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
. 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
. 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
where is additive noise. Also consider that is known. We want to estimate such that is minimized, where
Define the cost function , where , 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
The gradient descent algorithm to estimate the filter coefficients will thus be as follows:
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