Sigma Deltas

[Sanjayan Vinayagamoorthy]

Hello,

I have a couple of questions regarding Sigma Deltas again. From what I
understand the simplest digital filter available for the sigma delta
is the counter. That is you count the number of times the comparator
flips, and this is a representation of the input analog voltage.
However I think I have misunderstood something. For example if this is
the case, then digital audio, like a CD that used oversampling A/Ds
have to oversample a lot. For example, a 16 Bit A/D would have to
sample 65,536 * 44kHz. That's like 2.9 Gigasamples/s. That doesn't
make any sense! How is it that you can get the 16 bit digital out with
only 3 Megasample/s? Is it the digital filter or something else like
topology?

Is it possible for a 12 Bit A/D with only a 40 sample delay time with
the right kind of topology or digital filter?

Any help is much appreciated.

Thanks,
Sanjay

 

[Tim Wescott]

Sanjayan Vinayagamoorthy wrote:

Hello,

I have a couple of questions regarding Sigma Deltas again. From what I
understand the simplest digital filter available for the sigma delta
is the counter. That is you count the number of times the comparator
flips, and this is a representation of the input analog voltage.
However I think I have misunderstood something. For example if this is
the case, then digital audio, like a CD that used oversampling A/Ds
have to oversample a lot. For example, a 16 Bit A/D would have to
sample 65,536 * 44kHz. That's like 2.9 Gigasamples/s. That doesn't
make any sense! How is it that you can get the 16 bit digital out with
only 3 Megasample/s? Is it the digital filter or something else like
topology?

Is it possible for a 12 Bit A/D with only a 40 sample delay time with
the right kind of topology or digital filter?

Any help is much appreciated.

Thanks,
Sanjay

The counter acts like a 1st-order filter. If you use a higher-order
filter in the modulator, then you can follow it with a higher-order
low-pass. You can also claim that you don't care as much about
high-frequency accuracy (this is not a bad claim for audio and some
other things). I believe that most commercial S-D converters use 2nd-
or 3rd-order modulators and filters.

For a 2nd-order system the theoretical (and no doubt highly optimistic)
lower bound for oversampling would be 2^8, or about 12MHz; the same
lower bound for a 3rd-order system would be 2^5.333, or 1.8MHz.

--

Tim Wescott
Wescott Design Services
http://www.wescottdesign.com

 

[Rich Grise]

On Wed, 24 Nov 2004 10:58:11 -0800, Tim Wescott wrote:

Uh, Rich, he's talking about sigma-delta ADC's, which use a 1-bit ADC
(AKA a comparator) and some fancy signal processing on both the analog
and digital side to make a silk purse out of a sow's ear.

Parallel.

Except that's not what Sigma-Delta modulators are.

OK - blush. I knew it couldn't be _that_ simple.

Thanks!
Rich

 

[Lasse Langwadt Christense]

Rich Grise <rich@example.net> wrote in message news:<pan.2004.11.24.17.22.21.710866@example.net>...

snip

I'm sorry I couldn't be more helpful - I don't know sigma-delta, other
than that I've heard it called delta-sigma, which sounds like what one
records at a given sample time is the delta V since the last sample.

sigma-delta/delta-sigma it's called both. it basically integrates(sigma)
the error(delta) around a 1 bit ADC*. What this does is move the quantization
noise added when throwing away bits, to higher frequencies you can then later
remove by lowpass filtering.

Ideally doubling the sample frequency and filtering gives you an extra ~3dB
, if I remember correctly doing the same with a first-order delta-sigma
modulator gives you ~9dB

*it can be more that 1 bit

-Lasse

 

[Tim Wescott]

Sanjayan Vinayagamoorthy wrote:

Sanjayan Vinayagamoorthy wrote:

snip

The counter acts like a 1st-order filter. If you use a higher-order
filter in the modulator, then you can follow it with a higher-order
low-pass. You can also claim that you don't care as much about
high-frequency accuracy (this is not a bad claim for audio and some
other things). I believe that most commercial S-D converters use 2nd-
or 3rd-order modulators and filters.


Yes...most of the time I have used first order sigma-deltas since they
are very easy to setup and you don't have to worry about stability and
all that.


For a 2nd-order system the theoretical (and no doubt highly optimistic)
lower bound for oversampling would be 2^8, or about 12MHz; the same
lower bound for a 3rd-order system would be 2^5.333, or 1.8MHz.


How did you get those numbers by the way? The reason I don't want to
go to advanced order sigma-deltas is because the data I am sampling is
DC. As going to second order SDs would mean that my output would have
a low frequency tone. (I am not sure if this is correct, but for some
reason, I remember someone telling me this). Would it be okay to use a
2nd or higher order SD for sampling a DC signal?

From the theory that if you have a 2nd-order filter it'll be x^2 as

good, and a 3rd-order will be x^3 as good. In practice these are
absolute upper bounds, real performance will be much worse -- but the
ADI converters seem to approach this.

As far as the issues with tone generation etc., I don't know. If I were
going to make a S-D converter I'd have to do a lot of studying, so I
just scrutinize data sheets and look for issues.

(Technically I am wrong to call them SD, it should delta-sigma since
that's what the inventers Inose and Yasuda called them.)

Yes, well -- I use S-D because that's what seems to be more popular
today. You can invent something and call it whatever you like, whatever
the majority of folk call it is what its name ends up being.

--

Tim Wescott
Wescott Design Services
http://www.wescottdesign.com

 

[Sanjayan Vinayagamoorthy]

Sanjayan Vinayagamoorthy wrote:
snip
The counter acts like a 1st-order filter. If you use a higher-order
filter in the modulator, then you can follow it with a higher-order
low-pass. You can also claim that you don't care as much about
high-frequency accuracy (this is not a bad claim for audio and some
other things). I believe that most commercial S-D converters use 2nd-
or 3rd-order modulators and filters.

Yes...most of the time I have used first order sigma-deltas since they
are very easy to setup and you don't have to worry about stability and
all that.

For a 2nd-order system the theoretical (and no doubt highly optimistic)
lower bound for oversampling would be 2^8, or about 12MHz; the same
lower bound for a 3rd-order system would be 2^5.333, or 1.8MHz.

How did you get those numbers by the way? The reason I don't want to
go to advanced order sigma-deltas is because the data I am sampling is
DC. As going to second order SDs would mean that my output would have
a low frequency tone. (I am not sure if this is correct, but for some
reason, I remember someone telling me this). Would it be okay to use a
2nd or higher order SD for sampling a DC signal?

(Technically I am wrong to call them SD, it should delta-sigma since
that's what the inventers Inose and Yasuda called them.)

 

See the book "Digital Signal Processing A Pratical Guide for Engineers
and Scientists". Steven W. Smith explains on page 65 why a digital low
pass filter and decimator is equivalent to a counter for converting the
comparator's output into a 16 bit word. For the example you give yes
you would have to wait 65,536 samples fon the first valid output. But
there after with a 64 to 1 decimation and a 3 million samples per
second input stage sampling rate the A/D output rate would be 46,875 16
bit words per second. Note the digital filter output is a moving
average.

 

" How is it that you can get the 16 bit digital out with 3
Megasample/s? Is it the digital filter or something else"

You are right if you only have 1 counter. You couldn't use a Sigma
Delta A/D converter with audio. But what if you have 65,536 counters
and start the first counter counting with the first sample, start the
second counter counting with the second sample, start the third
counter counting with the third sample and so on until after 65,536
samples you have started all counters. Then after each counter has
counted for 65,536 samples read its output and restart it counting.
In this configuration you would have to wait for 65536 samples for the
first ouput from a Sigma Delta A/D converter. But after the first
output the data rate would be 3 million 16 bit words per second for
your example. Conceptually this is what the digital filter in a Sigma
Delta A/D converter does when it calculates a moving average of a Sima
Dela comparator's output. Then when you decimate the digital filter's
output by 64 to 1 you would only need 1024 counters and there would be
a 64 sample delay between the start of each counter if you implemented
in hardware what Sigma Delta manufacturers do in software.