The Karhunen–Loève Expansion

Abu Bucker Siddik

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New Mexico State University

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Presentation Objectives

• Introduce Karhunen–Loève (KL) expansion

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• Present expression and parameters for Karhunen–Loève expansion

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• Illustrate Karhunen–Loève expansion with MATLAB code and plots

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• Provide some insights about Karhunen–Loève expansion ����

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Karhunen–Loève Expansion

• Karhunen–Loève expansion is a representation of a random process as an infinite linear combination of orthonormal functions.
• The expansion is named after Kari Karhunen and Michel Loève who developed it as a series of expansion method for continuous random process.
• Because of the large number of computations required to find the orthonormal functions, KL expansion was virtually unused until the middle of the 20th century.
• Radical changes came with the appearance of powerful computers and development of efficient algorithms to compute the eigenfunctions.
• KL expansion is used extensively in the fields of detection, estimation, pattern recognition, and image processing as an efficient tool to store random processes.

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From https://en.wikipedia.org/wiki/Karhunen–Loève_theorem

Karhunen–Loève Expansion

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From Statistical Optics, Joseph W. Goodman

(1)

Karhunen–Loève Expansion

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From Statistical Optics, Joseph W. Goodman

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Karhunen–Loève Expansion

For simplicity, it is assumed that the random process U(t) has zero mean for all time.

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For expansion coefficients to be uncorrelated, we require that

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From Statistical Optics, Joseph W. Goodman

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(5)

Karhunen–Loève Expansion

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From Statistical Optics, Joseph W. Goodman

(6)

Karhunen–Loève Expansion

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From Statistical Optics, Joseph W. Goodman

(7)

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Karhunen–Loève Expansion

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From Statistical Optics, Joseph W. Goodman

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Application of KL expansion

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From Statistical Optics, Joseph W. Goodman

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Matlab Code for KL Expansion

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N=100; % number of samples for each sample function

sd=1; % standard deviation

mu=0; % mean

Nave=100; % number of sample functions

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for nave=1:Nave

R=(sd*randn(N,1))+ mu; % real part

I=(sd*randn(N,1))+ mu; % imaginary part

u=complex(R, I); % complex sample function

z(:,nave)=u; % realizations

end

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%Autocorrelation function (zero mean)

B=(z*z')/Nave;

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Matlab Code for KL Expansion

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% Eigen functions and Eigen Values

[phi,lamda] = eig(B);

% Expansion coefficients

Bn=phi’*z;

% sample functions using Eigen functions and Expansion coefficients

reaz1=phi*Bn;

% Correlation of expansion coefficient

EE=(Bn*Bn')/Nave;

% Linear combination of orthonormal functions

y=0;

for b=1:100;

y1=Bn(b,1)*phi(1,b);

y=y+y1;

end

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Matlab Code for KL Expansion

% display

disp("using Kl at time t1, u(t1)for 1st sample function:");

disp(y)

disp("from realization at time t1, u(t1)for 1st sample function:");

disp(z(1,1));

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Result Analysis

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Result Analysis

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Figure: Comparison of actual first sample function and first sample function obtained from KL expansion

Result Analysis

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Figure: Comparison of actual second sample function and second sample function obtained from KL expansion

Conclusion

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