# Linear phase

Linear phase is a property of a filter, where the phase response of the filter is a linear function of frequency, excluding the possibility of wraps at $\pm\pi$. In a causal system, perfect linear phase can be achieved with a discrete-time FIR filter.

Since a linear phase (or generalized linear phase) filter has constant group delay, all frequency components have equal delay times. That is, there is no distortion due to the time delay of frequencies relative to one another; in many applications, this constant group delay is advantageous. By contrast, a filter with non-linear phase has a group delay that varies with frequency, resulting in phase distortion.

A filter with linear phase may be achieved by an FIR filter which is either symmetric or anti-symmetric. A necessary but not sufficient condition is $\sum_{n =-\infty}^\infty h[n] \cdot \sin(\omega \cdot (n - \alpha) + \beta)=0$ for some $\alpha, \beta$. [1]

Some examples of linear and non-linear phase filters are given below. The plots below represent the phase response as a function of frequency in Hertz.

## Generalized linear phase

Systems with generalized linear phase have an additional frequency-independent constant added to the phase. Because of this constant, the phase of the system is not a strictly linear function of frequency, but it retains many of the useful properties of linear phase systems.[2]

## Linear phase approximation

Linear phase can be achieved with the help of Finite Impulse Response (FIR) filters having a symmetric coefficient sequence.

Nevertheless, FIR filters require high orders and thus more hardware than Infinite Impulse Response (IIR) filters. IIR filters can approximate a linear phase response with: