270 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 45, NO. 3, MARCH 1997

Probability of Error for Equal-Gain Combiners overRayleigh Channels: Some Closed-Form Solutions

Q. T. Zhang

Abstract—The probability of error of a general equal-gaincombiner (EG) in closed form isnot available in the literature. Inthis letter, by formulating the problem in the framework of signaldetection and evaluating the probability of error directly from thecharacteristic function (CF) of the signal at the combiner output,we obtain closed-form solutions for the probability of error ofthe EG combiner with two or three branches operating over aRayleigh-fading channel.

Index Terms— Bit-error performance, diversity reception,equal-gain combiner, FSK, PSK, Rayleigh fading.

I. INTRODUCTION

REGARDLESS OF ITS practical and theoretical impor-tance, the error performance of the equal-gain combiner

(EGC) in closed form isnot available in the literature. Thecrucial issue is to determine the probability density function(pdf) of the signal component at the combiner output. For amaximum ratio combiner operating in a Rayleigh or Nakagamienvironment, this corresponds to seeking the pdf of a sum ofGamma-distributed variables. It is a relatively easy task sincethe Gamma distribution has finite poles in its characteristicfunction. For an EG combiner, however, one has to determinethe pdf of a sum of Rayleigh or Nakagami amplitudes, whichdoes not share the desirable properties of the Gamma distribu-tion. Consequently, the residue theorem becomes completelyuseless in the performance evaluation. Indeed, determining thepdf of the sum of multiple Rayleigh amplitudes alone has beena dilemma for many years, dating back to Rayleigh [1]–[3].

In this correspondence, we derive a closed-form solutionfor the error rate to the problem of order-three diversitywith coherent detection by formulating it in the frameworkof statistical decision theory and evaluating the probabilityof error directly from the characteristic function (CF) of thecombiner output.

II. SYSTEM DESCRIPTION

Consider the widely used predetection diversity systemusing EG combining [3], [4], as illustrated in Fig. 1. It isassumed that the signals received at theantennas areindependent and jointly Gaussian distributed. The signalsare corrupted by zero-mean, white Gaussian noise, which is

Paper approved by N. C. Beaulieu, the Editor for Wireless CommunicationTheory of the IEEE Communications Society. Manuscript received August 8,1995; revised April 30, 1996, and August 19, 1996. This paper was presentedin part at the 1996 International Conference on Communication Technology,Beijing, China, May 5–7, 1996.

The author is with the Department of Electrical Engineering, Ryer-son Polytechnic University, Toronto, Ont., M5B 2K3 Canada (e-mail:kzhang@ee.ryerson.ca).

Publisher Item Identifier S 0090-6778(97)01963-6.

Fig. 1. Predetection EG diversity system withgk = exp(�j�k).

uncorrelated between different branches. The input at thethantenna over can be written as

(1)

In this expression, is zero-mean, white Gaussian noise atthe branch with one-sided power spectral density;is its complex representation, having a zero-mean complexGaussian distribution with variance . is the carrier angu-lar frequency and is the information signal depending onthe transmitted symbol . and represent, respectively,the symbol duration and the random phase uniformly dis-tributed over . The cosine function has been normalizedso that the square of the magnitude represents thesymbol energy. is the signal component received at the

th branch; in a complex form, it can be expressed as

(2)

The amplitude remains constant within a symbol du-ration but changes from symbol to symbol, following theRayleigh distribution

(3)

with denoting the average signal energy at branch. It follows from Fig. 1 that the combiner output is given by

(4)

Here, we continue using the same symbols for the noisecomponents since the process of co-phasing does not changetheir statistical characteristics.

0090–6778/97$10.00 1997 IEEE

IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 45, NO. 3, MARCH 1997 271

III. COHERENT DETECTION

For coherent detection, the signal received at each branch isco-phased, summed, and coherently demodulated. For coherentbinary phase shift keying (BPSK), the signal space is one-dimensional (1-D) and the resultant decision variable is

(5)

where is the noise projection of onto the base andhas the same statistical properties as the latter. The first termassumes the positive sign if the transmitted symboland assumes the negative sign if . The decision rule is

(6)

For orthogonal BFSK, the decision variable and the de-cision rule are the same except that the signal space istwo-dimensional (2-D) and, hence, the noise variance is twicethat of BPSK.

To determine the error performance, we begin with the CFof the decision variable. By definition, the CF ofis given by

(7)

with denoting the expectation operator. Applying theassumption of independence between the signal and noisecomponents to (5), we find that

(8)

where the CF of the Rayleigh-signal component, , canbe expressed by virtue of the sine and cosine transforms

(9)

On the last line, we have used formulas (14) and (19) of[6, pp. 15 and 73]. Here, denotes the confluenthypergeometric function. The CF of the Gaussian noiseiswell known, given by

(10)

Let be the power spectral density function of the totalnoise such that

(11)

and let be the average bit energy to average noise densityratio (signal-to-noise ratio, SNR) at theth branch such that

(12)

We substitute (9) and (10) into (8) to obtain

(13)Having obtained the CF of the decision variable, we may,

in principle, determine its pdf through the use of inverseFourier transform, from which the probability of error can becalculated. We note, however, that the calculation of such aninverse Fourier integral is not a simple matter. It is, therefore,preferable to determine the probability of error directly fromthe CF of the decision variable. To this end, we invoke thefollowing lemma originated by Gil-Palaez (see [5]).

Lemma 1 (Gil-Palaez):Let be a 1-D cumulative dis-tribution and , the corresponding characteristic functionwith real variable . Then

(14)

where denotes the imaginary part of.It follows from (13) and Lemma 1 that the probability of

error equals

(15)

This formula provides a general basis for the performanceevaluation of an EGC with coherent detection. It applies to,though not addressed here, arbitrary fading channels as longas their characteristic functions exist. The solution takes theform of a one-fold integral, the calculation of which relieson numerical methods. For the special cases considered inthis correspondence, however, we can obtain a closed-formsolution.

Now let us consider the case . In this case, the CFin (13) becomes

(16)

Define such that

(17)

and such that

(18)and, hence, we write

(19)

272 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 45, NO. 3, MARCH 1997

Inserting (19) into (15) and using of [7, Formula 7.622(1)],we obtain the following.

Theorem 1: Consider the problem of coherent detection ofbinary signals using a three-branch EGC. Then, the probabilityof error can be expressed as

(20)

where the constant equals for BPSK and equals forBFSK. We can alternatively express in terms of SNR’s atindividual branches

(21)

The function is defined as

(22)

with denoting the Gauss hypergeometric function.We observe that depends on the signal energy and

noise power density only through the SNR’s of the individualchannels.

We obtain for the case by letting andin (20)

(23)

In this expression, if we further let , the resultfor a nondiversity receiver follows:

(24)

which is identical to the well known result in the literature [9,p. 717]. The Gauss hypergeometric functions in (22) can beevaluated using the series expansion [8]

(25)

which converges very fast in our case since .Fig. 2 shows the error performance of a three-branch re-

ceiver using BPSK with equal and unequal SNR’s. ComparingBPSK and coherent BFSK for different is shown in Fig. 3.As expected, the error performance improves with increasedorder of diversity.

Fig. 2. Error performance of BPSK with third-order diversity for differentSNR setting.

Fig. 3. Performance variation with the order of diversity. All branches havethe same SNR�: BPSK versus coherent BFSK.

IV. NONCOHERENT DETECTION

For noncoherent detection, the combined signal is appliedto a noncoherent detector. Consider the case . Let

and let denote its pdf. Given, the probabilityof error equals , where ,and for DPSK and for noncoherent BFSK.The average probability of error is the statistical average of

over . We express in terms of andreorganize, obtaining

(26)

where can be found from (9). We next write the secondintegral as sine and cosine transforms and determine them

IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 45, NO. 3, MARCH 1997 273

using formulas 3.896(3-4) of [7, p. 480]. It ends up with theexpression of as the sum of four integrals. These integralscan further be simplified by using formula 7.622(1) of [7] andthe recursive relation (13.4.6) of [8]. The result is summarizedas follows.

Theorem 2: The error performance of a two-branch EGcombiner, when used for noncoherent detection of binaryDPSK and FSK signals, is given by

(27)

The error performance for nondiversity reception [9, p. 718]can be obtained from (27) by letting .

V. CONCLUSION

In this letter, we have examined the probability of error( ) of EGC’s with two or three branches operating in aRayleigh-fading environment. We tackled the problem bydirectly determining from the relevant characteristic func-

tions, ending up with some closed-form solutions which areexpressed in terms of Gauss hypergeometric functions.

The new formulas include the classical results for non-diversity reception in a Rayleigh fading environment as aspecial case. Furthermore, the assumption of equal SNR ateach branch is not necessary for the use of the new formulas.

REFERENCES

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[2] D. G. Brennan, “Linear diversity combining techniques,”Proc. IRE,vol. 47, pp. 1075–1102, June 1959.

[3] S. W. Halpern, “The effect of having unequal branch gains in practicalpredetection diversity systems for mobile radio,”IEEE Trans. Veh.Technol., vol. 26, pp. 94–105, 1977.

[4] N. C. Beaulieu and A. Abu-Dayya, “Analysis of equal gain diversityon Nakagami fading channels,”IEEE Trans. Commun., vol. 39, pp.225–234, Feb. 1991.

[5] J. P. Imhof, “Computing the distribution of quadratic forms in normalvariables,”Biometrika, vol. 48, nos. 3–4, pp. 419–426, 1961.

[6] A. Erdelyi, Ed., Tables of Integral Transforms. New York: McGraw-Hill, 1954.

[7] I. S. Gradshteyn and I. M. Ryzhik,Table of Integrals, Series, andProducts. New York: Academic, 1980.

[8] M. Abramowitz and I. A. Stegun,Handbook of Mathematical Functions.New York: Dover, 1972.

[9] J. Proakis,Digital Communications. New York: McGraw Hill, 1995,3rd ed.