The least mean squares (LMS) algorithm is attributed to Widrow and Hoff, and was introduced in 1960. It is the commonest form of filter adaption used in things like modem line equalisers and line echo cancellers. There it works very well. However, it only works well for signals of constant amplitude. It works very poorly for things like speech echo cancellation, where the signal level varies widely. This is quite easy to fix. If the signal level is normalised - similar to applying AGC - LMS can work as well for a signal of varying amplitude as it does for a modem signal. This normalised least mean squares (NLMS) algorithm is the commonest one used for speech echo cancellation. Many other algorithms exist - e.g. RLS (essentially the same as Kalman filtering), FAP, etc. Some perform significantly better than NLMS. However, factors such as computational complexity and patents favour the use of NLMS.
A simple refinement to NLMS can improve its performance with speech. NLMS tends to adapt best to the strongest parts of a signal. If the signal is white noise, the NLMS algorithm works very well. However, speech has more low frequency than high frequency content. Pre-whitening (i.e. filtering the signal to flatten its spectrum) the echo signal improves the adapt rate for speech, and ensures the final residual signal is not heavily biased towards high frequencies. A very low complexity filter is adequate for this, so pre-whitening adds little to the compute requirements of the echo canceller.
An FIR filter adapted using pre-whitened NLMS performs well, provided certain conditions are met:
The difficulty is that neither of these can be guaranteed.
If the adaption is performed while transmitting noise (or something fairly noise like, such as voice) the adaption works very well. If the adaption is performed while transmitting something highly correlative (typically narrow band energy such as signalling tones or DTMF), the adaption can go seriously wrong. The reason is there is only one solution for the adaption on a near random signal - the impulse response of the line. For a repetitive signal, there are any number of solutions which converge the adaption, and nothing guides the adaption to choose the generalised one. Allowing an untrained canceller to converge on this kind of narrowband energy probably a good thing, since at least it cancels the tones. Allowing a well converged canceller to continue converging on such energy is just a way to ruin its generalised adaption. A narrowband detector is needed, so adapation can be suspended at appropriate times.
The adaption process is based on trying to eliminate the received signal. When there is any signal from within the environment being cancelled it may upset the adaption process. Similarly, if the signal we are transmitting is small, noise may dominate and disturb the adaption process. If we can ensure that the adaption is only performed when we are transmitting a significant signal level, and the environment is not, things will be OK. Clearly, it is easy to tell when we are sending a significant signal. Telling, if the environment is generating a significant signal, and doing it with sufficient speed that the adaption will not have diverged too much more we stop it, is a little harder.
The key problem in detecting when the environment is sourcing significant energy is that we must do this very quickly. Given a reasonably long sample of the received signal, there are a number of strategies which may be used to assess whether that signal contains a strong far end component. However, by the time that assessment is complete the far end signal will have already caused major mis-convergence in the adaption process. An assessment algorithm is needed which produces a fairly accurate result from a very short burst of far end energy.