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Correlation and Spectral Analysis.ppt

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					Correlation and Spectral
        Analysis
       Application 4
Review of covariance
Autocorrelation (Autocovariance)
Noise Power
Zero-Mean Gaussian Noise
                 Power Spectrum
E{Pn(k)} = s2 = 1.12 = Rn(0)
                      Auto-correlation


                                     Rn(0) = s2 = 1.12




>> for j = 1:256,
R(j) = sum(n.*circshift(n',j-1)');
end
Window Selection: Hamming
y = filter(Hamming,1,n);
Hamming Filtered Power
     Spectrum
White Noise Auto-Covariance
vs. Hamming Filtered Noise
Filtered
    Noiseimage = imnoise(I,’gaussian’,0,10);
    N_autocov = xcorr2(Noiseimage);
    figure;imagesc(N_autocov/(128*128));colormap(gray);axis('image')
     Image Noise Field                          Autocovariance
    Unfiltered

figure;imagesc(fftshift(abs(fft2(N_autocov/(128*128)))));colormap(gray);axis('image')

          Image Noise Field                            Power Spectrum
Filtered (wc = 0.6; order 20; Hamming Window)
   N_autocov = xcorr2(Noiseimage_filtered);
   figure;imagesc(N_autocov/(128*128));colormap(gray);axis('image')

     Image Noise Field                          Autocovariance
Filtered (wc = 0.6; order 20; Hamming Window)
   N_autocov = xcorr2(Noiseimage_filtered);
   figure;imagesc(N_autocov/(128*128));colormap(gray);axis('image')

     Image Noise Field                          Power Spectrum
fMRI Simulation
        Windowing vs. Filtering
• “Window” applied in temporal or spatial domain
  to reduce spectral leakage and ringing artifact
   – Windows fall into a specialized set of functions
     generally used for spectral analysis


• “Filter” applied to reduce noise, i.e. noise
  matching, or to degrade or improve spatial
  resolution
   – Some cross-over: one method of filter design is the
     “window” method which uses window functions for
     frequency space modulating functions.
      Windowing vs. Filtering
• Mathematically,
Filtering

 MP 574
                   Outline
• Review of FIR/IIR Filters
  – Z-transform
  – Difference Equation
  – Filter Design by Windowing
• Power Spectra
  – Correlation and Convolution
  – Example from Prof. Holden’s Notes
• Windowing and Spectral Estimation
• Weiner/Adaptive Filters
• Deconvolution
z-Transform as an Analysis Tool

• Sampled version (discrete version) of
  the Laplace transform:
• z ®esT, where T is the sampling period.
• DFT and z-transform are related:
          z = eiwT where s ® eiwT
Laplace to z-Transform


                            Im(z)
               Non-causal
  iw                        unit circle
               signals
                                    0        Re(z)
           s                        ws


                                        Discrete FT

       Continuous FT
    z-Transform and Linear
           Systems
• Stated more generally:
                 T{f(n)}
   f(n)                    g(n)

                  h(n)
        Difference Equation
          Implementation
• Shift theorem of z-transform:
        Difference Equation
          Implementation
• Shift theorem of z-transform:




                              FIR
  FIR Coefficients and Impulse
           Response
• FIR filter:
           FIR vs. IIR filters
• Finite impulse response (FIR) implies a
  linear system that is always stable
  – There are no poles
• Infinite impulse response (IIR) is only
  stable if poles are inside the unit circle or
  pole coincides with a zero.
                  IIR System
                                   Im(z)
Zeros (o) at: -1, 2
Poles (x) at: 0.5±0.5j, 0.75       unit circle
                                    x                Re(z)
                               o        x        o
                                    x
IIR Stability
fvtool(B,A)




  B = [1 -1 -2];
  A=[1 -1.75 1.25 -0.375]
fvtool(B,A)
fvtool(B,A)
                        Unstable
B = [1 -1 -2];
A=[1 -1.75 1.25 -0.6]
                        Unstable
B = [1 -1 -2];
A=[1 -1.75 1.25 -0.6]
 Finite impulse response (FIR)
B = [1 -1 -2];
A=[1]
Definition of Stability
 FIR filter Design by Windowing
• Simply truncate IIR filter
• Rectangular Window:
Matlab: fdatool
                 filter() in Matlab
FILTER One-dimensional digital filter.
  Y = FILTER(B,A,X) filters the data in vector X with the
  filter described by vectors A and B to create the filtered
  data Y. The filter is a "Direct Form II Transposed"
  implementation of the standard difference equation:


  a(1)*y(n) = b(1)*x(n) + b(2)*x(n-1) + ... + b(nb+1)*x(n-nb)
                - a(2)*y(n-1) - ... - a(na+1)*y(n-na)
Exporting Filter Coefficients
                Extension to 2D
• Radial Transform
  – H(k)-> (H(k1)2+H(k2)2)1/2=T(k1,k2)
  – See Matlab script on filter design using radial
    transformation to 2D: Filter Design
      • http://zoot.radiology.wisc.edu/~fains/Code/MP574_FilterDesign.m
• Parks-McClellan Transformation
  – Step 1: Translate specifications of H(w1,w2) to H(w)
  – Step 2: Design 1D filter H(w)
  – Step 3: Map to 2D frequency space
  cosw = - ½ + ½ cosw1 + ½ cosw2 + ½ cosw1 cosw2 = T(w1,w2)
  - Step 4: determine h(n1,n2) by 2D FT.
Hamming Window Example
     Hamming Window Example

>> w1 = -pi:0.01:pi;
>> w2 = -pi:0.01:pi;
>> [W1,W2] = meshgrid(w1,w2);
>> H_2d = 0.54+0.46.*(-0.5+0.5.*cos(W1)+0.5.*cos(W2)+0.5.*cos(W1).*cos(W2));
>>figure;mesh(H_2d)

filter2()
2D FIR Filter Design,
  Parks-McClellan
“firdemo”

				
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posted:1/9/2014
language:English
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Lingjuan Ma Lingjuan Ma
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