现代数字信号处理AdvancedDigitalSignalProcessingch5wavelet

Advanced

Digital

SignalProcessing(Modern

Digital

Signal

Processing)Chapter

5

Time-FrequencyAnalysis

and

Wavelet

TransformGeneral

expressionInner

product5.1

Linear

TransformWave

&

Wavelet

TransformWavesWaves

are

non-compact

(infinite)

support

funcNon-compact

support

function:The

functions

extend

to

infinity

in

both

directWaveletsWavelets

are

compact

(finite)

support

functiThey

vary

with

frequency

as

well

as

positionCompact

support

function:The

functions

are

in

a

limit

duration.Wave

&

wavelet

transformwaveswaveletsWave

transform(wide-sense)

wavelettransformOrthogonal

transformOrthogonal

basis

functionIf

c=1,

then

g(ω,t)

is

orthonormal

basis

functiOrthogonal

(orthonormal)

transformOn

the

other

hand,

the

components

of

f(t)corresponding

to

the

basis

g(ωi,t)

is

orthogwith

any

basis

g(ωj,t),

i≠j

and

will

contribnothing

to

F(ωj).Understanding

of

the

(orthogonal)

transformIntuitive

interpretation

of

orthogonal

traFor

a

given

ωi,If

f(t)

is

orthogonal

with

g(ωi,t),

then

F(i.e.

there

is

no

component

corresponding

to

tbasis

g(ωi,t)

in

f(t).

Otherwise,

the

componf(t)

corresponding

to

the

basis

g(ωi,t)

willecompositiconmpose

F(ωi)

in

ω

space.effectGeometric

interpretation

of

orthogonaltransformOrthogonal

ProjectionThe

orthogonal

transform

of

f(t)

is

aprojection

of

f(t)

into

a

orthogonal

basis

sformed

by

{g(ωi,t)},

i=1,2,…Non-orthogonal

transformNon-Orthogonal

ProjectionThe

same

component

of

f(t)

may

projectinto

different

bases.

Redundancy

willprobably

exist

in

the

transform

results.Fourier

transform

(FT)i.e.

the

FT

is

an

orthonormal

wave

transform.Non-Stationary

(Time-Variant)

SignalStationary

(time-invariant)

signalNon-stationary

(time-variant)

signalx1(t)x2(t)x3(t)x4(t)FT

of

non-stationary

(time-variant)

signalSignals

are

different,but

spectrums

are

similarDeficiency

of

wave

transform

(e.g.

FT)Wave

transforms

are

not

suitable

for

timevariant

signal

since

they

don’t

includeposition

(time)

information

in

the

transforesults

(e.g.

FT

analyzes

the

globalfrequency

distribution

of

a

signal,

but

itnot

characterize

the

local

behavior

of

thesignal).5.2

Time-Frequency

AnalysisBasic

IdeaIn

FT,

the

local

behavior

of

a

signal

is

notrepresented

in

the

signal’s

frequency

spectruThe

FT

is

not

the

most

proper

representation

forthe

time-variant

signals

or

the

signals

containtransient

or

localization

componentsTime-frequency

analysis:

characterizing

the

timfrequency

information

of

a

signal

simultaneouslits

spectrumExamples

of

Time-Frequency

AnalysisMain

Tools

of

Time-Frequency

AnalysisShort

time

Fourier

transform

(STFT)Wavelet

transformWigner

distribution

(WD)Quadric

transform

(non-linear

transform)Time-frequency

distributionWiger-Ville

distribution

(non-stationaryrandom

signal)DefinitionSTFT

of

continuous

time

signal

x(t)where

w(t)

is

a

real

finite-width

windowfunction

which

slides

along

x(t)STFT

of

discrete

time

signal

x(n)where

w(n)

is

a

real

finite-length

windowsequence

which

slides

along

x(n)5.3

Short

Time

Fourier

TransformThe

result

of

STFT

is

a

2-D

function

whichreflects

the

signal

spectrum

varied

with

time.FT

of

Windowed

x(t)Wide-sense

wavelet

transformis

a

compact

support

function

(wavelet),

andThe

STFT

isThe

support

width

of

the

wavelet

(i.e.

thewidth

of

the

window)

is

constant

for

allfrequency

components.

The

Conflicting

Requirements

betweenthe

Frequency

Resolution

&

the

TimeResolution

in

STFTFrequency

resolution

requirementThe

window

width

T

should

be

wide

enough

togive

the

desired

frequency

resolution.Time

resolution

requirementThe

window

width

T

should

be

narrow

enough

soas

not

to

blur

the

time

dependent

events,

i.e.signal

segment

included

in

the

window

can

betreated

as

stationary

approximately.Partition

of

time-frequency

plane

in

STFTtProblems

of

STFTHeisenberg

uncertainty

principleWe

cannot

perfectly

localize

events

in

timeand

frequency

simultaneously!

STFT

is

redundant

representation

Notgood

for

compression

The

same

and

t

throught

the

entireplane!Multi-Resolution

Analysis

(MRA)Basic

ideaThe

high

frequency

components

vary

rapidly

intime.

A

relatively

short

signal

segment

cancharacterize

them

properly,

hence

a

relativelynarrow

time

window

can

be

used

(high

timeresolution

and

low

frequency

resolution).On

contrary,

the

low

frequency

components

varyslowly

in

time

anda

relatively

wide

time

windowshould

be

used

(high

frequency

resolution

andlow

time

resolution).HigherfrequencyMore

narrowtime

windowLowerfrequencyWider

timewindowHigher

timeresolutionHigher

frequencresolutionPartition

of

time-frequency

planeMRADifferent

time

and

frequency

resolutions

areadopted

to

the

different

frequency

(scale)componentsof

signal

at

same

time5.4

Continuous

WaveletTransform

(CWT)DefinitionCWTis

mother

(basis)

waveletwherewhich

satisfiesScaling

&

translation

of

mother

waveletwhere

a

is

scaling

(dilation)

parameter

and

b

itranslation

(shifting)

parameter.

is

thebasis

function

of

CWT.

It

is

called

the

analysiwavelet.Scaling

(dilation)Translation

(shifting)Scaling

and

translationRepresenting

CWT

inFrequencyDomainis

ω0,

andband

width

is

B,

then

the

central

frequency

and

band

wof

the

FT

ofare

ω0/aand

B/a

respectively,

i.e.Constant

quality

factorIf

the

central

frequency

of

the

FT

ofProperties

of

waveletFrequency

spectrum

analysis

abilityIf

the

wavelet

is

a

band-pass

filter

withrelatively

narrow

passband,

then

the

waveletwith

different

a

can

characterize

the

differentfrequency

componentsof

a

signal.Inverse

CWT

(ICWT)Admissible

conditionwhere

is

the

FT

ofThe

satisfies

the

admissible

condition

is

aadmissible

wavelet.A

basic

restriction

forconstructing

a

mother

waveletICWTExamples

of

Mother

WaveletsProperties

of

CWTLinearityTime

shiftingScalingMoyal

theorem

(inner

product

theorem)Energy

of

WTReproducing

kernel

equationICWTReproducing

kernel:

the

dependence

betweenRedundancyof

CWTReproducingkernel

equation5.5

Discrete

Wavelet

Transform(DWT)Definition

Discretizing

of

the

Scaling

&

TranslationFactormother

(basis)

waveletBasis

withlarger

scaleLowersampling

rateDWT:

DWT

or

wavelet

seriesUsually,are

adopted,

thenWavelet

FrameRequirements

for

Discrete

Wavelet

BasisCompletenessCancompletely?Reversibilitycharacterize

the

x(t)Can

x(t)

be

restored

fromstably?UniversalityWhether

any

x(t)

can

be

represented

bya

linear

combination

of

the

wavelet

basisCompletenessUniquenesscontinuityReversibilityUniquenesscontinuityFrameLet

be

a

cluster

offunctions

in

Hilbert

space

H,if

for

anyfunction

,

it

is

held

thatthen

is

a

frame.Moreover,

if

A=B,

thenframe

andis

a

tightIf

A=B=1,

then,henceis

a

set

of

orthogonal

bases

in

Hspace.

Such

a

set

of

bases

is

orthonormal

ifwheresatisfiesDual

frameRestoringand

it

is

called

the

dual

frame

of,

itFor

convenience,

when

A≠B

butis

usually

approximated

aswhereIf

A=B,

thenWavelet

frameIf

for

any

function

x(t),

the

wavelet

basisfunction

satisfiesthen

is

a

waveletframe.Its

dualwavelet

frame

iswhich

satisfiesIf

A=B,

thenorandIf

A≠B,

theni.e.is

an

admissible

wavelet.If

is

a

wavelet

frame,then

it

meets

the

three

requirements

fordiscrete

wavelet

basis

proposed

before,and

Designing

Orthonormal

Wavelet

Basiswith

MRA

Orthogonal

wavelet

basis:

removing

theinformation

redundancy

in

the

data

aftertransformationReproducing

kernel

equation

in

DWT

with

tight

fra