Short note: Causal and non-causal systems

A causal system is one whose output at any time depends only on the present and past values of the input, never on future values. A non-causal system has an output that depends on at least one future input value.

Causal system

• $y(t)$ depends on $x(\tau)$ only for $\tau \le t$.
• It does not respond before the input is applied (non-anticipative).
• All physically realizable real-time systems are causal.
• Examples: $y(t) = x(t) + 2x(t-1)$; an RC circuit; $y[n] = x[n] - x[n-1]$.
• LTI condition: $h(t) = 0$ for $t < 0$ (or $h[n] = 0$ for $n < 0$).

Non-causal system

• The output uses future input, e.g. $y(t) = x(t+1)$, $y[n] = x[n+1] + x[n]$, $y(t) = x(-t)$.
• It cannot work in real time, but can be used when the whole signal is stored (image processing, offline audio smoothing).
• An ideal low pass filter is non-causal because its $h(t)$ is a sinc that exists for $t < 0$.

Anti-causal system: output depends only on present and future inputs, i.e. $h(t) = 0$ for $t > 0$.