jinyuli - ISYE8843 Final Jinyu Li 1. Question 1 I use...

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ISYE8843 Final Jinyu Li 1. Question 1 I use Bayesian Wavelet Shrinkage to denoise. I choose a simple signal as following: t=linspace(0,1,1024); sig = (sin(5*pi*t)+2.0*cos(10*pi*t)+3.0*sin(15*pi*t)).*exp(-t); The noise I add has different size, as 0.2, 0.4 and 0.6: sigma = 0.2*n; % n= 1:3 randn('seed',1) sign = sig + sigma * randn(size(sig)); 1.1 Matlab code % Bayesian Wavelet Shrinkage clear all close all % (i) Make a Signal on [0,1] t=linspace(0,1,1024); sig = (sin(5*pi*t)+2.0*cos(10*pi*t)+3.0*sin(15*pi*t)).*exp(-t); % and plot it figure(1) plot(t, sig) for n = 1: 3 % (ii) Add noise of size n*0.2. Make sure the noise is fixed % by fixing the seed sigma = 0.2*n; randn('seed',1) sign = sig + sigma * randn(size(sig)); % (iii) plot the noisy signal here. figure(n+1) subplot(2,1, 1) plot(t, sign) % (iv) Take the filter H, in this case this is SYMMLET 4 filt = [ -0.07576571478934 -0.02963552764595 ... 0.49761866763246 0.80373875180522 ...
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0.29785779560554 -0.09921954357694 ... -0.01260396726226 0.03222310060407]; % (v) Transfer the signal in the wavelet domain. % Choose L=8, eight levels of decomposition sw = dwtr(sign, 5, filt); % At this point you may view the sw. Is it disbalanced? % Is it decorrelated? %(vi) Let's now apply Bayesian Shrinkage. % Assume that the likelihood of a detail wavelet coefficient is % normal (theta, sigma^2) and that the prior on theta % is also normal (0, tau^2). The Bayes rule is: %tau^2/(tau^2 + sigma^2) d. Take $tau^2=0.01 and %sigma^2=0.1. swt = sw; swt(2^5+1:end) = swt(2^5+1:end).*0.01/0.11; %swt = swt.*0.01/0.11; % (vii) Return now thresholded object back to the time % domain. Of course with the same filter and L. a=idwtr(swt,5, filt); % (viii) Check if made a good estimate. .. subplot(2,1,2); plot(t, a, '-') end 1.2 Output Result The original signal:
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The noise signal with noise size 0.2, and the denoised signal. The noise signal with noise size 0.4, and the denoised signal.
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The noise signal with noise size 0.6, and the denoised signal. As we see, all the denoised signals have nearly the same structure, and are similar to the original signal. The effect of Bayesian Wavelet Shrinkage is good for such signal.
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2. Question 2 2.1 Exact Calculating ∑∑ = = = = AE A E A E E B A P E P A M P A J P A M P A J P E B A P E P B P B P M J A E B P B P M J B P B P B M J P ) , 1 ( * ) ( * ) 1 ( * ) 1 ( ) 1 ( * ) 1 ( * ) , 1 ( * ) ( * ) 1 ( ) 1 ( 1 ) 1 , 1 , , , 1
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This note was uploaded on 10/23/2011 for the course ISYE 8843 taught by Professor Vidakovic during the Spring '11 term at Georgia Tech.

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jinyuli - ISYE8843 Final Jinyu Li 1. Question 1 I use...

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