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.,

$$h(n) = h(N-1-n), \quad n=0,1,2,\ldots,N-1.$$

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 $e^{-j\alpha\omega T}$ , where $\alpha = (N-1)/2$ is the slope of the linear phase. Linear phase is often ideal because a filter phase of the form $\Theta(\omega) = - \alpha \omega T$ corresponds to phase delay

$$P(\omega) \isdef - \frac{\Theta(\omega)}{\omega} = - \frac{-\alpha\omega T}{\omega} = \alpha T = \frac{(N-1)T}{2}$$

and group delay

$$D(\omega) \isdef - \frac{\partial}{\partial \omega}\Theta(\omega) = - \frac{\partial}{\partial \omega}\left(-\alpha\omega T\right) = \alpha T = \frac{(N-1)T}{2}.$$

That is, both the phase and group delay of a linear-phase filter are equal to $(N-1)/2$ samples of plain delay at every frequency. Since a length $N$ FIR filter implements $N-1$ samples of delay, the value $(N-1)/2$ 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.