Approximation of a Linear Shift–Variant System bya Set of Linear Shift–Invariant SystemsVasile Buzuloiu*, Marius Malciu*†, Sanjit K. Mitra‡

* University “Politehnica” of Bucureşti, România† CERN, Geneva, Switzerland‡ University of California at Santa Barbara, USA

Outline

Introduction Our method Application to one-dimensional systems

Abstract

We present a method to approximate the impulse response of a LSV (Linear Shift-Variant) system by the impulse responses of a set of LSI (Linear Shift-Invariant) systems which process in parallel on various windowed versions of the input signal

The method is outlined for one-dimensional systems

The extension to the multidimensional case is straightforward

Motivation

The interest for such a subject There are enough examples for which the linearity is

an acceptable hypothesis for the practical range of the variables, but the shift-invariance is not

The LSI property is a very useful one as it allows easy analysis and design of the systems

The approximation is useful for image restoration

Characterization of LSI systems

dtftf )()()(

dthftg )()()(

Characterization of LSV systems

dtftf )()()(

dthftg ),()()(

Decomposing h(t,τ) in bricks

),(),(),(1

thththN

kk

N

kk thth

1

),(),(

Decomposing h(t,τ) in bricks (2)

N

kk ththIf

1

),(),(

dfthtgtgtg )(),()(,)()(

A 1-D example

How we choose ),( thk

)()(

),()(),(

thw

thwth

kk

kkk

otherwise

ifw kkk ,0

,,1)( 211

Consequence

N

kkk dthftg

1

)()()(

)()()( fwf kk

Equivalent block diagram

Remark

The windows are not LSI blocks Nevertheless this gives a standard

structure for separating the LSI and LSV parts of the system

The N-dimensional case

Nxxxt ,,, 21

Nxxx ,,, 21