4.7 Robust Group-Blind Multiuser Detection

4.7 Robust Group -Blind Multiuser Detection

Consider the received signal of (4.97). As noted in Chapter 3, in group-blind multiuser detection, only a subset of the K users' signals need to be demodulated.

Specifically, suppose that the first ( K ) users are the users of interest. Denote and as matrices containing, respectively, the first and the last ( K “ ) columns of S . Similarly define the quantities , , and . Then (4.97) can be rewritten as

Equation 4.123

Equation 4.124

Let the autocorrelation matrix of the received signal and its eigendecomposition be

Equation 4.125

We next consider the problem of nonlinear group-blind multiuser detection in non-Gaussian noise. Denote and . Then (4.124) can be written as

Equation 4.126

Equation 4.127

for some . The basic idea here is to get an estimate of the sum of the undesired users signals, , and to subtract it from r . This effectively reduces the problem to the form treated in Section 4.6. To that end, denote

Equation 4.128

Equation 4.129

Next, we outline the method for estimating the signal in (4.127). Denote

Equation 4.130

In what follows we assume that the Huber penalty function is used. We first obtain a robust estimate of f by the following iterative procedure:

Equation 4.131

Equation 4.132

The robust estimate of f translates into a robust estimate of z , which by Proposition 4.2, in turn translates into a robust estimate of , as

Equation 4.133

Using the above-estimated , the desired users' signals are then subtracted from the received signal to obtain

Equation 4.134

Next, the subspace components of the undesired users' signals are identified as follows. Let

Equation 4.135

where the dimension of the signal subspace in (4.135) is K “ . We then have

Equation 4.136

Equation 4.137

for some or in its real-valued form,

Equation 4.138

Equation 4.139

A robust estimate of is then obtained from (4.139) using an iterative procedure similar to (4.131)-(4.132). Finally, the estimated undesired users' signals are subtracted from the received signal to obtain

Equation 4.140

Equation 4.141

Note that in order to form , the complex amplitudes of the desired users, , must be estimated, which can be done based on , as discussed in Section 3.4. Note also that such an estimate has a phase ambiguity of p , which necessitates differential encoding and decoding of data. The signal model (4.141) is the same as the one treated in Section 4.6. Accordingly, define the following cost function based on the Huber penalty function:

Equation 4.142

where denotes the j th row of the matrix . Let be the stationary point of which can also be found using an iterative method similar to (4.105)-(4.106). The Hessian of at the stationary point is given by

Equation 4.143

with

Equation 4.144

The estimate of the desired users' bits based on the slowest-descent search is now given by [let sign ( )]

Equation 4.145

with

Equation 4.146

The robust group-blind multiuser detection algorithm for synchronous CDMA with non-Gaussian noise is summarized below.

Algorithm 4.5: [Robust group-blind multiuser detector ”synchronous CDMA]

• Compute the sample autocorrelation matrix of the received signal and its eigen-decomposition.

• Compute the robust estimate of f using (4.130)-(4.132); compute the robust estimate of using (4.133).

• Compute the estimate of the complex amplitudes based on the robust estimate of using (3.127)-(3.129) [cf. (3.134)-(3.140)].

• Obtain the robust estimate of the undesired users signals' according to (4.134)-(4.139), by applying an iterative procedure similar to (4.130)-(4.132); subtract the undesired users' signals from the received signal to obtain in (4.141).

• Compute the stationary point from using an iterative procedure similar to (4.105)-(4.106); compute the Hessian using (4.143) and (4.144).

• Solve the discrete optimization problem defined by (4.145) and (4.146) using an exhaustive search [over ( Q + 1) points]; perform differential decoding.

Simulation Examples

We consider a synchronous CDMA system with a processing gain N = 15, the number of users K = 8, and random phase offset and equal amplitudes of user signals. The number of desired users is = 4. Only the spreading waveforms of the desired users are assumed to be known to the receiver. The noise parameters are = 0.01, k = 100. The BER curves of the robust blind detector of Section 4.5 (Algorithm 4.2) and the slowest-descent-search robust group-blind detector with Q = 1 and Q = 2 are shown in Fig. 4.13. It is seen that significant performance improvement is offered by the robust group-blind local-search-based multiuser detector in non-Gaussian noise channels over the (nonlinear) blind robust detector discussed in Section 4.5.

Figure 4.13. BER performance of a slowest-descent-based group-blind multiuser detector in non-Gaussian noise: synchronous case. N = 15, K = 8, = 4, = 0.01, k = 100.