You can cancel the nonlinear phase effects of an IIR filter by following the process shown in Figure 1331(a). The y(n) output will be a filtered version of x(n) with no filterinduced phase distortion. The same IIR filter is used twice in this scheme, and the time reversal step is a straight leftright flipping of a timedomain sequence. Consider the following. If some spectral component in x(n) has an arbitrary phase of a degrees, and the first filter induces a phase shift of –b degrees, that spectral component's phase at node A will be a–b degrees. The first time reversal step will conjugate that phase and induce an additional phase shift of –q degrees. (Appendix C explains this effect.) Consequently, the component's phase at node B will be –a+b–q degrees. The second filter's phase shift of –b degrees yields a phase of –a–q degrees at node C. The final time reversal step (often omitted in literary descriptions of this zerophase filtering process) will conjugate that phase and again induce an additional phase shift of –q degrees. Thankfully, the spectral component's phase in y(n) will be a+q–q = a degrees, the same phase as in x(n). This property yields an overall filter whose phase response is zero degrees over the entire frequency range.
Figure 1331. Two, equivalent, zerophase filtering techniques.
An equivalent zerophase filter is presented in Figure 1331(b). Of course, these methods of zerophase filtering cannot be performed in real time because we can't reverse the flow of time (at least not in our universe). This filtering is a block processing, or offline process, such as filtering an audio sound file on a computer. We must have all the time samples available before we start processing. The initial time reversal in Figure 1331(b) illustrates this restriction.
There will be filter transient effects at the beginning and end of the filtered sequences. If transient effects are bothersome in a given application, consider discarding L samples from the beginning and end of the final y(n) time sequence, where L is 4 (or 5) times the order of the IIR filter.
By the way, the final peaktopeak passband ripple (in dB) of this zerophase filtering process will be twice the peaktopeak passband ripple of the single IIR filter. The final stopband attenuation will also be double that of the single filter.
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