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Rochester Institute of Technology

RIT Scholar Works

Theses

Thesis/Dissertation Collections

5-1-2003

Construction of M - Band bandlimited wavelets for

orthogonal decomposition

Bryce Tennant

Follow this and additional works at:

http://scholarworks.rit.edu/theses

This Thesis is brought to you for free and open access by the Thesis/Dissertation Collections at RIT Scholar Works. It has been accepted for inclusion in Theses by an authorized administrator of RIT Scholar Works. For more information, please contactritscholarworks@rit.edu.

Recommended Citation

(2)

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

(3)

Construction of M - Band Bandlimited Wavelets

for

Orthogonal Decomposition

Bryce Tennant

(4)

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.

(5)

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,

Tom

Kenney,

Mark

Chamberlain,

William

Furman,

Tibor

Dobri,

and the countless others at Harris Corporation that have

providedmoral,financialand academicsupportfrom

day

oneofmythesis.

They

are

truly

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 many

things fromthem

both,

all ofwhichhavegottenme towhereI amtoday.

Special thanks to

Greg

Smith who has always been my best academic sounding

board,

our informal technical conversations have always been some of the most

enjoyable.

Most

importantly,

thanks to my

loving

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 my
(6)

Abstract

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

(7)

Contents

Chapter 1 Introduction 1

Chapter 2 GeneralFilter

Theory

5

2.1 M-Band Filter Banks 5

2.2

Paraunitary

Filter Banks 1 1

Chapter 3 Wavelets 13

3.1 Basic Wavelet

Theory

14

3.2 The Discrete Wavelet Transform 17

3.3 Approximation of aSignal

f(t)

with

Scaling

Function (j)(t) 20

3.4 Bandlimited Wavelets 25

3.5 The Meyer Wavelet 27

Chapter 4 M

-BandWavelet Development 34

4.1 M-Band

Theory

ofWavelets 34

4.2 M- Band Bandlimited OrthogonalWavelet Design 40

4.3 The Composite WaveletandEmbedded Wavelets 44

4.4 Phase Response ofthe

Scaling

FunctionandWavelets 54

Chapter5 Examples 61

Chapter6Conclusion 77

Appendix A The

Relationship

ofthe aParameterto theRolloffFactor intheSquare
(8)

Appendix B -DefinitionoftheComposite Wavelet 82

AppendixC- Phase Response

ofM- Band Bandlimited Wavelets 84

(9)

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 27

Figure3.5-Meyerwavelet

frequency

response 29

Figure3.6-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

conditionsforthe

H(co)

filter

frequency

response 42

Figure4.3a,4.3b- Band

extremesfortheMeyer scaling function 42

Figure 4.4

-Scaling

functionand compositewavelet 46

Figure

4.5-Scaling

functionandtheM

-Scale relationshipwiththecompositewavelet..47

Figure 4.6

-Themagnitude squared

frequency

responseofthecompositewavelet 49

Figure 4.7- The

composite waveletand theembedded waveletfunctions 52

Figure 4.8

-Overlap

of selected regionsinthe

frequency

domain 57

Figure 4.9- The

progressive phase cancellation ofthescaling functionandwavelet 58

Figure 4.10 - Phases requiredforcancellation inthefirstandsecond wavelet pair

(10)

Figure 4.11 - Phases

requiredforcancellationinthesecond andfinalwaveletpair 60

Figure 5.1

-7(00)and y

{co)

for M=4anda=7t/7 62

Figure

5.2-M=4 Meyer scalingfunction,<_>(co) 63

Figure 5.3

-Scaling

functionand compositewaveletfor Meyer 4- Band filter bank 63

Figure 5.4

-Scaling

functionand embedded wavelets withinthecomposite wavelet 64

Figure 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 69

Figure 5.8

-Corresponding

filters for

H(<d)

and

Gm()

70

Figure5.9- 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

(11)

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 major

difficulty

in

designing

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 synthesis

process be minimized or at

best,

completelyeliminated. When considering the design of

orthogonal

filters,

both the magnitude response and phase response need to be closely

examined. When

decomposing

a signal into two bands it has been shown that satisfying

both magnitude and phase requirements simultaneously yields a unique set of filters.

Although these filters provide for perfect reconstruction,

they

do not provide for good

frequency

separation. Ifone oftheconditionsisrelaxed, manyuseabletwo- Bandfilters

can be generated. Quadrature Mirror Filters

(QMF)

are a set of such a filter banks.
(12)

filters,

called paraunitary filter

banks,

can yield perfectreconstruction while maintaining

individual filter integrity.

Althoughthe wavelettransformwas

initially

introduced as a moreflexible analysistool

forsignalrepresentation than the Short Time Fourier

Transform,

itwas quickly linkedto

the

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 works

by

Gopinath[3],

Vetterli[ll], Nguyen[35],

Selesnick[33], Walter[37]

and others have taken the wavelet

transformand combineditwiththe

theory

ofM- Band

perfect 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],

and

Jones[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

(13)

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

the

theory

ofthe M - Band

wavelet 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,

mother

wavelet 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

the

theory

ofthe bandlimited orthogonal M - Band

wavelet transform. We will

develop

the

constraints andequationsthatneedtobesatisfiedfortheM- Band

perfect reconstruction

filters, thereby

linking

the

theory

introduced in Chapter 2 with that of wavelets. With a

firm mathematical foundation provided, we will then proceedto

develop

our orthogonal

bandlimited M - Band

scaling

function,

and wavelets. As a direct result of the

development 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

(14)

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))

with

a 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} and

y/m{t)

where me

{0,1...M

-2} in the

time domain. The corresponding filters foreachofthese wavelets willbe given as

Gm(co)

for the

frequency

domain representation and gm(n) for the time domain representation

over the same range of m. In some special cases we will substitute

Pi(co)

for the above
(15)

Chapter 2

General Filter

Theory

X(n) H0(z) *0M vQM

t=

y0M W

H^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)

and

Hi(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 ifeach

of the signals bandwidth is half of the input signals

bandwidth,

the Nyquist sampling

theorem 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 are

never ideal and always contain some energy in the stop band past the cutofffrequency.

This implies that whenthe

downsampling

ofour signal occurs, frequencies greaterthan
(16)

of aliasing is of primary concern when

designing

filter banks. The output of these

decimators,

vo(n) and vi(n), are signals that can be encoded or analyzed prior to

transmission 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 summed

together to get a reconstructed version of the original input signal.

Ideally

this

reconstructionshould be a perfect replicaofinput signalwithapossible unit sampleshift

and constant gainmultiplier.

Working

backwards from Figure

2.3,

we see that the output of the reconstruction

portionofthebankis given as

X(z)

=

Y0(z)F0(z)

+

Y,(z)F,(z)

Where both

Y0(z)

and

Yi(z)

are the outputs of the two interpolators. In the z domain

interpolation 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" filtered

signal 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)
(17)

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 causes

distortion in thereconstructed signal and shouldbeeliminated wherepossible.

Using

the

following

notation,

T(zyFo(z)H0{z)+F,{z)H,{z)

A(z)

=

F0{z)H0{-z)+F,{z)H,{-z)

2

wecanrewrite

(2.4)

as

X{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 the

expression 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 theform

T{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 linear

phase to generate perfect reconstruction ofour input signal. If in the above system we

(18)

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,

introduced

by

Smith and Barnwell in

1986,

is commonly referred to as a

Quadrature Mirror Filter (QMF). Although

by

definition aQMF is a set offour

filters,

the term has become synonymous with any set ofM filters that act as an analysis and

synthesis pair.

Typically

they

are denoted

by

the numberofchannels inthefilter

bank,

or

asaM-channel QMF. Such asystemis shownin Figure 2.4.

H0.z) xoM

Im

v0(n)

|m

yoN *bl*) kT

xl(n' *M yjM

? HjM

Im

I'M

^(z)

^

*

fc

y2(n)

*yz)

|m

I'M

F2(z)

1

^ w

i

m

,

Hi

fc

H [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 in
(19)

desire 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 mth

analysis

filter, Hm(co),

outputin terms of

the systeminput.

Therefore,

foreach subband signalpriortothedecimationwecanwrite

Xm(z)=X(z)Hm{z)

where me

{0,1... M-l}

The output

frequency

response of each analysis filter is such that the resulting

bandwidth is a factorofM less than the input signals bandwidth. Withthe signal

Xm(z)

being

bandlimited

by

%/M, wecan passthe resultinto adecimatorthatdown-samples the

signal

by

afactorofM withoutany loss in signalintegrity. However becausewe arenow

decimating

into M bands asopposedto twoweneedto accountfortheadditionalalaising

termsthatareintroduced dueto thenon- ideal

nature ofthefilter Hm(co). Inthezdomain

the

down-sampling

results in anewsignal

Vm(z)

definedas

Vm{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 in

between 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>)

(20)

ina subband output. The final output ofthesystemis the sumof all theresulting subband

signals as shownin

(2.8)

M-l M-l

*(*)=Z^(*(*)

= XrE

4zWl)ZFm(z)Hm(zW!)

m=0 (2.8) M-l Ifwe now define a newfunction

A,

(z)

=

j]

Fm

{z)Hm

(zWl

)

wecansimplify

(2.8)

tobe

m=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 a

function 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 whichwe

can 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

the

appropriate conditions on Ain

(2.11)

allowsus to generate afilter bankdefined

by

H(co)

that is representative of a QMF. We therefore define

A(z)

such that all aliasing

components are

identically

zero andthe only contributingportion ofthe signalis a result
(21)

A(z)

=

M\{z)

0

0

and solve for the proper unknowns in

(2.11)

to obtain the final filter design. The

H(z)

matrixshownin

(2.10)

is commonlyreferredtoastheAlias Component

(AC)

matrix.

2.2

Paraunitary

Filter

Banks

Paraunitary

filter banks are a special class of perfect reconstruction filter banks.

They

have the addedbenefitof allowingthe designerto generate synthesisfilters

directly

from

thedesignoftheanalysisfilters. Thecausaltransfermatrixforaparaunitary

filter,

H(z),

satisfies H

(z)H

(z)

=cifor all z. These conditions come

directly

from the orthogonal

nature ofthe analysis filters. One useful property ofthe paraunitary matrix is thatfor all

analysis

filters, [H_(z) Hi(z)...Hmi(z)]

wehave

M-l

Y_\Hk{a)]~=c

(2.12)

k=0

This is knownasthepowercomplementary

(PC)

conditionandwillbepivotalin future

sections 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 filter
(22)

found

by

solving for F in (2.11). It can be shown that in the dyadic case the synthesis

filters are suchthat F0{z)=z~NH0{z)and F,(z)=z~NH,{z) [46]. Inthe timedomainthis is

representedasthecomplex conjugate ofthemirrored analysis

filter,

or

f0{n)

=

h^{L-n)

(2.14)

f,(n)

=

hl{L-n)

(2.15)

Utilizing

theparaunitaryconditions aboveanentiretwo

-channel perfectreconstructive

(23)

Chapter 3

Wavelets

Wavelets haveemerged as a powerful field in signal analysisover thepast

thirty

years.

Key

results introduced

by

Meyer[27,

40],

Daubechies[39],

Vetterli

[11,42]

and many

others have allowed this branch of mathematics to grow into a mature field of study.

Introduced

initially by

the Hungarian born Mathematician Aldred

Haar,

wavelets started

out 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

the

theory

of wavelets. He developed the

theory

to

overcome 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 was

formally

developed

by

Meyer and Mallat in the late 1980s. This provided further

theory

for

developing

specific wavelets andscaling functions basedonthe application ofinterest. It

was not until Ingrid Daubechies made the connection between wavelets and filter bank

(24)

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 a

fairly

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 thesis

development. 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)
(25)

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 signal

decomposition. The

(<__>)

termis the

frequency

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 have

advantages and disadvantages in implementation. The literature found throughout this

field

typically

focuses on compactly supported wavelets. This is primarily due to the

well-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 infinite

frequency

response, causing signal leakage into the designed

filters stop band. In addition these wavelets tend notto have smoothband edges and can

result in choppy

looking

functions. This phenomenon can clearly be seen when

examining 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

(26)

appliedto thefunction inthe

frequency

domainthatgovernsthepass

band,

stop bandand

transitionbandofthescalingfunctionsFouriertransform.

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 function

f(t)

willbe generated using aweighted sum ofthebasis functions wherethe weights are

determined as the projection of the basis functions with the function

f(t)

itself. Because

wavelets 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 this

function,

\|/(t),

by

factors a and b

respectively. When afunction

f(t)

is projected onto these dilated functions the result is a

twodimensionaltime

[image:26.513.171.345.141.286.2]
(27)

W{a,b)

=

-L]

f(t)*

dt (3.4)

V a J

From

(3.4)

we seethatour waveletbasis functionsaredefinedas

1

(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 weget

afinal 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 the

mother wavelet respectively. It is these two parameters that provide for the

time-frequency

representation offered

by

the wavelet transform. If our wavelet satisfies the

admissibilityconditionin

(3.3)

thenwe candefinetheinversewavelettransformas

1 ~

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

(28)

can be removed and a discrete time equivalent can be developed [39]. We start

by

defining

a new function <|)(t)which mustsatisfythe

following

properties

$0{t)dt

=l

\\<p(tf=]\<p(tfdt

=l

(</>(t),0{t-n))

=

d{n)

This new

function,

called the scaling

function,

musthave unit energy and be orthogonal

to integertranslates ofitself. This implies thateach translate is

linearly

independent and

the set \(f){t-n)} where n is an

integer,

forms abasis for alinear vector space. We will

denote this vector space as V0. If we dilate the scaling function

by

a factor of 2~k

we

generate,

(<p(2-kt),0(2~kt-n))

=

2kd{n)

(3.8)

Equation

(3.8)

defines alinearvectorspace, Vk. Thisdilation implies thatfor

increasing

k

the corresponding vector space,

Vk,

generates coarser and coarser approximation of

L2{R}. Hence for any function

f(t)

that resides within

Vk,

f(2t)

must completely be

contained 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 linear

combination ofbasis functions of the next finer level. More concisely, if

V0

c

V_,

then
(29)

0(0=

c(nM2r-n)

(3-9)

We now

develop

a

frequency

domain representation of the results provided in (3.9). It

was definedthatourscaling functionmustsatisfythe

following

orthogonalityconstraint

(<p{t),</)(t-n))

=

d{n)

(3.10)

Taking

the FT of

(3.10)

results in thecommonlyreferred to Poissonsummationformula

shownin

(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 k

integer}

forms a nested subspace

[40],

then

the c(n) in

(3.9)

satisfies the powercomplementarycondition givenin (2.12). We define

theFouriertransformofthediscretesequencec(n) as

C(co)=

c(/i)e"-''fl"

(3.12)

Ifwelet t= andsubstitute thisinto

(3.9)

wearrive at 2

0-=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

the
(30)

2(2co+4nk)=C{a)+2nk)(a)+

2nk)

(3-15)

Taking

themagnitude squaredofbothsides and summingoverkgives

4^>(>

+

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 Poisson

Summation formula defined in (3.1

1)

reducingthe expressionto

45>(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) t

Ifweaddboth

(3.17)

and

(3.18)

togetherweget

4^\{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)

are

nothing more that the even and odd terms of k in the Poisson summation formula

reducing

(3.19)

to ourfinal formof

4=

|c(fi>)|2+|c(__>

+ (3.20)

3.3

Approximationofa

Signal

f(t)

with

Scaling

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
(31)

/*(0=

X0(2"^-"M^")

(3-2D

Therefore at some arbitrary resolution k and the previous coarser resolution

k+1,

we

define 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 finer

approximation

fk(t)

onto space Vk+i. We transition from signal space

Vk

to signal space

Vk+i

with some error signal defined as gk+1{t)=fk{t)-fk+l(t). The error, gk+i(t), is

containedcompletely within the signal space

Vk

andis orthogonal to the signal fk+i(t). It

would be plausible to reconstruct some signal

fk(t)

be retaining a single course

approximation and a series of detail functions captured when moving from vector space

Vn

toVn+i.We willdefine thewaveletfunction \|/(t) thatwillbeusedto capturethe detail

lost when moving to a coarser subspace. This wavelet

function,

as with the scaling

function,

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

(32)

(>(0.^-n))

=0

(3#24)

Equation

(3.24)

allows us to define \|/(t) in terms oftranslates ofthe scaling function in

thepreviouslyfinerdomain.

p{t)=

^din^t-n)

(3.25)

The coefficients,

d(n),

satisfy similar properties tothose inthe scalingfunction definition

andusingsimilardevelopmentasin

(3.20)

wecan write

d(n)

inthe

frequency

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

responses

C(co)

and

D(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 ofthe

matrix multiplication withoutaffectingthe

integrity

oftheresult

C{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 some

interesting

implications

thatrelate the

theory

of wavelets to thatoffilter banks. Ifwelook backto the previous

section onM- Band

filtering

we see that the above equation defines aparaunitary filter

bank. This implies that the two functions

C(co)

and

D(_o)

are actually low pass and high
(33)

approximation function and the

D(co)

is developed from the detail information.

Typically

low pass

filtering

provides an approximation or

averaging ofthe original signal whereas

thehighpass

filtering

preserves the signals detail information. From

(3.29)

we can write

three

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)whicharesimply

h(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 can

quickly determine the analysis and synthesis filters

H(co)

as a prototype function. A

simple 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

(34)

AnalysisFilters Synthesis Filters

Low Pass

h(n)

determined

by

governingrulesofscaling

function.

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,

we

can 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 the

scaling filterandwavelet

filter,

as shownin Figure

3.2,

we canfollowthe

filtering

with a

decimation.Thisgenerates a course approximation,ai(n) andadetail

function,

di(n). The

low 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

(35)

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

Wavelets

Typically

wavelets developed for signal processing have compact support in time

resulting in FIR filters.

Many

recentpapershave been writtenonthe developmentofFIR

waveletfilter 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 Figure

3.3,

results in a compactly supported

filter,

clearly

implying

that the

frequency

response is without bound. The

frequency

response ofthe

Daubechies scaling functionandwaveletcan beseenin Figure 3.4.

Compactly

supported
(36)
[image:36.513.56.471.72.298.2]

Daubechies 4

Tap Scaling

Function Daubechies4

Tap

Wavelet Function

Figure 3.3- The Daubechies 4

tapscaling functionand wavelet intime

Depending

on the application, the smoothness of the wavelet can be critical in the

implementation 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 differentiable

only once. The implication is thatthere is a slow

decay

in the

frequency

domain which

does notallow for

frequency

localizationof events. Althoughthere aremany Daubechies

wavelets ofvarying

lengths,

we need alarge number of filter taps to generate a smooth

wavelet that is M times differentiable. Our design will focus on wavelets that are

(37)

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

Meyer

Wavelet

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 apparent

disadvantage,

the filter is practical and offers both good time and

frequency

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 in

communication 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]

showedthat

theMeyerwaveletnaturally arises insubsampledbandlimitedprocesses.

They

illustrated

that the reconstruction filter banks can optimally recreate a sub sampled bandlimited

(38)

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 the

bandlimited Meyer wavelet can be

directly

designed to optimally match the magnitude

and 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 as

being

the positive

bandwidthofthe

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;r

1

2 =1 for 0<co<2k

(3.37)

Ifthe half band

frequency

of

O(oo)

is defined to be c_m, then from our discussion above

com

> n. Ifwedefinethe excessbandwidththatis greaterthan n as awecan write

o)m=n + a

(3i38)

where a>0.

Utilizing

the reducedPoisson Summation formula given in

(3.37)

we can

see 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

(39)

Figure3.5-Meyerwavelet

frequency

responseThe

structureisaboveis 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 easy

toshowthat

H{cd).

0>{2co)

(3.40)

This equation allows us to

develop

the

frequency

response of the associated low pass

filter with the knowledge of the

frequency

response of the scaling

function,

<&(c_).

Knowing

that <_>(2co) is a compressed version of

<_>(co),

we can generate regions of

supportforthefilter

H(co)

withthe definitionin (3.40). Recall that

h(n)

is adiscretetime

filter,

which implies that the

frequency

response,

H(co),

is 2ii periodic. Because the [image:39.513.183.360.75.288.2]
(40)

H{o))

=Ofarco> n+a

Theperiodic nature ofthe

filter, H(),

allows ustowrite

H 2n- co+a \

=H

(3x-a

=0

Graphically

we showthese twoperiods of

H(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 filter

H(co)

is exactly zero in the

n +a

region <co< k+ aand the shifted version is

identically

zero in the range defined

2>7t a

by

7t-cx<co< . Ifwe takethe bestcase scenario wherethese twobands coincide

we seethat

K+ a

=n-a

(41)

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 design

now reduces to

determining

appropriate transition bands that satisfy the Poisson

summation

formula,

graphically depicted in Figure 3.6. We can do this

by

finding

a

function

f(co)

andg(co) suchthat

f(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 is
(42)

fticof

=

0,

for

0<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 of

the 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 the

FT ofthetwo scalerelationship found in

(3.25)

we get an equivalent

frequency

domain [image:42.513.176.349.280.470.2]
(43)

2V{2a>)

=

D{co){co)

(3.44)

This in conjunction with the definition of the function g(n) in

(3.33)

provides an

expressionforthewaveletsfilteras

G{co)= v / (3.45)

<__>()

The above development does not illustrate the phase solution of the wavelet. This is

(44)

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 the

theory

for wavelet analysis. A synthesis routine for the Meyer scaling

function,

wavelet function and the corresponding filters have been presented. We will

now 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 the

scaling 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-Band

Theory

of Wavelets

Assume that we are given some signal,

/(f)

e

V_,

that we wish to break up into M

separatebanks. It is desirablethattheoriginalfilteredsignal canbereconstructedwithout

errordue to aliaising oramplitude distortion. We know that

Vk

is the space thatcontains
(45)

detailed space to Vk. Ifwe define

Wk

as the detail lost in going from

Vk_]

to

Vk

then we

can write

Vk=@Wj

(4.1)

j=k

and

Wk

e Vk_,. Here wehave used

Wk

to representthe total

detail 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

from

Vk_i

to Vk.

Rewriting

(4.1)

to

accountforthis 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

(46)

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 be

orthogonaltointegertranslatesofitselfacross 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 integershiftsfromthepreviousfinerresolution

0{t)=

Y,c(n)0(Mt-n)

(4.5)

forscalars, c(n

),

ne Z. This becomes inthe

frequency

domain

M(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-l
(47)

MVm(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)

and

4/m(co),

to the digital filter

frequency

responses,

C(co)

and

Dm(co)

respectively.Forourdevelopmentwe normalizethefilters

by 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 the

expressionco=co+2nk

.

Taking

the magnitude squared of the result and summing both

sidesoverkwe 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 summation

formula,

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 at

YJ\{Mco

+Mn+

2nMkf

=\H{co+

nf

(4-13)

k

(48)

forall co

2nm M

me

{0.

. .M

-1}.

Summing

theresultantMexpressions togethergives

M-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 wherethereis

a 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 results

thatweobtainforthewaveletfilters

Gm(co)

are

M-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 with

one another in the

frequency

domain. We know that each wavelet function can be

expressedas 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

(49)

{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)

mustbe

identically

equal tozeroor

J'"(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 that

we 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)

canbewritteninthe

frequency

domain as

M-l

I

M-l

I

?=0 co+^-)h M

)

2nq

co+ V M G\ f

2V

f co+ V M

2nq

co + V M (4.22)

=0 foreach me

{0,l...Af-2}

=0 for each m,pe

{0,1

...M

-2}

where m p

where we haveusedtheformula in

(4.8)

toputtheresults intermsifthe

H(co)

and

Gm(co)

filters. This development leadsusto acompact resultthatcanbewrittenina matrixform.

Combining

theresults of

(4.9),

(4.15)

and

(4.22)

we can write the

following

summation

form shown in

(4.23),

which provides the properties of the scaling and wavelet filters that

become crucial to our development. Define

P0(O))

=

H{0))

and
(50)

ie

{1,2.

..M

-1},

where

H(co)

and

Gm(co)

arethe

frequency

responses ofthe discretetime filters

in(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) M

PAa) PAa>) Pu., [co)

R\(D+171 M

M.M-\)

M P,\co+2n P,\0)+ M

JjJM-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 multiplication

while maintaining the equations integrity.

Performing

this operation andexpanding the results,

generates the final governing

filtering

equations that we are interested in for the M - Band

system.

I|/.M2=i

(4.25)

4.2 M- BandBandlimitedOrthogonal WaveletDesign

We startwiththebasic filterequation givenin

(4.26)

where wehavenormalized

(4.6)

usingthe

expressionin(4.8).

{Mo))=H(co){g))

(426)

Assume

O(co)

is bandlimited to a

frequency

com, i.e. O(co)=0for|co|>cOm. Then, from

(4.26),

(51)

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+a

We turn now to the filter that is defined

by

the scaling function in (4.26). The

filter,

H(oo), is

discrete which forcesthe

frequency

response to be 2uperiodic. Forour purposes wewill write

thisas

H(a))

=H(a)+

2n)

=

H*(2n-co)

(430)

This,

withthe Poisson Summation

formula,

allows us to define regions of supportforthe filter

itself.

Using

Figure 4.2 and Figure 4.3 as a visual aid and from

(4.27),(4.28)

and

(4.30)

we

define,

H(a>)*0 for \aJl</r+ a M

(4.31) i Tt-va

2k

H[co)=Q for <_y<

M

I

\ M

Theperiodicnatureof

H(co)

impliesthatwe can writeatthe

boundary

of(4.3

1)

T*-^r-J=T

m

r

(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 the

shifted version ofthe filter

by

2n. It shouldbe recognized that this is simplya visual aid

in

deriving

anexpression for a.

Clearly,

asillustrated

below,

these two functions do not
(52)

H(a>) H\_7--__.)

>2M- H-- -a AI

\ \\

n +o.

\

M

A'

\

Figure4.2

-BoundaryconditionsfortheH(co)filterfrequencyresponse.Theextreme caseis determinedwhenthe twobandedges ofH(co)andH(2ti-Q))coincide.

In the ideal

(non-realizable)

case a would be

identically

zero. This would place the

transition band for

H(co)

exactly at 7i/M and the defined stop band at n. This results

situationasin 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 bandofthe

H(co)

filter lines up

directly

[image:52.513.152.372.84.274.2] [image:52.513.91.431.410.590.2]
(53)

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 the

regions shown in gray.

They

are

identically

zero where the boundaries are zero and

identically

one in the enclosed white regions. From Figure 4.3b we can determine the

limitonouracondition giventhescaling function as

(M-i\

a=n

{M

+\j

Our scaling functioncanbegenerated

by

choosinganatermintherangedefined

by

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 Summation

formula

+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)

to

generate an orthogonal

(54)

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- Band

case,we increasethe

range underwhichtheSRRCroll-offfactorgenerates avalidscaling function.

4.3 The Composite WaveletandEmbedded Wavelets

In the previous secti

Figure

Figure 2.3- The two- Band decomposition and reconstruction filter bank.
Figure 2.4- M- Band decomposition and recomposition filter bank
Figure 3.1The time domain representation of the Meyer Wavelet
Figure 3.3- The Daubechies 4 tap scaling function and wavelet in time
+7

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