160 IRE TRANSACTIONS ON COMMUNICATIONS SYSTEMS June

CONCLUSIONS

From .the results of performance calculations presented here, it appears that scatter circuits with severe frequency- spreading and time-dispersion can be used effectively for commun.ication purposes. There will be a need for com- munication techniques that depart appreciably from conventional practice, but the resulting information- handling capacity per unit of received power should compare favorably with that normally achieved on other kinds of channels. Larger frequency allocations will be

needed for eEicient use of these scatter channels than are required on many other types of communication circuits.

ACKNOWLEDGME:NT The author gratefully acknowledges the assistance of

D. G. Drath in programming and running the computa- tions necessasy for this report. In addition, appreciation is expressed to Professors 8 . M. Peterson and V. R.. Eshleman of Stanford University, and to Dr. F. F. Fulton, Jr. for helpful discussions concerning the use of scatter- communication channels.

The Influence of Fading Spectrum on the Binary Error Probabilities of Incoherent and Diiff erentially Coherent

Matched Filter Receivers*

Summary-Previous derivations of the influence of fading on the error probabilities of binary data transmission systems have assumed that the fading rate is so slow that fluctuations within a bit may be ignored. This slow fading assumption is removed .in the present paper which derives general expressions for the binary error probabilities of incoherent and differentially coherent matched filter receivers employing post-detection diversity combining. In the analysis it is assumed that the transmitted signals occupy a band- width much smaller than the coherence bandwidth of the medium so that flat, fading may be assumed. In addition, it is assumed that the amplitude and phase fluctuations produced by the medium have the same statistical character as those of narrow-band Gaussian noise.

The general analytical results are specialized to the cases of frequency shift keying using incoherent detection, and phase shift keying using differentially coherent detection, and to the cases of exponential and Gaussian fading correlation functions. For these special cases, signal-to-noise degradation curves are given as a function of fading bandwidth. The PSK system is degraded more rapidly with increasing fading bandwidth than is the FSK system. Curves are given which show the error probabilities and correspond- ing fading bandwidths for which the noncoherent FSK and the differentially coherent PSK systems break even. For lower error probabilities or higher fading bandwidths, the FSK system becomes superior to the PSK system in the sense of being able to provide the same error probability with less signal-to-noise ratio.

The existence of an irreducible error probability is demonstrated for the incoherent and differentially coherent matched filter re- ceivers. Thus, in general, an increase in transmitted signal power cannot reduce the error probability below a certain value depending upon the fading spectrum and order of diversity. Theoretical

* Received January IO? 1962. This work reported in this paper was performed a t ADCOM, Inc., under subcontract with TTT Com- munication Systems, Inc., Paramus, N. J.

f ADCOM, Inc., Cambridge, Mass.

curves of irredlucible error probability are given for the incoherent FSK and differentially coherent PSK systems.

An important result of the analysis is that the shape of the fading spectrum can make a significant difference in the amount of signal- to-noise degradation.

The results of the analysis also indicate that care must be exer- cised in employing a slow fading assumption since, if low bit error probabilities are desired, significant degradations in per- formance can occur even though the fading rate is quite low relative to the data rate.

M I. INTRODUCTION

,4Ff PAPERS have appeared in the past dealing wit,h the evaluation of error probabilities in the transmission of binary signals through noisy,

fading communication channels. Invariably, it is assumed that, the received signal strength, although fluctuating, may be regarded as constant over the duration of a bit. Voelcker has considered the effect of nonzero fading band- widths on differentially coherent PSK. Although he did not consider fading within a bit, he did allow a (discrete) change in signal amplitude and phase between adjacent bits. He t h w obtained approximate expressions for the bit error probabilities in differentially coherent PSK. It appears, intuitively, that this discrete fading approxima- tion should be accurate for sufficiently small (but nonzero) fading bandwidths. However, our analysis, which is exact,

IEE , pt. B, vol. 107, pp. 31-38; January, 1960. 1 H. B. Voelcker, Phase shift keying in fading channels, PTOC.

1962 Bello and Nelin: InJluence of Fading Spectrum on Binary Error Probabilities 161

indicates that Voelckers approximation is invalid for some fading correlation functions.

The present analysis will be primarily concerned with the determination of binary error probabilities for two general classes of matched filter receivers. These are the incoherent and the differentially coherent matched filter receiver. The incoherent receiver is well known and employs matched filters followed by envelope detection. The differentially coherent matched filter receiver is a generalization of the differentially coherent PSK to the transmission of arbitrary binary waveforms. This generali- zation appears not to have been pointed out before.

Although the zero fading-bandwidth restriction will be removed in the present analysis, i t will be assumed that the transmitted signal is well within the coherent band- width of the medium, Le., that the fading is not frequency selective. In addition, it will be assumed that the trans- mission medium imparts to the transmitted signal an amplitude and phase modulation of the same statistical characteristics as those of narrow-band Gaussian noise.

The above assumptions may be represented in a compact form by using the complex representation of waveforms. Thus, let s( t ) , the received signal for a unity gain channel, be represented as

s(t) = Re {S(t)efunf] (1)

where Re { } is the usual real part notation and w0 is the center frequency of the received signal. The com- plex time function S( t ) , called the complex envelope of s( t ) , has a very simple interpretation when s ( t ) is a narrow- band process: the magnitude of S(t) is the conventional envelope of s( t ) and the angle of S(t) is the phase of s ( t ) with respect to carrier phase wot. A unique definition of S(t) in the non-narrow-band case is obtainable via the use of Hilbert We shall not consider such a generalization since i t is not needed for the analysis in this paper.

Let y k ( t ) , the fading received signal in the kth diversity receiver, be denoted by

y k ( t ) = Re { Yk(t)eiuof } (2)

where Y,(t) is the complex envelope of y k ( t ) . Then our assumptions with regard to the channel fading may be simply embodied in the equation

Yk(t) = Z,(t)S(t) (3)

where Z,( t ) has the statistical properties of the complex

McGraw-Hill Book Go., Inc., New York, N. Y.; 1953. 2 P. M. Woodward, Probability and Information Theory,

IRE TRANS. ON INFORMATION THEORY, vol. IT-4, pp. 53-57; March, J. Dugundji, Envelopes and pre-envelopes of real waveform,

1958. R. Arens, Complex processes for envelopes of normal noise,

September, 1957;; IRE TRANS. ON INFORMATION THEORY, vol. IT-3, pp. 204-207;

93, pp. 429-457; November, 1946. 5 1). Gabor, Theory of communications, J . IEE, pt. 111, vol.

envelope of narrow-band Gaussian noise. Thus, in complex notation, the fading in the transmission channel is repre- sented as multiplication by a complex Gaussian process. The utility of the complex notation lies not only in the simplicity with which one may represent the operation of a non-frequency-selective fading channel, but also in the fact that error probabilities can often be evaluated without ever leaving the compact complex notation.

The usual approach to the evaluation of binary error probabilities for conventional systems when fading is present is to evaluate first the bit error probabilities as- suming the signal strength is constant over a bit and then to average this probability with respect to a probability density function for signal strength. This method pre- supposes the existence of slow fading. In the method of approach used here, the error probabilities are calculated in a one-step, rather than two-step, procedure, and as a result, no assumption need be made a t the outset as to the rate of fading involved.

The error probabilities will be evaluated under the as- sumption that the receivers use linear post-detection combining and that the fading variables { zlc( t )} are mutually independent.

11. ERROR PROBABILITIES FOR INCOHERENT MATCHED FILTER RECEIVERS

A simplified block diagram of an incoherent matched filter receiver is shown in Fig. 1. Two matched filters are indicated, each matched to one of the two possible wave- forms that are successively repeated to form the trans- mitted binary communication signal. The outputs of the matched filters are envelope detected and subtracted. Diversity combining is performed by summing the outputs for all the diversity receivers. It is assumed that bit synchronism exists. Thus the output of the subtractors may be assumed to be sampled optimally, i.e., at the trailing edges of successive input pulses. The output of the sampler is shown compared to a nonzero threshold K.

The fading statistics are assumed stationary; i.e., Z,(t) is assumed to be a stationary complex-valued Gaussian process. Thus the bit error probability will be independent of bit location. It is then sufficient to evaluate the error probability for a received binary waveform occupying the time interval 0 < t < T , where T is the bit duration. Fig. 2 shows, in block diagram form: the detailed mathe- matical operations involved in determining the receiver output decision (1 or 0) for a received binary wave- form occupying the time interval 0 < t < T. To simplify the subsequent derivations and discussion, complex representations have been used for all waveforms, as may be seen in Fig. 2 .

Let the complex envelope of the additive Gaussian noise in the kth diversity channel be defined as N k ( t ) . Then the complex envelope of the received signal is Z, ( t )S ( t )+N, ( t ) . In the absence of additive and multiplicative noise ( Z k ( t ) = 1 will signify the absence of mult,iplicative noise),

162 IRE TRANSACTIONS ON COMMUNICATIONS SYSTEMS June

I W Y l Ill*.L+

0

Fig. 1-Block diagram of incoherent matched filter receiver.

Fig. 2-Functional diagram of incoherent matched filter receiver.

the complex envelope of the received waveforms will be defined as S,(t) and So(t) depending on whether the corre- sponding symbol was "1" or "O", respectively. At the trailing edge of the input pulse, the complex envelopes a t the outputs of the two matched filters are given by

,T

where t,he asterisk is the usual complex-conjugate notation. When S(t) is specified, Z,(t)S(t) + Nk(t ) is a complex- valued nonstationary Gaussian process. Thus U , and V , are complex normal random variables since a linear opera- tion on a Gaussian process always produces a Gaussian process. The quantities IUkj2 and IV,Iz are the squared envelopes a t the outputs of filters 1 and 0, respectively, evaluated at the trailing edge of the input pulse. Thus, although simple envelope detectors are shown in Fig. 1, square-law envelope detectors are used in the analysis. This change is made for ease of analysis, and actually results in no loss in generality for the zero threshold case (K = 0) or the "on-off" transmission case (S,(t) = 0, say).

where p o is the probability that a ((I." is printed given that a "0" is sent and pl is the probability that a "0" is printed given that a '' 1" is sent. In a previous paper6 the authors .have computed the probabilities Pr (q > 0) and Pr (q < 0) for the more general quadratic form q given by

It was shown (see (47)-(50) of Bdlo and Nelin') that

where the parameter is given by'

- P I 1 + P o 0 det P

Y = - (9) ,i[p" + Pool ' 1 + ~ 1 1 + p o n 2 det P d e t P 2 d e t P

in which p,, and poo are the diagonal elements and det P is the determhant of a matrix P given by the product of the matrix of the elementary quadratic form in (7) by the moment matrix for the pair U,, V,. This momellt :matrix is independent of k since it has been assumed that the fading statistics are the same for all diversity channels. Thus, P is given by

== k;; :::I = [::; :::][: ;] (lo) where

m11 = I Uk 1 2 , m10 = U*,V,, m o o = I V k l 2 (11) -- --

in which the overline indicates an ensemble- average. In our prokllem a = 1, b = - 1, and c = 0 which leads

to the simpler expression for y

Y = I- 2(mll - moo) (12) v(m11 - f 4(mIlmOO - I m10 1') - ( ~ I I - m o o )

From Fig. 2 we find that the input to the threshold To evaluate y we need the moments device is a random variable q given by

L

q = c [ I Uk l 2 - I v, I 7 (5) mT8 = kT s,' LZ*,(tl)S*(t,) + N*,(t,)][Z,(t,) + N k ( t , ) ] k=l

where L is the order of diversity. For simplicity of presen- -S?(tJs,(t,) dt, dt,; r , s = 0, 1. (13) tation we will assume a zero threshold, although the inclusion of a nonzero threshold involves no analytical difficulties. The two error probabilities of concern are then 6 p. A. Bello and B. D. p ~ ~ l i ~ , "predetection diversity

p o = Pr [ q > O/S(t) = Xo(t); 0 < t < T ] with selectively fading channels," IRE TRANS. ON COMMUNICATION

p l = Pr [ q < O/S(t) = S,(tj; 0 < t < 2'1 ' Symbol defi.nitions have been changed to conform to the present paper's definitions. (6) SYSTEMS, vol. CS-lo, pp. 00-00; March, 1962.

1962 Bello and Nelin: InJluence of Fading Spectrum on Binary Error Probabilities 163

Upon carrying through the averages in (13), defining the fading autocorrelation function

and making the change of variable r = tz - t,, one finds that

mv8 = 1-1 R ( T ) L ( . ~ > d7 + 4NoE,, L ( 7 ) = l S*(t)X,(t)S(t + 7)ST(t + 7) d t ;

T , s = 0, 1 (15)

E,, = S*,(t)S,(t) d t LT where N o is the spectral intensity of the additive noise.

In order to evaluate the error probabilities p o and pl it is necessary to evaluate y and thus the moments mra for the two cases S(t) = &(t ) , 0 < t < T and S(t) = S,(t), 0 < t < T. Let these values of y be denoted by yo and y,, respectively. Then po is obtained by substituting y = yo in (8a) and p , is obtained by substituting y = y , in (8b). Thus

po = FL(yo), P , = G L ( Y ~ ) . (16)

It is quickly verified that the functions FL(y ) and GL(y) are related in the following way:

In general p o # p,. Thus although binary symmetric operation may take place for slow fading rates, an increase in fading rates may bring nonsymmetric operation. How- ever examination of the moment expression reveals one interesting case in which binary symmetric operation exists a t all fading rates, namely, when the fading spectrum is symmetrical and the binary waveforms have the same envelopes, i.e., 1 So(t) I = 1 &(t ) I.

When the fading is slow enough so that R(T) becomes essentially constant over the interval - T < t < T , one may set

R(7) = R(0) = 1 z, ( 2 = 2 2 in (15) which results in

mrs = 2a:t ~x , ( t )X*( t ) dt S*,(t)X(t) dt s, s,

+ 4No 1 x:(t)x,(t) d t . (19) If the signals are of equal energy it is readily shown that for zero fading bandwidth p , = p,. Then we need consider only the case S ( t ) = &(t ) , which yields

mAo = 8aE2 1 p l 2 + 8NoE mi, = 8a2Ez + 8NoE (20) mio = 8a2Ezpp* + 8NJ@*

where the superscript 1 denotes that mrs is being evaluated for S(t) = S,(t), and

where E is the integral of the square, Le., the energy, of the real binary pulse waveforms, and p is the correlation coefficient between the mark and space waveforms. Using (20) in (12) we find that y1 is given by

in which

a2E Signal Power in Dat.a Channel = = Noise Power in Data Channel

(23)

where we define the noise power in the data channel as the noise power in a rectangular band-pass filter of unity gain and bandwidth 1/11. Minimum error probability is obtained for 101 = 0, i.e., uncorrelated binary signals. I n this case y1 = p. Substitution of y1 = p into (8b) yields the slow fading error probability for orthogonal, equal energy binary signals in a matched filter detection system employing square-law envelope detection and linear diversity combining (or square-law combining as it has been called). This expression has previously been derived by Pierce (identify our p with his SoT/no).

For the case of general fading and general binary signals one may determine p , by using an equivalent signal- to- noise ratio p, in the zero fading-bandwidth expression for p , derived by Pierce, which in our notation is given by G, ( p ) . This equivalent SNR is given by

P1 = Y1 (24)

From (17) it follows that to determine p o for general fading one may use an equivalent SNR po in the slow fading expression for p,, where

Po = -. - 7 0 1 + Yo

Binary symmetric operation results when po = p, . A study of the expressions po and p1 in terms of the

relevant moments shows that as the additive noise vanishes (No -+ 0) and thus the SNR p -+ the equiva- lent SNRs approach maxima which are finite. Thus there exists an irreducible error probability (or if one wishes an irreducible noise-to-signal ratio) caused by the fading. This might have been expected since the perturbation caused by fading, being due to a multiplicative rather than additive mechanism, are insensitive to signal ampli- tude changes.

shift keying, PROC. IRE, vol. 46, pp. 903-910; May, 1958. * J. N. Pierce, Theoretical diversity improvement in frequency

164 IRE TRANSACTIONS ON COM&IUNICATIONS SYSTEMS June

We will now specialize the general expressions above equivalent SNR po. This expression is quite cumbersome for the case of FSK transmission, i e . , and will not be presented here. A much simpler expression

valid for 2~2; '

1966 Bello and Nelin: Influence of Fading Spectrum on Binary Error Probabilities 165

lo-'

10-1

I ~ - 1 0 ' ~ $ 10-1 I 9 8 3.10-5 a

B RELATIVE FADING BANDIlDTH BT

. ..

Fig. 3-Signal/noise degradation for incoherent FSK with fre- quency spacing equal to reciprocal bit duration and 10-4 error probability; E = exponential fading autocorrelation, G = Gaus- sian fading autocorrelation, L = order of diversity.

I I I I I J 3.10.' 10-1 3.10-l 1

RELATNE FADWG BANDWlDTH BT 3

Fig. 4-Signal/noise degradation for incoherent FSK with infinite frequency spacing; E = exponential fading autocorrelation, G = Gaussian fading autocorrelation.

beyond that required in the zero fading-bandwidth case to attain the same error probability a t a normalized fading bandwidth b. In Fig. 3 the FSK system is assumed to have frequencies spaced 1/T apart, i.e., n = 1. It is interesting to note that an exponential fading correlation function yields significantly worse degradation than a Gaussian fading correlation function possessing the same half-power bandwidth. Because of the presence of an irreducible error probability, the degradation goes t80 infinity when the fading bandwidth and order of diversity are such that the irreducible error probability is Fig. 4 shows degradation curves for exponential and Gaussian fading correlation functions for the case of infinite separation between the mark and space frequencies. These curves are valid for all error probabilities and all orders of diversity because po becomes directly proportional to p when n = m . A comparison of Figs. 3 and 4 demon- ,strates that wide spacing of the mark and space frequencies

RELATNE FADING BANDmDTH BT

,Fig. 5-Irreducible error probability for incoherent FSK with fre- quency spacing equal to reciprocal bit duration; E = exponential fading autocorrelation, G = Gaussian fading autocorrelation, L = order of diversity.

is beneficial in reducing degradation caused by nonzero fading bandwidth, especially for the lower orders of diversity. Fig. 5 present's curves of irreducible error prob- ability as a function of relative fading bandwidth for the FSK system employing a mark-space frequency separation of l/T.

111. ERROR PROBABILITIES FOR DIFFERENTIALLY COHERENT MATCHED FILTER RECEIVERS

A simplified block diagram of the differentially coherent matched filter receiver is shown in Fig. 6. It is assumed that only two possible pulses S , ( t ) , S,(t) are transmitted as with the incoherent matched filter receiver. However, the transmitted bit is encoded into the change or lack of change of successive adjacent pulses. Thus a trans- mission of the pairs S,(t), So(t) or S,(t), &( t ) denotes a "O", say, while the pairs S , ( t ) , S , ( t ) and S,( t ) , S,(t) denote a "1." The receiver consists in part of two separate coherent matched filter receivers with filters matched to waveforms X, and So but differing by an input delay of one baud duration. Thus, the two receivers are operating upon adjacent bauds. The actual detector output is obtained by using the output of one coherent receiver as a reference for the other. This is done, as shown in Fig. 6, by a multiplication followed by extraction of the base- band component. In the special case when the pulses So(t) , X, ( t ) are =k1 over the baud duration, the differ- entially coherent system specializes to the differentially coherent phase-reversal Kineplex system.

Fig. 7 shows the detailed mathematical operations in- volved in determining the kth diversity receiver output for a pair of received binary waveforms occupying the

166 IRE TRANSACTIONS ON COMMUNICATIONS SYSTEMS June

U

Fig. 7-Functional diagram of differentially coherent matched filter receiver.

Since the quadratic form (37) is the special case of (7) that results when we set a = b = 0 and c = 1, me can determine each of t.he four Probabilities in (39) by using

Fig. G-Block diagram of differentially coherent matched filter (8) with appropriate values of y. When a = b = 0, c = I, receiver. the general expression (9) for y simplifies to

J o J o

time intervals - T < t < 0 and 0 < t < T: Again the use of complex notation simplifies the presentation consider- ably. The output of the diversity combiner is a quadratic form q with

L q = [ukv*, + u*,vk] (37)

k=l

where

uk = j r [zk(t)s(t) + Nk(t)][sT(t) - S%(t)] dt v, = 1' [zk(t - T)X(t - T ) + Nk(t - T ) ]

-[ST(t) - S*,(t)] dt. (38)

Because the source bit is encoded into changes in successive transmitted pulses, the expressions for error probability are somewhat more involved than in the previous section, namely,

p o = 3 Pr [q > O/S(t) = So(t>, S( t - T ) = S,(t); 0 < t < T ]

+ 4 Pr [p > O/S(t) = Sl(t), S(t - T ) = S,(t); 0 < t < TI (39)

S(t - T ) = So(l); 0 < t < TI p , = +Pr [ q < O/S(t) = So(t),

+ 4 Pr [ q < O/S(t) = S,(t), X(t - T) = S1(t); 0 < t < TI

where it has been assumed t,hat 0, 1 have equal a pyiori probabilities.

.[S1(t,) - S,(t,)][S:(tJ - S*,(tJ] dt, dtz. (41)

Upon carrying through the average inside the integral in (41) and making the change of variable T = tz - t,, one finds tha.t

m,, = [ - : Z ~ ( T - s!!' + rT)p7..(r) d r + 4N0 d A,, (42) where

A,, = i 1 for r = s 0 for r # s IL,,(T) = lT [s,(t) - so(t)ls*(t - rT)

*[sT(t $- T ) - s*,(t -f T)]s(t f 7 - S!!') d t

d = lT I Sl(t) - S,(t) l 2 dt. (43) In order t80 evaluate the error probabilities p o and pl

it is necessa:ry to evaluate y for the four cases resulting when S(t) a:nd S(t - T ) become equal to S,(t) or So(t) for 0 < t '

1962 Bello and Nelin: Influence of Fading Spectrum on Binary Error Probabilities 167

initial analysis that only the case of antipodal signals, i e . , So(t) = -Sl(t), yields binary symmetric operation independent of fading spectrum.

When the fading is slow enough so that R(T) becomes essentially constant over the interval -2T < t < 2T one may set R(T) = 2u2 in the expressions defining the moments. This results in the zero fading-bandwidth moment expressions

m,, = 2a2 lT [s1(t) - ~ , ( t ) ] ~ * ( t - r ~ ) dt .I [ST(t) - S*,(t)]S(t - ST) d t

+ 4Ai0 a,, I 1 X,(t) - So(t) I d t . (45) It is readily determined by using (45) in (40) that for

zero fading bandwidths

Yo0 = 7 1 1 , Yo1 = 7 1 0 (46)

when the binary signals have equal energy. Moreover, it may readily be shown that binary symmetric operation results in the equal energy, zero fading-bandwidth case when the correlation coefficient (see (21)) is real. As- suming equal energy signals, real p, and zero fading bandwidth

po = pl = G t i ~ l l ) (47) The relevant moments are

mi: = 8u2E2(1 - P ) ~ + 16N0E(1 - P) = m:: m:: = 8a2E2(1 - P ) ~

(48)

which, from (40), leads to

Y11 = 2P(l - PI. (49) The superscripts in (48) denote the fact that S ( t ) = Sl( t ) , S ( t - T ) = S l ( t ) for 0 < t < T, and the input SNR p is given by (23). Minimum error probability is attained for @ = -1, i.e., antipodal signals. In this case yll = 2p. Thus antipodal equal energy signals and differentially coherent matched reception yield 3-db signal-to-noise improvement over orthogonal equal energy signals and incoherent matched filter reception.

For the case of general fading and general binary signals, one miy determine p l and pa by using (equivalent SNRs in GL(p), Pierces expression for error probability in noncoherent FSK reception with square-law diversity combining and slow fading (zero fading bandwidth). Thus upon defining the four equivalent SXRs

P11 = 7 1 1 Po0 = Yo0

lo This has been proved by J. G. Lawton, Theoretical error rates of differentially coherent binary and Kineplex data trans-

334; February, 1959. mission systems, PROC. IRE (Correspondence), vol. 47, pp. 333-

one h d s

P O = 3LG:GL(P1o) + Gt(~od1 PI = 3[GL(P11) + GL(~oo)l. (51)

From an examination of the expressions for phi; h, I = 0, 1, in terms of the relevant moments it can be shown readily that as the additive noise vanishes (No -+ 0) and thus the input SNR p ---f m, the equivalent SNRs ap- proach maxima which are finite. Thus, just as for the incoherent detection case, there exists an irreducible error probability caused by the fading.

We shall consider now the special case

which defines the DPSK or phase-reversal Kineplex system. Since this pair of binary waveforms is antipodal, binary symmetric operation will exist for all fading cor- relation functions. It is readily demonstmted that for antipodal signals p l l = poo. Thus, the error probabilities for DPSK are given by

PO = pl = G ~ ( ~ 1 1 ) . (53)

To calculate p l l we need the moments mt:, which in turn require the p functions (see (43)). It is readily determined that the p functions are

where, in (54) the superscript on .ptt signifies that S(t) = S(t - T ) = &(t) for 0 < t < 1.

For the exponential fading correlat.ion function (28), substitution of (54) in (42) shows that

Substitution of the moments (55) in (40) gives the required equivalent SNR p l l . We present here only the following approximate expression obtained by expanding t.he ex- ponential in (55) ill a Taylor series,

P11 = 2P

1 + p43rb/3 2ab

1965 Ames, et al.: Communications via Rock Strata 169

[41

[51

[71

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McGraw-Hill Book Co., Inc., New York, N. Y., pp. 51-52; M.. B. Dobrin, Introduction to Geophysical Prospecting, 1933. R. A. Daly, Igneous Rocks and the Depths of the Earth, McGraw-Hill Book Co., Inc., New York, N. Y., pp. 51-52; 1933.

Determining Some Reservoir Characteristics, A.I.M.E., G. E. Archie, The Electrical Resistivity Log as an Aid in

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University Press, Cambridge, Mass., ch. 7; 1956. R. W. P. King, The Theory of Linear Antennas, Harvard

[SI R. W. P. King, Electromagnetic Engineering, McGraw-Hill

[9] J. T. deBettencourt and R. A. Sutcliffe, Studies in Deep Strata Book Co., Inc., New York, N. Y., vol. 1, Appendix 11; 1945.

Radio Communications, Raytheon Company, Norwood, Mass., Final Report under Contract No. AF19(604)-8359 to AF Cambridge Research Labs., October, 1962; U. S. A F Rept. NO. AFCRL-63-60.

1101 See [9], Section I1 and Appendix F. [ l l ] R. K. Moore, The Theory of Radio Communication Between

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[12] For a center fed dipole of length 2h

where I(z) is the current distribution on the antenna, I (0) the input current, and k the complex phase constant of the medium.

Correction

P. A. Bello and B. D. Nelin, authors of The Influence of Fading Spectrum on the Binary Error Probabilities of Incoherent and Differentially Coherent Matched Filter Receivers, which appeared in the June, 1962, issue of these TRANSACTIONS, have called the following to the attention of the Editor:

On page 162, in (10): pf, should be pol; and in (13): [zk(t2) + Nk(tZ)]Sf(tz)Ss(tl) should be

[ z k ( t Z ) s ( t 2 ) + Nk(62)]S?(tZ)Sr(tl)- On page 163, in (15): S*(t)S,(t)S (t + T)S: (~ + T)

should be

S*(t)s,(t)S(t + T)S:(t + 7); andin (19): Ji h,(t)S*(t) dt J A #:(t)s(t) dt should be Jh S,(t)S*(t) dt Ji Sr(t)S(t) dt.

On page 164, the last sentence of the left column should be: The expression for the equivalent SNR,, is obtained by substitution of the moments (30) in (12) and then applying (25). In the expression for m:, in

(36): -a($ - +n2rr2) should be -u(+ + $n2rZ). On page 166, in (38) and Fig. 7 : Interchange uk and vk.