Maximal-ratio combining over Nakagami fading channels with an arbitrary branch covariance matrix

IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 48, NO. 4, JULY 1999 1141

Maximal-Ratio Combining over NakagamiFading Channels with an Arbitrary

Branch Covariance MatrixQ. T. Zhang,Senior Member, IEEE

Abstract—The error probability of maximal ratio combiners(MRC’s) in a correlated Nakagami environment with an arbi-trary branch covariance matrix is not available in the literaturealthough some work has been done for two special cases with con-stant and exponential correlations. Correlation structures of thistype, though of theroetical interest, may not match to practicalsituations, even for an antenna array of a totally symmetricalconfiguration. In this paper, we tackle the general problemby virtue of characteristic functions, avoiding the difficulty ofexplicitly obtaining the probability density function (PDF) forthe signal-to-noise ratio (SNR) at the MRC output. We derivea simple closed-form solution for arbitrarily correlated channelswith an integral fading parameter and a solution in the formof a one-fold integral for a fading parameter of nonintegralvalues. Simple algorithms have also been developed for theirefficient implementation. The formulas are then used to analyzetwo possible antenna configurations for a base station, ending upwith some findings of interest to system design.

Index Terms—Correlated Nakagami fading, land mobile diver-sity systems, maximal-ratio combining.

I. INTRODUCTION

M ULTIPATH fading is a main obstacle for cellular radioto achieve reliable communication; an effective means

to combat multipath fading is diversity combining, which relieson a simple principle. Namely, if a number of well-separatedantennas are used to receive the same signal, it is unlikelythat all received signals fade at the same time. Recently,diverisity reception through a Nakagami channel has capturedthe attention of many researchers [1]–[7] for several reasons.First, Nakagami distribution has a relatively simple analyticalform, making it attractive in performance analysis. The moreimportant feature, however, is its flexibility in that it can beused to account for both severe and weak fading and includesthe classical Rayleigh fading as a special case. Finally, theacceptance of Nakagami model lies in its physical justification.Usually, statisitical models describe the large-sample behaviorof a fading mechanism without going to the details of theblack box. However, recent research [8] investigates into thephysical propagation mode of mobile communication channelsand ends up with Nakagami distribution.

Manuscript received June 22, 1997; revised September 15, 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)05711-4.

The early study of the error performance of diversity com-bining systems in a Nakagami environment was concentratedon some simple cases, assuming independent branch signals[6], [7]. This implies that the antennas must be separated atleast 50 wavelengths so that their correlation is negligible. Inthe real world, correlation between two antennas used in abase station is typically 0.7 or even higher due to the spacelimitation. The research focus was therefore gradually shiftedto diversity systems with correlated Nakagami distribution inthe subsequent study [1]–[3]. The crucial issue is to determinethe probability density function (PDF) of the signal-to-noiseratio (SNR) at the combiner’s output, which is the sumof a number of correlated gamma variables. This problem,stand-alone, is a difficult issue in mathematics. Accordingly,[1] and [2] deal only with dual-branch combiners. Maximalratio combiners (MRC’s) with multiple correlated Nakagamichannels were recently tackled by Aalo [3]. The state-of-artis that only a couple of closed-form PDF’s of relevance todiversity reception, are available in the literature for the powersum of multiple correlated Nakagami variables. They are theexact and approximate PDF’s corresponding to an antennaarray with constant cross correlation and with exponentialcorrelation, respectively. These two PDF’s were used in [3],leading to many interesting results for MRC’s with differentmodulation schemes. Though useful in mathematics and somesituations in engineering, the assumption of constant or expo-nential correlation generally does not match to the practicalenvironment in mobile communications. One may expectthat an antenna array with a totally symmetrical triangularconfiguartion would have a constant correlation. This is nottrue since the signal incident angle also has impact on thebranch correlation, and it is never symmetrical to all threebroadsides of a triangle [14]. On the contrary, the assumptionof exponential correlation is somewhat close to the situationin a linear array, but it requires the antennas to be placedunevenly. Clearly, engineering practice calls for performanceformulas to account for widespread covariance structures in aNakagami environment.

In this paper, we will provide such general solutions toMRC’s. We identify two situations for which different strate-gies will be used. In the first case, we assume that the fadingparameter is an integer and show that for an arbitrary branchcovariance matrix, the PDF of the SNR at the combiner’soutput can always be reprsented as finite summation of gammadistributions. This finite sum is then used to obtain the

0018–9545/99$10.00 1999 IEEE

1142 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 48, NO. 4, JULY 1999

Fig. 1. Maximal ratio combining system for diversity reception.

error probability in a simple closed form. In the secondcase, we tackle MRC’s with a nonintegral fading parameterfollowing the method of characteristic function. The remainderof the paper is organized as follows. We briefly describe thesystem model and some necessary background for correlatedNakagami distribution in Section II. Performance analysis forintegral and nonintegral fading parameters is conducted inSections III and IV, respectively. Section V illustrates thetheory by applying it to the evaluation of two antenna config-urations commonly used in base stations. Section VI finishesthe paper with concluding remarks.

II. SYSTEM MODELS

Consider a maximal rario combining system of order L, asillustrated in Fig. 1. The signal received at theth antennaover the symbol duration can be written as

(1)

where , is the carrier frequency, andthe information signal depending on the transmitted symbolThe influence of the th fading channel is characterized by itschannel gain which, in a complex form, is given by

(2)

with denoting its phase and its amplitude. Certainly,is a Nakagami variable. We define the channel gain vector

for the subsequent use. The faded signalis further corrupted by the Gaussian thermal noise withmean zero and variance Namely

(3)

From the signal model given in (1), we obtain the instanta-neous signal-to-noise ratio at theth branch as

(4)

The diversity system performs cophasing, maximal ratio com-bining and coherent demodulation; the combiner output is thengiven by

(5)

The positive sign is taken when bit one is transmitted and thenegative sign is taken for bit zero. For coherent binary phase-shift keying (PSK), the signal space is one-dimensional (1-D)and is the projection of the noise onto the basefunction and hence has the same statisticalproperties as the latter. It follows that the combiner outputSNR is

(6)

For orthogonal binary frequency-shift keying (FSK), the onlydifference is that the signal space is two-dimensional (2-D)and thereby the noise variance is twice that of binary PSK.

For Nakagami fading, is a gamma-distributed variate.The output SNR is thus the sum of -correlated gammavariables. To characterize, we need some background knowl-edge. Correlated gamma variables are usually defined by theirjoint characteristic function (CF) [9], [10]. Let denote thecovariance matrix of the SNR vector anddefine as Then, the joint CF of theinstantaneous SNR is given by [12]

(7)

where the fading parameter is assumed to be identical forall faded paths and is the positive definite matrix determinedby the branch covariance matrix Derivation of the relationbetween the two matrices is straightforward [11]. Let us brieflyoutline the process and results for the subsequent use. Denote

and Using the rules (10.16) and(10.19–10.21) for derivatives involving matrices [13], we canshow that

(8)

with denoting the th entry of The first equationimplies that the average SNR at theth branch equalstimes the th entry of the matrix Scaling the matrixby a factor does not change its correlation structure but doeschange the average SNR for each branch. Thus, for a givenaverage SNR, we can always achieve the desired SNR simply

ZHANG: MAXIMAL-RATIO COMBINING OVER NAKAGAMI FADING CHANNELS 1143

by approprate scaling. With the symmetrical property ofinmind, the entry of the covariance matrix ofis obtained as

(9)

Hence, is related to the branch covariance matrix by

(10)

It is clear that the matrix is uniquely defined by thecovariance matrix of

Return to the CF of the output SNR, Replacing in(7) with we obtain the CF of

(11)

This is an important relation in our subsequent analysis.

III. I NTEGRAL FADING PARAMETER

For binary PSK, an erroneous decision occurs if bit one(phase zero) is transmitted but Given channel gainvector , is distributed as

(12)

Consequently, we can express the conditional error probability

error (13)

where for BPSK and for coherent BFSK. Hence

(14)

where signifies the PDF of The calculation ofdepends on the value of We have two cases: integraland nonintegral Two situations have different methods tohandle. The following derivation will focus on BPSK since theperformance of coherent BFSK can be easily obtained fromits BPSK counterpart by reducing the the SNR by half.

A. Error Probability

Let us begin with partial expression of the CF of the totalSNR, We rewrite

(15)

Here again, stand for the eigenvalues of For correlatedbranch outputs, we assume, without loss of generality, that all

these eigenvalues are distinct. We may thus express (15) as

a partial sum

(16)

where the coefficients for can be determined by

(17)

and for

(18)

Each term in (16) corresponds to a Gamma distribution.

Accordingly, the PDF of can be considered as the linear

combination of the distributions of independent gamma

variables. Specifically, the PDF corresponding to the CF

is

(19)

and, hence, the variable has the same distribution as

(20)

We next invoke the results of ([15], pp. 722–723) to write

(21)

with defined by

(22)

Combining the above results, it follows that

(23)


In the above, we assume that branch outputs are correlated.If all the signals received by different antennas withstandindependent and identical fading. In this simple case,reducesto a diagonal matrix of equal element. This means that allbranches have the same average SNR, i.e.,

Then, (23) is simplified to

(24)

where equals in (22) with replaced byThe above formulas are derived for coherent BPSK. The

extension to coherent BFSK is straightforward. What we needis to simply replace (22) by

(25)

while the remaining formulas still apply.

B. Calculation of

As shown above, coefficients in the partial sum repre-sentation for involve the calculation of higherorder differentiation of a product function. Differentiation ofthis type is often accounted in communications engineeringand in the past, they are suggested to be evaluated manually.However, such manual calculation is very time consumingand even prohibitive, especially when the order is large. Itis therefore desirable to derive an algorithm, which can beperformed recursively for any order using a computer. To thisend, let us denote

(26)

The difficulty in determining higher order derivatives oflies in the functional form of as a product. Such anobstacle can be circumvented if we can convert to aform of summation. We therefore consider its logarithm. Notethe first derivative of takes the form of a product, asshown by [11]

(27)

For higher order derivatives with we may invoke theformula (14) of ([13], p. 230), yielding

(28)

where the derivative with order zero simply represents theoriginal function. The derivatives of the logarithm of is

easy to obtain. Using the method of deduction, it follows that

(29)

Note that the derivative of order dependsonly on lower order derivatives, up to th. In theother words, (28) and (29) constitute a recursive algorithm forhigher order derivative calculation. An-file program, called

, is provided in Appendix A for the implementationof the algorithm using matlab. To determine the coefficients

set and make it global, assignvalues to and and then call the -file program using thefollowing syntax:

if=

elseifprod(1:

lam

end.

Although the algorithm was derived in the context of cel-lular mobile communication in a Nakagami environment, itcan be easily adapted to obtain higher order derivatives ofcomplex functions, as accounted in many other communicationproblems.

IV. NONINTEGRAL FADING PARAMETER

The method described in the previous section exploitedthe fact that for an integral fading parameter, the CF of thetest variable can be expressed as a partial sum. This ideais obviously not applied to the case involving a nonintegralfading parameter. Therefore, we have to take a differentapproach.

Let us begin with (14). Note the PDF can be ex-pressed as the inverse Fourier transform of its CF. Thus,inserting (15) into (14), we obtain

(30)

We recognize that the second integral on the last line is theFourier transform of and can be evaluated usingformula (9) of ([17], p. 177)

(31)

It follows that

(32)


with the integrand given by

(33)

It is easy to examine the imaginary part of is an oddfunction of whereas the real part is even; hence, we have

(34)

After some manipulations, we obtain

(35)

Here and are the magnitude and phase of thefactor , given by (36) at the bottom ofthe page. This solution involves an integral and its evaluationrelies on numerical methods. The typical behavior of theintegrand is demonstrated in Fig. 2 for a triangular antennaarray. As regard to array configurations, we have more to sayin Section V. As shown by the curves, the integrand is alwaysa smoothed function of the integration variable, in spite ofthe value of SNR. The nonzero value of the integrand appearsaround the origin, and an increase in SNR tends to narrowdown the pulse width of the waveform. Such integration canaccurately be evaluated by using the method of piecewiseGaussian quadrature. The observation suggests that it sufficesto calculate the integral over a finite interval, say Wetake three steps. First, partition the interval nonuniformlyinto a number of subintervals such that smalleris assignedwith a smaller subinterval width. Second, use the method ofGaussian quadrature to determine the the integral for eachsubinterval. Finally, add them to obtain the result. We suggestusing the ten-point quadrature. To choose abscissas and weightfactors for Gaussian integration, reader is referred to ([19], pp.916–919).

It is interesting to examine the accuracy and the compu-tational efficiency of the integration method introduced inthis section. To this end, the Gaussian quadrature method isemployed to determine the error performance of three diversitysystems with a triangular array and linear arrays of three

Fig. 2. Typical behavior of the integrand as a function oft:

and five antennas, respectively. The antenna configuationswill be described in detail in the next section. The resultsare shown in Fig. 3, in which the curves obtained using theclosed-form formula (23), labeled with , are also includedfor comparison. We have chosen an integral fading parameter

so that both formulas are applicable. The curvesobtained using the numerical method coincide with the exactresults obtained from the closed-form formula in all the cases,demonstrating the high accuracy of the formula (32). Recallthat the two formulas were dereived by taking completelydifferent approaches, but now they end up with the sameresults. This fact verifies, at least in one way, the correctness ofour theoretical analysis. Numerical calculation was performedon a 386 computer of 33 MHz. Each curve obtained by thenumerical method contains 31 SNR points. The partitioningof integration interval we used, in an -file format, was[0 : 0.05 : 0.5, 0.5 : 0.1 : 1, 1.2 : 0.4 : 6, 6 : 20, 20 : 5 :60]. Thecentral processing unit (CPU) time required to obtain curves

and were 68.27, 68.05, and 77.89 s, respectively. Thus,on average, the CPU time for each SNR point is about 2.3s. We observe that as the diversity orderincreases fromthree to five, the CPU time increases only by 14.5%. We cantherefore conclude that the formula (32) can be implementedaccurately and efficiently using a personal computer.

Though derived for binary transmission, the performanceformulas presented in this paper can be directly applied toQPSK systems. They can also be applied to-DQPSK cel-

(36)


Fig. 3. Illustrating the accuracy of the formula given in (34)–(36): solidcurves are determined by using (23); results labeled with+ are determinedby using (34)–(36).

lular systems with coherent detection. During the 1980’s it wasbelieved that the differential demodulation might be requiredfor relatively high Doppler shift, fast faded mobile cellularapplications. During the 1990’s, however, it was discoveredthat the delay-spread immunity and the carrier-to-interferenceratio of coherent systems are considerably more robust thanthose of differentially demodulated systems. As a consequence,manufacturing companies have been implementing coherent(not differentially coherent) demodulators ([16], p. 156).

V. DIVERSITY RECEPTION IN BASE STATION

As an illustration of the theory, we study some feasiblediversity systems in a base station. Two kinds of antenna con-figurations will be conisdered here, due to their relative easeof installation. They are uniform linear and triangular arrays,as sketched in Fig. 4. The correlation matrix of an antennaarray depends not only on the array configuration but also onthe incident angle of the incoming signal. Different parameteraffects the correlation in a different way. Branch correlationdepends on the antenna height and antenna separation throughtheir ratio [14]. For a given antenna array, correlation betweentwo antennas varies with the signal incident angle. It reachesits maximum value when the signal comes from the endfiredirection and reduces gradually as the signal direction movestoward the broadside.

Suppose now that an antenna array is used to receive signalsfrom a mobile unit operating at 850 MHz. The signal suffersfrom Nakagami fading. As a typical value, we assume thatthe antenna is 100 ft in height. A signal at 850 MHz impliesthat its wavelength is m, or 1.16 ft. For both linearand triangular arrays, any two adjacent antennas are assumedto have equal separation, denoted here byThe two arrayconfigurations are illustrated in Fig. 4. To be representative ofpractical operational conditions with a linear array, we assumethat signals impinge upon the array at 45relative to thebroadside of the array. For a triangular array, we assume that

(a) (b)

Fig. 4. Two antenna configurations. (a) Triangular array. (b) Linear array.

signals come from the broadside of the line linking antennasA and B (refer to Fig. 4). Then, the incident angles to ACand to BC are both equal to 60We therefore expect that thecorrelation between antennas A and C (or between B and C)is higher than that between A and B, regardless of their equalseparation.

We first study the influence of the fading parameter onthe error performance using linear and triangular arrays, bothhaving three antennas and with antenna spacing

ft. This separation is close to the practical number usedin the real world [14]. With this parameter setting, we candetermine the correlation matrix of an antenna array using theempirical curves of Lee ([14], p. 203). The covariance matrixfor the triangular array, in a normalized form, is given by

(37)

and for the linear array, it is given by

(38)

Here for simplicity, we have assumed that all branches havethe same average SNR. We then determinefrom andscaling it by a scaling factor (SNR/m) so that we can write

SNR(39)

where denotes the matrix with its entries equal tothe square root of the corresponding entries of Notethe correlation matrix of the linear array has a Toeplitzstructure whereas that of the triangular array does not. In theprevious study [3], antenna array with equal cross correlationwas considered. From the above discussion, we see thatsuch a statistical structure is hardly achieved even for anantenna array with a completely symmetrical configuration.The incident angle breaks down the parity.

The results for linear and triangular arrays are graphedin Fig. 5(a) and (b), respectively. As expected, the errorperformance improves with increased fading parameter since


(a) (b)

Fig. 5. Variation of the error performance with the fading parameterm: (a) Linear array. (b) Triangular array.

(a) (b)

Fig. 6. Variation of the error performance with the diversity orderL:

small corresponds to severe fading. In particular,corresponds to the situation of Rayleigh fading. Sinceis not a linear function of , we do not expect a uniformchange of with This becomes evident in the curves.We also see that the linear array has a better performancethan the triangular array, at the cost of a larger array dimen-sion.

We next study the way the diversity order affects theperformance. We consider two linear array, both of fiveantennas. The only difference is the antenna spacing, one with

ft and the other with ft. Thefading environment is featured by Suppose, again,the signal incident angle is The correlation matrices can

be determined as

(40)

for and as

(41)


(a) (b)

(c)

Fig. 7. Variation of the error performance with the antenna separationd: d is measured in wavelength. (a) Dual-antenna array. (b) Linear array ofthree antennas. (c) Triangular array.

for [11]. The corrleation matrices for andcan all be obtained from the matrices shown above. The resultsare shown in Fig. 6. From the curves, we see that increase ofthe diversity order is a powerful means to improve the systemperformance over fading channels. The improvement is moresignificant when an appropriately large separation, , isutilized.

The use of a large diversity order to improve the systemimmunity to multipath fading, however, is often restricted bythe system complexity. It is therefore interesting to see how theerror perofrmance is affected by the antenna separation. Thevariation of system error performance with antenna separationis shown in Fig. 7 for different array structures; these struc-tures include dual antennas, a linear array of three antennasand a triangular array. Here again, was used. Increasein the antenna separation reduces the correlation between

antennas, thus a better diversity performance. Some obser-vations can be made: 1) there is little room for performanceimprovement for a dual-antenna array and 2) an increase in

significantly improves the performance of the three-antennaarray, no matter they are configured on a line or as a triangle.The improvement is significant whenincreases from to

and Further increase of the separation is rewardedby a little improvement. This suggests that an appropriateantenna separation should be such that the correlation fallsinto the range from 0.6 to 0.65. In Fig. 7(a), two linear three-antenna arrays are included for comparison; their performanceis indicated by the dash-dotted curves. The curves with labelsa and b correspond to the array with the aperture equal to

and , respectively. Comparing them with the curvesfor the dual-antenna array with the same aperture, we seethat increasing the diversity order is more effective. Taking


Fig. 8. Performance comparison between BPSK and coherent BFSK basedon a triangular array.

both system complexity and antenna dimension into account,the best choice would be the triangular array with antennaseparation ranging from to (i.e., 8.12–11.6 ft).

The results shown above are all for BPSK systems. Acomparison between BPSK and coherent BFSK are illustratedin Fig. 8 for a triangular array with spacing The casesof and are considered. Three-dB difference isobserved between the PSK and FSK schemes.

VI. CONCLUSION

In this paper, we have examined the error performanceof MRC diversity systems over Nakagami channels with anarbitrary covariance matrix. We tackled the problem followingtwo different strategies, depending on the values of the fadingparameter . For an integral , we expressed the CF of thetest variable into a partial sum, resulting in a simple formulain closed form. A recursive algorithm has been developed forcalculating the coefficients for the partial sum. For of anonintegral value, we employed the method of characteristicfunction and obtained a formula in the form of one-foldintegration, which can be efficiently calculated using piecewiseGaussian quadrature.

The formulas were then used to study two types of diversityschemes that can be used in a base station, one with linearantenna arrays and the other with triangular arrays. Computerresults reveal that increase of the diversity order can signif-icantly improve the error performance, but is subject to thespace limitation. Properly increasing the antenna separationcan also help to improve the performance. However, whenthe correlation reaches 0.6, further increasing the separationwill basically trade array dimension for a little improvenment.As an overall consideration, the triangular array with adjacentseparation ranging from seven to ten wavelengths is the bestcandidate for practical application.

APPENDIX ATHE -FILE FOR HIGHER ORDER DIFFERENTIATION

function s df(l,m,r)l: the lth eigenvaluem: fading parameterm-r: the order of defferentiationL: order of diversitylam: eigenvalues of Gamma matrix, global

L length (lam);ff 1;

for k 1:Lif k l

v lam(k)/lam(l);ff ff ((1 v) m);end

endf 1/ff;s f; f blm f(s) at s 1/lam(l)

if r mF

for n 1:m-racc f dlf(l,m,n); the term for k 0

for k 1:n 1a prod(n k:n 1)/prod(1:k);acc acc a F(k) dlf(l,m,n-k);

endF [F;acc]; store derivatives of orders

1 to (m r).end

s F(m r);end

function g dlf(l,m,q)m: fading parameterl: the lth eigenvalueq: qth derivative

L length(lam);if q 1

qq 1;else

qq [1:q 1]end

cc m prod(qq);

acc 0;for k 1:L

if k lx 1/lam(k);y 1/lam(l);a (x-y) q;acc acc (1/a);

endendg cc acc.


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Q. T. Zhang (S’84–M’85–SM’95) received theB.Eng. degree from Tsinghua University, Beijing,China, and the M.Eng. degree from the South ChinaUniversity of Technology, Guangzhou, China, bothin radio communications, and the Ph.D. degree inelectrical engineering from McMaster University,Hamilton, Ont., Canada.

He was with the Communications Research Lab-oratory, McMaster University, as a Senior ResearchEngineer and Adjunct Professor conducting researchon communications and signal processing. In Jan-

uary 1992, he joined the Satellite and Communication Systems Division,Spar Aerospace Ltd., Montreal, Canada, as a Senior Member of TechnicalStaff. At Spar Aerospace, he was first responsible for the image perfor-mance analysis of the Radar Satellite and was subsequently engaged in thedevelopment of the advanced satellite communication systems for the nextgeneration. He was a tenured Associate Professor with the Department ofElectrical Engineering, Ryerson Polytechnic University, Toronto, Ont. He iscurrently with the Department of Electrical Engineering, National Universityof Singapore, Singapore. His research is primarily in the areas of wirelesscommunications and ATM networks with a current emphasis on frequencyand interference management, handoffs, reliable reception over various fadingchannels, intelligent beamforming, ATM switches, video traffic modeling, andsource compression.

Maximal-ratio combining over Nakagami fading channels with an arbitrary branch covariance matrix

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