Physical meaning of convolution integral

The convolution integral

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

says that the output of an LTI system at time $t$ is the sum of the responses to all past (and present) input values, each weighted by how strongly the system still "remembers" it.

Physical meaning:

• The input is thought of as a chain of very narrow impulses; the impulse at time $\tau$ has strength $x(\tau)\,d\tau$.
• Each impulse produces its own response $h(t-\tau)$, which starts at $\tau$ and dies out according to the system's memory.
• The output is the superposition (linearity) of these shifted copies (time invariance).
• $h(t - \tau)$ acts as a weighting function: for a system with short memory, it is small when the age $t - \tau$ of the input is large.

Example – RC circuit charging: $h(t) = \frac{1}{RC}e^{-t/RC}u(t)$. The capacitor voltage now is

$$y(t) = \int_{-\infty}^{t} x(\tau)\frac{1}{RC}e^{-(t-\tau)/RC}\,d\tau$$

Recent inputs (small $t - \tau$) count almost fully; inputs older than about $5RC$ have almost no effect, because that charge has leaked away. Another everyday example is an echo in a hall: what you hear is the present sound plus fading copies of earlier sounds.