XVII. Communications Systems Research: Coding and Synchronization Studies TELECOMMUNICATIONS DIVISION A. Performance of a low-Rate Command Data link, S. Farber 1. Introduction This article gives the performance of an orthogonal signal frequency-shift-keyed command link. This ground- to-spacecraft link will code low-rate binary information into one of two frequency-modulated tones modulated onto the carrier. The purpose of this article is to examine the error performance capabilities of such a scheme under the assumptions (1) that the phase error out of the space- craft tracking loop does not vary significantly over a bit time and (2) that the phase error out of the spacecraft tracking loop does vary significantly over a bit time. 2. System Model The transmitted signal is assumed to be of the form s (t) = (2P)w cos [ w e t + k sin (ut + e) + *I, O = Wg,6J1 where represents a one being transmitted. The angles on the interval -'IT, T radians. is the carrier frequency, k the index of modulation, and = wo represents a zero being transmitted while o = O, and 6 are assumed to be uniformly distributed random variables defined The signal s ( t ) can be expanded in a Fourier series about we to yield s (t) = ( 2 P ) W J o ( k ) COS (met + *) - ( ~ P ) M J , (k) {cos + ( ~ P ) W J , ( k ) {COS Eoct - 2 (@t + e) + *I + COS IOct + 2 - ( 2 ~ ) s ~ ~ (k) {cos - + e) + *.I> - 3 (Ot + e ) + *I - COS 10ct + 3 (wt + e) + SI} + e ) + *I - COS LOet + (,t + e) + *I> + ( 2 P ~ ~ J J , ( k ) { ~ ~ ~ [ o , t - 4 ( ( o t + ~ ) +*] COS[^,$+^(^^+^)+*]} _ . . . JPL SPACE PROGRAMS SUMMARY 37-53, VOL. 111 1 76 https://ntrs.nasa.gov/search.jsp?R=19690007160 2020-03-12T08:14:50+00:00Z where ] I , is the Bessel function of order k. An indication of the behavior of J i (k), i = 0,1,2, can be seen in Fig. 1 for values of k satisfying 0 6 k 4 2. If the tracking loop works on the fundamental com- ponent, it will form an estimate !$ (t) of k (t) 2 that the received signal mixed with (2)*; sin [met + k ( t ) ] and filtered will yield r l ( t ) = (P)ssin[ksin(ot + e ) + + ( t ) ] +n,( t ) while the received signal mixed with (2)scos [ w e t + $ (t)] and filtered will yield r z ( t ) =(P)Mcos[ksin(Ot+O) + + ( t ) ] +nz( t ) where + (t) = $ (t) - and n1 (t) and n, (t) represent inde- pendent white gaussian noise of single-sided spectral den- sity N o (Ref. 1). If the tracking loop is a phase-locked loop preceded by a bandpass limiter, then the distribution on + as given by Lindsey (Ref. 2) using DSN parameters is where * 3x 1 + 0 . 3 4 5 ~ ~ r = 0.862 + 0.690 xy "= ++:)' and yo = 4. (It should be noted that the usual DSN pa- rameters are y = 1/400 and yo = 2.) 1% is the modified Bessel function of order k. P, represents the power in the 1.0 0.8 0.6 0.4 0.2 0.0 L??fL 0.0 t k Fig. 1. Plot of JZ, 2Jf, and 2Jf, showing division of power between fundamental and other components carrier, bLO the loop design bandwidth, and N o the single- sided spectral density of the noise. For the above signal, we find P, = PI: (k). 3. Error Rates for Various Detectors For convenience, let us define the random variables k, = $lTb cos + (t) dt where T b is the time per bit. If the data is extracted using only the component of r, (t) at frequency 0, namely, (P)" cos + (t) 21, (k) sin (ut + e) + 721 (t), 0 < t L Tb; o = wo, 01 JPL SPACE PROGRAMS SUMMARY 37-53, VOL. 111 1 77 then the problem is essentially to decide which of two signals is present. Hence, an incoherent phase receiver using orthogonal signals can be used to obtain a bit probability of error (Ref. 3) of 1 P i = E { f exp [ - 2 k$ R ] } where E is the expectation operation, R = STb/No, and S = 21: (k) P is the power in the data. If the data is extracted from the fundamental components of both rl (t) and r, (t), namely, (P)%cos+(t)2J1{k)sin(wt + 0) + n,(t), O < t < T b and (PI% sin 4 (t) 2Jl (k) sin ( w t + 0) + n, (t), 0 = iJJo,w1 then by using the doubly incoherent receiver discussed in Subsection 8, it is possible to obtain a probability of error of Pg= min E { M c ( / 3 ) e x p [ - ? h ( k $ + A $ ) R l } 0 ~ 5 ~ 1 where 1 - p ” and p is an arbitrary gain factor, 0 4 /3 4 1. We note that when p = 0, the doubly incoherent re- ceiver degenerates to the incoherent receiver so that we always have Pg L PL for a given index of modulation k. Since, as the index of modulation increases from zero, the amount of power in the data will increase, causing R to increase, while the amount of power in the carrier will decrease, causing p L to decrease, there will be an opti- mum value of k, corresponding to an optimum division of power. In particular, p L depends on where R = - = - . STb PTb 2J4(k) No No By letting 8 = 1/2bLoTb and &? = PTi,/No, we can write z = 2 & 8 J f ( k ) [It should be noted that Lindsey (Ref. 2) uses 8 l / b ~ o T b . ] 4. Extremely low Data Rates When the data rate is extremely low, corresponding to 8 << 1, it is appropriate (Ref. 1) to use the approxi- mations k+ = $ - ~ T ‘ ’ ~ ~ ~ + (t) d t E {cos +} and A+ = ks.’” sin+ (t) dt N E {sin +} Using Lindsey’s model for the density on 9 as given above, we find A + ” 0 1 78 JPL SPACE PROGRAMS SUMMARY 37-53, VOL. 111 The optimum value of p for the doubly incoherent receiver then occurs at /3 = 0 so that the doubly incoherent receiver reduces to the incoherent receiver with probability of error given by P," = Pi = Mexp(--Mq2R) The resulting minimum value of the probability of error is plotted in Fig. 2a versus W for several values of 6, while the optimum values of k are plotted in Fig. 2b and the resulting values of pL are plotted in Fig. 2c. In order that 1 oo 10-2 1 o - ~ - -Y ( P 1 o-6 1 o-8 10-lC T I the tracking loop acquire frequency lock, it is necessary to require that p L 1 6 . 5. Moderate Data Rates Moderate data rates occur when the phase does not 1 and the vary significantly over a bit time so that 6 approximations k@ = $lTb cos 4 (t) dt g cos 4 2. 1. 1. Y 0, 0. 0 / f- 1/11 7 1 80 1 at / f a" Fig. 2. Plots of behavior of incoherent receiver under the assumption of non-constant phase, showing (a) probability of error, Ibl optimal value of modulation index k, and (cl resulting value of p L I 200 JPL SPACE PROGRAMS SUMMARY 37-53, VOL. 111 1 79 and A$ = +JTb sin 4 ( t ) dt sin + are valid. Under these circumstances, we find the probability of error for the incoherent receiver is The minimum value of Pi is plotted in Fig. 3a, the optimum value of k to give this value of Pk is plotted in Fig. 3b, and the resulting value of pL is plotted in Fig. 3c. The probability of error for the doubly incoherent receiver is where We note that, when p -+ 1, we can evaluate c ( p ) by L’Hospital’s rule to find R 8 c(1) = 1 + - which is independent of 4. Combining this with the fact that the performance of the doubly incoherent detector can- not be better than the performance of the incoherent detector with + = 0, we find 2 1 exp (- R ) A P ; L + ( l + t) exp (- -j- 1 R ) so that P,” must be exponentially asymptotic to the optimum receiver performance for the given signaling scheme. The minimum value of P,” is plotted in Fig. 4a, the optimum value of k to give this value of P i is plotted in Fig. 4b, the resulting value of pL is plotted in Fig. 4c, and the optimum value of p is plotted in Fig. 4d. 6. In Between Rates For rates between those discussed in Subsections 3, 4, and 5, we expect the probabilities of error to fall some- where in between the probabilities of error obtained above. This would imply that as the rate increases from ex- tremely low to moderate, the probability of error for the incoherent detector would increase from the values in Fig. 2a to the much larger values in Fig. 3a. The proba- bility of error for the doubly incoherent detector, how- ever, would decrease from the values in Fig. 2a to the slightly lower values in Fig. 4a. The desirability of using a doubly incoherent receiver, which is somewhat more complicated to implement as opposed to the simpler inco- herent receiver, would, of course, depend on the exact probability of error for the rate under consideration. 180 J P l SPACE PROGRAMS SUMMARY 37-53, VOL. 111 1 oo 10-2 1 o - ~ - w CL 1 o-6 1 o-8 1 o-'( 25 20 15 10 5 0 40 80 \ \ \ \ 7 / / P 1/32 1 160 2( 2. c 1 .c 1.2 Y 0. f 0.4 ( I I Fig. 3. Plots of behavior of incoherent receiver under the assumption of constant phase, showing la1 probability of error, (b) optimal value of modulation index k, and (cl resulting value of pL JPL SPACE PROGRAMS SUMMARY 37-53, VOL. 111 181 1 oo 1 o-2 I o - ~ nu L 1 o-6 1 o-8 lo-]( 0 40 I \ 1.0 0.8 9 0.6 0.4 0.2 ) 1 Fig. 4. Plots of behavior of doubly incoherent receiver under the assumption of constant phase, showing (a) probability of error, (bl optimal value of modulation under k, (cl resulting value of pL, and (d) optimal value of ,8 182 JPL SPACE PROGRAMS SUMMARY 37-53, VOL. 111 7. Using the Second Harmonic The doubly incoherent detector may also be used to extract information about the data from the first harmonic of r, (t) and the second harmonic of r, (t); namely, and ( P ) s cos + (t) ZJ, (k) sin (ut + e) t n, (t), ( ~ 1 % COS + (t) I , (k) COS 2 + e ) + % (t), 0 = 6JO,W1 O < t L T b This would yield a probability of error of P: = min E {W c (p) exp [ - % k$ (R, + R,)] } O L f 3 6 1 where and R, = 2.E (k) a This yields an improvement in signal-to-noise ratio of about I;(k)/J:(k) over the incoherent receiver. An indica- tion of this ratio can be obtained from Fig. 1. For values of k near 1, the improvement is about 10%. 8. Description and Analysis of the Doubly Incoherent Receiver We assume two signals of the form and r l ( t ) = (P)]hsin[ksin(~t + e,) ++] +n,(t) T , ( t ) = (P)scos [ksin(Wt + e,) + +] + n, (t) , 0 < t L T ; o coo, o1 where the angles one-sided spectral density N o . and 8, are arbitrary and n, (t) and n, ( t ) are the independent white gaussian noise process of the The doubly incoherent receiver then consists of two sections of the form shown in Fig. 5, one with 0 = Wo and one with w = w ~ . The variable p is an arbitrary gain factor which is to be chosen to minimize the probability of error. If we define the random variables and then the output of the Wo section when sin& was transmitted is JPL SPACE PROGRAMS SUMMARY 37-53, VOL. 111 183 Fig. 5. Diagram of one section of the doubly incoherent receiver where u1 = ( z ) " k @ c o s O , + %, u3 = ($)Ih A+ cos e2 + n3, u2 = ( z ) " k , sin 8, + n2 u4 = (E)" sin e, + n, and the output of the m1 section when sinwot was transmitted is where Q1 = v: + v.: + p(v; + 0:) v1 = ml, u2 = m,, 0 3 = m3, v4 = m4 The noises ni and mi, i = 1 to 4, are mutually independent gaussian random variables of unit variance. Similar vari- ables are defined in a symmetrical way when s h a l t was transmitted. The estimate of which value of o was sent is taken to correspond to the section with the largest output. Thus, the probability of error is given by P$= min Pc(Qo i (if i = i, the result is trivial) then Thus, Theorem 1 shows that the order of any two successive stages may be interchanged, and thus any possible permu- tation of the stages may be realized, without changing the final output. Also, the commutivity shown implies that we need not keep track of order when discussing matrix products of the M:) . Theorem 2 186 JPL SPACE PROGRAMS SUMMARY 37-53, VOL. 111 Proof. For m = 1, we have Assume the result is true for m, then prove for m + 1: Thus, by induction, the result is true for any m between 1 and n. In particular, we see from Theorem 2, letting m = n, that Also, as in the previous article on this decoder, it can be shown that where ,P, and R, are defined inductively for n l 1 by and with P, = I , and P , = ( P i i ) . Here P11 = P,, = P,, = P,, = 1 and otherwise Pi j = 0; R, = H,. Thus, we see that con- necting the n decoder stages M g ) through M Z ) in any order whatever performs the operation H, x. Further- more, if the stages are connected in numerical order, MF) first, MP) second, etc., the output at any inter- mediate stage, say the jth stage, provides a decoder for Hi. Thus, the algorithm has multiple-mission capability. Reference 1. Bellman, Richard, Introduction to Matrix Analysis. McGraw- Hill Book Co., Inc., New York, 1980, C. Optimal Codes and a Strong Converse for Transmission Over Very Noisy Memoryless Channels, A. J. Viterbil 1. Introduction Wyner (Ref. 1) has obtained the following lower bounds on the asymptotic performance of the optimal codes for the additive white gaussian channel, where T is the message duration, R is the rate in nats/s, and C is the channel capacity: PE > exp { --T [ E (R) + o ( T ) ] } where C O L R 4 - 4 C 2 E ( R ) =-- R , (1) C ---AR C (2) The second bound on the probability of correct decision for rates above capacity is generally referred to as a “strong converse.” It is well known (Ref. 2) that equal-energy orthogonal signals are asymptotically optimum because they achieve the error probability ‘Consultant, University of California at Los Angeles. JPL SPACE PROGRAMS SUMMARY 37-53, VOL. 111 187 where E (R) is given by Eq. (1). We begin by showing, through an application of extreme value theory (Ref. 3), that for rates above capacity orthogonal signals yield where {zm} are independent normalized gaussian vari- ables, since the covariance matrix for orthogonal signals is the identity matrix. 1 - PE > exp{-T[ E*(R) + 0 In- Equation (6) can be rewritten as ( ;)]}. R > C (4) l - P E = e x p [ - T ( C + R)] which proves their asymptotic optimality2 above as well as below capacity. d X l I e x p [(2CT)%x] [F(x)IeRTdx We extend these results by showing the essential channels to the white gaussian channel and thus extend equivalence of all memoryless input-discrete very noisy the strong converse to this wider class of channels. exp [(2CT)%x] [F(x)]""'-ld [F(x)] (7) where 2. The Additive White Gaussian Channel Balakrishnan (Ref. 4) has shown that for any equal- energy, a priori equiprobable set of M signals used on the white gaussian channel, the probability of correct decision using the optimum (maximum likelihood) deci- sion rule is is the (cumulative) gaussian distribution. 1 - Pa = M-le-h E (exp [(2h)% max zm]} (5) where A = CT, C = S/N,, the ratio of received signal power to one-sided noise spectral density, while {z,} is variables with a covariance matrix whose elements are the normalized integral inner products among signals. We proceed to evaluate Eq. (7) by applying a tech- nique from extreme value theory due to Cram& (Ref. 3). Consider the transformation 1 L m L d l a set of M zero-mean, unit-variance, gaussian random 1 - f = F ( x ) (8) which has the inverse (Ref. 3) Let R = In M/T and restrict to orthogonal signals; = F-l (1 - e R T ) Eq. (5) becomes 1 - P , = e x p [ - T ( C + R)] 1 G m L e R T '-1) (6) Substituting Eqs. (8) and (9) into Eq. (7), we obtain *It has long been conjectured that regular simplex signals are globally optimum for alI rates on the white gaussian channel, and this obvi- ously implies the asymptotic optimality of orthogonal signals. However, only the local first- and second-order conditions of optimality of regular simplex signals have been shown (Ref. 4) and all attempts at proving global optimality at all rates have met with failure. 188 JPL. SPACE P R O G R A M S S U M M A R Y 37-53, VOL. 111 The last integral is bounded from below by {+- ~ ) " I } q (y) corresponding to the likelihood function for a zero- signal hypothesis reduces Eq. (14) to Eq. (5) (cf Helstrom, Ref. 5). For memoryless time-discrete channels, for R > C. Thus, it follows that for orthogonal signals on the white gaussian channel N P ( Y I x ' ~ ) ) = IT p ( ~ n l ~ ~ m ) ) n = 1 1 - PE > exp (- T { [ ( R Y ~ - (c)%I~ + o - (InyT)D> and specializing to the independent output measure, R > C (11) N which proves Inequality (4). 4tY) = rI (Yh) n=1 3. Input-Discrete Very Noisy Memoryless Channels The error probability expression for the white gaus- sian channel, Eq. (5), can be generalized to any memory- less finite-dimensional (or time-discrete) channel. For any set of M equally likely messages and a maximum likeli- hood decision rule, for any set of N-dimensional channel input sequences {x(i); i = 1,2, . . * , M } , and for y, an N-dimensional output sequence, we have where Then, since Y u Df=Y, the N-dimensional output space, we can rewrite Eq. (12) as j=1 where q(y) is an arbitrary probability measure on the output space and E , is the expectation with respect to this measure. Substitution of the appropriate likelihood functions for the white gaussian channel and for the Eq. (14) becomes where Such channel is said to be very no@ if where E (x, y) + 0 uniformly in x and y, and q (yn) is an arbitrary probability density or distribution. It follows that for any n and m We now restrict attention to a discrete input alphabet of K symbols, so that each xLm) is taken from the set {x1,x2, . . , xK}. For this class of channels, we need only consider the class of fixed composition codes, which are characterized by the property that each code word is some permutation of the same sequence of N symbols, since Shannon, Gallager, and Berlekamp (Ref. 6) have shown that the asymptotic performance of the best code in the restricted subclass is the same as for the best code in the unrestricted class. Thus, given that the relative frequency of the symbol x k in each code word of the fixed composition code is A number of occurrences of xk 2 k = 1,2, * . . ,K N pk JPL SPACE PROGRAMS SUMMARY 37-53, VOL. Ill 189 we have that the means of the random variables zm(y ) relative to the output measure are all equal to where we have used Eq. (18) and also Condition (17) to neglect .all terms above quadratic in E. Similarly, since the channel is memoryless, vary [x, ( y ) ] = N z = 1 But again neglecting terms above quadratic in E, Also, the capacity of a very noisy input-discrete mem- oryless channel is given by X y) - dy II Thus, choosing the relative frequencies {pk} correspond- ing to the maximizing distribution for capacity, we have from Eqs. (19), (20), and (21) Furthermore, since x, ( y ) is the sum of N independent random variables, by the central limit theorem it must be asymptotically gaussian. In fact, if we normalize by letting it follows from Eqs. (23) and (24) that v m ( y ) is a zero- mean, unit-variance, random variable and by the Berry- Esseen theorem (cf LoBve, Ref. 7) we have that Pu(x) , the distribution function of the normalized variable v,, differs from the normalized gaussian distribution F (x) by no more than when we neglect all terms in E of order higher than quadratic. Thus, all the variables v, are asymptotically gaussian with zero means and unit variances. Applying Eq. (25) to Eq. (15) and letting R’ = (In M ) / N nats/symbol, 1 - PE = exp [ - N (R’ + C)] This formula is identical to the form of Eq. (6) for orthogonal signals on white gaussian channels, except that the variables om are not necessarily independent. However, for rates below capacity it is well known (Ref. 6) that the error probability for the best code on memoryless very noisy input-discrete channels behaves asymptotically exactly as that for orthogonal signals in the white gaussian channel [i.e., Expressions (1) and (3) hold with T replaced by N and R replaced by R’]. For 190 JPL SPACE PROGRAMS SUMMARY 37-53, VOL. 111 this to be the case, the best code on memoryless very noisy input-discrete memoryless channels must asymp- totically lead to independent 0, (y) in Eq. (26), since any other covariance matrix would lead asymptotically to a greater PE below capacity. Thus, Eq. (26) reduces to Eq. (6), and the asymptotic behavior above capacity given by Expressions (2) and (4) must hold also for the best code on this class of channels. References 1. Wyner, A. D., “On the Probability of Error for Communication in White Gaussian Noise,” IEEE Trans. on Inform. Theory, Vol. IT-13, pp. 8590, Jan. 1967. J P l SPACE PROGRAMS SUMMARY 37-53, VOL. 111 2. Wozencraft, J. M., and Jacobs, I. M., PTinciples of Communi- cation Engineering. John Wiley & Sons, Inc., New York, 1965. 3. Cram&, H., Mathematical Methods of Statistics, pp. 374-376. Princeton University Press, Princeton, N. J., 1946. 4. Balakrishnan, A. V., “A Contribution to the Sphere-Packing Problem of Communication Theory,” J . Math. Analysis and App l . , Vol. 3, No. 3, pp. 485-506, Dec. 1961. 5. Helstrom, 6. W., Editor’s Note, IEEE Trans. Inform. Theory, Vol. IT-14, No. 2, p. 311, Mar. 1968. 6. Shannon, C. E., Gallager, R. G., and Berlekamp, E. R., “Lower Bounds to Error Probability for Coding on Discrete Memoryless Channels, I,” Infomn. Contr., Vol. 10, No. 1, pp. 81-83, Jan. 1967. 7. Lodve, M., Probability Theory, p. 288. Van Nostrand, New York, 1965. 191

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