Analysis of a Novel Density Matching Criterion Within the ITL Framework for Blind Channel Equalization

Abstract

In blind channel equalization, the use of criteria from the field of information theoretic learning (ITL) has already proved to be a promising alternative, since the use of the high-order statistics is mandatory in this task. In view of the several existent ITL propositions, we present in this work a detailed comparison of the main ITL criteria employed for blind channel equalization and also introduce a new ITL criterion based on the notion of distribution matching. The analyses of the ITL framework are held by means of comparison with elements of the classical filtering theory and among the studied ITL criteria themselves, allowing a new understanding of the existing ITL framework. The verified connections provide the basis for a comparative performance analysis in four practical scenarios, which encompasses discrete/continuous sources with statistical independence/dependence, and real/complex-valued modulations, including the presence of Gaussian and non-Gaussian noise. The results indicate the most suitable ITL criteria for a number of scenarios, some of which are favorable to our proposition.

Appendix: QD-D and QD-DR Gradient-Based Algorithms

The stochastic gradient-based algorithms rely on adaptive filters whose weights are updated in every iteration according to

$$\begin{aligned} {\mathbf {w}}(n{+}1) = {\mathbf {w}}(n) - \mu \nabla J({\mathbf {w}}(n)), \end{aligned}$$

(33)

where $\nabla J({\mathbf {w}}(n))$ is the gradient vector of $J({\mathbf {w}}(n))$ and $\mu$ is the step size.

Hence, for the QD-D cost, given by Eq. (19), the gradient is

$$\begin{aligned} \begin{aligned} \nabla _{{\mathbf {w}}} \hat{J}_{\mathrm{QD}{-}\mathrm{D}} =&\frac{1}{2\sigma ^{2}L^2}\Bigg (\sum _{i=1}^{L}\sum _{j=1}^{L}G_{2\sigma }\left( y(n{-}i)-y(n{-}j)\right) \\&\cdot (y(n{-}j){-}y(n{-}i))\left( {\mathbf {x}}(n{-}i){-}{\mathbf {x}}(n{-}j)\right) \Bigg )\\&{-} \frac{2}{\sigma ^{2}L}\sum _{i \in {\mathcal {A}}} \left[ p_{S}(s_i)\left( \sum _{j=1}^{L}G_{\sigma }\left( s_i-y(n{-}j)\right) (s_i-y(n{-}j)){\mathbf {x}}(n{-}j) \right) \right] \end{aligned} \end{aligned}$$

(34)

which can be directly replaced in Eq. (33) to form the QD-D gradient-based algorithm. The term $1/\sigma ^{2}$ is a common factor that usually is disregarded.

The QD-DR gradient is basically the second term of Eq. (34)—and the factor $2/\sigma ^{2}$ can be disregarded.

For the continuous counterpart, the samples of s(n) are used instead of the source PMF, and the QD-D (and QD-DR) gradient only differs on the second term, as shown in [12].