Overview of Fourier Series and Fourier Transforms and Their Applications | Chapter 10 | Mathematics and Computer Science: Research Updates Vol. 5

In Mathematical Analysis, Signal processing, and Physics, the Fourier Series and the Fourier transform are crucial tools that offer reliable methods for evaluating and expressing functions, especially periodic and non-periodic signals. These ideas, which were first presented by Jean-Baptiste Joseph Fourier in the early 1800s, are the basis of modern harmonic analysis. A periodic function can be represented as an infinite summation of sine and cosine waves using the Fourier series. Because of this dissection, complex periodic waveforms can be broken down into a set of more straightforward trigonometric elements, each of which has its own distinct frequency, amplitude, and phase. The Fourier series is particularly helpful in domains like electrical engineering, acoustics, vibration analysis, and image processing because of its ability to differentiate between various components. By establishing a link between time-domain and frequency-domain representations, experts may examine signal behaviours, remove extraneous elements, and develop effective systems for signal analysis and transmission. The Fourier Transform expands the frequency analysis to periodic functions by applying the Fourier series' restriction to periodic functions to non-repeating signals. By converting a time-domain signal into a continuous array of frequency components, the Fourier Transform helps to reveal signal properties that could otherwise be obscured. In fields like quantum mechanics, telecommunications, and control systems, this transformation is essential because it makes tasks like modulation, spectrum analysis, and system identification easier. Both approaches have the major benefit of being linear and having the ability to simplify differential equations, which makes them essential for resolving boundary value issues in both engineering and physics.

 

 

Author (s) Details

S.K. Mohapatra
Department of Mathematics, Kalinga Institute of Social Sciences (KISS) Deemed to be University, Bhubaneswar-751024, Odisha, India.

Prashna Naik
Department of Mathematics, Kalinga Institute of Social Sciences (KISS) Deemed to be University, Bhubaneswar-751024, Odisha, India.

 

Dhaneswar Beherdalai
Department of Mathematics, Kalinga Institute of Social Sciences (KISS) Deemed to be University, Bhubaneswar-751024, Odisha, India.

M.R. Mohapatra
Department of Mathematics, KIIT Deemed to be University, Bhubaneswar-751024, Odisha, India.

 

M.R. Mohapatra
Department of Mathematics, KIIT Deemed to be University, Bhubaneswar-751024, Odisha, India.

 

 

Please see the book here:- https://doi.org/10.9734/bpi/mcsru/v5/5624

 

Advancements in Holographic Display Utilizing Digital Light Field | Chapter 1 | Current Research Progress in Physical Science Vol. 4

Holography is still well known as the ultimate science and technology to achieve the ideal 3D display until today. Four historical important stages in display holography are mentioned explicitly in this paper. While the first three is hard to achieve dynamic optical field display because of the redundant information for human vision possessed by the hologram itself, the fourth could be thought the most efficient one to bring up the holographic video by making use of the digital light field.

 

Author (s) Details

Frank C. Fan
AFC Technology Co., Ltd., No.12 Songshan West Road, Baoan District, Shenzhen 518104, China.

 

Sam Choi
AFC Technology Co., Ltd., No.12 Songshan West Road, Baoan District, Shenzhen 518104, China.

 

C. C. Jiang
AFC Technology Co., Ltd., No.12 Songshan West Road, Baoan District, Shenzhen 518104, China.

 

Please see the book here:- https://doi.org/10.9734/bpi/crpps/v4/2327

A Concise Study about Fourier Transform and Principles of Quantum Mechanics| Chapter 7 | Research Highlights in Mathematics and Computer Science Vol. 2

 Based on the theory that the position-representation of a tangible state a> is the Fourier transform of its push-representation a> what the time-likeness a> is the inverse Fourier transform of allure energy-likeness a>, we are able to find the Planck connection E = hv, the de Broglie relation P=. Afterward, utilizing the Dirac arm of the sea function (x) and the property FTxf(x) = -iho, pFTxf(x) = -iho, pF(p) we get the links between drivers in ordinary and Hilbert spaces chief to the Dirac fundamental commutation connection [p, 2] = -thÎ, the Schrödinger equations, the Heisenberg principle of indeterminacy in quantum mechanics, the annihilation & invention of a photon from excitation & de-excitement of an atom according to Bohr.

Author(s) Details:

Do Tan Si,
HoChiMinh-City Physical Association, HoChiMinh-city, Vietnam.

Please see the link here: https://stm.bookpi.org/RHMCS-V2/article/view/8574

A Novel Approach for Predicting Disease in Plant Using Hybrid Deep Convolutional Neural Network | Chapter 5 | Current Approaches in Science and Technology Research Vol. 8

 Agriculture is India's most important economic industry. The detection of plant diseases is a key concern in the agriculture industry. An accurate and speedier identification of plant diseases, resulting in significant reductions in economic losses. Manually monitoring plant leaf disease is a highly important task that takes a long time. As a result, an automated solution is required for plant disease detection. Deep learning is quickly becoming the industry standard for picture classification. Researchers have been able to improve the accuracy of object detection thanks to recent advances in Deep Neural Networks. Researchers have already built a few architectures for effective plant disease classification, including the Faster Region-based Convolutional Neural Network (Faster R-CNN), Region-based Fully Convolutional Network (R-FCN), and Single Shot Multibox Detector (SSD). The well-known Deep Convolutional Neural Network architectures for generic Image categorization are Alex Net, Google Net, and VGG-16. When many diseases afflict the same leaf, however, this architecture does not perform well.

To address this, this study proposes a Hybrid Deep Convolutional Neural Network architecture with segmentation, which consists of five convolutional layers, five pooling layers, and two fully connected layers. Deep convolutional neural networks are extensively employed to assess visual imagery and are frequently utilised in image categorization behind the scenes. The input photos are fed to the CNN in the proposed Faster-RCNN system, increasing the accuracy of images to forecast illnesses from plant leaves.

Author (s) Details

R. Jayavadivel
Department of Computer Science and Engineering, Lovely Professional University, Jalandhar-Delhi, G. T. Road, Phagwara, Punjab, India.

V. Chandrasekar
Department of Computer Science and Engineering, Malla Reddy College of Engineering and Technology, Secunderabad, Telangana State, India.

V. Shanmugavalli
Department of Computer Science and Engineering, Vivekanandha College of Engineering for Women, India.

View Book :- https://stm.bookpi.org/CASTR-V8/article/view/1946

Recent Study on Barker Coded Modulated Thermal Wave Imaging for Defect Detection of Glass Fiber Reinforced Plastic | Chapter 12 | Advanced Aspects of Engineering Research Vol. 8

 Infrared active thermography offers subsurface details of the test item based on the thermal inhomogeneity of the constituent material. For successful analysis of diverse faults occurring at different depths in realistic objects, novel processing methodologies to enhance detectability and excitations permitting depth analysis with constituent band of frequencies are required. This paper demonstrates the depth analysis capabilities of phase modulated coded stimulation for infrared imaging, which has been confirmed using a glass fibre reinforced plastic plate with embedded Teflon inserts. Experiments indicated that using cumulative energy instead of distributed energy in pulse compression improved flaw detection in the current phase analysis. In addition to defect shape preservation, correlation-based processing was discovered to reduce non-uniform radiation/emissivity.

Author (s) Details

Md. M. Pasha
Department of Electronics and Communication Engineering, Koneru Lakshmaiah Education Foundation, Infrared Imaging Center, Green Fields, Vaddeswaram, Guntur, Andhra Pradesh, India and K. S. R. M College of Engineering Kadapa, Andhra Pradesh, India.

Dr. B. Suresh
Department of Electronics and Communication Engineering, Koneru Lakshmaiah Education Foundation, Infrared Imaging Center, Green Fields, Vaddeswaram, Guntur, Andhra Pradesh, India.

K. Rajesh Babu
Department of Electronics and Communication Engineering, Koneru Lakshmaiah Education Foundation, Infrared Imaging Center, Green Fields, Vaddeswaram, Guntur, Andhra Pradesh, India.

Sk. Subhani
PACE Institute of Technology and Science, India.

Dr. G. V. Subbarao
Department of Electronics and Communication Engineering, Koneru Lakshmaiah Education Foundation, Infrared Imaging Center, Green Fields, Vaddeswaram, Guntur, Andhra Pradesh, India.

View Book :- https://stm.bookpi.org/AAER-V8/article/view/1107

Causality Principle- Simplified Deliberation and Explanation of Its Complex Mathematics | Chapter 1 | Theory and Practice of Mathematics and Computer Science Vol. 7

Strike a bell, and then the sound of the gong is heard, and not until it strikes. This is the Universal Phenomenon assertion of Causality; and it's a 'matter of fact'- taken very lightly. The knowledge on causality theory is too fragmented and not concise, and the 'complex analysis theory' is lost, rendering the interpretation of this 'matter-of-fact' phenomenon very difficult. Authors use the mathematical formulas in the available literature without adequately describing them, and their functional usefulness seems to be missing; and in the complexities it gets lost. The formulas used do not perform very detailed and elaborate measures that provide readers with jitters. The aim of this chapter and its deliberation with thorough derivations is to present the Concepts of Causality in a strict manner and to establish mathematics in a simpler way, while still taking into account the purpose of applications. A basic definition of nature that is:' the result can only arise after the cause', i.e. called causality has excellent mathematical treatment and development that we name as the relationship between Kramer-Kronigs, analyticity, the theory of Titchmarsh, etc. Like the sentence in the upper half of the complex plane,' a causal answer mechanism is analytical,' sounds very complicated and abstract. We try to provide elaborate care here in this chapter on all the seemingly complicated and abstract mathematical statements and expressions. Although the Causality Concepts look very complicated and too abstract in terms of 'complex-analysis, here in this chapter we simplify the derivation of Kramer-Kronigs relationships of analyticity and obtain these expressions in domains of time and frequency. We start from the basics of the Impulse Response Function or the Function of Green and then describe the function of generalized susceptibility. The Kramer-Kronig relationships in the frequency domain and later in the time domain are then established using Fourier transformation techniques. This approach is used in various fields, such as impedance studies, dielectric relaxation/retardation studies, refractive index studies, electric polarization studies, studies of magnetic systems, studies of stress-strain relaxation, etc. Even if we render an artificial material with negative permittivity and negative permeability (thus showing negative refraction), the statistical causality checks, the Kramer-Kronigs relationship, can and must be observed. The examples we take into account in this chapter are for simple Debye systems, but the theory and concepts we intentionally apply can also be applied to non-Debye systems. We are not aware of the theory of causality formed and discussed here whether it can be extended to non-differentiable systems, i.e. the fractal support response function? In this respect, maybe a new formal mathematics must be developed. Our debate is just about continuous and distinguishable structures. We make a point that the contents of this chapter have not been fresh since the 1930's, but that the theory and its mathematics have been difficult to understand and also to teach. That's because the data on the information is too dispersed. This chapter will assist students in physics, engineering and mathematics as a teaching subject, and readers will find the descriptions and detailed derivations helpful in their academic work.

Author (s) Details

Shantanu Das
BARC, Mumbai, India (Reteired).

View Book :- https://stm.bookpi.org/TPMCS-V7/issue/view/18

Obtaining Differential Transforms: Applications to the Case of the Fourier Transform | Chapter 8 | New Insights into Physical Science Vol.3

In this paper is proven that all relations between a couple of dual operators ) B,A( i.e. operators obeying the commutation relation   I B,A  are invariant under substitution of ) B,A( with any another dual couple. From this property are obtained many differential operators realizing transformations in space and phase space such as translation, dilatation, hyperbolic, … , fractional order Fourier transforms and Fourier transform itself. Transforms of arbitrary functions and operators and geometric forms by these differential operators are given. The kernel of the integral transform associated with a differential transform is found. As case study the differential Fourier transform is highlighted in order to see how it is possible to get in a concise manner the known properties of the Fourier transform without doing integrations.  

Author(s) Details

Do Tan Si
HoChiMinh-City Physical Association, 40 Dong Khoi, Q1, TP.HCM, Vietnam and Université libre de Bruxelles and UEM, Belgium.

View Book :- http://bp.bookpi.org/index.php/bpi/catalog/book/214