Causality

A system is causal if its zero-state response at time $t$ depends only on the input at times $\le t$. The output now depends only on what the input has done up to now, not on what it will do in the future.

Formally: if two inputs $x_1(t)$ and $x_2(t)$ agree for all $t\le t_0$, then for a causal system $y_1(t_0)=y_2(t_0)$. The output at time $t_0$ cannot know anything about the future.

Every physical system that operates in real time is causal: effects don’t precede their causes. So when modeling a real-time signal-processing system, causality comes for free.

Non-causal systems exist mathematically

A mathematical operation like $y(t)=x(t+1)$ (the output now is the input one second from now) is a valid signal-processing operation, just not causal. You can implement it in offline processing by looking ahead in a recorded signal, but not in a real-time system.

Ideal filters are non-causal: the impulse response of an ideal lowpass filter is a sinc, which extends to $t=-\infty$. The system would need to know the entire future of its input to compute its output. Practical filters approximate the ideal with causal versions whose impulse responses are zero for $t<0$ and decay to zero as $t\to \infty$.

Test from the impulse response

For an LTI system, causality has a simple test:

$h(t)=0\text{ for }t<0⟺\text{system is causal}.$

The proof is just the definition. If $h(t)=0$ for $t<0$, then in the convolution $y(t)=\int x(\tau )h(t-\tau )d\tau$, the factor $h(t-\tau )$ is zero whenever $t-\tau <0$, i.e. $\tau >t$. So the integral runs only over $\tau \le t$, meaning $y(t)$ depends only on values of $x$ at times $\le t$. Conversely, if $h(t)$ is nonzero for some $t_0<0$, you can construct an input (an impulse at zero, say) that produces output before the input arrives.

Practical consequence

Any impulse response written with a $u(t)$ factor in front, and not with $u(t+\text{something positive})$, corresponds to a causal system. So $e^{-3t}u(t)$ is causal; $e^{-3t}u(t+1)$ is not. The unit step “switches on” the impulse response at $t=0$, ensuring it’s zero before that.

Causality and the s-plane

For an LTI system, causality and ROC of the Laplace transform are connected: a causal LTI system’s impulse response has Laplace transform whose ROC is a right half-plane $\mathrm{Re}(s)>\alpha$ for some $\alpha$. Stability requires the ROC to include the imaginary axis, which means $\alpha <0$, i.e. all poles are in the open left half-plane. So causality + stability ⟺ all poles in the open left half-plane.