By J. F. James
This re-creation of a winning textbook for undergraduate scholars in physics, laptop technological know-how and electric engineering describes very important modern rules in sensible technological know-how and data know-how at an comprehensible point, illustrated with labored examples and copious diagrams. the sphere is roofed widely instead of extensive, and comprises references to extra prolonged works on a variety of issues. This new version is somewhat multiplied, and comprises extra new fabric within the functions sections.
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Extra info for A Student's Guide to Fourier Transforms: With Applications in Physics and Engineering
3. A pair of o-functions and its transform. 3) The third of these is illustrated in Fig. 3 Convolutions and the convolution theorem Convolutions are an important concept, especially in practical physics, and the idea of a convolution can be illustrated simply by an example. Imagine a 'perfect' spectrometer, plotting a graph of intensity against wavelength, of a monochromatic source of light of intensity S and wavelength Aa. Represent the power spectral density ('the spectrum') of the source by S8(A - Aa).
Suppose that the intensity of the source as a function of wavelength is SeA). We assume that a monochromatic line at any wavelength A1 will be plotted as a similarly shaped function kI(A - AI). Then an infinitesimal interval of the spectrum can be considered as a monochromatic line, at AI, say, and of intensity SeA 1)dA 1 and it is plotted by the spectrometer as a function of A: and the intensity apparently at another wavelength: A2 is: The total power apparently at A2 is got by integrating this over all wavelengths: i: or, dropping unnecessary subscripts: D(A) = k S(Al)I(A - Al) dA I and the output curve, D(A) is said to be the convolution of the spectrum SeA) with the instrumental function I(A).
The spectrum of the signal can be regarded as the product of the periodic function with a top-hat function of width 2vf. In the signal, each sample is replaced by the convolution of the sinc-function with the corresponding 8-function. Each sample, an8(t - tn) is replaced by the sinc-function, an sincrr V f and each sinc-function conveniently has zeros at the positions of all the other samples (this is hardly a coincidence, of course) so that the signal can be reconstructed from a knowledge of its samples which are the coefficients of the Fourier series which form its spectrum.