4.9 Robust Multiuser Detection in Stable Noise

So far in this chapter we have modeled non-Gaussian ambient noise using a mixture Gaussian distribution. Recently, the stable noise model has been proposed as a statistical model for the impulsive noise in several applications, including wireless communications [357, 491]. In this section we first give a brief description of the stable distribution. We then demonstrate that the various robust multiuser detection techniques discussed in this chapter are also very effective in combating impulsive noise modeled by a stable distribution.

4.9.1 Symmetric Stable Distribution

A symmetric stable distribution is defined through its characteristic function as follows .

Definition 4.1: [Symmetric stable distribution] A random variable X has a symmetric stable distribution if and only if its characteristic function has the form

Equation 4.165

where

Thus, a symmetric stable random variable is completely characterized by three parameters, a , g , and q , where:

• a is called the characteristic exponent , which indicates the "heaviness" of the tails of the distribution ”a small value of a implies a heavier tail. The case a = 2 corresponds to a Gaussian distribution, whereas a = 1corresponds to a Cauchy distribution.

• g is called the dispersion . For the Gaussian case (i.e., a = 2), g = ½ Var( X ).

• q is a location parameter , which is the mean when 1 < a 2 and the median when 0 < a < 1.

By taking the Fourier transform of the characteristic function, we can obtain the probability density function (pdf) of the symmetric stable random variable X :

Equation 4.166

No closed-form expressions exist for general stable pdf's except for the Gaussian ( a = 2) and Cauchy ( a = 1) pdf's. For these two pdf's, closed-form expression exist:

Equation 4.167

Equation 4.168

It is known that for a non-Gaussian ( a < 2) symmetric stable random variable X with location parameter q = 0 and dispersion g , we have the asymptote

Equation 4.169

where C ( a ) is a positive constant depending on a . Thus, stable distributions with a < 2 have inverse power tails, whereas Gaussian distributions have exponential tails. Hence the tails of the stable distributions are significantly heavier than those of the Gaussian distributions. In fact, the smaller is a , the slower does its tail drop to zero, as shown in Figs. 4.15 and 4.16.

Figure 4.15. Symmetric stable pdf's for different values of a . g = 1.

Figure 4.16. Tails of the symmetric stable pdf's for different values of a . g = 1.

As a consequence of (4.169), stable distributions do not have second-order moments except for the limiting case of a = 2. More specifically , let X be a symmetric stable random variable with characteristic exponent a . If 0 < a < 2, then

Equation 4.170

If a = 2, then

Equation 4.171

for all m 0. Hence for 0 < a 1, stable distributions have no finite first- or higher-order moments; for 1 < a < 2, they have the first moments; and for a = 2, all moments exist. In particular, all non-Gaussian stable distributions have in finite variance. The reader is referred to [357] for further details of these properties of a -stable distribution.

Generation of Symmetric Stable Random Variables

The following procedure generates a standard symmetric stable random variable X with characteristic exponent a , dispersion g = 1 and location parameter q = 0 (see [357]):

Equation 4.172

Equation 4.173

Equation 4.174

Equation 4.175

Equation 4.176

Equation 4.177

Equation 4.178

Equation 4.179

Now in order to generate a symmetric stable random variable Y with parameters ( a , g , q ), we first generate a standard symmetric stable random variable X with parameters ( a , 1, 0), using the procedure above. Then Y can be generated from X according to the following transformation:

Equation 4.180

4.9.2 Performance of Robust Multiuser Detectors in Stable Noise

We consider the performance of the robust multiuser detection techniques discussed in previous sections in symmetric stable noise. In particular, we consider the performance of the linear decorrelator, the maximum- likelihood decorrelator, and the Huber decorrelator, as well as their improved versions based on local likelihood search. First, the y functions for these three decorrelative detectors are plotted in Fig. 4.17. For the Huber decorrelator, the variance s ² , the original definition of y _H ( ·) in (4.112), is replaced by the dispersion parameter g . Note that since the pdf of the symmetric stable distribution does not have a closed form, we have to resort to a numerical method to compute y _ML ( x ) given by (4.108). In particular, we can use discrete Fourier transform (DFT) to calculate samples of f ( x ) and f '( x ), as follows. Recall that the characteristic function is given by

Figure 4.17. The y functions for a linear decorrelator, Huber decorrelator, and maximum-likelihood decorrelator under symmetric stable noise. g = 0.0792.

Equation 4.181

The pdf and its derivative are related to the characteristic function through

Equation 4.182

Equation 4.183

Hence by sampling the characteristic function f ( t ) and then perform (inverse) DFT, we can get samples of f ( t ) and f '( t ), which in turns give y _ML ( x ).

First we demonstrate the performance degradation of the linear decorrelator in symmetric stable noise. The BER performance of the linear decorrelator in several symmetric stable noise channels is depicted in Fig. 4.18. Here the SNR is defined as . It is seen that the smaller is a (i.e., the more impulsive is the noise), the more severe is the performance degradation incurred by the linear decorrelator. We next demonstrate the performance gain achieved by the Huber decorrelator. Figure 4.19 shows the BER performance of the Huber decorrelator. It is seen that as the noise becomes more impulsive (i.e., a becomes smaller), the Huber deccorrelator offers more performance improvement over the linear decorrelator. Finally, we depict the BER performance of the linear decorrelator, the Huber decorrelator, and the ML decorrelator, as well as their their improved versions based on the slowest-descent search, in Fig. 4.20. It is seen that the performance of the improved/unimproved linear decorrelator is substantially worse than that of the Huber decorrelator and the ML decorrelator. The improved Huber decorrelator performs more closely to the ML decorrelator.

Figure 4.18. BER performance of a linear decorrelator in a -stable noise. N = 31, K = 6. The powers of the interferers are 10 dB above the power of user 1.

Figure 4.19. BER performance of a Huber decorrelator in a -stable noise. N = 31, K = 6. The powers of the interferers are 10 dB above the power of user 1.

Figure 4.20. BER performance of three decorrelative detectors and their local-likelihood-search versions. N = 31, K = 6, a = 1.2. The powers of the interferers are 10 dB above the power of user 1.