《OFDM技术开发系列》（英文版） LDPC-Based space—Time coded OFDM Systems over correlated fading channels Performance analysis and receiver design

IEEE TRANSACTIONS ON COMMUNICATIONS. VOL 50. NO. 1 JANUARY 2002 LDPC-Based Space-Time Coded OFDM Systems Over Correlated Fading Channels: Performance Analysis and Receiver design Ben Lu, Student Member, IEEE, Xiaodong Wang, Member, IEEE, and Krishna R Narayanan, Member: IEEE AbstrackWe consider a space-time coded (STC) orthogonal systems integrate the techniques of antenna array spatial diver- frequency-division multiplexing (OFDM)system with multiple sity and channel coding and can provide significant capacity and time-selective fading channels. It is shown that the product gains in wireless channels. Howea rovide significant capacity ny wireless channels of the time-selectivity order and the frequency-selectivity order are frequency-selective in nature, for which the STC design is a key parameter to characterize the outage capacity of the problem becomes a complicated issue. On the other hand, the correlated fading channel. It is also observed that STCs with large orthogonal frequency-division multiplexing(OFDM) technique effective lengths and ideal built-in interleavers are more effective transforms a frequency-selective fading channel into parallel in exploiting the natural diversity in multiple-antenna correlated fading channels. We then propose a low-density paritv-check correlated flat-fading channels. Hence, in the presence of fre- (DPC-code-based STC-OFDM system. Compared with the quency selectivity, it is natural to consider STC in the OFDM conventional space-time trellis code(STTC), the LDPC-based context. The first STC-OFDM system was proposed in [4]. In STC can significantly improve the system performance by ex- this paper, we provide system performance analysis and receiver ploiting both the spatial diversity and the selective-fading diversity design for anew STC-OFDM system over correlated frequency- turbo-code- based STC scheme, LDPC-based STC exhibits lower and time-selective fading channels receiver complexity and more flexible scalability. We also consider We first analyze the STC-OFDM system performance in receiver design for LDPC-based STC-OFDM systems in unknown correlated fading channels in terms of channel capacity and last fading channels and propose a novel turbo receiver employing pairwise error probability(PEP). In[5], information-theoretic demodulator and a soft LDPC decoder, which can significantly aspects of a two-ray propagation fading channel are studied reduce the error floor in fast fading channels with a modest com- More recently, in [6] and [7] the channel capacity of a mul putational complexity. With such a turbo receiver, the proposed tiple-antenna system in fading channels is investigated, and in LDPC-based sTc-OFDM system is a promising solution to highly [8] the limiting performance of a multiple-antenna system in channels block-fading channels is studied, under the assumption that the fading channels are uncorrelated and the channel state Index Termms-Correlated fading, iterative receive information(CSI is known to both the transmitter and the frequency-division multiplexing (OFDM), space-time code(STC). receiver. Here, we analyze the channel capacity of a mul tiple-antenna OFDM syste ver correlated frequency- and time-selective fading channels assuming that the csi is known . INTROdUCTION only to the receiver. As a promising coding scheme to approach A considerable amount of recent research has addressed the the channel capacity, STC is employed as the channel code design and implementation of space-time coded(STC) in this system. The pairwise error probability(PEP) analysis systems for wireless flat-fading channels, e. g, [11-3. The STC of the STC-OFDM system is also given, which follows the analysis for coded modulation systems [11[91,[10].Moreover, based on the analysis of the channel capacity and the PEP, e StC design principles for the system under consideration Coding of the IEEE Communications Society. Manuscript received September are suggested. Since the stC based on the state-of-the-art supported in part by the National Science Foundation under Grant CAREER low-density parity-check (LDPC)codes [11F-113] turns out to CCR-9875314 and CCR-9980599 The work of K.R. Narayanan was sup- be a good candidate to meet these design principles, we then and an ATP grant from the Texas higher education coordination board. This propose an LDPC-based STC-OFDM system and develop a resented in part at the 2001 IEEE International Symposium turbo receiver for this system. (Note that the design issues of Information Theory, Washington, DC, June 2001 STC in broad-band OFDM systems have been independently B. Lu and K.R. Narayanan are with the Department of Electrical Engi- neering, Texas A&M University, College Station, TX 77843 USA (e-mail: addressed in [14].) benlu@ee. tamu.edu; krishna a ee. tamu.edu) With ideal CSI. the iterative receiver based on the turbo X. Wang was with the Department of Electrical Engi niversity, College Station, TX 77843 USA. He is now with the Department of &M ciple [15] is shown to be able to provide the near-maxin Electrical Engineering, Columbia University, New York, NY 10027 USA kelihood performance in STC systems [16,[17]. wher Publisher Item Identifier S 0090-6778(02)00521-4 CSI is not available, a receiver structure consisting of a decision- 0090677802s170082002IEEE

74 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 50, NO. 1, JANUARY 2002 LDPC-Based Space–Time Coded OFDM Systems Over Correlated Fading Channels: Performance Analysis and Receiver Design Ben Lu, Student Member, IEEE, Xiaodong Wang, Member, IEEE, and Krishna R. Narayanan, Member, IEEE Abstract—We consider a space–time coded (STC) orthogonal frequency-division multiplexing (OFDM) system with multiple transmitter and receiver antennas over correlated frequency￾and time-selective fading channels. It is shown that the product of the time-selectivity order and the frequency-selectivity order is a key parameter to characterize the outage capacity of the correlated fading channel. It is also observed that STCs with large effective lengths and ideal built-in interleavers are more effective in exploiting the natural diversity in multiple-antenna correlated fading channels. We then propose a low-density parity-check (LDPC)-code-based STC-OFDM system. Compared with the conventional space–time trellis code (STTC), the LDPC-based STC can significantly improve the system performance by ex￾ploiting both the spatial diversity and the selective-fading diversity in wireless channels. Compared with the recently proposed turbo-code-based STC scheme, LDPC-based STC exhibits lower receiver complexity and more flexible scalability. We also consider receiver design for LDPC-based STC-OFDM systems in unknown fast fading channels and propose a novel turbo receiver employing a maximum a posteriori expectation-maximization (MAP-EM) demodulator and a soft LDPC decoder, which can significantly reduce the error floor in fast fading channels with a modest com￾putational complexity. With such a turbo receiver, the proposed LDPC-based STC-OFDM system is a promising solution to highly efficient data transmission over selective-fading mobile wireless channels. Index Terms—Correlated fading, iterative receiver, low-den￾sity parity-check (LDPC) codes, multiple antennas, orthogonal frequency-division multiplexing (OFDM), space–time code (STC). I. INTRODUCTION Aconsiderable amount of recent research has addressed the design and implementation of space–time coded (STC) systems for wireless flat-fading channels, e.g., [1]–[3]. The STC Paper approved by R. Raheli, the Editor for Detection, Equalization, and Coding of the IEEE Communications Society. Manuscript received September 25, 2000; revised May 28, 2001. The work of X. Wang and B. Lu was supported in part by the National Science Foundation under Grant CAREER CCR–9875314 and CCR–9980599. The work of K.R. Narayanan was sup￾ported in part by the National Science Foundation under Grant CCR–0073506 and an ATP grant from the Texas higher education coordination board. This paper was presented in part at the 2001 IEEE International Symposium on Information Theory, Washington, DC, June 2001. B. Lu and K. R. Narayanan are with the Department of Electrical Engi￾neering, Texas A&M University, College Station, TX 77843 USA (e-mail: benlu@ee.tamu.edu; krishna@ee.tamu.edu). X. Wang was with the Department of Electrical Engineering, Texas A&M University, College Station, TX 77843 USA. He is now with the Department of Electrical Engineering, Columbia University, New York, NY 10027 USA. Publisher Item Identifier S 0090-6778(02)00521-4. systems integrate the techniques of antenna array spatial diver￾sity and channel coding and can provide significant capacity gains in wireless channels. However, many wireless channels are frequency-selective in nature, for which the STC design problem becomes a complicated issue. On the other hand, the orthogonal frequency-division multiplexing (OFDM) technique transforms a frequency-selective fading channel into parallel correlated flat-fading channels. Hence, in the presence of fre￾quency selectivity, it is natural to consider STC in the OFDM context. The first STC-OFDM system was proposed in [4]. In this paper, we provide system performance analysis and receiver design for a new STC-OFDM system over correlated frequency￾and time-selective fading channels. We first analyze the STC-OFDM system performance in correlated fading channels in terms of channel capacity and pairwise error probability (PEP). In [5], information-theoretic aspects of a two-ray propagation fading channel are studied. More recently, in [6] and [7], the channel capacity of a mul￾tiple-antenna system in fading channels is investigated, and in [8] the limiting performance of a multiple-antenna system in block-fading channels is studied, under the assumption that the fading channels are uncorrelated and the channel state information (CSI) is known to both the transmitter and the receiver. Here, we analyze the channel capacity of a mul￾tiple-antenna OFDM system over correlated frequency- and time-selective fading channels, assuming that the CSI is known only to the receiver. As a promising coding scheme to approach the channel capacity, STC is employed as the channel code in this system. The pairwise error probability (PEP) analysis of the STC-OFDM system is also given, which follows the analysis for coded modulation systems [1], [9], [10]. Moreover, based on the analysis of the channel capacity and the PEP, some STC design principles for the system under consideration are suggested. Since the STC based on the state-of-the-art low-density parity-check (LDPC) codes [11]–[13] turns out to be a good candidate to meet these design principles, we then propose an LDPC-based STC-OFDM system and develop a turbo receiver for this system. (Note that the design issues of STC in broad-band OFDM systems have been independently addressed in [14].) With ideal CSI, the iterative receiver based on the turbo prin￾ciple [15] is shown to be able to provide the near-maximum￾likelihood performance in STC systems [16], [17]. When the CSI is not available, a receiver structure consisting of a decision- 0090–6778/02$17.00 © 2002 IEEE

LU et aL: LDPC-BASED SPACE-TIME CODED OFDM SYSTEMS OVER CORRELATED FADING CHANNELS R An Fred 排3 Fig 1. System description of a multiple-antenna STC-OFDM system over correlated fading channels. Each STC code word spans A subcarriers and P time ots in the syst a particular subcarrier and at a par time slot, STC symbols are transmitted from N transmitter antennas and received by M receive directed least-square estimator and a data detector is introduced At the receiver, the signals are received from M receiver in[18]. For the system considered here, the receiver in [18]per- antennas. After matched filtering and sampling, the discrete forms well at low to medium Doppler frequencies, but exhibits Fourier transform(DFT)is applied to the received discrete-time an irreducible high error floor in fast fading channels. a receiver signal to obtain employing the expectation-maximization(EM) algorithm has recently been proposed for STC systems [19][201, which ex- 3,=H[m,k],k+2,, hibits a good performance, but, on the other hand, its complexity is relatively high for the LDPC-based STC-OFDM systems Here, we develop a novel turbo receiver structure employing where H[p, k]E CAXN is the matrix of complex channel fre- a maximum a posteriori expectation-maximization(MAP-EM) quency responses at the hth subcarrier and at the pth time slot, demodulator and a soft LDPC decoder, which can significantly which is explained below, alp, h]EC andy, k] re- reduce the error floor in fast fading channels with a modest com- spectively the transmitted signals and the received signals at the putational complexity. (A similar iterative receiver structure is hth subcarrier and at the pth time slot, and alp, ]E is the developed for static MIMO channels in [21]) ambient noise, which is circularly symmetric complex Gaussian The rest of this paper is organized as follows. In Section Il, with unit variance multiple-antenna STC-OFDM system over correlated fre- Consider the channel response between the th transmitter an- quency- and time-selective fading channels is described. In tenna and the ith receiver antenna. Following [22], the time-do- Section IlL, the outage capacity of this system is analyzed In main channel impulse response can be modeled as a tapped Section IV, the PEP analysis is given. Based on the analysis delay line. With only the nonzero taps considered, it can be ex in Sections Ill and IV, in Section V, an LDPC-based STC pressed as osed for the OFDM system under consideration. In Section VI, a novel turbo receiver is developed. In Section VIl computer simulation results are given. Section VIll contains the conclusion h;(x;t=∑00(-( where 8( is the Dirac delta function, Lf denotes the number SYSTEM MODEL of nonzero taps, and c (; t)is the complex amplitude of the We consider an STC-OFDM system with K subcarriers, delay is n/(K△r)when N transmitter antennas, and M receiver antennas, signaling integer and Ay is the tone spacing of the OFDM system. In through frequency- and time-selective fading channels, as mobile channels, for the particular(2,3)th antenna pair, the illustrated in Fig. 1. Each STC code word spans P adjacent time-variant tap coeficients ai, i(; t ), V, vt, can be modeled OFDM words, and each OFDM word consists of (NK) STC as wide-sense stationary random processes with uncorrelated symbols, transmitted simultaneously during one time slot. Each scattering (wSSUS)and with band-limited doppler power STC symbol is transmitted at a particular OFDM subcarrier spectrum [22]. For the signal model in(1), we only need and a particular transmitter antenna to consider the time responses of ai, i (l; t) within the time It is assumed that the fading process remains static during interval t E [0, PT] where T is the total time duration of one each OFDM word (one time slot)but varies from one OFDM OFDM word plus its cyclic extension and PT is the total time word to another, and the fading processes associated with involved in transmitting P adjacent OFDM words different transmitter-receiver antenna pairs are uncorrelated. [23], for the particular lth tap of the (i,j)th antenna pair, (However, as will be shown below, in a typical OFDM system, the dimension of the band- and time-limited random process for a particular transmitter-receiver antenna pair, the fading ai, i ( l; t),te [ 0, Pt (defined as the number of significant processes are correlated in both frequency and time. eigenvalues in the Karhunen-Loeve expansion of this random

LU et al.: LDPC-BASED SPACE–TIME CODED OFDM SYSTEMS OVER CORRELATED FADING CHANNELS 75 Fig. 1. System description of a multiple-antenna STC-OFDM system over correlated fading channels. Each STC code word spans K subcarriers and P time slots in the system; at a particular subcarrier and at a particular time slot, STC symbols are transmitted from N transmitter antennas and received by M receiver antennas. directed least-square estimator and a data detector is introduced in [18]. For the system considered here, the receiver in [18] per￾forms well at low to medium Doppler frequencies, but exhibits an irreducible high error floor in fast fading channels. A receiver employing the expectation-maximization (EM) algorithm has recently been proposed for STC systems [19], [20], which ex￾hibits a good performance, but, on the other hand, its complexity is relatively high for the LDPC-based STC-OFDM systems. Here, we develop a novel turbo receiver structure employing a maximum a posteriori expectation-maximization (MAP-EM) demodulator and a soft LDPC decoder, which can significantly reduce the error floor in fast fading channels with a modest com￾putational complexity. (A similar iterative receiver structure is developed for static MIMO channels in [21].) The rest of this paper is organized as follows. In Section II, a multiple-antenna STC-OFDM system over correlated fre￾quency- and time-selective fading channels is described. In Section III, the outage capacity of this system is analyzed. In Section IV, the PEP analysis is given. Based on the analysis in Sections III and IV, in Section V, an LDPC-based STC is proposed for the OFDM system under consideration. In Section VI, a novel turbo receiver is developed. In Section VII, computer simulation results are given. Section VIII contains the conclusion. II. SYSTEM MODEL We consider an STC-OFDM system with subcarriers, transmitter antennas, and receiver antennas, signaling through frequency- and time-selective fading channels, as illustrated in Fig. 1. Each STC code word spans adjacent OFDM words, and each OFDM word consists of ( ) STC symbols, transmitted simultaneously during one time slot. Each STC symbol is transmitted at a particular OFDM subcarrier and a particular transmitter antenna. It is assumed that the fading process remains static during each OFDM word (one time slot) but varies from one OFDM word to another, and the fading processes associated with different transmitter-receiver antenna pairs are uncorrelated. (However, as will be shown below, in a typical OFDM system, for a particular transmitter–receiver antenna pair, the fading processes are correlated in both frequency and time.) At the receiver, the signals are received from receiver antennas. After matched filtering and sampling, the discrete Fourier transform (DFT) is applied to the received discrete-time signal to obtain 0 1 1 (1) where is the matrix of complex channel fre￾quency responses at the th subcarrier and at the th time slot, which is explained below, and are re￾spectively the transmitted signals and the received signals at the th subcarrier and at the th time slot, and is the ambient noise, which is circularly symmetric complex Gaussian with unit variance. Consider the channel response between the th transmitter an￾tenna and the th receiver antenna. Following [22], the time-do￾main channel impulse response can be modeled as a tapped￾delay line. With only the nonzero taps considered, it can be ex￾pressed as (2) where is the Dirac delta function, denotes the number of nonzero taps, and is the complex amplitude of the th nonzero tap, whose delay is , where is an integer and is the tone spacing of the OFDM system. In mobile channels, for the particular ( )th antenna pair, the time-variant tap coefficients can be modeled as wide-sense stationary random processes with uncorrelated scattering (WSSUS) and with band-limited Doppler power spectrum [22]. For the signal model in (1), we only need to consider the time responses of within the time interval 0 , where is the total time duration of one OFDM word plus its cyclic extension and is the total time involved in transmitting adjacent OFDM words. Following [23], for the particular th tap of the ( )th antenna pair, the dimension of the band- and time-limited random process 0 (defined as the number of significant eigenvalues in the Karhunen–Loeve expansion of this random

IEEE TRANSACTIONS ON COMMUNICATIONS. VOL 50. NO. 1 JANUARY 2002 process), is approximately equal to L,=[2j aPT+1, where as the mutual information conditioned on the correlated fading a is the maximum Doppler frequency. Hence, ignoring the channel values Hp, AIlP.Kl, is computed as([51, [8] edge effects, the time response of ci, i (; t) can be expressed terms of the Fourier expansion as In(r=i(yp, kpk;t=p, k]p,k[7,y) ;/(;t)≈ ,1(,n)c2t( ∑∑∑l(1+Mn6)3)m( where ij (L, n)m is a set of independent circularly symmetric where memin(N, M) and A,(p, k)is the ith nonzero eigenvalue of the nonnegative definite Hermitian matrix For OFDM systems with proper cyclic extension and sample H[p, k]"lp, k ]. The maximization of In(y)is achieved between the jth transmitter antenna and the zth receiver antenna complex Gaussian random variables with identical variances at the pth time slot and at the kth subcarrier, which is exactly the [51, i8). ( When the CSI is known to both the transmitter and the receiver, the instantaneous channel capacity is maximized by "water-filling[25]) The ergodic channel capacity is defined Hi,[p, =Hi, i(pI, kAn)=Eai,(L: pT)e-2nkmu/k as I(y= En(Im(7). In the system considered,the concept of ergodic channel capacity I(r)is of less interest, because the hi(pwf(h) (4) fading processes are not ergodic due to the limited number of antennas and the limited Lf and Lt ai (LS;Pr)]h is the Since lin(m)is a random variable, whose statistics are jointly Lf-sized vector containing the time responses of all the determined by (y, N, M)and the characteristics of correlated nonzero taps: wy()=[e-2mkma/K,.,e-22-kn /k fading channel we turn to another important concept-outage contains the corresponding DFT coefficients capacity, which is closely related to the code word error prob- ability, as averaged over the random coding ensemble and over Using(3), ai, i (; pr) can be simplified as all channel realizations[8]. The outage probability is defined as the probability that the channel cannot support a given informa- Cii (L;pr)= Bi, (l,n)ej2mmp/P=A u(p) tion rate R (5) Bi (=Bij(,-faPT), ..,Bi, (4,0), ...,Bi, (, faPT) Since it is difficult to get an analytical expression for(8),we Le-sized and resort to Monte Carlo integration for its numerical evaluation ()全[=2f, 2pf contains the corresponding inverse DFT coefficients. Substituting (5) A. Numerical Results to (4), we obtain In this subsection, we give some numerical results of the Hii[p, k]=gi wt(p)u (k) outage probability in( 8)obtained by Monte Carlo integration For simplicity, we assume that all elements in g; i Jij have the same variances. Define the selective-fading diversity order L as 9[r()…mL小 the product of the number of nonzero delay taps Lf and the di- L×1 mension of Doppler fading process L, i.e., L=LyLt. The fol wI(p=diaglwt(p),.,wt(p))LxL.(6)lowing observations tions of ( 8) From (6), it is seen that, due to the close spacing of OFDM sub- arriers and the limited Doppler frequency, for a specific an- 1)From Figs. 2 and 3, it is seen that at a practical outage tenna pair(i,,), the channel responses (Hi Dp, kIbo k are dif- probability (e.g, Pout 1%), for fixed (N, M, 7) ferent transformations [specified by w(p)and w(k)] of the the highest achievable information rate increases as same random vector gi i and hence they are correlated in both the selective-fading diversity order L increases, but the frequency and time ncrease slows down as L becomes larger. Eventually as L-o. the highest achievable information rate IIL. CHANNEL C converges to the ergodic capacity. [Note that the ergodic capacity is the area above each curve in the figure In this section, we consider the channel capacity of the system I(n)=Jo P(I/n()> R)dRI described above. Assuming that the channel state information 2) Fig. 3 compares the impacts of the frequency-selectivity (CSI) is only known at the receiver and the transmitter power order Lf and the time-selectivity order lt on the outage is constrained as trace{E[rx[小}≤r, the in- capacity. It shows that the frequency selectivity and the stantaneous channel capacity of this system, which is defined me selectiv essentially equivalent in terms of their

76 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 50, NO. 1, JANUARY 2002 process), is approximately equal to 2 1 , where is the maximum Doppler frequency. Hence, ignoring the edge effects, the time response of can be expressed in terms of the Fourier expansion as (3) where is a set of independent circularly symmetric complex Gaussian random variables, indexed by . For OFDM systems with proper cyclic extension and sample timing, with tolerable leakage, the channel frequency response between the th transmitter antenna and the th receiver antenna at the th time slot and at the th subcarrier, which is exactly the ( )th element of in (1), can be expressed as [24] (4) where is the -sized vector containing the time responses of all the nonzero taps; contains the corresponding DFT coefficients. Using (3), can be simplified as (5) where is an -sized vector, and contains the corresponding inverse DFT coefficients. Substituting (5) into (4), we obtain with (6) From (6), it is seen that, due to the close spacing of OFDM sub￾carriers and the limited Doppler frequency, for a specific an￾tenna pair ( ), the channel responses are dif￾ferent transformations [specified by and ] of the same random vector and hence they are correlated in both frequency and time. III. CHANNEL CAPACITY In this section, we consider the channel capacity of the system described above. Assuming that the channel state information (CSI) is only known at the receiver and the transmitter power is constrained as , the in￾stantaneous channel capacity of this system, which is defined as the mutual information conditioned on the correlated fading channel values , is computed as [5], [8] bit/s/Hz (7) where and is the th nonzero eigenvalue of the nonnegative definite Hermitian matrix . The maximization of is achieved when consists of independent circularly symmetric complex Gaussian random variables with identical variances [5], [8]. (When the CSI is known to both the transmitter and the receiver, the instantaneous channel capacity is maximized by “water-filling” [25].) The ergodic channel capacity is defined as . In the system considered, the concept of ergodic channel capacity is of less interest, because the fading processes are not ergodic due to the limited number of antennas and the limited and . Since is a random variable, whose statistics are jointly determined by ( ) and the characteristics of correlated fading channels, we turn to another important concept—outage capacity, which is closely related to the code word error prob￾ability, as averaged over the random coding ensemble and over all channel realizations [8]. The outage probability is defined as the probability that the channel cannot support a given informa￾tion rate (8) Since it is difficult to get an analytical expression for (8), we resort to Monte Carlo integration for its numerical evaluation. A. Numerical Results In this subsection, we give some numerical results of the outage probability in (8) obtained by Monte Carlo integration. For simplicity, we assume that all elements in have the same variances. Define the selective-fading diversity order as the product of the number of nonzero delay taps and the di￾mension of Doppler fading process , i.e., . The fol￾lowing observations can be made from the numerical evalua￾tions of (8). 1) From Figs. 2 and 3, it is seen that at a practical outage probability (e.g., 1 ), for fixed ( ), the highest achievable information rate increases as the selective-fading diversity order increases, but the increase slows down as becomes larger. Eventually, as , the highest achievable information rate converges to the ergodic capacity. [Note that the ergodic capacity is the area above each curve in the figure as .] 2) Fig. 3 compares the impacts of the frequency-selectivity order and the time-selectivity order on the outage capacity. It shows that the frequency selectivity and the time selectivity are essentially equivalent in terms of their

LU et aL: LDPC-BASED SPACE-TIME CODED OFDM SYSTEMS OVER CORRELATED FADING CHANNELS Outage Probability in Freq-Selective Fading Channel, SNR=20dB 0.8 0.7 05 0.4 0.3 02 0 Fig. 2. Outage probability versus informa 256.P= 1. SNR= 20 dB. where dashed lines present the m with one transmitter antenna(N= 1)and solid lines represent the system with four transmitter antennas( N=4). The vertical dash-dotted ne represents the AwGN channel capacity(when SNr= 20 dB). The fading channels are frequency-selective and time-nonselective with lt =lL=L {1.2.3.6} Outage Probability in Freq Time-Selective Fading Channel, SNR=20dB 05 03 0.1 Information Rate. bit/sec/Hz Fig 3. Outage probability versus information rate in a correlated fading OFDM system with N=2,M=1,A present the frequency-selective and time-nonselective channels with Lt=l,L=Lf=(2,6. 10). Dotted lin esent the and time selective channels with L =2,L= 2L=(2.6. 10). Note that, for the same L, the dashed lines and the dotted lines c each other, which shows the equivalent impacts of the frequency- and time-selective fading on the outage probability impacts on the outage capacity. In other words, the selec- in determining the correlation characteristics of the fading tive-fading diversity order L= Lflt ultimately affects channels)and it is determined only by the spatial diversity the outage capacity order(N, M) and the transmitted signal power y [6] [7] 3)From Fig. 2, it is seen that, as the area above each ci Moreover, it is seen that both the outage capacity and the he ergodic channel capacity is irrelevant of the ergodic capacity can be increased by fixing the number tive-fading diversity order L(which is the key parameter of receiver antennas and only increasing transmitter an

LU et al.: LDPC-BASED SPACE–TIME CODED OFDM SYSTEMS OVER CORRELATED FADING CHANNELS 77 Fig. 2. Outage probability versus information rate in a correlated fading OFDM system with M = 1, K = 256, P = 1, SNR= 20 dB, where dashed lines represent the system with one transmitter antenna (N = 1) and solid lines represent the system with four transmitter antennas (N = 4). The vertical dash–dotted line represents the AWGN channel capacity (when SNR= 20 dB). The fading channels are frequency-selective and time-nonselective with L = 1; L = L = f1; 2; 3; 6g. Fig. 3. Outage probability versus information rate in a correlated fading OFDM system with N = 2, M = 1, K = 256, P = 10, SNR= 20 dB. Dashed lines represent the frequency-selective and time-nonselective channels with L = 1, L = L = f2; 6; 10g. Dotted lines represent the frequency- and time-selective channels with L = 2, L = 2L = f2; 6; 10g. Note that, for the same L, the dashed lines and the dotted lines overlap each other, which shows the equivalent impacts of the frequency- and time-selective fading on the outage probability. impacts on the outage capacity. In other words, the selec￾tive-fading diversity order ultimately affects the outage capacity. 3) From Fig. 2, it is seen that, as the area above each curve, the ergodic channel capacity is irrelevant of the selec￾tive-fading diversity order (which is the key parameter in determining the correlation characteristics of the fading channels) and it is determined only by the spatial diversity order ( ) and the transmitted signal power [6], [7]. Moreover, it is seen that both the outage capacity and the ergodic capacity can be increased by fixing the number of receiver antennas and only increasing transmitter an-

IEEE TRANSACTIONS ON COMMUNICATIONS. VOL 50. NO. 1 JANUARY 2002 tennas(or vice versa),(e.g, by fixing M= I and let where the minimization is over all possible Stc codewordS= N-o, the ergodic capacity converges to the capacity Eipp, kipk. Assuming equal transmitted power at all trans- of AwGn channels [26]) mitter antennas, using the Chernoff bound, the PEP of trans- In summary, we have seen the different impacts of two di- mitting r and deciding in favor of another codeword I at the versity resources-the spatial diversity and the selective-fading decoder is upper bounded by diversity--on the channel capacity of a multiple-antenna cor related fading OFDM system. Increasing the spatial diversity P(x→21)≤e(~F(x,) (10) order(i.e, N, M) can always bring capacity(outage capacity and or ergodic capacity increase at the expense of extra phys- where y is the total signal power transmitted from all N trans- ical costs. By contrast, the selective-fading diversity is a free re- mitted antennas(recall that the noise at each receiver antenna source, but its effect on improving the channel capacity becomes is assumed to have unit variance). Using(4) -(6), d2(a, r)is less as L becomes larger. Since both diversity resources can im- 13), shown at the bottom of the page. In(12), prove the capacity of a multiple-antenna OFDM system, It is (ep, ke[p, k] is a rank-one matrix, which equals to a zero crucial to have an efficient channel coding scheme, which can matrix if the entries of codewords r and T corresponding to the take advantage of all available diversity resources of the system. kth subcarrier and the pth time slot are the same. Let D denote the number of instances when ep, keh[p, A+0, vp, Vh, IV PAIRWISE ERROR PROBABILITY similarly, as in [101 Deff, which is the minimum D over every two possible codeword pair, is called the effective length In the previous section, the potential information rate of a of the code. Denoting T=rank(D), it is easily seen that multiple-antenna OFDM system in correlated fading channels minr, r< min( Deff, NL). Since wf (k:)and w(p)vary with is studied. In order to obtain more insights on coding design, in different multipath delay profiles and Doppler power spectrum this section, we analyze the pairwise error probability(PEP)of shapes, the matrix D is also variant with different channel this system with coded modulation. environments. However it is observed that D is a nonnegative With perfect CSI at the receiver, the maximum likelihood definite Hermitian matrix; by an eigendecomposition, it can be ML) decision rule of the signal model (1)is given by wrItten as argon 9-∑HJ where V is a unitary matrix and/dag{入1,…,,,0,……,0} with Aili= being the positive eigenvalues of D.Moreover, as (9)assumed in Section Ill, all the (NML)elements of . iJi j are d(x)=∑∑∑∑H小小小 i=1=1k=0=1 Wi(P)w/(kelp, kle"p, Aw(kw!(p) NL)×1 P=1k= LX(NL) ∑D lx,-, e4]千[y4],,N1 W八(k)=dig{w()…,/(k)(NL)xN W()三dieg{w()…,W(p)} W(p)W/ (k:)ep, k]e [p, k/()Wt(p) (NLX(NL 9全H…,gLx1

78 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 50, NO. 1, JANUARY 2002 tennas (or vice versa), (e.g., by fixing 1 and let , the ergodic capacity converges to the capacity of AWGN channels [26]). In summary, we have seen the different impacts of two di￾versity resources—the spatial diversity and the selective-fading diversity—on the channel capacity of a multiple-antenna cor￾related fading OFDM system. Increasing the spatial diversity order (i.e., ) can always bring capacity (outage capacity and/or ergodic capacity) increase at the expense of extra phys￾ical costs. By contrast, the selective-fading diversity is a free re￾source, but its effect on improving the channel capacity becomes less as becomes larger. Since both diversity resources can im￾prove the capacity of a multiple-antenna OFDM system, it is crucial to have an efficient channel coding scheme, which can take advantage of all available diversity resources of the system. IV. PAIRWISE ERROR PROBABILITY In the previous section, the potential information rate of a multiple-antenna OFDM system in correlated fading channels is studied. In order to obtain more insights on coding design, in this section, we analyze the pairwise error probability (PEP) of this system with coded modulation. With perfect CSI at the receiver, the maximum likelihood (ML) decision rule of the signal model (1) is given by (9) where the minimization is over all possible STC codeword . Assuming equal transmitted power at all trans￾mitter antennas, using the Chernoff bound, the PEP of trans￾mitting and deciding in favor of another codeword at the decoder is upper bounded by (10) where is the total signal power transmitted from all trans￾mitted antennas (recall that the noise at each receiver antenna is assumed to have unit variance). Using (4)–(6), is given by (11)–(13), shown at the bottom of the page. In (12), ( ) is a rank-one matrix, which equals to a zero matrix if the entries of codewords and corresponding to the th subcarrier and the th time slot are the same. Let denote the number of instances when ; similarly, as in [10], , which is the minimum over every two possible codeword pair, is called the effective length of the code. Denoting , it is easily seen that . Since and vary with different multipath delay profiles and Doppler power spectrum shapes, the matrix is also variant with different channel environments. However, it is observed that is a nonnegative definite Hermitian matrix; by an eigendecomposition, it can be written as (14) where is a unitary matrix and 0 0 , with being the positive eigenvalues of . Moreover, as assumed in Section III, all the ( ) elements of are (11) with (12) (13)

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)

IEEE TRANSACTIONS ON COMMUNICATIONS. VOL 50. NO. 1 JANUARY 2002 1000001000100 Fig 4. Example of a parity-check matrix P for an(n, k.t.j)=(20. 5. 3. 4)regular LDPC code with code rate 1/4, block length n= 20, column weight t=3, and row weight 工FFT LDPC Coded MPSK Bits Encoder Bits Modulator symbola/S/p 工F LETT Fig. 5. Transmitter structure of an LDPC-based STC-OF DM system with multiple antennas. The LDPC codes have the following advantages for the to minimize the loss in the effective length between the binary STC-OFDM system considered here. 1)the LdPC decoder LDPC code bits and the modulated StC code symbols, which usually has a lower computational complexity than the is caused by the MPSK(or MQAM)modulation turbo-code decoder. In addition to this, since the decoding As an example, consider a regular binary LDPC code with complexity of each iteration in an LDPC decoder is much column weight t =3, rate R= 1/2 and block-length n less than a turbo-code decoder, a finer resolution in the per- 1024, the minimum distance is around 100 [11]. The STC based formance-complexity tradeoff can be obtained by varying the on this LDPC code is configured with a QPsk modulator and maximum number of iterations. Moreover, the decoding of two transmitter antennas, therefore the effective length of this LDPC is highly parallelizable. 2)The minimum distance of LDPC-based STC is at least 25, which is more than enough binary LDPC codes increases linearly with the block length to satisfy the minimum effective length requirement for a two with probability close to 1 [11]. 3)It is easier to design a transmitter antenna (N=2)OFDM system in a Six-tap(L=6) competitive LDPC code with any block-length and any code frequency-selective fading channel. Together with its built-in rate, which makes it easier for the LDPC-based STC to scale random interleaver, this LdPC code can well satisfy the two according to different system requirements (e.g, different coding design principles mentioned earlier and therefore is an number of antennas or different information rate). 4)LDPC empirically good STC for the OFDM system considered in this codes do not typically show an error floor, which is suitable for paper. Since the minimum distance of binary LDPC codes in short-frame applications. 5)Due to the random generation of crease linearly with the block length, further performance im parity-check matrix(or equivalently the encoder matrix), the provement is possible by increasing the block length. Note that, coded bits have been effectively interleaved; therefore, no extra we do not claim the optimality of the proposed LDPC-based interleaver is needed STC, but rather, we argue that with its low decoding complexity, The transmitter structure of an LDPC-based STC-OFDM flexible scalability and high performance, the LDPC-based STO system is illustrated in Fig. 5. Denote the set of all possible is a promising coding technique for reliable high-speed data STC symbols, which is up to a constant Vy of the traditional communication in multiple-antenna OFDM systems with fre constellation, e.g., MPSK or MQAM (recall that the additive quency-and time-selective fading noise is assumed to have unit variance). The(PKlog2l e)c Data Burst sructure encoder into(NPKlog2 12) coded bits and then the binary As in a typical data communication scenario, communication LDPC coded bits are modulated into(NPK) STC symbols is carried out in a burst manner. a data burst is illustrated in by an MPSK (or MQAM)modulator. These(NPK STC Fig. 6. It spans(Pa+ 1)OFDM words, with the first OFDM into N streams; the (PK)STC symbols of each stream are oFD( ntaining known pilot symbols.The mbols, which correspond to an STC code word, are split word words contain q STC code words transmitted from one particular transmitter antenna at K subcarriers and over P adjacent OFDM slots. Note that, in such VI TURBO RECEIVER a bit-interleaved coded-modulation system proposed above, the In this section, we consider receiver design for the proposed built-in random interleaver of the LDPC codes is also helpful LDPC-based STC-OFDM system. Even with ideal CSl, the

80 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 50, NO. 1, JANUARY 2002 Fig. 4. Example of a parity-check matrix P for an (n; k; t; j)=(20; 5; 3; 4) regular LDPC code with code rate 1=4, block length n = 20, column weight t = 3, and row weight j = 4. Fig. 5. Transmitter structure of an LDPC-based STC-OFDM system with multiple antennas. The LDPC codes have the following advantages for the STC-OFDM system considered here. 1) the LDPC decoder usually has a lower computational complexity than the turbo-code decoder. In addition to this, since the decoding complexity of each iteration in an LDPC decoder is much less than a turbo-code decoder, a finer resolution in the per￾formance-complexity tradeoff can be obtained by varying the maximum number of iterations. Moreover, the decoding of LDPC is highly parallelizable. 2) The minimum distance of binary LDPC codes increases linearly with the block length with probability close to 1 [11]. 3) It is easier to design a competitive LDPC code with any block-length and any code rate, which makes it easier for the LDPC-based STC to scale according to different system requirements (e.g., different number of antennas or different information rate). 4) LDPC codes do not typically show an error floor, which is suitable for short-frame applications. 5) Due to the random generation of parity-check matrix (or equivalently the encoder matrix), the coded bits have been effectively interleaved; therefore, no extra interleaver is needed. The transmitter structure of an LDPC-based STC-OFDM system is illustrated in Fig. 5. Denote the set of all possible STC symbols, which is up to a constant of the traditional constellation, e.g., MPSK or MQAM (recall that the additive noise is assumed to have unit variance). The ( ) information bits are first encoded by a rate 1 LDPC encoder into ( ) coded bits and then the binary LDPC coded bits are modulated into ( ) STC symbols by an MPSK (or MQAM) modulator. These ( ) STC symbols, which correspond to an STC code word, are split into streams; the ( ) STC symbols of each stream are transmitted from one particular transmitter antenna at subcarriers and over adjacent OFDM slots. Note that, in such a bit-interleaved coded-modulation system proposed above, the built-in random interleaver of the LDPC codes is also helpful to minimize the loss in the effective length between the binary LDPC code bits and the modulated STC code symbols, which is caused by the MPSK (or MQAM) modulation. As an example, consider a regular binary LDPC code with column weight 3, rate 1 2 and block-length 1024, the minimum distance is around 100 [11]. The STC based on this LDPC code is configured with a QPSK modulator and two transmitter antennas, therefore the effective length of this LDPC-based STC is at least 25, which is more than enough to satisfy the minimum effective length requirement for a two transmitter antenna ( 2) OFDM system in a six-tap ( 6) frequency-selective fading channel. Together with its built-in random interleaver, this LDPC code can well satisfy the two coding design principles mentioned earlier and therefore is an empirically good STC for the OFDM system considered in this paper. Since the minimum distance of binary LDPC codes in￾crease linearly with the block length, further performance im￾provement is possible by increasing the block length. Note that, we do not claim the optimality of the proposed LDPC-based STC; but rather, we argue that with its low decoding complexity, flexible scalability and high performance, the LDPC-based STC is a promising coding technique for reliable high-speed data communication in multiple-antenna OFDM systems with fre￾quency- and time-selective fading. C. Data Burst Structure As in a typical data communication scenario, communication is carried out in a burst manner. A data burst is illustrated in Fig. 6. It spans ( 1) OFDM words, with the first OFDM word containing known pilot symbols. The remaining ( ) OFDM words contain STC code words. VI. TURBO RECEIVER In this section, we consider receiver design for the proposed LDPC-based STC-OFDM system. Even with ideal CSI, the op-

LU et aL: LDPC-BASED SPACE-TIME CODED OFDM SYSTEMS OVER CORRELATED FADING CHANNELS q STc words ·, P【q-1) P⊥1ot ne sTc word one src word Fig. 6. OFDM time slots allocation in data burst transmission. a data burst consists of(Pq+ 1)oFDM words, with the first OFDM word containing known pilot symbols. The remaining(Pq)OFDM words contain q STC code words Turbo⊥ terative detect⊥on& decodin Soft MAP-EM LDPC 工nf,Biti Dem。d Decoder FaT e⊥s⊥on Pilot 工ni七ia1 MAP-EM Fig. 7. The turbo receiver structure, which employs a MAP-EM demodulator and a soft LDPC decoder, for multiple-antenna LDPC-based STC-OFDM systems timal decoding algorithm for this system has an exponential computes as output the extrinsic LLRs of the LDPC coded complexity. Hence the near-optimal turbo receiver based on the bits, as well as the hard decisions of the information bits at urbo principle [15] becomes attractive. As a standard proce- the last turbo iteration. It is assumed that the q Stc words in dure, such as in [16], in order to demodulate each STC code a data burst are independently encoded. Therefore, each STC word, the turbo receiver consists of two stages, the soft demod- word (consisting of POFDM words)is decoded independently ulator and the soff LDPC decoder and the so-called"extrinsic" by turbo processing. We next describe each component of the information is iteratively exchanged between these two stages receiver in Fig. 7 to successively improve the receiver performance However, in practice, the CsI must be estimated by the re- B. MAP-EM Demodulator ceiver. In the rest of this section, we develop a novel turbo re- 2For notational simplicity, here we consider an LDPC-based ceiver for unknown fast fading channels STC- OFDM system with two transmitter antennas and one A. Receiver Structure receiver antenna. The results can be easily extended to a system with N transmitter antennas and M receiver antennas. Note The proposed turbo receiver for the LDPC-based STC- that, for the purpose of performance analysis, the hi, (p) de- OFDM system is illustrated in Fig. 7. It consists of a soft fined in(4)only contains the time responses of L nonzero taps maximum a posteriori expectation-maximization (MAP-EM) whereas for the purpose of receiver design, especially when the demodulator and a soft LDPC decoder, both of which are CSI is not available, the hi, (p)needs to be redefined to contain iterative devices themselves. The soft MAP-EM demodulator the time responses of all the taps within the maximum mul tennas and the extrinsic log likelihood ratios (llrs) of the tipath spread. That is, h,0)全[小… LDPC coded bits 2) [cf(26)](which is fed back by the with L=[mKA+11> Ly and Tm being the maximum soft LDPC decoder). It computes as output the extrinsic a multipath spread; and wr(k) is correspondingly redefined posteriori LLRs of the ldpC coded bits i! fcf, (26).(As w, (k=Teo, e-2k(/-1/k]". The received signal MAP-EM demodulator will be specifically discussed later in during one data burst can be written this section. ) The soft LDPC decoder takes as input the llrs of the ldpc coded bits from the map-em demodulator and yp]=xpWhp+zp], p=0,1,., Pq

LU et al.: LDPC-BASED SPACE–TIME CODED OFDM SYSTEMS OVER CORRELATED FADING CHANNELS 81 Fig. 6. OFDM time slots allocation in data burst transmission. A data burst consists of (P q + 1) OFDM words, with the first OFDM word containing known pilot symbols. The remaining (P q) OFDM words contain q STC code words. Fig. 7. The turbo receiver structure, which employs a MAP-EM demodulator and a soft LDPC decoder, for multiple-antenna LDPC-based STC-OFDM systems in unknown fading channels. timal decoding algorithm for this system has an exponential complexity. Hence the near-optimal turbo receiver based on the turbo principle [15] becomes attractive. As a standard proce￾dure, such as in [16], in order to demodulate each STC code word, the turbo receiver consists of two stages, the soft demod￾ulator and the soft LDPC decoder and the so-called “extrinsic” information is iteratively exchanged between these two stages to successively improve the receiver performance. However, in practice, the CSI must be estimated by the re￾ceiver. In the rest of this section, we develop a novel turbo re￾ceiver for unknown fast fading channels. A. Receiver Structure The proposed turbo receiver for the LDPC-based STC￾OFDM system is illustrated in Fig. 7. It consists of a soft maximum a posteriori expectation-maximization (MAP-EM) demodulator and a soft LDPC decoder, both of which are iterative devices themselves. The soft MAP-EM demodulator takes as input the FFT of the received signals from receiver antennas and the extrinsic log likelihood ratios (LLRs) of the LDPC coded bits [cf. (26)] (which is fed back by the soft LDPC decoder). It computes as output the extrinsic a posteriori LLRs of the LDPC coded bits [cf. (26)]. (As an important issue in the EM algorithm, the initialization of the MAP-EM demodulator will be specifically discussed later in this section.) The soft LDPC decoder takes as input the LLRs of the LDPC coded bits from the MAP-EM demodulator and computes as output the extrinsic LLRs of the LDPC coded bits, as well as the hard decisions of the information bits at the last turbo iteration. It is assumed that the STC words in a data burst are independently encoded. Therefore, each STC word (consisting of OFDM words) is decoded independently by turbo processing. We next describe each component of the receiver in Fig. 7. B. MAP-EM Demodulator 2For notational simplicity, here we consider an LDPC-based STC- OFDM system with two transmitter antennas and one receiver antenna. The results can be easily extended to a system with transmitter antennas and receiver antennas. Note that, for the purpose of performance analysis, the de￾fined in (4) only contains the time responses of nonzero taps; whereas for the purpose of receiver design, especially when the CSI is not available, the needs to be redefined to contain the time responses of all the taps within the maximum mul￾tipath spread. That is, , with 1 and being the maximum multipath spread; and is correspondingly redefined as . The received signal during one data burst can be written as

EEE TRANSACTIONS ON COMMUNICATIONS. VOL- 50. NO. 1 JANUARY 2002 XpFEX1P1. x2pllKx(2k XwHXOHXOWEH X, diag=[, 0], a Pp,]. p, K-1JKXK where e and E denote respectively the covariance ma. 1.2 trix of the ambient white Gaussian noise z and channel W三 diaglWf, wrI2x)×(x2) ns in Section Il both of them are diagonal matrices as Z=E(zz)=Iand W三四r(O),wf(1)…,f(K-1) E(M)=dig21,,on3x…, hp FMi(p), hi 2 (p)2a/>x (17,, is the average power of the lth tap related with the jth where yp] and alp] are K-sized vectors which contain respec- is zero. Assuming that Eh is known(or measured with the aid K subcarriers and at the pth time slot; the diagonal elements of defined as the pseudo inverse of >h as/'23 72,L」1s xlp] are the K STC symbols transmitted from the jth trans- 2t≠0 mitter antenna and at the pth time slot. 7,t Lf,j=1,2.(2) Without CSI, the maximum a posteriori(MAP) detection problem is written as Using(17)and(21),Q(XXO)is computed as shown in(23) XTp]=argmax logp(XpllyIp)), p=1, 2, .., Pq.(18)at the bottom of the next page, where [WEnw Iaij) denotes (Recall that X[o] contains pilot symbols. The optimal solution Next, based on(23), the M-step proceeds as follows to(18)is of prohibitive complexity. We next propose to use the expectation-maximization(EM)algorithm 31] to solve(18) ri+ y rargmax e (xIx)+log P(X) The basic idea of the MAP-EM algorithm is to solve(18) teratively according to the following two steps(for notational -argan x[) og P(EkD convenience, we temporarily drop the time index p, with the 人=0 understanding that the MAP-EM algorithm discussed below is applied to each OFDM word in the data burst) K-1 argmin dalk)-log P(akI 1. E-step: Compute Q(xIxo) k=0 =E[ogp(yX, m)lv,x(Oy (19 or 2(+ [v =arg zin a (e= k-J)-log P(=k 2. M-step: Solve X(i+-1 augmaxQ(xx)+logP(x)(20) where(24)follows from the assumption that X contains inde- pendent symbols. It is seen from(25)that the M-step can be where xo denotes hard decisions of the data symbols at the ith decoupled into K independent minimization problems, each of EM iteration and P(X represents the a priori probability of x which can be solved by enumeration over all possible T E Q2x Q which is fed back by the LDPC decoder from the previous turbo total complexity of the maximization step is O(KQ2).Note that, unlike in [19], here the maximization in the M-ster is nondecreasing and under regularity conditions the EM algo- ried out without taking the LDPC coding constraints into con- rithm converges to a local stationary point [321 iderations, i.e., the symbols in X are treated as uncoded sym In the E-step, the expectation is taken with respect to the bols. The LDPC coding structure is exploited by the turbo iter- hidden"channel response h conditioned on y and X It ation as well as the ldpc decoder is easily seen that, conditioned on y and x h is complex Within each turbo iteration, the above E-step and M-step are iterated I times. At the end of the Ith EM iteration, the ex h(y xo)Ne(h, En) trinsic a posteriori LLRs of the LDPC code bits are computed and then fed to the soft ldpc decoder. at each ofdm sub with h(wxoHz-xOw+Eh) carrier, two transmitter antennas transmit two STC symbols, which correspond to(2 log2 2)LDPC code bits Based on(25 ), after I EM iterations, the extrinsic a posteriori llr of the jth WHxOHxow+Eh)WHxaHy, d() is computed at the output of the map-em demodulator L,eEh-wHXOHx-Ixow+Er) as follows: WHx()z-X(wΣ =kg()=+-P((

82 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 50, NO. 1, JANUARY 2002 with (17) where and are -sized vectors which contain respec￾tively the received signals and the ambient Gaussian noise at all subcarriers and at the th time slot; the diagonal elements of are the STC symbols transmitted from the th trans￾mitter antenna and at the th time slot. Without CSI, the maximum a posteriori (MAP) detection problem is written as 1 2 (18) (Recall that 0 contains pilot symbols.) The optimal solution to (18) is of prohibitive complexity. We next propose to use the expectation-maximization (EM) algorithm [31] to solve (18). The basic idea of the MAP-EM algorithm is to solve (18) iteratively according to the following two steps (for notational convenience, we temporarily drop the time index , with the understanding that the MAP-EM algorithm discussed below is applied to each OFDM word in the data burst): E-step: Compute (19) M-step: Solve (20) where denotes hard decisions of the data symbols at the th EM iteration and represents the a priori probability of , which is fed back by the LDPC decoder from the previous turbo iteration. It is known that the likelihood function is nondecreasing and under regularity conditions the EM algo￾rithm converges to a local stationary point [32]. In the E-step, the expectation is taken with respect to the “hidden” channel response conditioned on and . It is easily seen that, conditioned on and , is complex Gaussian distributed as with (21) where and denote respectively the covariance ma￾trix of the ambient white Gaussian noise and channel responses . According to the assumptions in Section II, both of them are diagonal matrices as and , where is the average power of the th tap related with the th transmitter antenna; 0 if the channel response at this tap is zero. Assuming that is known (or measured with the aid of pilot symbols), is defined as the pseudo inverse of as 1 0 0 0 1 1 2 (22) Using (17) and (21), is computed as shown in (23), at the bottom of the next page, where denotes the ( )th element of the matrix . Next, based on (23), the M-step proceeds as follows: (24) or (25) where (24) follows from the assumption that contains inde￾pendent symbols. It is seen from (25) that the M-step can be decoupled into independent minimization problems, each of which can be solved by enumeration over all possible (recall that denotes the set of all STC symbols). Hence, the total complexity of the maximization step is . Note that, unlike in [19], here the maximization in the M-step is car￾ried out without taking the LDPC coding constraints into con￾siderations, i.e., the symbols in are treated as uncoded sym￾bols. The LDPC coding structure is exploited by the turbo iter￾ation as well as the LDPC decoder. Within each turbo iteration, the above E-step and M-step are iterated times. At the end of the th EM iteration, the ex￾trinsic a posteriori LLRs of the LDPC code bits are computed and then fed to the soft LDPC decoder. At each OFDM sub￾carrier, two transmitter antennas transmit two STC symbols, which correspond to (2 ) LDPC code bits. Based on (25), after EM iterations, the extrinsic a posteriori LLR of the th ( 1 2 ) LDPC code bit at the th subcarrier is computed at the output of the MAP-EM demodulator as follows:

LU et a: LDPC-BASED SPACE-TIME CODED OFDM SYSTEMS OVER CORRELATED FADING CHANNELS ∑cPx]=xyl channel estimates. The procedure is listed in Table I. In Table I lo [(k Freg-filter denotes either the least-square estimator (Lse) or the minimum mean-square-error estimator(MMSE)as ∑c-xp|-g(x)+ log P(ac LSE:Freq-filter y, x=(w xXw-l ∑xep|-dx)+logP(x X WHXH MMSE: Freqfilter(3. x=(wHx'xw+ET APEd(k) (26) x wx y (27) where C; is the set of r for which the jth LDPC coded bit is where X represents either the pilot symbols or x( provided "+1" and C is similarly defined. The extrinsic a priori LLRs by the MAP-EM demodulator Comparing these two estimators Aep di(k)I i k are provided by the soft LDPC decoder at the the lse does not need any statistical information of h, but the previous turbo iteration(where P denotes the previous turbo it- MMSE offers better performance in terms of mean-square-error eration; at the first turbo iteration, A2[d(k)]=O). Finally, (MSE). Hence, in the pilot slot, the LSE is used to estimate chan- he extrinsic a posteriori LLRS Aid(k li k are sent to the nels and to measure >h, and in the rest of data slots the MMSE soft LDPC decoder, which in turn iteratively computes the ex- is used. In Table L, Temp-filter denotes the temporal filter trinsic LLRs A2[d (k)]i k and then feeds them back to the which is used to further exploit the time-domain correlation of MAP-EM demodulator and thus completes one turbo iteration. the channel At the end of the last turbo iteration, hard decisions of the infor mation bits are output by the LDPC decoder. For details of the Temp-filterhlp-11 hp-21 soft LDPC decoder, see [111 全∑a- C. Initialization of MAP-EM Demodulator The performance of the MAP-EM demodulator(and hence where hp-j, j= 1, ..,t, is computed from(**)[cf. the overall receiver)is closely related to the quality of the initial Table I]; laj -I denotes the coefficients of an L-length yalue ofXoplfcf. (19)]. At each turbo iteration, x()[p] needs (l s Pa) temporal filter, which can be obtained by solving to be specified to initialize the MAP-EM demodulator. Except the Wiener equation or from the robust design as in [33 and for the first turbo iteration, xop] is simply taken as x(p| [34]. From the above discussions, it is seen that the compu- given by(24)from the previous turbo iteration. We next discuss tation involved in initializing Xo[] mainly consists of the the procedure for computing Xo)lp] at the first turbo iteration. ML detection of xo[p] in(*)and the estimation of h[p] in The initial estimate of X(o] is based on the method pro-(*). In general, for an STC-OFDM system with parameters osed in [33] and[34], which makes use of pilot symbols and (N,M,K, Lf), the total complexity in initializing x Lp] is decision-feedback as well as spatial and temporal filtering for O[(K )+M(NL1 Q(xixo)=-Eml(3. xD)lly-XWll?)+const Eml(v x o)(y-XWh)(XWh-XWh)I)+const -xP(x)(④a-)wxxw+cm Iy-XWh 12-traceXwsnwHx)+const. ∑{[x+21时}+om with a[k=[-1[k], x2[k-J52x W() (k) wF(h) WEnW]o+1 +1) [WE,wHK K+k+1,+ WWk+k++1)W立W1k+k+15++)2×2

LU et al.: LDPC-BASED SPACE–TIME CODED OFDM SYSTEMS OVER CORRELATED FADING CHANNELS 83 (26) where is the set of for which the th LDPC coded bit is “ ” and is similarly defined. The extrinsic a priori LLRs are provided by the soft LDPC decoder at the previous turbo iteration (where denotes the previous turbo it￾eration; at the first turbo iteration, 0). Finally, the extrinsic a posteriori LLRs are sent to the soft LDPC decoder, which in turn iteratively computes the ex￾trinsic LLRs and then feeds them back to the MAP-EM demodulator and thus completes one turbo iteration. At the end of the last turbo iteration, hard decisions of the infor￾mation bits are output by the LDPC decoder. For details of the soft LDPC decoder, see [11]. C. Initialization of MAP-EM Demodulator The performance of the MAP-EM demodulator (and hence the overall receiver) is closely related to the quality of the initial value of [cf. (19)]. At each turbo iteration, needs to be specified to initialize the MAP-EM demodulator. Except for the first turbo iteration, is simply taken as given by (24) from the previous turbo iteration. We next discuss the procedure for computing at the first turbo iteration. The initial estimate of is based on the method pro￾posed in [33] and [34], which makes use of pilot symbols and decision-feedback as well as spatial and temporal filtering for channel estimates. The procedure is listed in Table I. In Table I, - denotes either the least-square estimator (LSE) or the minimum mean-square-error estimator (MMSE) as LSE: - MMSE: - (27) where represents either the pilot symbols or provided by the MAP-EM demodulator. Comparing these two estimators, the LSE does not need any statistical information of , but the MMSE offers better performance in terms of mean-square-error (MSE). Hence, in the pilot slot, the LSE is used to estimate chan￾nels and to measure , and in the rest of data slots the MMSE is used. In Table I, - denotes the temporal filter, which is used to further exploit the time-domain correlation of the channel - (28) where 1 is computed from ( ) [cf. Table I]; denotes the coefficients of an -length ( ) temporal filter, which can be obtained by solving the Wiener equation or from the robust design as in [33] and [34]. From the above discussions, it is seen that the compu￾tation involved in initializing mainly consists of the ML detection of in ( ) and the estimation of in ( ). In general, for an STC-OFDM system with parameters ( ), the total complexity in initializing is . with (23)