The Hilbert transform based on the FRFT

One of the important properties of the analytic signal is to reconstruct the real signal from its analytic version, which is an unresolved problem in the published papers about the Hilbert transform (HT) based on the fractional Fourier transform (FRFT). The definition and properties of a HT and analytic signal based on the FRFT were proposed, which are connected with the convolution and product theorem for the FRFT, the theoretic analysis and deducing were given. The analytic version contains no negative spectrum while maintaining the essential information of the real signal. The simulated result shows that the theory can enhance the security of the single-sideb and communication (SSB) system and improve the encryption for the real image. The theory is the generalization of the HT based on the FRFT.