Fourier analysis uses Fourier series and Fourier integrals to represent functions. Fourier series represent periodic functions as the sum of sines and cosines, while Fourier integrals extend this to non-periodic functions using integrals of sines and cosines instead of series. Orthogonal function systems like sines and cosines allow determining the coefficients using formulas like the Euler formulas. Extensions include replacing sines and cosines with other orthogonal functions and applying Fourier methods to functions of any period by changing variables.
Topic from the course of linear algebra and differential equation. The slide carries definition of Fourier series with graphical representation of applications of F-series and an example.
Topic from the course of linear algebra and differential equation. The slide carries definition of Fourier series with graphical representation of applications of F-series and an example.
Fuzzy random variables and Kolomogrov’s important resultsinventionjournals
:In this paper an attempt is made to transform Kolomogrov Maximal inequality, Koronecker Lemma, Loeve’s Lemma and Kolomogrov’s strong law of large numbers for independent, identically distributive fuzzy Random variables. The applications of this results is extensive and could produce intensive insights on Fuzzy Random variables
Its states Periodic function, Fourier series for disontinous function, Fourier series, Intervals, Odd and even functions, Half range fourier series etc. Mostly used as active learning assignment in Degree 3rd sem students.
In this slide fourier series of Engineering Mathematics has been described. one Example is also added for you. Hope this will help you understand fourier series.
International Journal of Mathematics and Statistics Invention (IJMSI) inventionjournals
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
Fuzzy random variables and Kolomogrov’s important resultsinventionjournals
:In this paper an attempt is made to transform Kolomogrov Maximal inequality, Koronecker Lemma, Loeve’s Lemma and Kolomogrov’s strong law of large numbers for independent, identically distributive fuzzy Random variables. The applications of this results is extensive and could produce intensive insights on Fuzzy Random variables
Its states Periodic function, Fourier series for disontinous function, Fourier series, Intervals, Odd and even functions, Half range fourier series etc. Mostly used as active learning assignment in Degree 3rd sem students.
In this slide fourier series of Engineering Mathematics has been described. one Example is also added for you. Hope this will help you understand fourier series.
International Journal of Mathematics and Statistics Invention (IJMSI) inventionjournals
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
Hierarchical Digital Twin of a Naval Power SystemKerry Sado
A hierarchical digital twin of a Naval DC power system has been developed and experimentally verified. Similar to other state-of-the-art digital twins, this technology creates a digital replica of the physical system executed in real-time or faster, which can modify hardware controls. However, its advantage stems from distributing computational efforts by utilizing a hierarchical structure composed of lower-level digital twin blocks and a higher-level system digital twin. Each digital twin block is associated with a physical subsystem of the hardware and communicates with a singular system digital twin, which creates a system-level response. By extracting information from each level of the hierarchy, power system controls of the hardware were reconfigured autonomously. This hierarchical digital twin development offers several advantages over other digital twins, particularly in the field of naval power systems. The hierarchical structure allows for greater computational efficiency and scalability while the ability to autonomously reconfigure hardware controls offers increased flexibility and responsiveness. The hierarchical decomposition and models utilized were well aligned with the physical twin, as indicated by the maximum deviations between the developed digital twin hierarchy and the hardware.
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Indigenized remote control interface card suitable for MAFI system CCR equipment. Compatible for IDM8000 CCR. Backplane mounted serial and TCP/Ethernet communication module for CCR remote access. IDM 8000 CCR remote control on serial and TCP protocol.
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• Remote control system for accessing CCR and allied system over serial or TCP.
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Technical Specifications
Indigenized remote control interface card suitable for MAFI system CCR equipment. Compatible for IDM8000 CCR. Backplane mounted serial and TCP/Ethernet communication module for CCR remote access. IDM 8000 CCR remote control on serial and TCP protocol.
Key Features
Indigenized remote control interface card suitable for MAFI system CCR equipment. Compatible for IDM8000 CCR. Backplane mounted serial and TCP/Ethernet communication module for CCR remote access. IDM 8000 CCR remote control on serial and TCP protocol.
• Remote control: Parallel or serial interface
• Compatible with MAFI CCR system
• Copatiable with IDM8000 CCR
• Compatible with Backplane mount serial communication.
• Compatible with commercial and Defence aviation CCR system.
• Remote control system for accessing CCR and allied system over serial or TCP.
• Indigenized local Support/presence in India.
Application
• Remote control: Parallel or serial interface.
• Compatible with MAFI CCR system.
• Compatible with IDM8000 CCR.
• Compatible with Backplane mount serial communication.
• Compatible with commercial and Defence aviation CCR system.
• Remote control system for accessing CCR and allied system over serial or TCP.
• Indigenized local Support/presence in India.
• Easy in configuration using DIP switches.
Water scarcity is the lack of fresh water resources to meet the standard water demand. There are two type of water scarcity. One is physical. The other is economic water scarcity.
Hybrid optimization of pumped hydro system and solar- Engr. Abdul-Azeez.pdffxintegritypublishin
Advancements in technology unveil a myriad of electrical and electronic breakthroughs geared towards efficiently harnessing limited resources to meet human energy demands. The optimization of hybrid solar PV panels and pumped hydro energy supply systems plays a pivotal role in utilizing natural resources effectively. This initiative not only benefits humanity but also fosters environmental sustainability. The study investigated the design optimization of these hybrid systems, focusing on understanding solar radiation patterns, identifying geographical influences on solar radiation, formulating a mathematical model for system optimization, and determining the optimal configuration of PV panels and pumped hydro storage. Through a comparative analysis approach and eight weeks of data collection, the study addressed key research questions related to solar radiation patterns and optimal system design. The findings highlighted regions with heightened solar radiation levels, showcasing substantial potential for power generation and emphasizing the system's efficiency. Optimizing system design significantly boosted power generation, promoted renewable energy utilization, and enhanced energy storage capacity. The study underscored the benefits of optimizing hybrid solar PV panels and pumped hydro energy supply systems for sustainable energy usage. Optimizing the design of solar PV panels and pumped hydro energy supply systems as examined across diverse climatic conditions in a developing country, not only enhances power generation but also improves the integration of renewable energy sources and boosts energy storage capacities, particularly beneficial for less economically prosperous regions. Additionally, the study provides valuable insights for advancing energy research in economically viable areas. Recommendations included conducting site-specific assessments, utilizing advanced modeling tools, implementing regular maintenance protocols, and enhancing communication among system components.
Student information management system project report ii.pdfKamal Acharya
Our project explains about the student management. This project mainly explains the various actions related to student details. This project shows some ease in adding, editing and deleting the student details. It also provides a less time consuming process for viewing, adding, editing and deleting the marks of the students.
Student information management system project report ii.pdf
PS.pptx
1. Fourier Analysis
Dmitriy Sergeevich Nikitin
Assistant
Tomsk Polytechnic University
email: NikitinDmSr@yandex.ru
Lecture-8
Additional chapters of mathematics
1
2. 2
The central starting point of Fourier analysis is Fourier series.
They are infinite series designed to represent general periodic
functions in terms of simple ones, namely, cosines and sines.
This trigonometric system is orthogonal, allowing the
computation of the coefficients of the Fourier series by use of
the well-known Euler formulas. Fourier series are very
important to the engineer and physicist because they allow
the solution of differential equations in connection with
forced oscillations and the approximation of periodic
functions.
Fourier Analysis
3. 3
The underlying idea of the Fourier series can be extended in
two important ways. We can replace the trigonometric
system by other families of orthogonal functions, e.g., Bessel
functions and obtain the Sturm–Liouville expansions. The
second expansion is applying Fourier series to nonperiodic
phenomena and obtaining Fourier integrals and Fourier
transforms.
Both extensions have important applications to solving
differential equations. In a digital age, the discrete Fourier
transform plays an important role. Signals, such as voice or
music, are sampled and analyzed for frequencies. An
important algorithm, in this context, is the fast Fourier
transform.
Fourier Analysis
4. 4
A function f(x) is called a periodic function if f(x) is defined for
all real x, except possibly at some points, and if there is some
positive number p, called a period of f(x), such that
f(x+p) = f(x) (and also f (x + np) = f (x))
The graph of a periodic function has the characteristic that it
can be obtained by periodic repetition of its graph in any
interval of length p. The smallest positive period is often
called the fundamental period.
Fourier Series
5. 5
Our problem will be the representation of various functions of
period in terms of the simple functions
1 cos x sin x cos 2x, sin 2x, …, cos nx, sin nx, … .
All these functions have the period 2π. They form the so-
called trigonometric system. The series to be obtained will be
a trigonometric series, that is, a series of the form
𝐚𝟎 + 𝐚𝟏𝐜𝐨𝐬 𝐱 + 𝐛𝟏𝐬𝐢𝐧 𝐱 + 𝐚𝟐𝐜𝐨𝐬 𝟐𝐱 + 𝐛𝟐𝐬𝐢𝐧 𝟐𝐱 + ⋯
= 𝐚𝟎 +
𝐧=𝟏
∞
𝐚𝐧𝐜𝐨𝐬 𝐧𝐱 + 𝐛𝐧𝐬𝐢𝐧 𝐧𝐱 (1)
𝐚𝟎, 𝐚𝟏, 𝐛𝟏, 𝐚𝟐, 𝐛𝟐, … are constants, called the coefficients of
the series. We see that each term has the period 2π. Hence if
the coefficients are such that the series converges, its sum
will be a function of period 2π.
Fourier Series
6. 6
Now suppose that f(x) is a given function of period 2π and is
such that it can be represented by a series (1), that is, (1)
converges and, moreover, has the sum f(x). Then, using the
equality sign, we write
𝐟(𝐱) = 𝐚𝟎 +
𝐧=𝟏
∞
𝐚𝐧𝐜𝐨𝐬 𝐧𝐱 + 𝐛𝐧𝐬𝐢𝐧 𝐧𝐱 .
and call this formula the Fourier series of f(x). The coefficients
are the so-called Fourier coefficients of f(x), given by the Euler
formulas
𝐚𝟎 =
𝟏
𝟐𝛑
−𝛑
𝛑
𝐟 𝐱 𝐝𝐱
𝐚𝐧 =
𝟏
𝛑
−𝛑
𝛑
𝐟 𝐱 𝐜𝐨𝐬 𝐧𝐱 𝐝𝐱 𝐛𝐧 =
𝟏
𝛑
−𝛑
𝛑
𝐟 𝐱 𝐬𝐢𝐧 𝐧𝐱 𝐝𝐱
Fourier Series
8. 8
Clearly, periodic functions in applications may have any
period, not just 2π as in the last section (chosen to have
simple formulas). The transition from period to be period
p = 2L is effected by a suitable change of scale, as follows.
Let f(x) have period. Then, using the equality sign, we write
𝐟(𝐱) = 𝐚𝟎 +
𝐧=𝟏
∞
𝐚𝐧𝐜𝐨𝐬
𝐧𝛑
𝐋
𝐱 + 𝐛𝐧𝐬𝐢𝐧
𝐧𝛑
𝐋
𝐱 .
with the Fourier coefficients of f(x) given by the Euler
formulas
𝐚𝟎 =
𝟏
𝟐𝐋
−𝐋
𝐋
𝐟 𝐱 𝐝𝐱
𝐚𝐧 =
𝟏
𝐋
−𝐋
𝐋
𝐟 𝐱 𝐜𝐨𝐬
𝐧𝛑𝐱
𝐋
𝐝𝐱 𝐛𝐧 =
𝟏
𝐋
−𝐋
𝐋
𝐟 𝐱 𝐬𝐢𝐧
𝐧𝛑𝐱
𝐋
𝐝𝐱
From Period 2π to Any Period p = 2L
9. 9
If is an even function, that is, f(-x) = f(x), its Fourier series
reduces to a Fourier cosine series
𝐟(𝐱) = 𝐚𝟎 +
𝐧=𝟏
∞
𝐚𝐧𝐜𝐨𝐬
𝐧𝛑
𝐋
𝐱 .
with the Fourier coefficients
𝐚𝟎 =
𝟏
𝐋
𝟎
𝐋
𝐟 𝐱 𝐝𝐱, 𝐚𝐧 =
𝟐
𝐋
𝟎
𝐋
𝐟 𝐱 𝐜𝐨𝐬
𝐧𝛑𝐱
𝐋
𝐝𝐱
If is an odd function, that is, f(-x) = -f(x), its Fourier series
reduces to a Fourier sine series
𝐟(𝐱) = 𝐚𝟎 +
𝐧=𝟏
∞
𝐚𝐧𝐜𝐨𝐬
𝐧𝛑
𝐋
𝐱 .
with the Fourier coefficients
𝐛𝐧 =
𝟐
𝐋
𝟎
𝐋
𝐟 𝐱 𝐬𝐢𝐧
𝐧𝛑𝐱
𝐋
𝐝𝐱
From Period 2π to Any Period p = 2L
10. 10
The Fourier coefficients of a sum f1 + f2 are the sums of the
corresponding Fourier coefficients of f1 and f2.
The Fourier coefficients of cf are c times the corresponding
Fourier coefficients of f.
Sum and Scalar Multiple
T H E O R E M 1
11. 11
The idea of the Fourier series was to represent general
periodic functions in terms of cosines and sines. The latter
formed a trigonometric system. This trigonometric system has
the desirable property of orthogonality which allows us to
compute the coefficient of the Fourier series by the Euler
formulas. The question then arises, can this approach be
generalized? That is, can we replace the trigonometric system
by other orthogonal systems (sets of other orthogonal
functions)? The answer is “yes” and leads to generalized
Fourier series, including the Fourier-Legendre series and the
Fourier-Bessel series.
Orthogonal systems
12. 12
Functions y1(x), y2(x), … defined on some interval 𝐚 ≤ 𝐱 ≤ 𝐛
are called orthogonal on this interval with respect to the
weight function r(x)>0 if for all m and all n different from m,
𝐲𝐦, 𝐲𝐧 =
𝐚
𝐛
𝐫 𝐱 𝐲𝐦 𝐱 𝐲𝐧 𝐱 𝐝𝐱 (𝐦 ≠ 𝟎)
𝐲𝐦, 𝐲𝐧 is a standard notation for this integral.
The norm ||𝐲𝐦||of is defined by
||𝐲𝐦|| = 𝐲𝐦, 𝐲𝐦 =
𝐚
𝐛
𝐫 𝐱 𝐲𝐦
𝟐
𝐱 𝐝𝐱
Orthogonal functions
13. 13
Fourier series are made up of the trigonometric system,
which is orthogonal, and orthogonality was essential in
obtaining the Euler formulas for the Fourier coefficients.
Orthogonality will also give us coefficient formulas for the
desired generalized Fourier series, including the Fourier–
Legendre series and the Fourier–Bessel series. This
generalization is as follows.
Let y0, y1, y2, … be orthogonal with respect to a weight
function r(x) on an interval 𝐚 ≤ 𝐱 ≤ 𝐛, and let f(x) be a
function that can be represented by a convergent series
𝐟 𝐱 =
𝐦=𝟎
∞
𝐚𝐦𝐲𝐦(𝐱) = 𝐚𝟎𝐲𝟎 𝐱 + 𝐚𝟏𝐲𝟏 𝐱 + ⋯ .
This is called an orthogonal series, orthogonal expansion, or
generalized Fourier series.
Orthogonal series
14. 14
Given f(x), we have to determine the coefficients 𝐚𝐦, called
the Fourier constants of f(x) with respect to y0, y1, y2, …
𝐚𝐦 =
(𝐟, 𝐲𝐦)
||𝐲𝐦||
=
𝟏
||𝐲𝐦||𝟐
𝐚
𝐛
𝐫 𝐱 𝐟(𝐱)𝐲𝐦 𝐱 𝐝𝐱 𝐧 = 𝟎, 𝟏, … .
Orthogonal systems
15. 15
Fourier series are powerful tools for problems involving
functions that are periodic or are of interest on a finite
interval only. Since, of course, many problems involve
functions that are nonperiodic and are of interest on the
whole x-axis, we ask what can be done to extend the method
of Fourier series to such functions. This idea will lead to
Fourier integrals.
The main application of Fourier integrals is in solving ODEs
(Ordinary differential equations) and PDEs (Partial differential
equations).
Fourier Integral
16. 16
We now consider any periodic function fL(x) of period 2L that
can be represented by a Fourier series
𝐟𝐋 𝐱 = 𝐚𝟎 +
𝐧=𝟏
∞
𝐚𝐧𝐜𝐨𝐬 𝐰𝐧𝐱 + 𝐛𝐧𝐬𝐢𝐧𝐰𝐧𝐱 . 𝐰𝐧 =
𝐧𝛑
𝐋
and find out what happens if we let L→∞. We should expect
an integral (instead of a series) involving cos wx and sin wx
with w no longer restricted to integer multiples w = wn = nπ/L
of π/L but taking all values.
Fourier Integral
17. 17
After some transformations we get
𝐟 𝐱 =
𝟎
∞
𝐀 𝐰 𝐜𝐨𝐬 𝐰𝐱 + 𝐁 𝐰 𝐬𝐢𝐧 𝐰𝐱 𝐝𝐰 ,
where
𝐀 𝐰 =
𝟏
𝛑
−∞
∞
𝐟 𝐯 𝐜𝐨𝐬 𝐰𝐯 𝐝𝐯 , 𝐁 𝐰 =
𝟏
𝛑
−∞
∞
𝐟 𝐯 𝐬𝐢𝐧 𝐰𝐯 𝐝𝐯 .
This is called a representation of f(x) by a Fourier integral.
Fourier Integral
18. 18
Sufficient conditions for the validity of a formula for Fourier
integral are as follows
T H E O R E M 1
If f(x) is piecewise continuous in every finite interval and has a
right-hand derivative and a left-hand derivative at every point
and if the integral −∞
∞
|𝐟 𝐱 | 𝐝𝐱 exists, then f(x) can be
represented by a Fourier integral with A and B given by the
above formulas. At a point where f(x) is discontinuous the
value of the Fourier integral equals the average of the left-
and right-hand limits of f(x) at that point.
Fourier Integral
19. 19
Just as Fourier series simplify if a function is even or odd, so
do Fourier integrals. Indeed, if f has a Fourier integral
representation and is even, then B(w) = 0. This holds because
the integrand of B(w) is odd. Then f(x) reduces to a Fourier
cosine integral
𝐟 𝐱 = 𝟎
∞
𝐀 𝐰 𝐜𝐨𝐬 𝐰𝐱 𝐝𝐰 , 𝐀 𝐰 =
𝟐
𝛑 −∞
∞
𝐟 𝐯 𝐜𝐨𝐬 𝐰𝐯 𝐝𝐯
Similarly, if f has a Fourier integral representation and is odd,
then A(w) = 0. This is true because the integrand of is odd.
Then f(x) becomes a Fourier sine integral
𝐟 𝐱 = 𝟎
∞
𝐁 𝐰 𝐬𝐢𝐧 𝐰𝐱 𝐝𝐰 , 𝐁 𝐰 =
𝟐
𝛑 −∞
∞
𝐟 𝐯 𝐬𝐢𝐧 𝐰𝐯 𝐝𝐯
Fourier Integral
20. 20
Just as Fourier series simplify if a function is even or odd, so
do Fourier integrals. Indeed, if f has a Fourier integral
representation and is even, then B(w) = 0. This holds because
the integrand of B(w) is odd. Then f(x) reduces to a Fourier
cosine integral
𝐟 𝐱 = 𝟎
∞
𝐀 𝐰 𝐜𝐨𝐬 𝐰𝐱 𝐝𝐰 , 𝐀 𝐰 =
𝟐
𝛑 −∞
∞
𝐟 𝐯 𝐜𝐨𝐬 𝐰𝐯 𝐝𝐯
Similarly, if f has a Fourier integral representation and is odd,
then A(w) = 0. This is true because the integrand of is odd.
Then f(x) becomes a Fourier sine integral
𝐟 𝐱 = 𝟎
∞
𝐁 𝐰 𝐬𝐢𝐧 𝐰𝐱 𝐝𝐰 , 𝐁 𝐰 =
𝟐
𝛑 −∞
∞
𝐟 𝐯 𝐬𝐢𝐧 𝐰𝐯 𝐝𝐯
Fourier Integral