6.1 Introduction to Turbo Processing

In recent years , iterative (turbo) processing techniques have received considerable attention, stimulated by the discovery of powerful turbo codes [35, 36]. The turbo principle can be applied successfully to many detection/decoding problems, such as serial concatenated decoding, equalization, coded modulation, multiuser detection, and joint source and channel decoding [169, 379, 448]. We start the discussion in this chapter by illustrating the general concept of turbo processing for concatenated systems using a simple example.

A typical communications system in general consists of a cascade of subsystems with different signal processing functionalities. Consider, for example, the simple communication system employing channel coding and signaling over an intersymbol interference (ISI) channel, as shown in Fig. 6.1. In a conventional receiver, the demodulator makes hard decisions about the transmitted bits { b [ i ]} based on the received signal r ( t ), which are then passed to the channel decoder to decode the transmitted information. The problem with this approach is that by making hard decisions of the bits, information loss is incurred in each subsystem (i.e., demodulator and decoder). This is because while the subsystem only indicates whether it believes that a given bit is a 0 or a 1, it usually has sufficient information to estimate the degree of confidence in its decisions. One straightforward way to reduce the loss of information, and the resulting loss in performance, is to pass the confidence level along with the decision (i.e., to render soft decisions). This is often done when passing information from a demodulator to a channel decoder, which is known to result in approximately a 2-dB performance gain in the additive white Gaussian noise (AWGN) channel [396]. However, even if optimal bit-by-bit soft decisions are passed between all the subsystems in the receiver, the overall performance can still be far from optimal. This is due to the fact that while later stages (e.g., the channel decoder) can use the information gleaned from previous stages (e.g., the demodulator), the reverse is not generally true. While the optimal performance can be achieved by performing a joint detection, taking all receiver processing into account simultaneously (e.g., maximum likelihood detection based on the supertrellis of both the channel code and the ISI channel), the complexity of such a joint approach is usually prohibitive. This motivates an iterative (turbo) processing approach which allows earlier stages (e.g., the demodulator) to refine their processing based on information obtained from later stages (e.g., the channel decoder).

Figure 6.1. Coded communication system signaling through an intersymbol interference (ISI) channel.

To employ turbo processing in the system shown in Fig. 6.1, both the demodulator and channel decoder are of the maximum a posteriori probability (MAP) type. The function of a MAP demodulator is to produce soft decisions which reflect the probability that a given bit is a 0 or a 1. At the l th iteration, the information available to the MAP demodulator consists of the received signal r ( t ) and the a priori probabilities of the input bits, the latter of which are obtained by the MAP channel decoder based on its output from the ( l “ 1)th iteration. The MAP demodulator uses this information, combined with knowledge of the chosen modulation and of the channel structure, to produce the a posteriori probabilities (APPs) of the channel bits:

Equation 6.1

Equation 6.2

for all { b [ i ]} _i . Consider the log-likelihood ratio (LLR) formed from the a posteriori probabilities of (6.1) and (6.2):

Equation 6.3

It is seen from (6.3) that the LLR is the sum of two distinct quantities . The first term , ( b [ i ]), is the extrinsic information produced by the first-stage subsystem in the receiver (i.e., the MAP demodulator), which is information that the MAP demodulator gleans about b [ i ] from the received signal r ( t ) and the a priori probabilities of the other transmitted bits, without using the a priori probability of b [ i ]. The second term, ( b [ i ]), contains the a priori probability of b [ i ]. Note that typically, for the first iteration ( l = 1), we set P ( b [ i ] = 1) = P ( b [ i ] = 0) = ½ [i.e., for all i ]. The extrinsic information { ( b [ i ])} produced by the MAP demodulator is sent to the second-stage subsystem (i.e., the MAP channel decoder) as the a priori information for channel decoding.

Based on the a priori information provided by the MAP demodulator, and the channel code constraints, the MAP channel decoder computes the a posteriori LLR of each code bit:

Equation 6.4

The factorization (6.4) will be shown in Section 6.2. Here again we see that the output of the channel decoder is the sum of the extrinsic information obtained by the second-stage subsystem (i.e., the MAP channel decoder) and the prior information ( b [ i ]) delivered by the preceding stage (i.e., the MAP demodulator). The extrinsic information is then fed back to the MAP demodulator as the a priori information in the next [i.e., ( l + 1)th] iteration. It is important to note that (6.3) and (6.4) hold only if the inputs to the demodulator or the decoder are independent. Since both the ISI channel and the channel encoder have memory, this independence assumption will not be valid; therefore, interleaving (i.e., permutation of time order) must be present between the demodulator and the decoder in order to provide approximate independence. Finally, the turbo receiver structure for the coded ISI system is illustrated in Fig. 6.2. This scheme was introduced in [105] and is termed a turbo equalizer . The name turbo is justified because both the demodulator and the decoder use their processed output values as a priori input for the next iteration, similar to the operation of a turbo engine. Application of the turbo processing principle for joint demodulation and decoding in fading channels can be found in [139, 181].

Figure 6.2. Turbo receiver for coded communication over ISI channel.

The turbo principle can similarly be applied in coded multiple-access channels, resulting in procedures known as turbo multiuser detectors . In this chapter we discuss applications of such techniques in a variety of multiple-access communication systems with different coding schemes (convolutional codes, turbo codes, space-time codes), signaling structures [CDMA, TDMA, space-division multiple-access (SDMA)] and channel conditions (AWGN, fading, multipath).

The rest of this chapter is organized as follows . In Section 6.2 we present a maximum a posteriori (MAP) decoding algorithm for convolutional codes. In Section 6.3 we discuss turbo multiuser detectors in synchronous CDMA systems. In Section 6.4 we treat the problem of turbo multiuser detection in the presence of unknown interferers. In Section 6.5 we discuss turbo multiuser detection in general asynchronous CDMA systems with multipath fading channels. In Section 6.6 we discuss turbo multiuser detection for turbo-coded CDMA systems. In Sections 6.7 and 6.8 we discuss applications of turbo multiuser detection in space-time block-coded systems and space-time trellis -coded systems, respectively. Some mathematical proofs and derivations are given in Section 6.9.

The following is a list of the algorithms appearing in this chapter.

• Algorithm 6.1: MAP decoding algorithm for convolutional codes

• Algorithm 6.2: Low-complexity SISO multiuser detector ”synchronous CDMA

• Algorithm 6.3: Group-blind SISO multiuser detector ”synchronous CDMA

• Algorithm 6.4: SISO multiuser detector ”multipath fading channel