# 4.10 Appendix

### 4.10 Appendix

#### 4.10.1 Proof of Proposition 4.1 in Section 4.4

We follow the technique used in [203] by defining the function

Equation 4.184

Notice that

Equation 4.185

Equation 4.186

Equation 4.187

Equation 4.188

where (4.188) follows from the assumption that y '( x ) m . In (4.188), denotes that the matrix ( A - B ) is positive semidefinite. It then follows from (4.185), (4.187), and (4.188) that d ( t ) 0, for any t . Now on setting

Equation 4.189

we obtain

Equation 4.190

Assume that the penalty function r ( x ) is convex and bounded from below; then the cost function C ( q ) is convex and has a unique minimum C ( q *). Therefore, q * is the unique solution to (4.15) such that z ( q *) = . Since the sequence C ( q l ) is decreasing and bounded from below, it converges. Therefore, from (4.190) we have

Equation 4.191

Since for any realization of r , the probability that z ( q l ) falls in the null space of the matrix ( SR -1 S T ) is zero, then (4.191) implies that z ( q l ) with probability 1. Since z ( q ) is a continuous function of q and has a unique minimum point q *, we then have q l q * with probability 1, as l .

#### 4.10.2 Proof of Proposition 4.2 in Section 4.5

Denote . Then (4.75) can be written in matrix form as

Equation 4.192

Denote . Then from (4.73) and (4.74) we obtain

Equation 4.193

Using (4.192) and (4.193), we obtain

Equation 4.194

Equation 4.195

Equation 4.196

where in (4.194) denotes the Moore “Penrose generalized matrix inverse [189]; in (4.195) we have used the fact that , which can easily be verified by using the definition of the Moore “Penrose generalized matrix inverse [189]; in (4.196) we have used the facts that ; and (4.196) is the matrix form of (4.76). Finally, we notice that

Equation 4.197

It follows from (4.197) that the k th diagonal element of the diagonal matrix A -2 satisfies

Equation 4.198

## Chapter 5. Space-Time Multiuser Detection

Section 5.1.   Introduction

Section 5.2.   Adaptive Array Processing in TDMA Systems

Section 5.3.   Optimal Space-Time Multiuser Detection

Section 5.4.   Linear Space-Time Multiuser Detection

Section 5.5.   Adaptive Space-Time Multiuser Detection in Synchronous CDMA

Section 5.6.   Adaptive Space-Time Multiuser Detection in Multipath CDMA

### 5.1 Introduction

It is anticipated that transmit and receive antenna arrays together with adaptive space-time processing techniques will be used in future high-capacity cellular communication systems, to combat interference, time dispersion and multipath fading. There has been a significant amount of recent interest in developing adaptive array techniques for wireless communications (e.g., [48, 154, 356, 570, 571]). These studies have shown that substantial performance gains and capacity increases can be achieved by employing antenna arrays and space-time signal processing to suppress multiple-access interference, co-channel interference, and intersymbol interference, and at the same time to provide spatial diversity to combat multipath fading. In this chapter we discuss a number of signal processing techniques for space-time processing in wireless communication systems. We first discuss adaptive antenna array processing technques for TDMA systems. We then discuss space-time multiuser detection for CDMA systems.

Due to multipath propagation effects and the movement of mobile units, the array steering vector in a multiple-antenna system changes with time, and it is of interest to estimate and track it during communication sessions. One attractive approach to steering vector estimation is to exploit a known portion of the data stream (e.g., the synchronization data stream). For instance, the TDMA mobile radio systems IS-54/136 use 14 known synchronization symbols in each time slot of 162 symbols. These known symbols are very useful for estimating the steering vector and computing the optimal array combining weights. We discuss a number of approaches to adaptive array processing in such systems.

Many advanced signal processing techniques have been proposed for combating interference and multipath channel distortion in CDMA systems, and these techniques fall largely into two categories: multiuser detection (cf. Chapters 1 “4) and space-time processing [370]. Recall that multiuser detection techniques exploit the underlying structure induced by the spreading waveforms of the DS-CDMA user signals for interference suppression. In antenna array processing, on the other hand, the signal structure induced by multiple receiving antennas (i.e., the spatial signatures) is exploited for interference suppression [34, 231, 273, 348, 467, 505, 574]. Combined multiuser detection and array processing has also been addressed in a number of works [90, 121, 142, 202, 347, 504, 505]. In this chapter we provide a comprehensive treatment of space-time multiuser detection in multipath CDMA channels with both transmitter and receiver antenna arrays. We derive several space-time multiuser detection structures, including the maximum likelihood multiuser sequence detector, linear space-time multiuser detectors, and adaptive space-time multiuser detectors.

The rest of this chapter is organized as follows . In Section 5.2 we discuss adaptive antenna array techniques for interference suppression in TDMA systems. In Section 5.3 we treat the problem of optimal space-time processing in CDMA systems employing multiple receive antennas. In Section 5.4 we discuss linear space-time receiver techniques for CDMA systems with multiple receive antennas. In Section 5.5 we discuss space-time processing methods in synchronous CDMA systems that employ multiple transmit and receive antennas, and their adaptive implementations . Finally, in Section 5.6 we present adaptive space-time receiver structures in multipath CDMA channels with multiple transmit and receive antennas.

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

• Algorithm 5.1: LMS adaptive array

• Algorithm 5.2: DMI adaptive array

• Algorithm 5.3: Subspace-based adaptive array for TDMA

• Algorithm 5.4: Batch blind linear space-time multiuser detector ”synchronous CDMA, two transmit antennas, and two receive antennas

• Algorithm 5.5: Blind adaptive linear space-time multiuser detector ”synchronous CDMA, two transmit antennas, and two receive antennas

• Algorithm 5.6: Blind adaptive linear space-time multiuser detector ”asynchronous multipath CDMA, two transmit antennas, and two receive antennas