The Global Optimization of Phase-Incoherent Signals Citation Schaffner, Charles Albert (1968) The Global Optimization of Phase-Incoherent Signals. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/t799-rn87. https://resolver.caltech.edu/CaltechTHESIS:12212015-144937012 Abstract The problem of global optimization of M phase-incoherent signals in N complex dimensions is formulated. Then, by using the geometric approach of Landau and Slepian, conditions for optimality are established for N = 2 and the optimal signal sets are determined for M = 2, 3, 4, 6, and 12. The method is the following: The signals are assumed to be equally probable and to have equal energy, and thus are represented by points ṡ i , i = 1, 2, …, M, on the unit sphere S 1 in C N . If W ik is the halfspace determined by ṡ i and ṡ k and containing ṡ i , i.e. W ik = {ṙϵC N :| ≥ | ˂ṙ, ṡ k ˃|}, then the Ʀ i = ∩/k≠i W ik , i = 1, 2, …, M, the maximum likelihood decision regions, partition S 1 . For additive complex Gaussian noise ṅ and a received signal ṙ = ṡ i e jϴ + ṅ, where ϴ is uniformly distributed over [0, 2π], the probability of correct decoding is P C = 1/π N ∞/ʃ/0 r 2N-1 e -(r 2 +1) U(r)dr, where U(r) = 1/M M/Ʃ/i=1 Ʀ i ʃ/∩ S 1 I 0 (2r | ˂ṡ, ṡ i ˃|)dσ(ṡ), and r = ǁṙǁ. For N = 2, it is proved that U(r) ≤ ʃ/C α I 0 (2r|˂ṡ, ṡ i ˃|)dσ(ṡ) – 2K/M. h(1/2K [Mσ(C α )-σ(S 1 )]), where C α = {ṡϵS 1 :|˂ṡ, ṡ i ˃| ≥ α}, K is the total number of boundaries of the net on S 1 determined by the decision regions, and h is the strictly increasing strictly convex function of σ(C α ∩W), (where W is a halfspace not containing ṡ i ), given by h = ʃ/C α ∩W I 0 (2r|˂ṡ, ṡ i ˃|)dσ(ṡ). Conditions for equality are established and these give rise to the globally optimal signal sets for M = 2, 3, 4, 6, and 12. Item Type: Thesis (Dissertation (Ph.D.)) Subject Keywords: (Electrical Engineering) Degree Grantor: California Institute of Technology Division: Engineering and Applied Science Major Option: Electrical Engineering Thesis Availability: Public (worldwide access) Research Advisor(s): Grettenberg, Thomas L. Thesis Committee: Unknown, Unknown Defense Date: 23 April 1968 Record Number: CaltechTHESIS:12212015-144937012 Persistent URL: https://resolver.caltech.edu/CaltechTHESIS:12212015-144937012 DOI: 10.7907/t799-rn87 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 9342 Collection: CaltechTHESIS Deposited By: INVALID USER Deposited On: 22 Dec 2015 17:37 Last Modified: 05 Apr 2024 21:49 Thesis Files Preview PDF - Final Version See Usage Policy. 7MB Repository Staff Only: item control page

Copyright ~ by Charles Albert Schaffner l968 THE GLOBAL 0¥.rIMIZATION OF PHASE-INCOHERENT SIGNALS Thesis by Charles Albert Schaffner In Partia l Fulfillment of the Requir ements For the Degree of Doctor of Philosophy California Institute of Te chnology Pasadena, California l968 (Submitted April 23, l968) ii ACKNCJ;'7LEDGEMENT I would like to thank Dr. T. L, Grettenberg, my research advisor, for his advice, guidance and discussions during this research. I am also indebted to Dr. H. A. Krieger for his time and guidance in parameterizing the problem in a form which led to this solution, · I would also like to thank the Hughes Aircraft Company for their support as a Hughes Doctoral Fellow during the cours e of research presente d in this Thesis. Also, I would like to thank Mrs . Dor is Schlicht for typing this Thesis. Finally I wish to express my appreciation to my wife for her patience and understanding during this research. ABSTRACT The problem _o f global optimization of . M phase-incoherent signa ls in N complex dimensions is formulated. Then, by using the geometric approach of Landau and Slepian, conditions for optimality are estabN= 2 lished for and the optimal signal sets are determined for M = 2, 3, 4, 6, and 12. The method is the following: The signals are assume d to be equally probable and to have equal energy, and thus are represented by points Si' i = l, 2, ... ' M, Wik is the halfspace determined by i.e. Wik i = l, 2, s . 1 s.l. trEcN: I <r-, s i > I :::: I <r, sk> I}, ••• , M, and If sk and containing Si' in s1 then the~. l. n w.k, = k7h l. the maximum likelihood decision re g ions, partition For additive complex Gaussian noise r = -sie je + n, CN. on the unit sphere where e n and a received signal [0, 2 ~], is uniformly distributed over f the co probability of correct decoding is r2N-le-(r2+l)U(r)dr, 0 where 1 U(r) -M For N = 2, f; . f i=l~in s 1 r 0 (2r I (8, S) I )dcr(S), it is proved that u(r) ~ J and r = \\r\I. I (2rl (s,s) I )dcr(s) 0 ca 2K M h(~K [Mo(Ca)-o(s 1 )]), where ca= [sES 1 :l<s,si)I :::: a}, the total numbe r of boundaries of the net on decision regions, and function of h s1 K is determined by the is the strictly increasing strictly convex (where W is a halfspace not containing s.l. ), iv given by h = ~ r 0 (2rl(s,si)l)d0 (s). Conditions for equality are ~w established and these give rise to the globally optimal s ignal sets for M = 2, 3, 4, 6, and 1 2. v TABLE OF CONTENTS Topic Chapter I TABLE OF ILLUSTRATIONS iii ABSTRACT iv INTRODUCTION • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • l THE OPTJMln-1 RECEIVER AND THE E Q.UIVALENT VECTOR ClIA.NNE L • • .. • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • 3 II FORMUh~TION OF THE PROBLEM ••••••••••• : •••••• 7 III THE METHOD OF h~NDAU AND SLEPIAN • • •• ••• ••• •• lO IV THE CASE OF M ~ 2 ..•.............. . l5 v SQ\fE NECESSARY CONDITIONS ••.•••••••••••••••• 23 VI A TRANSFORMATION INTO THREE - SPACE ••••• •• •••• 29 VII SOLUTIONS FOR N = 2 •••••••••••••••••••••• • • 34 VIII CALCULATION OF THE PROBABILITY OF ERROR FOR PHASE - INCOHERENT ORTHffiONA.L SIGNALS •••••••• 46 CCMPARISON OF THE PROBAB ILITY OF ERROR 2 BE1WEEN THE GLOBALLY OPTIMAL SIGNALS IN c AND ORTHffiONA.L SIGNALS ••••••••••••••••••••• 49 CONCLUSIONS •• .••• • •••••••••••• ••••• •• • • • •••• 53 REFERENCES • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • 54 IX x N = 2, vi TABIE OF ILLUSTRATIONS Figure VI-l VII-l VII-2 Title The Co-ordin~te Transformation from the Unit · Sphere in C onto the Unit Sphere in Three Dim.ens iona l Real Space ( o = 0) . . .........• . .... 31 The Optimal Signal Set for M = 2, K = l Shown in the u, x and x, y, z Co-ordinate System . . . . . . . • . • . . . . . . . . • . . . . . . . . . . . . . . . . . . . . . . 41 The Optimal Signal Set for M = 3, K = 3 Shown in the u, x and x, y, z Co-ordinate System .. . ...................................... . 42 VII-3 The Optimal S i gnal Set for M = 4, K = 6 Shown in the u, x and x, y, z Co-ordinate System ........................................ . VII-4 The Optimal Signal Set for M = 6, K = 12 Shrn{n in the u, x and x,y,z Co-ordinate System ..... ... ..................... . ...... .. .. . VII-5 44 The Optimal Si gnal Set for M = 1 2, K = 30 Sho1m in the u, x and x, y, z Co-ordinate System ........................................ . IX-l IX-2 Comparison of the Probability of Error for Orthogonal Si~nals and Globally Optimum Signals in C ....•...........•..•............. 51 Comparison of the Probability of Error for Globally Opti mum Signal s in c2 as a Function of the Number of Signals for Fixed Signal-toNoise Rat i o (A 2) .........•................. • . . 52 l INTRODUCTION The problem of optimal (minimizing the probability of error) signa l selection for transmission of messages over phase-coherent and phase-incoherent channels has been a subject of many investigations. Under the assumption of additive white Gaussian noise, equal energy, and equiprobable signal sets, Balakrishnan [l] showed in 1961 that with no bandwidth constraint the regular simplex is globally optimal for small and large signal-to-noise ratios for the phase - coherent channel. Landau and Slepian [2] established in 1966 that, in fact, the regular simplex code is globally optimal for the phase-coherent channel independent of the signal-to-noise ratio and for a larger class of probability density functions. Also in 1966, using the approach of Balakrishnan, Scholtz and Web er [ 3] proved that the orthogonal signal set is locally optimal for the phase-incoherent channel under no bandwidth constraint. phase-incoherent signals in straint, the signals with M-1 For M dimensions, i.e., a bandwidth con- l<s.,s.J >I = Mll were established as locally l. optimal by Weber [4] in 1967 . Using the geometric approach of Landau and Slepian, we formulate a condition for global optimality of phase-incoherent signals in N complex dimens ions . M equi-probable In t he geometric approach, the length of the signal vectors is proportional to energy ; and the dime nsionality of the space is analogous to bandwidth [8] . For the set of probability densities which are monotone decreasing away from the signa l vectors (of which the Gaussian is a member ), we prove the validity of the s e conditions for N= 2 along with some r e lated 2 necessary conditions. the unit sphere in Euclidean space. We then perform a transformation which maps c 2 onto the unit sphere in three-dime nsional With this transformation, we _a re able to use Euler's formula to show that the global solutions obtainable by this method are M = 2, 3, 4, 6, and l 2 ; and these have respect ively l, 2, 3, 4, and 5 hyperplanes form ing the boundary of their decision regions . obtain the globally optimal signal sets for these We then M's. In particular, we demonstrate that the signal sets which are globally optimal in t wo complex dimensions are, in fact, the above mentioned signal sets for <si, sj > = o signal set signals). M=2 and M 3 for two signals a nd (i.e., the orthogonal \ <si, 8 0 > I = ~ for three For four s i gnals, the globally optimal signal set has I(si, sj) \ = '3 For six and twelve signals, the inner product b etween the signal vector and t he ones determining the decision region are given by 1 \G.,s.)\ = J. J J2 for twelve signals. for six signals and \<s.,s. >I = J. J ~ \/ 5--~-J5 3 CHAPTER I THE OPTJMUM RECEIVER A.ND THE EQUIVALENT VECTOR CHANNEL In this Chapter we derive the optimum receiver for the transmission of messages over a phase-incoherent channel. In this deriva- tion, it is assumed that the noise is additive white Gaussian and that the messages are all equi-probable a.nd have equal energy. (A.1.x.1. ( t), i Let = l, 2, ••• ' M} be the set of real messages to be transmitt ed where Aixi (t ) is defined on 0 .~ t ~ T and has energy A. 2 1. Let y. (t) be the Hilbert transform of xi ( t), i.e ., -21. y . (t) = ltGx.(t). n: 1. s. (t) Let jy. (t) 1. 1. 1. be the complex message defined by s.(t) = x.(t) + 1. 1. which has spectrum f 2X.1. (f) ~ 0 s.l (f) (1) f < 0 0 and having unit energy. Let n(t) be complex white noise with zero mean and power spectrum f :2: 0 s n (f) = (2) 0 f < 0 Next we assume that the received signal is of the form ·e + n(t) r(t) = A.s.(t) eJ 1. 1. (3) 4 where e represents the phase of the r-f probability density define d on t'he interval l carrier and has a uniform 0 ~ e ~ 2rr,. i. e ., e E[o, 2rr ] 2rr (4) p(e) elsewhere 0 rrn (t)} be a s et of complex orthonor mal lTi i=l,2,•••,N We now let basis functions for the linear space spanned by the [si(t)}i=l, 2, •• ·,M· th component of the r vector· as Then we define the k J T rk = r(t) cpk*(t) dt (5) 0 and similarly -l(- ( 6) cpk (t) dt T and nk = f n(t) cpk * (t) dt . (7) 0 This yields for the kth component the equation r k = A. s. l. l. k j9 e + n k and hence we obta in the vector e quation - j8 r=A.s.e +n J. l. (8) The minimum probability of error receiver is then to sele ct the ith mess ages as b e ing tra nsmitte d when p Crl A.s.) p (s.) = max p (rl Aksk) p (sk) l.l. l. k ( 9) 5 As suming complex Gaussian di stributed nois e with zero mean and variance 2N , 0 we obtain 11-r-A.s- .e j - 2NO i i 1 1 P (rl A J_.s J_., e) 9 2 \l (10) --Ne ( 2JrN ) 0 Then J 2 = p(r lA.s.) J_ J_ 1( p(rl A.s., e) . p(e) de J_ J_ (ll) 0 - j - 2N 11-r-A.s.e 0 J_ J_ 1 1 But where 1 2Jr 9 2 ll d9 de is known to be I is the modified Bessel funct ion of the first kind. The 0 minimum probabil ity of error decision rule i s to select the ith message as being transmitted such that s J_. maximizes 6 A. 2 J. I I ) - 2N0 A.1 \ (r, s.1 > 0 ( e N 0 p(si. ) (13) We now as sume that the messages ar e a ll e qui-probable and h ave and equal en ergy probability decis ion rule reduces to selecting s. J. Ai 2 2 =A . Then the optimum to maximize Now x 2n (14) n=O which i s a monotone increasing function of decis ion rule is to select s. J. !x i. Ther efore the optimum such that (15) max k For convenience in l ater sections, we l et N 1 0 = -2 then sents the signal-to-noise power rat i o of the complex signal . using this notation, we see t hat all signal vectors A2 repre- Also, [si }i =l, 2, •••,M have unit energy and may be considered as points on the unit sphere in CN . 7 CHAPTER II FORMULATION OF THE PROBLEM In this Chapter, we use the equivalent vector channe l presented in Chapter I. In this formulation, the set of signal vectors rs.} all have unit energy (i.e.. \\s.\\ = l),· and hence we l i i=l, 2, • • • , M ' i represent them as points on the unit sphere in CN. The dimensionality of the complex space i s proportiona l to the band;·Tidth of the communication system. [8] -r The received vector is assumed to be of the form. j9 + -n where s. ( t) = x. ( t ) + jy. ( t ) and x.(t) = -s.e l. l. l. l. l. message transmitted where -n The noise yi(t) is the Hilbert transform of e represents the unknown phase of the carrier and is assumed to be uniformly distribute d on The probability density of rece iving a vector l. xi(t). is assumed to be complex additive Gaussian noise with zero mean and variance one. x. (t) is the real (x. (t) _, s. (t)) l. l. r, r-f [0, 2rr ]. given that was transmitted, is then seen to be (Chapter I, Eq. (12)) 2rr ~rr J 0 e 2 \. (r, s.l. >Icos e d9 2 1 -(r +1 ) \ (r I =~ e I 0 ( 2r r , si) ) (16) rr where r \\rll . P(-;\s.) l. where Thus, we can write 2 1 Ne rr -(r +1) P C\<E. ,s.)\) r r i (17) 8 I Pr < <E. r , r 0 ( 2r I(E. , s.1 ) \) r s ·>I) 1 r > 0 is for each fixed We may partition CN (18) a strictly increasing function on [O,l]. int~ M decision regi~ns (19) and each~. region conta ins 1 i = 1, 2, error • • • , M. ( Q.) s.eje 1 for all 0 ~ e.~ 2~, We may now write the probability of no decoding as M =[ Pr <l<E.r ,s.> I) dm(r) 1 1 =~ J ~ 0 co 2 r 2N-l e -(r +1) U ( r ) dr (20) where M U(r) I: i=l p(s.) 1 f ~.n sr i r and s ro pr trECN: \Ir\\ = ro}. <I <s, s. >I) do(s) 1 (21) 9 Assuming equa lly probable s i gnals, Eq.( 21) c an be rewritten as M U(r) = J ~-[ i=l~n sr i P r cI (8, s.i >I ) da (s) ( 22 ) r Q is maximized if and, clearly, U(r) is maximized for each r > O. Now, we l et (23) Note, also, if we l et containing s. ]. Wik be a h a lf-spa ce determined by s.)_ and and defined by , (24 ) then ( 25) Consequently, our problem is to find a condition on the location of the points s , s , 1 2 • • • , sM on the unit sphere i n is maximized . P Ri ~n s ; 1 r s uch that U(r) i s an increasing function of and the decision reg ions finite number of half- spaces of Where CN CN Ri [o, l], are the intersection of a determined by points on s1 . 10 CHAPI'ER III THE METHOD OF LANDAU AND SIEPIAN In this Chapter, we present the method of Landau and Slepian modified for the phase-incoherent optimization problem. o <a < 1, For of size a s1 we define the "cap" of centered at s. l. and to be (27 ) We let a(ca ) a(c.i,a ), i = 1, 2, • • • , M denote the common value of a(ca ) and further restrict such that (28) If W is a half space which does not contain h::: c. si , let Jw (\<s,s.>I) da(s) . i, a n Pr ( 29) J. The method of Landau and Slepian is based on proving the following properties of (A) h h which we shall prove in Chapter DI for is a function only of a(C. n W) i,a for fixed N = 2: a and, in fact, is a strictly increasing strictly convex function. (B) If V is the intersection of a finite number of half-spaces, at least one of which does not contain then ll f P <l<s,s.)\) dcr(s);;:: h(cr(c . i,o:n v)) i r c.i , a n v with equality if and only if Assuming follows. h For is a single half- space. has the properties (A) and let i = l, 2, • • • , M, such that~i k. l. (B), we may proceed as be the smallest integer is the intersection of distinct half-spaces ki i .e ., • • • Wik. '• V (30) wil' wi2' i s the number of boundaries of the net on l. determined by K b e the total number of boundaries on the Let Ri. s1 M K = ~ L ki (31) i=l Let be the portion of the boundary of boundary of Ri. Ti 2 ' l. • • • , T .k , through i s.. Then Ric where each is bounded by T .. l.J c l.J, which is a B .. 1.J Til' and "hyperplanes" Hence, if we l et l. T .. l.J can be partitioned into regions C. E.i, a = R.n l. i,o: and W. . T .. n C. a l.J i,o: (32) , (33 ) we have the identity fJ i=l R . l. f . do + l. f E. i, a l2 Letting we have f . .= l, l. o(T. . iJ, ex + a(E. ) . i,ex If we next let f P r (35) M ki M = Mo(cex ) + )lJ L .. ) o(E. ) - \ i,ex L \L o(Tl.J,ex i=l i=l j=l f.l. = Pr (\(8,8.)j) l. (36) and from Eq.(30), · (I (8, 8.) I) do(8) ;;:: h(a(T.. )) i iJ, ex with equality if and only if T .. l.J' ex Tij,ex = ci,exnwij" We may then write the inequality L f I: L \h(cr(T .. )) l.J,ex i=l j=l (37) i , ex function. l. M i=l c. and, from property A, k. M h is a strictly increasing strictly convex Therefore (38) with equality if and only if and i I: f has the same value for all j. Substituting M o (T. . ) l.J' ex M the inequality given in (38) into Eq.(37 ) yields f M 2= f fido ~ [ fido + fid<J i=l E. · i=l R . i =l c. l. i, ex i, ex M - 2K h(.!c_ 2K k. l. LL i=l j=l o(T. . )) l.J' ex (39 ) 13 Now, from Eq. (36 ) we note that M L (40) cr(E. i,a ) - i=l We will now place a restriction on the Eq. (39 ). Let w1 w1 n c1 ,a c and w w2 n c1 cr(E. i, a ) portion of be two h a lf-spaces such that 2 (41) ,a and (42) (43 ) The n we may wr ite h(cr(w 2 n c 1 ,a)) = h(o(w1 n c 1 ,a)) + J f 1 dcr (44) A where A = (W -W ) n cl a 2 1 , (45) and M l cr(A) = 2K L i =l o(E. i,a ) (46) l4 Pr cI <s, s.l. >I ) Now, since is monotone increas ing in its argument, we have that J P Cl <s,s . ) I) d<J(s) r p (a:) o(A) ;;e: i r ~ A f 2K Pr( l (s, si ) I ) do(s) ;;e: 2KPr (o:) o(A) (48) P (a:) o(E . (49) A r i , o: ) i=l But , s i nce == R. n C. E. i , a: c we have that i ,o: l. M M [ p (a:) o(E . a:) r i, ~ i=l J cI [ Pr <s, s.i >I) do(s) (50 ) i=l E . i , a: wi th equality if and only if ) = 0 (5l) a: Combining the results from ( 26 ), (39), (40), and (5l), we obtain an i nequal ity for i, U(r) f U( r ) ~ o(E . I p r c <s, s.J. >I ) (52 ) Ci a: ' and, furthermore, there is equality if and only if such a cap size exists with the additional properties . l. T. . =C . and j . J.J ) a: nw l.J .. i , a: , ,where w. . is a half' space for a ll i J. J 2. 3. for all i and j . l5 CHAPTER IV THE CASE OF N = 2, M ;;:: 2 In this Chapter the validity of properties N = 2. III are proved for the case of (zl' z2 ) = (xl + jyl' x2 + jy2) ·e 2 j iP 0 ~ p ~ 1, zl = rpeJ , z2 = rJl-p e ' dm = r 3 drdo, where do 1( 1( -1( s 0 =e O <a< l (r,p,9, <.P ) -rr < e ~ rr, -1( r3p - 1( where < cp ~ rr, so that pdpd9d<.P Thus, the unit sphere in pdpd9d<.P 2 2rr • c 2 has l f ff If into The jacobian of this transformation is r > O. B of Chapter We consider the transformation which sends and A and 0 jy 0 (l,o) and then for p the cap equation i s (53) For later convenience, we introduce the notation 2 t3 = l - - M' l V=a2 - 2 and then · 1 JJf 1( 1( (54) pdpd9d<I? - rr - rr a and the requirement (Eq.(28)) 1 2 1 ~ a2 ~ 1 - M 1 1 M o(S ) ~ o(Ca) ~ 1 2 cr(S1 ) becomes o ~ v ~ ~2 t3 (55) 16 Now, suppose i.e., s sl r..1. ej1) s2 s -s 1 2 = e 1 and and s -- e jy2 ( s e j02 2 a r e linearly independent points on s 'v s/) , j yl( j 11. s e , 1 2 Ji - 1 2 ) (56) (57) l - then the "hyperplane" equat ion s1 ; \(p,s1 >\ = \(p,s2 ) \ becomes If we let e j 6i (59) (60) and (61) ' then the "hyperplane" equat i on can b e rewritten as 2 l 2 ( p - - )(s 2 2 2 - sl ) (62) l7 If we let Isl t s 2 l and we assume without los s of generality l then that -s 2 ' t is well-defined for the fol lowing cases for Eq . (62 ). = ro, if t we hav e either if 0 <t < ro ' (3 = o, (3 CP' and we have =l, or l -- -2'· 2 the equation i s p cos ( e- <P-o) = O; and u where for convenience we have let \u = o, t ~ we have cos ( 9- <P - 0) for If 0 ~ t I~T = (63) u 2 p l and - 2 is de fined ~~ . t 2 2J1 + t Proof of (A) . We now us e t hes e transformed equations to establish the convexity of h. We first let Wt and not ~ontaining s measure. s1 ~ s That is, 0 be a half- space, determined by which inte rsects 2 Ca and in a set of positive and (64) where for v < T < ~ 2 define I where J 'r do (p) = kt (u) du v ( 65) 18 t v =ex 2 l - 2 , and u - u 2 then cmv (t) at J o T + = 2rc arcos gt(T) -dT dt 2rc- (arcos gt(u)) du at v T 0 + f . 2rc ~t (arcos gt (u)) du = 4rcT t2 v J V JT u 2 du - u 2 __ 4rc T J~2 ' - v2 > 0 . 2 t Thus , for f ixed Now, for v, mv(t) ( 66 ) i s a strictly increas i ng function of 2 l which is the range of 0 :s;; m < re (2 - v), be the inverse function of mv, we let t. t (m) v Next, l et T Hv (t) = f kt(u) Pr(Ju ~) du ' + v and let hv (m) = Hv ( t v (m)) oHv (t) at and so that Oh (m (t)) v v 00) Hv (t) = hv (mv (t)) . O:Dv (t) ot Then (68) 19 oH (t) v = J ~t kt(u) Pr (Ju+ ~) du v (69) We may now write as l - v (70) 2 now integrating by parts yields. = Pr (ex) + f,.JR2 ,. v ,. - u 2 - v (71) 2 which is a positive strictly increasing function of Therefore, we have proved that for each fixed v, h t, v and hence of w. is a strictly increasing strictly convex function. Proof of (B) . We now prove the conjecture that if V is the intersection of a finite number of half spaces, at l east one of which does not contain then J Pr(\<S,si)\) dcr(S) ~h(cr(ci,cxnv)) ci,cxn v with equality if and only if recalling tha t V is a single half space. kt(u) = 2:rr arcos gt(u) where We proceed by gt(u) = J,Iu 4 - u 2 20 Therefore, we observe that - re l 2- rt::"2 2(1+ - u >J(i;)- - (t u always negative. 1= for 2 (72) is a strictly increasing function of t and is Now, let n v = . nl wt . + l) u -rcT = which for fixed 2 intersects Wt , Wt , • • • W be half-spaces such that l 2 ' tn Ca in a set of positive measure and 8 q wt. 0 ]_ ]_ i = l, 2, • • • , m ~ n. Now, define gt (u) = l and kt(u) = 0 for l T < U < 2· Then l 2,,.. j kt(u) du v o(wtn ca) = (73 ) l 2 f [ 2n:2 - kt (u) ]du v Ther efore, l 2 o(v n ca) = f (74) k(u) du v where we describe k(u) as follows. I.et 21 i + l = l, 2, ... ' m = nH-l , .. . d(i) -l i n < ••• <~ then there is a partition for ' (75) ' of such that uE[ u. 1 , u . J , JJ (76) k(u) where ~ (i~) is a constant and distinct elements of [l, 2, • • • belong to ' m} • [l, 2, is a collection of not necessarily n} ' such that at most two of them In particular, this description shows that differentiable in (u. 1 ,u.) J- J k is continuous on and has right- and left- hand derivatives at the left and right end points, respectively. In fact, these der i vaties are given by ok dk(u) l \ du = 2 (__, d(i~) If we now let Wt cr(wtn ca) = cr(v n ca) cr(Wt n C) = m(t.) i a i l, 2, • • · , m. dk(u) ~ du for v ~ u ~ T for ' t . ( u) J.~ (77) OU be a half space such t hat 8 0 ~ Wt and then mv (t) = cr(wtn ca ) cr(V n Ca) ~ i = l, 2, • • • , m -+ t. > t for J. i = Consequently, from Eq . (7 2 ) okt(u) ou with equality if a nd only if (78) V = wt. J. for some i 22 with t.J. = t l l 2 2 I k(u)du = f and l ::; i ::; m. But dk ( u) kt(u) du and Ok t ( u) -du - '2: -ou -- for v ::; u ::; T implies v v such that there is a point Since for for P u ::; u 0 and is monotone increasing, we may r therefore write l 2 - hv(wv(t))= J [k(u)-kt( u)] Pr(Ju + ~) ~u v ~] l [k(u)-kt(u)J Pr(Ju + ~) du + v [k(u)-kt(u)J Pr(Ju + ~) du uo ~ P~[? l [k(u)-kt(u)] du+ with equality if and only if J [ k(u)-kt(u) ] dul 0 uo v J j k( u) = kt(u) for all Pr(\ 0i,s0 ) \) do(p) :;:: hv(o(v n ca:)) v n ca: with equality if and only i f V is a singl e half- s pace . J (79) Hence (80) 23 CHAPI'ER V SOME NECESSARY CONDITIONS In this Chapter we establish some necessary conditions that must be satisfied for the existence of an optimal signal set. the case in which Mrr . rr 2K::;; 2 k ].. ~ 2 for i We consider = 1, 2, ••• , M ~ 2K ~ 2M ~ x = . For a given allowable cap size, i.e., one such necessary condition is the existence of a half-space Wt such that (81) We now define 1 2rrx(- f:)-v) (82) 2 which is in the domain of t v' the inverse function of Hence, we define T (v) x t (W (v)) (83) = Wx (v) (84) v x then ru (T (v)) v x Thus, the half-space determined by necessary condition t T (v) x must satisfy the 24 ~ v (T (v)) x (85) + Now, from Eq.(82), we have that CW (v) x (86) - 2:n:x Now, for convenience of notation, we define 1" x T (v) x ( v) Then we may write k ru (T (v)) v x T x v (v) (u) du (88) which yields anv (Tx (v)) ov k_ - -or (v) x (v) = - 2~ arcos gT (v) x (v) (89) . and, from EQ. ( 66) we have that anv (Tx (v)) ot Now, substituting Eqs.(86), (89), and (90) into Eq.(85) and solving oTx (v) for we obtain ' (90) 25 2 Tx (v)[arcos gT (v)(v) - x] x - v (9l) 2 Now, from Eq. (52 ), we have Ux(v) = f Pr(\(p,s0 )\) do(p) - ~K hv(~K [Mo(c0 )-o(sl)]) ca l 2 = 2rc2 f Vu+ Pr ~) du - ~ h (W (v)) 2 x v x v l 2 = 2n 2 J Pr(Ju + ~) du - ~ Ry(Tx(v)) (92) v Then, = - 2n2P (o:) r ~~01\r(;(v)) + OHy(Tx(v)) oTx(v)J XL ,. (v) gT (v)(v)] } x v Another property of the U (v) the convexity property of hv . x av ot 2 2 ,. (v) - u f1 _x_2_ _ __.,,. 2 dPr (Vu + ~) . ,- (v) - v x (93) equation is obtained with the aid of This is that (94) If we naw consider requirement 2, Chapter III, i.e., 26 o(E. i,a c ) = 0, i,a ) = o(R.n C. i we find some additional requirement for the existence of the cap is the existence of a 0 s v s l 2 t3 v =a 2 l 2 , such that 2n: 2 arcos g'rx(v)(v) = 2 K/M We shall show, in fact, that there is exactly one such it and V(x), U n: x- -2 1( V(¥) = O; for T x occurs in the interval x First of all, for let call is the unique point at which the maximum of V(x) and the minimum of v, x< 2 ' gT (v) (v) = cos x ~ v = o, so we x we have gT (v) (v) = cos x x T x v (v) (96) - v On the 2 other hand, ,. ro)t) = J kt(u) du v J arcos(21") = 2n: [ arcos arcos ( 2v) and integrating by parts, we obta in t t!n.n.] d(~ cos .n. ) (W) . 27 CJ.) v (t) (98) where (99) - v Hence, the defining equation for T x (v) = Wx (v) m (T (v)) v 2 x becomes 1 1 2 arcos jTx (v)(v) - v arcos gT x (v)(v) = x( 2 ~ - v) Thus, we are looking for a value of v v which satisfies the system (101) cos x - v (100) 2 1 cos ~ x (102) The solution is easily found to be 1 tan ~ x V(x) = 2 tan x and (103 ) 28 T x (V(x)) To show that 2 tan x (~2 t3 = sin t3 x (lo4) 2 2 Jcos t)x - cos x l V(x) ~ 2 t3 for 11: 0 <x < 2} V(x)) = t3 tan x - tan t)x. and has derivative . 2 we consider This function is 2 t)(sec x - sec t)x) and since 0 for O<t)~l t3 tan x - tan t)x ~ 0 (105) is the value of which will be U x attaine d if an optimal signa l set occurs for a given value of x} we OU (V(x)) have Uc) (x) = _x~Ox,..--- ~ o. Thus} for fixed MJ u is a decreasing 0 function of x· and} hence, the maxinum possible value of ' obtained for ~K = M - l } i.e 'J 11: x = M- l u0 is 29 CHAPTER VI A TRANSFORMATION INTO THREE-SPACE In this Chapter we :perform a transformation that maps the unit sphere in c2 onto the unit sphere in three-dimensional real Euclidean space. In particular, we apply this transformation to the hyperplane equation (Eq.(62)) and to the equation for the boundary of the cap. The equation (Eq. (62 )) of the hyperplane between two signals, say is and 2 l 2 2 (p - -)(s -s l ) 2 2 = (l06) where we have defined u = p 2 - 1 2 and boundary of the cap about signal s1 * 8-ii?-6. The equation for the is =2s u + 2JiJ:i:' - s 2·r;--:;, Vij:- - u- cos 1 I (107) ~ v (a) and a give n set of signal vectors, functions only of the two variables x = 8-ii? and where V=Ci. 2 1 and - 2 s l - 2 . Now, Eqs. (lo6 ) and (lo7) are for a g i ven cap s i ze u. Solving for u as a function of x, we obtain from (106) 30 (l08) and from (l07) vs ' ±J u 2 '2 '2 l '2 2 2 l vs - (s + ( 4 - s ) cos (x--o))(v - ) 2 2s '2 l + (4 - s '2 Using the coordinate transformati on shovm in Figure VI-l for we transform the u,x (l09) 2 ) cos (x-o) o = o, equations into points on the unit sphere in three r eal dime nsions with x = 2u y =Vi- 4u2 z =Jl - 4u 2 cos ( x) (llO) sin ( x) I n Section VII, Eqs . (l08) and (l09) are used to i l lustrate the conf igurations of the caps and hype rplanes . These figures are then trans- formed by (llO) and plotted on the surface of a unit sphere in three dimensions . Substituting the transformation (llO) into the hyperplane Eq . (lo6), we see that the hyperplane for t = -'-' I s"-'1-_2 2 6 = 0 is given by x = ty where and the intersection with the unit sphere is g iven by the s2 -sl set of equations 31 y Unit sphere . in three dimensions .Jl - 4u 2 sin(x) Point determined x, u by Jl - 4 2 U COS ( X) x x z FIGURE VI-l The co-ordinate transformation from the unit sphere in 2 c onto the unit sphere i n three-dimensional r eal space (& = o). 32 x = ty z =Jl - x (ill) sin ~ 2 That is, the hyperplane in c2 is transformed into a t wo-dimens ional real plane which intersects the sphere and passes through the origin . The decision regions in three dimensional real space are now the intersection of half spaces determined by these transformed hyperplanes. t denote the hyperplane in three-space be tween s. and Let H . . J. J.J t Then let W.. b e the half-space containing s .. The decision region J.J J. ~t is then the intersection of M-l of these h alf spaces, i.e. i = n w t kti (ll2) ik and is therefore a convex region bounded by a certain number of hyperplanes that pass through the or igin. Now, by using this transformation, we note that the maximum likelihood reg ions forming the net on the three-dimensional sphere can be composed of regular spherical polygons. r:D t . Since the~.J. are convex, a vertex on the surface of the sphere must be . formed by at l east three edges . Thus, we have that is the total number of vertice s on the net and of edges on the ne t. 3V ~ 2K where K is the total number We may now apply Euler's formula [5], for the net and obtain an inequality for V V-K+M = 2, K K ~ 3 (M- 2 ) Since we require the same number of boundaries on each of the (ll3) 33 decision regions, the total number of boundaries integer K must be an (I) ti.mes the total number of signals divided by 2; K = IM (ll4) 2 We may therefore rewrite (65) as an inequality for I i.e., I M as or 6M - 12 M (ll5) 12 (ll6) ~ ---­ or M ~ 6- I Now, as shown in Chapter v, ing function of x and hence a monotone increasing function in Therefore, for a given as possible . u0 (r) Eq. (52) is a monotone decreas- M, we wish to choose K and thus I K. as large Consequently, we wi sh to obtain equality in inequalities (ll5) and (ll6) since these codes will have the maxi mum number of boundaries for each decision region and the same number of boundaries for each region. The only cases of equality are The corresponding values of K are M = 3, 4, 6, and 12. K = 3, 6, 1 2, 30 . When M = 2, the above inequalities do not hold since there are no vertices. CHAPTER VII SOLUTIONS FOR N 2 In this Chapter we use the results of Chapters IV, V, and VI to construct the globall y optimum signal sets for N = 2, M = 2, 3, 4, I (Sp s j ) \ for all i, j 6, and l 2 . We obtain the value of of these cas es . for each We then graphically present the r esults for these cases showing the location of the signal vectors, the caps, and the hyperplanes . Two Signals: If M = 2, f3 = o, then and l a: =ff . Also, for sl must b e l and the r equirement 2K s; M-1 s; - M ~ K = l. l 0 S:VS:2f3=:>V = 0 Thus the decision region (ll7) o. iyl That is, and e (O,l) Thre e S i gnals : Then X= ~ M 3, K 3 =:> V(x) 0 and a: = 72- ' 2 ' l l f3 =3 t = T (V(x )) x s i n f3x Jcos 2 f3x-cos 2~ :rr tan b 35 l VJ . sin ~x Hence sin x l = 2' and trere - and Four Signals: 4, M = Then, V(x) = ~ tan rc/6 = tan rc/3 rc/6 i and a =J23 ' t = / sin 2rc 2rc=J2 l vcos Therefore we have ' - - s2 - e iy2(!._ i( &t 2rc) e 3 J'3 {fl 'V ·~f ' 6 - cos 3 resulting in l J3 Six Signals : M = 6, K = 12 ~ x 1l = '4 2 (3 = '3 , t V(x) sin rr./6 ~~~----'-'--~~~~ ~cos 2 rr./6 - cos 2 rr./4 s 0 -- l sin rr. 6 l sin rr = r:: V2 Select s 5 = O. Therefore we h ave the following set of signals. _ i y 4( l s4 = e J2 e i (6 + '!?-) l , ) J2 , l 37 resulting in l<SO'si>I = \<so,82> \ = \<so,s3> I = l<"so,s4>\ = .l<si,82> 1 = \<Si,s3>I = l<si,s4>l = l<si,s5 >I l<s2,s3> l = \<s2,s4>1 = l<s2,s5>\ = \<s3,s4 >1 = l<s3,s5>\ Twelve Signals: M = l2, V(x) and = ~ tan 1( 6 2 tan 1( 5 ~ o: == K = 30 => X (1 1 3 == ~2 J~ ~ ~ 0.39733 J5 ) ~ 0. 947 27 3(5 - J5) t = = ~ ~ l.6180 sin 1(/6 Jcos 2 16 1( then + f3 = ~ =; , - cos 2 1( I5 V3 - J5 s0 == l := ~ ~ 0.85065 V575 Jsin 21(/5 - sin21(/6 sin 1(/5 g = 5 C:If 5 ~ 0.52573 and sll == O. The optimum signal set is therefore s 0 _ s 1 = e iy = e iy1 (l,O) (J2i 5-15 iy2(J s2 = e _ s =e 3 __ s - e 4 2 e r-; e i(o + ; ) ,j J5 ) 3 _ 5-J5 i( 6 + ;1!) , 5 -v 5 (J5J5 iy 3 e , J J5 ) 3 - Ir 5 -v5 i(o + 1!) , ~i v~ 5 - iy4 (~ e i(O + 7;) , 5-J5 J5-/5.J5 ) 3 _ , , , - - iy7(J3-J5 s7 - e 5-15 i(O + 2;) e . ' J 2 5-J5 ) ' 39 - s 8 = e - i y~J 3 - J5 i ( 6 + i") r;: e , - i y -e s 9()3 -J5 9 - slO = e 8) 5 -v 5 5-/5 e i (6 + J5 e i y i o f ; i - i(y + 5-/5 g·) J , 2 ) 5-15 ~re) , ~ 5-J5 , , , and which results in \<ss,s9> \ = \<ss,s11>I = \<s9,s1o>I = \<s9,sll> I = \ <s1o~ll>I J5 ~15 40 I(so,s6> I == I<so,s7 >I == I<so, ss> I \<sl's3> I == I<sl, 84> I I<8l, 8s> I == \<8l , 8l o> I == I<8l' 8l l> I \<82,s4>1 I<82, 85 >I == I <82, 86> I I<83, 85 >I I<83, s-1 >I I <84, 8s> I == I <84, 8l l> I I<so, s9> I == I<so, slo>I I <s2, 89>I I<s3, 8l o> I == I <s3, 8ll>I \<82' 8n >I == I<s4, 8 6> I == I <85, 81 >I == I <85, 89> I == I <85, 8n> I == \<86' 88 >I == I<s6, 89> I == I<81, s-9> I == I (87' sl o> I == I<8s' 8l o> I "J35 -J5 -15 And Using the signal sets just presented, we substitute in hyperpl ane e quation (Eq . (l08)) (Eq . (l09)) . and the equation for the boundary of the caps We now graphically present the results for M == 2, 3, 4, 6, and l 2, showing the r e l ationship of the hyperpl anes and the caps both in the u, x coordinate system and in the x,y,z coordinate system . 41 ------- hyperplane and cap ( c oi ncide for M = 2, K = 1) . 1( 2 x 0 ,_ sl so 1( - 2 - I 1( 1 - 2 I 1 - 4 0 u I 1 1 4 2 y Lx z FIGURE VII-1 The opt i mal s i gnal set for the M = 2, K=1 shown in u, x and the x, y, z co-or dinate system. hyperplane cap 1( 1( 2 0 x - 1( 1 1 1 ° u - 4 I / I ............... ..... / / / / II / ,/ I/// , ,. I ............... -----------------------?:( / ,,,,, I "'- . . . ._ 1 4 , / \"" "" /I\' I\ . . . ._ I \ \ / \ I I / \ s0 ..... \,\ I \ \ F IGURE VII- 2 The optimal signa l s e t for M = 3, K= 3 shown i n the u,x and the x, y,z co- ordinate s ystem. hyperplane cap ------ ',~ \ I • s2 re 2 l .\ - -· 1/) · - I '\ \ --------==~-~:~~ x 0 I' , , / /8 0 • sl __ _ ------- re - - 2 --- ·~ ,,,,...,. -.:.:.:::-..:::::.::.::-=-~' '\ I :1 • s3 . / - re - 4 2 \ ,, I 1 2 u ------- - --- ~ ------------~ ~,,.,,--- / / ,~ ?//-- \~\ \ / \ \ / // \ I/ \ I \ } I I \',, ' "~ ~ ' I //I I "-.... I , . . .-.._. ....___ . ._ ----=---If/' 1{·____.;, . =--_: .: .: .- -- ~' ', --- I ~ FIGURE VII- 3 The optimal s i gnal set for u,x and the M = 4, x,y, z K =6 shown in the co-ordina te system . 44 hyperplane c ap 1( 2 x 0 1( 2 - 1( l l -4 - 2 0 l l 4 2 u I • sl I ' \\! ,\I s 5 I ! ) --- Ii; "j~~-------------==~/ s 1 II i/II ------- I\ \\ \ /''.......... _,.,'•\'" 0 III ~: J \\ {\ II I '\\ II S • 3 I FIGURE VII- 4 The optimal s i gnal set for u,x and the M = 6, x,y, z K =l 2 shown in the co - ordi nate system . hyperpla ne cap n: 2 x 0 n: - 2 - n: 1 2 1 0 u - 4 1 1 4 2 FIGURE VII-5 The optimal signal set for M = 1 2, u,x and the x,y,z K = 30 shown in the co-ordinate system. 46 CHAPTER VIII CALCUIATION OF THE PROBABILITY OF ERROR FOR PHASE-INCOHERENT ORTHOOONAL SIGNAIS Using the notation of Chapter I and assuming eq_ual energy and eq_ui-probable signals, we have (118) The probability of correct decoding assuming orthogonal signals is then P[l(r,s.)\ >max \(r,sk)\J 1 kri ·e + (n, SJ.. ) I2 > max I(n, sk) I2 ] = P[ \AeJ kf i (119) Since we are using orthogonal signals, let the orthonormal basis functions b e cpi(t) = si(t). Then the probability of correct decoding can be written as (120 ) where the i n are independent complex normal distributed with zero mean and variance one. Since t he statistics of the noise do not depend upon which signal was transmitted) we have that the probability of being correct is independent of which signal was transmitted. (121) Now) \nk\2 i s equal to the sum of the squares of its real and imaginary parts, and these parts are each identically d i stributed independent) normal) mean zero) and va riance one ha l f. \nk\ 2 Therefore) has a chi-squared density of rank two and mean one. e - \nk\ 2 \nkl 2 > 0 0 el sewhere 2 P( \nk\ )- (122) (123) Result ing in the probability of correct being Pc= E (1-e -\Ae [ j8 i 21N-l] + n \ (124 ) 48 i n == n e Let -je then n and n i random variables since ni are identically distributed is invariant under phase translations. Therefore, we obtain (125) By expanding in a binomial series, we obtain (126) Now, since n is a complex norma l random variable with zero mean and variance one, we have E[e-k\A+n\2]-! fe-[{n)2+ k(A+n)'\dn = _!_ne k+l -\~:[! kA 2 l ] k+l e - (k+l) [n + k+l . dnj :n: (127) n We see that the above integral in brackets is the integral of a complex normal density of mean to unity. kA k+l and variance k:l and is therefore e~ual We therefore have that the probability of be ing correct is kA2 N-1 pc == L - (-1) k k+l (.N1l) _e- - k =0 The probability of error is (128) k + 1 1 - P c and may be written as (129) CHAPTER IX COMPARISON OF THE PROBABILITY OF ERROR BE'IWEEN THE GLOBALLY OPTIMAL SIGNAIS JN c2 AND ORTHOGONAL S I GNAIS In the previous Chapter we obtained the probability of corre ct decoding for orthogonal phase-incohe rent signals. In this section, we obtain the probability of correct decodi ng for the globally optimal phas e - incoherent signal sets in complex two space. We then calculate and present the probabil ity of error for both these cases as a function of signal-to-noise ratio . By using Eqs.( 20) , (52) , and (67) we may obtain the probabili ty of correct decoding for unit of signal- to- noise ratio -J~ (A = 1). ro p where c - 2 r3e - (r2+1) U(r) dr (130) 0 rr '2 U(r) = 2n: 1/2 f f 'f 1 2K Pr(Ju + 2 ) du - M kt (u) Pr(Ju + ~) du v v but :. p c (131) (132) du - 2~xf kt(u)I0 ( 2,Ju + v ~)Jdr lc133) In order to calculate the probability of being correct for any signalto- noise ratio, we see from Eq. (12), Chapter I, that the above equation must be modified to become du - 2~xJ v kt(u)I0 (2Aq)didr (134) 50 where 2rc areas u (135 ) and t (136) The probability of error can now be calculated from the above equation by p e 1 - p c for the signal-to-noise ratios desired. Figure DC-1 shows a performance comparison of the orthogonal signal (M = N) and the globally optimal signals in signal-to-noise ratio 2 A . c2 as a function of Figure DC-2 shows the degradation in probability of error as the number of signals increases for fixed signal-to-noise .ratios . 5l M signals in t wo dimensions ------= - 0 5 M orthogonal signa l s in M dimensions lO l5 Signal to noise ratio 25 FIGURE IX-l Comparison of the probability of error for orth~g onal signals and globally optimum signals in C . 52 l ,,,,..--,,--£:2=4 . . . =0 - - - - 2 -- -- - / ~ ---,,,,----- / / - A =l6 I :>, I •ri rl ------ / / I /.L G-i 0 ~ / I l o- 3 rz:I - / 2 !-! !-! .-- / / s --------- --------------- __.....-- ------- - - ,,.,. ........'--. A =l 2 - A =9 ..µ -- 2 A "25 . I I I I .I ·ri ~ .g l O~ I 2 4 6 8 Number of s i gnals lO F IGURE IX-2 Compar i son of t he p r obability o f error for gl obally optimum s i gnal s in c 2 as a f unction of the number of s i gnal s for fixed sign a l t o noi se rat i o (A 2 ). l2 53 CHAPTER X CONCLUSIONS We have formulated a set of conditions for the global optimality of M equalJ..y probable, equal energy phase-incohe r ent signals in complex dimensions. N In this method, we consider the signal vector as points on the unit sphere in C2· ' and, by means of a geometric argu- ment similar to that of Landau and Slepian, we proved the validity of N = 2. these conditions for We then perform the unit sphere in c 2 onto the unit sphere in three real-dimensional Euclidean space. Using this transformation, we map the hyperplane equation and the equation for the boundary of the cap. We then establish that the only signal sets in be shovm to be globally optimal by this method are and 12. c2 that can M = 2, 3, 4, 6, We determine that for the globally optimal signal sets there are respectively l, 2, decision regions. 3, 4, and 5 hyperplanes determining the optimum Next, we determine what these globally optimal signal sets are for these values of M's. These sets are those for which the inner products between a given signal vector and the ones making up the decision region about the given vector are for \<S.,s. >I J. J M = 2, 1 J2 for M l 2 for 6, and 1 \<s.,s.>\ = c 1 J V3 I<s., s. >I = er_ i J V5-J5 M = 3, <'Si' sj > = o for M 4, for M 12. We then compare the probability of error pe rformance between the globally optimal signals in c2 and the orthogona l signal sets in as a function of the signa l-to-noise ratio. CM 54 REFERENCES l. A. V. Balakrishnan, "A Contri bution of the Sphere Packing Problem of Communication Theory", Journal Math. Anal. and Appl., Vol. 3, p. 485-506, December 1961. 2. H. J. Landau and D. Slepian, "On the Optimality of the Regular Simplex Code", Bell System Technical Journal, Vol. XLV, October l966, p. 1247 to 1272 . 3. R. A. Scholtz and C. L. Weber, "Signal Design for Phase-Incoherent Communication", IEEE Transactions on Information Theory, Vol, ITl2, No. 4, October 1966, p . 456- 4-63 . 4. C. L. Weber, "Signal Design for Phase-Incoherent Communications II", presented at the International Symposium on Information Theory, September _ll-15, San Remo, Italy. 5. H. S. M. Coxeter, "Regular Polytopes", Macmillan Co., New York, l 963 . 6. J. M. Wozencraft and I. M. Jacobs, "Principles of Communications Engineering", John Wiley and Sons, Inc., New York, New York, 1965. 7. C. W. Helstrom, "The Resolution of Signals in White, Gauss ian Noise", Proceedings of the IRE, Vol. 43, September 1955, p . l lll- ll18. 8. H. J . Landau and H. 0 . Pollak, "Prolate Spheroidal Wave Functions, Fourier Analysis and Uncertainty - III; the Dimension of the Space of Essentia lly Time- and Band- Limited Signals", Bell System Technical Journal, July 1 962, p. 1295-1336 . 9. T. L. Grettenberg, "A Representation Theorem for Complex Normal Processes ", IEEE Transactions on Information Theory, Vol . IT- ll, April 1965. lO. C. A. Schaffner and H. A. Krieger, "The Global Optimi zation of Two and Three Phase - Incoherent Signals", California Institute of Technology, Communications Laboratory Technical Report No . 3, January 1968 . ll. T. L. Grettenberg, unpublished notes .