An acoustic echo signal (AES) is a time delayed version of a signal which in general is undesired. The presence of AES can degrade the quality of a signal significantly hence the vast interest in acoustic echo cancellation (AEC) algorithms that reduce significantly AES presence in transmitted speech.

Consider the systems depicted in Figure 1 below:

Figure 1: Single line AEC architecture

The problem then is to correctly find the filter that will estimate correctly with minimum error

$f(h[.,k],x[.]) = \sum\limits_{i = 1}^M h[i] x[n-i]$

. The presence of the signal $s[n]$ may cause the adaptive system to cancel out the speech signal, thus leading to most AECs updating the filter coefficients only when there is no speech detected. Now consider a received signal

$r[n] = \sum\limits_{i=1}^M h[i]x[n-i] + v[n]$

.  The presence of the signal $s[n]$ may cause the adaptive system to cancel out the speech signal, thus leading to most AECs updating the filter coefficients only when there is no speech detected. Now consider a received signal

$r[n] = \sum\limits_{i=1}^M h[i]x[n-i] + v[n]$

where $v[n]$ is additive noise. Also consider that $x[n-i], \forall i \in \{1, \cdots, M\}$ is known. We want to estimate $\hat{h}[i], \forall i \in \{1, \cdots, M\}$ such that $|e[n]|^2$ is minimized, where

$e[n] = \sum\limits_{i=1}^M (h[i]-\hat{h}[i])x[n-i] + v[n]$

Define the cost function $J(\hat{{\bf h}}) = \sum\limits_{k=1}^N |e[k]|^2$, where ${\bf \hat{h}}= [\hat{h}[1], \cdots,\hat{h}[M]]^T$, and consider the noise signal being i.i.d. Gaussian. Then the gradient of the cost function for a frame of length N in the noise free case can be given as

$\frac{\partial J(\hat{{\bf h}})}{\partial \hat{h}[i]} = - 2 \sum\limits_{k=1}^N \left(r_k-\sum\limits_{i=1}^M \hat{h}[i]x[k-i] \right) x[k-i] , \forall i$

The gradient descent algorithm to estimate the filter coefficients will thus be as follows:

$\hat{h}[i]_{n+1} = \hat{h}[i]_{n} + 2 \mu \sum\limits_{k=1}^N \left(r_k-\sum\limits_{i=1}^M \hat{h}[i]_{n}x[k-i] \right) x[k-i] , \forall i$

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