DAC interpolation 
DAC interpolation 
Mar 2 2013, 03:25
Post
#1


Group: Members Posts: 2 Joined: 2March 13 Member No.: 106971 
The NyquistShannon 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 WhittakerShannon interpolation theorem, computing the sum of the infinite series of normalized sinc functions for each sample. (Well,
CODE $x(t)=\sum_{\infty}^{\infty}x[n]\cdot{\rm sinc}(\frac{tnT}{T})$ 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? 


Mar 2 2013, 04:45
Post
#2


Group: Members Posts: 208 Joined: 31July 08 Member No.: 56508 
The formula of WhittakerShannon contains infinite sums. Practical DACs typically evaluate this sum over 20–100 samples which is a good approximation: it allows for nearly perfect reconstruction of signals up to 0.45•Fs.



Mar 2 2013, 11:11
Post
#3


Group: Members Posts: 423 Joined: 16December 10 From: Palermo Member No.: 86562 
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  ... I live by long distance.



Mar 2 2013, 11:27
Post
#4


Group: Developer Posts: 3523 Joined: 2December 07 Member No.: 49183 
E.g.: actual characteristics of PCM1794 can be found in http://www.ti.com/lit/ds/symlink/pcm1794.pdf (page 8) 


Mar 3 2013, 12:52
Post
#5


Group: Members Posts: 41 Joined: 7February 13 Member No.: 106471 
Noteworthy is that finite "sinc interpolation" is usually windowed to reduce ripple and improve stopband attenuation. A common window is the Kaiser window.



Mar 3 2013, 21:26
Post
#6


Group: Members Posts: 1418 Joined: 9January 05 From: In the kitchen Member No.: 18957 
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 timelimited 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.  
J. D. (jj) Johnston 


Mar 4 2013, 04:10
Post
#7


Group: Members Posts: 2 Joined: 2March 13 Member No.: 106971 
Thanks, I think I understand a bit better now. Especially how the use of a lowpass filter approximates the use of the sinc function in the sampling theorem.



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