A Gaussian Channel With Slow Fading

8/12/2019 A Gaussian Channel With Slow Fading

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35ORRESPONDENCEprobability of error receiver, and since finally p = l/2 whenx = 0, we have the identity

from which (5) follows.Using the inequality (4) and the identity (5) in (1) givesthe asserted inequality (3).ROBERT 0. HARGERDept. Elec. Engrg.University of MarylandCollege Park, Md. 20742

REFERENCES[l] J. N. Pierce, "Theoretical diversity improvement n fr equency-shift eying,Proc. IRE, vol. 46. pp. 903-910.May 1958.[2] J. M. Wozencraft nd I. M. Jacobs, rinciples of Communication Engineering.New York: Wiley, 1965,sec. 7.4.

A Gaussian Channel With Slow FadingAbstract-An interleaved fading channel whose state is knownto the receiver is analyzed. The reliability function E(n) is obtainedfor rates R in the range R, 5 R 5 C. The capacity s shown to beC = EA { 4 ln (1 + A%)) where A is a factor describing the fadingmechanism and u is the signal-to-noise ratio per dimension.

INTRODUCTIONThis correspondence is concerned with the analysis of afading channel that can be described as a Gaussian channelwith a slowly varying signal level. The following assumptionsare made.

The fading is slow enough to make it possible for the re-ceiver to observe the instantaneous value of the signallevel.Scrambling units are used at both ends of the channel. Thescramblers, which are regarded as parts of the channel,are constructed to make it possible to neglect the effectof memory in the channel.The channel is band limited, making a vector notationnatural.

As a consequence, our channel is a time-discrete amplitude-continuous memoryless channel. For each transmitted symbol X,the receiver observes a pair of symbols Y and A, where A is themomentary signal level. The channel-transition probabilityfunction p(y, a ] Z) is given byp(y, a 1 z) = p(u)(2~)-~ exp ( -*(y - a$}. 0)

Where explicit assumptions about the probability distributionof the fading variable A are used, we assume A to be Rayleighdistributed,p(u) = 2ae-a a 1 0. (59

GENERAL EXPRESSION FOR THE RELIABILITY FUNCTIONSuppose each input signal vector x = (x1, x2, **. , 2,) isconstrained to satisfy an energy constraint of the formxv f(x,) < 0, where f(z) = ~2 - 7. For R, 2 R 2 C, theManuscript eceivedMmoh 3, 1969; &Bed July 18, 1969.Thia work w&sdone * R. G. Gallager,nformation Theory and Reliable Communication.New Yorkat t he Researchnstitute of National Defence,Sweden. Wiley, 1968,pp. 323-333.

reliability function E(R) is given by*E(R) = max ~-UP, p(x):), 4 - PRI,

where in our case,J-UP, Pc4, 4

S(s ) +P= -1n p(x)p(y, a 1 x)~(+) exp rj(x) dx du da(3and where the maximum is over all input probability distri-butions p(x), all p E [0, I], and all r 2 0. For the Gaussiachannel with this input constraint, the maximum over p(xis known to be given by

pa(x) = (2~17)~~ exp - $ *1 1Our channel is a Gaussian channel with a varying signal levelAs the distribution (4) is optimum for each value of the signalto-noise ratio in a Gaussian channel, it follows that it is optimumin our channel too. With the distributions given in (1) and (4the integrals in (3) can be evaluated. The result is&(P, Pdd, 4 = ~(1 + ~>?t+ + In (1 - 24

- In E,((l - 2q + A27j(1 + p)-y).At this point let us introduce the new variable /3 = 1 - 2rqand let us define

dP, PI = Eo(P, P&l, 4,6, P>= EAiW+ A2170 PY-11-p21.

In our new notation E(R) takes the form

(5

E(R) = max { -PR -I- m;x dp, P>l ,P (6

dp, P> = $41 + p)(l - P) + 3 ln P - In NP, PI.In order to guarantee (5) to be meaningful, we must restrictp to satisfy /3 > 0, so the maximum in (6) is over all p, p E[0, I]. For each p let g(p, 0) be maximized over p by /3* = p*(p).Using the general relation dlS/dR = -p, we alternatively have

E(R) = gb, P*> - PC g dP* s7ap + dp apIID8e> (7.8-P

In Appendix I we show that g(p, p) is a convex-upwardfunction of /I for each p E [O, 11.Thus, if the equation ag/afl= has a solution in the interval 0 5 /3 5 1, it i s unique anequals fl*.

RAYLEIGH-DISTRIBUTED SIGNAL LEVELWe now turn to the case of Rayleigh fading, i.e., we assump(u) to be given by (2). From (5) we have

h(p, p) = s,- 2ue-[p + u2z-1]-p2 da,


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354where we have introduced z = q-1(1 + p). Integrating byparts and rearranging the terms, we obtain the differentialequation ah/a/3 - zh = -sZp+*, which has the solution

IEEE TRANSACTIONSON INFORMATION THEORY, MAY

hb, P> = ezB h(p, 0) - 2 sup z-p2e-zz dz] . (8)By direct computation we have h(p, 0) = &2I(l - p/2)which inserted in (8) gives

smh(p, 0) = eza~pz tmpemtdt. (9)4

We thus see hat for /3 2 0 and for each p E [0, l] h(p, 0) is adecreasing function of /3 taking values in the bounded interval[O, ZNY(l - p/2)].In order to find fi* we differentiate g(p, /3) as given in (6) Fig. 1. h(p, 8) and k@, 9) 88 funotions of 6.with respect to /3, Observing that cYg/a/3= 0 for p = 0, p = /3*(O), we thus at7=w -$(I + p)+ +/j-_ h-$. (10) C = Ml - P> EAI-4 ln (P + A2d11p=ILet us define = EA(3 ln(1 -I- Av)}.

k(p, p> = p1-P2[(1 + +,)/I - +z-1-l. (11) DISCUSSIONWith the aid of (10) and from the fact that g(p, 0) is convexupward in p, we realize that if the equation

~XP, -9 = hh PI (12)has a solution in the interval 0 2 0 < 1, it i s unique andequals fl*.The functions k and h are shown in Fig. 1. The value PO,defined in the figure, is obtained from (11) and is &, =[2(2 + q)]-l 5 1, which shows that for each q 2 0 and eachp E [0, l] there exists a ,8 E [0, l] such that k(p, /3) > h(p, ,6?).Thus, in order to show that (12) has a solution in the interval0 5 /3 5 1, it is now sufficient to show that h(p, 1) 2 k(p, 1)for all p E [0, 11. This is shown in Appendix II, and thus wehave shown that when the fading is of the Rayleigh type, thereliability function E(R) can be represented in the parametricform (7), where /3* is the solution to (12).

CAPACITYFor the capacity it is possible to find a simple expressionwithout making the special assumption that the fading i s ofthe Rayleigh type. Generally we have

ag/a/3 is given in (10) and from (5) we realize that :

Thus we haveagap p-o = -4 + ;p-1,

which shows that p*(O) = 1. From (6) we haveat7=3P 30 - 0) - h- $

and from (5) straightforward computations giveah

Ip p-o = Ed{-+ In@ + A2rl)l.

In order to visualize how E(R) depends on the signal-to-ratio 17, e have plotted in Fig. 2 its two related parameteand Ro, the capacity and the exponential error-bound paramas functions of g for the case of Rayleigh fading. R. is deas R. = g(1, p*(l)). For comparison we have also plottedcorresponding functions for a nonfading channel, i.e., a chawith p(a) = 6(a - 1). As shown, the degradation due to famay be described as a power loss of between 1 and 3 dB. Wesee that the degradation is almost the same whetherwe uor R. as the measure of pe rformance.APPENDIX I

We are going to show that g(p, ,L?) s given in (6) is a coupward function of , I for each p E [0, I]. The logarithm increasing convex-upward function, and thus it is sufficieshow that h-1 is a convex-upward function of 0. This is trivtrue for p = 0, so let us assume 0 < p 5 1. We have

$ (h-l) = h-f2($) - h $1and as h(p, ,8) 2 0, it is sufficient to show that

Let us define the random variablesX2 = [/j + AZZ-1]--p2Y2 = $p(+p + I)[@ + A2.z-1]-p2-2 .

From Schwarzs inequality we havef (; + 1)($)2 = [EAXY-j2 5 EAX2EA I = h $

h$ > (1 + $$$$2 > 2($$2. Q.EAPPENDIX II

From (9) and (11) we have h(p, 1) - k(p, 1) = e~.where we have introducedf(z) = Jm fY2e- dt - [ez(l + +pz-)I-.


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CORRESPONDENCEA C.R,(nats/dim)

355interval length) are obtained from this waveform to form thevector sample

Fig. 2. Channel capacity and exponentialbound parameter z+s unctions ofsignal-to-noise atio. @ Capacity. Gaussianchannel; @ capacity, fadingchannel; 0 exponentialbound, Gaussianchannel: @ exponentialbound,fading channel.Straightforward computations give

4f=dz -2p.z -e-(p + 2~)~ < 0p; j(Z) = 0

from which we conclude that f(z) 2 0. This implies h(p, 1) 2MP, 1). Q.E.D.

ACKNOWLEDGMENTThe author wishes o thank his teacher Prof. L. H. Zetterberg,who read the manuscript and with whom he had many helpfuldiscussions.

THOMAS ERTCSONRes. Inst. National Defence104 50 Stockholm, Sweden

Nonparametric Detection Using Dependent SamplesAbstract-A new general approach o the formulation of a non-parametric detector using dependent samples is introduced andapplied to a space-diversity system employing dc signaling. Acomparison based on a form of asymptotic relative efficiency ismade between the new detector and a Mann-Whitney detector.Under certain conditions the new procedure demonstrates an

improvement in transmission efficiency.INTRODUCTION

A binary signal detector using a reference noise waveformcan be formulated in terms of a multivariate nonparametrichypothesis test to allow for the use of dependent samples.Assume that the detector has available a sample waveform N(t)from the stationary noise process and that a group of p equallyspaced dependent samples over a time period T (decision-Manuscript receivedMay 9, 1968; evised March 12, 1969.This work w&8supported n part by the Doctoral SupportPlan of Bell Telephone aboratories.Inc., and in part by the National Science oundation,under Grant GK-1075.This paper s based n part of & thesissubmitted o the GraduateDivision of theSchoolof Engineerin and Science f New York Uni\versity n partial fulfillmentof the requirementsor the degree f Doctor of Philosophy.

( yy, y;y . . . ) yi) = y,This process s repeated n times with sufficient delays betweensampling groups so that finally n independent vector samplesyw, . . . ) yw, . . . , Y(n)are obtained. The vector i ndependenceis not a severe limitation since the noise-vector samples aretaken on a periodic reference interval that comprises a smallpart of the total transmission period. During normal trans-mission, p equally spaced dependent samples (X1, . . . , X,) = Xare obtained on the decision nterval T. The following hypothesis-testing problem results:H: noise: Y(l), . * . , Yen, X E F(zl, . * * , z,)K: signal and noise: yl , s. . , Y E F(+ . . . ,:z,j

X E Gh, - . - , 4,where F(zl, . . . , s,) and G(zl, . . . , 2,) are p-variate cumulati vedistribution functions. The detector is then equivalent to aa-sample p-variate nonparametric hypothesis test. During thedecision interval, m independent vector samples X(l), * . * ,X(i), . . . ) X(m) can be obtained if an m-channel space- or time-diversity system is used. In a space-diversity system a referencenoise sample Y(i) could be obtained separately on each channel,thereby eliminating the spacing problem in obtaining theindependent reference vector samples when n = m. For somesituations, however, such as in an active sonar array, spacediversity does not insure independent noise processeson eachchannel.This general formulation has been introduced to stimulatefurther study of nonparametric methods with dependence byfollowing a multivariate approach. Very little work has beendone on the problem of dependence [l] in signal detection.Some multivariate nonparametric tests have been consideredin the mathematical literature [2], [3]. However, the forms ofthese tests are such that their implementation for a signal-detection system would be quite complicated and, possibly, oflimited use. In the present work a simple procedure that trans-forms the multivariate data to univariate data and then appliesa univariate nonparametric test is applied to signal detection.

CONVERSION TO UNIVARIATE DATALet L be a fixed transformation from p-dimensional spaceto one-dimensional space. Then

x = q X(i))and yci) = q Y)are all univariate random variables. Since L is a fixed trans-formation, if

XCi and Y( E F(zl, . . . , 3,then X(i) and Y(t) have the same univariate distribution, sayJ&). After the transformation, the multivariate problemreduces to a univariate a-sample nonparametric problem withindependent samples, and a suitable 2-sample detector can beused. The overall detector then consists of a transformationor predetector L that combines the dependent samples on thedecision interval and a nonparametric detector that uses thecombined samples. The overall operation is nonparametric withrespect to the false-alarm rate 01.The transformation L reducesthe data, and in some cases this lowers the efficiency. Theproper selection of L should minimize this effect. A reasonablechoice for the predetector L is that device that is optimumwhen the noise is Gaussian and white, for then there is at

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A Gaussian Channel With Slow Fading