Advisor : Yung-An Kao Student : Chih-Wei Chen 2006/05/05

Eigenfilters: A New Approach to Least-SquaresFIR Filter Design and ApplicationsIncluding Nyquist Filters Advisor : Yung-An Kao Student : Chih-Wei Chen 2006/05/05

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

Outline • Introduction • Linear phase FIR Low-Pass eigenfilters • Example

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

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

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

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

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

Linear phase FIR Low-Pass eigenfilters • Matrix from

Linear phase FIR Low-Pass eigenfilters • Stopband error

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

Linear phase FIR Low-Pass eigenfilters

Linear phase FIR Low-Pass eigenfilters • Total measure to be minimized is

Linear phase FIR Low-Pass eigenfilters

Linear phase FIR Low-Pass eigenfilters • The solution • Step1:Given ωp、 ωs、αcomputeP • Step2: Compute the eigenvalue and eigenvector of P • Step3: Find smallest eigenvalue corresponding eigenvector

Example

The end

Advisor : Yung-An Kao Student : Chih-Wei Chen 2006/05/05