Linear Group-Blind Multiuser Detection for Synchronous CDMA

where, as before, K is the total number of users; A_k, b_k[i], and s_k are, respectively, the complex amplitude, ith transmitted bit, and signature waveform of the kth user; n[i] ~ N_c(0, s² I_N) is a complex Gaussian noise vector; ; ; and . In this chapter it is assumed that the receiver has knowledge of the signature waveforms of the first users (), whose data bits are to be demodulated, whereas the signature waveforms of the remaining users are unknown to the receiver. Denote

The problem of linear group-blind multiuser detection can be stated as follows. Given prior knowledge of the signature waveforms of the desired users, find a weight vector w_k for each desired user k, 1 k , such that the data bits of these users can be demodulated according to

and

or

The basic idea behind the solution to the problem above is to suppress the interference from known users based on the signature waveforms of these users and to suppress the interference from other unknown users using subspace-based blind methods. We first consider the linear decorrelating detector, which eliminates the multiple-access interference (MAI) completely, at the expense of enhancing the noise level. To facilitate the derivation of its group-blind form, we need the following alternative definition of this detector. In this section we denote as a -vector with all elements zeros, except for the kth element, which is 1.

Definition 3.1: [Group-blind linear decorrelating detector—synchronous CDMA] The weight vector d_k of the linear decorrelating detector for user k is given by the solution to the following constrained optimization problem:

Under the constraint , we have . It then follows that for w range(S), ||w^H SA||² is minimized subject to if and only if w^H s_l = 0 for l = + 1, . . ., K. Since rank(S) = K, such a w range(S) is unique and is indeed the linear decorrelating detector.

The second linear group-blind detector considered here is a hybrid detector that zero-forces the interference caused by the known users and suppresses the interference from unknown users according to the MMSE criterion.

Definition 3.2: [Group-blind linear hybrid detector—synchronous CDMA] The weight vector w_k of the group-blind linear hybrid detector for user k is given by the solution to the following constrained optimization problem:

where . It is then easily seen that the matrix has an eigenstructure of the form

Proposition 3.2: [Group-blind linear hybrid detector (form I)—synchronous CDMA] The weight vector of the group-blind linear hybrid detector for user k is given by

Proposition 3.3: [Group-blind linear MMSE detector (formI)—synchronous CDMA] The weight vector of the group-blind linear MMSE detector for user k is given by

Therefore, we have

Based on the results above, we can implement the linear group-blind multiuser detection algorithms based on the received signals and the signature waveforms of the desired users. For example, the batch algorithm for the group-blind linear hybrid detector (form I) is summarized as follows.

Algorithm 3.1: [Group-blind linear hybrid detector (form I)—synchronous CDMA]

Hence

Hence

where Q_s is an N x matrix, R_s is a nonsingular upper triangular matrix, and P is a permutation matrix. Then the columns of Q_s form an orthogonal basis of range().

Proposition 3.6: [Group-blind linear MMSE detector (form II)—synchronous CDMA] The weight vector of the group-blind linear MMSE detector for user k is given by

Furthermore, we have

In summary, for both the group-blind zero-forcing detector and the group-blind hybrid detector, the interfering signals from known users are nulled out by a projection of the received signal onto the orthogonal subspace of these users' signal subspace. The unknown interfering users' signals are then suppressed by identifying the subspace spanned by these users, followed by a linear transformation in this subspace based on the zero-forcing or MMSE criterion. In the group-blind MMSE detector, the interfering users from the known and unknown users are suppressed separately under the MMSE criterion. The suppression of the unknown users again relies on identification of the signal subspace spanned by these users.