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Convolution is a mathematic operation that can be performed on two functions, which produces a third output function which is a “blend” of the two inputs.

Convolution can be thought of as a measure of the amount of overlap of one function as it is shifted over the other function

Formal Definition

$$ f \ast g = \int_{-\infty}^{\infty} f(\tau)\ g(t - \tau) d \tau $$

Mathematical Properties

Convolution is commutative:

$$ f \ast g = g \ast f $$

Convolution is associative:

$$ (f \ast g) \ast h = f \ast (g \ast h) $$

Convolution is distributive:

$$ f \ast (g + h) = f \ast g + f \ast h $$

These other properties also hold true:

$$ a (f \ast g) = (af) \ast g $$

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