Causality condition for CT LTI system

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, not on future values. Example: $y(t) = x(t) + 2x(t-1)$. All physical real-time systems are causal.
• A non-causal system's output depends on future input values as well. Example: $y(t) = x(t+1)$, $y(t) = x(-t)$. Such systems can be used only on stored (recorded) data, e.g. image processing.

Condition for a CT LTI system to be causal

For an LTI system,

$$y(t) = \int_{-\infty}^{\infty} h(\tau)\,x(t-\tau)\,d\tau$$

Split the integral at $\tau = 0$:

$$y(t) = \underbrace{\int_{0}^{\infty} h(\tau)\,x(t-\tau)\,d\tau}_{\text{uses } x \text{ at times} \le t} + \underbrace{\int_{-\infty}^{0} h(\tau)\,x(t-\tau)\,d\tau}_{\text{uses } x \text{ at times} > t}$$

The second integral uses future inputs $x(t - \tau)$ with $t - \tau > t$. For $y(t)$ to be independent of every future input, this term must vanish for all $x$, which requires

$$h(t) = 0 \quad \text{for } t < 0$$

Then

$$y(t) = \int_{0}^{\infty} h(\tau)\,x(t-\tau)\,d\tau = \int_{-\infty}^{t} x(\tau)\,h(t-\tau)\,d\tau$$

Condition: a CT LTI system is causal if and only if its impulse response $h(t) = 0$ for $t < 0$. (Physically, the system cannot respond before the impulse is applied at $t = 0$.)

Examples: $h(t) = e^{-2t}u(t)$ is causal; $h(t) = e^{-2|t|}$ or $h(t) = \sin(\omega_c t)/\pi t$ is non-causal.