Ideal LPF realizability; step response

Ideal LPF is not physically realizable

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

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

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

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

A realizable (causal) LTI system needs $h(t) = 0$ for $t < 0$. Here $h(t)$ is an even sinc, non-zero on both sides; e.g. $h(-\pi/2\omega_c) = 2\omega_c/\pi^2 \ne 0$. It responds before the impulse arrives, so it is non-causal. Adding delay $t_0$ gives $\dfrac{\sin\omega_c(t-t_0)}{\pi(t-t_0)}$, still non-zero for $t < 0$. Also, $|H| = 0$ over a whole band violates the Paley–Wiener condition $\int \frac{|\ln|H(j\omega)||}{1+\omega^2}d\omega < \infty$. Hence it is not physically realizable.

Step response of the ideal LPF

The step response is the running integral of $h(t)$:

$$\begin{aligned} s(t) &= \int_{-\infty}^{t} h(\tau)\,d\tau = \int_{-\infty}^{t}\frac{\sin\omega_c\tau}{\pi\tau}\,d\tau \\ &= \int_{-\infty}^{0}\frac{\sin\omega_c\tau}{\pi\tau}\,d\tau + \int_{0}^{t}\frac{\sin\omega_c\tau}{\pi\tau}\,d\tau \end{aligned}$$

Using $\int_0^\infty \frac{\sin\omega_c\tau}{\tau}d\tau = \frac{\pi}{2}$, the first integral is $\frac{1}{\pi}\cdot\frac{\pi}{2} = \frac12$. In the second, put $x = \omega_c\tau$:

$$s(t) = \frac12 + \frac{1}{\pi}\int_0^{\omega_c t}\frac{\sin x}{x}\,dx = \frac12 + \frac{1}{\pi}\,\text{Si}(\omega_c t)$$

where $\text{Si}(z) = \int_0^z \frac{\sin x}{x}dx$ is the sine integral.

 s(t)        overshoot ~9%
  1  |          _/\_ _
     |         /     ' '-----
 0.5 |- - - - /
     |  _    /
  0  +-' '--/---------------> t
           0   pi/wc

• $s(0) = 0.5$; $s(-\infty) = 0$; $s(\infty) = 1$.
• Maximum at $t = \pi/\omega_c$: $s = 0.5 + \text{Si}(\pi)/\pi \approx 1.09$ (about 9% overshoot, Gibbs phenomenon).
• The response starts before $t = 0$ (non-causal) and rises from 0 to 1 in a time of order $\pi/\omega_c$.