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We start the discussion by stating the difference equation which defines how
the input signal is related to the output signal:
where:
P is the feedforward filter order
bi are the feedforward filter coefficients
Q is the feedback filter order
ai are the feedback filter coefficients
x[n] is the input signal
y[n] is the output signal.
A more condensed form of the difference equation is:
which, when rearranged, becomes:
if we let a0 = 1.
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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