LU et aL: LDPC-BASED SPACE-TIME CODED OFDM SYSTEMS OVER CORRELATED FADING CHANNELS id(independent and identically distributed) circularly sym sible. However, as observed in [1], the space-time trellis netric complex Gaussian with zero-means. Then(10)can be codes (STTCs) with higher state numbers(and essen tially larger effective length) have better performance hich suggests that increasing the effective length of the Px→)≤exp (15) TC beyond the minimum requirement(e. g, NL, in our stem)may help to improve the factor li=l Aj 3)Also as seen from(7), to achieve the channel capacity, all whereB:(Avg., is the jth element ofVg:Since V is the(NKP) transmitted STC symbols are required to be independent. Therefore, after introducing the coding con- unitary,Bi Cfi, are also i i.d. circularly symmetric complex straints to the coded symbols, an interleaver is needed to Gaussian with zero-means and their magnitudes Bi Oi. are scramble the coded symbols in order to satisfy the inde- i.i.d. Rayleigh distributed. By averaging the conditional PEP in pendence condition From the standpoint of PEP analysis, (15)over the Rayleigh probability density function(pdf), the such an interleaver helps to improve the factor -A PEP of a multiple-antenna STC-OFDM system over correlated as well fading channels is finally written as In summary, in the system considered here, because of the di- verse fading profiles of the wireless channels and the assump- P(x→2)≤ tion that the Csi is known only at the receiver, the systematic Ir=1(1+) coding design(e.g, by computer search)is less helpful; instead, two general principles should be met in choosing STC codes n order to robustly exploit the rich diversity resources in this (16) system, namely, large effective length and ideal interleaving STTCs have been proposed for multiple-antenna systems over flat-fading channels [1]. However, the complexity of the It is seen from(16)that the highest possible diversity order the STTC increases dramatically as the effective length increases STC-OFDM system can provide is(NMD), i.e., the product of and therefore it may not be a good candidate for the OFDM the number of transmitter antennas, the number of receiver an- system considered here. Another family of STCs is turbo-code tenna, and the number of selective-fading diversity order in the based STCs[27],[28], but their decoding complexity is high channels. In other words, the attractiveness of the STC-OFDM and they are not flexible in terms of scalability(e.g,when system lies in its ability to exploit all the available diversity re- employed in systems with different requirements of the infor- mation rate). Here, we propose a new STC scheme: low-density However, note that, although in the analysis of PEp the parity-check(LDPC)based STC three parameters(N, M, L)appear equivalent in improving the system performance, they actually play different roles from the B. ldPC- Based sTc apacity viewpoint, as indicated in Section Ill First proposed by Gallager in 1962[11] and recently reex V LDPC-BASED STC-OFDM SYSTEM amined in[12],[13]and [29], low-density parity-check (LDPC codes have been shown to be a very promising coding technique In this section, we consider coding design for STC-OFDM for approaching the channel capacity in AWGN channels. For systems. As in Section we assume that the CSI is known only example, a carefully constructed rate 1/2 irregular LDPC code with long block length has a bit error probability of 10 at just 0.04 dB away from Shannon capacity of AWGN channels [30] An LDPC code is a linear block code characterized by a very The PEP analysis of a general STC-OFDM system in Sec- sparse parity-check matrix, as seen in Fig. 4. The parity check tion IV, as well as the channel capacity analysis in Section I, matrix P of an(n, k;, t, j LDPC code of rate R=k/n is an sheds some lights on the STC coding design problem (n-kx n matrix, which has t ones in each column and j>t 1)The dominant exponent in the PEP(16)that is related ones in each row. Apart from these constraints, the ones are to the structure of the code is r, the rank of the matrix placed at random in the parity check matrix. When the number D. Recall that mmnzI n( Deff, NL), in order to of ones in every column is the same, the code is known as a achieve the maximum diversity(NML), it is necessary regular LDPC code; otherwise, it is called irregular LDPC code that De> NL, i. e, the effective length of the code must In contrast to P, the generator matrix G is dense. Consequently be larger than the dimension of matrix D in(12). Since the number of bit operations required to encoder is O(n2)which is associated with the channel characteristic, which is not is larger than that for other linear codes. Similar to turbo codes known to the transmitter(or the STCencoder)in advance, LDPC codes can be efficiently decoded by a suboptimal iterative it is preferable to have an STC code with a large effective belief propagation algorithm which is explained in detail in [11] At the end of each iteration, the parity check is performed. Ifthe 2) Another factor in the PEP is I=1 Aj the product of parity check is correct, the decoding is terminated,otherwise eigenvalues of matrix D. Since d changes with different the decoding continues until it reaches the maximum number of hannel setups, the optimal design ofII=1A; is not fea- iterations(e.g, 30)LU et al.: LDPC-BASED SPACE–TIME CODED OFDM SYSTEMS OVER CORRELATED FADING CHANNELS 79 i.i.d. (independent and identically distributed) circularly sym￾metric complex Gaussian with zero-means. Then (10) can be rewritten as 8 (15) where is the th element of . Since is unitary, are also i.i.d. circularly symmetric complex Gaussian with zero-means and their magnitudes are i.i.d. Rayleigh distributed. By averaging the conditional PEP in (15) over the Rayleigh probability density function (pdf), the PEP of a multiple-antenna STC-OFDM system over correlated fading channels is finally written as (16) It is seen from (16) that the highest possible diversity order the STC-OFDM system can provide is ( ), i.e., the product of the number of transmitter antennas, the number of receiver an￾tennas, and the number of selective-fading diversity order in the channels. In other words, the attractiveness of the STC-OFDM system lies in its ability to exploit all the available diversity re￾sources. However, note that, although in the analysis of PEP the three parameters ( ) appear equivalent in improving the system performance, they actually play different roles from the capacity viewpoint, as indicated in Section III. V. LDPC-BASED STC-OFDM SYSTEM In this section, we consider coding design for STC-OFDM systems. As in Section II, we assume that the CSI is known only at the receiver. A. Coding Design Principles The PEP analysis of a general STC-OFDM system in Sec￾tion IV, as well as the channel capacity analysis in Section III, sheds some lights on the STC coding design problem. 1) The dominant exponent in the PEP (16) that is related to the structure of the code is , the rank of the matrix . Recall that , in order to achieve the maximum diversity ( ), it is necessary that , i.e., the effective length of the code must be larger than the dimension of matrix in (12). Since is associated with the channel characteristic, which is not known to the transmitter (or the STC encoder) in advance, it is preferable to have an STC code with a large effective length. 2) Another factor in the PEP is , the product of eigenvalues of matrix . Since changes with different channel setups, the optimal design of is not fea￾sible. However, as observed in [1], the space–time trellis codes (STTCs) with higher state numbers (and essen￾tially larger effective length) have better performance, which suggests that increasing the effective length of the STC beyond the minimum requirement (e.g., , in our system) may help to improve the factor . 3) Also as seen from (7), to achieve the channel capacity, all the ( ) transmitted STC symbols are required to be independent. Therefore, after introducing the coding con￾straints to the coded symbols, an interleaver is needed to scramble the coded symbols in order to satisfy the inde￾pendence condition. From the standpoint of PEP analysis, such an interleaver helps to improve the factor as well. In summary, in the system considered here, because of the di￾verse fading profiles of the wireless channels and the assump￾tion that the CSI is known only at the receiver, the systematic coding design (e.g., by computer search) is less helpful; instead, two general principles should be met in choosing STC codes in order to robustly exploit the rich diversity resources in this system, namely, large effective length and ideal interleaving. STTCs have been proposed for multiple-antenna systems over flat-fading channels [1]. However, the complexity of the STTC increases dramatically as the effective length increases and therefore it may not be a good candidate for the OFDM system considered here. Another family of STCs is turbo-code based STCs [27], [28], but their decoding complexity is high and they are not flexible in terms of scalability (e.g., when employed in systems with different requirements of the infor￾mation rate). Here, we propose a new STC scheme: low-density parity-check (LDPC)-based STC. B. LDPC-Based STC First proposed by Gallager in 1962 [11] and recently reex￾amined in [12], [13] and [29], low-density parity-check (LDPC) codes have been shown to be a very promising coding technique for approaching the channel capacity in AWGN channels. For example, a carefully constructed rate 1 2 irregular LDPC code with long block length has a bit error probability of 10 at just 0.04 dB away from Shannon capacity of AWGN channels [30]. An LDPC code is a linear block code characterized by a very sparse parity-check matrix, as seen in Fig. 4. The parity check matrix of an ( ) LDPC code of rate is an matrix, which has ones in each column and ones in each row. Apart from these constraints, the ones are placed at random in the parity check matrix. When the number of ones in every column is the same, the code is known as a regular LDPC code; otherwise, it is called irregular LDPC code. In contrast to , the generator matrix is dense. Consequently, the number of bit operations required to encoder is which is larger than that for other linear codes. Similar to turbo codes, LDPC codes can be efficiently decoded by a suboptimal iterative belief propagation algorithm which is explained in detail in [11]. At the end of each iteration, the parity check is performed. If the parity check is correct, the decoding is terminated; otherwise, the decoding continues until it reaches the maximum number of iterations (e.g., 30)