Approximation of a Linear Shift–Variant System by a Set of Linear Shift–Invariant Systems

1. Approximation of a Linear Shift–Variant System bya Set of Linear Shift–Invariant Systems Vasile Buzuloiu*, Marius Malciu*†, Sanjit K. Mitra‡ * University “Politehnica” of Bucureşti, România † CERN, Geneva, Switzerland ‡ University of California at Santa Barbara, USA

2. Outline • Introduction • Our method • Application to one-dimensional systems

3. Abstract • We present a method to approximate theimpulse response of a LSV (Linear Shift-Variant) system by the impulseresponses of a set of LSI (Linear Shift-Invariant) systems which processin parallel on various windowed versions of the input signal • Themethod is outlined for one-dimensional systems • The extensionto the multidimensional case is straightforward

4. Motivation • The interest for such a subject • There are enough examples for which the linearity is an acceptable hypothesis for thepractical range of the variables, but the shift-invariance is not • The LSI property is a very useful one as it allows easyanalysis and design of the systems • The approximation is useful for image restoration

5. Characterization of LSI systems

6. Characterization of LSV systems

7. Decomposing h(t,τ) in bricks

8. Decomposing h(t,τ) in bricks (2)

9. A 1-D example

10. How we choose

11. Consequence

12. Equivalent block diagram

13. Remark • The windows are not LSI blocks • Nevertheless this gives a standard structure for separating the LSI and LSV parts of the system

24. The N-dimensional case