Editing Digital filter (section)

====Impulse response====

The [[impulse response]], often denoted <math>h[k]</math> or <math>h_k</math>, is a measurement of how a filter will respond to the [[Kronecker delta]] function.<ref>{{cite web |title=Lab.4&5. Introduction to FIR Filters |url=http://www.just.edu.jo/~hazem-ot/Lab.4&5.%20FIR%20Filters.pdf |archive-url=https://ghostarchive.org/archive/20221009/http://www.just.edu.jo/~hazem-ot/Lab.4&5.%20FIR%20Filters.pdf |archive-date=2022-10-09 |url-status=live |publisher=Jordan University of Science and Technology-Faculty of Engineering |access-date=13 July 2020}}</ref> For example, given a difference equation, one would set <math>x_0 = 1</math> and <math>x_k = 0</math> for <math>k \ne 0</math> and evaluate. The impulse response is a characterization of the filter's behavior.  Digital filters are typically considered in two categories: [[infinite impulse response]] (IIR) and [[finite impulse response]] (FIR).
In the case of linear time-invariant FIR filters, the impulse response is exactly equal to the sequence of filter coefficients, and thus:

:<math>\ y_n= \sum_{k=0}^{N} b_{k} x_{n-k} =\sum_{k=0}^{N} h_{k} x_{n-k}</math>

IIR filters on the other hand are recursive, with the output depending on both current and previous inputs as well as previous outputs. The general form of an IIR filter is thus:
:<math>\ \sum_{m=0}^{M} a_{m}y_{n-m} = \sum_{k=0}^{N} b_{k} x_{n-k}</math>

Plotting the impulse response reveals how a filter responds to a sudden, momentary disturbance.
An IIR filter is always recursive. While it is possible for a recursive filter to have a finite impulse response, a non-recursive filter always has a finite impulse response. An example is the moving average (MA) filter, which can be implemented both recursively{{citation needed|date=May 2019}} and non recursively.