Fractional convolution and correlation via operator methods and an application to detection of linear FM signals

Using operator theory methods together with our previously introduced unitary fractional operator, we derive explicit definitions of fractional convolution and correlation operations in a systematic and comprehensive manner. Via operator manipulations, we also provide alternative formulations of those fractional operations that suggest efficient algorithms for discrete implementation. Through simulation examples, we demonstrate how well the proposed efficient method approximates the continuous formulation of fractional autocorrelation. It is also shown that the proposed fractional autocorrelation corresponds to radial slices of the ambiguity function (AF). We also suggest an application of the fast fractional autocorrelation for detection and parameter estimation of linear FM signals.