《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)