A Bias Removal Technique for Blind Source Separation With Noisy Measurements

In blind source separation measurement noise introduces biases in the esti mated coe cients of the separation matrix In this letter a simple modi cation to an adaptive algorithm for blind source separation is derived that mitigates these bi ases for reasonable signal to noise ratios when the correlations of the noises are known Index Terms adaptive algorithm bias removal blind source separation to appear in ELECTRONICS LETTERS Accepted for publication June Please address correspondence to Scott C Douglas Department of Electrical Engineering School of Engi neering and Applied Science Southern Methodist University P O Box Dallas TX USA Voice FAX Electronic mail address douglas seas smu edu World Wide Web URL http www seas smu edu ee Introduction Blind source separation of instantaneous signal mixtures has received much attention recently In this task a set of signal vectors x k x k xn k T is generated from a set of source signal vectors s k s k sm k T m n as x k Hs k where H is an n m dimensional mixing matrix These measured signals are processed by an m n dimensional demixing matrix W k such that y k W k x k where y k y k ym k T contains the source estimates If W k is adjusted such that lim k W k H PD where P is an m m dimensional permutation matrix and D is a diagonal matrix of scaling factors fdiig i m then a scaled version of each source signal appears in the vector y k One useful algorithm that has been developed for the above task is W k h I k n I f y k y k oi W k where k is a learning rate parameter f y k f y k fm ym k T and the each fi y depends on the source signal sj k extracted at the ith output In many practical situations x k is more accurately modeled as x k Hs k k where k k n k T contains zero mean noise terms Even a small amount of noise can signi cantly bias W k away from that speci ed by for algorithms such as In this letter a novel modi cation of is derived that reduces the bias in the estimated matrix W k if the measurement noise is small and the correlation of k is known The new algorithm employs a bias compensating correction term Simulations indicate that the modi ed algorithm signi cantly reduces the coe cient biases due to measurement noise Bias Removal in Blind Source Separation For this derivation it is assumed that k is a zero mean jointly Gaussian random vector with R Ef k T k g whose i j th element is r ij Ef i k j k g although this assumption shall be relaxed in the sequel Each fi y is twice di erentiable An unbiased estimate of W k is obtained by the algorithm W k h I k n I f b y k b y k oiW k where b y k W k Hs k Since f y k y k in is biased due to noise a matrix M k is sought such that Eff y k y k g EfM k g Eff b y k b y k g Substituting into and into Eff y k y k g respectively one obtains Eff y k y k g Eff b y k W k k b y k T k W k g The i j th element of is Effi yi k yj k g E fi b yi k n X l wil k l k b yj k E fi b yi k n X