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 linear-phase filters must be FIR filters, because a causal recursive filter cannot have a symmetric impulse response.
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