Turbo Receiver

Turbo Receiver

We next consider receiver design for the proposed LDPC-based STC-OFDM system. Even with ideal CSI, the optimal decoding algorithm for this system has exponential complexity. Hence we apply the turbo receiver structure. As a standard procedure, to demodulate each STC code word, the turbo receiver consists of two stages, the soft demodulator and the soft LDPC decoder, and the extrinsic information is iteratively exchanged between these two stages to successively improve the receiver performance.

However, in practice, the channel state information must be estimated by the receiver. In the following we discuss a turbo receiver for unknown fast-fading channels based on the MAP-EM algorithm. The turbo receiver for the LDPC-based STC-OFDM system is illustrated in Fig. 10.28. 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 M receiver antennas, and the extrinsic log-likelihood ratios of the LDPC coded bits {l₂} [cf. (10.62)] (which are fed back by the soft LDPC decoder). It computes as output the extrinsic a posteriori LLRs of the LDPC coded bits [cf. (10.62)]. (As an important issue in the EM algorithm, the initialization of the MAP-EM demodulator will be 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 q STC words in a data burst are encoded independently. Therefore, each STC word (consisting of P OFDM words) is decoded independently by turbo processing. We next describe each component of the receiver shown in Fig. 10.28.

Figure 10.28. 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.

MAP-EM Demodulator

Here we apply the MAP-EM algorithm discussed in Section 10.4.3. For notational simplicity, here we consider an LDPC-based STC-OFDM system with two transmitter antennas and one receiver antenna. The results can easily be extended to a system with N transmitter antennas and M receiver antennas. Note that for the purpose of performance analysis, the h_i,j(p) defined in (10.66) contains only the time responses of L_f nonzero taps; whereas for the purpose of receiver design, especially when channel state information (CSI) is not available, the h_i,j(p) needs to be redefined to contain the time responses of all the taps within the maximum multipath spread. That is, , with and t_m being the maximum multipath spread; and w_f(k) is correspondingly redefined as . The received signal during one data burst can be written as

Equation 10.82

with

where y[p] and z[p] are Q-sized vectors which contain, respectively, the received signals and the ambient Gaussian noise at all Q subcarriers and at the pth time slot; the diagonal elements of X_j[p] are the Q STC symbols transmitted from the jth transmitter antenna and at the pth time slot.

Without CSI, the MAP detection problem is written as,

Equation 10.83

(Recall that X[0] contains pilot symbols.) As in Section 10.4, we use the EM algorithm to solve (10.83).

In the E-step, the expectation is taken with respect to the "hidden" channel response h conditioned on y and X⁽ⁱ⁾. It is easily seen that conditioned on y and X⁽ⁱ⁾, h has a complex Gaussian distribution given by

Equation 10.84

with

where S_z and S_h denote, respectively, the covariance matrix of the ambient white Gaussian noise z and channel response h. As before, by assumption, both of them are diagonal matrices as and , where is the average power of the lth tap related with the jth transmitter antenna; if the channel response at this tap is zero. Assuming that S_h is known (e.g., measured with the aid of pilot symbols), is defined as the pseudo-inverse of S_h as

Equation 10.85

Using (10.82) and (10.84), Q(X|X⁽ⁱ⁾) is computed as

Equation 10.86

with

where denotes the (i,j)th element of the matrix .

Next, based on (10.86), the M-step proceeds as follows:

Equation 10.87

or

Equation 10.88

where (10.87) follows from the assumption that X contains independent symbols. It is seen from (10.88) that the M-step can be decoupled into Q independent minimization problems, each of which can be solved by enumeration over all possible x W x W. (Recall that W denotes the set of all STC symbols.) Hence the total complexity of the maximization step is .

Within each turbo iteration, the E-step and M-step above are iterated I times. At the end of the Ith EM iteration, the extrinsic a posteriori LLRs of the LDPC code bits are computed and then fed to the soft LDPC decoder. At each OFDM subcarrier, two transmitter antennas transmit two STC symbols, which correspond to 2 log₂ |W| LDPC code bits. Based on (10.88), after I EM iterations, the extrinsic a posteriori LLR of the jth (j = 1, ..., 2 log₂ |W|) LDPC code bit at the kth subcarrier d^j[k] is computed at the output of the MAP-EM demodulator as follows:

Equation 10.89

where is the set of x for which the jth LDPC coded bit is "+1" and is defined similarly. The extrinsic a priori LLRs {l₂(d^j[k])}_j,k are provided by the soft LDPC decoder at the previous turbo iteration. Finally, the extrinsic a posteriori LLRs {l₁(d^j[k])}_j,k are sent to the soft LDPC decoder, which in turn iteratively computes the extrinsic LLRs {l₂(d^j[k])}_j,k 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 information bits are output by the LDPC decoder.

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 X⁽⁰⁾[p] [cf. (10.44)]. At each turbo iteration, X⁽⁰⁾[p] needs to be specified to initialize the MAP-EM demodulator. Except for the first turbo iteration, X⁽⁰⁾[p] is simply taken as X^(I)[p] given by (10.87) from the previous turbo iteration. We next discuss the procedure for computing X⁽⁰⁾[p] at the first turbo iteration.

The initial estimate of X⁽⁰⁾[p] is based on the method proposed in [260, 263], 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 10.2, where Freq-filter denotes either the least-squares estimator (LSE) or the MMSE estimator as

Equation 10.90

where X represents either the pilot symbols or X^(I) provided by the MAP-EM demodulator. Comparing these two estimators, the LSE does not need any statistical information of h, but the MMSE offers better performance in terms of mean-square error (MSE). Hence, in the pilot slot, the LSE is used to estimate channels and to measure S_h; and in the remaining data slots, the MMSE is used. In Table 10.2, Temp-filter denotes the temporal filter, which is used to further exploit the time-domain correlation of the channel:

Equation 10.91

where , is computed from (**) (cf. Table 10.1); denotes the coefficients of an I-length (I Pq) temporal filter, which can be obtained by solving the Wiener equation or from robust design as in [260, 263]. From the discussion above, it is seen that the computation involved in initializing X⁽⁰⁾[p] consists mainly of the ML detection of X⁽⁰⁾[p] in (*) and the estimation of in (**). In general, for an STC-OFDM system with parameters , the total complexity in initializing X⁽⁰⁾[p] is .

Simulation Examples

In this section we provide simulation results to illustrate the performance of the proposed LDPC-based STC-OFDM system in frequency- and time-selective fading channels. The correlated fading processes are generated by using the methods in [180]. In the following simulations the available bandwidth is 1 MHz and is divided into 256 subcarriers. These correspond to a subcarrier symbol rate of 3.9 kHz and OFDM word duration of 256 ms. In each OFDM word, a guard interval of 40 ms is added to combat the effect of intersymbol interference; hence T = 296 ms. For all simulations, 512 information bits are transmitted from 256 subcarriers at each OFDM slot, and therefore the information rate is 2 x 256/296 = 1.73 bits/s per hertz. Unless otherwise specified, all the LDPC codes used in simulations are regular LDPC codes with column weight t = 3 in the parity-check matrices and with appropriate block lengths and code rates. The modulator uses the QPSK constellation. Simulation results are shown in terms of the OFDM word-error rate (WER) versus the SNR g.

Table 10.2. Procedure for Computing X⁽⁰⁾[p] for the MAP-EM Demodulator (at the First Turbo Iteration)

Performance with Ideal CSI Figures 10.29 and 10.30 show the performance of multiple-antenna (N transmitter antennas and one receiver antenna) LDPC-based STC-OFDM systems by using turbo detection and decoding with ideal CSI. Performance is compared for systems with different fading profiles and different numbers of transmitter antennas. Namely, Ch1 denotes a channel with a single tap at 0 ms, Ch2a denotes a channel with two equal-power taps at 0 and 5 ms, Ch2b denotes a channel with two equal-power taps at 0 and 40 ms, and Ch6a denotes a channel with six equal-power taps equally spaced from 0 to 40 ms. Suffix N2 denotes a system with two transmitter antennas (N = 2), and similarly for N3; suffix P1 denotes that each STC code word spans one OFDM slot (P = 1), and similarly for P5 and P10. Unless otherwise specified, all the STC-OFDM systems are assumed to use two transmitter antennas (N = 2) and each STC code word spans one OFDM slot (P = 1).

First, Fig. 10.29 shows the performance of the LDPC-based STC-OFDM system in frequency- and time-nonselective channels. The dash-dotted curves represent the performance after the first turbo iteration; and the solid curves represent the performance after the fifth iteration. It is seen that the receiver performance is improved significantly through turbo iteration. During each turbo iteration, in the LDPC decoder, the maximum number of iterations is 30; and as observed in simulations, the average number of iterations needed in LDPC decoding is less than 10 when WER is less than 10^-2. Compared with the conventional trellis-based STC-OFDM system (see figures in [8]), the LDPC-based STC-OFDM system improves performance significantly (e.g., there is around a 5-dB performance improvement in Ch2a/Ch2b channels and even more improvement in Ch6a channels). Moreover, due to the inherent interleaving in the LDPC encoder, the proposed LDPC-based STC narrows the performance difference between Ch2a and Ch2b channels (essentially the outage capacity of these two channels are the same). As the selective-fading diversity order L increases from Ch1 to Ch6a, LDPC-based STC can efficiently take advantage of the available diversity resources and hence can significantly improve the system performance. Moreover, in a highly frequency-selective channel, Ch6a, the LDPC-based STC performs only 3.0 dB away from the outage capacity of this channel (at a high information rate, 1.73 bits/s per hertz) at WER of 2 x 10^-4.

Figure 10.29. Word-error rate of an LDPC-based STC-OFDM system with multiple antennas (N = {2,3}, M = 1) in frequency- and time-nonselective fading channels, with ideal CSI.

Next, Fig. 10.30 shows the performance of the LDPC-based STC-OFDM system in frequency- and time-selective fading channels. The maximum Doppler frequency is 200 Hz (i.e., the normalized Doppler frequency is f_dT = 0.059). Again, it is seen that the performance of the system improves as the selective-fading diversity order L (including both frequency and time selectivities) increases.

Figure 10.30. Word-error rate of an LDPC-based STC-OFDM system with multiple antennas (N = 2, M = 1) in frequency- and time-selective fading channels, with ideal CSI.

Finally, Fig. 10.29 also compares the performance of LDPC-based STC-OFDM systems with the same multipath delay profiles (Ch2a) but with different numbers of transmitter antennas (N = 2 or N = 3). Since Ch2bN3 has a larger outage capacity than Ch2bN2, it is seen that at medium to high SNRs, Ch2bN3 starts to perform better than Ch2bN2 with a steeper slope, which shows that the LDPC-based STC can be flexibly scaled according to a different number of transmitter antennas and can still improve the performance by exploiting the increased spatial diversity, especially at low WER (which is attractive in data communication applications).

Performance with Unknown CSI In the following simulations, the receiver performance with unknown CSI is shown. The system transmits in a burst manner, as illustrated in Fig. 10.13. Each data burst includes 10 OFDM words (q = 9, P = 1), the first OFDM word contains the pilot symbols, and the remaining nine OFDM words contain the information data symbols. Simulations are carried out in two-tap (two equal-power taps at 0 and 1 ms) frequency- and time-selective fading channels. The maximum Doppler frequency of the fading channels is 50 or 150 Hz (with normalized Doppler frequencies 0.015 and 0.044, respectively). Note that in Figs. 10.31 and 10.32, the energy consumption of transmitting pilot symbols is not taken into account in computing SNRs.

The turbo receiver performance of a regular LDPC-based STC-OFDM system is shown in Fig. 10.31, whereas that of an irregular LDPC-based STC-OFDM system is shown in Fig. 10.32. (The average column weight in the parity-check matrix of the irregular LDPC code is 2.30.) TurboDD denotes the turbo receiver as before, except that the perfect CSI is replaced by the pilot/decision-directed channel estimates as proposed in [262], and TurboEM denotes the turbo receiver with the MAP-EM demodulator as proposed in Section 10.5.4. The temporal filter parameters are taken from [260]. The performance of these two receiver structures is compared when using either the regular LDPC codes or the irregular LDPC codes. From the simulations it is seen that with ideal CSI the receiver performance of regular and irregular LDPC-based STC-OFDM systems is quite similar. When CSI is not available, the proposed TurboEM receiver reduces the error floor significantly. Moreover, it is observed that by using the irregular LDPC codes, both the TurboDD and TurboEM receivers improve their performance, and the TurboEM receiver can even approach the receiver performance with ideal CSI in low to medium SNRs. A possible reason for the better performance of irregular LDPC-based STC than that of regular LDPC-based STC in the presence of nonideal CSI is the better performance of the irregular LDPC codes at low SNRs. In simulations, the turbo receiver takes three turbo iterations, and at each turbo iteration, the MAP-EM demodulator takes three EM iterations. At the cost of 10% pilot insertion and a modest complexity, the turbo receiver with the MAP-EM demodulator is a promising receiver technique, especially for application in fast-fading channels.