## 3 An Efficient Speech CBSS Algorithm in Multipath Channels
### Fast IVA Algorithm  
The Fast IVA algorithm uses the negative entropy criterion to measure the strength of the non-Gaussian and minimizes the contrast function over each frequency band to complete the estimation of the source signals. The advantage of this algorithm is that it performs CBSS in order at the ordered frequency points, thus solving the permutation ambiguity problem. On this basis, this paper uses Taylor series in the algorithm optimization process to approximate the contrast function of Fast IVA and then performs fast and ordered CBSS of speech signals at each frequency point. Here, the Kullback-Leibler divergence [34] between the two functions is defined as a measure of high-order dependence, which is the cost function of multivariate random variables. The relative entropy [24] is expressed as follows:  
\[ \begin{split} C&=KL\Big{(}p(s_{1},s_{2},...,s_{N}) \Big{\|}\prod_{i}q(s_{i})\Big{)}\\ &=const.+\sum_{k}\log\Bigl{|}\det(H^{(k)})\Bigr{|}-\sum_{i}E_{s_ {i}}(\log q(s_{i}))^{\prime}\end{split}  \tag{7}\]  
where \(p(s_{1},s_{2},...,s_{N})\) represents the exact joint probability density function of a single source vector; \(\prod_{i}q(s_{i})\) represents the product of approximate marginal probability distribution functions; \(k\) is the number of frequency points; \(det(\cdot)\) represents the matrix determinant operator; \(E(\cdot)\) denotes the expectation, and \(KL(\cdot)\) denotes the Kullback-Leibler scatter calculation. For simplicity, suppose that \(x_{i}\) is zero-mean and the separated signal \(y_{i}\) is pre-whitened so that the rows of the mixing matrix \(H^{(k)}\) or the separation matrix \(W^{(k)}\) are orthogonal in each dimension, i.e., \({A^{(k)}}^{T}A^{(k)}=I\) or \({W^{(k)}}{W^{(k)}}^{T}=I\). Therefore, Equation (7) can be simplified as:  
\[C=const.+\sum_{i}E_{s_{i}}(\log q(s_{i})). \tag{8}\]  
In this paper, the fast fixed-points method is chosen to optimize the contrast function. Compared with the natural gradient method, the Fast IVA algorithm avoids the selection of learning rate and has the advantage of fast convergence. During the optimization of the algorithm, we introduce a Taylor series approximation to the contrast function of Lagrangian residue type in the notation of the complex variables in order to be able to quickly obtain the optimal approximate solution. Therefore, the contrast function of the Fast IVA algorithm with Lagrange multipliers \(\beta\) is obtained as follows:  
\[C=-\sum_{i}E_{s_{i}}(\log q(s_{i}))-\sum_{k}\beta\Bigl{(}{W^{(k)}}^{T}W^{(k)}- I\Bigr{)}. \tag{9}\]  
Since the Hessian matrix of the contrast function [35] is a diagonal matrix under the whiteness constraint, the following simple learning rule can be obtained by applying [PERSON]'s method:  
\[w_{i}^{(k)}\gets w_{i}^{(k)}-\frac{E\Bigl{[}\varphi^{(k)}\Bigl{(}y_{i}^{(1 )},y_{i}^{(2)},...,y_{i}^{(K)}\Bigr{)}x^{(k)}\Bigr{]}+\beta w_{i}^{(k)}}{E \Bigl{[}\varphi^{(k)^{\prime}}\Bigl{(}y_{i}^{(1)},y_{i}^{(2)},...,y_{i}^{(K)} \Bigr{)}\Bigr{]}+\beta}. \tag{10}\]  
It can be known that the equilibrium point of Equation (10) above is the local minima of the contrast function, since the demixing matrix is no longer updated. When the equation holds, the equilibrium point is found. The Lagrange multiplier is eliminated to obtain the following fixed point iteration algorithm:  
\[\left\{ \begin{array}{l}w_{i}^{(k)}\gets E\Bigl{[}\varphi^{(k)^{\prime} }\Bigl{(}y_{i}^{(1)},y_{i}^{(2)},...,y_{i}^{(K)}\Bigr{)}\Bigr{]}w_{i}^{(k)}-E \Bigl{[}\varphi^{(k)}\Bigl{(}y_{i}^{(1)},y_{i}^{(2)},...,y_{i}^{(K)}\Bigr{)}x^ {(k)}\Bigr{]}\\ \varphi^{(k)^{\prime}}\Bigl{(}y_{i}^{(1)},y_{i}^{(2)},...,y_{i}^{(K)}\Bigr{)} =-\frac{\partial\varphi^{(k)}\Bigl{(}y_{i}^{(1)},y_{i}^{(2)},...,y_{i}^{(K)} \Bigr{)}}{\partial y_{i}^{(k)}}\end{array} \right.. \tag{11}\]  
In addition to normalization, the rows of demixing matrix \(w_{i}^{(k)}\) need to be decorrelated.