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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?enter image description here

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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 $\omega=\pi$.

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  • $\begingroup$ Thank you very much. $\endgroup$ – Ece Su Ildiz Jun 11 at 8:36

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