1. ECE4270 Fundamentals of DSP Lecture Chebyshev Approximation Algorithm for FIR Design
J McClellan
School of Electrical and Computer Engineering
Center for Signal and Image Processing
Georgia Institute of Technology

2. Overview of Lecture
Read Chapter 7 (Oppenheim-Schafer 2010): 7.7 and 7.8
The Remez exchange algorithm
a.k.a. “Parks-McClellan” algorithm (1972)
OPTIMAL approximation of the FIR filter
Chebyshev norm approximation
Alternation Theorem
Equiripple Frequency Response
Examples of FIR filters
Lowpass, Highpass, Bandpass, Differentiators
Comparison to Kaiser Window Design Method

3. General Frequency Selective Filter
Transition width is same
at each discontinuity
and ripple is scaled by
size of discontinuity.
Can be thought of as a sum and difference ideal lowpass filters.

4. Example with Rectangular Window

5. Minimize an Error Norm
Find the impulse response that minimizes the approximation error
Least-squares or L2 Lp L1
L-infinity
Not differentiable

6. Approximation in Different Norms
Least-squares norm produces
filters with large overshoots
near the transition region.
Minimizes energy in the “error”
Chebyshev norm produces
“equiripple” filters.
Also called min-max norm

7. Parks and McClellan, 1972
T. W. Parks and J. H. McClellan, IEEE Trans. Circuit Theory,
CT-19, pp , March, 1972.

8. Remez
“Remez exchange algorithm” for Chebyshev approximation was developed in the mid 1930’s.
Atomic Energy Commission translated his work into english; also that of many other Russian mathematicians

9. The Parks McClellan Algorithm
Uses the Remez exchange algorithm to iteratively find the impulse response that minimizes the maximum approximation error over a set of closed intervals in the frequency-domain.
Leads to equiripple approximations that are optimum in the sense of having the smallest approximation error for a given transition width.
In MATLAB, the function is remez( ), or firpm()

10. The Alternation Theorem
Weighted approximation error:
Minimize the maximum error over a disjoint set of frequencies:
The optimum approximation alternates between +d and -d at least L+2 times in F. The maximum number of alternations (for LPF) is L+3.

11. Optimum FIR Filter Count Extremal points Chebyshev norm produces
“equiripple” filters.
Error “alternates” between +/- maximum error
Count Extremal points

12. Chebyshev Polynomials
Sum of cosines = Polynomial

13. Linear Phase Type I FIR Filter
Zero-phase impulse response:
Frequency response:
Causal version:
Four Types

14. Weighted Approximation Error

15. Exchange Algorithm Duality
Search for frequencies where the error alternates between +/- maximum error. These are the EXTREMAL Frequencies
d is always increasing
Duality

16. Remez (PMc-FIR) Demo

17. Remez (PMc-FIR) Demo

18. Remez (PMc-FIR) Demo

19. Remez (PMc-FIR) Iteration

20. Remez (PMc-FIR) Demo

21. Remez FIR Design M=17, Length=18

22. Remez FIR Design M=18, Length=19

23. Remez (PMc-FIR) Bandpass Filter

24. A Design Example The C-T specifications are (1/T=2000 Hz):
The corresponding D-T specifications are:

25. Design Formula
Kaiser obtained the following design formula for PMc-FIR by curve fitting many examples:
MATLAB example:
» [M,Fo,Mo,W] = remezord( [.4,.6], [1,0], [0.01,0.001], 2 );
» [h,delta]=remez(M,Fo,Mo,W);
firpm

26. Remez (PMc-FIR) Lowpass Design

27. Remez (P-Mc FIR) Differentiator
Length=6, M=5
Type-IV Filter
Odd symmetry
RED=desired
BLUE=design
Hilbert Transformer also Type-III or IV with ODD symmetry

28. Compare Remez & Kaiser Design formula “estimate” for REMEZ
Design formula for KAISER
Example:

29. END
Questions?

30. Comparison of Filter Structures
Complexity is proportional to amount of computation and storage plus program storage and computational cycles.
FIR direct form - (M+1) coefficients
(M+1) multiplications, M additions
(M+1) coefficients, M delays (registers)
M = Filter Order
IIR cascade form - Ns second-order sections.
5Ns multiplications, 5Ns additions
5Ns coefficients, 5Ns delays

31. Lowpass Filter Implementations
Specifications of lowpass filter
These specs met by the following approx.
Butterworth - 12th-order
Chebyshev - 8th-order
Elliptic - 6th-order
Kaiser window - 37 sample impulse response
Parks-McClellan - 27 sample impulse response

32. Comparison of Lowpass Filters