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Description of block diagram
Simple IIR filter block diagramA typical block diagram of an IIR filter looks
like the following. The Z − 1 block is a unit delay. The coefficients and number
of feedback/feedforward paths is implementation-dependent.
Stability
The transfer function allows us to judge whether or not a system is
bounded-input, bounded-output (BIBO) stable. To be specific, the BIBO stability
criteria requires all poles of the transfer function to have an absolute value
smaller than one. In other words, all poles must be located within a unit circle
in the z-plane.
The poles are defined as the values of z which make the denominator of H(z)
equal to 0:
Clearly, if then the poles are not located at the origin of the z-plane. This is
in contrast to the FIR filter where all poles are located at the origin, and is
therefore always stable.
IIR filters are sometimes preferred over FIR filters because an IIR filter can
achieve a much sharper transition region roll-off than FIR filter of the same
order.
Example
Let the transfer function of a filter H be
with ROC a < | z | and 0 < a < 1
which has a pole at a, is stable and causal. The time-domain impulse response is
h(n) = anu(n)
which is non-zero for n > = 0.
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