2.4 Blind Multiuser Detection: Subspace Methods

2.4 Blind Multiuser Detection: Subspace Methods

In this section we discuss another approach to blind multiuser detection, which was first developed in [549] and is based on estimating the signal subspace spanned by the user signature waveforms. This approach leads to blind implementation of both the linear decorrelating detector and the linear MMSE detector. It also offers a number of advantages over the direct methods discussed in Section 2.3.

Assume that the spreading waveforms of K users are linearly independent. Note that C _r of (2.27) is the sum of the rank- K matrix S A ² S ^H and the identity matrix s ² I _N . This matrix then has K eigenvalues that are strictly larger than s ² and ( N - K ) eigenvalues that equal s ² . Its eigendecomposition can be written as

Equation 2.72

where L _s = diag ( l ₁ , ..., l _K ) contains the largest K eigenvalues of C _r , U _s = [ u ₁ , ..., u _K ] contains the K orthogonal eigenvectors corresponding to the largest K eigenvalues in L _s ; and U _n = [ u _{K + 1} , ..., u _N ] contains the ( N - K ) orthogonal eigenvectors corresponding to the smallest eigenvalue s ² of C _r . It is easy to see that range ( S ) = range ( U _s ). The column space of U _s is called the signal subspace and its orthogonal complement, the noise subspace , is spanned by the columns of U _n . We next derive expressions for the linear decorrelating detector and the linear MMSE detector in terms of the signal subspace parameters U _s , L _s , and s ² .

2.4.1 Linear Decorrelating Detector

The linear decorrelating detector given by (2.13) is characterized by the following results.

Lemma 2.1: The linear decorrelating detector d ₁ in (2.13) is the unique weight vector w range ( U _s ), such that w ^H s ₁ = 1, and w ^H s _k = 0 for k = 2, ..., K .

Proof: Since rank ( U _s ) = K , the vector w that satisfies the foregoing conditions exists and is unique. Moreover, these conditions have been verified in the proof of Proposition 1 in Section 2.2.2.

Lemma 2.2: The decorrelating detector d ₁ in (2.13) is the unique weight vector w range ( U _s ) that minimizes subject to w ^H s ₁ = 1.

Proof: Since

Equation 2.73

it then follows that for w range ( U _s ) = range ( s ), f ( w ) is minimized if and only if w ^H s _k = 0 for k = 2, ..., K . By Lemma 2.1 the unique solution is w = d ₁ .

Proposition 2.3: The linear decorrelating detector d ₁ in (2.13) is given in terms of the signal subspace parameters by

Equation 2.74

with

Equation 2.75

Proof: A vector w range ( U _s ) if and only if it can be written as w = U _s x , for some . Then by Lemma 2.2 the linear decorrelating detector d ₁ has the form d ₁ = U _s x ₁ , where

Equation 2.76

where the third equality follows from the fact that

which in turn follows directly from (2.27) and (2.72). The optimization problem (2.76) can be solved by the method of Lagrange multipliers. Let

Since the matrix L _s - s ² I _K is positive definite, L ( x ) is a strictly convex function of x . Therefore, the unique global minimum of L ( x ) is achieved at x ₁ , where L ( x ₁ ) = 0, or

Equation 2.77

Therefore, , where a _d is determined from the constraint ( U _s x ₁ ) ^H s ₁ = 1; that is, . Finally, the weight vector of the linear decorrelating detector is given by .

2.4.2 Linear MMSE Detector

The following result gives the subspace form of the linear MMSE detector, defined by (2.32).

Proposition 2.4: The weight vector m ₁ of the linear MMSE detector defined by (2.32) is given in terms of the signal subspace parameters by

Equation 2.78

with

Equation 2.79

Proof: From (2.34) the linear MMSE detector defined by (2.32) is given by

Equation 2.80

By (2.72),

Equation 2.81

Substituting (2.81) into (2.80) and using the fact that , we obtain (2.78).

Since the decision rules (2.7) and (2.9) are invariant to a positive scaling, the two subspace linear multiuser detectors given by (2.74) and (2.78) can be interpreted as follows. First, the received signal r [ i ] is projected onto the signal subspace to get , which clearly is a sufficient statistic for demodulating the K users' data bits. The spreading waveform s ₁ of the desired user is also projected onto the signal subspace to obtain . The projection of the linear multiuser detector in the signal subspace is then a signal such that the detector output is the data bit is demodulated as for coherent detection, and for differential detection. According to (2.74) and (2.78), the projections of the linear decorrelating detector and that of the linear MMSE detector in the signal subspace are given, respectively, by

Equation 2.82

Equation 2.83

Therefore, the projection of the linear multiuser detectors in the signal subspace is obtained by projecting the spreading waveform of the desired user onto the signal subspace, followed by scaling the k th component of this projection by a factor of ( l _k - s ² ) ^-1 (for linear decorrelating detector) or (for linear MMSE detector). Note that as s ² 0, the two linear detectors become identical, as we would expect.

Since the autocorrelation matrix C _r , and therefore its eigencomponents, can be estimated from the received signals, from the discussion above we see that both the linear decorrelating detector and the linear MMSE detector can be estimated from the received signal with the prior knowledge of only the spreading waveform and the timing of the desired user (i.e., they both can be obtained blindly). We summarize the subspace blind multiuser detection algorithm as follows.

Algorithm 2.5: [Subspace blind linear detector ”synchronous CDMA]

• Compute the detector:

Equation 2.84

Equation 2.85

Equation 2.86

Equation 2.87

• Perform differential detection:

Equation 2.88

Equation 2.89

2.4.3 Asymptotics of Detector Estimates

We next examine the consistency and asymptotic variance of the estimates of the two subspace linear detectors. Assuming that the received signal samples are independent and identically distributed (i.i.d.), then by the strong law of large numbers , the sample mean converges to C _r almost surely (a.s.) as the number of received signals M . It then follows [521] that as a.s., for k = 1, ..., K . Therefore, we have

Equation 2.90

Equation 2.91

Similarly, a.s. as M . Hence both the estimated subspace linear multiuser detectors based on the received signals are strongly consistent . However, it is in general biased for finite number of samples. We next consider an asymptotic bound on the estimation errors.

First, for all eigenvalues and the K largest eigenvectors of , the following bounds hold a.s. [521, 609]:

Equation 2.92

Equation 2.93

Using the bounds above, we have

Equation 2.94

Note that , , and are all bounded. On the other hand, it is easily seen that

Equation 2.95

Equation 2.96

Therefore, we obtain the asymptotic estimation error for the linear MMSE detector, and similarly that for the decorrelating detector, given, respectively, by

2.4.4 Asymptotic Multiuser Efficiency under Mismatch

We now consider the effect of spreading waveform mismatch on the performance of subspace linear multiuser detectors. Let with be the assumed spreading waveform of the desired user and s ₁ be the true spreading waveform of that user. can then be decomposed into components of the signal subspace and the noise subspace; that is,

Equation 2.97

with

Equation 2.98

Equation 2.99

For simplicity, in the following we consider the real-valued signal model [i.e., A _k > 0, k = 1, ..., K , and n [ i ] ~ N ( , s ² I _N )]. [Here N ( ·, ·) denotes a real-valued Gaussian distribution.] The signal subspace component can then be written as

Equation 2.100

for some with a ₁ > 0. A commonly used performance measure for a multiuser detector is the asymptotic multiuser efficiency (AME) [520], defined as ^[2]

^[2] P ₁ ( s ) is the probability of error of the detector for noise level s ; .

Equation 2.101

which measures the exponential decay rate of the error probability as the background noise approaches zero relative to that of a single-user system having the same signal-to-noise ratio. A related performance measure, the near “far resistance , is the infimum of AME as the interferers' energies are allowed to vary arbitrarily.

Equation 2.102

Since as s 0, the linear decorrelating detector and the linear MMSE detector become identical, these two detectors have the same AME and near “far resistance [296, 307]. It is straightforward to compute the AME of the linear decorrelating detector, since its output consists of only the desired user's signal and the ambient Gaussian noise. By (2.15) “(2.17), we conclude that the AME and the near “far resistance of both linear detectors are given by

Equation 2.103

Next we compute the AME and the near “far resistance of the two subspace linear detectors under spreading waveform mismatch. Define the N x N diagonal matrices

Equation 2.104

Equation 2.105

Denote the singular value decomposition (SVD) of S by

Equation 2.106

where the N x K matrix G = [ g _ij ] has g _ij = 0 for all i j and g ₁₁ g ₂₂ ... g _KK . The columns of the N x N matrix W are the orthogonal eigenvectors of s s ^T , and the columns of the K x K matrix V are the orthogonal eigenvectors of R = S ^T S . We have the following result, whose proof is given in the Appendix (Section 2.8.2).

Lemma 2.3: Let the eigendecomposition of C _r be C _r = U L U ^T . Then the N x N diagonal matrix in (2.105) is given by

Equation 2.107

where G is the transpose of G in which the singular values are replaced by their reciprocals.

Using the result above, we obtain the AME of the subspace linear detectors under spreading waveform mismatch, as follows.

Proposition 2.5: The AME of the subspace linear decorrelating detector given by (2.74) and that of the subspace linear MMSE detector given by (2.78) under spreading waveform mismatch is given by

Equation 2.108

Proof: Since d ₁ and m ₁ have the same AME, we need only to compute the AME for d ₁ . Because a positive scaling on the detector does not affect its AME, we consider the AME of the following scaled version of d ₁ under the signature waveform mismatch:

Equation 2.109

where the second equality follows from the fact that the noise subspace component is orthogonal to the signal subspace U _s . Substituting (2.106) and (2.107) into (2.109), we have

Equation 2.110

Equation 2.111

Equation 2.112

The output of the detector is given by

Equation 2.113

where . The probability of error for user 1 is then given by

Equation 2.114

It then follows that the AME is given by (2.108).

It is seen from (2.114) that spreading waveform mismatch causes MAI leakage at the detector output. Strong interferers ( A _k >> A ₁ ) are suppressed at the output, whereas weak interferers ( A _k << A ₁ ) may lead to performance degradation. If the mismatch is not significant, with power control, so that the open -eye condition is satisfied (i.e., ), the performance loss is negligible; otherwise , the effective spreading waveform should be estimated first. Moreover, since the mismatched spreading waveform is first projected onto the signal subspace, its noise subspace component is nulled out and does not cause performance degradation; whereas for the blind adaptive MOE detector discussed in Section 2.3, such a noise subspace component may lead to complete cancellation of both the signal and MAI if there is no energy constraint on the detector [183].