Novel generalized Fourier representations and phase transforms

The Fourier representations (FRs) are indispensable mathematical formulations for modeling and analysis of physical phenomena and engineering systems. This study presents a new set of generalized Fourier representations (GFRs) and phase transforms (PTs). The PTs are special cases of the GFRs and true generalizations of the Hilbert transforms. In particular, the Fourier transform based kernel of the PT is derived and its various properties are discussed. The time derivative and integral, including fractional order, of a signal are obtained using the GFR. It is demonstrated that the general class of time-invariant and time-variant filtering operations, analog and digital modulations can be obtained from the proposed GFR. A narrowband Fourier representation for the time-frequency analysis of a signal is also presented using the GFR. A discrete cosine transform based implementation, to avoid end artifacts due to discontinuities present in the both ends of a signal, is proposed. A fractional-delay in a discrete-time signal using the FR is introduced. The fast Fourier transform implementation of all the proposed representations is developed. Moreover, using the analytic wavelet transform, a wavelet phase transform (WPT) is proposed to obtain a desired phase-shift in a signal under-analysis. A wavelet quadrature transform (WQT) is also presented which is a special case of the WPT with a phase-shift of $$ radians. Thus, a wavelet analytic signal representation is derived from the WQT. Theoretical analysis and numerical experiments are conducted to evaluate effectiveness of the proposed methods.