Linear phase

Linear phase is a property of a filter, where the phase response of the filter is a linear function of frequency. The result is that all frequency components of the input signal are shifted in time (usually delayed) by the same constant amount, which is referred to as the group delay. And consequently, there is no phase distortion due to the time delay of frequencies relative to one another.

Perfect linear phase is easily achieved with a discrete-time FIR filter. But FIR filters require high orders and thus more hardware than Infinite Impulse Response (IIR) filters. IIR filters can approximate a linear phase response with:

• a Bessel transfer function which has a maximally flat group delay
• a maximally flat group delay approximation function
• a phase equalizer

Examples

When a sinusoid$,\ \sin(\omega t+\theta ),$  passes through a filter with group delay $\tau ,$  the effect on its phase is:

$\sin(\omega (t-\tau )+\theta )=\sin(\omega t+\theta -\omega \tau ),$

where the phase shift $\omega \tau$ is a linear function of frequency (ω). An additive phase component of π radians at some frequencies is equivalent to a negative amplitude factor, which does not affect the group delay. Also, phase response graphs customarily depict the principle value of $\omega \tau ,$  which may contain meaningless discontinuities of 2π.

Some examples of linear and non-linear phase filters are given below.

phase response vs normalized frequency (ω/π)

Bode plots. Phase discontinuities are π radians, indicating a sign reversal.

Phase discontinuities are removed by allowing negative amplitude.

Two depictions of the frequency response of a simple FIR filter

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 $\alpha ,\beta$.

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.

See also

• Minimum phase