Infinite impulse response

An infinite impulse response (IIR) filter, as suggested by its name, has an impulse response which lasts forever (or since the signal processing began). An IIR is infinite because feedback is used by saving a history of prior calculations. This means that IIR filters are causal, or that the output depends on past inputs. Filtering methods like IIR that are causal will have a phase shift (delay) in the resulting filtered output.

An IIR filter is typically characterized by its order, which is the number of feedback stages required. An infinite impulse response filter having P feed-forward stages and Q feedback stages has the following form:

y_n = b_0 x_n + b_1 x_{n-1} + \cdots + b_P x_{n-P} + a_1 y_{n-1} + a_2 y_{n-2} + \cdots + a_Q y_{n-Q}

The z-transform of the above Qth-order filter is given as

H(z) = \frac{\sum_{k=0}^P b_k z^{-k}} {1 - \sum_{k=1}^Q a_k z^{-k}}

If the poles of H(z) are located inside the unit circle then the filter is stable. If any pole is outside the unit circle, the filter will be unstable. If a filter is unstable, its impulse response will be unbounded.

Infinite impulse response filters are sometimes preferred to finite impulse response(FIR) filters because an IIR filter can achieve a much sharper transition region roll-off than an FIR filter of the same order.


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