Linear fractional shift invariant (LFSI) systems

In this paper, we formulate continuous time linear fractional shift invariant (LFSI) systems that generalize the well-known linear time invariant (LTI) systems by means of an angle parameter φ. LTI systems are obtained as a special case of LFSI systems for φ = 0. LFSI systems belong to the large class of time-varying systems. Whereas LTI systems commute with time shifts, LFSI systems commute with fractional shifts defined on the time-frequency plane. Just as the conventional Fourier transform (FT) diagonalizes LTI systems, an LFSI system associated with angle φ is diagonalized by the fractional Fourier transform (FrFT) defined at the perpendicular angle φ + (φ/2). We show that the eigen-functions of an LFSI system at angle φ are linear FM (chirp) signals with a sweep rate of tan φ. Finally, we demonstrate via a simulation example that, in certain cases, LFSI systems can outperform LTI systems.