Blind Multiuser Detection in Correlated Noise

Blind Multiuser Detection in Correlated Noise

So far in developing the subspace-based linear detectors and the channel estimation methods, the ambient channel noise is assumed to be temporally white. In practice, such an assumption may be violated due to, for example, the interference from some narrowband sources. The techniques developed under the white noise assumption are not applicable to channels with correlated ambient noise. In this section we discuss subspace methods for joint suppression of MAI and ISI in multipath CDMA channels with unknown correlated ambient noise, which were first developed in [551]. The key assumption here is that the signal is received by two antennas well separated so that the noise is spatially uncorrelated.

We start with the received augmented discrete-time signal model given by (2.169). Assume that the ambient noise vector n[i] has a covariance matrix . Then the Pm x Pm autocorrelation matrix C_r of the received signal r[i] is given by

Equation 2.218

The linear MMSE detector m₁ for user 1 is given by (2.177) with C_r replaced by (2.218). As before, we must first estimate the desired user's composite signature waveform given by (2.198). Notice, however, that when the ambient noise is correlated, it is not possible to separate the signal subspace from the noise subspace based solely on the autocorrelation matrix C_r.

To estimate the channel in unknown correlated noise, we make use of two antennas at the receiver. Then the two augmented received signal vectors at the two antennas can be written, respectively, as

Equation 2.219

Equation 2.220

where H₁ and H₂ contain the channel information corresponding to the respective antennas. It is assumed that the two antennas are well separated so that the ambient noise is spatially uncorrelated. In other words, n₁[i] and n₂[i] are uncorrelated, and their joint covariance is given by

Equation 2.221

where S₁ and S₂ are unknown covariance matrices of the noise at the two antennas. The joint autocorrelation matrix of the received signal at the two antennas is then given by

Equation 2.222

where the submatrices are given by

Equation 2.223

Equation 2.224

Equation 2.225

We next consider two methods for estimating the noise subspaces from the signals received at the two antennas.

Singular Value Decomposition

Assume that both H₁ and H₂ have full column rank ; then the matrix C₁₂ also has rank r. Consider the singular value decomposition of the matrix C₁₂,

Equation 2.226

The Pm x Pm diagonal matrix G has the form G = diag(g₁, ..., g_r, 0, ..., 0), with g₁ · · · g_r > 0. Now if we partition the matrix U_j as U_j = [U_j,s U_j,n] for j = 1, 2, where U_j,s and U_j,n contain the first r columns and the last Pm - r columns of U_j, respectively, the column space of U_j,n is orthogonal to the column space of H_j:

Equation 2.227

where denotes the orthogonal complement space of range(H_j). User 1's channel corresponding to antenna j, f_j,1, can then be estimated from the orthogonality relationship

Equation 2.228

Canonical Correlation Decomposition

Assume that the matrices C₁₁ and C₂₂ are both positive definite. The canonical correlation decomposition (CCD) of the matrix C₁₂ is given by [18]

Equation 2.229

or

Equation 2.230

The Pm x Pm matrix G has the form G = diag(g₁, ..., g_r, 0, ..., 0), with g₁ ··· g_r > 0. Let

Equation 2.231

Partition the matrix L_j such that

Equation 2.232

where L_j,s and L_j,n are the first r columns and the last Pm - r columns of L_j, respectively. The matrix U_j are similarly partitioned into U_j,s and U_j,n. We have [580].

Equation 2.233

Note, however, that L_j,s does not necessarily span the signal subspace range(H_j) [580].

Now suppose that we have estimated the composite signature waveform of the desired user , using the identified noise subspace L_j,n. Since , we have

Equation 2.234

where the second equality in (2.234) follows from (2.231) and the fact that U_j is a unitary matrix; and the third equality follows from the fact that .

Let the estimated weight vectors of the linear MMSE detectors at the two antennas be . To make use of the received signal at both antennas, we use the following equal-gain differential combining rule for detecting the differential bit b₁[i]:

Equation 2.235

Equation 2.236

We next summarize the procedures for computing the linear MMSE detector in unknown correlated noise based on the discussion above. Let

Equation 2.237

be the matrix of M received augmented signal sample vectors at antenna j corresponding to one block of data transmission.

Algorithm 2.9: [Blind linear MMSE detector in multipath CDMA with correlated noise—SVD-based method]

• Compute the auto- and cross-correlation matrices:

Equation 2.238

• Perform an SVD on to get the noise subspace .

• Compute the composite signature waveforms by solving

Equation 2.239

• Form the linear MMSE detectors:

Equation 2.240

• Perform differential detection according to (2.235)–(2.236).

Algorithm 2.10: [Blind linear MMSE detector in multipath CDMA with correlated noise—CCD-based method]

• Perform QR decomposition:

Equation 2.241

• Perform an SVD on :

Equation 2.242

• Compute

Equation 2.243

where is an upper triangular matrix.

• Partition . Compute the composite signature waveforms by solving

Equation 2.244

• Form the linear MMSE detectors:

Equation 2.245

• Perform differential detection according to (2.235)–(2.236).

The procedure above is based on the fast algorithm for computing CCD given in [580].

Note that the two methods above operate on the Pm-dimensional signal vectors r_j[i], j = 1,2. The same procedures can be applied to the decimated received signal vectors to operate on the Nm-dimensional signal vectors r_j,q[i], j = 1,2, q = 0, ..., p - 1. As before, such decimation-combining approach reduces the computational complexity by a factor of . It also reduces the number of users that can be accommodated in the system by a factor of p.

Simulation Examples

We illustrate the performance of the detectors above via simulation examples. The simulated system is the same as that in Section 2.7.3, except that the ambient noise is temporally correlated. The noise at each antenna j is modeled by a second-order autoregressive (AR) model with coefficients a_j = [a_j,1, a_j,2]; that is, the noise field is generated according to

Equation 2.246

where n_j[i] is the noise at antenna j and sample i, and w_j[i] is a complex white Gaussian noise sample with unit variance. The AR coefficients at the two arrays are chosen as a₁ = [1, -0.2] and a₂ = [1.2, -0.3].

We first consider a five-user system. In Fig. 2.18 the performance of the Pm-dimensional blind linear MMSE detectors is plotted for both SVD- and CCD-based methods. The corresponding performance by the decimation-combining receiver structure is plotted in Fig. 2.19. Next a 10-user system is simulated and the performance of the Pm-dimensional blind linear MMSE detectors is plotted in Fig. 2.20.

Figure 2.18. Performance of Pm-dimensional blind linear MMSE detectors in a five-user system with correlated noise.

Figure 2.19. Performance of decimation-combining blind linear MMSE detectors in a five-user system with correlated noise.

Figure 2.20. Performance of Pm-dimensional blind linear MMSE detectors in a 10-user system with correlated noise.

It is seen from Figs. 2.18–2.20 that CCD-based detectors are superior in performance to SVD-based detectors. It has been shown that the CCD has the optimality property of maximizing the correlation between the two sets of linearly transformed data [18]. Maximizing the correlation of the two data sets can yield the best estimate of the correlated (i.e., signal) part of the data. CCD makes use of the information of both and together with and creates the maximum correlation between the two data sets. On the other hand, SVD uses only the information and does not create the maximum correlation between the two data sets, and thus yields inferior performance.