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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Trigonometric Identities, Continued
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-Phase Filters