##
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