Echo Cancellation Demystified
It is worth mentioning that such an echo cancellation scheme is naturally linear and is very sensitive to the nonlinearities in the echo path. This linear system will not be able to match the impulse response of a nonlinear echo path and therefore effectively remove the echo.
Good echo cancellation performance can be achieved by using the NLMS (Normalized Least Mean Squares) algorithm, which is also known as the normalized stochastic gradient algorithm, or its many variations. The NLMS algorithm is the most widely used one and it provides a low cost way to determine the optimum filter coefficients. The algorithm minimizes the mean square of the residual echo error signal at each adaptation step (e.g. at each sample), hence the name of the algorithm. Normalization by the signal power is used because speech is a highly non-stationary process.
Without derivations, which you can find elsewhere in the literature on adaptive signal processing, we give the general formula for the coefficient adaptation for the NLMS algorithm:
i is the sample number
ak is the k-th coefficient of the filter
N is the number of filter coefficients
b is the adaptation step, which controls the convergence time and adaptation quality
e is the residual echo error signal
y is the far-end talker signal
s2 is the reference signal power
The number of coefficients in the filter should be large enough to cover the echo path delay and all additional delays due to the lines and circuitry between the echo canceller and the place where the echo is created (e.g. the total delay between points A and B of the echo canceller as per Figure 5). This should also include the dispersion time due to the network elements.
The hybrid echo path delay is known to be short. The time span over which its impulse response is significant, is typically 2 to 4 milliseconds, but usually when canceling the hybrid echo, the number of filter coefficients is chosen to cover hybrid echo path delays up to 16 milliseconds, which is usually the upper bound of the hybrid echo path delay. The 16 ms path needs 128 coefficients at the sampling rate of 8 KHz.
The length of the acoustic echo path, as has already been pointed out, depends on the size of the room, where it exists. So, without going into room measurements, for acoustic echo path delays of up to 256 ms we will have to have 2048 coefficients at the sampling rate of 8 KHz.
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