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lEEE TRANSACTIONS ON INSTRUMENTATION A ND MEAS UREME NT. VOL. 38. NO . 3. JUNE 198918

Validity of Uniform Quantization Error Model for

Sinusoidal Signals Without and With Dither

Abstract-A technique is presented for determining the probabilit!

density function (PDF) nd variance of the quantization error of a si-

nusoidal signal applied to a uniform quant izer, namel!, an ideal A / D

converter. Th e results are the basis for determining the validit! of the

uniform quantization noise model for this class of signals.

When dither is employed it tends to decorrelate quantization error

and the input signal. Effect of added uniform dither on the PD F of the

quantizer input is investigated, as d ell as the PD F of the quantization

error corresponding to the dithered sinusoid. The results prove the

validity of the uniform quantization model in this case.

I . I N T R O D U C T I O N

MPLITUDE quantization has been extensively in-A estigated in the past four decades. Widrow [ ] con-

structed the probability density function (PDF) of quan-tization noise from the PD F of the quantizer input. He

also [2] evaluated quantitatively the distortion resulting

from rough quantization. and made a statis tical analysis

of amplitude quantized sampled data systems. More re-

cently, Sr ipad and Snyder [3] gave a necessary and suffi-

cient condition for quantization errors to b e uniform and

white. Since quantization noise constitutes a major source

of errors in instrumentation, especially with A/D con-

verters of I O bits or less , it is important to develop m oreaccurate quantization error models . Since the most com-

mon signal is the sinusoidal on e, a major objective of this

paper is to investigate the PDF and variance of the cor-

responding quantization error in terms of the amplitude of

the sinusoid.

Dithering is an effective tool for reducing the effects ofamplitude quantization. Schuchman [4] realized the con-

ditions for a dither signal so that the quantizer noise be-

comes independent of the input s ignal. Vanderkooy and

Lipshitz [ 5 ] used dither to resolve audio signals smaller

than the quantizing step. The Japanese paper by Yama-

saki [6] illustrated the effects of large amplitude dither onthe spectrum of the quantizer output. Z ames and Schney-

d or [7] showed that the behavior of a nonlinear system

employing dither depends on the P DF of the dither s ignal.

Lotto and Paglia [8] showed that dithering improves both

differential and integral linearities of A/D converters .

Again , due to the importance of the sinusoidal signal in

instrumentation and measurement, another objective of

Manuscript received August 5 . 1988: revised January I . 1989.The authors are with the Electrical Engineering Depar tment. University

o f Lowell . Lowell , MA 01854.

IEEE Log Number 8926903.

this paper is to investigate the effect of different ampli-

tudes of uniform dither on t h e P D F of the dithered sinu-

soid at the quantizer input. The PDF of the quantization

error , in the presence of dither , is also obtained.

11 P D F A N D V A R I A N C EF S I N U S O I D A LI G N A L

Q U A N T I Z A T I O NR R O R

For the sinusoidal s ignal

x = A sin w t

A is the amplitude, w is the radian frequency, and x lies

in the dynamic range of the quantizer (bipolar A/D con -

verter). x is regarded by the quantizer as a random vari-

able. From (A-IO) the PDF of x is given by

( 1 )

1f r ( x ) = T J m 7 - A I I . ( 2 )

F o r e x a mp le , th e P D F o f x for two different s inusoids is

shown in Fig. 1 . A is a multiple of the quantizing step A ,

thus A = 15.5 on the f igure means A = 15.5A. hrough-

out the paper the quantizer input, x, is assumed to be a

sequence of statistically independent random variables

with PDF given by (2) . However, x is actually obtained

by sampling a s inusoidal function. I t should be noted that

these two methods give similar results only when the sam -

pling frequency is not near to a harmonic or subharmonic

of the sinusoid.

The P DF of the quantiza tion e r ror e is [3]

( 3 )

LO, otherwise

where C $ r ( U ) is the characteristic function of x and given

by

m

~ J U )= 1 ~ ( x )ex p ( j . x )h. (4 )-m

S i n c e f , ( x ) o f (2 ) is an even function, then

00 I 8-9456/89/0600-07 18$01 OO O 1989 I EEE

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WA G D Y A N D N G . S I N U S O I D A L S IG N A L S W I T H O U T A N D W I T H D I T H E R 719

L

6.

7.

6.

II I I

i k l Q 5

fxoo 5. ; I l i I I3. I l l l i I

X1G-2 4 . I! I \ I / l I I I

I I l l I 1 1I:: 1 I I k=i21.51 i/

OLJ.5 -1.0-0.5

6.0 0.5:.O

1. 5X Xl62

Fig. I . PDF o f tw o example s inusoids.

Using Fourier transform tables [9], it can be shown that

&(U? = J o ( A u ) ( 6 )

where J O ( z ) s a Bessel function of zeroth order. By com-

bining (3) and (6 ) , and normalizing A and e with respect

to A (which is equivalent to letting A = 1 ), then

h(e)= I + C ~ ~ ( 2 7 r n ~ )ex p ( - j 2 a n e ) ,I1 #0

( 7 )1 1_ -2 - < - . 2

Tosimplify this expression, we consider the Bessel func-tion of order U , given by [9]

where I is the Gamma function, and ( m ! ) s factorial in .

Consequently

which indicates that .lo(7rnA) is proportional to ( r ~ ) * ' ~ ,

and hence, an even function of n . Thus (7) can b e rewrit-

ten asW

h(e)= 1 +2 C ~ ~ ( 2 7 r n ~ )co s (27rne)n = I

( 10)1 1-_2 - - 2

Using (10) the PD F of e is shown in Fig. 2 fo r A = 1 .5 ,

3 . 5 , 7 . 5 , a n d 15.5. Computations revealed inflection

points at I e I = 0.475 and 0.485 . How ever , i t is obvious

that increasing A flattens the P DF of e and makes i t closer

to the rectangular-shaped quantization error model found

in most of the relevant literature.

To determine the variance 0; of e we utilize the fact

t h a t f , ( e ) o f ( I O ) is an even function, and hence, has a

zero mean, thus

U: =

I / 2

- I / 2 11 = 1e2de +2 5 J0(27rnA)

. i l l 2 e 2 . co s (27rne)de . ( 1 1 )- 1 / 2

1 J

e XIO-~

O b . 4. -3. -2. -1 . 0. 1. 2. 3. 4 5.

Fig. 2. PDF o f quantization error for s inusoids using a new method based

on Beasel functions.

Fig. 3 . Variance of quantization error versus s inusoid amplitudes

Using a handbook of mathematical tables and formulas

[ l o ] , it can be shown that

e 2 - co s (27rne) de = ( - 1 ) " / 2 7 r 2 n 2 . ( 1 2 )K*From (1 1) an d ( 1 2 ) it follows that:

2 1 1 cc ( - 1 ) "a , =- +~C - * Jo(27rnA). (13)

12 w n = l n 2

The variat ion of 0: versus A is shown in Fig. 3 .  It is ob-

vious that as A increases, IJ; approaches a normalized

value of 1/ 1 2 . Given an error bound on U: we can easily

use (13) to determine the minimum amplitude of a sinu-

soidal signal in terms of the A/D converter quantizing

step, A . Also, the propagation of quantization noise power

in subsequent hardware stages, o r software processing al-gori thms would be much more accurate for small sinu-

soids if (13) is used as opposed to the well-known value

of A2/12.

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720 I E E E T R A N S A C T I O N S O N I N S T R UM E N T AT I O N A N D M E A S U R E M E N T . VOL. 38. NO . 3. J U N E 19x9

111. EFFECT F D I T H E R I N GN QUANTIZERN P U T N D

OUTPUTN CASE F A S I N U S O I D x L O ~ID CONVERTER

ii- y i e

The effect of changing c on the PD F of the quantizer input

is shown in Fig. 6 for the case A = 1 5 .5 A , for example.

It is obvious that as c increases, f\ ( y ) becomes less non-

linear and more flat; an advantage of larger amplitude

dither.

When the dither amplitude is greater than the sinusoidal

signal amplitude, i .e. , c >A , it can be shown that (14)

yields:

The effect of changing c on the PD F of the quantizer input

is shown i n Fig. 7, for the case A = 0 . 5 A . for example.

Again, A .( y ) becomes m ore flat as c increases.

Le t us' now obtain the PDF of the quantization error in

the presence of dither. To determinef,( e ) using the same

technique that led to ( l o ) , which is a closed form solu-

tion, would be rather difficult. Instead, we resort to Wid-

row's method [11 to derive the PDF of quantization noisevia an additive linear process. J;. ( e ) s the sum of the dis-

tributions of noise corresponding to the constituents that

are added to get f,.( ) , the PDF of the quantizer input.This can be formulated as

-'. -1. 0.

y x101

Fig. 6. PDF of the quantizer input for different dither amplitudes( c <A ) .

Y

Fig . 7 . PDF of the quantizer input for different dither amplitudes( A < c ) .

where 2 k + 1 is the number of quantization slots that canrepresent y .

With no ditherf , .( y ) s given by ( 2 ) and f e t e ) , calcu-

lated via (17) , is shown in Fig. 8  f o r A = 1 .5 , 3.5, 7 . 5 ,

an d 15.5. T he P D F of e is obviously not flat-topped since

f, y ) does not satisfy Widrow's quantization theorem [2],or the weaker condit ion obtained by Sripad and Snyder

[3].f,( e ) of Fig. 8  is monotonic for 1 e 1 >0, and is little

different from Fig. 2 due to the difference in the me thods.

When dither is present ,&,( y ) s given by ( 1 5 ) o r (16),

thus

k = ( A + c - $)/A.

k- A A

& ( e ) = n = C _ k & . ( n ~ e ) , 5 e 5 -2 2

(17)It is obvious that if 2 ( A + c ) is an odd number of A's ,

then y is represented by all the 2 k + 1 quantization slots,

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WA G D Y A N D N G : S IN U S O ID A L S IG N A L S WIT H O U T A N D WIT H D IT H E R 72

6.I I I ~

+

f d e )

2

0

-5. -& -3. Q. -1. 0. 4 . 2. 3. 4. 5.

e MO-1

Fig 8 PDF o f quan t i zat ion e r ro r fo r mu so i d s u s ing Wid row' s method

e x :c - !

Although dithering is a known concept, there are st i l l

some gaps in the l iterature on the subject . Equations (15)

and (16) give m ore insight into the effect of uniform d ither

on the quantizer input. The computer simulations based

on (17) truly verify that uniform dithering, of proper mag-

nitude, renders the uniform q uantization error model per-

fectly suitable for any sinuso id regardless o f its amp li-

tude.

A P P E N D I X

D E T E R MI N I N GHE P D F OF A S I G N A L

The determination of the PD F of a signal is known by

many researchers. However, it is worth presenting the

PDF general formula along with a formal proof and an

example application. To avoid generali t ies, we assume

that the equation

x = s ( 4 ('4-1 1

has three roots as in Fig. 10. The roots are denoted by t i ,

i ' = 1 , 2 , 3 , i . e . ,

Fig . 9. PD F of quantization error for dithered sinusoids.Our objective is to find the set of values t , such that x I

otherwise the highest quantization slots on both sides of

the quantizer characteristics (bipolar A/D converter) willbe partially filled. Several cases were investigated withthe following normalized A an d c: 1) A = 2 , 4 , 8, 16 with

C = 1 / 2 , 2 ) A = 1 - 1 / 2 , 2 . 2 , 3 - 1 / 2 , 7 - 1 / 2 , 1 5 - 1 /2 w ith

C = 1, and 3 ) A = 3- 1 / 2 , 7 - 1 / 2 , 15 - 1 / 2 wi th C = 2 .

In these cases we consistently obtained a f lat- topped PD F

for the quantization error, as shown in Fig. 9.   For the

cases: 4) A = 1.75, C = 0 .25 , and 5) A = 2 .25 , C =

0 . 7 5 , f , ( e ) s not flat-topped even tho ugh it is finite at 1 e I= 0.5. Cases 1)-5) indicate that for a f lat- toppedf,(e),

the peak-to-peak dither magnitude should be an integral

multiple, M , of the quantization slot , A, i .e . ,

AC = M -

2

This indicates that proper uniform dithering al lows the

well-known quantization error uniform model to be al-

ways valid.

I V . C O N C L U S I O N S

It is intuitively known that as the amplitude of a sinu-

soid increases the uniform quantization error model be-

come s more valid. Equation (10) is a closed-form formula

which quantifies this intuitive observation. It is also

known that the quantization noise variance A 2/12 is more

suitable for sinusoids when amplitudes are sufficientlylarge. Equation (13) is another closed-form formu la which

accurately quantifies noise variance for any value of the

signal amplitude. This al lows very accurate analysis and

design for a wide variety of applications, especially those

involving very small sinusoids.

g ( t ) I +dx , and the probability of this set in an ar-

bitrary duration T . As we see from Fig . 10 this set con-

sists of the following three intervals: ti I I i +d t i ,

i = 1 , 2, 3 where d t , >0 an d dt 3 >0, bu t d t2 <0. It

follows that:

P ( x I I +dr) = P ( t , I I l +d t , )

2+ P ( t 2 I I 2 +dt2)

+ P ( t 3 I I 3 +d t , ) .

( A - 3 1

Thus

dtl dt

T T T( x I I + dx) =-+ + ( A - 4)

and since [111

P ( x I I +dx) = f , ( ~ )d x ( A - 5)

where f , ( x ) is the PDF of x, then from (A-4) and (A-5)

we deduce the following general theorem:

(A - 6)

where n is the number of real roots of the inverse function

t = $ ( x ) derived from (A-1).

As an example, consider the sinusoid:

x = A sin wt, 0 I t I 7r ( A - 7 )

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72 2I E E E T R A N S A C T I O N S O N I N S T R U M E NT A T I O N A N D M E A S U R E M E N T . V OL

3X . NO 3. J U N E 19x9

1

Fig. 10. The signal .'i= g ( ) n the duration T .

Denoting wt by 8 , then

1(for both 8 ranges).

( A - 9 1

Using (A-6)- (A-9) it follows that:

- A I I . ( A - 1 0 )1

fr(X) =TJm'

REFERENCES

[I] . Widrow, " A study of rough amplitude quantization by m eans ofNyquist sam pling theory." IR E Trans. Circuit Theory. vo l . CT-4 ,

pp. 266-276 , Dec . 1956 .[2 ] -, "Statist ical analysis of amplitude quantized sampled- data sys-tems," A f E E Trans. (Ap p l . Ind usr . ) , vol. 81. pp. 555-568, Jan. 1961.

131 A . B. Sripad and D. L. Snyder. " A necessary and sufficient condition

for quantization errors to be uniform and white." fEEE Truns.Acousr., S p e e c h . Sigt iul Proc.essitig, vo l . ASSP-25 , pp . 442-448. Oct.

1977 .[4] L . Schuchman . "Dither s ignals and their etfects o n quantization

noise." fEEE Traris. Cotnrnun. T e c h n o / . , vol. COM;12. pp . 162-165. 1964.

[SI 1. Vanderkooy and S . P. Lipshitz. "Resolution helow the least s ig-

nificant bit in digital systems with dither, ' ' J . Au dio €rig. S o c . , vol.32. no. 3 . p p . 106-113. Mar. 1984.

[6 ] Y . Yamasaki. "The application o f large amplitude dither to the quan-tization of wide range audio signals ." J . Acorcstic S oc . J q ~ n n , ol .

39. no. 7 . pp . 452-462. July 1983.17) G . Zames and N . A. Shneydor. "D ither in nonlinear systems," I EEE

Trcitis. . . l ~ / o ~ t i t ~ t .oirrr.. vol. AC-2 I. p. 660-667. Oct. 1976.

[SI I . D. Lotto and G. E . Paglia. "Dithering improves A I D converterlinearity." / E € € Trutis . fmrrrot i . Me n s . . vol. 1M-35. pp . 170-177.

June 1986.191 F . Oberhett inger. Tobelleti ;iir Fourier Trons fo r tmt ion . Berlin ,

German): Springer-Verlag. 1957.

1101 R . S . Burington. H ~ t ~ d b ~ df Murhet~i trriculTuhles und Formulas.3rd ed. . Sandusky. OH: Handbook Publishers , 1949.

[ 1 1 1 A . Papoulis . Probobiliry, Ro,idoni Voriuhlev . arid Stoc/7crsric Pro-

cc'.s.se.\. 2nd ed. New York: McGraw-Hill . 1984.