1. Eigenfilters: A New Approach to Least-Squares FIR Filter Design and Applications Including Nyquist Filters Advisor : Yung-An Kao Student : Chih-Wei Chen 2006/05/05

2. IEEE Transaction on circuits and system, vol. CAS-34, NO. 1, January 1987 P.P. VAIDYANATHAN, and TRUONG Q. NGUYEN

3. Outline Introduction Linear phase FIR Low-Pass eigenfilters Example

4. Introduction A new method of designing linear-phase FIR filter is proposed, the method is based on the computation of an appropriate real, symmetric, and positive-definite matrix. The proposed design procedure is general enough to incorporate both time and frequency domain constraints Application  Nyquist filter  Equiripple filter

5. Introduction The desired response is The amplitude response of H(z) is Type I filter

6. Introduction The least-squares (LS) approach Linear equation, LSE solution can be express matrix from

7. Linear phase FIR Low-Pass eigenfilters We wish minimizing an error measure using another method If error measure can be expressed the from

8. Linear phase FIR Low-Pass eigenfilters The FIR linear phase filter frequency response Type I filter Type II filter

9. Linear phase FIR Low-Pass eigenfilters Matrix from

10. Linear phase FIR Low-Pass eigenfilters Stopband error

11. Linear phase FIR Low-Pass eigenfilters Passband error It cannot be written in the form Change, derive zero-frequency response is given by

12. Linear phase FIR Low-Pass eigenfilters

13. Total measure to be minimized is

14. Linear phase FIR Low-Pass eigenfilters

15. The solution  Step1:Given ω p 、 ω s 、 α compute P  Step2: Compute the eigenvalue and eigenvector of P  Step3: Find smallest eigenvalue corresponding eigenvector

16. Example

18. The end