IEEE TRANSACTIONS
COMMUNICATIONS,
ON
VOL.NO.
COM-33,
3,
249
MARCH 1985
A Use of Double Integration in Sigma Delta Modulation
JAMES C . C A h Y , FELLOW,IEEE
1
'
Abstract-Sigmadeltamodulation
is viewedas
a technique that
employs integration andfeedbacktomove
quantizationnoise outof
baseband. This techniquemaybeiterated
by placing feedback loop
around feedback loop, but when three or more loops are used the circuit
canlatch into undesirable overloading modes. In the, desired mode, a
simple linear theorygives a good description the
of modulation even when
the quantization has only two levels. A modulator that employs double
integration andtwo-level quantizationis easy to implement andtolerant
is
of parameter variation. Atsampling ratesof 1 MHz it provides resolution
are
equivalent to 16 bit PCM for voiceband signals. Digital filters that
suitable for converting the modulation to PCM are also described.
multipleintegration,andcalculatethesignal-to-noiseratios.
Theresultsareconfirmedbysimulation,andexperimental
measurements. We show that when more than two integrators
are used, the circuit can latch into undesirable modes where
its performance is ruined. Finally, we give a design for a digital
processor for constructing PCM from this modulation.
11. QUANTIZATIONWITH FEEDBACK
Fig. 1 (a) shows the circuit of a differential quantizer which
is a form of the well-known delta modulator; we will use this
circuit t o explain our view of feedback modulation. Fig. l(b)
shows a sampled data model of the circuit; it assumes that the
A/D ahd D/A conversion are ideal and that signals are random,
I. INTRODUCTION
so that the quantization may be represented by added noise e
and linear gain G (level-spacing/threshold spacing). AccumulaHIS paper describes the design of a digital modulator that
isinteti'ded for use in oversampled
PCM encoders. These
tion A representstheintegration.Mathematicaldescriptions
of related circuits have been presented in several places
[ 11,
encodersmodulatetheiranaloginputsintoasimpledigital
[ 3 1 , [ 41, [ 121, [ 141 -[ 161. They show that the presence of
format
highspeed;thendigitalprocessingtransformsthe
feedback around the quantizer has three uses, which are summodulationto PCM sampledattheNyquistrate
[ 11-[9].
marized below.
Propertiesofthepreliminarymodulation
have stronginPrediction and Preemphasis: The modulated signal A4 comfluence on the design-of the entire encoder. For example, the
isp.roporresolution of the PCM can be no better than that. of the modu- prisesanoisecomponentandacomponentthat
tional to the rate of change of input amplitude. Modulating a
lation, and the complexity and speed of the digital processor
rateofchangecanbemoreefficientthanmodulatingthe
depends on the kind of modulation used and its resolution.
amplitudedirectly,particularlyforvideoandaudio
signals
The tolerance of the analog circuits employed in the modulator
can determine the suitability of the design for integrated
cir- whosespectraldensitiesfallwithincreasingfrequencyand
cuit implementation and the power consumed by these circuitswhosesamplevaluesarehighlycorrelated
[ 151, [ 161. This
can be a large part of the power used
b y the entire encoder.
improved efficiency can result in decreased sampling rate or a
There is, therefore, much incentive to find an efficient modureduction in the number of quantization
levelsneeded for a
lator,onethatprovideshighresolution(idlechannelnoise
given resolution.
morethan 80 dBbelowpeaksignal)atmoderatesampling
Control of Overloading: Ordinary PCM quantization overrates (less than 1 MHz for 4 kHz telephone signals) yet employs loads by clippingsignal amplitudes directly. When this happens
simple robust circuits (tolerances no tighter than +3 percent).
t o t h e signal applied to the A/D in Fig. 1, it is t h e derivative
Early work on oversampled encoders [ 11 , [ 21 was mostly
of the input signal that is clipped, resulting in slope overloadtheoreticalandbasedondeltamodulation.Later,practical
ing of the output signal. Distorting the slope of video [ 171 and
realization preferred sigma delta modulation but modified it to audio signals can be less disturbing than clipping their amplilower t h e samplingrateandsimplifythedigitalprocessing.
tudes directly. This also can lead to improved efficiency of the
For a video application, multilevel quantization was used [3]
modulation.
toreducethemodulationrate.Fortelephoneapplications,
NoiseShaping: Placing thequantizerinafeedbackloop
some modulators [4], [5] achieve
highresolutionbybiasing
with a fiiter shapes the spectrum of the modulation noise[ 3 1 ,
the modulator to an especially quiet state for idle channel op[ 131, and at the same time it can decorrelate the noise from
eration. One [ 71 employed triple integration in the sigma delta t h e signal. If we assume
that the quantization noise e in Fig.
modulator;another [ 81 employeddigitalaccumulationand
l ( b ) is white,thenthespectraldensityofthenoiseinthe
companded quantizationlevels in the feedback path.
modulated signalrises withfrequency;butafterintegration
Recentadvancesindigitalintegratedcircuittechnology
t h e noise is white again
at the output. We willsee that other
have greatly reduced the need to have simple digital processciicuit configurations [ 131 -[ 181 can shape the output noise
ing; indeed, now it is feasible [ 101 to have digital line equalispectrum to suit particular applications.
on
zation,echocanceling,digitalhybrids,andconferencing
When higher order filters are used in place
of simple integrathechipwiththecodec.Theseapplications,however,place
tion [ l l ]-[ 141, the propertiesof the modulation are modified.
to be Themodulated signalincludescomponentsthatareproporstringentdemandsonresolutionanddynamicrange
provided by the modulator.
tional to high-order derivatives of the signal. Overloading limits
The present work explores the advantages of having double not only the slope but also the rate of change of slope of the
integration in a sigma delta modulator.
We demonstrate that
signal, and modulation noiserises more steeply with frequency.
a particular class of circuits can provide high resolution and be
The restriction on the design of these high-order filters is the
toleranttoimperfection.
We explainthereasonsforusing
need to keep the feedback stable.
All these properties of feedback quantization can influence
t h e design of modulators for oversampled codecs, and in some
Paper approved by the Editor for Signal Processing and Communication
Electronics of the IEEE Communications Society for publication without oral applications the requirements are in conflict. For example, oppresentation. Manuscript received March28, 1984; revised October 16,1984. timum design of predictors usually calls for leaky integration,
The author is with AT&T Bell Laboratories, Holmdel, NJ 07733.
but optimum noise shaping calls for long-time constant inte-
T
0090-6778/85/0300-0249$Ol.OO 0 1985 IEEE
250
IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. COM-33, NO.
DEMODULATOR
3,
MARCH 1985
MODULATOR
r------1
I QUANTIZER
I
1'"
I
Lc7-'
DELAY
DELAY
"n-l+""-l
I
A
I
ACCUMULATOR
(b)
Fig. 1. (a) An exampleof a differential modulator and demodulator. @) A
sampled data representation of the differential modulation.
MODULATOR
INTEGRATOR
A
,
DEMODULATOR
LOW PASS
AID
FILTER
1
1
lxh+en-e.-~)
(b)
Fig. 2. (a) A sigma delta modulator and demodulator. (b) A sampleddata
representation of sigma delta modulation.
.
.
gration [ 15 1 . It is the different empGasis given to these separ- has gained favor bcause it is 'very tolerant of imperfection and
mismatch of the two D/A circuits. [3]1 The structure.of Fig. 1
ate prop.erties that accounts for the different filters that have
is not so tolerant ,because its D/A imperfections are multiplied
beenproposedforfeedbackquantizers.
Our design forthe
the integrating filter at the remodulator will be based on the requirements of telephone toll-by the large baseband gain of
ceiver.
networks.
The next section of this work will be directed atthe, task of
111. REQUIREMENT
O F GENERAL PURPOSE MODULATORS
generalizing the filter A used in this modulator for the purpose
FOR TOLL NETWORK USE
of moving quantization noise out of t h e signal band. Applicasignal prediction and special overload
Digital ,codecs used in the telephone toll-network must ac- tions that want to use
cept a wide range of signals, and their design may not r,ely
o n characteristics could do so by providing preemphasis and deemphasis filters external to the modulator[ 151 .
properties of restricted classes of signals nor.properties of special receivers. We may not rely on there'being high correlation
between Nyquist samples, nor assume t,hat slope overloading
IV. SIGMADELTA MODULATION
isanymoreacceptablethanclippingamplitudesorthata
The
modulator
shown in Fig. 2 generates a quantized signal
colored noise is less objectionable than white noise.
We may take advantage of the fact that the
signal is band that oscillates between levels, keeping its average equal to the
average input. It is easy to show [ 31 that for active inputs, the
limited, however, by moving quantization noise out of band
spectral density of the noise in the quantizedsignal is given by
whereitcanberemovedbyappropqatefilters.Thesigma
delta modulator shown in Fig. 2 does this without differentiating the signal: it eliminates the need for integration at the receiverbecauselow-frequencycomponents
of themodulated
signal representtheinputamplitudedirectly.Thisstructure
25 1
CANDY: DOUBLE INTEGRATION IN SIGMA DELTA MODULATION
QUANTIZER
A2
DELAY
INNER LOOP
,
SECOND LOOP
Fig. 3. A quantizerwith two feedbackloopsaround
elements are nominally unity.
20 kHz
it. Thegainsofall
40 MHz
4 MHz
400 kHz
SAMPLINGRATE
(4lr)
Fig. 4. Quantizationnoiseplottedagainstsamplingfrequencyforvarious
numbers of feedback loops. Baseband is 3.5 kHz and the noise is referred
to the step size: the noise of ordinary PCM is - 10.8 dB on this scale.
The circuits that can be derived by reiterating simple feedwhere a is the quantization step size and
7 the sampling period.
class of feedback
0 < f < fo is then given approxi- backloopsofthiskindareaveryuseful
The net noise in baseband
mately by
quantizers with high-order filters
[ 141, [ 181. Ritchie points
out the penalties that must be paid for using feedback, and
they are summarized in the next section.
providedthat f0.r Q 1and eo = a/&
is t h e averagenoise
generated by the quantizer alone. Thus, the resolution can be
greatly increased by oversampling and feedback.
For example,
quantizinga 3.5 kHzsignal at16 MHz withthisfeedback
quantizer reduces the noise eo by 96 dB, which is equivalent
to a 2
reduction of the step size.
This procedure for increasing the resolution with feedback
can be reiterated as illustrated in Fig.3 . Here we have included
gains G, in cascade with each integrator in order to describe
circuit imperfections that cause the loop gains to be other than
unity. Appendix A shows that the spectral densityof the noise
in the modulated signalgeneratedbytwofeedbackloops
is
given by
~ ~ (= (1
f -) z - ~ ) ~ E ( =
z >2 e 0 f i ( 1 - cos ( 2 n f ~ ) ) ( 3 )
when G, = 1. The in-band noise is given approximately by
This double feedback increases the resolution by 95 dB for a
samplingrateofonly1
MHz and 3.5 kHzbaseband.Fig.
4
compares the resolution of PCM to that obtainable with one,
two, and three feedback loops. Measurements on circuits with
active input signals agree with these calculated values.
V. PENALTIES
FOR USINGFEEDBACK
In Fig. 2 t h e signal applied to the quantizer can be expressed
as the input less the noise from the previous cycle: w, = x, e,andthisnoiseusesupsomeofthedynamicrangeof
the quantizer. Overloading may be avoided by adding one extra
level to the quantizer because e spans + a / 2 . In a similar manner it may be shown that, when the number of feedback loops
is L > 0, the range of signalsapplied to the quantizer is increased by 2L- a. If this signal exceeds the range of the quantizer, the modulation noise increases, as is illustrated by Fig. 5 .
For many applications, the increase in noise for large
signal
valuescanbetolerated,providedanadequatesignal-to-noise
ratio is maintained.
Besidesrequiringadditionalquantizationrange,feedback
demands increased precision in the gains of the circuits. Gains
in the range +lo percent are usually acceptable for quantizers
having a single feedback loop, but more precise gains are needed
when additional feedback loops are used. In Fig. 6 we plot the
calculated and measured signal-to-noise ratio against values of
gains G, placed in series with each integrator for one, two, and
three feedback loops. The measured change of noise with gain
is larger and more variable than predicted by calculation. This
isbecausethenoise
iscorrelatedwithsignalamplitudeina
way that depends on the gains G,, and this correlation is ignored in the calculations.
The third penalty for using feedback concerns the depend-
'
252
IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. COM-33, NO. 3, MARCH 1985
POSITION OF QUANTIZATION LEVELS
-30
I
fi.5
20.5
t2.5
6
Z
-40
-30
-40
'0
W
-0
-50
-50
-60
i.o
0.5
0
-70
2.5
2.0
i.5
-80
AMPLITUDED OF SINEWAVE
Fig. 5 . The noise introduced into sinusoidal signals
of various amplitudes by
modulation with two feedback loops placed around a quantizer having the
stated number of levels. Sampling is at 128 kHz.
-30
-40
m
9
W
-50
-60
z
W-
s
8
z
6
0
-70
-80
-
-
U
0
4
\!
W
m
z
1.1
4
I
u
0.5
0.6
0.7
0:8
0,'9
.
4:O
1.2
\
GAIN
1
'
I
I
1
I
0
0.2
0.5
0.8
i.o
01AS. a
(C)
Fig. 7. Graphs of idle channel noise plotted against input bias. (a) Ordinary
PCM. (b) Simple sigma delta modulation. (c) Two-level quantization with
double feedback. Noise is referred to the step size. The sampling rate is
256 kHz.
tx: .
t
0.5
0.6
0.7
0.8
0.9
GAIN
, i,'O\
4
ence of modulation noise on signal amplitude.
A determination of this dependence for ordinary sigma delta modulation
in [ 191 shows that it can have an important influence on the
performance of oversampled codecs. Fig. 7 is a graph of noise
plotted against the dc bias, x, applied to the modulator. Fig.
7(a) is the quantization error without feedback; the error
is
zero for x = 0 or 1 which corresponds to the position of t w o
adjacent levels, and elsewhere the error is proportional to t h e
distance to a level. Fig. 7(b) shows the idle channel noise of a
quantizer with a single feedback loop. The noise
is gathered
into a series of narrow peaks, and
t h e large peaks can be an
embarrassment in the designof oversampled codecs.
VI. TWO-LEVEL
QUANTIZATION
We will demonstrate how the three penalties, described in
the previous section, can be avoided by using
just two feedback loops and degrading the quantizer to a single threshold
circuit that generates a two-level output.
With a singlethresholdtheinconvenience
of establishing
threshold spacing is removed, and the concept of gain
of t h e
quantizer becomes unreal unless other circuit properties provide a calibration for the amplitudes applied to the threshold.
We find by experiment that signal
levels adjust themselves so
that the effectivegainof the quantizer compensates for changes
O
f
I
I
1I
in the values of circuit gains G,. The measured noise shown in
-2
0.5
0.6
0.7
0.8
0
Fig. 8 is almost independent of these gains and corresponds t o
GAIN
-4
thecalculatedvalueforloopgains
of unity.Likewise,the
(C)
change in the attenuation of the signal Y ( o ) / X ( o is
) less t h a n
Fig. 6. The change in noise with gain placed in cascade with each integrator
50.05
dB
for
values
of
gain
0.5
<
G,
<
2.
We
find
that the
for multilevel quantization. The reference noise is that measured at unity
penalty of having to establish quantizer gains in one and two
gain value. The curve gives calculated values that apply for sampling rates
feedback loops is substantially eliminated by the use of twothat are at least eight times the Nyquist rate. (a), (b), and (c) are for one,
level quantization: signal levels in the circuit automatically adtwo,andthreefeedback
loops, respectively.Thesecircuitsbecome
just themselves to make the effective loop gains unity. With
unstable at gains 2, 1.236, and 1.087.
-
L
253
CANDY: DOUBLE INTEGRATION IN SIGMA DELTA MODULATION
0.5
I
I
I
I.o
1.5
2.0
GAINS G i AND G2
Fig. 8. The noise indoublefeedback,two-levelmodulationplottedagainst
gains G, at various sampling rates. The input was excited with a random
signalspanning *O.la. Thehorizontallines mark calculatedvalues of
noise for unity gain and multilevel quantization with the same step size.
more than two feedback loops, undesirable oscillations spoil
the functioning of the circuit when the quantizer has only two
levels. We discuss these oscillations in the next section.
The second penalty concerns the loss of quantization range
because feedback increases the amplitudes that are applied to
the quantizer. With a single threshold there is no limitation on
the range of its input amplitude, only the output levels are defined, but we see in Fig.
5 that for two-level quantization the
noiseincreasesrapidlywithsignalamplitude.Fig.
9 presents
similar data plotted on logarithmicscales.
The third penalty concerned the correlation
of noise with
input level. Fig. 7(c) shows a graph of modulation noise plotted against input bias for a modulator utilizing two feedback
loops. Comparing it to the graphs in Fig. 7(b), we see that use
of two loops substantially decorrelates the noise except at the
ends of the range, where the modulation noise peaks in the
L
same fashion that it does in modulators that utilize only one
- 35
feedback loop. Reference
[ 191 shows that the amplitude of
these peaks of noise is given by
Nmax
=fi(fo+
which can be large compared to the calculated noise
their width u is narrow:
(5)
(4). But
50
40
30
20
p
io
I
- 30
I
I
I
I
1
-25
-20
-15
-10
-5
INPUTAMPLITUDE /STEPSIZE. dB
Fig. 9. The dependenceof noise on the amplitudeof a sinusoidal input signal
for two-level modulation with double feedback. The noise
is referred to its
small signal value. This result applies to sampling rates that are eight or
more times the Nyquist rate.
of the quantizer. In particular, when the quantizer has only
t w o levels, the undesirable modes are easily excited and difficult t o extinguish. The susceptibility of the circuit to enter unIt appears that the use of two-level quantization and double desirable modes has prevented measurement of the resolution
integration could be the basis of a useful modulator. The next
of circuits that have triple integration and two-level quantizasection explains why it is wise to use no more than two intetion except for gains G 2 , G 3 less than 0.55. Even for gains as
grators.
low as 0.1, theunwantedmodeswereself-sustainingafter
beingexcited.Whentwofeedbackloopsareused,however,
VII. LIMIT-CYCLES
THAT OVERLOADTHE QUANTIZER
the contents of the integrators always decay to
a small value
gains G ,
The feedback quantizers that we have described cause their whenexcitationsareremoved,providedthatthe
are less than 1.23 for multilevel quantization and less than 2
outputs to oscillate between levels in a way that keeps their
for two-level quantization.
average value equal t o t h e average input. In the desired mode
of operation the signalsheld in the integrators are comparaWhen modulators function in an undesirable mode, the sigtively small, but when three or more feedback loops are present,
nalsintheintegratorsare
so large,compared t o t h e largest
quantization levels, thattheinnerfeedbacksareineffective.
other modes can be excited [ 141. These modes are characterThebehavior of thecircuit is dominatedbytheoutermost
ized by being very noisy and having large-amplitude, low-frefeedback, which, if it contains more than two integrations, is
quency oscillations in the integrators, which exceed the range
u = fora.
(6 1
254
IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. COM-33, NO. 3, MARCH 1 9 8 5
unstable.Stabilityisregainedwhentheinnerfeedbacksare
made effective by clipping the amplitudes
ofsignalsheldin
the integrators, or by nonlinear feedback [ 141. But it is questionable whether the extra resolution obtained in practice can
justifytheuse
of thesemorecomplicatedcircuitsandthe
tighter tolerances that they demand.
VIII. IMPLEMENTATION
OF THE MODULATORSUSING
Two INTEGRATORS
,
x(tl
The circuit shown in Fig. 3 is a sampled data model of t h e
modulator; its signals are represented by impulse sample values.
This circuit could well be implemented using switched capacitorsforaccumulation.Implementationsinabipolartechnology, however, would prefer t o use continuous signals such
as those in the circuit of Fig. 10. The analysis of the switched
and the continuous circuits in Appendix C shows that their
operation is equivalent when the feedback signal
y ( t ) is held
constantthroughoutthesampleinterval,thetimeconstant
R C equals 1.57, and the two inputs are related by the expression
Xn7
-InT
‘-
THRESHOLD
~ ~ _ _ _ _
y(tl
Fig. 10. Ananalogversion of thedoubleintegratingmodulator.
30
50
(2x(t)
-
x(t - 7)) dt.
(7)
(n-1)r
The analog circuit in Fig. 10 is relatively easy to construct
becausethereareonlytwomainconstraintsonitsdesign.
There is the need to keep signals small in order to conserve
power yet have signal levels large enough
to swamp noise and
imperfectioninthethresholdcircuit,andthere
is theneed
t o set the time constant R C with sufficient precision. Fig. 1 1
showsgraphs of themodulator’sresolutionplottedagainst
input amplitude for three values of the time constant. Time
constants changing in the range
1.27 to 1.87 give less than 1
dBvariationinnoiselevel;thisshouldsatisfymostapplications.
Equation (7) can be used to define the frequency response
of the filter that should be placed in cascade with the input
3 in order to make its reof the sampled data circuit in Fig,
sponse identical to that of the analog circuit in Fig.
8 . That
frequency response is given by
’ ( 0 ) = ( 2 - f - w ) smc
G(0)=X
.
(f7).
X(w>
(8)
This low-pass filter, inherent
to the circuit of Fig. 10, is usefulforreducingaliasingdistortion.Forexample,when
3.5
kHzsignalsarebeingmodulatedat
512 kHz, anyspurious
signals in the range 508-5 16 kHz alias into band. But the distortion is small because I G(508) I < -40, dB. Table I lists the
attenuation of t h e signalsaliasedintobandforvarioussampling frequencies; also listed is gain introduced into baseband.
This gain could be equalized in the digital processor.
Simulations show that signal amplitudes in the integrators
of these two-level feedback modulators can be very large, and
realimplementationsneedtolimittheirsize.Clippingtheir
amplitudesspeedstherecoveryfromoverloadbutmayincrease the noise. Fig. 12 shows that essentially full resolution
of two-level quantization is obtained by allowing the integrated signals t o swing through at least k l . 0 step sizes. Variation
of themodulator’snetgainwithinputamplitudewas
less
than +0.025 dB.
Fig. 13 shows that leakage in the integration has negligible
effect on noise, provided its time constant exceeds
1/2fo seconds. The data in this figure agree with calculations that represent quantization by added noise, and assumes unity gain in
the feedback loops.
IX. DESIGNO F
THE
DIGITAL
PROCESSOR
A digital processor will convert the output of the modulator
into PCM by smoothing the signal with a digital low-pass filter
40
%
-
9
5
Y
-
u)
30
5
z
0
-
-70
-60
-50
-40
-20
-30
SIGNALAMPLITUDE,
-
10
-
0
0
-10
dB
Fig. 11. Thesigna-to-nolse
ratio
plotted
against
the
amplitude
of a
sinusoidal input signal for the circuit in Fig. 10. The sampling rate is 256
kHz with 3.5 kHz baseband; the time constantRC, is (a) 1.5 7;(b) 27; (c)
7.
TABLE I
GAINS OF THEFILTER (2 - exp (7
j 2 x f 7 ) ) sinc
SAMPLINGFREQUENCY
G A I N A T 3.5 k H z
I/TkHz
dB
cf7)
GAIN AT
(Ih-3.5)
kHz. de
64
0.87
-23 9
I28
0 24
-30.8
256
0.06
-37.1
512
0.01 5
-43 2
1024
0.004
-49.3
2048
0.001
-55.3
4096
0 00
-6 I .J
8192
0.00
-67 4
255
CANDY: DOUBLE INTEGRATION IN SIGMA DELTA MODULATION
quency response
(1 0)
and impulse response
n(n
h , =-
+ 1)
for 1 < n < N
2
0
forN<n<2N
I
0.1
I
I
1
I
I
0.2
0.3
0.4
0.5
0.6
h, =
AMPLITUDE OF INPUTISTEPSIZE
250
150
500
6400
0
ps
80
25
50
I
I
I
I
I
I
I
I
o
0.5
?.a
i5
2.0
2.5
3.0
35
'INTEGRATIONLOWERCUTOFF-FREQUENCY,
-
1)(3N - n )
for2N<n<3N
,
l
L
Fig. 12. The effect on noise of clipping the amplitude of signals in the two
integrators for various amplitudes of a sinusoidal input. W is the ratio of
the clipping level to the step size. Quantization levels are at k0.5.
I N T E G R A T I O N TIME-CONSTANT,
(3N - n
I I
4.0
kHz
Fig. 13. Thedependence of noise on leakage in bothintegratorsfortwolevel quantization and double feedback. Basebandfo
= 3.5 H z .
andresamplingitattheNyquistrate.Thefilterattenuates
spurioushigh-frequencycomponentsofthe
signal andthe
high-frequency components of the modulation noise, so that
resampling does not alias significant noise into baseband.
Reference 191 describes t h e design of adigitalprocessor
that is suitable for use with ordinary sigma delta modulation.
It reduces the sampling rate in
stages, employing 32 kHz as
anintermediatefrequencyforreachingthe
8 kHzNyquist
rate of telephone signals. This technique leads
to simple and
efficient circuits. Use of double integration in the modulator
does not influence the design of the filter with respect to outof-band signal components, but it does influence the requirementsforattenuatingmodulationnoise,becausethenoise
now rises more rapidly with increasing frequency.
The filter used in [91 for lowering the sampling rate to 3 2
kHz has a triangular-shaped impulse response, and frequency
response given by
sin fNr
sin fr
Calculations of noise, in Appendix B, show that such filtering
does not provide sufficient attenuation for modulations generated by means
of doubleintegration.It
is shown that the
modulation frequency would need to be raised from
1 to 2.5
MHz in order to make up for
use of such inadequate filtering.
Afilterthatdoesprovideadequateattenuationhasfre-
where N is the number of input sample values that occur in
one period of the resampling. The duration
of the filter impulseresponse is threeresamplingperiods.Itsfrequency
response has triple zeros at the 32 kHz resampling rate and all harmonics of it. When this filter
is used, the noise, aliased into
band, resu1t.s in less than 0.5 dB loss of resolution. This filter
can be implemented as a n FIR structure that does not require
full multipliers because the modulated signal is a 1 bit code. In
t h e finalstageofdigitalprocessingasharpcutofffilteris
needed. Analysis in Appendix C shows that at least 25 dB of
attenuation is needed,which
is provided bythefilterdesecscribedin [9] ; itcomprisestwosecond-orderlow-pass
tions followed by an accumulation and dump.
X. CONCLUSION
We have demonstrated that
a sigma delta modulator with
double
integration
can
be
designed
t o provide
resolution
equivalent to that of 16 bit PCM when modulating 4 kHz signals a t 1 MHz, with simple robust circuits. A somewhat higher
modulation rate, 2.5 MHz, would permit use of-simpler digital
processing given by (9) for converting the modulation to
PCM.
It is instructive to compare this modulator to the interpolating modulator described in
[ S I . Theyprovidecomparable
9 bit
resolution when the interpolating modulator generates
words at 256 kHz and the sigma delta modulator generates 1
1 MHz. Theinterpolativemodulatorhasmore
bitwordsat
complex circuits, but the tolerances of the two modulators to
imperfection are roughly equivalent. The overriding adva,ntage
of the modulator described here is the fact that its quantization is uniform. To obtain comparable resolution from the interpolatingmodulatorwithoutcompandingitsquantization
levels would require that it generate 4 bit words at 2MHz.
There isneed t o have uniform quantization in order that
the, encoder can accept the sum ofseveral independent signals
without having interaction between them. For example, when
digital hybrids and digital conferencing are to be provided, the
sumofthreeormorespeech
signals maybepresentinthe
modulator at one time; then we require that the quantization
noisebeindependentofthe
signal amplitude.Thepossible
use of triple integration has been rejected because it can latch
into noisy modes of operation.
APPENDtX A
MODULATION NOISE
QuantizationinthemodulatorshowninFig.
3 is represented by additive noise e. We assume it is white with spectral
power density 2reO2, where eo is the noise power in the band
of frequencies below
t h e half sampling rate. When the gains
G,, areunity, we candescribethemodulated
signal bythe
256
IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. COM-33, NO. 3, MARCH 1985
Thus, the noise (13) is increased by the root
of t h e decimation ratio N . This loss of resolution could be made up for
by increasing the input modulation rate from / r1 t o N 1 / 4 / ~ .
A better decimating filteris
z-transform expression
Y'(Z) =Z-lX'(z)
-t(1
-
z-')~E(z).
1)(1
The noise in this signal has spectral density
N M ( ~ =
) 2(1 - cos ( 0 7 ) ) e o f i .
(12)
If NMOis the component of noise in baseband, 0 < f < f o and
w0 = 27rf0, then
-
8 sin (
+
~ ~ 7sin
)
(2w07)].
) be approximated
When f07 is small enough that sin( 2 ~ 0 7 can
by the first five terms in its Taylor expansion,
This noise can be made very small by using high modulation
rates. Subsequent digital processing, called decimation, lowers
the rate without increasing the noise.
N
L(=N)3
3 1 -Z-l
It modifies the modulation noise to be
The spectral density of the noise following resaypling can be
obtained by reverting to the time domain. Let E (z) represent
the accumulated noise
which is equivalent to
N
A ~ P E N D IBX
DESIGNOF
THE
DECIMATOR
The decimator uses 1,ow-pass,filters to attenuate out-of-band
components of the modulation that, will be aliased into band
by the resampling. TO simplify the design of these filters, there
is much advantage in lowering the sampling rate in stages. In
theinitialstageswherethesamplingrate
isstilllargecompared to the Nyquist rate, it
is wise to place zeros of the filters
atthenewsamplingrateandharmonics
of it.Suitable sequences of,evenly spaced zeros occur in trigonometric functions, and a particularly easy spectral response to implement is
(sin wNT/sin cj7). In the time domain it is an averaging o f N
samples with ,T being the peripd of the input samples and NT
the period of the output samples.
In [ 91 a cascade of two such filters having response
When this is resampled with period N T , i.e., every Nth sample
is retained, there will b e n o cGrrelation between the samples,
if t h e original samples are uncorrelated. The resampled noise
e'(iN7) will be white with spectral density N e o f i . It follows
that the noise at the decimator output after resampling can be
described as
and its baseband component forN7fo 2- 1 is approximated by
eo n2
N~~ = -(2f07)5/2.
6
(14)
There is no change from (1
3). All of the penalty for decimating
is represented by the in-band attenuation
of the decimating
filter which, when equalized, increases the noise.
was used to lower sampling rates of sigma delta modulation to
This decimating filter can be used successfully to lowerSam'32 kHz. This filter is inadequate for use with modulators em- pling rates to about four times the Nyquist rate, and its in-band
ploying double integration, as the following analysis
willdeattenuation will be less than 3 dB. Filters with much sharper
monstrate.Afterfilteringby
(14), themodulationnoisein
cutoff characteristics are needed in the final stage of decima(1 1) becomes
R mustbesufficienttomakethe
tion.Theirattenuation
modulation noise that is aliased into band small with respect
t o N M Oin (13). That is,
Resampling with period NT results in
eo
(16) (1
N D ( 2 ) = ___
-
Z-')2
N2
where Z - = z - represents
~
aone-perioddelayatthenew
rate.Followingargumentssimilarto(1
1)-( 13),thein-band
noise can be approximated by
which is satisfied if
257
CANDY: DOUBLE INTEGRATION IN SIGMA DELTA MODULATION
When 1/Nr is 32 kHz and f o is 4 kHz, this requires R < -25
way in the sample time( n
dB. This is easier than the requirement for antialiasing filters
in D-channel banks, R < -32 dB. The low-pass filter described
in [ 91 is adequate for use with double integrating modulators.
APPENDIXC
COMPARISONOF A MODULATOR THAT INTEGRATES
ANALOGSIGNALS
WITH ITS SAMPLED
DATA
-
EQUIVALENT
+i)~
we get
,
y(.- (. +)+ 7 )
+
y n 7 . (33)
n=O
When the modylator in Fig.
3 has input samples x , ’ and
output samples y n and the signal applied to the quantizer is
w,- 1 ‘, a relationship between their values can be expressedas
We areprimarilyinterestedinmodulatorsthataresampled at very high frequency compared to baseband so that the
signal varies little during the sampling interval. To simplify the
analysis, we shall
now assume that the center of area of the
inputwaveformforeachsample
liesmidwayintheinterval
t n = (n *)T. Then
+
or
N-1
N- 1
N-1
provided all signals are initially zero.
Ananalog.circuitthatcan
haveequivalentresponse
is
shown in Fig. 10. Here x ( t ) is a continuous signal and the feed- Yn=O
(N-n+(ficI!-))ytlT.
back signal y ( t ) is held constant throughout each sample in(34)
terval. To analyze this circuit we make use of the following relationshipsbetweenintegrationandsummation
of sample
Comparing this result to (27) we see that they can be equivavalues.
lent, with the right initial conditions and
Lemmas: F o r f ( t )= 0 when t < 0 and
r 2 C 1 C 2= r 2 ,
R C , = 1.57,
Yn’=Yn7
and
This leads t o t h e final result
(29)
I
= 2xn7
- x(i2-1)7
and
which has spectral equivalent
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isappliedtothe
quantizer in Fig. 3 at the Nth sample time as
y ( t ) is
Applyingresults(13)-(15)andnotingthatbecause
heldconstant,thecenter
of area of itswaveformliesmid-
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VOL. COM-33, NO. 3, MARCH 1985