4.3 Asymptotic Performance of Robust Multiuser Detection

4.3 Asymptotic Performance of Robust Multiuser Detection

4.3.1 Influence Function

The influence function (IF) introduced by Hampel [170, 203], is an important tool used to study robust estimators. It measures the influence of a vanishingly small contamination of the underlying distribution on the estimator. It is assumed that the estimator can be defined as a functional T , operating on the empirical distribution function of the observation F _n , [i.e., T = T ( F _n )] and that the estimator is consistent as n [i.e., , where F is the underlying distribution]. The IF is defined as

Equation 4.31

where D _x is the distribution that puts a unit mass at x . Roughly speaking, the influence function IF ( x; T, F ) is the first derivative of the statistic T at an underlying distribution F and at the coordinate x . We next compute the influence function of the nonlinear decorrelating multiuser detectors defined by (4.15).

Denote the j th row of the matrix S by (i.e., [ c _j , ₁ . . . c _j , _k ] ^T ). Assume that the signature waveforms of all users are chosen randomly and let q ( x ) be the probability density function of x _j . Assume further that the noise distribution has marginal density f . Denote the joint distribution of the received signal r _j and the chip samples of the K users x _j under the true parameter q by G _q ( r , x ), with density

Equation 4.32

If G _n is the empirical distribution function generated by the signal samples { r _j , x _j } , the solution to (4.15) can also be written as (G _n ), where is the K -dimensional functional determined by

Equation 4.33

for all distributions G for which the integral is defined. Let the distribution be

Equation 4.34

Substituting this distribution into (4.33), differentiating with respect to t , and evaluating the derivative at t = 0, we get

Equation 4.35

where by definition,

Equation 4.36

Note that by (4.33) the second term on the right-hand side of (4.35) equals zero:

Equation 4.37

Now assume that the functional is Fisher consistent [170] [i.e., ], which means that at the model the estimator asymptotically measures the right quantity when applied to the model distribution. We proceed with (4.35) to obtain

Equation 4.38

where

Equation 4.39

is the cross-correlation matrix of the random infinite-length signature waveforms of the K users. From (4.38) we obtain the influence function of the nonlinear decorrelating multiuser detectors determined by (4.15) as

Equation 4.40

The influence function above is instrumental to deriving the asymptotic performance of the robust multiuser detectors, as explained below.

4.3.2 Asymptotic Probability of Error

Under certain regularity conditions, the M -estimators defined by (4.14) or (4.15) are consistent and asymptotically Gaussian [170]; that is (here we denote as the estimate of q based on N chip samples),

Equation 4.41

where the asymptotic covariance matrix is given by

Equation 4.42

and where (4.42) follows from (4.32) and (4.40).

We can also compute the Fisher information matrix for the parameters q at the underlying noise distribution. Define the likelihood score vector as

Equation 4.43

The Fisher information matrix is then given by

Equation 4.44

It is known that the maximum likelihood estimate based on i.i.d. samples is asymptotically unbiased and the asymptotic covariance matrix is J ( q ) ^“1 [377]. As discussed earlier, the maximum likelihood estimate of q corresponds to having y ( x ) = “ f '( x )/ f ( x ). Hence we can deduce that the asymptotic covariance matrix = J ( q ) ^“1 , when y ( x ) = “ f '( x )/ f ( x ). To verify this, substitute y ( x ) = “ f '( x )/ f ( x ) into (4.42); we obtain

Equation 4.45

where we have assumed that f '( “ ) = f '( ) = 0.

Next we consider the asymptotic probability of error for the class of decorrelating detectors defined by (4.15) as the processing gain N . Using the asymptotic normality condition (4.41), . The asymptotic probability of error for the k th user is then given by

Equation 4.46

where u is the asymptotic variance given by

Equation 4.47

Hence for the class of M -decorrelators defined by (4.15), their asymptotic probabilities of detection error are determined by the parameter u . We next compute u for the three decorrelating detectors discussed in Section 4.2.3, under the Gaussian mixture noise model (4.3).

Linear Decorrelating Detector

The asymptotic variance for the linear decorrelator is given by

Equation 4.48

That is, asymptotically, the performance of the linear decorrelating detector is determined completely by the noise variance, independent of the noise distribution. However, as will be seen later, the noise distribution does affect substantially the finite sample performance of the linear decorrelating detector.

Maximum-Likelihood Decorrelating Detector

The variance of the estimate used in the maximum-likelihood decorrelating detector achieves the Fisher information covariance matrix, and we have

Equation 4.49

In fact, (4.49) gives the minimum achievable u ² . To see this, we use the Cauchy “Schwarz inequality, to yield

Equation 4.50

where the last equality follows from the fact that y ( u ) f ( u ) 0, as u . To see this, we use (4.3) and (4.21) to obtain

Equation 4.51

Hence it follows from (4.50) that

Equation 4.52

Minimax Decorrelating Detector

For the minimax decorrelating detector, we have

Equation 4.53

Equation 4.54

where 1 _W ( x ) denotes the indicator function of the set W , and d _x denotes the Dirac delta function at x . After some algebra, we obtain

Equation 4.55

Equation 4.56

The asymptotic variance of the minimax decorrelating detector is obtained by substituting (4.55) and (4.56) into (4.47).

In Fig. 4.2 we plot the asymptotic variance u ² of the maximum-likelihood decorrelator and the minimax robust decorrelator as a function of and k under the Gaussian mixture noise model (4.3). The total noise variance is kept constant as and k vary [i.e., ]. From the two plots we see that the two nonlinear detectors have very similar asymptotic performance. Moreover, in this case the asymptotic variance u ² is a decreasing function of either or k when one of them is fixed. The asymptotic variances of both nonlinear decorrelators are strictly less than that of the linear decorrelator, which corresponds to a plane at u ² = s ² = (0.1) ² . In Fig. 4.3 we plot the asymptotic variance u ² of the three decorrelating detectors as a function of k with fixed and in Fig. 4.4 we plot the asymptotic variance u ² of the three decorrelating detectors as a function of with fixed k . As before, the total variance of the noise for both figures is fixed at s ² = (0.1) ² . From these figures we see that the asymptotic variance of the minimax decorrelator is very close to that of the maximum-likelihood decorrelator for the cases of small contamination (e.g., 0.1), while both of the detectors can outperform the linear detector by a substantial margin.

Figure 4.2. Asymptotic variance u ² of (a) a minimax robust decorrelating detector, and (b) a maximum-likelihood decorrelating detector, as a function of and k , under the Gaussian mixture noise model, with variance of the noise fixed at .

Figure 4.3. Asymptotic variance u ² of three decorrelating detectors as a function of k with fixed parameter . The variance of the noise is fixed at .

Figure 4.4. Asymptotic variance u ² of three decorrelating detectors as a function of with fixed parameter k . The variance of the noise is fixed at .