convolution

Convolution is the operation that computes an LTI system's output by blending the input signal with the system's impulse response. Mechanically, you flip one signal backwards, slide it across the other, and at every position multiply-and-add the overlap. The intuition: each instant of the input fires off a scaled, delayed copy of the impulse response, and convolution adds up all those overlapping echoes to get the total output. It is written y = x * h (the star is convolution, NOT multiplication).

Once you see the world through convolution, smoothing a noisy graph, blurring a photo, and adding reverb to audio are all the same operation with different impulse responses. The single most useful fact in the entire subject: convolution in the time domain becomes simple multiplication in the frequency domain. That is why we so often hop into the Fourier or Laplace world — a messy slide-and-add integral collapses into Y(f) = X(f)·H(f).