Exact Pairwise Error Probability for the MIMO Block Fading Channel. Zinan Lin, Elza Erkip and Andrej Stefanov

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1 International Symposium on Information Theory and its Applications, ISITA004 Parma, Italy, October 0 3, 004 Exact Pairwise Error Probability for the MIMO Block Fading Channel Zinan Lin, Elza Erkip and Andrej Stefano Department of Electrical and Computer, Brooklyn, NY 0, USA zlin03@utopiapolyedu, elza@polyedu, stefano@polyedu Abstract In this work, we present a closed form of the exact pairwise error probability PEP) for coded block Rayleigh fading channels with multiple transmit and receie antennas Based on the alues of the signal-to-noise ratio and the eigenalue products in each block and each transmit antenna, where the eigenalues are obtained from the product of the signal-difference matrix and its complex conjugate transpose, we classify the calculation of PEP into two groups: the symmetric case in which the mentioned products are all eual and the asymmetric case We proide a new techniue to calculate the PEP in the symmetric case In the asymmetric case, we apply the finite integral form of Q-function and our results from the symmetric case to obtain the PEP Introduction The presence of block fading see [] and references herein) proides an effectie form of diersity in communication systems For multiple input and multiple output MIMO) block fading channels, both spatial and temporal diersity can be exploited when we use a properly designed channel code Block fading model with multiple antennas also arises in cooperatie systems [] with MIMO terminals, where partners help the original mobile transmit parts of the codeword Howeer, unlike regular MIMO block fading channels, for cooperatie MIMO systems, the receied SNR from different users blocks) could be different and the different users blocks) could hae different number of transmit antennas For coded systems, the pairwise error probability PEP) forms the basic structure for the union bound calculation of the error probability and is used as the main criteria for code design The PEP performance for MIMO systems has been widely analyzed in uasistatic and fast fading channels, ie [3, 4, 5, 6, 7] In this work, we obtain the exact analytical expression of PEP for Rayleigh block fading channels for MIMO systems We classify the PEP calculation into two types based on the product of the signal to noise ratio SNR) and the eigenalues, which are obtained from the product of the codeword difference matrix and its complex conjugate transpose First is the symmetric case repeatedpoles case of [4]), where all the mentioned products are eual In this case, we define new random ariables to arrie at the PEP expression in an easier fashion The second is the asymmetric case, where the products are not necessarily identical In this case, we make use of the finite integral form of Q-function [8] and the results from the symmetric case to derie the exact PEP Compared with [4], all the deriations shown in our paper inole only real computations, thus aoiding complex analysis Especially in the symmetric case, we directly compute the aeraged PEP with respect to the newly defined random ariables Also, unlike the PEP expressions gien in [4] and [6], our exact PEP form can be easily used with transfer function of the code to get an upper bound on the bit/frame error probability and distance spectrum of trellis codes The PEP deried in this paper is useful for analyzing and designing coded cooperatie systems It is demonstrated in [0] that the exact PEP proides a good guideline for the study of cooperation benefits in cooperatie systems The paper is organized as follows Section presents the general analysis of PEP for MIMO systems in block fading channels Methods for symmetric and asymmetric cases are described respectiely Thereafter, we show the numerical examples in Section 3 Section 4 concludes our work Exact Pairwise Error Probability for MIMO Systems In this section, we derie MIMO PEP expression for L-block fading channels Assume that there are M i antennas at the transmitter and V i antennas at the receier for block i, where i =,, L For regular block fading channels, since there is a single transmitter and a single receier, the number of transmitter

2 antennas for different blocks are the same In a coded space-time cooperatie system [], parts of the codewords are transmitted by different users which hae independent fading towards destination This results in an euialent block fading model Since different users could hae different number of transmitter antennas, M i could be different Howeer, all i are eual to V Also due to the possible different locations of the users, the receied SNR s from different blocks users) could be different in a cooperatie system We denote the total receied SNR in block i as Es,i N o We denote αm,i as the fading alue from transmit antenna m to receie antenna in block i We assume that αm,i s are iid complex Gaussian random ariables Without loss of generality, we assume that the total number of time slots N is a multiple of L Let c n m,i and en m,i be the elements of the codewords c and e, which are transmitted from antenna m at time n of block i Then the signal difference matrix B i with size M i N/L is composed of elements, δm,i n = cn m,i en m,i, B i = δ,i δ,i δ N/L,i δ,i δ,i δ N/L,i δm i,i δm i,i δ N/L M i,i ) Denote C i = B i B i) H, where operator H represents the complex conjugate transpose of the matrix Let m, i m = M i represent the eigenectors of C i with size M i and λ i m represent the eigenalues of matrix C i, i =,, L, m =,, M i Obiously, matrix C i is a Hermitian matrix and therefore its eigenalues are real [3] Then the Euclidean distance between codewords c and e oer N time slots can be expressed as [4] d c, e) = L V M i i= = m= λ i E s,i m w M m,i ) i where w,i, w,i, w M i,i) = α,i, α,i,, α M i,i) ) i,, i, M i i and w m,i are iid complex Gaussian random ariables with zero-mean Therefore, the PEP between two arbitrary codewords c and e oer N time slots in the Rayleigh L-block fading channels is gien as [] P EP c e) = E w Q L V M i i= = m= a i m wm,i 3) where is Gaussian noise ariance, a i m = E s,iλ i m 4M i and wm,i = r m,i is distributed as Rayleigh with parameter Based on the alues of a i m s, we classify PEP calculation into two types, one is our symmetric case where all a i m are eual; the other is asymmetric case where a i m are not necessarily identical Symmetric Case In the symmetric case, we assume that a i m s are all eual to a = E sλ 4M We propose newly defined random ariables to simplify the calculation of the exact PEP We will also use the symmetric result for the calculation of the exact PEP in the more general case Denote r = r, and z m,i = r m,i r, i =,, L, =,, V, m =,, M, i,, m),, ) Note that r and zm,i are correlated Their joint PDF, fr, z, zm,l V ) can be obtained from their cumulatie density function F r, z, zm,l V ) and expressed as fr, z,,, z V M,L) = r LV M V L M = i= m= i,,m),,) zm,ie r + ) V L M = i= m=z m,i ) i,,m),,) 4) By making use of 4) and the integral property of the Q-function [, En3-63)], we can easily obtain the PEP for the symmetric case P EP c e) = ) Λ Λ ) u Λ + k Λ k + k 5) k u) k=0 where u = + a = + Esλ, Λ = LMV 4M In L-receier diersity model with one transmitter and one receier antenna, the same parity bits are repeated L times, hence E s, = = E s,l = E s and λ = = λ L = d, where d is the Euclidean distance of the codeword pair Obiously, it is one example of the symmetric case Therefore, the formula gien in [9, En4-4-5)] is identical to 5) Note, howeer, that the symmetric case is applicable to a non-repetitie code and MIMO system as well Asymmetric Case When all the a i m are not identical, it is hard to get the PEP expression using the simple method gien in Section In this section we will ealuate PEP with the aid of the alternate finite integral form of Q- ) function [8], Qx) = π π 0 exp x dθ, x 0 sin θ

3 After substituting this form of Q-function in 3), we change the integration order of rm,i and θ and first integrate with respect to rm,i Then we obtain P EP c e) = π Define Ψξ) = π 0 L M i i= m= L i= Mi m= + ai m sin θ ξ + a i m dθ 6), 7) where ξ = sin θ Note that since ai m is same for each receier antenna, V is the minimum number of repeated poles in Ψξ) Hence Ψξ) can be expanded as n V terms with V repeated poles, n V + terms with V + repeated poles,, n k terms with k repeated poles, where k L+M +V is the maximum number of the repeated poles Applying partial fraction expansion on Ψξ), we can write Ψξ) as the sum of the following terms, Ψξ) = AV ), ξ + a V ) + + A ξ + + V +) n V +, + V +) a n V + A V ), ξ + a V ) V +) A n V +, ξ + V +) a n V Ak) n k, ξ a k) n k ) + + A V ),V ξ + a V ) ) + + ξ + A k) n k,k ξ + a k) n k V +) A n V +,V + + V +) a n V + ) k, 8) where A p),j represents the jth residue associated with the th p-repeated pole a p), p = V,, k, j =,, p and =,, n p Using 7), A p),j can be found as follows, A p),j = Ψξ) ξ + d p j)! [ Ψξ) a p) ) p ξ= a ) p) p j ] ξ+ a p) dξ p j ξ= a p) if j = p if j < p 9) as, P EP c e) = u p) k p=v np ) j j l=0 where u p) = + a p) p A p),ja p) ) j = j= j k p=l r np = a p) ) p ) j + l l + l u p) ) l, 0) Example of the PEP Calculation for Cooperatie Space-time Systems We will gie an example to illustrate the calculation of the exact PEP for MIMO systems Consider a 64-State conolutional code, [33, 7, 7, 65, 5, 37] for cooperatie space-time Rayleigh fading channels We use BPSK modulation and assume there is one transmit antenna at the first user block), two transmit antennas at the second user block) and one receie antenna That is L =, M =, M = and V = The first user utilizes [33, 7] conolutional code and the second user uses the generator [33, 7, 7, 65] We calculate the PEP between the all zero codeword and the one with the shortest error path We find the eigenalue of C is 40 and the eigenalues of C are and 6433 hae the same receied SNR and fix γ = E s Hence, a ) = λ γ 4M = 0, a ) = λ γ 4M a ) 3 = λ 3γ 4M Ψξ) = = We assume both users as 0 db = 9586 and = 8044 Then Ψξ) can be expressed as ) ξ + a ) A ), ξ + a ) ) + ξ + a ) A ), ξ + a ) ) ) ξ + a ) 3 ) + A ) 3, ξ + a ) 3 )) Using 9), we compute the residues A p),j s as 00, and , respectiely Then the resulting PEP at 0 db for cooperatie space-time fading system becomes We will show in Section 3 that the PEP simulation matches this calculation exactly 3 Numerical Example Combining 6) and 8), we can find the PEP for the asymmetric case is the sum of the integrations oer the indiidually expanded terms, where the integration of each term corresponds to a symmetric PEP calculation Making use of the result obtained from the symmetric case, we derie the aeraged PEP for MIMO systems In this section, we simulate PEP performance of Tarokh et al s codes [4] for general MIMO block fading channels We also study conolutional codes [] for the space-time coded cooperatie channels We assume that all zero codeword is transmitted and compute the PEP between the transmitted all-zero codeword and

4 the shortest error path [6] Our goal is to illustrate that the exact PEP calculations of Section indeed match the simulation results 0 0 Tx, Rx, QPSK 4States Tarokh s Codes Theoretical PEP Simulated PEP 0 3 General MIMO Block Fading Channels Figure shows the PEP results for -block fading channels Same SNR is assumed in each block We use QPSK-4 State trellis code gien in [4] repeated in each block and consider the shortest error eent The resulting eigenalues of C and C are both eual to 4, which corresponds to the symmetric case calculation We can obsere from Figure that the simulated PEP matches the theoretical PEP exactly as expected 3 Space-time Coded Cooperatie Systems For the cooperatie space-time block fading channel [], besides the different eigenalues in each user/block, the receied SNR from different users could be different leading to different SNR in each fading block Both of these factors result in different a i s for blocks Here we consider the systems with two users, the original user has one transmitter antenna and the partner has two transmitter antennas We use [33 7] conolutional code for the original user and [ ] conolutional code for the partner Then the euialent code for this -block fading channel is [33 7, ] In Figure, we show the PEPs for two different cases One is the case in which both users hae the same receied SNR and the other is that the partner has fixed SNR at 0 db and the SNR for the original mobile changes which is illustrated in the x-axis From Figure, we can also find that our analytical PEP matches the simulated PEP exactly Notice the PEP for the eual receied SNR case of the coded cooperatie system corresponds to the example in Section 4 Conclusion In this work, we describe a simple techniue to ealuate the PEP and present the exact PEP expression for MIMO systems in block fading channels We classify the PEP calculation into two groups based on the alues of the products of the SNR and the eigenalue of the product of the code difference matrices in each block In the symmetric case, where all the products are eual, we simplify the calculation of the PEP by defining the new random ariables The other group is asymmetric case, which is more general and includes symmetric case as its specific example In the calculation of PEP for this group, we apply more general approach by using the alternatie finite integral form PEP SNRdB) Figure : PEP for QPSK-4 State trellis code [4] repeated in -block fading channel PEP Tx Antenna for User and Tx Antennas for User, [ ] codes Simulated PEP,SNR = SNR Theoretical PEP, SNR = SNR Simulated PEP,SNR=0dB Theoretical PEP, SNR = 0dB SNRdB) Figure : PEP for -user coded cooperatie systems User with one transmitter antenna uses [33, 7] conolutional code and user with two transmitter antennas uses [7, 65, 5, 37] of Q-function and the results obtained in symmetric case In the numerical examples, we illustrate the accuracy of our closed form expression References [] E Biglieri, J Proakis and S Shamai, Fading channels: information-theoretic and communications aspects, IEEE Trans Inform Theory, ol 44, no 6, pp 69-69, Oct 998 [] A Stefano and E Erkip, Cooperatie Space- Time Coding for Wireless Networks, in Proceedings of IEEE Information Theory WorkshopITW), La Sorbonne, Paris, France, April 003

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