Forward-Backward Filtering

There are no linear-phase recursive filters because a recursive filter
cannot generate a symmetric impulse response. However,
it *is* possible to implement a zero-phase filter *offline*
using a recursive filter twice. That is, if the entire input signal
is stored in a computer memory or hard disk, for example,
then we can apply a recursive filter both forward and backward in
time. Doing this *squares* the amplitude response of the filter
and *zeros* the phase response.

To show this analytically, let denote the output of the first filtering operation (which we'll take to be ``forward'' in time in the normal way), and let be the impulse response of the recursive filter. Then we have

where is the input signal at sample . For the second pass, we ``flip'' to obtain and apply the filter again:

FLIP

The final output is then this result flipped:

FLIP FLIP FLIP FLIP

where the last simplification tells us that flipping the input and output signals is equivalent to flipping the impulse response instead. Putting all these operations together, we have

FLIP FLIPFLIP

By the

Using this result and applying the convolution theorem
(§6.3) twice gives the
*z* transform

On the unit circle, this reduces to, for real filters ,

If the filter were complex, then we would need to conjugate its coefficients when running it backwards.

In summary, we have thus shown that forward-backward
filtering *squares* the amplitude response and *zeros* the
phase response.^{11.3}Note also that the phase response is truly zero,
never alternating between zero and
. No matter what nonlinear
phase response
a filter may have, this phase is
completely canceled out by forward and backward filtering. The
amplitude response, on the other hand, is squared. For simple
bandpass filters (including lowpass, highpass, etc.), for which the
desired gain is 1 in the passband and 0 in the stopband, squaring the
amplitude response usually *improves* the response, because the
``stopband ripple'' (deviation from 0) is squared,
thereby *doubling* the stopband attenuation in dB. On the other
hand, passband ripple (deviation from 1) is only doubled by the
squaring (because
).

A Matlab example of forward-backward filtering is presented in §11.6 (in Fig.11.1).

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