Mathematical Methods for Physics and Engineering : A Comprehensive Guide

13.1 FOURIER TRANSFORMS

The inverse of convolution, called deconvolution, allows us to find a true

distributionf(x) given an observed distributionh(z) and a resolution function

g(y).

An experimental quantityf(x)is measured using apparatus with a known resolution func-

tiong(y)to give an observed distributionh(z).Howmayf(x)be extracted from the mea-

sured distribution?

From the convolution theorem (13.38), the Fourier transform of the measured distribution
is

̃h(k)=

√

2 π ̃f(k) ̃g(k),

from which we obtain

̃f(k)=√^1

2 π

̃h(k)

̃g(k)

.

Then on inverse Fourier transforming we find

f(x)=

1

√

2 π

F−^1

[

̃h(k)

̃g(k)

]

.

In words, to extract the true distribution, we divide the Fourier transform of the observed
distribution by that of the resolution function for each value ofkand then take the inverse
Fourier transform of the function so generated.

This explicit method of extracting true distributions is straightforward for exact

functions but, in practice, because of experimental and statistical uncertainties in

the experimental data or because data over only a limited range are available, it

is often not very precise, involving as it does three (numerical) transforms each

requiring in principle an integral over an infinite range.

13.1.8 Correlation functions and energy spectra

Thecross-correlationof two functionsfandgis defined by

C(z)=

∫∞

−∞

f∗(x)g(x+z)dx. (13.40)

Despite the formal similarity between (13.40) and the definition of the convolution

in (13.37), the use and interpretation of the cross-correlation and of the convo-

lution are very different; the cross-correlation provides a quantitative measure of

the similarity of two functionsfandgas one is displaced through a distancez

relative to the other. The cross-correlation is often notated asC=f⊗g, and, like

convolution, it is both associative and distributive. Unlike convolution, however,

it isnotcommutative, in fact

[f⊗g](z)=[g⊗f]∗(−z). (13.41)