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I'm either being pedantic or still missing the point. Given a choice,
I'd rather not miss the point.
You can't have a sampled input without having the output of the
transform be periodic. The proof is simple:
Understood. The way I had always looked at this was as follows
(hoping I can make this clear in my non-UTF8 client, and hopefully
without typos). Casting a sampled x(t) as
sx(t) = [Sum_over_all_n] x[n] delta-function[t - nT]
where T is the sampling interval. Then, fourier-transform sx(t):
X(f) = [Integral over +- infinity] sx(t) exp(-2pi*i*f*t) dt
X(f) = [Sum_over_n] x[n] exp(-2pi*i*f*n*T)
which is a fourier series representation of X(f), which by definition
is periodic. Is there anything mathematically fishy about looking at
it in this slightly more formal way?