New Kurtosis Optimization Schemes for MISO Equalization

Abstract

Abstract-This paper deals with efficient optimization of cumulant based contrast functions. Such a problem occurs in the blind source separation framework, where contrast functions are criteria to be maximized in order to retrieve the sources. More precisely, we focus on the extraction of one source signal and our method applies in deflation approaches, where the sources are extracted one by one.We propose new methods to maximize the kurtosis contrast function. These methods are intermediate between a gradient and an iterative "fixed-point" optimization of so-called reference contrasts. They rely on iterative updates of the parameters which monotonically increase the contrast function value: we point out the strong similarity with the EM (Expectation-Maximization) method and with recent generalizations referred to as MM (Minimization-Maximization). We also prove the global convergence of the algorithm to a stationary point. Simulations confirm the convergence of our methods to a separating solution. They also show experimentally that our methods have a much lower computational cost than former classical optimization methods. Finally, simulations suggest that the methods remain valid under weaker conditions than those required for proving convergence.

An Efficient Algorithm by Kurtosis Maximization in Reference-Based Framework

“…Recently, contrast functions referred to as "referencebased" have been proposed [6], [7] which are based on crossstatistics or cross-cumulants between the estimated outputs and reference signals [6]- [12]. Due to the indirect involvement of reference signals in the iterative optimization process, these reference-based contrast functions have an appealing feature in common: the corresponding optimization algorithms are quadratic with respect to the searched parameters.…”

“…Due to the indirect involvement of reference signals in the iterative optimization process, these reference-based contrast functions have an appealing feature in common: the corresponding optimization algorithms are quadratic with respect to the searched parameters. Taking advantage of this quadratic feature, a maximization algorithm based on singular value decomposition (SVD) has been proposed [6], [7] and was shown to be significantly quicker than other maximization algorithms. However, this method generally requires an additional "fixpoint" like iteration to improve the separation quality and often suffers from the need to have a good knowledge of the filter orders due to its sensitivity on the rank estimation.…”

“…Similarly, an improved algorithm with reference signals update after each one-dimensional optimization has been presented in [12], which shows better performance. Based on [11] and [12], a new family of algorithms to maximize a kurtosisbased contrast function is proposed [7], whose global convergence to a stationary point is proved. And a tradeoff can be adjusted between performance and speed of the optimization method.…”

“…Inspired by [7] and [12], a similar reference-based contrast function is constructed, based on which a novel algorithm is proposed for complex-valued signals. More precisely, the cross-cumulant between estimated signals and reference signals is utilized as the contrast criterion in terms of kurtosis.…”

New Negentropy Optimization Schemes for Blind Signal Extraction of Complex Valued Sources

“…They are particularly appealing because the corresponding maximization problem can be evolved into a quadratic optimization problem with respect to the searched parameters. Tak-ing advantage of this quadratic feature, a few maximization algorithms based on kurtosis contrast function have been proposed and significantly quicker than traditional kurtosis-based algorithms [21][22][23]. However, these methods generally require searching for the optimal step-size and behave sensitive to outliers.…”