Linear Group-Blind Detection in Correlated Noise

Linear Group-Blind Detection in Correlated Noise

The problem of blind linear multiuser detection in unknown correlated noise is discussed in Section 2.7.5. In this section we consider the problem of group-blind linear multiuser detection in the same environment, which was first treated in [551]. Recall that in this case it is assumed that the signal is received by two well-separated antennas, so that the noise is spatially uncorrelated. The two augmented received signal vectors at the two antennas are given, respectively, by

Equation 3.187

Equation 3.188

where H₁ and H₂ contain the channel information corresponding to the respective antennas; n₁[i] and n₂[i] are the Gaussian noise vectors at the two antennas with the following correlations:

Equation 3.189

Equation 3.190

Equation 3.191

Define

Equation 3.192

Equation 3.193

Equation 3.194

The canonical correlation decomposition (CCD) of the matrix C₁₂ is given by

Equation 3.195

Equation 3.196

The Pm x Pm matrix G has the form G = diag(g₁, . . . , g_r, 0, . . . , 0), with g₁ . . . g_r > 0. Define L_j,s and L_j,n as, respectively, the first r columns and the last Pm – r columns of L_j, j = 1, 2. It is known then that

Equation 3.197

As discussed in Section 2.7.5, the composite signature waveform of the desired user k, 1 k , can be estimated based on the orthogonality relationship .

We next consider the group-blind linear detector in correlated ambient noise based on the CCD method. Since the signal subspace cannot be identified directly in the CCD, we will not consider the group-blind linear zero-forcing or MMSE detectors, which require the identification of some signal subspace. Nevertheless, the form II group-blind linear hybrid detector can easily be constructed for correlated noise, as given by the following result.

Proposition 3.13: [Group-blind linear hybrid detector in correlated noise (form II)] The weight vector of the group-blind linear hybrid detector for the kth user at the jth antenna in correlated noise is given by

Equation 3.198

Proof: By definition, the group-blind linear hybrid detector is given by the following constrained optimization problem:

Equation 3.199

Using the method of Lagrange multipliers to solve (3.199), we obtain

Equation 3.200

where is the Lagrange multiplier, and . Substituting (3.200) into the constraint that , we obtain

Hence

Equation 3.201

Moreover, by definition,

Equation 3.202

Substituting (3.202) into (3.201), and using the fact that , we obtain (3.198).

The group-blind linear multiuser detection algorithm in multipath channels with correlated noise is summarized as follows.

Algorithm 3.8: [Group-blind linear hybrid detector—multipath CDMA and correlated noise]

• Compute the CCD: Let

  Equation 3.203

  

  
  

  Equation 3.204

  

  
  

  Equation 3.205

  

  
  

• Estimate the desired users' channels (cf. Section 2.7.3):

  Equation 3.206

  

  
  

  Equation 3.207

  

  
  

  Form using .

• Form the detector:

  Equation 3.208

  

  
  

• Perform differential detection:

  

Simulation Example

The simulated system is the same as that described in Section 3.5.1. The noise at each antenna j is modeled by a second-order autoregressive (AR) model with coefficients [a_j,1, a_j,2]; that is, the noise field is generated according to

Equation 3.209

where v_j[n] is the noise at antenna j and sample n, and w_j[n] is a complex white Gaussian noise sample. The AR coefficients at the two antennas are chosen as [a_1,1,a_1,2] = [1, –0.2] and [a_2,1,a_2,2] = [1.2, –0.3]. The performance of the group-blind linear hybrid detector is compared with that of the blind linear MMSE detector. The result is shown in Fig. 3.20. It is seen that similar to the white noise case, the proposed group-blind linear detector offers substantial performance gain over the blind linear detector.