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I have a general question regarding Least mean squares adaptive filters.

Using the example of noise cancellation, I understand that if you have a set of reference signals (S) and corrupted signals (S+N) then you can perform minimisation of the error to find the optimal weights for the set of inputs.

I am wondering if this is normally performed 'offline' or if this can be performed on the fly with the weights changing over time?

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Adaptive means that you try to find a "best" filter, locally, according to some criterion. Using it online is not mandatory. Indeed on images the requirements for "onlinity" or causality are often less stringent than for signals.

However, for efficiency, it can be beneficial to update coefficients every time, or even more perform some approximation.

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That really depends on context, but generally adaptive implies that the calculations are done on-line / on the fly.

In some applications, the filter is updated for a while, then the adaptation is turned off and the last lot of weights are used.

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To expand on what Peter K. has said, if the signals being used by the filter are stationary, then the filter weights or coefficients can be determined and the filter operates as it was designed without further updates to the filter weights. However, if the signals change, or become quasi-stationary, the filter will adapt continuously.

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