By J. F. James
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On Friday, February 20, 1980, I had the excitement to be current on the inaugural lecture of my colleague Jan Reedijk, who had simply been named on the Chair of Inorganic Chemistry of Leiden college. in accordance with culture, the rite came about within the outstanding corridor of the previous college Academy construction.
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Additional info for A Student's Guide to Fourier Transforms: With Applications in Physics and Engineering, Third edition
If the frequency band stretches from ν0 to ν1 , the empty frequency band between ν0 and 0 can be divided into a number of equal frequency intervals each less than 2(ν1 ν0 ). The sampling interval then need be only 1/[2(ν1 ν0 )] instead of 1/(2ν1 ). This is a way of demodulating the signal, and the spectrum that is recovered appears to occupy the first alias even though the original occupied a possibly much higher one. The process is illustrated in Fig. 8. 1 The interpolation theorem This too comes from Whittaker’s interpolary function theory.
This is much used in practical physics, where digital recording of data is common, and generally the signal at a point can be well enough recovered by a 36 Useful properties and theorems sum of sinc-functions over twenty or thirty samples on either side. The reason for this is that, unless there is a very large amplitude to a sample at some distant point, the sinc-function at a distance of 30π from the sample has fallen to such a low value that it is lost in the noise. It depends obviously on practical details such as the signal-to-noise ratio in the original data and, more importantly, on the absence of any power at frequencies higher than the folding frequency.
Convolution of a function with a sinusoid. (p) is the Fourier transform of f (x) and the two δ-functions are the Fourier transform of cos(2π rx), the other partner in the convolution. The product is the pair of δ-functions modified in height by the appropriate Fourier component of (p). e. φ(p) δ(p r) D φ(r)δ(p r). Thus (p) D 1 [φs (r)δ(p r) C φs ( r)δ(p C r) 2 C iφa (r)δ(p r) C iφa ( r)δ(p C r)] and, since φs (r) D φs ( r) and φa (r) D φa ( r), we have 1 fφs (r)[δ(p r) C δ(p C r)] C iφa (r)[δ(p 2 and on transforming back we find (p) D r) δ(p C r)]g C(x) D φs (r)cos(2π rx) C φa (r)sin(2π rx) so that C(x) is a sinusoid of amplitude φs (r)2 C φa (r)2 and phase-shifted by comparison with the original sinusoid by an angle α, given by α D tan 1 [φa (r)/φs (r)].