Why shouldn't the magnitude of a practically realizable frequency selective filter be constant for a range of frequencies?

Question

According to Paley-Wiener theorem, a practically realizable frequency selective filter shouldn't have a constant magnitude for a range of frequencies. Why is this so? How do we prove and understand it?!

To summarize, causality has very important implications in the design of frequency-selective filters. These are: (a) the frequency response $H(\omega)$ cannot be zero, except at a finite set of points in frequency; (b) the magnitude $|H(\omega)|$ cannot be constant in any finite range of frequencies and the transition from passband to stopband cannot be infinitely sharp [this is a consequence of the Gibbs phenomenon, which results from the truncation of $h(n)$ to achieve causality]; and (c) the real and imaginary parts of $H(\omega)$ are interdependent and are related by the discrete Hilbert transform. As a consequence, the magnitude $|H(\omega)|$ and phase $\Theta(\omega)$ of $H(\omega)$ cannot be chosen arbitrarily.

Point b in the quote. Source: Proakis ch.10 4th edition

Answer 1

Note that all-pass filters with a perfectly constant magnitude response over the whole frequency range are realizable. A first-order discrete-time all-pass filter has the following transfer function:

$$H(z)=\frac{1-az^{-1}}{a^*-z^{-1}}\tag{1}$$

If $|a|>1$, the system $(1)$ is causal and stable. It is straightforward to show that

$$\left|H\left(e^{j\omega}\right)\right|=1,\qquad\forall\omega\tag{2}$$

i.e., the system's magnitude response is constant.

What is not possible is to have a constant magnitude response over only a part of the possible frequency range $[0,2\pi]$ (or, in continuous time, $(-\infty,\infty)$).

This can be seen as follows. First, design an ideal frequency-selective filter by computing the IDFT of some ideal frequency response (with a constant gain in the pass band(s), and a gain of zero in the stop band(s)). Let's consider discrete-time filters, but the discussion is basically the same for continuous-time filters. Let the resulting ideal impulse response be $h_{id}[n]$. First, we shift that impulse response to the right to have most of its energy at positive indices $n>0$:

$$\tilde{h}_{id}[n]=h_{id}[n-n_0],\qquad n_0>0\tag{3}$$

In order to obtain a causal filter, we need to multiply that shifted impulse response by a sequence $f[n]$ satisfying

$$f[n]=\begin{cases}0,&n<0\\1,&n>n_1\ge 0\end{cases}\tag{4}$$

with some non-negative $n_1$. The values of $f[n]$ between $n=0$ and $n=n_1$ can be chosen arbitrarily, but in practice we would choose them such that $f[n]$ increases monotonically in the interval $[0,n_1]$. The final impulse response of our causal filter is

$$h[n]=\tilde{h}_{id}[n]f[n]=h_{id}[n-n_0]f[n]\tag{5}$$

Note that $h[n]$ is still infinitely long.

In the frequency domain this corresponds to

$$H(e^{j\omega})=\left(H_{id}(e^{j\omega})e^{-jn_0\omega}\right)\star F(e^{j\omega})\tag{6}$$

where $\star$ denotes convolution. This convolution destroys the piecewise constant property of the original frequency response $H_{id}(e^{j\omega})$, and the magnitude of the resulting causal frequency response $H(e^{j\omega})$ is neither piecewise constant, nor does it have infinitely sharp transitions.