Linear phase in FIR filter

The filters which have constant (independent of frequency) Phase delay and / or Group delay are called as Linear Phase Filters. In Linear Phase Filters the phase response is a linear (straight-line) function of frequency.

By Definition, Phase Delay $\tau_{p}=\frac{-\theta(\omega)}{\omega}$ and Group Delay $\tau_{g}=\frac{-d}{d \omega} \theta(\omega)$

Whenever a signal is passed through a device or is propagating through space or a medium, its frequency components are delayed. This signal delay is different for different frequencies. For a signal consisting of multiple frequency components, the delay variation will cause distortion as each frequency component is delayed by different amount of time at the output of the device. This changes the shape of the signal in addition to any constant delay or scale change. A sufficiently large delay variation will cause problems such as poor fidelity in audio.

When a signal is passed through a Linear Phase filter, every frequency component of the output signal is delayed by an equal amount. Thus the output signal is delayed, but not distorted. So Linear Phase filter do not cause "Phase Distortion" or "Delay Distortion". The wave shape is preserved as much as possible for a given amplitude response. For a linear-phase filter, group delay and phase delay are of the same value. So linear-phase filters are also called Constant Time Delay Filters.

A FIR filter is linear-phase if its coefficients are symmetrical or anti symmetrical around the center coefficient. For a FIR system to have a linear phase the condition for impulse response is $h(n)=\pm h(M-1-n) \cdot$ If $h(n)=h(M-1-n)$ then the FIR is symmetrical around and if $h(n)=-h(M-1-n)$ then the FIR is anti symmetrical around the centre coefficient.

Lack of phase / delay distortion is a major advantage of the FIR filters the over IIR and analog filters.

Given: Phase response is $\theta(\mathrm{w})=-\alpha \mathrm{w},$ where $\alpha$ is constant.

$\therefore$ Phase Delay $\tau_{p}=\frac{-\theta(\omega)}{\omega}=\frac{-(-\alpha \omega)}{\omega}=\alpha,$ which is a constant.

$\therefore$ Group Delay $\tau_{g}=\frac{-d}{d \omega} \theta(\omega)=\frac{-d}{d \omega}(-\alpha \omega)=\alpha,$ which is a constant.

$\therefore \tau_{p}=\tau_{g}$

$\therefore$ The given filter is a linear-phase.

Hence, a filter is said to have lincar phase response if its phase response is $\theta(w)=-\alpha$ w.