1. 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 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 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

4. 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

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