LDPC-Coded MIMO Receiver Design Over Unknown Fading Channels

LDPC-Coded MIMO Receiver Design Over Unknown Fading Channels Jun Zheng and Bhaskar D. Rao University of California at San Diego Email: juzheng@ucsd.edu, brao@ece.ucsd.edu Abstract We consider an LDPC-coded MIMO system comosed of M transmit and N receive antennas oerating in a flat fading environment. The channel state information is assumed to be unavailable both to the transmitter and the receiver. A soft iterative receiver structure is develoed which consists of three main blocks, a soft MIMO detector and two LDPC comonent soft decoders. Without forming any secific channel estimate, we roose several soft MIMO detectors at the comonent level that offer an effective tradeoff between comlexity and erformance. At the structural level, the LDPC-coded MIMO receiver is constructed in a unconventional manner where the soft MIMO detector and LDPC variable node decoder form one suer soft-decoding unit, and the LDPC check node decoder forms the other comonent of the iterative decoding scheme. By exloiting the roosed receiver structure, tractable extrinsic information transfer functions of the comonent soft decoders are obtained, which further lead to a simle and efficient LDPC code degree rofile otimization algorithm with roven global otimality and guaranteed convergence from any initialization. Finally, numerical and simulation results are rovided to confirm the advantages of the roosed design aroach for the coded system. I. INTRODUCTION Communication systems using multile antennas at both the transmitter and the receiver have recently received increased attention due to the caability of roviding great caacity increases in a wireless fading environment 1]. However, the caacity analysis and MIMO system design is often based on the underlying assumtion that the fading channel coefficient between each transmit and receive antenna air is erfectly known at the receiver. This is not a realistic assumtion for most ractical communication systems esecially in fast fading channels. For communication systems with unknown channel state information CSI at both ends, conventional receivers usually have a two-hase structure, channel estimation using the reset training symbols followed by coherent data detection. Due to the imortance of the channel estimator, which directly determines estimation quality and hence the overall system erformance, various MIMO channel estimation algorithms have been studied 2] 3]. However, conventional channel estimators form estimates based only on the training symbols, thereby failing to make use of the channel information contained in the received data symbols. Consequently, the twohase model limits the erformance and can not aroach This research was suorted by CoRe grant No.02-10109 sonsored by Ericsson. the MIMO channel caacity or the maximum achievable information rate. Possible solutions to the above roblem include use of blind source signal searation algorithm 4] 5], MIMO differential modulation 6] 7], and unitary sace-time modulation USTM 8] 9]. However, none of these schemes can aroach the non-coherent MIMO caacity limit due to their sub-otimal code structure, and in the case of USTM, only asymtotic or the diversity otimality is achieved in high SNR regimes and the aroach suffers from exonential decoding comlexity. In this aer, we focus on the design of ractical LDPCcoded MIMO systems emloying a soft iterative receiver structure consisting of three comonent soft decoding blocks, a soft MIMO detector and two soft LDPC comonent decoders variable node and check node decoders. At the comonent level, we roose several soft MIMO detectors, which can generate soft log likelihood ratio LLR of each coded bit under the condition of unknown CSIR without forming any exlicit channel estimate. At the structural level, the LDPCcoded MIMO receiver is constructed in an unconventional manner where the soft MIMO detector and LDPC variable node decoder form one suer soft-decoding unit and the LDPC check node decoder forms the other comonent of the iterative decoding scheme. Utilizing the receiver structure, tractable extrinsic information transfer functions EXIT of the comonent soft decoders are obtained, which further lead to a simle and efficient LDPC code degree rofile otimization algorithm. This algorithm is shown to have global otimality and guaranteed convergence from any initialization, which is in contrast to the sub-otimal manual curve fitting technique roosed in 10]. Fig. 1. II. SYSTEM MODEL Transmitter model of LDPC-coded MIMO systems We consider a MIMO system with M transmit antennas and N receive antennas signaling through a frequency flat fading channel with i.i.d channel coefficients between the transmit

and receive antenna airs. As illustrated in Fig. 1, a block of k binary information bits denoted d = {d 1,,d k } is first encoded by an outer LDPC encoder with code rate R = k/n into a codeword c = {c 1,,c n } of length n. The codeword c is further segmented into L consecutive sub-blocks C i of length K. Each sub-block C i is then encoded by the inner sace-time encoder into a coherent sace-time sub-frame X i. This encoder is comosed of an interleaver, modulator, serialto-arallel converter, and a ilot insertion oerator. Within each sub-frame, the first = MT τ symbols are training ilots, followed by MT d data symbols with T d + T τ = T. Hence, the transmitted signal X i can be stacked and searated into two sub-matrices, i.e. X i = T, ρ/m X T τ, Xd,i] T 1 where X τ C Tτ M are the fixed ilot symbols sent over T τ time intervals and X d,i C T d M are the information bearing data symbols sent over T d transmission intervals, with the following ower constraints, tr X H τ X τ = MTτ, E Xd, i tr X H d, i X d, i ] = MT d. 2 Therefore, ρ is the average signal to noise ratio SNR at each receive antenna. Each element of the transmitted data signal X d,i comes from a finite comlex alhabet X of size X. One entire MIMO codeword X consists of l = LTM comlex symbols, which are transmitted from M transmit antennas and across L consecutive coherent sub-frames of length T M symbols each. It is assumed that the fading coefficient matrix H i remains static within each coherent sub-block and varies indeendently from one sub-block to another. Hence, the signal model can be written as Y i = X i H i + w i, 1 i L, 3 where Y i is a T N received comlex signal matrix, X i is a T M transmitted comlex signal matrix, H i is an M N comlex channel matrix, and w i is a T N matrix of additive noise. Both matrices H i and w i are assumed to have zero mean unit variance indeendent comlex Gaussian entries. III. SOFT-INPUT SOFT-OUTPUT MIMO DETECTOR Conventional channel estimators erform estimation only based on the training ilots, thereby failing to make use of the channel information contained in the data symbols. Due to the mismatch between the actual and estimated channel, system erformance of the two-hase receiver structure channel estimation followed by coherent data detection suffers severe degradation esecially in low SNR regimes or fast fading channels. In this section, several novel MIMO detectors which include the otimal soft MIMO detector as well as two modified sub-otimal detectors are roosed that offer an effective tradeoff between detection comlexity and erformance. For the sake of simlicity, subscrit i, denoting the i th coherent block, is droed in this section while describing the soft MIMO detection ] algorithms. To be secific, we denote X = X T τ, X T T ] d, H, and Y = Y T τ, Yd T T as the transmitted signal, channel matrix, and received signal in each coherent block, resectively. Furthermore, sub-matrices X τ, X d, Y τ, and Y d have the following structures, i.e. ] T, ] T X τ = x T τ,1,,x T τ,t τ Xd = x T d,1,,x T d,t d, ] T, ] T Y τ = yτ,1, T,yτ,T T τ Yd = yd,1, T,yd,T T d, 4 where x τ,k, x d,k, y τ,k, and y d,k reresent comlex row vectors of size 1 M. Similarly, the binary sub-codeword C that mas to the transmitted signal X can also be decomosed into C = c T 1,,c T T d ] T, ck T d k=1 B1 M log 2 X, 5 where B is binary set {0,1} and each row c k reresents the corresonding binary information that mas to x d,k. A. Otimal soft MIMO detector First, according to the channel model 3, the conditional robability density of the received signal matrix Y given the transmitted signal matrix X is given by 11] } Y X = ex tr{ I T + XX H ] 1 YY H π TN det N I T + XX H ]. 6 It is evident from the above transitional robability that the unknown MIMO channel is actually a memoryless vector channel and hence the otimal MIMO detector does not necessarily need to form a secific channel estimate. In order to obtain the a osteriori robability of each coded bit, the a riori robability of the inut signal matrix X is first calculated as X = T d X d = C = k=1 T d c k = k=1 M log 2 X j=1 c k,j, 7 where each element of matrix X d is a member of a comlex alhabet X of size X, each corresonding to log 2 X LDPCcoded bits. Therefore, the log likelihood ratio of each LDPC coded bit is given by L os c k,j = log ΣX D + k,j Σ X D k,j Y X Y X X X 1 k T d, 1 j M log 2 X, 8 where D + k,j D k,j is the set of X for which the k,jth bit c k,j of the LDPC coded sub-block C is +1 1. Finally, by subtracting the inut a riori information from the obtained a osterior log likelihood ratio, the soft extrinsic information of each coded bit is obtained as, = 1 L ext c k,j = L os c k,j log c k,j = 0, 9,

Notice that there is no channel estimation stage in the soft MIMO detector described above, and therefore the roosed detection algorithm does not deend on the unknown channel state H but only on its underlying statistical distribution. B. Sub-otimal soft MIMO detector The otimal soft MIMO detection algorithm roosed in Section III-A rovides the otimal extrinsic LLR values of each coded bit. However, the summation in both the numerator and the denominator of equation 8 consists of 2 K1 items, with K =T d M log 2 X increasing linearly with number of data slots T d or coherence time T. It has an unaffordable exonential comlexity for ractical communication systems, esecially when the coherence time T is large. Hence, we roose a sub-otimal MIMO detector in this section with comlexity increasing linearly with T d. Instead of erforming soft MIMO detection by one oeration, we can extract artial extrinsic information by rocessing only two rows of the data matrix X d at a time, and then combining different artial extrinsic information to form the final extrinsic LLR. In order to combine information from coded rows x d,k and x d,k, we first erform the otimal MIMO detection algorithm on the following reduced size sub-coherent block X k,k ] = ] T, ] T X T τ,x T d,k, x T d,k Yk,k ] = Yτ T,yd,k T, yd,k T. 10 Therefore, the artial extrinsic LLR value L ext, k of bit c k,j obtained from the a riori information of row c k, c k, and channel observation Y k,k ] is given by L ext, k = log log ΣXk,k ] D+ k,j Σ Xk,k ] D k,j = 1 c k,j = 0 Y k,k ] Xk,k ] Y k,k ] Xk,k ] X k,k ] X k,k ], 1 k, k T d, 11 where D + k,j D k,j is the set of X k,k ] for which bit c k,j is + 1 1. By the same reasoning, artial extrinsic information of bit c k,j, related to and contained in the a riori information of c k and channel observations Y τ and y d,k can also be obtained by erforming otimal detection on the following sub-coherent block X k] = X T τ, x T d,k] T, Yk] = Y T τ, y T d,k] T, 12 with the corresonding extrinsic LLR value given by ΣXk] D + Y Xk] k] X k] k,j L ext- = log Σ Xk] D Y Xk] k,j k] X k] = 1 log c k,j = 0, 1 k T d. 13 Having obtained extrinsic information L ext, k and L ext-, one can obtain by the following substraction, L ext-d, k = Lext, k Lext-, 14 the extrinsic information of bits c k,j extracted solely from the channel observation y d,k and the a riori information of c k. In contrast to the situation of erfect channel state information at the receiver CSIR where L ext c k,j only deends on the a riori knowledge of c k and observation y d,k, a non-zero extrinsic information of c k,j can be obtained from the a riori knowledge of c k and observation y k with k k in an unknown MIMO fading environment. An intuitive exlanation of above difference can be made by viewing c k as artially fixed ilots based on the inut a riori information. Therefore, better channel knowledge is learned although no exlicit channel estimation exists, which translates into a better a osterior robability of c k,j. Hence, a non-zero artial extrinsic information solely from the a riori robability of c k and the channel observation y k is obtained. Due to the assumtion that the inut a riori information of different bits are indeendent, all the artial extrinsic information L ext-d,k c k,j and L ext- c k,j can be viewed as being close to indeendent. The final outut extrinsic information L ext c k,j is obtained by summing all the indeendent artial extrinsic information obtained from different coded rows c k and ilot observations, i.e. T d L ext = = T d k =1 k k k =1 k k L ext-d, k + Lext- L ext, k Td 2 L ext-, 15 A summation of 2 2M log 2 X terms is required to extract the artial extrinsic information L ext,k c k,j in equation 11 and 2 M log 2 X terms for L ext- c k,j in equation 13. Therefore, in order to obtain the outut soft extrinsic LLR values, a total number of T d 1 2 2M log 2 X + 2 M log 2 X terms of robability summation is required for each coded bit, as oosed to 2 T dm log 2 X terms in the original otimal soft MIMO detector. C. Sub-otimal butterfly soft MIMO detector Motivated by the fast Fourier transform FFT algorithm, we can further reduce the comlexity of the soft MIMO detector to log 2 T d 2 2M log 2 X + 2 M log 2 X terms of summation er coded bit by using a sub-otimal butterfly MIMO detector structure. It is first assumed that the number of the data slots T d = 2 m is ower of 2. If not, we can aroriately zero-ad the transmitted signal matrix X. The sub-otimal butterfly detection algorithm obtains the extrinsic information through a multi-level structure similar to the fast Fourier transform, where the extrinsic information is accumulated from level to level. Secifically, if the artial extrinsic LLR value of coded

bit c k,j at the n th level is Lext-d n, then the extrinsic LLR value of the n + 1 th level is udated as L n+1 ext-d = L n ext-d + L n+1 ext-d, 0 n m 1, 16 where the second term L n+1 ext-d of equation 16 reresents the additional artial extrinsic information obtained from the information of coded bits c k, with sub-codeword row index k given by { k k + 2 mn1 if = k 2 mn1 if k mod 2 mn < 2 mn1 k mod 2 mn 2 mn1. 17 Similar to the extraction algorithm rovided in 14, L n+1 ext-d is given by the following form = L n+1 ext Lext-, 18 L n+1 ext-d where L ext- is given by equation 13, and artial extrinsic information L n+1 ext is obtained by erforming otimal soft MIMO detection on the sub-coherent block X k,k ] and Y k,k ] with modified inut a riori information, i.e. L n+1 ext = log ΣXk,k ] D+ k,j Σ Xk,k ] D k,j = 1 Y k,k ] Xk,k ] Y k,k ] Xk,k ] n+1 a n+1 a Xk,k ] Xk,k ] log c k,j = 0. 19 Furthermore, the modified a riori robability n+1 a Xk,k ] in equation 19 is a combination of the a riori robability of c k and c k as well as the n th level extrinsic information of c k, which can be reresented as M log n+1 2 X a Xk,k ] = n c k,j ck,j ext ck,j, 20 j=1 where ext n ck,j is given by n ex c k ext,j Lext-d n ck,j ck,j = 1 + ex Lext-d. 21 n ck,j Therefore, L n+1 ext-d can be viewed as the artial extrinsic information obtained solely from the a riori information of c k, channel observation y k, and its extrinsic information at the n th level. Starting from the initial condition L 0 ext-d c k,j = 0, the extrinsic information L n ext-d c k,j of each coded bit is accumulated at each level by absorbing additional artial extrinsic information through the sub-coherent block combining rocess. The final soft extrinsic LLR value of each coded bit is formed by combining the extrinsic LLR information at the m th lowest level with the extrinsic information obtained from ilot observations, which is given by L ext = L m ext-d + Lext- 1 k T d. 22 D. Discussion Note that both the sub-otimal structure in Section III-B as well as the sub-otimal butterfly MIMO detector in the revious subsection are modifications of the otimal soft MIMO detection algorithm rovided in Section III-A. The two subotimal MIMO detection algorithms rovided in Section III-B and III-C have the following structural differences. First, the sub-otimal MIMO detector in Section III-B forms extrinsic information through a linear combining structure, where there are a total of T d 1 artial extrinsic information terms each corresonding to the artial extrinsic LLR obtained from other rows k ; each term is comuted by erforming otimal detection on the sub-coherent block given by 11-14. On the other hand, the sub-otimal butterfly MIMO detector in Section III-C erforms data detection by emloying a multilevel structure, where the extrinsic information is distributed at succeeding levels until all the inut a riori information and the channel observations are combined and exchanged between all different rows. IV. RECEIVER DESIGN OF THE CODED MIMO SYSTEM A. New receiver structure of the LDPC-coded MIMO system Y SoftIn SoftOut MIMO Detector I A,DET I E,DET F. I A,DET Variable node decoder VND VND SUPER SOFT DECODER MIMO Detector & LDPC Variable Node Decoder Fig. 2. I E,VND. f s I A,VND Edge Interleaving 1 Π Iterative Loo Π I A,CND. f c I E,CND Receiver structure of LDPC-coded MIMO systems Check node decoder CND LDPC CHECK NODE DECODER Conventionally a coded MIMO receiver is obtained by connecting the inner soft MIMO detector and the outer LDPC decoder to form one large iterative decoding loo. The overall MIMO receiver actually consists of two iterative decoding loos. The soft extrinsic information, which describes the uncertainty of each coded bits, is iteratively exchanged in the outer loo between the MIMO detector and LDPC decoder as well as in the inner loo between variable node and check node decoders inside the LDPC decoder. In this aer, we structure the MIMO receiver differently by combining the soft MIMO detector and LDPC variable node decoder together as a suer soft decoder, a form also suggested in 10]. As illustrated in Fig. 2, the decoding loo is formed by exchanging extrinsic information between the suer decoder and the LDPC check node decoder iteratively. Comared with the conventional iterative MIMO receiver named as bit-interleaved coded modulation with iterative decoding BICM-ID algorithm, the new receiver structure has two advantages. First, the new receiver structure has only one iterative decoding loo and hence has lower decoding comlexity comared to the conventional BICM-ID receiver structure. Second, the roosed structure has the advantage of

enabling the EXIT function of the soft comonent decoders to have tractable forms. By fully exloiting the closed form EXIT functions, a simle iterative aroach for the LDPC code degree rofile otimization with guaranteed convergence and global otimality is roosed in Section IV-B, which is suerior to the sub-otimal manual curve fitting technique 12] 10]. B. LDPC code otimization Following the methodology given in 10] 12], the EXIT functions of the suer MIMO soft decoder combination of the LDPC variable node decoder and soft MIMO detector can be obtained as dv, I E,VND = f s IA,VND = λ i J i 1 J 1 I A,VND +J 1 F ρ J d v, i J 1 I A,VND, 23 where λ i is the fraction of the variable nodes having edge degree d v, i, and D v is the number of different variable node degrees. The function J is given by Jσ 2 A = 1 ln 2 1 σa 2 σ A ln coshy 2π ex σ2 A y 12 dy, 24 2σ 2 A and the maing F ρ reresents the inut-outut relations of the MIMO detector between I E,DET and I A,DET at SNR ρ, i.e. I E,DET = F ρ IA,DET. 25 The check nodes of the LDPC code have a transfer characteristic given by the following 12] I E,CND = f c IA,CND D c dc, 1 ρ i J i 1 J 1 1 I A,CND, 26 where ρ i is the fraction of the check nodes having edge degree d c, i, and D c is the number of different check node degrees. Following the successful decoding convergence criterion rovided in 12], the degree rofile otimization roblem can be reduced to the following maximization roblem by taking the LDPC code rate R as the objective max R outer = max {λ i,ρ i} {λ i,ρ i} 1 under linear constraints given by Dc ρ i/d c,i Dv λ i/d v,i, 27 I E,VND I A,VND I A,CND I E,CND = I A,CND I A,VND, D c λ i = 1, ρ i = 1, 0 λ i,ρ i 1. 28 Utilizing the closed form EXIT functions of the comonent soft decoders given by 23 and 26, we roose an efficient LDPC code degree rofile otimization algorithm in the following, which is comosed of two simle linear otimization stes. Variable node degree rofile otimization: For a fixed check node degree rofile {ρ k i } from the kth iteration, the otimal variable node degree rofile {λ k+1 i } is given by } = arg max λ i /d v,i, 29 {λ k+1 i under the constraints f s f c a n a n, {λ i} λ i = 1, 0 λ i 1, 1 n N, 30 where { a n an 0,1] } is a set of secified constraint oints, and N is the total number of constraints on the curve. Check node degree rofile otimization: For a fixed variable node degree rofile {λ k+1 i } from the k+ 1 th iteration, the otimal check node degree rofile {ρ k+1 i } is given by D c } = arg min ρ i /d c,i, 31 {ρ k+1 i under the constraints f c f s a n a n, {ρ i} ρ i = 1, 0 ρ i 1, 1 n N, 32 where a n and N are defined as before. Initializations: In general, we can start with any feasible degree rofiles. Based on our exerience from numerical simulations, we find that it is always a good choice to start with a regular check node degree d c. If we stack the LDPC code degree rofile {λ i,ρ i } into a suer vector η = λ 1,,λ Dv,ρ 1,,ρ Dc ] T. We can see that the objective R outer given in equation 27 is a concave function with resect to η and that all the constraints given in 28 are linear. Hence, the above degree otimization roblem has only one unique otimal solution. Due to the non-decreasing roerty of the roosed iterative maximization algorithm, it is guaranteed to converge to the global maximum solution η from any initialization oint. Therefore, in contrast to the subotimal manual curving fitting technique roosed in 10], the above iterative LDPC otimization algorithm rovides much better erformance and can serve as an efficient tool for coded MIMO system design. V. NUMERICAL AND SIMULATION RESULTS The robability of bit error of a 2 2 MIMO system over unknown fading channel with coherence time T = 6, training number T τ = 2, and BPSK modulation is demonstrated in Fig. 3. The outer LDPC code is a regular 3,6 code with code rate R = 1/2, and codeword length 8 10 4. As can be

Probability of bit error, P b 10 1 10 2 10 3 2by2 Regular 3,6 LDPCcoded MIMO System, T=6, T τ =2 Otimal soft detector Subotimal detector Subotimal butterfly detector MMSEbased detector 10 4 1.5 2 2.5 3 3.5 4 Signal to noise ratio, E /N s 0 Fig. 3. Probability of bit error of a 2 2 regular 3, 6 LDPC-coded MIMO system over a unknown fading channel with coherence time T = 6 and training number T τ = 2 using several different soft MIMO detectors. Probability of bit error, P b 10 1 10 2 10 3 10 4 2by2 Otimized rate 1/2 LDPCcoded MIMO System, T=6, T τ =2 Otimal soft detector Subotimal detector Subotimal butterfly detector MMSEbased detector 1 1.5 2 2.5 3 3.5 Signal to noise ratio, E /N s 0 Fig. 4. Probability of bit error of a 2 2 otimized LDPC-coded MIMO system over a unknown fading channel with coherence time T = 6 and training number T τ = 2 using several different soft MIMO detectors. observed from the above lot, over 1.5dB erformance gain can be achieved by using otimal soft MIMO detectors rather than the simle MMSE-based detector. The two sub-otimal MIMO detectors rovide significant erformance gain, and at the same time maintain affordable decoding comlexity. Using the otimization algorithm rovided in Section IV-B, the otimal LDPC code degree rofiles with outer code rate R outer = 1/2 for the coded MIMO system using the different soft MIMO detection algorithms are obtained and used in the overall erformance simulation. We consider the same 2 2 coded MIMO system used in Fig. 3 that transmits over the same unknown fading channel with coherence time T = 6 and ilot number T τ = 2 for simulations. The robability of bit error of the LDPC-coded MIMO system with otimized LDPC code degree rofile is shown in Fig. 4. Comared with Fig. 3, we can achieve about 0.6dB erformance gain by using the otimized LDPC degree rofile as oosed to the simle regular 3, 6 LDPC code. VI. CONCLUSION In this aer, we develoed a ractical LDPC-coded MIMO system over a flat fading wireless environment with no channel state information neither at the transmitter nor at the receiver. We first roosed several soft MIMO detectors, including one otimal soft MIMO detectors and two simlified subotimal detectors, that offer an effective tradeoff between comlexity and erformance. A coded MIMO receiver is constructed in an unconventional manner, where the soft MIMO detector and LDPC variable node decoder form one suer soft-decoding unit, and the LDPC check node decoder forms the other comonent of the iterative decoding scheme. By exloiting the roosed receiver structure, tractable extrinsic information transfer functions of the comonent soft decoders are obtained. Based on the closed form EXIT functions, a simle and efficient LDPC code degree rofile otimization algorithm is roosed. The roosed otimization algorithm is shown to have global otimality and guaranteed convergence from any initialization, which is suerior to the sub-otimal manual curve fitting technique in revious work. Numerical and simulation results of the unknown LDPC-coded MIMO system using the otimized degree rofile further confirm the advantage of using the roosed design aroach for the coded MIMO system. REFERENCES 1] E. Telatar, Caacity of multi-antenna Gaussian channels, Euroean Trans. Telecomm. ETT, vol. 10, no. 6,. 585 595, Nov 1999. 2] J. H. Kotecha and A. M. Sayeed, Transmit signal design for otimal estimation of correlated MIMO channels, IEEE Trans. on Signal Processing, vol. 52,. 546 557, Feb 2004. 3] D. Samardzija and N. 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