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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

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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) ⇐⇒ G(f ) ∗ H(f ) g(t) ∗ h(t) ⇐⇒ G(f ) × H(f ) g(αt) ⇐⇒

F f 1 |α| G( α )

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...chemist Julius Lothar Meyer published a table of the 49 known elements arranged by valency. The table revealed that elements with similar properties often shared the same valency. English chemist John Newlands produced a series of papers in 1864 and 1865 noting that when the elements were listed in order of increasing atomic weight, similar physical and chemical properties recurred at intervals of eight; he likened such periodicity to the octaves of music.This Law of Octaves, however, was ridiculed by Newlands' contemporaries, and the Chemical Society refused to publish his work.[7] Newlands was nonetheless able to draft a table of the elements and used it to predict the existence of missing elements, such as germanium. The Chemical Society only acknowledged the significance of his discoveries five years after they credited Mendeleev.[8] Mendeleev's table Dmitri Mendeleev Mendeleev's 1869 periodic table; note that his arrangement presents the periods vertically, and the groups horizontally. Russian chemistry professor Dmitri Mendeleev and German chemist Julius Lothar Meyerindependently published their periodic tables in 1869 and 1870, respectively. They both constructed their tables by listing the elements in rows or columns in order of atomic weight and starting a new row or column when the characteristics of the elements began to repeat. The success of Mendeleev's table came from two decisions he made. The first was to leave gaps in the table when it......

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...|||| Fourier Series When the French mathematician Joseph Fourier (1768–1830) was trying to solve a problem in heat conduction, he needed to express a function f as an inﬁnite series of sine and cosine functions: 1 f x a0 n 1 a n cos nx a1 cos x b1 sin x bn sin nx a3 cos 3x b3 sin 3x a0 a2 cos 2x b2 sin 2x Earlier, Daniel Bernoulli and Leonard Euler had used such series while investigating problems concerning vibrating strings and astronomy. The series in Equation 1 is called a trigonometric series or Fourier series and it turns out that expressing a function as a Fourier series is sometimes more advantageous than expanding it as a power series. In particular, astronomical phenomena are usually periodic, as are heartbeats, tides, and vibrating strings, so it makes sense to express them in terms of periodic functions. We start by assuming that the trigonometric series converges and has a continuous function f x as its sum on the interval , , that is, 2 f x a0 n 1 a n cos nx bn sin nx x Our aim is to ﬁnd formulas for the coefﬁcients a n and bn in terms of f . Recall that for a power series f x cn x a n we found a formula for the coefﬁcients in terms of derivn atives: cn f a n!. Here we use integrals. If we integrate both sides of Equation 2 and assume that it’s permissible to integrate the series term-by-term, we get y f x dx y a 0 dx y n 1 a n cos nx bn sin nx dx 2 a0 n 1 an y cos nx dx n 1 bn......

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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......

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...vi List of Tables Table 1 Benchmark 18 Table 2 Versus Table 35 Table 3 Operational Yearly Expenses (Existing System) 39 Table 4 Operational Expenses for Proposed System 40 (Alternative 1 : Client/Server) Table 5 Operational Expenses for Proposed System 41 (Alternative 1 : Peer-to-Peer) Table 6 System’s Hardware (Alternative 1 : Client/Server) 42 Table 7 System’s Software (Alternative 1 : Client/Server) 43 Table 8 System’s Hardware (Alternative 1 : Peer-to-Peer) 44 Table 9 System’s Software (Alternative 1 : Peer-to-Peer) 45 Table 10 Network Connection Equipment 45 Table 11 Development Cost (Alternative 1 : Client/Server) 46 Table 12 Development Cost (Alternative 1 : Peer-to-Peer) 46 vii Table 13 Savings (Alternative 1 : Client/Server) 47 Table 14 Savings (Alternative 1 : Peer-to-Peer) 47 Table 15 Payback Period (Alternative 1 : Client/Server) 48 Table 16 Payback Period (Alternative 1 : Peer-to-Peer) 49 Table 17 Versus Table Operational/Organizational Analysis 50 viii List of Figures Figure 1 Company’s Organizational Chart 3 Figure 2 System Design Paradigm 5 Figure 3 Modified Waterfall 11 Figure 4 Existing System Context Diagram 27 Figure 5 Top Level Diagram 28 Figure 6 Exploded Diagram Process 3 29 Figure 6.1 Exploded...

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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......

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...f ( t ) = L -1 {F ( s )} 1. 3. 5. 7. 9. 11. 1 t n , n = 1, 2,3,K t sin ( at ) t sin ( at ) sin ( at ) - at cos ( at ) cos ( at ) - at sin ( at ) sin ( at + b ) sinh ( at ) e at sin ( bt ) e at sinh ( bt ) t ne at , n = 1, 2,3,K uc ( t ) = u ( t - c ) Heaviside Function F ( s ) = L { f ( t )} 1 s n! s n +1 Table of Laplace Transforms f ( t ) = L -1 {F ( s )} F ( s ) = L { f ( t )} 1 s-a G ( p + 1) s p +1 1 × 3 × 5L ( 2n - 1) p 2n s 2 s 2 s + a2 s2 - a2 2 n+ 1 2. 4. 6. 8. 2 e at t p , p > -1 t n- 1 2 p 2s a 2 s + a2 2as 2 3 2 , n = 1, 2,3,K cos ( at ) t cos ( at ) sin ( at ) + at cos ( at ) cos ( at ) + at sin ( at ) cos ( at + b ) cosh ( at ) e at cos ( bt ) e at cosh ( bt ) f ( ct ) (s + a2 ) 10. 12. (s + a2 ) 2 13. 15. 17. 19. 21. 23. 25. 27. 29. 31. 33. 35. 37. (s + a ) s(s - a ) (s + a ) 2 2 2 2 2 2 2 2 2a 3 14. 16. 18. 20. 22. 24. 26. 28. 30. 32. 34. 36. (s + a ) s ( s + 3a ) (s + a ) 2 2 2 2 2 2 2 2 2as 2 s sin ( b ) + a cos ( b ) s2 + a2 a 2 s - a2 b s cos ( b ) - a sin ( b ) s2 + a2 s 2 s - a2 s-a (s - a) 2 + b2 -b n +1 2 (s - a) 2 + b2 - b2 b s-a (s - a) 2 (s - a) 2 n! (s - a) 1 æsö Fç ÷ c ècø e - cs e - cs L { g ( t + c )} uc ( t ) f ( t - c ) ect f ( t ) 1 f (t ) t e - cs s - cs e F (s) F ( s - c) ¥ s d (t - c ) Dirac Delta Function uc ( t ) g ( t ) t t n f ( t ) , n = 1, 2,3,K ( -1) T 0 n F ( n) ( s ) ò F ( u......

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...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......

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...STUDY IS IN PROCESS. Following component needed:- 1. Induction coil or magnetic coil 2. Capacitor (various value ceramic) 3. Resistance 4. Diode 1n4148 , 4007 5. Voltage regulator 6. Oscilloscope 7. Transistor 2N222A 8. Inductor(various value) 9. Led lamp(input minimum 5v) 10. Mobile battery 11. Other necessary component DETAIL DESCRIPTION OF THE WORK TO BE DONE- The output which is obtained with the help of electrodynamics induction is sufficient to charge the battery whose requirement is 5 volts and 80-120mA current rang e.g. mobile battery, Trimmer, laptop battery, table lamp bulb, wall clock, motor controls wirelessly, electric Toothbrush. Electrodynamics induction- is wireless transmission technique of electrical energy between TWO COIL that is tuned to resonate at the same frequency. The equipment do this is sometimes called RESONANT or RESONANCE TRANSFORM. Resonant transform work by making coil ring with an Oscillating current. This generates an oscillating magnetic field. Because the coil is highly resonate any energy placed in coil is dies away relatively Slowly over many cycles; BUT IF SECOND COIL is brought near it, the coil can pick up most of the energy before it lost, even if it is some distance away. The range is up to 2meters Working explained in the following point below with proper circuit diagram: Ac supply for source is primary part of our project which consists of oscillator circuit. For ac supply we use COLPITTS oscillator......

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...LIFE TABLES Life tables are statistical tools constructed for measuring the mortality of a population. They provide a description of the most important aspects of the state of human mortality. They try to show, for a person at a particular age, what the probability is that they die before their next birthday. TYPES OF LIFE TABLES Life tables have been classified into several types: 1. Conventional/period/cross-sectional life tables- these tables are based on age specific mortality rates for a specific period of time for the whole population. These rates are derived from deaths occurring over a selected period of time and the population at the mid-point of this period, a period of one year is mostly considered. 2. Generation/cohort life tables- these are based on a cohort throughout its life. We should remember a cohort is a group of individuals all born during the same time interval. The table shows the probability of death of people from a given cohort over the course of their lifetime. 3. Complete life tables- these are tables in which the mortality experience is considered in each single year throughout the life span. It is a more detailed method of measuring mortality since these tables are constructed on yearly basis. 4. Abridged life tables- are tables in which the measures are given for age groups and not for every single year of age. Its values are mostly in general terms and it is used to simplify the complete life tables whose construction is very laborious....

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...the Laplace transform. It is very effective in the study of initial value problem involving linear differential equation with constant coefficient. Laplace transform was first introduced by a French mathematician called Pierre Simon Marquis de Laplace about 1780’s. This method associated with the isolation of the original problem that is function ƒ(t) of a real variable and some function ƒ(s) of a complex variable so that the ordinary differential equation for the function ƒ(t) is transformed into an algebraic equation for ƒ(s) which in most cases can readily be solved. The solution of the original differential equation can be arrived at by obtaining the inverse transformation. The transformation and its inverse can be derived by consulting already prepared table of transform. This method is particularly useful in the solution of differential equation and has more application in various fields of technology e.g. electrical network, mechanical vibrations, structural problems, control systems. Meanwhile in this research work, I shall look into the Laplace transform, the properties of the Laplace transform and the use of this technique in solving delay differential equation will be looked into. 1.2 Statement of the Problem There are so many engineering and other related problems that can be expressed in the form of ordinary differential equations. But such problems cannot easily be solved using the elementary method of solution. In such cases, the Laplace transform......

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...University of Phoenix Material Website Review Find national, state, and local governmental health agency websites that address the issue you selected in your Week One assignment. Follow the example to document your findings in the table. Health issue: _Teen Pregnancy__________________________ |Website |Website Information Type |Type of Surveillance: |Data changes over time: |Identify two interventions | | | |survey, self report, |Has the incidence |that affect the issue. | | | |statistics, case report, |increased or decreased? | | | | |and so forth | | | |Example: http://www.cdc.gov/media/h1n1flu/index.htm#FAS |Table of contents, search, |Statistics and |Swine flu incidences have |Vaccine, hand washing, and | | |print, videos, surveillance |self-reporting |decreased. There are now |isolation masks ......

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...organization ensure that data quality is maintained and improved using the extract, transform and load (ETL) process? Be sure to support your position with specific examples. In order for a healthcare organization (HCO) to manage its data as an asset one can employee a data warehouse (DW). Data that is used in a DW must be extracted from operational systems. Operational data houses disparate data from multiple source systems that must be integrated prior to loading into DW, e.g. clinical, financial, registration, on-line transaction processing (OLTP), etc. (Anonymous, 2000). Since you shouldn’t directly work with operational data, a working copy of the data will be needed for manipulation without impacting other systems. Extraction, transform, load (ETL) systems will extract from operational systems and create a fixed-in-time snap shot of the data (Miron, 2011). ETL is one of the most challenging and risky steps in quality data management but one that should never be overlooked. The goal of the data extraction process is to bring all source data into a common, consistent format so it can be made ready for loading into the data warehouse. This stage is so crucial to the DW as this is where most of the data is cleaned, as different source systems can have variation in format, different source codes for the same kind of data, invalid characters, etc., it is these issues that make ETL necessary to transform data into useable, consistent and reliable form for loading......

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...Appendix B: Tables TABLE 1 CUMULATIVE PROBABILITIES FOR THE STANDARD NORMAL DISTRIBUTION Entries in the table give the area under the curve to the left of the z value. For example, for z = –.85, the cumulative probability is......

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...CONVERSION TABLES HOBBY CONVERSION CHART SCALE EQUIVALENTS COMPUTED TO THE NEAREST 1/64” 1” = 100’ 1” = 75’ 1” = 60’ 1” = 50’ 1” = 40’ 1” = 30’ 1” = 20’ 1” = 10’ 1/32” = 1’ 1/16” = 1’ 3/32” = 1’ 1/8” = 1’ 3/16” = 1’ 1/4” = 1’ 3/8” = 1’ 1/2” = 1’ 3/4” = 1’ 1” = 1’ 1:1200 1:900 1:700 1:600 1:500 1:400 1:250 1:125 1:400 1:200 1:125 1:100 1:75 1:48 1:32 1:24 1:16 1:12 ACTUAL SIZE 1” = 1’ 1:12 G 1:24 #1 1:32 2” 4” 6” 8” 10” 1’ 2’ 3’ 4’ 5’ 10’ 5/32” 11/32” 1/2” 21/32” 27/32” 1” 2” 3” 4” 5” 10” 5/64” 11/64” 1/4” 21/64” 27/64” 1/2” 1” 1-1/2” 2” 2-1/2” 5” 1/16” 1/8” 3/16” 1/4” 5/16” 3/8” 3/4” 1-1/8” 1-1/2” 1-7/8” 3-3/4” O 1:48 S 1:64 HO 1:87 TT 1:120 N 1:160 Z 1:250 3/64” 5/64” 1/8” 11/64” 13/64” 1/4” 1/2” 3/4” 1” 1-1/4” 2-1/2” 1/32” 1/16” 3/32” 1/8” 5/32” 3/16” 3/8” 9/16” 3/4” 15/16” 1-7/8” 1/64” 3/64” 1/16” 3/32” 7/64” 9/64” 9/32” 13/32” 35/64” 11/16” 1-3/8” 1/64” 1/32” 3/64” 1/16” 5/64” 3/32” 13/64” 19/64” 13/32” 1/2” 1” 1/64” 1/64” 1/32” 3/64” 1/16” 5/64” 5/32” 15/64” 19/64” 3/8” 3/4” 1/64” 1/64” 1/64” 1/32” 3/64” 1/16” 7/64” 5/32” 7/32” 17/64” 35/64” FRACTION TO DECIMAL CONVERSION CHART FRA. DEC. FRA. DEC. FRA. DEC. FRA. DEC. 1/64 1/32 3/64 1/16 5/64 3/32 7/64 1/8 9/64 5/32 11/64 3/16 13/64 7/32 15/64 1/4 .0156 .0312 .0468 .0625 .0781 .0937 .1093 .125 .1406 .1562 .1718 .1875 .2031 .2187 .2343 .2500 17/64 9/32 19/64 5/16 21/64 11/32 23/64 3/8 25/64 13/32 27/64 7/16 29/64 15/32 31/64 1/2 .2656 .2812 .2968 .3125 .3281 .3437 .3593 .3750 .3906 .4062...

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...Laplace Transform The Laplace transform can be used to solve diﬀerential equations. Besides being a diﬀerent and eﬃcient alternative to variation of parameters and undetermined coeﬃcients, the Laplace method is particularly advantageous for input terms that are piecewise-deﬁned, periodic or impulsive. The direct Laplace transform or the Laplace integral of a function f (t) deﬁned for 0 ≤ t < ∞ is the ordinary calculus integration problem ∞ 0 f (t)e−st dt, succinctly denoted L(f (t)) in science and engineering literature. The L–notation recognizes that integration always proceeds over t = 0 to t = ∞ and that the integral involves an integrator e−st dt instead of the usual dt. These minor diﬀerences distinguish Laplace integrals from the ordinary integrals found on the inside covers of calculus texts. 7.1 Introduction to the Laplace Method The foundation of Laplace theory is Lerch’s cancellation law ∞ −st dt 0 y(t)e = ∞ −st dt 0 f (t)e (1) L(y(t) = L(f (t)) implies or implies y(t) = f (t), y(t) = f (t). In diﬀerential equation applications, y(t) is the sought-after unknown while f (t) is an explicit expression taken from integral tables. Below, we illustrate Laplace’s method by solving the initial value problem y = −1, y(0) = 0. The method obtains a relation L(y(t)) = L(−t), whence Lerch’s cancellation law implies the solution is y(t) = −t. The Laplace method is advertised as a table lookup method, in which the solution y(t) to a diﬀerential......

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