Linear-Phase Filters

(Symmetric Impulse Responses)

A *linear-phase filter* is typically used when a *causal*
filter is needed to modify a signal's magnitude-spectrum while
preserving the signal's time-domain waveform as much as possible.
Linear-phase filters have a *symmetric impulse response*, *e.g.*,

The symmetric-impulse-response constraint means that

We will show that every real symmetric impulse response corresponds to
a *real* frequency response times a
*linear phase term*
, where
is the *slope* of the linear phase. Linear phase is
often ideal because a filter phase of the form
corresponds to phase delay

and group delay

That is, both the phase and group delay of a linear-phase filter are equal to samples of plain delay

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Center for Computer Research in Music and Acoustics (CCRMA), Stanford University