On 28/07/2016 12:04 AM, Ethan Fenn wrote:
Because I don't think there can be more than one between any two
adjacent sampling times.
This really got the gears turning. It seems true, but is it a theorem?
If not, can anyone give a counterexample?
I don't know whether it's a classical theorem, but I think it is true.
Define the normalized sinc function as:
sinc(t) := sin( pi t ) / (pi t)
sinc(0) = 1. the signal is analytic everywhere.
A bandlimited, periodically sampled discrete-time signal {x_n} can be
interpolated by a series of time-shifted normalized sinc functions, each
centered at time n and scaled by amplitude x_n. This procedure can be
used to produce the continuous-time analytic signal x(t) induced by
{x_n}. We want to know how many peaks (direction changes) there can be
in x(t) between x(n) and x(n+1).
Sinc is bandlimited and has no frequencies above the Nyquist rate
(fs/2). A sum of time shifted sincs is also bandlimited and therefore
has no frequencies above the Nyquist rate.
Now all you need to do is prove that a band-limited analytic signal
whose highest frequency is fs/2 has no more than one direction change
per sample period. I can't think how to do that formally right now, but
intuitively it seems plausible that a signal with no frequencies above
the nyquist rate would not have time-domain peaks spaced closer than the
sampling period.
Ross.
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