ESSA TR ERL 178-ITS 113 A UNITED STATES DEPARTMENT OF COMMERCE PUBLICATION f' "'Co ESSA Technical Report ERL 178-ITS 113 U.S. DEPARTMENT OF COMMERCE Environmental Science Services Administration Research Laboratories A Two-Signal Primer for Fourier Analysis of a Random Access Communication System M. NESENBERGS ESSA RESEARCH LABORATORIES The mission of the Research Laboratories is to study the oceans, inland waters, the lower and upper atmosphere, the space environment, and the earth, in search of the under- standing needed to provide more useful services in improving man's prospects for survived as influenced by the physical environment. Laboratories contributing to these studies are: Earth Sciences Laboratories: Geomagnetism, seismology, geodesy, and related earth sciences; earthquake processes, internal structure and accurate figure of the Earth, and distribution of the Earth's mass. 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White, Administrator RESEARCH LABORATORIES Wilmot N. Hess, Director ESSA TECHNICAL REPORT ERL 178-ITS 113 A Two-Signal Primer for Fourier Analysis of a Random Access Communication System M. NESENBERGS INSTITUTE FOR TELECOMMUNICATION SCIENCES BOULDER, COLORADO July 1970 For sale by the Superintendent of Documents, U. S. Government Printing Office, Washington, D. C. 20402 Price 40 cents. Digitized by the Internet Archive in 2012 with funding from LYRASIS IVIembers and Sloan Foundation http://archive.org/details/twosignalprimerfOOnese TABLE OF CONTENTS Page ABSTRACT 1 1. INTRODUCTION 1 2. STATEMENT OF PROBLEM 2 3. FOURIER COEFFICIENTS 8 4. DERIVATION OF MOMENTS 9 4.1 Means 9 4.2 Noise Variance 11 4. 3 Signal Variance 12 4.4 Hypothesis Ho 14 4. 5 Hypothesis H^ 18 5. DETECTABILITY PARAMETER 20 5.1 Definitions 20 5. 2 Example 23 5. 3 Error Probability 25 6. CONCLUSIONS 28 7. REFERENCES 29 A TWO -SIGNAL PRIMER FOR FOURIER ANALYSIS OF A RANDOM ACCESS COMMUNICATION SYSTEM M. Nesenbergs The detectability parameter, i.e., the acquisition signal-to- noise ratio, is derived for an elementary two -signal multiple access channel. The basic Fourier series approach requires no approximations, and the methodology should be useful in a forthcoming rigorous analysis of a nnore realistic random access satellite repeater. Key Words: Detectability parameter, ideal hard limiter, random access, satellite repeater. 1. INTRODUCTION A communications satellite with multiple, and in particular with random, access capability is of interest to various data collection and transmission networks. General aspects of such systems have been explored (Schwartz, Aein, and Kaiser, 1966), and specific questions dealing with synthesis and analysis of multiple access have been answered (Jones, 1963; Aein, 1964; Shaft, 1965; SoUfrey, 1969; Anderson and Wintz, 1969). Still, doubt remains about some of the statistical arguments used and the results so obtained. For instance, a new tractable and reliable derivation of the effective acquisition signal-to-noise ratio (SNR) or, as it is often called, "the detectability paramieter" is needed. Our long-range goal is to analyze a multiple (e.g., M-customer) random access repeater with an ideal hard limiiter. To accomplish this in a concerted single effort, unfortunately, appears too complex a task. Consequently, a short-range goal is defined and solved in this report, limited to only two signals {M=2), using appropriate signal design and word correlation detection, and establishing useful statistical properties of the correlator outputs. The results of this study are to be used later both as a guide and as a tool to treat the far nnore difficult M-signal case. Because a full distribution of these random variables is too cumbersome to derive, we are content to obtain the first two moments (i. e. , means and variances) without a need for so-called judicious and reasonable approximations. AH signals are of equal amplitude (see Jones, 1963 and Shaft, 1965 for effects due to unequal amplitudes), the modulation is to n phase shift keying (PSK), and the individual carriers deviate in frequency and possess random phases. Some coding is likely to be used to design the modulating waveformis. This will cause no complication, as the present treatment permits arbitrary codes, be they pseudo noise (PN) sequences, orthogonal codes, or what have you. 2. STATEMENT OF PROBLEM Consider a communication system as shown in figure 1, The channel consists ot an ideal hard limiter (IHL) plus additive white Gaussian noise, n(t). A bandpass filter (BPF) is used to reject bother- some higher harmonics. By observing its distorted and noisy input, the receiver tries to detect the message of one of the transmitters, say 0. The receiver nnust first decide whether transmitter is actually transmitting (hypothesis H ), or some other transmitter, say 1, trans - o mits instead (hypothesis H ). i The received signal plus noise is given by x(t) = sgn I cos e (t) + cos 02(t)] + n(t), v = 0, 1. (1) The function sgn x denotes the nonlinear IHL characteristic sgnx = + l ifx>0, = if X = 0, (2) = - 1 if X < 0, and n(t) is the white Gaussian noise with autocorrelation function R(T) = No 6(T), -oo cpi , and cp2 are assumed mutually independent random variables with a common uniform distribution over (0, 2tt). The binary i access (or address) modulations 3o (t), P^ (t), Pa (t) perform phase keying to tt , as shown in figure 2. The correlating (i.e. , reference) signal is assumed to be perfectly matched to transmitter r(t) = cos eo(t). (6) and represents the most advantageous acquisition situation. The other modulations are delayed by T^ and Tg respectively from the reference (fig. 2). We assume ^ T^, Tg ^ T. The receiver correlates the stored waveform r(t) with the received waveform x(t), and eventually chooses between the two alternative hypotheses Ho and H]_ . The statistical situation is therefore as shown in figure 3, where a is the correlator output if we assume H to be true: V a V T ^ I cos eo(t) sgnTcos e (t) + cos e2(t)l dt T (7) •^/. cos 00 (t) n(t) dt, V = 0, L Because of the random phases, a is in general not a Gaussian random variable. The distribution of a is not known, nor is it easily com- V puted. We will be content to derive the first two moments, Ea and vara , averaged over the independent phases and noise. <5 00. (/) Ld O -z. LU Z) o LU C/) CD =) Q O 3 CP o o (/I n m^ - n" ^m m 2: n > Two useful properties follow readily from (10), (a) If m+n is even, the C =0, The largest coefficients are mn C , nonzero for all nn = 0, 1, Z, ... . m+1, m (b) The Bessel's equality applies and through the well known identity z- m=l TT 90 (11) yields (see C balance sheet in fig. 4) mn ° ~ "^--1 2 Z^ ^mo " 2^^ Z-/ <^mn " 3 nn=l m = Z n=l (12) 4. DERIVATION OF MOMENTS 4.1 Means In this section we find the average Ea of the random variable a , V = 0, 1. We substitute the Fourier series (9) into (7), and con- clude that most terms must vanish in the averaging process. The 3] ^ O 4 (Converges \ Rapidly ^ ) Figure 4. BALANCE SHEET FOR SUMS OF SQUARES, C^mn. 10 noise term has zero mean and can be promptly ignored. Conside: - C / Ea = 5^ -^^ Z /coseo{t)cosme (t)cos nGs (t)dt. (13) m, n-0 J If V = 0, only m = 1 can produce a non-zero integrand. Likewise, n = is the only way to dispose of cps • We conclude that (m, n) = (1, 0) is the only nonvanishing term T ^^e/. EOo = ^ E / cos'^eo(t)dt I Cio (14) ^0. 405 284). 2 ^10 4 n If V = 1 , no choice of (in, n) disposes of cpo and Eai = 0. (15) 4. 2 Noise ,\ riance To derive var Qi , we separate the mean square into noise and signal ternns, *Here and elsewhere, uu T need not be an integer. However, the error is of the order of ^ and can be ignored for (JU > 10 Hz. 11 V 1 [y /cos0o{t)n(t)dtj (16) 00 J- + Er ^^Q -^^ / coseo(t)cosme (t)cos nBg (t)dtl It is clear that the first entity does not depend on v and is by far the simpler. We consider it next (see (3)): T a^ = e[y /coseo(t)n(t)dt]^ TT = e[y^ / / cos0o{t)coseo(T)n(t)n(T)dtdTj T I -// -) (17) T T E[coseo(t)coseo(T)] E [n(t)n(T)]dtdT ^ I E[cos^eo(t)]dt No 2T 4. 3 Signal Variance The second quantity in (16) is evaluated next. To make things come out neatly, i. e. , 12 let us write a^ +. Ea ; = E vara ^ o^ + o^ , V = 0, 1, (18) V n V T C I • ii.u. J - ii.\ / ^ — - — / cos9o (t)cos m6 (t)cos n92 (t) dt i EC C, ,1 (m,n;k,^). mn k-L v' m, n, k, -t=0 (19) where T T I (m, n;k, t) = — ^ E / / cos9o (t)cosme (t)cosn92(t) =y/- (20) • cos9o (t)cos k9 (t)cos ^92(t) dtdT T T -^^ 2 E f Ccos [Bo (t) ±m9^(t) ± nB^ (t) ± 9o (t) ± k9 (t) ± I 92(t) j dt dT. The second line in (20) is based on the trigonometric identity (proof by induction) J /I cos X. = — 2_\ cos(xo ±Xi ± . . . ± X ) (21) j=0 J 2^ ± 13 that holds for all integers J ^ and all arguments Xq , x-^ , . . . , x , J subject to proper interpretation of symbol Z. All 2 ± connbinations (distinct or not, zero or not) are included in the sum. Let us review the conditions for a nonvanishing term I (m, n; k, -t). The indices must be such that (a) the coefficients of uu t and c uu T vanish (to assure a spectrum passage through BPF before correl- ation), and (b) the coefficients of cpo , cp^ , and cpg vanish (to avoid averaging to zero). These coefficient constraints are summarized as follows: V = V = 1 uj t: c 1 ± m ± n = (ju T: c ± 1 ± k ± ^ = cpo: 1 ± m ± 1 ± k = cpi: cps: ± n ± -t = 1 ± m ± n = ± 1 ± k ± ^ = 1 ± 1 = (22) ± m ± k = ± n ± ^ = In both cases there are three linearly independent equations for four unknowns, and multiple solutions cannot be avoided. We proceed by treating the case v = in section 4.4 and case y = 1 in section 4. 5. 4.4 Hypothesis Ho If the hypothesis Ho is true, v = may be substituted in (18) - (20) . Define the quantity e(t, T) = 00 (t) - e2(t) - 00 (T) + e2(T) (23) and observe that the argument of (20) that satisfies (22) for v = must be a multiple of e(t,T). Therefore, only (m,n,k,-t) values giving rise to 14 1 1 I(J) = ^ /* A°^ T T .s j 0(t, t) dtdT, j = 0, 1, 2, can contribute to (20). The lo (m, n, ;k, l)'s that do not vanish are (24) ]o(l,0;l,0) = 2 1(0), ]o(0,l;0,l) = 2"^ 1(1), 1^(0, 1;2,1) = ^,(2,1;0,1) - z'^l{l), ]o(2, 1;2,1) = 2''^I(1), (25a) and four equal terms for all j ^ 2, ]^(j ± l,j; j± l,j) = 2 I(j). (2 5b) It remains to evaluate the integral I(j) in (24). Let j be even, j - 2k (k - 0,1,2,...), and cos2k6(t, T) = cos 2k(iiUo -(D2) (t - T) = cos 2k(a)o -^3 )t cos 2k((jOo -uug ) T + sin2k(aJo -uu2)t sin2k(uUo -^s) T, (26) T _2 I(2k) = 1-^ / cos 2k(ajo -aJ2)tdt -| I cos2k(ajo-aj2)tdt + Pf / ^ [ sin 2k(uJo "^^2)^ 1 -cos 2k(iiJo """2)^ 2k(ajo-uu2)T J I 2k(uJo-iU2)T J T ^2 sin 2k(uJo -uU2)t dt r sink(uUo -U02)t "| I k(UUo-UJ2)T (27) 15 For odd j = 2k - 1 (k = 1,2,...) we introduce the followrn^ LOtation. For 1-1=1, 2, let P(^,t) = %{t) - p^(t - T^), (28) a function that is either or n. Consequently cos p(|a, t) must equal ±1 for all ^ t ^ T. We expand this function in Fourier series (with i =/^), ■ 2Tr^ cosp(^,t) = 2 y U-j:\ 2tt^ 1 / "'^~^ \,,i,n— ^ = - /cosp(|j, t)e dt (29) r 2tt^, 1 f This is a line spectrum with line separation 2n/T. The power in the spectrum can be summarized as T 2 ^ A ^ Icosuutcos p(^, t)dt + U; I sinojt cos P(|_L, t) dt where (30) (31) = otherwise. Return to the evaluation of I(2k - 1), k = 1 , 2, . . . . As in (26), 16 i cos(2k - 1) 9(t, t) = cos[(2k-l){'jUo-aj2)(t-T) + P(2 .; ■■ p(2,T)] (32) = cos(2k-l)(a}o -'JU2) t cos(2k-l)(aJo-uU2) T cos p(2,t) cos P(2, t) + sin(2k -l)((iJo -'-^2) t s in( 2k - 1 ) ((JUq -0^2) T cos 3(2, t) cos P(2, t), and T I(2k-1) = ' ;^ I cos(2k-l)(iiio -uu2)tcos p(2, t) dt ^ 1 f -^Jco: T Ya(^(2k-l)(aJo -^2)y T + 1^ I sin(2k-l)(aJo-uu2)tcos p(2,t)dt 1 (33) The major distinction between I(2k) and I(2k-1) must lie in their dependence on niodulation (e. g. , coding). I(2k) does not depend on miodulation at all. I(2k-1) does depend through (28)-(31). We may- collect all I(2k)'s into a variance term a ^ that is divorced from 00 crossmodulation effects; and all I(2k-l)'s into a variance term a ^ ox that does depend on crossmodulation between ^Qcind ^z{see (19)): a ^ :^ a ^ + a ^ . (34) o 00 ox The component terms depend on (10), (19), (20), (27), and (33). 17 = 2",ECSk-l,2k+Sk+1.2j^'2'^' k=l 2 V^ 1 I sink(ajo > a)3)T ]■• aj = 2 (2Coi+ Csi)^ 1(1) (35) + 2"' ,^, vC2k_2, 2k-l + ^2k, 2k-iy ^^^^"^^ k=2 n^ k=l [{2k-l)^- i? Y2 (^(2k-l)(ujo-uu2)y 4. 5 Hypothesis H^ Let V = 1 and proceed as before. There are two arguments in (20) that contribute for all m = 0, 1 , 2, . . . , and 3o(t) - 01 (t) +m[e2(t) - 01 (t)] )o(T) + 01 (T) - m[e2(T) - 01 (T)], )^(t) - 02 (t) + m[9,(t) - 02 (t)] )o(T) + 02(T) - m[9i(T) - 02(T)] (36) 18 Therefore in (19) 2 ^ m = m+1 , m 1 1 (m+1, m;m+l, m) + li (m,m+l;iTi, m+1) '(37) where for j = 1,2,... define li(l,0;l,0) + li(0, 1;0, 1) = 2 J(0), ii(j+i J;j+i.j) + ii(j,j+i;JJ+i) = 2"^j(j). Each J(j) is a sum of four terms. Let j = 2k(k=0, 1, 2, . . . ), T j(2k) =1^ j cos^aJo-(2k+l)aJi+2kuj2^t cos P(l,t)dtJ + lij''"0 T + l_x I ^°^ V ° 'p ojo -(2k+l)uJi+2kuj2)t cos p(l,t)dt (39a) +2kuJi -(2k+l)uj2) t cos p(2, t)dt ;38) uUo+2kuUi -(2k+l)uJ2 Jt cos p(2,t)dtl Yi No -(2k+l)uji+2ka]2 i + Y2 ^ ujq +2kiJi -(2k+l)aj2^ 19 Next, let j = 2k -1 (k=l, 2, . . . ), J(2k 1) = Ij^ I cosi^ajo+{2k-l)uji T 2kaj2 )t cos p(l, t)dt -i2 \- I sinruJo+(2k-l)uJi -2kaj2) t COS p(l, t)dt I r.. ^ + T- I cos( (jUo -2kaJi +{2k-l)(jU2 H cos P(2, t)dt ' (39b) T A + 1—1 sini oJo -2kaj-L +(2k-l)ai2 jt cos P{2, t)dt = Yi ( uuo +(2k-l)(jUi -2kuu2 y + Ys ( ""o -^kuj;^ +(2k-l)(JU2 J • Apparently, both J (2k) and J(2k-1) do depend on cross -modulation. The variance (let a,^ ~ ^i ^ ^° agree with (34) and to emphasize this dependence) follows from (10), (37), (38), and (39): 00 8 V^ 1 r ^ ' "^ r ^ l^~) ^Ix"^ ^ (4k+l)^ I Yii^wJo-(2k+l)aji+2kuj2y + Y2 Q^o +2kiiji -(2k+l)aj2^| j (40) ^^ ^1 (4k-l)^ {[vi(a;o +(2k-l)(jji -2kuu2^ + Y2 (oJo -2kuJi +(2k-l)aj2 }^ 5. DETECTABILITY PARAMETER 5. 1 Definitions Aein (1964), Anderson and Wintz (1969) and others use a detectability paranaeter var Oo (41) 20 This definition ignores the false alarm probability connpletely (Helstrom, I960), and is suited to measure the message output SNR after acquisition. The acquisition model (fig. 3) suggests an alternative definition var Oo + var tti Rudnick (1962) gives Neyman -Pearson arguments for use of D , and in fact shows that D should be maximized. Even in the present case of E a ]^ = (15), the two definitions become equivalent only in the extreme var Oi^ « var Qfo . If the two variances are of the samie order of magnitude, then D — g d . We have as our main result 2a^ +a ^ +(a ^ + a, '^) ' ^ ^ n 00 ox ix where EOo is given in (14), a ^ in (17), a and a in (35), and ^^ (=> n 00 ox a ^ in (40). All the quantities are exact for the assumed system Ix model (fig. 1); no approximations or bounds have been used so far. To elaborate nnore on the a 's in (43), one needs to invoke additional properties of the channel. These properties are apt to be based partly on measurement, and partly on engineering inference. Consider a ^ in (35). The only unknown entity is the frequency offset OJo -0)2 . By (5), the difference should be small but not necessarily 21 zero. The variance term can be upperbounded by setting (juq equal to cug (see Abramowitz and Stegun, 1964), 'o:"?-ia (4k^-i)^ "°-°°^^^^- (44) The crossmodulation-dependent quantities a ^ (35) and o ^ ox Ix (40) also depend on the frequency spacings. The physical interpretation of the variances amounts to weighted sums of crossnaodulation energy at specified spectral lines. There is a great variety of methods for estinaation and bounding of spectra; the difficulty must clearly lie elsewhere. Consider our knowledge, or lack of same, about the following: (a) frequencies uuq , uu^ , uUg , (b) delays Ti , Tg , (c) modulations po (t), Pi(t), Pgi'^)- Quantity a (44) was upperbounded by setting all frequencies equal. This appears uncertain for o and a, , Also since ox Ix message FM is certain to be nnuch slower than address modulation, we nnust decide whether there is practical justification for treating the carrier frequencies (jOq , (ju^, and uug as deterministic constants (functions) or as random variables (functions). In common practice all possible delays O^T ^T(|a=l,2) can occur. We may wish to single out the pair (Ti,T2) that has the worst effect on a ^ + a, ^. The final answer depends on the code ox Ix ^ (e. g. , signal design) used. From figure 2 we can observe super- ficially that maximum variance must occur at bit sync. 22 The choice of code, as indicated, offers an area of concern, especially for large M. For M = 2, the three waveforms Po(t)> 3i(t), and P2 (t) give little substance to a multiple access argument involving crosstalk and/or lack of orthogonality. 5. 2 Example Consider the following, atypically simple example. Set T^ = To = 0, T = 2n, and for \i = 1,2, let uUq - uu = 1/M , where M » 1 is a large integer. Pick the codewords as Po(t) = (0, 0,0,n,n,TT), Pi(t) = (0,0,TT,0,n,n). (45) p2(t) = (0,n,0, 0,n, 0). By (28) and by (29) :os P(l,t) = (+1, +1, -1, -1, +1, +1), :os P(2,t) = (+1, -1, +1, -1, +1, -1). Yi(o) -~ (^°^ orthogonal), Y2(o) = (orthogonal), nrr L J (46) (47) 23 We denote by ='= the constraint that (2k-l)/M2 rnust be an integer and write (35) as ^ ,^. [(2k-l)--i]- k=M< ,^2k-l- 2 r ^2(1) (48) ^f^MT 0. 002 080 (uuo -ojg)' Similarly, from (40), 2 ^ 8 r Y> 1 I A2k+1 2k ^ Ix ~ tF I ^,. (4k+l)^ I ^^ V Ml " Ms^ ,^„.. Tik+rp "^^K /'2k+l) 2k ^ k=0 M2 MiV (49) k=r + ^-...TikTI)^ ^-2k 2k-l -s ^^^Mi " M2 ^ i^Ti* (4k- D' 2k 2k- r Y2^ TT VM2 Ml y 24 where '■'■' denotes a constraint that the argiu.ieit of y (• • • ) nnust be an integer, [i = 1 , Z, ^ 2 _ .4 r |Yx(1)|" I |-Y2(i)l^ + Iyi(Q)I^^ + ] Ix rF L Mi^ Ms"" 4(Mi+M2)^ ' ' ' J ^ 4 r 3 1 n^ f = 0. 012 482 (ojo -uui)^ + 0. 004 161 (% -ajg)^ + 0. 000 114 (2a)o -OJi-uUs)^. The total variance (denominator in (43)) contains N /T plus (44), (48), and (50). For M^ » 1, a ^ can be ignored in comparison with '^ O 2 T^A ^ ^ 2 a . Moreover, a is quite likely to be negligible in respect to 5. 3 Error Probability In this brief section we introduce elementary error probabilities pertinent to the system (fig. 1). Quite typically a threshold device follows the correlator. Some level a is used to decide which of the hypotheses is likely to be true. Thus, if a > a^ holds, Ho is accepted as true; and if a ^ a '^i is accepted. The nature of such binary hypothesis testing is well understood (Crame'^r, 1961; Helstrom, I960). Two types of error are possible, 25 and occur with probabilities (51) P = Pr (select H^lH is true) = Pr (a^ >a^), P = Pr (select H JH^ is true) Pr (a^^ a^). If the distributions of a^ and a were accurately known, the error probabilities could be evaluated exactly. However, we know the means and variances exactly, and nothing beyond that about the distributions. We present two idealizations that are easy to use but inaccurate in practice. Nominally, the correct values will lie between them. The Gaussian Model Pretend that the random variable a (v = 0, 1) is Gaussian with mean and variance given above. Then , a P., = 1 - erf ^ 01 /var tti P , ^ - e rf ^ ^ /va r a o where the error function is defined as in Viterbi (1966) 1 r -^' / e ~2" du, -a> the set relationships (55) c "o^^JU^'^o} hold for all a . Regardless of threshold setting, a union bound yields 27 ^Ol + ^IO^P'^'^^"!'- <5^' We interpret Pr(a^ a ) as an "irreducible error probability" of sorts, and note that Pr (a ^ a ) = erf D (Gaussian model), 1 '"' <■ ^ (Chebyshev bound). The second line follows from the Chebyshev bound (54) applied to Pr [(ao - Eao) - (c^i - Ea^) ^ Ea^ - Ec^o] . 6. CONCLUSIONS A direct second order analysis of an extremely simplified two- signal multiple access channel has been carried out. The effects of an ideal hard limiter are worked out with the aid of standard Fourier series. This approach is conceptually quite simple (Anderson and Wintz, 1969), as it avoids the formalism of hypergeometric functions (Jones, 1963; Sollfrey, 1969). The model, the techniques, and, in a qualitative way, the results are intended to guide us in a forthcoming computer analysis of multiple (M » 2) randora access channels. To retain trust in the basic solution, the method presented does not require any approximations or bounds . The channel model consists of an ideal hard limiter and Gaussian noise. Carrier phases are mutually independent and uniform over (0, 2tt). The relative delays and frequency deviations can be treated as deterministic or as random variables. We have derived second order statistics (means and variances) for the word correlation out- puts. The underlying distributions are non-Gaussian and too cumber- some to derive. 28 Three properties that emerge in the above two -signal case and may very well be valid in the general M signal case, are the following. First, reasonable frequency drifts and departures from a true carrier do not seem to increase the distortion variance in a drastic fashion. The same conclusion appears to be valid for slow message FM or PM. Second, the variance of the correlator outputs contains a substantial term that is entirely independent of crossmodulations for the assumed to n modulations. Third, waveform and coding departures from orthogonality do not necessarily affect the detectability parameter by robustly scaling down the signal mean. 7. REFERENCES Abramowitz, M- , and I. A. Stegun, Eds, (1964), Handbook of Mathe - matical Functions (National Bureau of Standards, Applied Mathematics Series 55, Washington, D. C. ). Aein, J. M. (1964), Multiple access to a hard-limiting communication- satellite repeater, IEEE Trans. Space Electronics and Tele- metry SET-IQ, No. 4, 159-167. Anderson, D. R. , and P. A. Wintz (1969), Analysis of a spread- spectrum multiple-access system with a hard limiter, IEEE Trans. Communication Technology COM-17 , No. 2, 285-290. Cramer, H. (1961), Mathematical Methods of Statistics (Princeton University Press, Princeton, N. J. ). Helstrom, C. W. (I960), Statistical Theory of Signal Detection (Pergamon Press, New York, N. Y. ) Jones, J. J. (1963), Hard-limiting of two signals in random noise, IEEE Trans. Information Theory IT-9, No. 1, 34-42. 29 Rudnick, P. (1962), A signal-to-noise property of binary decisions, Nature 193, 604-605. Schwartz, J.W., J. M. Aein, and J. Kaiser (1966), Modulation techniques for nnultiple access to a hard -limiting satellite repeater, Proc. IEEE 54, No. 5, 763-777. Shaft, P. D. (1965), Limiting of several signals and its effect on comimunication system performance, IEEE Trans. Communi- cation Technology COM-U, No. 4, 504-512. Sollfrey, W. (1969), Hard limiting of three and four sinusoidal signals, IEEE Trans. Information Theory IT-15 , No. 1, 2-7. Viterbi, A. J, (1966), Principles of Coherent Communications (McGraw-Hill Book Company, New York, N. Y. ), 30 GPO 830 - 546 ■fpiiir