Mathematical Methods for Physics and Engineering - Matematica.NET

INTEGRAL TRANSFORMSg(y)(a)(b)(c)(d)0yFigure 13.5 Resolution functions: (a) ideal δ-function; (b) typical unbiasedresolution; (c) and (d) biases tending to shift observations to higher valuesthan the true one.even function, i.e. one for which f(t) =f(−t), we can define the Fourier cosinetransform pair in a similar way, but with sin ωt replaced by cos ωt.13.1.7 Convolution and deconvolutionIt is apparent that any attempt to measure the value of a physical quantity islimited, 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 afunction of an independent variable, x say, i.e. the true function to be measuredtakes the form f(x). On the other hand, the apparatus we are using does not givethe true output value of the function; a resolution function g(y) is involved. Bythis we mean that the probability that an output value y = 0 will be recordedinstead as being between y and y+dy is given by g(y) dy. Some possible resolutionfunctions of this sort are shown in figure 13.5. To obtain good results we wishthe resolution function to be as close to a δ-function as possible (case (a)). Atypical piece of apparatus has a resolution function of finite width, although ifit is accurate the mean is centred on the true value (case (b)). However, someapparatus may show a bias that tends to shift observations to higher or lowervalues than the true ones (cases (c) and(d)), thereby exhibiting systematic error.Given that the true distribution is f(x) and the resolution function of ourmeasuring apparatus is g(y), we wish to calculate what the observed distributionh(z) will be. The symbols x, y and z all refer to the same physical variable (e.g.446