📄 an introduction to delta sigma converters.mht
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Subject: An Introduction to Delta Sigma Converters
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<META content=3D"Uwe Beis" name=3DAuthor>
<META=20
content=3D"An introduction to delta sigma converters without much theory =
and mathematics"=20
name=3Ddescription>
<META=20
content=3D"delta sigma, sigma delta, converter, ADC, DAC, bitstream, =
PCM, PPM, pulse proportion modulation, conversion noise, low pass =
filter, modulator, oversampling rate, second order, first order, =
Decimation, Direct Stream Digital, DSD"=20
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<P></P>
<H2>
<CENTER>An Introduction to Delta Sigma Converters</CENTER></H2>
<P>
<CENTER><IMG height=3D16 =
src=3D"http://www.beis.de/Elektronik/Fahne_de.gif" width=3D28=20
align=3Dmiddle border=3D0 NATURALSIZEFLAG=3D"3"> German version: =
<B><A=20
href=3D"http://www.beis.de/Elektronik/DeltaSigma/DeltaSigma_D.html">Eine =
Einf=FChrung in Delta-Sigma-Wandler</A></B></CENTER>
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<TABLE cellSpacing=3D0 cellPadding=3D3 width=3D"50%" border=3D1>
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<TD width=3D"100%">
<P>
<CENTER> Update as of August, 2007: Chapters "<A=20
=
href=3D"http://www.beis.de/Elektronik/DeltaSigma/DeltaSigma.html#MultiBit=
">Multi-Bit=20
Converter</A>" and "<A=20
=
href=3D"http://www.beis.de/Elektronik/DeltaSigma/DeltaSigma.html#Bitstrea=
mMath">Mathematical=20
Operations with Bitstream Signals</A>" added as well as some more =
minor=20
additions</CENTER>
<P></P>
<P>
<CENTER>For questions and comments of public interest also visit =
the <A=20
href=3D"http://www.beis.de/Forum/index.html" =
target=3D_blank>Forum</A> I=20
established</CENTER></TD></TR></TBODY></TABLE></CENTER>
<P></P>
<P>When looking for an introduction to delta sigma conversion I found =
that most=20
explanations were from a very theoretical point of view. It took me a =
while to=20
understand how Delta Sigma converters really work. So I decided to write =
this=20
introduction for people who prefer circuit diagrams to reading abstract=20
equations.</P>
<P>To understand what I'm talking about you should at least be familiar=20
with:</P>
<P>- Standard analogue techniques (op-amps, comparators etc.)<BR>- =
Standard=20
digital techniques (latches, binary codes etc.)<BR>- Standard ADCs and =
DACs=20
(resolution, speed)<BR>- What a low pass filter is (at least an analogue =
one)<BR>- The sampling theorem (sample frequency > 2 x input =
bandwidth, alias=20
effects)</P>
<P>Delta sigma converters are different from other converters. Note that =
I do=20
not make a difference between analogue-to-digital (ADC) and =
digital-to-analogue=20
converters (DAC). Both are very similar and what is realized in one of =
them=20
using analogue signal processing circuitry is implemented in the other =
one using=20
digital signal processing and vice versa. I will explain the delta sigma =
technique with the analogue-to-analogue delta sigma converter as the =
first=20
object.</P>
<P>
<CENTER><IMG height=3D136 alt=3D"Block Diagram of a Delta Sigma =
Converter"=20
src=3D"http://www.beis.de/Elektronik/DeltaSigma/DeltaSigma.GIF" =
width=3D667=20
align=3Dbottom border=3D0 NATURALSIZEFLAG=3D"3"></CENTER>
<P></P>
<P>
<CENTER><B>Figure 1 - Block Diagram of a Delta Sigma =
Converter</B></CENTER>
<P></P>
<P>A delta sigma ADC or DAC always consists of a <B>delta sigma =
modulator</B>=20
which produces the <B>bitstream</B> and a <B>low pass filter</B>.</P>
<P>The modulator will be implemented with digital technology if you have =
a=20
digital signal source and in analogue technique in case of an analogue =
signal=20
source. The same applies to the low pass filter: You will use an =
analogue low=20
pass filter if you need an analogue signal output. A digital low pass =
filter=20
will be implemented if you want a digital output. The digital low pass =
filter=20
will probably be realized by a digital circuit or by an algorithm within =
a=20
signal processor.</P>
<P>Before I proceed to the delta sigma modulator I would like to have a =
closer=20
look to the bitstream and the low pass.</P>
<H3>The Bitstream</H3>
<P>The bitstream can be regarded either as a digital or an analogue =
signal. The=20
bitstream is a one-bit serial signal with a bit rate much higher than =
the data=20
rate e.g. of the ADC. Its major property is that <B>its average level =
represents=20
the average input signal level</B>. A digital "high" represents the =
highest and=20
a "low" represents the lowest possible output value.</P>
<UL>
<LI>Analogue output: The bitstream will be converted to an analogue =
signal by=20
a one bit DAC that converts the logic information (low / high or 0 / =
1) to two=20
precise analogue voltage levels, e.g. -1V and +1V.=20
<LI>Digital output: A "high" (or "low" resp.) in the bitstream =
represents the=20
highest (lowest) digital output value, e.g. hexadecimal FF (00) in an =
8 bit=20
system. </LI></UL>
<P>You can find a similar bitstream in a <B>pulse width modulated</B> =
(PWM)=20
system but it has some disadvantages compared to the bitstream of a =
delta sigma=20
modulator. The delta sigma kind of bitstream is also known as a <B>pulse =
proportion modulated</B> (PPM) signal. The serial transmission of =
numerically=20
represented signal values (e.g. the serial output of a conventional ADC) =
is=20
called <B>pulse code modulation</B> (PCM).</P>
<H3>The Low Pass Filter</H3>
<P>The low pass filter at the output is required, because you have to =
gain the=20
average signal level out of the bitstream. You can regard the bitstream =
as a=20
signal with its information in the lower frequency band and lots of =
noise above=20
it. I presume low pass filters to be known and will not go into further =
deatils=20
here.</P>
<H3>The Delta Sigma Modulator</H3>
<P>The delta sigma modulator is the core of delta sigma converters. As =
mentioned=20
above it produces a bitstream. The average level of this bitstream =
represents=20
the input signal level. A simple analogue <B>first order delta sigma=20
modulator</B> block diagram looks like this:</P>
<P>
<CENTER><A name=3DBlockDiag1stA></A><IMG height=3D208=20
alt=3D"Block Diagram of a First Order Analog Delta Sigma Modulator"=20
src=3D"http://www.beis.de/Elektronik/DeltaSigma/DeltaSigma1BlockDiagram.G=
IF"=20
width=3D535 align=3Dbottom border=3D0 NATURALSIZEFLAG=3D"3"></CENTER>
<P></P>
<P>
<CENTER><B>Figure 2 - Block Diagram of a First Order Analogue Delta =
Sigma=20
Modulator</B></CENTER>
<P></P>
<P>Please notice that due to the negative feedback loop the average(!) =
output=20
level at the 1-Bit DAC must always be equal to the input signal =
level.</P>
<P>The digital counterpart looks just as simple:</P>
<P>
<CENTER><A name=3DBlockDiag1stD></A><IMG height=3D196=20
alt=3D"Block Diagram of a First Order Digital Delta Sigma Modulator"=20
src=3D"http://www.beis.de/Elektronik/DeltaSigma/DeltaSigmaDBlockDiagram.G=
IF"=20
width=3D524 align=3Dbottom border=3D0 NATURALSIZEFLAG=3D"3"></CENTER>
<P></P>
<P>
<CENTER><B>Figure 3 - Block Diagram of a First Order Digital Delta Sigma =
Modulator</B></CENTER>
<P></P>
<P>The comparator, just like in the analogue version, decides whether =
its input=20
value is higher or lower than a certain threshold and puts out a single =
bit=20
signal, the bitstrem. BTW, due to the preceding integrator this =
threshold is=20
arbitrary. In order to obtain the bitstream in the digital modulator it =
is=20
sufficient to strip off the comparator's input MSBit.</P>
<P>A 1-Bit DAC can output two different values only. They are termed =
VRef- and=20
VRef+ and those of the 1-Bit DDC (digital-to-digital converter) DRef- =
and DRef+=20
correspondingly. In both types of modulators they determine its input =
range.=20
Examples:</P>
<P>In the analogue modulator input ranges result out of the reference =
voltages=20
as follows:</P>
<P>
<TABLE cellSpacing=3D0 cellPadding=3D3 border=3D1>
<TBODY>
<TR>
<TD>VRef-</TD>
<TD>VRef+</TD>
<TD>Input Range</TD></TR>
<TR>
<TD>0 V</TD>
<TD>+1 V</TD>
<TD>0 to +1 V</TD></TR>
<TR>
<TD>-10 V</TD>
<TD>+10 V</TD>
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