Convolution

Convolution combines two signals, f and g, into a third. One signal is flipped and slid across the other; at each position the products of the overlapping values are summed. For discrete signals:

(f ★ g)[n] = Σₘ f[m] · g[n − m]

The star operator ★ is the standard notation for convolution, much like + for addition: (f ★ g) is read "the convolution of f and g". The middle dot · is ordinary multiplication, a different operation.

Convolution describes how a system responds to an input, for example how an echo spreads a sound. It is also the basis of image filters, such as blurring and edge detection.

Convolving two rectangular pulses of lengths a and b samples produces a trapezoid of a + b − 1 samples (in continuous time the width is a + b).