Blind Adaptive Implementations

Blind Adaptive Implementations

We next develop both batch and sequential blind adaptive implementations of the linear space-time receiver. These implementations are blind in the sense that they require only knowledge of the signature waveform of the user of interest. Instead of the decorrelating detector used in previous sections, we will use a linear MMSE detector for the adaptive implementations because the MMSE detector is more suitable for adaptation and its performance is comparable to that of the decorrelating detector. We consider only the environment in which we have two transmit antennas and two receive antennas. The other cases can be derived in a similar manner. Note that inherent to any blind receiver in multiple transmit antenna systems is an ambiguity issue. That is, if the same spreading waveform is used for a user at both transmit antennas, the blind receiver cannot distinguish which bit is from which antenna. To resolve such an ambiguity, here we use two different spreading waveforms for each user (i.e., s_j,k, j {1, 2} is the spreading code for user k for the transmission of bit b_{j, k}).

There are two bits, b_{1, k}[i] and b_{2, k}[i], associated with each user at each time slot i, and the difference in time between slots is 2T, where T is the symbol interval. The received signal at antenna 1 during the two symbol periods for time slot i is

Equation 5.178

Equation 5.179

and the corresponding signals received at antenna 2 are

Equation 5.180

Equation 5.181

We stack these received signal vectors and denote

Then we may write

Equation 5.182

where

The autocorrelation matrix of the stacked signal [i], C, and its eigendecomposition are given by

Equation 5.183

Equation 5.184

where L_s = diag{l₁, l₂, . . . , l_{2 K}} contains the largest (2K) eigenvalues of C, the columns of U_s are the corresponding eigenvectors, and the columns of U_n are the 4N – 2K eigenvectors corresponding to the smallest eigenvalue s².

The blind linear MMSE detector for detecting [b[i]]₁ = b_1,1 [i] is given by the solution to the optimization problem

Equation 5.185

From Chapter 2, a scaled version of the solution can be written in terms of the signal subspace components as

Equation 5.186

and the decision is made according to

Equation 5.187

and

Equation 5.188

or

Equation 5.189

Before we address specific batch and sequential adaptive algorithms, we note that these algorithms can also be implemented using linear group-blind multiuser detectors instead of blind MMSE detectors. This would be appropriate, for example, in uplink environments in which the base station has knowledge of the signature waveforms of all of the users in the cell, but not those of users outside the cell. Specifically, we may rewrite (5.182) as

Equation 5.190

where we have separated the users into two groups. The composite signature sequences of the known users are the columns of . The unknown users' composite sequences are the columns of . Then, from Chapter 3, the group-blind linear hybrid detector for bit b_1,1 [i] is given by

Equation 5.191

This detector offers a significant performance improvement over (5.186) for environments in which the signature sequences of some of the interfering users are known.

Batch Blind Linear Space-Time Multiuser Detection

To obtain an estimate of g₁, we make use of the orthogonality between the signal and noise subspaces [i.e., the fact that . In particular, we have

Equation 5.192

Equation 5.193

In (5.193), specifies g₁ up to an arbitrary complex scale factor a (i.e., ). The following is a summary of a batch blind space-time multiuser detection algorithm for the two transmit antenna/two receive antenna configuration.

Algorithm 5.4: [Batch blind linear space-time multiuser detector—synchronous CDMA, two transmit antennas, and two receive antennas]

• Estimate the signal subspace:

  Equation 5.194

  

  
  

  Equation 5.195

  

  
  

• Estimate the channels:

  Equation 5.196

  

  
  

  Equation 5.197

  

  
  

  Equation 5.198

  

  
  

  Equation 5.199

  

  
  

• Form the detectors:

  Equation 5.200

  

  
  

  Equation 5.201

  

  
  

• Perform differential detection:

  Equation 5.202

  

  
  

  Equation 5.203

  

  
  

  Equation 5.204

  

  
  

  Equation 5.205

  

  
  

A batch group-blind space-time multiuser detector algorithm can be implemented with simple modifications to (5.200) and (5.201).

Adaptive Blind Linear Space-Time Multiuser Detection

To form a sequential blind adaptive receiver, we need adaptive algorithms for sequentially estimating the channel and the signal subspace components U_s and L_s. First, we address sequential adaptive channel estimation. Denote by z [i] the projection of the stacked signal [i] onto the noise subspace:

Equation 5.206

Equation 5.207

Since z[i] lies in the noise subspace, it is orthogonal to any signal in the signal subspace, and in particular, it is orthogonal to (). Hence g₁ is the solution to the following constrained optimization problem:

Equation 5.208

To obtain a sequential algorithm to solve the optimization problem above, we write it in the following (trivial) state space form:

The standard Kalman filter can then be applied to the system above as follows. Denote . We have

Equation 5.209

Equation 5.210

Equation 5.211

Once we have obtained channel estimates at time slot i, we can combine them with estimates of the signal subspace components to form the detector in (5.186). Since we are stacking received signal vectors, and subspace tracking complexity increases at least linearly with signal subspace dimension, it is imperative that we choose an algorithm with minimal complexity. The best existing low-complexity algorithm for this purpose appears to be the NAHJ subspace tracking algorithm discussed in Section 2.6.3. This algorithm has the lowest complexity of any algorithm used for similar purposes and has performed well when used for signal subspace tracking in multipath fading environments. Since the size of U_s is 4N x 2K, the complexity is 40 · 4N · 2K + 3 · 4N + 7.5(2K)² + 7 · 2K floating point operations per iteration.

Algorithm 5.5: [Blind adaptive linear space-time multiuser detector—synchronous CDMA, two transmit antennas, and two receive antennas]

• Using a suitable signal subspace tracking algorithm (e.g., NAHJ), update the signal subspace components U_s [i] and L_s [i] at each time slot i.

• Track the channel g₁ [i] and according to the following:

  Equation 5.212

  

  
  

  Equation 5.213

  

  
  

  Equation 5.214

  

  
  

  Equation 5.215

  

  
  

  Equation 5.216

  

  
  

  Equation 5.217

  

  
  

  Equation 5.218

  

  
  

  Equation 5.219

  

  
  

  Equation 5.220

  

  
  

• Form the detectors:

  Equation 5.221

  

  
  

  Equation 5.222

  

  
  

• Perform differential detection:

Equation 5.223

Equation 5.224

Equation 5.225

Equation 5.226

A group-blind sequential adaptive space-time multiuser detector can be implemented similarly. The adaptive receiver structure is illustrated in Fig. 5.13.