Introduction

In Sec. 5.7 we described decision feedback equalizers (DFEs). These equalizers use past decisions on symbols to refine the decision on the present symbol. In this chapter we extend this idea to decision feedback equalization in memoryless MIMO channels. Here past decisions within a block are used to improve future decisions in that block, at a given block-time instant. The basic idea is introduced in Sec. 19.2. The decision feedback transceiver has a precoder F, feedforward equalizer G, and a feedback matrix B. In Sec. 19.3 we show how these matrices can be jointly optimized to minimize the mean square reconstruction error, under the zero-forcing constraint. We will see that in DFE systems the minimized mean square error depends on the geometric mean of 1/σh,k rather than on their arithmetic mean (where σh,k are the channel singular values). This transition from AM to GM is a fundamental property of DFE systems, and results in substantial decrease of the mean square error. In Sec. 19.4 we extend the joint optimization to the case where there is no zero forcing. DFE systems with minimum symbol error rate are presented in Sec. 19.5. Error probability plots are presented in Sec. 19.6, showing the superiority of DFE transceivers over linear transceivers. It turns out that optimal DFE systems are related to systems optimizing mutual information; this is elaborated in Sec. 19.7. A brief description of DFE receivers based on QR decomposition, and the VBLAST receiver are given in Sec. 19.8.

Introduction

In this chapter we review a number of well established ideas in digital communication. Section 5.2 revisits the matched filter introduced in Sec. 2.5 and discusses it from the viewpoint of information sufficiency and reconstructibility. We establish the generality of matched filtering as a fundamental front-end tool in receiver design. It is often necessary to make sure that the noise sequence at the input of the detector is white. The digital filter that ensures such a property, the so-called sampled noise whitening filter, is described in Sec. 5.3. A vector space interpretation of matched filtering is presented in Sec. 5.4, and offers a very useful viewpoint.

Estimation of the transmitted symbol stream from the received noisy stream is one of the basic tasks performed in any communication receiver. The foundation for this comes from optimal sequence estimation theory, which is briefly discussed in Sec. 5.5. This includes a review of maximum likelihood and maximum a posteriori methods. The Viterbi algorithm for sequence estimation is described in Sec. 5.6. While this is one of the most well known algorithms, simpler suboptimal methods, such as decision feedback equalizers (DFE), have also become popular. The motivation for DFE, which is a nonlinear equalizer, is explained in Sec. 5.7. A nonlinear precoder called the Tomlinson-Harashima precoder can be used as an alternative to the nonlinear equalizer at the receiver, and is described in Sec. 5.8.

Introduction

In this chapter we consider the effect of introducing redundancy into the symbol stream at the transmitter of a digital communication channel. The introduction of a sufficient amount of redundancy helps to equalize the channels more easily. For example FIR channels can be equalized with FIR filters without the need for oversampling at the receiver as in Secs. 4.7–4.8. Two types of redundancies will be discussed here. The first one, discussed in Sec. 7.2, is zero padding (ZP), where a block of zeros is inserted between adjacent blocks of input samples. The second one, studied in Sec. 7.3, is called cyclic prefixing (CP), where a subset of input samples is repeated in each block. In Sec. 7.4 we show how the CP system can be represented in terms of a circulant matrix. Important variations of cyclic prefix systems such as single-carrier (SC-CP) systems and multicarrier systems, also called orthogonal frequency division multiplexing (OFDM) systems, are discussed in Sec. 7.5. Cyclic prefixing is commonly employed in OFDM systems, and in discrete multitone (DMT) systems used in DSL technology. Some details about the DMT system are discussed in Sec. 7.6.

Zero padding

Figure 7.1 explains the zero-padding operation on a discrete-time signal s(n) (the symbol stream to be transmitted). We divide the signal into blocks of length M and insert L zeros at the end of each block to obtain the zero-padded result x(n), which is then sent over the channel.

Introduction

The digital communication system described in Chap. 2 is reproduced in Fig. 4.1. This system contains both continuous-time and discrete-time quantities. The signal xc(t) entering the channel is a continuous-time interpolated version of the sequence s(n), whereas the signal ŝ(n)at the receiver is a sampled version of the filtered received signal. It is often convenient to represent the communication system entirely in terms of equivalent discrete-time quantities such as digital filters. This is indeed possible based on the relationship between continuous-time signals and their uniformly sampled versions. In this chapter we first describe this interconnection (Secs. 4.2 and 4.3). This is done both for SISO channels and MIMO channels. The raised cosine function, which arises in the context of pulse shaping, is described in Sec. 4.4. The multiuser communication system is briefly described in Sec. 4.5. In Secs. 4.6–4.8 we discuss equalization of the digitized channel using digital filters, under the so-called zero-forcing constraint. Section 4.9 contains further remarks on the digital design of pre and postfilters. Digital equalizers without zero forcing (so-called minimum mean square error equalizers) will be introduced in Sec. 4.10.

Notations. Whenever it is necessary to distinguish continuous-time quantities from discrete-time we use the subscript c (continuous) or d (discrete). Where no subscript is used, the distinction will be quite clear from the context. The notation δc(t) denotes the continuous-time Dirac delta function, and δ(n) denotes the discrete-time impulse.