Linear fractional shift invariant (LFSI) systems

is obtained as another special case' 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 4. LTI systems are obtained as a special case of LFSl systems for 4 = 0. LFSl systems belong to the large class of time-varying systems. Whereas LTI systems commute with time shifts, LFSl systems commute with fractional shifts defined on the time-frequency plane. Just as the conventional Fourier transform (FT) diagonalizes LTI systems, an LFSl system associated with angle 4 is diagonalized by the fractional Fourier transform (FrFT) defined at the perpendicular angle 1 + (r/Z). We show that the eigenfunctions of an LFSI system at angle 4 are linear FM (chirp) signals with a sweep rate of tan 4. Finally, we demonstrate via a simulation example that, in certain cases, LFSI systems can outperform LTI systems. Additionally, in the limit as 4 approaches TT or odd multiples of K , the FrFT reduces to a simple axis-reversal operation, (lF"s)(t) = s ( t ) . (2) The FrFT can alternatively be interpreted as a unitary signal representation with respect to a fractional domain. r, as illustrated in Fig. I . I '