1. Digital filters: Design of FIR filters
احسان احمد عرساڻي
Lecture 23-24

2. Introduction to FIR filters
These have linear phase
No feedback
Output is function of the present and past inputs only
These are also called ‘all-zero’ and ‘non-recursive’ filters
These do not have any poles

3. Applications Where: highly linear phase response is required
Need to avoid complicated design

4. FIR Filter Design Methods
Windows
Frequency-sampling

5. FIR Filter Design: Windows Method
Start from the desired frequency response Hd(ω)
Determine the unit (sample) pulse reponse
hd(n)=F-1{Hd(ω)}
hd(n) is generally infinite in length
Truncate hd(n) to a finite length M

6. Truncating hd(n)
Take only M terms
N=0 to N=M-1
Remove all others

7. Truncating hd(n) Take only M terms Remove all others
N=0 to N=M-1
Remove all others
Multiplying hd(n) with a rectangular window

8. Determine H(ω) Take Fourier transform of h(n) Therefore, compute:
Hd(ω) and
W(ω)
Hd(ω) depends on the required response hd(n)

9. Computing W(ω)
W(ω)=F{w(n)}
w(n) is a rectangular pulse

10. Example
A low-pass linear phase FIR filter with the frequency response Hd(ω) is required
Hd(n) happens to be non-causal having infinite duration

11. The impulse response hd(n)

12. Windowing the hd(n)

13. The truncated hd(n)

14. Example
A low-pass linear phase FIR filter with the frequency response Hd(ω) is required

15. Frequency of oscilation increases with M
Magnitude of oscillation doesn’t increase or decrease with M
Oscillations occur due to the Gibbs phenonmenon caused by the multipli- cation of the rectangular window with Hd(ω)

16. Frequency of oscilation increases with M
Magnitude of oscillation doesn’t increase or decrease with M
Oscillations occur due to the Gibbs phenonmenon caused by the multipli-cation of the rectangular window with Hd(ω)

17. Other windows

18. Other windows

19. Spectrum of Kaiser window
(Cycles per sample)

20. Spectrum of Hanning window

21. Spectrum of Hamming Window
(Cycles per sample)

22. Spectrum of Blackman Window
(Cycles per sample)

23. Spectrum of Tukey Window
(Cycles per sample)

24. Windows’ characteristics

25. The FIR filter’s response with Rectangular window
M=61

26. FIR filter’s response with Hamming window

27. FIR filter’s response with Blackman window

28. FIR filter’s response with Kaiser window
M=61

29. Using the FIR filter

30. Blackman’s filter output