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DAC interpolation
SAI Peregrinus
post Mar 2 2013, 03:25
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The Nyquist-Shannon theorem (the sampling theorem) states that if a function x(t) contains no frequencies greater than B hertz, it is completely characterized by measuring its amplitudes at a series of points spaced 1/(2B) seconds apart. It further states that to reconstruct the original signal one must use the Whittaker-Shannon interpolation theorem, computing the sum of the infinite series of normalized sinc functions for each sample. (Well,
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$x(t)=\sum_{-\infty}^{\infty}x[n]\cdot{\rm sinc}(\frac{t-nT}{T})$
where n is the sample number, t is the sample time, T is 1/(sampling rate), and sinc(x) is the normalized sinc function.)

I've never (in my somewhat limited experience) seen an audio DAC that does that, yet they seem to have pretty reasonable output. How accurate are the standard approximations, really? What is the interpolation error? How does one characterize it?
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Woodinville
post Mar 3 2013, 21:26
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What the OP fails to note is that if we allow a transition band between the highest frequency captured and fs/2, then we don't have to use a sinc function any longer, and we can also use a finite length filter.

Perhaps the person who wrote about the Shannon Theorem could read the proof of the Nyquest conjecture and by doing so, see how the statement above is incontrovertably proven.

Likewise the people who argue that all time-limited signals have infinite bandwidth, again, strictly speaking, this is true, however, we can trivially estimate the actual error and know its magnitude exactly, in terms of filtering, and easily move it well below the noise floor of any possible system.

Again, people need to understand the whole of the theorem and the paper, not just part of it.

Likewise people who argue from a statistical point of view, but who don't calculate error bounds.


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