approximation of a linear shift–variant system by a set of linear shift–invariant systems
DESCRIPTION
Approximation of a Linear Shift–Variant System by a Set of Linear Shift–Invariant Systems. Vasile Buzuloiu * , Marius Malciu * †, Sanjit K. Mitra‡ * University “Politehnica” of Buc u re ş t i , Rom â nia † CERN, Geneva , Switzerland ‡ University of California at Santa Barbara , USA. Outline. - PowerPoint PPT PresentationTRANSCRIPT
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