Convolution.rst

.. _func-Convolution:

===========
Convolution
===========

.. index:: Convolution

Description
-----------

Convolution is an extension of :ref:`func-CompositeFunction`
which performs convolution of its members using Fast Fourier Transform.

.. math:: f(x)=\int\limits_{A}^{B}R(x-\xi)F(\xi)\mbox{d}\xi

Here :math:`R` is the first member function and :math:`F` is the second
member. A Convolution must have exactly two member functions. The
members can be composite if necessary. Interval :math:`[A,B]` is the
fitting interval. The function is evaluated by first transforming
:math:`R` and :math:`F` to the Fourier domain, multiplying the
transforms, then transforming back to the original domain. The GSL FFT
routines are used to do the actual transformations.

It should be noted that the two functions (:math:`R` and :math:`F`) are
evaluated on different intervals. :math:`F` is computed on :math:`[A,B]`
while :math:`R` is computed on :math:`[-\Delta/2, \Delta/2]`, where
:math:`\Delta=B-A`.

In the following example a :ref:`func-Convolution` is convolved with a
box function:

.. figure:: /images/Convolution.png
   :alt: Convolution.png

Note that the box function is defined on interval [-5, 5]:

.. figure:: /images/Box.png
   :alt: Box.png

.. attributes::

.. properties::

.. categories::

.. sourcelink::
   :h: Framework/CurveFitting/inc/MantidCurveFitting/Functions/Convolution.h
   :cpp: Framework/CurveFitting/src/Functions/Convolution.cpp