Some studies on nonpolynomial interpolation and error analysis

2017

Self Cite

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

Section: Methodsmentioning

confidence: 99%

Section: Methodsmentioning

confidence: 99%

Section: Methodsmentioning

confidence: 99%

See 1 more Smart Citation

Breaking the Limits: Redefining the Instantaneous Frequency

2017

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“…denote real part of analytic signal (AS) and subscripts denote the quadrants of the Fourier domain considered in AS representation. The DTFT of these AS, with their frequency supports, can be written as (16).…”

Section: The 2d-dtft and Associated Analytic Signalmentioning

confidence: 99%

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

Joshi

2019

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In this paper, we propose the Fourier frequency vector (FFV), inherently, associated with multidimensional Fourier transform. With the help of FFV, we are able to provide physical meaning of so called negative frequencies in multidimensional Fourier transform (MDFT), which in turn provide multidimensional spatial and space-time series analysis. The complex exponential representation of sinusoidal function always yields two frequencies, negative frequency corresponding to positive frequency and vice versa, in the multidimensional Fourier spectrum. Thus, using the MDFT, we propose multidimensional Hilbert transform (MDHT) and associated multidimensional analytic signal (MDAS) with following properties: (a) the extra and redundant positive, negative, or both frequencies, introduced due to complex exponential representation of multidimensional Fourier spectrum, are suppressed, (b) real part of MDAS is original signal, (c) real and imaginary part of MDAS are orthogonal, and (d) the magnitude envelope of a original signal is obtained as the magnitude of its associated MDAS, which is the instantaneous amplitude of the MDAS. The proposed MDHT and associated DMAS are generalization of the 1D HT and AS, respectively. We also provide the decomposition of an image into the AM-FM image model by the Fourier method and obtain explicit expression for the analytic image computation by 2D-DFT.Index Terms-The Fourier and Hilbert transform, analytic signal (AS), Fourier frequency vector (FFV), (time) frequency (rad/s) and spatial frequency (rad/m), single orthant Fourier transform (SOFT).

“…The value of MF and RBW is estimated for IBFs of each channel (denoted as I μ and I σ ) using (3) and (4), respectively. To capture the slowly varying features of the IBFs, we obtain upper envelope of each IBFs, using nonpolynomial cubic spline interpolation [20], and estimate the value of MF and RBW of mean removed upper envelopes (denoted as E μ and E σ , respectively).…”

Section: Filter Bank Processing Mean Frequency and Rms Bandwidth Calmentioning

confidence: 99%

Fourier-Based Feature Extraction for Classification of EEG Signals Using EEG Rhythms

Joshi

Patney

et al. 2015

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In this paper, we propose a method for the analysis and classification of electroencephalogram (EEG) signals using EEG rhythms. The EEG rhythms capture the nonlinear complex dynamic behavior of the brain system and the nonstationary nature of the EEG signals. This method analyzes common frequency components in multichannel EEG recordings, using the filter bank signal processing. The mean frequency (MF) and RMS bandwidth of the signal are estimated by applying Fouriertransform-based filter bank processing on the EEG rhythms, which we refer intrinsic band functions, inherently present in the EEG signals. The MF and RMS bandwidth estimates, for the different classes (e.g., ictal and seizure-free, open eyes and closed eyes, inter-ictal and ictal, healthy volunteers and epileptic patients, inter-ictal epileptogenic and opposite to epileptogenic zone) of EEG recordings, are statistically different and hence used to distinguish and classify the two classes of signals using a least-squares support vector machine classifier. Experimental results, with 100 % classification accuracy, on a real-world EEG signals database analysis illustrate the effectiveness of the proposed method for EEG signal classification. B Pushpendra Singh