For FIR filter,

$H(z)=∑_{n=0}^{M-1}h(n) z^{-n}$

To obtain magnitude and phase response put

$z=e^{jω}$

$∴H(e^{jω})∑_{n=0}^{M-1}h(n) z^{-jωn}$

Here phase response is given by:

$ϕ(ω)=tan^{-1}⁡\frac{(Im[H(e^{jω} )])}{(Re[H(e^{jω} )])}$

The group delay is the delayed response of filter as a function of frequency ω.

The phase delay (T_p) and group delay (T_g)are given by,

$T_p=\frac{(-ϕ(ω))}{ω}$ and

$T_g=\frac{(-dϕ(ω))}{dω}$

The parameter T is constant phase delay parameter and it is given by $\frac{(M-1)}{2}$

If the phase delay and group delay are constant then such filters are called as linear phase filters. The condition for linear phase in terms of delay parameter is:

ϕ(ω)=-ωT

Similarly in terms of filter length, condition for Linear phase is

h(n)=h(M-1-n)

If only constant group delay is considered then the Linear phase condition is,

h(n)=-h(M-1-n)