Theoretical foundations of digital vector Fourier analysis of two-dimensional signals padded with zero samples

Introduction: The practice of using Fourier-processing of finite two-dimensional signals (including images), having confirmed its effectiveness, revealed a number of negative effects inherent in it. A well-known method of dealing with negative effects of Fourier-processing is padding signals with zeros. However, the use of this operation leads to the need to provide information control systems with additional memory and perform unproductive calculations. Purpose: To develop new discrete Fourier transforms for efficient and effective processing of two-dimensional signals padded with zero samples. Method: We have proposed a new method for splitting a rectangular discrete Fourier transform matrix into square matrices. The method is based on the application of the modulus comparability relation to order the rows (columns) of the Fourier matrix. Results: New discrete Fourier transforms with variable parameters were developed, being a generalization of the classical discrete Fourier transform. The article investigates the properties of Fourier transform bases with variable parameters. In respect to these transforms, the validity has been proved for the theorems of linearity, shift, correlation and Parseval's equality. In the digital spectral Fourier analysis, the concepts of a parametric shift of a two-dimensional signal, and a parametric periodicity of a two-dimensional signal have been introduced. We have estimated the reduction of the required memory size and the number of calculations when applying the proposed transforms, and compared them with the discrete Fourier transform. Practical relevance: The developed discrete Fourier transforms with variable parameters can significantly reduce the cost of Fourier processing of two-dimensional signals (including images) padded with zeros.

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More on algebraic properties of the discrete Fourier transform raising and lowering operators

4open ◽

10.1051/fopen/2018010 ◽

2019 ◽

Vol 2 ◽

pp. 2 ◽

Cited By ~ 1

Author(s):

Mesuma K. Atakishiyeva ◽

Natig M. Atakishiyev ◽

Juan Loreto-Hernández

Keyword(s):

Fourier Transform ◽

Discrete Fourier Transform ◽

Fourier Transforms ◽

Difference Operators ◽

Discrete Fourier Transforms ◽

Raising And Lowering Operators ◽

Integral Fourier Transform ◽

Symmetrical Form ◽

Algebraic Properties ◽

Lowering Operators

In the present work, we discuss some additional findings concerning algebraic properties of the N-dimensional discrete Fourier transform (DFT) raising and lowering difference operators, recently introduced in [Atakishiyeva MK, Atakishiyev NM (2015), J Phys: Conf Ser 597, 012012; Atakishiyeva MK, Atakishiyev NM (2016), Adv Dyn Syst Appl 11, 81–92]. In particular, we argue that the most authentic symmetrical form of discretization of the integral Fourier transform may be constructed as the discrete Fourier transforms based on the odd points N only, while in the discrete Fourier transforms on the even points N this symmetry is spontaneously broken. This heretofore undetected distinction between odd and even dimensions is shown to be intimately related with the newly revealed algebraic properties of the above-mentioned DFT raising and lowering difference operators and, of course, is very consistent with the well-known formula for the multiplicities of the eigenvalues, associated with the N-dimensional DFT. In addition, we propose a general approach to deriving the eigenvectors of the discrete number operators N(N), that avoids the above-mentioned pitfalls in the structure of each even-dimensional case N = 2L.

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Detection of engine misfire using characteristic harmonics of angular acceleration

Proceedings of the Institution of Mechanical Engineers Part D Journal of Automobile Engineering ◽

10.1177/0954407019834104 ◽

2019 ◽

Vol 233 (14) ◽

pp. 3816-3823

Author(s):

Qixin Song ◽

Wenzhi Gao ◽

Pan Zhang ◽

Jiankang Liu ◽

Ziqing Wei

Keyword(s):

Fourier Transform ◽

Diesel Engine ◽

Discrete Fourier Transform ◽

Torsional Vibration ◽

Fourier Transforms ◽

Angular Acceleration ◽

Operating Conditions ◽

Discrete Fourier Transforms

A new torsional vibration–based method for the detection of engine misfires was proposed based on the discrete Fourier transform of angular acceleration of the crankshaft. By analysis of the sensitivity of the discrete Fourier transform to fluctuations in speed and load of the engine, the characteristic harmonics and characteristic discrete Fourier transforms of a cylinder were defined. Then cylinder misfires under any operating conditions were diagnosed by checking the characteristic discrete Fourier transforms of the cylinder at its characteristic harmonics. An experiment on a four-stroke, six-cylinder diesel engine showed that this method accurately identified misfire faults and the misfiring cylinders.

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Sliding Space-Frequency Processing of Finite Two-Dimensional Real Discrete Signals Based on the Fourier Transform with Variable Parameters

Intellekt Sist Proizv ◽

10.22213/2410-9304-2021-4-138-147 ◽

2021 ◽

Vol 19 (4) ◽

pp. 138-147

Author(s):

A. V. Ponomarev ◽

O. V. Ponomareva

Keyword(s):

Fourier Transform ◽

Spatial Frequency ◽

Discrete Fourier Transform ◽

Point Of View ◽

Two Dimensional ◽

Variable Parameters ◽

One Dimensional ◽

Computational Costs ◽

Discrete Signals ◽

Frequency Processing

In the field of Fourier processing of finite signals, three main directions of scientific research have been identified: Fourier processing of one-dimensional finite signals - processing of scalar functions of a scalar argument, Fourier processing of two-dimensional finite signals - processing of scalar functions of a vector argument, multichannel Fourier processing - processing of vector functions of a scalar argument. As part of the creation of a generalized theory of Fourier processing of finite signals, the authors proposed: the theory of spectral analysis of discrete signals at finite intervals in the bases of parametric exponential functions and the theory of two-dimensional digital signal processing in Fourier bases with variable parameters. The developed theories, generalizing the theory of Fourier processing of one-dimensional and two-dimensional signals, are based: on the introduction of new concepts of the shift of finite discrete signals in one-dimensional and two-dimensional cases and the introduction of new basic Fourier processing systems of discrete signals, which have the properties of multiplicativity, functions in the system. The mathematical apparatus of two-dimensional discrete Fourier transform with variable parameters in matrix and algebraic form is considered. A new method for processing finite two-dimensional real discrete signals in the spatial-frequency domain based on the discrete Fourier transform with variable parameters, the method of sliding spatial-frequency processing, has been introduced. An efficient method and algorithm for fast diagonal sliding spatial-frequency processing of finite two-dimensional real discrete signals based on the discrete Fourier transform with variable parameters has been developed. The estimation of the efficiency and effectiveness of the algorithm of the diagonal sliding two-dimensional discrete Fourier transform with variable parameters from the point of view of computational costs is carried out. As a result of experimental studies on model two-dimensional discrete finite signals, the validity, efficiency and reliability of the proposed method of sliding spatial-frequency processing of finite two-dimensional real discrete signals based on the discrete Fourier transform with variable parameters have been proved. A comparison (from the point of view of computational costs) of the developed method of sliding spatial-frequency processing of finite two-dimensional real discrete signals based on the discrete Fourier transform with variable parameters with the standard method of sliding processing of this type of signals is carried out.

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