Optimal Receiver for MPSK Signaling with Imperfect Channel Estimation
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1 This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the WCNC 27 proceedings. Optimal Receiver for PSK Signaling with Imperfect Channel Estimation Himanshu Tyagi Department of Electrical Engineering Indian Institute of Technology - Delhi Hauz Khas New Delhi 6, India tyagi@yahoo.com Ranjan K. allik Department of Electrical Engineering Indian Institute of Technology - Delhi Hauz Khas New Delhi 6, India rkmallik@ee.iitd.ernet.in Sumit Raina Lehman Brothers Winchester Building, Hiranandani Powai umbai 476, India sumit.raina@gmail.com Abstract We derive the structure of the optimal receiver for PSK signaling over a correlated Rayleigh fading channel. The channel is estimated using the minimum mean square error criterion by means of pilot symbols. For the case of high signalto-noise ratio, an approximate expression for the symbol error probability of this scheme is obtained as an integral, and compared with the of a which uses maximalratio combining. Numerical results show that the performance gap between the optimal and suboptimal receivers increases with increase of the channel correlation and the number of diversity branches, whereas it decreases with increase of pilot-to-signal ratio. Keywords Correlated Rayleigh fading, PSK, optimal receiver, symbol error probability. I. Introduction Communication systems using channel estimation at the receiver have gained lots of importance with the advent of third-generation wireless systems. ost of the third generation systems use PSK and QA signaling over a fading channel with diversity. In multipath diversity techniques it is assumed that many independent observations are available for the same signal which do not fade simultaneously. Therefore an improved performance can be expected if the outputs from all diversity branches are used jointly. The important issue here is how the weights of the signals from different fading channels are combined. The combination, ensuring maximum signal-to-noise ratio SNR at the output of the combiner, is called maximalratio combining RC. This construct, though ensures maximum average SNR at the output when the channel is known perfectly, is still suboptimal for correlated channels in the presence of channel estimation errors. As the channel gain in a fading environment is random, the receiver estimates the channel gain after each fixed interval. A minimum mean square error SE estimation method using pilot symbols is an effective method of obtaining the channel state information. Analytical expressions for the symbol error probability in case of BPSK signaling using practical channel estimation have been derived in [, 2]. In this paper we find the structure of an for PSK signaling over a correlated Rayleigh fading channel, when SE channel estimation is used by means of pilot symbols. For the case of high SNR, an approximate expression for the of this scheme is obtained as an integral, and compared with the of a which uses RC. II. System odel Consider an L-branch diversity reception system using - PSK signaling in flat fading. The complex baseband sampled received signal vector is given by r = hs + n, where s is the complex information bearing symbol with energy s 2 = 2E s, h is the random complex channel gain vector, and n is the additive white Gaussian noise AWGN vector independent of h, which is a zero-mean complex circular Gaussian random vector distributed as CN L, 2N I L, L denoting the L vector of zeros and I L the L L identity matrix. The channel gain vector h is a zero-mean complex circular Gaussian random vector distributed as CN L, K h, implying correlated Rayleigh fading with channel covariance matrix K h. A. Channel Estimation The channel is estimated using pilot symbols over N time indices. Let s p =[s p,...,s pn ] T be the N pilot symbol vector used for channel estimation, where T denotes transpose. The L N received pilot signal matrix R p can be written as R p = hs T p + N p, 2 where N p is the L N AWGN matrix, with independent and identically distributed i.i.d. elements, each having a CN, 2N distribution. We consider the SE estimate of the channel, which is given by ĥ = R pa, where a is an N vector. Using the orthogonality principle to minimize ĥ h 2, the SE estimate can be expressed as ĥ = hs T p a + N p a, /7/$ IEEE 483
2 This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the WCNC 27 proceedings. where a = E [ h 2] s p 2 E [ h 2 ]+2N L s p, 4 E[ ] denoting the expectation operator. Substituting 4 in 3 we get where α = ĥ = αh + e, 5 s p 2 E [ h 2] s p 2 E [ h 2 ]+2N L, 6 and e is the estimation error vector which is distributed as CN L,σ 2 ei L, with B. Optimal Receiver Structure σ 2 e = 2N a 2. 7 Let S denote the data symbol set for PSK signaling, given by S = { } 2Es e j2πk, k =,...,, 8 where j =. Assuming equiprobable data symbol transmission following the pilot symbol transmissions, the optimal in the maximum likelihood sense uses the decision rule ŝ = argmax s S fr ĥ,s, 9 where fr ĥ,s is the conditional probability density function p.d.f. of r, conditioned on ĥ and s. We find from and 5 that for the zero-mean jointly complex circular Gaussian random vectors r and ĥ, the covariance matrix of r, the covariance matrix of ĥ, and the cross-covariance matrix of r and ĥ, are given respectively as noting that s 2 =2E s K r = 2E s K h +2N I L, Kĥ = α 2 K h + σ 2 ei L, K r ĥ = αsk h. The received vector r, conditioned on ĥ and s, is therefore distributed as CNm, K, where m = K r ĥ K ĥ ĥ = αsk h α 2 K h + σei 2 L ĥ, a K = K r K r ĥ K ĥ KH rĥ = 2E s K h +2N I L 2α 2 E s K h α 2 K h + σei 2 L K h, b H denoting the Hermitian conjugate transpose operator, with a conditional p.d.f. fr ĥ,s= π L detk exp { r m H K r m. 2 Since K does not depend on s, substituting 2 in 9 and taking the logarithm results in the decision rule ŝ = argmax s S Re { m H K r } { = argmax s S Re s ĥ H α 2 K h + σei 2 K h } K r, 3 where K is given by b. III. Error Analysis Let the matrix A be defined as This can be simplified as A =α 2 K h + σ 2 ei K h K. 4 A = 2 [ E s σe 2 + α 2 ] N IL +2N σek 2 h. 5 Note that A is a Hermitian matrix. Define the average SNR per branch and the average pilot-to-noise ratio PNR per branch as = E se [ h 2], N L = α2 E [ h 2] σel 2 This gives from 6 α = = s p 2 E [ h 2] 2N L, Without loss of generality, consider the case when symbol s = 2E s is transmitted. The decision variable in 3 can now be written as z = ĥh A 2E s h + n = α 2E s h H Ah + 2E s e H Ah + αh H An +e H An. 8 The probability of correct decision is the probability that the phase of z lies between π/ and π/, which, subtracted from unity, gives the. When the condition [ E 2Es e H Ah + αh H An 2] [ E e H An 2] 9 484
3 This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the WCNC 27 proceedings. is satisfied, the double noise term e H An in 8 can be neglected. Noting that E [ h 2] =trk h, we can rewrite the condition 9 using the statistics of h, e, andn as + tr K htr A 2 L tr K h A 2. 2 which is a high SNR condition. Under the condition 2, the decision variable in 8 can be approximated as z α 2E s h H Ah + 2E s e H Ah + αh H An. 2 The instantaneous SNR SNR conditioned on h for this decision variable can be written as α 2 2E s h H Ah γ opt = [ E 2Es e H Ah + αh H An 2 ] h = 2 α 2 E s h H Ah. 22 E s σe 2 + α 2 N h H A 2 h Let Ψ γopt jω =E [e jωγopt ] denote the characteristic function c.f. of γ opt. The is approximated in terms of this c.f. as [3] P e π π Ψ γopt sin2 π dθ. 23 However, obtaining the exact c.f. of γ opt is a difficult task. We therefore obtain an approximate c.f. by applying the condition 2. Denoting tr K h =ΩL, 24 we can rewrite 5 as A = 2[E s σe 2 + α 2 I L + N ] + ΩK h. 25 Further, denoting the Hermitian matrix F as F = + ΩK h, 26 the instantaneous SNR can be written as α 2 E s h H I L + F h γ opt =. 27 E s σe 2 + α 2 N h H I L + F 2 h Using the first order approximation 2 I L + F k I L kf, 28 which holds for the high SNR condition 2, and the definitions of and in 6 along with 24, we can express γ opt approximately as γ opt + Ω hh I L +2F h. 29 Since the right-hand side of 29 is a Hermitian quadratic form in h, the c.f. of γ opt can be approximated as [4] Ψ γopt jω. det I L jω Γp + Ω K h I L +2F 3 Substituting 3 in 23, an approximate expression for the, valid for the high SNR condition 2, is given by P e π det π I L + sin2 π dθ Γp + Ω K h I L +2F. 3 Let λ,...,λ K denote the K distinct eigenvalues of the normalized channel covariance matrix Ω K h, λ i having multiplicity q i for i =,...,K. The can be expressed in terms of these eigenvalues as P e π π K i= +sin 2 π [ + λ i +2] + 2 q i dθ.32 The integral given by 32 can be evaluated in closedform. In case of the which uses RC, the decision variable z subopt is given by putting A = I L in 8. Under the high SNR approximation + 33 obtained by putting A = I L in 2, we get from 27 γ subopt = + Ω h 2, 34 and from 3 and 32 the approximate expressions P e,subopt π det π dθ I L + sin2 π Γp + Ω K h
4 This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the WCNC 27 proceedings. QPSK, L=4, ρ h =.5, / simulation approximation approximation2 QPSK, L=8, / ρ h =.8 5 ρ h = average SNR per branch db Fig.. versus for obtained by simulation and approximations when L =4,ρ h =.5, and / =5 db QPSK, L=8, = db, average SNR per branch db Fig. 3. versus with varying ρ h when L =8and / =5dB QPSK, L=8, ρ h =.5 / = db / = db 5 / channel correlation coefficient ρ h Fig. 2. Variation of performance of optimal and suboptimal receivers with ρ h when L =8, =db,and = 5dB and P e,subopt π K respectively. i= π +sin 2 π + λi IV. Numerical Results and Conclusion qi dθ,36 For a uniformly correlated Rayleigh fading channel having channel correlation coefficient ρ h, which represents, for example, a space diversity system with closely spaced receive antennas, the in the case of QPSK signaling is computed using the approximate expressions 23 and average SNR per branch Γ tot db Fig. 4. versus with varying / when L =8and ρ h = The channel covariance matrix is given by ρ h... ρ h. K h =Ω ρ h , 37.. ρ h ρ h... ρ h L L where /L <ρ h <. Fig. compares the of the computed by simulation using 6 runs with that computed by onte Carlo simulation of the approximation expression 23 and the approximation2 expression 35. We find that as the average SNR per branch and the average PNR per branch increase, the two curves tend to converge, showing the validity of the approximate expression for high values of +. For further evaluation we have used the approximate expression 35 due 486
5 This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the WCNC 27 proceedings. L = 2 QPSK, ρ h =.5, / L = 4 L = 6 L = average SNR per branch db Fig. 5. versus with varying L when ρ h =.5 and / =5dB to its computational simplicity. The variation of the error performance of the optimal and s with ρ h is shown in Fig. 2. We find that the has superior performance, with the performance gap increasing with increase of ρ h. Figs. 3, 4, and 5 give plots of the versus with channel correlation coefficient ρ h, average pilot-to-signal ratio per branch /, and number of branches L, respectively. We find that the performance gap between the optimal and s increases with increase of ρ h and L, but decreases with increase of /. References [] L. Cao and N. C. Beaulieu, Closed-form BER results for RC diversity with channel estimation errors in Ricean fading channels, IEEE Trans. Wireless Commun., vol. 4, no. 4, pp , July 25. [2] W.. Gifford,. Z. Win, and. Chiani, Diversity with practical channel estimation, IEEE Trans. Wireless Commun., vol. 4, no. 4, pp , July 25. [3] C. Tellambura, A. J. ueller, and V. K. Bhargava, Analysis of -ary phase-shift keying with diversity reception for landmobile satellite channels, IEEE Trans. Veh. Technol., vol. 46, no. 4, pp , Nov [4] G. L. Turin, The characteristic function of Hermitian quadratic forms in complex normal variables, Biometrika, vol. 47, pp. 99 2,
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