Table Of ContentApplications of Random Process Excursion Analysis
Applications of Random
Process Excursion Analysis
Irina S. Brainina
PovolzhyeStateUniversityofTelecommunications
andInformatics(PSUTI),
Russia
Translated by
Dmitri Arch
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Preface
The author of this monograph took interest in the theory of excursions for random
processes and its applications in early 1970s. The research on an adaptive radio
communication system with a shortwave channel affected by multipath frequency-
selective fading revealed the need to estimate efficiency loss resulting from finite
fading rate. The problem of finding out, by way of periodic sounding of the com-
municationchannel,whichofthetwofrequencysubchannelsavailableprovidesthe
best quality of transmission was naturally reduced to the estimation of the distribu-
tion law for time intervals preceding the first deterioration in the condition of the
previouslyselectedsubchannel.
However, the literature dedicated to the theory of excursions for random pro-
cesses does not seem to provide final solutions to the problem. The distribution of
intervals betweenlevelcrossingssuggested in1945 by S.O. Rice turned outtobe a
slowly converging series of multiple integrals whose multiplicity grows without
limitation. Even in case of Gaussian random processes for which only rough
approximations were obtained, those approximations are only true for the initial
partsofthedistribution.
The problems related to excursions of broadband random processes are
extremely complicated; modern computers are not powerful enough to obtain accu-
rategeneralsolutions,especiallyincaseofnon-Gaussianprocesses.
Under such circumstances, it is appropriate to try and solve the problems by
using approximate methods suitable for engineering applications. The approxima-
tions that have been proposed so far provide low accuracy or require cumbersome
calculations. Besides, in most cases, the authors have not estimated approximation
errors,nor have they determined the areas for which the errors remain within given
limits.
At the same time, determining probabilistic properties of excursions in random
processes is a highly topical issue for experts not only in radio engineering and
communicationbutalsoinotherfields,sinceknowingsuchpropertiesenablessolv-
ingalargenumberofappliedproblems.
This monograph presents the results of the author’s many years of research into
thetheory ofexcursions.Amongotherthings,theauthormanagedtoapplycalcula-
tion methods chosen for Gaussian processes to a larger class of non-Gaussian ran-
domprocesses.
The formulae obtained—both precise and approximate—are rather simple and
have a wide range of applications for engineering. For example, they can be used
for the implementation of adaptive signal processing devices capable of overcom-
ing a considerable uncertainty at the beginning of a self-instruction process. Many
xii Preface
problemsrelatedtothedetectionorrecognitionofrandomanddeterministicsignals
against the background of noise with a priori unknown statistic properties are
solved with the help of special devices which adapt to the most informative para-
metersoftheexcursionsofreceivedsignal-plus-noisewaveform.
This monograph contains the results of many years of the author’s work in the
Research Department of Samara Institute of Telecommunications, presently the
Povolzhye State University of Telecommunications and Informatics (PSUTI). The
research was, for the most part, carried out at the chairs of multichannel telecom-
munication and theoretical foundations of radio engineering and communication
(TFRC).
In 2002, Moscow-based Radio i sviaz publishing house published the author’s
monograph entitled Adaptive Digital Processing of Telecommunication Signals,
Using the Applications of Random Process Excursion Theory. The first two chap-
ters of this book also describe some of the new results obtained in the field of the
excursion theory, which have been expanded and updated by the author in this
monograph.
The author is heartily grateful to her teacher D. D. Klovsky, D. Sc.
(Engineering), full Professor, Head of Chair (TFRC), for his interest in this work
andcontinuoussupport.
Big thanks go to Dmitri Arch who translated the book from the Russian and
madeitaccessible toaninternationalaudience.
The author is grateful to associate Professor Oleg Petrov of TFRC chair, who is
a talented programmer, for his large contribution to the development of a suite of
programsforsimulatingrandomprocessexcursion.
Deep gratitude is also due to Professor Vladimir Lyandres of Ben-Gurion
University of the Negev, Department of Electrical and Computer Engineering, and
Dr. Roman Goot of Academic Technological Institute in Holon, Israel, for their
interestinthisworkandtheirsupport.
The author also expresses her thanks to the unknown benevolent reviewer for
giving useful comments and recommendations regarding this monograph; all of
them were taken into consideration in the process of preparing the book for
publishing.
IrinaS.Brainina
Introduction
In spite of the fact that for decades excursions of stationary random processes have
been the focus of attention of a number of researchers, the development of excur-
sion theory is far from its completion. The reason for that is the difficulty of
describing the subtle nature of random processes by means of mathematics, which
would allow us to estimate a variety of distribution functions, for example, the dis-
tribution of widths of above-threshold excursions, the distribution of areas envel-
oped by the signal excursions, the distribution of time periods required for a signal
toreachcertainlimits,andthedistributionofextremaamplitudes.
Of all characteristics of random process excursions, the first approximations of
excursion duration distribution density, the mean duration of excursions, and, in
some cases, variance in their duration are, perhaps, the ones that have been studied
themostthoroughly.
The researchers who made the largest contributions to the development of
random process excursion theory include S.O. Rice, H. Cramer, M.R. Leadbetter,
J.A. McFadden, D. Slepian, R.L. Stratonovich, V.I. Tikhonov, B.R. Levin, and
Yu.K.Belyaev,tomentionafew.
The first chapter of this monograph starts with a historical sketch of methods
applied for studying parameters of excursions in broadband random processes (the
methodsarebrieflydescribedinSection1.1).
Section 1.2 of the book provides some examples of using excursion statistics for
practicalpurposes.
The random process excursions theory has already proved very useful and the
numberofitsapplicationsislikelytogrowinthefuture.
In view of the difficulty of finding strict solutions of many problems in the field
of excursion theory, we found particularly useful the approximations that are accu-
rateenoughforengineeringapplications.
A number of new results obtained in the excursion theory are presented in the
Chapters2(cid:1)6ofthismonograph.
Some methods of both estimation and accurate calculation of informative para-
meters for excursions in Gaussian stationary random processes have been devel-
oped. Such parameters include, for example, the variance in the number of points
where the signal crosses the predetermined level, or the variance in the number of
above-threshold excursions within the time interval of length τ (both variances are
definedinChapter2ofthisbook—seeSection2.1).
In Sections 2.2 and 2.3 we obtain upper and lower estimates for distribution
functions of time intervals between an arbitrarily chosen moment and the first
zero crossing of the random process (Section 2.2) or the first moment when zero
xiv Introduction
leveliscrossedinagivendirection(Section2.3).Similarestimatesforexcursion
durationdistributiondensitiesareobtainedinSection3.1(Chapter3).
Having assumed that any two successive excursions above the predetermined
threshold are statistically independent, we obtained recurrent formulae that allow
calculating moreaccurately the densities ofdistributionfor excursiondurations and
for durations of intervals between excursions (Section 3.2). The same assumption
was used as a basis for the estimation of variances in excursion durations and in
thelengthofintervalsbetweenexcursions(Section4.1).
Methods of extending application of the results obtained to certain classes of
non-Gaussian random processes are presented in Section 3.3. They suggest using
nonlinear noninertial transformation or a non-Gaussian process with a predeter-
mined correlation function into the equivalent Gaussian process (it will have a dif-
ferentcorrelationfunction,butthesamethreshold-crossingparameters).Itisshown
that the distribution law for momentary values of centered non-Gaussian processes
does not have any significant effect on the characteristics of excursions, which are
mainlydeterminedbythewaveformandroot-mean-squarefrequencyoftheprocess
powerspectrum.
InSection4.2weestimateparametersforexponentialtailsofdistributioncurves
representing the distribution of such excursions that are much longer than the pro-
cess time correlation interval (in this case, we are dealing with the tails of distribu-
tion density curves representing the distribution of durations of excursions above a
givenstandardizedlevel).
Section 5.1 explains how the generation sequence obtained by bilateral clipping
onlevelzeroofacentered randomprocesscanbeused.Thepointswherethepulse
sequence changes its sign from negative to positive or vice versa coincide with
those of crossing level zero by the process. This pulse sequence, therefore, carries
all the information on the distribution law for time intervals between successive
crossings of level zero by the process. It is called “the generation sequence”
because we can obtain from it a family of pulse sequences by adding up any num-
ber i52,3,4,...k of successive intervals between zero crossings. Each sequence so
obtained carries the information on the duration of a combination of i successive
intervals. This additional information allowed us to reduce as much as we wanted
the error of calculating the parameters of zero crossings, by increasing the total
numberkofsequences.
Formulae are provided for estimating the cumulative function and distribution
density of zero-crossing intervals. In those formulae, the functions estimated are
expressed through correlation functions. Using the readily available software for
computing correlation functions saves us a lot of time and efforts that would be
necessaryifweweretosimulatezero-crossingstatistics.
In Chapter 5 (Section 5.2), we show how the method based on the use of a
generating pulse sequence can be extended to random processes of any kind
including the noncentered and even non-Gaussian ones. Our approach allows
choosing any threshold for the analysis of above-threshold excursions, as long
as it remains within the range of momentary values of the process under
analysis.
Introduction xv
In order to obtain the distribution laws for durations of above- or below-
threshold excursions in a broadband random process for the case of a randomly
chosen standardized threshold ðX =σÞ6¼0 (here σ is the root-mean-square value of
0
the process), the generating sequence is formed from the flow of positive pulses
whichrepresentexcursionsabovethegivenlevelðX =σÞ.
0
After inverting every other pulse and removing spaces between pulses, we
obtained a symmetrical bipolar pulse sequence. The time intervals between zero
crossings in the generating sequence carry all the information we need on the dura-
tionoftheprocessexcursionsabovethepredeterminedrelativevalue.
Similarly, the generating sequence can be used to obtain a family of i auxiliary
sequences and then apply the same method for calculating the parameters of excur-
sionsastheonethatwasappliedinthecaseoflevelzero.
By inverting pulses representing the random signal excursions above the prede-
termined level another generating sequence can be obtained which carries informa-
tion on the time intervals during which the signal level remains below the
predeterminedlevel.
For any given two threshold levels one can make six generating pulse
sequences,twoofwhichwillrepresentexcursionsaboveandbelowthefirstthresh-
old, respectively, two others will represent excursions above and below the second
threshold,andthe last twowillrepresent time periods during which the signal level
remainswithinoroutsidethecorridorlimitedbythetwothresholds,respectively.
The problems such as the one mentioned above often arise when there is a need
to control tolerances and check for compliance various technical specifications and
norms. Information on the parameters of excursions may also become useful, for
instance, in the processing of medical data such as cardiograms, encephalograms,
tomographic images, in recognition of images and sounds, the studies of the distri-
bution of intervals between events when analyzing the type of telecommunication
traffic,thestudiesofsignalfadingcausedbymultipath propagationofradiowaves,
andinmanyotherfields.
What is especially important is that the proposed method of calculating distribu-
tion laws for the durations of random process excursions above or below any arbi-
trarily chosen relative value is applicable to any processes, and not just to centered
Gaussian ones. For example, it can be applied to Rayleigh distribution, Rice distri-
bution,distributionχ2,anduniformorrectangulardistribution,tonameafew.
InSection5.3,weshowhowthedistributionofthetotaldurationofisuccessive
intervals is calculated. The distribution function is expressed through the correla-
tionfunctionsofafamilyofpulsecurvesobtainedfori52,3,4,...k.
Chapter 6 of this monograph focuses on two hitherto poorly studied characteris-
ticsofrandomprocessexcursions.
Section 6.1 describes one method allowing to estimate, on the basis of results
obtained in Chapter 5, the cumulative function and distribution density of time
intervals between an arbitrarily chosen moment and the next moment when the sta-
tionary differentiable random process reaches the set level (whether it reaches it
from above or from below is irrelevant). So far this problem has only been solved
fornondifferentiableMarkovprocesses.
xvi Introduction
Computer simulation results presented in Appendix 2 prove that the above
method of estimating the cumulative function and distribution density of time to
thenextlevelcrossingisprettyaccurate.
Section 6.2 studies areas enveloped by the random process excursion curves.
Some sources [4, 6, 53] give approximate probability distribution of areas under
the curves that represent short excursions above high standardized levels. Due to
their complicated nature, the generic problems of calculating the distribution of
areasenvelopedbyexcursioncurvesstillremaintobesolved.
ThetablesinSection6.2showtheestimatesobtainedforaverageareasenveloped
by excursion curves (both above- and below-threshold excursions are considered).
Thevalues presentedinthetables were obtainedbymultiplyingthe knownaverage
durations of excursions by their average amplitudes. The calculated average values
of all excursion parameters are given for two kinds of random processes—a
GaussianoneandaRayleighone.
The calculated average areas closely match the results obtained by computer
simulation(seeAppendix3).
Section 6.2 describes one method (also proved accurate through simulation) of
estimating exponential tail parameters for distribution of areas enveloped by rela-
tivelylongexcursions.
Chapter 7 of this monograph presents algorithms for building certain digital
adaptive analyzers of stationary random process characteristics and provides esti-
mates ofstatisticalerrorsandmeasurementtimes.Instrumentalanalysisofrealran-
dom processes combined with computer simulations allows conducting
experimental studies of a number of characteristics of excursions whose theoretical
analysis still presents major problems. One example of such studies is the analysis
of the distribution of areas under excursions above a given threshold. Due to a
complicated nature of the problem, the theoretical solution to it has not yet been
found.
All the adaptive digital analyzers that we proposed function on the basis of the
same principle. In the process of adaptation of a particular analyzer, the average
valuesofitssettingsarequicklyestimated,whichthenserveasabasisfortheauto-
mated process of finding the analyzer’s optimal settings. The settings that are sub-
ject to optimization include some important parameters such as sampling rate,
amplitude quantization increment, signal gain/attenuation factor, the number of
channels available in the analyzer, and measurement duration for the given maxi-
mumacceptablevaluesofstatisticalandinstrumentalerrors.
By doing preliminary “training” which does not take very long, we managed
to build our analyzers so that their adaptation (“training”) time is automatically
set in inverse proportion to the effective width of the power spectrum of the pro-
cess. This is achieved by accumulating the predetermined number of excursions
during adaptation or measurement (N and E, respectively). The counter capacity
E is chosen on the basis of a compromise between the given maximum statistical
error of measurement and the maximum acceptable measurement duration. As in
most cases, the adaptation does not have to be very accurate; it does not take a
long time in proportion to the total measurement session duration (the adaptation
