A straightforward deconvolution method for use in small computers
NUCLEAR INSTRUMENTS AND METHODS 163 (1979) 5 1 9 - 5 2 2 , (~) N O R T H - H O L L A N D PUBLISHING CO
A STRAIGHTFORWARD DECONVOLUTION METHOD FOR USE...
NUCLEAR INSTRUMENTS AND METHODS 163 (1979) 5 1 9 - 5 2 2 , (~) N O R T H - H O L L A N D PUBLISHING CO
A STRAIGHTFORWARD DECONVOLUTION METHOD FOR USE IN SMAIJ~ COMPUTERS E SJONTOFT
H C Orsted lnsntut, Copenhagen, Denmark Recewed 6 July 1978 and in rewsed form 25 January 1979 The subject of this paper is the solution of the convolution (or folding) integral, which relates an observed energy spectrum to the true (or onglnal) energy spectrum The techmque presented ns neither of ~terat~ve nature nor does ~t need Fourier transforms of the observed spectrum It works dnrectly on the observed data The method therefore should be statable to small computers
1. Introduction It 1s well known that repeated measurements of a fixed quantity results m a d~strlbuUon of measurements due to the inexactness of the measunng process Often the resulting d~stributlon is the "normal" or the "Gausslan" distribution. When a great number of different quantities are to be measured in the same experiment the resultlng distributions may overlap and it IS often ~mposslble to recognize the locations of the original quantities Many deconvolutlon procedures L) have been made in order to extract ongmal data from measured overlapping distributions Most of them use a great amount of computauonal effort because they work in Fourier space or are lteratlve Consequently they are not very suitable for small computers. This paper presents a method which works in real space and makes a very modest demand on computing capacity and storage The method is based on the folding theorem of Fourier transforms and demands that the Maclaunn series of the reciprocal function of the Fourier transformed convolution function can be calculated 2. Fourier transforms
Before giving the proof it may be useful to state the following concepts from the Fourier transform theory2) ' If f(x) is a function of the variable x, its Fourier transform- if it exists- is defined as the function F(~) = (1/2n) ~
exp(--l~X) f(x)dx
f(x) is deduced by the inverse Fourier transformation f ( x ) = (1/2n) ~
exp0~x) F(c0dc~ -oe
By definition, the folding of two functions, if it exists, is the expression f *h =
f(x-t)
h(t)dt
-a-j
The interest in the folding concept comes from the following theorem:
Theorem: Provided that it exists, the folding of two functions f(x) and h(x) has its Fourier transform (2n) t F(o~)H(oO, an expression-m which F(o0 and H(~z) are the Fourier transforms of f(x) and h(x) respectively; this transformation is reciprocal In order to clarify this the following Fourier transformations are stated without proof y = f f x ) ~ Y = f(cQ,
y = f ' ( x ) ~ Y = I~F(~), 1 Y - tT x/(-~-~) exp ~ - 2 - a J ~