I have one small doubt. I am studying filter banks, and hit upon by the frequency response of Haar wavelets. Which has the following form:
$$\frac{1}{2} + \frac{1}{2}e^{j\omega}$$
This drops to to zero at $\omega=\pi$, and we call it low pass filter. But the $\omega$ has a period of $2\pi$. Hence the frequency response has to gradually come down to zero at $\omega=2\pi$ not at $\pi$. Because if I plot the response it has large response factors between $\pi$ and $2\pi$. Hence it would pass majority of high frequencies lying between $\pi$ and $2\pi$.
Though people plot the response between $-\pi$ and $\pi$, I would like to see between $0$ and $2\pi$.
So my question is why frequency has to drop to zero at $\pi$ but not at $2\pi$.
Since discrete-time filters have a $2\pi$-periodic frequency response, the frequency $2\pi$ corresponds to frequency $0$. So if you had a zero response at frequency $2\pi$ you would also have a zero response at frequency $0$, which cannot be the case for a lowpass filter. On the other hand, the frequency $\pi$ is the highest possible frequency for a discrete-time system, so it is this frequency that a lowpass filter must suppress.