3.2 Linear Group-Blind Multiuser Detection for Synchronous CDMA

3.2 Linear Group -Blind Multiuser Detection for Synchronous CDMA

We start by considering the following discrete-time signal model for a synchronous CDMA system:

Equation 3.1

Equation 3.2

where, as before, K is the total number of users; A _k , b _k [ i ], and s _k are, respectively, the complex amplitude, i th transmitted bit, and signature waveform of the k th user ; n [ i ] ~ N _c (0, s ² 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

It is assumed that the users' signature waveforms are linearly independent (i.e., S has full column rank). Hence both and also have full column ranks. Then (3.2) can be written as

Equation 3.3

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

Equation 3.4

and

Equation 3.5

or

Equation 3.6

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 k th 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:

Equation 3.7

This definition is equivalent to the one given in Section 2.2.2. To see this, it suffices to show that , and for l k . Since contains the first columns of S , then for any w we have

Equation 3.8

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:

Equation 3.9

Another form of linear group-blind detector is analogous to the linear MMSE detector introduced in Section 2.2.3. It suppresses the interference from the known users and that from the unknown users separately, both in the MMSE sense. First define the following projection matrix:

Equation 3.10

which projects any signal onto the subspace null ( ). Recall that the autocorrelation matrix of the received signal in (3.1) is given by

Equation 3.11

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

Equation 3.12

where , with ; and the columns of form an orthogonal basis of the subspace range ( S ) null ( ). We next define the linear group-blind MMSE detector. As noted in Chapter 2, any linear detector must lie in the space . The group-blind linear MMSE detector for the k th user has the form , where and , such that suppresses interference from known users in the MMSE sense, and suppresses interference from unknown users in the MMSE sense. Formally, we have the following definition.

Definition 3.3: [Group-blind linear MMSE detector ”synchronous CDMA] Let be the components of the received signal r [ i ] in (3.3) consisting of the signals from known users plus the noise. The weight vector of the group-blind linear MMSE detector for user k is given by , where range ( ) and range ( ) such that

Equation 3.13

Equation 3.14

Note that in general the linear group-blind MMSE detector m _k defined above is different from the linear MMSE detector defined in Section 2.2.3, due to the specific structure that the former imposes.

We next give expressions for the three linear group-blind detectors defined above in terms of the known users' signature waveforms and the unknown users' signal subspace components and defined in (3.12).

Proposition 3.1: [Group-blind linear decorrelating detector (form I) ”synchronous CDMA] The weight vector of the group-blind linear decorrelating detector for user k is given by

Equation 3.15

Proof: Decompose d _k as , where and . Substituting this into the constraint in (3.7), we have

Equation 3.16

Hence d _k has the form for some . Substituting this into the minimization problem in (3.7), we get

Equation 3.17

Equation 3.18

Equation 3.19

Equation 3.20

where (3.17) follows from (3.11); (3.18) follows from the fact that ; (3.19) follows from (3.12); and (3.20) follows from the fact that . Hence

Equation 3.21

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

Equation 3.22

Proof: Decompose w _k as , where range ( ) and range ( ). Substituting this into the constraint in (3.9), we have

Equation 3.23

Hence for some . Substituting this into the minimization problem in (3.9), we get

Equation 3.24

Equation 3.25

where (3.24) follows from the fact that , and (3.25) follows from (3.12). Hence

Equation 3.26

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

Equation 3.27

Proof: We first solve for in (3.13). Since range ( ), and has full column rank , we can write for some . Substituting this into (3.13), we have

Equation 3.28

Next we solve in (3.14) for some . Following the same derivation as that of (3.25), we obtain

Equation 3.29

Therefore, we have

Equation 3.30

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]

• Compute the unknown users' signal subspace:

  Equation 3.31

  

  
  

  Equation 3.32

  

  
  

  where is given by (3.10).

• Form the detectors:

  Equation 3.33

  

  
  

• Perform differential detection:

Equation 3.34

Equation 3.35

The group-blind linear decorrelating detector and the group-blind linear MMSE detector can be implemented similarly. Note that both of them require an estimate of the noise variance s ² . A simple estimator of s ² is the average of the N “ K eigenvalues in . Note also that the group-blind linear MMSE detector requires an estimate of the inverse of the energy of the desired users, , as well. The following result can be found in Section 4.5 (cf. Proposition 4.2):

Equation 3.36

Hence diag can be estimated by using (3.36) with the signal subspace parameters replaced by their respective sample estimates.

In the results above, the linear group-blind detectors are expressed in terms of the known users' signature waveforms and the unknown users' signal subspace components and defined in (3.12). Let the eigendecomposition of the autocorrelation matrix C _r in (3.11) be

Equation 3.37

The linear group-blind detectors can also be expressed in terms of the signal subspace components L _s and U _s of all users' signals defined in (3.37), as given by the following three results.

Proposition 3.4: [Group-blind linear decorrelating detector (form II) ”synchronous CDMA] The weight vector of the group-blind linear decorrelating detector for user k is given by

Equation 3.38

Proof: Using the method of Lagrange multipliers to solve the constrained optimization problem (3.7), we obtain

Equation 3.39

where . Substituting (3.39) into the constraint that , we obtain

Equation 3.40

Hence

Equation 3.41

where (3.41) follows from (3.11), (3.37), and the fact that .

Proposition 3.5: [Group-blind linear hybrid detector (form II) ”synchronous CDMA] The weight vector of the group-blind linear hybrid detector for user k is given by

Equation 3.42

Proof: Using the method of Lagrange multipliers to solve the relaxed optimization problem (3.9) over , we obtain

Equation 3.43

where is the Lagrange multiplier and . Substituting (3.43) into the constraint that we obtain

Equation 3.44

Hence

Equation 3.45

where (3.45) follows from (3.11), (3.37), and the fact that . It is seen from (3.45) that w _k range ( U _s ) = range ( S ); therefore, it is the solution to the constrained optimization problem (3.9).

To form the group-blind linear MMSE detector in terms of the signal subspace U _s , we first need to find a basis for the subspace range ( ). Clearly, range ( ) = range ( ). Consider the (rank- deficient ) QR factorization of the N x K matrix :

Equation 3.46

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

Equation 3.47

Proof: Since the columns of Q _s form an orthogonal basis of range ( ), following the same derivation as (3.30), we have

Equation 3.48

Furthermore, we have

Equation 3.49

Equation 3.50

Equation 3.51

Equation 3.52

where (3.49) follows from , (3.50) follows from and (3.51) follows from (3.46). Substituting (3.52) into (3.48), we obtain (3.47).

Based on the results above, we can implement the form II 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 linear hybrid group-blind detector (form II) is as follows. (The group-blind linear decorrelating detector and the group-blind linear MMSE detector can be implemented similarly.)

Algorithm 3.2: [Group-blind linear hybrid detector (form II) ”synchronous CDMA]

• Compute the signal subspace:

  Equation 3.53

  

  
  

  Equation 3.54

  

  
  

• Form the detectors:

  Equation 3.55

  

  
  

• Perform differential detection:

  Equation 3.56

  

  
  

  Equation 3.57

  

  
  

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.