Novel generalized Fourier representations and phase transforms

Abstract

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 2 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.

The Generalized Fourier Transform: A Unified Framework for the Fourier, Laplace, Mellin and Z Transforms

“…We, hereby, observe that when σ = 0 (or lim σ → 0), then GFT will become the FT with s = jω, s * = −jω and thus, FT can be used for finding the solution of IVPs by adopting the differentiation property discussed in (26), provided the FT of the functions under consideration exist. Thus, to obtain the solution of IVPs, we derive and propose the following derivative property of the FT as…”

“…There Similar to the GFT ( 5) and (53), we can extend the proposed methodology and include that class of signals which are not in L p (R), p ∈ (0, ∞). For example, the CWT of a signal x(t) is defined if x(t) ∈ L 2 (R) and the original can be recovered by inverse CWT [22,23,24,25,26]. We hereby define the CWT of a signal x(t) such that x(t) ∈ L 2 (R) and x(t) exp(−σ |t|) ∈ L 2 (R) for some σ > σ 0 as…”

High-gain Spontaneous Parametric Down-conversion in Dispersive and Absorbing Nanostructured Systems

“…Now we examine the time integral t −∞ dt ′ e i(ωp−ω ′ −ν)t ′ and introduce a substitution t ′ = t − τ which allows us to rewrite it as e i(ωp−ω ′ −ν)t ∞ 0 dτ e i(ω ′ +ν−ωp)τ ≡ e i(ωp−ω ′ −ν)t ζ(ω ′ + ν − ω p ), where ζ(ω) is a generalised function proportional to the Fourier transform of the Heaviside step function and is closely related to the analytical properties of the GF [75,76]. One notable analytical property of the GF is [46,76]:…”

Identification Strategy Design with the Solution of Wavelet Singular Spectral Entropy Algorithm for the Aerodynamic System Instability

“…Chen [15] used wavelet transform to decompose the data signal to obtain necessary frequency details, improving the accuracy of classification and detection for fault. Because of its time-frequency function and refinement capability, the wavelet transform is widely used in the analysis of non-stationary signals and processing with its nature of being adaptive to the signal [16][17][18]. However, under the only consideration with the wavelet transform, it may cause the problem of overlapping information features.…”