## 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

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 *flip theorem for*, we have that the

*z*transforms*z*transform of FLIP is :

*z*transform

*squares*the amplitude response and

*zeros*the phase response. 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).

**Next Section:**

Phase Distortion at Passband Edges

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Antisymmetric Linear-Phase Filters