Capacity Considerations for STC-OFDM Systems

Capacity Considerations for STC-OFDM Systems

System Model

We consider an STC-OFDM system with Q subcarriers, N transmitter antennas, and M receiver antennas, signaling through frequency- and time-selective fading channels, as illustrated in Fig. 10.22. Each STC code word spans P adjacent OFDM words, and each OFDM word consists of (NQ) STC symbols, transmitted simultaneously during one time slot. Each STC symbol is transmitted on a particular OFDM subcarrier and a particular transmitter antenna.

Figure 10.22. 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; on a particular subcarrier and in a particular time slot, STC symbols are transmitted from N transmitter antennas and received by M receiver antennas.

As in Section 10.4, 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 M receiver antennas. After matched filtering and sampling, the DFT is applied to the received discrete-time signal to obtain

Equation 10.63

where is the matrix of complex channel frequency responses at the kth subcarrier and at the pth time slot, which is explained below; and are, respectively, the transmitted and received signals at the kth subcarrier and at the pth time slot; is the ambient noise, which is circularly symmetric complex Gaussian with unit variance.

Consider the channel response between the jth transmitter antenna and the ith receiver antenna. As before, the time-domain channel impulse response can be modeled as a tapped-delay line. With only the nonzero taps considered, it can be expressed as

Equation 10.64

where d(·) is the Dirac delta function; L_f denotes the number of nonzero taps; a_i,j(l; t) is the complex amplitude of the lth nonzero tap, whose delay is n_l/(QD_f), where n_l is an integer and D_f is the tone spacing of the OFDM system. In mobile channels, for the particular (i,j)th antenna pair, the time-varying tap coefficients a_i,j(l;t) can be modeled as wide-sense stationary random processes with uncorrelated scattering (WSSUS) and with bandlimited Doppler power spectrum [396]. For the signal model in (10.63), we need only consider the time responses of a_i,j(l; t) within the time interval t [0, PT], where T is the total time duration of one OFDM word plus its cyclic extension and PT is the total time involved in transmitting P adjacent OFDM words. Following [568], for the particular lth tap of the (i, j)th antenna pair, the dimension of the band- and time-limited random process a_i,j(l;t), t [0, PT] (defined as the number of significant eigenvalues in the Karhunen-Loeve expansion of this random process) is approximately equal to where f_d is the maximum Doppler frequency. Hence, ignoring edge effects, the time response of a_i,j(l;t) can be expressed in terms of the Fourier expansion as

Equation 10.65

where {b_i,j(l, n)}_n is a set of independent circularly symmetric complex Gaussian random variables, indexed by n.

For OFDM systems with proper cyclic extension and sample timing, with tolerable leakage, the channel frequency response between the jth transmitter antenna and the ith receiver antenna at the pth time slot and the kth subcarrier, which is exactly the (i,j)th element of H[p, k] in (10.63), can be expressed as [506]

Equation 10.66

where is the L_f-sized vector containing the time responses of all the nonzero taps, and contains the corresponding DFT coefficients.

Using (10.65), a_i,j(l; pT) can be simplified to

Equation 10.67

where is an L_t-length vector and contains the corresponding inverse DFT coefficients. Substituting (10.67) into (10.66), we have

Equation 10.68

with

From (10.68) it is seen that due to the close spacing of OFDM subcarriers and the limited Doppler frequency, for a specific antenna pair (i,j) the channel responses {H_i,j[p, k]}_p,k are different transformations [specified by w_t(p) and w_f(k)] of the same random vector g_i,j, and hence they are correlated in both frequency and time.

Channel Capacity

In this section we consider the channel capacity of the system described above. Assuming that the channel state information is known only at the receiver and the transmitter power is constrained as trace {E[x[p, k]x^H[p, k]]} g, the instantaneous channel capacity of this system, which is defined as the mutual information conditioned on the correlated fading channel values , can be computed as [39, 363]

Equation 10.69

where , and l_i(p, k) is the ith nonzero eigenvalue of the nonnegative definite Hermitian matrix H[p, k]H^H[p, k]. The maximization of is achieved when {x[p, k]}_p,k consists of independent circularly symmetric complex Gaussian random variables with identical variances [39, 363]. (When the CSI is known to both the transmitter and the receiver, the instantaneous channel capacity is maximized by "water filling" [40].) The ergodic channel capacity is defined as . In the system considered, the concept of ergodic channel capacity I(g) is of less interest, because the fading processes are not ergodic, due to the limited number of antennas and the limited L_f and L_t.

Since is a random variable, whose statistics are determined jointly by (g, N, M) and the characteristics of correlated fading channels, we turn to another important concept—outage capacity, which is closely related to the code word error probability, as averaged over the random coding ensemble and over all channel realizations [39]. The outage probability is defined as the probability that the channel cannot support a given information rate R,

Equation 10.70

Since it is difficult to get an analytical expression for (10.70), we resort to Monte Carlo integration for its numerical evaluation.

In the following, we give some numerical results of the outage probability in (10.70) obtained by Monte Carlo integration. For simplicity, we assume that all elements in {g_i,j}_i,j have the same variance. Define the selective-fading diversity order L as the product of the number of nonzero delay taps L_f and the dimension of Doppler fading process . The following observations can be made from the numerical evaluations of (10.70).

1. From Figs. 10.23 and 10.24, it is seen that at a practical outage probability (e.g., P_out = 1%), for fixed (N, M, g) the highest achievable information rate increases as the selective-fading diversity order L increases, but the increase diminishes as L becomes larger. Eventually, as L , the highest achievable information rate converges to the ergodic capacity. [Note that the ergodic capacity is the area above each curve in the figure:

   Figure 10.23. Outage probability versus information rate in a correlated fading OFDM system with M = 1, Q = 256, P = 1, SNR = 20 dB, where dashed lines represent a system with one transmitter antenna (N = 1) and solid lines represent a 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- and time-nonselective with L_t = 1, L = L_f = {1, 2, 3, 6}.

   

   Figure 10.24. Outage probability versus information rate in a correlated fading OFDM system with N = 2, M = 1, Q = 256, P = 10, SNR = 20 dB. Dashed lines represent frequency-selective and time-nonselective channels with L_t = 1, L = L_f = {2, 6, 10}. Dotted lines represent frequency- and time-selective channels with L_f = 2, L = 2L_t = {2, 6, 10}. Note that for the same L, the dashed and dotted lines overlap, which shows the equivalent impacts of frequency- and time-selective fading on the outage probability.

   

2. Figure 10.24 compares the effects of the frequency-selectivity order L_f and the time-selectivity order L_t on the outage capacity. It shows that the frequency and time selectivity are essentially equivalent in terms of their effects on the outage capacity. In other words, the selective-fading diversity order L = L_fL_t ultimately affects the outage capacity.

3. From Fig. 10.23 it is seen that as the area above each curve, the ergodic channel capacity is invariant to the selective-fading diversity order L (which is the key parameter in determining the correlation characteristics of the fading channels) and it is determined only by the spatial diversity order (N, M) and the transmitted signal power g [122, 478]. Moreover, it is seen that both the outage and ergodic capacities can be increased by fixing the number of receiver antennas and increasing the number of transmitter antennas (or vice versa) (e.g., by fixing M = 1 and letting N , the ergodic capacity converges to the capacity of AWGN channels [353]).

In summary, we have seen the different effects of two diversity resources, spatial diversity and selective-fading diversity, on the channel capacity of a multiple-antenna correlated fading OFDM system. Increasing the spatial diversity order (i.e., N, M) can always bring capacity (outage capacity and/or ergodic capacity) increase at the expense of extra physical costs. By contrast, the selective-fading diversity is a free resource, but its effect on improving the channel capacity becomes less as L becomes larger. Since both diversity resources can improve 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.

Pairwise Error Probability

To obtain further insight into coding design, it is of interest to analyze the pairwise error probability of this system with coded modulation.

With perfect CSI at the receiver, the maximum likelihood decision rule for the signal model (10.63) is given by

Equation 10.71

where the minimization is over all possible STC code words . Assuming equal transmitted power at all transmitter antennas, using the Chernoff bound [396], the PEP of transmitting x and deciding in favor of another code word at the decoder is upper bounded by

Equation 10.72

where g is the total signal power transmitted from all N transmitted antennas. (Recall that the noise at each receiver antenna is assumed to have unit variance.) Using (10.66)-(10.68), is given by

Equation 10.73

with

Equation 10.74

Equation 10.75

In (10.74), is a rank-one matrix, which is equal to a zero matrix if the entries of code words x and corresponding to the kth subcarrier and pth time slot are the same. Let D denote the number of instances when ; similar to [438], D_eff, which is the minimum D over every two possible code word pair, is called the effective length of the code. Denote ; it is easily seen that . Since w_f(k) and w_t(p) vary with different multipath delay profiles and Doppler power spectrum shapes, the matrix D also varies with different channel environments. However, since D is a nonnegative definite Hermitian matrix, by an eigendecomposition, it can be written as

Equation 10.76

where V is a unitary matrix and , with being the positive eigenvalues of D. Moreover, by assumption, all the (N M L) elements of {g_i,j}_i,j are i.i.d. circularly symmetric complex Gaussian with zero means. So (10.72) can be rewritten as

Equation 10.77

where is the jth element of . Since V 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 (10.77) over the Rayleigh pdf (probability density function), the PEP bound for a multiple-antenna STC-OFDM system over correlated fading channels can finally be written as

Equation 10.78

It is seen from (10.78) that the highest possible diversity order the STC-OFDM system can provide is N M L: the product of the number of transmitter antennas, the number of receiver antennas, and the selective-fading diversity order of the channels. In other words, the attractiveness of the STC-OFDM system lies in its ability to exploit all the available diversity resources. However, note that although in the analysis of PEP the three parameters (N, M, L) appear equivalent in improving the system performance, they actually play different roles from the capacity viewpoint, as indicated above.