IEEE COMMUNICATIONS LETTERS, VOL. 11, NO. 5, MAY 2007 393

Simple SER Formulas for Noncoherent MFSK withSquare-Law Diversity Combining in Generic Nakagami-m Fading

Qinghua Shi and Q. T. Zhang, Senior Member, IEEE

Abstract— The error performance of M -ary frequency shiftkeying (MFSK) with square-law diversity combining (SLDC) oncorrelated Nakagami-m fading channels was previously analyzedin the literature. The results are either in the form of anintegral, or a closed form involving the calculation of higher-order derivatives, or other closed forms subject to the constraintof integer-valued fading parameters. In this letter, we formulatethe problem in the framework of Lauricellar hypergeometricfunctions leading to a simple symbol-error-rate formula involvingonly some elementary functions for generic correlated Nakagamifading environments.

Index Terms— M -ary frequency shift keying, square-law di-versity combining, Nakagami-m fading.

I. INTRODUCTION

M -ARY frequency shift keying (MFSK) with square-law diversity combining (SLDC) is a simple and ef-

fective transmission technique, and its error performance hasbeen analyzed for various operational environments includingNakagami-m fading. In particular, two closed-form solutionsto the symbol error rate (SER) of MFSK with post-detectionSLDC for Nakagami fading with an integer fading parameterare obtained in [1] and [2]. The unified closed form SERexpression given in [3] allows for generic Nakagami fadingwith arbitrary correlation and real fading parameters. Theonly drawback is that the solution involves some higher orderderivatives of the characteristic function (CHF) of the decisionvariable, requiring a recursive algorithm for its efficient cal-culation. An alternative solution for generic Nakagami fadingis subsequently given in [4], taking the form of one integral.

A new probability density function (PDF) was recentlyderived in [5] by virtue of confluent Lauricellar multivariatehypergeometric functions (LMHFs) [6], providing anotherpowerful means for tackling the performance issue of MFSK-SLDC. Radaydeh et al. subsequently show that this PDF canbe used alongside the classical conditional SER to obtainelegant performance results for MFSK-SLDC in Rayleighfading [7]. Though interesting, the approach of [7] is notextendible to other fading environments. In this letter, weshow how to combine Aalo’s PDF with the conditional SER

Manuscript received November 28, 2006. The associated editor coordinat-ing the review of this letter and approving it for publication was Prof. GeorgeK. Karagiannidis. This work was supported by the Dean’s Grant from theFaculty of Sciences and Engineering, City University of Hong Kong.

Q. Shi was with the Department of Electronic Engineering, City Univer-sity of Hong Kong, Hong Kong. He is now with Nanyang TechnologicalUniversity, Singapore (e-mail: qhshi@ntu.edu.sg).

Q. T. Zhang is with the Department of Electronic Engineering, CityUniversity of Hong Kong, 83 Tat Chee Avenue, Kowloon, Hong Kong (e-mail: eekzhang@cityu.edu.hk).

Digital Object Identifier 10.1109/LCOMM.2007.061944.

obtained in [4] to produce a simple solution for MFSK-SLDCon generic correlated Nakagami fading channels.

II. AVERAGE SER

Consider an L-branch SLDC system for the detection ofMFSK signals in Nakagami-m fading. The system model isthe same as that in [4]. Let Es denote the symbol energy,let N0 denote the power spectral density of additive whiteGaussian noise (AWGN), and let αl and ml ≥ 0.5 denotethe channel gain and fading parameter of the lth diversitybranch, respectively. With these notations, we can representthe instantaneous signal-to-noise ratio (SNR) at branch l asγl = α2

l Es/N0, and denote the total instantaneous SNR at theSLDC output by

γ =L∑

l=1

γl. (1)

We will start our derivation from the special case of indepen-dent branches, and then extend the results to the general caseof correlated channels.

A. Independent Nakagami Fading

A novel conditional (given γ) SER for MFSK-SLDC isgiven by of [4, eq. (9)], which, by using the Kummer trans-formation [8, eq. (13.1.27)], can be expressed as

Pe(γ) =M−1∑n=1

(M − 1

n

)(−1)n+1e−

nn+1 γ

·n(L−1)∑

k=0

σk,n(L)k

(1 + n)k+L

k∑ν=0

(−k)ν

(L)ν

(− γn+1 )ν

ν!(2)

where, for a real number a, the Pochhammer symbol (a)n isdefined by (a)n � a(a+1) · · · (a+n−1) with (a)0 � 1, andthe coefficient σk,n by [4]

σk,n ={

1 (k = 0)1k

∑kj=1(jn − k + j)ajσk−j,n (k ≥ 1)

(3)

with aj = 1j! for j ≤ l − 1 and aj = 0 otherwise.

This conditional SER, when directly averaged over the usualPDF of γ, leads to the average SER in the form of oneintegral [4]. The situation, however, is quite different if wecombine (2) with a new PDF expression of γ by Aalo [5] toexploit properties of LMHFs. By defining the order-r confluentLMHF as [6]

Φ(r)2 (b1, · · · , br; c;x1, · · · , xr)

�∞∑

n1,··· ,nr=0

(b1)n1 · · · (br)nr

(c)n1+···+nr

xn11

n1!· · · xnr

r

nr!, (4)

1089-7798/07$25.00 c© 2007 IEEE

394 IEEE COMMUNICATIONS LETTERS, VOL. 11, NO. 5, MAY 2007

the new PDF of γ can be written as [5]

fγ(γ) =

[L∏

i=1

(mi

γi

)mi]

γmΣ−1

Γ(mΣ)

·Φ(L)2

(m1, · · · ,mL;mΣ ;−m1

γ1

γ, · · · ,−mL

γL

γ

)(5)

with mΣ �∑L

l=1 ml and γl signifying the average SNR atbranch l.

We first average (2) over (5) to obtain

Pe =

[L∏

i=1

(mi

γi

)mi]

M−1∑n=1

(M − 1

n

)(−1)n+1

·n(L−1)∑

k=0

σk,n(L)k

(1 + n)k+L

k∑ν=0

(−k)ν

(L)ν

(− 1n+1 )ν

ν!I(n, ν) (6)

where I(n, ν) is defined by

I(n, ν) � 1Γ(m∑ )

∫ ∞

0

γν+mΣ−1e−n

n+1 γ

·Φ(L)2

(m1, · · · ,mL;mΣ ;−m1

γ1

γ, · · · ,−mL

γL

γ

)dγ (7)

and the integral can be simplified by using [7, eq. (12)] or [6,eq. (2.4.10)]. It turns out that

I(n, ν) = (mΣ)ν

(n + 1

n

)ν+mΣ

·F (L)D (ν + mΣ ;m1, · · · ,mL;mΣ ;−ξ1, · · · ,−ξL) (8)

where

ξl � (n + 1)ml

nγl

, l = 1, 2, · · · , L (9)

and F(r)D denotes the rth order LMHF of the fourth type [6],

given by

F(r)D [a, b1, · · · , br; c;x1, · · · , xr]

�∞∑

n1,··· ,nr=0

(a)n1+···+nr

(c)n1+···+nr

(b1)n1

n1!· · · (br)nr

nr!xn1

1 · · ·xnrr . (10)

We further denote

θk �[1 +

nγk

(n + 1)mk

]−1

, k = 1, 2, · · · , L, (11)

and invoke [5, eq. (19)] to simplify (8) yielding

I(n, ν) = (mΣ)ν

(n + 1

n

)ν+mΣ

[L∏

l=1

(1 + ξl)−ml

]

·F (L)D (−ν;m1, · · · ,mL;mΣ ; θ1, · · · , θL) . (12)

Since −ν is a non-positive integer, it follows from (10) thatF

(L)D in (12) reduces to a finite sum. Specifically, all terms

satisfying n1 +n2 + · · ·+nL > ν in the multiple sum of (10)become zero according to the definition of the Pochhammersymbol. As such, (12) simplifies to

I(n, ν) = (mΣ)ν

(n + 1

n

)ν+mΣ

[L∏

l=1

(1 + ξl)−ml

]

·ν∑

n1,··· ,nL=0n1+···+nL≤ν

(−ν)n1+···+nL

(mΣ)n1+···+nL

L∏k=1

(mk)nkθnk

k

nk!. (13)

The average SER given by (6) and (13) involves only elemen-tary functions. It is applicable to independent Nakagami-mfading channels with arbitrary fading parameters and branchpowers.

Let us check the new SER formula by considering thespecial case of L = 1 which implies k = ν = 0 in (6) and(13). It follows that

Pe =M−1∑n=1

(M − 1

n

)(−1)n+1 1

1 + n

[1 +

nγ

(n + 1)m

]−m

(14)which is identical to the well-known result given in, forexample, [9, eq. (4)].

B. Correlated Nakagami Fading

The results obtained above can be easily extended to thecases of correlated Nakagami fading.

1) Identical Fading Parameters: It is commonly assumedthat L correlated diversity branches are subject to identicalNakagami-m fading with arbitrary real-valued fading indexes,i.e., m1 = m2 = · · · = mL � m ≥ 0.5. Let R denotethe normalized covariance matrix of {γl}L

l=1. The momentgenerating function (MGF) of γ is given by [10]

φγ(s) =L∏

l=1

(1 +

s

mλlγl

)−m

(15)

where {λl}Ll=1 are L distinctive eigenvalues of

√R and

√R

denotes a matrix with its entries equal to the square root ofthe corresponding entries of R. In other words, the correlatedNakagami model is statistically equivalent to an independentone with average branch powers {λlγl}L

l=1 and identicalfading parameters m. Inserting these equivalent parametersinto (6) and (13) allows us to readily determine the averageSER for correlated Nakagami fading.

2) Arbitrary Fading Parameters: A generic correlated Nak-agami fading channel model with arbitrary covariance matrixand arbitrary real-valued fading parameters is described in[11], where the channel is fully determined by the joint CHF[11, eq. (6)]. With the help of Property 4 of [11], the CHFof γ = γ1 + · · · + γL can be written in a form similar to(15), thereby the equivalent fading parameters and branchpowers can be identified. These equivalent parameters, wheninserted into (6) and (13), produce the average SER for thecorresponding correlated Nakagami fading channels.

C. Numerical Results

We investigate the system performance on the basis of biterror rate (BER) versus bit energy (Eb). The BER and Eb arerelated to the SER and Es by

Eb =Es

log2 M

Pb =M

2(M − 1)Pe,

respectively. The results for 4FSK in independent and identi-cally distributed (i.i.d.) Nakagami fading are graphed in Fig.1 where BER is shown as a function of Eb/N0 for different

SHI and ZHANG: SIMPLE SER FORMULAS FOR NONCOHERENT MFSK WITH SQUARE-LAW DIVERSITY COMBINING 395

Fig. 1. BER performance of 4FSK-SLDC with i.i.d. branches.

Fig. 2. BER performance of MFSK-SLDC with correlated and unbalancedbranches (L = 3).

settings of m and L. As expected, the BER performanceimproves significantly as the fading parameter m or diversityorder L increases. It is also observed that simulation resultsfor the Rayleigh fading case agree with their theoreticalcounterparts.

The BER results for using an order-3 (L = 3) SLDCfor non-coherent detection of MFSK signals in correlatedNakagami fading are depicted in Fig. 2, where it is assumedthat the three branches have the same fading parameters, i.e.,m1 = m2 = m3 = m, and the normalized covariance matrix

of {γl}Ll=1 is given by

R =

⎛⎝ 1 0.795 0.605

0.795 1 0.6050.605 0.795 1

⎞⎠ . (16)

We also set different average SNRs for different branches,such that γ1 : γ2 : γ3 = 1 : 0.5 : 1.5.

III. CONCLUSION

In this letter, an elegant combining of the conditional SERexpression given in [4] and a novel PDF of the combineroutput SNR recently given in [5] enabled us to exploitthe properties of the Lauricellar multivariate hypergeometricfunctions [6] to produce a simple closed-form solution tothe average SER for square-law diversity combining withnon-coherent MFSK in Nakagami fading. The results aresimple, involving only elementary functions and applicableto generic correlated Nakagami fading environments witharbitrary covariance matrix and fading parameters.

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