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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



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



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



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



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



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)



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)



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).



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.



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


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


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



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)



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.

[15]

[16]

1171

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.

1261

[27]

[28]

1291

[30]New York: Dover, 1950.

1311 A. Zygmund, Tigonometrical Series. New York: Dover, 1955.[32] I. Katznelson, An Introduct ion to Harmonic Analysis. New York

Dover, 1976.

[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



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.

sigma delta exact

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