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Output from a digital filter is made up from previous inputs and previous outputs, using the operation of convolution:
Two convolutions are involved: one with the previous inputs, and one with the previous outputs. In each case the convolving function is called the filter coefficients.
If such a filter is subjected to an impulse (a signal consisting of one value followed by zeroes) then its output need not necessarily become zero after the impulse has run through the summation. So the impulse response of such a filter can be infinite in duration. Such a filter is called an Infinite Impulse Response filter or IIR filter.
Note that the impulse response need not necessarily be infinite: if it were, the filter would be unstable. In fact for most practical filters, the impulse response will die away to a negligibly small level. One might argue that mathematically the response can go on for ever, getting smaller and smaller: but in a digital world once a level gets below one bit it might as well be zero. The Infinite Impulse Response refers to the ability of the filter to have an infinite impulse response and does not imply that it necessarily will have one: it serves as a warning that this type of filter is prone to feedback and instability.
The filter can be drawn as a block diagram:
The filter diagram can show what hardware elements will be required when implementing the filter:
The left hand side of the diagram shows the direct path, involving previous inputs: the right hand side shows the feedback path, operating upon previous outputs.
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