Commutative property of CT LTI systems

Statement: Convolution of continuous-time signals is commutative:

$$x(t) * h(t) = h(t) * x(t)$$

For LTI systems, this means the output is the same whether the input is $x(t)$ applied to a system with impulse response $h(t)$, or $h(t)$ applied to a system with impulse response $x(t)$.

Proof:

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

Substitute $\lambda = t - \tau$, so $\tau = t - \lambda$ and $d\tau = -d\lambda$. When $\tau \to -\infty$, $\lambda \to \infty$; when $\tau \to \infty$, $\lambda \to -\infty$:

$$\begin{aligned} x(t) * h(t) &= \int_{\infty}^{-\infty} x(t - \lambda)\,h(\lambda)\,(-d\lambda) \\ &= \int_{-\infty}^{\infty} h(\lambda)\,x(t - \lambda)\,d\lambda = h(t) * x(t) \end{aligned}$$

$\blacksquare$

x(t) --> [ h(t) ] --> y(t)   ==   h(t) --> [ x(t) ] --> y(t)

Use: we can flip whichever signal is simpler when evaluating the integral; and for cascaded LTI systems, the order of the systems can be swapped.