Uncertainty of mean value and variance o

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IEEE Instrumentation and Measurement
Technology Conference
Anchorage, Alaska, USA, 21-23 May, 2002

Uncertainty of Mean Value and Variance Obtained from Quantized Data
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

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Abstract This paper aaiiresses the problem of recovering the mean value
and the variance of an input signalfrom the ou@utahla of a granularquantizer. Particular emphasis is dedkated to the measurement uncertainly with
and without dither. Known geneml results are applied to random Gaussian
and uniformly distributed signals and to deterministic sinusoidal signals.

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 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
A
~ ( x Q / x=)~ (-xX 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

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Index Terms - Quantkation noise d e l , quanfization e m r , dikher, analoglo-digifalconversion, measurement uncertainty.

I. INTRODUCTION

The effect of a deterministic quantizer with quantization steps
q is usually modeled by an additive, signal-independentnoise,
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 :
A

~

X

=
Q E{XQ}

(1)

Such an approximation simplifies the analysis, but it is applicable 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. Section I1 resumes the state of the art of the statistical analysis
of the quantization and Section I11 presents the mathematical
frameworkwith some examples.

11. STATISTICALANALYSIS OF QUANTIZATION:
STATE OF THE ART
(XQ)

e

k=+m

as(w - 2 n k / q ) r ( w - 2 n k / q ) (3)

k=-

00

and, respectively,

k=+m

WU)

( ~ ~ ( 2 ~ rk( iwq-) 2 7 ~ 7 )

5

(4)

k=-m

where (Pz( w ) is the characteristic function of the input signal
x and r ( w ) 5 ~ ~ ~ - y ( x ) e j w [3],
z d ;[4],
c [9].

and, respectively,

Let x

(PxQ ( U )

represent the input (output, respectively) signal of
A

the quantizer, let v = X Q - x represent the quantization error, and let ~ ( x Q / xrepresent
)

the channel profile [9], i.e. the

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. Rectangularprofile: the deterministic quantizer

The rectangular profile represents a deterministic, uniform
quantization law [101 where the input x is compared to a certain number of transition levels which partition the input range
in disjoint intervals, so that only one digital code X Q corresponds 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. However, because of the ADC's internal noise, the deterministiclaw
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
A
r ( w ) = sinc(wq/2) = sin(wq/2)/(wq/2), it can be shown

0-7803-7218-2/02/$10.00' 2002 IEEE

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that, for a symmetric distribution of the input signal, the mean
and the mean square values of v are given by

where U;

A

E { v 2 } - (E{v})' and uxv= E{zv} - pxE{v}.

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

and, respectively,by

111.

where R{s} ( S { s } ) is the real (respectively, the imaginary)
part of s.
Similarly, E { ~ Q=} E { z }
E { v 2 } + 2 E { z v } , where

and &,(U)

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

4daX(w)/dw.

A

By indicating with px the mean value of z, and assuming s =
z - px,it can be shown that [7]

w { ax

(T)}

= COS

(2nk$)

as

(y)

ANALYSIS OF THE UNCERTAINTY

In general epT and eu2 can not be c o m t e d , because of their
dependenceon 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 systematiccontributionto 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 considereduniformly distributed between zero and the maximum.

m(y))' dpx) represent the mean value (the variance, respectively) of a generic functiony = y(px). It can be easily demonstrated that
m(e,,) = 0 7
(14)

(9)

and that

and

mku;:)= 4 e p , ) .
(16)
The expression of .(e,,:) is much more complicated but, for
the cases of interest, it can be approximated by

and that,

8

{ 6, (y )}

(y )-

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= cos ( 2 7 r k F )

68

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.

where Q S ( w ) is the characteristicfunction 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
A

ell, = jlx - pz = E{v}

and, respectively,by

(12)

If @?(2nk/q) 1, .(ep2) can be approximated by

Similarly, if Q z ( k ) 1, where Qs(lc)
a8(27rk/q)- ( 2 ~ k / qa8(2nk/q),
)
v(e+) becomes

Finally, m(.)and U ( . ) have to be combined as d
in order to obtain the type B uncertainty U B ( . ) [131.
Let consider some practical cases.

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Fig. 1. Type B uncertainty in the measurementof the mean value. of a
Gaussian distribution.

Fig. 2. Systematic contributions to the measurementuncertainty of 02 for a
Gaussiannoise.

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A. Gaussian noise

The characteristic function of a random Gaussian noise with
zero mean value and variance U: is

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

(20)

s G (U)= e - O . 5 0 2 ~ ~ .

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

and, respectively,by

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

m,and the approximatevalue obtainedfrom
(21). The maximum bias error is plotted in the same figure:

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u ~ ( p ~ )

notice that the ratio between the maximum value and the uncertaity u g ( p z )is equal to &for ux > 0.25q and tends to fi
only when uxtends to zero.
h

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 approximatelygiven by "(e,?)
M
&(U:)

+ dm,
and that, for ux < 0.3q, this expression is

1

2

3

4

more accurate than those reported in the cited literature.
B. Uniform noise

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

The characteristic function of the uniformly distributed noise

The asymptotic envelopes of the type B uncertainty of
is
approximatelyue(px) 5
q2/J87r4D2 12q2, as

~ W M+

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0

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Fig. 5.

Fig. 4. Systematic contributions to the measwemat uflceftainty of
U: = D2/3
for the uniformlydistributed 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
for D > 412.

F
w(e

2)

is given by

1 0 y

B Uncertainty in the measmment of the mean value of a
deterministic sinewave.

"

'

I

" "

I

"

' ' I

" "

I

"

'

q2/(R2fi)

C. Sinewave

A sinusoidal signal with amplitude A is characterizedby the
characteristicfunction
@ss =

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 bandlimited, i.e. its characteristicfunction does not vanishes for
Iw( 2 2 ~ / q E (E > 0) and, therefore, the quantizationnoise
model cannot be applied. The proposed model, approximates
the variance of the error e,, as

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
and the variance of the error e,; as
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,
becomes
the characteristicfunction of the input signal, Q8(u),
Q 8 ( w )= Q U ( w )Gd(w), where U and d account, respectively,
for the original signal and for the additive dither contribution.

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where J1(.) is the first-order Bessel function of the first kind.

Even in this case (18) and (19) allows determiningwith 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 m a x ( u B ( p z ) )
q2/(J2Aq.rr4+12q2), while that of
qm/n2.

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It is known that an uniform dither with a peak-to-peak amplitude equal to an integer number of quantizationsteps nulls the
expected value of total quantizationerror E{v}, but it cannot
render E { z v }independentof pz. Recently, it has been proved
that, in order to make E{v}, E { v 2 } and E { m } independent

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of the input, the dither signal must have a triangular probability density function of peak-to-peak amplitude equal to 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.

IV. CONCLUSIONS

A simple mathematical framework for estimating the systematic contributionto the measurementuncertainty of mean value
and variance has been provided. It has been verified for some
typical distributions.

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The reported equations allows quantifying the optimum value
of the dither for a given signal, without referring to nonpractical hypotheses. Let consider as an example the sine wave
of amplitude A with superimposeda nonsubtractivedither.

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-

References

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Organizationfor 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.

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and, respectively,

+

where U; = U$ A2/2.

Similarly, it is possible to determine u g { p z } and v(e+) for
the uniform and triangular random dither. From the analytical
expressionof the maximum values of ue(pZ}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

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2

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Fig. 7. Contour plot of the type B uncertainty in the measurement of the
mean value pz of a d i t h d sinewave.

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