E R~ t NFO l.OR ENT Z lA N . l 1-' LCI T TH f LE A.S l - S O llAR l; S FUNC T J fJ f\4 . I F PL 0 Tf 2) 1 1 3 ) • 0 Utl MO T P LOl F i i l ). 1 f<' L O T F J f ·1 ) • 2 P LO l AN D LAHE L Fl f T) . JF P LO T (3 1 I J 3 ) a. 0 DO NO T 1->UJl S I T). l P U i l S t T ). 2 Y L OT ANO LA Rf: L <. ( T J . t FP L OTl 4 ) ( J ) ) Cl f)(I NflT PL OT tHT I • l PL O T HIT l • 2 ... LO T AN D LA R E L HI T I . J F PLO T l 'i Jfl 3 ) • 0 !)() NO T fl LO T F llTJ - F lf ) . C AR O AZ • • • • I F PL O Tfl J fl3 J l P U il Fl l Tl - ~ f T J. C 2 ._, L O T ANO LAP. El FI ( T, _ f- f T J . Cl D( I NOT PL O T F DITI. 1 PL OT F OIT). C 2 P L.OT ANO LA 8H FIJI 11. IFPLOTU1 l!J 3 J a IF SN AR SNAk SNAR0 05 E SNAH005 F SNAR00 5 G SNA K005'1 SNAR005 1 SN A• OOl>C SNAk $ NAk SI'! AR S NAR OObR SNA ROObC ~N 6 W00 1-, D $ 1.it-Jo<t' (•bf 5 1, A~ f1(1f.. F SN A ~OOb ~ S ""· !ik Jfl6.H !;.t.i.'. Rf)Q •_. J ~ ,,,a .:._ on7r. S "t U0) 1;H. c;r-..A i.ll) O H~ SNld c rl()'t (. \ N AMOl'Hll ~ NAIHH 1 7 E- . \ f1rU HOfJ If. !, NMH1() 7(; S. N AM.OU I H SNAM00 71 206 XLNGTHCF7.2) THF. LtNGTH OF THE X-AXI S IN INCHES. SNAkOO"t YLNGTHIF7.?I SNAROOA A ~ r. t CAJ<O A3 K• •• NPO INTI 151 TH ~ L~ NGTH Of THE Y-AXIS IN INCHES. THE NIJ MijER OF DA TA POINTS ltit THIS SCAN. SNAM.0088 THEkE ARE A TOTAL Of NSTITL SCANS TO BE SNAR SNAR ~ l'JUlCE SH> AS 'A', fACH OF WHICH IS PRECEOfO l\Y A CARO BK. NYOINT WILL BE LEH THAN t N~WEEP C t C. ( IF ANY PART Of THE S•ECTRUM IS A•BITk•RILY nMITTEU DR OMITTED BECAUSE Of SNAR SHAR SN•R ~f 1 J SF.- , SNA R Sl-' 110-S, '-TC . 'IALl <»H l~I TH f NUM BER O~ POINTS TU THE tEFT lHI S SC AN JS HI Rf: SHIF-H:O. <; r. C C C C C C C C C C THE SNAR008C SC.AN i,.jHf) St MA X IMUM I SSNAtl.008 0 FAklHE S I rn THE LEFT ..UST l\ E READ IN FIRST . NALI GN • 0 NO SHIFT OR HRS! SCAN. USE A PnS ITIVE VALUE FOR NAL!GN WHEN THE RE IS AMBIGll ITY Al\OlJT WHICH PEAK S SHIJllL D llE ALIGNED. MALI GN . LE.-1 THf MAX(HA WI LL Bf LINf O U~ . CARDS IAl41K ••• AINPO!NT/121Kl ••••• DATA POINTS (OX.1 2061 THE FOLLOWING CARil~ ARE US ED ONLY I F NSIG • +l. CARO A3K• ••• NSPOINT l1 51 THE NUMBER Of DATA POINTS IN THE SI GNAL SHAP E. CARDS IA l41K'•••AINSPO !NT/1 2 1K'l ••••• S I GNAL SHAPE DATA POIN TS (RX, L21J6 J. SNAR006E SNAR OOllF SNAR SNAR SNAR SNAROOAG SNAROORH SNAROORI SNIM009t SNAR009A SNAM009R SNAHooqc c C------------------------------- - ---- ---- ------------------------------ -SN6R0090 C CARDS A3K • •• AINPO JNT /L 2 JK ARE MEPE AT ED K= L,NST ITL SNAR009F c CARO~ R0 •• •• ,.8CNPOJNT/L21K AR ~ REPEATED 8 •••• N SPfC l S~A~on~F C-----------------------------------------------------------------------SNAM009G DIMENSION TIT LE (3J, VC1 024), XC1024 J, Lll024J, tH5l ,A(5, 5 J, SOMCSNAR009t-4 l l 024 , 's' 1024, 'FI c 1024,. FO ( 1024 , 'H ( l 024) 'SSQ I 1 024 I' nsc RPT 120 )' SN.t.Rooq I 2XAXISC4J,YAXISl3J, FMTHl 2 l, FMTVl 21, IF PLUTl6) SNAROlOt COMMOM/COMCPL/ITEST,XLNGTH,YLNGTH SNARO!OA REAL LA8EL15l SNAROlOB DATA FMTV/4HIFB.,4H31 SNAROlOC DATA FHTH/4H(f8.,4Hll I SNAROlOD DATA LABEL/4HSPCT,4HSGNLo4HRMOV,4HFLTR,4HHITI/ SNAROlOF. DATA XAXIS/4HFRE0 0 4HUENC,4HY IN,4H Hl I SNAROIOF DATA YAXIS/4HAB S 0,4H~PT1 0 4 HON I SNAROIOG C-----------------------------------------------------------------------SHAROlOH C READ THE NUMBER OF COM~LETE CYCLES. SNAROIDI C---- - ----------------------- --------- -- ---._--------------------------SNAR01l& REAO ( 5 , l ) Pi SP'ECT l FORMAT 1151 DO 100 IS•loNSPECT RF.AO 15,1 02 ) IO SCfitPTCt J.t = l, 20 ) 102 FORMA T 120A41 S NA ~O llA SNARO ll B SN AROllC SNARO ilO SNAR Oll E C -------------------------------- -~---------------------------- -~-- -- --SNARO llF C READ IN THE PARAMETERS. SNAROl l G C---------------------------------------------------------~------------ SNAkO ll H REAOC 5, 2 ) TITLEl 2) ,NSWEEP,NSTITL,IPSOL, NSIG, NR FLCT, NSTON, l SWEEP, FIT, VSCAL E ,A6NONC,SEPR1E 2 FORMA T IA 4 ,2J 5 , 12 ,1 3t2!5 9 3 Fl~.5,F 7.5 ,Flt.5• REA015,899) (IFPlOT(J),J =l,bl,XLNGTH.YLNGTH eqq FORHAT l6! 3 , 2 F7.21 ITEST • 1 IFIXLNGTH.GT,1 2 .51 XLNGTH•l2.5 !FI YLNGTH.GT.1 0.01 YLNGTH=lO.O XLNG = XLNGTH YLNG : YLNGTH XLNGTH • 1. 2*XLNGTH NWDTH • (6* NSWEEP1/ 5 no 3 I= l, 1024 Y{))=O. SNAMO lll SN AR0 1 7f. SNAR 0 12A SN &R0 1 2R SNA~Ol 2C SN AR0 120 SNA~Ol2F SNAR0 1 2f SNA<O l 2G SNAR0l2~ SNAR0121 SNAR0 13t SNAROl3 A SNMW 13 K 3X(ll=I SNA <Ol3C NREAO 'Cl: O SN AR O 130 NPT • .1024 SNAR 0!3F. C-------------------------------------------------- --------------------- SN AR0 !3F C READ IN THE SPECTRUM . SN A•Ol 3G C-----------------------------------------------------------------------SNAROl3H 4 READ 15 0 51 HPOINT , NA LI GN SNAR0 13 1 5 FORMAT (2151 SNAR014t CALL OCTHEX(L,NPOINTI SNAR0 14A SNAR0148 IFINRFLCT .EQ.01 CALL RFLCT! IL,NPO!NTI c ----------------~------------------------------------------------------SNAROl4C C ALIGN THE SCANS. SNAR014D C-----------------~--------------------~------~--~~-~-------------SNAR014E l.ft lf (NALIGN) 7 ,. lO e l O 7 LBIG 00 9 If =O I ~ 1, CLBJG - LI 11) 5,9,q 8 !BIG =I LBIG • SNAROl.,.F SNAR014G MPOIHT S NAA O I ~~ SN AR0 14 1 SNAR015t LCI) SN &RO l ~A q CONTINUE ShAR015 8 NALIGN • 1 81G - H81 G SNAROL 5C 400 fORMATt//,5X,69HTHE SCAN WHO SE MAXIMUM IS FARTHEST TO THE LEFT MUSSNAR01 50 LT BE READ IN F IRST . I SNA• 0 15 F. lFINALIGN.GE.01 GO TO 10 SNAR0 15 F WRITE (b,4001 GO TO 100 10 CONTINUE IMAX • NPOINT - NALI GN SN AMOl~G S NAR0l ~ t1 SNAMOl5l SNAR016f. LO • Q SNAR0 16A LOMAX • IHAX120 00 80 l .• l,LJ!.M.U S"AROl68 SN ARO l6C 207 80 LO• LO+Llll SN••OlbO , LO • LO/lO ..AX SN•ROll>E t--------------------------------------------------------------------SN•ROlbF C &00 THE SCANS. SNAROlbG C------------.:..--------------------- ---J--------------------------------5NAR016H DO II I • I, '""X SNAROl•I K • I + MALIGN SNAROllt Yl • tlKl-tO SNAROllA 11 'f( II • ·y 111 + YI. SNAll.0118 NPT • "INOINPT,l"AXI SNAll.Ollt t--------------------------------~-----SH&llOl 70 t l·F THIS IS THE FIRST SCAN FIND THE MAlll-..:; :· ·., :, .' SH&llOl7E .,_--------Slllt.11Dl lf'llllll!AOI u.1s.u ,•.· ,.,, , . SUllOllCI C----.,-----------------------15 ff!!:~:;_~~:';~~·~ •. n.n •••I• •= r . YBIG Y l l) 7F ;• ;.~)iD_i\. :~.~iii ~ SNAFt0lH8 17 CONTINUE SNAR018C 18 NREAO • NREAO + I IF IN STlll - lf!IE&OI 19.19, ,. SN&ROl8E SNAROl~D C---------- ------------------------------------ ---------------------- ---SN•ROl~F c ALL SCANS HAVE BEEN READ IN, ALIGN F. U , AND ADDED. we NOW FlNOSNAROlRG t THE MAXIMUM Of THE COMBINED S"ECTRUH. SN&ROIRH t--------- --------------------------------------------------------------SNAR0181 19 Y81G : O. 00 Ill l •l 9 NPT IFCV81G-Yfl)l 116,117,117 SNAM0l9f. SNAR019A SNAR019~ lib NBIG • I Y81G • YCll 117 CONTINUE IFCNSIGI 20,200,22 SNARDl9C SNAR0\90 SN ARO l 9E SNAR019F C---- ---·'------------ -------------------- ------------ -- ------------------SN AR 0 l % C USE THE ENTIRE SPEC TRUM FOR THE LINE SHAPE . SNAROl9H C---- ------------- ---- --------- ----- - ------------- - ---------- ----- ------SNAROl91 20 00 21 I • I , NPT SNU020f, 11 S Cl I • YCII SNAR020A NSPT • NPT Sr<AR0206 NS..t.X • NBIG SNAR020C SHAX • YIN81Gl SNAR0200 GO TO 27 SNAR020E C-------------..,-------------------------------------------------------SNAROZOF C SELECT PART Of THE SPECTRUM FU~ THE LINE SHAPE. SNAii.OZOG C--------------------------------~-- ·----~- -----------------------------SHAR020H 200 YFIT • FIT•YCNBIGI 00 201 1•1.NBIG IFCYFIT-Yllll 202,201,201 20 1 COllTINUE Z02 NF IT • I NN • NBIG-NFIT NSFIT • Nfll-NN NSFIT • "AXOC1,NSF!TI NSPT • nH11 . .TEMI' • Nf!T-NSFJT NSPT • MINOCNSPT,NTEMPI NSMAX • Nft!G-NSFIT SMAX • YCNBIGI 00 203 l •I ,NSPT K • l +NSFIT 203 Siii • YiK) GO TO 27 SNAll.0201 SNAR021t SNAR021A SN00218 SNAR02 IC SNAAOZID . SNAAOZIE SltAll&ztF S..AR021H SNAR02 I l SNAR022t SNAR022A SNAR0228 C-------------------------------------~--------------------------------SNAR022C READ IN THE LINE SHAPE. SNAR022D C-- ---------------------------------- --------•------------------------- -SNAR022E 22 READ C5ill NSPT SNAR022F CALL OCTHEXJL,NSPTI SNAR022G C SMAX • o. OD 24 I • 1, MSPT Sill• Lill - ll ll IF I SMAX - S Cl I I 23 ,24,24 23 NSMAX • I SMAX • SI I ) 24 CONTINUE lFfNRFLCT.EQ. OJ CALL RFLCTRCSrNSPT> SNAR02?H SNAR0 22 1 SNAR023' SNAR02H SN AR 0 2 3~ SNAR023C .SNAR0230 SN•~023E C-----------------------------------------•-----------------------------S NAft023f C FILTER THE SPECTRUM. NAR023G C------------------------------------------------- - ---------------- -----SNAR 023H SHAR0231 27 IF CNSIGI 28,29,29 28 CALL REMOVE CNB TG, x,Y,FtT,F J,8 ,NP T, NFJT,ABNONC ,N S W E EP,SWEEP,SEPRTE,SNAR024~ lNWOTH,NRFLCT,A,PHJ,JfPLOTflJJ SNAR014A CALL FllTERCNSTON,S,NSPT,NFIT,NSMAX,SMAX•fl,fO,H,SOH,SWEEP,NSWEEP,SNAR0248 IHElP,IPSDll SNAR02"C NFIPT • NFIT SNA•0240 GO TO 30 SNAR024E 29 CALL FILTER CNSTON,S,NsPT,NPT,NSMAX,SHAx.v,Fo,H,SOM,SWEEP,NSWEEP, SNAR024F IHELP,JPSOLJ SNAR02"G C----------------------------------------------------------------------SNAA024H C SET THE PARAHETeMS FOR PLOTTING FICT). SNHOZ41 c---------~--------~--------------------------------------------------SNAAOl5l NflPT • NPT lO TITLf(J) • 1 . T!fl!Cll • LA!l_!Ltl.! SNAAOZ'A SNAAQISI SN.A~.,Jt 208 VMIN • -o.B•VINBIGI VMAX • 1.2•YIN81GI NOUT • NFIPT NXO • Nlll'i IFINSTDN.~T.OI GD TO Joo SNAROZ50 SNAROZ5f SNAM015f SllAR025G $NAR025H C•-~----~--------------------------------------------------------------SNAA02'1 C CALCULATE THE AVERAGE ENHANCEM.ENT Of THE SIGNAL- TD-NOISE SNAR026t C ~ATID WHICH WOULD RESULT FROM A MATCHED FILTER. SNAR026A C-- -·------------ ----------------------------------------------·-SNAR02b8 SIGMA • o. StlAA02bC SNAR026D SSQ 111 • I Sil 11 . . 2 lllJ 66 l•Z.NSPT . SNAR026E 46 5$0111 • SSQ 11•11 + tsllll••2 SNAa02bf I)!] 64 J•l.NFIPT SN&R02Mi K· • NSMAX • J SNAR026tt IMAX • NS~~ • It SNAR0261 llllAX • MlliOINf'IPT,IMUI SNAROZU SNAR027& IMIN • l·K IMIN • MAX()lhlMllO SNAA0278 SNAROZ7C IXK • ltlAX+lt itttc. '• I MIN+K SNAA0270 44 .UGM • '1GllA+SQRT I SSOllXK l·SSOCINK II SNAR027E •f'llll'T • NFll'T SNAA027F SN&R021'4> il'lllCMT • ·IJ$1UVl$MAX•F I NPTI c~-~~~~~~~~~~!~~~~-------------------------..:.----:~::~w. . . lil!fUil l f Mio Sffl FT THE SPEC'*'*• Sll. .Qalt. " ~~~,..~..f"".--. ..;""._.~'-!'._.;,,..;~--. ..,...... . __ ,.,. _ _ _ _ _ _ __ ,._7 ·1t&1t.. • lll'l~IG+l \., . {;~'.ll. !lfl.1;;111'.tVtflPT I - _ _ _ _ ____ ,... • . u . ~· • .<'ALL. ,_,..UJws. 11&t11 t · ,· . . . . ,,. · .,'~ ·. ::,; . - SMA~I S#AA. tlllAll . , • , Slit · CALL R~L C 1R!FO,NFIPTI SNAROZRE IF(NPT,EQ , NF!Pll GO TO 51 SNARO ZAF 00 SOl Jsl,NFIPT SN•R028G 501 FO!NPT-1•11 • FO!NF!PT-1•1> SNAR028H K • NPT-NFIPT SNAR0281 IO.•~INBIGI SNAR029£ BIG DO 502 l • l,K SN&R029A 502 FOi 11 • BIG SNAR0 29B NFIPT • NPT SNAR029C C---~--------- --------------- - --------------------------- ---------------SNAR0290 C PLOT Fl ( T) , SNAR029E C---------- ------ ------ --- - ---------------------------- ---- - --------- ---SNAR029F 51 IFll FPLOTC21.E0.21 SNAR0 29G = l t ALL VLA8EL(O.,o •• -. e,1 . 2,vLNG,10,YAXIS,10,1 , FMTV,8t SNAR029H I FC I FPLOT 121.EQ,2 l SNAR 0.291 lCALL VLABELIO.,o.,o • • swEEP, XLNG,5,XA XIS,15,o,f"TH,8) SN AR030& IFCIFPLOT!21.GE . ll SNAR030A lCALL CPLOTCY 1 NPT , NWOTH,TJTL E,YHAX,YHIN,1) SNAR0 3 08 IFCNSTON.lT.01 GO TO 500 SNAR03 0C C----------------------------------------------------------- -----------SN AR0300 C PLOT SI Tl , SNAR030E C------------------------------------------------------- - ---------------SNAR030F TITLECll • LA8EL(21 SN AR030G TM l N • -o.&•SHAX SNAR030H TMAX • l.2•SMAX SNAR030 1 I FI I FPlOTC31 . E0,2 1 SN A~031' lC ALL VLA8ELIO.,o.,- . a,1.2, YLNG,tO,YAXIS,10 , 1,FMTV,8I SHAR03 1 A IFllFPLOT! 3 1.EQ, 21 SNAR03 18 lCALL VLABELCO.,o . ,o ., sweeP,XLNG,5,XAXIS,1 5 ,0,FMTH.8) SNAR031C IFCIFPLOTC31,GE . ll SNAR0310 lCALL CPLOT( S ,NSPT,NWOTH,TITLE,TMAX,TM I N,11. SNAR031E IFINS TON.LE.01 GD TO 500 SNAR031f C------------------------------------------~~--------------- -- ------S N &R031 G C PLOT HIT) , . SNAR03 1H C---------------------------------------------------~---~-------------SN AR03ll IFINRFLC1 , GE . Ol CALL RFLCTRIH , NSPTI . SNAR 0 32' 30 5 TITLE l ll • LA 8 EL151 SNAR03Z4 HMAX • O, SNAR0328 00 707 I • I , 11~PT SNARn3 2( AB SOL • A8S IHCI II SN AR0 3 20 707 HMA X • AMA Xl lHMAX , ABSOLI SNAR03 2E HHIN • -HMA X SNAR03 2F SNAR03 2G ! F llF PLOTl .. l . EQ , 21 lCALL VLA6EL~o.,o ., -1 . ,1.o ,v L N G,10,v A x 1s .10.1 , FHTV ,8 ) SNAR 03 2H 1Ffl FPL OTt4J. EQ, 2 ) S NAR0,21 lCALL VLABEL( O. ,o.,o.,sweEP,XLNG, 5 . XAXJS,1 5,0 ,FHTH.e) SNAR033t lf!IFPLOTC4l,GE.11 SNAR0 3 3 l lCALL CPLOTCH,NS PT,NWOTH,TITLE.HMl X.HMIN.lJ S~AR 033A SOO CONTINUE SNAR033C IFf YSCALEl31, 3 1 , 33 SNAR03 30 C--------------- --------------------------------------- -- - ---- ---------SNAR03 3E C DETERMINE THE SC ALE FUR THE INPUT AND OUT PUT SPEC TRA, SNA~033F C----·------------------------------------- ----------------------------- S N A~0 ~ 3f. 3 1 FOMIN = 0 . SN A R03~H FOMA X • 0• MINI • I IFCNRFLCT .GE.QI MINI • NFIPT-NOUT•l 00 32 l•MINl.NFIPT FOMIN • AMINICFOHIN , FOClll 32 FOMAX • AMAXllFOMAX , FOllll !00 TO 34 • . 33 FOMIN • YSCALE•VMIN FD)4A~ ".. V.S CAl.~•YMAX. _ - ·-·-- · -· SN Afl03 3 I SNAR034& SN AR03 0 SNA R0 348 SNAR0 34C SNAR 03"0 SNAR0 34E SN AR034F SNARQ34G 209 34 IFINSIGI 35,36,36 SNAR034tt C----------------------~---------------- ----~~---------~------------SNAR0341 C PLOT FJITI - Fill . SNAR035& t-•--------------~-----------------------------------------------------~SNAR035A 35 TITLElll • LA8!Ll31 SNAR0358 FIMIN • lO.•fOlllN 'INAK • lO,•l'Ol'AX lFl . . fLCT.GJ.01 CALL AFLCTAlfJ,NPTI '" UN • 'IMIN/'rCllelCI XHX • FIMAX/YINBIGI IFllFPLOTl51,E0,21 SNAR035t SNAR0350 5NAR035E SNU035F SNAR035G SNAR035H le.ALL VLAAEL f o. ,o •• IC.HJit, XMK .YLNC. .10. VAlC I s.10' 1,FHTY .e) SNAR03!5 I IFUFPLOTl51 , EQ,21 SNAR036& lCALL VLA8tL l 0 • ,O • ,O. ,SWEEP, XLNG ,~ ,X AXIS, }S tO, FMTH ,8, SNAR036A IFllFPLOTl51,GE.11 !CALL CPLOTtFl,HPT,NWOTH,TITLE,FIMAX,F IMJN,l) SNAR0 368 SNAR036C C---------~----------~-------------------- ----------------------------SNAR0360 PLOT FOi Tl WITH OR WITHOUT ERROR BARS, C SNU03bE C---- ---------------~--------------------------~----~----------------SNAR036F 36 TITLEll) LA6EL(4) lfCNSTOMI 38,37,38 SHAR036G c SNAR036H C-----~-----~---------------~----------------------------------------SNAR03bl C PLOT THE ERROR BARS, SNAR037t C~-------- -------------------~-----------------~-------~------~----SNAR037A 37 WOTH • NWOTH 00 45 J • l,NOUT, 20 IFINRFLCTI 56,55,55 · SNAR0378 SNAR037C SNAR0370 55,~0·T~F~;T-J+l ~~::g:;; 56 K ~ J 15 T • K WIDE • WOTH•ll5.IXLNGTHI HIGHT• lfOMlX-FOMINl•llO./YLNGTHl+FOMIN 45 CALL EBPLOT IT,FOIKl,SOHIJl,O.,WIOE,FOMIN,HIGHT,0.131 38 Xllll • FOH I N/YINBIGI XMX • FOMAX/YIN81GI IF CIFPLOTl61,EQ,21 lClLL VlABELlo,,o ,,xHN,XMX,YLNG,10,YAXIS,10,1.FMTV,81 lf11FPLOTC61.E0.21 1C4ll VLA8ELCO.,o •• o.,swEEP,XLNG,5,XAXlS,lS,o,f"™· ·· IFllFPLOTl61,GE , 11 · .. ... , ltALL CPLOTC FO,NFIPT ,NwOTH, TITLE ,FOMAX ,fONIN•lt ,. . , :. SNAR037f. SNAR03 7H SNAR0371 SNAR038& SNAR038A SNAR0388 SNAR038C SNAR0380 t---:-------------------------------..,-- .. . SNAR03~E SNARo38F SNAR038G SNAR038" SNAR0391 -------------SNAR039' ~---..:----~~~!~~-~~-~~~m~~~~~-~~~-NUNE~IC.AL ~~~r·· -------------~=::g::: ' ' ::~= :::!~~IT:~~~~;~:~~~:;!:~:~ITL,NSIG,:NAFLC·~~QN,SWEEP,FIT, ~:::::::; IYSCALF,ENHCMT,AKNONC, SEPRTE,l•SOL WRITE 16,3021 HELP IFINSIGI 39 ,40,40 3q AA -= A(l1/YCN8tGJ 0 • 8121 /SQRTl81311 SNAR 0 39E SNAR039F SNAR039G SNAR03fJtt SNAR0391 WOTH • NSWEEP SNAR040E YO • IWOTH/SWEEPI 88• 8131 • 1Y0••21 SNAR040A SNAR040B XO .,. NXO SH AR040C XO• IX0-81411/YO •YO • 8151/YINBIGI WRITEf 6,43) lA,o,ee,xo ,vo SNAR0400 SNAR04 0E PHI • PHl•CSMAX**2J/ISSOINSPTl•llYINBIGl•.5•A8NONCl•*2l) SN AR040 f SNAR040G WRITE (6,9981 PHI SNA R040H WRITE 16,999) SNARO°'fOI CIACJ,J) , J a l, 5 ),J•l,51 40 CONTINUE SNAR041& WRITE 16.171 SNAR 041A 77 FORMlT(IHll SNAR0418 101 FORMAT 11Hl,IX,20A41 SNAR04IC 42 FORMAT~lXrA4,/,/,}X,9HNSW EE P . ,1s,1,1x,9HNSTITL • ,1s.1,3X,7HNSIGSNAR0410 1 • ,1s,1,1x,9HNRFLCT • ,1s,1,2x,8HNSTON. ,1s,1,2X,8HSWEEP • ,Ftl.SNAR041E 25, /,4X,6HFIT c , Fll.5,l,1X,9HYSCALE • ,Fl l. 5,/ 9 /,lX,9HENHCMT • 9 F5SHARO•lF 3.1.1,1,1x,9HA8MONC • ,F1.s,1,1X,9HSEPRTE • • F11.s,1, 2x ,8HIPSOL • • sNAR041G 4151 Sl>IAR041H 43 FORMAT (/,24X,3Hl/2 , / 9 l5X,2 0HC l - D*S • CX-XO) ) , / , t X. 34HLIX> SNAROltll l• YO+ A•-------------------- , / ,30X , 1H2,/,16X,17H( l + .B*IX-XOI )SfltAR0'42l 2,1,1.zx, 4HA • ,El6.s,1,2x,4HO s ,Elb.8 , 4X,1H2,/,2X,4H8 = .Et6.8 ,4HSNAA042A 3 /HZ,1,1x.5HXO • tEl6.8,3H Hz,1.1 x.SHYO s ,e16.8) SHAAOo\28 302 FORMATl1 X, 32HTHE IMPROVEMENT IN NOJSE LEVEL •,f7.3,/,lH+t3lX,lH:,/SNARo-\ZC 11 998 FORMAT1//, 5X,6HPHI • ,Eu•• e,bH EIFI, ,//) 999 FORMATl40X,l8HCORRELATION MAJRIX,51/,1X,5F19,bll 100 CONTINUE $TOI' ENO SMAR.,..ZD SIUA04rE SNARD42F SNAR842G SNARO'-'tt< ~N~R042 1 REMOV E ( N81 G, X, Y, FIT, F J, 8,NPT ,NF l T , A8'.IDNC ,NSWEf ll, S WEEPN. E MOOOOA l ,SEPRTE ,NWOTH,NRFLCT ,A ,P ... J , fH" L0T) RE"'OOOOA C----- --------------------------------------------------------------- - --kE MOOOOC C THIS SUBROUTINE REMOVES THE PAKENT PEAK, REMOOOOO C-------------- ------------ ---- -----------------------------------------Ri MOOOOE SU 8ROllTI NE OlMENSION X(l024~,VC1024>,fl(l024),FPRIME(SJ,XFJT(5ltYFITl5),6f5) REHOOOOF DIMENSION 0013>.A~~.5),SORlC«S•tOfLTA(5),8TEST45>,AT~ST(5.5) AE"OOOOG COMMON /COMCPI_ / 1 TE s T 'XLNG TH. YLNG TH OOU6LE PRE CI S ION CHT X OillENSICIN CMTXIS,51 fl e--ooOOH Hf HOOOOI REHOOOIL RFAL LAMDlZ,LAH80A R~MOOO lA 001 l I • O, Rf.11000111 210 DOt 3) • l. kt:MOOOl C FITT • •ll•YIN81GI Rf.MnOO!O 00 l REMOOOIE I 1, NBJr; 1& If IYlll - FITTI 1,1, 2 CONTI NU~ 2 NftT • RFMOOOI• REMnOOI G I R EMOOO lH C. --------------------------------- ----------------- ---------------- -----R EMOOOll C SUBTRACT THE SATELLITE PEAK FROM THE PARENT P EAK FOR THE REM000 2t C LEASl - S~U AKES FIT. REM0002A C-- ---------------------------------------------------------------------REM00028 ~WEEPN • NSWEEP REM0002C JHALF • SEPRTE•ISWEEPN/SWEEPI REM0002D IFIJHALF.LE,01 GO TO 23 REM0002E ADO • O.S•ABNDNC REMD007 > IMAX • NPT-JHALF REM0002G MAX • "lNOtMFJT,tMAXt REM0007.H 00 21 l • l,MAX REM00021 2 1 VIII • Vlll-AOD•Vll•JHALFI RFM0003i C -----------~----------------------------------------------------- -----REMU003A C DftTA I N AN APPROXIMA'ff FIT USING ONLV FIVE DATA POINTS, (.-----------------------------------------------..... ,. .. . ;:, XflT Ill = l . REMOee:H -~-'!""·---·-----------!~. YFIT Ill • Ylll 00 3 1 · • 2 t 'S J Cl-l)•(NFIT/4) REM000 30 RE M0003E R EM0003F = Rt M0003G XFIT Ill • J REM0003H 3 YFIT Ill • YIJI REM00031 C------------ -----------------------------------------------------------REMOOO"t C S ET THE INITIAL VALUE S OF THE PAKAHETER S, B. REH000"A C-- - ------- - - --------------------------- ---- ------------ ------------- - - -REM0004R Bl 11 = YI NBI GI REHll004C 8(2 ) = 0. R f":- M00041J 8141 NBIG 8151 O. 8131 • IB!ll/IYFIT( 5 11 -l.ll(X F IT151 - 6 14 11**2 PHI O ~ 5. • 8111 * *2 TEST : Cl.05*ARNONCl * * 2 1* PHIO lAMOAZ = ' O.Ol CALL PHICLC(PHIZ , 5, XFJT,Vftl,FJ,8,51 CALL LSQFITC5,5, XFtT, YflT,fl, Bt l • PHl,FPRIM f , LAMOA Zt PHIZ,LA"80 A, IA, S QRTC,l,~TE S T, OELT A , B T E ST) REM0004E REM0004F REM0004G AfM(J004H REM00041 REM0005 t Rf M0005A RfM00058 REM0005 C REM0 005 0 REM0005E RFM0005 F REM0005G PHI O • PHI 00 4 l • lt5 LAMOAZ • LAMBDA PHIZ • PHI CALL LSOFITCS,5 , XFIT,Y f JT,FI,8, 3 ,PHl ,F P R IMEtLAMDAl , PHJZ,LA~BOA, REMOOO~H lA,SQRTC, 1 ,ATEST,OfLTA, BTEST> Rf MOOOS I DELPHI • PHIO - PHI REM0006t IF ITES T - OELPHI I 4 ,4,S REl<0006 A 4 PH10 • PHI REM00068 C---------------·------------------------------------------------- -------REM0006C C FIND THE LEA ST- SQUARE S LORENTZIAN US ING ALL DATA POINTS. REM00060 C- -------- --------------- - ---- - -- -- ---- --- --- ---------- ----~------------RFM0006f REMOO Ob f ~)T = N F IT PHI O = l ~ I Y IN R I GI lt~ l ~ ) ~•7 It . ()~ • A~~ON tl • *2 l * ~HI O L A.-11lA/ = O,O J C ALL Ptt l CLC IPMJZ , NFt T .x , v ,Fl , 8. 5 1 CA l l L SUF IT ( Nf l T ,';),X , y , f I ,A, I , P H I . f PM. I MF , 1~ A M OAZ , J.I H I Z , LA MB(JA , J A , SOR TC , l,AT FS T, O E LT A,l~ l ~ S Tt PHIO H ~ M 0 006 1 R f ~fl00 7 t M: f "40 007 b R fMfl 0 07 A Rt; Mf)(l (17f. P HI :::i K "l>\rl 0 06G R f.M00 0 6'H on 6 1 .. i . lo IH·Mil00 70 LA MOA Z ~ "." r.tf l 0 0 "fl- . J)H J Z a LAMR DA PH I :r R~ ~ nn n7F CA LL tS(J f I TINFJ T,~,X , V , F I ,B, 3, 1-'H I ,.-:PR I Mf:. LAMIJA l ,P lf tZ , U\""}( (.IA , l A , S <-lk TC, l ,A TE'Sl , O f L T A,~ T t-~ r ) -= OF.LPH r JJH J O - VH J I F(T FS T- ll f LPH l)h ,b,7 l'HIO PHI I I F ll . E0.1 0 1 ~RI TE l~,9) = ~. 1-lr. l•U IOO/ b f< E: 1"111'1 07H ~H-M11007i Mf M(l f)Oft f, RfMOO OM A R E~ O lJOk A 1 :- - - --- - ------- - -- ------- ---- --------- - --- - -------- - - ----------- - -- -- -~-~ E MtlCO ~( C ALC IJLA T E TH E CfJ KRl:L AT lf lN fol .6 1fU X REf'ol 0 0 0 8r1 t r.---- - - --------- -------------- - -- - - ------ ---------- --- - -------- - - -- --- --~ f~noo R E . D040l• l, S Rf,..OOOAF 40 J = l , 5 Rf- MtJCl OAC 1)(1 ~0 CMT~(J , J) = A ll, JI HNO J NY ( CHT X: , 5 ,llE TF R,.1,(1 . , IT E $ 0) I F \ Il f· Sn . ,.1E. O I '"'l-t1T F l 6 ,4 ') ) JTF SO I f-f I TF\ll . NF..0 ) 1;0 rn '+-4 00 ~l I • l , 5 C ALL C.HT = CHTX CI , t ) 41 SURTC(I I • SQRl( CMTI 00 4 2 l • l. 5 DO ~2 J • ..l t l 42 A(f , JI • CMTX(!,Jl/I SORl C lll •SQRTC I J ll 00 4 3 l • lt4 IPL :a: l+l AFMOOQR H -t t: M00081 Rf-Ml100 9 f, R1- M1J()t)'4A R F.l-l(J009ti R. ~ M fl0 09(. RfMOOOQO RE MOOOl./C R l::M0009f RF ><OOOQG REM0009H RfM00091 00 43 J • !Pl,5 RfMQOlOL 43 Afl,J) : AIJ 9 l} R EMOOlOA 4 4 CONTINUE REMOOIOR C--------------------------------- - - ------------------------------------REMOOlOC REMOOIOO C ADO THE SATELL I TE PEAK BACK TO THE PARENT PEAK AFTER THE C LE AS T-SOUAKES FIT. REMOO I Df .-~ ----::-.=::=:::.--=.:-==.::::::-."C"C:----- ,- -...,-------=:::-::::-:-:-.--.~.:-:--:::-_-::: '.".:'.~::-.:--~~ EM.O OI O F :. 211 IFIJHALF.L E.01 GO TO 24 REMOOIOG MAXPI = MAX+l REMOOIOH 00 22 l•l,MAX REIOOOIOI K = MAXPl-1 RE.,001 lt 22 VIKI • YfKl+AOO*YIK+JHALFI RtMOOlt)i' 24 CONTINUE REMOOllR C-----------------------------------------------------------------------REMOOllC C COMPLETE THE LORENTZIAN IN THE PART OF THE SPEC TRUM NOT USED REMOOl lO C FOR THE LEAST-SQUARES FIT. REMllO ll F C:-- -------- ------------------ --- ----------------------- --- -- - - - - - - - - ----R EMfJO l l F 00 10 l=NFIT,NPT XI • XI 11-IH"I 10 Fl flt• 8(5}+8(1J•Cl.-Al 2J•)( l)/(l .+fH3 l •X l**2l REMOOllG Rl-MOOl IH R~MllOlll ·C.- - - - - --------- ----------- -- --------- --- ------- ----- --- --- -- - - ----------RE MOO I 2 r. C ( SHIFT THE BASELINE OF THE SPtC TRUM ANO THE LEAST-SQUARES LOkHITZ lAflil. REM0012A REMOO J 2R r,- --------~---------- --- - -------------------------------------- ---- ----R EH0012C 00 8 l•l,NPT REM00 12 0 8 VIII• Y~ll-Bl51 YMIN • -.e•YINBIGJ YMAX • l.2•YIN81GI 00 12 I •l , NPT 1 2 fl(IJ • FHll-Bl51 IFINRFLCT .GE.OJ CALL RFLCT RIFI ,NPT) REM0012E REMOOJ2F REMOOl2G REMllOl 2H REM00121 REM0013t C---------~------~---------------- -------------------- - ---------------R E M 11013A C PLOT THE LORENTZIAN DETERMINED AV LEA Sl-SlllJARE S . REM00 13B C-----------------------------------------------------------------------R EM0 013C 15 I F( IFPLOT ,EQ.11 REM00130 ltALl CPLOT(F[,NPT,NWOTH,00,YMAX,YMIN,Ol REM0013E IFINRFLCT.GE.OI CALL RFLCTRIFl,NPTJ REM0013f 35 CONTINUE REHUOl3G C- --------- -------------------------------------------------------------REMOOl3H REMOVE THE PARENT PEAK 11.E. SUBTRACT THE LEAST-SQUARES REMOOl31 C C DETERMINED LORENTZIAN FROM THE INPUT SPECTRUM!. REMOOl4& C------------ --------------------------- - -------------------------------REMOOl4 A 00 11 l • l,NPT REM0014B 11 fllll • YllJ-fllll REMOOl4C 9 FORMlT llX,38HTHE LEAST-SQUARE S FIT 010 NOT CONVERGE,/,JX,24HMORE REMOOl40 IS.CANS ARE NECESSARY) RE MOOlltE 45 FOl<MATllX,30HCORR ELATION MATRIX IS SINGULAR,/,IX,8HITEST • ,151 REM00 14F RETURN REMOO l't(i ENO REMOOl~ SU8ROUTINE RFLCTRIS,NSPTI RFLROOOA C-----------------------------------------------------------------------RFLROOOB C THI S SUBROUTINE REFLECTS A REAL VECTOR ABOUT THE CENTER. RFLROOOC c----------------------------------------------------------------- ------RFLROOOD DIM ENS ION SINSPTJ · RFLROOOE 25 J = NS PT /2 RFLROOOF 00 26 l =J,J RFLROOOG K = NSPT + l - I RFLROOOH SI • S 111 RF LROOOI SI 11 • SIKI RFLR OOI& 26 SIKI = SI RFLROOIA RETURN RFLROOIB END RFLROOlC SUBROUTINE OCTHEXIL,NPOINTI OCTHOOOA C-----------------------------------------------------------------------OCTHOOOR C THIS SUBROUTINE CONVERTS SX,1206 INPUT TO THE PROPER VALUES OC THOOOC WHEN READ IN AS 8X, 72Z l , OCTHOOOD C C-----------------------------------------------------------------------OCTHOOOE OIMENS!ON Lll0241 OClHOOOF INTEGER LREAD* 2 1721 OCTHOOOG FORMAT I BX,72Z IJ OCTHOOOH NCARO • NPOINT/12 OCTHOOOI !POINT = 0 DC TH001£ DD 2 ICARD•l,NCARD OCTHOOIA READ (5 ,J I ILREADl!l,l •l,72 1 OCTHOOl8 IOIGIT • -6 OCTHOOIC 00 2 l • l , 12 OCTHOOIO IOIGJT ~ IOIG!T+6 OCTHOOIE IPOINT • IPOINT+l OCTHOOI F LllPOINTI • LREAOllOIGIT+lJ OCTHOOIG OCTHOOlH 00 2 Js::2•6 2 LllPOINTI = 8•LllPOINTl+LREADllDIGIT+J) NLEFT = NPOJNT-12•NCARD OCTHOOll OCTH002L IF4NLE F T) 5,5,3 3 NREAO • 6•NLEFT OCTttOOZA OCTHOOZa RElO 15,lJ · IL~EoDIJl,l•l,NREAOI !DIGIT • -6 00 4 l•l,NLffT I OIGIT • IDIG!T+b lPOINT • IPOINT+l ltlPOINT) • 00 4 J,.2,6 4 LllPOINll • ') RETURN l~EA D CI DIGtT~l l R•LllPOINTl+LREAOllOIGIT+~I OCTHOOZC OCTHOO:!b OCTH00 2E OCTH00 2F OCTH002G OCTH002H OCTH0021 OCTH003& ~ IC lHf'lO.lA ENO OC. THOO"H" SlJf\ROIJTJ NE LSOF IT ( NOl:\S,Nf>Af{frll, )C • V, F. 8. N IT..:~ .~ti I, H 1 1.( l MF. 1lAMlJA 1., 1,MI 7 I. "IUHlOOA 1 tlAMKOA, A, SqR T A .1 CALC • fl. Tf ~ f ,llt". 1. 1' A ,Al~~ T) I. SIJFOOOA C tSCHOllOC c C C THIS SIJ8MmHINF Pt-Mt-llftMS A Lt"A~l-S(.)lJAFH·S Flf (Jf THI-: NllNt.!)Or-oono LINEAR PAHAME TEA.S,H• TO I MF ORSf:- RV F.tl WllAN T l r1t:~ Vt I) AT H1f: l'lllNfS1 . ~oi:ooot~ KCTJ, USING THE MFTHOO (If- l.\AM OOAR IH, ,l .SllC. I NIHJ~T .. AVVL.MATH.1ll• L!'iltFOOOfl 431 fl9h'31. 1.SUF-000(, C. 212 C LS<J>OOOH C C NOH\ 11 JHf HUHRf.M OF 08\EHVATlflNS Of YIXI. NPl\HM • rHF. NIJMHt-: f< OF PAP.Afri'lfTfMS IN F, WHERF (, )f lHf • INOEPENOENT VAR.fA8LF. - LS OFOOOI LSOfOOtr. Y=fl)(J. Olf'IENSION lCINOf\SJ. LSOFOOIA Y •THE MEASURED VALU~S OF THE UfPENOENT VARIABLE - OIM Y(NOBSI. LSOFOOIB F ~ THE CALCULATED VALIJfS OF THE OE~ENOENT VA~IABL~ - h(M FCNOBSl .LSOFOOlC H • THE PARAMETERS - DIMENSION BINPARMI. LSOFOOIO C NITER• THE NUMBER OF ITERATIONS RfUUESTEO. LSOFOOIE r, PHI • THE SlHI OF THE SOUARES OF THE RESIDUES. LSOFOOIF C FPRIMEIJ I • THE PARTIAL OF F WITH RES~ECT TO ~lJI. LSOFOOIG C LAMDAZ • THE INITIAL VALUE Of LAMBDA. LSOFOOIH C eHIZ • THE INITIAL VALUE OF PHI. LSOFOOll C LAM80A • THE FINAL VALUE OF LAMSDA ·, LSQF002& C ICALC • lo PHICLC ANO FCALC ARE USED. LSOF002A C 2. PHICL2 ANO FCALC2 ARE USED. LSQF0028 r, 6 = SCRATCH LSOF002C C ATEST • SC RATCH l SOF0020 C L SUF002E C-----------------------------------------------------------------------LSOF002F C THE SUBROUTINES PHICLC ANO FCALC MUST BE PROVIDED! LSQF002G C-----------------------------------------------------------------------LSOF002H C LSOF0021 DIMENSION AINPARM,NPARM),GI 5),FPRJMf(NPARM},SORTAINPARH), LSOF003t I DELTA!NPARMl.BTEST(NPARMl,ATEST(NPARM.NPARMI LSOF003A DIMENSION XINOBSJ,YINOBSl,FINOBSl,BINPARMI LSOF003B REAL LAM8DA 9 LAMTST 9 LAH0AZ LSOF003C MAX = 5 LSQF0030 v • 10.0 LS0F003E lAHTST • LAHDAZ LSOF003F PHITST • PHIZ LS0F003G 00 5 ITER•l,NITER LSOF003H LAMBDA = LAMTST LS0F0031 PHI • PHITST LS0F004t CALL AGCALCIA,G,NPARM,Noas.x,e.F,Y.SQRlA,FP~IME,JCALC) LSOF004A IFILAMBOA.LE.0 .000011 GO TO l LSOF0048 LAMTST LAMBOA/V LSOF004C C C C C = l• = L SQ F 0040 CALL SCALCCNPARH,A,LAMT ST,G,OELT A,MAX,JT,e, soRT A,C , BTEST,ATEST) lf tlT.EQ.HAXl GO TO 2 I F llCALC.E0.11 CALL PHICLCIPHITST,NOBs.x ,v .F,BTEST,NPARHl LSOF004E LSOF004F LSOF004G Jf(ICALC.EQ.2) CALL PH1CL21PHITST,NORS,x,v,F,8TEST,NPARH) LSOF004H IFIPHITST.LE .PHll GO TO 4 CALL GAMCLCINPARM,DELTA,GoGAMMAI IFIGAMHA .GE.0. 78541 GO TO 3 LSQF 0 041 LSOF005t LSQF005A LSOF005B ' L SQFOOH.; LSOF005D LSOF005E LSOF 005 F LSOF005G l SOF005H LSOF005 1 LSllFOOl>t LSOF006A C• o.s•c GO TO 1 WRITE(h,61 NPARM,LAHTST LAMTST = LAMTST•V GO TO l 4 CONTINUE 00 5 t=l,NPARM 5Blll•BTESTlll PHI = PHIHT LAMBOA • LlMTST 6 FORMATIJX,l~HWARNING JT aMAX,/,lX,6HNPARM•,tb,/,lX,7 HLAM80A•,FlS.7lLSOFOOb ~ RETURN LSOF006C ENO LSOF0060 SUBROUT INE PHtCLCIPHl,NOBS,x,y,f,B,NPARMI PHICOOOA C PH I COOO~ C c THIS SUBROUTINE CALCULAT~S THE SUM OF THE SOllARES Of THE Rf510UfS WHEN A LORENTZ IAN I S USEO. PHICOOOC P~l r nnoo C PHICOOOE C----------------------------------------------------------------------- PH l COOOf C ~HI COOOG DIMENSION XtNORS)tYINOttSltFtNf>ASJ,B(HYAkMJ PHICOOOH PHI z:: O. PHI COOOI 00 l l• l ,NOBS PHICOOlt Xl • XII 1-11141 PHI COOlA fill • Bl5l+81ll*ll.·8121•Xll / ll . +Sl3l•Xl••2l eHIGOOl~ PHI • PHl+IVO I-Fl 111**2 PHICOO I C RETURN ENll Ptilf.0010 PHTGOOlE FCLCOOOA SUBROUTINE FCALCcx,s,F.P.ItNOBSI G-------------------~---------------------------------------------------FCLGOOO& C C THIS SUBROUTINE CALCULATES THE VALUES Of THE LORENTZIAN ANO ITS DERIVATIVES FOR THE LEAST SQUARES FIT. FCLCOOOC FCLCOOOD C-----------------------------------------------~---------~------------FCLCOOOE DIMENSION XlNOBSl,flNOBSl,Bl51,Pl51 XI • XII 1-8141 BX2 • l ./ll .+Bl31•Xl••21 Pill • tl.-8(21*Xll * BX2 Pf21 ~ FCLCOOOF FCLCOOOG FCLCOOOH FCLCOOOI FCLCOOl& fCLCOOll -~ (l)*Xl*8X2 PT • Plll•Pl21 Pf3J • Xl•P T :Pf-Id z:: Rtll•9f2l•8X2-2.•Sl.,)•PT '• 151 • I. Fiii • B<ll*Plll+~l~I FCl.C0018 fCLCOUIC l'CLCOOlD FGLCOlllf FCLCOOlf R~TURN ENO SU8i:t.OUTINE ~CLCOUIG PHICL2(PHJ,NOSS,x,v,~.n.~WAR.M) PHli'OOOA PHJ2000~ C c C THIS Sll8RflUTINf. CALCULATtS THE SUI< (II' !Ht •F.Sl~llES WHEti AN EXPllNfNTIAL 15 u~rn. SQllA•~S llf IHE PHl l OOOC PHl10000 PHl2000E 213 C----------------------------------------- ----------------------------- -PHl2000F C PH I 2000G DIM ENSI ON XINDBSl,Y CNOttSl, FCNOBSl , BCNP ARMI PH l 20DOH flHI • O. l=-l,N06S FtI J • RCl J+RC21* E XPf-lH3)•Xf I ) ) PHI 20001 PH1200lt PH l 7.00lA PHl200lR PHIZ OO l C on l I PHI 11: PHl+IYfl)-FCJ)l•• 2 RETURN ENO P~12 001 0 SlJ8HOUTINE FCALC2CX,8,F,P,l,NOBS) fCL2000A C-----------------------------------------------------------------------FCL2000R C THIS SUBROUTINE CALCULATES THE VALUES OF TH E EXPONE NTIAL ANO FCL2000C C ITS OERIVATIVE S FOR THE LEA ST SOUAR ES FIT. FCL2000 0 C----------------------- --- ------------- --------------------------- ---- - FCL2 000E Dl~· ENSION XINOBSI ,F INOBS I ,6131 ,PC31 FCLZOOOF P C! ) • I. FCL2 000G PC21 EXPl-BC3l* Xl l l l FC L2000H Fill • 6l21*Pl21 FCL2000 I POI • -XII l*FI 11 FCL20nlt Fiii • Bl ll+FCl l FCL200IA RETURN FCL200 11' ENO FCL200lC SUBROUTINE AGCALC(A ,G,NPARM,NOBS , X, 8 , F, Y,SDRTA, FPR1ME ,ltALC J AGCAOOOA C AGC A0008 C THIS SUBROUTINE CALCULATES TH E A MATRIX AND THE G VECTOR. AGCAOOOC C AGCAOOOO c -- - ----------- --------------- -----------------------~-------------~-AGCAOOOE AGCAOOOF C O I NE"'-S I cm A I NP ARM ,N P AH.M ) .Gt f'I PMtr-::) ,x ( NORS) . 11 ( Nt'ARM I , F (N011 S ) ,y ( NUS S ) A (;CA llOOG l ,SORl AINPARM l,FP~ l~ ~ I NPA~~) Oil l GtJI AGCAOOOH J"'l , NPAR M AGCAOOO l = O. AGCAOOI~ 00 l K=l,J AGCAOO lA AGCAOO l A = O. DO 2 J:l,NOBS IFCICALC.ea.11 CALL FCALCtx.~.F . FPRIMEo lo NOMSI IF(ICALC.f0.2) CALL FCA LC21X , B t F , FPR 1 M~, l.~ O H S J ACJ,K) A~tAOntt ACC•ootn A~C:AOO lF DO 2 J = l t NPAR~ AGCAOOl F GLJ) = GIJl+IY<t 1-F ( 11 l •FPR!pi.!f- IJ I DO 2 K= l,J At,f.AOOl<; 2 ACJ,KI >'\ GC: A OOt~ = AI JoKl+F PRIME t. ll •FP Rl >IF CKI Ai;C•OO ll C- ---- ---------------------------- - --------------------------------- - - - - AGC AOOZi C SCA LE A AND G AGCAOO?t C ----- ------------ -- -- -~- ------- -- - -------- - - ----------- - - ----- -------- - A GC A 0 0 2P hO 3 J=l,NPA~M SORTACJI = SQR TC ACJ,Jl l 00 4 GIJ J AGCA0 0 2C • ~C A OOl D J: \ ,NPARM AC(. fl.IH)l F = GI Jl/SQRTA(J) A ~Cft 002F K=l ,J At·f-, A00 2f. 4 AIJ,K I = AIJ,KJ/ISO ~ T A IJ };.'IS 'J R TAIK)) AGCAOO?I... C-- -- - - -------------------- - ---- --- -- --- -------------------------- ------AGCh002 1 ( COMP L ~Tt THE UPP~R HALF O F A. AliC A00 3& C- - - ----------------- ------------ - -- - ----- ---------------------------- --AGCA00 3• JMAX = NPAMM- 1 AC,h003!\ 00 5 J=ltJMAX AGC A00 3C KMIN = J+ l AGC AOO)O on 5 K=KM I N.NPARM ~~ C A00 3 E ~ A(J,KI = AIK, JJ AGCA003F R f TUR N Af,f.AOll;:\(; DO 4 ENO SllRROlJT I NE BCALC t NP ARM • A• LJ!iMT S T, (; ,llEL TA ,M AX t IT, 8 ,SORT A .C ,HT E S T, IAT ESTI L c c c THI S SIJttROU TINE CAl. CIJ LATE S Tll E NEW EST!MAH PARAMETERS, k. . Al;( A(Hl3H BCtCOOOA " CLCOOO" BCLCOOllC AC LCOOOl> BC LC OOOE OF THF Rr: :.cnooF ( - - - - - - - -- - - - - - - - - - - - -- - - - - - - -- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - ... ( L .: I ' ." n(~ r .:Lr.co!•- h(.L ,: nr1 n I D I P-1 E ~ S l tll<i AtNP A1<~·.1ll1-'ARfo• J . {. ( •'l l-' t. k,.. , , r• ~·l TA!NP"° k "' l • ttf'·ll<~~ ,..~ . I ~1..1 ~ 1 A I NP M-04 I • Bi 1: ~ T I '<il-'A k I"' I •AT ~ 5 1 I N ~ L-· .. • I\ · · t. .:.. ~q r- : L :.Ol) l f. e1:u:. ci·:. 1 ! WEAL LAMTST on 0 (J i 1 !='.-.:. r.<• n 1;.. J =l ,t..1Pl\k.11 "':L~.- (• l ( K z ) , f'.I PARM = AIJ , KI l AT~ST ( J . KJ 0 0 2 J =l, NP Aq M "CL C( r l t 7 ATESTl .l .J I ~ :v:r)(l) i;: CALL ~-Lcn~11~· 1 ,. Alt S l I J.J ) <t- L U 'lS T E OS OLVINPA R~ . Alf~ T. G. MA), 0 . ~0 1 . Of L l A. J T . QJ St L ~,0 1 ~ f\ .:.Lcr ·.· J--· J F f IT. EC . ,..,AXJ GO TIJ 4 ~ DO 3 .!=l , NPAQ.M f' C.L ( ,.1\ J 1 BTESTIJ ) P~Lc c:r•~·. ~ tLCO ~ ;· ! = R l.ll• C~ OELlA( .11/SO k T A IJ J ·• RETtlRN ENfl SU8ROllT1NF f.. AM C LCINPAR"', [')tL l-" , b , (~A HMA I t c c c Wf THJ~ Si lf\ROlll l NF f. l· LCllLtiTf-~ r•.A.MMA . 1H t' ll EL lti ci r- TA!Nt l1 ;·. -<. r 1M 1 Mf llSf Af\JP f\ •: L CJ10? 1.. THF ll !'d ·Lf 1 \ ~"- l w !: tr·· c. ~:ll{i}>N l ~~·111n n .. (.t:.1•.J ·OO-" G 1. M :c 100 ~ l•·•I. 1.. f L TA(.ti 11.;:onnc :.r .. Cll (\f\ ': ( !>t t ()(1 f} f C. -- -- - - --- ---- ---~--------- - -- - - - -- ---- -- -- - - ----------- ---- -- - --- - - - -- - C· b~ ((1 0f1 f OJMENstnN OE LM Ar. • G~.4G '"' [)fLTA ( N pAqM) , GINP ti~~ ) O. O. DEL TA G= 0,. _Q._~~-2..._I_~~- (i. t."1 (. ()()(\11 1~.6,..(. ()() (J J C A MCO O ll. '~ AMC llO ll\ --·-- --·-·--··-----~!..~~ ~ !?~ .t ¥' 214 I l•f:I MAI. ~ (., M l!(~ f.MM~ •r:,( 11"~'? 1:' GAMtOO lt 1 1t= Lfii'M~• I J~l l ' Al J ) >:- ~· z (·,AM((lO J [ 1 (.A M( (l(i l f- r1 f tlttr , • l1ft.l/\f, .. fHLlAC l ) l)F l r, • l>l"I. MA li*l?Ml\f, C l l'>LAH • Ot-L T AC./(SQJITIH ~ Lldl 1~1 l·.fll 1,J\,.'11'. ( H)l I= R ... lUMN (,l\MCOOI r. (,J\MCOOt t·t f~AM( OO 1 I fNll ( , flMrJ) fl'J I GA MMA .. AMCW> I C IJSC.AM ) su e~OU T IN E Flll~k IN STlJN.S .NSP l.N F I Pl,JSMAX,SMAX,FJ, FD.H.SlJ", SWEEPflLTOOOA l,NSWEEP , HELP,IPSDLl F I LT OOOk C-----------------------------------------------------------------------FILTOOOC C THI S SUllROUTINF. SELECTS THE FILTER OF YOUR CHOI CE . F ILTOOOO c-----~~~--------------------------------------------------------- - --FILTOOOf. DI MENSI ON S 11024 I ,FI 11024 l ,FOi 1024 l ,HI 1024l•SOM1102" l F IL TOOO F IF (NSTON1 3,1,2 CALL MATCH IS , NSPT,NFIPT,JS MAX, SHAX,F l,FO tSOM,H) FILTooo<: GO TO 4 FILTOOOI FJ LTOO OH 2 CALL RESOLV tNSTON,S,NSPT,NFIPT,JSMAX,SMAX,FI ,FlJ,H,JPSOL) FILTOOlt: GO TO 4 3 CALL RCFLTR INSTON,NFIPT,Fl,FO,SWEEP,NSWEEP,H,NSPTI FILTOOIA FILTOOIR C-------~-------------- ---------------------- -------------•~----------FILTOOI C C C CALCULATE THE MAXI MUM ENHANCENENT OF THE SI GNAL-TO-NOISE RATIO WHICH RESULTS FROM THE FILTER USED ABOVE. FI LT001 0 FILTOOIE C~----------~--------------------------------------~---~------------FILlOOI• ~ FOM&X • O. F ILTOOl G FILTOOIH FILTOOll FILT002 t f 1Ll 002A F ILT002" FIMAX • O. DO 5 l • l,NFIPT FONAX • AMAXllFOMAX,FOllll FIMAX = AMAXl(FlMAX,Flll I ) HE LP = O. b 00 A 1=1,NSPT f I L·roo7C HELP~ ~ f ! LTOO?F H EL P+Hl ll ~*2 F OMAX/ ( F1MAX•SPRTfHE~Pl) HE LP ~IL T 002~ RETURN FltTOO?F ENO F l LT OO?C, SUBHOIJTINE I S , NSP T, NF l ~T,J S MAX,SMAX,FJ,flJ ,SOM ,H) MATCH µJ(HOOO h C-- -- ------------------- --- ---- - ------ -- --- - ---------- - - ---- - ------ - ---- MTCHO(}O~ . C THIS S URROUTINE PHI.FORMS A MATCH F.11 FJLTFfi n~ F. RAT I ON J N THH: MT ( HOOflC C SPACE. Ml(1-1 noru 1 t-----------------------------------------------------------------------MTC Hl)(lflf D I MENS ION SSO (102 41. 5 11 02 41, flt1 024 J, F0tl024 ), SOHl l 02'd , "'l(MOOOF 1H(l 024) 00 l Ml (. HOOni, MT C: HOOOI... = ( S (l)l**2 SSO lll 1 = 2 , NSPT SSO( l,l a S SO MfCHO(lOI + (I-!l) = J,NF I PT DO 4 ·J IS( 111**2 r-1 f (.HOOlf. MTCH n O l A t4TC ~+ 001M K = · JSMAX - J IMAX NSPT - K IMA X = MJNO<NFJPT,JMAX I IMIN = 1-K J MIN = MAXQ(l,JMI ~ I IXK = IMAX+K INK = IMIN+K C = SMAX/ISSO IJX Kl-SSWllNKll S I GMA = lJ. DO 2 I K IC I MT C ~l (l 0l ( MlCHOOl D ~T C HOOlf MTCHOOlF Mlf. HOOIG Jio!TCHOOJH MTCHOOll ~T C "O<!, ,. .: IMIN,JMAX "ITC H007 h t +K MT r: H002R SIGMA • SIGHA+Sttk)•f llll FOCJ I = C•S IGHA ~TCH002C MT C H00111 IJ ~ 20•I J /2<>) -J+l t F ( l .1 ) t, , 5,4 ~TCH002F MT f: l.{f"O?F· (. - ----- -- - ---- ---------- - --- --- ----- ------ - ----- ---- ------ - --- - ------- - - Ml (HDO?~ (. CALCULATF. THE SIZf: OF THf f-tRnR b ARS. ~T(.~00?~1 C-----------------------------------------------------------------------M1 CH0021 '"' S IGMA 2 DO 3 IK = 3 4 o. MJ(.1·H)03f. = lt-'I N. J M/\X ;J.l (. t·H~ 03:\ I I+ '( SIGMA= ~Tf. i-! (}Q3!t S 1GMA+IFOl.IH• S ll Kl-t-11 1 J t::S M .\X l ~ ~? ,.. l·':. HOO:iC: = (>llSO~ T(S!f.MAI SD,..IJl CONl TNllF t-'H. 1t11il ~if '• c = S M/\X/ SS() (r~Sl-' 1 J ,..rc ..... on~~ nn ' ·' TC •i003(, f, 1-'I U• ll f·"i ~ I s I ,14 0.:.PT 1-. HI I) = C•C:, l J ) R f- T llR N l.• f f, H()03H Ml f. ~1{10~1 f NO 1-· Tf.H004r. S llf-\K(lllllf'll F KC FLTR IN $ l11N . Nfo t P r,F-t,F11. sw 1-1-P ~ /11C:, ~f. l- 1-'.H.N S ~ TI ' ' L - L O f lOt\ f -- ---- - - --- --- - --- -- - --- - ------ ---- -- - -------- -------- --- - ---- --- -- ---- ~ C F- trl on~ Tl-llS ~. Wil<lllll T NF ()111-'L J C Al l· S THf C . 1:-H; l-C: f t.., lff A lHlltt\Lf SbC:Tirl N 11c1·u;r111C kC: I-IL Tl-It. 1~ 1; 1- 1 .000 1 1 1• - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - ---:-- - - - - - - - - - - - - - - - - - - - - - - -1-l: 1. i:- L n 1 r-·1-NS I t1N Fl ( 101'41.Fllt 1 0('41 , t~( JIJ,?4) l~ f. 1- t~I O O I· noo1- t~S WE: l:P• t...STON KCI = RCJ "' -1 00 . •sw t: EY/><i.: I EX = EXP l-RCI I FO(ll : RC H=FJlll I !<C: l-LllOO(. k Cl- LOO(•H k C f- Lf)001 r.ic ~1. o n1r. no l N=2 ,NFIPT ~t:l-L <I C)IA Fll l NJ FOi ll .M(F--L OfllC = EXICl:F(lfN - ll+ RC J l);FJINI = R(. ) OF(t ( JI no 2 N=2 . NF I PT 2 FO!Nl = EX11<-F O IN- l )+R C l *f-lllN I 1-{[ ~lOOlA ~CFLOO]n f.4 LF-LOO H = t. NSP T )024 RCI? = KC l •4? M.f.f-LOO lf; ~C ~ LOO \ ~ 00 ..\ N=l,NS\.'T M<.t-1.00 l l t:XH i::: i<( F 1. nn1..-: 215 1-:N = N RCFlOO?.f. RC:F LOOi'" = M(l7*~N#EXH l~(N J ~ 1-)t'H "' FX•FXH k<.:~1.no711 kF ltJk/1.1 l"CFL002C Mrr1.0020 FNO K~LVOOOA ~ - --- ----- - ------- -- ---- - --- - ------- ----- -- - -- ---- ---------- --- ----- ---- N ~ ~ V0(){)~ f. THI'. ', IHHHllH J N~ i,FJH-1 1111-'. '> 1. J f·lt- /1.k A'll• · l'~~ll Ull-llf'lr. /\P J.ILH-~ f Nh !-t\t.VOllO(. , I t• '1.-!FUlll "IC Y 'd•hl"J . K~.1 .. \/ 0 0fHJ L- - - -·· -- -- --·- -·· - - - - - - - --- - - - - - ·-··- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - RC..l_VOOOJ. fl I ,.. f. "P· I rm ~ f 1 Oi''o) <t t- I I I IJ? 1,) • .; 111 I Oi''t I , 111 I !1;14 I ' I NV l ;..;,.It M( '\ ) t Ill L 1 A I \ ll•' t, 1_\I(\ :HIF I • II r I '. r I ·1) ' I· I ., I i' ) 'h I ..~ ) t f-1' M I ..~ t· ( 'i ) • A I ''\' i ) • \()U 'I /\ ( -~ ) ' I .I ( 11 I ;i ) • /\. T f 'i l I . .. '\ ) ,, (:, I_ \1110 n(' ~. f· /. I_ Li\Hf1A/,l. AM l1IJA H\l. V fH1(1H H1klHtl lt1~ '· ), (. f JM,J (.IJM~Ll·J: 1-·~1.V O OO I (. -- - - - - - __ - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -- - - - - - - - - - - - - - - - -- - -- - - - - - - - - --f1 \1. VOii it~ (. (.FNTH< ~. M AX µs1 .vno1.-i (- --- - -- -------- -- ------ --- ----- -- -------------------- -------- ------- - --llSLV OlllH ';o TIJ }~\ l f- lJ S.f>IAX .L t:.5 111 IS HJF T JSM A X"' ;1 1 1-{\J, \l( Hll '· = i\1 5' 1-'T- I SH!Fl NS~ T t\tl R~L\.'11\l l ( M..:.. 1. ·JOOl ~ l J C.M AX-'ll l ':>l l ?1 S iii 1-\1 , '.'llOJ ~ 1 -= l . t J<. P! '' 7~ ~SLVOll JI lF ll T. U· .Slll hfl Tll 7.h M.~•.Vtl O?f. NSP T = NSP T-IT• ">l l NCENTM; : 2 * JSN AX - NSPT JHNCENTRI 3 .n,l ~IN = NSPT+l NSPT NSPl+NCENfR M~L VO Ol/\ l =M J N ,NS~T R S l.V0Cl7 f M~LVO~/~ R~t.v~ 102f. R ~ LV002[1 = DO 2 RSI.YOO?~ = O. GU TO n MAX = NSPT-NCENTR no 4 1=1,N S Pl 4 SIMAX- 1+' I = S IN SPJ-1 ,11 ~ 1 \ ' ('fllc . W~l.V1Hl l 1 • ::- :-. 11•1!-.H I FTI ?'i I T-: NSPT-.1-..l"ll\X S I 11 RSLVOO?G A~LV002H RSLV0021 ~SLV0 03(. RS LV003A RSLV 0038 NS.-T = MAX JS.MAX !:' J C, MAX-NCENTR P 0:, 1. voo "c MAX' -::: -NCF.N 'f~ r1r1 'i 1 ""1 ,MAX S l l l = U. f.f <) LV00~1 i ~(,. 1. vno3r: 'i ,., cnNT I Nllf R Sl.V00 3 F f.( "'- VOCHf> t--~ -- - - ----- --------- ---- ------ -- ---- - ------ ------- ------- ------- ---- --k~LV0 03H c N SPT t'otl S T EUIJAL 2*"'M R.s1.vno·1 1 Fun Ht.RM C------ --------------------- ------- - ------- -------- -- - ----- - - ---- - - --- - -R S ~V004& M12I o , f<SLvno4A =o Ml31 R. ~1.v no1.1-1. NSPT ~ NSV TT ~S L V004 C M ll> " " l H2 = 2 u c;i.vo o 40 qs t. V0 04 t- = NSPTT/2 IFINSPTT.LT .ll Gn TO A 7 NSP TT R~LV004F ~SLV004G M i l l : Mll)+l M2 2*M2 r;n TO 'f k 5 •_ V004H kSl.VO O'd i.t •, 1_ l fl011l = l1 NSOlFF = M2-NSPT MAX = M2-NS(lJFF/2 0 0 9 l=l,NS PT q SIMAX- 1+1 I = SCNSPT-t+ l I RC\1.v110"Jb. RSt_vno">q R<;l VOO~( RS1.voo5n !MI N = MAX+l 00 10 l=IMI N, M2 10 S t I l = o. MAX MAX - NSP I RStVOO">tf<~LVOO ')F ~!~ 1.V0051 ~ = on 11 R ~LV 005H R SLV!}()5J 1~1.MA X o. R S!.,V OObf, NSP T = M2 ~SLVOOf. J-1 11 Si l l = J SMAX = JSHAX+ MAX ~5LVn06A c - ---- -------- ---c---------- - --- -- ------ -------- --------- - --------- -----RSLVOOb ( f. CALCULATE S I IW'1E\, h) RSL\I OOliol'J C-----------------------------------------------------------------------RSLV(l06f DO 12 l =l,NSPT R'\1.V006F 12 FOR ! ER I 11 = S il I CALL l1ARM(FURIER . M,rNv.Fo. 1 , 1ffHR) NSPT2 = NSPT/2 RSLV006G RSl. VO ObH kSLVOObl IF I IP SOl• • Nf.O> r.n Tn ?I RSL V007r. ( - . .. . - - ---- ---- -- ---- ----- --------- -------- --- ------------ - -- ---------- - P\I V00 7 b <: nr:n: HM JNF THE f. XPIJNHlT!AI_ DEC: AV !IF ~ I l llMF:Cf\ I I',\' 1. Ft.•,1 - ~. t-1 \IA ld· ', kC.,I VOff/f•, (. F ITllNf. . 1~~.1.·v •071_ C - - - - - - - -- -- - ·· - - - - - - - - - - - - - - - -- - - - - - -- - - - - - - - - - - - - - -- - - - - - - --- - -- - - -- - - - R ';L •: rio 711 nn 11 7 J 2 }, MS P T ? HSl.'/<:Off lJIJI .-: J HIJ) 1 1 ·1 HLll Rill R l ?I .. \ct;.l,Vf)07F C AB S IFOR IE ~ IJ) l R~LV 0 071~ Hl.ll•H(.J) HC:.. 1 vrin7 ... 1.1•,1_vnnr 1 o. = Hl l ) f-\ ( ·~ ) "' Lf\M/IA./ M \l. '.'On ~ f. I Al 11(. ( H I ;> ) I H I l Cl I ) I I I .1 1 I o I ..- 0.111 CA I.. L PM I C:. L?. I ~HI l • NS P I ,I • I ,I, H • i: , fl • i I f.A l.l L~IH 1 T INSPT ?, .i.,7J . li. I· , ti . '• • l'HI ,t: l'H JM V ,1_M . • fl{ • ''il l/ ol. f\,.ll·W· i\, l'I , 1 ~PH I A,•' •A 1 1· S ( tl>t· l I fl., HI I \ I I k ~I. V01) !-':fl M.';!_ VO!liHI 1{',I VllPtlt. I ~" I.\' Oll.-1 fl I ! ~ I \,\l tHH r--- - -- -- ···-· ··---------· --------·----· -- -···----. ·- · · -- -· ... ·- ·-- --·· ... l' ' l V lltHH· 1:- -· -- ---- ·- ------ --------- ---·· · ··-- · - . · R •.t \I0 1l~U1 11 I M \ Nf\lt- 1111 Hll~ ll I Hl t.'llHH Y N t l\ •. 1 IN '-.ltll•tll.t1). 1: ' ' \ 1ltHH ~ t~ ~I \: 1 · t1~ I 216 15 TART un = t lS PT 2 -q 011 103 J s J S TA k T, NS Pl 2 f. llT OFF = CllTl1 FF+CAASfFOR I EIH I~ I CUTOFF = CllTOFF/5. 00 17 I ~ l ,NSP T2 FOM. I ~ CABS IFO RIE~fl) ) IFIFOR l. LE.CllTOFFl GO TO J R RS l.V0 0 9 £ R S LVOOqA RSLVOO<H\ ~SLV004(. RSl.V0040 RSLV009F RSLV009F 17 COt.ITlNUF M.SL V 0 0 9C, r. 11 TO 20 R SLVOOYH lk .IMAX ~ N ~ Pl-1+1 on l9 J r. J,JMAX l f.f FOklERLJI ( 0 . , 0.) = 1 11 r.nrH HlllF RS LV(J0 91 RSLV O l Ot: ~S LVO lO.A k:$ •.·: 0 1 r.b C----------------------------------------------------------------------RSLVOlf1( r. CALCIJLATE HfCJM. .:: r;A J C-----------------------------------------------------------------------HSLV(l )[1f = ~ S L V010'1 N ~ Tf1N ') f< 11'l S TOh = U.l>ll'i TIJ1·1 STON=EXP I S TONl-1. JF( IPS~L .F 0 .01 U = 1 ~7 . ~'llS T ON - .ll *J";ll /ttCAA S CF OM. I ER f\))) c:~7. J C = O. !f (I PSOL .N E. 01 G<J TO 22 Bl = Q+l. Kl = Bf3)/ll.+0.5 ? 7Si:itALOGI B l)) 81 0 = - 0. 3 ~*1Bl~ ~2 J = O•B t 2 J RSLVO l OG kSLVO ll~ RSLVO l l B l< S t.V O ll C µ SLV" llD RSl.VO ll t= no 1 3 J=l,NSP T ~SL VO ll F CONJ M.Si. VO ll{> RSLVO l l H R:; Lvn 1 t i HSl. V0 1 2 £ XZ = CONJG IFO Rt f: ~(.1 1 l = FOHJERfJl*CO~ J CJ = J lf(J.~T. NS PT?l CJ ~ NSPT -J+l 0 ~ O•I EXPl-Rt3J*C JJJ+l . CJ = CJ -I . = Hl • I CJ••2) EJ = EX P IEJ) C = (+(X 2/0)*f.J FOR IEK(J ) = ICO NJ/Ol•EJ EJ l l H '; l_ VOtO F RSLVO l OH RSLVO J o J kSLVO l t f, r,n TO 21, 22 CON TINUE 00 23 J • \, NSPT CONJ• CONJGIFORIE RIJ ll X2 = FORIER(Jl •CONJ f) = l.+O*X2 C • C+X ZIO 73 FOAIERI J ) • CDNJ/ 0 7 4 RNSP T 2 NSPT C = RNSPT•SM4X/ ( DO 14 J=l.NSPT 14 ~ORIERIJ) • C•FORIERIJ) RSLV0 12A R~l_\1017 ~ K~L V0 12C H:S !. VO l.' ' l RSLV012F. R SL VOl~F RSLV0 1 2C. RSLVO l ? H RSLV0 12 1 RS~V0 13 & RSLV0131 ~SLVO l lH MSL V0 1 3C RSLV0130 RSLV0\3F RSLV0 13F RSLV013(, RSLVO J3H r ----- ------------------- -- ----- ---------------------------------- -- ----~~tV r. !31 CALC ULATE H(l J C l~SLV0 14f, C----- ----- - - ---------- - ----------------- -- ----------------- - - - - --- -----RSLV0 )4A CALL HA~MI FOR I ER 9 H, INVTH!,-2. I FEHRJ l< S LV01 4H 00 15 l ? H(l) 1 ~ 1,NSPT = FORJER(JI k SLVOl4C RSLV014f) C-- -- --------------------------- - --------- - -- ---------------------------R SLV 0 14 E C CAl CULAT E FO(TJ R SL V0 14F C----------- - ------------------------------------- ---------------- - -----RSLV0 14 G 00 16 N=l,NFIPT k SLvn t4H FO(N) MAX = O. N-l+J SM AX MAX c HJNQ(MAX,NSPTI MIN = N-NFJPT+JSMAX MIN = MAXO tMlNtl) 00 \ 6 l=MIN,HAX 16 FO IN l • FOINl+HI J l •F I (N-)+J SMAX l RE TURN ENO R SL V0\4! R5 LV0l5l RS LV0l5A K S LV015 ~ RSL V0 15 C RSL V0150 RSL VOJ5f RSLV(l l')F f:!S L V0 15G 217 APPENDIX C A SEMIEMPIRICAL CALCULATION OF SPIN-SPIN COUPLING CONSTANTS 218 We have used the extended Hlickel theory type orbitals of 51 R. Hoffmann and W. :N. Lipscomb to compute coupling constants for e. p. r. and n. m. r. spectra. Only the e. p. r. first-order Fermi52 contact coupling, and the n. m. r. second-order Fermi-contact coupling are considered. Thee. p. r. coupling constants of nuclei which lie at a node of the odd electron's orbital in a 7T-electron radical could not be calculated. The results for e. p. r. are similar to those obtained by the hyperconjugation approach. The computer programs used make the method easy to apply, and the results show substantial agreement with experiment. A description of the calculation of the e. p. r. coupling constants has been published. 53 The semiempirical method used attempts to guess the results which would be obtained from a single-determinant minimum basis set Hartree-Fock calculation. The inner-shell electrons are not explicitly taken into account in these calculations. All overlap integrals are computed. This is quite simple, because they have been evaluated analytically. 54 Extended Htickel theory approximates the diagonal elements of the Hamiltonian matrix by valence-state ionization potentials, and uses the Wolfsberg-Helmholz 55 approximation, Eq. (Cl), for the off-diagonal element.s. H.. l] .. + H .. )S ... = -21 K (H 11 ]] l] (Cl) The constant K, Eq. (Cl), is. set equal to 1. 75. Slater 56 ls orbitals are used for the hydrogens and Slater 2s and 2p orbitals are 219 used for the first row elements. These approximations give Hamiltonian matrix elements quite different from the calculated elements for small molecules. The resulting occupied orbitals and the associated eigenvalues are, however, an excellent approximation to those obtained by ab initio calculations on small molecules. The unoccupied orbitals and associated eigenvalues bear no obvious relation to the Hartree-Fock quantities. Computation of the contact coupling of the first-row elements requires a 2s function which does not have a node at the nucleus. Since the Slater 2s function has such a node, a function of the form 1 2s' = (1 - S2 f 2 (2s - S ·ls), (C2) where S is the l/Jis - tJ;2 s overlap integral, was used. In the compu- tation of overlap integrals the 2s overlap was substituted for the 2s' overlap. This did not cause any serious error in the overlap 54 integrals . The value of these orbitals at the nucleus is similar to that obtained by S. C. F . calculations for the atoms. 57 The Fermi-contact operator for electron-spin resonance is JC " = (C3 ) where o is the Dirac delta operator, S and IN are electron and nuclear spin operators, and (3 , {3N, and gN have their usual meanings. is a perturbation on the electronic Hamiltonian. This By eva luating this 220 perturbation to first order in terms of the above orbitals, the coupling constant may be expressed as (C4) where ¢N is the value of the orbital of the unpaired electron at nucleus N. Although Hoffmann and Lipscomb 51 used a Slater exponent of 1. 0 for hydrogen, S. C. F. calculations indicate that an exponent of about 1. 2 would be preferable. This contraction of hydrogen ls functions would increase ¢~in Eq. (C4) by a factor of 1. 728. The relative sizes of the splittings are not affected, since the Hij and Sij in Eq. (Cl) ~re insensitive to the exponent. The S. C. F. calculations included first-row ls basis functions, so we varied the exponent to obtain agreement with experiment. A value of 1. 15 reproduces the experimental results for the vinyl and 1-methylvinyl radicals reasonably well. Table I gives the results of this treatment of several radicals. The values obtained by Dixon 58 are listed for comparison. Dixon used a hydrogen exponent of 1. 20 and slightly different geometries. We used the experimental geometry of the parent compound for the present set of calculations except in those cases for which special notes appear in the Table. Since the bond angles and bond lengths of most radicals are not known, a useful semiempirical scheme for correlating coupling 22 1 TABLE I. Comparison of expe rimental a nd calculated coupling constants. A(G) Radical Experimental Calculated Extended ~Hickel Ref. Hyperconjugation Formyl• l36 74.3 VinyJb (gem) 13.4, 15.7 34.2 68.5 13.6 32.9 66.8 19.48 32 . 92 57.89 18. 2 28.4 59 . 8 18.1 6.4 0.0 12.5 .5.3 11.0 23.3 4.2 17.9 6.51 23.42 6.5 16.8 24.1 Ethyl• ({J) 26.87 21.4 27 .1 2-Propylf.• ({J) 24.68 18 .8 22.8 lerl-Butyl1 22 . 72 16.8 20.3 (;') 36 .66 1.12 29.6 1.46 38.2 2.9 Cyclopentyl'·• (/3) (i') : 35. 16 0 . 53 30.4 0.42 37 .2 0.57 Cyclohexadienyl'·h (CH2 ) 47.71 42.0 54 Ethynyl 16.1 74 62 N itrogen dioxide 107 74 63 (cis) (trans) 1-Methylvinyl• (CH1 ) (cis) ( trans) Phenyl (ort/10) (meta) (para ) CyclopropyJ<l (a) (fJ) Cyclobutyl'·• ({J) 59 17 .3h 49. 7b 8 1. 6b 60 60 61 60 ... • L (H-C=O) = 120 deg.M b L (H-C=C) = 146 deg. • L (C-C=C) = 146 deg. d The a hydrogen is 28.6° out of the carbon plane. • The C-CH2 group is planar. f The carbon skeleton is planar. • The a hydrogen is in t he carbon plane. h The single and conjugated bond lengths a re 1.51 and 1.34 A, respect ively. TABLE I I. Variation of parameters. -----·--A(G) Parameter Hydrogen exponent Values (1) 1.15 (2) 1. 20 - ----·---- -- - - -- - Radical Value (1) Value (2) 2-Propyl (/3) lerl-Butyl (/3) 13.6 32.9 66.8 2 1. 4 18.8 16. 8 14.5 35.0 74.6 23.6 21.1 19.0 Vinyl (gem) (cis) (trans) Ethyl (fJ) C-C hon<l lt•ngth (1) 1.54 (2) 1.44 Ethyl (fJ) 2-Propyl (/3) lerl-Butyl (/3) 21.4 18 . 8 16. 8 29 .0 24.9 21 .8 Ca out of the plane of the ring (1) oo (2) 10° Cyclopentyl (/3 ) 30.4 0.42 29 . 7 0.40 (-y) 222 constants cannot be very sensitive to small changes in geometry and hydrogen exponent. Our results in Table II show that the coupling constants computed from extended Htickel wave functions generally satisfy these requirements. The coupling constants of the vinyl radical have a strong dependence on the bond angle . However, Adrian . and Karplus, 65 and Cochran, Adrian, and Bowers 62 have also obtained optimum agreement with experiment for an angle of 140° to 150° using a valence-bond approach. The low value for nitrogen coupling is to be expected, since no ls first-row orbitals were used. Ethynyl radical may be a 7r radical, in which case these calculations would not apply . The complete disagreement for the para coupling in phenyl radical is not understood at this time. The results obtained by computing e. p. r. spin-spin coupling constants from extended Hi.ickel theory molecular orbitals show general agreement with experiment. The theory predicts changes within a given group of radicals very well. For example, the computed coupling constants of aliphatic radicals are all 5. 7 ± 0. 2 gauss lowe r than the experimental values. The contact operator for n. m. r. is . JC /I where yN is the nuclear gyromagnetic ratio. (C5) The first-order pe rturba tion term vanishes, and the second- order t erm 223 _ L; I (O I JC " In) I 2 nlO (C6) En - Eo when applied to wave functions of the type used gives JkL = - 8,q 2 h [ ~3 ] yk YL occ. unocc. ,./., ~ -1.J El. ,./., ,./., ,./., 'f'ik 'f'"L 'f'"k 'f'·L 1 ] E.] - E·1 where ¢jk is the value of M. 0. j at nucleus K. ] (C7) Because the energies could not be evaluated by the extended Hi.ickel method, the molecular orbital eigenvalues were used instead. The coupling constants obtained by this procedure are given in 66 Table III. Fahey, Graham, and Piccioni have obtained essentially the same results using hydrogenic 2s orbitals instead of orthogonalized Slater 2s orbitals. The approximations involved in computing the coupling constants were evaluated by making the same approximation for the energies, but using the Hartree-Fock orbitals and eigenvalues 72 for formaldehyde which were obtained by Foster and Boys. This gave a coupling constant for formaldehyde of 40. 7 Hz . compared to an experimental value 71 of between 40. 2 and 42. 4, and a Hi.ickel value of -12. 7. A set of ethane orbitals and eigenvalues similar to those of R. M. Pitzer also gave coupling constants in agreement with experiment. On this basis, it appears that the major error in these calculations is that the extended Hi.ickel method does not reproduce the Hartree-Fock results with sufficient accuracy. 224 TABLE III Molecule Jin Hz Exp. Cale. Ref. 1. H1 - H1 A. Saturated molecules 280 316 Hydrogen -12.4 -11. 3 Methane Ethane (vie) 3.3 8.0 Methanol (gem) -10.8 -14.4 B. Unsaturated molecules Ethylene (gem) 2.3 -10.1 (cis) 11. 5 4.4 (trans) 19.1 10.7 Acetylene 9.8 5. 1 Formaldehyde 40.2-42.4 * -12 . 7(40. 7)+ 67 68 48 69,70 48 48 48 48 71 2. C13 _ H1 Ethane (short) (long) Ethylene (short) (long) Acetylene (short) (long) 125.0 -4. 8 156.2 -2.4 248.7 49 . 7 74. 8 - 3.2 98.5 -6.7 160.4 -3.6 48 48 48 48 48 48 34.6 67.2 170.6 27.7 48 48 48 3. c13 _ c13 Ethane Ethylene Acetylene * Solvent dependent. + Hartree-Fock orbitals were used. 60.0 119.1 225 The extended Hilckel approach reproduces the general trends for both vicinal and directly bonded couplings reasonably well. This is hardly a triumph, because these trends are easily predicted within the framework of an average excitation energy approximation. 73 The calculated values for geminal spin-spin couplings bear little resemblance to the experimental values. Because the basic purpose of this approach was to remove the average excitation energy approximation, we must consider it a failure. The results of these calculations for e. p . r. coupling constants are in excellent agreement with experiment, but the agreement for n. m. r . is poor. In hindsight, we recall the remarks about the accuracy of occupied and unoccupied orbitals that we made previously, and note that we could have expected the relative accuracy of the computed e. p. r. and n. m. r. coupling constants. We would expect extended Hilckel theory to accurately predict n. m. r. couplings only when the contributions from the various excited states do not involve small differences between large numbers, and precise information about the excited states is unimportant. Comparison with successful applications of the average excitation energy approximation indica tes that this is the case. It appears that a method, such as Lipscomb's N. E . M. 0. calculations, which gives results similar to those from a rigorous Hartree-Fock calculation, in those cases for which the rigorous calculations have been done, would be successful in predicting n. m. r . spin-spin couplings . 226 REFERENCES 1. J. A. Pople, W. G. Schneider, and H. J.. 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