An Interleave-Division-Multiplexing MISO System With Partial CSI at ...

IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 56, NO. 3, MAY 2007

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An Interleave-Division-Multiplexing MISO System With Partial CSI at Transmitter Chuxiang Li, Kai Li, Xiaodong Wang, Senior Member, IEEE, and Li Ping, Member, IEEE

Abstract—We treat the performance analysis and optimization of an interleave-division-multiplexing (IDM) multiple-input– single-output (MISO) system with partial channel state information (CSI) at the transmitters and an iterative (turbo) receiver. We propose a general methodology to analyze and evaluate the performance of such an IDM-MISO system. In particular, an SNR tracking method is proposed to calculate the output SNR of the soft-input soft-output IDM detector. The extrinsic mutual information transfer chart technique is employed to analyze the performance of the entire system. More specifically, the repetition decoder and the low-density parity check decoder are considered for the uncoded IDM system and the coded IDM system, respectively. Furthermore, based on the above analytical framework, we present the optimal transmit strategy for the IDM-MISO system with an imperfect CSI at the transmitter. Extensive simulations are performed to demonstrate the performance of the proposed IDM schemes under the MISO channels with different correlation statistics. Our results show that the proposed IDM-MISO scheme can outperform the existing space–time block code schemes in the low-rate transmission scenario. Index Terms—Interleave-division-multiplexing (IDM), iterative receiver, low-density parity check (LDPC), multiple-input– single-output (MISO), power allocation.

I. I NTRODUCTION

T

RANSMIT diversity in multiple-input–multiple-output (MIMO) systems is an effective technique to combat wireless fading environments [2]. One of the main challenges for transmit diversity techniques is the decoding complexity, which arises from the fact that the received signal at each receive antenna is the combination of the signals from all transmit antennas. Accordingly, effective and efficient cancellation of the cross-antenna interference is crucial to achieve diversity transmission. Orthogonal design of space–time block codes (STBCs) is an important transmit diversity technique with low decoding complexity [1], [8], [15], [16]. However, it is well known that the rate-one orthogonal STBC only exists for the case of two transmit antennas, i.e., the Alamouti code [1]. Manuscript received October 8, 2004; revised December 20, 2005 and June 15, 2006. This work was supported in part by the U.S. National Science Foundation (NSF) under Grant CCR-0207550 and in part by the U.S. Office of Naval Research (ONR) under Grant N00014-03-1-0039. The review of this paper was coordinated by Prof. H. Leib. C. Li was with the Department of Electrical Engineering, Columbia University, New York, NY 10027 USA. He is now with Marvell Semiconductor, Inc., Santa Clara, CA 95054 USA. K. Li and X. Wang are with the Department of Electrical Engineering, Columbia University, New York, NY 10027 USA (e-mail: wangx@ee. columbia.edu). L. Ping is with the Department of Electrical Engineering, City University of Hong Kong, Kowloon, Hong Kong. Digital Object Identifier 10.1109/TVT.2007.895495

On the other hand, iterative processing techniques, i.e., the turbo principle, have been successfully applied to many systems involving interference cancellation [20] due to its excellent performance. In [11] and [12], a turbo-BLAST scheme is proposed, which is closely related to the turbo multiuser detection method for CDMA systems [20]. In [6] and [21], an interleavedivision-multiplexing (IDM)-based MIMO system is proposed. Although the IDM scheme provides a simple and flexible transmission strategy for arbitrary MIMO configurations, its inherent properties have not been well studied yet in current literature. Another extensively analyzed problem in the current MIMO literature is the optimal transmit strategy when imperfect channel state information (CSI) is available at the transmitter [e.g., only the covariance matrix of the multiple-input–singleoutput (MISO) channel] [5], [7], [13], [19]. It has been shown that to maximize the capacity (either the ergodic capacity [19] or the outage capacity [13]), the signals should be transmitted along the directions of the eigenvectors of the covariance matrix of the MISO channel, and the optimization problem is thus reduced to power allocation over different substreams [7]. Note that Gaussian constellation is assumed to achieve the capacity in existing work [5], [7], [13], [19]; however, in practice, only a finite-constellation signaling can be employed. The main purpose of this paper is to analyze and optimize the performance of an IDM-MISO system with partial CSI at the transmitter. We present a general methodology for the performance analysis and optimization of a practical IDM-MISO system employing an iterative receiver and finite-constellation signaling. In particular, an SNR tracking method is proposed to quickly calculate the output SNR of the soft-input–soft-output (SISO) IDM detector. The extrinsic mutual information transfer (EXIT) technique is employed to analyze the performance of the entire system. More specifically, the repetition code and the low-density parity check (LDPC) code are considered for the uncoded IDM system and the coded IDM system, respectively. Based on the proposed analytical framework above, the optimal power allocation can be derived for the IDM-MISO system. Extensive simulations are carried out to demonstrate the performance of the proposed IDM-MISO system under the MISO channels with different correlation statistics. Our results show that the proposed IDM-MISO scheme can outperform the existing STBC schemes in the low-rate transmission scenario. The remainder of this paper is organized as follows. Section II describes the general IDM-MISO system, including the IDM transmitter and the iterative receiver. Section III presents the analytical framework for performance analysis

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IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 56, NO. 3, MAY 2007

Fig. 1. Diagram of an IDM-MISO system.

and the optimal IDM-MISO transmission strategy. Simulation results are given in Section IV. Section V contains the conclusions. II. S YSTEM D ESCRIPTIONS Consider a MISO system with T transmit antennas. Denote h as the complex Gaussian MISO channel response with zero 

mean, i.e., h ∼ Nc (0, Σ), where Σ = E{hhH } denotes the covariance matrix of h. Fig. 1 shows the diagram of a generalized IDM-MISO system. Note that in this paper, perfect CSI and partial CSI (only Σ) are assumed to be available at the receiver and the transmitter, respectively. A. IDM Transmitter As shown in Fig. 1, after the serial-to-parallel (S/P) conversion, the coded source bits {cj } are allocated to T parallel substreams {ci,j }Ti=1 . Each parallel substream {ci,j } is interleaved by an independent interleaver πi , 1 ≤ i ≤ T . Then, a linear processing (U ) and a power allocation (P ) are performed on the interleaved coded bits x to obtain the signals to be transmit˜ = U P 1/2 x, ted. Hence, the transmitting signal is given by x 

where P = diag{P1 , P2 , . . . , PT } denotes the power allocation diagonal matrix, with Pi indicating the transmit power at the ith antenna (1 ≤ i ≤ T ), and the unitary precoding matrix U is characterized as follows. It has been shown that to achieve the capacity (either the ergodic capacity [19] or the outage capacity[13]), the optimal transmission strategy should

satisfy the following property: Σ and the covariance matrix 

˜ H } = E{U P 1/2 xxH P 1/2 U H }, should be ˜ , Q = E{˜ xx of x simultaneously diagonalizable. That is, for Σ = U ΛΣ U H , where ΛΣ and U contain the eigenvalues and the corresponding eigenvectors of Σ, respectively, we have Q = U ΛQ U H , where ΛQ = P E{xxH } is a diagonal matrix. Therefore, the received signal in the IDM-MISO system, as shown in Fig. 1, can be written as 1

H 2 ˜ +n=h y = hH x
U P x + n = ˜H h

T 

˜i h



Pi xi + n

(1)

i=1

where n ∼ Nc (0, σ 2 ) denotes the complex additive white  H ˜H = ˜ 1, h ˜2, . . . , h ˜T ] Gaussian noise (AWGN) and h h U = [h denotes the equivalent MISO channel response. Interleaver design for the MIMO systems employing iterative receivers is an important issue. In general, the space–time interleaver design principle is to guarantee that each subchannel is occupied by each independently coded substream in an equal manner [11]. As shown in Fig. 1, each codeword for sj is first equally allocated to different transmit antennas by the S/P processor, and the interleaver πi over each subchannel i (1 ≤ i ≤ T ) further ensures that the codewords simultaneously transmitted from different antennas are independent. Note that such a simple interleaver design is essentially equivalent to the space–time interleaver design presented in [11] and [12]. Later in Section IV, we will show some results of the effects of the interleaver design on system performances.

LI et al.: INTERLEAVE-DIVISION-MULTIPLEXING MISO SYSTEM WITH PARTIAL CSI AT TRANSMITTER

B. Iterative (Turbo) Receiver

and

It is shown in Fig. 1 that the iterative receiver mainly consists of a SISO detector and a SISO channel decoder, which work together in a turbo manner. In particular, the SISO detector performs the soft detection that is based on the received signal y and the prior information provided by the SISO decoder, denoted by {λdec (xi,j )}Ti=1 . Also, the output of the SISO detector is the log-likelihood-ratio (LLR) of the transmitted bits, denoted by {λdet (xi,j )}Ti=1 . Based on the LLRs passing from the SISO detector {λdet (cj )}, the SISO channel decoder outputs the a posteriori LLRs of the information bits, denoted by {Λdec (cj )}. Furthermore, the extrinsic LLR is updated by λdec (cj ) = Λdec (cj ) − λdet (cj ), and then, the extrinsic LLR {λdec (xi,j )}Ti=1 will be fed back to the SISO detector as the updated prior information. 1) Basic SISO Detector: Here, for notational simplicity, we drop the time index j from {xi,j } and {ci,j }, 1 ≤ i ≤ T . Suppose that BPSK signaling is employed and the channel ˜ ∈ RT . Then, the coefficients are real, i.e., xi ∈ {1, −1} and h received signal in (1) can be rewritten as ˜i y=h



T 

Pi xi +

˜j h

j=1,j =i






Pj xj + n



(2)

T 

˜i h



zi

Pi E{xi }

i=1

˜i =h



Pi E{xi } +

T 

˜j h

j=1,j =i




 Var{xi } = 1 − |E{xi }|2 = 1 − tanh2



Pj E{xj }



λdec (xi ) 2

 .

2) Generalized SISO Detector: ˜ ∈ CT and a) Complex channel and real signal: When h x contains BPSK symbols, the real part and the imaginary part of the received signal can be expressed, respectively, by  ˜ i } Pi xi + {zi } yR = {y} = {h and  ˜ i } Pi xi + {zi } yI = {y} = {h both of which are equivalent to (2). Then, yR and yI can be separately treated in the same manner as that in (7), i.e., the reception at the single receive antenna is equivalent to that over two virtual receive antennas. Therefore, for the case of complex channels and BPSK signaling, the MISO system employing T transmit antennas is equivalent to a MIMO system employing T transmit antennas and two virtual receive antennas [21]. Note that for the case of more than one (virtual) receive antennas, the overall output LLR can be written as

(3) λdet (xi ) =



nR  j=1

˜ } E{zi |h,P

(j)

λdet (xi )

(j)

and ˜ P }= Var{y|h,

T  i=1

T  j=1,j =i




(8)

where λdet (xi ) denotes the output LLR at the jth (virtual) receive antenna and can be computed using (7). ˜ ∈ CT b) Complex channel and complex signal: When h and x contains QPSK symbols, the real part and the imaginary part of the received signal can be rewritten, respectively, as

˜ i |2 Pi Var{xi }+σ 2 |h

˜ i |2 Pi Var{xi }+ = |h

(6)

Using (3)–(6), the output LLR of the IDM SISO detector can be written as

˜ P) p(y|xi = 1, h,  λdet (xi ) = log ˜ P) p(y|xi = −1, h, ˜ P}  y − E{zi |h, = 2hi Pi ˜ P} Var{zi |h, √ T ˜ k Pk E{xk }  y − k=1,k =i h . (7) = 2hi Pi T 2 ˜ 2 k=1,k =i Pk |hk | Var{xk } + σ



where zi can be approximated by a Gaussian random variable. Denote E{xi } and Var{xi } as the mean and the variance of xi , respectively. Then, the conditional mean and variance of the received signal y in (2) are given, respectively, by ˜ P} = E{y|h,

1199

˜ j |2 Pj Var{xj }+σ 2 |h 



˜ } Var{zi |h,P

(4) ˜ P } and Var{zi |h, ˜ P } denote the conditional where E{zi |h, mean and variance of zi , respectively. Denote λdec (xi ) as the extrinsic LLR, which is obtained by the SISO decoder as the prior for the SISO detector. Then, E{xi } and Var{xi } can be computed, respectively, as   λdec (xi ) E{xi } = Pr{xi = 1} − Pr{xi = −1} = tanh 2 (5)

yR = {y}

 ˜ i }, − {h ˜ i } [ {xi }, {xi }]T + {zi } = Pi {h  
 
¯ x i

˜ (R) h i

=

 ˜ (R) x ¯ i + {zi } Pi h i

and yI = {y}

 ˜ i }, {h ˜ i } [ {xi }, {xi }]T + {zi } = Pi {h  
 
¯ ˜ (I) h i

=

 ˜ (I) x Pi h i ¯ i + {zi }.

xi

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IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 56, NO. 3, MAY 2007

It is observed that the reception at each virtual receive antenna is equivalent to that over 2T virtual transmit antennas and one virtual receive antenna. Thus, for the case of complex channels and QPSK signaling, the MISO system employing T transmit antennas is equivalent to a MIMO system employing 2T virtual transmit antennas and two virtual receive antennas [21]. Remark 1: Due to inherent property of the IDM detector (7), QPSK is the only complex signaling which can be employed here. It is worth noting that the IDM scheme in Fig. 1 is more suitable for low-rate transmission. Later in Section IV-C, we will further discuss the IDM schemes for high-rate transmission. Also note that the main purpose of this paper is to analyze and optimize the performance of the IDM-MISO system in low-rate transmission scenario, which is the basis for the IDM schemes in high-rate transmission scenario and, thus, can be easily extended and applied in high-rate IDM schemes. III. O PTIMAL IDM-MISO S TRATEGY In this section, we propose the general methodology to evaluate and optimize an IDM-MISO system when partial CSI is available at the transmitter. In particular, in Section III-A, we propose a method to evaluate the output SNR of the SISO detector. Then, in Section III-B, we employ the EXIT technique to analyze the performance of the iterative receiver. The repetition decoder and the LDPC decoder are treated for the uncoded IDM system and the coded IDM system, respectively. Based on the above analytical framework, in Section III-C, we discuss the optimal power allocation for the IDM-MISO system. A. Output SNR Evaluation for SISO Detectors ˜ and P , the output LLR of the SISO Note that given h detector (7) for each transmit antenna i is typically assumed to be a consistent Gaussian random variable [10], [17], i.e., 2 2 /2), σdet,i ). Then, the conditional output λdet (xi ) ∼ N ((σdet,i SNR of the SISO detector (7) for the ith transmit antenna can be written as γdet,i =

2 σdet,i 4

= k=T

˜ i |2 Pi |h

k=1,k =i

˜ k |2 Pk E{Var{xk }} + σ 2 |h

,

1≤i≤ T

where using (6), E{Var{xk }} can be expressed by    λdec (xk ) E {Var{xk }} = E 1 − tanh2 2   ∞  ξ 1 2 = 1 − tanh 2 2πσ 2 dec,k −∞



1≤k≤T

γdet,i = k=T

k=1,k =i

˜ i |2 Pi |h ˜ k |2 Pk g(σdec,k ) + σ 2 |h

,

1 ≤ i ≤ T. (10)

Since {σdec,i }Ti=1 depends on both the channel code employed and the SISO detector output SNR {γdet,i }Ti=1 , g(σdec,i ) de˜ and fined in (9) is also a function of {γdet,i }Ti=1 . Then, given h P , in general, the output SNR of the iterative receiver can be obtained using the following algorithm. The quantities with the superscript (n) denote those in the nth iteration. Algorithm 1 [Iterative SNR evolution for the IDM SISO detector] ˜ and P , do the following. Given h (n) (1.a) n = 0 and g(σdec,k ) = 1 for all 1 ≤ k ≤ T , i.e., no prior information at the beginning. (n+1) (1.b) Compute the SNR using (10), i.e., γdet,i =  (n) 2 ˜ i |2 Pi ) / ( k = T ˜ 2 ( |h k = 1,k =i |hk | Pk g ( σdec, k ) + σ ), 1 ≤ i ≤ T. (n+1) (n+1) (1.c) Update {σdec,i }Ti=1 using {γdet,j }Tj=1 , i.e., update the priors using the new output LLRs. (n+1) (1.d) Calculate g(σdec,k ) using (9), 1 ≤ k ≤ T . (n+1)

(n)

If |γdet,i − γdet,i | ≤  for all 1 ≤ i ≤ T , then stop the iteration; otherwise, n ⇐ n + 1 and then go back to step (1.b). ∗ = (1.f) The final achievable SNR is given by γdet,i

(1.e)

(n)

γdet,i , 1 ≤ i ≤ T . In step (1.c), the distribution of the extrinsic SISO decoder output ({σdec,i }Ti=1 ) is required to be updated using the SISO detector output ({γdet,i }Ti=1 ). For the uncoded system where the repetition codes are employed, the closed-form expression is available for {σdec,i }Ti=1 with respect to {γdet,i }Ti=1 , and thus, the corresponding g(·) can be obtained analytically. On the other hand, for coded systems where general forward error correcting (FEC) codes are employed, the closed-form expression is not always available for {σdec,i }Ti=1 with respect to {γdet,i }Ti=1 , nor is the corresponding g(·), but they can be obtained via some numerical procedures [22]. B. Output Evaluation for SISO Decoders

  2   2   ξ − σdec,k /2  dξ × exp − 2   2σdec,k   = g(σdec,k ),

where the second equality follows from the consistent Gaussian assumption for the extrinsic LLR of the SISO decoder out2 2 /2, σdec,k ). Using (9), put [17], [18], i.e., λdec (xk ) ∼ N (σdec,k the conditional output SNR of the SISO detector can be rewritten as

(9)

We next discuss the output evaluation for SISO decoders. Specifically, an analytical approach is presented for the repetition decoder employed by uncoded IDM systems, and a semianalytical approach is developed for the LDPC decoder as a special case of coded IDM systems. 1) Repetition Decoders: Denote S(s) = {xi,j , 1 ≤ i ≤ T ; 1 ≤ j ≤ L} as the set of the transmitted signal corresponding to the source bit s, where L is the length of the transmission frame. In particular, xi,j denotes the signal transmitted from the

LI et al.: INTERLEAVE-DIVISION-MULTIPLEXING MISO SYSTEM WITH PARTIAL CSI AT TRANSMITTER

Fig. 2.

1201

Diagram of an iterative LDPC decoder.

ith antenna and located in the jth position of the frame. After the repetition-based a posteriori probability (APP) decoding, the APP LLR of the source bit s is given by  Λdec (s) = λdet (xi,j ) (i,j)∈S(s)

and then the extrinsic LLR of each xi,j is given by

2 2 I [(i, j) ∈ S(s)] σdet,i − σdet,i

i=1 1≤j≤L

=4

T  

1

(11)

which will be fed back to the SISO detector as the updated prior information. From (11), the variance of the extrinsic LLR of the repetition decoder output can then be expressed as



(V )



∞ −∞

  (ζ −σ 2 /2)2 log2 (1 + e−ζ )exp − dζ. 2σ 2 (13)

 =J

i=1 1≤j≤L

 2 (dv − 1) [J −1 (IV )]2 + σdet . (14)

(12)

where I(·) is the indicator function. Accordingly, g(σdec,i ) with respect to {γdet,i }Ti=1 can be analytically obtained using (9) and the closed-form expression of {σdec,i }Ti=1 in (12). 2) LDPC Decoders: As mentioned in Section III-A, the transfer function of {σdec,i }Ti=1 with respect to {γdet,j }Tj=1 depends on the channel codes and, in general, has no closedform expression. In this section, we develop a semianalytical approach to predict the output of the iterative LDPC decoder based on message passing. Fig. 2 shows the diagram of an iterative LDPC decoder [18], which consists of a variable node decoder (VND) and a check node decoder (CND). a) Inner VND: The variable node with a degree of dv has dv + 1 input messages, where specifically, one is from the channel and the other dv is from the edge interleaver. The output LLR for each edge i can be written as [18]  (V ) (V ) Li,out = λdet + Lj,in , 1 ≤ i ≤ dv j =i

where λdet denotes the LLR coming from the SISO detector (V ) output, Lj,in is the jth APP LLR coming into the VND, and



Note that the function J(σ) is a monotonically increasing function of σ, and thus, we have σV = J −1 (IV ) [17], [18]. 2 2 /2, σdet ) as the input of the VND. The Denote λdet ∼ N (σdet output mutual information of the VND is then given by   2 2 (dv − 1)σV + σdet IE,VND (σdet , IV , dv ) = J

I [(i, j) ∈ S(s)] γdet,i − γdet,i  , 1≤i≤T

(V )

tween the BPSK input and Lj,in is given by Ij,in = J(σV ) = IV , where J(·) is defined by J(σ) = 1 − √ 2πσ 2

1 ≤ i ≤ T and 1 ≤ j ≤ L

T  

(V )

AWGN channel with BPSK input and assumed as Lj,in ∼ N (σ 2 /2, σ 2 ), 1 ≤ j ≤ dv . Then, the mutual information be-



λdec (xi,j ) = Λdec (s) − λdet (xi,j ),

2 σdec,i =

(V )

Li,out is the ith extrinsic LLR going out of the VND. Note (V ) that Lj,in can be modeled as the output LLR of an equivalent

b) Outer CND: The CND with a degree of dc corresponds to the decoding of a single parity check code with length dc . The output LLR of the CND is then given by [4] 1−

!

(C)

Li,out = ln 1+ (C)

(C) L j,in

j =i

!

1−e

(C) L j,in

1+e

(C) L j,in

j =i

1−e

,

1 ≤ i ≤ dc

(C) L j,in

1+e

(C)

where Lj,in and Lj,out denote the jth input LLR and the jth output LLR of the CND, respectively. With the equivalent (C)



AWGN channel treatment, i.e., Lj,in = J(σC ) = IC , the average output mutual information of the CND can be approximated by that of a repetition code [18]. That is IE,CND (IC , dc ) ≈ 1 − IE,REP (1 − IC , dc )   dc − 1J −1 (1 − IC ) . =1 − J

(15)

c) Mutual information tracking procedure: In the iterative LDPC decoding process, the message is iteratively passed

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between the VND and the CND. Denote the quantities with the superscript (n) as those in the nth iteration. Given the input distribution σdet , the output mutual information of a (dv , dc ) LDPC decoder can then be obtained as follows. Algorithm 2 [Mutual information tracking for a (dv , dc LDPC decoder] Denote N as the total number of iterations of message passing. Do the following. (n) (2.a) Set n = 0, IE,CND = 0. (2.b)

(n)

(n)

(n)

Set IA = IE,CND , then compute IE,VND (σdet , (n)

IA , dv ) using (14). (n) (n) (n) (n) (2.c) Set IA = IE,VND , then compute IE,CND (IA , dc ) using (15). (2.d) If n ≥ N , then stop; otherwise, n ⇐ n + 1 and go to step (2.b). (2.e) The standard deviation (std) of the extrinsic LLR output by the LDPC decoder is then σdec = √ (n) dv J −1 (IE,CND ). Although the output LLR of the SISO detector for one transmit antenna is Gaussian distributed, the actual LLR coming into the VND is the mixture of the output LLRs for all T transmit antennas, and thus, it is not Gaussian distributed as required by (14). We can track the output mutual information for each antenna i using step (2.b) in Algorithm 2, denoted by {IE,VND,i }Ti=1 . Then, in step (2.c), the mutual information of the actual output LLR can be roughly approximated as IA = IˆE,VND ≈ 1/T Ti=1 IE,VND,i . Even though g(·) can always be obtained numerically [22], Algorithm 2, together with the above approximation method, provides a semianalytical way to calculate the output {σdec,i }Ti=1 of the LDPC decoder. C. Power-Optimized IDM-MISO Strategy As pointed out in [13] and [19], to achieve optimal transmission, the signals should be transmitted along the directions of the eigenvectors of Σ, and thus, the optimization problem is reduced to the power allocation over different substreams [7]. The corresponding power allocation problem for the IDMMISO system can be formulated as P ∗ = arg min Φ(P ), P

subject to Tr{P } ≤ PT

(16)

where PT denotes the total transmit power constraint and Φ(P ) denotes the objective function which will be detailed later. To solve problem (16), some optimization methods, e.g., the differential evolution method [9], can be adopted. During the differential evolution, we need to obtain the value of the objective function Φ(P ) for each given P . We next define Φ(P ) for both uncoded systems and coded systems. Note that, without loss of generality, BPSK modulation is assumed. 1) Power Allocation for Uncoded Systems: For the uncoded system, the objective function Φ(P ) is defined as the bit error rate (BER). In particular, with the consistent Gaussian assumption, the achievable BER of an uncoded IDM-MISO system is

equivalent to that of an uncoded system with BPSK modulation in an AWGN channel. That is M 1  φ(hm , P ) M m=1   M 1  ∗ (h , P ) = Q γdet m M m=1 

Φ(P ) =

M

=

1  Q M m=1 $  % T %  ∗ × & I ((j, n) ∈ S(s)) γdet,j (hm , P ) j=1 1≤n≤L

(17) )∞  ∗ where Q(x) = 1/2π x exp{−ζ 2 /2}dζ; γdet,j (hm , P ) denotes the finally achievable SNR for the jth transmit antenna for the given hm and P (1 ≤ m ≤ M ), which can be calculated ∗ (hm , P ) denotes the corresponding using Algorithm 1, and γdet overall SNR which is an averaged value over M channel realizations. 2) Power Allocation for LDPC-Coded Systems: For the LDPC-coded system, the objective function Φ(P ) is defined as the frame error rate (FER). In particular, for given h and P , the total throughput at the receiver is given by I(h, P ) =

T 

J (σdet,i (h, P ))

i=1

where σdet,i (h, P ) denotes the std of the SISO detector output corresponding to the ith transmit antenna and can be calculated using the Algorithms 1 and 2. Then, the objective function Φ(P ) is defined as the outage probability corresponding to a certain rate R, i.e., T

  Φ(P ) = Pr{I(h, P ) ≤ R} = Pr J(σdec,i (h, P )) ≤ R i=1

(18)

where the probability Pr{·} can be calculated numerically. Note that Algorithm 2 provides a semianalytical approach to compute σout , which treats the ideal LDPC code with infinite length; in practice, however, the LDPC code has finite length. Although g(·) can always be numerically obtained for finitelength LDPC codes, the corresponding performance predicted using Algorithm 2 can be viewed as an upper bound of the actual performance. IV. S IMULATION R ESULTS It is well known that insufficient antenna spacing and lack of scattering cause the individual antennas to be correlated. One way to model the correlation of MISO channels is to take into account the antenna correlation at the transmitter. In particular, the correlated MISO channel response vector h can be modeled by H hH = hH wB

(19)

LI et al.: INTERLEAVE-DIVISION-MULTIPLEXING MISO SYSTEM WITH PARTIAL CSI AT TRANSMITTER

1203

Fig. 3. Comparison between the proposed interleaver design in Fig. 1 and the referred interleaver design in Section IV-A: T = 2; BPSK; 1/T repetition code; interleaver length L for the proposed scheme and T L for the referred scheme.

where hw ∈ CT and hw ∼ Nc (0, I T ); B can be obtained from the channel covariance matrix Σ = E{hhH } = BB H . The (m, n)th entry of Σ can be written as T

T

T

2

1 x x x Σm,n ∼ = e−j2π(n−m)∆ cos(φ0 ) · e− 2 [2π(n−m)∆ sin(φ0 )σ0 ] (20)

where the antenna spacing is assumed to be half of the wavelength, i.e., ∆ = 1/2; φT0 x denotes the mean for angle of arrival; σ0Tx denotes the angle spread. Note that the antenna correlation here is determined by the parameters of the angle distribution, φT0 x and σ0Tx . In the simulations, we consider three different cases [14]: high antenna correlation (φT0 x = 20◦ and σ0Tx = 6◦ ), moderate antenna correlation (φT0 x = 20◦ and σ0Tx = 20◦ ), and low antenna correlation (φT0 x = 20◦ and σ0Tx = 85◦ ). Then, the MISO channel response vectors with different correlation statistics can be generated using (19) and (20). Note that the block-fading channel model is employed in this paper. A. Interleaver Issues As discussed in Section II-A, the interleaver design for IDM-MISO systems is an important issue [11]. We now show some results of the effects of different interleaver design on system performances. As a comparison, we consider another interleaver design, where instead of employing T independent interleavers πi (1 ≤ i ≤ T ), one single interleaver is employed, and it is located between the S/P processor and the channel encoder. Fig. 3 shows the comparison between the proposed interleaver design in Fig. 1 and the above referred interleaver design. In particular, the MISO channels with low antenna correlation are treated here; T = 2; BPSK and the repetition code with a rate of 1/T are employed. The interleaver length for the proposed scheme in Fig. 1 is L, and to guarantee a fair comparison, the interleaver length for the above referred scheme is then T L. It is shown in Fig. 3 that the proposed

Fig. 4. Performance of the iterative SNR prediction algorithm (Algorithm 1): T = 2 and T = 4; QPSK; 1/2T repetition code; interleaver length L = 256.

interleaver design is better than the referred interleaver design. The rationale here is that in the referred scheme, the interleaver cannot guarantee an equal allocation of different independently coded substream over each antenna. Moreover, we can also see in Fig. 3 the effects of the interleaver length on the system performance. Intuitively, the larger the interleaver length is, the better the performance is, because longer interleaving can more probably guarantee that the codewords simultaneously transmitted from different antennas are independent. As shown in Fig. 3, the performance gain, due to the interleaver length, is not evident because L ≥ 256 is already long enough for the proposed IDM-MISO system. B. Output Evaluation of IDM SISO Detectors Next, we evaluate the performance of Algorithm 1, i.e., the iterative approach to predict the output performance of the IDM SISO detectors. The BER performance is referred to demonstrate its effectiveness. Fig. 4 shows the simulated BER results and the BER predictions semianalytically

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calculated using Algorithm 1, with the closed-form expressions in (12) and (17). In particular, the MISO channels with different correlation statistics are treated; both T = 2 and T = 4 cases are considered here; QPSK and the repetition code with a rate of 1/2T are employed; the frame length is L = 256. It is shown in Fig. 4 that the analysis results well match the simulated results, i.e., the simulated results at the fifth iteration for the case T = 2 and the simulated results at the tenth iteration for the case T = 4. Moreover, it is seen that the simulated BER performance converges quickly within a small number of iterations. Thus, Algorithm 1 provides a fast and reasonably accurate approach to predict the asymptotic performance of the iterative receiver. C. Performance of Uncoded IDM-MISO Systems As an alternative of the STBC schemes, the uncoded IDM strategy is a simple and flexible design methodology for MISO systems with arbitrary configuration. We next present some comparisons between the existing STBC schemes and the uncoded IDM-MISO systems. 1) Low-Rate Transmission—IDM Versus Alamouti Code: It is well known that the Alamouti code is the only orthogonal STBC with full rate using complex signaling [1], [15]. Fig. 5 shows the comparison between the Alamouti code and the IDM-MISO for the case of one bit per transmission. Specifically, two IDM-MISO schemes can be employed, i.e., BPSK with 1/2 repetition code and QPSK with 1/4 repetition code; the MISO channels with different correlation statistics are considered. It is shown in Fig. 5 that even without power allocation, i.e., equal power at different transmit antennas, the IDM-MISO employing QPSK and 1/4 repetition code can achieve similar performance as the Alamouti code, and the performance of the IDM-MISO employing BPSK and 1/2 repetition code is only a little worse than that of the Alamouti code. Moreover, it is seen that the power-optimized IDM-MISO can outperform the Alamouti code. Note that channel correlation does not have evident effects on the performance comparison between the Alamouti code and the IDM-MISO scheme, as shown in Fig. 5. 2) Low-Rate Transmission—IDM Versus Rate-One STBC Scheme: For the cases of T > 2, several nearly orthogonal STBC schemes with full rate have been proposed in current literature. In [8], a rate-one STBC scheme is proposed for the case of T = 4. Fig. 6 shows the BER performance of the rateone STBC with the linear mmse decoding scheme [2], [3] for the case of one bit per transmission. It is shown in Fig. 6 that under the MISO channels with different correlation statistics, even without power allocation, the IDM-MISO scheme employing QPSK and 1/8 repetition code can outperform the rate-one STBC or achieve similar performance, and the IDM-MISO scheme employing BPSK and 1/4 repetition code is only a little worse than the rate-one STBC. Similarly, as in Fig. 5, it is also shown in Fig. 6 that the power-optimized IDMMISO outperforms the rate-one STBC, and the channel correlation does not have evident effects on the above comparisons. 3) High-Rate Transmission Scheme: Only low-rate transmission, e.g., one bit per transmission, is treated in Figs. 5 and 6. We next demonstrate the IDM-MISO system performance for high-rate transmission. Fig. 7 shows the comparison between

Fig. 5. Comparison between the Alamouti code and IDM-MISO: T = 2; one bit per transmission; BPSK with 1/T repetition code and QPSK with 1/2T repetition code; interleaver length L = 256.

the Alamouti code and the IDM-MISO scheme for the case of two bits per transmission. Fig. 8 shows the similar comparison between the rate-one STBC and the IDM-MISO scheme. In particular, QPSK signaling and 1/T repetition code are employed in both of the above two IDM schemes. It is shown in Figs. 7 and 8 that the performance of the IDM-MISO is worse than those of the Alamouti code and the rate-one STBC in the cases of T = 2 and T = 4, respectively. It should be noted that the IDM system, as shown in Fig. 1, is more suitable for low-rate transmission. To support high-rate transmission, instead of employing higher constellation signaling (e.g., QAM), a multilayer (ML) IDM scheme is presented in [22]. The ML-IDM scheme consists of the superposition of multiple layers; each of which is equivalent to the IDM scheme shown in Fig. 1, but with equal power allocation over different transmit antennas. Hence, the analysis and optimization of one single layer transmission treated in this paper is essentially the basis for the ML-IDM scheme, and the corresponding approaches and results in this

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Fig. 8. Comparison between the rate-one STBC and IDM-MISO: T = 4; two bits per transmission; QPSK; 1/T repetition code; interleaver length L = 256.

Fig. 6. Comparison between the rate-one STBC and IDM-MISO: T = 4; one bit per transmission; BPSK with 1/T repetition code and QPSK with 1/2T repetition code; interleaver length L = 256.

paper can be easily extended and applied to the high-rate ML-IDM scheme. Specifically, in addition to the power allocation among different layers, as treated in [22], we can further treat the joint power optimization among different transmit antennas for each layer. Since the main purpose of this paper is to analyze and optimize one single layer transmission, we will not include the detailed analysis for ML-IDM systems, which is beyond the main scope of this paper. Remark 2: From the above comparisons, some remarks about the uncoded IDM-MISO strategy can be drawn as follows. As shown in Figs. 5 and 6, QPSK is the best signaling for the IDM system, and the IDM system employing the QPSK signaling can approach the performance of orthogonal or quasiorthogonal codes over arbitrary number of transmit antennas in the case of one bit per transmission. As shown in Fig. 8, in addition to offering certain performance gain, power allocation can also help to eliminate the error floor in the high SNR regions. As shown in Figs. 7 and 8, the IDM system presented in Fig. 1 is more suitable for low-rate transmission, e.g., one bit per transmission. D. Performance of LDPC-Coded IDM-MISO Systems

Fig. 7. Comparison between the Alamouti code and IDM-MISO: T = 2; two bits per transmission; QPSK; 1/T repetition code; interleaver length L = 256.

1) Case of Two Transmit Antennas: Fig. 9 shows the FER performances of the LDPC-coded IDM-MISO system. In the simulations, the QPSK signaling is used for the IDM scheme; T = 2; (3, 4) regular LDPC code is employed, and the code length is 4096; the target rate is one bit per transmission (R = 1). In particular, both the FER performances under equal power transmission and that under the optimal power strategy are shown. For the case of equal power transmission, in addition to the simulation results obtained by using the practical LDPC decoder with finite length, the results predicted using Algorithm 2 are also shown in Fig. 9. As we mentioned in Section III-C, the predicted performance can be treated as a bound for the actual performance. It is shown in Fig. 9 that the predicted performance and the simulated performance are very close, and

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Fig. 9. Outage probability of power-optimized and equal-power IDM transmission: T = 2; QPSK; R = 1; (3, 4) regular LDPC code.

Fig. 10. Diagram of an LDPC-coded Alamouti STBC system.

the gap is normally less than 1 dB in the high SNR region. Moreover, it is shown in Fig. 9 that the power allocation strategy can offer about 0.5–0.8-dB performance gain. 2) Coded IDM Schemes Versus Coded STBC Schemes: To demonstrate the advantage of the IDM-MISO strategy over the existing STBC schemes in the low-rate transmission scenario, the FER performance of the LDPC-coded Alamouti STBC system is also shown in Fig. 9, where the same (3, 4) regular LDPC code is used, and 16-QAM signaling is employed. The corresponding receiver for the LDPC-coded Alamouti STBC system is shown in Fig. 10. Note that the Alamouti code can be treated as a rate-one inner code here. It is shown in Fig. 10 that different from the iterative IDM receiver shown in Fig. 1, no iterative LLR exchanging does exist between the inner decoder (i.e., the Alamouti code decoder) and the outer decoder (i.e., the LDPC decoder). To guarantee R = 1, we have to employ 16-QAM signaling instead of QPSK. On the one hand, it is shown in Fig. 9 that the proposed IDM-MISO strategy can achieve a little better performance than the coded Alamouti scheme. On the other hand, it should be noted that compared with the lower constellation signaling (e.g., QPSK), the highconstellation signaling (e.g., 16-QAM) inevitably increases the implementation complexity, e.g., synchronization, detection,

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Fig. 11. Outage probability of power-optimized and equal-power IDM transmission: T = 4; QPSK; R = 1.

and amplification. As pointed out in [18], such issues will be much more severe for the cases of T > 2. For instance, consider the coded nearly orthogonal STBC scheme for the case of T = 4 [8], where a rate-1/8 outer code is used. To guarantee R = 1, 256-QAM signaling has to be employed. 3) Case of Four Transmit Antennas: Fig. 11 shows the similar comparisons for the case of T = 4 as those in Fig. 9, and similar conclusions can be drawn. In Fig. 11, the QPSK signaling is employed in the IDM scheme. To guarantee R = 1, the code with a total rate of 1/8 should be used, and the specific scheme is detailed below. Note that the code design for blockfading MIMO channels remains an open problem. We next present some remarks about the code design issue for the IDMMISO system under the block-fading channel conditions. It is worth noting that it is infeasible to simply choose an LDPC code with the required rate. This is due to the inherent property of the IDM detector (7), especially for the case of the large number of transmit antennas. Since the method to characterize the detector property under the block-fading MIMO channel condition remains open, we treat the nonfading channel as a reference to gain some insight. Fig. 12 shows the EXIT chart [18] for the IDM-MISO system, where the QPSK signaling and different regular LDPC codes are employed. In particular, the solid lines denote the EXIT curves of joint IDM detection and VND process, i.e., IE,VND = fVND (IA,VND ); the dashed lines denote the EXIT curves of the CND process, i.e., IE,CND = fCND (IA,CND ) [and thus, −1 (IE,CND )]. It is shown in Fig. 12 that the IDMIA,CND = fCND MISO cannot work together with (7, 8) regular LDPC code which has a rate of 1/8, and this has also been verified by the simulation results. The rationale behind is that for the joint IDM detection and VND curve (fVND (·)), the output extrinsic mutual information (IE,VND ) is very low in the region of low input prior mutual information (IA,VND ). Then, in the corresponding region for the CND curve (i.e., low IA,CND region), the output extrinsic mutual information (IE,CND ) is even smaller than the input prior mutual information (IA,CND ). In other words,

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mitter. We have proposed a general methodology for the performance analysis of the practical iterative receivers. In particular, an SNR tracking approach has been proposed to predict the output SNR of SISO detectors, and the EXIT chart methods have been developed to analyze the entire system performance for both uncoded and LDPC-coded IDM-MISO systems. Based on the above analytical framework, we have further presented the optimal transmit power allocation strategy. Extensive numerical results have been provided to verify the performance of the proposed approaches under the MISO channels with different correlation statistics. R EFERENCES

Fig. 12. EXIT chart for IDM-MISO system: T = 4; QPSK; R = 1; nonfading channels; regular LDPC codes; Eb /N0 = 12 dB; solid lines denote the EXIT curves for joint IDM detection and VND; dashed lines denote the EXIT curves for CND.

the inherent property of the IDM detector presents the VND curve a very low starting point for the iterative process, and such a low start point makes the CND curve fall into the corresponding region where the iterative process between the two curves cannot achieve the convergence. Intuitively, at the initial stage of the iterative interference cancellation process, the detector curve is interference-dominated instead of noise-dominated, i.e., increasing the input SNR cannot result in any performance gain. Simultaneously, however, the LDPC decoder cannot offer any help due to its property in this region, as we discussed above. Accordingly, to make the LDPC codes cooperatively work with the IDM detector, we need to alter the property of the LDPC codes within the initial region. One possible solution is to introduce the repetition codes into the IDM-MISO system. In Fig. 11, to achieve a total rate of 1/8, we employ parallel codes, including a rate-1/4 LDPC code and a rate-1/4 repetition code. On the one hand, such a parallel scheme does not change the IDM-MISO system structure, as shown in Fig. 1, because the repetition code and the LDPC code are independent of each other. On the other hand, the property of the repetition code in the low IA region (i.e., IE > IA ) can offer evident contribution to the interference cancellation process so that it can achieve the convergence. Note that a similar scheme has been presented in [6], where the parallel codes include a turbo code and a repetition code. Also note that Figs. 11 and 12 only show some hints of the code design for IDM-MISO systems, and the main conclusion drawn from Fig. 11 is similar to that from Fig. 9. That is, the proposed power allocation procedure can result in about 0.5–1-dB performance gain. V. C ONCLUSION We have analyzed and optimized the performances of the IDM-MISO systems when partial CSI is available at the trans-

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[20] X. Wang and H. V. Poor, “Iterative (turbo) soft interference cancellation and decoding for coded CDMA,” IEEE Trans. Commun., vol. 47, no. 7, pp. 1046–1061, Jul. 1999. [21] K. Y. Wu, W. K. Leung, and L. Ping, “A simple approach to near-optimal multiple transmit antenna space–time codes,” in Proc. IEEE ICC, Anchorage, AK, May 2003, pp. 2603–2607. [22] K. Y. Wu, L. Ping, and W. K. Leung, “Multi-layer turbo space–time codes for high-rate applications,” in Proc. IEEE GLOBECOM, Dallas, TX, Dec. 2004, pp. 3758–3762.

Chuxiang Li received the B.S. and M.S. degrees in electronic engineering from Tsinghua University, Beijing, China, in 1999 and 2002, respectively, and the Ph.D. degree in electrical engineering from Columbia University, New York, NY, in 2006. From January to June 2002, he was with Microsoft Research Asia, Beijing, China, as a Visiting Student. From February to December 2005, he was with NEC Laboratories America, Inc., Princeton, NJ, as a Research Assistant. In October 2006, he joined Marvell Semiconductor, Inc., Santa Clara, CA, as a Senior Design Engineer. His research interests include the general areas of wireless communications and signal processing, specifically in space–time coding and MIMO processing, OFDM, wireless radio resource scheduling, and LDPC coding.

Kai Li received the B.S. degree in electronics and information science from Peking University, Beijing, China, in 2000, and the M.Eng. degree in electrical and computer engineering from National University of Singapore, Singapore, in 2002. He is currently working toward the Ph.D. degree in the Department of Electrical Engineering, Columbia University, New York, NY. His research interests include wireless communications, signal processing, information theory, codes on graphs, and iterative algorithms.

Xiaodong Wang (S’98–M’98–SM’04) received the Ph.D. degree in electrical engineering from Princeton University, Princeton, NJ. He is currently in the faculty of the Department of Electrical Engineering, Columbia University, New York, NY. His research interests fall in the general areas of computing, signal processing, and communications, and he has published extensively in these areas. Among his publications is a recent book entitled Wireless Communication Systems: Advanced Techniques for Signal Reception (Prentice Hall, 2003). His current research interests include wireless communications, statistical signal processing, and genomic signal processing. Dr. Wang received the 1999 NSF CAREER Award and the 2001 IEEE Communications Society and Information Theory Society Joint Paper Award. He currently serves as an Associate Editor for the IEEE TRANSACTIONS ON COMMUNICATIONS, IEEE TRANSACTIONS ON W IRELESS C OMMUNICATIONS , IEEE T RANSACTIONS ON S IGNAL PROCESSING, and IEEE TRANSACTIONS ON INFORMATION THEORY.

Li Ping (S’87–M’91) received the Ph.D. degree from Glasgow University, Glasgow, Scotland, in 1990. He lectured in the Department of Electronic Engineering, Melbourne University, Richmond, Vic., Australia, from 1990 to 1992, and was a Research Staff with Telecom Australia Research Laboratories from 1993 to 1995. Since January 1996, he has been in the Department of Electronic Engineering, City University of Hong Kong, Kowloon, Hong Kong, where he is currently a Professor. His research interests include mixed analog/digital circuits, communications systems, and coding theory. Dr. Ping received a British Telecom-Royal Society Fellowship in 1986, the IEE J. J. Thomson Premium in 1993, and the Croucher Foundation Award in 2005.