Mathematical Methods for Physics and Engineering : A Comprehensive Guide

INTEGRAL TRANSFORMS

g(y)

(a)

(b)

(c)

(d)

0 y

Figure 13.5 Resolution functions: (a) idealδ-function; (b) typical unbiased

resolution; (c) and (d) biases tending to shift observations to higher values

than the true one.

even function, i.e. one for whichf(t)=f(−t), we can define theFourier cosine

transform pairin a similar way, but with sinωtreplaced by cosωt.

13.1.7 Convolution and deconvolution

It is apparent that any attempt to measure the value of a physical quantity is

limited, to some extent, by the finite resolution of the measuring apparatus used.

On the one hand, the physical quantity we wish to measure will be in general a

function of an independent variable,xsay, i.e. the true function to be measured

takes the formf(x). On the other hand, the apparatus we are using does not give

the true output value of the function; a resolution functiong(y) is involved. By

this we mean that the probability that an output valuey= 0 will be recorded

instead as being betweenyandy+dyis given byg(y)dy. Some possible resolution

functions of this sort are shown in figure 13.5. To obtain good results we wish

the resolution function to be as close to aδ-function as possible (case (a)). A

typical piece of apparatus has a resolution function of finite width, although if

it is accurate the mean is centred on the true value (case (b)). However, some

apparatus may show a bias that tends to shift observations to higher or lower

values than the true ones (cases (c)and(d)), thereby exhibiting systematic error.

Given that the true distribution isf(x) and the resolution function of our

measuring apparatus isg(y), we wish to calculate what the observed distribution

h(z) will be. The symbolsx,yandzall refer to the same physical variable (e.g.