Time–Frequency Localization for the Fractional Fourier Transform in Signal Processing and Uncertainty Principles

2021 ◽  
Author(s):  
Zaineb Aloui ◽  
Kamel Brahim
Keyword(s):  
Signal Processing ◽  
Fourier Transform ◽  
Fractional Fourier Transform ◽  
Uncertainty Principles ◽  
Time Frequency ◽  
Time Frequency Localization

Identifying the lithology of volcanic rocks by using the time-frequency features of array acoustic logging data

2020 ◽  
Vol 8 (3) ◽  
pp. SL127-SL136
Author(s):  
Wenhua Wang ◽  
Pujun Wang ◽  
Zhuwen Wang ◽  
Min Xiang ◽  
Jinghua Liu
Keyword(s):  
Signal Processing ◽  
Fourier Transform ◽  
Fractional Fourier Transform ◽  
Processing Method ◽  
P Wave ◽  
S Wave ◽  
Acoustic Logging ◽  
Time Frequency ◽  
Signal Processing Method ◽  
Component Wave

The traditional acoustic logging signal processing method is computing the slowness of each component wave by time-domain or frequency-domain methods. But both of the two methods are limited. To combine the signals’ times, frequencies, or amplitudes, we have analyzed the array acoustic logging signals by the fractional Fourier transform and the Choi-Williams distribution. First, we apply the fractional Fourier transform on an array acoustic logging waveform with proper [Formula: see text], then the Choi-Williams distribution analysis method is used to process the signal in the fractional Fourier domain, and finally the result will show in the fractional Fourier time-frequency domain. The results show the following. The array acoustic logging signal is received earlier in the mudstone and diabase formation than in the tuff and breccia formations. The basic frequencies of the compressional wave (P-wave) are not very different, but the basic frequency of the shear wave (S-wave) is highest in the tuff formation and is lowest in the diabase formation. The relative energies of each component wave in the diabase, mudstone, tuff, and breccia formation can be summarized as: for the P-wave, diabase > mudstone ≈ tuff ≈ breccia; for the S-wave, diabase ≈ mudstone > breccia > tuff; and for the Stoneley wave, diabase > mudstone > tuff > breccia. The signal processing method combining the fractional Fourier transform and the Choi-Williams distribution can comprehensively research the time, frequency, and amplitude, thereby improving the segmentation of the time and frequency domains and providing a new method for interpretation of array acoustic logging.

LFM Signal Detection Method Based on Fractional Fourier Transform

2014 ◽  
Vol 989-994 ◽  
pp. 4001-4004 ◽  
Author(s):  
Yan Jun Wu ◽  
Gang Fu ◽  
Yu Ming Zhu
Keyword(s):  
Fourier Transform ◽  
Signal Detection ◽  
Frequency Analysis ◽  
Experimental Analysis ◽  
Fractional Fourier Transform ◽  
Detection Method ◽  
Time Frequency Analysis ◽  
Time Frequency ◽  
Frequency Information ◽  
Strong Impulse

As a generalization of Fourier transform, the fractional Fourier Transform (FRFT) contains simultaneity the time-frequency information of the signal, and it is considered a new tool for time-frequency analysis. This paper discusses some steps of FRFT in signal detection based on the decomposition of FRFT. With the help of the property that a LFM signal can produce a strong impulse in the FRFT domain, the signal can be detected conveniently. Experimental analysis shows that the proposed method is effective in detecting LFM signals.

scholarly journals Compressive Sensing in Signal Processing: Algorithms and Transform Domain Formulations

2016 ◽  
Vol 2016 ◽  
pp. 1-16 ◽  
Author(s):  
Irena Orović ◽  
Vladan Papić ◽  
Cornel Ioana ◽  
Xiumei Li ◽  
Srdjan Stanković
Keyword(s):  
Signal Processing ◽  
Fourier Transform ◽  
Compressive Sensing ◽  
Signal Reconstruction ◽  
Signal Acquisition ◽  
Transform Domain ◽  
Hermite Transform ◽  
Time Frequency ◽  
Reconstruction Methods ◽  
Fourier Transform Domain

Compressive sensing has emerged as an area that opens new perspectives in signal acquisition and processing. It appears as an alternative to the traditional sampling theory, endeavoring to reduce the required number of samples for successful signal reconstruction. In practice, compressive sensing aims to provide saving in sensing resources, transmission, and storage capacities and to facilitate signal processing in the circumstances when certain data are unavailable. To that end, compressive sensing relies on the mathematical algorithms solving the problem of data reconstruction from a greatly reduced number of measurements by exploring the properties of sparsity and incoherence. Therefore, this concept includes the optimization procedures aiming to provide the sparsest solution in a suitable representation domain. This work, therefore, offers a survey of the compressive sensing idea and prerequisites, together with the commonly used reconstruction methods. Moreover, the compressive sensing problem formulation is considered in signal processing applications assuming some of the commonly used transformation domains, namely, the Fourier transform domain, the polynomial Fourier transform domain, Hermite transform domain, and combined time-frequency domain.

scholarly journals Nyquist Zone Index and Chirp Rate Estimation of LFM Signal Intercepted by Nyquist Folding Receiver Based on Random Sample Consensus and Fractional Fourier Transform

Sensors ◽  
2019 ◽  
Vol 19 (6) ◽  
pp. 1477 ◽  
Author(s):  
Xinqun Liu ◽  
Tao Li ◽  
Xiaolei Fan ◽  
Zengping Chen
Keyword(s):  
Fourier Transform ◽  
Random Sample ◽  
Fractional Fourier Transform ◽  
Estimation Method ◽  
Ultra Wideband ◽  
Estimation Methods ◽  
Radar Signal ◽  
Chirp Rate ◽  
Time Frequency ◽  
Random Sample Consensus

The Nyquist folding receiver (NYFR) can achieve a high-probability interception of an ultra-wideband (UWB) signal with fewer devices, while the output of the NYFR is converted into a hybrid modulated signal of the local oscillator (LO) and the received signal, which requires the matching parameter estimation methods. The linear frequency modulation (LFM) signal is a typical low probability of intercept (LPI) radar signal. In this paper, an estimation method of both the Nyquist Zone (NZ) index and the chirp rate for the LFM signal intercepted by NYFR was proposed. First, according to the time-frequency characteristics of the LFM signal, the accurate NZ and the rough chirp rate was estimated based on least squares (LS) and random sample consensus (RANSAC). Then, the information of the LO was removed from the hybrid modulated signal by the known NZ, and the precise chirp rate was obtained by using the fractional Fourier transform (FrFT). Moreover, a fast search method of FrFT optimal order was presented, which could obviously reduce the computational complexity. The simulation demonstrated that the proposed method could precisely estimate the parameters of the hybrid modulated output signal of the NYFR.

scholarly journals Helgason–Gabor–Fourier transform and uncertainty principles

2020 ◽  
pp. 2050056
Author(s):  
Mustapha Boujeddaine ◽  
Mohammed El Kassimi ◽  
Saïd Fahlaoui
Keyword(s):  
Fourier Transform ◽  
Uncertainty Principle ◽  
Plancherel Formula ◽  
Riemannian Symmetric Space ◽  
Uncertainty Principles ◽  
Reconstruction Formula ◽  
Time Frequency ◽  
Local Uncertainty Principle ◽  
Local Uncertainty ◽  
Parseval Formula

Windowing a Fourier transform is a useful tool, which gives us the similarity between the signal and time frequency signal, and it allows to get sense when/where certain frequencies occur in the input signal, this method was introduced by Dennis Gabor. In this paper, we generalize the classical Gabor–Fourier transform (GFT) to the Riemannian symmetric space calling it the Helgason–Gabor–Fourier transform (HGFT). We prove several important properties of HGFT like the reconstruction formula, the Plancherel formula and Parseval formula. Finally, we establish some local uncertainty principle such as Benedicks-type uncertainty principle.

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