By J. F. James

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**Example text**

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)].