Condition for LTI system to be causal

An LTI system is causal if and only if its impulse response is zero for negative time:

$$h[n] = 0 \quad \text{for } n < 0 \qquad \left(\text{CT: } h(t) = 0 \text{ for } t < 0\right)$$

Reason: the output of a DT LTI system is the convolution sum

$$y[n] = \sum_{k=-\infty}^{\infty} h[k]\,x[n-k] = \underbrace{\sum_{k=-\infty}^{-1} h[k]\,x[n-k]}_{\text{future inputs}} + \underbrace{\sum_{k=0}^{\infty} h[k]\,x[n-k]}_{\text{present and past}}$$

For $k < 0$, $x[n-k]$ is an input at a time later than $n$ (a future value). A causal system's output cannot depend on future inputs, so every weight $h[k]$ for $k < 0$ must be zero. Then

$$y[n] = \sum_{k=0}^{\infty} h[k]\,x[n-k] = \sum_{k=-\infty}^{n} x[k]\,h[n-k]$$

Physically: $h[n]$ is the response to $\delta[n]$ applied at $n = 0$, and a causal system cannot respond before the input arrives.

Examples:

• $h[n] = (0.5)^n u[n]$: causal.
• $h[n] = u[n] - u[n-4]$: causal.
• $h[n] = \delta[n+1] + \delta[n]$, i.e. $y[n] = x[n+1] + x[n]$: non-causal (uses the future input $x[n+1]$).
• $h[n] = (0.5)^{|n|}$: non-causal.

Related terms: a sequence that is zero for $n < 0$ is called a causal sequence. If $h[n] = 0$ for $n > 0$ instead, the system is anti-causal. Causality is needed for real-time operation; non-causal systems can still be used for stored data (e.g. image processing).