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 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 samples of plain 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.

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written by Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.