Forums

sigma-delta : unintuitive

Started by kbc July 28, 2003
I am finding the sigma-delta adc difficult to understand.

The only way, i feel, to reduce the quantization noise ,
given the bit-resolution , is to sample non-uniformly.

Is that what happens ?

       shankar
Hi Shankar -

depends on the modulator and filter that are used - but if you sample at 
ramdon intervals, this should take the 1/f away. One of the  ideas of 
the sigdelt is to push the quantization noise to higher frequencies.  I 
think you should look at the quantization noise as a first and second 
order modulation function, and using a good algorithm, you should be 
able to get rid of most of this.

Andrew

kbc wrote:

>I am finding the sigma-delta adc difficult to understand. > >The only way, i feel, to reduce the quantization noise , >given the bit-resolution , is to sample non-uniformly. > >Is that what happens ? > > shankar > >
kbc wrote:
> > I am finding the sigma-delta adc difficult to understand. > > The only way, i feel, to reduce the quantization noise , > given the bit-resolution , is to sample non-uniformly. > > Is that what happens ?
No. The quantization noise power in a digital signal is constant no matter what the sample rate is. Thus if you oversample, you can filter out some of the quantization noise because part of the digital spectrum contains noise that is not part of the signal. Thus just by oversampling you can gain bits of resolution. Delta sigma goes further and puts feedback around the quantizer so that the quantization noise is shaped to place most of it at the higher frequencies. Thus when you oversample and then lowpass filter, more of the noise is filtered off. -- % Randy Yates % "...the answer lies within your soul %% Fuquay-Varina, NC % 'cause no one knows which side %%% 919-577-9882 % the coin will fall." %%%% <yates@ieee.org> % 'Big Wheels', *Out of the Blue*, ELO http://home.earthlink.net/~yatescr
"Randy Yates" <yates@ieee.org> wrote in message
news:3F25E9A0.6651F163@ieee.org...
> The quantization noise power in a digital signal is constant no > matter what the sample rate is.
I have tried, but I haven't been able to show this for >= 2 stage S-D modulators. The trouble is that, while the digital stream has limited power, there doesn't seem to be any bound on the *error* signal it's digitizing, and so there is no limit on the difference between the two! I know they don't go too unstable in practice, but how do we get an estimate on the maximum error signal?
Matt Timmermans wrote:
> > "Randy Yates" <yates@ieee.org> wrote in message > news:3F25E9A0.6651F163@ieee.org... > > The quantization noise power in a digital signal is constant no > > matter what the sample rate is. > > I have tried, but I haven't been able to show this for >= 2 stage S-D > modulators. The trouble is that, while the digital stream has limited > power, there doesn't seem to be any bound on the *error* signal it's > digitizing, and so there is no limit on the difference between the two! > > I know they don't go too unstable in practice, but how do we get an estimate > on the maximum error signal?
Fools rush in, and all that. At the risk of piling foolishness on ignorance, I hazard a guess. Although it may not be possible to know the upper bound of the error in any one sample (other than that imposed by dynamic range), the feedback and the band-limited input together bound the short-term average error to a reasonable amount. After all, the buggers do work! Jerry -- Engineering is the art of making what you want from things you can get. &#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;
Think of a slowly varying signal. Instead of quantizing each sample of
this signal how about quantizing the difference of two samples. The
dynamic range is lower and you would need fewer bits. Reconstruct the
sampled version of the original signal by doing the inverse of the
difference.

This is in essence what sigma-delta converters do.

Jerry Avins <jya@ieee.org> wrote in message news:<3F267E28.CBE24859@ieee.org>...
> Matt Timmermans wrote: > > > > "Randy Yates" <yates@ieee.org> wrote in message > > news:3F25E9A0.6651F163@ieee.org... > > > The quantization noise power in a digital signal is constant no > > > matter what the sample rate is. > > > > I have tried, but I haven't been able to show this for >= 2 stage S-D > > modulators. The trouble is that, while the digital stream has limited > > power, there doesn't seem to be any bound on the *error* signal it's > > digitizing, and so there is no limit on the difference between the two! > > > > I know they don't go too unstable in practice, but how do we get an estimate > > on the maximum error signal? > > Fools rush in, and all that. At the risk of piling foolishness on > ignorance, I hazard a guess. > > Although it may not be possible to know the upper bound of the error in > any one sample (other than that imposed by dynamic range), the feedback > and the band-limited input together bound the short-term average error > to a reasonable amount. After all, the buggers do work! > > Jerry
"Jerry Avins" <jya@ieee.org> wrote in message
news:3F267E28.CBE24859@ieee.org...
> Although it may not be possible to know the upper bound of the error in > any one sample (other than that imposed by dynamic range), the feedback > and the band-limited input together bound the short-term average error > to a reasonable amount. After all, the buggers do work!
Interesting that you mention a band-limited input. I have a feeling that this is important for their stability and good performance. Even though it isn't a part of the standard analysis, a band-limited input rules out adversarial sorts of input signals that might try to maximize the magnitude of the feedback term.
You know, I think S-D modulators are designed incorrectly.

The goal should be to minimize the power of the real quantization noise,
defined as modulator output minus input signal, after passing through a
reconstruction filter.

This can be viewed as an adversarial game played between the modulator,
which gets to choose the sign of the noise pulse produced in every sample
period, and the input signal, which gets to (within bounds) choose the
magnitude.

In general, I don't think that the modulator's optimal strategy would be to
simply quantize a LTI function of the input signal.  Instead, a more
compilcated function would partition the state space of the reconstruction
filter into 1 and -1 output regions.  The modulator's best strategy would be
equivalent to simulating the reconstruction filter and applying this
decision procedure on its state.

Hmmm...

Matt


Matt Timmermans wrote:
> > "Randy Yates" <yates@ieee.org> wrote in message > news:3F25E9A0.6651F163@ieee.org... > > The quantization noise power in a digital signal is constant no > > matter what the sample rate is. > > I have tried, but I haven't been able to show this for >= 2 stage S-D > modulators. The trouble is that, while the digital stream has limited > power, there doesn't seem to be any bound on the *error* signal it's > digitizing, and so there is no limit on the difference between the two! > > I know they don't go too unstable in practice, but how do we get an estimate > on the maximum error signal?
Hi Matt, I wasn't clear when I stated "digital signal." What I meant was a signal that is digitized (or re-digitized) using a simple N-bit quantizer. The noise is about 6*N dB below the full-scale signal. I believe what you are talking about is the stability of the modulator due to the feedback loop. That's a whole other subject and one that is quite complex from what I've gathered. I think they've analyzed first- and second-order loops analytically for stability, but higher-order loops are still a shot in the dark (I think). I have some material on analyzing stability at work - I'd be happy to provide the references when I get in tomorrow if you're interested. -- % Randy Yates % "...the answer lies within your soul %% Fuquay-Varina, NC % 'cause no one knows which side %%% 919-577-9882 % the coin will fall." %%%% <yates@ieee.org> % 'Big Wheels', *Out of the Blue*, ELO http://home.earthlink.net/~yatescr
Matt Timmermans wrote:
> > You know, I think S-D modulators are designed incorrectly. > > The goal should be to minimize the power of the real quantization noise, > defined as modulator output minus input signal, after passing through a > reconstruction filter.
Yeah, well that's a heckuva minimization problem. Is there only one minimum? What does the performance surface look like? Interesting idea - lots of questions in the how-to-do category.
> This can be viewed as an adversarial game played between the modulator, > which gets to choose the sign of the noise pulse produced in every sample > period, and the input signal, which gets to (within bounds) choose the > magnitude. > > In general, I don't think that the modulator's optimal strategy would be to > simply quantize a LTI function of the input signal. Instead, a more > compilcated function would partition the state space of the reconstruction > filter into 1 and -1 output regions. The modulator's best strategy would be > equivalent to simulating the reconstruction filter and applying this > decision procedure on its state. > > Hmmm... > > Matt
Hmmm... -- % Randy Yates % "...the answer lies within your soul %% Fuquay-Varina, NC % 'cause no one knows which side %%% 919-577-9882 % the coin will fall." %%%% <yates@ieee.org> % 'Big Wheels', *Out of the Blue*, ELO http://home.earthlink.net/~yatescr