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5-1-2003
Construction of M - Band bandlimited wavelets for
orthogonal decomposition
Bryce Tennant
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Recommended Citation
Construction of M - Band Bandlimited Wavelets for Orthogonal
Decomposition
by
Bryce Tennant
A Thesis Submitted
in
Partial Fulfillment
of the
Requirements for the Degree of
MASTER OF SCIENCE
In
Electrical Engineering
Approved by:
Prof. RM. Rao
(Thesis Advisor)
Prof. S.A. Dianat
Prof. T
.M. Rao
Prof. V.I. Amuso
Prof. R.I. Bowman _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
_
( Department Head)
DEPARTMENT OF ELECTRICAL ENGINEERING
COLLEGE OF ENGINEERING
ROCHESTER INSTITUTE OF TECHNOLOGY
ROCHESTER, NEW YORK
Construction of M - Band Bandlimited Wavelets
for
Orthogonal Decomposition
Bryce Tennant
Permission granted
ntle of thesis: Construction of M - Band Bandllmited Wavelets for Orthogonal Decomposition
I, author's name, hereby grant permission to the RIT Ubrary of the Rochester Institute of Technology to reproduce my thesis in whole or in part. Any reproduction will not be for commerdal use or profit.
Acknowledgements
Thanks are not enough to express the true gratitudethatI have for Dr. Raghuveer
Rao andall that he has presented to me in my eleven years ofworking withhim. There
has beenno otherperson in my lifewho hashad such anincredibleeffectonmypassion
for engineeringand mathematics.
I would like to thank Adam
Templeton,
TomKenney,
MarkChamberlain,
William
Furman,
TiborDobri,
and the countless others at Harris Corporation that haveprovidedmoral,financialand academicsupportfrom
day
oneofmythesis.They
aretruly
exceptionalengineers, andIamfortunatetowork withallofthemona
daily
basis.Without of course my parents, none of this would be possible. I would like to
thankmymotherfor her undying love andsupport, agiftforwhichIamforever grateful.
To my
father,
whose work ethic and drive I can only attempt at. I have learned so manythings fromthem
both,
all ofwhichhavegottenme towhereI amtoday.Special thanks to
Greg
Smith who has always been my best academic soundingboard,
our informal technical conversations have always been some of the mostenjoyable.
Most
importantly,
thanks to myloving
wife, Julie.Who,
second only to myself,has hadto make sacrificesto gethere. I wouldliketorecognize her for all she has
done,
every
day
since we have been together, to so strongly support me with this and all myAbstract
While bandlimited wavelets and associated IIR filters have shown serious potential in
areas of pattern recognition and communications, the dyadic Meyer wavelet is the only
known approach to construct bandlimited orthogonal decomposition. The sine scaling
function and wavelet are a specialcase oftheMeyer. Previous workshave proposed aM
- Band
extension of the Meyer wavelet without solving the problem.One
key
contribution of this thesis is the derivation of the correct bandlimits for the scaling
function and wavelets to guarantee an orthogonal basis. In addition, the actual
construction of the wavelets based upon these bandlimits is developed. A composite
wavelet willbe derived basedontheM scale relationshipsfromwhichwe will extractthe
wavelet functions. A proper solution to this task is proposed which will generate
associated filters with the knowledge of the scaling function and the constraints for
Contents
Chapter 1 Introduction 1
Chapter 2 GeneralFilter
Theory
52.1 M-Band Filter Banks 5
2.2
Paraunitary
Filter Banks 1 1Chapter 3 Wavelets 13
3.1 Basic Wavelet
Theory
143.2 The Discrete Wavelet Transform 17
3.3 Approximation of aSignal
f(t)
withScaling
Function (j)(t) 203.4 Bandlimited Wavelets 25
3.5 The Meyer Wavelet 27
Chapter 4 M
-BandWavelet Development 34
4.1 M-Band
Theory
ofWavelets 344.2 M- Band Bandlimited OrthogonalWavelet Design 40
4.3 The Composite WaveletandEmbedded Wavelets 44
4.4 Phase Response ofthe
Scaling
FunctionandWavelets 54Chapter5 Examples 61
Chapter6Conclusion 77
Appendix A The
Relationship
ofthe aParameterto theRolloffFactor intheSquareAppendix B -DefinitionoftheComposite Wavelet 82
AppendixC- Phase Response
ofM- Band Bandlimited Wavelets 84
List
of
Figures
Figure 2.1 - Thetwo- Band decomposition
and reconstructionfilter bank 5
Figure 2.2- M- Band decomposition
and recompositionfilter bank 8
Figure 3.1 - Thetimedomain
representation ofthe Meyer Wavelet 16
Figure 3.2- Analysis
and synthesis waveletfilterbanks 25
Figure 3.3- The Daubechies 4
tapscalingfunctionand waveletintime 26
Figure3.4-Magnitude
frequency
response ofthe Daubechiesfourtapwavelet 27Figure3.5-Meyerwavelet
frequency
response 29Figure3.6-Regionsof supportfor theMeyer lowpassfilter 30
Figure3.7-The Meyerwavelet spectrum magnitude 32
Figure 4.1 - M - Band details
and approximation off(t)k 36
Figure 4.2
-Boundary
conditionsfortheH(co)
filterfrequency
response 42Figure4.3a,4.3b- Band
extremesfortheMeyer scaling function 42
Figure 4.4
-Scaling
functionand compositewavelet 46Figure
4.5-Scaling
functionandtheM-Scale relationshipwiththecompositewavelet..47
Figure 4.6
-Themagnitude squared
frequency
responseofthecompositewavelet 49Figure 4.7- The
composite waveletand theembedded waveletfunctions 52
Figure 4.8
-Overlap
of selected regionsinthefrequency
domain 57Figure 4.9- The
progressive phase cancellation ofthescaling functionandwavelet 58
Figure 4.10 - Phases requiredforcancellation inthefirstandsecond wavelet pair
Figure 4.11 - Phases
requiredforcancellationinthesecond andfinalwaveletpair 60
Figure 5.1
-7(00)and y
{co)
for M=4anda=7t/7 62Figure
5.2-M=4 Meyer scalingfunction,<_>(co) 63
Figure 5.3
-Scaling
functionand compositewaveletfor Meyer 4- Band filter bank 63Figure 5.4
-Scaling
functionand embedded wavelets withinthecomposite wavelet 64Figure 5.5- Timedomain
representation oftheMeyerwaveletsasscaling function 66
Figure 5.6
-Sample filters for M=4andoc=n/7 68
Figure 5.7
-Scaling
functionand waveletsfortheM=4case anda=7i.40 69Figure 5.8
-Corresponding
filters forH(<d)
andGm()
70Figure5.9- M
-Band scaling functionsand waveletsinthe timedomain 71
Figure 5.10- Original input
signal composed of5sine waves ofarbitraryphase 73
Figure5.11 - Low
passapproximation andthreeband details fortheorthogonal M=4,
a=7t/40, bandlimitedwaveletdecomposition 74
Figure5.12,5.13- The
reconstructedsignal andtheerror signal 75
Figure 5.14- Magnitude
Chapter 1
Introduction
Proper design ofM - Band filters for
signal decomposition has provedto be one ofthe
mostimportant designproblems in the signal processing field inthe three decades since
their introduction. A proper design promotes the optimal decomposition and
reconstruction of a signal into different spectral bands for the analysis of subbands.
Encoders that take advantage of the signals characteristic energies can be
designed,
allowing for a better overall signal representation.
Historically,
a majordifficulty
indesigning
these filters arises primarily due to their non idealcharacteristics.Typically
M- Band filters
generate aliasing that results in a distorted representation ofthe original
signal. It is certainly desired that any distortion introduced
by
the analysis and synthesisprocess be minimized or at
best,
completelyeliminated. When considering the design oforthogonal
filters,
both the magnitude response and phase response need to be closelyexamined. When
decomposing
a signal into two bands it has been shown that satisfyingboth magnitude and phase requirements simultaneously yields a unique set of filters.
Although these filters provide for perfect reconstruction,
they
do not provide for goodfrequency
separation. Ifone oftheconditionsisrelaxed, manyuseabletwo- Bandfilterscan be generated. Quadrature Mirror Filters
(QMF)
are a set of such a filter banks.filters,
called paraunitary filterbanks,
can yield perfectreconstruction while maintainingindividual filter integrity.
Althoughthe wavelettransformwas
initially
introduced as a moreflexible analysistoolforsignalrepresentation than the Short Time Fourier
Transform,
itwas quickly linkedtothe
theory
ofM-Band Filters. The pioneeringefforts of
Grossman,
Morlet, Daubechies,
andMeyer allowed forthe analysis tool to establish a firmmathematical standing while
the groundbreaking works ofDaubechies and Mallat provided for a connectionbetween
the wavelet transform and subband
filtering
[39,
40]. Recent worksby
Gopinath[3],
Vetterli[ll], Nguyen[35],
Selesnick[33], Walter[37]
and others have taken the wavelettransformand combineditwiththe
theory
ofM- Bandperfect reconstructionfilter banks
to generate a powerful signal processing tool. The advantage of the wavelet transform
overtraditionalfilters becomes apparentwith the spectral characteristics ofthe transform
andthe fundamentalproperties oftheresulting basis functions.
The recent works of
Vaidyanathan[19], Stedden,
Heller,
Gopinath,
Burrus[3]
Vetterli,
Herkey
[11],
andJones[4]
have treated this subject in detail.Typically
theresulting M-Band wavelets are generated using polyphase methods or optimization procedures. The
results areparaunitary FIR filterbanks thatprovide forperfectreconstruction. Allofthese
filters offer compact support andthereforeresult in a
frequency
response withoutbound.In contrast, this thesisaddressestheproblem of
designing
an orthogonal bandlimitedM-Band perfectreconstructionfilter bank. Thisproblem wasfirst touchedupon inthework
restrictionshe placedonthedesignofthe scaling functionareinsufficienttoguarantee an
orthogonal decomposition. We will generate tighter constraints on our scaling function
thatwillresultina new filter bank designandthereforeproducevery differentresults.
This thesis will
develop
thetheory
ofthe M - Bandwavelet transform and use it to
generate a new class ofwavelets based on abandlimited function. In Chapter 2 we will
provide a general background on M - Band
filter banks and filter theory. Chapter 3 will
be dedicated to the introduction of the wavelet transform, the scaling
function,
motherwavelet and associated filters. We will introduce the
theory
in the context of a two-Band signal analysis tool. The results will be extended in Chapter 4 so as to
develop
thetheory
ofthe bandlimited orthogonal M - Bandwavelet transform. We will
develop
theconstraints andequationsthatneedtobesatisfiedfortheM- Band
perfect reconstruction
filters, thereby
linking
thetheory
introduced in Chapter 2 with that of wavelets. With afirm mathematical foundation provided, we will then proceedto
develop
our orthogonalbandlimited M - Band
scaling
function,
and wavelets. As a direct result of thedevelopment the corresponding filter bank will also be provided. We will introduce the
concept of a composite wavelet that will act as a governing shell to all the remaining
wavelets embedded within the system once the scaling function has been determined.
This section will focus both on the magnitude and the phase response of each derived
wavelet. Chapter 5 will illustrate several design examples of the scaling
function,
wavelets andassociatedfilters forvarious values ofM. We willuse on oftheseresultsto
A briefword on notation. We will commonly replace PR for perfectreconstruction, FT
for Fourier transform and QMF for Quadrature Mirror Filters. We will use O(_y)to
represent the scaling function Fourier transform and (j>(t) to represent the time domain
equivalent. The corresponding filterto the scaling function willbe denoted as
H(G))
witha time domain equivalent of h(n). The M-l wavelet filter Fourier transforms will be
described as
Ym(<y)
where me{0,l..._Vf
-2} andy/m{t)
where me{0,1...M
-2} in thetime domain. The corresponding filters foreachofthese wavelets willbe given as
Gm(co)
for the
frequency
domain representation and gm(n) for the time domain representationover the same range of m. In some special cases we will substitute
Pi(co)
for the aboveChapter 2
General Filter
Theory
X(n) H0(z) *0M vQM
t=
y0M WH^z] *!..
i'
\W
!'
y^nj PiN [image:15.513.52.467.229.339.2]>
Figure 2.3- The two- Band decompositionand reconstructionfilter bank.
2.1 M-Band Filter Banks
Foratwo
-channelfilter bankthe general representation forsignal decomposition and
reconstructionis shownin Figure 2.3. A signal ofinterestis passed through two separate
filters
Ho(co)
andHi(co)
corresponding to a low pass and a high pass filter respectively.Eachfilter output constitutes a
frequency
band of the input signal. It is clearthat ifeachof the signals bandwidth is half of the input signals
bandwidth,
the Nyquist samplingtheorem implies that we can resample the filtered outputs at half the rate while still
maintaining the original signals spectral integrity.
However,
in practice our filters arenever ideal and always contain some energy in the stop band past the cutofffrequency.
This implies that whenthe
downsampling
ofour signal occurs, frequencies greaterthanof aliasing is of primary concern when
designing
filter banks. The output of thesedecimators,
vo(n) and vi(n), are signals that can be encoded or analyzed prior totransmission or storage. When the signal is to be reconstructed, it will be done using a
reverseprocess of expansion. Eachsubband signalis upsampled and passedthrougha set
of reconstructive
filters, Fo(z)
and Fi(z). The outputs of these two filters are summedtogether to get a reconstructed version of the original input signal.
Ideally
thisreconstructionshould be a perfect replicaofinput signalwithapossible unit sampleshift
and constant gainmultiplier.
Working
backwards from Figure2.3,
we see that the output of the reconstructionportionofthebankis given as
X(z)
=Y0(z)F0(z)
+Y,(z)F,(z)
Where both
Y0(z)
andYi(z)
are the outputs of the two interpolators. In the z domaininterpolation is givenas
Y0(z)
=V0(z2)
providinganoutputof
X(z)
=V0(z2)F0(z)
+V1(z2)F,(z)
(2.1)
The
Vn(z)
terms in(2.1)
consists of a primary term that represents the "true" filteredsignal and an aliasing term that represents that component of the signal that is greater
thanthecutoff
frequency,
(Oc.Thesignaljustpriorto thedecimatorisrepresentedas,Xn(z)
=X(z)Hn(z)
where ne{0,l}
(2.2)
Theoutput ofany givendecimatoristhesumoftheprimaryand aliasedterms
Vn
(Z)
=-(x
(zy>
K
(z/2
)+x(-z/2
K
(-
zVl))
where ne{0,l}
(2.3)i(z)
=^(F0(zK(z)+F1(z)//1(z))
+^)(F0(z)//0(-z)+F1(z)//1(-z))
(2.4)The second term in equation
(2.3)
is the aliasing term. Itis clear that this term causesdistortion in thereconstructed signal and shouldbeeliminated wherepossible.
Using
thefollowing
notation,T(zyFo(z)H0{z)+F,{z)H,{z)
A(z)
=F0{z)H0{-z)+F,{z)H,{-z)
2wecanrewrite
(2.4)
asX{z)=X{z)T{z)+X{-z)A{z)
(2.5)To completely remove the component of the signal attributed to aliasing we should
force the term
A(z)
in(2.5)
to be zero. With this accomplished we are left with theexpression X(z)=X(z)T(z). Recall thatour goalis perfectreconstruction where wehave
allowed ourselves a constantgainmultiplier and some integersample delay. This implies
thatour
T(z)
functionmust reduceto theformT{z)-c-z
Ifwe examinereal coefficientcaseof
T(z)
thenwehave(2.6)
T{co)
=c-e-'!iai*
(2.7)
The expression given in
(2.7)
implies that we require a constant multiplier and linearphase to generate perfect reconstruction ofour input signal. If in the above system we
H,{z)=H0(-z)
F0(z)
=H0{z)
F,(z)
=-H,(z)
then we have a set of filters that completely satisfy our PR constraints. In addition all
analysisand synthesisfilters can allbe determined fromthe singleprototype
filter,
Ho(co).This
form,
introducedby
Smith and Barnwell in1986,
is commonly referred to as aQuadrature Mirror Filter (QMF). Although
by
definition aQMF is a set offourfilters,
the term has become synonymous with any set ofM filters that act as an analysis and
synthesis pair.
Typically
they
are denotedby
the numberofchannels inthefilterbank,
orasaM-channel QMF. Such asystemis shownin Figure 2.4.
H0.z) xoM
Im
v0(n)|m
yoN *bl*) kTxl(n' *M yjM
? HjM
Im
I'M
^(z)^
*fc
y2(n)
*yz)
|m
I'M
F2(z)1
^ w
i
m
,Hi
fcH [z) M-l
nI-1 ,
^M
|m
F (z)' ?
Figure2.4- M- Band decompositionand recompositionfilter bank
One advantageofthis systemthat doesnotreadily occurinthe two
-Band case is that
perfect reconstruction canbeobtained while still
designing
afilter bankwith"good" pass
[image:18.513.52.460.360.542.2]band and stop band characteristics. Vaidyanathan
[46]
showed that the filter bank indesire the output of the reconstruction filter bank to be a shifted and possibly scaled,
version ofthe filter bankinput. To derivetherequirements forperfectreconstruction we
will start
by
creating an expression forthe mthanalysis
filter, Hm(co),
outputin terms ofthe systeminput.
Therefore,
foreach subband signalpriortothedecimationwecanwriteXm(z)=X(z)Hm{z)
where me{0,1... M-l}
The output
frequency
response of each analysis filter is such that the resultingbandwidth is a factorofM less than the input signals bandwidth. Withthe signal
Xm(z)
being
bandlimitedby
%/M, wecan passthe resultinto adecimatorthatdown-samples thesignal
by
afactorofM withoutany loss in signalintegrity. However becausewe arenowdecimating
into M bands asopposedto twoweneedto accountfortheadditionalalaisingtermsthatareintroduced dueto thenon- ideal
nature ofthefilter Hm(co). Inthezdomain
the
down-sampling
results in anewsignalVm(z)
definedasVm{z)
=^H^z^W'jX^W1)
where wedefine W=e'm. M l=0
In a real world system it is these signals that get encoded for storage or transmission.
Figure 2.4 shows the steps required to reconstruct the original signal from the M
decimatedrepresentations. These signals,
Vm(z),
arepassedthrough abankofexpanders,up-sampling the signal
by
a factor of M.Up-sampling
essentially inserts M-l zeros inbetween each valid sample of vm(n) to generate new signals ym(n). The act of
up-samplingcanberepresentedinthez domainas
YJz)
=Vm(zM)
=jfXHm(zW!)x(zW>)
ina subband output. The final output ofthesystemis the sumof all theresulting subband
signals as shownin
(2.8)
M-l M-l
*(*)=Z^(*(*)
= XrE4zWl)ZFm(z)Hm(zW!)
m=0 (2.8) M-l Ifwe now define a newfunctionA,
(z)
=j]
Fm
{z)Hm
(zWl)
wecansimplify(2.8)
tobem=0
M-l
X{z)=^x(zWl)Al(z)
(2.9)/=o
Clearly
the equation in(2.9)
allows us to represent the reconstructed output as afunction of the original input andM-l terms of signal aliasing. We can express the sum
givenin
(2.9)
as amatrix equation. This allows ustogenerate an expressionin whichwecan remove each of the undesired aliasing components while enforcing our PR
requirements. Wewilltherefore write ourdefinitionof
Ai(z)
as(2.10)
"
4>(z)
"
H0{z)
Hx{z)
-Hm-i(z)
\
FQ(z)
1
M
A(z)
=H0{zW)
H:{ZW)
-.
Hm-i(zW)
Fi(z)
A-.(z)_
H0{zWM-1)
H,(zWM~l)
-HUzWM-l)_
_FM-l(z)_orin simplernotation
A{z)
=H{z)F{z)
(2.11)where we have absorbed the M scaling factor into our desired A(z).
Forcing
theappropriate conditions on Ain
(2.11)
allowsus to generate afilter bankdefinedby
H(co)
that is representative of a QMF. We therefore define
A(z)
such that all aliasingcomponents are
identically
zero andthe only contributingportion ofthe signalis a resultA(z)
=M\{z)
0
0
and solve for the proper unknowns in
(2.11)
to obtain the final filter design. TheH(z)
matrixshownin
(2.10)
is commonlyreferredtoastheAlias Component(AC)
matrix.2.2
Paraunitary
FilterBanks
Paraunitary
filter banks are a special class of perfect reconstruction filter banks.They
have the addedbenefitof allowingthe designerto generate synthesisfilters
directly
fromthedesignoftheanalysisfilters. Thecausaltransfermatrixforaparaunitary
filter,
H(z),
satisfies H
(z)H
(z)
=cifor all z. These conditions comedirectly
from the orthogonalnature ofthe analysis filters. One useful property ofthe paraunitary matrix is thatfor all
analysis
filters, [H_(z) Hi(z)...Hmi(z)]
wehaveM-l
Y_\Hk{a)]~=c
(2.12)k=0
This is knownasthepowercomplementary
(PC)
conditionandwillbepivotalin futuresections withthedesignof ourM- Band
orthogonalbandlimitedwaveletconstruction. In
the dyadic case, the high pass filter can be written as afunction ofthe low pass filter in
theanalysisfilter bankas
h,{n)
={-l)"h;{L-n)
(2.13)
The equation given in
(2.13)
provides us with a method to calculate thehigh pass filterfound
by
solving for F in (2.11). It can be shown that in the dyadic case the synthesisfilters are suchthat F0{z)=z~NH0{z)and F,(z)=z~NH,{z) [46]. Inthe timedomainthis is
representedasthecomplex conjugate ofthemirrored analysis
filter,
orf0{n)
=h^{L-n)
(2.14)f,(n)
=hl{L-n)
(2.15)Utilizing
theparaunitaryconditions aboveanentiretwo-channel perfectreconstructive
Chapter 3
Wavelets
Wavelets haveemerged as a powerful field in signal analysisover thepast
thirty
years.Key
results introducedby
Meyer[27,
40],
Daubechies[39],
Vetterli[11,42]
and manyothers have allowed this branch of mathematics to grow into a mature field of study.
Introduced
initially by
the Hungarian born Mathematician AldredHaar,
wavelets startedout as a strictly mathematical curiosity. Published as an appendix in his doctoral thesis,
Haar showed that a dilated and shifted function could generate orthogonal systems of
functions. It was not until the mid seventies, while working at an oil company, did Jean
Morlet actually
formally develop
thetheory
of wavelets. He developed thetheory
toovercome the shortcomings of the STFT. Morlet was the first to generate a "mother
wavelet"
andapplythe
theory
toanalyze signalsusingmulti-resolution analysis.Working
with Alex Grossman the two showed that the transform could decompose a signal and
recompose it while maintaining signal integrity. This was a critical step in the
developmentofthe wavelet theory. The multi-resolution
theory
of wavelets wasformally
developed
by
Meyer and Mallat in the late 1980s. This provided furthertheory
fordeveloping
specific wavelets andscaling functions basedonthe application ofinterest. Itwas not until Ingrid Daubechies made the connection between wavelets and filter bank
In their most basic
form,
wavelets are signal analysis tools, similar to Fourier analysis,that decompose a signal into spectral components. The wavelet transform is unlike its
Fourier transform cousin, in that it provides time domain information while also
providing for
frequency
content. Unlike the Fourier's e~"*, the wavelet transform does
nothavepredefined basis functions. Insteadwavelets and scaling functions are designed
as orthogonal functions basedon a strict setofrules that, inthe end, governthe shape of
each function.
Determining
a wavelet basis has proved to be afairly
difficult task.Daubechies developed a set of rules that can be applied to generate wavelets with
compact support[39]. Meyer applied the rules to generate some of the first known
bandlimited wavelet functions
[40],
which are of particular interest to this thesisdevelopment. There is abundant literature available in this area, interested readers are
referredto the works [2][3][31]-[38],[39]-[43].
3.1 Basic Wavelet
Theory
The wavelet, y/(t), is definedas a continuous time function with zero mean and finite
energy. Expressed in mathematical notation, this function therefore must satisfy the
conditions imposed in both
(3.1)
and(3.2)
below.\y/(t)dt
=0 (3.1)\\y/{t\
-c where c< (3.2)approaches infinity. In addition wavelets must satisfy the so called
Admissibility
Condition in (3.3). The condition guarantees that any wavelet transform will have an
inverse,
allowing the developer to completely reconstruct the signal from the signaldecomposition. The
(<__>)
termis thefrequency
responseofthe waveletfunction.r
ft
w
\
' , /'=C (3.3)
i
H
Wavelet and scalingfunctions canbe either bandlimitedorcan have compact supportin
time. These two possibilities result in functions that behave
differently
and haveadvantages and disadvantages in implementation. The literature found throughout this
field
typically
focuses on compactly supported wavelets. This is primarily due to thewell-understood and simple means of FIR filter implementation. These filter
implementations,
however,
do have a pitfall due to their compactly supported nature.They
will have an infinitefrequency
response, causing signal leakage into the designedfilters stop band. In addition these wavelets tend notto have smoothband edges and can
result in choppy
looking
functions. This phenomenon can clearly be seen whenexamining anyoftheDaubechies waveletfunctions. Thealternative approachisto utilize
abandlmitedwavelet. This results in asmooth functionthatdoes not pass signalintothe
filters stop band. The cleardisadvantageis in the complexityofthe filterimplementation
due to the IIR nature ofthe inverse Fourier transform. Figure 3.1 provides one possible
bandlimited wavelet function commonly referred to as the Meyer wavelet. In order to
appliedto thefunction inthe
frequency
domainthatgovernsthepassband,
stop bandandtransitionbandofthescalingfunctionsFouriertransform.
Meyer Wavelet
!
!
1
!
!
_tti
JV4*
1
i
!
60 40 -20 0 20 40 E0
Figure3.1 Thetimedomainrepresentation oftheMeyer Wavelet
Like the Fourier transform, the wavelet transform uses a set of basis functions to
decompose a signal into its constituent parts. These basis functions form a tight
frame[39,40]. For our purposes we willdefine atightframe tobe aset offunctions such
that any
f(t)
can be composedusing thefunctions that span the tight frame. The functionf(t)
willbe generated using aweighted sum ofthebasis functions wherethe weights aredetermined as the projection of the basis functions with the function
f(t)
itself. Becausewavelets form a nested subspace the transform must be expressed in terms of dilated
versions of a single prototype
function,
commonly referred to a the mother wavelet.Conceptually
the transform dilates and shifts thisfunction,
\|/(t),by
factors a and brespectively. When afunction
f(t)
is projected onto these dilated functions the result is atwodimensionaltime
[image:26.513.171.345.141.286.2]W{a,b)
=-L]
f(t)*
dt (3.4)
V a J
From
(3.4)
we seethatour waveletbasis functionsaredefinedas1
(t-b^
(3.5) V a )
The constant factor -j- in each basis function is
present so as to normalize the analysis
function tounit energy.
Substituting
(3.5)
back into the definitionofthe transform wegetafinal expressionforthecontinuous wavelettransform.
W{a,b)
=-^]f{tyab(t)dt (3.6)The variables of the transform, a and
b,
control the dilation and shift amount of themother wavelet respectively. It is these two parameters that provide for the
time-frequency
representation offeredby
the wavelet transform. If our wavelet satisfies theadmissibilityconditionin
(3.3)
thenwe candefinetheinversewavelettransformas1 ~
1
f(r)=^
I I
ri2"wMK_,(0<k<* o.7)
ca=_Li=L|fl|
whereC istheconstantdefined in (3.3).
3.2 The Discrete Wavelet Transform
The continuous transform that is described above fits nicely into a situation where we
wanttovisualizethestructure ofa signal. Itdoesnothowever readily lend itselfto digital
can be removed and a discrete time equivalent can be developed [39]. We start
by
defining
a new function <|)(t)which mustsatisfythefollowing
properties$0{t)dt
=l\\<p(tf=]\<p(tfdt
=l(</>(t),0{t-n))
=d{n)
This new
function,
called the scalingfunction,
musthave unit energy and be orthogonalto integertranslates ofitself. This implies thateach translate is
linearly
independent andthe set \(f){t-n)} where n is an
integer,
forms abasis for alinear vector space. We willdenote this vector space as V0. If we dilate the scaling function
by
a factor of 2~kwe
generate,
(<p(2-kt),0(2~kt-n))
=2kd{n)
(3.8)Equation
(3.8)
defines alinearvectorspace, Vk. Thisdilation implies thatforincreasing
kthe corresponding vector space,
Vk,
generates coarser and coarser approximation ofL2{R}. Hence for any function
f(t)
that resides withinVk,
f(2t)
must completely becontained withinVk-i- Ifwe further define the vector space such that every vectorin
Vk
must also be in
Vk-i,
then we can write the basis function (|)(t) in terms of a linearcombination ofbasis functions of the next finer level. More concisely, if
V0
cV_,
then0(0=
c(nM2r-n)
(3-9)
We now
develop
afrequency
domain representation of the results provided in (3.9). Itwas definedthatourscaling functionmustsatisfythe
following
orthogonalityconstraint(<p{t),</)(t-n))
=d{n)
(3.10)Taking
the FT of(3.10)
results in thecommonlyreferred to Poissonsummationformulashownin
(3.11)
2
Yp{co+2M}
=1 (3.11)This formula will be used in many locations through out the remainder of our
development andshouldbe noted as adirectresult ofthe orthogonalityconstraint placed
on ourbasis functions (j)(t). We will now showifwesatisfytheorthogonalityconstraints
onthe scaling function andif
{(p(t-k):
for kinteger}
forms a nested subspace[40],
thenthe c(n) in
(3.9)
satisfies the powercomplementarycondition givenin (2.12). We definetheFouriertransformofthediscretesequencec(n) as
C(co)=
c(/i)e"-''fl"
(3.12)Ifwelet t= andsubstitute thisinto
(3.9)
wearrive at 20-=2>W(r-n)
(3.13)\2J ____,
whichhas aFTof
20(2tf>)
=C(_tf)l>M
(3.14)
The discrete nature of c(n) results in the function
C(co)
being
27.periodic.Shifting
the2(2co+4nk)=C{a)+2nk)(a)+
2nk)
(3-15)
Taking
themagnitude squaredofbothsides and summingoverkgives4^>(>
+4;^2
=
\c{a)fYJ\{<
+2nkf
(3.16)k k
The summation on the right-hand side of
(3.16)
is nothing more than the PoissonSummation formula defined in (3.1
1)
reducingthe expressionto45>(2_+
4>*)|2
=\C{cof
{3.17)k
We nowmake thesubstitutionof co'
=co + kanddistributethis through
(3.17)
4|o(2fi/+2tf+ 4;rt)|2 = \c{co' +
nf
(3-18) tIfweaddboth
(3.17)
and(3.18)
togetherweget4^\{2co+
4nkf
+4j]|0(2+2^+4^3.f
=\c{tvf+\C{tv +frf
(3.19)
_ it
Close examination reveals that the two summations on the left-hand side of
(3.19)
arenothing more that the even and odd terms of k in the Poisson summation formula
reducing
(3.19)
to ourfinal formof4=
|c(fi>)|2+|c(__>
+ (3.20)3.3
ApproximationofaSignal
f(t)
withScaling
Function <jtt)In the above section wehave developed an orthogonal set ofbasis functions that will
approximateany function in L2{R}. We will now usethis<|>(t)toapproximate anarbitrary
functionf(t). We start
by
assuming knowledge ofasetofcoefficients a(k,n) suchthatfor/*(0=
X0(2"^-"M^")
(3-2D
Therefore at some arbitrary resolution k and the previous coarser resolution
k+1,
wedefine the difference betweenthese two approximations as the detail lost in
transitioning
from the finer space to the coarser space. This is the same as projecting the finerapproximation
fk(t)
onto space Vk+i. We transition from signal spaceVk
to signal spaceVk+i
with some error signal defined as gk+1{t)=fk{t)-fk+l(t). The error, gk+i(t), iscontainedcompletely within the signal space
Vk
andis orthogonal to the signal fk+i(t). Itwould be plausible to reconstruct some signal
fk(t)
be retaining a single courseapproximation and a series of detail functions captured when moving from vector space
Vn
toVn+i.We willdefine thewaveletfunction \|/(t) thatwillbeusedto capturethe detaillost when moving to a coarser subspace. This wavelet
function,
as with the scalingfunction,
mustobeysome simplerulesof construction.Firstly
itmustintegrateto zero\y/(t)dt-0
In additionthewavelet musthaveunitenergy
$\y/{tfdt
=l (3.22)andits integertranslates mustbeorthogonaltoshifted versions ofitself.
(y/{t),y(t-n))
=d(n)
(3.23)An additional constraint is that this new function must also be orthogonal to shifted
versions of the scaling function. This is because \j/(t) exists completely within the
(>(0.^-n))
=0(3#24)
Equation
(3.24)
allows us to define \|/(t) in terms oftranslates ofthe scaling function inthepreviouslyfinerdomain.
p{t)=
^din^t-n)
(3.25)
The coefficients,
d(n),
satisfy similar properties tothose inthe scalingfunction definitionandusingsimilardevelopmentasin
(3.20)
wecan writed(n)
inthefrequency
domain as\D{cof+\D{6)
+xf
=4(3.26)
The orthogonal nature of equation
(3.24)
along with the definitions in(3.20)
and(3.26)
letus write afinalequationrelatingthe two
frequency
responsesC(co)
andD(co)
C{tv)D'
{a)
+C{cO+ 7l)D*{(Q+7r)=0(3.27)
where the * denotes the complex conjugate. The results of
(3.20),
(3.26)
and(3.27)
can becombinedintoone compact matrix equation.C{a>)
C{o)+7r)D{co)
D{co+n)=4/
(3.28)
C*{co)
D*{co)
+k) D*{a>+7i;)
The commutative nature of
(3.28)
providesus with amechanismto swap theorder ofthematrix multiplication withoutaffectingthe
integrity
oftheresultC{co)
D*{a>)
C*{co+7r) D*{co+7t)
C{co)
C{co+D{co)
D{co+7t)41 (3.29)
Equation
(3.29)
defines our system in detail and provides someinteresting
implicationsthatrelate the
theory
of wavelets to thatoffilter banks. Ifwelook backto the previoussection onM- Band
filtering
we see that the above equation defines aparaunitary filterbank. This implies that the two functions
C(co)
andD(_o)
are actually low pass and highapproximation function and the
D(co)
is developed from the detail information.Typically
low pass
filtering
provides an approximation oraveraging ofthe original signal whereas
thehighpass
filtering
preserves the signals detail information. From(3.29)
we can writethree
filtering
equationsfoundin(3.30)-(3.32).\C{o)f+\D{cof=4
(3.30)\c{co
+nf
+\D{co
+xf
=4 (3.31)C{a))C*{a)+7r)+D{co)D*{co+7r)=0
(332)
Forourdevelopmentwe willdefinetwo newfunctions
h(n)
andg(n)whicharesimplyh(n)
=&
V '
2
,( s (3.33)
( \
d\n)
g\n)=^L
This simplifies ourformula in
(3.27),(3.30),(3.31)
to\H{0)f
+\G{0)f=l (3.34)\H{co
+xf
+\G{co+xf
=1 (3.35)H{6))G*{a))+H{co+ x)G*{cD +x)=0
(3.36)
We have definedacomplete QMFBthatsatisfies the paraunitarycondition. This implies
that if we create a prototype filter
H(co)
from a valid scaling function (|)(t) then we canquickly determine the analysis and synthesis filters
H(co)
as a prototype function. Asimple tableis providedbelowtoillustrate the developmentofthe analysis and synthesis
filters. The analysis filters are in thefirst column andthe corresponding synthesis filters
are inthesecondcolumn. The only filterthatneeds tobe determined usingthe governing
AnalysisFilters Synthesis Filters
Low Pass
h(n)
determinedby
governingrulesofscalingfunction.
h{n)
=h{-n)
High Pass g(n)
=
{-lj-nh{l-n)
g{n)=g{~n)We can now reducethe process ofperforming the wavelet transform to atechnique of
low pass and highpass filtering. As discussed in the section on general filter
banks,
wecan down-sample the signal output of a filter whose bandwidth is half of the original
signals bandwidth
by
afactor oftwo without loss of signalinformation. Ifwe apply thescaling filterandwavelet
filter,
as shownin Figure3.2,
we canfollowthefiltering
with adecimation.Thisgenerates a course approximation,ai(n) andadetail
function,
di(n). Thelow pass approximation is then passed through the same bank of filters and again
decimated. This process continues untilthe desired resolutionis reached. The end result
is a series of detail functions and a single coarse approximation function. The reverse
process is used to recombine the signal from its constituent parts. A schematic of the
analysis and synthesisbanks arefound inFigure3.2.
Analysis Bank
__(__)
> r ? 8(n)
h(n) g(n)
h(n) g(n)
d2.
d.3
__(n) ?
V
SynthesisBank
2g(n)
M
*
f,
-+2g(n) ->1 -. 2h(n)
_i_L
f,
2g(n)2h(n)
2h(n) >
Figure3.2- Analysis
and synthesis waveletfilterbanks
3.4
Bandlimited
WaveletsTypically
wavelets developed for signal processing have compact support in timeresulting in FIR filters.
Many
recentpapershave been writtenonthe developmentofFIRwaveletfilter banks[5][9-10][12-13],[32],[34]. The Daubeichies wavelethas become one
of the most recognized wavelets in the signal processing community. This wavelet and
scaling
function,
seen in Figure3.3,
results in a compactly supportedfilter,
clearlyimplying
that thefrequency
response is without bound. Thefrequency
response oftheDaubechies scaling functionandwaveletcan beseenin Figure 3.4.
Compactly
supportedDaubechies 4
Tap Scaling
Function Daubechies4Tap
Wavelet FunctionFigure 3.3- The Daubechies 4
tapscaling functionand wavelet intime
Depending
on the application, the smoothness of the wavelet can be critical in theimplementation ofthe design. A smooth wavelet helps isolate edges and features in a
signal. We tend to desire wavelets thathave some degree ofsmoothness overthose that
appearchaotic
by
nature. One common metric of smoothnessis the Holderexponent[43]
which is essentially a measurement of local differentiability. We can conclude that the
four
tap
Daubechies wavelet has a low Holder exponent and is actually differentiableonly once. The implication is thatthere is a slow
decay
in thefrequency
domain whichdoes notallow for
frequency
localizationof events. Althoughthere aremany Daubechieswavelets ofvarying
lengths,
we need alarge number of filter taps to generate a smoothwavelet that is M times differentiable. Our design will focus on wavelets that are
FrequencyResponseofphi
30 Frequency
Responseof psi
25
I
20
15
-10"
5
A
n .^/.aaA/iI
i
IMAa
/n.a ^ .-2-10123 -3-2-10123
Figure3.4-Magnitude
frequency
response oftheDaubechies fourtapscaling functionand wavelet.3.5
The
MeyerWavelet
The Meyer wavelet is a continuous time wavelet that is band-limited over the range
0<(H<4k/3. This band limitedness causes the wavelet to be
IIR,
but despite this apparentdisadvantage,
the filter is practical and offers both good time andfrequency
resolution[2]. The Meyer wavelet has also been shown to be closely related to the raised cosine
function prominent in communications
[31]
[37]. This naturally leads to designs incommunication systems that take advantage ofthis relationship.W.W. Jones proposed a
M
-Band filter based on theMeyerwavelet that was usedfor orthogonally multiplexed
communications or multicarrier modulation.
Rao,
Bopardikar,
andAdiga[1]
showedthattheMeyerwaveletnaturally arises insubsampledbandlimitedprocesses.
They
illustratedthat the reconstruction filter banks can optimally recreate a sub sampled bandlimited
wavelet. It is also wellknownthatincommunication systemsthatafiltercan provide for
optimum signaldetection andenergycompaction whenthe filters spectrumis matchedto
that ofthe signal that is
being
detected. Rao and Chapa further illustratedin[2]
that thebandlimited Meyer wavelet can be
directly
designed to optimally match the magnitudeand phase of aknown signal while stillpreserving thewavelets orthogonality. A detailed
development of the Meyer wavelet and scaling function can be found in their paper,
primaryresults thatpertainto thiswork are reproducedbelow.
We start
by defining
the half band of the scaling function asbeing
the positivebandwidthofthe
frequency
response <_>(co), whichclearly mustbe less than k.Therefore,
it must be that the half band of
|<_E>(2_z)|
< n. The Poisson Summation formula in(3.11)
reducestobe
ft{(o\
2 +|<_>(<y
+2;r1
2 =1 for 0<co<2k(3.37)
Ifthe half band
frequency
ofO(oo)
is defined to be c_m, then from our discussion abovecom
> n. Ifwedefinethe excessbandwidththatis greaterthan n as awecan writeo)m=n + a
(3i38)
where a>0.
Utilizing
the reducedPoisson Summation formula given in(3.37)
we cansee that the bandlimited nature of the scaling function forces us to write the
following
restrictions on<_>(to)tosatisfythesummation.
|0(/y)|2
=1 for
\co\
<{n-a)
, (3.39)
\&{cof
=1 for{n
Figure3.5-Meyerwavelet
frequency
responseThestructureisaboveis definedas a result ofthePoisson
summationformula
Figure 3.5 illustrates how these two functions overlap and interact with one another to
enforce theresults in (3.39).
Combining
theresults from both(3.14)
and(3.33)
it is easytoshowthat
H{cd).
0>{2co)
(3.40)
This equation allows us to
develop
thefrequency
response of the associated low passfilter with the knowledge of the
frequency
response of the scalingfunction,
<&(c_).Knowing
that <_>(2co) is a compressed version of<_>(co),
we can generate regions ofsupportforthefilter
H(co)
withthe definitionin (3.40). Recall thath(n)
is adiscretetimefilter,
which implies that thefrequency
response,H(co),
is 2ii periodic. Because the [image:39.513.183.360.75.288.2]H{o))
=Ofarco> n+aTheperiodic nature ofthe
filter, H(),
allows ustowriteH 2n- co+a \
=H
(3x-a
=0Graphically
we showthese twoperiods ofH(co)
asfollows(3.41)
FilterTransition Regions
H(ffl+2n)
it-a.
2 2
3x-ct 2
2n
Figure3.6-Regionsof supportfor the Meyer lowpassfilter.Gray
regions represent regions oftransitionfrompassband to stop band.
Limiting
ourselves to the range 0<ox27iwe see the filterH(co)
is exactly zero in then +a
region <co< k+ aand the shifted version is
identically
zero in the range defined2>7t a
by
7t-cx<co< . Ifwe takethe bestcase scenario wherethese twobands coincidewe seethat
K+ a
=n-a
conditions
n
a=
(3.42) 3
This valueis absolutelycrucialtothedevelopmentoftheMeyerwaveletand will prove
to be even moreimportantwhenwe expandour developmentto the M- Band
case.
Any
value of a that exceeds this will cause filter overlap and violate the rule dictated
by
(3.26). It should be noted that the Poisson Summation formula operates only on the
magnitudes ofthescaling functionand places norestrictions onthe phase ofthe solution.
This implies that a phase solution to the scaling function needs to be developed as a
separate step in the development process. Now thatalimitation has be derived forthe a
term, a scaling function can be generated
directly
from the results in (3.39). The designnow reduces to
determining
appropriate transition bands that satisfy the Poissonsummation
formula,
graphically depicted in Figure 3.6. We can do thisby
finding
afunction
f(co)
andg(co) suchthatf(co)2 +
g(co)2 =1
and
g(co)2 =
f(2n-co)2
in region
{n-a)<co<{n
+a)Furtherdevelopmentin
[2]
shows that the correspondingwavelet in the twobandcase isfticof
=0,
for0<co<{n-a)
g(co), for
{n-a)<co<{n
+a)
1,
for{n
+a)<co<(2_t
-2a)
(3.43)CO f\
, for
{2n-2a)<co<{2n
+2a)
2
J
0,
for w<{2n+2a)
The FT of the wavelet defined in
(3.43)
is shown in the figure below. Make notice ofthe locations of the 3dB points in this figure as it becomes necessary to utilize this in
subsequent sections oftheM- Band design.
Meyer Wavelet Spectrum Magnitude
1
0.9
0.8
0.7
O.B
0.5
04
0.3
0.2
01
:::i:::i::0::i:::
i
jJ
-____#
'--
t
n \ \ In \\ 8 10
Figure3.7-The Meyerwaveletspectrum magnitude.
Noticethe3dBpointsoccurat ji and2%respectively.
As in
(3.40)
thereis acorrespondingfilter equationforthe wavelet\|/(t). Ifwetake theFT ofthetwo scalerelationship found in
(3.25)
we get an equivalentfrequency
domain [image:42.513.176.349.280.470.2]2V{2a>)
=D{co){co)
(3.44)This in conjunction with the definition of the function g(n) in
(3.33)
provides anexpressionforthewaveletsfilteras
G{co)= v / (3.45)
<__>()
The above development does not illustrate the phase solution of the wavelet. This is
Chapter 4
M
-Band
Wavelet Development
We now have developed much of the foundation that is required for the M-Band
wavelet design. We have illustrated the basic principles that govern M
-Band filter
theory
and thetheory
for wavelet analysis. A synthesis routine for the Meyer scalingfunction,
wavelet function and the corresponding filters have been presented. We willnow generate a M - Band
wavelet filter bank based on a non-trivial extension of the
results in the previous sections. We will first
develop
the magnitude response for thescaling function andwavelets in similar waythatwas presented above. Oncethe Poisson
summation formula has been appropriately satisfied, we will generatephasing details for
thecorresponding waveletsso asto maintainorthogonalityacross allfilters.
4.1
M-BandTheory
of WaveletsAssume that we are given some signal,
/(f)
eV_,
that we wish to break up into Mseparatebanks. It is desirablethattheoriginalfilteredsignal canbereconstructedwithout
errordue to aliaising oramplitude distortion. We know that
Vk
is the space thatcontainsdetailed space to Vk. Ifwe define
Wk
as the detail lost in going fromVk_]
toVk
then wecan write
Vk=@Wj
(4.1)j=k
and
Wk
e Vk_,. Here wehave usedWk
to representthe totaldetail lost in moving from a
finerresolution to thenext coarser one. There arehoweverM-l bands ofdetailspacethat
we need to consider. Each band will contain some detail information about the original
signal that will need to becaptured when
translating
fromVk_i
to Vk.Rewriting
(4.1)
toaccountforthis generates
^=8^
(4-2)m=0j=k
We have used mtorepresentthedetail spaceinquestion. This implies that there areM-l
signals thatneed tobe determined in order to capture all the detail lostwhen performing
this projection. We will denote these signals as ekm(t) where k is the level of the
decomposition and m represents the band that we are decomposing. These functions
represent the signal lost in each band when the transform is made. A simple symbolic
k-l
Figure 4.1- M- Band details
andapproximationf(t)kusedto
generatethenextfinerresolutionapproximationoff(t)
We let (j)(t) be the scaling function and i|/m(t), m=0,l,...M-2, be the wavelets in a M - Band
orthogonal decomposition. For M - Band
systems the scaling
function,
tyt), is designed to beorthogonaltointegertranslatesofitselfacross all scales.This implies
{(/>(t\<p(t-n))
=d(n)
(4J)
Taking
theFourierTransformof(4.3),
providesthewellknownPoissonsummationformula.^\{co+2nkf
=1 -~<fc<oo(4>4)
k
To generate a multi-resolution
decomposition,
we must allowfor <|)(t) to bereconstructedfrom integershiftsfromthepreviousfinerresolution0{t)=
Y,c(n)0(Mt-n)
(4.5)
forscalars, c(n
),
ne Z. This becomes inthefrequency
domainM(Mco)
=C(o))o(ta)
(46)Where C(co)- y\c(n)e~'" and
O(co)
is the FTof c()(t). The corresponding equation for the M-lMVm(Ma))
=Dm(co)(<o), m=0,l...M-2(4J)
These relationships, referred to as the 2 - Scale
relationships, relate the FT's of the scaling
function and wavelets, namely
O(co)
and4/m(co),
to the digital filterfrequency
responses,C(co)
and
Dm(co)
respectively.Forourdevelopmentwe normalizethefiltersby defining
M M
Wewillnow use theresults oftheprevious sectionsto show
2nm
M-l
m=0
H co+
-M
=1 (4.9)
We start with the results in
(4.6),
simplified with the use of(4.8),
and substitute in theexpressionco=co+2nk
.
Taking
the magnitude squared of the result and summing bothsidesoverkwe obtain
Y\{Mco
+2nMkf
=\H{cofY,ft{co+2nkf (4-10)k k
Theresult on theleft hand side of
(4.10)
can be simplified usingthePoisson summationformula,
Y}&{Mco
+2nMkf
=\H{cof
(4>11)
k
WealsoknowfromthePoissonsummationformulathat
Y\{Mco
+2nkf
=1(4.12)
k
Returning
to(4.11)
we see thatthis equation provides us withthe 2rckMterms(0,
2kM,
4kM...)of(4. 12). Ifwesubstitute co
= cot-.,into (4. 1
1)
we arrive atYJ\{Mco
+Mn+2nMkf
=\H{co+nf
(4-13)
k
forall co
2nm M
me
{0.
. .M
-1}.
Summing
theresultantMexpressions togethergivesM-l
H\co+
2nm M
(4.14)
Y^{Mco
+2nk}
=k m=0
Recognizing
that the first summation is nothing more than what we presented in(4.12)
providesus withtheresultthatwe set outtoprovein (4.9).
Similarmethodscanbe usedtoprovide equivalent results forthewavelet functions and
filters
frequency
responses,^(co)
and Gm(co). Unlikethe two- Band case wherethereisa single scaling function and a corresponding wavelet, the M - Band
system has M-l
wavelet functions that we will reference as
^(co)
where me{0,1...M
-2}. The resultsthatweobtainforthewaveletfilters
Gm(co)
areM-l
p=0
G\co+
2np
M
=1 where me
{0,1...
M-2} (4.15)
Finally
we are interested in how the scaling filter and each waveletfilter interact withone another in the
frequency
domain. We know that each wavelet function can beexpressedas aweighted sum ofthe scaling functionat previous resolution. Thisis simply
theM - Bandextension ofthe two dimensional rule that we established in the previous
section.
WnM)=
2X(nMM'
~") (4-16)
n
We also haveestablished that thescaling function mustbe orthogonal to each waveletin
{Wm(t),0{t
-1))
=X<(n)(^(Mf
-n),<f>{t
-1))
n
The dotproduct ontherighthandside of
(4.17)
canbereducedusing(4.3)
and(4.5)
tobe(0{Mt-n),<p{t-l))
=?^M)M (4.18)
allowingustowrite
(4.17)
as(r.,*-i))=Z-W^
(4..9,As statedabove, thescaling function and each wavelet mustbe
orthogonal, whichimplies
that
(4.19)
mustbeidentically
equal tozeroorJ'"(n)
m
w/-_T.ne{0.1...M-2}
(4.20)n *"
The same rules apply to the wavelet functions themselves, each wavelet should be
orthogonaltoeveryother wavelet intheM
-Band system.
Using
the samemethods thatwe usedtoarrive at
(4.20),
itcaneasily be shownthat^--, d
(n-Ml)
,2_,dm(n)^- =0 where m,pe
{0,1...
M-2) and m* p (4.21)
Theresults in
(4.19)
and(4.21)
canbewritteninthefrequency
domain asM-l
I
M-lI
?=0 co+^-)h M)
2nq
co+ V M G\ f2V
f co+ V M2nq
co + V M (4.22)=0 foreach me
{0,l...Af-2}
=0 for each m,pe
{0,1
...M
-2}
where m pwhere we haveusedtheformula in
(4.8)
toputtheresults intermsiftheH(co)
andGm(co)
filters. This development leadsusto acompact resultthatcanbewrittenina matrixform.
Combining
theresults of(4.9),
(4.15)
and(4.22)
we can write thefollowing
summationform shown in
(4.23),
which provides the properties of the scaling and wavelet filters thatbecome crucial to our development. Define
P0(O))
=H{0))
andie
{1,2.
..M
-1},
whereH(co)
andGm(co)
arethefrequency
responses ofthe discretetime filtersin(4.8).Theorthogonalityconditionsonthescalingfiltersand waveletfilters leadto
(4.23) [46]
i
_,.,
2mn\n(
Iran | CO +\P\
(0+M
\
M-.Si.
fori, j
e{0,1... M-l}
(4.23)Wecan expandtheseresultsintothesingle matrixformshownin
(4.24)
PM pja+
PM P\ca+
_hI.PM-\>+ 2tf M
2__.
M _____ M Pn\0)+ P,\co+ PmrJOH-M 24m-\) M 2n(M-i) MPAa) PAa>) Pu., [co)
R\(D+171 M
M.M-\)
M P,\co+2n P,\0)+ MJjJM-t,
M Pu., 0)+ 2jt pu-i <+ M 2a(M-l) M (4.24)The commutative nature of
(4.24)
allows us to reverse the order ofthe above multiplicationwhile maintaining the equations integrity.
Performing
this operation andexpanding the results,generates the final governing
filtering
equations that we are interested in for the M - Bandsystem.
I|/.M2=i
(4.25)4.2 M- BandBandlimitedOrthogonal WaveletDesign
We startwiththebasic filterequation givenin
(4.26)
where wehavenormalized(4.6)
usingtheexpressionin(4.8).
{Mo))=H(co){g))
(426)Assume
O(co)
is bandlimited to afrequency
com, i.e. O(co)=0for|co|>cOm. Then, from(4.26),
CO <Mn
(4.27) The Poisson Summation formula clearly indicates that the scaling function must have a
bandwidththatis greaterthanti.Wethereforerewritetheoomas
}-=7C + CC
(4.28)
Whereais some value greaterthanzero.From (4.26)we can alsodeterminethat
\<b(a)f
=\ for Q<a<7r-a\<t>(a>f
=0 fora>7i+aWe turn now to the filter that is defined
by
the scaling function in (4.26). Thefilter,
H(oo), isdiscrete which forcesthe
frequency
response to be 2uperiodic. Forour purposes wewill writethisas
H(a))
=H(a)+2n)
=H*(2n-co)
(430)This,
withthe Poisson Summationformula,
allows us to define regions of supportforthe filteritself.
Using
Figure 4.2 and Figure 4.3 as a visual aid and from(4.27),(4.28)
and(4.30)
wedefine,
H(a>)*0 for \aJl</r+ a M
(4.31) i Tt-va
2k
H[co)=Q for <_y<
M
I
\ MTheperiodicnatureof
H(co)
impliesthatwe can writeattheboundary
of(4.31)
T*-^r-J=T
mr
(4-32)We have attempted todepict the boundaries defined fromthe above conditions in Figure
4.2 below. The solid line shows the "ideal" filter
H(co)
and the dotted line shows theshifted version ofthe filter
by
2n. It shouldbe recognized that this is simplya visual aidin
deriving
anexpression for a.Clearly,
asillustratedbelow,
these two functions do notH(a>) H\_7--__.)
>2M- H-- -a AI
\ \\
n +o.
\
MA'
\
Figure4.2
-BoundaryconditionsfortheH(co)filterfrequencyresponse.Theextreme caseis determinedwhenthe twobandedges ofH(co)andH(2ti-Q))coincide.
In the ideal
(non-realizable)
case a would beidentically
zero. This would place thetransition band for
H(co)
exactly at 7i/M and the defined stop band at n. This resultssituationasin Figure 4.3a.
Extremecase Idealcase H(at) 2M -1k Hi2x-w) M \
\
,M\
. 1 1 2M-1te -a
/M
-n-a \
Figure4.3a,4.3b- BandextremesfortheMeyer scaling function
Theother
boundary
conditionis whenthestop bandoftheH(co)
filter lines updirectly
[image:52.513.152.372.84.274.2] [image:52.513.91.431.410.590.2]limit that is necessary forthe development of our a term and is shown in Figure 4.3b.
The regions shaded in gray indicate the transition bands of our filters. As mentioned
abovethesediagrams are merelytools forvisualization and are not intendedto depictthe
actual filters themselves. The specified regions illustrated are simply regions of support
for the
filters,
this idea is that the filter is not equal to zero and less than one in theregions shown in gray.
They
areidentically
zero where the boundaries are zero andidentically
one in the enclosed white regions. From Figure 4.3b we can determine thelimitonouracondition giventhescaling function as
(M-i\
a=n{M
+\jOur scaling functioncanbegenerated
by
choosinganatermintherangedefinedby
0<a<7T
M-n
(4.33)
(4.34)
M+
\y
To guarantee a valid scaling function the transitionbandmust satisfy the Poisson Summation
formula. Wewritethisas
1 0<co<n-a
\<$>{co\
=y{co) n-a<co<K+ a (4.35)
0 otherwise
The y(<>) is a function that satisfies the transition band defined
by
the Poisson Summationformula
+y{2n-co)2=l (4.36)
If we let M=2 in
(4.34)
we see that it does indeed reduce to the expected value in the two-Band Meyerwavelet case which was the result ofW.W. Jones in [4],however
we will show in
subsequent sections that tighterbounds on must be placed on
(4.33)
togenerate an orthogonal
relatedto theroll-offfactor ofthe square root raised cosine
(SRRC)
function, asshown in [31].Theexactrelationshipoftheaparameter andtherollofffactor is described in Appendix A. The
functionin
(4.33) incorrectly
impliesthatwhenweextendtotheM- Bandcase,we increasethe
range underwhichtheSRRCroll-offfactorgenerates avalidscaling function.
4.3 The Composite WaveletandEmbedded Wavelets
In the previous secti