Short time coupled fractional fourier transform and the uncertainty principle

2021 ◽  
Vol 24 (3) ◽  
pp. 667-688
Author(s):  
Ramanathan Kamalakkannan ◽  
Rajakumar Roopkumar ◽  
Ahmed Zayed
Keyword(s):  
Fourier Transform ◽  
Time Delay ◽  
Frequency Shift ◽  
Uncertainty Principle ◽  
Fractional Fourier Transform ◽  
Short Time ◽  
Addition Formulas

Abstract In this paper, we introduce a short-time coupled fractional Fourier transform (scfrft) using the kernel of the coupled fractional Fourier transform (cfrft). We then prove that it satisfies Parseval’s relation, derive its inversion and addition formulas, and characterize its range on ℒ 2(ℝ2). We also study its time delay and frequency shift properties and conclude the article by a derivation of an uncertainty principle for both the coupled fractional Fourier transform and short-time coupled fractional Fourier transform.

Novel Short-Time Fractional Fourier Transform: Theory, Implementation, and Applications

2020 ◽  
Vol 68 ◽  
pp. 3280-3295 ◽  
Author(s):  
Jun Shi ◽  
Jiabin Zheng ◽  
Xiaoping Liu ◽  
Wei Xiang ◽  
Qinyu Zhang
Keyword(s):  
Fourier Transform ◽  
Fractional Fourier Transform ◽  
Short Time

A Novel Artificial Intelligence Technique for Analysis of Real-Time Electro-Cardiogram Signal for the Prediction of Early Cardiac Ailment Onset

2020 ◽  
pp. 42-66
Author(s):  
Dinesh Bhatia ◽  
Animesh Mishra
Keyword(s):  
Feature Extraction ◽  
Fourier Transform ◽  
Fractional Fourier Transform ◽  
Mapping Technique ◽  
Self Organizing Maps ◽  
Map Generation ◽  
Unsupervised Training ◽  
Cardio Vascular ◽  
Short Time

The role of ECG analysis in the diagnosis of cardio-vascular ailments has been significant in recent times. Although effective, the present computational algorithms lack accuracy, and no technique till date is capable of predicting the onset of a CVD condition with precision. In this chapter, the authors attempt to formulate a novel mapping technique based on feature extraction using fractional Fourier transform (FrFT) and map generation using self-organizing maps (SOM). FrFT feature extraction from the ECG data has been performed in a manner reminiscent of short time Fourier transform (STFT). Results show capability to generate maps from the isolated ECG wavetrains with better prediction capability to ascertain the onset of CVDs, which is not possible using conventional algorithms. Promising results provide the ability to visualize the data in a time evolution manner with the help of maps and histograms to predict onset of different CVD conditions and the ability to generate the required output with unsupervised training helping in greater generalization than previous reported techniques.

Time delay estimation using fractional Fourier transform

2007 ◽  
Vol 87 (5) ◽  
pp. 853-865 ◽  
Author(s):  
Kamalesh Kumar Sharma ◽  
Shiv Dutt Joshi
Keyword(s):  
Fourier Transform ◽  
Time Delay ◽  
Fractional Fourier Transform ◽  
Time Delay Estimation ◽  
Delay Estimation

Time Delay Model of Fractional Fourier Transform and the Application in Signal Filtering

2011 ◽  
Vol 121-126 ◽  
pp. 3637-3641
Author(s):  
Yuan Gan Wang ◽  
Hong Lin Yu ◽  
Xin Yu Liang
Keyword(s):  
Fourier Transform ◽  
Time Delay ◽  
Fractional Fourier Transform ◽  
Transform Domain ◽  
Original Signal ◽  
Delay Model ◽  
Traditional Methods ◽  
Noise Component ◽  
Signal Filtering ◽  
Actual Signal

Various frequency bands of noises are contained in the actual signal. And it's difficult to eliminate the noise portion, which has a time delay and same spectrum with the original signal with conventional filtering methods. Based on the time delay and the multiplication delay characteristics of the fractional Fourier transform (FRFT), we put forward a FRFT time delay model, which can increase distance between the signal and the noise component. Through corresponding Fractional Fourier Transform to noise-containing signal, the distance between the signal and common frequency noises can be constantly increased within the transform domain, thus easily separating the noise component. The algorithm of the model can be simply deduced, easy realized and converged fast. In the experiment, we simulated the separating characteristics of the transform, and used the method to de-noise the grating signal. Compared with other traditional methods, we find that the FRFT acquired a better result.

Synchrosqueezing with short-time fourier transform method for trinary frequency shift keying encoded SSVEP

2020 ◽  
pp. 1-15
Author(s):  
Dechun Zhao ◽  
Xiaoxiang Li ◽  
Xiaorong Hou ◽  
Mingyang Feng ◽  
Renping Jiang
Keyword(s):  
Fourier Transform ◽  
Frequency Shift ◽  
Recognition Algorithm ◽  
Short Time Fourier Transform ◽  
Frequency Shift Keying ◽  
Transform Method ◽  
Coherent Demodulation ◽  
Reconstructed Signal ◽  
Shift Keying ◽  
Short Time

BACKGROUND: The frequencies that can evoke strong steady state visual evoked potentials (SSVEP) are limited, which leads to brain-computer interface (BCI) instruction limitation in the current SSVEP-BCI. To solve this problem, the visual stimulus signal modulated by trinary frequency shift keying was introduced. OBJECTIVE: The main purpose of this paper is to find a more reliable recognition algorithm for SSVEP-BCI based on trinary frequency shift keying modulated stimuli. METHODS: First, the signal modulated by trinary frequency shift keying is simulated by MATLAB. At different noise levels, the empirical mode decomposition, singular value decomposition, and synchrosqueezing with the short-time Fourier transform are used to extract the characteristic frequency and reconstruct the signal. Then, the coherent method is used to demodulate the reconstructed signal. Second, in the paradigm of BCI using trinary frequency shift keying modulated stimuli, the three methods mentioned above are used to reconstruct EEG signals, and canonical correlation analysis and coherent demodulation are used to recognize the BCI instructions. RESULTS: For simulated signals, it is found that synchrosqueezing with short-time Fourier transform has a better effect on extracting the characteristic frequencies. For the EEG signal, it is found that the method combining synchrosqueezing with short-time Fourier transform and coherent demodulation has a higher accuracy and information translate rate than other methods. CONCLUSION: The method combining synchrosqueezing with short-time Fourier transform and coherent demodulation proposed in this paper can be applied in the SSVEP system based on trinary frequency shift keying modulated stimuli.

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