7/31/2019 Optimal 2003

1/26

OPTIMAL GLOBAL PROJECTIONDENOISING ALGORTHIM

BY

SANDESH KUMAR B V

M.TECH(SIGNAL PROCESSING AND VLSI)DEPT. OF ELECTRONICS AND

COMMUNICATIONS

SCHOOL OF ENGINEERING AND

TECHNOLOGY.

JAIN UNIVERSITY.

7/31/2019 Optimal 2003

2/26

CONTENTS OBJECTIVE

ABSTRACT

INTRODUCTION

THE OPTIMAL GLOBAL PHASE SPACE PROJECTIONALGORTHIM

GLOBAL PHASE SPACE PROJECTION AND SUBSPACEDECOMPOSITION FOR NOISE REDUCTION.

SELECTION OF EMBEDDING DIMENSION AND TIMEDELAY

IMPLEMENTATION PROCEDURE.

EXPERIMENTAL VERIFICATION.

APPLICATIONS.

CONCLUSION.

7/31/2019 Optimal 2003

3/26

OBJECTIVE

FOR REDUCING THE NOISE IN

DIAGNOSIS OF THE FAULT.

7/31/2019 Optimal 2003

4/26

ABSTRACT

Noise reduction is a main step in fault diagnosis .However, it is not effective enough to purify the

nonlinear fault features using the traditional signal

denoising techniques.

This Work improved the global projection denoising

algorithm via calculating the optimal embedding

dimension m and considering optimal time delay =1

The denoising effects are very effective and reliable inreducing the noise and reconstructing the signals.

7/31/2019 Optimal 2003

5/26

INTRODUCTION

To reduce the noise resident in the signals, manymethods had been studied, such as wavelet

analysis, and numerical filters.

Problem with wavelets? Problem with numerical filters such as Kalman

filter and Weiner filter?

In global projection , There are two important

parameters in the phase space reconstruction, that

is, the time delay and the embedding dimension

m for embedding.

7/31/2019 Optimal 2003

6/26

The selection of m and has much influence on

the effect of denoise.

Developed a method to calculate the optimal

embedding dimension m called caos method.

Optimal time delay is fixed as 1 and verified.

The denoising effects of the lorenz signal addedwith white noise are simulated with Matlab.

7/31/2019 Optimal 2003

7/26

THE OPTIMAL GLOBAL PHASE SPACEPROJECTION ALGORITHM

According to the Takens Theorem an equivalentdynamical system can be constructed usingdelay embedding methods from time series.

For an m-dimensional system, there exists anembedding representation of a time series:

7/31/2019 Optimal 2003

8/26

Hence, a reconstructed phase space (RPS) matrix

Z of embedding dimension m and time delay is

called a trajectory matrix and is defined by:

where the row vectors zn,with n=1+(m-1),.N

7/31/2019 Optimal 2003

9/26

An observed time series signal with additive

noise is given by:

Where represents the observed

signal, and the unknown clean signal x andadditive noise w are assumed to be independent

of each other.

7/31/2019 Optimal 2003

10/26

This directly equates to the trajectory matrix

relationship:

where Z, X, and W are the corresponding time delay

RPS of each signal.

7/31/2019 Optimal 2003

11/26

GLOBAL PHASE SPACE PROJECTIONAND SUBSPACE DECOMPOSITION.Let

Applying Karhunen-Loeve Transform to theabove equation.

Compute the covariance matrix of Z i.e. Rz and perform the

Eigen decomposition to find the Eigen vectors and Eigen

values.

7/31/2019 Optimal 2003

12/26

Let U=[U1 U2] where U1 denotes the K x M

matrix of principal eigenvectors of Rz

The space spanned by U1 is called the signal

subspace, and the complementary space spanned

by U2 is called the noise subspace

7/31/2019 Optimal 2003

13/26

of X where H is a KK matrix

The residual signal obtained in this case is given by

7/31/2019 Optimal 2003

14/26

The estimator is obtained from

is a diagonal matrix of modified eigenvalues called the weighting matrix, which

can filter the noise mixing in signal.

7/31/2019 Optimal 2003

15/26

The final equation of GP algorithm is given by:

7/31/2019 Optimal 2003

16/26

The selection of and m

We will apply the Caos method to choose the optimal m

7/31/2019 Optimal 2003

17/26

To investigate its variation from m to m+1,We define

Choose time delay = 1

7/31/2019 Optimal 2003

18/26

The implementation procedure

For time-varying signals, an embedding of the entire time series requires amuch higher dimension. In order to reduce the embedding dimension, theoriginal time series can be divided into windows, each of which can beindividually projected.

Hence the process of optimal global projection is as follows:

Step 1. Divide the signal with additive noise into windows, each of which can beindividually projected. Since disjoint windows result in edge effects, thisarticle uses an overlap-add method to overlap the windows.

Step 2. Select the optimal time delay and embedding dimension m usingmutual information and Caos method, respectively. Then reconstruct thephase space of each window and calculate the covariance matrix of the datain each window .

7/31/2019 Optimal 2003

19/26

Step 3. Convert each window via the linear

transform method and compute the weighting

matrix . Then reconstruct the signal from highdimension to low dimension.

Step 4. Re-join adjacent windows by applying theoverlap-add method and obtain the denoised

signal.

7/31/2019 Optimal 2003

20/26

Experimental VerificationTo illustrate the effect of the proposed method, the Lorenz

signal with additive noise is considered. The Lorenz signal

is produced via the following equation:

7/31/2019 Optimal 2003

21/26

7/31/2019 Optimal 2003

22/26

The denosied result with different m and

7/31/2019 Optimal 2003

23/26

APPLICATIONS

To analyze the machinery faults buried with

noise.

To analyze any signals which is buried with

noise.

7/31/2019 Optimal 2003

24/26

CONCLUSION

An optimal GP denoising method using noise reduction

is introduced for denoising in large rotating

machinery, the Lorenz cases showed the selection of

the optimal and m can improve the denoised Lorenzsignal, which contains the rich nonlinear components.

As a result, the proposed method is a promising new

addition to use for nonstationary, nonlinear fault

signal in the large rotating machinery as well as in theother kinds of machinery.

7/31/2019 Optimal 2003

25/26

REFERENCES

[1]Ephraim Y and Trees HLV. A signal subspace approach forspeech enhancement. IEEE T. Speech Audi P 1995; 3: 251266.

[2] Johnson MT and Povinelli RJ. Generalized phase spaceprojection for non-linear noise reduction. Physica D 2005;

201: 306317.[3]Mees AI, Rapp PE and Jennings LS. Singular value

decomposition and embedding dimension. Phys Rev A 1987;36(1): 340346.

[4]Cao LY. Practical method for determining the minimumembedding dimension of a scalar time series. Physica D 1997;110: 4350.

7/31/2019 Optimal 2003

26/26