Convolution

 Đây là định nghĩa:
The convolution of ƒ and g is written ƒ∗g, using an asterisk or star. It is defined as the integral of the product of the two functions after one is reversed and shifted. As such, it is a particular kind of integral transform:

	(commutativity)

While the symbol t is used above, it need not represent the time domain. But in that context, the convolution formula can be described as a weighted average of the function ƒ(τ) at the moment t where the weighting is given by g(−τ) simply shifted by amount t. As t changes, the weighting function emphasizes different parts of the input function.
More generally, if f and g are complex-valued functions on R^d, then their convolution may be defined as the integral:

Tuy nhiên chúng ta chủ yếu quan tâm đến tích chập rời rạc (discrete convolution)

      (commutativity)

Một số tính chất của convolution

Commutativity

Associativity

Distributivity

Associativity with scalar multiplication

for any real (or complex) number .

Multiplicative identity

No algebra of functions possesses an identity for the convolution. The lack of identity is typically not a major inconvenience, since most collections of functions on which the convolution is performed can be convolved with a delta distribution or, at the very least (as is the case of L¹) admit approximations to the identity^{[disambiguation needed]}. The linear space of compactly supported distributions does, however, admit an identity under the convolution. Specifically,

where δ is the delta distribution.

Inverse element

Some distributions have an inverse element for the convolution, S⁽⁻¹⁾, which is defined by

The set of invertible distributions forms an abelian group under the convolution.

Complex conjugation

Integration

If ƒ and g are integrable functions, then the integral of their convolution on the whole space is simply obtained as the product of their integrals:

This follows from Fubini's theorem. The same result holds if ƒ and g are only assumed to be nonnegative measurable functions, by Tonelli's theorem.

Differentiation

In the one-variable case,

where d/dx is the derivative. More generally, in the case of functions of several variables, an analogous formula holds with the partial derivative:

A particular consequence of this is that the convolution can be viewed as a "smoothing" operation: the convolution of ƒ and g is differentiable as many times as ƒ and g are together.
These identities hold under the precise condition that ƒ and g are absolutely integrable and at least one of them has an absolutely integrable (L¹) weak derivative, as a consequence of Young's inequality. For instance, when ƒ is continuously differentiable with compact support, and g is an arbitrary locally integrable function,

These identities also hold much more broadly in the sense of tempered distributions if one of ƒ or g is a compactly supported distribution or a Schwartz function and the other is a tempered distribution. On the other hand, two positive integrable and infinitely differentiable functions may have a nowhere continuous convolution.
In the discrete case, the difference operator D ƒ(n) = ƒ(n + 1) − ƒ(n) satisfies an analogous relationship:

Nguồn: http://en.wikipedia.org/wiki/Convolution