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956 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 31, NO . 9, SEFTEMBER 1989
Quantization Noise in Single-Loop Sigma-DeltaModulation with Sinusoidal Inputs
ROBERT M . GRAY, FELLOW, IEEE, wu CHOU, STUDENT MEMBER, IEEE, AND PING w. WONG, STUDENT MEMBER, IEEE
Abstract-An exact nonlinear difference equation is derived and solved
for a simple sigma-delta modulator consisting of a discrete time integrator
and a binary quantizer inside a single feedback loop and an arbitrary
input signal. It is shown that the system can be represented as an affineoperation (discrete-time integration of a biased input) fo llowe d by a
memoryless nonlinearity. An extension of the transform method for the
analysis of nonlinear systems is applied to obtain formulas for first- and
second-order time-average moments of the binary quantization noise,
including the sample mean, energy, and autocorrelation. The results are
applied to the special case of a sinusoidal input signal to evaluate these
time averages and the power spectrum. In the limit of large oversampling
ratios, the marginal moments behave as if the quantization noise had a
uniform distribution. The spectrum is neither white nor continuous,
however, even in the limit of large oversampling ratios.
I. INTRODUCTION
VERSAMPLED sigma-delta (CA) (or delta-sigma)
0 odulation provides a promising architecture for high
resolution analog-to-digital converters (ADC's) because it is
robust against circuit imperfections and hence is amenable to
LSI and VLSI implementation. [1]-[6]
The simplest discrete time E A modulator is shown in Fig. 1 and consists of a discrete time integrator and a binary
quantizer inside a single feedback loop. It is described by the
nonlinear difference equations in the next section. The basic
idea of such a system is that instead of sampling the input
signal at the Nyquist rate, the minimum rate required to
accurately reproduce the original signal from its samples, the
input signal is first sampled at many times the Nyquist rate
(oversampled) and then quantized at a very low resolution
(such as one bit per sample) inside a feedback loop. The
resulting binary sequence is then low-pass filtered and
downsampled to produce an approximate reproduction of the
original discrete time sequence. The overall system approxi-mates the action of a single high resolution quantizer at the
original rate. Thus, one can reduce the need for the multiple
thesholds of high resolution quantization and the associated
precise tolerances needed (a difficult goal in VLSI) in trade for
timing accuracy (easier in VLSI). Although more elaborate
systems are possible (and yield better performance), we here
focus on a simple single-loop modulator as a basic first step in
any analysis technique.
During recent years considerable work has been devoted
toward developing the tradeoffs between system complexity
and performance for such systems. [1]-[lo].Most of the performance analyses of such systems have
Paper approved by the Edi tor for Speech Processing of the IEEECommunications Society. Manuscript received November 15, 1987; revisedJuly 15, 1988. This research was supported by the National Science
Foundation and by a Stanford University Center for Integrated Systems SeedGrant. This paper was presented in part at the 1988 International Symposiumon Information Theory, Kobe, Japan, June 19-24, 1988.
The authors are with the Information Systems Laboratory, Department ofElectrical Engineering, Stanford, CA 94305.
IEEE Log Number 8929596.
= en-l + '
Fig. 1. Basic E A modulator.
made two basic assumptions. The first is that the input
waveform or sequence is either a dc (quiet input) or a sinusoid.
This is done both for simplicity and because these two inputsare important as they typify two aspects of more general
sources. If the original input is oversampled at many times the
Nyquist rate, then for relatively short periods of time the input
does stay relatively constant and an analysis for dc inputs
provides a potentially useful approximation for more general
slowly varying inputs. Unfortunately, however, the dc input
does not capture all important attributes of general inputs,
e.g., how the rate of change of the input affects the output and
intermodulation products. In addition, it is standard practice to
quantify the quality of an ADC by means of its response to
sinusoids.
The second assumption is that the binary quantizer noise can
be modeled as a signal independent white uniform noise
source, thereby linearizing the highly nonlinear system and
permitting one to use standard Fourier techniques to find the
various moments (such as noise mean and power) and derive
the spectral densities of the various signals. It was shown in [8]
that the conditions required for the use of the white noiseassumptions are violated in a fundamental way in C Amodulators. It was further shown in [6], [111, and [9] that even
in the dc source case, the quantization noise in an ideal C Amodulator is decidedly nonwhite and that, in contrast to the
white noise assumption, the spectrum of the quantization no ise
is purely discrete. Consistent with the traditional assumptions,
however, it was proved in [8] that for dc unputs the time
average mean and energy of the binary quantizer error
sequence indeed behave as if the marginal distribution of the
error were uniform. Fig. 2 provides some examples of the
binary quantizer noise sequence (normalized for convenience)
generated by the system of Fig. 1 with a full scale discrete time
sinusoid as input. Each graph depicts 1024 samples of the
binary quantizer error sequence. The frequencies of the input
sinusoids were selected uniformly at random from the range
[0, 11 (where frequencies are normalized to lie within [0, 13).
The error sequences were then generated directly from the
nonlinear difference equations describing Fig. 1 and plottedusing ProMatlab on a Sun 3/50. Fig. 3 shows that correspond-
ing power spectrum formed by squaring the magnitudes of the
coefficients of the FFT of the error sequence. As in the dc
case, the sequences do not resemble white noise and the
0090-6778/89/0900-0956$01.OO 0 989 IEEE
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957RA Y et al. : SIGMA-DELTA MODULATION WITH SINUSOIDAL INPUTS
Sigma-DeltaError: R= 10, N= 1024, f = 0.069070Hz Sigma-DeltaError: R= ,N= 024, f = 0.300446Hz
0 4
0 3
0 2
0 1
0
-0 1
0 2
0 3
0 4
I ' I I
200 400 600 800 1000 1200
mean= 0.000092,variance = 0.080464 n
Sigma-DeltaError: R = 10,N= 1024, f = 0.071948Hz0.5 I
0 4
0.3
0.2
0.
0
-0.
-0.2
-0.3
-0.4
1
I1 , I200 400 600 800 loo0 1200
-0.5 ['I
mean= 0.000219,variance = 0.077062n
0 800 1000 120000 400 600
mean= 0.000920,variance = 0.095491 n
Sigma-DeltaFmx R= IO,N= 12. f = 0.083603Hz0.5 I
I ' , I ' 1 1 , '-0.5' ' ' ' J100 200 300 400 500 600
mean= 0.025509,variance = 0.080308 n
Fig. 2. Examples of normalized Z A binary quantizer error sequences.
spectrum does not display a uniform shape. The marginal
characteristics do, however, look uniform with a mean of
approximately zero and a second moment of approximately1/12. The principal goal of this work is to provide a means of
accurately predicting the marginal moments and the location
and amplitudes of the spectral coefficients.
There have been some attempts to avoid this assumption by
using Fourie r's series or continuous time approximations [ 1 11,
[6]. These results produced useful expressions only in the case
of the dc source, however, and the results for sinusoidal inputs
are multiple sums of weighted Bessel functions which do not
appear to yield useful expressions or bounds. Variations on the
white noise approach have also been attempted for the
sinusoidal driver by Ardalan and Paulos [lo] who used the
describing function analysis technique of control theory [121-[16] to first replace the nonlinear quantizer by a linear gain
chosen to minimize the mean squared error for a particular
sinusoidal input and then model the resulting error (between
the original nonlinear quantizer output and the linear gain
output) by a signal-independent white noise term. This
approach still attributes uniform white noise behavior to the
quantization noise, although it tries to improve the model by
the inclusion of the additional linear gain. This approach still is
based on an assumed approximation and does not address the
issue of deriving the behavior of the quantization noise (rather
than assuming it to have a specific form) and the meaning of a
system gain in such a highly nonlinear system is not clear.
An alternative approach was introduced for E A modulation
in [8], [9] and a similar approach developed in unpublished
work of Kim and Neuhoff. Instead of assuming a noisedistribution, nonlinear difference equations were derived for
the system and the sample average moments were derived
exactly using limiting results from ergodic theory and Bohr-
Fourier analysis. This development was valid, however, only
for the case of a dc input.
In this paper, the results of [8] and [9] on the moments and
power spectrum of the binary quantizer noise in a sigma-delta
modulator with a dc input are extended by developing and
solving nonlinear difference equations for the operation of the
single-loop E A modulator with general input signals. The
transform method of Rice [17] and Davenport and Root [18]
for finding moments of nonlinear systems is extended to
sample averages and applied to evaluate the time average
mean, energy, and autocorrelation of the binary quantization
noise. The results resemble Davenport and Root's solution
with time average characteristic functions replacing theirprobabilistic averages.
The results are then specialized to the case of a sinusoidal
input. The time average moments are given in terms of single
sums of weighted Bessel functions. I t is shown that in the limit
of large oversampling ratio, the first-order (marginal) mo-
ments approximate those of a uniform distribution, but that the
second order moments do not approximate those of a white
process. The spectrum is shown to be purely discrete, as in the
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95 8 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 31 , NO . 9, SEPTEMBER 1989
0.007
E 0.006-
g 0.0056
k 0.004-
N= 1024, f=0.069070
o : /
-
-
0.007'008i% 0.005v,
E o . 0 0 4 -
0.003
0.002
0.001-
-
+
- +
-
+
spectral energy= 0.080464 frequency (Hz)
0.018 -
0.008 I
0.01
&* ,&e?+ 4-'0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
-
spectralenergy= 0.077062 frequency (Hz)
N= 12, f=0.0836030.01
0.008. 7
0.003
0.002t0
0.001
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
frequency (Hz)spectral energy= 0.080958
I.016
0.014
0.002
0
*,*.t R
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
frequency (Hz)spemal energy= 0.095492
Fig. 3 . FFT's of examples of Fig. 2.
dc input case, and the output frequencies and the correspond-
ing amplitudes are found.
11. DIFFERENCEQUATIONS
The basic difference equations describing the behavior of
the Z A modulator can be written as follows [18]:
where x, € [- b, b]; = 1, 2, * * * is the discrete time input
signal (usually formed by sampling a continuous time signal),
U, is the intergrator state, U* is the initial integrator state, and
q(u) is a binary quantizer defined by
where b is chosen as the maximum absolute input value and
hence the binary quantizer has the full input extremes as its
possible outputs. We will assume for convenience that U* =0. This corresponds physically to resetting the integrator at the
beginning of each sampling period of the original signal
(assumed to be sampled at the Nyquist rate). Mathematically,
however, the assumption has no effect on the asymptotic
results to be obtained for the averages of a large number of
samples, and the assumption greatly simplifies the algebra.
A process of fundamental importance is the binary quantizer
error sequence defined by
(2.2)n = O
q(u , , ) -u ,=x , -~ ,+~ ; n = l , 2 ,
where the definition for eo follows from those of u0and q and
where the second equality follows from (2.1). The error is
written in this particular way with the quantized value
appearing first so that one can write q(un)= U, + E,, thereby
expressing the output of the binary quantizer as an additive
combination of the input and a noise term. In most treatments
of E A modulation the binary quantizer error sequence E,, is
assumed to be independent of the signal, uniformly distrib-
uted, and white. In fact, the noise is a deterministic function of
the input signal and system initial state and in principle it can
be derived exactly from the input sequence. One can try to
approximate the actual quantizer noise behavior by the action
of an additive input-independent noise source. A useful form
of approximation is to have the model exhibit the same long
term time average behavior as the true quantization noise,e.g., have the same long term mean, power, sample autocorre-
lation, and spectra. The white noise assumption does not in
general accomplish this goal, however. For example, it was
shown in [9] that even for a simple dc input, the sample
autocorrelation and spectra of the actual binary quantizer noise
sequence differ markedly from white noise: the spectra is
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GRAY et al.: SIGMA-DELTA MODULATION WITH SINUSOIDAL INPUTS
purely discrete and has components whose amplitudes and
frequencies depend on the input. It was, however, also shown
that the marginal time averages agree with the uniform
assumption for the dc input case.
Instead of assuming a solution to the above difference
equation for a given input signal, we recast the difference
equation into alternative forms that permit us to solve the
equation and to directly evaluate limiting sample averages,
e.g., the sample mean and power of the binary quantization
noise. Toward this end, it is convenient to consider instead the
normalized and shifted binary quantizer error sequence:
9 5 9
1 E , - 1 U, 1 X , - lY ; n = l , 2 , - . a . (2 . 3 )
, - 2 2 b 2 b 2 2 b
The following relations summarize how each sequence of
interest can be recovered from the yn and some of the useful
equalities relating the sequences:
(2 . 4 )
If one considers yn - 1/2 to be a noise term (it is a scaled
and shifted version of the quantizer noise), then (2.5) has the
interpretation of dither ing the input signal by this noise prior
to quantization since the integrator state U , is the input to thebinary quantizer. Thus, a Z A modulator can be interpreted as
a quantizer dithered by past quantization noise. Equation (2.6)has the interpretation that the binary quantizer output is the
input plus difference of two noise terms. Intuitively, the binary
quantizer noise is (discrete time) differentiated and added to
the input signal to form the binary quantizer ostput.
The following are two basic properties of the difference
equations. The first property follows easily from induction(the details are in the Appendix) and the second follows from
the first and (2.5) and (2.7). Property 1 is also physicallyimmediate from the operation of the feedback loop. We use the
usual definition of the half-open interval [c, 6) = { x : cI <dl
Property I: If uo = 0, then
U, E [ ~ , - ~ - b ,, _ ~ + b ) ;= l , 2 , . * - ( 2 . 7 )
and
U, E - 2 b , 2 b ); n = l , 2 , * e - . ( 2 . 8 )
Property 1 states that the integrator st ate always lies within a
range twice that of the input range. In additio?, the integrator
state must lie within a range equal to the irput range, but
centered at the previous input sample value.
Property 2: The sequence y, satisfies
O s y , < l ; n = 1 , 2 , ( 2.9 )
and hence
(2 . 1 0 )
The above properties can be viewed as a form of stability
property. The binary quantizer input never exceeds the
overload range of the binary quantizer and hence the magni-
tude of the binary quantizer error is uniformly bounded by b .We close this section with a difference equation for the y,
sequence in terms of the input sequence. Rewriting (2.6)yields
Y1= o
(2 . 1 1 )
Equation (2.11) is the basic difference equation to be solved.
IU . SOLUTION OF THE DIFFERENCEQUATION
We next develop an alternative recursive description for y,
and use it to express yn as a function of the input signal alone.
The result combined with (2.4)-(2.6) provides an extension of
[8, Theorem 11 from the special case of a dc input signal to a
general input sequence. First we introduce the notation (y ) for
the fractional part of y (or y mod 1). This is defined for all real
numbers y by the unique representationy = M + (y) where
LYJ s the greatest integer less than or equal t o y and 0 I y)
< 1 . The solution to the difference equation given below is
proved by induction (see the Appendix for details).
Property 3: The sequence yn satisfies the following
recursion withy , = 0:
and the sequence y, and the input sequence x, are related by
n = 1 , 2 , 3 , (3 . 2 )
where the sum is considered to be 0 if the lower index exceeds
the upper, that is if n = 1.Comment: The sequencey, is formed by a combination of
a simple affine operation with memory followed by a
memoryless nonlinearity. The normalized input sequence hasa bias of 1/2 added, the sum is integrated (discrete time), and
the fractional part of the result is taken. The first operation is
affine and not linear in the strict sense because of the addition
of the dc value. In summary,
Y n = s n) ( 3 . 3 )
where
s,=% (-+-)xk .k= O 2 2 b
( 3 . 4 )
This result holds for arbitrary input sequencesx, E [- b, b].
In the special case where x, = x , a dc or constant value,
then the property states that
( 3 . 5 )
which agrees with [8, Theorem 11 for the dc input case.
Combining the property with (2.4)-(2.6) yields the integrator
state sequence, error sequence, and binary quantizer se-
quence. We note in particular that the normalized binary
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960 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL 37 , NO 9, SEPTEMBER 1989
quantizer error sequence is given by
This exact representation for the normalized binary quantizer
error holds for an arbitrary input sequence bounded in [ - b,
b] and is the basis of the subsequent results for the particular
example of sinusoidal input sequences.
IV. TIMEAVERAGE OMENTS
In their classic works on random processes, Rice [17] and
Davenport and Root [18] introduced and developed a tech-
nique called the transform method for finding expectations orprobabilistic moments of the output of a system consisting of a
linear component followed by a memoryless nonlinearity and
having a random process with known moments as input. Their
approach provides a “formal” technique for the analogous
problem considered here of finding time average moments in a
discrete time system consisting of an affine operation followed
by a memoryless nonlinearity. The qualifier “formal” is used
because several limits must be interchanged in order to mimic
the Davenport and Root approach. T hese interchanges, how-
ever, do not immediately follow from known results when
dealing with time averages instead of ordinary integrals.
Hence, such interchanges must be proved correct, a nontrivial
technical detail which is treated in the Appendix.A primary goal of this paper is to evaluate the sample
average mean and power of the binary quantizer error
sequence, that is, to evaluate the limits such as
1 N
and
1 N
called the time average mean and energy (or sample mean and
power), respectively, and the sample autocorrelation
assuming for the moment that the limits exist (a fact we must
prove). If the {, were drawn from a stationary and ergodic
process with a uniform marginal density function f ( z )= 1 for
z E [- 1/2 , 1/21, then the sample mean and power would
with probability one be 0 and 1/12, respectively. If in addition
the process were white or memoryless, the sample autocorre-
lation would with probability one be simply r f (0 )= 1/12 and
rf(k) = 0 for k # 0. These values would be consistent with
the usual assumption that ln s uniform and white. Here,
however, the goal is to prove or disprove rather than assume
this sample average behavior. In addition, there is no question
of stationarity or ergodicity here because the entire system is
deterministic if the input sequence and initial integrator state
are both specified.
We again focus on the sequence y , for convenience. Sample
averages for yn will yield those for {, with a little algebra:
1M{rnl = i - M { Y n + ,1 (4.1)
To evaluate time average limits depending on yn ,which is a
memoryless nonlinear function of s,, as in (3.3)-(3.4), we
consider a discrete time analog to the nonlinear systems treated
in [18, Davenport and Root, ch. 131 which consist of a linear
filter followed by a memoryless nonlinearity:
Y = g ( S n )
where g(x ) = ( x ) and s, is given in (3.4). Since g is a well-
behaved periodic function with period 1 , it can be expanded in
a Fourier series
m
g ( x ) = C g ( l ) e 2 x j ‘ x (4.4a)
/= - m
where
g( )= 1 d x g ( x ) e - 2 x ” x (4.4b)
and the equality in (4.4a) holds almost everywhere. In the case
g(x ) = ( x ) he coefficients are
Combining preceding formulas gives
which provides an exponential expansion for yn . This is a
discrete time analog to (13-3) of Davenport and Root [181 with
a Fourier series replacing a Fourier transform.
Suppose now that we can interchange the sample average
operation and the infinite sum:
(4.7)This interchange must be verified for a particular application
before the formula can be taken as valid. T he quantity
I N
can be viewed as a simple average characteristic function.
Observe that
+(O)= 1.
Thus, if the limits of (4.7) exist and their order can be
interchanged, then the sample mean can be evaluated as
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96 1RA Y et al.: SIGMA-DELTA MODULATION WITH SINUSOIDAL INPUTS
and therefore
1 j 1
2a / # 0
M{y, ,}=-+- - @ ( I ) . (4.10)
A simple means of computing the sample power of the
sequence is to repeat the above derivation of (4.9) with the
memoryless nonlinearity g being replaced by h(x) = (x)’
which has Fourier coefficients
I = 01
(4.11)
* 1rO
h(1 )=
-+-2al 2a212’
Thus, if the limits can be interchanged as in (4.7); that is, if
(4.12)
then
The autocorrelation is found in a similar manner
r y (k )= M { yny n + k }
again assuming the existence of the limits and the validity of
their interchange. Equation (4.14) is a discrete time analog to
(13-40) of Davenport and Root [181, their “fundamental
equation of the transform method” for evaluating an autocor-
relation in terms of joint characteristic function. If we define
the sample joint characteristic function
(4.15)
In words, the relative frequency of the sequence sn places nomass at the endpoints of the unit interval.
B) The limiting characteristic function @ ( I ) defined by (4.8)
exists for all 1.
Then (4.9) and (4.13) are valid. Suppose that in addition the
following condition holds.
C) The limiting joint characteristic function of (4.15) exists.
Then (4.16) is valid.
Finally, a sufficient condition for A) to hold is the
following:A‘) Condition B) holds and lim,+- + ( I ) = 0.
This is as far as we can go without assuming a particular
input and evaluating the sample characteristic functions. Note
that if the limits defining the sample characteristic functions
exist and the marginal characteristic functions g o to 0, then the
preceding formulas provide exact values for the desired
sample averages.
V. SINUSOIDALNPUTS
Suppose now that the input sequence has the form
(5.1),,= a cos n$ T=a cos nw
where $ = 2 a f is the continuous time frequency, T he
sampling period of the continuous time signal, f , = 1/T is the
sampling frequency, w = 2 a f / f s s the angular frequency of
the discrete time input, and la1 I . We assume the If I I ,
some fixed maximum frequency (usually the Nyquist fre-
quency of a real data source such as speech). The oversam-
pling ratio is given by R = f s / 2 W , half the ratio of the
sampling frequency to the maximum frequency of the sinusoi-
dal input (the input “bandwidth”). Thus, w 5 2 a W / f , =a/R nd increasing the oversampling ratio corresponds to
decreasing the discrete time frequency. When we speak of
oversampled C A quantization we mean that R %- 1 and hence
that w 4 a.We shall require in the derivation that f / f , be an
irrational number so that the actual frequency and sampling
frequency are not rationally related. This is reasonable, for
example, if the input frequency is selected according to a
probability density function from the set of possible frequen-
cies [0, m. ith probability one the selected frequency will
be irrational. Furthermore, as discussed in [9], the assumption
of an irrational number can be considered as an approximation
to the case of a rational number with a large denominator
relatively prime to the numerator. We shall emphasize the case
of a full scale sinusoid, that is a = 6, but the basic results willbe developed without this assumption.
If x,,s given by (5.1), then from Gradshteyn and Ryzhik
[I91 (P. 30)
n- 7 -
then
and therefore
We close this section by giving a sufficient condition under
which the assumed limit interchanges are valid and hence the
preceding formulas can be used to evaluate the sample
average moments. The proof is given in the Appendix.
Define for any set B C [0,1) the indicator function le(x) as1 if
x E Band
0otherwise. n - 1
aProperty 4: Suppose that the following conditions hold.
A) Given any E > 0 there is a 6 > 0 such that if G = {r:O
1 x;
2 2bi = O
si n (nu-3 i)
s r < 6 o r l - - 6 < r < l} , then
n - 1 a
2 4b-- + - + a s i n (5.3)
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96 2 IEEE TRANSACTIONS O N COMMUNICATIONS, VOL. 31 , N O. 9, SEPTEMBER 1989
where we have defined
0
4 b sin -2
Thus, for @ ( l )defined in (4.8) we have that
1 N
( 5 . 4 )
This limit is evaluated using the well-known fact from ergodic
theory (due to H. Weyl) that for any integrable function h and
any irrational y we have for any y that
(See, e.g., [20]). In order to invoke this result we now assume
that W / T = fs/2f is irrational. The limit of (5.4) is then found
to be
I odd
o ( 2 ~ l a ) ; 1 even
@ ( I ) = [ j r / ( u / 2 b ) ~ 9 (5 .6)
where Jo is the Bessel function of order 0,
J o ( 2 T / a )= 5' du eJ2*/ain 2*".
Since Jo(0)= 1, @ ( O ) = 1, as it should by direct calculation.
By evaluating the limit we have proved that Condition B)
holds. Observe also that since Bessel functions go to zero as
their argument becomes large, @( / ) + 0 as I + 03. Thus,Condition A ' is satisfied which in turn implies Condition A).
Thus, we can apply Property 4 to conclude that the following
formula holds if the rightmost sum is well defined:
0
1 1 m 1 a
b-
2J o ( 4 a l a )sin T I- . (5.7)
From the linearity of sample averages and (3.6)
(5.10)
and in the general case we have for large oversampling ratiosand hence large a that
M { n } t . (5.11)
To find the second moment, w e combine (4.13), (5.7), and
Property 4 to obtain
(5.12)
Again invoking the linearity of sample averages and (2.7) we
have that
(5.13)
and hence
a
b
1 1 - 1M{l i}=-+- - o ( 4 ~ l a )O S T I - . (5.14)
12 4T2 / = ' 1 2
In the special case a = b we have
1 1 - 1M{{i}=-+- 1 - l ) ' Jo (4~h ) . (5.15)
12 4T2 / = ' I
This is approximately 1/12 when a is large. Observe,
however, that unlike the sample mean the approximation is not
exact when a = b.We now evaluate the sample joint characteristic function in
orde r to find the sample autocorrelation. Combining (4.15)
and (5.3),
1, (5 . 1 6 )j w ( i + /)(?I- ) e j 2 a a [ sin (nw - 3 / 2 ) w )+ / sin (nw- (3 /2 )w + k w ) ]
1k ( i , [)=,f{ ejr(i+/)(n-I)ej2,a[isin(nw-(3/2)w)i-Isin ( n w - ( 3 / 2 ) w + k w ) ]
(5.17)9
- M { e j m ( i + / ) ( n - l ) e j 2 n a [ i sin ( n w - ( 3 / 2 ) w ) + / c o s ( k w ) sin ( n w - ( 3 / 2 ) w ) + / sin ( k w ) cos (nw- (3 /2 )w) l-
Observe that if a = b, then sin (7rIa/b) is 0 and we have so that
exactly that M { y n } = 1/2! More generally, since the Bessel
functions vanish as their argument gets large, in the limit of ak(i , ) = e J H ( i + ' ) ( u / 2 b ) e j H ' k 4 k ( i ,). (5.18)- - -large oversampling 'atios (and hence and large a ) we
~ $ ~ ( i ,) is evaluated in a manner similar to that used for thehave ordinary sample characteristic function @ ( l ) to obtain
i + l odd
du e j2*a[ i sin ~ s u + /in ( 2 m + k o ) ] ; i + 1 even
(5.8) d'k(ir I ) = [M { y n l t j
0
consistent with a uniform asymptotic distribution assumption. (5.19)
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GRAY et al.: SIGMA-DELTA MODULATION WITH SINUSOIDAL INPUTS 9 6 3
In order to proceed further the above integral must be
evaluated. To do this we use the Jacobi-Anger formula
ejz sin 4 = i , ( 2 ) jm 4 (5.20)
where Jm s the Bessel function of order m . Substituting (5.20)
into (5.19) and using the orthonormality of complex exponen-
tials yields
m= - m
= 2 Jn(27rai) m(27ral)eJmkwn m
= e j m k w J - , ( 2 n a i ) J m ( 2 a a I ) . ( 5. 21 )m
Using the fact that J-,(z) = (- 1)"Jrn(z)we have that
q5k(i,I)=cj m k w ( - ) m J m ( 2 ? r a i ) J m ( 2 a a l ) . ( 5 .2 2 )
Equation (5.22) is the most useful expression for our pur-
poses, but an interesting and apparently simpler alternative
expression follows from Watson [21, p. 3591:
m
+ k ( i , 1 )= e J m ( k w - " ) J m ( 2 ~ a z ~ J m ( 2 n ( u l )m
= J o ( 2 a a J i 2 + 1 2 - 2 i i l cos k w ) . (5.23)
the advantage of (5.22) over (5.23) is the factoring of theterms dependent on i and I.
Combine (4.16), (5.22), (5.18), and invoke Property 4 to
obtain
m
r y ( k ) = C e j m k w ( - (2 ) e ja i (a /2b )Jm2 x a ) )m = m i , / : i + e v e n
(2 )ej*/ ( ' /2b)Jm2 n a )e j*Ik ) . (5.24)
The sum can be evaluated as the sum over all i and I for which
both are even plus the sum over all i and I for which both are
odd to obtain
m
where
c1 1 J o ( 4 7 ~ ~ 1 )- - _ i2 = I = , 21
c e ( m ) = --$-J m ( 4 a a I ) ( ;)
Jm (4a(u l )
sin a1- ;21
a = II 21
m=O
m even
m odd
(5.26)
= I
(5.27)
Note that c,(O) = M { y , } .Equations (5.25)-(5.27) give the sample autocorrelation
function of the sequence yn . An important fact about this
expression is that it can be written in the form
m
ry(k)= s I e j2akX / (5 .28)
where the hl can be considered as normalized frequencies in
[O, 1). The easiest means of indexing the frequencies and
spectral amplitudes is to consider the indexes I in (5.28) to
havetheformI = ( m , i ) ; m= 0 . . , - l ,O , 1 , , i = 1,2, and
I = - m
(5.30a)
(5.30b)
Equation (5.28) defines the extended Fourier series or the
Bohr-Fourier series of the sequence r y ( k ) . t is an extension of
a Fourier series because ry (k)need not be periodic for the
series to converge in an appropriate sense. In fact, if asequence has exponential expansion of the form (5.28), then it
is almost periodic in the sense of Bohr. (For a discussion of
almost periodic functions, see, e.g., [22]-[24] and the
discussion in [20] and the references therein.) For our
purposes, however, it is enough to observe that the sequence sl
can be interpreted as the power spectrum of y n since it is the
exponential transform of the autocorrelation function, that is,
the spectrum of y , is purely discrete and has amplitude s/ t the
frequency A l. This spectrum is extremely nonwhite since it isnot continuous and not flat. The output frequencies depend on
the input frequency w and comprise all harmonics of the input
frequency w as one might expect with a nonlinear device. It is
interesting to observe that not only are all harmonics of the
input frequency contained in the output signal, but also all
shifts of these harmonics by R (when computed in radians).
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9 6 4 IEEE TRANSACTIONS ON COMM UNICATIONS, VOL. 31 , NO. 9, SEPTEMBER 1989
In order to simplify the expressions and ease discussion, we
now assume that a = b, that is, that a full scale sinusoid is the
input. In this case, (5.26)-(5.27) reduce to
1
1 ; ; m=O
m even (5.31)
Jm(4aaf)( - 1 ) l ; m odd
21I = 1
Jm (27ra (2 f - 1))( - 1 ) l ; m even
[ ; m odd
(5.32)
Thus, in this case, we can simplify (5.28)-(5.30) and write
where
r
m=O
I - m even
m odd
(5 .34)
and
( ( m5-i)m even
A =
[ (m f - ) ;m o d d
(5.35)
The autocorrelation and spectrum for < then follow from
(4.3).
The above formulas were used to compute the Bohr-Fourier
power spectrum for the input frequencies used in the examples
in Fig. 2 and the results are depicted (by 0 ’ s ) along with the
FFT spectrum (the + ’s) in Fig. 4. In addition, the sample
Bohr-Fourier spectra for the sequences of Section I1 are
depicted (the * ’ s ) . It is important to note the differences
between the FFT and the sample Bohr-Fourier spectra. The
FFT is computed for a finite sample using uniformly spacedfrequencies. The Bohr-Fourier spectra is also an exponential
transform, but it uses the frequencies predicted by the theory
developed here and not the uniformly spaced frequencies. As
the sample sequence becomes longer, the uniformly spaced
frequencies should provide better and better approximations to
the frequencies predicted by the theory and the ergodic
theorem implies that the amplitude at these frequencies should
similarly approach those predicted by the theory. For finite
samples, however, the FFT and the sample Bohr-Fourier
spectra will only approximate the theoretically predicted
values. The approximations depicted in Fig. 4 re reasonable
considering the sequences use only 1024 samples for the
computation. Note the sensitivity of amplitude to frequency.
The Bohr-Fourier spectra are usually closer to the predicted
spectra than are the FFT spectra and the FFT spectral
amplitudes can differ from both the others by a significantamount even though the frequencies are close to the true Bohr-
Fourier frequencies.
VI. COMMENTS
An exact derivation of the sample moments and powerspectrum for a single-loop sigma-delta modulator has been
given and the formulas evaluated the important special case of
a sinusoidal input. This provides a new example of a case
where spectral analysis can be accomplished for a highly
nonlinear system, an analysis made possible by the fact that the
system was shown to have the special structure of an affine
operation followed by a memoryless nonlinearity. This struc-
ture permits an extension of the classical transform method of
Rice and of Davenport and Root to be applied. As in the
analysis of the same system for a dc input 191, the marginal
moments behave like a uniformly distributed sequence of
random variables, but the spectrum is purely discrete and it is
not flat. The result are complicated in that they involve singlesums of weighted Bessel functions, bu t previous developments
have resulted in more complicated formulas (triple sums of
Bessel functions in Iwersen [ l l ] ) . While the results for dc
inputs reported in [9] had been predicted using continuous
time approximation arguments by Candy and Benjamin [6],the results reported here have not previously appeared to our
knowledge.
A potential shortcoming of the analysis presented here is
that its difficulty might preclude its extension to more
interesting sigma-delta modulators involving multiple loops or
interpolation filters. Although the difference equation solution
may not help in the analysis of all multiloop systems,
preliminary results suggest that it can provide an exact analysis
of ideal cascade or multistage sigma-delta quantizers with the
architecture proposed by Uchimara et al. [25] and Matsuya et
al. [26]. This work is reported in [27] and [28]. Extensions to
higher order sigma-delta systems with nonbinary quantizers
have been reported by He, Buzo, Kuhlman. [29]
APPENDIX
A . Proof of Property I
The second relation follows from the first since x, E [- b,b]. The proof follows by induction as follows. For n = 1 we
have ICI = (xo - b ) E [xo - b, xo + 6). Assume that (2.7)holds for n - 1 and hence that
- bI , - 2- bI , - 1 <X, 2 + bI b. (A . 1)
If U, 2 0 then this implies that
u , = u , - 1 +x,- 1- bZX , - 1- b
and that
U,= U,- 1 +x,- 1 - b< 2 b +x,- 1-b =x,- I + b,
proving (2.7) for the case of nonnegative U,-]. f IC,-, 0,
thenU, = U,- 1 +x,- 1 + b<x,_ 1 +b
and
U,= U,- 1+X , - 1 + b Z - b +x,- 1 +b=x,- 1- b,
completing the proof.
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GRAY et al.':SIGMA-DELTA MODULATION WITH SINUSOIDAL INPUTS
0.008
0.006
965
-
-
Bohr-Fourier S p t r u m : N= 024. f= 0.0690700.012
0.004-
0.002
Bohr-Fourier Spectrum:N= 12, f= 0.0836030.012
0.008
;
- + e 0.002
P
+-
0.008
0.006
-
~
t
00 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 I
frequency (Hz)spectral energy= 0.080958
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
frequency(Hz)spectral energy= 0.080464
Bohr-Fourier Spectrum:N= 024,f= 0.071948
O ' O 1 2 J
Bohr-Fourier Specbum: N= 024,f= 0.3004460.025
0.020.011 :
0.015 -
0.01 -1
0.005 -0.002o'l+
2 8
0 A0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45
fiquency (Hz)spectral energy= 0.095492
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
frequency (Hz)spectral energy= 0.077062
Fig. 4. Predicted spectra, sample Bohr-Fourier spectra, and FFT's of
examples of Fig. 2.
B. Proof of Property 3From (2.11) nd the definition of q( - )
(1/2 + x0/2b). rom (3.1)
I 1 xn--3\
( y n - l + 2 + f ; ifyn-l<--- x r l - 2
2 2 b We have, however, that for any real numbers a and b
( ( a )+b )= ( a + b) ( A . )
since removing the integer part of cy does not affect the
fractional part of the sum. Thus, (A.3) ecomes
First suppose that
1 x n - 2y n - l + - + - < l .
2 2 b
Since the left-hand side is nonnegative [from Property 1 and
(2.3)] nd strictly less than 1, its fractional part is itself, that
is, ( Y , - ~ + 112 + ~ , - ~ / 2 b )yn-l + 1/2 + x,-2/2b,
proving the (3.1) nder the assumption of (A.2).Next supposethat (A.2) oes not hold, that is, that y n - l + 112 + xn-2/2b2 1. Since it is also true from Property 1 that yn- + 1/2 +x,-2/2b < 1 + b/2b + 112 = 2, we must have that ( y n - l
+ ~ , _ ~ / 2 b112,which proves (3.1).+ 1/2 + xn-2/2b) = yn-1 + ~,-2/2b 1/2 - 1 = y n - l
Equation (3.2) s proved by induction. We have that y2 =
C. Proof of Property 4
Define the partial Fourier Sum
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1 M
G(L, n)-- G ( L ,n )Mn=l
simple discontinuities at the endpoints, the convergence of the
above limit is uniform for ( sn ) n any closed subset of [O, 11not
Thus, there is an N, such that if N 2 N,, hen
5 6 . (A.8)
Since the sums G(L,n ) converge uniformly for (s,) E Gf,
= [ a m , 1 - a m ] , here is an L, such that for all L 2 L, we
have for all n such that (s,) E GC,
I N
+- (g(Sn)-G(L, n ) ) l G r n ( ( s n ) )
JG(L,n)-g(sn)I<E,- (A.5)
Since both bounds (A.3) and (AS) hold uniformly in n, we
have for all N 2 N , and L 1 , that
proving (4.9). Note that by identifying the two limits, the fact
that g(s,) is bound above by 1 implies that M {g ( s , ) } is finite
5 ( y + 2)E,+ E + (y+2)Em,
5 € ( 2 ( 7 + 2 ) + ),
which proves, that N-l g (sn ) is also a Cauchy sequence
and hence must have a limit, M { g ( s n ) } . his fact with (A.6)
and (A.7) imply that L > L,
IM{g(sn))-M{ G(L, n>>l (Y+~)E,*A.9)
Thus letting m + 00 hence L, .+ 00 yields
and therefore
L
n = lv (A. 10)
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GRAY et al. : SIGMA-DELTA MODULATION WITH SINUSOIDAL INPUTS 967
Thus, for N h N,,,e have analogous to (A.6) that
1 N
The remainder of the proof of (4.13) follows as in the
The fact that A‘) implies (A) follows from Katznelson [32],previous proof.
p. 42. (See also Lyons [33] for a discussion of this fact.)
ACKNOWLEDGMENT
The first author gratefully acknowledges the helpful com-ments of Dr. J . Candy of AT&T Bell Labs., Holmdel. We arealso indebted to Dr. S . Shitz of AT&T Bell Labs., Murray
Hill, NJ, for suggesting the transform method as an alternativeto the original proofs, which were based on Weyl’s criterion.The new proofs are both simpler and more general.
131
141
151
1101
1141
REFERENCES
H. ho se and Y. Yasuda, “A unity bit coding method by negativefeedback,” Proc. ZEEE, vol. 51, pp. 1524-1535, Nov. 1963.J. C. Candy, “A use of limit cycle oscillations to obtain robust analog-to-digital converters,” ZEEE Trans. Commun., vol. COM-22, pp.298-305, Mar. 1974.-, “A use of double integration in sigma delta modulation,” IEEETrans. Commun., vol. COM-33, pp. 249-258, Mar. 1985.-, “Decimation for sigma delta modulation,” IEEE Trans.Commun., vol. COM-34, pp. 72-76, Jan. 1986.J. C. Candy, Y. C. Ching, and D. S. Alexander, “Using triangularlyweighted interpolation to get 13-bit PCM from a EA modulator,”ZEEE Trans. Commun. , pp. 1268-1275, Nov. 1976.J. C. Candy and 0. . Benjamin, “The structure of quantization noisefrom Sigma-Delta modulation,” ZEEE Trans. Commun., vol. COM-29, pp. 1316-1323, Sept. 1981.G. R. Ritchie, “Higher order interpolative analog to digital con-verters,” Ph.D. dissertation, Univ. Pennsylvania, 1977.R. M. Gray, “Oversampled sigma-delta modulation,” IEEE Trans.Commun., vol. COM-35, pp. 481-489, Apr. 1987.-, “Spectral analysis of quantization noise in a single-loop sigma-delta modulator with dc input,” IEEE Trans. Commun., vol. 37, pp.588-599, June 1989.S . H. Ardalan and J. J. Paulos, “An analysis of nonlinear behavior indelta-sigma modulators,” IEEE Trans. Circuits Syst., vol. CAS-34,pp. 593-603, June 1987.J. E. Iwersen, “Calculated quantizing noise of single-integration delta-modulation coders,” Bell Syst. Tech. J. , pp. 2359-2389, Sept. 1969.M. Vidyasagar, Nonlinear System Analysis. Englewood, Cliffs,NJ: Prentice Hall, 1978.A. Gelbe and W . E. Vander Velde, Multiple-Znput DescribingFunctions and Non-linear Systems Design. New York: McGraw-Hill, 1968.D. P. Atherton, Nonlinear Control Engineering. New York: VanNostrand Theinhold, 1982.
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D. P. Atherton, Stability of Nonlinear Systems. New York: Wiley,1981.A. R. Bergens and R. L. Franks, “Justification of the describingfunction method,’’ SIA MJ . Contr., vol. 9, pp. 568-589, 1971.S . 0. Rice, “Mathematical analysis of random noise,” in SelectedPapers on Noise and Stochastic Processes, N. Wax, Ed. NewYork: Dover, 1954, pp. 133-294. (Reprinted from Bell Syst. Tech. J. ,vols. 23 and 24).
1181 W. B. Davenport, Jr., and W. L. Root, An Introduction to theTheory of Random Signals an d Noise. New York: McGraw-Hill,
1958.1191 I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, an dProducts. New Yor k Academic, 1965.
[20] K. Petersen, Ergodic Theory. Cambridge, England: CambridgeUniversity Press, 1983.
1211 G. N. Watson, A Treatise on the Theory of Bessel Functions.Cambridge, England: Cambridge University Press, 1980, second ed.
1221 S . Bochner, “Beitrage zur Theorie der fastperiodischen Funktionen I,11,” Math. Ann., vol. 96, pp. 119-147, 1927.
1231 S. Bochner and J . von Neumann, “Almost periodic functions ingroups, II,” Trans. Amer. Math. Soc., vol. 37, pp. 21-50, 1935.
[24] H. Bohr, Almost Periodic Functions. New York: Chelsea, 1947.Translation by H. Cohn.
1251 K. Uchimura, T. Hayashi, T. Kimura, and A. Iwata, “VLSI-A to Dand D to A converters with multi-stage noise shaping modulators,” inProc. 1986ZCASSP, Tokyo, Japan, 1986, pp. 1545-1548.Y. Matsuya, K. Uchimura, A. Iwata, T. Kobayashi, and M. Ishikawa,“A 16b oversampling conversion technology using triple integrat ionnoise shaping,” in Proc. 1987IEEE Int. Solid-stat e Circuits Conf.,Feb. 1987, pp. 48-49.
P.W.
Wong and R. M. Gray, “Two-stage sigma-delta modulation,”IEEE Trans. Acoust., Speech Signal Processing, to be published.W . Chou, P. W . Wong, and R. M. Gray, “Multi-stage sigma-deltamodulation,” ZEEE Trans Znform. Theory, to be published.N.He, A. Buzo, and F. Kuhlmann, “Multi-loop sigma-delta quantiza-tion,” submitted for publication.H. S. Carslaw, An Introduction to the Theory of Fourier’s Seriesand Integrals.
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1311 A. Zygmund, Tigonometrical Series. New York: Dover, 1955.[32] I. Katznelson, An Introduct ion to Harmonic Analysis. New York
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[33] R. Lyons, “Fourier-Stieltjes coefficients and asymptotic distributionmodulo 1,” Annals Math., vol. 122, pp. 155-170, 1985.
*Robert M. Gray (S’68-M’69-SM’77-F’80) wasborn in San Diego, CA, on November 1, 1943. Hereceived the B.S. and M.S. degrees from M.I.T. in1966 and the Ph.D. degree. from U.S.C. in 1969, allin electrical engineering.
Since 1969 he has been with Stanford University,where his is currently a Professor of ElectricalEngineering. His research interest are the theoryand design of data compression and classificationsystems, oversampled analog-to-digital conversion,speech and image coding and recognition, and
ergodic and information theory.Dr. Gray was a member of the Board of Governors of the IEEE Information
Theory Group (1974-1980, 1985-1988). He was an Associate Editor of theIEEE TRANSACTIONS ON INFORMATION THEORY from September 1977through October 1980 and Editor of that journal from October 1980 throughSeptember 1983. He is currently as Associate Editor of Mathematics ofControl, Signals, an d Systems. He was an IEEE delegate to the Joint IEEE/U.S.S.R. Workshop in Information Theory in Moscow in 1975. He is thecoauthor, with L. D. Davisson, of Random Processes, Prentice Hall, 1986,and the author of Probabili ty, Random Processes, an d Ergodic Properties,Springer-Verlag, 1988. He was corecipient with L. D. Davisson of the 1976IEEE Information Theory Group Paper Award and corecipient with AndresBuzo, A. H. Gray, Jr., and J. D.Markel of the 1983 IEEE ASSP SeniorAward. He was a fellow of the Japan Society for the Promotion of Science(1981) and the John Simon Guggenheim Memorial Foundation (1981-1982).In 1984 he was awarded on IEEE Centenial medal. He is a member of SigmaXi, Ete Kappa Nu, SIAM, IMS, AAAS, AMS, and the Societe des Ingenieurs
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9 6 8 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 31 , NO . 9, SEPTEMBER 1989
et Scientifiques de FiLicense (KB6XQ).
‘ance. He holds an Advanced Class Amateur Radio
*Wu Chou (S’87) received the M.S. degree in
electrical engineering and the M.S. degree instatistics in 1987 and 1988, respectively, fromStandord University. He is currently a Ph.D.candidate at Stanford University.
His main research interests are in the areas ofdigital signal processing, nonlinear effects in digitalfilters, data compression, vector quantization, andimage processing.
Ping W. Wong (S’84) was born in Hong Kong onOctober 9 , 1954. He received the BSc. (Eng.)degree from the Universityof Hong Kong in 1977,and the M.S.E.E. degree from the University ofMichigan-Dearborn in 1985.
He is currently a student at the InformationSystems Laboratory, Stanford University, Stanford,CA. From 1977 to 1982, he was an electricalengineer at Coronet Industries Limited, HongKong, where he was involved in designing radio
frequency circults. From 1981 to 1983, he workedon automatic train control systems at Mass Transit Railway Corporation,Hong Kong. His research interest is in digital signal processing, quantizationeffects, communication and oversampled analog-to-digital conversion.