Convolution sum; ideal LPF is non-causal

Derivation of the convolution sum

Any discrete-time signal can be written as a sum of shifted, scaled unit impulses (sifting property):

$$x[n] = \sum_{k=-\infty}^{\infty} x[k]\,\delta[n-k]$$

e.g. $x[n] = \{2, 3\}$ (starting at $n=0$) is $2\delta[n] + 3\delta[n-1]$.

Let the system be LTI with impulse response $h[n]$, i.e. $\delta[n] \to h[n]$.

1. Time invariance: $\delta[n-k] \to h[n-k]$.
2. Scaling (homogeneity): $x[k]\,\delta[n-k] \to x[k]\,h[n-k]$ ($x[k]$ is just a number for fixed $k$).
3. Additivity: sum over all $k$:

$$y[n] = \sum_{k=-\infty}^{\infty} x[k]\,h[n-k] = x[n] * h[n]$$

This is the convolution sum. By substituting $m = n - k$, $y[n] = \sum_m h[m]x[n-m]$ (commutative).

Procedure: fold $h[k]$ to $h[-k]$, shift by $n$, multiply with $x[k]$, add the products; repeat for each $n$.

Example: $x[n] = \{1, 2\}$, $h[n] = \{1, 1\}$ (both starting at 0): $y[0] = 1$, $y[1] = 1\cdot1 + 2\cdot1 = 3$, $y[2] = 2$, so $y[n] = \{1, 3, 2\}$.

Ideal LPF is non-causal

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

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

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

Causality test: an LTI system is causal if and only if $h(t) = 0$ for $t < 0$ (the output cannot depend on future input, since $y(t) = \int h(\tau)x(t - \tau)d\tau$ uses $x$ at $t - \tau > t$ whenever $h(\tau) \ne 0$ for $\tau < 0$).

For the ideal LPF, $h(t) = \dfrac{\sin\omega_c t}{\pi t}$ is an even function, non-zero on both sides of $t = 0$. For example,

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

So $h(t) \ne 0$ for $t < 0$ and the ideal LPF is non-causal. Even with a delay $t_0$, $h(t) = \dfrac{\sin\omega_c(t-t_0)}{\pi(t-t_0)}$ still has a tail reaching $t = -\infty$, so no finite delay makes it causal. (The discrete-time ideal LPF, $h[n] = \dfrac{\sin\Omega_c n}{\pi n}$, is non-causal for the same reason.) Hence it cannot be realized in real time.