A design challenge in AEC algorithms is the dual between fast convergence and cancellation accuracy in terms of ERLE. Whilst choosing a small step size will enhance the accuracy results, the convergence rate will be long which makes it quite impractical for RTOSs. A compromise has been the variable descent step size affine projection based algorithms. Consider the systems depicted in Figure 1 below:


Single line AEC architecture

Figure 1: Single line AEC architecture

Consider a classical affine projection algorithm which proceeds as:

{\hat{\bf e}[n]}= {\bf d}[n] - {\bf X}^T \hat{{\bf h}}[n-1]

\hat{{\bf h}}[n] = \hat{{\bf h}}[n-1] + {\bf \mu}[n] {\bf X} [ {\bf X}^T {\bf X}]^{-1} \hat{\bf e}[n]

where {\bf d}[n] = [d[n], \cdots, d[n-L+1]]^T is the desired signal, {\bf X} = [{\bf x}[n], \cdots, {\bf x}[n-L+1]]^T is the input matrix with each vector {\bf x}[k], k \in \{n, \cdots,n-L+1\} of length K. \hat{{\bf h}}[n] is a vector of the filter weights and {\bf \mu}[n] is the variable descent step size. It should be noted that \hat{{\bf e}[n]} is the apriori error and the output of the system is the a posterior error defined as:

{\bf e}[n]= {\bf d}[n] - {\bf X}^T \hat{{\bf h}}[n]

Replacing \hat{{\bf h}}[n] in (2) with (1) we get:

{\bf e}[n]= ({\bf 1} - {\bf \mu}[n]) \hat{\bf e}[n]

In the case where there is no near end speech, {\bf \mu}[n] = 1 \forall n, which is the classical set up. However, in a more practical scenario, there is always near end signal. Denote the near end signal as {\bf \nu}[n], then, using the second order statistics,{\bf \mu}[n]= 1 \pm \sqrt{\frac{\sigma_{{\bf \nu}[n]}^2}{\sigma_{{\bf e}[n]}^2}}
A heuristic is employed in estimating the second order statistics of both the near end speech and the error signal.

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