Review of Filter Design MP574 April 14, 2006

Difference Equation Implementation Shift theorem of z-transform:

Difference Equation Implementation Shift theorem of z-transform: FIR

Filter Design: IIR vs. FIR Rules of thumb: Use IIR when the only important requirements are sharp cutoff and high throughput Use FIR in any application where linear phase is required

FIR Filter Design Phase Delay 4 Types of Linear Phase Tp = -q(w)/w Tg= -dq(w)/dw 4 Types of Linear Phase Positive symmetry, odd number of coefficients Tp = (N-1)/2 Tsample Positive symmetry, even number of coefficients Tp = (N/2-1/2)Tsample Zero at w = p Negative, odd Negative, even Tg = (N-1-p)/2 Tsample p/2 phase shift Zero at w = 0

Design Steps Filter specification Coefficient calculation Realization dp peak passband deviation (or ripple) ds stopband deviation fp passband edge frequency fs stopband edge frequency Fs sampling frequency Coefficient calculation Coefficients of H(z) Realization Implementation

Matlab: fdatool

IIR: Chebyshev Type II Order 10

IIR: Chebyshev Type II Order 10

FIR: Hamming Window Order 30

FIR: Hamming Window Order 30

Sptool Import

Sptool Apply Filter

Summary of Filter Design Range of polynomial forms of different orders IIR converges to specifications more rapidly Non-linear phase Can be unstable in implementation FIR most commonly used for medical implementations Linear phase: signal preserved without distortion More constraints require higher order fits Stable and robust to quantization errors Matlab tools for filter design Take some time to familiarize yourself with them

Extension to 2D 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”

Improvement After Affine Registration to Mask Frame

T1 fitting There are two slices only one of which contains the lesion (arrow)