LTI Systems and Convolution

A system is linear (superposition holds) and time-invariant (a delayed input gives a delayed output). For LTI systems, knowing the impulse response $h(t)$ tells you everything:

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

Discrete form: $y[n] = \sum_k x[k]\, h[n-k]$.

Convolution in time becomes multiplication in frequency: $Y(f) = X(f)H(f)$ — which is why engineers so often work in the frequency domain.