Adaptive short-time Fourier transform and synchrosqueezing transform for non-stationary signal separation

On fractional Langevin equation involving two fractional orders

In this work, we investigate existence and uniqueness of solutions for nonlinear Langevin equation involving two fractional orders with anti-periodic boundary conditions. Our analysis relies on the coupled fixed point theorems for mixed monotone mappings in partially ordered metric spaces. An example illustrating our approach is also discussed.

Fractional Langevin equation with anti-periodic boundary conditions

The offset linear canonical transform (OLCT) plays an important role in many fields of optics and signal processing. In this paper, we address the problem of signal filtering and reconstruction in the OLCT domain based on new convolution theorems. Firstly, we propose new convolution and product theorems for the OLCT, which state that a modified ordinary convolution in the time domain is equivalent to simple multiplication operations for the OLCT and the Fourier transform (FT). Moreover, it is expressible by a one dimensional integral and is easy to implement in designing filters. The classical convolution theorem in the FT domain is shown to be a special case of our derived results. Then, a practical multichannel sampling expansion for band-limited signal with the OLCT is introduced. This sampling expansion constructed by the new convolution structure can reduce the effect of spectral leakage and is easy to implement. By designing OLCT filters, we can obtain derivative sampling and second-order derivative interpolation. Furthermore, potential applications of the multichannel sampling are discussed. Last, based on the new convolution structure, we investigate and discuss several applications, including swept-frequency filter analysis, image denosing and image encryption. Some illustrations and simulations are presented to verify the validity and effectiveness of the proposed method.

Convolution and Multichannel Sampling for the Offset Linear Canonical Transform and Their Applications

The linear canonical transform (LCT) has been shown to be a powerful tool for optics and signal processing. Many theories for this transform are already known, but the uniform sampling theorem, as well as the sampling rate conversion theory about arbitrary lattices sampling in the LCT domain are still to be determined. Focusing on these issues, this paper carefully investigates arbitrary lattices sampling, the sampling with separable matrices and nonseparable matrices, to obtain uniform sampling theorem and the sampling rate conversion theory in the LCT domain. Firstly, the spectral expression of the discrete-time signal sampled via arbitrary lattice is deduced in the LCT domain. Based on it we propose the alias-free sampling relationship between two matrices and present the perfect reconstruction expressions for bandlimited signals in the LCT domain. Secondly, for further research on discrete signals to obtain sampling rate conversion theory, we define the multidimensional discrete time linear canonical transform (MDTLCT), as well as the convolution for the MDTLCT. Thirdly, the formulas of multidimensional interpolation and decimation via integer matrices in the LCT domain are derived. Then, based on the results of interpolation and decimation, we make analyses of the sampling rate conversion via rational matrices in the LCT domain, including spectral analyses and the formulas in time domain. Finally, simulation results and the potential applications of the theories are also presented.

Lattices sampling and sampling rate conversion of multidimensional bandlimited signals in the linear canonical transform domain

Existence and uniqueness of solutions of initial value problems for nonlinear langevin equation involving two fractional orders

Stochastic resonance of two coupled fractional harmonic oscillators with fluctuating mass

This paper studies the resonance behavior in two coupled harmonic oscillators with fluctuating mass. Firstly, the statistic synchronization between the two particles is obtained and defined. Then, the analytical expression of the output amplitude gain is derived by using the stochastic averaging method. Based on the analytical result and some corresponding numerical results, we analyze the coupling’s influence on the resonance behaviors of the output amplitude gain, including the parameter-induced stochastic resonance, the bona fide resonance, and the stochastic resonance. In weak coupling region, the coupling can not only enhance or weaken the resonance behaviors, but also change the resonance forms. In strong coupling region, the two particles are forced together by the coupling force and move synchronously; thus, the coupling’s influence will vanish. Finally, some numerical simulations are performed to provide a verification and an intuitive understanding of the theoretical results.

The resonance behavior in two coupled harmonic oscillators with fluctuating mass

We introduce a new instantaneous frequency (IF) estimator investigating the resulting process of the IF estimation within the framework of the signal's phase derivative and the linear canonical transform. We define a quantitative performance index for computational speed to show that the new estimator can outperform others within that framework in optimising computational speed (minimising the performance index). The authors investigate the relationship between the new estimator and the previous one proposed by Li et al.
, and also compare their estimation accuracy and computational speed. The authors further verify the theoretical analyses by applying the new and Li's estimators to capture the IF of some linear and non-linear frequency-modulated signals.

https://ietresearch.onlinelibrary.wiley.com/doi/epdf/10.1049/iet-spr.2017.0469

Estimating instantaneous frequency based on phase derivative and linear canonical transform with optimised computational speed

Focusing on two issues associated with the existing multichannel sampling expansions (MSEs) of the linear canonical transform (LCT), those are, the implicit expression of system functions possibly leads to the inconvenience of reconstructing signals in practical situations, and the reconstruction of the original signal from its finite samples, the authors first propose a novel MSE in the Fourier transform domain, providing an explicit expression for the response function of the reconstruction filter. Moreover on this basis, they formulate two kinds of LCT-type of MSEs related, respectively, to the modified convolution structure and the generalised convolution structure of the LCT. For these MSEs, though there is an explicit expression for system functions, the number of the signal's samples takes infinity. They then obtain multichannel interpolation formulae that interpolate a finite set of uniform samples through applying the derived MSEs to the LCT-band-limited, chirp periodic signals. They further present some possible applications of their proposals to show the advantage of the theory. Finally, the simulations are also performed to verify the correctness of the derived results.

https://ietresearch.onlinelibrary.wiley.com/doi/pdf/10.1049/iet-spr.2016.0680

Tao Yu

Papers

Existence and uniqueness of solutions of initial value problems for nonlinear langevin equation involving two fractional orders

Stochastic resonance of two coupled fractional harmonic oscillators with fluctuating mass

The resonance behavior in two coupled harmonic oscillators with fluctuating mass

Estimating instantaneous frequency based on phase derivative and linear canonical transform with optimised computational speed

Multichannel sampling expansions in the linear canonical transform domain associated with explicit system functions and finite samples