Convolution

The convolution $h$ of two functions $f$ and $g$ is defined by
$$h(x) \equiv \int_{-\infty}^\infty f(u) g(x - u) du$$
and the convolution operator is often indicated by the symbol $\otimes$, so the above equation can be written
$$h = f \otimes g$$


illustration of convolution

This figure indicates that convolution involves taking the mirror image of $g$, overlaying it on $f$, sliding $g$ from left to right, and measuring the overlap of $f$ and $g$. Convolution is a smoothing operation that smears one function with another. It is often used in conjunction with Fourier transforms because the convolution theorem can simplify the calculation of Fourier transforms.