Introduction to Applied Statistical Signal Analysis. Guide to Biomedical and Electrical Engineering Applications

406 Pages·2007·6.972 MB·English

by  Richard Shiavi (Auth.)

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PREFACE
T hisbookpresentsapracticalintroductiontosignalanalysistechniquesthatarecommonlyusedin
abroadrangeofengineeringareassuchasbiomedicalengineering,communications,geophysics,
speech, etc. In order to emphasize the analytic approaches, a certain background is necessary.
Thebookisdesignedforanindividualwhohasabasicbackgroundinmathematics,science,andcomputer
programming that is required in an undergraduate engineering curriculum. In addition one needs to have
an introductory-level background in probability and statistics and discrete time systems.
Thesequenceofmaterialbeginswithdefinitionsoftermsandsymbolsusedforrepresentingdiscrete
datameasurementsandtimeseries/signalsandapresentationoftechniquesforpolynomialmodelingand
datainterpolation.Chapter3focusesonthewindowingandthediscreteFouriertransform.Itisintroduced
by presenting first the various definitions of the Fourier transform and harmonic modeling using the
Fourier series. The remainder of the book deals with signals having some random signal component
and Chapter 4 reviews the aspects of probability theory and statistics needed for subsequent topics. In
addition, histogram fitting, correlation, regression, and maximum likelihood estimation are presented. In
the next chapter these concepts are extended to define random signals and introduce the estimation of
correlationfunctionsandtestsofstationarity.Chapter6reviewslinearsystemsanddefinespowerspectra.
Chapter 7 presents classical spectral analysis and its estimators. The periodogram and Blackman-Tukey
methodsarecoveredindetail.Chapter8coversautoregressivemodelingofsignalsandparametricspectral
estimation.Chapter9presentstheclassicalusesofcrosscorrelationandcoherencefunctions.Inparticular,
the practical techniques for estimating coherence function are presented in detail. Chapter 10 is a new
chapter for the third edition and covers envelope estimation and kernel functions. Presentation of these
topics is motivated by the growth in usage of these techniques. Envelope estimation is important not
only for signals such as electromyograms but also when using high frequency carrier signals such as in
ultrasoundapplications.ThefundamentalsofHilberttransforms,analyticsignals,andtheirestimationare
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PREFACE
presented. Kernel functions appear in the neuromuscular literature dealing with point processes such as
action potentials. The main purpose is to create a continuous amplitude function from a point process.
A summary of kernel functions and their implementation is presented.
The material in Chapter 10 is drawn with permission from the doctoral dissertation work of Robert
Brychta and Melanie Bernard. They are both graduate students in Biomedical Engineering at Vanderbilt
University. Robert’s research is being done in the General Clinical Research Center and Melanie’s is
being done in the Visual System’s laboratory.
Thepresentationstyleisdesignedfortheindividualwhowantsatheoreticalintroductiontothebasic
principles and then the knowledge necessary to implement them practically. The mode of presentation is
to: define a theoretical concept, show areas of engineering in which these concepts are useful, define the
algorithmsandassumptionsneededtoimplementthem,andthenpresentdetailedexamplesthathavebeen
implemented using FORTRAN and more recently MATLAB. The exposure to engineering applications
will hopefully develop an appreciation for the utility and necessity of signal processing methodologies.
The exercises at the end of the chapters are designed with several goals. Some focus directly on the
materialpresentedandsomeextendthematerialforapplicationsthatarelessoftenencountered.Thedegree
of difficulty ranges from simple pencil and paper problems to computer implementation of simulations
and algorithms for analysis. For an introductory course, the environment and software recommended are
those that are not overly sophisticated and complex so that the student cannot comprehend the code or
script. When used as a course textbook, most of the material can be studied in one semester in a senior
undergraduate or first year graduate course. The topic selection is obviously the instructor’s choice.
Most of the examples and many of the exercises use measured signals, many from the biomedical
domain. Copies of these are available from the publisher’s Website.3 Also available, for interactive
learning, are a series of MATLAB notebooks that have been designed for interactive learning.1(cid:6)2 These
notebooks are written in the integrated environment of Microsoft Word and MATLAB. Each notebook
presentsaprincipleanddemonstratesitsimplementationviascriptinMATLAB.Thestudentisthenasked
toexerciseotheraspectsoftheprincipleinteractivelybymakingsimplechangesinthescript.Thestudent
then receives immediate feedback concerning what is happening and can relate theoretical concepts to
real effects upon a signal. The final one or two questions in the notebooks are more comprehensive
and ask the student to make a full implementation of the technique or principle being studied. This
requires understanding all of the previous material and selecting, altering, and then integrating parts of
the MATLAB script previously used.
1Shiavi,R.,LearningSignalProcessingUsingInteractiveNotebooks.IEEETransactionsonEducation;42:355-CD,1999.
2Shiavi,R.“TeachingSignalProcessingUsingNotebooks”.ASEEAnnualConference;CharlotteNC,June,1999.
3Http://books.elsevier.com/companions/9780120885817
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ACKNOWLEDGMENTS
The author of a textbook is usually helped significantly by the institution by which he is employed
and through surrounding circumstances. In particular I am indebted to the Department of Biomedical
Engineering and the School of Engineering at Vanderbilt University for giving me some released time
and for placing a high priority on writing this book for academic purposes. This being the third edition,
There have been three sets of reviewers. I would like to thank them because they have contributed to the
book through suggestions of new topics and constructive criticism of the initial drafts. In addition, I am
very grateful to Robert Brychta and Melanie Bernard, both graduate students in Biomedical Engineering
at Vanderbilt University. Their doctoral research provided the basis for the topics in Chapter 10.
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LIST OF SYMBOLS
ENGLISH
a(cid:4)i(cid:5)(cid:6)b(cid:4)i(cid:5) parameters of AR, MA, and ARMA models
A polynomial coefficient
m
B bandwidth
B equivalent bandwidth
e
c (cid:4)k(cid:5) sample covariance function
x
c (cid:4)k(cid:5) sample cross covariance function
yx
C coefficients of trigonometric Fourier series
n
C (cid:4)k(cid:5) autocovariance function
x
C (cid:4)k(cid:5) cross covariance function
xy
Cov[ ] covariance operator
d(cid:4)n(cid:5) data window
D(cid:4)f(cid:5) data spectral window
e error in polynomial curve fitting
i
E[ ] expectation operator
E sum of squared errors
M
E total signal energy
tot
f cyclic frequency
f frequency spacing
d
f folding frequency, highest frequency component
N
f sampling frequency
s
f(cid:4)t(cid:5) scaler function of variable t
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LISTOFSYMBOLS
f (cid:4)(cid:7)(cid:5)(cid:6)f(cid:4)x(cid:5) probability density function
x
f (cid:4)(cid:7)(cid:6)(cid:8)(cid:5)(cid:6)f(cid:4)x(cid:6)y(cid:5) bivariate probability density function
xy
F (cid:4)(cid:7)(cid:5)(cid:6)F(cid:4)x(cid:5) probability distribution function
x
F (cid:4)(cid:7)(cid:6)(cid:8)(cid:5)(cid:6)F(cid:4)x(cid:6)y(cid:5) bivariate probability distribution function
xy
g loss coefficient
g sample coefficient of skewness
1
h(cid:4)t(cid:5)(cid:6)h(cid:4)n(cid:5) impulse response
H(cid:4)f(cid:5)(cid:6)H(cid:4)(cid:9)(cid:5) transfer function
I(cid:4)f(cid:5)(cid:6)I(cid:4)m(cid:5) periodogram
Im( ) imaginary part of a complex function
(cid:3)[ ] imaginary operator
K2(cid:4)f(cid:5)(cid:6) K2(cid:4)m(cid:5) magnitude squared coherence function
L(cid:4)x(cid:5) Lagrange coefficient function
i
m mean
N number of points in a discrete time signal
p(cid:6)q order of AR, MA, and ARMA processes
P signal power, or signal duration
P[ ] probability of [ ]
P (cid:4)x(cid:5) polynomial function
m
q-q quantile-quantile
r correlation coefficient for q-q plot
Q
Re( ) real part of a complex function
R (cid:4)k(cid:5) autocorrelation function
x
R (cid:4)k(cid:5) cross correlation function
yx
(cid:4)[ ] real operator
s2 variance of linear prediction error
p
S(cid:4)f(cid:5)(cid:6)S(cid:4)m(cid:5) power spectral density function
S (cid:4)f(cid:5)(cid:6) S (cid:4)m(cid:5) cross spectral density function
yx yx
T sampling interval
U(cid:4)t(cid:5) unit step function
Var[ ] variance operator
w(cid:4)k(cid:5) lag window
W(cid:4)f(cid:5) lag spectral window
x(cid:4)t(cid:5)(cid:6)x(cid:4)n(cid:5) time function
X(cid:4)f(cid:5)(cid:6)X(cid:4)m(cid:5)(cid:6)X(cid:4)(cid:9)(cid:5) Fourier transform
z coefficients of complex Fourier series
m
GREEK
(cid:7) significance level
(cid:10) (cid:4)t (cid:6)t (cid:5)(cid:6) (cid:10) ensemble autocovariance function
x 0 1 x(cid:4)K(cid:5)
(cid:10) coefficient of skewness
1
(cid:11)(cid:4)t(cid:5) impulse function, dirac delta function
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LISTOFSYMBOLS
(cid:11)(cid:4)n(cid:5) unit impulse, Kronecker delta function
(cid:12)(cid:4)n(cid:5) linear prediction error
(cid:13) energy in a function
i
(cid:14) (cid:4)f(cid:5) co-spectrum
yx
(cid:15) kth central moment
k
(cid:16)(cid:4)n(cid:5) white noise process
(cid:17)(cid:4)(cid:18)(cid:5) ensemble normalized autocovariance function
(cid:19) Gaussian probability distribution function
(cid:20) correlation coefficient
(cid:20) (cid:4)k(cid:5) normalized autocovariance function
x
(cid:20) (cid:4)k(cid:5) normalized cross covariance function
yx
(cid:21)2 variance
(cid:21) standard error of estimate
e
(cid:21)2 covariance
xy
(cid:22)(cid:4)f(cid:5) phase response
(cid:22) (cid:4)f(cid:5)(cid:6)(cid:22) (cid:4)m(cid:5) cross phase spectrum
yx yx
(cid:3) orthogonal function set
n(cid:4)t(cid:5)
(cid:23) (cid:4)t (cid:6)t (cid:5)(cid:6) (cid:23) (cid:4)k(cid:5) ensemble autocorrelation function
x 0 1 x
(cid:24) (cid:4)f(cid:5) quadrature spectrum
yx
(cid:9) radian frequency
(cid:9) radian frequency spacing
d
ACRONYMS
ACF autocorrelation function
ACVF autocovariance function
AIC Akaike’s information criterion
AR autoregressive
ARMA autoregressive-moving average
BT Blackman-Tukey
CCF cross correlation function
CCVF cross covariance function
cdf cumulative distribution function
CF correlation function
CSD cross spectral density
CTFT continuous time Fourier transform
DFT discrete Fourier transform
DTFT discrete time Fourier transform
E energy
erf error function
FPE final prediction error
FS Fourier series
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LISTOFSYMBOLS
IDFT inverse discrete Fourier transform
IDTFT inverse discrete time Fourier transform
LPC linear prediction coefficient
MA moving average
MEM maximum entropy method
MLE maximum likelihood estimator
MSC magnitude squared coherence
MSE mean square error
NACF normalized autocovariance function
NCCF normalized cross covariance function
pdf probability density function
PL process loss
PSD power spectral density
PW power
TSE total square error
VR variance reduction
WN white noise
YW Yule-Walker
OPERATORS
∗
X(cid:4)f(cid:5) conjugation
x(cid:4)n(cid:5)∗y(cid:4)n(cid:5) convolution
ˆ
S(cid:4)m(cid:5) sample estimate
(cid:2)
S(cid:4)m(cid:5) smoothing
x(cid:4)n(cid:5) periodic repetition
FUNCTIONS
sgn(x) signum(cid:4)x(cid:5)=1(cid:6) x>0
=−1(cid:6) x<0
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1
INTRODUCTION AND TERMINOLOGY
1.1 INTRODUCTION
H istorically,asignalmeantanysetofsigns,symbols,orphysicalgesticulationsthattransmitted
information or messages. The first electronic transmission of information was in the form of
Morse code. In the most general sense a signal can be embodied in two forms: (1) some
measuredorobservedbehaviororphysicalpropertyofaphenomenonthatcontainsinformationaboutthat
phenomenon or (2) a signal that can be generated by a manufactured system and have the information
encoded.Signalscanvaryovertimeorspace.Ourdailyexistenceisrepletewiththepresenceofsignals,
and they occur not only in man-made systems but also in human and natural systems. A simple natural
signalisthemeasurementofairtemperatureovertime,asshowninFigure1.1.Studyofthefluctuationsin
temperatureinformsusaboutsomecharacteristicsofourenvironment.Amuchmorecomplexphenomenon
isspeech.Speechisintelligencetransmittedthroughavariationovertimeintheintensityofsoundwaves.
Figure1.2showsanexampleoftheintensityofawaveformassociatedwithatypicalsentence.Eachsound
has a different characteristic waveshape that conveys different information to the listener. In television
systemsthesignalisthevariationinelectromagneticwaveintensitythatencodesthepictureinformation.
Inhumansystems,measurementsofheartandskeletalmuscleactivityintheformofelectrocardiographic
and electromyographic voltages are signals. With respect to these last three examples, the objective of
signalanalysisistoprocessthesesignalsinordertoextractinformationconcerningthecharacteristicsof
the picture, cardiac function, and muscular function. Signal processing has been implemented for a wide
varietyofapplications.Manyofthemwillbementionedthroughoutthistextbook.Goodsourcesforother
applications are Chen (1988) and Cohen (1986).
1
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CHAPTER1 INTRODUCTIONANDTERMINOLOGY
FIGURE 1.1 The average monthly air temperature at Recife, Brazil. [Adapted from Chatfield, fig. 1.2, with
permission]
FIGURE 1.2 An example of a speech waveform illustrating different sounds. The utterance is “should we
chase (cid:2)(cid:2)(cid:2) .”[AdaptedfromOppenheim,fig.3.3,withpermission]
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1.2 SIGNALTERMINOLOGY
A time-dependent signal measured at particular points in time is synonymously called a time series.
The latter term arose within the field of applied mathematics and initially pertained to the application of
probabilityandstatisticstodatavaryingovertime.Someoftheanalyseswereperformedoneconomicor
astronomicdatasuchastheBeveridgewheatpriceindexorWolfer’ssunspotnumbers(Anderson,1971).
Manyofthetechniquesthatareusedcurrentlyweredevisedbymathematiciansdecadesago.Theinventionof
thecomputerandnowthedevelopmentofpowerfulandinexpensivecomputershavemadetheapplicationof
thesetechniquesveryfeasible.Inaddition,theavailabilityofinexpensivecomputingenvironmentsofgood
qualityhasmadetheirimplementationwidespread.Alloftheexamplesandexercisesinandrelatedtothis
textbookwereimplementedusingsubroutinelibrariesorcomputingenvironmentsthataregoodforabroad
varietyofengineeringandscientificapplications(Ebel,1995;Foster,1992).Theselibrariesarealsogood
fromapedagogicalperspectivebecausethealgorithmsareexplainedinbookssuchasthosewrittenbyBlahut,
1985;IngleandProakis,2000;Pressetal.,2002;andStearns,2003.Beforebeginningadetailedstudyofthe
techniquesandcapabilitiesofsignalortimeseriesanalysis,anoverviewofterminologyandbasicproperties
ofsignalwaveformsisnecessary.Aswithanyfield,symbolsandacronymsareamajorcomponentofthe
terminology,andstandarddefinitionshavebeenutilizedasmuchaspossible(Granger,1982;Jenkinsand
Watts,1968).Othertextbooksthatwillprovidecomplementaryinformationofeitheranintroductoryoran
advancedlevelarelistedinthereferencesection.
1.2 SIGNAL TERMINOLOGY
1.2.1 Domain Types
Thedomainofasignalistheindependentvariableoverwhichthephenomenonisconsidered.Thedomain
encountered most often is the time domain. Figure 1.3 shows the electrocardiogram (ECG) measured
fromtheabdomenofapregnantwoman.TheECGexistsateveryinstantoftime,sotheECGevolvesin
the continuous time domain. Signals that have values at a finite set of time instants exist in the discrete
time domain. The temperature plot in Figure 1.1 shows a discrete time signal with average temperatures
given each month. There are two types of discrete time signals. If the dependent variable is processed
in some way, it is an aggregate signal. Processing can be averaging, such as the temperature plot, or
summing, such as a plot of daily rainfall. If the dependent variable is not processed but represents only
FIGURE 1.3 TheabdominalECGfromapregnantwomanshowingthematernalECGwaveform(m)andthefetal
ECGwaveform(f).[AdaptedfromInbar,p.73,fig.8,withpermission]
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