Ideal LPF impulse response; realizable?

Impulse response of an ideal low pass filter

An ideal low pass filter (LPF) passes all frequencies below the cutoff $\omega_c$ without distortion and completely blocks higher frequencies:

$$H(\omega) = \begin{cases} K\,e^{-j\omega t_0}, & |\omega| < \omega_c \\ 0, & |\omega| > \omega_c \end{cases}$$

(constant gain $K$ and linear phase, i.e. distortionless in the pass band).

 |H(w)|
   K  +-----------+
      |           |
 -----+-----+-----+-------> w
    -wc     0     wc

The impulse response is the inverse Fourier transform:

$$\begin{aligned} h(t) &= \frac{1}{2\pi}\int_{-\omega_c}^{\omega_c} K\,e^{-j\omega t_0}\,e^{j\omega t}\,d\omega \\ &= \frac{K}{2\pi}\left[\frac{e^{j\omega(t-t_0)}}{j(t-t_0)}\right]_{-\omega_c}^{\omega_c} \\ &= \frac{K}{2\pi}\cdot\frac{2j\sin\omega_c(t-t_0)}{j(t-t_0)} \\ &= \frac{K\sin\omega_c(t-t_0)}{\pi(t-t_0)} \\ &= \frac{K\omega_c}{\pi}\,\mathrm{sinc}\left[\omega_c(t-t_0)\right] \end{aligned}$$

where $\mathrm{sinc}(x) = \sin x / x$. The peak value $K\omega_c/\pi$ occurs at $t = t_0$, and zeros occur at $t = t_0 \pm n\pi/\omega_c$.

            h(t)
             /\    peak K*wc/pi at t0
   .  ..   /  \   ..  .
 --'--''--/----\--''--'--> t
          |  t0 |
   tails extend to t -> -inf

Is it practically realizable?

No. A physically realizable system must be causal, i.e. $h(t) = 0$ for $t < 0$. The sinc response extends from $t = -\infty$ to $+\infty$, so it is non-zero for $t < 0$ for any finite delay $t_0$: the filter would respond before the impulse is applied. Hence the ideal LPF is non-causal and not realizable. (The Paley–Wiener criterion also fails, because $|H(\omega)|$ is exactly zero over a band.) Practical filters only approximate it: Butterworth and Chebyshev filters accept a finite transition band and non-linear phase, while FIR filters use a large delay and truncate (window) the sinc tails.