Unitary and Hermitian fractional operators and their extensions: Fractional Mellin transform, joint fractional representations and fractional correlations

We give an overview of the fractional Fourier transform (FrFT), summarize some fundamental properties of the FrFT and demonstrate its relationships with signal transforms such as the Wigner distribution (WD), the ambiguity function (AF) and the Radon transform (RT).; We also provide a review of Hermitian and unitary operators and their properties with regard to their use in theoretical signal analysis and especially joint signal representations. We study the equivalence relation between unitary and Hermitian operator representations by reviewing the theory developed by Sayeed and Jones. We review Scully and Cohen's characteristic function operator method to derive joint signal representations of arbitrary variables.; Inspired by the recent popularity of the (FrFT) and motivated by the use of Hermitian and unitary operator methods in signal analysis, we introduce a new unitary fractional-shift operator. The unitary fractional-shift operator generalizes the well-known time-shift and frequency-shift operators by describing shifts at arbitrary orientations of the time-frequency plane. We establish the connection with the FrFT by deriving two signal representations, one invariant and the other covariant, to the newly introduced unitary fractional-shift operator. Via Stone's theorem and the duality concept, we also derive the new Hermitian fractional operator which generalizes the fundamental Hermitian time and frequency operators.; We suggest an application of the fast fractional autocorrelation for detection and parameter estimation of linear FBI (chirp) signals.; Using the new Hermitian fractional operator within the characteristic function operator method, we also derive joint fractional representations (JFRs) of signals. We compute the fractional AF and the fractional WD of some simple functions and provide the plots for a Gaussian amplitude-modulated linear FM signal.; We extend the FrFT and related fractional concepts of the time-frequency phase space into a new fractional Mellin transform. We give examples of how the FrMT a better analysis tool than the conventional Fourier transform or the FrFT for certain types of nonlinear FM signals.; We suggest the use of FrFT in excising broadband, linear FM interferences in spread spectrum communication systems. We propose a preprocessing of the received signal by an FrFT-based excision scheme prior to demodulation. Our simulations demonstrate that, for reasonably strong interferences, this technique often improves the bit error rate performance of the receiver. (Abstract shortened by UMI.)...