Ideal LPF is not realizable

The ideal low-pass filter passes all frequencies below the cutoff $\omega_c$ with unit gain and rejects all others:

$$H(j\omega) = \begin{cases} 1, & |\omega| \le \omega_c \\ 0, & |\omega| > \omega_c \end{cases}$$

A physically realizable system must be causal: for an LTI system this means $h(t) = 0$ for $t < 0$.

Step 1: Impulse response

The impulse response is the inverse Fourier transform of $H(j\omega)$:

$$\begin{aligned} h(t) &= \frac{1}{2\pi}\int_{-\infty}^{\infty} H(j\omega)e^{j\omega t}\,d\omega = \frac{1}{2\pi}\int_{-\omega_c}^{\omega_c} e^{j\omega t}\,d\omega \\ &= \frac{1}{2\pi}\cdot\frac{e^{j\omega_c t} - e^{-j\omega_c t}}{jt} = \frac{1}{2\pi}\cdot\frac{2j\sin\omega_c t}{jt} \\ &= \frac{\sin\omega_c t}{\pi t} = \frac{\omega_c}{\pi}\,\text{sinc}\!\left(\frac{\omega_c t}{\pi}\right) \end{aligned}$$

 H(jw)                     h(t)
   1 ______                    wc/pi
    |      |                    /\
    |      |           _  _    /  \    _  _
----+------+--> w    -' '-' '-/----\-' '-' '-> t
  -wc  0   wc               -pi/wc pi/wc

Step 2: Check causality

$h(t)$ is an even sinc function. It has its peak $\omega_c/\pi$ at $t = 0$ and oscillating tails on both sides, extending to $t = -\infty$. For example, at $t = -\pi/(2\omega_c)$:

$$h\!\left(-\frac{\pi}{2\omega_c}\right) = \frac{\sin(-\pi/2)}{\pi(-\pi/2\omega_c)} = \frac{2\omega_c}{\pi^2} \neq 0$$

So $h(t) \ne 0$ for $t < 0$: the filter must produce output before the impulse is applied at $t = 0$.

Step 3: Delay does not help

With linear phase, $H(j\omega) = e^{-j\omega t_0}$ for $|\omega| < \omega_c$, giving

$$h(t) = \frac{\sin\omega_c(t - t_0)}{\pi(t - t_0)}$$

This is centred at $t_0$, but its tail still extends to $-\infty$, so $h(t) \ne 0$ for some $t < 0$ for every finite $t_0$.

Step 4: Paley–Wiener criterion

A causal filter with magnitude $|H(j\omega)|$ must satisfy

$$\int_{-\infty}^{\infty}\frac{\big|\ln|H(j\omega)|\big|}{1 + \omega^2}\,d\omega < \infty$$

For the ideal LPF, $|H| = 0$ over the whole stopband, so $|\ln|H|| = \infty$ there and the integral diverges. A causal filter can have zero gain only at isolated frequencies, never over a band.

Hence the ideal low-pass filter is non-causal and not physically realizable. Practical filters such as Butterworth and Chebyshev only approximate it, accepting a finite transition band, some passband ripple or stopband leakage, and a non-linear phase.