1. 1 BIEN425 – Lecture 11 By the end of the lecture, you should be able to: –Design and implement FIR filters using frequency-sampling method –Compare the advantages / disadvantages of FIR filter design using windowing versus frequency-sampling methods.

2. 2 Alternative for designing FIR given A r (f) Supposed N frequency samples equally spaced Recall: Note then h(k) = IDFT{H(f i )} To ensure that filter is linear-phase, H(f) will have to be of the form

3. 3 Frequency sampling method

4. 4

5. 5 Example Choosing A r (f) to be ideal

6. 6

7. 7 Example: optimizing trans. band h x (k) will be different depending on the value of x

8. 8 Strategy: –Find the value of x so that the stopband attenuation A s (x) is maximized –Find A s (x) for 3 arbitrarily chosen x values –Assume A s (x) follows a quadratic polynomial –Solve for c to fit the data points –Solve for x x AsAs

9. 9 Example

10. 10 Another example

11. 11 Extension using FFT

12. 12 With Blackman windows

13. 13 Kaiser and Chebyshev windows Don’t worry about the complexity Just have to know the characteristics

14. 14

15. 15 So far, we have only deal with low-pass WHY? –Because it’s straight-forward? Of course! What if we want other filters –We can translate low-pass into any other filter types –Highpass? Very simple –Bandpass? Here is the example

16. 16 Let’s design a bandpass filter Want: bandpass 50-100Hz Procedure: –Create lowpass filter with the width in the passband (i,e. Fc = 24Hz) –Compute h(k) – introduce concept of taps –Apply windows if necessary –Shift to desired frequency s_shift=sin(2*pi*76/fs*(0:30)); h_shifted=h.*s_shift;

17. 17 Ideal lowpass –(25-24hz)

18. 18 Actual lowpass – 1024 points

19. 19 Tap-31 (take middle 31 points of h(k))

20. 20 With Blackman window

21. 21 Apply shift of 75hz

22. 22 Other options Least-square method In a sense, we would like to design a filter so that its actual response Ar(f) matches the desired response Ad(f) by minimizing the objective function J So far, every frequency is treated with the same weight (or importance)

23. 23 We could specify weighted importance for particular frequency bands with the variable w(i) Note that this is not the window parameters As a result, given the distribution of the weights we could find h(i) so that J is minimized.