# How should we read FIR filter phase response graph?

What should we see from the FIR phase response graph, as an example for a linear phase? And what does this straight line at $$0.5\pi$$ mean?

First of all you see that the phase is a piecewise linear function, so it's a linear phase FIR filter. There's a phase jump at half the Nyquist frequency, which shows that the filter has a zero at that frequency. Note that the phase jumps by $$\pi$$ corresponding to sign inversion.
You can also see what type of linear phase FIR filter it is. Since the phase is zero at $$\omega=0$$ it must be either type I or type II (i.e., even symmetry). If it were type III or type IV (odd symmetry), the phase would jump at $$\omega=0$$, and the phase at $$\omega=0^+$$ would equal either $$\pi/2$$ or $$-\pi/2$$. You can distinguish type I and type II by looking at the phase at the Nyquist frequency. A type I filter (odd filter length) must have a phase of either zero or $$\pi$$ at Nyquist, whereas the phase of a type II filter jumps at Nyquist because a type II filter always has a zero at Nyquist. You can't see the phase jump here (because the figure doesn't go beyond Nyquist) but you can see that the phase equals $$-\pi/2$$ just before Nyquist. A type II filter's phase always equals $$\pm\pi/2$$ at $$\omega=\pi^-$$, and it jumps by $$\pi$$ at $$\omega=\pi$$.