The Sampling Theorem

This more general property of a band-limited signal (one with zero power outside a bandwidth $B$) goes by the name of the ``Shannon Sampling Theorem''. It states that a set of samples separated by $1/2B$ is sufficient to reconstruct the signal. One can obtain a preliminary feel for the theorem by counting Fourier coefficients. The number of parameters defining our signal is twice the number of frequencies, (since we have an $a$ and a $b$, or an  $r$ and a $\varphi$, for each $\omega_n$). Hence the number of real values needed to specify our signal for a time $T$ is

$$2\times\frac{\Delta\omega T}{2\pi} = 2\left(\frac{\Delta\omega}{2\pi}\right)T = 2B T$$

This rate at which new real numbers need to be measured to keep pace with the signal is $2B$. The so called ``Nyquist sampling interval'' is therefore $(2B)^{-1}$. A real proof (sketched in Section 1.8) would give a reconstruction of the signal from these samples!

In words, the Shannon criterion is two samples per cycle of the maximum frequency difference present. The usual intuition is that the centre frequency $\nu_0$ does not play a role in these considerations. It just acts a kind of rapid modulation which is completely known and one does not have to sample variations at this frequency. This intuition is consistent with radio engineers/astronomers fundamental right to move the centre frequency around by heterodyning^1.2 with local (or even imported^1.3) oscillators, but a more careful examination shows that the centre frequency should satisfy $\nu_0=(n+\frac{1}{2})B$ for the sampling at a rate $2B$ to work.

Footnotes

... heterodyning^1.2

see Chapter 3

... imported^1.3

aaaaagggh! beware of weak puns. (eds.)