Convolution

Convolution is a mathematical operation performed on two functions. It has numerous applications in digital signal processing (e.g. impulse response) and DSP for music (e.g. convolution reverb).

Continuous Convolution¶

For two continuous functions $x(t)$ and $h(t)$, the convolution is defined with the shift operator $\tau$ as:

$$ y(t) = a(t) * b(t) = \int_{-\infty}^{\infty} a(t-\tau) b(\tau) d \tau$$

The follwing animation shows what is happening during convolution - one signal is shifted alongside the other signal, while their overlap is multiplied and summed up to create the output: