### 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].

#### -Phase Filters

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

#### Phase in the Stopband

Practical zero-phase filters are zero-phase in their passbands, but may switch between 0 and 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 over various intervals within stopbands. Similarly, practical linear phase'' filters are typically truly linear phase across their passbands, but typically exhibit discontinuities by radians in their stopband(s). As long as the stopbands are negligible, which is the goal by definition, the -phase regions can be neglected completely.

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Example Zero-Phase Filter Design
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Butterworth Lowpass Filter Example