Exact analysis of postdetection combining for DPSK and NFSK systems over arbitrarily correlated...

IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 46, NO. 11, NOVEMBER 1998 1459

Exact Analysis of Postdetection Combiningfor DPSK and NFSK Systems Over

Arbitrarily Correlated Nakagami ChannelsQ. T. Zhang,Senior Member, IEEE

Abstract— Postdetection combining is a popular means toimprove the bit error performance of DPSK and noncoherentFSK (NFSK) systems over fading channels. Nevertheless, theerror performance of such systems in an arbitrarily correlatedNakagami environment isnot available in the literature. The dif-ficulty arises from inherent nonlinearity in noncoherent detectionand from attempts to determine explicitly the probability densityfunction of the total signal-to-noise ratio at the combiner output.In this paper, we directly determine the error probability from thecharacteristic function of decision variables, resulting in closed-form solutions involving matrix differentiation. The performancecalculation is further simplified by developing a recursive tech-nique. The theory is illustrated by analyzing two feasible antennaarrays used in base stations for diversity reception, ending upwith some findings of interest to system design.

Index Terms—DPSK, diversity reception, Nakagami channels,noncoherent FSK.

I. INTRODUCTION

DIVERSITY reception is an effective means to com-bat multipath fading. The fading environment of multi-

channels is statistically described by their joint distribution.Nakagami distribution is one such model in common use.This is largely attributable to its flexibility in that it can beused to account for both fast and slow fading, and includesthe classical Rayleigh fading as a special case. Furthermore, arecent study of propagation in mobile channels further justifiesthe use of such a model from the physical point of view [8].

Various combining schemes in a Nakagami environmenthave been studied [1]–[7]. Most of these studies focus onvarious predetection combining schemes operating in mutuallyindependent branch environments [6]–[7]. The independent as-sumption greatly simplifies the analysis. However, the resultsso far obtained may fail to reflect the suituations in the realworld. For example, typical correlation between two antennasused in a base station is 0.7 or even higher as a result ofspace limitation [15]. Correlation among branch antennas hasbeen taken in account in some recent publications [1]–[3]. The

Paper approved by N. C. Beaulieu, Editor for Wireless CommunicationTheory of the IEEE Communications Society. Manuscript received August 18,1997; revised March 25, 1998 and June 25, 1998. This work was supportedin part by the Natural Sciences and Engineering Research Council of Canada.

The author is with the Department of Electrical Engineering, Ryer-son Polytechnic University, Toronto, Ont., Canada M5B 2K3. (e-mail:[email protected]).

Publisher Item Identifier S 0090-6778(98)08136-7.

difficult issue is to determine the probability density function(PDF) of the instantaneous signal-to-noise ratio (SNR) at thecombiner’s output, and the state of the art is that only acouple of relevant PDF’s are available in the mathematicalliterature. As a consequence, the aforementioned studies eitherconcentrate on dual-branch combiners [1], [2] or multibranchcombiners with special correlation matrix [3]. The statisticalstructure assumed in [3] is constant or exponential correlationamong the antennas. In reality, correlation among the antennascan take on an arbitrary symmetrical structure, since it dependsnot only on the antenna array configuration, but also on theincident angle of the incoming wavefront. Analytical resultsare still not available for this general case.

The situation for postdetection combining is even worse, andits performance over correlated Nakagami channels remainsunknown. The difficulty in the past arose from the attempt toexpress the conditional error probability in terms of a totalsignal-to-noise ratio at the combiner’s output, and then tofind an explicit expression for probability density function ofthe total SNR. Both issues proved to be difficult due to thenonlinearity involved in postdetection combining and to thecurrent status in multivariate analysis. In this paper, we derivegeneral solutions to postdetection combiners for noncoherentdetection of DPSK and FSK signals over arbitrarily correlatedNakagami channels by taking a completely different approach.We directly work on the domain of characteristic functions(CF), thereby eliminating the need to explicitly determine thePDF of the SNR at the combiner’s output. The remainderof the paper is organized as follows. We briefly describe thenecessary background for correlated Nakagami distribution inSection II, followed by an exposition of the CF method fordetermining error probability in Section III. Error performanceof FSK and DPSK systems is derived in Sections IV andV, respectively. Section VI addresses the issue of efficientcalculation of the error performance. The analytical results arethen applied in Section VI to study two possible antenna arrayconfigurations for base stations. Section VII finishes the paperwith concluding remarks.

II. CORRELATED NAKAGAMI CHANNELS

We are interested in diversity reception systems which com-bine branches with joint correlated Nakagami distribution.For many combining schemes in use, the bit error performancedepends on the SNR’s at different diversity branches only

0090–6778/98$10.00 1998 IEEE

1460 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 46, NO. 11, NOVEMBER 1998

through their summation

(1)

where represents the SNR at theth branch antenna.Nakagami amplitude distribution implies that the variables

are jointly gamma distributed. Correlated gammavariables are usually defined by their joint characteristic func-tion [9], [10]. Let denote the covariance matrix of theSNR vector and define as

Then, the joint CF of the instantaneous SNRis given by [13]

(2)

where is the fading parameter andis the positive definitematrix determined by the branch covariance matrix asshown below. Denote and Usingthe rules (10.16) and (10.19–10.21) for derivatives involvingmatrices [14], we can show that

(3)

with denoting the th entry of The first equationimplies that the average SNR at theth branch equals m timesthe th entry of the matrix Scaling the -matrix bya 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 simplyby approprate scaling. With the symmetrical property ofinmind, the entry of the covariance matrix ofis obtained as

(4)

Hence, is related to the branch covariance matrix by

(5)

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

Return to the CF of the output SNR, Replacing in(2) with we obtain the CF of

(6)

This is an important relation for our subsequent analysis.

III. T HE CF METHOD

Various methods can be used to determine the bit errorperformance of a signaling scheme. But not all of themcan lead to a solution in a simple form, depending on thedecision variable and the fading environment. In this paper,we will use the method of characteristic function (CF), whichis particularly powerful for signals suffering from Nakagamifading. The idea of the CF method is described below. Suppose

is the decision variable with probability density function(PDF) The CF of is defined asWithout loss of generality, we assume that an erroneousdecision corresponds to The error probability can bedirecly determined from the CF, as shown by

(7)

(8)

Note that the integral inside the squared brackets on the lastline can be obtained from the Fourier transform of a unit-stepfunction; the result is It follows that

(9)

Here we have used the fact that represents the totalprobability and is therefore equal to unity.

The solution to the integral in (9) depends on the channelconditions. We identify two cases. Let denote the ratio

In the first case, the denominator of is atleast two orders higher than its numerator and, furthermore,all poles of on the right-half complex plane or on theleft-half plane are of integer order. In this case, we can relatethe path integral in (9) to a contour integral and determine itsvalue using the residue theorem. It turns out that

(10)

The first line should be used if all poles in are ofintegral order, and the second line should be used if all polesin are of integral order. In any event, we have aclosed-form solution. If, as encountered in some situations, allpoles in the complex plane are of integral order, then the half-plane having a fewer number of poles should be chosen so asto simplify the calculation.

In the second situation, not all poles in or inare of integral order. Note that the real part of

is an even function of since it is the Fourier transformof a real valued PDF. Thus, we can simplify (9) as

(11)

whose calculation relies on numerical methods.

ZHANG: EXACT ANALYSIS OF POSTDETECTION COMBINING 1461

Fig. 1. Postdetection combiner for FSK.

IV. NFSK SYSTEMS

Consider an order- diversity system for noncoherent detec-tion of FSK (NFSK) signals, as illustrated in Fig. 1. The signalcomponents received at different antennas are jointly Nak-agami distributed with an arbitrary covariance matrix, whereasthe noise components at different branches are Gaussian andspatially independent. We will first obtain the decision variableand then derive its error performance.

A. Decision Variable

The received signal at theth antenna in the symbolduration can be written as

(12)

where and and is zero-meanwhite Gaussian noise process with one-sided spectral density

The carrier frequency depends on the message withcorresponding to symbol 1 and to symbol 0. The influenceof the kth fading channel is characterized by the channel gain

and phase delay which remain unchanged over thesymbol interval In a complex form, the influence of thefading channel can be compactly expressed as

(13)

Without loss of generality, suppose symbol 1 is transmitted,and let us consider the branch The upper bandpass filters(BPF) is centered at the carrier frequency, thereby allowingthe desired signal to pass through; while the lower BPF iscentered at , hence blocking the signal component. Thenoise output components from the upper and lower BPF’scan also be represented in a complex form, denoted here by

and , respectively. They are mutually independentand each follows zero-mean complex Gaussian distributionwith variance equal to twice that for The BPF outputsare further applied to the square-law detectors. The diversitysystem combines the outputs from all upper branches andaccumulates information from all lower branches to enhance

symbol detection. These two combined outputs, when ex-pressed in terms of the complex channel gains and noisecomponents, are given by

(14)

respectively. The decision variable is then taken as theirdifference

(15)

Since symbol 1 is transmitted, an erroneous decision is madewhen , and a correct event otherwise. The situation isreversed when symbol 0 is sent.

B. Bit Error Performance

To determine the error probability, we begin with theconditional CF of To simplify notation, we use todenote a chi-square variable withdegrees of freedom andnoncentrality parameter For a given set of channel gains

the variables andhave the following distributions [11]:

(16)

Since the noise components are independent for alland it follows [12, p. 134] that the conditional CF ofequals

(17)

where is defined by

(18)

and denotes the SNR at the combiner’s output, given by

(19)


where is the instantaneous SNR at branchItis clear that the conditional CF depends on the channel gainsonly through the SNR We next average the conditional CFover the PDF of yielding

(20)

We recognize that the last integral is exactly the CF ofexcept that is replaced by Thus, we can write

(21)

which, when combined with (2), produces

(22)

To use residue to determine the error probability, we need toknow the location of poles introduced by the determinant. Ifwe denote the eigenvalues ofby it is easy to show

(23)

Clearly, all the poles introduced by the determinant are locatedon and, moreover, these poles can be of nonintegralorder, depending on the value of the fading parameter. Onthe other hand, has only one pole of integral orderon the left-half plane Consequently, we choose tocalculate the error probability using the pole onThe use of the second line of (10) yields

(24)

The residue represents the closed-loop integration ofover variable Change of variable

allows us to write

(25)

with denoting

(26)

This is the basic formula for diversity reception of NFSK overNakagami fading channels with an arbitrary branch covariancematrix. We only need to calculate the th derivative. Itis apparent that in noncoherent BFSK, the noise comoponentintroduces poles of integral order, making it possible to obtaina closed-form solution. If we put and then(25) reduces to

(27)

which is the result for noncoherent FSK system on Rayleighchannel [16, p. 200].

V. DSPK

The diversity system we will study consists ofbranches,as illustrated in Fig. 2. The received signal at theth branch,over the symbol interval is given by

(28)

where and signify the gain and the phase delay of thethchannel; is the magnitude of the transmit signal, andthecarrier frequency; is a zero-mean white Gaussian processwith one-sided spectral density The nth message symbolis encoded into the phase difference Consider theth branch. Let

(29)

denote its complex channel gain, and assume that the channelis slowly fading so that remains unchanged over twosuccessive symbol intervals. Let denote the complexoutput from the two samplers at theth branch. Then, thecomplex outputs over two successive symbols are

(30)

where is a white complex Gaussian sequence with meanzero and variance With these notations, it follows fromFig. 2 that the final output of theth branch equals

(31)

from which the decision variable follows:

(32)

where The use of the scaling factor of 2 isfor theoretical convenience and has no effect on the errorperformance.

To determine the bit error performance, we assume, withoutloss of generality, that the symbol is such that no phase changehappens over the th and th symbol intervals. Then, anerror event occurs when We take two steps to analyzethe bit error performance of DPSK. First, we determine theconditional CF of by assuming that all channel gainsare given. Then, we average the conditional over the jointNakagami distribution of the channel gains to obtain the finalresult. Let us begin with the conditional CF’s of Tosimplify notation, define

(33)

Then, can be written in a quadratic form, as shown by

(34)


Fig. 2. Postdetection combiner for DPSK.

Given the channel gains are mutually in-dependent. Under this condition, has the complex Gaussiandistibution

(35)

from which the conditional CF of the quadratic formfollows:

(36)

Note that due to the special structure of the matrixtheexponent in (36) can be greatly simplified. Denote

(37)

The properties of to be used are and its specialeigenstructure. Specifically, has eigenvalues and 1 withthe corresponding eigenvectors given by

(38)

Using this eigenstructure, we can easily simplify (37) to obtain

(39)

We can also show that

(40)

The results obtained above, together with the mutual indepen-dence of for a given set of channel gains, allow us towrite

(41)

It is clear that the conditional CF for DPSK has exactly thesame form as that for noncoherent FSK, except that is

replaced by The same is true of the unconditional CF;thus, we have

(42)

Comparing to (22), we see that the CF for DPSK has exactlythe same as that for noncoherent FSK, except thatin (22)is replaced by Consequently, the formula (25) remainsapplicable and the only change is that

(43)

For and the above formula reduces to

(44)

which is identical to the well-known result for nondiversity re-ception of DPSK signals suffering from Rayleigh fading [16].

From the above analysis, we see that it is the specialeigenstructure of that introduces a pole of order on

making it possible to use the residue theorem.In the performance analysis of DPSK systems subject tosmall Doppler frequency shift, it is common practice toassume that the channel gain remains unchanged over twosuccessive symbol intervals. This assumption makes it possibleto use a joint gamma distribution to obtain a closed-formsolution for DPSK in a Nakagami environment. Without thisassumption, the entries of vector in the exponent of(36) would take the form of instead. Thislatter vector consists of both amplitude and phase variation,thereby requiring a joint complex Nakagami distribution forits characterization. Unfortunately, such a distribution is notavailable in the literature. We therefore remark that onlywhen Doppler shift is small, can the problem of DPSK ina Nakagami environment be resolved leading to a closed-formsolution in form similar to that for NFSK.


VI. EFFICIENT EVALUATION

The closed-form solution provided in (25) involves a deriva-tive of order Determination of such a derivative canbe quite complicated or even prohibitive whenis large, sincethe function to be differentiated is the product of three factors.Furthermore, the function includes a matrix as its argument,thereby further complicating the calculation. It is thereforenecessary to develop an efficient technique to resolve theproblem. To this end, we denote

(45)

To avoid differentiating a product, we consider the use of thelog function to convert as a summation. Thus, we startfrom the relation

(46)

and invoke the formula (14) of [14, p. 230]

(47)

to rewrite the error probability as

(48)

As a convention, the derivative with order zero simply repre-sents the original function. We note that in (47), the derivativeof order depends only on the derivatives of lower orderup to Thus, we can calculate the derivative of order

recursively by starting from the original functionand increasing the order by one in every iteration until

We need to determine the derivatives ofwhich is explicitly expressible as

(49)

It remains to determine the last term on the right. By usingthe formulas (10.19–10.21) for matrix differentiation in [14,p. 45] and (47) above, we are able to obtain

(50)

Here, is defined as Whenis a product of three matrices. For we need to continuethe computation and the result is the summation of matrix

(a)

(b)

Fig. 3. Two antenna configurations. (a) Triangular array. (b) Linear array.

products. The coefficients in the sum have the same form asthose in a multinomial expansion. The explicit expression is

(51)

In this expression,

Equations (48)–(51) constitute efficient formulas for recursiveevaluation of the bit error performance.

VII. I LLUSTRATING EXAMPLES

As an illustration of the theory, we study some feasiblediversity systems for a base station. Two kinds of antennaconfigurations will be conisdered here, due to their relativeease of installation. They are uniform linear and triangulararrays, as sketched in Fig. 3. The correlation matrix of anantenna array depends not only on the array configuration butalso on the incident angle of the incoming signal. Differentparameters affect the correlation differently. Increase in theantenna separation between two antennas tends to reducetheir correlation. The antenna height behaves in a similarmanner; the higher the antennas, the lower the correlation.The correlation matrix of a given antenna array varies with


the signal incident angle. Correlation between two antennasattains its maximum value when the signal arrives from thebroadside, and reduces gradually as the signal direction movestoward the endfire direction.

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 that theantenna is 100 ft in height. A signal at 850 MHz impliesthat its wavelength is m or 1.16 ft. For bothlinear and triangular arrays, any two adjacent antennas areassumed to have equal separation, denoted here byTobe representative of practical operational conditions with alinear array, we assume that in all cases treated below, signalsimpinge upon the array at relative to the broadside of thearray. For a triangular array, we assume that signals come fromthe broadside of the line linking antennasand ; refer toFig. 3. Then, the incident angles to and to are bothequal to We therefore expect that the correlation betweenantennas and (or between and ) is higher than thatbetween and , regardless of their equal separation.

Figs. 4–6 are shown for DPSK; the discussion of NFSK isdeferred to Fig. 7. We first study the influence of the fadingparameter on the error performance using linear and triangulararrays, both having three antennas and with antenna spacing

ft. This separation is close to the practicalnumber used in the real world [15]. With this parameter setting,we can determine the correlation matrix of an antenna arrayin accordance with empirical results of Lee [15, p. 203]. Thecorrelation matrix for the triangular array is given by

(52)

and for the linear array, it is given by

(53)

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

(54)

where denotes the matrix with its entries equal to thesquare root of the corresponding entries of Note thecorrelation matrix of the linear array has a Toeplitz structure,whereas that of the triangular array does not.

The results for linear and triangular arrays are graphedin Fig. 4(a) and (b), respectively. As expected, the errorperformance improves with increased fading parameter sincesmall corresponds to fast fading. In particular,corresponds to the situation of Rayleigh fading. Sinceisnot a linear function of we do not expect a uniform changeof with This becomes evident in the curves. We alsosee that the linear array has a better performance than thetriangular array, at the cost of a larger array dimension.

(a)

(b)

Fig. 4. Variation of Pe withm: (a) linear array, (b) triangular array.

Fig. 5. Variation of Pe with diversity order.


Fig. 6. Variation of Pe withd for triangular array.

We next study the way the diversity order affects theperformance. The correlation matrix for a five-antenna lineararray with ft is

(55)

The correlation matrices for 3, and 4 are the submatrixof The fading parameter we used was and theresults are shown in Fig. 5. The curves show that increaseddiversity order significantly improves the system performanceover fading channels.

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 performance is affected by the antenna separation. Thevariation of system error performance with antenna separation,

is shown in Fig. 6 for a triangular array. The fadingparameter was The correlation matrix still takesthe form

(56)

The values for and vary with For 8, 10, 12, 16,and 20, equals (0.815, 0.994), (0.727, 0.913), (0.620,0.802), (0.543, 0.723), (0.450, 0.672), and (0.360, 0.527),respectively. The error performance improves with increasedantenna separation at a varying degree, depending on the valueof The results suggest that an appropriate antenna separationshould be such that the correlation falls into the range from0.6–0.65.

The results shown above are all for DPSK systems. Acomparison between DPSK and nonherent FSK are illustratedin Fig. 7 for a triangular array with spacing The casesof and are considered. It is observed that theDPSK scheme is about 3 dB better than NFSK.

Fig. 7. Performance comparison between DPSK and FSK.

VIII. C ONCLUSION

In this paper, we have examined the error performance ofpostdetection combiners for noncoherent detection of DPSKand FSK signals over Nakagami channels with an arbitrarycovariance matrix. We tackled the problem in the characteristicfunction domain and ended up with closed form solutionsin terms of differentiation. We also developed an efficienttechnique to simplify the error performance calculation. Itwas revealed that if channel gains remain unchanged over twosuccessive symbol intervals, the error probability of DPSK hasthe same expression as NFSK except that the-matrix usedfor NFSK is replaced by for DPSK. Without this condition,the error probability of DPSK would take a different form.

The formulas were then used to study two types of diver-sity schemes that can be used in a base station, one withlinear antenna arrays and the other with triangular arrays.Computer results show that increase of the diversity ordercan significantly improve the error performance at the cost ofincreased antenna site. Properly increasing antenna separationcan also help improve the performance. However, when thecorrelation reaches 0.6, further increasing the separation willbasically trade array dimension for little improvement. Asan overall consideration, the triangular array with adjacentseparation ranging from 7–10 wavelengths is a good candidatefor practical application.

REFERENCES

[1] Y. Miyakgaki, N. Morinaga, and T. Namekawa, “Error probabilitycharacteristics for CPSK signal through m-distributed fading channel,”IEEE Trans. Commun., vol. COM-26, pp. 88–89, Jan. 1978.

[2] E. K. Al-Hussaini and A. M. Al-Bassiouni, “Performance of MRCdiversity systems for the detection of signals with Nakagami fading,”IEEE Trans. Commun., vol. 33, pp. 1315–1319, Dec. 1985.

[3] V. A. Aalo, “Performance of maximal-ratio diversity systems in corre-lated Nakagami-fading environment,”IEEE Trans. Commun., vol. 43,pp. 2360–2369, Aug. 1995.

[4] G. Efthymoglou and V. Aalo, “Performance of RAKE receivers inNakagami fading channel with arbitray fading parameters,”Electron.Lett., vol. 31, no. 18, pp. 1610–1612, Aug. 1995.

[5] M. Blanco, “Diversity receiver performance in Nakagami fading,” inProc. IEEE Southeastern ’83 Conf., Orlando, FL, pp. 529–532.


[6] T. Eng and L. B. Milstein, “Coherent DS-CDMA performance inNakagami-fading environment,”IEEE Trans. Commun., vol. 43, nos.pp. 1134–1143, Feb./Mar./Apr. 1995.

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

[8] W. R. Braun and U. Dersch, “A physical mobile radio channel mode,”IEEE Trans. Veh. Technol., vol. 40, pp. 472–282, Feb. 1991.

[9] J. Curland, “Distribution of the maximum of the arithematic mean ofcorrelated random variables,”Ann. Math. Statist., vol. 26, pp. 294–300,1955.

[10] S. Kotz and J. W. Adams, “Distribution of a sum of identicallydistributed exponentially correlated Gamm-variables,”Ann. Inst. Statist.Math., vol. 35, pp. 277–283, 1964.

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[12] N. L. Johnson and S. Kotz,Continuous Univariate Distributions, vol.2. New York: Wiley, 1976.

[13] M. D. Springer,The Algebra of Random Variables. New York: Wiley,1979.

[14] W. H. Beyer, CRC Standard Mathematical Tables, 28th ed. BocaRaton, FL: CRC Press, 1987.

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Q. T. Zhang (S’84–M’85–SM’95) received theB.Eng. degree from Tsinghua University, Beijing,and the M.Eng. degree from South China Universityof Technology, Guangzhou, China, both in radiocommunications, and the Ph.D. degree in electricalengineering from McMaster University, Hamilton,Ontario, Canada.

After graduation from McMaster in 1986, hespent a few years at the Communications ResearchLaboratory of the same institution, as a Senior Re-search Engineer and Adjunct Professor conducting

research on communications and signal processing. In January 1992, he joinedthe Spar Aerospace Ltd., Satellite and Communication Systems Division,Montreal, as a Senior Member of Technical Staff. At Spar Aerospace,he was first responsible for the image performance analysis of the RadarSatellite and was subsequently engaged in the developemnt of the advancedsatellite communication systems for the next generation. Since September1993, he has been a tenured Associate Professor with the Department ofElectrical Engineering, Ryerson Polytechnic University, Toronto. His researchis primarily in the areas of wireless communications and ATM networkswith current emphasis on frequency and interference management, handoffs,reliable reception over various fading channels, intelligent beamforming, ATMswitches, video traffic modeling, and source compression.

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