ChapterZero

Perfect Reconstruction, but not really

Interesting seminar today, given by Bernhard, one of the guys in the Functional Analysis group here at UH: it was about the $\Sigma \Delta$ method of sampling used in the SACD (super accurate CD ?) encoding method.

What was most interesting was what he mentioned about the perfect reconstruction guaranteed by the Sampling Theorem. Recall that the Sampling Theorem says if you sample a signal at twice the maximum frequency present in the signal, then you can reconstruct the function (well, a function which differs from the original function over a set of measure 0 at most) using the formula

$\displaystyle f(t) = \sum_{n=-\infty}^\infty f\left(\frac{n\pi}{\omega}\right) \frac{\sin \omega( t - \frac{n \pi}{\omega})}{\omega( t - \frac{n \pi}{\omega})},$

where $\omega$ is the maximum frequency present in the signal.

In practical (i.e. engineering) uses, the function values $f_n \equiv f\left(\frac{n\pi}{\omega}\right)$ are quantized to values p_n so that $|f_n - p_n| \leq \epsilon$ for some error $\epsilon$ which can be decreased by increasing the number of bits used for the quantization. Using PCM (pulse code modulation), the current technique for rounding used on CDs, one rounds the value of f_n to the nearest quantization value. The idea is that if this holds true, then as the number of samples increases (since we calculate the sum in the above formula over a finite range to ), the approximation obtained for becomes more accurate. This idea is actually wrong, in general!

The error in reconstruction is

$\displaystyle \left| \sum_{n=-N}^N (f_n - p_n) \frac{\sin \omega( t - \frac{n \pi}{\omega})}{\omega( t - \frac{n \pi}{\omega})} \right|,$

so if it happens that using PCM $f_n - p_n = (-1)^{N-n} \epsilon$ , the error terms add up (since the sign switching of the $\sin$ term is canceled), and it becomes

$\displaystyle \left| \sum_{n=-N}^N (f_n - p_n) \frac{\sin \omega( t - \frac{n \pi}{\omega})}{\omega( t - \frac{n \pi}{\omega})} \right| = \epsilon (2N+1) \sin(wt)\sum_{n=-N}^N \left|\frac{1}{\omega t - n \pi}\right|,$

which clearly increases as increases. So instead of decreasing as the number of samples increases, the error increases!

In the remainder of the seminar, he talked about two things: a smarter way to quantize, $\Sigma \Delta$ , and a way to choose a better reconstruction function $s_\Omega$ (used in construction with oversampling above the ratio of 2) with better properties, which guarantees that the error can’t blow up in such a manner.

I believe somewhere I got his arguments mixed up, because looking at what I have here, I don’t see how $\Sigma \Delta$ would help avoid the problem of blowing up errors, but it was clear from his presentation. There are probably a few other errors too (for instance, his error term turned out to be asymptotic to $\ln$ , but mine is to $\cot$ if anything).
But the message is there.

The problem of finding an optimal $s_\Omega$ is interesting in it’s own right. Here are the required properties: $\int_\R |s_\Omega^\prime(x)| \, dx$ should be as small as possible, and $\hat{s}_\Omega$ should be 1 on the interval $[-\Omega, \Omega$ , be at least C^0 , and have as small support as possible.

This entry was posted on Friday, April 15th, 2005 at 4:11 pm and is filed under Mathematics. You can follow any responses to this entry through the RSS 2.0 feed. You can leave a response, or trackback from your own site.

Perfect Reconstruction, but not really

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