1. EEE4176 Application of Digital Signal Processing
Lecture: FIR Filter Design
Assistant Prof. Yangmo Yoo Dept. of Electronic Engineering SOGANG UNIVERSITY
2011.9
EEE4176 Application of Digital Signal Processing

2. EEE4176 Application of Digital Signal Processing
FIR Filters
Linear Phase FIR Filters
Generally, both the constant phase and constant group delay filters are referred to as LINEAR PHASE FILTER.
Impulse response of FIR filter
Order of the filter = N
Length of the filter = N+1
Cf)
2011.9
EEE4176 Application of Digital Signal Processing

3. Linear Phase FIR Filters
Type-I
, : even order / odd length;
: Even Symmetric
Frequency Response
Amplitude function:
: even symmetric, period
2011.9
EEE4176 Application of Digital Signal Processing

4. Linear Phase FIR Filters
Type-II
, : odd order / even length;
: Even Symmetric
Frequency Response
Amplitude function:
: even symmetric, period
※ : Not suitable for HPF and BRF
2011.9
EEE4176 Application of Digital Signal Processing

5. Linear Phase FIR Filters
Type-III
, : even order / odd length;
: Odd Symmetric
Frequency Response
Amplitude function:
: off symmetric, period
※ : Not suitable for LPF, HPF and BRF
※ Used for differentiators and Hilbert transformers.
2011.9
EEE4176 Application of Digital Signal Processing

6. Linear Phase FIR Filters
Type-IV
, : odd order / even length;
: Odd Symmetric
Frequency Response
Amplitude function:
: off symmetric, period
※ : Not suitable for LPF, and BRF
※ Used for differentiators and Hilbert transformers.
2011.9
EEE4176 Application of Digital Signal Processing

7. Linear Phase FIR Filters
Summary of Linear-Phase filter types
2011.9
EEE4176 Application of Digital Signal Processing

8. FIR filter design: Impulse Response Truncation Method
IRT filter design procedure
Write the desired (or ideal) amplitude response
Phase: either 0(symmetric) or (anti-symmetric)
: Symmetric about n=0
Choose the filter's phase characteristics
integer or fractional group delay
initial phase 0 or
Choose the filter order N, then the Ideal desired filter is described as\
Compute the impulse response of the ideal filter response using the IDFT: In most cases, can be expressed in a closed form.
Truncate the impulse response to get
2011.9
EEE4176 Application of Digital Signal Processing

9. FIR filter design: Impulse Response Truncation Method
Differentiators
: Type III or IV
※ Odd N is preferred !! (See Fig. 9.9), Type IV
※ T = 1 => Frequency range limited due to the finite sampling rate
2011.9
EEE4176 Application of Digital Signal Processing

10. FIR filter design: Impulse Response Truncation Method
Hilbert Transformers
Note:
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EEE4176 Application of Digital Signal Processing

11. FIR filter design: Impulse Response Truncation Method
Digital Hilbert transformer
Ideal Hilbert transformer
Causal filter design
2011.9
EEE4176 Application of Digital Signal Processing

12. FIR filter design: Impulse Response Truncation Method
Digital Hilbert transformer
N: Even vs. Odd ?
※ Type III is preferred.
Ideal noncausal impulse response: h(n) = 0 for n = 0 for even n
h(n) = -h(n) (h(n) = -h(N-n)
2011.9
EEE4176 Application of Digital Signal Processing

13. Hilbert Transformer Applications: PW Doppler
2011.9
EEE4176 Application of Digital Signal Processing

14. Hilbert Transformer Applications: PW Doppler
2011.9
EEE4176 Application of Digital Signal Processing

15. FIR filter design: Window Method
Design procedure
Determine the spec
Design the ideal impulse response using IRT method.
Band edge freq of ideal filter =
N ?
Windowing
The length of is
Window selection
Select a Window with properties meeting your requirements.
Depends also on Filter spec: Limitations in length, ripple, attenuation, etc.
=> How to determine N ?
2011.9
EEE4176 Application of Digital Signal Processing

16. FIR filter design: Window Method
Filter Length
For sharper transition and/or lower attenuation and ripple, use longer window
Choose N -> design and check response -> change N and repeat the design
2011.9
EEE4176 Application of Digital Signal Processing

17. FIR filter design: Window Method
Kaiser windows
Empirical formulas for design parameters
Matlab approach to FIR filter design using a Kaiser window: See pp. 298
FIR filter design examples
 Read text (pp )
2011.9
EEE4176 Application of Digital Signal Processing

18. FIR filter design: Window Method
Advantages of window method
Simple, robust, good performance
Disadvantages of window method
Not optimum in the sense that
often over-spec
ripples are not uniform
• Filter length N is not optimum, i.e., minimum
2011.9
EEE4176 Application of Digital Signal Processing

19. FIR filter design: Least-Squares Design
Formulation
Define a weight function and the weighted freq-domain error as
Define the integral of weighted square freq-domain error as
Find 's for minimum
(D)
(F)
(E)
2011.9
EEE4176 Application of Digital Signal Processing

20. FIR filter design: Least-Squares Design
Formulation
Substituting (D) and (F) in (E) gives
Solve for
2011.9
EEE4176 Application of Digital Signal Processing

21. Least-Squares Linear-Phase FIR Filter Design 1/3
FIR Filter with length of L+1
Even symmetry in the impulse response  sum of cosines:
Matrix form
2011.9
EEE4176 Application of Digital Signal Processing

22. Least-Squares Linear-Phase FIR Filter Design 2/3
Design problem can be written as
Desired frequency response at the specific frequencies
2011.9
EEE4176 Application of Digital Signal Processing

23. Least-Squares Linear-Phase FIR Filter Design 3/3
Least Squares Optimization
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EEE4176 Application of Digital Signal Processing

24. EEE4176 Application of Digital Signal Processing
OPTIMAL FIR FILTERS
Chebyshev approximation
Optimum design criterion in the sense that the weighted approximation error between the desired freq response and the actual freq response is spread evenly across the passband and evenly across the stopband of the filter minimizing the maximum error.
Equi-ripple FIR filter
Optimal filter in the sense stated above
Minimize maximum error. Same error in pass and stop bands.
Sharpest FIR filter or Smallest length filter for a given spec.
Precise control of the critical filter frequencies
Frequency sampling method
You specify the frequency response for a set of frequencies
Computer gives the impulse response for that freq. response
2011.9
EEE4176 Application of Digital Signal Processing

25. EEE4176 Application of Digital Signal Processing
OPTIMAL FIR FILTERS
Mathematical background
1. Define the desired amplitude response and the weighting function on a compact subset of
, i.e., the set (union of passbands and stopbands).
2. Define the weighted error function as
3. Choose that minimizes
2011.9
EEE4176 Application of Digital Signal Processing

26. EEE4176 Application of Digital Signal Processing
OPTIMAL FIR FILTERS
Parks-McClellan algorithm
Minmax solution : are fixed are variable.
Use Remez exchange algorithm as a computational procedure based on the alteration theorem
Used to design Multi-band filters (LPF, HPF, BPF, BRF), differentiator, Hilbert
Inputs :
Filter length , filter type, # of bands, edge frequencies,
Desired filter response, weighting factors
Alteration theorem
The function is optimal in the minimax sense if and only if there exist at least K+2 frequencies in such that
At least K + 2 extrema in sign.
2011.9
EEE4176 Application of Digital Signal Processing

27. EEE4176 Application of Digital Signal Processing
OPTIMAL FIR FILTERS
Parks-McClellan algorithm
Example) K = 7 case
2011.9
EEE4176 Application of Digital Signal Processing

28. Practical FIR Filter Design with MATLAB
From Practical FIR Filter Design in MATLAB by Ricardo A. Losada (Mathworks)

29. Practical FIR Filter Design with MATLAB
2011.9
EEE4176 Application of Digital Signal Processing

30. Practical FIR Filter Design with MATLAB
2011.9
EEE4176 Application of Digital Signal Processing

31. Practical FIR Filter Design with MATLAB
2011.9
EEE4176 Application of Digital Signal Processing

32. Practical FIR Filter Design with MATLAB
2011.9
EEE4176 Application of Digital Signal Processing

33. Practical FIR Filter Design with MATLAB
2011.9
EEE4176 Application of Digital Signal Processing

34. Practical FIR Filter Design with MATLAB
2011.9
EEE4176 Application of Digital Signal Processing