A GRAPHICAL REVIEW OF NOISE-INSTABILITY

CHARA

CTERIZATION IN ELECTRONIC SYSTEMS

Juan Jos

´

e Gonz

´

alez de la Rosa

Univ. C

´

adiz. Electronics Area, EPSA. Ram

´

on Puyol S/N. E-11202-Algeciras-Spain

Isidro Lloret Galiana

Univ. C

´

adiz. Dpt. Computer Science, EPSA. Ram

´

on Puyol S/N. E-11202-Algeciras-Spain

Carlos Garc

´

ıa Puntonet

Univ. Granada. Dept. ATC, ESII. Periodista Daniel Saucedo. E-18071-Granada-Spain

V

´

ıctor Pallar

´

es L

´

opez

Univ. C

´

ordoba. Dept. Electronics, Campus Rabanales. A. Einstein. C-2. E-14071. C

´

ordoba. Spain

Keywords:

Allan deviation, enveloping curves, frequency stability, GPS-receiver, noise processes, traceable calibration.

Abstract:

A thorough study of the noise processes characterization is made with simulated data by means of our non-

classical estimators. Individual and hybrid noise sequences, previously generated by seed functions, have

been used to obtain a set of characterization graphs identifying the noise type by mean of the enveloping

curve. It is also shown the case of a hidden noise. An real test situation is presented which involves a traceable

characterization via a GPS receiver.

1 INTRODUCTION

Noise affects short-term stability of clocks and oscil-

lators of systems in a broad range of technical ﬁelds

like communications, instrumentation and medicine.

The relationship between the causes and the differ-

ent noise processes is continuously being reviewed

(Vig, 2001). Random oscillator instabilities are linked

to the environment (temperature changes, vibrations,

shock and electromagnetic ﬁelds) and to the guts of

the electronics components (thermal noise, internal ir-

regularities in the xtal and the semiconductor devices,

and surface imperfections).

Numerous works have provided the users with use-

ful analytical tools with the aim of getting the com-

pleteness of the calibration procedure (Allan, 1987),

(Rutman and Walls, 1991). Multivariance analysis al-

lows getting higher measurement accuracy (Vernotte,

1993) in estimating the uncertainties in order to better

distinguish the different types of noises.

In this work we have summarized the subject to

have an uniﬁed practical frame, understandable in

multidisciplinary engineering projects. An analysis

of the noise processes is performed to provide an

easy-going review of short term instability character-

ization, by analyzing the slopes of the AVAR

1

and

1

Allan variance or two-sample Allan variance

MVAR

2

in the log-log curves. Noise time series have

been simulated and estimators of the variances have

been programmed with the aim of having a thorough

vision of the time-domain slopes when compared to

former works: (Howe et al., 1999), (Allan, 1987),

(Rutman and Walls, 1991), (Vernotte, 1993), (Vig,

2001).

The paper is structured as follows: in Section 2

we review the oscillators independent noise processes

and the methods used to identify them; Section 3

shows the simulation results concerning time-domain

stability characterization. The concept of enveloping

curve is introduced herein withe the objective of char-

acterizing the effects of different simultaneous noise

processes. A real case of short-term characteriza-

tion procedure and its associated conclusions are pre-

sented in Section 4. In this section, we consider a

precision function generator as device under test in a

GPS-traceable characterization.

2 CLASSICAL NOISE MODELS

It is a customary situation to deal with unperfect sig-

nals which contain additive noise. The instantaneous

2

Modiﬁed Allan variance

310

José González de la Rosa J., Lloret Galiana I., García Puntonet C. and Pallarés López V. (2005).

A GRAPHICAL REVIEW OF NOISE-INSTABILITY CHARACTERIZATION IN ELECTRONIC SYSTEMS.

In Proceedings of the Second International Conference on Informatics in Control, Automation and Robotics - Signal Processing, Systems Modeling and

Control, pages 310-315

DOI: 10.5220/0001154903100315

Copyright

c

SciTePress

output voltage of an oscillator can be expressed as:

v(t) = [V

o

+ ε(t)] sin [2πν

0

t + φ(t)] , (1)

where V

o

is the nominal peak voltage amplitude, ε(t)

is the deviation from the nominal amplitude, ν

0

is the

name-plate frequency, and φ(t) is the phase deviation

from the ideal phase 2πν

0

t. Changes in the peak value

of the signal is the amplitude instability. Fluctuations

in the zero crossings of the voltage is the phase insta-

bility. The so-called frequency instability is depicted

by the ﬂuctuations in the period of the voltage. The

situation is depicted in ﬁgure 1

3

.

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

Time (s)

Voltage (V)

Amplitude instability

Phase instability

Frequency instability

Figure 1: Simulated types of instabilities in a 25 Hz si-

nusoidal output with additive noise. The noise process has

a power spectral density proportional to the inverse of the

frequency (ﬂicker phase modulation).

The impacts of oscillator noise and the causes of

short term instabilities have been described in many

research works and tutorials like (Howe et al., 1999),

(Vig, 2001) and (Sullivan et al., 1990). The short-term

stability measures most frequently found on oscillator

speciﬁcation sheets is the two-sample deviation, also

called Allan deviation, σ

2

y

(τ).

Classical variance in non-stationary noise

processes don’t converge to concrete values. It

diverges for some noise processes, such as random

walk, i.e., the variance increases with increasing

number of data points. This is the reason whereby

non-classical statistics are used to characterize short

term instability.

AVAR and MVAR have proven their adequacy

in characterizing frequency phase and instabilities.

These easy-to-compute variances converge for all

noise processes observed in precision frequency

sources, have a straightforward relationship to power

3

A similar example was provided by Prof. Eva Ferre-

Pikal (Univ. of Wyoming) and used by John R. Vig in (Vig,

2001).

law spectral density of noise processes, and are faster

and more accurate in than the FFT (Lesage and Ayi,

1984).

The estimates of AVAR and MVAR for a given cali-

bration time τ for a m-data series of phase differences,

x, are given by equations 2 and 3, (Greenhall, 1988):

AV AR ≡ σ

2

y

(τ, m) =

1

2(m − 1)

m

X

j=2

y

j

− y

j−1

2

=

1

2τ

2

(m − 1)

m

X

j=2

∆

2

τ

x(jτ )

2

(2)

MV AR ≡

1

2τ

2

h∆

2

τ

xi

2

, (3)

where the bar over x denotes the average in the

time interval τ (averaging time), and ∆

2

τ

x = x

i+2

−

2x

i+1

+ x

i

, is the so called second difference of

x. The fractional frequency deviation is the relative

phase difference in an interval τ. It is deﬁned by equa-

tion 4:

y =

1

τ

Z

t

t−τ

y(s)ds =

x(t) − x(t − τ )

τ

=

∆

τ

x(t)

τ

.

(4)

Non-classical statistics estimators, deﬁned above, in

equations 2 and 3, for non-stationary series charac-

terization, give an average dispersion of the fractional

frequency deviation due to the noise processes cou-

pled to the oscillator. As a consequence time do-

main instability (two-sample variance) is related to

the noise spectral density via (Rutman and Walls,

1991):

σ

2

y

(τ) =

2

(πν

0

τ)

2

Z

f

h

0

S

φ

(f)sin

4

(πfτ)df, (5)

where ν

0

is the carrier frequency and f is the Fourier

frequency (the variable), and f

h

is the band-width of

the measurement system. S

φ

(f) is the spectral den-

sity of phase deviations, which is in turn related to

the spectral density of fractional frequency deviations

by(Rutman and Walls, 1991):

S

φ

(f) =

ν

2

0

f

2

S

y

(f), (6)

The classical power-law noise model is a sum of the

ﬁve common spectral densities. The model can be de-

scribed by the one-sided phase spectral density S

φ

(f)

via (IEEE, 1988), (Greenhall, 1988):

S

φ

(f) =

ν

2

0

f

2

2

X

α=−2

h

α

f

α

= ν

2

0

4

X

β=0

h

β

f

β

, (7)

for 0 ≤ f ≤ f

h

. Where, again, f

h

is the high-

frequency cut-off of the measurement system (the

A GRAPHICAL REVIEW OF NOISE-INSTABILITY CHARACTERIZATION IN ELECTRONIC SYSTEMS

311

Table 1: Noise processes characterized by the time and fre-

quency domain slopes. Up to bottom: random walk fre-

quency modulation, ﬂicker frequency modulation, white

frequency modulation, ﬂicker phase modulation, white

phase modulation.

AVAR MVAR

S

y

(f) S

φ

(f) σ

y

(τ) ∼ |τ|

µ

2

σ

y

(τ) ∼ |τ|

µ

′

α β = α − 2

µ

2

µ

′

−2 −4 0.5 1 (0.5)

−1 −3 0 0 (0)

0 −2 −0.5 −1 (−0.5)

1 −1 −1 −2 (−1)

2 0 −1 −3 (−1.5)

band-width); h

α

and h

β

are constants which rep-

resent, respectively, the independent characteristic

models of oscillator frequency and phase noise (Al-

lan, 1987), (IEEE, 1988), (Greenhall, 1988).

For integer values (the most common case) we have

the following approximate expression:

σ

y

(τ) ∼ τ

µ/2

, (8)

where µ = −α − 1, for −3 ≤ α ≤ 1; and µ ≈ −2

for α ≥ 1. In the case of the modiﬁed Allan variance,

the time-domain instability can be approximated via:

Modσ

y

(τ) ∼ τ

µ

′

(9)

Hereinafter we use expressions 8 and 9 for analyzing

noise in these work.

3 TIME DOMAIN STABILITY

CHARACTERIZATION CURVES

Equations 8 and 9 are used to make the graphical

representation of σ

y

(τ) vs. τ, and lets us infer the

noise processes which causes frequency instability by

means of measuring the slope in a log-log graph (Rut-

man and Walls, 1991), (Wei, 1997). These functional

characteristics of the independent noise processes are

widely used in modelling frequency instability of os-

cillators. Table 1 shows the experimental criteria

adopted in the main references. In the second column

or MVAR we have picked up two different criteria ac-

cording to the references (Rutman and Walls, 1991)

and (Lesage and Ayi, 1984), respectively. We have

kept the notation in the works (Rutman and Walls,

1991) and (Lesage and Ayi, 1984) for µ/2 and µ

′

,

respectively.

The ﬁve noise processes have been modelled and

VAR and MVAR have been calculated. Hereinafter

we show the simulation results of the time-series and

their associated VAR and MVAR graphs. From this

simulations we adopt the criteria depicted in the sec-

ond column of MVAR in table 1. Figures 2-6 show

the simulations results. Each data sequence contains

4096 points for a time resolution of τ = 10

−4

s. Al-

lan deviation curves have been depicted for averaging

times τ = n × τ

0

, with n ∈ [1, 500].

0.02 0.04 0.06 0.08 0.1

-0.05

-0.04

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

0.04

0.05

Time (s)

Amplitude (V)

White phase modulation

10

-4

10

-3

10

-2

10

-2

10

-1

10

0

10

1

10

2

log(tau)

log(sigma)

AVAR and MVAR; beta=0

Figure 2: Characterization of a noise process corresponding

to β = 0.

0.02 0.04 0.06 0.08 0.1

-2

-1

0

1

2

3

x 10

-3

Time (s)

Amplitude (V)

Flicker phase mod.

10

-4

10

-3

10

-2

10

-2

10

-1

10

0

log(tau)

log(sigma)

AVAR and MVAR; beta=-1

Figure 3: Characterization of a noise process corresponding

to β = −1.

In many practical situations two or more noise

processes simultaneously affect clocks performance.

In this cases instability of the device under test is ex-

plained away through the behaviour of the upper en-

veloping curve. If the individual variance curves cross

each other, it is possible to see the slope changes in the

variance curve, for a time-series which includes sev-

eral types of noise(Vernotte, 1993). This situation is

shown in ﬁgures 7 and 8.

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312

0.1 0.2 0.3 0.4

-6

-4

-2

0

2

4

6

8

x 10

-4

Time (s)

Amplitude (V)

White freq. mod.

10

-4

10

-3

10

-2

10

-2

10

-1

log(tau)

log(sigma)

AVAR and MVAR; beta=-2

Figure 4: Characterization of a noise process corresponding

to β = −2.

0.1 0.2 0.3 0.4

-3

-2

-1

0

1

2

x 10

-4

Time (s)

Amplitude (V)

Flicker freq. mod.

10

-4

10

-3

10

-2

10

-2.4

10

-2.3

10

-2.2

log(tau)

log(sigma)

AVAR and MVAR; beta=-3

Figure 5: Characterization of a noise process corresponding

to β = −3.

In ﬁgure 7, the individual variance curves cross. So

the enveloping curve characterizes the short-term in-

stability. By the contrary, in ﬁgure 8 the β = 0 noise

processes has a variance greater than the β = −4 per-

turbation. In this case the enveloping curve is the ﬁrst

(upper) AVAR curve.

4 EXPERIMENTAL RESULTS

AND CONCLUSIONS

A high resolution function generator is chosen as de-

vice under test. It is set up to deliver a 1.1 Hz TTL

signal. The experimental arrangement is depicted in

0.1 0.2 0.3 0.4

-1

0

1

x 10

-4

Time (s)

Amplitude (V)

Random walk freq. mod.

10

-4

10

-3

10

-2

10

-3

log(tau)

log(sigma)

AVAR and MVAR; beta=-4

Figure 6: Characterization of a noise process corresponding

to β = −4.

10

-4

10

-3

10

-2

10

-4

10

-3

10

-2

log(tau)

log(sigma)

AVAR

10

-4

10

-3

10

-2

10

-6

10

-5

10

-4

10

-3

10

-2

log(tau)

MVAR

Figure 7: Noise processes corresponding to β = 0 and β =

−4. Situation of changing slope.

ﬁgure 9. The measurement system comprises a TIC

4

,

a GPS receiver and the frequency source under test.

These instruments have been connected via GPIB to

the computer. Data points are captured every 1 s.

Time interval counters (TICs) and GPS receivers

are widely used in traceable frequency calibrations.

A transfer standard receives a signal that has a cesium

oscillator as source (Lombardi, 1999). This signal de-

livers a cesium derived frequency to the user, who is

beneﬁted as not all laboratories can afford a cesium

(Lombardi, 1996). These instruments differ in speci-

ﬁcations and details regarding the time base, the main

gate and the counting assembly. Furthermore, manu-

facturers tend to omit the conditions under these spec-

4

Time Interval Counter

A GRAPHICAL REVIEW OF NOISE-INSTABILITY CHARACTERIZATION IN ELECTRONIC SYSTEMS

313

10

-4

10

-3

10

-2

10

-3

10

-2

10

-1

10

0

10

1

10

2

10

3

10

4

log(tau)

log(sigma)

AVAR

10

-4

10

-3

10

-2

10

-3

10

-2

10

-1

10

0

10

1

10

2

10

3

10

4

log(tau)

log(sigma)

MVAR

Figure 8: Noise processes corresponding to β = 0 and β =

−4. The upper noise process is the enveloping curve.

iﬁcations have been provided or measured.

GPIB

GPS Receiver

TIC

UTC 1 pps

CHA

CHB

10 MHz

Ext. Ref.

Input

oscil

lator

50

Ω

output

1.1 Hz

Figure 9: Experimental arrangement.

Figure 10 shows the signals involved in the mea-

surement process. Each measurement cycle corre-

sponds to 1 s. The bottom graph corresponds to

the instantaneous phase-deviation series, which com-

prises m = 898 data. These data are the result of ﬁl-

tering the spiky time-series of phase differences, and

are used to perform the calibration. These data are

supposed to be corrupted by white noise, with a rec-

tangular probability density function. This is corrob-

orated later by means of AVAR and MVAR.

The ratio of the classical variance (VAR) to the Al-

lan variance (AVAR) provides a primary test for white

noise. This quantity (0.672) is less than 1 + 1/

√

m ≃

1.033; thus it is probably safe to assume that the data

set is dominated by white noise, and the classical sta-

tistical approach can safely be used. Failure of the

test does not necessarily indicate the presence of non-

white noise (Fluke, 1994).

A slope test (based in AVAR and MVAR curves)

has been developed to conﬁrm the presence of white

noise. AVAR and MVAR curves are depicted in ﬁgure

11.

Measures of the slopes over the log-log graphs in

100 200 300 400 500 600 700 800 900 1000

0.2

0.4

0.6

0.8

From GPIB (sec.)

Signals in the measurement system

100 200 300 400 500 600 700 800 900 1000

-80

-60

-40

-20

0

Phase shift (sec.)

100 200 300 400 500 600 700 800 900

0.1

0.12

0.14

0.16

0.18

x(i) (sec.)

100 200 300 400 500 600 700 800

0.0906

0.0908

0.091

0.0912

Measurement cycles

Filtered x(i), (sec.)

Figure 10: Signals in the measurement chain. From top to

bottom: original data from the TIC and the GPIB interface,

accumulated phase shift, spiky phase differences, ﬁltered

phase differences.

10

0

10

1

10

2

10

-9

10

-8

10

-7

10

-6

10

-5

10

-4

log(tau)

log(VAR), log(MVAR)

Figure 11: AVAR (upper) and MVAR (lower) log-log

curves. The ﬁnal calibration period is τ = 500 × τ

0

for

τ

0

= 1 s.

ﬁgure 11 offer the results -1 and -1.5 for log(AV AR)

vs. log(τ), and log(M V AR) vs. log(τ), respec-

tively; which indicate that a white phase modulation

process is coupled to the frequency source under test

(see table 1).

ACKNOWLEDGEMENT

The authors would like to thank the Spanish Min-

istry of Education and Science for funding the project

DPI2003-00878 which involves noise processes mod-

elling and time-frequency calibration.

ICINCO 2005 - SIGNAL PROCESSING, SYSTEMS MODELING AND CONTROL

314

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