...Fourier Transform and its applications Jatin Kumar Murray State University Abstract It has been widely recognized that waveforms are an integral part of the various universe phenomenon. Waveforms can be used to represent almost everything in the world. Therefore it is understandable that concepts related to waveforms or signals are extremely important as their applications exist in a broad variety of fields. The processes and ideas related to waveforms play a vital role in different areas of science and technology such as communications, optics, quantum mechanics, aeronautics, image processing to name a few. Even though the physical nature of signals might be completely different in various disciplines, all waveforms follow one fundamental principle; they can be represented by functions of one or more independent variables. This paper would focus on the concept of Fourier Transform, the technique through which signals can be deconstructed and represented as sum of various elementary signals. It briefly describes Linear Time Invariant systems and their response to superimposed signals. Fourier transform has many applications in physics and Engineering. This paper would also cover some of Fourier Transform applications in telecommunication and its impact on society. Introduction Some of the basic signals that exist in the world and are useful in various technology fields are continuous and discrete time...
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...ENTS 699R: Lecture 1d support ENTS 699R Lecture 1d support: Fourier Transform Tables Alejandra Mercado June, 2013 1 Transform Pairs The following is a table of basic transform pairs that can be used as building blocks to derive more complicated transform pairs: Time domain function, with dummy variable t 1 2 3 4 5 6 7 F F F Frequency domain function, with dummy variable f δ(t) ⇐⇒ 1 1 ⇐⇒ δ(f ) δ(t − t0 ) ⇐⇒ e−j2πf t0 sin (2πf0 t + φ) ⇐⇒ F F j −jφ δ(f 2 [e 1 −jφ δ(f 2 [e + f0 ) − ejφ δ(f − f0 )] cos (2πf0 t + φ) ⇐⇒ + f0 ) + ejφ δ(f − f0 )] 1 |t| ≤ T F t 2 rect ( T ) = ⇐⇒ T sinc(f T ) = T sin(πf T ) πf T 0 o.w. sinc(βt) ⇐⇒ F 1 β f · rect ( β ) = 1 β ·1 |f | ≤ o.w. β 2 0 Page 1 ENTS 699R: Lecture 1d support 2 Properties For the table of Fourier Transform properties, assume that we already know that: g(t) ⇐⇒ G(f ) h(t) ⇐⇒ H(f ) and that α, β, T, φ, f0 , t0 are all arbitrary constants. Time domain function, with dummy variable t A B C D E F F G H I F F F F F Frequency domain function, with dummy variable f Property name time/frequency reversal duality time shift frequency shift linearity g(−t) ⇐⇒ G(−f ) G(t) ⇐⇒ g(−f ) g(t − t0 ) ⇐⇒ g(t)ej2πf0 t F F F e−j2πf t0 G(f ) ⇐⇒ G(f − f0 ) 1 2 j 2 αg(t) + βh(t) ⇐⇒ αG(f ) + βH(f ) g(t) cos(2πf0 t) ⇐⇒ g(t) sin(2πf0 t) ⇐⇒ F F F (G(f − f0 ) + G(f + f0 )) modulation (G(f + f0 ) − G(f − f0 )) modulation multipl. in time domain convolution in time domain time scaling g(t) × h(t)...
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...Serie de Fourier El análisis de Fourier fue introducido en 1822 en la “Théorie analyitique de la chaleur” para tratar la solución de problemas de valores en la frontera en la conducción del calor. Más de siglo y medio después las aplicaciones de esta teoría son muy bastas: Sistemas Lineales, Comunicaciones, Física moderna, Electrónica, Óptica y por supuesto, Redes Eléctricas entre muchas otras. Series de Fourier. 1 Funciones Periódicas Una Función Periódica f(t) cumple la siguiente propiedad para todo valor de t. f(t)=f(t+T) A la constante mínima para la cual se cumple lo anterior se le llama el periodo de la función Repitiendo la propiedad se puede obtener: Series de Fourier. f(t)=f(t+nT), donde n=0,1, 2, 3,... 2 Funciones Periódicas t t Ejemplo: ¿Cuál es el período de la función cos( 3 ) cos( 4 )? f(t) Solución.- Si f(t) es periódica se debe cumplir: t t f(t T) cos( t T ) cos( t T ) f(t) cos( 3 ) cos( 4 ) 3 4 Pero como se sabe cos(x+2kp)=cos(x) para cualquier entero k, entonces para que se cumpla la igualdad se requiere que T/3=2k1p, T/4=2k2p Es decir, T = 6k1p = 8k2p Donde k1 y k2 son enteros, El valor mínimo de T se obtiene con k1=4, k2=3, es decir,T=24p Series de Fourier. 3 Funciones Periódicas Gráfica de la función 3 2 1 t t f(t) cos( 3 ) cos( 4 ) T f(t)=cos(t/3)+cos(t/4) f(t) 0 -1 -2 24p -3 0 50 100 150 200 t Series de Fourier. 4 Funciones Periódicas Podríamos pensar que cualquier suma de funciones...
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...1 | (Radians/Real Numbers) | Input | 26·3π | Output | 1π | Decimal Output | 0.3183098861838 | 2 | (Radians/Real Numbers) | Input | 26·3π | Output | 1π | Decimal Output | 0.3183098861838 | 3 | (Radians/Real Numbers) | Input | 1π 4 sinπ3·2+6π cosπ3·2-6π | Output | 2 3-9π | Decimal Output | -1.7621311848105 | 4 | (Radians/Real Numbers) | Input | 1π 4sinπ3·2+6πcosπ3·2-1π 6π | Output | 2 3π-9π2 | Decimal Output | 0.1907671380625 | 5 | (Radians/Real Numbers) | Input | 1π 6-2sinπ3·2-3πcosπ3·2-1π 6-6sinπ3·6-3πcosπ3·6 | Output | 92 π2+2 3π | Decimal Output | 1.5586031172341 | 6 | (Radians/Real Numbers) | Input | 1π 4sinπ3·2+6πcosπ3·2-1π 6π+1π 6-2sinπ3·2-3πcosπ3·2-1π 6-6sinπ3·6-3πcosπ3·6 | Output | 4 3π-92 π2 | Decimal Output | 1.7493702552966 | 7 | (Radians/Real Numbers) | Input | 26 6-x sinπ3 x π3 -sinπ3 x π3 -1ⅆx | Output | 6-xsinπ x3π+9cosπ x3π2+x-С | 8 | (Radians/Real Numbers) | Input | 26 6-2 sinπ3·2 π3 -sinπ3·2 π3 -1ⅆx | Output | -3 3 x2+2 3π+x-С | 9 | (Radians/Real Numbers) | Input | 1π 6-xsinπ3 x-3πcosπ3 x limx→2-6-1π 6-xsinπ3 x-3πcosπ3 x | Output | -32 π+5 3 6-xsinπ x3-3cosπ x3ππ2 | 10 | (Radians/Real Numbers) | Input | 1π 6-2sinπ3·2-3πcosπ3·2-1π 6-6sinπ3·6-3πcosπ3·6 | Output | 92 π2+2 3π | Decimal Output | 1.5586031172341 | 11 | (Radians/Real Numbers) | Input | 1π 4sinπ3·2+6πcosπ3·2-6π+1π 6-2sinπ3·2-3πcosπ3·2-1π 6-6sinπ3·6-3πcosπ3·6 | Output | 4 π 3+92-9 ππ2 | Decimal Output...
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...questions carry equal marks. PAGE 2 ME3291 QUESTION 1 The heat conduction equation in 1D is given by T/ t = b 2 T/ x2. Here T is the temperature and b is the thermal conductivity. You are interested to use the DuFort & Frankel discretization scheme to obtain the finite difference equation of the governing equation because you have heard of its inherent stable properties. The DuFort & Frankel scheme is given as: (Tpq+1 - Tpq-1)/(2 t) = (b / ( x)2) [Tp+1q – (Tpq-1 + Tpq+1) + Tp-1q]. where Tpq = T (p x, q t) is the finite difference representation. You are interested to use the von Neumann (Fourier) stability analysis to determine if it is inherently stable or otherwise. If otherwise, then you show the criterion for the limit of stability. You may assume that Tpq = q ei ph where is the amplification factor, is a particular spatial Fourier mode, i is the complex number (-1)0.5, and h x in the analysis. (a) Determine the (quadratic) equation for (10 marks) (b) Hence or otherwise, determine the possible range of r ( t/( x)2) so that there is no source or sink in the governing heat conduction equation. ≤ 1.0 since (15 marks) PAGE 3 ME3291 QUESTION 2 The one-dimensional wave...
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...PVC, PE, etc. Bio-composite materials that can be decomposed naturally will be a solution to solve these problems. The aim of this study was to utilize the waste shredded coconut after the coconut milk was extracted, and used as reinforcing agent bio-composite material with matrix of Polylacticacid(PLA) that can decompose naturally. During this time, grated coconut is simply discarded without further exploited. Alkalization chemical treatment with 5% NaOH 1 hour to Grated Coconut Milk Residue (GCMR) was done to improve compatibility with PLA matrix. Bio-composite material was made with a fraction of 0%, 15%, 30% w/w using compression molding technique. The chemical structure of GCMR before and after chemical treatment was observed with Fourier Transform Infrared. The mechanical property of bio-composite material was observed with tensile test. Whereas the morphological characteristics of GCMR and fracture on the surface of bio-composite observed using Field Emission Scanning Electron Microscope. Then, thermal properties of bio-composite observed with Simultaneous Thermal Analysis. Result showed the addition of grated coconut fibers improve the mechanical properties, thermal stability, speed recrystallization PLA and PLA interfacial bonding between the filler GCMR. Thus, this material has the potential to reduce the environmental crisis caused by non-biodegradable material and non-renewable. Kata Kunci—Instruksi; Poly(latic acid)(PLA), Reinforcing Agent, Grated Coconut Residue...
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...[Fourier analysis of Control System] [Fourier analysis of Control System] Submitted to: Dr. S. K. Raghuwanshi Submitted By: Rishi Kant Sharan Semester: V Branch: Electronics & Communication Engineering Submitted to: Dr. S. K. Raghuwanshi Submitted By: Rishi Kant Sharan Adm. No: 2010JE1117 Semester: V Branch: Electronics & Communication Engineering Abstract The assignment focuses on the Fourier analysis of Control System. Which leads to frequency domain analysis of control system. The scope of estimation and controlling the behavior a system by means of Fourier transformation of its transfer function and analyzing its frequency response. Abstract The assignment focuses on the Fourier analysis of Control System. Which leads to frequency domain analysis of control system. The scope of estimation and controlling the behavior a system by means of Fourier transformation of its transfer function and analyzing its frequency response. ACKNOWLEDGEMENT There is an old adage that says that you never really learn a subject until you teach it. I now know that you learn a subject even better when you write about it. Preparing this term paper has provided me with a wonderful opportunity to unite my love of concept in CONTROL SYSTEM. This term paper is made possible through the help and support from everyone, including: professor, friends, parents, family, and in essence, all sentient beings. Especially, please allow me to dedicate...
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...MEEN 260 Introduction to Engineering Experimentation Homework 10: Laplace Transform, and Frequency Response Solution Assigned: Thursday, 9 Apr. 2009 Due: Thursday, 16 Apr. 2009, 5:00pm Learning Objectives: After completing this homework assignment, you should be able to: 1) 2) 3) 4) 5) Determine the Laplace Transform of a signal using the definition, tables, or properties of the Laplace Transform Utilize the Laplace Transform to find the Transfer Function of a dynamic system represented by a system of differential equations Utilize the Laplace Transform to solve for the transient response of a dynamic system Discuss the difference between the Laplace and Fourier Transforms and their respective uses Using the Transfer Function of a system, determine and plot the associated frequency response, and determine the steady state response of a system to a harmonic input signal Homework Problems: Problem 1) Definition of Laplace Transform Using the mathematical definition, compute the Laplace transform for the function: f (t ) = 3t + t cos(2t ) Solution: From the mathematical definition, we split the function into two pieces: 3 3 · We use u-v substitution (u=3t, dv=e-st) to get: ∞ 3 ∞ 3 3 0 0 The second piece of the function is more complicated. Recall that: · So we find: cos 2 We use u-v substitution (u=e-st, dv=cos(2t)) and get: sin 2 2 This does not give us a useful answer, so we perform a u-v substitution to the right hand side of the equation to obtain: cos 2 By rearranging...
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...2007-2008 JAWAHARLAL NEHRU TECHNOLOGICAL UNIVERSITY, HYDERABAD B.TECH. ELECTRONICS AND COMMUNICATION ENGINEERING I YEAR COURSE STRUCTURE |Code |Subject |T |P/D |C | | |English |2+1 |- |4 | | |Mathematics - I |3+1 |- |6 | | |Mathematical Methods |3+1 |- |6 | | |Applied Physics |2+1 |- |4 | | |C Programming and Data Structures |3+1 |- |6 | | |Network Analysis |2+1 |- |4 | | |Electronic Devices and Circuits |3+1 |- |6 | | |Engineering Drawing |- |3 |4 | | |Computer Programming Lab. |- |3 |4 | | |IT Workshop |- |3 |4 | | |Electronic Devices and Circuits Lab |- |3...
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...Vojta Rousar Mrs. Rothstein Comp/Lit 11 (2) 13 October 2009 Global Warming The Earth's climate is constantly changing over time. Many climatologists believe that the temperature of the Earth slowly fluctuates over time. In fact, several scientists estimate that between 15,000 and 30,000 years ago the Earth was covered by large sheets of ice. This period of time was known as the Ice Age. As the temperature of the Earth began to rise 7,000 years ago, the Ice Age came to an end. The first theory of global warming came in 1824 when French mathematician Jean Baptiste Joseph Fourier discovered that the Earth's temperature was slowly increasing. Fourier argued that the earth's atmosphere traps solar radiation and reflects it back toward the earth. In the late 19th century Fourier's theory was labeled the "greenhouse effect" when Nobel Laureate Svante Arrhenius coined the term to explain how carbon dioxide traps heat in the Earth's atmosphere. Today, scientists disagree on the effects of global warming while some deny the phenomena all together. Despite these arguments many historians point out the direct relationship between man and the environment, often referencing the American Dust Bowl of the 1930s, where large scale soil erosion reduced parts of Colorado, Kansas, New Mexico, Oklahoma and Texas to arid deserts. Currently, many governments and corporations are working to reduce fuel emissions and produce "Earth friendly" products such as hybrid cars. Yet, many scientists...
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...Lovely Professional University, Punjab Course Code MTH251 Course Category Course Title FUNCTION OF COMPLEX VARIABLE AND TRANSFORM Courses with Numerical focus Course Planner 16423::Harsimran Kaur Lectures 3.0 Tutorials Practicals Credits 2.0 0.0 4.0 TextBooks Sr No T-1 Title Advanced Engineering Mathematics Reference Books Sr No R-1 R-2 Other Reading Sr No OR-1 Journals articles as Compulsary reading (specific articles, complete reference) Journals atricles as compulsory readings (specific articles, Complete reference) , Title Higher Engineering Mathematics Advanced Modern Engineering Mathematics Author Grewal, B. S. Glyn James Edition 40th 3rd Year 2007 2011 Publisher Name Khanna Publishers Pearson Author Jain R. K. and Iyenger S. R. K. Edition 3rd Year 2007 Publisher Name Narosa Relevant Websites Sr No RW-1 RW-2 (Web address) (only if relevant to the course) www2.latech.edu/~schroder/comp_var_videos.htm freescienceonline.blogspot.com/2010_04_01_archive.html Salient Features Topic videos available Complex Analysis Reference Material Available LTP week distribution: (LTP Weeks) Weeks before MTE Weeks After MTE Spill Over 7 6 2 Detailed Plan For Lectures Week Number Lecture Number Broad Topic(Sub Topic) Chapters/Sections of Text/reference books Other Readings, Lecture Description Relevant Websites, Audio Visual Aids, software and Virtual Labs Introduction Functions of a Complex Variable Learning Outcomes Pedagogical Tool Demonstration/ Case Study...
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...in textbooks on Fourier theory. It seems that there is little that can be done with wavelets that cannot be done with traditional Fourier analysis. Stephane Mallat was not the father of wavelet theory, but he is certainly an evangelist. His textbook on the subject, A Wavelet Tour of Signal Processing [1], contains proofs about the theory of wavelets, and a summation about what is known about them with applications to signal processing. One of his many papers, Characterization of Signals from Multiscale Edges [2], is frequently cited as a link between wavelets and edge detection. Mallat’s method not only finds edges, but classifies them into different types as well. Mallat goes on to describe a method of recovering complete images using only the edges, but we will not implement it in this project. In this project, we study this paper, and implement the method of Mallat to multiscale edge detection and analysis. We will first present a short background on wavelet theory. Then we will describe the different types of edges that exist in images, and how they can be characterized using a Lipschitz constant. Next, we describe the algorithm for the wavelet transform, from the Mallat paper. Finally, we show the results of applying the algorithm to a test image, and a real image. wave with the signal. When the results high valued, the coefficients of the Fourier transform will be high. Where the signal or the wave is close to zero, the coefficients will be low. Fourier analysis has a...
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...UNIVERSITY OF MIAMI TONALITY ESTIMATION USING WAVELET PACKET ANALYSIS By Vaibhav Chhabra A Research Project Submitted to the Faculty of the University of Miami in partial fulfillment of the requirements for the degree of Master of Science Coral Gables, Florida May 2005 UNIVERSITY OF MIAMI A research project submitted in partial fulfillment of the requirements for the degree of Master of Science TONALITY ESTIMATION USING WAVELET PACKET ANALYSIS Vaibhav Chhabra Approved: ________________ Ken Pohlmann Professor of Music Engineering _________________ Dr. Edward Asmus Associate Dean of Graduate Studies ________________ Colby Leider Assistant Professor of Music Engineering _________________ Dr. Paul Mermelstein Professor of Electrical Engineering DEDICATION They say that one’s experience is what defines an individual. After all, you are what you are because of your experiences. On that note I would like to dedicate this work to all those who have contributed to my experience in this journey. For what I have learned has laid the foundation for what I will learn. I would also like to thank my family who has always been supportive of me, my brother Ruchir who is a natural send-master, Papa and Ma thanks for keeping the faith. All the Chacha’s, Chachi’s and cousins, thank you all for the support. Next on my thank you list are my Tae Kwon Do buddies. Sensei Jeff thanks for all of your advice, some day I’ll be teacher like you. Rico, training...
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...SUPLEMEN Pemodelan Sistem / Pengolahan Sinyal / Metode Kuantitatif TUTORIAL SINGKAT MATLAB oleh: Judi Prajetno Sugiono Sekolah Tinggi Teknik Surabaya (2005, 2008, 2011) judi@stts.edu ©2005 p. 1 of 40 MATLAB Short Tutorial Reserve word (don’t used it as variable's name) · · · · · ans pi nan inf eps Special sign · · · · · % [] ; ' : line comment begin - end of matrix row separation, or not echoed command if place in the end of a statement begin - end of string indexing sign Variable is assume as matrix % empty matrix A=[] A = [] % matrix 1x1 or a constant A=[0] A = 0 % same with A=0 A = 0 % complex number: use i or j to express imaginary part z=3+4j z = 3.0000 + 4.0000i Entry a matrix % use as column separation and or as row separation A=[1 2 3; 4 5 6; 7 8 9] A = 1 4 7 2 5 8 3 6 9 Last saved by jpsugiono 9/23/2011 judi@stts.edu ©2005 p. 2 of 40 How to point element of matrix % A(row,column) A(1,3) ans = 3 % sign use as get all row or column A(2,:) ans = 4 5 6 % sign use as get from m to n cell in row or colomn A(1:2, 2:3) ans = 2 5 3 6 row and column vector % row vector a=[0 1 2 3 4 5] a = 0 1 2 3 4 5 % column vector b=[0; 1; 3; 4; 5] b = 0 1 3 4 5 % Shortcut to build a vector % init:step:final a=0:0.2:1 a = 0 0.2000 0.4000 0.6000 String % begin and end with < ' >, and act like a matrix of character ...
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...REACTION: (Paget’s Disease) This article entitled “A Tour Around Paget’s Disease of Bone” basically talks about everything you need to know about this certain disease. First, it talks about the origin of this disease and who discovered it. According to this article, it is discovered by Sir James Paget and that the first recorded evidence of this disease happened about 1000BC and it is the second leading skeletal disorder affecting the aging population that time. It is delightful to know the etymology of this disease and who discovered it because it gives us more knowledge about the disease. Second, it talks about the pathophysiology of the disease, its diagnostic evaluation and the clinical manifestations of the disease. These are important to know for us to able to assess the patient very well in order for us to give the appropriate nursing interventions for the patient and for us to establish a proper and efficient nursing care plan for the patient. Third, it mentioned the causative microorganisms that cause the disease. This is also important to know especially for the doctors for them to be able to evaluate the disease better and for them give the appropriate medical interventions for their patients. With this, they would be able to provide the proper medications their patients’ need that would react effectively on a certain microorganism. Lastly, it talks about the different kinds of treatment the patient will undergo or the pharmacological therapy that the patient...
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