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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.
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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.
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OBJECTIVE
FOR REDUCING THE NOISE IN
DIAGNOSIS OF THE FAULT.
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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.
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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.
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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.
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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:
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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
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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.
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This directly equates to the trajectory matrix
relationship:
where Z, X, and W are the corresponding time delay
RPS of each signal.
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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.
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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
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of X where H is a KK matrix
The residual signal obtained in this case is given by
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The estimator is obtained from
is a diagonal matrix of modified eigenvalues called the weighting matrix, which
can filter the noise mixing in signal.
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The final equation of GP algorithm is given by:
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The selection of and m
We will apply the Caos method to choose the optimal m
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To investigate its variation from m to m+1,We define
Choose time delay = 1
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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 .
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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.
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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:
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The denosied result with different m and
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APPLICATIONS
To analyze the machinery faults buried with
noise.
To analyze any signals which is buried with
noise.
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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.
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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.
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