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I have just started learning some DSP, I keep seeing things like memory in regard to filters. I am unsure exactly what this means and why it is useful. I understand that increasing the number of taps increases the memory etc.. and a tap is essentially just a coefficient for changing the signal.

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Filter memory refers to the effect of past input on the current input. Infinite impulse response (IIR) filters and finite impulse response (FIR) filters are fundamentally different from the perspective of their memory. The memory of an FIR will essentially be the number of taps, while IIR has theoretically infinite memory (although when they are stable, the memory passes under the numerical accuracy of the system beyond a certain point).

More on this http://gregstanleyandassociates.com/whitepapers/FaultDiagnosis/Filtering/Filter-Memory/filter-memory.htm.

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Energy stored in capacitors and inductors, lumped or distributed, within the filter, whose current state constrains what could have happened in the past to allow said current state, which thus can be called a form of memory of that past. Or the digital state equivalent.

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The output of a FIR filter depends on how many coefficients it has. For example, a 64-tap filter's output will be calculated from its 64 coefficients, and the latest input sample + the 63 ones before, so we can say it has 63 memory units.

This is related to the use of shift registers in digital filters (one per coefficient), who effectively are memories.

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