a short introduction on wavelet transformation and … · a short introduction on wavelet...

A short introduction on Wavelet Transformation

and applications in Electric DrivesDamian Giaouris BEng, BSc, PG Cert, MSc, PhD

Senior Lecturer in Control of Electrical Systems

Electrical Power Research Group

School of Electrical and Electronic Engineering,

Merz Court,

Newcastle University,

Newcastle Upon Tyne,

NE1 7RU,

United Kingdom

[email protected]

https://www.staff.ncl.ac.uk/damian.giaouris/

TUM – Jan 2016

Outline

What is a

Transformation?

What is the

CWT?

What is the

DWT?

Drives

Applications

What is a

Transformation?

The FT is NOT

an integral

What is the

CWT?

What is the

DWT?

Drives

Applications

What is a

Transformation?

The FT is NOT

an integral

What is the

CWT?

Why/How

What is the

DWT?

Drives

Applications

What is a

Transformation?

The FT is NOT

an integral

What is the

CWT?

Why/How

What is the

DWT?

Why/HowDrives

Applications

Vector Spaces

Vector Space - A set of objects (vectors) that satisfy some properties:

A1 closure

For all v1, v2 ∈ V , v1 + v2 ∈ V.

A2 identity For each v ∈ V , there is a zero element 0 ∈ satisfying v + 0 = 0 + v = v.

A3 inverses

For each v ∈ V , there is an element −v (its additive inverse) such that

v +(−v)= −v + v = 0.

A4 associativity For all v1, v2, v3 ∈ V , (v1 + v2)+ v3 = v1 +(v2 + v3).

A5 commutativity For all v1, v2 ∈ V ,v1 + v2 = v2 + v1.

S1 closure For all v ∈ V and α ∈ R,αv ∈ V.

S2 associativity For all v ∈ V and α, β ∈ R,α(βv)=(αβ)v.

S3 identity For all v ∈ V ,1v = v.

D1 distributivity For all v1, v2 ∈ V and α ∈ R,α(v1 + v2)= αv1 + αv2.

D2 distributivity For all v ∈ V and α, β ∈ R,(α + β)v = αv +βv.

a2

a1

a1+a2

a1

a2

Orthonormal vectors

Vector space S Basis:

A subset of vectors that can describe any other vector in S

21 bba 21 cc

Cartesian plane The x-y axes

22

1

ba

ba 1

c

c

?ic21111 bbbbab 21 ccOrthonormal

01 21 ccab1

Not Orthonormal

abb 11

221 ''

T

cc

b1

b2

a

b1

b2b'2

b'1

a

211 bbabbaa 2

Transformation

Inverse transformation

sinsin

coscos

22

1

aba

aba 1

c

c

A vector space = The set of all 2x2 matrices

A vector space = The set of all polynomials of order n or less

General VS

xixp2

31 2

22

5xxp

1 2

23 53 6

2 2p p

i x x p x

Inner product=

ax

ax

dxxpxpxpxppp 2*

12121, Length=

ax

ax

dxxpxp 11*

02

5

2

31

1

221

x

x

dxxxixpxp

12

3

2

31

1

11

x

x

dxxixixpxp lorthonorma, 21 xpxp

Rx

dxxpxpRL *2The set of all square integrable functions:

12

5

2

51

1

2222

x

x

dxxxxpxp

Transform or correlation?

21 bba 21 cc

xpdxxpxpxp i

i Rx

i

*

i

ii bbaa

correlation of and when i i

x R

p x p x dx p p x R

Transformation = Correlation

i

i

Rx

i dixpdxxpxpxp *

iic ba

Rx

ii dxxpxpc *

Correlation

x(t)=[1 2 3 4]

y(t)=[0.8 1.8 3.1 4.1]

How “similar” is x with y and z?

z(t)=[-0.81 -2.3 1 10]

sum(x.*y)/norm(y)=5.468

sum(x.*z)/norm(z)=3.634

sum(signal 1 x signal 2)=Inner product => Correlation/Comparison

corr x t y t dt

1corr x t y t dt

y t

T(y)

yy1 y2 yn

T(y1)

T(yn)

T(y2)

Transformation

A detailed comparison (inner product) with a series of “test” signals

1 1 1c T y x t y t dt

x(t)

Any set of signals y can be used

For the inverse transformation, specific rules must be applied on the signals y (form a basis

in the vector space)

, {y1(t), y2(t),…}

T(y)

yy1 y2 yn

T(y1)

T(yn)

T(y2)

T(y)

yy1 y2 yn

T(y1)

T(yn)

T(y2)

T(y)

yy1 y2 yn

T(y1)

T(yn)

T(y2)

2 2 2c T y x t y t dt

n n nc T y x t y t dt

Fourier Transform

Transformation coefficient:

Inverse Transformation:

t

t

i dttXtfic*

ditXictf i

tji etXtX

t

t

tj dtetfc

dectf tj

t

t

tj dtetfF

deFtf tj

2

1

A comparison of our signal with a series of sinusoids

Fourier Transform

t

t

tj dtetfF

deFtf tj

2

1

A comparison of our signal with a series of sinusoids

sum(sin(0.5*t).*sin(t))=-5.281796023015190e-07sum(sin(2*t).*sin(0.5t))=2.112762663746470e-06sum(sin(0.5*t).*sin(0.5*t))=6.283185309820478e+02

t

Fourier Transform

1c 2c

complexci

12

Fourier Transform

cRe

cIm

c

cArg

Well known theory of FT…

Fourier Transform as a Flow Chart

Choose Frequency ω

Compare x(t) with ejωT and get the

corresponding coefficient

Problems with the FT


t

Signal

t

Signal

Original signal

Rectangular

window

otherwise

atatrect

,0

,1,

atjtj dtetfdtetrecttfF

0


2

2

t

2

2

1

t

Rectangular

window

t


0 1 2 3 4 50

0.5

1

1.5

2

2.5

3x 10

7

frequency, Hz

2Hz @ 12.1Hz @ 0.5

0 1 2 3 4 50

0.5

1

1.5

2

2.5

3x 10

7

frequency, Hz

2Hz @ 12.1Hz @ 0.25

0 1 2 3 4 50

0.5

1

1.5

2

2.5

3x 10

7

frequency, Hz

2Hz @ 12.1Hz @ 0.1

To reduce the main lobe of the sinc() we must increase the length of the time window.


ttty 100sin20cos10cos

5 f, Hz

|F(f)|

10 50

0 0.1 0.2 0.3 0.4 0.5 0.6-3

-2

-1

0

1

2

3

0 10 20 30 40 50 600

2

4

6

8

10x 10

4

frequency, Hz


ttty 100cos20cos10cos

0 0.1 0.2 0.3 0.4 0.5 0.6-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

0 10 20 30 40 50 600

0.5

1

1.5

2x 10

4

We need another transformation!

Windowed complex exponentials

t

Signal

w

FT

t

Signal

w

FT

t

Signal

w

FT

dtetwtfF tjW

,

Short Time FT

STFT as a Flow Chart

Choose Frequency ω



Choose Shift τ

Capture each non stationary event in separate time windows

Time resolution

t

Time width

of window

Event 1 Event 2

t

Time width

of window

Event 1 Event 2

Frequency resolution

2

2

t

2

2

1

2

2

t

2

2

1


Capture each non stationary frequency in separate frequency windows

so that it will not be covered by the main lobe of the sinc function

f

Frequency width

of window

Event 1 Event 2

f

Frequncy 1

Frequency 2

Frequency width

of window


2

2

t

2

2

1

2

2

t

2

2

1

Decreasing time window=>Increase frequency window

Time/Frequency coverage

t

Time width

of window

f

Frequency width

of window

t

Time width

of window

f

Frequency width

of window

t

Time width

of window

f

Frequency width

of window

Time/Frequency Plane

t

t

f

f

4

1 ft

Heisenberg uncertainty theory

t

t

f

f

t

t

f

f

t=0:0.001:10; x=sin(2*pi*50*t);x(5147)=-x(5147); plot(t,x)axis([5.1 5.2 -1 1]);

5.1 5.12 5.14 5.16 5.18 5.2-1

-0.5

0

0.5

1

0 20 40 60 80 1000

0.5

1

1.5

2

2.5x 10

7

STFT example

specgram(x,100,1/0.001,[],100-1) % 100 is the w lengthspecgram(x,1000,1/0.001,[],1000-1)

Time

Fre

quency

0 1 2 3 4 5 6 7 8 90

50

100

150

200

250

300

350

400

450

500

Time

Fre

quency

0 1 2 3 4 5 6 7 8 90

50

100

150

200

250

300

350

400

450

500

Heisenberg uncertainty theory

STFT

Wavelet transformation

Flow chart (?)

Choose Frequency ω



Choose Shift τ

Choose length of window a

Varying windows

t

Mother wavelet

“3D Transformation”: frequencies, scales, shifts and coefficients

Combine scales with frequencies => 2D Transformation

Low frequency components => long duration High frequency components => short duration

Wavelet - Correlation

a

b

c

a

bt

atba

1)(,

dt

a

bttx

abaT *)(

1,

Wavelet Transform – Time Domain

2/2 2

1 tett 2/2

2

11

a

bt

ea

bt

aa

bt

-8 -6 -4 -2 0 2 4 6 8-1

-0.5

0

0.5

1

1.5

a=1

a=2

a=0.5

5.0a

1a

2a

2/22

11

a

t

ea

t

aa

t

Wavelet Transform – Frequency Domain

0 0.2 0.4 0.6 0.8 10

1

2

3

4

5

6

7x 10

4

a=1

a=2

a=0.5

2a

1a

5.0a

-8 -6 -4 -2 0 2 4 6 8-1

-0.5

0

0.5

1

1.5

a=1

a=2

a=0.5

5.0a

1a

2a

Wavelet => bandpass filter

Low frequency components => long duration High frequency components => short duration

Logarithmic coverage

f0

2f0

4f0

8f0

f

f0

2f0

4f0

8f0

f3f0 5f

06f

07f

0

Time – Frequency Domain

Time – Frequency Domain

HF events (small duration)

LF events (long duration)

Wavelet Transform - Reconstruction

0

2,,1)(

a

dbdaTCtx baba

dC12

2

Admissibility constant

0)( dtt Wave…

dtt)(2 …let

Wavelet

Example

-5 0 5-1

-0.5

0

0.5

1

-8 -6 -4 -2 0 2 4 6 8-1

0

1a

-8 -6 -4 -2 0 2 4 6 8-1

0

1b

-8 -6 -4 -2 0 2 4 6 8-1

0

1c

cos(1.5t)

Example cont…

-5 0 5-1

0

1

-5 0 5-1

0

1

-5 0 5-1

0

1

c=cwt(x,scales, wavelet,plot);

Example cont…

Example

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

t1=0:0.001:0.3-0.001;t2=0.3:0.001:0.6-0.001;t3=0.6:0.001:1-0.001;y1=cos(5*pi*t1);y2=cos(25*pi*t2);y3=cos(50*pi*t3);t=[t1 t2 t3];y=[y1 y2 y3];>> plot(t,y)>> c=cwt(y,0.1:60,'mexh','plot');

Example

Absolute Values of Ca,b Coefficients for a = 0.1 1.1 2.1 3.1 4.1 ...

time (or space) b

scale

s a

100 200 300 400 500 600 700 800 900 1000 0.1

3.1

6.1

9.1

12.1

15.1

18.1

21.1

24.1

27.1

30.1

33.1

36.1

39.1

42.1

45.1

48.1

51.1

54.1

57.1

Example

t=0:0.001:10; x=sin(2*pi*50*t);x(5147)=-x(5147); plot(t,x)axis([5.1 5.2 -1 1]);

5.1 5.12 5.14 5.16 5.18 5.2-1

-0.5

0

0.5

1

>>c=cwt(x,1:50,'mexh','plot');

Absolute Values of Ca,b Coefficients for a = 1 2 3 4 5 ...

time (or space) b

scale

s a

1000 2000 3000 4000 5000 6000 7000 8000 9000 10000

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

Edge Detection

+

- +

- +

+

- -t

t

t

-+

- +t

-

+ +t

-

WT coefficient=0

WT coefficient=“Max”

Increase

Suddenly

WT coefficient=0

WT coefficient=“Min”

Decrease

WT coefficient=0

Increase

Edge Detection

b

b

a1

a2

a2>a

1

Absolute Values of Ca,b Coefficients for a = 1 11 21 31 41 ...

time (or space) b

scale

s a

1000 2000 3000 4000 5000 6000 7000 8000 1

31

61

91

121

151

181

211

241

271

301

331

361

391

421

451

481

Fast Wavelet Transformation

Discrete WT: Frames

baba xdttxbaT ,*, ,)(),(

Orthonormal transformation?

We have to use every possible scale from 0 to

and every possible shift from to

“Two times infinite fundamental components”

Can we choose only a set of scales and shifts

so that we will have a true orthonormal transformation?

Yes but we need more restrictions, i.e. “better wavelets”

mm anbbaa 000 ,

m

m

mnmbaa

anbt

att

0

00

0

,,

1)()(

nmm

m

mxdt

a

anbt

atxnmT ,

0

00

0

,*1

)(),(

Tm,n are called wavelet

or detail coefficients

Discrete WT: Frames

22

,

2, txBxtxA

m n

nm

m n

nmnmTBA

tx ,,

2)(

m n

nmnmTA

txBA ,,

1)(

Tricky analysis: In that case the value of A gives the redundancy, i.e.

if A=2 then I have two times the number of necessary fundamental

components to accurately describe the function. It is like having 6

vectors that span a 3D space.

The energy of the WT coefficients

must be within a range of the energy of the original signal

For a special choice of wavelets

Discrete WT: Dyadic Grid

1,2 00 ba

log2(a)

ntnt

t mm

m

m

mnm

22

2

2

2

1)( 2/

,

A=B=1

m n

nmnmTtx ,,)(

0',',

dtnmnm Orthonormal= ',' if nnmm

1,,

dtnmnm

For a special choice of wavelets

Special Wavelets???

No big problem as Matlab has many wavelets that can be used to DWT

Daubechies (DB) wavelets

Our problems just began!

Paradox by Zeno of Elea (~450 BC)

A B

A BA1

A BA1 A2

A BA1 A2

A BA1 A2

A BA1 A2

Scaling Function

2

0B

2

0f

1m

t12

0B

0f

0m

t02

4

0B

4

0f

2m

t22

m m

0B

0f

20B

20f

40B

40f

0m1m

20 mm

t02 t12 t22

m m

t22

0

0 ,,)()(m

m n

nmm nmTtxtx

Note: WT coefficients = Output of a BP filter

Note: SF coefficients = Output of a LP filter

DWT: Multiresolution

30m 2

0m 1

0m

0mm

0md10md20md30md

0mS

10md20md30md

20 mS 20 mT

20md30md

10 mS 10 mT

)(0 nxS

LP 2

2

2

2

nT ,1

nS ,1

nT ,2

nS ,2

HP

LP

HP

DWT: Reconstruction

x(t)

HP

LP

HP

LP

HP

LP

c1

c2

c3

c4 Wa

ve

let co

effic

ien

ts

DWT: Reconstruction

0mS

knkmkSc 2,0

knkmkSb 2,0

10mS

knkmkSc 2,10

knkmkSb 2,10

10mT

20mS

20mT

kkmkn Sc ,2 0

kkmkn Sb ,2 0

knkmkSb 2,0

+

+ +

+

0mS

More levels (more scales) => Better analysis

But we add delays!!!

Discrete WT: Dyadic Grid

0 10 20 30 40

0

50

500

1000

1500

2000

2500

3000

waveletlevel

de

lay

Applications

Applications

SignalDWT

Wavelet coefficientsProcess

Useful

information

ApplicationsIEEE XPLORE (wavelet AND electric drive): 163 papers

Oldest:

“Power disturbance and power quality-light flicker voltage requirements”

Industry Applications Society Annual Meeting, 1994

Newest:

“Implementation of Wavelet-Based Robust Differential Control for Electric

Vehicle Application”, IEEE Transactions on Power, 2015

1995 2000 2005 20100

10

20

30

Year

Papers

Controller DesignIEEE IA, Khan et al, 2008“Implementation of a New Wavelet Controller for Interior Permanent-Magnet Motor Drives”

Electric PowerIEEE PD, Morsi et al, 2009“Fuzzy-Wavelet-Based Electric Power Quality Assessment of Distribution Systems Under Stationary

and Nonstationary Disturbances”

Power ElectronicsS. A. Saleh: Experimental Performances of the Single-Phase, Wavelet-Modulated Inverter, TPEL,

2011

Power ElectronicsAmer. J. Appl. Sci., Mamat et al, 2006“Fault detection of 3-phase VSI using wavelet-fuzzy algorithm”

Fault DetectionIEEE IA, Douglas et al, 2004

“A New Algorithm for Transient Motor Current Signature Analysis Using Wavelets”

Denoising

NoiseNoiseNoise

Denoising

SignalDWT IDWT

New

SignalIf |ci|<k

No

Yesci=0

++

Noise

x(t)

f

ab

s(f

ft)

DenoisingIEEE IE Giaouris at el, 2008,

“Wavelet denoising for electric drives”

Denoising

DWT and KF

DWT and KF

i. r=0.1 and the KF has the correct information directly

ii. r=0.1 and the KF has the correct information from the WT

iii. r=0.1 and the KF assumes r=0.01

Any questions?

Millennium Bridge – Newcastle upon Tyne

a short introduction on wavelet transformation and … · a short introduction on wavelet...

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