1. Correlation and Spectral Analysis
Application 4

2. Review of covariance

3. Autocorrelation (Autocovariance)

4. Noise Power

5. Zero-Mean Gaussian Noise

6. Power Spectrum
E{Pn(k)} = s2 = 1.12 = Rn(0)

7. Auto-correlation Rn(0) = s2 = 1.12 >> for j = 1:256,
R(j) = sum(n.*circshift(n',j-1)');
end

8. Window Selection: Hamming
y = filter(Hamming,1,n);

9. Hamming Filtered Power Spectrum

10. White Noise Auto-Covariance vs. Hamming Filtered Noise

11. 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

12. Unfiltered
figure;imagesc(fftshift(abs(fft2(N_autocov/(128*128)))));colormap(gray);axis('image')
Image Noise Field
Power Spectrum

13. 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

14. 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

15. fMRI Simulation

16. 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.

17. Windowing vs. Filtering
Mathematically,

18. Filtering
MP 574

19. Outline Review of FIR/IIR Filters Power Spectra
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

20. 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

21. Laplace to z-Transform
Im(z)
Non-causal signals iw s
unit circle
Re(z) ws
Discrete FT
Continuous FT

22. z-Transform and Linear Systems
Stated more generally:
T{f(n)}
f(n)
g(n)
h(n)

23. Difference Equation Implementation
Shift theorem of z-transform:

24. Difference Equation Implementation
Shift theorem of z-transform:
FIR

25. FIR Coefficients and Impulse Response
FIR filter:

26. 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.

27. 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

28. IIR Stability

29. fvtool(B,A)
B = [ ];
A=[ ]

30. fvtool(B,A)

31. fvtool(B,A)

32. Unstable
B = [ ];
A=[ ]

33. Unstable
B = [ ];
A=[ ]

34. Finite impulse response (FIR)
B = [ ];
A=[1]

35. Definition of Stability

36. FIR filter Design by Windowing
Simply truncate IIR filter
Rectangular Window:

37. Matlab: fdatool

44. 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)

45. Exporting Filter Coefficients

46. Extension to 2D Radial Transform Parks-McClellan Transformation
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
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.

47. Hamming Window Example

48. Hamming Window Example
>> w1 = -pi:0.01:pi;
>> w2 = -pi:0.01:pi;
>> [W1,W2] = meshgrid(w1,w2);
>> H_2d = *( *cos(W1)+0.5.*cos(W2)+0.5.*cos(W1).*cos(W2));
>>figure;mesh(H_2d)
filter2()

49. 2D FIR Filter Design, Parks-McClellan

50. “firdemo”