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 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

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 signals iw s unit circle Re(z) 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 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 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”