On the blind implementation of the ultimate source separators for arbitrary noisy mixtures☆
Introduction
Since more than a decade, the SO [1], [2], [16], [17], [18] and HO [3], [6], [8], [9], [10], [11], [12], [14] blind source separation methods are strongly developed to process, in particular, instantaneous mixtures of statistically independent NB sources. Noting N and P the number of sensors and the number of sources received by the latter respectively, such separators, assumed linear and time invariant (TI), are characterized by an (N×P) complex matrix W and aim at outputting, for a given (N×1) complex observation vector , the (P×1) vector whose components are as independent as possible in terms of a given statistical independence criterion, where † means transposition and conjugation. Among the widely used independence criteria, we find in particular the cancellation or the minimization of the SO correlation between the outputs, either at the same time [1] or at a different time [2], [16], [17], [18], or of some functions of the FO cross-cumulants of the outputs [3], [6], [9], [10], [12], [14] which is sometimes equivalent to maximizing some FO contrast functions [6], [7], [9], [13], [15]. In this context, it seems important to wonder whether these particular SO or HO blind methods may implement, in situations of practical interest, some ultimate, informed or optimal linear and TI source separators, corresponding either to the SMF or to the WLS source separator, whose properties and performances have been recently described in [5]. The lack of awareness to this important problem, in the general context of arbitrary noisy mixtures of sources, may limit the use of these blind methods in operational situations.
In any case, a necessary condition for a given SO or FO method to blindly implement the SMF or the WLS source separator is that the optimization of the SO or FO independence criterion used at the outputs of this method accurately interprets some SO or FO properties of the ultimate separators. This is the reason why, assuming instantaneously mixed NB sources, the purpose of this paper is to enlighten the previous important problem through the analysis of the SO correlation and the FO cross-cumulants at the SMF and WLS separators outputs for arbitrary noisy mixtures of sources.
Section snippets
Hypotheses and notations
Assuming that the N sensors of the array receive a noisy and instantaneous mixture of P stationary and statistically independent NB sources, the observation vector can be written as follows:where is the noise vector, assumed stationary, mi(t) and are the complex envelope, assumed stationary, and the steering vector of the source i respectively, is the vector whose components are the mi(t) and A is the (N×P) matrix of the source steering
The SMF and WLS NB source separators
The SMF, Wsmf, and the WLS, Wwls, NB source separators, defined byhave been shown in [5] to correspond to the optimal and quasi-optimal (optimal interference canceller or (OIC)) NB source separators, respectively, in terms of an output signal to interference plus noise ratio (SINR) performance criterion maximization. Using expression (1), their output vector can be written as
In the
Arbitrary noisy situations
From (6) and (7), we deduce the expression of the correlation matrix, , of the output vector for the SMF and the WLS separators, given bywhere is a diagonal matrix and .
Firstly, expressions (8) and (9) show that if Rm(τ)=0 and Rb(τ)=0 (white sources, white noise and τ≠0 for example), then Ry,smf(τ)=Ry,wls(τ)=0 and the
Arbitrary noisy situations
Noting Qy the quadricovariance of the vector , such that , , , yl(t)), where yi(t) is the component i of , we deduce, from (6) and (7), the quadricovariance at the output of the SMF and the WLS separator, given by
Conclusion
In this paper, the SO correlation and the FO cross-cumulants at the output of the SMF and the WLS separators have been discussed.
If the outputs of these informed source separators are both SO and FO decorrelated in the absence of noise, it is generally no longer true in the presence of an arbitrary background noise, which makes outputs SO and FO correlation appear and which generally prevent these ultimate separators to be blindly implemented from the SO and FO statistics of the observations.
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This work was supported by the Direction des Recherches et Etudes Techniques (DRET) of the French Defence Office.