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Propeller Head Plaza Technical and scientific discussion of amps, cables and other topics. |
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Propeller Head Plaza Technical and scientific discussion of amps, cables and other topics. |
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In Reply to: RE: Somebody do the math posted by Todd Krieger on July 27, 2009 at 18:16:54:
"the ringing to a degree "synthesizes" the HF...."
Still don't understand anti-aliasing, sampling, and reconstruction?
The ringing fills in the gaps *exactly* as they were before the sampling was done, but after the band-limiting.
assume that f(t) is band-limited to below frequency B/2, thus
f(t)'s Fourier transform conforms to
F(f(t)) = F(W) = 0 for W >= B/2.
If f(t) is now sampled at discrete intervals t=n/B, n=0,1,2,3,...
then (it seems like magic!) the discrete values in this sample
train happen to be the coefficients of the Fourier *series* of
f(t)'s Fourier *transform* F(W) when F(W) is infinitely imaged
around integer multiples of B (making F(W) periodic so that its
Fourier series exists indeed).
The coefficients in the series (i.e. the samples)
determine F(W) unambiguously, F(W) determines f(t) unambiguously, ergo
the series coefficients (the samples) determine f(t) unambiguously.
In order to reconstruct f(t) from the samples we need an interpolating
function, let's call it S. S has to pass exactly through all sample points, for obvious reasons (1). (2) S's amplitude/phase vs frequency spectrum has to be 100% below B/2, and 0% above B/2, lest S modifies f(t)'s spectrum and reconstruction fails (S).
Sinc(Pi*B*t-n) satisfies (1), as at the n-th sample its value
is unity, and at all other sample times it is zero.
Sinc(...) satisfies (2) as its Fourier transform is the brickwall function with transission at B/2.
Sinc(...) is unique as there is no other time function that satisfies (2): the brickwall has exactly one inverse Fourier transform.
So the Sinc() function reconstructs the samples into f(t), QED.
And then you wake up and realise that your classmates of old ... are running most of the TV shows.
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Follow Ups
- RE: Somebody do the math - Werner 23:41:41 07/27/09 (0)
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