In The Name of God

The Compassionate The

Merciful

Wavelet Based Methodsfor

System Identification

Nafise Erfanian Saeedi

Presentation Agenda

Introduction to wavelets General applications for wavelets Application of wavelets in system

identificationSimulation Example Comparison with conventional methodsConclusions

Introduction to wavelets

A wavelet is a waveform of effectively limited duration that has an average value of zero

Wavelet Analysis

Comparing wavelet analysis to Fourier analysis

Introduction to wavelets

Continues Wavelet Transform (CWT)

Wavelet Transform

Discrete Wavelet Transform (DWT)

Introduction to waveletsIntroduction to wavelets

`

Continues Wavelet Transform

Introduction to waveletsIntroduction to wavelets

Five Steps to CWT1- Take a wavelet and compare it to a section at the start of the

original signal.

2- Calculate a number, C, that represents how closely correlated the wavelet is with this section of the signal. Note that the results will depend on the shape of the wavelet you choose.

Introduction to waveletsIntroduction to wavelets

3- Shift the wavelet to the right and repeat steps 1 and 2 until you've covered the whole signal.

Introduction to waveletsIntroduction to wavelets

4- Scale (stretch) the wavelet and repeat steps 1 through 3.

5- Repeat steps 1 through 4 for all scales.

Introduction to waveletsIntroduction to wavelets

Results

Introduction to waveletsIntroduction to wavelets

Time

Scale

Small Coefficients

Large Coefficients

Low scale >> Compressed wavelet >> Rapidly changing details >> High frequency

High scale >> Stretched wavelet >> Slowly changing, coarse features >> Low frequency

Introduction to waveletsIntroduction to wavelets

An Example from Nature: Lunar Surface

Introduction to waveletsIntroduction to wavelets

Discrete Wavelet Transform

Approximations and Details

One Stage Filtering

Problem: Increasing data volume

Introduction to waveletsIntroduction to wavelets

Filtering with down sampling

Introduction to waveletsIntroduction to wavelets

Multi Stage Decomposition

Introduction to waveletsIntroduction to wavelets

Different Mother wavelets

Introduction to waveletsIntroduction to wavelets

Haar Mexican hat PDF’s Derivative Morlet

Mayer Symlet Coiflet Daubechies

1) Detecting Discontinuities and Breakdown Points

Freqbrk.mat

db5 level 5

Introduction to waveletsGeneral Applications for wavelets

2) Detecting Long-Term Evolution

Cnoislop.mat

db3 level 6

Introduction to waveletsGeneral Applications for wavelets

3) Detecting Self-Similarity

vonkoch.mat

coif3 continues

Introduction to waveletsGeneral Applications for wavelets

4) Identifying Pure Frequencies

sumsin.mat

db3 level 5

Introduction to waveletsGeneral Applications for wavelets

2 Hz

200 Hz

20 Hz

5) De-Noising Signals

noisdopp.mat

sym4 level 5

Problem: Loss of Data

Introduction to waveletsGeneral Applications for wavelets

Solution: Special Algorithms

Introduction to waveletsGeneral Applications for wavelets

Other Applications:

• Biology for cell membrane recognition, to distinguish the normal from the pathological membranes

• Metallurgy for the characterization of rough surfaces

• Finance (which is more surprising), for detecting the properties of quick variation of values

• Detection of short pathological events as epileptic crises or normal ones as evoked potentials in EEG (medicine)

• Study of short-time phenomena as transient processes

• Automatic target recognition

Introduction to waveletsGeneral Applications for wavelets

Here, we consider wavelet approaches to

analyze signals that are a (linearly) filtered version of some source signal with the purpose

of identifying the characteristics

of the filtering system.

Introduction to waveletsWavelets in system identification

System Identification Methods:

Parametric

Non parametric

Introduction to waveletsWavelets in system identification

Solution one:For a causal system

Problem: Round-off errors accumulate with larger time indices, making this approach impractical for slowly decaying

(i.e., infinite) impulse response functions.

Introduction to waveletsWavelets in system identification

Solution two:Frequency-domain methods for linear systems based on

coherence Analysis

Usually with pseudorandom noise as input

Introduction to waveletsWavelets in system identification

Wavelet representation of signalsFor a finite energy signal:

discrete parameter

wavelet transform (DPWT)

analyzing functions

scale index k

translation index m

Introduction to waveletsWavelets in system identification

Dyadic Sampling:compression/dilation in the DPWT is by a power of two

with

Introduction to waveletsWavelets in system identification

DPWTs are calculated from Analysis equation

For orthogonal wavelets

An interesting observation

Introduction to waveletsWavelets in system identification

For a source-filter model

Introduction to waveletsWavelets in system identification

Using orthogonality property

Introduction to waveletsWavelets in system identification

It is proved that k=0 is the best choice to prevent aliasing without wasting resources

Introduction to waveletsWavelets in system identification

Discrete time signalsDiscrete Wavelet Transform (DWT)

Introduction to waveletsWavelets in system identification

System identification using DWT

Introduction to waveletsWavelets in system identification

x[n]

excitation

y[n]=h[n]*x[n]System

under testD W T

hestimated[n]

i) Choice of excitation

System under test: Chebyshev,IIR,10th order high pass filter

with 20db ripple

Excitations:

Introduction to waveletsSimulation Example

Results for different excitations

Introduction to waveletsSimulation Example

Haar and Daubechies excitations give very good identification

Results of changing the coefficients number for Daubeshies

Introduction to waveletsSimulation Example

ii) Different Systems

wavelet used as excitation and analysing function:

Daubechies D4

Introduction to waveletsSimulation Example

System 1:FIR band-stop filter

(a) Frequency response

(b) Error variation with

frequency

Introduction to waveletsSimulation Example

System 2:Butterworth IIR,

10th order

Band-stop

(a) Frequency response

(b) Error variation with

frequency

Introduction to waveletsSimulation Example

System 3:Chebyshev IIR,

10th order

Band-stop

(a) Frequency response

(b) Error variation with

frequency

Introduction to waveletsSimulation Example

System 4:Elliptic IIR,

10th order

Band-stop

(a) Frequency response

(b) Error variation with

frequency

Introduction to waveletsSimulation Example

1) Chirp method System under test:

Chebyshev high-pass filter

Introduction to wavelets Comparison with conventional methods

2) Time domain recursionIntroduction to wavelets Comparison with conventional methods

System under test: Chebyshev high-pass filter

3) Inverse filteringIntroduction to wavelets Comparison with conventional methods

System under test: Chebyshev high-pass filter

4) CoherenceIntroduction to wavelets Comparison with conventional methods

System under test: Chebyshev high-pass filter

A new method for non-parametric linear time-invariant system identification based on the discrete wavelet transform (DWT)

is developed.

Identification is achieved using a test excitation to the system under test, that also acts as the analyzing function for the

DWT of the system’s output.

The new wavelet-based method proved to be considerably better than the conventional methods in all cases.

Introduction to waveletsConclusions

1- R.W.-P. Luk a, R.I. Damper b, “Non-parametric linear time-invariant system identification by discrete wavelet transforms”, Elsevier Inc,2005

2- M. Misiti, Y. Misiti, G. Oppenheim, J. M. Poggi, “Wavelet Toolbox for use with matlab” Mathworks Inc., 1996.

حامد،- 3 “کاشانی، ؛” سيستم شناسايي در موجک مدلسازی، کاربرد درس 1383سمينار

Introduction to waveletsRefrence

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