In general, the object of deconvolution is to find the solution of a convolution equation of the form:

Thus, our convolution equation is

By the convolution theorem, this equation may be Fourier transformed to

This is where the convolution equation above comes from.

If the kernel is a function only of the difference of its arguments, namely, and the limits of integration are, then the right hand side of the equation can be rewritten as a convolution of the functions K and f and therefore the solution will be given by

The spreading Gaussian is the propagation kernel for the diffusion equation and it obeys the convolution identity:

In its general nature, it is a singular integral operator, defined by convolution with a function having a mathematical singularity at the origin, the Newtonian kernel Γ which is the fundamental solution of the Laplace equation.

This process is usually formulated by a convolution equation.

With this condition, the solution to the inhomogeneous Helmholtz equation is the convolution

For an arbitrary response, this involves a computationally expensive time convolution, although in most cases the time response of the medium ( or Dispersion ( optics )) can be adequately and simply modeled using either the recursive convolution ( RC ) technique, the auxiliary differential equation ( ADE ) technique, or the Z-transform technique.

* If G is a locally compact Hausdorff topological group and μ its Haar measure, then the Banach space L < sup > 1 </ sup >( G ) of all μ-integrable functions on G becomes a Banach algebra under the convolution xy ( g ) = ∫ x ( h ) y ( h < sup >− 1 </ sup > g ) dμ ( h ) for x, y in L < sup > 1 </ sup >( G ).

* Measure algebra: A Banach algebra consisting of all Radon measures on some locally compact group, where the product of two measures is given by convolution.

The area under the resulting product gives the convolution at t. The horizontal axis is for f and g, and t for.

In mathematics and, in particular, functional analysis, convolution is a mathematical operation on two functions f and g, producing a third function that is typically viewed as a modified version of one of the original functions, giving the area overlap between the two functions as a function of the amount that one of the original functions is translated.

In particular, the circular convolution can be defined for periodic functions ( that is, functions on the circle ), and the discrete convolution can be defined for functions on the set of integers.

Computing the inverse of the convolution operation is known as deconvolution.

The light diffracted by a grating is found by summing the light diffracted from each of the elements, and is essentially a convolution of diffraction and interference patterns.

which is the convolution of the sequence with a sequence extended by periodic summation:

:: which is the circular convolution of and.

A linear time-invariant ( LTI ) filter can be uniquely specified by its impulse response h, and the output of any filter is mathematically expressed as the convolution of the input with that impulse response.

Mathematically this is described as the convolution of a time-varying input signal x ( t ) with the filter's impulse response h, defined as:

However the duration of the filter's impulse response, and the number of terms which must be summed for each output value ( according to the above discrete time convolution ) is given by where T is the sampling period of the discrete time system ( N-1 is also termed the order of an FIR filter ).

Another prime-size FFT is due to L. I. Bluestein, and is sometimes called the chirp-z algorithm ; it also re-expresses a DFT as a convolution, but this time of the same size ( which can be zero-padded to a power of two and evaluated by radix-2 Cooley – Tukey FFTs, for example ), via the identity.

where * denotes the Dirichlet convolution, and 1 is the constant function.

The theorem follows because * is ( commutative and ) associative, and 1 * μ = i, where i is the identity function for the Dirichlet convolution, taking values i ( 1 )= 1, i ( n )= 0 for all n > 1.

* As a convolution operator: Via the formalism of incidence algebras, difference operators and other Möbius inversion can be represented by convolution with a function on the poset, called the Möbius function μ ; for the difference operator, μ is the sequence ( 1, − 1, 0, 0, 0, ...).

This formulation is especially useful for implementing a numerical convolution on a computer: The standard convolution algorithm has quadratic computational complexity.

A convolutional encoder is called so because it performs a convolution of the input stream with the encoder's impulse responses:

Where the associated time series flips the order of the coefficients because the linear filter is a convolution, and so both have the same index in this sum.

However, because of the b < sub > k – n </ sub > term in the convolution, both positive and negative values of n are required for b < sub > n </ sub > ( noting that b < sub >– n </ sub >

It is called the unit function because it is the identity element for Dirichlet convolution.

The averaging is often done by convolution with a Gaussian filter, which, at every spatial point, weights neighboring voxels by their distance, with the weights falling exponentially following the bell curve.

In practice, the O ( N ) additions can often be performed by absorbing the additions into the convolution: if the convolution is performed by a pair of FFTs, then the sum of x < sub > n </ sub > is given by the DC ( 0th ) output of the FFT of a < sub > q </ sub > plus x < sub > 0 </ sub >, and x < sub > 0 </ sub > can be added to all the outputs by adding it to the DC term of the convolution prior to the inverse FFT.

Another class of common 2D operations called image convolution are often used to reduce or enhance image details.

Though we most often express filters as the impulse response of convolution systems, as above ( see LTI system theory ), it is easiest to think of the matched filter in the context of the inner product, which we will see shortly.

Due to the computational expense of the convolution operation, the Voigt profile is often approximated using a pseudo-Voigt profile.

Methods have also been developed to use circular convolution as part of an efficient process that achieves normal ( non-circular ) convolution with an or sequence potentially much longer than the practical transform size ( N ).

Consider two waveforms f and g. By calculating the convolution, we determine how much a reversed function g must be shifted along the x-axis to become identical to function f. The convolution function essentially reverses and slides function g along the axis, and calculates the integral of their ( f and the reversed and shifted g ) product for each possible amount of sliding.

This is like convolution used in LTI systems to find the output of a system, when you know the input and impulse response.

So we generally find the output of the system convolutional encoder, which is the convolution of the input bit, against the states of the convolution encoder, registers.

That is the convolution integral and is used to find the convolution of a signal and a system ; typically a