The functions described in this section initialize an infinite impulse response (IIR) filter and perform a zero-phase digital filtering of input data in both forward and backward directions. The formulas below explain why the filtered signal has zero-phase distortion. Consider the following case in the frequency domain: if

x(n)

is the input sequence and

h(n)

is the IIR filter's impulse response, then the result of the forward filter pass is:

where

X(e
iφ
)
is the Fourier transform of
x(n)
H(e
iφ
)
is the Fourier transform of
h(n)
Y
1
(e
iφ
)
is the Fourier transform of the forward filter pass

Backward filtering corresponds to filtering of time-reversed signal. Time reversal corresponds to replacing φ with -φ in the frequency domain, so the result of time reversal is:

When the filter is applied for the second time, the above formula is multiplied by the Fourier transform of the filter's impulse response function

H(e

iφ

)

:

The final time reversal in the frequency domain results in:

You can see from the resulting equation that:

To initialize and use an IIRIIR filter, follow this general scheme: