##
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.*,

*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

*at every frequency*. Since a length FIR filter implements samples of delay, the value is exactly half the total filter delay. Delaying all frequency components by the same amount

*preserves the waveshape*as much as possible for a given amplitude response.

**Next Section:**

Zero-Phase Filters (Even Impulse Responses)

**Previous Section:**

Pole-Zero Analysis Problems