A recursive least-squares extension of the natural gradient algorithm for blind signal separation of audio mixtures
Maha Mohamed Elsabrouty, T. Aboulnasr, Martin Bouchard · Canadian acoustics · 2004
this paper. 2. EXISTING MAXIMUM-LIKELIHOOD BASED ALGORITHMS Let N i n s i , , 2 , 1 ), ( K = be scalar inputs (or sources) to the blind signal separation model at a time n . For simplicity, it is assumed that the mixing is linear and that the mixing matrix is square, i.e. the number of inputs N is equal to the number of mixtures N i n x i , , 2 , 1 ), ( K = . Therefore, the mixing matrix A is a square matrix of size N N . The mixing model can be expressed as: ) ( ) ( n n s A x = (1). The mixture x is then applied to a whitening matrix V . The resulting whitened mixtures in z are expressed as: ) ( ) ( ) ( ) ( n n n n s B s A V x V z = = = (2), where B is the resulting mixing matrix after the whitening stage. The purpose of the blind signal separation algorithms is to estimate a matrix W such that N N = I B W , where I is an identity matrix. Then the outputs of the separation process referred to as ) (n y would be identical to the source inputs ) (n s . Maximum Likelihood targets a separation via increasing the likelihood between the outputs ) (n i y and the inputs ) (n i s [5]. In the case of pre-whitened inputs, the cost function of the log-likelihood ) (W L of the de-mixing matrix W can be expressed as: # # # # # # # # # = # i i i p E L ) ( log ) ( z w W (3), where {} E refers to the expected value, i w is the th i row of the matrix W and () i p is a probability density function. The above cost function has the gradient ) (W L # as: # # # # # # # # # = = # # i T E L ) ( g log ) ( z y W (4), where i p y g = ) ( and is usually set to ) tanh( 2 i y for supergaussian data, such as audio data. Pre-whitening also constrains the matrix W to be orthogonal, meaning that N N = I W W . This constraint places the optimization of the cost func...