I just have a question about using an least-mean-squares algorithim adaptive filter for system identification. Consider the following enter image description here

I am told that as the error converges to a small value, the adaptive filter coefficients w[k] will indeed repersent the unknown system h[k]. Now, that doesn't make sense to me since the n[k] is being used in the error calculation.

Won't the adaptive filter coefficients w[k] now repersent the unknown system coefficients h[k] AND the noise n[k]?


Because an LMS estimator will, over time, "average out" uncorrelated zero-mean noise. It's pretty much in the name.

But yes, you're right, there is a noise component in the estimate; one of the qualities of an estimator is how little the noise variance influences the estimate variance after a given length of observation.

That's the case, however, for all estimators: you measure signal + noise, and you estimate parameters from that. The parameters must be somewhat noisy; otherwise, there's something broken with your noise model.

  • $\begingroup$ Ah. I assume, this would work without the noise too then. $\endgroup$ – AlfroJang80 Dec 5 '18 at 16:55
  • $\begingroup$ it probably does in this case, but there's estimators which need the noise to work. $\endgroup$ – Marcus Müller Dec 5 '18 at 16:57

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