The document discusses Fourier series and their applications. It begins by introducing how Fourier originally developed the technique to study heat transfer and how it can represent periodic functions as an infinite series of sine and cosine terms. It then provides the definition and examples of Fourier series representations. The key points are that Fourier series decompose a function into sinusoidal basis functions with coefficients determined by integrating the function against each basis function. The series may converge to the original function under certain conditions.
Mathematics (from Greek μάθημα máthēma, “knowledge, study, learning”) is the study of topics such as quantity (numbers), structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope and definition of mathematics
Mathematics (from Greek μάθημα máthēma, “knowledge, study, learning”) is the study of topics such as quantity (numbers), structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope and definition of mathematics
Mathematics (from Greek μάθημα máthēma, “knowledge, study, learning”) is the study of topics such as quantity (numbers), structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope and definition of mathematics
Mathematics (from Greek μάθημα máthēma, “knowledge, study, learning”) is the study of topics such as quantity (numbers), structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope and definition of mathematics
Changing variable is something we come across very often in Integration. There are many
reasons for changing variables but the main reason for changing variables is to convert the
integrand into something simpler and also to transform the region into another region which is
easy to work with. When we convert into a new set of variables it is not always easy to find the
limits. So, before we move into changing variables with multiple integrals we first need to see
how the region may change with a change of variables. In order to change variables in an
integration we will need the Jacobian of the transformation.
Changing variable is something we come across very often in Integration. There are many
reasons for changing variables but the main reason for changing variables is to convert the
integrand into something simpler and also to transform the region into another region which is
easy to work with. When we convert into a new set of variables it is not always easy to find the
limits. So, before we move into changing variables with multiple integrals we first need to see
how the region may change with a change of variables. In order to change variables in an
integration we will need the Jacobian of the transformation.
What is Fourier Transform
Spatial to Frequency Domain
Fourier Transform
Forward Fourier and Inverse Fourier transforms
Properties of Fourier Transforms
Fourier Transformation in Image processing
A signal is a pattern of variation that carry information.
Signals are represented mathematically as a function of one or more independent variable
basic concept of signals
types of signals
system concepts
Presentation on Fourier Series
contents are:-
Euler’s Formula
Functions having point of discontinuity
Change of interval
Even and Odd functions
Half Range series
Harmonic analysis
CFD Simulation of By-pass Flow in a HRSG module by R&R Consult.pptxR&R Consult
CFD analysis is incredibly effective at solving mysteries and improving the performance of complex systems!
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Work done in cooperation with James Malloy and David Moelling from Tetra Engineering.
More examples of our work https://www.r-r-consult.dk/en/cases-en/
About
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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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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.
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• 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.
Courier management system project report.pdfKamal Acharya
It is now-a-days very important for the people to send or receive articles like imported furniture, electronic items, gifts, business goods and the like. People depend vastly on different transport systems which mostly use the manual way of receiving and delivering the articles. There is no way to track the articles till they are received and there is no way to let the customer know what happened in transit, once he booked some articles. In such a situation, we need a system which completely computerizes the cargo activities including time to time tracking of the articles sent. This need is fulfilled by Courier Management System software which is online software for the cargo management people that enables them to receive the goods from a source and send them to a required destination and track their status from time to time.
Final project report on grocery store management system..pdfKamal Acharya
In today’s fast-changing business environment, it’s extremely important to be able to respond to client needs in the most effective and timely manner. If your customers wish to see your business online and have instant access to your products or services.
Online Grocery Store is an e-commerce website, which retails various grocery products. This project allows viewing various products available enables registered users to purchase desired products instantly using Paytm, UPI payment processor (Instant Pay) and also can place order by using Cash on Delivery (Pay Later) option. This project provides an easy access to Administrators and Managers to view orders placed using Pay Later and Instant Pay options.
In order to develop an e-commerce website, a number of Technologies must be studied and understood. These include multi-tiered architecture, server and client-side scripting techniques, implementation technologies, programming language (such as PHP, HTML, CSS, JavaScript) and MySQL relational databases. This is a project with the objective to develop a basic website where a consumer is provided with a shopping cart website and also to know about the technologies used to develop such a website.
This document will discuss each of the underlying technologies to create and implement an e- commerce website.
Industrial Training at Shahjalal Fertilizer Company Limited (SFCL)MdTanvirMahtab2
This presentation is about the working procedure of Shahjalal Fertilizer Company Limited (SFCL). A Govt. owned Company of Bangladesh Chemical Industries Corporation under Ministry of Industries.
Vaccine management system project report documentation..pdfKamal Acharya
The Division of Vaccine and Immunization is facing increasing difficulty monitoring vaccines and other commodities distribution once they have been distributed from the national stores. With the introduction of new vaccines, more challenges have been anticipated with this additions posing serious threat to the already over strained vaccine supply chain system in Kenya.
TECHNICAL TRAINING MANUAL GENERAL FAMILIARIZATION COURSEDuvanRamosGarzon1
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The Single Aisle is the most advanced family aircraft in service today, with fly-by-wire flight controls.
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Automobile Management System Project Report.pdfKamal Acharya
The proposed project is developed to manage the automobile in the automobile dealer company. The main module in this project is login, automobile management, customer management, sales, complaints and reports. The first module is the login. The automobile showroom owner should login to the project for usage. The username and password are verified and if it is correct, next form opens. If the username and password are not correct, it shows the error message.
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Also when the user tries to sale items which are not in stock, the system will prompt the user that the stock is not enough. Customers of this system can search for a automobile; can purchase a automobile easily by selecting fast. On the other hand the stock of automobiles can be maintained perfectly by the automobile shop manager overcoming the drawbacks of existing system.
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.
1. 10 Fourier Series
10.1 Introduction
When the French mathematician Joseph Fourier (1768-1830) was trying to
study the flow of heat in a metal plate, he had the idea of expressing the
heat source as an infinite series of sine and cosine functions. Although the
original motivation was to solve the heat equation, it later became obvious
that the same techniques could be applied to a wide array of mathematical
and physical problems.
In this course, we will learn how to find Fourier series to represent periodic
functions as an infinite series of sine and cosine terms.
A function f(x) is said to be periodic with period T, if
f(x + T) = f(x), for all x.
The period of the function f(t) is the interval between two successive repe-titions.
10.2 Definition of a Fourier Series
Let f be a bounded function defined on the [−, ] with at most a finite
number of maxima and minima and at most a finite number of discontinuities
in the interval. Then the Fourier series of f is the series
f(x) =
1
2
a0 + a1 cos x + a2 cos 2x + a3 cos 3x + ...
+ b1 sin x + b2 sin 2x + b3 sin 3x + ...
where the coefficients an and bn are given by the formulae
a0 =
1
Z
−
f(x)dx
an =
1
Z
−
f(x)cos(nx)dx
bn =
1
Z
−
f(x)sin(nx)dx
2. 10.3 Why the coefficients are found as they are
We want to derive formulas for a0, an and bn. To do this we take advantage
of some properies of sinusoidal signals. We use the following orthogonality
conditions:
Orthogonality conditions
(i) The average value of cos(nx) and sin(nx) over a period is zero.
Z
−
cos(nx)dx = 0
Z
−
sin(nx)dx = 0
(ii) The average value of sin(mx)cos(nx) over a period is zero.
Z
−
sin(mx)cos(nx)dx = 0
(iii) The average value of sin(mx)sin(nx) over a period,
Z
−
sin(mx)sin(nx)dx =
if m = n6= 0
0 otherwise
(iv) The average value of cos(mx)cos(nx) over a period,
Z
−
cos(mx)cos(nx)dx =
8
:
2 if m = n = 0
if m = n6= 0
0 if m6= n
Remark. The following trigonometric identities are useful to prove the
above orthogonality conditions:
cosAcosB =
1
2
[cos(A − B) + cos(A + B)]
sinAsinB =
1
2
[cos(A − B) − cos(A + B)]
sinAcosB =
1
2
[sin(A − B) + sin(A + B)]
3. Finding an
We assume that we can represent the function by
f(x) =
1
2
a0 + a1 cos x + a2 cos 2x + a3 cos 3x + ...
+ b1 sin x + b2 sin 2x + b3 sin 3x + ... (1)
Multiply (1) by cos(nx), n 1 and integrate from − to and assume it is
permissible to integrate the series term by term.
Z
−
f(x) cos(nx)dx =
1
2
a0
Z
−
cos(nx) + a1
Z
−
cos x cos(nx)dx + a2
Z
−
cos 2x cos(nx)dx + ...
+ b1
Z
−
sin x cos(nx)dx + b2
Z
−
sin 2x cos(nx)dx + ...
= an, because of the above orthogonality conditions
an =
1
Z
−
f(x) cos(nx)dx
Similarly if we multiply (1) by sin(nx), n 1 and integrate from − to
, we can find the formula for the coefficient bn. To find a0, simply integrate
(1) from − to .
4. 10.4 Fourier Series
Definition. Let f(x) be a 2-periodic function which is integrable on [−, ].
Set
a0 =
1
Z
−
f(x)dx
an =
1
Z
−
f(x)cos(nx)dx
bn =
1
Z
−
f(x)sin(nx)dx
then the trigonometric series
1
2
a0 +
X1
n=1
(ancos(nx) + bnsin(nx))
is called the Fourier series associated to the function f(x).
Remark. Notice that we are not saying f(x) is equal to its Fourier Series.
Later we will discuss conditions under which that is actually true.
Example. Find the Fourier coefficients and the Fourier series of the square-wave
function f defined by
f(x) =
0 if − x 0
1 if 0 x
and f(x + 2) = f(x)
Solution: So f is periodic with period 2 and its graph is:
5. Using the formulas for the Fourier coefficients we have
a0 =
1
Z
−
f(x)dx
=
1
Z 0
(
−
0dx +
1
Z
0
1dx)
=
1
()
= 1
an =
1
Z
−
f(x) cos(nx)dx
=
1
Z 0
(
−
0dx +
1
Z
0
cos(nx)dx)
=
1
(0 +
sin(nx)
n
0 )
=
1
n
(sin n − sin 0)
= 0
bn =
1
Z
−
f(x) sin(nx)dx
=
1
Z 0
(
−
0dx +
1
Z
0
sin(nx)dx)
=
1
cos(nx)
(−
n
0 )
= −
1
n
(cos n − cos 0)
= −
1
n
((−1)n − 1), since cos n = (−1)n.
=
(
0, if n is even
2
, if n is odd
n
6. The Fourier Series of f is therefore
f(x) =
1
2
a0 + a1 cos x + a2 cos 2x + a3 cos 3x + ...
+ b1 sin x + b2 sin 2x + b3 sin 3x + ...
=
1
2
+ 0 + 0 + 0 + ...
+
2
sin x + 0 sin 2x +
2
3
sin 3x + 0 sin 4x +
2
5
sin 5x + ...
=
1
2
+
2
sin x +
2
3
sin 3x +
2
5
sin 5x + ...
Remark. In the above example we have found the Fourier Series of the
square-wave function, but we don’t know yet whether this function is equal
to its Fourier series. If we plot
1
2
+
sin x
2
1
2
+
2
sin x +
2
3
sin 3x
1
2
+
2
sin x +
2
3
sin 3x +
2
5
sin 5x+
1
2
+
2
sin x +
2
3
sin 3x +
2
5
sin 5x +
2
7
sin 7x+
1
2
+
2
sin x +
2
3
sin 3x +
2
5
sin 5x +
2
7
sin 7x +
2
9
sin 9x +
2
11
sin 11x +
2
13
sin 13x,
we see that as we take more terms, we get a better approximation to the
square-wave function. The following theorem tells us that for almost all
points (except at the discontinuities), the Fourier series equals the function.
Theorem (Fourier Convergence Theorem) If f is a periodic func-tion
with period 2 and f and f0 are piecewise continuous on [−, ], then
the Fourier series is convergent. The sum of the Fourier series is equal to
f(x) at all numbers x where f is continuous. At the numbers x where f is
discontinuous, the sum of the Fourier series is the average value. i.e.
1
2
[f(x+) + f(x−)].
7. Remark. If we apply this result to the example above, the Fourier Series
is equal to the function at all points except −, 0, . At the discontinuity 0,
observe that
f(0+) = 1 and f(0−) = 0
The average value is 1/2.
Therefore the series equals to 1/2 at the discontinuities.
8. 10.5 Odd and even functions
Definition. A function is said to be even if
f(−x) = f(x) for all real numbers x.
A function is said to be odd if
f(−x) = −f(x) for all real numbers x.
Example. cos x, x2, |x| are examples of even functions. sin x, x, x3 are
examples of odd functions. The product of two even functions is even, the
product of two odd functions is also even. The product of an even and odd
function is odd.
Remark. If f is an odd function then
Z
−
f(x)dx = 0,
while if f is an even function then
Z
−
f(x)dx = 2
Z
0
f(x)dx.
(i) If f(x) is odd, then
• f(x)cos(nx) is odd hence an = 0 and
• f(x)sin(nx) is even hence bn = 2
R
0 f(x)sin(nx)dx
(ii) If f(x) is even, then
• f(x)cos(nx) is even hence an = 2
R
0 f(x)cos(nx)dx and
• f(x)sin(nx) is even hence bn = 0
9. Example. 1. Let f be a periodic function of period 2 such that
f(x) = x for − x
Find the Fourier series associated to f.
Solution: So f is periodic with period 2 and its graph is:
We first check if f is even or odd.
f(−x) = −x = −f(x), so f(x) is odd.
Therefore,
an = 0
bn =
2
Z
0
f(x)sin(nx)dx
Using the formulas for the Fourier coefficients we have
bn =
2
Z
0
x sin(nx)dx
=
2
− x
(
cos nx
n
0
−
Z
0
(−
cos nx
n
)dx)
=
2
(
1
n
[− cos n] + [
sin nx
n2 ]0
)
= −
2
n
cos n
=
−2/n if n is even
2/n if n is odd
The Fourier Series of f is therefore
f(x) = b1 sin x + b2 sin 2x + b3 sin 3x + ...
= 2 sin x −
2
2
sin 2x +
2
3
sin 3x −
2
4
sin 4x +
2
5
sin 5x + ...
10. 2. Let f be a periodic function of period 2 such that
f(x) = 2 − x2 for − x
Solution: So f is periodic with period 2 and its graph is:
We first check if f is even or odd.
f(−x) = 2 − (−x)2 = 2 − x2 = f(x), so f(x) is even.
Since f is even,
bn = 0
an =
2
Z
0
f(x) cos(nx)dx
11. Using the formulas for the Fourier coefficients we have
an =
2
Z
0
f(x) cos(nx)dx
=
2
Z
0
(2 − x2) cos(nx)dx
=
2
(2 − x2)
(
sin nx
n
0
−
Z
0
−2x
sin nx
n
dx)
=
2
([(2 − 2)
sin n
n
− (2 − 0)
sin 0
n
] +
Z
0
2x
sin nx
n
dx)
=
2
2
n
Z
0
x sin(nx)dx
=
2
2
n
− x
(
cos nx
n
0
−
Z
0
(−
cos nx
n
)dx)
=
2
2
n
1
n
([− cos n] + [
sin nx
n2 ]0
)
= −
4
n2 cos n
=
−4/n2 if n is even
4/n2 if n is odd
It remains to calculate a0.
a0 =
2
Z
0
(2 − x2)dx
=
2
[2x −
x3
3
]0
=
42
3
The Fourier Series of f is therefore
f(x) =
1
2
a0 + a1 cos x + a2 cos 2x + a3 cos 3x + ...
b1 sin x + b2 sin 2x + b3 sin 3x + ...
=
22
3
+ 4(cos x −
1
4
cos 2x +
1
9
cos 3x −
1
16
cos 4x +
1
25
cos 5x + ....)