# how to understand noise shaping of sigma_delta in time domain?

Started by November 12, 2008
```let's think of the noise shaping function for sigma-delta modulators. In
frequency domain, we say quantization noise is pushed to higher frequency
and then removed by lowpass filter.

what I wana do is to find a time domain understanding of this noise
shaping function...

start from the white assumption of quantization noise. I can imagine this
is because we assume the quantization error randomly distributed in the
range from -delta/2 to delta/2.

Then after oversamping and negative feedback, what mechanism in time
domain changes the noise's distribution? and what in time domain
corresponds to the low/high probability of low/high frequency noise?

thank you very much~
```
```On Nov 12, 11:27&#2013266080;am, "lxx.helen" <lxx.he...@gmail.com> wrote:
> let's think of the noise shaping function for sigma-delta modulators. In
> frequency domain, we say quantization noise is pushed to higher frequency
> and then removed by lowpass filter.
>
> what I wanna do is to find a time domain understanding of this
> noise shaping function...
>
> start from the white assumption of quantization noise. I can imagine this
> is because we assume the quantization error randomly distributed in the
> range from -delta/2 to delta/2.

well, it isn't evenly distributed as such, and we don't assume that
for 1-bit quantization.  for a quantizer with more steps (the "N-bit
quantizer), we make that assumption for no dither (might not be such a
good assumption), and for triangular p.d.f. dither (and a multi-step
quantizer) we *do* assume that even distribution (has variance of
delta^2/12) and that the additional triangular p.d.f. dither adds
another delta^2/6.

> Then after oversamping and negative feedback, what mechanism in time
> domain changes the noise's distribution? and what in time domain
> corresponds to the low/high probability of low/high frequency noise?

it's a frequency-domain question.  to answer it with time-domain
explanation gets pretty clumsy.  it's like the wiggles in the noise
signal that wiggle a little slower are attenuated in the math that
describes the sigma-delta modulator.

to get to the concept of noise-shaping, it really should be done in
the frequency domain.  even if, in the 1-bit case, a linear system
model is problematic (but it's not too far off, at least if you model
the "gain of the comparator" correctly).

r b-j

```
```>let's think of the noise shaping function for sigma-delta modulators. In
>frequency domain, we say quantization noise is pushed to higher
frequency
>and then removed by lowpass filter.
>
>what I wana do is to find a time domain understanding of this noise
>shaping function...
>
>start from the white assumption of quantization noise. I can imagine
this
>is because we assume the quantization error randomly distributed in the
>range from -delta/2 to delta/2.
>
>Then after oversamping and negative feedback, what mechanism in time
>domain changes the noise's distribution? and what in time domain
>corresponds to the low/high probability of low/high frequency noise?
>
>thank you very much~
>

I have kind of a direction for this problem. let's say we apply a DC input
to a 1st order sigma_delta modulator.In this case, the
modulator(quantizer)output is periodic, so is the quantization input(or
integrator output). And quantization error is the difference between this
two values, so it's  periodic too.

Then i don't konw how to explain further...
```
```On Nov 13, 5:27 am, "lxx.helen" <lxx.he...@gmail.com> wrote:
> let's think of the noise shaping function for sigma-delta modulators. In
> frequency domain, we say quantization noise is pushed to higher frequency
> and then removed by lowpass filter.
>
> what I wana do is to find a time domain understanding of this noise
> shaping function...
>
> start from the white assumption of quantization noise. I can imagine this
> is because we assume the quantization error randomly distributed in the
> range from -delta/2 to delta/2.
>
> Then after oversamping and negative feedback, what mechanism in time
> domain changes the noise's distribution? and what in time domain
> corresponds to the low/high probability of low/high frequency noise?
>
> thank you very much~

Forget the time domain. I spend much of my time in the frequency
domain nowadays - much more peaceful and you can understand things
better.

hardy
```
```On Wed, 12 Nov 2008 11:13:54 -0800 (PST), HardySpicer
<gyansorova@gmail.com> wrote:

>On Nov 13, 5:27 am, "lxx.helen" <lxx.he...@gmail.com> wrote:
>> let's think of the noise shaping function for sigma-delta modulators. In
>> frequency domain, we say quantization noise is pushed to higher frequency
>> and then removed by lowpass filter.
>>
>> what I wana do is to find a time domain understanding of this noise
>> shaping function...
>>
>> start from the white assumption of quantization noise. I can imagine this
>> is because we assume the quantization error randomly distributed in the
>> range from -delta/2 to delta/2.
>>
>> Then after oversamping and negative feedback, what mechanism in time
>> domain changes the noise's distribution? and what in time domain
>> corresponds to the low/high probability of low/high frequency noise?
>>
>> thank you very much~
>
>Forget the time domain. I spend much of my time in the frequency
>domain nowadays - much more peaceful and you can understand things
>better.

I find that I frequent the time domain - it's much more eventful.
--
John
```
```On Nov 12, 2:48&#2013266080;pm, John O'Flaherty <quias...@yeeha.com> wrote:
> On Wed, 12 Nov 2008 11:13:54 -0800 (PST), HardySpicer
>
>
>
>
>
> <gyansor...@gmail.com> wrote:
> >On Nov 13, 5:27 am, "lxx.helen" <lxx.he...@gmail.com> wrote:
> >> let's think of the noise shaping function for sigma-delta modulators. In
> >> frequency domain, we say quantization noise is pushed to higher frequency
> >> and then removed by lowpass filter.
>
> >> what I wana do is to find a time domain understanding of this noise
> >> shaping function...
>
> >> start from the white assumption of quantization noise. I can imagine this
> >> is because we assume the quantization error randomly distributed in the
> >> range from -delta/2 to delta/2.
>
> >> Then after oversamping and negative feedback, what mechanism in time
> >> domain changes the noise's distribution? and what in time domain
> >> corresponds to the low/high probability of low/high frequency noise?
>
> >> thank you very much~
>
> >Forget the time domain. I spend much of my time in the frequency
> >domain nowadays - much more peaceful and you can understand things
> >better.
>
> I find that I frequent the time domain - it's much more eventful.
> --
> John- Hide quoted text -
>
> - Show quoted text -

You can consider sigma-delta to be "waveform approximation" instead of
"sample approximation". If you have, say, a third-order feedback loop
then the first three integrals of the output bitstream will match the
first three integrals of the input waveform.

Common misunderstanding;

"If I am oversampling by 64X, then I just count the number of ones and
-1's in each 64-clock period and this is all the information that I
can get. So this is only a 6-bit system!"

This is wrong. Firstly, the decimation filters used in these
converters are often thousands of taps long, so this means that
```
```On Nov 12, 7:16&#2013266080;pm, Robert Adams <robert.ad...@analog.com> wrote:
> On Nov 12, 2:48&#2013266080;pm, John O'Flaherty <quias...@yeeha.com> wrote:
>
>
>
>
>
> > On Wed, 12 Nov 2008 11:13:54 -0800 (PST), HardySpicer
>
> > <gyansor...@gmail.com> wrote:
> > >On Nov 13, 5:27 am, "lxx.helen" <lxx.he...@gmail.com> wrote:
> > >> let's think of the noise shaping function for sigma-delta modulators. In
> > >> frequency domain, we say quantization noise is pushed to higher frequency
> > >> and then removed by lowpass filter.
>
> > >> what I wana do is to find a time domain understanding of this noise
> > >> shaping function...
>
> > >> start from the white assumption of quantization noise. I can imagine this
> > >> is because we assume the quantization error randomly distributed in the
> > >> range from -delta/2 to delta/2.
>
> > >> Then after oversamping and negative feedback, what mechanism in time
> > >> domain changes the noise's distribution? and what in time domain
> > >> corresponds to the low/high probability of low/high frequency noise?
>
> > >> thank you very much~
>
> > >Forget the time domain. I spend much of my time in the frequency
> > >domain nowadays - much more peaceful and you can understand things
> > >better.
>
> > I find that I frequent the time domain - it's much more eventful.
> > --
> > John- Hide quoted text -
>
> > - Show quoted text -
>
> You can consider sigma-delta to be "waveform approximation" instead of
> "sample approximation". If you have, say, a third-order feedback loop
> then the first three integrals of the output bitstream will match the
> first three integrals of the input waveform.
>
> Common misunderstanding;
>
> "If I am oversampling by 64X, then I just count the number of ones and
> -1's in each 64-clock period and this is all the information that I
> can get. So this is only a 6-bit system!"
>
> This is wrong. Firstly, the decimation filters used in these
> converters are often thousands of taps long, so this means that- Hide quoted text -
>
> - Show quoted text -

Sorry, hit the wrong button, continued here ...

You can consider sigma-delta to be "waveform approximation" instead
of
"sample approximation". If you have, say, a third-order feedback loop
then the first three integrals of the output bitstream will match the
first three integrals of the input waveform.

Common misunderstanding;

"If I am oversampling by 64X, then I just count the number of ones
and
-1's in each 64-clock period and this is all the information that I
can get. So this is only a 6-bit system!"

This is wrong. Firstly, the decimation filters used in these
converters are often thousands of taps long, so this means that EACH
BIT CONTRIBUTES TO MANY OUTPUT SAMPLES.

Secondly, THE ORDER OF THE BITS COUNT (A LOT). If you were only
interested in the first integral, then it's true that the order would
not matter too much, but once you go to higher order then there are
complex correlations between all the bits and the order matters very
much.

So you need to analyze the error of these systems not in terms of
single-sample input-to-output error, but rather in terms of waveform
error between a discrete-time filtered version of the input, and the
digitally-filtered version of the bit-stream (with the discrete-time
filter matching the digital filter exactly); this way, you can start
to think about the error waveform. However, note that since the
filters have a lot of time dispersion, you can only talk about the RMS
Waveform error, not an individual sample error.

This is somewhat related to the concept of sub-radix-2 conversion; if
you have a number system with a radix of less than two, than there are
MANY ways that you can obtain an output value within a particular
error tolerance. This means it is possible that a single bit can play
a role in many different output samples; in some samples it will play
a more "major" role than in others, depending on the alignment of the
bit with that particular filter tap for that particular output time.