🚀 go-pugleaf

sza@physics.phy.duke.edu (Eric Szarmes) writes:

>...
>   This convolution essentially replaces each delta function with a copy 
>of the signal spectrum, and if the sampling rate is more than twice the 
>highest frequency component, we see (as below) that there is no aliasing, 
>or overlap, of spectral components. Filtering the sampled signal with the 
>appropriate lowpass filter then exactly reproduces the original signal. 
>...

Note:  that "_more_than_" is crucial.  If we sample a sinewave at
exactly twice its frequency, and happen to sample it at its zero
crossings, we get a stream of zeros out.  And if we sample it at any
other phase, we still have no way of knowing its amplitude.  Note also
that to satisfy the Nyquist theorem, the sinewave must go on forever,
else it will have other (including higher) frequency components.  

Note also that you can extend the Nyquest theorem to allow `perfect'
sampling of any band whose _width_ is less than 1/2 the sampling
frequency, so long as it does not contain a frequency that is a harmonic
of 1/2 the sampling frequency (since all such harmonics suffer the same
problem).  This assumes instantaneous, jitter-free sampling, of course.