Mathematical Problems in Engineering

Volume 2017 (2017), Article ID 2546838, 9 pages

https://doi.org/10.1155/2017/2546838

## Blind Separation of Cyclostationary Sources Sharing Common Cyclic Frequencies Using Joint Diagonalization Algorithm

^{1}Université de Lyon, UJM-Saint-Etienne, LASPI, IUT de Roanne, 42334 Roanne, France^{2}Université Sidi Mohamed Ben Abdellah, FSTF, LSSC, BP 2202, Route d’Immouzzer, Fès, Morocco

Correspondence should be addressed to Amine Brahmi

Received 9 October 2016; Accepted 31 January 2017; Published 22 February 2017

Academic Editor: Thomas Schuster

Copyright © 2017 Amine Brahmi et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We propose a new method for blind source separation of cyclostationary sources, whose cyclic frequencies are unknown and may share one or more common cyclic frequencies. The suggested method exploits the cyclic correlation function of observation signals to compose a set of matrices which has a particular algebraic structure. The aforesaid matrices are automatically selected by proposing two new criteria. Then, they are jointly diagonalized so as to estimate the mixing matrix and retrieve the source signals as a consequence. The nonunitary joint diagonalization (NU-JD) is ensured by Broyden-Fletcher-Goldfarb-Shanno (BFGS) method which is the most commonly used update strategy for implementing a quasi-Newton technique. The efficiency of the method is illustrated by numerical simulations in digital communications context, which show good performances comparing to other state-of-the-art methods.

#### 1. Introduction

The target of blind source separation (BSS) is to retrieve the input signals called sources from their mixtures coming to multiple sensors without any preknowledge about the mixing process. BSS is a major problem of signal processing which has been addressed in the last three decades (see [1] for a review). In literature, many approaches have been developed in order to figure out this issue using statistics of second and fourth order, namely, Second-Order Blind Identification (SOBI) [2] and Joint Approximate Diagonalization of Eigen matrices (JADE) [3]. These approaches have proved to establish some limitations in a wide scope of practical situations where the source signals are nonstationary and very often cyclostationary such as radiocommunications, telemetry, radar applications, and mechanics [4]. In fact, according to Ferreol in [5], the stationarity assumption of source signals performs ineffectively BSS problem.

Cyclostationarity is a subclass of nonstationarity which distinguishes stochastic processes whose statistics change periodically with time. Thus, it is necessary to consider cyclostationarity to perform BSS. Many methods have been proposed to blindly achieve the separation for cyclostationary sources. Brahmi et al. have solved the problem of blind identification of FIR MIMO systems driven by cyclostationary inputs whose cyclic frequencies are pairwise distinct using joint block diagonalization based on BFGS method in [6]. Liang et al. in [7] use the information provided from the cyclic frequencies so as to separate source signals. Abed-Meraim et al. [8] address the problem of BSS assuming that the source signals are cyclostationary based on an iterative algorithm using the cyclic correlation function of observation signals. This method is useful when each source signal has only one cyclic frequency and the number of the source signals which share a common cyclic frequency is known. Ghennioui et al. [9] have proposed a new approach combining a nonunitary joint diagonalization algorithm to a general automatic matrices selection procedure for the case of unknown and different cyclic frequencies. Ghaderi et al. present in [10] a method for blind source extraction of cyclostationary sources, whose cyclic frequencies are known and share some common ones. Jafari et al. in [11, 12] propose an adaptive blind source separation algorithm for the separation of convolutive mixtures of cyclostationary signals based on natural gradient algorithm. The aforementioned algorithm requires estimating the cycle frequencies of source signals. Boustany and Antoni [13] propose a method for blind extraction of one cyclostationay signal using a subspace decomposition of the observation signals via their cyclic statistics. Capdessus et al. [14] propose an algorithm for the extraction a signal of interest which is cyclostationary one and its cyclic frequency is a priori known. This algorithm relies on second-order statistics of the observation signals. Rhioui et al. propose in [15] a method for the mixing matrix identification for underdetermined mixtures of cyclostationary signals with different cyclic frequencies. Jallon and Chevreuil [16] have come up with a justification for using the common algorithm for the cyclostationary context despite the fact that it has been originally developed for the stationary one. Pham [17] has proposed a new approach based on joint diagonalization of a set of cyclic spectral density of observation matrices. The two last approaches are addressed in the simplest mixture model (noise-free data). Despite the fact that these algorithms are successful under assumed conditions, they have diverse limitations, since, in front of real situations, the cyclic frequencies in most of cases are unknown and may be shared by source signals.

The main purpose of this work is to perform blind separation of instantaneous mixtures of cyclostationary source signals which may share one or more common cyclic frequencies whose preknowledge is not needed. By exploiting the particular structure of cyclic correlation matrices of source signals, we show that the considered problem can be rephrased as a problem of joint diagonalization of matrices that have been automatically selected using a new procedure. Then, the joint diagonalization algorithm based on BFGS method [18] is applied on this set of matrices. The rest of this paper is organized as follows. Section 2 formulates the problem of interest. Section 3 puts forward some theoretical preliminaries related to cyclostationarity and NU-JD. Section 4 describes the proposed method for BSS. In Section 5, the performances of the proposed method are numerically evaluated and compared with other existing methods in digital telecommunications context. Finally, in the Section 6, conclusions are drawn.

#### 2. Problem Statement

The BSS problem can be modelled in a simple linear instantaneous mixture of emitted source signals that are received by sensors (see Figure 1). The relationship of mixing system is given bywhere is the observations vector, is the vector of unknown source signals, is the additive noise vector, and is the unknown mixing matrix. The purpose of BSS is to find an estimate of and retrieve from only, aswhere , denotes pseudoinverse matrix, is a permutation matrix (corresponding to an arbitrary order of restitution of the sources), is a diagonal matrix (corresponding to arbitrary scaling for the recovered sources), , represents the phase vector (corresponding to phase shift ambiguity in complex domain of the source signals), and is square diagonal matrix containing the elements of the vector . Thus, one should look for a separating matrix such that