Convolution Integrals

A convolution is simply a way of blending two functions: one is an input, the other says how strongly the system still remembers each past moment of it. "Convolution" comes from the Latin for rolling together, which fits the picture of one function being flipped and slid across the other. It's good for finding a system's output from its input, and for inverting products of transforms. The integral $(f * g)(t) = \int_0^t f(t - \tau)\,g(\tau)\,d\tau$ looks like a lot, but it's just three simple parts: $g(\tau)$ (what came in at some earlier time $\tau$), $f(t - \tau)$ (how much that still matters, $t - \tau$ seconds later), and $\int_0^t$ (add up every earlier moment). So the output now is all the past inputs, each weighted by how well it's remembered. It's not as bad as it looks; bear with it.

Inverting products. Back in the computing transforms lesson we warned that the transform of a product isn't the product of the transforms. Convolution is the operation that does turn into a product. So if $Y(s) = F(s)G(s)$ and you know each inverse, $y = f * g$, even when partial fractions would be messy.

Meaning. Think about the impulse response from the last lesson. Any input $g$ can be pictured as a long string of tiny kicks, one at each time $\tau$, with size $g(\tau)\,d\tau$. Each kick sets off a copy of the impulse response $h$, started at time $\tau$. Because the system is linear, the output is the sum of all those copies, and that sum is exactly the integral $y = h * g$. So a linear system's output at time $t$ is a weighted sum of all past inputs, with weights given by its impulse response. This is how audio effects, image blurs and signal filters work.

Convolution is commutative: $f * g = g * f$.

A common mistake. It's tempting to think $(f * g)(t)$ is just $f(t)g(t)$, because the convolution of two functions is "some kind of product". It isn't: it's an integral over the whole past, and $1 * 1 = t$, not $1$. The ordinary product $f(t)g(t)$ has its own transform, and it isn't $F(s)G(s)$.

Reverb plugins convolve dry audio with a hall's impulse response; image blur and sharpening filters are 2D convolutions; neural networks for images are built from convolutions.