Mathematical Methods for Physics and Engineering - Matematica.NET

13.1 FOURIER TRANSFORMSThe inverse of convolution, called deconvolution, allows us to find a truedistribution f(x) given an observed distribution h(z) and a resolution functiong(y).◮An experimental quantity f(x) is measured using apparatus with a known resolution functiong(y) to give an observed distribution h(z). Howmayf(x) be extracted from the measureddistribution?From the convolution theorem (13.38), the Fourier transform of the measured distributionis˜h(k) =√2π ˜f(k)˜g(k),from which we obtain1 ˜h(k)˜f(k) = √2π ˜g(k) .Then on inverse Fourier transforming we find[f(x) = √ 1 ˜h(k)F −1 .2π ˜g(k)In words, to extract the true distribution, we divide the Fourier transform of the observeddistribution by that of the resolution function for each value of k and then take the inverseFourier transform of the function so generated. ◭This explicit method of extracting true distributions is straightforward for exactfunctions but, in practice, because of experimental and statistical uncertainties inthe experimental data or because data over only a limited range are available, itis often not very precise, involving as it does three (numerical) transforms eachrequiring in principle an integral over an infinite range.]13.1.8 Correlation functions and energy spectraThe cross-correlation of two functions f and g is defined byC(z) =∫ ∞−∞f ∗ (x)g(x + z) dx. (13.40)Despite the formal similarity between (13.40) and the definition of the convolutionin (13.37), the use and interpretation of the cross-correlation and of the convolutionare very different; the cross-correlation provides a quantitative measure ofthe similarity of two functions f and g as one is displaced through a distance zrelative to the other. The cross-correlation is often notated as C = f ⊗ g, and, likeconvolution, it is both associative and distributive. Unlike convolution, however,it is not commutative, in fact[f ⊗ g](z) =[g ⊗ f] ∗ (−z). (13.41)449