Performance Analysis and Optimal Filter Design for Sigma-Delta Modulation via Duality with DPCM

0.22 MB·English

by  Or Ordentlich

1
Performance Analysis and Optimal Filter Design for
Sigma-Delta Modulation via Duality with DPCM
Or Ordentlich and Uri Erez, Member, IEEE
Abstract—Sampling above the Nyquist rate is at the heart of (orD/A)convertertranslatestorequiringthatitinducesasmall
sigma-delta modulation, where the increase in sampling rate is average distortion simultaneously for all processes within our
translated to a reduction in the overall (mean-squared-error)
compound model.
reconstruction distortion. This is attained by using a feedback
Sigma-deltamodulationisa widelyusedtechniqueforA/D
5 filter at the encoder, in conjunction with a low-pass filter at the
1 decoder. The goal of this work is to characterize the optimal as well as D/A conversion. The main advantage offered by
0 trade-off between the per-sample quantization rate and the re- thistype ofmodulationisthe ability to trade-offthesampling
2 sultingmean-squared-errordistortion,undervariousrestrictions rate and the numberof bits per sample required for achieving
onthefeedbackfilter.Tothisend,weestablishadualityrelation a target mean-squared error (MSE) distortion. The input to
n between the performance of sigma-delta modulation, and that
u of differential pulse-code modulation when applied to (discrete- the sigma-delta modulator is a signal sampled at L times
J time) band-limited inputs. As the optimal trade-off for the the Nyquist rate (L > 1). This over-sampled signal is then

9 latter scheme is fully understood, the full characterization for quantized using an R-bit quantizer. In much of the literature
sigma-delta modulation, as well as the optimal feedback filters, aboutsigma-deltamodulation,nostochasticmodelisassumed

] immediately follow. fortheinputsignal.However,whensuchamodelisassumed,
T
the benefit of over-sampling can be easily understood from
I
. basic rate-distortion theoretic principles: the (per-sample)rate
s I. INTRODUCTION
c required to achieve distortion D for the over-sampled signal
[ Analog-to-digital (A/D) and digital-to-analog (D/A) con- is L times smaller than the rate required to achieve the same

2 verters are essential in modern electronics. In many cases, distortion for the signal obtained by sampling at the Nyquist
v it is the quality of these converters that constitutes the main rate. Thus, in principle, increasing the sampling rate should
9 bottleneck in the system, and consequently, dictates its entire allow one to use quantizers with lower resolution, which is
2 performance. On the other hand, as digital circuits are now desirable in many applications.
8
considered relatively cheap to implement, the interface be- However, the rate-distortion theoretical property that guar-
1
0 tween the analogand digitaldomainsis oftenoneof the most antees a constant product of the number of bits per sample
. expensivecomponentsinthesystem.DevelopingA/DandD/A needed to achieve distortion D, and the over-sampling ratio
1
0 components that are on the one hand relatively simple, and L, is only valid when a very long block of samples is vector-
5 on the other hand introduce little distortion, is therefore of quantized. In A/D and D/A conversion, vector-quantization
1 interest. in high dimensions is a prohibitively complex operation, and
v: Often, the same A/D (or D/A) component is applied to quantization is invariably done via scalar uniform quantizers.
i a variety of signals with distinct characterizations. For this Scalar quantizers alone cannot translate the increase of sam-
X
reason,it isdesirabletodesignthedataconverterto berobust plingratetoasignificantreductioninthenecessaryresolution,
r to the characteristics of the input signal. One assumption butfortunatelythis problemcan be circumventedwith the aid
a
that cannot be avoided is the bandwidth of the signal to be of appropriate signal processing.
converted,whichdictatestheminimalsamplingrate,according In sigma-delta based converters, the quantization noise
toNyquist’stheorem.Beyondbandwidth,however,onewould is shaped using a causal shaping filter embedded within a
like to assume as little as possible about the input signal. A feedbackloop, see Figure 1. The filter coefficientsare chosen
reasonable modelfor the input signal is thereforea stochastic inamannerthatensuresthatmostoftheenergyoftheshaped
one, where the input signal is assumed to be a stationary quantizationnoiseliesoutsidethefrequencybandoccupiedby
Gaussian process with a given variance and an arbitrary the over-sampled signal. At the decoder, the quantized signal
unknown power spectral density (PSD) within the assumed is low-pass filtered, cancelling outthe high-frequenciesof the
bandwidth, and zero otherwise. In this paper, we adopt this quantization noise process without effecting the signal, such
compound model which is rich enough to include a wide that the decoder’s output is composed of the original signal
varietyofprocesses.TherobustnessrequirementfromtheA/D corrupted by a low-pass noise process.
Another technique for compressing sources with memory,
The work of O. Ordentlich was supported by the Adams Fellowship
which explicitly models the source as a stochastic process,
ProgramoftheIsraelAcademyofSciencesandHumanities,afellowshipfrom
TheYitzhakandChayaWeinsteinResearchInstituteforSignalProcessingat is differential pulse-code modulation (DPCM). In DPCM, a
TelAviv University andtheFeder FamilyAward. TheworkofU.Erezwas prediction filter is applied to the quantized signal. The output
supportedbybytheISFunderGrant1557/13.
of this filter is then subtracted from the source and the result
O. Ordentlich and U. Erez are with Tel Aviv University, Tel Aviv, Israel
(email:ordent,[email protected]). is fed to the quantizer, see Figure 2. At the decoder, the
2
quantized signal is simply passed through the inverse of the addition,sincedataconvertersoftenoperateatveryhighrates,
prediction filter. The well-known “DPCM error identity” [1] it makes sense to impose various constraints on the sigma-
states that the output of the decoder is equal to the source delta feedback filter C(Z), such as confining it to be a finite
plus the quantization error, just like in simple non-predictive impulse response (FIR) filter with a limited number of taps.
quantization.ThebenefitofusingDPCM, however,isthatthe ForagivendesiredMSEdistortionlevel,ourgoalistofindthe
signalfedto the quantizeris theerrorinpredictingthesource constrained sigma-delta feedback filter C(Z) that minimizes
from its quantized past, rather than the source itself. If the the quantization rate w.r.t. all sources in the compound class,
coefficients of the prediction filter are chosen appropriately, andtocharacterizetheattainedrate.Thisgoalisdifferentthan
the variance of this error should be smaller than the variance theonepursuedin[8],wheretheoptimalunconstrainedfilters
of the original source, which translates to a reduction in the w.r.t. a known PSD were found.
number of bits required from the quantizer for achieving a The problem of finding the optimal N-tap FIR sigma-
certain distortion. delta feedback filter C(Z) for a compound family of sources
The performance of DPCM under the assumption of high- similar to ours, was considered in [6]. The optimal filter was
resolution quantization is well understood since as early as claimed in [6] to be the Nth order MMSE prediction filter
the mid 60’s [1]–[3]. Under this assumption, the prediction C(Z) = (1 Z 1)N of a bandpass stationary process from
−
−
filter should be chosen as the optimal linear minimum mean- its past, and for a fixed target MSE distortion the required
squared-error (MMSE) prediction filter of the source process quantizationratewasfoundtodecreaselinearlywithN.Such
from its past [1], and the effect of the filtered quantization as statement is obviously inaccurate, as it violates Shannon’s
noisecanbeneglectedinthepredictionprocess.Whileinmost rate-distortion theorem. The major drawback of [6] is that
cases where DPCM is traditionally used, the high resolution it (implicitly) makes the high-resolution assumption that the
assumptioniswelljustified,ittotallybreaksdownfortheclass varianceofthequantizer’sinputissolelydictatedbythetarget
of band-limitedprocesses, which includesthe inputsignals to signal X , whereas the contribution of the quantization
n
{ }
sigma-delta modulators. Indeed, the prediction error of such noise to this variance can be neglected. As discussed above,
a process from its infinite past has zero-variance, rendering for over-sampled processes this assumption may not be valid
theDPCMhigh-resolutionrate-distortionformulascompletely evenwhenthequantizer’sresolutionisveryhigh.Inparticular,
useless. usingthefilterC(Z)=(1 Z 1)N from[6],theenergyofthe
−
−
quantizationnoise within the frequencyband occupiedby the
signal indeed decreases exponentially with N. However, the
A. Connection to Previous Work
noise’senergyoutsidethisbandincreasesrapidlywithN,and
The connection between DPCM and sigma-delta modula- for any quantization resolution it will become much greater
tion, as two instances of predictive coding, was known from than σ2 for N large enough, making the high-resolution
X
the outset. Indeed, both paradigms emerged from two Bell- assumption inapplicable. In this case, the dynamic range of
LabspatentsauthoredbyCCCutler[4],[5]in1952and1954. the quantizer will be exceeded and overload errors would
In fact, by adding appropriate pre- and post-filters to the frequently occur.
sigma-delta modulator, as depicted in Figure 3, the input to It therefore follows that in the analysis of sigma-delta
the quantizer, as well as the final reconstructionof the signal, modulatorsoneshouldnotmake high-resolutionassumptions,
becomeidenticaltothoseintheDPCMarchitecture[6,Section but rather must take into account the effect of the filtered
II], [7, Chapter 3.2.4]. For this reason, it has become folklore quantization noise on the variance of the quantizer’s input.
that the two architectures are equivalent. When a sigma-delta Fortunately, in the analysis of DPCM modulators the high-
modulator is used for compression of digital discrete-time resolutionassumptionhasbeenovercomein[9].Itwasshown
signals, the pre-filtering can be performed digitally and the that for any distortion level and any stationary Gaussian
additionalcomplexityofthearchitecturedepictedin Figure3, source, the DPCM architecture induces a rate-distortion op-
w.r.t. that in depicted in Figure 1, may be acceptable. This is timaltestchannel,providedthatthe predictionfilter ischosen
however not the case for data converters, as the input to the astheoptimalfilterforpredictingthesourcefromitsquantized
latter is analog and pre-filtering must be done in continuous- past, and in addition water-filling pre- and post-filters are
time,whichismorechallenging.Themotivationforthiswork applied. The analysis of [9], which takes into account the
is understanding the performance limits of A/D and D/A effectofthequantizationnoise,canthereforebeusedtoobtain
conversionbasedonthesigma-deltaarchitecture,andtherefore the optimal feedbackfilter and its correspondingperformance
pre-filteringis precluded.Thus, the architectureis confinedto for a DPCM system applied to an over-sampled stationary
that depicted in Figure 1. Gaussiansource.Inthispaper,weleveragetheresultsfrom[9]
Another important aspect of our interest in sigma-delta to the analysis of sigma-delta modulators, by establishing an
modulators as a mean of data-conversion rather then data- appropriate duality between the two architectures.
compression, is that it dictates that the assumptions one can
make on the statistics of the input signal must be minimal.
B. Contributions
Consequently, we consider a compound class of sources that
consistsofallstationaryGaussianprocesseswithvarianceσ2 Let be the compoundclass of all discrete-timestationary
X S
whose PSD is limited to some predefined frequency band. In Gaussian sources with variance σ2 and PSD that is zero for
X
3
all ω / [ π/L,π/L], L 1. Note that this class corresponds quantization. The purpose of this excess-rate is to ensure
∈ − ≥
to uniformly sampling a compound class of continuous-time that an overload event, which jeopardizes the stability of the
stationaryGaussian processeswith varianceσ2 and PSD that system, occurswith low probability.The stochastic modelwe
X
is zero for all f > f , at a sampling rate of 2Lf sam- assume for the input process allows us to tackle the issue of
max max
| |
ples/per second. Let XDPCM be a discrete-time stationary stability in a systematic and rigourousmanner, and the trade-
{ n }
Gaussian process with PSD off between the excess-rate and the overload probability is
analytically determined.
Lσ2 for ω π/L
SDPCM(ω)= X | |≤ , (1) Clearly, a sigma-delta modulator can only perform well
X (0 for π/L< ω <π if overload errors are rather rare. Our stability analysis in
| |
Section III is based on avoiding overload events w.h.p., and
and note that XDPCM .
{ n }∈S does not aim to consider the effect of such events on the
Ourmainresult,derivedinSectionII,isthatforanyprocess
distortiononcetheyoccur.Ingeneral,theoverloadprobability
XΣ∆ from the compoundclass , the test channelinduced
{ n } S of the scheme described in Section III decreases double
by the sigma-delta modulator (Figure 1) achieves exactly the
exponentially with the excess-rate of the quantizer w.r.t. the
samerate-distortionfunctionasthatoftheDPCMtestchannel
mutual information. Thus, taking an excess rate of 1 2
(Figure 2) with input XDPCM . More specifically, for such −
{ n } bits will usually yield a sufficiently low overload probability.
processes,for anychoiceof σ2 andpredictionfilter C(Z)
DPCM However, sigma-delta quantizers are often employed with a
in the test channel of Figure 2, the same choice of C(Z)
one-bit quantizer. In this case, the overload error probability
together with the choice
cannotbe verylow.Consequently,thedesignerwouldneedto
σ2 guaranteethattheeffectofoverloaderrorsislocalintime,and
σ2 = DPCM (2)
Σ∆ L 1 π/L 1 C(ω)2dω does not drive the system out of stability. There are various
· 2π π/L| − | restrictionsonecanplaceonC(Z)inpursuitofthelattergoal.
−
in Figure 1, yields the samRe compression rate and the same The issue of maintaining stability when overload errors are
distortion. unavoidable is outside the scope of this paper. Nevertheless,
While this result is simple to derive, it has a very pleasing we stress that our main result is of great relevance to this
consequence: the problem of optimizing the filter C(Z) in setting, as it shows that the filter C(Z) should be chosen as
sigma-deltamodulationw.r.t.anysignalin ,underanysetof theoptimalMMSEpredictionfilterof XDPCM fromitsnoisy
S { n }
constraints,canbecastasanequivalentproblemofoptimizing past under the stability ensuring restrictions.
the DPCM prediction filter w.r.t. input XDPCM under the
{ n }
same set of constraints. Furthermore, in Section II-A, we II. MAIN RESULT
formalize a similar duality between DPCM and sigma-delta
We begin by introducing some basic notation that will be
modulation for a frequency-weighted-mean-squared-errordis-
usedinthesequel.Foradiscretesignal c ,theZ-transform
tortionmeasure.InthiscaseSDPCM(ω)isreplacedwithaPSD { n}
X is defined as
that depends on the distortion’s weight function.
Inprinciple,recastingthesigma-deltaoptimizationproblem C(Z), ∞ c Z n,
n −
as an MMSE prediction problem may be derived directly
n=
from the formulas characterizing its performance, as given X−∞
and the Fourier transform as
in Proposition 1. Nevertheless, establishing the equivalence
bfoertmweednesscirgibmead-daebltoavem,oidsuilnatsiiognhtafunldaDsPiCtMal,loiwnsthteo sbpoercriofiwc C(ω),C(Z)|Z=ejω = ∞ cne−jωn.
n=
known results from the literature about the latter. X−∞
Havingrecastthefilteroptimizationproblemforsigma-delta For a discrete (real) stationary process Xn with zero-mean
as that of optimal linear prediction, we can readily obtain the andautocorrelationfunctionRX[k],E{(Xn}+kXn) we define
solution under constraints for which an explicit solution was the power-spectral density (PSD) as the Fourier transform of
lacking in the literature, or was cumbersome to derive. the autocorrelation function
One may question the relevance of the test channel of
S (ω), ∞ R [k]e jωk.
Figure1anditsinformation-theoreticanalysistothepractical, X X −
resource limited, problem of A/D and D/A conversion via k=X−∞
sigma-deltamodulators.To thatend,in SectionIIIwe replace The PSD of a continuous stationary process is defined in an
theAWGNchannelfromFigure1withasimplescalaruniform analogous manner.
(dithered) quantizer of finite support, which is suitable for Assume XΣ∆(t) is a continuous stationary band-limited
implementation within A/D and D/A converters. As long as Gaussianprocesswithzeromeanandvarianceσ2 ,whosePSD
X
overload does not occur, the effect of applying the scalar is zerofor allfrequencies f >f , butotherwiseunknown.
max
| |
quantizerisequivalenttothatofanadditivenoisechannel.We The Nyquist sampling rate for this process is 2f samples
max
show that the rate-distortion trade-off derived for sigma-delta per second. Since our focus here is on quantization of over-
modulation in Section II remains valid with high probability, sampledsignals,weassumethatXΣ∆(t)issampleduniformly
with a constant additive excess-rate penalty for using scalar with rate of 2Lf samples per second for some L > 1.
max
4
NΣ∆ 0,σ2
n ∼N Σ∆
H(ω)
(cid:0) (cid:1)
1
UΣ∆ UΣ∆+NΣ∆
XnΣ∆ Σ n n n XˆnΣ∆
π π ω
−
−L L
C(Z) − Σ
NΣ∆
n
Fig.1. Thetestchannelcorrespondingtothesigma-deltamodulationarchitecture, withthesigma-deltaquantizerreplacedbyanAWGNchannel.Theinput
isassumedtobeover-sampled atLtimestheNyquistrate.
NDPCM 0,σ2
n ∼N DPCM H(ω)
(cid:0) (cid:1)
1
UDPCM UDPCM+NDPCM VDPCM
XnDPCM Σ n n n+ Σ n π π ω XˆnDPCM
−
−L L
C(Z)
Fig. 2. Thetest channel corresponding to the DPCM architecture, with the DPCMquantizer replaced byan AWGN channel. Theinput is assumed to be
over-sampledatLtimestheNyquistrate.
N 0,σ2
n
∼N
H(ω)
(cid:0) (cid:1)
1
U U +N
Xn 1−C(Z) Σ n n n 1−C1(Z) π π ω Xˆn
−
−L L
C(Z) − Σ
N
n
Fig.3. Atestchannelcorrespondingtothesigma-deltamodulationwithpre-filter1−C(Z)andpostfilter 1 .Thistest-channel isequivalent tothat
1−C(Z)
fromFigure2.However, thepre-filtermakes thisarchitecture unattractive fordataconverters.
The obtained sampled process XΣ∆ is therefore a discrete the additive white Gaussian noise (AWGN) channels embed-
{ n }
stationary Gaussian process with zero mean and variance σ2 ded within the two test channels.
X
whose PSD is zero for all ω / [ π/L,π/L], but otherwise
∈ − The test channels in Figure 1 and Figure 2 do not imme-
unknown.Our goalis to characterizethe rate-distortiontrade-
diately induce an output distribution from which a random
off obtained by a sigma-delta modulator, modeled as the test
quantization codebook with rate I(U ;U +N ) and MSE
channel from Figure 1, whose input is XΣ∆ . To that end, n n n
{ n } distortionDcanbedrawn.Thereasonforthisisthesequential
weestablishanequivalencebetweentheperformanceobtained
nature of the compression, which seems to conflict with the
by this test channel for any stationary band-limited Gaussian
need of using high-dimensional quantizers, as required for
processwithvarianceσ2 andtheperformanceobtainedbythe
X attaininga quantizationerrordistributedasN with compres-
test channel from Figure 2, which models a DPCM compres- n
sionrateI(U ;U +N ).Fortunately,thisdifficulty,whichis
sionsystem,forastationaryflatband-limitedGaussianprocess n n n
alsopresentindecision–feedbackequalizationforintersymbol
with variance σ2 . The performance of the latter is now well
X interference channels, can be overcome with the help of
understood [9], and, as we shall show, can be translated to
an interleaver [9]–[11]. Thus, the scalar mutual information
a simple characterization of the performance of sigma-delta
I(U ;U +N ) can indeedbe interpretedasthe compression
modulation. n n n
rate needed to achieve the distortion attained by the test
First, we recall the derivation of the distortions attained channelsin Figure 1 andFigure 2. We elaboratefurtherabout
by the test channels from Figure 1 and Figure 2, and this in subsection II-B. Moreover,in Section III we show that
the scalar mutual information I(UΣ∆;UΣ∆ + NΣ∆) and I(U ;U +N ) isclosely relatedtothe requiredquantization
n n n n n n
I(UDPCM;UDPCM+NDPCM) between the input and output of rate in a sigma-delta modulator that applies a uniform scalar
n n n
5
quantizer of finite support. the optimal MMSE infinite length prediction filter of XDPCM
n
WebeginwiththetestchannelinFigure1,thatcorresponds from all past samples of the process XDPCM+NDPCM . The
{ n n }
to sigma-delta modulation, with the sigma-delta quantizer following straightforward proposition characterizes the rate
replaced by an AWGN channel with zero mean and variance and distortion for any choice of the causal filter C(Z) and
σ2 . The filter C(Z) is assumed to be strictly causal. any value of σ2 .
Σ∆ DPCM
Proposition 1: ForanyGaussianstationaryprocess XΣ∆ Proposition 2: Fora Gaussianstationaryprocess XDPCM
{ n } { n }
withvarianceσ2 whosePSDiszeroforallω / [ π/L,π/L], with variance σ2 and PSD
X ∈ − X
the test channel from Figure 1 achieves MSE distortion
Lσ2 for ω π/L
D =σ2 1 π/L 1 C(ω)2dω, SXDPCM(ω)=(0 X for |π/|L≤< ω <π , (7)
Σ∆· 2π | − | | |
Z−π/L the test channel from Figure 2 achieves MSE distortion
and its scalar mutual information satisfies1
σ2
D = DPCM,
I(UΣ∆;UΣ∆+NΣ∆) L
n n n
1 1 π σ2 and its scalar mutual information satisfies
= 2log(cid:18)1+ 2π Z−π|C(ω)|2dω+ σΣ2X∆(cid:19). I(UnDPCM;UnDPCM+NnDPCM)= 21log 1+ 21π π |C(ω)|2dω
(cid:18) Z−π
Proof: From Figure 1, we have that Lσ2 1 π/L
+ X 1 C(ω)2dω .
UnΣ∆ =XnΣ∆−cn∗NnΣ∆, (3) σD2PCM2π Z−π/L| − | (cid:19)
and therefore Proof: From Figure 2, we have that
UnΣ∆+NnΣ∆ =XnΣ∆+(δn−cn)∗NnΣ∆, UnDPCM =XnDPCM−cn∗VnDPCM (8)
where δn is the discrete identity filter. Using the fact that VnDPCM =UnDPCM+NnDPCM+cn∗VnDPCM (9)
XΣ∆ is a low-pass process, passing it through the filter
{ n } Substituting (8) in (9) yields
H(ω) has no effect, and hence
VDPCM =XDPCM+NDPCM. (10)
XˆΣ∆ =h (UΣ∆+NΣ∆) n n n
n n∗ n n Using the factthat XDPCM is a low-passprocess, as before,
=XnΣ∆+hn∗(δn−cn)∗NnΣ∆. we obtain { n }
TheMSEdistortionattainedbythetestchannelfromFigure1
XˆDPCM =h (XDPCM+NDPCM)
is therefore n n∗ n n
=XDPCM+h NDPCM. (11)
1 π/L n n∗ n
D =E(XnΣ∆−XˆnΣ∆)2 =σΣ2∆· 2π Z−π/L|1−C(ω)|2dω. Sofintchee{fiNltenDrPeCdMp}roisceAsWs hGN wNitDhPCvMariiasnσc2e σD2P/CLM.,Tthheusv,ariance
Thescalarmutualinformationbetweenthe“quantizer’s”input n∗ n DPCM
UnΣ∆ and output UnΣ∆+NnΣ∆ is given by D =E(XDPCM XˆDPCM)2 = σD2PCM.
n − n L
I(UΣ∆;UΣ∆+NΣ∆)=h(UΣ∆+NΣ∆) h(NΣ∆) (4)
n n n n n − n AsintheanalysisofthetestchannelfromFigure1,thescalar
1 E(UΣ∆)2 mutual information between UDPCM and UDPCM+NDPCM is
= log 1+ n , (5) n n n
2 σ2 given by
(cid:18) Σ∆ (cid:19)
where (4), as well as (5), follow from the statistical indepen- 1 E(UDPCM)2
I(UDPCM;UDPCM+NDPCM)= log 1+ n .
dence of NΣ∆ and UΣ∆. Using (3), the variance of UΣ∆ is n n n 2 σ2
n n n (cid:18) DPCM (cid:19)
1 π (12)
E(UΣ∆)2 =σ2 +σ2 C(ω)2dω. (6)
n X Σ∆2π | | Now, substituting (10) in (8) gives
Z−π
Substituting (6) into (5) establishes the second part of the UDPCM =(δ c ) XDPCM c NDPCM,
n n− n ∗ n − n∗ n
proposition.
and the variance of U is therefore
n
Next, we analyze the test channel in Figure 2, that cor-
1 π
responds to DPCM compression with the DPCM quantizer E(UDPCM)2 = SDPCM(ω)1 C(ω)2dω
n 2π X | − |
rσeD2pPlCaMce.dAbsyiannthAeWtGesNt cchhaannnneell owfitFhigzeurroe m1,eathneanfidltevrarCia(nZce) + 1 Z−ππ SDPCM(ω)C(ω)2dω
is strictly causal. The distortion corresponding to this test 2π N | |
fcohuanndneiln,a[s9,wTehlleaosreIm(U1nD]PfCoMr;tUhneDPsCpMec+iaNl cnDaPsCeMw),hwereereCa(lrZe)adiys = LσX2 π/L 1Z−Cπ(ω)2dω+ σD2PCM π C(ω)2dω.
2π | − | 2π | |
Z−π/L Z−π (13)
1Alllogarithms inthispaperaretakentobase2.
6
Substituting (13) into (12) establishes the second part of the Theorem 2: Let XΣ∆ be a Gaussian stationary process
{ n }
proposition. withvarianceσ2 whosePSDiszeroforallω / [ π/L,π/L]
X ∈ −
Remark 1: In propositions 1 and 2 we derived the scalar and let be a family of strictly causal filters. Define the
C
mutual information between the input and output of the “virtual” process S as a Gaussian stationary process with
n
{ }
AWGN test channels embedded in Figures 1 and 2, respec- PSD as in (7), and the “virtual” process W as a Gaussian
n
{ }
tively. As will become clear in Section III, the scalar mutual i.i.d. random process statistically independent of S with
n
{ }
informationis closely related to the requiredquantizationrate variance L D, D >0. Let
·
whena scalarmemorylessquantizerisused withinthesigma-
delta or DPCM modulator. In [9], [11], the directed informa- σD∗2 =Cm(Zi)n E(Sn−cn∗(Sn+Wn))2
tion was shown to be related to the required quantization rate ∈C
C (Z)=argminE(S c (S +W ))2.
when the quantizer is followed by an entropy coder. Here, D∗ n− n∗ n n
C(Z)
we donotconsiderapplyingentropycodingtothe quantizer’s ∈C
output as we require that the designed modulator be robust IfthefilterC(Z)inthesigma-deltatestchannelfromFigure1
to the statistics of the input process, whereas entropy coding belongs to and the MSE distortion attained by this test
is very sensitive to the process statistics. Moreover, if the channel is DC, then
designofanA/D(orD/A)isconsidered,theappropriatemerit
for the modulator’s complexity is the number of quantization I(UΣ∆;UΣ∆+NΣ∆) 1log 1+ σD∗2 , (14)
levels within the scalar quantizer, which are not reduced by n n n ≥2 L D
(cid:18) · (cid:19)
incorporating an entropy coder.
with equality if C(Z)=C (Z).
Our main result now follows immediately from Proposi- D∗
tions 1 and 2. Theorem 2 states that for a target distortion D, the sigma-
Theorem 1: Let XΣ∆ beanyGaussianstationaryprocess delta filter which minimizes the required compression rate
withvarianceσ2 w{hosneP}SDiszeroforallω / [ π/L,π/L], is the optimal linear time-invariant MMSE estimator, within
let XDPCM bXe a flat low-pass Gaussian sta∈tio−nary process the class of constraints , for Sn from the past of the noisy
with{PSnD a}s in (7), and let C(Z) be a strictly causal filter. process Sn+Wn . ForCexample, if consists of all strictly
{ } C
The test channel from Figure 1 with causal finite-impulse response (FIR) filters of length p, the
optimal filter C(Z) is the optimal predictor of S from the
n
D
σ2 = , samples Sn 1 +Wn 1,...,Sn p +Wn p , which can be
Σ∆ 1 π/L 1 C(ω)2dω easily cal{cula−ted in clo−sed-form.− − }
2π π/L| − |
− The optimal sigma-delta filter design problem was studied
and the test channel fromRFigure 2 with
by several authors, under various assumptions [1], [6], [8],
[12]–[15]. However, to the best of our knowledge, the simple
σ2 =L D,
DPCM · expressionfromTheorem2fortheoptimalfilterastheoptimal
both achieve MSE distortion D and their scalar mutual infor- predictor of Sn from the past of Sn +Wn is novel. The
{ }
mation satisfy references most relevant to Theorem 2, are perhaps [14]
and [8], [15]. In [14], Spang and Schultheiss formulated an
I(UΣ∆;UΣ∆+NΣ∆)=I(UDPCM;UDPCM+NDPCM) optimization problem for finding the best FIR filter with p
n n n n n n
1 1 π coefficientsinasigma-deltamodulatorwithascalarquantizer,
= log 1+ C(ω)2dω
underafixedoverloadprobability.Theiroptimizationproblem
2 2π | |
(cid:18) Z−π can be solved numerically, but no closed form solution was
σ2 1 π/L
+ X 1 C(ω)2dω . given. In [8] and [15] the design of an optimal unconstrained
D 2π | − |
Z−π/L (cid:19) sigma-deltafilterwasstudied,undertheassumptionofafixed
scalar quantizer which can only be scaled in order to control
This theorem indicates that for any stationary band-limited the overload probability. Equations that characterize the op-
Gaussianprocesswithvarianceσ2 ,thesigma-deltatestchan- timal filter were derived. However, the obtained expressions
X
nel from Figure 1 achieves exactly the same rate-distortion usually yield filters with an infinite number of taps, and do
trade-off as that of the DPCM test channel from Figure 2 not provide the solution to the constrained problem. It is also
with a stationary flat band-limited Gaussian input with the worth mentioning that for the case of a stationary Gaussian
same variance, provided that the AWGN variances are scaled process Xn with L = 1 (sampling at the Nyquist rate)
{ }
according to (2). Thus, Theorem 1 provides a unified frame- and known PSD, the optimal infinite length filter under the
workforanalyzingtheperformanceofsigma-deltamodulation assumption of high-resolution quantization is known to equal
and DPCM. A great advantage offered by such a unified the optimal prediction filter of Xn from its (clean) past [12].
framework, is that any result known for DPCM can be trans- As already mentioned in the introduction, the high-resolution
latedtoacorrespondingresultforsigma-deltamodulation,and assumption never holds when L>1 and therefore this result
vice versa. Theorems 2 and Corollary 1 below constitute two is inapplicable for over-sampled signals.
important examples of such results.
ProofofTheorem2: ByProposition1,ifthetestchannel
7
from Figure 1 achieves MSE distortion D, we must have with equality if and only if C(Z) is a strictly causal filter
satisfying
D
σ2 = .
Σ∆ 1 π/L 1 C(ω)2dω 1+ σX2 −(L−1)/L ω [ π, π]
By Theorem 1, the2πcRo−rπr/eLsp|on−ding m|utual information |1−C(ω)|2 =(cid:16)1+ σDDX2 (cid:17)1/L ω ∈∈/ [−−LLπ,LLπ], (20)
I(UΣ∆;UΣ∆ + NΣ∆) is equal to the mutual information
I(UnDPCM;nUDPCM+nNDPCM) in the DPCM test channel from and (cid:16) (cid:17)
n n n
FTihguusr,e2withXnDPCM =Sn,NnDPCM =WnandσD2PCM =L·D. σΣ2∆ = 21π −ππ/L/L|1D−C(ω)|2dω = 1+ σDX2L·−D(L−1)/L.
I(UΣ∆;UΣ∆+NΣ∆)=I(UDPCM;UDPCM+NDPCM) R (cid:16) (cid:17)
n n n n n n
Remark 2: Note that the existence of a strictly causal filter
1 E(S c (S +W ))2
= log 1+ n− n∗ n n , (15) C(Z) which satisfies (20) is guaranteedby Wiener’s spectral-
2 L D
· ! factorization theory [16] due to the readily verified fact that
where we have used (8), (10), and (12), to arrive at (15). It 221π −ππlog|1−C(ω)|2dω =1.
followsthatamongallfiltersin ,thefilterthatminimizes(15) R
is C (Z), and that it attains (1C4) with equality. The optimal filter induces a two-level frequency response for
D∗ 1 C(ω)2. In [11] Østergaard and Zamir used sigma-delta
It is interesting to note [9] that since Wn is an i.i.d. m| o−dulation| to attain the optimal multiple-description rate-
{ }
process with variance L D and C(Z) is strictly causal, the distortionregion.Interestingly,theoptimalfilterC(Z)intheir
·
mutual information (15) can also be written as scheme also induced a two-level response for 1 C(ω)2.
| − |
We also note that the optimality of the unconstrained filter
I(UΣ∆;UΣ∆+NΣ∆)
n n n specified by (20) can be deduced as a special case of [8,
1 E(S +W c (S +W ))2 Section IV].
n n n n n
= log − ∗ .
2 L D ! Remark 3: Note that for the optimal unconstrained filter
·
(16) C(Z)specifiedby(20),thepre-andpost-filtersfromFigure3
have no effect as long as the PSD of the input signal XΣ∆
Thus,theoptimalpredictorofSn fromthepastof Sn+Wn is zero forall ω / [ π/L,π/L].However,filters with{a finnit}e
{ }
is identical to the optimal predictor of Sn+Wn from its past number of taps w∈ill−never incur a flat frequency response in
samples. When C(Z) is taken as the (unique) infinite order theinterval[ π/L,π/L],andforsuchfiltersthesystemsfrom
optimal one-step prediction filter of Sn +Wn from its past Figure 1 and−Figure 3 will not be equivalent.
samples, the prediction error variance is the entropy power of
Remark 4: The output of the test channel from Figure 1
the process S +W [16], which equals
{ n n} (as well as that from Figure 2) is of the form XˆΣ∆ =
n
221π −ππlog(SS(ω)+L·D)dω =(L·D) 1+ σDX2 1/L. (17) XannΣd∆is+stEatnΣis∆ti,cawllhyeriendEepnΣe∆ndhenast ozferoXnΣm∆ea.nThanisdevsatirmiaantceecDan,
R (cid:18) (cid:19) be further improvedby applyingscalar MMSE estimation for
Moreover, the infinite order prediction error XΣ∆ from XˆΣ∆. This boils down to producing the estimate
n n
Xˆ˜Σ∆ =αXˆΣ∆, where
Epred ,S +W c (S +W ) n n
n n n− n∗ n n σ2
α= X .
isinthiscaseawhiteprocess.This,togetherwith(17)implies σ2 +D
X
that for the optimal unconstrainedsigma-delta filter C(Z) we
Consequently, the obtained MSE distortion is reduced to
must have
σ2 D
SEpred(ω),|1−C(ω)|2(L·D+SS(ω)) D˜ =E(XnΣ∆−αXˆnΣ∆)2 = σX2X+· D.
σ2 1/L Itisstraightforwardtoverify[17] thatwiththisimprovement,
=(L D) 1+ X , ω [ π,π) (18)
· D ∀ ∈ − the sigma-delta test channel from Figure 1 with C(Z) and
(cid:18) (cid:19)
σ2 as specified in Corollary 1 attains
Combining(16), (17), and (18) yieldsthe followingcorollary. Σ∆
1 σ2
Corollary 1: Let XΣ∆ be a Gaussian stationary process I(UΣ∆;UΣ∆+VΣ∆)= log X ,
{ n } n n n 2L D˜
withvarianceσ2 whosePSDiszeroforallω / [ π/L,π/L]. (cid:18) (cid:19)
If the test chanXnel from Figure 1 attains MS∈E d−istortion D, which is the optimal rate-distortion function for a stationary
then Gaussiansource{XnΣ∆}withPSDasin(7).Itfollowsthatthe
sigma-deltatestchannelfromFigure1withC(Z)andσ2 as
1 σ2 Σ∆
I(UΣ∆;UΣ∆+NΣ∆) log 1+ X . (19) specifiedinCorollary1isminimaxoptimalfortheclassofall
n n n ≥ 2L D stationary Gaussian sources with variance σ2 and PSD that
(cid:18) (cid:19) X
8
equals zero for all ω / [ π/L,π/L], i.e., no other system It follows that the DPCM test channel for the process
can achieve MSE disto∈rtio−n D˜ with a smaller compression • S under MSE distortion is equivalent to the sigma-
n
{ }
rate, universally for all sources in this class. delta test channel with input XΣ∆ under FWMSE
{ n }
distortion,inthesensethatinbothchannelsiftheattained
A. Extension to Frequency-Weighted Mean Squared Error distortion is DFWMSE (under the appropriate distortion
Distortion measure), then
In many applications, higher values of distortion are ac- I(UΣ∆;UΣ∆+NΣ∆)=I(UDPCM;UDPCM+NDPCM)
n n n n n n
ceptable in certain frequency bands while smaller distortion
1 E(S c (S +W ))2
is permitted in other bands. The MSE distortion measure is = log 1+ n− n∗ n n .
inadequateforsuchscenarios,andacommonlyuseddistortion 2 L DFWMSE !
·
measure,that(partially)capturessuchperceptualeffects,isthe
frequency-weighted mean squared error (FWMSE) criterion.
Under this criterion, the distortion is measured as
B. Sigma-DeltaModulationwithanInterleavedVectorQuan-
1 π
D , P(ω)S (ω)dω, (21) tizer
FWMSE E
2π
Z−π The goal of this short subsection is to give the test channel
where P(ω) is a non-negativeweight function, and S (ω) is from Figure 1 an operational meaning, i.e., to show how
E
the PSD of the error process E , XΣ∆ XˆΣ∆. Note that the AWGN from the figure can be replaced with a lossy
n n − n
for P(ω) = 1, ω [ π,π), the FWMSE criterion reduces source code of rate R = I(UΣ∆;UΣ∆ + NΣ∆) whose
∀ ∈ − n n n
to the MSE one.The nexttheoremshowsthatthe constrained incurredquantizationnoise is distributed as NΣ∆. As already
n
optimal sigma-delta filter under the FWMSE criterion is the mentioned, the key idea is to use an interleaver [9]–[11], as
optimalconstrainedpredictionfilterofanoisyprocessdefined we now recall.
according to the weight function P(ω). Assume that XΣ∆ , the input process to the sigma-
{ n }
witThhveoarrieamnc3e:σL2ewt {hXosneΣ∆PS}DbeisazeGroaufossriaalnlωsta/tio[naπr/yLp,rπo/ceLs]s, edsesletantmiaolldyuilnatdoerp,ehnadsenatdoefcaayllinsgammpelmesoroyf,ssuufcfihcitehnattlyXdnΣis∆tanist
P(ω) a weightiXng function which forms a va∈lid−PSD, and samplingtimes.InordertocompressanN-dimensionalvector
C
a family of strictly causal filters. Define the “virtual” process xΣ∆ =[XΣ∆,...,XΣ∆],
S as a Gaussian stationary process with PSD 1 N
n
{ } containingN consecutivesamplesof the process XΣ∆ , we
Lσ2 P(ω) for ω π/L { n }
SFWMSE(ω)= X | |≤ , (22) first split it into K vectors
X (0 for π/L<|ω|<π xΣ∆ =[XΣ∆ ,...,XΣ∆], k =1,...,K,
k (k 1)M+1 kM
and the “virtual” process W as a Gaussian i.i.d. ran- −
n
dom process statistically ind{epen}dent of S with variance whereM ,N/K.Now,we canapplyK parallelsigma-delta
n
L D , D >0. Let { } modulators,oneforeachsuchvector,wheretheonlycoupling
FWMSE FWMSE
· between the K parallel systems is through the quantization
σD∗2FWMSE =Cm(Zi)n E(Sn−cn∗(Sn+Wn))2 step, which is applied jointly on all of them, as depicted
∈C in Figure 4. By our assumption that XΣ∆ has decaying
CD∗FWMSE(Z)=argminE(Sn−cn∗(Sn+Wn))2. memory, if M is large enough the{Kninp}uts that enter
C(Z)
∈C the quantizer Q() = [Q (),...,Q ()] are i.i.d. random
1 K
IfthefilterC(Z)inthesigma-deltatestchannelfromFigure1 variablesdistribut·edas UΣ∆·from Figure· 1. For large enough
belongs to and the FWMSE distortion w.r.t. P(ω) attained n
C K,standardrate-distortionargumentsimplythatthereexistsa
by this test channel is DFWMSE, then vector quantizer with rate I(UΣ∆;UΣ∆+NΣ∆) that induces
n n n
I(UΣ∆;UΣ∆+NΣ∆) 1log 1+ σD∗2FWMSE , quantization noise distributed as NnΣ∆.
n n n ≥2 L DFWMSE!
· III. SIGMA-DELTA MODULATIONWITHA SCALAR
with equality if C(Z)=CD∗FWMSE(Z). UNIFORM QUANTIZER
Sketch of proof:: The proof is fairly similar to that of TheprevioussubsectionshowedhowtoreplacetheAWGN
Theorem2. Thus,for brevity,we omitthe fullproofand only channel in Figure 1 with a vector quantizer whose rate is
highlight its main steps: arbitrarily close to R = I(UΣ∆;UΣ∆ + NΣ∆) and whose
n n n
Repeat the derivation of Proposition 1 where now the induced quantization noise is distributed as NΣ∆. The inputs
• n
MSE distortion is replaced by FWMSE distortion. Note to the vector quantizer are vectors of i.i.d. Gaussian compo-
that this has no effect on I(UΣ∆;UΣ∆+NΣ∆). nents. Thus,any “off-the-shelf”rate–distortionoptimalvector
n n n
Repeat the derivationof Proposition 2 where the PSD of quantizer for an i.i.d. Gaussian source can be used. The total
•
the input process is (22), rather than (7). Note that this sigma-delta compression system that is obtained is therefore
changes I(UDPCM;UDPCM +NDPCM), but has no effect simple in the sense that it only requires the vector quantizer
n n n
on the attained distortion. to be good for quantizing an i.i.d. Gaussian source, which is
9
H(ω)
Q()
· 1
UΣ∆ UΣ∆+NΣ∆
xΣ1∆ Σ 1,n Q1(·) 1,n 1,n xˆΣ1∆
π π ω
−
−L L
C(Z) − Σ
NΣ∆
1,n .
.
. H(ω)
1
UΣ∆ UΣ∆ +NΣ∆
xΣM∆ Σ K,n QK(·) K,n K,n xˆΣK∆
π π ω
−
−L L
C(Z) − Σ
NΣ∆
K,n
Fig.4. K parallel sigma-delta modulators coupled byanK-dimensional quantizer Q(·).
a standardtask,ratherthanrequiringitto be a goodquantizer QR,σ2(x)
3
for a band-limited Gaussian source.
However,thesigma-deltamodulationarchitectureis mainly
used for A/D and D/A conversion. In such applications,
1
vectorquantizationistypicallyoutofthequestion,andsimple
uniform scalar quantizers of finite support are used. For such x
−4 −2 0 2 4
quantizers, the quantization error is composed of two main −1
factors[1]:granularerrorsthatcorrespondtothequantization
error in the case where the input signal falls within the
quantizer’s support, and overload errors that correspond to −3
the case where the input signal falls outside the quantizer’s
support.Dueto thefeedbackloop,inherenttothe sigma-delta
Fig.5. Anillustration ofQR,σ2(·)forR=2andσ2=1/3.
modulator, errors of the latter kind, whose magnitude is not
bounded, may have a disastrous effect as they jeopardize the
system’sstability.Inordertoavoidsucherrors,thesupportof
Clearly,ifweemploythescalarsigma-deltamodulatorona
the quantizer has to be chosen appropriately. As the support
longenoughinputsequence,anoverloadeventwilleventually
of the quantizer determines its rate for a given quantization
occur. As discussed above, the effects of overload errors can
resolution, the overload probability can be controlled by
be amplified due to the feedback loop, and in this case the
increasing the quantization rate.2
average MSE may significantly grow. We therefore split the
Weshallshowthat,giventhatoverloaderrorsdidnotoccur,
inputsequenceintofiniteblocksoflengthN,andinitializethe
the quantization noise can be modeled as an additive noise.
memoryof the filter C(Z) with zerosbeforethe beginningof
Thus, the test channel from Figure 1 accurately predicts the
eachnewblock.Thismakessurethattheeffectofanoverload
totaldistortionincurredbyasigma-deltaA/D (orD/A)inthis
error in the original system is restricted to the block where it
case. Moreover, the overload probability is a doubly expo-
occurs.
nentially decreasing function of R I(UΣ∆;UΣ∆ +NΣ∆),
− n n n The analysis is made much simpler by introducing a sub-
where 2R are the number of levels in the scalar quantizer.
tractive dither [17]. Namely, let Z be a sequence of
n
Thus, fixing the desired overload error probability as P , we { }
ol i.i.d. random variables uniformly distributed over the interval
may achieve the MSE distortion predicted by the test channel [ 12σ2 /2, 12σ2 /2). In order to quantize UΣ∆, we
from Figure 1 (characterized in Proposition 1) with a scalar − Σ∆ Σ∆ n
add Z to it before applying the quantizer, and subtract Z
quantizerwhoserateisI(UnΣ∆;UnΣ∆+NnΣ∆)+δ(Pol),where aftperwnards, suchpthat the obtained result is QR,σ2 (UnΣ∆ +n
δ(Pol)=O loglog P1ol . Zn) − Zn. Adding and subtracting UnΣ∆, we gΣe∆t UnΣ∆ +
√1L2eσt2QaRn,σd2(cid:16)(2·R) bqeuaan(cid:16)utnizifaot(cid:17)iro(cid:17)mnqleuvaenltsiz,esruwchiththqautatnhteizaqtuioanntsitzeepr (cid:16)erQroRr,σisΣ2∆th(eUrenΣf∆or+e Zn)−(UnΣ∆+Zn)(cid:17), and the quantization
support is [ Γ/2,Γ/2), where Γ , 2R√12σ2, see Figure 5.
Our goal is−to analyze the distortion and overload probability Nn ,QR,σΣ2∆(UnΣ∆+Zn)−(UnΣ∆+Zn) (23)
attained by a sigma-delta modulator that uses a Q ()
quantizer, as a function of R and σ2 . R,σΣ2∆ · The main result in this section is the following.
Σ∆ Theorem 4: Let D be the MSE distortion attained by the
testchannelinFigure1withafilterC(Z)offinitelength,and
2As discussed in Section I-B, one can try to limit the effect of overload
I(UΣ∆;UΣ∆+NΣ∆) the scalar mutualinformationbetween
errorsbyplacing various constraints onC(Z).Here,werestrict attention to n n n
controlling theoverload probability. theinputandoutputoftheAWGNchannelinthesamefigure.
10
Z
n
H(ω)
+ 1
XnΣ∆ Σ UnΣ∆ Σ QR,σΣ2∆(·) −Σ UnΣ∆+NnΣ∆ π π ω XˆnΣ∆
−
−L L
C(Z) − Σ
NΣ∆
n
Fig. 6. A sigma-delta modulator with a dithered scalar uniform quantizer. The input is assumed to be over-sampled at L times the Nyquist rate, and the
dither sequence {Zn} is assumed to be an i.i.d. sequence of random variables uniformly distributed over the interval h−q12σΣ2∆/2,q12σΣ2∆/2(cid:17) and
statistically independent {XΣ∆}.
n
For any 0 < P < 1 the scalar sigma-delta modulator from theinterval[ 12σ2 /2, 12σ2 /2),statisticallyindepen-
ol − Σ∆ Σ∆
Figure6 appliedona sequenceof N consecutivesourcesam- dent of XΣ∆ . Note that N has zero mean and variance
{ n p} pn
pleswithquantizationrateR=I(UΣ∆;UΣ∆+NΣ∆)+δ(P ) σ2 . Following this reasoning, the reference sigma-delta
n n n ol Σ∆
attains MSE distortion smaller than data converter depicted in Figure 6 (with an infinite-support
D(1+o (1)) quantizer) is equivalent to the test channel from Figure 1
N
,
1−Pol withNnΣ∆ ∼Uniform [− 12σΣ2∆/2, 12σΣ2∆/2) instead
given that overload did not occur. In addition, the overload of NnΣ∆ ∼ N(0,σΣ2∆(cid:16)). Tphus, the aveprage MSE d(cid:17)istortion
probability is smaller than P , where o (1) 0 as N attained by the reference scalar sigma-delta modulator from
ol N
increases, and → Figure 6 is as given in Proposition 1 up to a multiplicative
factorof1+o (1)thataccountsforedgeeffects.Theseeffects
1 2 P N
δ(P ), log ln ol . (24) aretheby-productoftheoperationofnullingthefiltermemory
ol
2 −3 2N
(cid:18) (cid:19) at the beginning of each new block, which incurs temporal
Proof: Let Q˜ (x) be the operation of rounding x non-stationarities. In particular, if the filter C(Z) has L taps,
√12σ2Z
to the nearest pointin the (infinite) lattice √12σ2Z. It is easy then only after L samples within the block the statistics of
theprocess UΣ∆ willconvergetoitsstationarydistribution.
to verify that for any x [ Γ/2,Γ/2) we have { n }
∈ − However,iftheblocklengthissufficientlylargew.r.t.thefilter
√12σ2 √12σ2 length and the inverse of the MSE distortion, the influence of
Q (x)=Q˜ x+ . (25)
R,σ2 √12σ2Z 2 !− 2 these effects vanishes.
Next, we turn to analyze the probability that an overload
Applying (23) therefore yields that if overload did not occur
error occurs within a block of length N, as a function of R
in the nth sample, i.e., if UΣ∆+Z Γ/2, we have
| n n|≤ andI(UΣ∆;UΣ∆+NΣ∆).Sincethiseventisequivalenttothe
n n n
eventthat at the reference system some input to the quantizer
N =Q˜ UΣ∆+Z + 3σ2
n √12σ2 Z n n Σ∆ exceedsΓ/2inmagnitudewithintheblock,itsufficestoupper
Σ∆ (cid:18) q (cid:19) bound the probability of the latter event.
UΣ∆+Z + 3σ2 . (26)
− n n Σ∆ Assume the reference scalar sigma-delta modulator from
(cid:18) q (cid:19) Figure 6 is applied to a vector xΣ∆ = [XΣ∆,...,XΣ∆] of
Dealing with the overloadevent of the quantizer directly is 1 N
N consecutive samples of the process XΣ∆ , where the
rather involved. Instead, as done in [18], we first consider a { n }
memory of the filter C(Z) is initialized with zeros. Define
reference system with an infinite-support quantizer (R = )
∞ the event OL , UΣ∆ + NΣ∆ > Γ/2 and the event
and analyze its performance. If the magnitude of the input OL, NOLk. By t{h|e kunion bokund|, we have}
to the infinite-support quantizer never exceeds Γ/2 within ∪k k
the processed block, then clearly the reference system is
completelyequivalenttotheoriginalsystemwithinthisblock.
N
Thus, it sufficesto find the average distortionof the reference P ,Pr(OL) Pr(OL ). (27)
ol k
system and the probability that the input to its quantizer ≤
k=1
X
exceedsΓ/2withinablock.Inwhatfollowswewilltherefore
assume that the quantization noise is given by (26) regardless
of whether or not UΣ∆ +Z Γ/2, and account for the The random variable UΣ∆ +NΣ∆ = XΣ∆+(δ c )
| n n| ≤ k k k k − k ∗
overload probability later. NΣ∆ is a linear combination of a Gaussian random variable
k
Assuming that the dither sequence Z is drawn sta- XΣ∆ and statistically independent uniform random variables
{ n} k
tistically independent of the process XΣ∆ , the Crypto NΣ∆ . In [19, Lemma 4] the probability that a random
{ n } { k }
Lemma, see, e.g. [17, Lemma 4.1.1], implies that N is an variable of this type exceeds a certain threshold was bounded
n
{ }
i.i.d. sequence of random variablesuniformlydistributed over in terms of its variance. Applying this boundto UΣ∆+NΣ∆
k k