Some Studies on Multidimensional Fourier Theory for Hilbert Transform, Analytic Signal and AM–FM Representation

Digital Signal Processing

2020

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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.

“…In this appendix, we consider the PT of a 2D signal (e.g., image) which can be easily extended for multidimensional signals. Let

be a non-periodic and real valued function, then the 2D-FT is defined as

and the inverse 2D-FT is defined as

The 2D PT transfer function corresponding to 1D counterpart (27) can be written as

where the 2D analytic signal (2D-AS) is defined by considering the first and fourth quadrants of the 2D-FT plane as [6]

where

is the HT of

. The 2D-PT can be computed by considering the real part of the 2D-PT of 2D-AS which we defined as

and if

, then

, therefore 2D counter part of 1D PT (30) can be defined as

where

denotes real part of the function

.…”

Section: Multidimensional Ptmentioning

confidence: 99%

“…The 2D-PT can be computed by considering the real part of the 2D-PT of 2D-AS which we defined as

and if

, then

, therefore 2D counter part of 1D PT (30) can be defined as

where

denotes real part of the function

. As the 2D-AS (102) is not unique and it can also be defined by considering the first and second quadrants of the 2D-FT plane [6] , so corresponding modifications (i.e., integration limits would be

0

) can be easily applied to equations (101) , (102) and (103) .…”

Section: Multidimensional Ptmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Novel generalized Fourier representations and phase transforms

Digital Signal Processing

2020

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“…The FDM can decompose real as well as complex signals (which can be multichannel or multivariate) into a set of desired number of Fourier intrinsic band functions (FIBFs) with desired cut-off frequencies [41]. The FDM has demonstrated its efficacy for representation and analysis of nonlinear and non-stationary data in many applications [14,40,[42][43][44]. In this work, we consider the type-2 DCT [15], which is the most common variant of DCTs, to formulate the FDM.…”

Section: Introductionmentioning

confidence: 99%

Novel Fourier quadrature transforms and analytic signal representations for nonlinear and non-stationary time-series analysis

2018

R. Soc. open sci.

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The Hilbert transform (HT) and associated Gabor analytic signal (GAS) representation are well known and widely used mathematical formulations for modelling and analysis of signals in various applications. In this study, like the HT, to obtain quadrature component of a signal, we propose novel discrete Fourier cosine quadrature transforms (FCQTs) and discrete Fourier sine quadrature transforms (FSQTs), designated as Fourier quadrature transforms (FQTs). Using these FQTs, we propose 16 Fourier quadrature analytic signal (FQAS) representations with following properties: (1) real part of eight FQAS representations is the original signal, and imaginary part of each representation is FCQT of real part; (2) imaginary part of eight FQAS representations is the original signal, and real part of each representation is FSQT of imaginary part; (3) like the GAS, Fourier spectrum of all FQAS representations has only positive frequencies; however, unlike the GAS, real and imaginary parts of FQAS representations are not orthogonal. The Fourier decomposition method (FDM) is an adaptive data analysis approach to decompose a signal into a set Fourier intrinsic band functions. This study also proposes new formulations of the FDM using discrete cosine transform with GAS and FQAS representations, and demonstrates its efficacy for improved time-frequency-energy representation and analysis of many real-life nonlinear and non-stationary signals.

“…In order to avoid this problem, recently many nonlinear and nonstationary signal representation, decomposition and analysis methods, e.g. empirical mode decomposition (EMD) algorithms [1,[3][4][5][6][7][8]14], synchrosqueezed wavelet transforms (SSWT) [9], variational mode decomposition (VMD) [10], eigenvalue decomposition (EVD) [11], empirical wavelet transform (EWT) [12], sparse timefrequency representation [15], time-varying vibration decomposition [16], resonance-based signal decomposition [17] and Fourier decomposition methods (FDM) [13,18,19,21,22,24] based on the Fourier theory, are proposed. The Fourier theory is the only tool for spectrum analysis of a signal and the FDM has established that it is a superior tool for nonlinear and nonstationary time series analysis.…”

Section: Introductionmentioning

confidence: 99%

Breaking the Limits: Redefining the Instantaneous Frequency

Circuits Syst Signal Process

2017

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International audienceThe Carson and Fry (1937) introduced the concept of variable frequency as a generalization of the constant frequency. The instantaneous frequency (IF) is the time derivative of the instantaneous phase and it is well-defined only when this derivative is positive. If this derivative is negative , the IF creates problem because it does not provide any physical significance. This study proposes a mathematical solution and eliminate this problem by redefining the IF such that it is valid for all monocomponent and multicomponent signals which can be nonlinear and nonstationary in nature. This is achieved by using the property of the multivalued inverse tangent function which provides basis to ensure that instantaneous phase is an increasing function. The efforts and understanding of all the methods based on the IF would improve significantly by using this proposed definition of the IF. We also demonstrate that the decomposition of a signal, using zero-phase filtering based on the well established Fourier and filter theory , into a set of desired frequency bands with proposed IF produces accurate time-frequency-energy (TFE) distribution that reveals true nature of signal. Simulation results demonstrate the efficacy of the proposed IF that makes zero-phase filter based decomposition most powerful, for the TFE analysis of a signal, as compared to other existing methods in the literature