Eigenfilters: A New Approach to Least-SquaresFIR 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

• 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

R is the region 0 but excluding the transition

• The least-squares (LS) approach

• Linear equation , LSE solution can be express matrix from

and are quantities depending upon and p sc A

Linear phase FIR Low-Pass eigenfilters

• We wish minimizing an error measure using another method

• If error measure can be expressed the from

2

: real vector related to

: real, symmetric, and positive-definite matrix, about and

n

p s

h

v

P

Linear phase FIR Low-Pass eigenfilters

• The FIR linear phase filter frequency response

( 1) / 2 for even ( 1)

/ 2 for odd ( 1)

M N N

M N N

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

0 1

Linear phase FIR Low-Pass eigenfilters

Linear phase FIR Low-Pass eigenfilters

• The solution Step1:Given ωp 、 ωs、 α compute P Step2: Compute the eigenvalue and eigenve

ctor of P Step3: Find smallest eigenvalue correspondi

ng eigenvector

Example

Low-Pass Filter

parameter

-1=28

0.3 0.4

0.1,0.5

p s

N

Example

The end