scholarly journals The Generalization of the Poisson Sum Formula Associated with the Linear Canonical Transform

2012 ◽  
Vol 2012 ◽  
pp. 1-9 ◽  
Author(s):  
Jun-Fang Zhang ◽  
Shou-Ping Hou
Keyword(s):  
Fourier Transform ◽  
Canonical Transformation ◽  
Linear Canonical Transform ◽  
Original Signal ◽  
Periodic Property ◽  
Sum Formula ◽  
The Relationship

The generalization of the classical Poisson sum formula, by replacing the ordinary Fourier transform by the canonical transformation, has been derived in the linear canonical transform sense. Firstly, a new sum formula of Chirp-periodic property has been introduced, and then the relationship between this new sum and the original signal is derived. Secondly, the generalization of the classical Poisson sum formula to the linear canonical transform sense has been obtained.

scholarly journals A Version of Uncertainty Principle for Quaternion Linear Canonical Transform

2018 ◽  
Vol 2018 ◽  
pp. 1-7 ◽  
Author(s):  
Mawardi Bahri ◽  
Resnawati ◽  
Selvy Musdalifah
Keyword(s):  
Fourier Transform ◽  
Uncertainty Principle ◽  
Young Inequality ◽  
Linear Canonical Transform ◽  
Two Dimensional ◽  
Research Papers ◽  
Quaternion Fourier Transform ◽  
Quaternion Linear Canonical Transform ◽  
The Relationship

In recent years, the two-dimensional (2D) quaternion Fourier and quaternion linear canonical transforms have been the focus of many research papers. In the present paper, based on the relationship between the quaternion Fourier transform (QFT) and the quaternion linear canonical transform (QLCT), we derive a version of the uncertainty principle associated with the QLCT. We also discuss the generalization of the Hausdorff-Young inequality in the QLCT domain.

scholarly journals Relation between Quaternion Fourier Transform and Quaternion Wigner-Ville Distribution Associated with Linear Canonical Transform

2017 ◽  
Vol 2017 ◽  
pp. 1-10 ◽  
Author(s):  
Mawardi Bahri ◽  
Muh. Saleh Arif Fatimah
Keyword(s):  
Fourier Transform ◽  
Inversion Formula ◽  
Ambiguity Function ◽  
Alternative Proof ◽  
Basic Relation ◽  
Linear Canonical Transform ◽  
Quaternion Fourier Transform ◽  
Fundamental Relationship ◽  
The Relationship

The quaternion Wigner-Ville distribution associated with linear canonical transform (QWVD-LCT) is a nontrivial generalization of the quaternion Wigner-Ville distribution to the linear canonical transform (LCT) domain. In the present paper, we establish a fundamental relationship between the QWVD-LCT and the quaternion Fourier transform (QFT). Based on this fact, we provide alternative proof of the well-known properties of the QWVD-LCT such as inversion formula and Moyal formula. We also discuss in detail the relationship among the QWVD-LCT and other generalized transforms. Finally, based on the basic relation between the quaternion ambiguity function associated with the linear canonical transform (QAF-LCT) and the QFT, we present some important properties of the QAF-LCT.

Wavelet packets associated with linear canonical transform on spectrum

2021 ◽  
pp. 2150030
Author(s):  
M. Younus Bhat ◽  
Aamir H. Dar
Keyword(s):  
Fourier Transform ◽  
Multiresolution Analysis ◽  
Fourier Transforms ◽  
Fractional Fourier Transform ◽  
Integral Transforms ◽  
Wavelet Packets ◽  
Linear Canonical Transform ◽  
Fresnel Transform ◽  
The Fourier Transform ◽  
Vector Valued

The linear canonical transform (LCT) provides a unified treatment of the generalized Fourier transforms in the sense that it is an embodiment of several well-known integral transforms including the Fourier transform, fractional Fourier transform, Fresnel transform. Using this fascinating property of LCT, we, in this paper, constructed associated wavelet packets. First, we construct wavelet packets corresponding to nonuniform Multiresolution analysis (MRA) associated with LCT and then those corresponding to vector-valued nonuniform MRA associated with LCT. We investigate their various properties by means of LCT.

scholarly journals The Solvability of a Class of Convolution Equations Associated with 2D FRFT

Mathematics ◽  
2020 ◽  
Vol 8 (11) ◽  
pp. 1928
Author(s):  
Zhen-Wei Li ◽  
Wen-Biao Gao ◽  
Bing-Zhao Li
Keyword(s):  
Fourier Transform ◽  
Convolution Operator ◽  
Fractional Fourier Transform ◽  
Hankel Transform ◽  
Polar Coordinates ◽  
Two Dimensional ◽  
Spatial Shift ◽  
Fractional Hankel Transform ◽  
The Relationship ◽  
Convolution Equations

In this paper, the solvability of a class of convolution equations is discussed by using two-dimensional (2D) fractional Fourier transform (FRFT) in polar coordinates. Firstly, we generalize the 2D FRFT to the polar coordinates setting. The relationship between 2D FRFT and fractional Hankel transform (FRHT) is derived. Secondly, the spatial shift and multiplication theorems for 2D FRFT are proposed by using this relationship. Thirdly, in order to analyze the solvability of the convolution equations, a novel convolution operator for 2D FRFT is proposed, and the corresponding convolution theorem is investigated. Finally, based on the proposed theorems, the solvability of the convolution equations is studied.

Computers and infrared spectroscopy: evolution and revolution

1991 ◽  
Vol 69 (11) ◽  
pp. 1781-1785 ◽  
Author(s):  
D. J. Moffatt ◽  
J. K. Kauppinen ◽  
H. H. Mantsch
Keyword(s):  
Fourier Transform ◽  
Infrared Spectroscopy ◽  
Key Words ◽  
Maximum Entropy ◽  
Input Data ◽  
Maximum Entropy Method ◽  
Entropy Method ◽  
History Of ◽  
The Fourier Transform ◽  
The Relationship

A brief history of the relationship between computer and infrared spectroscopist is given with emphasis on the use of the Fourier transform. Subsequently, an algorithm is developed that may be used to devise an objective Fourier self-deconvolution procedure which depends only on the input data and is not subject to the biases that are often introduced in such subjective techniques. Key words: deconvolution, Fourier transform, maximum entropy method.

The Study of Tire Pattern Noise by Using Wavelet Transform

2011 ◽  
Vol 105-107 ◽  
pp. 267-270 ◽  
Author(s):  
Sung Wook Hwang ◽  
Jin Hyuk Han ◽  
Ki Duck Sung ◽  
Sang Kwon Lee
Keyword(s):  
Fourier Transform ◽  
Wavelet Transform ◽  
The Other ◽  
Road Noise ◽  
Signal Process ◽  
The Road ◽  
Pattern Noise ◽  
On The Road ◽  
Non Stationary Signal ◽  
The Relationship

Tire noise is classified by pattern noise and road noise in a vehicle. Especially pattern noise has impulsive characteristics since it is generated by impacting of tire’s block on the road. Therefore, a special signal process is needed other than traditional Fourier Transform, because the characteristic of signal is varying with time. On the other hand, the pattern noise is a kind of non-stationary signal and is related to the impulsive train of pitch sequence of a block. In this paper, Wavelet Transform is applied to verify the impulse signal caused by impact of block and groove and to verify the relationship between the pattern noise and the train of pitch sequence.

Correlation of Aggregate Form Properties with Fourier Frequencies

2019 ◽  
Vol 2674 (8) ◽  
pp. 405-419 ◽  
Author(s):  
Hongbin Xu ◽  
Jorge A. Prozzi
Keyword(s):  
Fourier Transform ◽  
Prediction Models ◽  
Elongation Ratio ◽  
Aggregate Form ◽  
Independent Components ◽  
Complex Series ◽  
Aggregate Morphology ◽  
Mean And Variance ◽  
Aggregate Shape ◽  
The Relationship

Aggregate morphology consists of three independent components: form, angularity, and surface texture. As these three components act in different ways in affecting the performance of pavement layer, it is vital to characterize them objectively and accurately. In this study, the coordinates of boundary pixels of aggregate from 2-D images were treated as an equally spaced complex series. The description of the aggregate shape in the frequency domain was obtained by analyzing this complex series with the discrete Fourier transform (DFT). Based on this, the reconstruction of aggregate shape at different frequencies was performed, which validated that the three elements of aggregate shape correspond to different frequencies. In addition, by fitting the relationship between DFT coefficients and form parameters (elongation ratio, sphericity, mean radius, variance of radius), the frequencies contributing to the form properties of aggregate were identified. Moreover, prediction models for the elongation ratio, sphericity, mean, and variance of radius based on the DFT coefficients at the contributing frequencies were developed separately.

The linear canonical wavelet transform on some function spaces

2018 ◽  
Vol 16 (01) ◽  
pp. 1850010 ◽  
Author(s):  
Yong Guo ◽  
Bing-Zhao Li
Keyword(s):  
Fourier Transform ◽  
Wavelet Transform ◽  
Continuous Operator ◽  
Function Spaces ◽  
Linear Continuous Operator ◽  
Convolution Theorem ◽  
Linear Canonical Transform ◽  
Equivalent Definition ◽  
New Space ◽  
Definition Of

It is well known that the domain of Fourier transform (FT) can be extended to the Schwartz space [Formula: see text] for convenience. As a generation of FT, it is necessary to detect the linear canonical transform (LCT) on a new space for obtaining the similar properties like FT on [Formula: see text]. Therefore, a space [Formula: see text] generalized from [Formula: see text] is introduced firstly, and further we prove that LCT is a homeomorphism from [Formula: see text] onto itself. The linear canonical wavelet transform (LCWT) is a newly proposed transform based on the convolution theorem in LCT domain. Moreover, we propose an equivalent definition of LCWT associated with LCT and further study some properties of LCWT on [Formula: see text]. Based on these properties, we finally prove that LCWT is a linear continuous operator on the spaces of [Formula: see text] and [Formula: see text].

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