Mathematical Methods for Physics and Engineering : A Comprehensive Guide

13.1 FOURIER TRANSFORMS

−a aa−a

x y z

f(x)

−b b

2 b 2 b

∗ g(y) = h(z)

1

Figure 13.6 The convolution of two functionsf(x)andg(y).

length or angle), but are denoted differently because the variable appears in the

analysis in three different roles.

The probability that a true reading lying betweenxandx+dx, and so having

probabilityf(x)dxof being selected by the experiment, will be moved by the

instrumental resolution by an amountz−xinto a small interval of widthdzis

g(z−x)dz. Hence the combined probability that the intervaldxwill give rise to

an observation appearing in the intervaldzisf(x)dx g(z−x)dz. Adding together

the contributions from all values ofxthatcanleadtoanobservationintherange

ztoz+dz, we find that the observed distribution is given by

h(z)=

∫∞

−∞

f(x)g(z−x)dx. (13.37)

The integral in (13.37) is called theconvolutionof the functionsfandgand is

often writtenf∗g. The convolution defined above is commutative (f∗g=g∗f),

associative and distributive. The observed distribution is thus the convolution of

the true distribution and the experimental resolution function. The result will be

that the observed distribution is broader and smoother than the true one and, if

g(y) has a bias, the maxima will normally be displaced from their true positions.

It is also obvious from (13.37) that if the resolution is the idealδ-function,

g(y)=δ(y)thenh(z)=f(z) and the observed distribution is the true one.

It is interesting to note, and a very important property, that the convolution of

any functiong(y) with a number of delta functions leaves a copy ofg(y)atthe

position of each of the delta functions.

Find the convolution of the functionf(x)=δ(x+a)+δ(x−a)with the functiong(y)

plotted in figure 13.6.

Using the convolution integral (13.37)

h(z)=

∫∞

−∞

f(x)g(z−x)dx=

∫∞

−∞

[δ(x+a)+δ(x−a)]g(z−x)dx

=g(z+a)+g(z−a).

This convolutionh(z) is plotted in figure 13.6.

Let us now consider the Fourier transform of the convolution (13.37); this is