[CST-2] IT&C Nyquist

[Matthew Lloyd]

You've hit the nail on the head with the ambiguity about offset and
amplitude, Martin.

All the DSP books I can dip my paws into definitely state that the
sampling frequency Fs must be strictly greater than twice the maximum
frequency component, so that may be a mistake in the notes. Fs>2B.

The reason for the ambiguity is simple. When the signal is sampled and
then reconstructed using the sinc function, the result is not exactly
the original signal but an aliased version of it. The spectrum will be
periodically repeated at intervals of Fs. A real-valued sine wave at
frequency Fs/2 will have two spectral components, one at Fs/2, say x+iy,
and another at -Fs/2, which by the properties of real-valued Fourier
transforms will be x-iy, i.e. the complex conjugate. When this signal is
aliased, the component at Fs/2 will interfere linearly with the
component at -Fs/2 shifted up by Fs, and so the reconstructed version
will have a component at Fs/2 expressed by x+iy + x-iy = 2x. In this way
we have been reduced from two degrees of freedom (x and y) to just one
(x); we know the value of x but y could be anything from -inf to +inf
and hence we have lost the magnitude and phase information about the
original sinusoid. In particular, if the sine wave is of the appropriate
phase so that x=0 but y=amplitude, the reconstructed version will have a
component at Fs/2 given by 2x=0!

In fact, even if the frequency of the sinusoid is just slightly less
than Fs/2, we will still have problems! Suppose it was at Fs/2-delta.
The aliased version will have components at Fs/2-delta and Fs/2+delta
and if delta is small enough, the two sinusoids will interfere giving a
beat frequency at 2*delta in the reconstructed version. This corresponds
to the sine wave going in and out of phase with the samples so that for
long stretches at a time the samples are all very nearly zero, while at
other stretches they oscillate at maximum amplitude pos neg pos neg pos
neg etc. This would not sound pleasant.

This is why in practice people bandpass filter their signals to 0.45Fs
before sampling!

Matthew