IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 48, NO. 4, JULY 1999 1151

A Simple Approach to Probability of Error forEqual Gain Combiners over Rayleigh Channels

Q. T. Zhang,Senior Member, IEEE

Abstract—A new technique for determining the probabilityof error of a general equal gain (EG) combiner over Rayleighchannels is presented in this paper. Unlike the existing methods,the new procedure takes a single step and directly derives theerror performance from the characteristic function of the decisionvariable, thereby avoiding the lengthy process of deriving thedistribution of the sum of Rayleigh variates and thus eliminatingall approximations required therein. The new solution is exactand much simpler compared to the existing results. It takes theform of a one-fold integral, which can be easily evaluated by usingthe Hermite’s numerical method.

Index Terms—Cellular mobile systems, diversity reception,equal gain combining, Rayleigh fading channels.

I. INTRODUCTION

EQUAL GAIN (EG) combining is a powerful means tocombat multipath fading and has been widely adopted

in practice due to its good performance and ease of imple-mentation [1]–[4]. In this paper, we are concerned with theerror performance of EG combiners (EGC’s) over Rayleighfading channels. Closed-form solutions for EGC with up tothree branches have recently been derived in [5]. For generalEG combining of an arbitrary diversity order, no closed-formsolutions exist, and certain types of approximations are usuallyused. One such useful solution is obtained by Beaulieu [6],[7] through a two-step procedure: determine the distributionor probability density function (pdf) of the Rayleigh sum andthen use it to obtain the average probability of bit error. Thecentral idea is to determine the distribution of a Rayleigh sumby using a Chernoff-bound type of approximation technique.Specifically, the Rayleigh sum is assumed to takenonzerovalues only over afinite domain so that the complementarydistribution of the Rayleigh sum can be expressed as theexpected value of a switch function. With the assumptionabove, the switch function is justified to be approximated as aFourier series whereby the expected values can be calculated.From the numerical point of view, the error probability soobtained is essentially in the format of a two-fold integral.

Now, a natural question is: Can we avoid the lengthyderivation of the distribution of the Rayleigh sum and theapproximations used therein in order to obtain the errorperformance? In this paper, we present a one-step procedure

Manuscript received January 29, 1998; revised July 16, 1998.The author was with the Department of Electrical Engineering, Ryerson

Polytechnic University, Toronto, Ont., M5B 2K3, Canada. He is now withthe Department of Electrical Engineering, National University of Singapore,Singapore, 119260, Singapore.

Publisher Item Identifier S 0018-9545(99)05737-0.

Fig. 1. Schematic diagram for an EG diversity receiver.

that leads to anexactsolution to the problem of EG combinersover Rayleigh channels in the form of a one-fold integral. Theintegral can be efficiently and accurately evaluated by usingthe Hermite method. The new technique distinguishes fromthe previous method in three aspects. First, we consider theRayleigh signal magnitudes and white Gaussian noise at thesame time, treating them as the constituents of a single entityfor decision making. Second, the exact probability of error isdirectly derived from the characteristic function (CF) of thedecision variable, thereby avoiding the unnecessary process ofdetermining the distribution or pdf of the Rayleigh sum. Third,the numerical calculation is much simpler.

II. COHERENT DETECTION

A typical EGC system with coherent detection is illustratedin Fig. 1, in which the signal received at each branch iscophased, summed, and coherently demodulated. The decisionvariable for a diversity system using coherent BPSK can beexpressed as

(1)

where is the signal amplitude at the output of branch.With slow Rayleigh fading assumption on the channels,

0018–9545/99$10.00 1999 IEEE

1152 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 48, NO. 4, JULY 1999

remains constant within a symbol duration but changes fromsymbol to symbol, following the Rayleigh distribution:

(2)

where is the average signal energy. The termrepresents the complex baseband Gaussian noise at the outputof branch , with zero mean and variance . The first term in(1) assumes the positive sign if the transmitted symboland assumes the negative sign if . The decision rule is

.(3)

For orthogonal BFSK, the decision variable and the decisionrule are the same except that the noise variance is doubled.As shown in [5], the error probability can be directly cal-culated from the characteristic function (CF) of the decisionvariable , thereby eliminating the unnecessary step fordetermining the pdf of . The result is

(4)

Since the branches are assumed to be mutually independent,the CF of simply equals the product of the CF’s of individualcomponents, leading to

(5)

The CF of the Rayleigh signal componentcan be calculatedby virtue of the sine and cosine transforms [8, pp. 15 and 73],ending up with

(6)

In this expression, denotes the confluent hyper-geometric function. The sum of Gaussian noise components in(1) can be treated as a single Gaussian noise with total power

. The CF of a Gaussian variable iswell known. Define the average noise power by

(7)

and denote the average bit energy to average noise densityratio (SNR) at the th branch by

(8)

Then, we can obtain the CF of

(9)

Note the integrand of (4) contains a Gaussian-shaped function.We therefore consider using the Hermite method for numericalcalculation. To this end, we change the variable

(10)

to obtain

(11)

where is defined by

Im

(12)

Alternatively, can be expressed in terms of the SNR’s,as shown by

Im

(13)

In the above expressions, the confluent hypergeometric func-tion can be evaluated using

(14)where is defined as and

. Using the Hermite method of integration [10, p.924], we have

(15)

where is the integration order. Note the integral is definedover the positive half axis. Therefore, only the abscissas ofpositive values are used in the calculation. The Gaussian-shaped function makes the integrand converge to zero veryfast. Accordingly, is sufficient to ensure good accu-racy; for details, see Abramowitz and Stegun [10].

ZHANG: APPROACH TO PROBABILITY OF ERROR FOR EQUAL GAIN COMBINERS 1153

For facilitating the use, the procedure is summarized asfollows.

1) For a given abscissa and SNR , evaluateusing

(16)

Note when using the Hermite method of integration, themaximum absolute value of is about five. The scalingfactor of is less than one for all possible value ofSNR . For small , it is much less than one and forlarge , it is approximately equal to . This impliesthat is at the order of .

2) Calculate the confluent hypergeometric functions

(17)

for . Since is a finite number atthe order of , for , ten terms should be enoughto ensure a good accuracy. For the case of , 15terms should be sufficient.

3) Determine

Im

(18)

4) Determine the error probability using (15).

Comparing (15) and (18) with [7, eqs. (22), (23), (27), and(29)], we can see that the formula presented here is muchsimpler. Furthermore, for each term in the integral sum, theprevious method needs to calculate two types of confluenthypergeometric function in contrast to only one in the presentmethod.

A. Special Case

The expression for the error probability can be simplified ifall branches have the same SNR. Namely,

. In this case, (13) reduces to

(19)

Fig. 2. Probability of error versus reference SNR.

B. Coherent BFSK

In the previous analysis, we were focused on coherentBPSK. Since the only difference between coherent BFSK andBPSK is the value of the signal-to-noise ratios, we can obtainformulas for coherent BFSK by simply replacing all in theabove expressions with .

III. N UMERICAL RESULTS

Let us consider a coherent BPSK system, which usesmultiple antennas to receive incoming signals over Rayleighchannels. The signal levels varies at different branches. With-out loss of generality, the signal-to-noise ratios at all branchesare measured with reference to the SNR at the first branch,

. Two diversity orders are considered: and .In the first case, the SNR’s at the five antennas are given

dB (20)

In the second case, we have three antennas with the SNR’sgiven by

dB (21)

The results are shown in Fig. 2.We next would like to see how the diversity order affects

the error performance. To this end, we change the diversityorder while maintaining that all branches have the same SNR.Probability of error as a function of SNR at the first branch forvarying diversity order is shown in Fig. 3. It is clear that theerror performance improves with increased diversity order. Wefurther observe that the error performance improves rapidly as

increases from two to four and the improvement graduallyslows down as further increases. This indicates that three orfour antennas are sufficient for good performance and furtherinvestment will end up with a little additional return.

1154 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 48, NO. 4, JULY 1999

Fig. 3. Variation of probability of error with the number of branches.

IV. CONCLUSION

In this paper, we have derived a general exact solution forthe problem of EG combining over Rayleigh fading channels,in the form of a one-fold integral. The approach we take is todirectly derive the solution from the characteristic functionof the decision variable, thereby eliminating the need forderiving the distribution of the sum of Rayleigh signals and,

hence, avoiding all approximations required therein. The newmethod has the advantage of simplicity in expressions andcomputational efficiency.

REFERENCES

[1] W. C. Jakes, Ed.,Microwave Mobile Communications.New York:Wiley, 1974.

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

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

[4] S. W. Halpern, “The effect of having unequal branch gains in practicalpredetection diversity systems for mobile radio,”IEEE Trans. Veh.Technol.,vol. 26, no. 1, pp. 94–105, 1977.

[5] Q. T. Zhang, “Probability of error for equal-gain combiners overRayleigh channels: Some closed form solutions,”IEEE Trans. Commun.,vol. 45, pp. 70–73, Mar. 1997.

[6] N. C. Beaulieu and A. Abu-Dayya, “Analysis of equal gain diversityon Nakagami fading channels,”IEEE Trans. Commun.,vol. 39, no. 2,pp. 225–234, 1991.

[7] N. C. Beaulieu, “An infinite series for the computation of the com-plementary probability distribution function of a sum of independentrandom variables and its application to the sum of Rayleigh randomvariables,”IEEE Trans. Commun.,vol. 38, no. 9, pp. 1463–1474, 1990.

[8] A. Erdelyi, Ed., Tables of Integral Transforms, vol. 1. New York:McGraw-Hill, 1954.

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

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

Q. T. Zhang (S’84–M’85–SM’95), for a photograph and biography, see thisissue, p. 1150.