## Convolution Theorem | CHM 621 Home Page Fourier Transform TOC < Derivative Theorem < > Autocorrelation Theorem > |

**Theorem.** Consider the following two Fourier Transform pairs.

Let a new function, *C*(*t*) be the convolution of *F*_{1}(*t*) and *F*_{2}(*t*).

The convolution theorem gives the following relationships between the convolution and multiplication operators.

**Proof.** Write out the Fourier transform for *C*(*t*).

Rewrite the double integral by letting *T* = *t* - *t'*, and *dT* = *dt*.

Finally, convert the exponential into a product and separate the two integrals.

**Properties of convolution.**

Commutation:

Distribution:

Association:

Combinations with multiplication: *Note that multiplication and convolution do not commute!*

More complicated examples:

**Common convolutions.**

Rectangle: The convolution of two equal-sized rectangles is a triangle,

where *t*^{0} is the width of the rectangle and the Full Width at Half Maximum of the triangle. The base of the triangle is 2*t*^{0}. The convolution of two different-sized rectangles is a trapezoid,

where the base of the trapezoid is (*t*_{1}^{0}+*t*_{2}^{0}) and the top is |*t*_{1}^{0}-*t*_{2}^{0}|.

Gaussian: The convolution of two Gaussian functions is a Gaussian function,

where,

Since the standard deviation and the Full Width at Half Maximum of a gaussian function are proportional to the *t*^{0} parameters, one also has the following two relationships.

Impulse: The convolution of an impulse with any other function replicates that function. The replicate is centered at the position of the impulse. *For this reason, the impulse function is called a replicator when used in convolution.*

Comb: The convolution of a comb with any function having a size smaller than the comb spacing, replicates the function at the center of each comb impulse. *The comb function replicates multiple copies when used in convolution.*