Representation of Composite Fading and Shadowing Distributions by using Mixtures of Gamma Distributions

Transcription

1 Representation o Composite Fading and Shadowing Distributions by using Mitures o Gamma Distributions Saman Atapattu, Chintha Tellambura, and Hai Jiang Department o Electrical and Computer Engineering, University o Alberta, Edmonton, AB, Canada {atapattu, chintha, hai.jiang}@ece.ualberta.ca Abstract The akagami-lognormal distribution is the commonly used composite distribution or modeling multipath ading and shadowing. In this paper 1, simple and new orm o distribution which can accurately represent both the mutlipath ading and shadowing eects is introduced. The signal-to-noise ratio SR) o the akagami-lognormal distribution ollows the gamma-lognormal distribution, which is accurately approimated by a weighted miture o gamma distributions. We show how the weights and other parameters o the summands are obtained. Further, accuracy o the miture distribution is compared with the K G distribution a popular approimation o the akagamilognormal distribution. Inde Terms Gaussian-Hermite integral, K G ading model, akagami-lognormal distribution. I. ITRODUCTIO Fading small-scale signal power luctuations) is a undamental characteristic o wireless channels. Several ading models such as Rayleigh, Rice and akagami distributions are widely used in wireless research. The large-scale signal power variation is shadowing, which is statistically modeled by the lognormal distribution [1]. Further, the composite ading and shadowing channel is modeled as Rayleigh-lognormal, Rice-lognormal, or akagami-lognormal distribution. However, such composite distributions do not lend themselves to perormance analysis readily. For eample, their cumulative distribution unction CDF) and moment generating unction MGF) may not epressible in simple closed-orm analytical ormulas. Consequently, in the recent literature, there has been an attempt to develop simpler approimations. Two more popular channel models as approimation or the Rayleigh-lognormal and akagami-lognormal distributions are the K and K G distributions, respectively. The K distribution is a miture o the Rayleigh and gamma distributions. In [], the Rayleigh-lognormal and K distributions are compared with respect to their probability density unctions PDF), and it is shown that both PDFs match well at the tail. However, there is a considerable deviation between two PDFs or small values o shaping actor o gamma distribution β in []). It can be seen that moments are matched only up to the second moment. In [3], error perormance is compared or the Rayleigh-lognormal and K distributions with dierential 1 This research is supported by the atural Science and Engineering Research Council SERC) o Canada and the Alberta Ingenuity Fund AIF), Alberta, Canada. phase-shit keying DPSK) and minimum shit keying MSK). It is shown that the two distributions do not match well in high signal-to-noise ratio SR) range with low shaping actor β. On the other hand, the K G distribution, which is a miture o the akagami and gamma distributions, has been introduced in [4] to approimate the akagami-lognormal distribution. However, no comparison is given or the two distributions. Although the K and K G distributions were introduced as simpler alternatives to the composite models, their amplitude and SR distributions include the modiied Bessel unction o the second kind K α )) [] [4] [5]. Further, their CDFs and MGFs include the generalized hypergeometric unction and the Whittaker unction, respectively. Thereore, average channel capacity and average symbol error probability over the K and K G ading channels end up with Meijer s G-unctions [5]. In [6], dierent diversity receptions under K G ading channels have be analyzed in terms o moments, average SR, and the amount o ading. The PDFs o end-to-end SR o each combing method are with complicated orms, or the corresponding MGFs do not have closed-orm epressions. For eample, the MGFs are approimated with the Padé approimation or maimal ratio combining MRC) and dual branch selection combining SC), and the outage probabilities are calculated with numerical integration [6]. Perormance o generalized SC over K ading channels is analyzed with numerically integrated marginal MGF o K channel when the ading parameter is equal to an integer plus one-hal [7]. Although K G distribution is considered as an approimation and a simple orm or the akagami-lognormal distribution, we note that most o the perormance results are derived using urther approimations to avoid mathematical diiculties. As a result, we introduce a better and simple orm o distribution in this research in order to approimate the akagami-lognormal amplitude) and the gamma-lognormal SR) distributions. The new distribution is derived by replacing the integral o the akagami-lognormal distribution with the Gaussian-Hermite quadrature sum [8] [9] [1]. The rest o the paper is organized as ollows. The approimation or the PDFs o the signal amplitude and SR with the akagami-lognormal channel is given in Section II and III, respectively. Statistical properties and perormance evaluation can be ound in Section IV and Section V, respectively. The /1/$6. 1 IEEE

2 concluding remarks are made in Section VI. II. APPROXIMATIO FOR SIGAL AMPLITUDE PDF For the akagami-lognormal channel, let X be the ading amplitude, which is a random variable. The composite akagami-lognormal distribution nl is a miture o akagami ading and lognormal shadowing. Its PDF can be epressed as an integral orm [1] nl = m m m 1 e m y 1 ln y μ) Γm)y m e λ dy 1) πλy where, m is the ading parameter in akagami-m ading, μ and λ are the mean and the standard deviation o lognormal shadowing, respectively, and Γ ) is the standard gamma unction. When m =1, distribution in 1) is Rayleighlognormal distribution which has severe ading. The ading eect and shadowing eect diminish or larger m m ) and smaller λ λ ), respectively. A closed-orm epression o the composite akagami-lognormal PDF is not available in the literature. In the ollowing, we introduce an accurately approimated PDF which can represent both ading and shadowing eects. ln y μ Using the substitution t = λ, nl in 1) can be written as nl = mm m 1 e t ht)dt ) π Γm) λt+μ+ where ht) = e m e ) λt+μ). The term ht)dt has the orm o the Gaussian-Hermite integration which can be approimated as e t ht) dt e t w iht i ) where t i and w i are abscissas and weight actors or the Gaussian-Hermite integration [11]. t i and w i or dierent values are available in [11, Table 5.1)] or can be calculated by a simple MATLAB program. Thereore, epression ) can be re-written as nl mm m 1 π Γm) w i ht i ). 3) Using this approimation, we give the epressions o the new PDF and statistical properties o the signal amplitude in the ollowing. The new PDF is denoted as X in the sequel. A. PDF X The PDF, X, is deined as X C m 1 e bi, 4) where = m m w i e m λt i+μ) / π Γm), = me λt i+μ), and C is the normalization actor to ensure X d =1. It can be shown that C = π/ w i. B. CDF F X The CDF F X can be evaluated as F X = X t)dt to yield F X = C m γ m, ) 5) where γ ; ) is the incomplete gamma unction deined as γa, ) ta 1 e φ dφ [1, eq ]. Further, F X can also be re-written as F X = C/ a i Γm) Γm, bi ) ) /b m i where Γ ; ) is the upper incomplete gamma unction deined as Γa, ) ta 1 e φ dφ [1, eq ]. ote that /b m i = w i / π Γm). C. MGF M X t) The MGF M X t) can be evaluated as M X t) = Ee t )= e t X d. HereE ) denotes the epectation operator. With the aid o [1, eq ], M X t) can be evaluated as M X t) = CΓm) ) m m e t 8b t i D m 6) bi where D p ) is the parabolic cylinder unction [1, eq. 9.4]. D. Amount o Fading The amount o ading AoF) is a statistical property which measures the severity o the ading channel. AoF can be epressed as AoF = EX 4 )/[EX )] 1 which can be calculated rom the second and the orth moments o the amplitude distribution or the irst and the second moments o the SR distribution. AoF o the new distribution given in 4) can be evaluated as πm +1) AoF = w ie λt i m w ie ) 1. 7) λt i It can be seen that AoF depends on ading parameter m o the akagami ading and standard deviation λ o the lognormal shadowing. However, AoF is independent o the mean μ) o the lognormal shadowing. When the shadowing eect diminishes, λ, AoF 1/m. It conirms that AoF =1 under Rayleigh ading with no shadowing. Further, AoF as λ. Thereore, AoF ranges within [1/m, ), which is similar to the AoF o the composite akagami-lognormal channel model. III. APPROXIMATIO FOR THE SR PDF When the ading envelop is X, the transmitted symbol energy is E s and single sided power spectral density o the comple additive white Gaussian noise AWG) is,the instantaneous SR per symbol is γ = X where = E s / is the unaded SR. Based on the amplitude distribution given in 4), the SR distribution γ can be epressed as γ = C m γ m 1 e γ 8)

3 where γ is equal to a Miture o Gamma Distributions [13]. Since the SR PDF o the akagami-lognormal channel is gamma-lognormal distribution, we can accurately approimate the gamma-lognormal distribution through a mitures o gamma distributions. The CDF o SR, F γ can be derived as F γ = C m γ m, b ) i. 9) The MGF o SR, M γ t), can be evaluated as M γ t) = CΓm) ) m m. 1) t + bi The main perormance measurement o wireless communication systems is the instantaneous SR and its statistical properties. Since the miture o gamma distributions has simple orms or PDF, CDF and MGF epressions, it is a powerul tool in analytical research studies. IV. COMPARISO WITH THE K G CHAEL MODEL The K G distribution KG has been introduced as an approimated and a simple channel model or the akagamilognormal distribution [4]. The K G channel model has been used in several research studies in the literature to analyze the eect o both ading and shadowing eects in wireless channels [3] [7]. Thereore, it is reasonable to compare X and KG in terms o how they match with nl. Inthe ollowing, the K G distribution is introduced irst. Then we compare the mean square errors MSE) o X and KG rom nl, the moments o the three distributions, and the PDFs o the three distributions. A. K G Distribution For the K G distribution, the PDF o X is given by [6]: 4m β+1)/ β [ KG = Γm)Γk)Ω K β+1)/ α m Ω ) 1/ ], 11) where m is akagami ading parameter, α = k m, β = k + m 1 and Ω=E[X ]/k is the mean signal power. Using the power series epansion o the logarithm MGFs o akagami-lognormal distribution and K G distribution, the relationships between parameters λ and μ) o akagamilognormal distribution and parameters α and β) ok G distribution can be derived [] as λ =Ψ β) and μ =Ψβ)+lnα) where Ψ ) is the irst derivative o the psi unction [1, eq ]. B. Mean Square Error MSE) The MSE between X and nl can be calculated numerically or dierent values in X. I we plot MSE verses, a lower bound o can be obtained, which guarantees a given MSE value. Fig. 1 shows the MSE o KG and the MSE o X versus. It can be seen MSE between KG and nl does not depend on. MSE Fig X & nl β = 1 β = 1.5 β = KG & nl MSE versus o X and MSE or KG distribution. that, i the value o slightly increases, the MSE decreases signiicantly. I the MSE requirement is set to be 1 4,theK G distribution does not meet this requirement when β =1, 1.5 or, while our new distribution can meet the requirement with 11, 7 or 5 when β =1, 1.5 or, respectively. This means that we can eiciently get a more accurate approimation or the akagami-lognormal distribution than the K G model. We assume that m =and α =4or the calculations. C. Moments The n th moment o a PDF X can be deined as MX n EX n ). Thereore the n th moment o X distribution, M n X, can be evaluated as m+n M n CΓ ) 1 X = m n. 1) b i By interchanging the order o integrations, The n th moment o nl distribution can be evaluated as Mnl n = Γ n + m)e nμ + n λ 8 ). m n Γm) The epression or the n th moment o K G channel model, MK n G, is available in [4] as M n K G = Γ n + β)γ n + m) Γβ)Γm) α m)n. Table I shows some numerical eamples to compare the moments n) o the three distributions. For the X, the value o is selected such that an MSE requirement say, < 1 4 ) is met. We assume that m =and α =4or the calculations. It can be seen that the moments n) o X and nl distributions are almost the same. On the other hand, the dierences between moments o nl and KG distributions increase or the third moment and beyond.

4 TABLE I THE FIRST FIVE MOMETS OF THREE DISTRIBUTIOS. β =1, =11 β =, =5 n X nl KG X nl KG X nl KG β = D. PDFs Fig. illustrates the three distributions or two cases β =1 and β =, with m =and α =4. X is almost the same as nl. The miture sizes o X or the two cases are =11and =5, respectively. On the other hand, KG has more deviation rom nl, particularly or small values o and β. V. PERFORMACE AALYSIS WITH UMERICAL EXAMPLES Since the miture o gamma distributions has a simple orm and accurately approimate the gamma-lognormal distribution, it is motivated to derive closed-orm epressions or some important perormance measures such as the channel capacity, outage probability, and symbol error rate as ollows. A. Average Channel Capacity Based on Shannon s theorem, the average channel capacity, C can be calculated by averaging the instantaneous channel capacity over SR as C = Blog 1 + ) γ d, where B is the signal transmission bandwidth. Using a similar approach to that in [14], C can be calculated or integer m as C = CBm 1)! m ln e m j=1 ) j Γj m, ). 13) B. Outage Probability The outage probability is the probability that the receiver SR is below a given threshold γ th. It can be straightorwardly calculated as P out γ th )=F γ γ th ), where F γ is in 9). C. Average Symbol Error Rate We present average symbol error rate SER) or M-PSK and M-QAM in the ollowing. 1) M-PSK: The average SER or M-PSK, P PSK e,isgiven in [15, eq. 9.15)]. With the MGF given in 1), P PSK e can be evaluated with the aid o [15, eq. 5A.17)], the average SER o M-PSK modulation can be evaluated in closed-orm or integer m..1 β = Fig.. Plots o three distributions. ) M-QAM: Square M-QAM signals that have constellation size M = k with an even integer k are considered. The average SER or M-QAM, P QAM e, over the generalized ading channels is given in [15, eq. 9.)]. With the MGF given in 1), P QAM e can be evaluated as [ ) m Pe QAM a π/ i sin θ =K m b i sin θ + gqam dθ 1 1 ) ) m π/4 sin θ dθ] M sin θ + gqam. 14) where K =CΓm)1 1/ M)/π and g QAM =3/M 1). The irst and the second integrals in epression 14) can be evaluated with the aid o [15, eq. 5A.3)] and [15, eq. 5A.13)], respectively. Thereore, average SER o M-QAM modulation can be evaluated in closed-orm or any m. D. umerical Results Figs. 3-5 show normalized average capacity, outage probability and SER, respectively. Continuous and dashed lines represent the perormance over the gamma-lognormal distribution, while discrete symbols represent the perormance over the miture o gamma distributions with = 15. With = 15, we can achieve MSE less than 1 6 or the ollowing eamples. In Fig. 4 and Fig. 5, σ denotes the standard deviation in db given as σ = 1/ ln 1)λ db. Further, σ = 4.5, 8, 13 db represents urban, typical macrocells and microcells, respectively in cellular mobile systems. We assume α = 4 or all the three igures and m = or outage and SER calculations. The perormance curves or gamma-lognormal are plotted by Monte-Carlo simulations. All igures show that both the gamma-lognormal distribution

5 ormalized average channel capacity b/s/hz) m = m = db) SER QAM BPSK db) Fig. 3. ormalized average channel capacity versus unaded SR. Fig. 5. SER versus unaded SR. Outage probability P out ) Fig ormalized outage threshold db) Outage probability versus normalized outage threshold. and the miture o gamma distributions have well-matched perormance curves. VI. COCLUSIOS This paper proposes a miture distribution to model both multipath and shadowing eects in wireless channels. In this new model, the SR distribution ollows a miture o gamma distributions, resulting in simple orms o PDF, CDF and MGF. This approach leads to ecellent accuracy in representing the composite distributions. For eamples, in terms moment matching, the new approach can be highly accurate match a large number o moments Table I), whereas the K G distribution can only match the irst two moments. We believe that the proposed distribution provides a highly leible and simpler alternative to several currently popular channel distributions. REFERECES [1] G.L.Stüber, Principles o mobile communication nd ed.). orwell, MA, USA: Kluwer Academic Publishers, 1. [] A. Abdi and M. Kaveh, K distribution: An appropriate substitute or Rayleigh-Lognormal distribution in ading-shadowing wireless channels, Electron. Lett., vol. 34, pp , Apr [3], Comparison o DPSK and MSK bit error rates or K and Rayleigh- Lognormal ading distributions, IEEE Commun. Lett., vol. 4, no. 4, pp. 1 14, Apr.. [4] P. M. Shankar, Error rates in generalized shadowed ading channels, Wireless Personal Communications, vol. 8, no. 3, pp , 4. [5] P. S. Bithas,. C. Sagias, P. T. Mathiopoulos, G. K. Karagiannidis, and A. A. Rontogiannis, On the perormance analysis o digital communications over generalized-k ading channels, IEEE Commun. Lett., vol. 1, no. 5, pp , May 6. [6] P. S. Bithas, P. T. Mathiopoulos, and S. A. Kotsopoulos, Diversity reception over generalized-k K G ) ading channels, IEEE Tran. Wireless Commun., vol. 6, no. 1, pp , Dec. 7. [7] P. Theoilakos, A. G. Kanatas, and G. P. Ethymoglou, Perormance o generalized selection combining receivers in K ading channels, IEEE Commun. Lett., vol. 1, no. 11, pp , ov. 8. [8] C. Tellambura and A. Annamalai, An uniied numerical approach or computing the outage probability or mobile radio systems, IEEE Commun. Lett., vol. 3, no. 4, pp , Apr [9] J. Wu,. B. Mehta, and J. Zhang, Fleible lognormal sum approimation method, in Proc. IEEE Globecom, vol. 6, Dec. 5, pp [1]. B. Mehta, J. Wu, A. F. Molisch, and J. Zhang, Approimating a sum o random variables with a lognormal, IEEE Trans. Wireless Commun., vol. 6, no. 7, pp , July 7. [11] M. Abramowitz and I. A. Stegun, Eds., Handbook o Mathematical Functions: with Formulas, Graphs, and Mathematical Tables. ewyork: Dover Publications, [1] I. S. Gradshteyn and I. M. Ryzhik, Table o Integrals, Series, and Products. ew York: Academic Press,. [13] M. Wiper, D. R. Insua, and F. Ruggeri, Mitures o gamma distributions with applications, Journal o Computational and Graphical Statistics, vol. 1, no. 3, pp , Sept. 1. [14] M. S. Alouini and A. J. Goldsmith, Capacity o Rayleigh ading channels under dierent adaptive transmission and diversity-combining techniques, IEEE Trans. Veh. Technol., vol. 48, no. 4, pp , July [15] M. K. Simon and M.-S. Alouini, Digital Communication over Fading Channels nd ed.), ew York: Wiley, 5.