Spider DSA User’s Manual
319
IIR Real Time Digital Filters
Infinite impulse response (IIR)filters have an impulse response that decays
very slowly and theoretically lasts forever. This is due to the fact that the filter
input includes the measured signal and also the filter output creating a feedback
path which results in the infinite impulse duration. This is in contrast to finite
impulse response filters (FIR) which have fixed-duration impulse responses.
The design procedure for IIR filters is somewhat more complicated than FIR filter
design because there is no direct design method like the data window method for
FIR filters. IIR filters are typically designed by starting with an ideal analog filter
in terms of the frequency response characteristics such as the Chebyshev,
Butterworth, or Bessel filter. Then the analog filter is converted into a digital filter
using a method known as the Bilinear transformation or the impulse invariance
method.
An IIR digital filter can be understood by considering the equation that defines
how the input signal is related to the output signal:
Where P is the feed-forward filter order,
are the feed-forward filter coefficients,
Q is the feedback filter order,
are the feedback filter coefficients,x[n] is the input
signal and y[n] is the output signal.
The previous equation can also be expressed as a convolution of the filter
coefficients and the input signal.
Which, when rearranged, becomes:
To find the transfer function of the filter, we first take the Z-transform of each side
of the above equation, where we use the time-shift property to obtain:
We define the transfer function to be:
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