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This more general property of a band-limited signal (one with zero
power outside a bandwidth ) goes by the name of the ``Shannon Sampling
Theorem''. It states that a set of samples separated by 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
and a , or an and a , for each ). Hence the
number of real values needed to specify our signal for a time is

This rate at which new real numbers need to be measured to keep pace with
the signal is . The so called ``Nyquist sampling interval'' is therefore
. 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 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
for the sampling at a rate to work.

#### Footnotes

- ...
heterodyning
^{1.2}
- see Chapter 3
- ...
imported
^{1.3}
- aaaaagggh! beware of weak puns. (eds.)

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