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An adaptive filter is a filter that self-adjusts its transfer function according to an optimization algorithm driven by an error signal.

Adaptive filters are classified into two main groups: linear, and non linear. Linear adaptive filters compute an estimate of a desired response by using a linear combination of the available set of observables applied to the input of the filter. Otherwise, the adaptive filter is said to be nonlinear. Adaptive filters are required for some applications because some parameters of the desired processing operation (for instance, the locations of reflective surfaces in a reverberant space) are not known in advance.

References:
1. Adaptive filter
2. Haykin, S., Adaptive filters (PDF)

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