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

graphics/03equ001.gif


Equation 3.2

graphics/03equ002.gif


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; graphics/110fig01.gif ; graphics/110fig02.gif ; and graphics/110fig03.gif . In this chapter it is assumed that the receiver has knowledge of the signature waveforms of the first graphics/ktilde.gif users ( graphics/110fig04.gif ), whose data bits are to be demodulated, whereas the signature waveforms of the remaining graphics/110fig05.gif users are unknown to the receiver. Denote

graphics/110equ01.gif

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

Equation 3.3

graphics/03equ003.gif


The problem of linear group-blind multiuser detection can be stated as follows . Given prior knowledge of the signature waveforms graphics/stilde.gif of the graphics/ktilde.gif desired users, find a weight vector w k graphics/034fig03.gif for each desired user k , 1 k graphics/ktilde.gif , such that the data bits of these users can be demodulated according to

Equation 3.4

graphics/03equ004.gif


and

Equation 3.5

graphics/03equ005.gif


or

Equation 3.6

graphics/03equ006.gif


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 graphics/etilde.gif as a graphics/ktilde.gif -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

graphics/03equ007.gif


This definition is equivalent to the one given in Section 2.2.2. To see this, it suffices to show that graphics/111fig01.gif , and graphics/111fig02.gif for l k . Since graphics/stilde.gif contains the first graphics/ktilde.gif columns of S , then for any w we have

Equation 3.8

graphics/03equ008.gif


Under the constraint graphics/111fig03.gif , we have graphics/111fig04.gif . It then follows that for w range ( S ), w H SA 2 is minimized subject to graphics/111fig03.gif if and only if w H s l = 0 for l = graphics/ktilde.gif + 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 graphics/ktilde.gif 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

graphics/03equ009.gif


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

graphics/03equ010.gif


which projects any signal onto the subspace null ( graphics/112fig07.gif ). Recall that the autocorrelation matrix of the received signal in (3.1) is given by

Equation 3.11

graphics/03equ011.gif


where graphics/112fig08.gif . It is then easily seen that the matrix graphics/112fig02.gif has an eigenstructure of the form

Equation 3.12

graphics/03equ012.gif


where graphics/112fig09.gif , with graphics/112fig03.gif ; and the columns of graphics/112fig04.gif form an orthogonal basis of the subspace range ( S ) null ( graphics/112fig07.gif ). We next define the linear group-blind MMSE detector. As noted in Chapter 2, any linear detector must lie in the space graphics/112fig11.gif . The group-blind linear MMSE detector for the k th user has the form graphics/112fig12.gif , where graphics/112fig13.gif and graphics/112fig14.gif , such that graphics/112fig05.gif suppresses interference from known users in the MMSE sense, and graphics/112fig05.gif 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 graphics/112fig06.gif 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 graphics/113fig01.gif , where graphics/mtildek.gif range ( graphics/stilde.gif ) and graphics/112fig05.gif range ( graphics/ubars.gif ) such that

Equation 3.13

graphics/03equ013.gif


Equation 3.14

graphics/03equ014.gif


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 graphics/stilde.gif and the unknown users' signal subspace components graphics/lambars.gif and graphics/ubars.gif 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

graphics/03equ015.gif


Proof: Decompose d k as graphics/113fig09.gif , where graphics/113fig10.gif and graphics/113fig11.gif . Substituting this into the constraint graphics/113fig12.gif in (3.7), we have

Equation 3.16

graphics/03equ016.gif


Hence d k has the form graphics/113fig05.gif for some graphics/113fig06.gif . Substituting this into the minimization problem in (3.7), we get

Equation 3.17

graphics/03equ017.gif


Equation 3.18

graphics/03equ018.gif


Equation 3.19

graphics/03equ019.gif


Equation 3.20

graphics/03equ020.gif


where (3.17) follows from (3.11); (3.18) follows from the fact that graphics/113fig07.gif ; (3.19) follows from (3.12); and (3.20) follows from the fact that graphics/113fig08.gif . Hence

Equation 3.21

graphics/03equ021.gif


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

graphics/03equ022.gif


Proof: Decompose w k as graphics/114fig01.gif , where graphics/wtilde.gif range ( graphics/stilde.gif ) and graphics/wbar.gif range ( graphics/ubars.gif ). Substituting this into the constraint graphics/114fig02.gif in (3.9), we have

Equation 3.23

graphics/03equ023.gif


Hence graphics/114fig03.gif for some graphics/114fig04.gif . Substituting this into the minimization problem in (3.9), we get

Equation 3.24

graphics/03equ024.gif


Equation 3.25

graphics/03equ025.gif


where (3.24) follows from the fact that graphics/114fig05.gif , and (3.25) follows from (3.12). Hence

Equation 3.26

graphics/03equ026.gif


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

graphics/03equ027.gif


Proof: We first solve for graphics/mtildek.gif in (3.13). Since graphics/mtildek.gif range ( graphics/stilde.gif ), and graphics/stilde.gif has full column rank graphics/ktilde.gif , we can write graphics/114fig06.gif for some graphics/114fig07.gif . Substituting this into (3.13), we have

Equation 3.28

graphics/03equ028.gif


Next we solve graphics/115fig01.gif in (3.14) for some graphics/115fig02.gif . Following the same derivation as that of (3.25), we obtain

Equation 3.29

graphics/03equ029.gif


Therefore, we have

Equation 3.30

graphics/03equ030.gif


Based on the results above, we can implement the linear group-blind multiuser detection algorithms based on the received signals graphics/115fig03.gif and the signature waveforms graphics/stilde.gif 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

    graphics/03equ031.gif


    Equation 3.32

    graphics/03equ032.gif


    where graphics/pbar.gif is given by (3.10).

  • Form the detectors:

    Equation 3.33

    graphics/03equ033.gif


  • Perform differential detection:

Equation 3.34

graphics/03equ034.gif


Equation 3.35

graphics/03equ035.gif


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 graphics/lambarcn.gif . Note also that the group-blind linear MMSE detector requires an estimate of the inverse of the energy of the desired users, graphics/115fig06.gif , as well. The following result can be found in Section 4.5 (cf. Proposition 4.2):

Equation 3.36

graphics/03equ036.gif


Hence graphics/115fig05.gif diag graphics/115fig04.gif 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 graphics/stilde.gif and the unknown users' signal subspace components graphics/lambars.gif and graphics/ubars.gif defined in (3.12). Let the eigendecomposition of the autocorrelation matrix C r in (3.11) be

Equation 3.37

graphics/03equ037.gif


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

graphics/03equ038.gif


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

Equation 3.39

graphics/03equ039.gif


where graphics/116fig03.gif . Substituting (3.39) into the constraint that graphics/116fig04.gif , we obtain

Equation 3.40

graphics/03equ040.gif


Hence

Equation 3.41

graphics/03equ041.gif


where (3.41) follows from (3.11), (3.37), and the fact that graphics/116fig05.gif .

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

graphics/03equ042.gif


Proof: Using the method of Lagrange multipliers to solve the relaxed optimization problem (3.9) over graphics/117fig06.gif , we obtain

Equation 3.43

graphics/03equ043.gif


where graphics/117fig02.gif is the Lagrange multiplier and graphics/117fig01.gif . Substituting (3.43) into the constraint that graphics/117fig03.gif we obtain

Equation 3.44

graphics/03equ044.gif


Hence

Equation 3.45

graphics/03equ045.gif


where (3.45) follows from (3.11), (3.37), and the fact that graphics/117fig04.gif . 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 ( graphics/ubars.gif ). Clearly, range ( graphics/117fig05.gif ) = range ( graphics/ubars.gif ). Consider the (rank- deficient ) QR factorization of the N x K matrix graphics/117fig05.gif :

Equation 3.46

graphics/03equ046.gif


where Q s is an N x graphics/ktilde.gif matrix, R s is a graphics/120fig01.gif nonsingular upper triangular matrix, and P is a permutation matrix. Then the columns of Q s form an orthogonal basis of range ( graphics/ubars.gif ).

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

graphics/03equ047.gif


Proof: Since the columns of Q s form an orthogonal basis of range ( graphics/ubars.gif ), following the same derivation as (3.30), we have

Equation 3.48

graphics/03equ048.gif


Furthermore, we have

Equation 3.49

graphics/03equ049.gif


Equation 3.50

graphics/03equ050.gif


Equation 3.51

graphics/03equ051.gif


Equation 3.52

graphics/03equ052.gif


where (3.49) follows from graphics/118fig01.gif , (3.50) follows from graphics/118fig02.gif 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 graphics/118fig03.gif and the signature waveforms graphics/stilde.gif 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

    graphics/03equ053.gif


    Equation 3.54

    graphics/03equ054.gif


  • Form the detectors:

    Equation 3.55

    graphics/03equ055.gif


  • Perform differential detection:

    Equation 3.56

    graphics/03equ056.gif


    Equation 3.57

    graphics/03equ057.gif


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.



Wireless Communication Systems
Wireless Communication Systems: Advanced Techniques for Signal Reception (paperback)
ISBN: 0137020805
EAN: 2147483647
Year: 2003
Pages: 91

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