Summary of Series/Parallel Filter Sections

In summary, we noted above the following general guidelines regarding series vs. parallel elementary-section implementations:

  • Series sections are preferred when all sections contribute to the same passband, such as in a lowpass, highpass, bandpass, or bandstop filter.
  • Parallel sections are usually preferred when the sections have disjoint passbands, such as a formant filter bank used in voice models. Another example would be the phase vocoder filter bank [21].

$ \pi $-Phase Filters

Under our definition, a zero-phase filter always has a real, even impulse response [ $ h(n) = h(-n)$], but not every real, even, impulse response is a zero-phase filter. For example, if $ h(n)$ is zero phase, then $ -h(n)$ is not; however, we could call $ -h(n)$ a ``$ \pi $-phase filter'' if we like (a zero-phase filter in series with a sign inversion).

Phase $ \pi $ in the Stopband

Practical zero-phase filters are zero-phase in their passbands, but may switch between 0 and $ \pi $ in their stopbands (as illustrated in the upcoming example of Fig.10.2). Thus, typical zero-phase filters are more precisely described as piecewise constant-phase filters, where the constant phase is 0 in all passbands, and $ \pi $ over various intervals within stopbands. Similarly, practical ``linear phase'' filters are typically truly linear phase across their passbands, but typically exhibit discontinuities by $ \pi $ radians in their stopband(s). As long as the stopbands are negligible, which is the goal by definition, the $ \pi $-phase regions can be neglected completely.

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Example Zero-Phase Filter Design
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Butterworth Lowpass Filter Example