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

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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. IEEE Transaction on circuits and system, vol. CAS-34, NO. 1, January 1987

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

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### Eigenfilters: A New Approach to Least-SquaresFIR Filter Design and ApplicationsIncluding Nyquist Filters

Student : Chih-Wei Chen

2006/05/05

• 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
• Passband error

It cannot be written in the form

Change, derive zero-frequency response is given by

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