Accurate MGF matching technique for diversity reception in correlated lognormal fading channels

Accurate MGF Matching Technique for Diversity Reception in Correlated Lognormal Fading Channels Cong Lam Sinh, Quoc Tuan Nguyen Dinh-Thong Nguyen University of Engineering and Technology, Vietnam National University, Hanoi - Vietnam Email: congls@vnu.edu.vn, tuannq@vnu.edu.vn Faculty of Engineering and Information Technology, University of Technology, Sydney – Australia Email: dinh-thong-nguyen@uts.edu.au Abstract –The two-point moment generating function (MGF) matching technique has been used with some success to approximate the output from an MRC diversity combiner operating in correlated lognormal fading channels The technique, however, is very sensitive to the choice of the location of the two matching points This paper proposes to apply the principle of power conservation across the combiner to control the accuracy of the location of the MGF matching points The technique is both novel and effective and this is backed by sophisticated simulation results To demonstrate the accuracy of the proposed MGF matching technique, the paper presents a closed-form expression for the estimation of BER of BPSK using MRC diversity reception in a correlated lognormal shadowing environment of MRC reception in correlated lognormal fading channels The underlining complexity in using the lognormal model for shadowing in MRC diversity reception is that it results in the well-known ‘sum of lognormal powers’ problem [3], [4] Some authors avoid this complexity by using a gamma pdf to approximate the shadowing as first proposed in [5] In [4] the authors propose to approximate the sum of lognormal random variables by a single lognormal random variable whose normal mean and variance parameters are found by a two-point matching of its MGF to that of the sum However, the result of the twopoint matching is very sensitive to the choice of the matching points This problem has been briefly addressed by our group in [6] in the context of MRC diversity reception in the simple case of independent lognormal fading channels, by evoking the power conservation principle across the combiner In this paper, we extend the problem of MRC diversity reception in a far more complex environment of correlated lognormal fading channels The power conservation principle is used to determine and to control the accuracy of the location of the two MGF matching points, using a simple search algorithm This technique is both innovative and effective and has not been done before to the best of our knowledge I INTRODUCTION In most realistic scenarios of wireless propagation between a base station and a receiver, the physics of radio wave propagation encountering random parallel multipaths and cascaded obstructions is not well understood The latter is commonly known as shadow fading, e.g [1], referring to the random fluctuation in the received average power as the mobile receiver moves in and out of the shadow of hills or buildings which obstruct the line-of-sight transmission The global path is usually modelled as a lognormal stochastic process while the local path is modelled as Rayleigh process In most realistic situations, fading is the result of a mixture of the two fading mechanisms, and fading mitigation requires both microdiversity techniques using multiple antennas or multiple OFDMA subchannels and macrodiversity techniques using multiple base stations Maximum ratio combining (MRC) is most effective for microdiversity while selective combining (SC) is more suitable for macrodiversity [2] However in this paper, for simplicity we assume a microdiversity environment and that MRC is used The rest of the paper is organized as follows Section II defines the signal model and briefly describes the maximum ratio combining (MRC) principle for diversity reception In Section III we present the derivation of the pdf of the correlated multivariate Gaussian vector Z from a given correlation matrix of the related multivariate lognormal vector p, i.e p=exp(Z) In Section IV we describe the estimation of the pdf of the sum of correlated lognormal powers using the current two-point MGF matching technique Section V is the main contribution of our paper in which we present an innovative technique to control the accuracy of the two-point MGF matching by invoking the power conservation principle across the MRC combiner Section VI briefly describes the essential steps in Monte Carlo simulation of BER of BPSK signal using MRC Shadowing has a much higher degree of correlation than short-term multipath fading, therefore the main objective of this paper is to formulate the performance 978-1-4799-2903-0/14/$31.00 ©2014 IEEE 140 reception in correlated lognormal fading channels, and finally a conclusion is presented in Section VII The cascade (product) model of shadowing implies that the path loss is exponentially proportional to the distance to a power α between to and that the standard deviation of shadow fading loss is independent of the distance and is in the range of to 12dB [8] The power gain of a shadow fading channel is usually modeled as a lognormally distributed random variable, II SIGNAL AND MRC DIVERSITY RECEPTION MODEL The effect of maximum ratio combining is to add up the powers, hence the signal-to-noise ratios, of the received signals to be combined Expressing it in matrix form, the diversity receiving system is described as, y = h*x + n distributed, i.e Z~ N(µZ,CZ) Since p is a correlated multivariate lognormal vector, let p=(p1, p2,…,pN) = (eZ1, eZ2,…., eZN) in which each component pi is a lognormal RV and Z=(Z1, Z2,…, ZN) is a correlated multivariate normal vector and its pdf is (1) where • • • • y = [y1, y2,… yN]T is the received symbol vector from all the diversity branches, h = [h1, h2,… hN]T is the channel gain vector on all the diversity branches, x is the transmitted BPSK symbol vector (complex signals) and n = [n1, n2,… nN]T is the AGWN noise vector on all the diversity branches f Z (z ) = yˆ (t ) = H y 2 = Var ( Z ) = (σ Z , σ Z , , σ ZN C Z (i , j ) = Cov ( Z i , Z j ) = σ Zij h Es N0 (5c) ρ ρ ρ N −1 ρ ρ N -2 ρ ρ N -3 ρ N −2 ρ N -3 σ p2 ,       (6) In which ρ is the correlation coefficient between any two successive pi and pi+1 In view of (3) where the transmit SNR, Es/No, may be assumed fixed, in this paper we use the term channel In general, the mean, variance and covariance matrix of p are, respectively: power gain, p = h , and signal-to-noise ratio, γ, (7a) μ p = E ( p ) = ( μ p1 , μ p , , μ p N ) interchangeably where it is appropriate σ p2 = Var ( p ) = (σ LOGNORMAL (5b) ) all pi variates have the same variance where the transmit signal energy is E s = E[ s (t )] III CORRELATED CHANNELS (5a) Here we use the simple decreasing correlation model by Gudmundson in [8] for the shadow fading p, then its covariance matrix is, assuming 1  ρ Cp = σ p  ρ  (3)  ρ N −1 The received SNR is then 1/ (z - μ Z ) T C-1Z (z - μ Z ) (4) ) 2 σZ and it is well-known that the SNR in this equalized signal is equal to the sum of the SNRs in all diversity branches at the input to the MRC combiner [7] γ = (2π ) N / CZ exp(− μ Z = E ( Z) = ( μ Z , μ Z , , μ ZN ) (2) H h h where the mean, variance and covariance matrix of Z are, respectively: The equalized received signal from an MRC combiner, used for detection/demodulation, is h Z i.e p ≡ h = e ~ LN(µZ,CZ), with Z being normally p1 ,σ p2 , , σ pN 2 FADING C P ( i , j ) = Cov ( pi , p j ) = σ pij = σ p ρ 141 (7b) ) i− j (7c) The relationship between the two parameter sets in (5) and (7) can be summarized as follows: integration, we make the variable transformation Cz-1/2(z-μZ)=√2u, i.e z=√2C1/2u + μZ, or μ p = e μ Z +σ Z / , 2 dz = ( C Z (8a,b) σ p = e 2μZ +σ Z (eσ Z − 1) = [ E ( p)]2 (e σ Z − 1) N j =1 where CZij is the (i,j) element of CZ1/2  μ pi μ Zi = ln  μ2 +σ pi  pi ( 2  ,   σ Zi = ln + σ pi / μ pi Then (11) becomes (9a) M Y ( s) = ) ( (9b) π N μ N T  Czij u j + i )] exp(−u u)du  ∏ exp[−s exp( = j −∞ i = ξ ξ ) The integral in (12) has a suitable form for GaussHermite expansion approximation for the MGF of the sum of N correlated lognormal SNRs, which is first taken with respect only to variable z1 as For simplicity, in this paper we assume all random variates of Z have the same μZ and σZ, and all random variates of p have the same μp and σp M Y (s) ≈ IV ESTIMATING PDF OF SUM OF CORRELATED LOGNORMAL POWERS USING MGF MATCHING TECHNIQUE ∞ −∞ N −∞ −∞ N C Zlj n1 =1 zj + j=2 ξ C Zl a n1 + μl )] d z d z N ξ (13) By proceeding in a similar way for the integrals with respect to other variables z2,…zN, we obtain −spi (10) ) f ( p)dp N exp[ − s  exp( i =1 l =1 Np n N =1 n1 =1 ξ N C j =1 Zlj w n1 w n N an j + π N /2 (14) μl )] ξ in which wn and an are, respectively, the weights and the abscissas of the Gauss-Hermite polynomial The approximation becomes more and more accurate with increasing approximation order Np Since fLN(p)dp = f(z)dz, the MGF of the combined SNR in (10) becomes (2π )L / Np   M Y ( s, μ Z , C Z ) ≈ where s is the variable in the Laplace transform domain MY (s) = i=2 N i i =1 =  (∏e −∞ ξ Np exp( −  z i )  w n1 N /2 N ψ Y (s) =   (∏e−sp ) f ( p1 , p2 , , pN )dp1dp2 .dpN +∞ N −∞ ∏ exp[ − s exp( Consider N correlated lognormal RVs, {pi}, with joint distribution f(p1, p2,… , pN), input to the MRC The MGF of the MRC output power, Y=∑ipi , is +∞ ∞   π l =1 +∞ N/2 ∞ C Z (i , j ) = Cov ( Z i , Z j ) = ln + σ pij / μ pi μ pj (9c) Finally, the sum of N correlated lognormal RVs can then be approximated by a single lognormal RV [4], ˆ Yˆ = 100.1 X where Xˆ ~ N ( μˆ , σˆ ) By matching the (11) z  (z - μZ )T C-Z1 (z - μZ ) L exp[−sexp( i )]exp(− dz 1/ ∏ ξ  i=1 Z C −∞ ) N du  C Zij u j + μ i zi = Or ∞ 1/ LN X X MGF of the approximation YˆLN with the MGF of the To decorrelate the above expression so that it can have a suitable form for Gauss-Hermite expansion for the lognormal sum in (14) at two different positive real values s1 and s2, we obtain a system of two 142 (12) simultaneous equations which can be used to solve for μˆ X and σˆ X using function fsolve in Matlab The two simultaneous equations, with RHS being completely known from (14), are: The assumption of a micro-diversity environment above may not be realistic because diversity paths have different distance and topography However in this paper we apply this assumption for the sake of simplicity of computation and simulation Np Thus by systematically searching for the two matching points (s1, s2) until the power estimation error is smaller than a specified percentage threshold, an accurate 2-point MGF matching can be achieved as evidenced in Figure In this figure, the MGF matching corresponding to SNR=7.36 clearly shows a significant improvement from the result using the two matching points proposed in [4]  wn exp[− si exp{( anσˆ X + μˆ X ) / ξ }] n =1 = π M YLN ( si , μ Z , C Z ), (15) i = 1,2 For a discussion on the choice of matching points (s1, s2), see [4][6] From Figure of [4] in which for N=4, μ=0dB and σ=8dB, i.e average SNR=7.36dB from (8a), it is recommended that (s1,s2)=(1.0, 0.2) for various different values of correlation coefficient ρ Once the estimated Gaussian parameters are found, the pdf of the estimated SNR from the output of the MRC combiner is (10 log10 γ − μˆ X ) (18) ξ exp(− fˆLN ,MRC (γ ) = ) γ σˆ X 2π 2σˆ X V ACCURATE MGF MATCHING USING POWER CONSERVATION PRINCIPLE The problem encountered in using the 2-point MGF matching technique proposed in [4] is that it is highly sensitive to the location of the two matching points and also to the initial starting values for μˆ X and σˆ X chosen for the Matlab function fsolve In [4], the values of the matching points are chosen in an ad-hoc manner to visually judge the accuracy of the match Furthermore, as clearly seen in Table 1, the technique does not guarantee conservation of signal power across the MRC combiner The power loss is as much as 25% The accuracy of the 2-point MGF matching in the preceding section can be greatly improved and controlled to a specified degree by reinforcing this ‘lossless’ principle This implies equal system average power on both sides of the combiner where the log conversion constant ξ=10/ln(10) The BER of BPSK in Gaussian channel with bit SNR γ is BERAWGN,BPSK (γ ) = Q( 2γ ) ∞ BERLN , BPSK =  BER AWGN , BPSK (γ ) fˆLN ,MRC (γ )dγ  μˆ 10 log10 γ − μˆ X 2σˆ X  = u ⇔ γ = exp X + u  ξ σˆ X   ξ (20) can be reduced to (16a) While the estimated average output power gain is Pˆout = exp( μˆ X + σˆ X / ) Pˆout − Pin Pin BER LN , BPSK = π ∞  BER (γˆ X (u )).e −u du , AWGN , BPSK where γˆ X ( u ) = exp( μˆ X / ξ + u σˆ X (16b) / ξ ) is the argument of BERAWGN,BPSK(.) in (19) The above expression for BER can then be accurately approximated by an Np-order Gauss-Hermite The percentage power estimation error is defined as % PEE = 100 (20) By a change of variable Since the average input power gain to the combiner, assuming a micro-diversity environment, is Pin = N μ p = N exp( μ Z + σ Z / ) (19) polynomial expansion as given in (21) (17) 143 BERLN ,BPSK ,MRC = Np BERLN,BPSK,MRC =  wn BERAWGN,BPSK(γˆ X (an )) π n=1 Np [  wn Q 2γˆ X (an ) π n=1 ] (21) Table1: Estimation result from two-point MGF matching for N=4 correlated diversity branches with ρ=0.3 (s1,s2); ( μˆ X , σˆ X )dB Output power/input power PEE (0.003, 0.104); (7.2567, 5.7083) 12.6131/ 12.6491 0.28% 7.36 (0.002 , 0.203); (9.6505, 5.6957) 21.8042/ 21.8002 0.019% From [4] 7.36dB (0.2, 1.0); (9.3283, 4.9462) 16.3868/ 21.8002 24.83% 39.8201/ 40.0000 0.450% 125.9572/ 126.4911 0.42% SNR_dB 10 15 (0.017, 0.098); (12.2157, 5.7340) (0.005, 0.017) (17.2240, 5.7287) Figure 1: Comparision between 2-point matching+power conservation and 2-point matching in [4] Finally the N correlated lognormal variates are generated as pi=eZi and the channel gain hi=eZi/2 VI SIMULATION SET-UP In the theory part of the paper, we plot BER of the MRC output versus average SNR per lognormal channel ( γ LN ≡μp in (8a)) for specified value of the variance σz =8dB and specified values of correlation coefficient, say ρ =0.3 Thus μz can be calculated from (8a), then σp can be calculated from (8b) and Cp(i,j) from (7c), and finally Cz(i,j) can be calculated from (9c) The intermediate correlated normal variates Z=(Z1, Z2,…,ZN) can now be generated as i Z i = μi +  cijU j for i, j = 1,2, ,N j =1 Figure 2: BER for BPSK in Correlated Lognormal Fading ( ρ = 0.3; σ Z = 8dB ) using N-branch MRC diversity reception (22) VII CONCLUSION in which Uj ~ N(0,1) are i.i.d unit normal variates and cij is the (i,j) element of Cz1/2, obtained from matrix Cz=Cz1/2(C1/2)T using Cholesky decomposition We have successfully presented an innovative and simple technique for accurate two-point matching of moment generating functions by evoking the principle of power conservation between the two matched MGFs The merit of the technique has been demonstrated in the accurate estimation of the ‘sum of lognormal powers’ of 144 the output signal from an MRC diversity combiner The accuracy of the proposed MGF matching technique is also backed by Monte Carlo simulation of the BER of BPSK signal in lognormal fading channel using MRC diversity reception ACKNOWLEDGEMENTS This work was supported by research grants from QG.12.45 Projects of the University of Engineering and Technology, Vietnam National University Hanoi REFERENCES [1] M Patzold, Mobile Fading Channels, Wiley & Sons 2002 [2] P.M Shankar, “Macrodiversity and Microdiversity in Correlated Shadowed Fading Channels,” IEEE Trans on Vehicular Technology, vol 58, no 2, pp.727-732, 2009 [3] M Di Renzo et al, “A general formula for log-MGF computation: Application to the approximation of Log-Normal power sum via Pearson Type IV distribution,” Proc IEEE Vehicle Technology Conference, vol 1, pp 999-1003, May 2008 [4] N.B Mehta et al., “Approximating a Sum of Random Variables with a lognormal,” IEEE Trans on Wireless Communications, vol 6, no 7, pp 2690-2699, July 2007 [5] A Abdi and M Caveh, “K distribution: an appropriate substitute for Rayleigh-lognormal distribution in fading-shadowing wireless channels,” Electronics Letters, vol 34, no 9, pp.851-852, 1998 [6] Dinh Thi Thai Mai et al., “BER of QPSK using MRC Reception in a Composite Fading Environment,” Proc 12th Int Symposium on Communications and Information Technology ISCIT 2012, 2-5 October, Gold Coast, Australia [7] D.G Brennan, “Linear diversity combining techniques,” Proceedings of the IEEE, vol 91, no 2, pp 331-356, 2003 [8] M Gudmundson, “A correlation model for shadow fading in mobile radio,” Electronics Letters, vol 27, pp.2146-2147, 1999 145 ... ,MRC (γ ) = ) γ σˆ X 2π 2σˆ X V ACCURATE MGF MATCHING USING POWER CONSERVATION PRINCIPLE The problem encountered in using the 2-point MGF matching technique proposed in [4] is that it is highly sensitive... MRC diversity combiner The accuracy of the proposed MGF matching technique is also backed by Monte Carlo simulation of the BER of BPSK signal in lognormal fading channel using MRC diversity reception. .. between 2-point matching+ power conservation and 2-point matching in [4] Finally the N correlated lognormal variates are generated as pi=eZi and the channel gain hi=eZi/2 VI SIMULATION SET-UP In the