Discussing first the *mathematical* aspect, let us assume the goal of an experiment is to measure a random variable *X*,
described by the probability density function *f*_{x}(*x*). Instead of *X*, however, the setup allows us to observe only the sum *U* = *X*+*Y* of two random variables, where *Y* has the probability density function *f*_{y} (*y*) (typically *Y* is a composite of the measurement error and acceptance functions).
The *(convolved* or *folded)* sum has the probability density *f*(*u*) given by the *
convolution integrals*

If *f*(*u*) and *f*_{y} (*y*) are known it may be possible to solve the above equation for *f*_{x} (*x*) analytically *( deconvolution* or *unfolding)*.

Most frequently, one knows the general form of *f*_{x} (*x*) and *f*_{y}(*y*), but wants to determine some open parameters in one or both functions. One then performs the above integrals and, from fitting the result *f*(*u*) to the distribution obtained by experiment, finds the unknown parameters. For a number of cases *f*(*u*) can be computed analytically. A few important ones are listed below.

The *convolution of two two normal distributions with* zero mean and variances and is a normal distribution with zero mean and variance .

The *convolution of two distributions* with *f*_{1} and *f*_{2} degrees of freedom is a distribution with *f*_{1} +*f*_{2} degrees of freedom.

The *convolution of two Poisson distributions* with parameters and is a Poisson distribution with parameter .

The *convolution of an exponential and a
and a normal distribution* is approximated by another exponential distribution. If the original exponential distribution is

and the normal distribution has zero mean and variance , then for the probability density of the sum is

In a semi-logarithmic diagram where is plotted versus *x*
and versus *u* the latter lies by the amount higher than the former but both are represented by parallel straight lines, the slope of which is determined by the parameter .

The *convolution of a uniform and a
and a normal distribution* results in a quasi-uniform distribution smeared out at its edges. If the original distribution is uniform in the region and vanishes elsewhere and the normal distribution has zero mean and variance , the probability density of the sum is

Here

is the distribution function of the standard normal distribution. For the function *f*(*u*) vanishes for *u*<*a* and *u*>*b* and is equal to 1/(*b*-*a*) in between. For finite the sharp steps at *a*
and *b* are rounded off over a width of the order .

Convolutions are also an important tool in the area of *digital signal or signal or image processing* . They are used for the description of the response of linear shift-invariant systems, and are used in many filter operations.

One-dimensional discrete convolutions are written

(often abbreviated to ).

Convolutions are commutative, associative, and distributive; they have as the identity operation

with for all , and . The figure shows a one-dimensional example of two sequences and their convolution.

Normally one uses a fast Fourier transform (FFT), so that the transformation becomes

For the FFT, sequences *x* and *y* are padded with zeros to a length of a power of 2 of at least *M* + *N* - 1 samples.
For more e.g. [Kunt80]details and more references ,
[Oppenheim75] or [Rabiner75].