1. Review of Filter Design
MP574 April 14, 2006

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

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

4. 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

5. 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

6. 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

7. Matlab: fdatool

8. IIR: Chebyshev Type II Order 10

9. IIR: Chebyshev Type II Order 10

10. FIR: Hamming Window Order 30

11. FIR: Hamming Window Order 30

12. Sptool Import

13. Sptool Apply Filter

14. 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

15. 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.

16. Hamming Window Example

17. 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()

18. 2D FIR Filter Design, Parks-McClellan

19. “firdemo”

20. Improvement After Affine Registration to Mask Frame

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