Digital Communications over Generalized-KFad ing Canne s

Petros S. Bithas*, Nikos C. Sagias*, P. Takis Mathiopoulos*, George K. Karagiannidist,and Athanasios Rontogiannis*

*Institute for Space Applications and Remote Sensing, National Observatory of Athens,Metaxa & Vas. Pavlou Street, 15236 Athens, Greece E-mail: {pbithas;nsagias;mathio;tronto}@space.noa.gr

tDivision of Telecommunications, Electrical and Computer Engineering Department,Aristotle University of Thessaloniki, 54124 Thessaloniki, Greece E-mail: geokarag@auth.gr

Abstract- In this work, a flexible distribution, called asGeneralized-K, is used for fading channel modeling. Startingfrom its probability density function (PDF), useful closed-formexpressions for the cumulative distribution function, moments-generating function, moments, and average Shannon's channelcapacity are derived. These expressions are used to study impor-tant performance criteria such as the capacity, amount of fading,outage performance and bit error probability for a great varietyof modulation formats. The proposed mathematical analysis isaccompanied with various performance evaluation results whichdemonstrate the usefulness and flexibility of the proposed model.

I. INTRODUCTION

Radiowave propagation through wireless channels is acomplicated phenomenon characterized by various effects in-cluding multipath fading and shadowing. Multipath fading isintroduced due to the constructive and destructive combinationof randomly delayed, reflected, scattered, and diffracted signalcomponents. Depending on the nature of the radio propaga-tion environment, there are different models describing thestatistical behavior of multipath fading [1]. Relatively newmodels in communications over fading channels are the Kand the Generalized-K distributions [2], which in the pasthave been widely used in radar applications [3], [4]. TheGeneralized-K distribution has two shaping parameters, andas a consequence includes the K distribution as a special case.Moreover, it is sufficiently generic as it is able to incorporatemost of the fading and shadowing effects observed in wirelesscommunication channels and hence seems to be appropriatefor the generic modeling of fading channels.

Representative past work conceming the K distribution canbe found in [5]-[7]. In [5], Iskander et al. have proposeda method that combines the maximum likelihood and themethod of moments for estimating the parameters of the Kdistribution. In [6], Abdi and Kaveh have shown that theK model provides similar performance to the well-knownRayleigh-Lognormal (R-L) model with the former being alsomathematically tractable. The same authors in [7] have derivedin closed-form the bit error rate of differential phase-shiftkeying (DPSK) and minimum-shift keying operating over Kfading channel. Moreover, in the same work they presented a

0-7803-9206-X/05/$20.00 ©22005 IEEE

close agrement between the derived results and those basedon the R-L distribution.

In [2], the generalized-K distribution was presented asanalytically simpler than the Nakagami-lognormal or Suzukidistributions and general enough to approximate them, aswell as several others including Rayleigh and Nakagami-m.Moreover, in the same work, the Amount-of-Fading (AF) andthe Average Bit-Error-Rate (ABER) for the special case of thebinary phase swift keying (BPSK) modulation, were derived.However, detailed performance analysis for the SNR statisticsof a receiver operating over Generalized-K fading channel hasnot yet been published in the open technical literature and thisis the topic of the current work.

In this paper, after this introduction, the Generalized-K sys-tem and channel model is studied and closed-form expressionsfor its statistics are derived in Section II. In Section III theperformance analysis of a single receiver is investigated, whilein Section IV several numerical evaluated results are discussed.

II. THE GENERALIZED-K FADING CHANNEL MODELThe probability density function (PDF), fx (.), of the en-

velope X under the Generalized-K fading conditions is [2]

2b bx k+m-1fx (x) =r (k) 2)

where k and m are distribution's shaping parameters,Kk-m (.) is the modified Bessel function of order (k - m)[8, eq. (9.6.1)], F (.) is the Gamma function [8, eq. (6.1.1)],b - 2/m/Q and Q is the mean power and can be derived byusing [2, eq. (7)] as Q = E (X2) /k. Since the Generalized-K is a two parameters distribution (1) is able to describe agreat variety of fading and shadowed models. For example,for m -- oc and k o00, (1) approaches hte Additive WhiteGaussian Noise (AWGN), no fading, channel, for k -3 00, itapproaches to the well-known Nakagami-m PDF, [2], whilefor m = 1, it reduces to the R-L PDF [6], [7].The instantaneous signal-to-noise (SNR) per symbol is

defined as -y = X2E /No, where Es = E(1s12), withE (.) denoting expectation and 1. absolute value and No isthe single-sided power spectral density of the AWGN. Usingthe instantaneous SNR, the corresponding average SNR 7 =

684

Kk m (b x) (1)

Qk EI/No, [9, eq. (03.04.21.0008.01)] and (1) the cumulativedistribution function (CDF) of -y can be easily derived as

F^1() =wr csc [7r (k-rm)][m(knmk1rrniF2(m;1 +m - k,1 +rm;mk-y/77)Lk FJ(k)F(I +m- k) F(I+m)

(mk k) kkF2(k;1-m+k,i +k; mk/V FJ (m)IF(1-m+k) F(I+k)

(2)where pFq (.) is the generalized hypergeometric function,[10, eq. (9.14/1)], with p, q integers. Differentiating (2) withrespect to x, the PDF of ^y can be derived as

( Y) _ r csc [,7r (k - m)]"Ya r(k) F(m)

x [(mk) moF (1 +m -k;mk)y/7)(m k)k ykoF (1-m + k; m ky/7)]V aJ F(l-m+k)

(3)By using (3) and [10, eq. (7.522/9)], the moments-

generating function (MGF) can be easily obtained asMa (s) =ir csc [7r (k - m)]

mmk m 1F1 [m; 1 +m - k;m kl(soy)]J\ ] sr F (k)F(l+rm- k)

_m k Fk1F [k; 1-m + k; mk/ (s7)]a skr(m)r(1-m+k)

(4)Moreover, starting with the definition of the nth order moment[11, eq. (5.38)] and using (3), integrals of the form I1 =f0'0 XMOF1 (b; x) dx need to be solved. 11 can be solvedby using [10, eq. (7.811/4)] and expressing the confluenthypergeometric function OF1 (.) as [9, eq. (07.17.26.0008.01)]

OF1 (b; x) =rE (b)G3 [XI o 1-b, 1/21 (5)

where Gpq [Xbbb2,' 'bq is the Meijer's G-function [10,eq. (9.301)]. Hence, the nth order moment of 7y output SNR,pw- (n), can be easily derived after some straight-forwardmathematical manipulations as

(im k -n csc [7r (k - m)](n) =rr2 (nFk

xr(n+m)

F(l-k-n)r(1/2-im-n)F(1/2+n+m)r(n+k)1

F(1-m-n)F(1/2-k-n)F(1/2+n+k)](6)

Using (3) in the definition of the average channel capacity C[12], in Shannon's sense, integrals of the form appear

2 =Jxm-10g2 (1+x)oF (1+im-k;ink) dx.

(7)

10

._

.0

m)loto 0-av

¢~

10-4-5 0 5 10 15

Average Input SNR per Bit (dB)20

Fig. 1. ABEP of DBPSK, BPSK, Gray-encoded 16-PSK and 16-QAMsignaling versus a for several values of k and m.

By expressing In (1 + x) = G"2 [xi 1' 1] and 0F1(.) as in(5), 12 can be solved with the aid of [13] and C can beobtained in closed-form as

- 7r csc [7r (k - m)] BWln 2F (k) F (m)

T{( k)mG3mk -mr, 1-rn-xlk G3,10_k -m -t ,, 2,4 L 7a -m,-m,k-m]

7(o) 2,4 [- - -k, -k, m-kjwhere BW is signal's transmission bandwidth.

III. PERFORMANCE ANALYSIS

In this section the performance of a single receiver operatingover the Generalized-K fading channel will be investigated.The received baseband signal is z = s h + n, where s is thetransmitted complex symbol, n is the complex AWGN, and his the channel complex gain. One of the most commonly usedperformance criterion for telecommunications systems operat-ing over fading channels is the bit-error probability. By using(4), and following the MGF based approach, [1], the average

bit-error probability (ABEP) can be readily evaluated for a

variety of modulation schemes [1]. Hence, the ABEP can becalculated i) directly for non-coherent binary frequency-shiftkeying and differential binary phase-shift keying (DBPSK)and ii) via numerical integration for binary phase-shift keying(BPSK), M-PSK, M-ary quadrature amplitude modulation(M-QAM) and M-DPSK since single integrals composed ofelementary (i.e., exponential and trigonometric) functions andwith finite limits are obtained.

685

(8)

"10.

.0

.0

0

to

10- -

C1io-

-15 S-10 -5 0

Normalized Outage Threshold (dB)

8

7

61-11

;._to5

0

o 3

2

0

Fig. 2. P,,,t as a function of the normalized outage threshold for severalvalues of k and m.

3.0Another standard performance criterion is the outage proba-

bility, P,ut which is defined as the probability that the outputSNR falls below a given threshold, Yth. Hence, Pout(7th) can

be simply obtained as

Po0t (-Yth) = F, ('7th)- (9)

Moreover, AF, which is another important statistical char-acteristic of fading channels, can be easily obtained by using,[1],

(10)AF= var (y) Hay (2) _1.A -72 - --2

Finally, the normalized average channel capacity is directlyrelated to the bandwidth efficiency and denotes the amountof data transmitted in a given spectrum allocation. It can beeasily obtained, by using (8), as C/BW, in terms of b/s/Hz.

IV. NUMERICAL RESULTS

In order to demonstrate the usefulness of the theoreticalvalue and based on the proposed formulation, the performanceof a receiver operating over the Generalized-K fading channelis presented in Figs. 1-4.More specifically, in Fig. 1, the ABEP is plotted for DBPSK,

BPSK, 16-PSK and 16-QAM signaling with Gray encoding, as

a function of the average input SNR per bit, yb = P7/ log(M),for several values of m and k. As expected, the ABEP im-proves as 7b increases, while for a fixed value Of 5b, ABEPalso improves with an increase of k and/or m. Fig. 2 showsPout versus normalized outage threshold for several valuesof k and m. Pout decreases (i.e., the outage performanceimproves) with an increase of k and/or m. However the gapamong the curves decreases as k and/or m increase. Notice

N .

a

.2.0

clV

r-0= 1.5C

0

C 1.00

0.5

2 4 6Shaping parameter, m

8 10

Fig. 3. AF as a function of m for several values of k.

-5.0 -2.5 0.0 2.5 5.0

Average Input SNR per Bit (dB)7.5 10.0

Fig. 4. C/BW versus average SNR per Bit, for several values of m and k.

that for m = 1, the Generalized-K distribution approachesthe R-L one.

The AF is presented as a function of the m parameter andfor several values of k. Note that as m and/or k increase,the AF decreases in an exponential way. Moreover, it is moreclear in Fig. 3 that the variation of m, k has greater influenceat the channel's performance when they have small values,i.e., m E [0,2] and k E (0.5,2). Finally, in Fig. 4 the

686

k=0.5k=k=2k=5k= 10 -

..

... ..

I

normalized average channel capacity is plotted as a functionof the average SNR per bit for several values of m, k. Forcomparison purposes, the normalized channel capacity for theAWGN, i.e., Cawgn/BW = log2(1 + 7Yb), and the Rayleighchannels are also plotted. As it was expected, the averagecapacity of the Generalized-K fading channel is always lessthan the capacity provided by the AWGN channel. Moreover,for m = k = 1 the provided capacity is less than Rayleigh's,and it improves as m and/or k increase.

ACKNOWLEDGMENTS

This work has been performed within the framework of theSatellite Network of Excellence (SatNEx) project, a Networkof Excellence (NoE) funded by European Committee (EC)under the FP6 program.

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