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I am approaching to the study of FIR systems. In particular, I was analyzing the graphs of amplitude and phase of the transfer function, when I had some trouble understanding how the phase behaves. Let me explain this with an example:

Suppose that our transfer function is $\displaystyle{H(z)=\frac{1-z^{-5}}{1-z ^ {-1}}}$. Then the DTFT is $\displaystyle{H(e^{i\omega}) = e^ {-2i\omega } \frac{\sin(\frac{5}{2}\omega)}{{\sin({\frac{1}{2}\omega }})}}$. If we set as $x$ coordinate the variable $\omega$ and as $y$ coordinate the phase, we observe that the graph is piecewise linear, with slope -2, and with jumps at the points where transfer function vanishes (in this case, $\omega = \pm\frac{2\pi} 5, \pm \frac{4\pi} 5$, taking as domain the interval $[-\pi , \pi ]$).

In the same way, if we consider $\displaystyle{H_1(z) = 1- z ^{-2}}$, therefore $\displaystyle{H_ 1 (e ^{i\omega}) = 2 ie ^ {-i\omega}\sin(\omega) }$ and we get a phase function piecewise linear, with slope $-1$ and jump for $\omega = 0$.

My question is: why are there jumps at these points? I see that the slope is given by the $\arctan\left(\frac{Im(H)}{Re(H)}\right)$, but I don't see why at these points the function jumps (and in particular, the correlation with the fact that it jumps in $\omega $ such that $H(e^{i\omega })= 0$ ).

If someone can give me an explanation, it would be really appreciated.

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It depends on how you define the phase. For a linear phase FIR filter you could write the frequency response as

$$H(e^{j\omega})=A(\omega)e^{j\phi(\omega)}\tag{1}$$

where $A(\omega)$ is a real-valued (or purely imaginary) function that can become positive and negative such that the phase $\phi(\omega)$ is a linear function without jumps. This is the case for the first frequency response in your question. Note that the term $\sin(5\omega /2)\,/\sin(\omega/2)$ takes both signs.

Another way to write the frequency response is

$$H(e^{j\omega})=\big|H(e^{j\omega})\big|e^{j\varphi(\omega)}\tag{2}$$

Now the phase $\varphi(\omega)$ jumps by $\pi$ at all frequencies where the corresponding amplitude function $A(\omega)$ changes sign. Since $|H(e^{j\omega})|$ can't change sign, the phase has to compensate for that and you get a piecewise linear phase with jumps at the zeros (of odd multiplicity) of $H(e^{j\omega})$.

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