Design of Space Time Spreading Matrices

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1 Design of Space Time Spreading Matrices Yibo Jiang, Ralf Koetter, and Andrew Singer Coordinated Science Laboratory Dept. of Electrical and Computer Engineering Univ. of Illinois at Urbana-Champaign Urbana, IL {yjiang, koetter, October 29, 2002 Abstract In this paper, we study the design of space time spreading matrices that are modulation matrices for communications over multiple-antenna block fading channels. We assume the channel is known to the receiver only and find necessary and sufficient conditions on the space time spreading matrices for the separation of demodulation and decoding at the receivers without loss of optimality. From the perspective of information theory, we design the space time spreading matrices to maximize the mutual information between the transmitted information vector and the channel output. We apply the designed space time spreading code as an inner code in a bit-interleaved serially concatenated coding scheme, and find the connection between the endpoints of the extrinsic information transfer curves and certain mutual information quantities. A design example and simulation results are given. 1 Introduction We consider a multi-input multi-output (MIMO) communication setup in which we assume there are t transmit antennas and r receive antennas. Results by Foschini and Gans [1] and Telatar [2] have shown that, for a multiple-antenna channel where each transmit and receive antenna pair This work was supported in part by the National Science Foundation under grants NSF CCR , NSF ITR and NSF CCR , and in part by the Motorola Center for Communications. 1

2 is subject to i.i.d. Rayleigh flat fading, the channel capacity increases linearly with min(t, r) in the high signal-to-noise ratio (SNR) regime if the channel state information (CSI) is known to the receiver only. This result has sparked tremendous interest and effort in the design of practical channel codes for communication over multiple-antenna channels (i.e., MIMO channels). Some prominent schemes include BLAST [3], space time trellis codes [4], space time block codes (STBCs) from orthogonal designs [5, 6], and linear dispersion codes (LD codes) [7]. BLAST has a high transmission rate and simple decoding algorithms, but suffers from low diversity gain. In [4], a rank criterion and a determinant criterion were proposed to design space time trellis codes. However, the decoding complexity of space time trellis codes [4] increases exponentially with the transmission rate or the number of transmit antennas. Space time block codes from orthogonal designs can achieve full diversity, i.e., in the high SNR regime, the pairwise error probability (PEP) decays like SNR tr. There exist simple maximum likelihood decoding algorithms for STBCs from orthogonal designs. However, there is also a rate limitation for such codes. For example, it was shown in [6] that no complex orthogonal design that achieves full diversity and full rate (one symbol per time slot) exists for more than two transmit antennas. It was also pointed out in [8] that STBCs from orthogonal designs maximize the SNR. In [9], design rules derived from minimizing the union bound on the error probability and maximizing the channel capacity were proposed. It was also mentioned that STBC from orthogonal designs is a solution that minimizes the union bound on the error probability. Later in the paper, we will compare the design rules in [9] to our results. Our notation will be as follows: Capital letters denote matrices; underscores indicate vectors; boldfaced letters denote random objects; daggers denote complex conjugate transpose; asterisks denote conjugation. The space time codes referred to as linear dispersion (LD) codes can be conveniently written as [7] Q S = (α k A k + iβ k B k ) where α 1,, α Q, β 1,, β Q are real symbols to be transmitted and A 1,, A Q, B 1,, B Q are t T complex matrices. Here T is used to denote the block length. The coefficient matrices A 1,, A Q, B 1,, B Q are chosen to maximize the mutual information between the vector 2

3 (α 1,, α Q, β 1,, β Q ) and the channel output Y (r T complex matrix) subject to some energy constraint, i.e., max A k,b k,,,q max I(α 1,, α Q, β 1,, β Q ;Y H, A k, B k, k = 1,, Q) (1) p(α k,β k,,,q) subject to some energy constraint (see [7]). The matrix H is a r t CSI matrix and is assumed known to the receiver. It should be emphasized that, although in practical applications α k, β k, k = 1,, Q are chosen from some constellations, real alphabets are assumed in Eq. (1). It was shown in [7] that maximizing the above mutual information has some connections to minimizing the average pairwise error probability. These LD codes can achieve high data rates and good BER performance. Fundamentally, there is a tradeoff between the transmission rate and the error probability. In [10], the tradeoff between multiplexing (corresponding to data rate) and diversity (corresponding to error probability) was studied in the high SNR regime, and different coding schemes were compared. In this paper, we consider the design of a class of block codes whose codewords are linear in each information bit, i.e., X(b) = K b ka k where b = (b 1,, b K ) { 1, +1} K is the binary information vector to be encoded and A k, 1 k K are t T complex modulation matrices. Since the information bits are spread throughout the codeword matrix, we call these bit-linear space time spreading codes (BL-STSC) and call A k, 1 k K space time spreading matrices. The code rate is K/T bits per time slot. We use a discrete-time block fading channel model Y = HX + N and assume the r t CSI matrix H is known to the receiver but not to the transmitter. If in a communication system, a BL-STSC is used as the modulation code and there is some channel code preceding it, joint maximum likelihood demodulation and decoding is optimal in the sense of minimizing the error probability. We prove that if and only if A i A j + A ja i = 0 for all i, j, i j, the probability distribution of b 1,,b K conditioned on channel output Y = Y and CSI H = H is a product distribution. Therefore, demodulation and decoding can be separated. The designs of the codes in [4, 5, 6] are essentially based on the maximization of the minimum distance since the PEPs over all distinct codeword pairs are minimized. However, it is in the high SNR regime that the minimum distance determines the BER performance of a code. Since our design is not limited to any specific SNR regime, we will not use PEP as a design metric, but rather we will use mutual information as the design metric. We consider I(b 1,,b K ;Y H) which is the mutual information between the input and output 3

4 b X(b) Y BL-STSC MIMO Channel Figure 1: A compound channel. b X MIMO Y X=bA Channel Figure 2: A simple compound channel. of the compound channel shown in Fig. 1. We assume the information vector (b 1,,b K ) { 1, +1} K and each vector is equiprobable. Thus rather than performing the double maximization as in Eq. (1), we assume the input alphabet is discrete and the input distribution is fixed. We want to choose the space time spreading matrices A k, 1 k K such that the mutual information I(b 1,,b K ;Y H) is maximized subject to bit energy constraints. We take two steps to attack the problem. First, we find an upper bound on the mutual information I(b 1,,b K ;Y H), and necessary and sufficient conditions to achieve the upper bound. The upper bound is equal to K I(b k;b k HA k + N H). Subject to a bit energy constraint, the second step is to maximize the mutual information I(b;bHA +N H) which is the mutual information between the input and output of the simplified compound channel shown in Fig. 2. We find a lower bound on I(b;bHA + N H) and show that if A satisfies AA = ε b t I t, the lower bound is maximized subject to a bit energy constraint E b = tr{aa } ε b. Combining the results from the two steps, we reach the design criteria for space time spreading matrices. To achieve the above mutual information, one useful coding technique is to use a binary channel code as an outer code, bit-interleave it and serially concatenate it with the BL-STSC. At the receiver, an iterative decoding technique can be used. The system is shown in Fig. 3. A number of tools to analyze iterative decoding have been developed in the literature. The extrinsic information transfer chart (EXIT chart) [11] is of particular interest here. We show that necessary and sufficient conditions for p(b 1 = b 1,,b K = b K Y, H) to be a product distribution are also the necessary and sufficient conditions for the EXIT curves of the BL-STSC decoders to be constant. Furthermore, We show that maximization of the mutual information I(b 1,,b K ;Y H) and I(b;bHA +N H) are closely related to maximization of the heights of left and right endpoints of the EXIT curve 4

5 Outer Code c n Interleaver b BL-STSC X MIMO Channel Y BL-STSC Decoder L e Deinterleaver L a Interleaver Outer Decoder Figure 3: A bit interleaved serially concatenated coding scheme and the corresponding iterative decoding diagram. The inner code is a bit-linear space time spreading code. of the BL-STSC decoder. Since maximization of the heights of two endpoints means better BER performance of the bit-interleaved serially concatenated coding schemes under iterative decoding, space time spreading matrices (equivalently a BL-STSC) designed to maximize I(b 1,,b K ;Y H) subject to bit energy constraints will have good performance as an inner code. Finally, we give a design example and simulation results. 2 Signal Model We use the following discrete-time channel model in this paper. In the MIMO channel in Fig. 3, there are t transmit antennas and r receive antennas. The coefficient h i,j denotes the complex channel gain between transmit antenna j and receive antenna i and experiences i.i.d. frequency flat fading for all i and j. The CSI matrix H = [h i,j ] C r t is known to the receiver only and does not change within a block of T symbols. From block to block, H changes independently (quasi-static fading). Let X C t T indicate the transmitted signal matrix, Y C r T be the received signal matrix such that the discrete-time baseband equivalent channel model is Y = HX + N (2) where N C r T is an additive noise matrix with i.i.d. entries n i,j CN(0, 2σ 2 ). In this paper, we focus on a class of linear block codes, called bit-linear space time spreading codes (BL-STSCs) whose codeword matrices are linear in the information bits. We have the following definition. Definition 2.1 A bit-linear space time spreading code (BL-STSC) C is a block code whose codeword 5

6 matrices are linear in the information bits K C = {X : X = b k A k, b k { 1, +1}, k}, (3) where the space time spreading matrices A k C t T, 1 k K are complex modulation matrices. The rate of code C is K/T bps/hz. In this paper, we assume that the information bits are independent and equiprobable. Both STBCs and LD codes [6, 7] can be described in the following way: M C = {X : X = (S k B k + Sk B k )} (4) where S k, 1 k M are the modulated symbols, B k, 1 k M and B k, 1 k M are complex coefficient matrices. If the modulation scheme is PAM, or QAM with each dimension as an independent PAM, the modulated symbol S k is linear in the information bits. Then after simple manipulation, we can convert the model in Eq. (4) into the definition of BL-STSCs. Thus with PAM or QAM, the STBCs and LD codes are also BL-STSCs. 3 Necessary and Sufficient Conditions for the Separability of Demodulation and Decoding Given the channel output, it is well known that joint maximum likelihood demodulation and decoding is optimal in the sense of minimizing the error probability. However, if the probability distribution of the modulator input b 1,,b K conditioned on channel output Y = Y and CSI H = H is a product distribution, i.e., K p(b 1 = b 1,,b K = b K Y, H) = p(b k = b k Y, H) (5) then demodulation and decoding can be separated without loss of optimality. Define log likelihood ratio (LLR) for bit b k as L e (b k ) ln p(b k = +1 Y, H) p(b k = 1 Y, H) (6) 6

7 The demodulator outputs the K dimensional LLR vector (L e (b 1 ),, L e (b K )) which contributes a sufficient statistic for decoding. Here we treat the decoding of bit-linear space time spreading codes as demodulation. The following theorem states necessary and sufficient conditions on the space time spreading matrices {A 1,, A K } for the separability of demodulation and decoding. Theorem 3.1 Let C be a bit-linear space time spreading code with spreading matrices A k C t T, 1 k K. The probability distribution of b 1,,b K conditioned on the channel output Y = Y and the channel state information H = H is a product distribution for all Y and H K p(b 1 = b 1,,b K = b K Y, H) = p(b k = b k Y, H) (7) if and only if A i A j + A ja i = 0 for all i, j, and i j. Proof Define the vector b as b = (b 1,, b K ). Since Y = K b kha k + N, we have p(b = b,y = Y,H = H) = p(b)p(h)p(y H, b) =2 K p(h)( { 2πσ) 2rT exp tr{(y K b kha k )(Y } K b kha k ) } 2σ 2 =2 K p(h)( { 2πσ) 2rT exp tr{y Y + K HA } ka k H } 2σ 2 exp { tr{ K b k(y A k H + HA k Y ) K 1 i=1 2σ 2 K j=i+1 b ib j H(A i A j + A ja } i )H } (8) K K 1 =α(h, Y )exp{ b k L k (Y, H) K i=1 j=i+1 b i b j G i,j (H)} where α(h, Y ) 2 K p(h)( 2πσ) 2rT exp{ tr{y Y + K HA ka k H } 2σ 2 } (9) L k (Y, H) tr{y A k H + HA k Y } 2σ 2 (10) G i,j (H) tr{h(a ia j + A ja i )H } 2σ 2 (11) 7

8 We first prove the direct part of the claim: If A i A j + A ja i = 0 for all i, j, and i j, we obtain G i,j(h) = 0 and hence K p(b, Y, H) = α(h, Y )exp{ b i L i (Y, H)} (12) i=1 Then p(b k = 1 Y, H) p(b k = 1 Y, H) = p(b k = 1, Y, H) p(b k = 1, Y, H) = b:b k =1 p(b, Y, H) b:b k = 1 p(b, Y, H) = e2l k(y,h) (13) It follows that K p(b k = b k Y, H) = K e b kl k (Y,H) e L k(y,h) + e L k(y,h) (14) On the other hand, we have p(b = b Y, H) = = p(b, Y, H) p(y, H) K = b1 { 1,+1} exp{ K b kl k (Y, H)} exp{ K b k L k (Y, H)} bk { 1,+1} exp{ K b kl k (Y, H)} exp{ b k L k (Y, H)} = b k { 1,+1} K e b kl k (Y,H) e L k(y,h) + e L k(y,h) (15) Comparing Eq. (14) and (15), we arrive at the product distribution. We now prove the only if part of the claim: If p(b 1 = b 1,,b K = b K Y, H) is a product distribution, we have p(b = (b 1,, b i 1, +1, b i+1,, b K ) Y, H) p(b = (b 1,, b i 1, 1, b i+1,, b K ) Y, H) = p(b i = +1 Y, H) p(b i = 1 Y, H) (16) According to Eq. (8), we have p(b = (b 1,, b i 1, +1, b i+1,, b K ) Y, H) p(b = (b 1,, b i 1, 1, b i+1,, b K ) Y, H) i 1 =exp{2l i (Y, H) 2 b j G j,i (H) 2 j=1 K j=i+1 b j G i,j (H)} (17) 8

9 After combining Eq. (16) and (17), we obtain i 1 exp{2l i (Y, H) 2 b j G j,i (H) 2 j=1 K j=i+1 b j G i,j (H)} = p(b i = +1 Y, H) p(b i = 1 Y, H) (18) Since the right hand side (RHS) of Eq. (18) is invariant to b j { 1, +1} for all j i and must hold for all H, it follows G j,i (H) = 0 if 1 j i 1 and G i,j (H) = 0 if i + 1 j K. According to the definition in Eq. (11), it follows that A i A j + A ja i = 0 for 1 i K, 1 j K and i j since H is arbitrary. Next, we derive necessary and sufficient conditions on the codeword matrices for the requirement A i A j + A ja i = 0, i, j, i j to hold. Lemma 3.2 Let C be a bit-linear space time spreading code C = {X(b) : X(b) = K b ka k, b = (b 1,, b K ) { 1, +1} K } with spreading matrices A k C t T, 1 k K. Matrices A i and A j satisfy the condition A i A j + A ja i = 0, i, j, i j if and only if the codeword matrix X(b) satisfies X(b)X (b) = C 0, b. The constant matrix C 0 must be equal to K A ka k. Proof we first prove the only if part of the claim: K K X(b)X(b) = ( b k A k )( b l A l ) = = l=1 K A k A k C 0 K 1 K l=k+1 K b k b l (A k A l + A la k ) + b 2 k A ka k (19) We now prove the direct part of the claim: Define ˆb to be the vector consisting of the elements of b except for b i and b j, and define ˆX(ˆb) = X(b) b i A i b j A j, then we have [ ˆX(ˆb) + b i A i + b j A j ][ ˆX(ˆb) + b i A i + b j A j ] = C 0 (20) After some manipulations, we obtain b i [ ˆX(ˆb)A i + A i ˆX(ˆb) ] + b j [ ˆX(ˆb)A j + A j ˆX(ˆb) ] + b i b j (A i A j + A ja i ) (21) =C 0 ˆX(ˆb)( ˆX(ˆb)) A i A i A ja j 9

10 If we fix ˆb, the above matrix equation should hold for arbitrary (b i, b j ) { 1, 1} 2. Before proceeding, we first consider the following one dimensional problem. If the equation α 1 b i +α 2 b j +α 3 b i b j = α 4 holds for all (b i, b j ) { 1, 1} 2, we obtain the following four equations by plugging in the values of (b i, b j ) α 1 + α 2 + α 3 = α 4, α 1 α 2 α 3 = α 4 α 1 + α 2 α 3 = α 4, α 1 α 2 + α 3 = α 4 It follows that the constant coefficients must satisfy α 1 = α 2 = α 3 = α 4 = 0. Applying this result we see A i A j + A ja i = 0, ˆX(ˆb)A i + A i ˆX(ˆb) = 0 ˆX(ˆb)A j + A j ˆX(ˆb) = 0, C 0 ˆX(ˆb)( ˆX(ˆb)) A i A i A ja j = 0 It then follows that C 0 = K A ka k. Now we obtain two different forms of necessary and sufficient conditions for the separability of demodulation and decoding. Corollary 3.3 Let C be a bit-linear space time spreading code C = {X(b) : X(b) = K b ka k, b = (b 1,, b K ) { 1, +1} K } with spreading matrices A k C t T, 1 k K. The following statements are equivalent 1. p(b 1 = b 1,,b K = b K Y, H) = K p(b k = b k Y, H); 2. A i A j + A ja i = 0, i, j, i j; 3. X(b)X (b) = C 0, b; Proof Follows from Theorem 3.1 and Lemma 3.2. As mentioned in Section 2, if the modulation scheme is PAM, or QAM with each dimension as an independent PAM, STBCs from orthogonal designs are BL-STSCs. The codeword matrix X of a STBC from orthogonal designs satisfies XX = ( M S k 2 )I t. If BPSK or QPSK is used, 10

11 since they are equal-energy constellations, we have M S k 2 = ME s and XX = ME s I t C 0. Applying Corollary 3.3, we conclude that for STBCs from orthogonal designs (with BPSK or QPSK modulations), the probability distribution of b 1,,b K conditioned on the channel output Y = Y and the channel state information H = H is a product distribution. In [12], for several STBCs from orthogonal designs, simple decoding algorithms were provided and it was shown that maximum likelihood decoding of the symbols S 1,, S M can be decoupled. It is equivalent to say the probability distribution of the symbols S 1,, S M conditioned on channel output and channel state information is a product distribution, i.e., M p(s 1 = S 1,,S M = S M Y, H) = p(s k = S k Y, H) (22) In this case, if each symbol is modulated by w bits, the STBC decoder outputs an M2 w dimensional probability vector. Now assume the demodulation and decoding can be separated. The following lemma states the properties of the probability distribution of the LLR vector. Lemma 3.4 Let C be a bit-linear space time spreading code C = {X(b) : X(b) = K b ka k, b = (b 1,, b K ) { 1, +1} K } with spreading matrices A k C t T, 1 k K. If A i A j + A ja i = 0, i, j, i j, the following two properties hold 1. p(l e (b 1 ) = L 1,,L e (b K ) = L K H, b 1,, b K ) = K p(l e(b k ) = L k H, b k ); 2. p(l e (b 1 ) = L 1,,L e (b K ) = L K H) = K p(l e(b k ) = L k H); where the LLR L e (b k ) is defined as L e (b k ) = ln p(b k=+1 Y,H) p(b k = 1 Y,H), 1 k K. Proof We have Y = HX + N = K b kha k + N. If A i A j + A ja i = 0, i j, from Eq. (13), we have L e (b k ) = tr{y A k H + HA k Y } σ 2 = tr{2ha ka k H } σ 2 b k + tr{na k H + HA k N } σ 2 = tr{ K i=1 b ih(a i A k + A ka i )H + NA k H + HA k N } σ 2 (23) 11

12 Conditioned on H = H, we have tr{n(ha k ) + HA k N } σ 2 = 1 σ 2 r i=1 j=1 T [n i,j (HA k ) i,j + (HA k ) i,j n i,j] (24) Thus the above term is a zero mean Gaussian random variable since n i,j, i, j are i.i.d. zero mean complex Gaussian random variables with variance 2σ 2. Hence conditioned on H and b k, L e (b k ) is a Gaussian random variable. Furthermore, conditioned on H and b 1,, b K, (L e (b 1 ),,L e (b K )) is a Gaussian random vector. In order to prove the first property in the Lemma, we need to show that L e (b i ) and L e (b j ) are conditionally uncorrelated. We have E[tr{N(HA i ) + HA i N }tr{n(ha j ) + HA j N }] r T r T =E[( [n k,l (HA i ) k,l + (HA i) k,l n k,l ])( [n m,q (HA j ) m,q + (HA j ) m,q n m,q])] l=1 m=1 q=1 r T =2σ 2 [(HA i ) k,l (HA j) k,l + (HA i ) k,l (HA j ) k,l ] l=1 (25) =2σ 2 [tr{(ha j )(HA i ) } + tr{(ha i )(HA j ) }] = 2σ 2 tr{h(a j A i + A ia j )H } = 0 Hence conditioned on H and b 1,, b K, L e (b i ) and L e (b j ) are uncorrelated and thus independent. It follows that L e (b 1 ),,L e (b K ) has a product distribution as p(l e (b 1 ) = L 1,,L e (b K ) = L K H, b 1,, b K ) K K = p(l e (b k ) = L k H, b 1,, b K ) = p(l e (b k ) = L k H, b k ) (26) Next, since the information bit vectors are equally probable, we have p(l e (b 1 ) = L 1,,L e (b K ) = L K H) =2 K p(l e (b 1 ) = L 1,,L e (b K ) = L K H, b 1,, b K ) b 1 b K =2 K K p(l e (b k ) = L k H, b k ) b 1 b K K =2 K p(l e (b k ) = L k H, b k ) = bk K p(l e (b k ) = L k H) (27) This concludes the proof. 12

13 From the above proof, we see that L e (b k ) = tr{y A k H +HA k Y } σ 2. 4 Design of the Space Time Spreading Matrices In this section, our objective is to design space time spreading matrices A 1,, A K to maximize the mutual information I(b 1,,b K ;Y H) subject to bit energy constraints. We assume the information vectors (b 1,,b K ) are equiprobable. It is difficult to directly optimize I(b 1,,b K ;Y H). We will first find an upper bound on the mutual information. Necessary and sufficient conditions are then given in order to achieve the upper bound. Lemma 4.1 Let C be a bit-linear space time spreading code with spreading matrices A k C t T, 1 k K. We assume the information bits b 1,,b K are independent and equiprobable. The following upper bound on the mutual information holds K I(b 1,,b K ;Y H) I(b k ;Y H,b 1,,b k 1,b k+1,,b K ) (28) Equality is achieved if and only if A i A j + A ja i = 0 for all i, j and i j. Proof By the chain rule for mutual information, we have K I(b 1,,b K ;Y H) = I(b k ;Y H,b 1,,b k 1 ) (29) Furthermore, for 1 k K, define v (k) = (b 1,,b k 1 ), and u (k) = (b k+1,,b K ). Again by the chain rule for mutual information, we have I(b k ;Y,u (k) H,v (k) ) =I(b k ;Y H,v (k) ) + I(b k ;u (k) H,v (k),y) =I(b k ;u (k) H,v (k) ) + I(b k ;Y H,v (k),u (k) ) (30) =I(b k ;Y H,v (k),u (k) ) In the third step, we use the independence of b k and u (k). Since I(b k ;u (k) H,v (k),y) 0, we have I(b k ;Y H,v (k) ) I(b k ;Y H,v (k),u (k) ) (31) 13

14 In Eq. (31), equality holds if and only if I(b k ;u (k) H,v (k),y) = 0, i.e., b k and u (k) are independent conditioned on H = H, Y = Y and v (k) = v (k). Combining Eq. (29) and Eq. (31), we have the following inequality K I(b 1,,b K ;Y H) I(b k ;Y H,b 1,,b k 1,b k+1,,b K ) (32) where the equality holds if and only if for 1 k K p(b k = b k,b k+1 = b k+1,,b K = b K H, Y, b 1,, b k 1 ) =p(b k = b k H, Y, b 1,, b k 1 )p(b k+1 = b k+1,,b K = b K H, Y, b 1,, b k 1 ) (33) Next we prove that the upper bound in Eq. (28) is achieved if and only if A i A j +A ja i = 0 for all i, j and i j. It is easy to see that the condition in Eq. (33) is equivalent to K p(b 1 = b 1,,b K = b K Y, H) = p(b k = b k Y, H) (34) By Theorem 3.1, it follows that the upper bound is achieved if and only if A i A j + A ja i = 0 for all i, j and i j. Next we will go on to maximize the upper bound over the space time spreading matrices subject to bit energy constraints. For a space time spreading code, the total energy in the codeword matrix X is K K K E[tr{XX }] = E[tr{ b i b j A i A j }] = tr{a i A i } (35) i=1 j=1 In the above derivation, we use the assumption that b i, 1 i K are i.i.d with zero mean. Thus, the bit energy of b i is E bi = tr{a i A i } and is constrained in the optimization. For any k, since Y = HX + N = K i=1 b iha i + N, we can subtract the contribution from all bits except b k and arrive at I(b k ;Y H,b 1,,b k 1,b k+1,,b K ) = I(b k ;b k HA k + N H). Thus the upper bound (28) is equal to i=1 K I(b k ;b k HA k + N H) (36) 14

15 Since each term I(b k ;b k HA k +N H) depends only on one spreading matrix A k, the maximization can be performed term by term. Although each mutual information term has a relatively simple form, it is still difficult to maximize directly. Our approach is to first derive a lower bound on I(b k ;b k HA k + N H), then maximize the lower bound over A k subject to a bit energy constraint. Strictly speaking, this does not lead to a lower bound on I(b 1,,b K ;Y H) except in the case A i A j + A ja i = 0 for all i, j and i j. For simplicity, we omit the subscripts. 4.1 A Lower Bound on I(b;bHA + N H) From now on, we focus our analysis on the case where the fading coefficients h i,j, i, j are i.i.d and subject to Rayleigh fading, i.e., h i,j CN(0, 1). The following theorem gives a lower bound on I(b;bHA + N H). Theorem 4.2 For the signal model Y = bha + N, where the channel state information matrix H C r t is a circular symmetric complex Gaussian random matrix with i.i.d. entries distributed with CN(0, 1), A C t T is the space time spreading matrix for the equiprobable information bit b { 1, +1}, and N C r T is the noise matrix with i.i.d. entries distributed with CN(0, 2σ 2 ), the following lower bound on the mutual information holds I(b;Y H) 1 h 1 2 t j=1 ( 1 + λ ) r j 2σ 2 (37) where h(x) xlog 2 (x) (1 x)log 2 (1 x), 0 x 1 and λ j, 1 j t are the eigenvalues of AA. Proof First we compute the LLR P(b = 1 Y = Y,H = H) L e (b) = ln P(b = 1 Y = Y,H = H) = ln exp{ tr{(y HA)(Y HA) } } 2σ 2 exp{ tr{(y +HA)(Y +HA) } } 2σ 2 = tr{y (HA) + HAY } σ 2 = 2 tr{haa H } σ 2 b + tr{n(ha) + HAN } σ 2 (38) γ(h)b + η(n, H) 15

16 Conditioned on H = H η = tr{n(ha) + HAN } σ 2 = 1 σ 2 = 1 σ 2 r j=1 T 2 Re{n j,k (HA) j,k } r j=1 T [n j,k (HA) j,k + (HA) j,kn j,k ] (39) Thus conditioned on H = H, η is a Gaussian random variable with zero mean and variance 2γ(H). Since the LLR L e (b) is a sufficient statistic for bit b, we have I(b;Y H) = I(b;L e (b)). Define ˆb = sgn(l e (b)) { 1, +1}, from the data processing theorem, we have [13] I(b;L e (b)) I(b; ˆb) = H(b) H(b ˆb) = 1 H(b ˆb) (40) where we have used that P(b = 1) = P(b = 1) = 0.5 and therefore H(b) = 1. By symmetry, H(b ˆb) = h(p(ˆb b)), therefore I(b;L e (b)) 1 h(p(ˆb b)) (41) Now P(b ˆb) = P(b = 1)P(b ˆb b = 1) + P(b = 1)P(b ˆb b = 1) = 1 2 [P(ˆb = 1 b = 1) + P(ˆb = +1 b = 1)] = 1 2 [P(L e(b) < 0 b = 1) + P(L e (b) > 0 b = 1)] (42) = 1 2 [P(γ + η < 0) + P( γ + η > 0)] = 1 [P(η < γ) + P(η > γ)] 2 = 1 2 p(h = H)[P(η < γ H = H) + P(η > γ H = H)]dH Since η is a Gaussian random variable with zero mean and variance 2γ(H) conditioned on H = H, we have P(b ˆb) = < 1 2 ( ) γ(h) p(h = H)Q dh = 2γ(H) ( ) γ(h) p(h = H)Q dh 2 p(h = H)exp{ γ(h) 4 }dh = 1 [ { 2 E exp γ(h) }] 4 16 (43)

17 where Q(x) is the usual Gaussian tail probability function, and the bound Q(x) < 1 2 e x2 /2, x 0 is used. Next we evaluate E[exp{ γ(h) 4 }] = E[exp{ tr{haa H } 2σ 2 }]. Since AA is nonnegative definite, it can be decomposed as AA = UΛU (44) where UU = U U = I and Λ = diag{λ 1,, λ t }, λ i 0, i. Then, we have [ { E exp γ(h) }] [ { }] [ { }] = E exp tr{(hu)λ(hu) } 4 2σ 2 = E exp tr{hλh } 2σ 2 (45) In the second step, we use the fact that HU has the same distribution as H since U is unitary and H is an i.i.d. circular symmetric complex Gaussian random matrix. After manipulation, we observe that [ t r ] tr{hλh } = λ j h k,j 2 (46) j=1 Since h k,j, 1 k r, 1 j t are i.i.d complex Gaussian random variables, we have [ { E exp γ(h) }] [ { }] = E exp tr{hλh } 4 2σ 2 [ t r ] = E exp λ j 2σ 2 h k,j 2 j=1 (47) = t r j=1 [ { E exp λ }] j 2σ 2 h k,j 2 Since h k,j 2 is a chi-square random variable with two degrees of freedom, we have the moment generating function E[exp{ s h k,j 2 }] = (1 + s) 1, Re{s} 1. Therefore [ { E exp γ(h) }] = 4 t j=1 r (1 + λ j 2σ 2) 1 = t (1 + λ j 2σ 2) r (48) j=1 Applying the above result in inequality (43), we have P(b ˆb) < 1 2 t (1 + λ j 2σ 2) r (49) j=1 17

18 The RHS of the above inequality is at most 0.5. Since the entropy function h(x) is monotonically increasing in the interval [0, 0.5], we have I(b;Y H) = I(b;L e (b)) 1 h(p(b ˆb)) 1 h 1 2 t (1 + λ j j=1 2σ 2) r (50) This concludes the proof. We remark that if σ 0, the lower bound approaches one, and if σ, the lower bound approaches zero. Hence the lower bound is asymptotically tight because I e 1/0 as σ 0/. It is also easy to see that as the number of receive antennas r, the lower bound approaches one. Thus as the number of receive antennas becomes large, I e approaches one. The product 1 2 t j=1 (1+ λ j 2σ 2 ) r in the lower bound in Theorem 4.2 has almost the same form as the upper bound on PEP (Eq. (9), [4]). It is not surprising since in both cases, only two codeword matrices are considered. 4.2 Maximizing the Lower Bound on I(b;bHA + N H) Subject to a Bit Energy Constraint In this section, we proceed to maximize the lower bound over the space time spreading matrix A subject to a bit energy constraint E b = tr{aa } ε b. Since tr{aa } = t j=1 λ j, the constraint becomes t j=1 λ j ε b. The following theorem finds the optimal eigenvalues and thus the optimal matrix A subject to this bit energy constraint. Theorem 4.3 Suppose the block length T is no less than the number of transmit antennas t. Subject to the bit energy constraint E b = tr{aa } = t j=1 λ j ε b where λ j, 1 j t are the eigenvalues of AA, the lower bound 1 h( 1 2 t j=1 (1 + λ j 2σ 2 ) r ) is maximized when λ j = ε b t, j. Therefore, the optimal A satisfies A opt A opt = ε b t I t where I t is the t t identity matrix. Proof Since 1 2 t j=1 (1 + λ j 2σ 2 ) r 0.5, and h(x) is monotonically increasing in the interval [0, 0.5], 18

19 the optimization problem becomes min (λ 1,,λ t) S 1 2 t j=1 ( 1 + λ ) r j 2σ 2 (51) where S {(λ 1,, λ t ) : t j=1 λ j ε b, λ j 0, 1 j t}. Or equivalently max t (λ 1,,λ t) S j=1 ( 1 + λ ) j 2σ 2 (52) Since the geometric mean is less than the arithmetic mean t j=1 ( 1 + λ ) j 2σ 2 ( t j=1 (1 + λ j t 2σ 2 ) ) t = ( 1 + t j=1 λ ) t j 2tσ 2 ( 1 + ε b 2tσ 2 ) t (53) The equality holds if and only if λ j = ε b t, 1 j t. It follows that the optimal matrix A satisfies A opt A opt = ε b t UI t U = ε b t I t. Since rank(aa ) rank(a) min(t, T) = t, the equation A opt A opt = ε b t I t has solutions. The criterion AA = ci t has been shown to be optimal with respect to different senses of optimality in the literature, e.g., the average matched filter bound in [9]. We conjecture that a matrix A satisfying AA = ε b t I t maximizes the mutual information I(b;bHA + N H) itself subject to the bit energy constraint tr{aa } ε b. We motivate this conjecture by the following argument. It is easy to show that, for the continuous input signal model Y = xha +N where x CN(0, 1), if a matrix A satisfies AA = P t I t, the mutual information I(x;Y H) is maximized subject to the energy constraint tr{aa } P. We assume T t from now on. 4.3 A Design Criterion for Space Time Spreading Matrices Our objective is to design the space time spreading matrices A 1,, A K to maximize the mutual information I(b 1,,b K ;Y H) subject to bit energy constraints. Based on the results 19

20 from previous sections, we know K I(b 1,,b K ;Y H) I(b k ;b k HA k + N H) (54) If and only if A i A j + A ja i = 0 for all i, j and i j, the upper bound is achieved. Furthermore, if A k A k = ε k t I t, we prove that a lower bound on I(b k ;b k HA k +N H) is maximized subject to the bit energy constraint tr{a k A k } ε k and we conjecture that the mutual information I(b k ;b k HA k + N H) itself is also maximized. Combining these results, we design space time spreading matrices based on the following criteria A k A k = ε k t I t, 1 k K (55) A i A j + A ja i 0, 1 i K, 1 j K, i j (56) In [9], similar design criteria were proposed by minimizing the union bound on error probability. The authors [9] defined average deviation from unitarity D 1 = 1 K K i=1 κ(a i) where κ is the condition number and average deviation from pairwise skew-hermitianity D 2 = 1 K K 1 K i=1 j=i+1 A i A j +A ja i 2 F. Their design criteria were to minimize D 1 and D 2 (i.e., minimize the union bound on error probability) subject to a capacity-efficiency criterion. 4.4 The Application in the Bit Interleaved Serially Concatenated Codes and the Iterative Decoding We have considered the design of space time spreading matrices based on the mutual information I(b 1,,b K ;Y H). To achieve this mutual information, a useful coding technique is bit interleaved serially concatenated coding. A binary channel code can be chosen as the outer code and the bitlinear space time spreading code (BL-STSC) can be used as the inner code. At the receiver, iterative decoding can be used. The system is shown in Fig. 3. We will show how our results from the previous sections can be used in this framework. We point out that a number of results have been obtained by applying various space time transmission schemes as inner codes in a bit interleaved serially concatenated code and simplifying the decoding algorithms (see [14] and the reference therein). 20

21 Y BL-STSC L e (b i ) Decoder Deinterleaver L a (b i) L a ( cn ) Interleaver L e ( c n ) Outer Decoder L a ( c n ) + L ( c n ) e Figure 4: The iterative decoding flow chart for the bit interleaved serially concatenated coding scheme. We first introduce some concepts of iterative decoding. The iterative decoding process is illustrated in Fig. 4. At the receiver, to decode bit b i, the soft-input soft-output (SISO) BL-STSC decoder takes the channel output Y and the a priori input L a (b k ), 1 k K and outputs the extrinsic information L e (b i ). The a priori input is used to determine the a priori probability in the computation of the extrinsic information. Since the information bits are approximately independent after bit-interleaving, the a priori probability is updated as P(b = b) = K P(b k = b k ), with P(b k = 1) = 1 P(b k = 1) = 1, k i (57) 1 + exp{l a (b k )} and P(b i = 1) = P(b i = 1) = 0.5. The extrinsic information is defined as the LLR L e (b i ) = ln P(b i = 1 Y = Y,H = H) P(b i = 1 Y = Y,H = H) (58) The extrinsic information sequence is interleaved and passed to the SISO outer decoder as the only input. The outer decoder evaluates the extrinsic information L e (c n ) for every codeword bit c n. Since this paper is focused on the BL-STSC inner code, we will not describe the computation of L e (c n ) here. The details can be found in the original paper on serially concatenated turbo codes [15]. The extrinsic information sequence from the outer decoder is interleaved and passed to the BL-STSC decoder as a priori input in the next iteration. A number of tools to analyze iterative decoding have been developed in the literature. The extrinsic information transfer chart (EXIT chart) [11] is of particular interest here. An EXIT chart consists of two mutual information transfer curves (called EXIT curves), each characterizing the flow of extrinsic information through the SISO constituent decoder of the iterative decoding system. We illustrate the idea of the EXIT chart by describing its application in the system in Fig. 21

22 I e,1 I a, QPSK demapper Outer decoder I I a,1 e,2 Figure 5: An illustrative EXIT chart. There are two transmit antennas and two receive antennas in the channel. The SNR is E b /N o = 0 db. The outer code is a rate 0.5 recursive systematic convolution (RSC) code with generators G1 = 31 and G2 = 27. The inner code is [s 1, s 2 ] T where s 1 and s 2 are independent QPSK (with Gray mapping) symbols. The staircase path is the predicted trajectory. 3. Define I a,1 = I(b i ;L a (b i )), I e,1 = I(b i ;L e (b i )), I a,2 = I(c n ;L a (c n )) and I e,2 = I(c n ;L e (c n )). It is possible that the mutual information is not same for all bits of the BL-STSC, for example, when the number of bits is quite small and the coefficient matrices A k, 1 k K are too irregular. In that case, we can take their average, defining I e,1 = K i=1 I(b i;l e (b i ))/K. If I(b i ;L e (b i )) is constant for all i, I e,1 = K i=1 I(b i;l e (b i ))/K = I(b i ;L e (b i )) for all i. Therefore we use this general definition I e,1 = K i=1 I(b i;l e (b i ))/K. Each decoder can be viewed as a nonlinear system mapping the input mutual information to the output mutual information. Given the SNR, I e,1 can be viewed as a function of I a,1, I e,1 = T 1 (I a,1 ), and similarly I e,2 = T 2 (I a,2 ). Except for a few special channels, it is intractable in general to derive the analytical form of an EXIT curve. Interchanging the axes of the second curve and plotting it together with the first curve on the same diagram, we arrive at the so-called EXIT chart (see Fig. 5). A staircase path on the EXIT chart (as illustrated in Fig. 5) predicts the trajectory of the iterative decoding process. Under a wide range 22

23 of conditions, the actual decoding trajectory matches well with the predicted one [11]. The closer the intersection point of the EXIT curves is to the (1, 1) point, the better the BER performance of the system can be. Given that the EXIT behavior of binary codes is well investigated, we focus on the EXIT curves of the BL-STSC decoders. The left endpoint of EXIT curve corresponds to I a,1 = 0, i.e., no side information is available to the BL-STSC decoder and equal priors are assumed. Hence the LLRs L e (b k ), 1 k K are sufficient statistics for the information bits b k, 1 k K respectively and we have I(b k ;Y H) = I(b k ;L e (b k ) I a,1 = 0), 1 k K. Before proceeding, we first prove a simple mutual information inequality. If x 1 and x 2 are independent, we have I(x 1,x 2 ;y) = I(x 1 ;y) + I(x 2 ;y x 1 ) = I(x 1 ;y) + H(x 2 x 1 ) H(x 2 x 1,y) I(x 1 ;y) + H(x 2 ) H(x 2 y) = I(x 1 ;y) + I(x 2 y) (59) The equality holds if and only if x 1 and x 2 are independent conditioned y = y. Since the information bits b 1,,b K are independent, we can apply the inequality (59) and obtain K K I(b 1,,b K ;Y H) I(b k ;Y H) = I(b k ;L e (b k ) I a,1 = 0) = K Ie,1 0 (60) where Ie,1 0 is the height of the left endpoint of the EXIT curve. The equality in Eq. (60) is achieved if and only if b 1,,b K are independent conditioned on Y = Y and H = H. By Theorem 3.1, it is equivalent to say that the equality is achieved if and only if A i A j + A ja i = 0 for all i, j and i j. Thus, the design criteria in Eq. (55) and (56) also tend to maximize the height of the left endpoint of the EXIT curve. Similarly, the right endpoint corresponds to I a = 1, which implies that all other bits are perfectly known when decoding bit b k. Hence we have I(b k ;L e (b k ) I a,1 = 1) = I(b k ;Y H,b 1,,b k 1,b k+1,,b K ) = I(b k ;b k HA k + N H) (61) Thus the design criterion in Eq. (55) tends to maximize the height of the right endpoint which is equal to K I(b k;l e (b k ) I a,1 = 1)/K. Through simulations, we find that the EXIT curves of the BL-STSC decoders are near-straight lines. The maximization of the heights of two endpoints will move the intersection point of the 23

24 EXIT curves of BL-STSC decoder and outer decoder closer to the (1, 1) point, which will tend to make the BER performance better. Next, we find necessary and sufficient conditions for a constant EXIT curve. Theorem 4.4 Let C be a bit-linear space time spreading code K C = {X : X = b k A k, b k { 1, +1}, k} with spreading matrices A k C t T, 1 k K. The EXIT curve I e,1 = T 1 (I a,1 ) is constant if and only if the space time spreading matrices satisfy A i A j + A ja i = 0, i, j, i j. Proof We first prove the direct part of the claim: Assume A i A j +A ja i = 0. Without loss of generality, we consider the SISO decoding of the bit b 1. Define the vector b as b = [b 2,, b K ]. Define the matrix X as X = K k=2 b ka k. This implies that the codeword matrix X for b satisfies X = X + b 1 A 1. We have L e (b 1 ) = ln P(b 1 = 1 Y = Y,H = H) P(b 1 = 1 Y = Y,H = H) = ln b:b 1 =1 P(b = b)p(y = Y H = H,b = b) = ln b:b 1 = 1 b:b 1 =1 P(b = b)exp{ tr{(y HX)(Y HX) } 2σ 2 } b:b 1 = 1 P(b = b)exp{ tr{(y HX)(Y HX) } 2σ 2 } P(b = b)p(y = Y H = H,b = b) (62) Because A 1 A j + A ja 1 = 0, j 1, we have HXX H = H( X + b 1 A 1 )( X + b 1 A 1 )H = H[ X X + b 1 ( XA 1 + A 1 X ) + A 1 A 1 ]H = H[ X X K (63) + b 1 b k (A k A 1 + A 1A k ) + A 1A 1 ]H = H[ X X + A 1 A 1 ]H k=2 Since the information bits are independent and P(b 1 = 1) = P(b 1 = 1) = 0.5, after some 24

25 manipulation, we obtain L e (b 1 ) = ln b:b 1 =1 b:b 1 = 1 P(b = b)exp{ tr{ H X X H +Y (H X) +(H X)Y +Y (HA 1 ) +(HA 1 )Y } 2σ 2 } P(b = b)exp{ tr{ H X X H +Y (H X) +(H X)Y Y (HA 1 ) (HA 1 )Y } 2σ 2 } = tr{y (HA 1) + (HA 1 )Y } σ 2 P( b = b)exp{ tr{ H X X H +Y (H X) +(H X)Y } b 2σ 2 } + ln P( b = b)exp{ tr{ H X X H +Y (H X) +(H X)Y } b 2σ 2 } (64) = tr{y (HA 1) + (HA 1 )Y } σ 2 Since Y = HX + N = K b kha k + N, we further observe that L e (b 1 ) = tr{ K b kh(a k A 1 + A 1A k )H + NA 1 H + HA 1 N } σ 2 = tr{2ha 1A 1 H } σ 2 b 1 + tr{na 1 H + HA 1 N } σ 2 (65) In the second step, we used the conditions A 1 A j + A ja 1 = 0, j 1. Hence the extrinsic information L e (b 1 ) is a function of b 1, H and N, and is independent of the a priori information L a (b k ), 2 k K which are used to determine the priors of b k, 2 k K. It follows that the extrinsic mutual information I e,1 = I(b 1 ;L e (b 1 ) I a,1 ) is constant for all values of a priori mutual information I a,1. For other bits b k, 2 k K, the proof is essentially same. Thus the EXIT curve I e,1 = T 1 (I a,1 ) is constant. Next we prove the only if part of the claim: Assume the EXIT curve I e,1 = T 1 (I a,1 ) is constant. Thus the heights of left and right endpoint are same. Since for 1 k K I(b k ;L e (b k ) I a,1 = 0) = I(b k ;Y H) (66) I(b k ;L e (b k ) I a,1 = 1) = I(b k ;Y H,b 1,,b k 1,b k+1,,b K ) (67) we have I(b k ;Y H) = I(b k ;Y H,b 1,,b k 1,b k+1,,b K ), 1 k K (68) 25

26 By the chain rule for mutual information, we have I(b k ;Y,b 1,,b k 1,b k+1,,b K H) =I(b k ;Y H) + I(b k ;b 1,,b k 1,b k+1,,b K H,Y) =I(b k ;b 1,,b k 1,b k+1,,b K H) + I(b k ;Y H,b 1,,b k 1,b k+1,,b K ) (69) =I(b k ;Y H,b 1,,b k 1,b k+1,,b K ), 1 k K In the third step, we use the independence of the information bits. Combining Eq. (68) and (69), we obtain I(b k ;b 1,,b k 1,b k+1,,b K H,Y) = 0, 1 k K (70) Therefore, for 1 k K, we have p(b 1 = b 1,,b K = b K H, Y ) =p(b k = b k H, Y )p(b 1 = b 1,,b k 1 = b k 1,b k+1 = b k+1,,b K = b K H, Y ) (71) Hence it must be true that K p(b 1 = b 1,,b K = b K Y, H) = p(b k = b k Y, H) (72) By Theorem 3.1, we require A i A j + A ja i = 0, i, j, i j. So far, we have shown the following statements are equivalent 1. p(b 1 = b 1,,b K = b K Y, H) = K p(b k = b k Y, H); 2. A i A j + A ja i = 0, i, j, i j; 3. X(b)X (b) = C 0, b; 4. I(b 1,,b K ;Y H) = K I(b k;y H,b 1,,b k 1,b k+1,,b K ); 5. I(b 1,,b K ;Y H) = K I(b k;y H); 6. The EXIT curve I e,1 = T 1 (I a,1 ) is constant; Applying these results to the STBCs from orthogonal designs, we obtain the following property. 26

27 Corollary 4.5 Under BPSK or QPSK modulation, the EXIT curve of the decoder of a STBC from orthogonal designs is constant. Proof Under BPSK or QPSK modulation, STBCs from orthogonal designs are BL-STSCs. Since the codeword matrix X satisfies XX = ( M S k 2 )I t and both BPSK and QPSK are equal-energy constellations, it follows that M S k 2 = ME s and XX = ME s I t. According to Lemma 3.2 and Theorem 4.4, the EXIT curve is constant. The question arises as to whether or not there is such a code that it is not from an orthogonal design but which has a constant EXIT curve. We will show that the answer is Yes. We can construct a simple code which is not from an orthogonal design, but which also satisfies Theorem 4.4. The following code under BPSK modulation is an example ( ) b 1 b 2 X = b 1 b 2 (73) Essentially, since b 1 and b 2 are decoupled in time, the a priori information of one bit is irrelevant for the decoding of the other bit. Thus the corresponding EXIT curve will be constant. 5 Code Design Examples In this section, we give a simple example to illustrate our design approach. First we consider the problem of determining how many t t complex matrices satisfy the conditions A i A i = I t, 1 i M A i A j + A ja i = 0, 1 i M, 1 j M, i j (74) The amicable orthogonal designs theory in [16] tells us that if t = 2 a b, b odd, a 0, then the number of such matrices M satisfies M 2a + 2 and the bound can be achieved. For example, if t = 2, 3, 4, then M = 4, 2, 6 respectively. We assume t = T = 3. The number of information bits K = 3. Thus we need to find three 3 3 complex matrices according to the design criteria in Eq. (55) and (56). Consider the following 27

28 three matrices A 1 = A 2 = A 3 = 0 i 0 (75) i It is easy to verify that A i A i = I 3 for 1 i 3, A 1 A 2 + A 2A 1 = 0, A 1A 3 + A 3A 1 = 0, and A 2 A 3 + A 3A 2 0. Written in a compact form, the BL-STSC is (denoted by BL-STSC 1 ) b 1 b 2 b 3 X(b) = b 2 b 1 + ib 3 0 (76) b 3 0 b 1 + ib 2 Next consider a second BL-STSC (denoted by BL-STSC 2 ) b 1 b 2 b X(b) = b 1 b 2 b 3 i.e. B 1 = B 2 = B 3 = (77) b 3 b 1 b This code has the same rate and bit energy as the first code. It is easy to verify that its coefficient matrices {B 1, B 2, B 3 } satisfy B i B i I 3, 1 i 3 and B i B j + B jb i 0, i j. Thus the second BL-STSC is inferior to the first BL-STSC according to our design criteria. Next, we use both codes in bit interleaved serially concatenated coding schemes. In the simulation, we assume the number of receive antennas is r = 3. To get the EXIT curve of a BL-STSC decoder, we randomly generate the information bit stream, the fading channel, and the additive noise according to their distribution. Furthermore, we assume the a priori information L a (b) is conditionally distributed with N( σ2 a 2 b, σ 2 a) (used in [11]) where the parameter σ a is determined by the a priori mutual information I a = I(b;L a (b)). Fig. 6 shows the heights of left and right endpoints of the EXIT curves for BL-STSC 1 and BL-STSC 2 with respect to different SNR. We point out that in the computation of the bit energy E b, we assume a rate 1 2 outer code is used. It is apparent that both endpoints of the first code (BL-STSC 1) are higher than those of the second code (BL-STSC 2). We use a rate 0.5 recursive systematic convolution (RSC) code with generators G1 = 31, G2 = 27 as the outer code. The EXIT curve for this code is shown in Fig. 5 (the axes of 28

29 I e, BL STSC 1, I a,1 =0 BL STSC 2, I a,1 =0 BL STSC 1, I a,1 =1 BL STSC 2, I a,1 = E /N (db) b o Figure 6: The left and right endpoints of the EXIT curves of two BL-STSC decoders with respect to different E b /N o. The number of transmit and receive antennas are both 3. The block length is T = 3 symbols. the curve are interchanged). We perform a closed loop simulation for two bit interleaved serially concatenated coding schemes (denoted by coding scheme 1 and coding scheme 2 in Fig. 7), which use the same RSC code as the outer code and use BL-STSC 1 and BL-STSC 2 as the inner code, respectively. We use four iterations in the simulation. Fig. 7 shows the BER curves of these coding schemes. The first coding scheme has better BER performance than the second. In our design example, while the performance improvement is fairly modest, it underlines the soundness of our design criteria. 6 Conclusion We have studied the design of space time spreading matrices. We use mutual information as the measure, which is more appropriate in the low to moderate SNR regime than PEP. We have found necessary and sufficient conditions on space time spreading matrices for the a posterior distribution of the information vector to be a product distribution, i.e., the separation of demodulation and decoding at the receiver without loss of optimality. The approach is to first find an upper bound 29

30 10 1 Coding scheme 1 Coding scheme BER E /N b o Figure 7: The BER curves for two different bit interleaved serially concatenated coding schemes. Both use a rate 0.5 recursive systematic convolution (RSC) code with generators G1 = 31, G2 = 27 as the outer code, but use different bit-linear space time spreading codes (BL-STSCs) as the inner codes. The number of iterations is 4. on the mutual information between the information vector and the channel output, and necessary and sufficient conditions to achieve the upper bound, then indirectly maximize the upper bound subject to a bit energy constraint. We apply the designed space time spreading code as an inner code in a bit-interleaved serially concatenated coding scheme, and find the connection between two endpoints of the EXIT curve and certain mutual information quantities. We also find necessary and sufficient conditions for a constant EXIT curve. We then give a design example and simulation results. References [1] G. J. Foschini and M. J. Gans, On limits of wireless communications in a fading environment when using multiple antennas, Wireless Personal Communications, vol. 6, pp , March

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