Uncertainty of mean value and variance o

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IEEE Instrumentation and Measurement Technology Conference

Anchorage, Alaska,

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USA, 21-23 May, 2002

Uncertainty of Mean Value and Variance Obtained from Quantized Data

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Giovanni Chiorboli

Dipartimento di Ingegneria dell'hformazione, University of Parma

Parco Area delle Scienze 181/A, Parma PR, 1-43100, ITALY

giovanni.chiorboli @unipr.it

Abstract

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-

This paper aaiiresses the problem of recovering the mean value

and the variance of an input signal from the ou@ut ahla

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of a granular quan-

tizer. Particular emphasis is

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dedkated to the measurement uncertainly with

and without dither. Known geneml results are applied to random Gaussian

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and uniformly distributed signals and to deterministic sinusoidal signals.

Index Terms

-

Quantkation noise d e l , quanfization e m r , dikher, analog-

lo-digifal conversion, measurement uncertainty.

I. INTRODUCTION

The effect of a deterministic quantizer with quantization steps

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q is usually modeled by an additive, signal-independent noise,

uniformly distributed in ( - q / 2 , q / 2 ) , with mean square value

of q2/12 and white spectrum [l] - [8]. Therefore, the mean

value

ps and the variance 0:of the

non

quantized input signal

x

are obtained from the mean value and the variance of the quantized data X Q :

(1)

A

~ X Q= E { X Q }

and, respectively,

Such an approximation simplifies the analysis, but it is appli- cable only if the quantization step is sufficiently small when compared to the standard deviation of the input signal. When this does not apply, the quantization error becomes apparently correlated with the input signal, since the distribution of the quantized output depends on the input offset position within the quantization cell [7], [12], so that the meaning of any measurement appears questionable.

In this paper it is proposed a general mathematical model for estimating the systematic contribution to the uncertainty. Sec- tion I1 resumes the state of the art of the statistical analysis of the quantization and Section I11 presents the mathematical framework with some examples.

11. STATISTICAL ANALYSIS OF QUANTIZATION:

STATE OF THE ART

Let

x

( X Q ) represent the input (output, respectively) signal of

the quantizer, let v = X Q

-

x

represent the quantization er-

ror, and let ~ ( x Q / x ) represent the channel profile [9], i.e. the

A

probability of X Q conditioned by x. By assuming P ( X Q / X )

independent of the channel position

x,

it can be shown that the quantization step, defined as q

e

S_+,"p(xQ/x)dx, does not depend on XQ. If, moreover, the profile is symmetrical with respect to the center of the channel, it is possible to write

~ ( x Q / x ) =

~ ( x

- X Q ) = ~ ( X Q -

x),

where y(x)

= p ( O / x ) . In this case, the characteristic functions of the output code and of

the quantization error, i.e. E{ej""Q} and E{ejwU}, are

A

k=+m

( P x Q ( U )

e

as(w

- 2 n k / q ) r ( w

-

2 n k / q ) (3)

k=- 00

and, respectively,

k = + m

W U )

5

( ~ ~ ( 2 ~ k i q )

r ( w

- 2 7 ~ 7 )

(4)

k=-m

where (Pz ( w ) is the characteristic function of the input signal

x and r ( w )

5

~ ~ ~ - y ( x ) e j w z d ; c [3], [4], [9].

It is well known that, for a given channel profile, the moments of X Q and v can be evaluated from the characteristic functions (3) and

(4)

U11.

A. Rectangular profile: the deterministic quantizer

The rectangular profile represents a deterministic, uniform quantization law [ 101 where the input

x

is compared to a cer- tain number of transition levels which partition the input range in disjoint intervals, so that only one digital code X Q corre-

sponds to a given input voltage range. In the particular case of a rounding quantizer, the quantization law is given by

( 5 )

where

1x1

is the greatest integer less than or equal to

x.

How- ever, because of the ADC's internal noise, the deterministic law represents only an ideal approximation, but it is convenient to ideally assume a noiseless quantizer with a noisy input signal.

Since the transformation of the rectangular channel profile is

(2)

that, for a symmetric distribution of the input signal, the mean

and the mean square values of

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v

are

given by

and, respectively, by

where R{s}

( S { s } ) is the real (respectively, the imaginary)

part of s.

Similarly,

E { ~ Q } = E { z }

+

E { v } and E { z $ } = E { z 2 }

+

E { v 2 }

+

2 E { z v } , where

and &,(U)

4

daX(w)/dw.

By indicating with

px the mean value of z, and assuming s =

z

-

px, it can be shown that [7]

A

w

{

ax

(T)

}

= COS ( 2 n k $ )

as

( y )

(9)

and

and that,

8

{

6,

(

y )

}

= cos ( 2 7 r k F )

68

(

y )

-

where Q S ( w )

is the characteristic function of s.

The above equations point out a well known dependence of the statistics of the quantized data on the position within the quantization channel of the mean value of the input distribution. When the quantization noise model represented by (1)

and (2) is applied, for instance, to the analysis of Gaussian input distributions with standard deviations sufficiently small, the estimates of px and ux can be affected by a relevant error since

aP(w)

does not vanishes for (wI 2 27r/q. The Sheppard's

approximations (1) and (2) are affected by bias errors, that are given by

(12)

A

ell, = jlx

-

pz = E { v }

A

where U; E { v 2 } -

(E{v})'

and uxv = E { z v }

-

pxE{v}.

The maximum values of the deviations ep= and eu; were inves- tigated in [7], [ 121, which derived a general expression of these deviations for sinusoidal, uniformly distributed and Gaussian signals.

111.

ANALYSIS

OF THE UNCERTAINTY

In general epT and eu2 can not be c o m t e d , because of their dependence on the unknown values of ux and pZ, so that they contribute to the measurement uncertainty together with the statistical fluctuations of pas and U&.

In the following paragraphs, the systematic contribution to the uncertainty will be directly calculated from the above equations rather than from the maximum values of the bias errors, since ell= and e,,2 can not be usually considered uniformly distributed between zero and the maximum.

m(y))' dpx) represent the mean value (the variance, respectively) of a generic function y = y(px). It can be easily demon- strated that

m(e,,) = 0 7 (14)

and that

The expression of .(e,,:) is much more complicated but, for the cases of interest, it can be approximated by

mku;:) = 4 e p , ) . ( 1 6 )

Although the above equations already represent a close-form expression of the bias error contribution, it is more practical to

use simpler but reasonable expressions.

If @ ? ( 2 n k / q )

<<

k2G,2(2.rr/q) V k

>

1, .(ep2) can be approximated by

Similarly, if Q z ( k )

<<

Ic4Q:(l) V k

>

1, where

Qs(lc)

a8(27rk/q)

-

( 2 ~ k / q ) a8(2nk/q), v(e+) becomes

Finally, m(.) and U ( . ) have to be combined as

d

m

in order to obtain the type B uncertainty U B ( . ) [131.

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Fig. 1. Type B uncertainty in the measurement of the

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mean value. of a

Gaussian distribution.

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A. Gaussian noise

The characteristic function of a random Gaussian noise with

zero mean value and variance U:

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is

s G (U) =

e - O . 5 0 2 ~ ~ .

(20)

Therefore,

v

( ep5 ) and v ( euz ) are approximately given by

Fig. 1 shows the exact value of the type B uncertainty of

E ,

u ~ ( p ~ )

m,

and the approximate value obtained from (21). The maximum bias error is plotted in the same figure: notice that the ratio between the maximum value and the un- certaity u g ( p z ) is equal to &for ux

>

0.25q and tends to

fi

only when ux tends to zero.

The two contributions to the type B uncertainty of uz are shown in fig. 2. It is interesting to observe that the maximum value of the bias error is approximately given by "(e,?) M

&(U:)

+

dm,

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and that, for ux

<

0.3q, this expression is

more accurate than those reported in the cited literature.

B. Uniform noise

The characteristic function of the uniformly distributed noise

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h

Fig. 2. Systematic contributions to the measurement uncertainty of

02

for a Gaussian noise.

that disappears much more slowly than for the Gaussian noise. The variance of the bias error e,, can be approximated by

1 2 3 4

Fig. 3. lLpe B uncertainty in the measurement of the mean value of an uniform distribution.

The asymptotic envelopes of the type B uncertainty of is

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l o 0 g

Fig. 4. Systematic contributions to

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the measwemat

uflceftainty of

U: = D2/3 for the

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uniformly distributed noise.

shown in fig. 3 and, obviously, the maximum value of the error

elm is given by

f i

ug(pCLa).

The standard deviation and the

mean value of e,; are shown in fig. 4, where it is apparent that

the asymptotic envelope of w(e 2 ) is given by q 2 / ( R 2 f i )

for D

>

412.

C.

Sinewave

A sinusoidal signal with amplitude

A

is characterized by the characteristic function

F

@ s s = Jo(wA) (26)

where Jo(.) is the zero-order Bessel function of the first kind. Even more than the previous signals, the sinewave is not band-

limited, i.e. its characteristic function does not vanishes for

Iw(

2

2 ~ / q - E (E

>

0) and, therefore, the quantization noise model cannot be applied. The proposed model, approximates the variance of the error e,, as

and the variance of the error e,; as

where

J1

(.) is the first-order Bessel function of the first kind. Even in this case (18) and (19) allows determining with a good agreement the two components of the systematic contribution to the uncertainty, as shown in figures

5

and 6. The upper

bound of the type B uncertainty of pCLa is

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m a x ( u B ( p z ) )

q2/(J2Aq.rr4+12q2), while that of

q m / n 2 .

Fig. 5. B Uncertainty in the measmment of the mean value of a

deterministic sinewave.

1 0 y " ' I " " I " ' ' I " " I " '

Fig. 6. Systematic contributions to the measurement u n c e h t y of

U: = A 2 / 2 for the sinewave.

D.

Dithered sine wave

An other interesting consequence of the proposed equations

(18) and (19) is that they can be also applied to nonsubtractive dithered quantizers. When dither is added to the input signal, the characteristic function of the input signal, Q8(u), becomes

Q 8 ( w ) = Q U ( w ) Gd(w), where U and d account, respectively,

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of the input, the dither signal must have a triangular probabil-

ity density function of peak-to-peak amplitude equal to

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29 (i.e.

@ ~ T ( w ) = sinc2(Aw))

[14].

However, in practical cases, the

peak-to-peak amplitude of the dither is only approximatively known and the transition levels of the quantizer are not equally spaced, so that making the peak-to-peak amplitude of the dither equal to the nominal value of q can be uneffective.

The reported equations allows quantifying the optimum value of the dither for a given signal, without referring to non-

practical hypotheses. Let consider as

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an example the sine wave

of amplitude A with superimposed a nonsubtractive dither.

Gaussian dither added to the signal, changes the characteristic function of the input signal from (26) to

so that the maximum value of u ~ { p , } and of ,/.(eu:) be- and, respectively,

where

U; = U $

+

A 2 / 2 .

Similarly, it is possible to determine u g { p z } and v(e+) for the uniform and triangular random dither. From the analytical expression of the maximum values of u e ( p Z } and v(eu:), it is easy to estimate the amount of dither that is necessary to

obtain

a given uncertainty. Fig. 7 shows, as an example, u ~ { p , } as a function of A and 0, for the Gaussian dither.

1oor----7

.-..-

.._...

'

1

'.

I

'.

!

I !

! I

1

'

' 2

'

' 3

'

' 4

'

' 5

IV. CONCLUSIONS

A simple mathematical framework for estimating the systematic contribution to the measurement uncertainty of mean value and variance has been provided. It has been verified for some typical distributions.

References

W.F. Sheppard. On the calculation of the most probable values of fkquencyconstants, for data arranged according to equidistant divisions

of a scale. Proc. of the London Mathematical Sociery,

zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA

29, Part 2:353-380,

1898.

W.R. Bennett. Spectra of quantized signals. Bell Syst. Tech. 1,27:446- 472,1948.

B. Widrow. Statistical analysis of amplitude quantized sampled data systems. Trans. Am,: Inst. Elec. Eng., Pt. 11, Applications and Industry,

79:555-568, January 1961.

G.A. KO" Hybridamputer techniques for measnring statistics from

quantized data. Simulation, 7~229-239,1965.

A. B. Sripad and D. L. Snyder. A necessary and sufii cient condition for quantization errors to be uniform and white. IEEE Trans. Acoust., Speech, Signal Processing, ASSP-25:442-448, October 1977. R. M. Gray. Quantization noise spectra. IEEE Trans. Information The-

ory, 36:1220-1244, November 1990.

I. Koll6I. Bias of Mean Value and Mean Square Value Measurements Based on Quantized Data. IEEE Trans. Instrumentation and Measure- ment, 43:733-739, October 1994.

B. Widrow, I. Koll6I and M.C. Liu. Statistical theory of quantiza- tion. IEEE Tmns. Instrumentation andMeasurement, 45353-361, April 1996.

I De. Lotto, S. Osnaghi. On random signal quantization

zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA

and its effects

in nuclear physics measuremen& Nuclear Instruments and Methods,

56:157-159, 1967.

Methods and draft standardr for the dynamic characterisation and test-

ing of Analogue to Digital converters. European project DYNAD - SMT4-CT98-2214, http://www.fe.up.pt/Ilsm/dynad, Draft standard 3.4, July 2001.

A. Papoulis. Probability. Random Variables and Stochastic Processes.

New York McGraw-Hill, 1965.

P. Carbone, D. Petri. Mean value and variance of noisy quantized data.

Measurement, 23: 13 1-144, 1998.

I S 0 Guide to expression of Uncertainty in Measurement. International

Organization for Stanahniizarion, ISO, Ginevra, CH, 1993.

R. A. Wannamaker, S. P. Lipshitz, J. Vanderkcmy, J. N. Wright A theory of nonsubtractive dither. IEEE Trans. on Signal Pmcessing, 48:49%516, Feb. u)oo.

A [LSB]

Fig. 7. Contour plot of the type B uncertainty in the measurement of the

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