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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,
CODE
$x(t)=\sum_{-\infty}^{\infty}x[n]\cdot{\rm sinc}(\frac{t-nT}{T})$
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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Nessuno
post Mar 2 2013, 11:11
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Consider that in strict theory the Shannon theorem conditions never apply in reality simply because mathematically a band limited signal cannot be also time limited, thus "having an infinite number of samples, etc etc...".
In real world, with a sufficiently large number of samples, as Alexey Lukin said, the truncation error rounds toward zero and other components (quantization, sampling time uncertainty etc...) prevail.

This post has been edited by Nessuno: Mar 2 2013, 11:12


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