Representing Periodic Functions by Fourier Series. (a n cos nt + b n sin nt) n=1


 Dwain Freeman
 1 years ago
 Views:
Transcription
1 Represening Periodic Funcions by Fourier Series 3. Inroducion In his Secion we show how a periodic funcion can be expressed as a series of sines and cosines. We begin by obaining some sandard inegrals involving sinusoids. We hen assume ha if f() is a periodic funcion, of period, hen he Fourier series expansion akes he form: f() = a + (a n cos n + b n sin n) Our main purpose here is o show how he consans in his expansion; a n,n=, 1,, 3... and b n,,, 3,... may be deermined for any given funcion f(). Prerequisies Before saring his Secion you should... Learning Oucomes Afer compleing his Secion you should be able o... 1 know wha a periodic funcion is be able o inegrae funcions involving sinusoids 3 have knowledge of inegraion by pars calculae Fourier coefficiens of a funcion of period calculae Fourier coefficiens of a funcion of general period
2 1. Inroducion We recall firs a simple rigonomeric ideniy: cos = 1+cos or equivalenly cos = cos (1) Equaion 1 can be inerpreed as a simple finie Fourier Series represenaion of he periodic funcion f() =cos which has period. We jus noe ha he Fourier Series represenaion conains a consan erm and a period erm. A more complicaed rigonomeric ideniy is sin 4 = cos + 1 cos 4 () 8 which again can be considered as a finie Fourier Series represenaion. (Do no worry if you are unfamiliar wih he resul ().) Noe ha he funcion f() =sin 4 (which has period ) is being wrien in erms of a consan funcion, a funcion of period or frequency 1 (he firs harmonic ) and a funcion of period or frequency (he second harmonic ). The reason for he consan erm in boh (1) and () is ha each of he funcions cos and sin 4 is nonnegaive and hence each mus have a posiive average value. Any sinusoid of he form cos n or sin n has, by symmery, zero average value as, herefore, would a Fourier Series conaining only such erms. A consan erm can herefore be expeced o arise in he Fourier Series of a funcion which has a nonzero average value.. Funcions of Period We now discuss how o represen periodic nonsinusoidal funcions f() ofperiod in erms of sinusoids, i.e. how o obain Fourier Series represenaions. As already discussed we expec such Fourier Series o conain harmonics of frequency n (n =1,, 3,...) and, if he periodic funcion has a nonzero average value, a consan erm. Thus we seek a Fourier Series represenaion of he general form f() = a + a 1 cos + a cos b 1 sin + b sin +... The reason for labelling he consan erm as a will be discussed laer. The ampliudes a 1,a,... b 1,b,... of he sinusoids are called Fourier coefficiens. Obaining he Fourier coefficiens for a given periodic funcion f() is our main ask and is referred o as Fourier Analysis. Before embarking on such an analysis i is insrucive o esablish, a leas qualiaively, he plausibiliy of approximaing a funcion by a few erms of is Fourier Series. HELM (VERSION 1: March 18, 4): Workbook Level 3.: Represening Periodic Funcions by Fourier Series
3 Consider he square wave of period one period of which is shown in Figure 1. 4 Wrie down Figure 1 i. he analyic descripion of his funcion, ii. wheher you expec he Fourier Series of his funcion o conain a consan erm, iii. any oher possible feaures of he Fourier Series ha you migh expec from he graph of he squarewave funcion. Your soluion (iii) Since he square wave is an even funcion (i.e. he graph in Figure 1 has symmery abou he y axis) hen is Fourier Series will conain cosine erms bu no sine erms because only he former are even funcions. (Well done if you spoed his a his early sage!) (ii) The Fourier Series will conain a consan erm (ofen referred o as he d.c. (direc curren) erm by engineers) since he square wave here is nonnegaive and canno herefore have a zero average value) <<, << 4 << f( +) = f() f() = (i) We have To be precise i is possible o show, and we will do so laer, ha he Fourier Series represenaion 3 HELM (VERSION 1: March 18, 4): Workbook Level 3.: Represening Periodic Funcions by Fourier Series
4 of his square wave is {cos 13 cos cos 5 17 cos i.e. he Fourier coefficiens are a =, a 1 = 8, a =, a 3 = 8 3, a 4 =, a 5 = 8 5,... Noe, as well as he presence of he consan erm and of he cosine (bu no sine) erms, ha only odd harmonics are presen i.e. sinusoids of period,,,,... or of frequency , 3, 5, 7,... imes he fundamenal frequency 1. We now show in Figure graphs (for <<only since he square wave and is Fourier Series are even) of (i) he square wave (ii) he firs wo erms of he Fourier Series (iii) he firs hree erms of he Fourier Series (iv) he firs four erms of he Fourier Series (v) he firs five erms of he Fourier Series (i) 4 (ii) (iii) + 8 cos + 8 (cos 1 cos 3 ) (iv) + 8 (v) (cos 1 3 cos cos 5 ) (cos 1 3 cos cos 5 1 cos 7 ) Figure We can clearly see from Figure ha as he number of erms is increased he graph of he Fourier Series gradually approaches ha of he original square wave  he ripples increase in number bu decrease in ampliude. (The behaviour near he disconinuiy, a =,isslighly more complicaed and i is possible o show ha however many erms are aken in he Fourier Series, some overshoo will always occur. This effec, which we do no discuss furher, is known as Gibbs Phenomenon.) HELM (VERSION 1: March 18, 4): Workbook Level 3.: Represening Periodic Funcions by Fourier Series 4
5 Orhogonaliy properies of sinusoids As saed earlier, a periodic funcion f() wih period has a Fourier Series represenaion f() = a + a 1 cos + a cos b 1 sin + b sin +... = a + (a n cos n + b n sin n) (3) To deermine he Fourier coefficiens a n, b n and he consan erm a use has o be made of cerain inegrals involving sinusoids, he inegrals being over a range α, α +, where α is any number. (We will normally choose α = ). Find sin n d and cos n d where n is an ineger Your soluion [ 1n cos n ] sin n d = [ 1 n = 1 n { cos n + cos n = n (4) cos n d = sin n ] In fac boh inegrals are zero for = n (5) As special cases, if n =he firs inegral is zero and he second inegral has value. N.B. Any inegraion range α, α +, would give hese same (zero) answers. These inegrals enable us o calculae he consan erm in he Fourier Series (3) as in he following guided exercise. Inegrae boh sides of (3) from o and use he above resuls. Hence obain an expression for a. 5 HELM (VERSION 1: March 18, 4): Workbook Level 3.: Represening Periodic Funcions by Fourier Series
6 Your soluion We ge for he lef hand side f()d (whose value clearly depends on he funcion f().) Inegraing he righ hand side erm by erm we ge 1 a d + { a n cos n d + b n sin n d = 1 [a ] + {+ (using he inegrals (4) and (5) shown above). Thus we ge f()d = 1 (a ) or a = 1 f()d (6) Key Poin The consan erm in a rigonomeric Fourier Series for a funcion of period is a = 1 f()d = average value of f() over 1period. This resul ies in wih our earlier discussion on he significance of he consan erm. Clearly a signal whose average value is zero will have no consan erm in is Fourier Series. The following square wave is an example. HELM (VERSION 1: March 18, 4): Workbook Level 3.: Represening Periodic Funcions by Fourier Series 6
7 1 f() 1 Figure 3 We now obain furher inegrals, known as orhogonaliy properies, which enable us o find he remaining Fourier coefficiens i.e. he ampliudes a n and b n (n =1,, 3,...)ofhe sinusoids. Recall, using a sandard rigonomeric ideniy ha sin n cos m = 1 {sin(n + m) + sin(n m) Hence evaluae where n and m are any inegers. sin n cos m d Your soluion We ge { sin n cos m d = 1 sin(n + m) d + sin(n m) d = 1 {+ = using he resuls (4) and (5) since n + m and n m are also inegers. This resul holds for any inerval α, α +. 7 HELM (VERSION 1: March 18, 4): Workbook Level 3.: Represening Periodic Funcions by Fourier Series
8 Key Poin For any inegers m, n, including he case m = n, wehave he orhogonaliy relaion sin n cos m d = We shall use his resul shorly bu need a few more inegrals firs. Consider nex cos n cos m d where m and n are inegers. Using anoher rigonomeric ideniy we have, for he case n m, cos n cos m d = 1 {cos(n + m) + cos(n m)d = 1 {+ = using he inegrals (4) and (5). For he case n = m we mus ge a nonzero answer since cos n is nonnegaive. In his case: cos n d = 1 = 1 (1 + cos n)d [ + 1 sin n n For he case n = m = we have ] = (provided n ) cos n cos m d = Proceeding in a similar way o he above, obain sin n sin m d for inegers m and n. Again consider separaely he cases n m, n = m and n = m =. Your soluion HELM (VERSION 1: March 18, 4): Workbook Level 3.: Represening Periodic Funcions by Fourier Series 8
9 Using he ideniy sin n sin m = 1 {cos(n m) cos(n + m) and inegraing he righ hand side erms, we ge, using (4) and (5) sin n sin m d = n, m inegers n m using he ideniy cos θ =1 sin θ wih θ = n gives for n = m sin n d = 1 (1 cos n)d = Of course, when n = m =, sin n sin m d =. We summarise hese resuls in he following key poin: For inegers n, m Key Poin sin n cos m d = cos n cos m d = sin n sin m d = All hese resuls hold for any inegraion range α, α +. n m n = m n = m = { n m, n = m = n = m 3. Calculaion of Fourier coefficiens Consider he Fourier Series for a funcion f() of period : f() = a + (a n cos n + b n sin n) (7) To obain he coefficiens a n (n =1,, 3,...), we muliply boh sides by cos m where m is some posiive ineger and inegrae boh sides from o : for he lef hand side we obain f() cos m d 9 HELM (VERSION 1: March 18, 4): Workbook Level 3.: Represening Periodic Funcions by Fourier Series
10 for he righ hand side we obain a cos m d + {a n cos n cos m d + b n sin n cos m d The firs inegral is zero using (5). Using he orhogonaliy relaions all he inegrals in he summaion give zero excep for he case n = m when, from he las key poin cos m d = Hence f() cos m d = a m from which he coefficien a m can be obained. Rewriing m as n we ge a n = 1 f() cos n d for n =1,, 3,... (8) Using (6), we see he formula also works for n =(bu we mus remember ha he consan erm is a.) From (8) a n = average value of f() cos n over one period. By muliplying (7) by sin m obain an expression for he Fourier Sine coefficiens b n ; n =1,, 3,... Your soluion HELM (VERSION 1: March 18, 4): Workbook Level 3.: Represening Periodic Funcions by Fourier Series 1
11 A similar calculaion o ha performed o find he a n gives f() sin m d = a sin m d { + a n cos n sin m d + b n sin n sin m d All erms on he righ hand side inegrae o zero excep for he case n = m where b m sin m d = b m relabelling m as n. b n = 1 (There is no Fourier coefficien b.) Clearly b n = average value of f() sin n over one period. f() sin n d n =1,, 3,... (9) Key Poin A funcion f() wih period has a Fourier Series f() = a + (a n cos n + b n sin n) The Fourier coefficiens are a n = 1 b n = 1 f() cos n d n =, 1,,... f() sin n d n =1,,... In he inegrals any convenien inegraion range α, α + may be used. 11 HELM (VERSION 1: March 18, 4): Workbook Level 3.: Represening Periodic Funcions by Fourier Series
12 4. Examples of Fourier Series We shall obain he Fourier Series of he halfrecified square wave shown. 1 f() period We have Figure 4 { 1 << f() = << f( +) = f() The calculaion of he Fourier coefficiens is merely sraighforward inegraion using he resuls already obained: in general. Hence, for our square wave a n = 1 f() cos n d a n = 1 (1) cos n d = 1 [ sin n n ] = provided n Bu a = 1 (1) d =1so he consan erm is a = 1. (The square wave akes on values 1 and over equal lengh inervals of so 1 value.) Similarly b n = 1 (1) sin n d = 1 [ ] cos n n is clearly he mean Some care is needed now! b n = 1 (1 cos n) n Bu cos n =+1 n =, 4, 6,..., However, cos n = 1 n =1, 3, 5,... b n = n =, 4, 6,... b n = 1 (1 ( 1)) = n n n =1, 3, 5,... i.e. b 1 =,b 3 = 3,b 5 = 5,... HELM (VERSION 1: March 18, 4): Workbook Level 3.: Represening Periodic Funcions by Fourier Series 1
13 Hence he required Fourier Series is f() = a + (a n cos n + b n sin n) in general f() = 1 + (sin + 13 sin ) sin in his case Noe ha he Fourier Series for his paricular form of he square wave conains a consan erm and odd harmonic sine erms. We already know why he consan erm arises (because of he nonzero mean value of he funcions) and will explain laer why he presence of any odd harmonic sine erms could have been prediced wihou inegraion. The Fourier series we have found can be wrien in summaion noaion in various ways: or, since n is odd, we may wrie 1 + (n odd) 1 sin n n n =k 1 k =1,,... and wrie he Fourier Series as 1 + k=1 1 sin(k 1) (k 1) Obain he Fourier Series of he square wave one period of which is shown: 4 Figure 5 13 HELM (VERSION 1: March 18, 4): Workbook Level 3.: Represening Periodic Funcions by Fourier Series
14 Your soluion a = 1 4d =4 a =is he consan erm as we would expec. Also [ sin n ] a n = 1 = 4 n ( sin 4 cos n d = 4 ( n ) sin ( n n )) = 8 ( n n sin ) n =1,, 3,... I follows from a knowledge of he sine funcion ha n =, 4, 6,... 8 a n = n =1, 5, 9,... n 8 n =3, 7, 11,... n Also b n = 1 4 sin n d = 4 [ cos n n ] = 4 n ( cos ( n We have, since he funcion is nonzero only for <<, ) cos ( n )) = Hence, he required Fourier Series is f() =+ 8 (cos 13 cos cos 5 17 ) cos which, like he previous square wave, conains a consan erm and odd harmonics, bu in his case odd harmonic cosine erms raher han sine. You may recall ha his paricular square was used earlier and we have already skeched he form of he Fourier Series for, 3, 4 and 5 erms. Clearly in finding he Fourier Series of square waves he inegraion is paricularly simple because HELM (VERSION 1: March 18, 4): Workbook Level 3.: Represening Periodic Funcions by Fourier Series 14
15 f() akes on piecewise consan values. For oher funcions, such as sawooh waves his will no be he case. Before we ackle such funcions however we shall generalise our formulae for he Fourier coefficiens a n,b n o he case of a periodic funcion of arbirary period, raher han confining ourselves o period. 5. Fourier Series for funcions of general period This is a sraighforward exension of he period case ha we have already discussed. Using x (insead of ) emporarily as he variable. We have seen ha a periodic funcion f(x) has a Fourier Series f(x) = a + (a n cos nx + b n sin nx) wih a n = 1 b n = 1 Suppose we now change he variable o where f(x) cos nx dx n =, 1,,... f(x) sin nx dx n =1,,... x = P Thus x = corresponds o = P and x = corresponds o = P. Hence regarded as a funcion of, wehave a funcion wih period P. Making he subsiuion x =, and hence dx = d, inhe expressions for a P P n and b n we obain a n = P b n = P P P P P ( ) n f() cos d P n =, 1,... ( ) n f() sin d P n =1,... These inegrals give he Fourier coefficiens for a funcion of period P whose Fourier Series is f() = a + [ ( ) ( )] n n a n cos + b n sin P P Various oher noaions are commonly used in his case e.g. i is someimes convenien o wrie he period P = l. (This is paricularly useful when Fourier Series arise in he soluion of parial differenial equaions.) Anoher alernaive is o use he angualr frequency ω and pu P =/ω. 15 HELM (VERSION 1: March 18, 4): Workbook Level 3.: Represening Periodic Funcions by Fourier Series
16 Wrie down he form of he Fourier Series and expressions for he coefficiens if (i) P =l (ii) P = ω. Your soluion d ) l f() cos(nω) d ( n f() cos l l ω ω [a n cos(nω)+b n sin(nω)] wih a n = ω wih a n = 1 l )] l ( n + b n sin ) l ( n a n cos [ (ii) f() = a + and similarly for b n. (i) f() = a + and similarly for b n. You should noe ha, as usual, any convenien inegraion range of lengh P (or l or ω be used in evaluaing a n and b n. ) can Example Find he Fourier Series of he funcion shown in Figure 6, viz a saw ooh wave wih alernaive porions removed f() Figure 6 Here he period P =l =4sol =. The Fourier Series will have he form f() = a + ( ) ( ) n n a n cos + b n sin The coefficiens a n are given by where Hence a n = 1 cos ( ) n d a n = 1 ( ) n f() cos d { << f() = << HELM (VERSION 1: March 18, 4): Workbook Level 3.: Represening Periodic Funcions by Fourier Series 16
17 The inegraion is readily performed using inegraion by pars: ( ) [ ( )] n n cos d = n sin ( ) n sin d n [ ( )] 4 n = cos n n Hence, since a n = 1 n cos( )d a n = The consan erm is a where Similarly where = ( ) n sin d = The second inegral gives zero. Hence b n = 1 = 4 {cos n 1. n n =, 4, 6,... 4 n n =1, 3, 5,... a = 1 [ n cos d =1. ( ) n sin d ( )] n + n b n = cos n n n =, 4, 6,... n + n n =1, 3, 5,... ( ) n cos d. Hence, using all hese resuls for he Fourier coefficiens, he required Fourier Series is f() = 1 4 ( ) (cos + 1 ( ) 3 9 cos + 1 ( ) ) 5 5 cos ( ( ) sin 1 ( ) sin + 1 ( ) ) 3 3 sin... Noice ha because he Fourier coefficiens depend on 1 (raher han 1 as was he case for n n he square wave) he sinusoidal componens in he Fourier Series have quie rapidly decreasing ampliudes. We would herefore expec o be able o approximae he original sawooh funcion using only a quie small number of erms in he series. 17 HELM (VERSION 1: March 18, 4): Workbook Level 3.: Represening Periodic Funcions by Fourier Series
18 Obain he Fourier Series of he funcion f() = 1 <<1 f( +) = f() f() 1 1 Figure 7 Firs wrie ou he form of he Fourier Series in his case Your soluion Since P =l =and since he funcion has a nonzero average value, he form of he Fourier Series is a + [a n (cos n)+b n sin(n)] Now wrie ou inegral expressions for a n and b n. Will here be a consan erm in he Fourier Series? Your soluion Because he funcion is nonnegaive here will be a consan erm. Since P =l =sol =1 we have a n = b n = cos(n)d n =, 1,,... sin(n)d n =1,,... The consan erm will be a where a = 1 1 d. HELM (VERSION 1: March 18, 4): Workbook Level 3.: Represening Periodic Funcions by Fourier Series 18
19 Now evaluae he inegrals. Try o spo he value of he inegral for b n so as o avoid inegraion. Noe ha he inegrand is an even funcions for a n and an odd funcon for b n. Your soluion The inegral for b n is zero for all n because he inegrand is an odd funcion of. (We shall cover his poin more fully in he nex uni.) Since he inegrand is even in he inegrals for a n we can wrie a n = The consan erm will be ao where 1 cos n d n =, 1,,... a = 1 d = 3. For n =1,, 3,... we mus inegrae by pars (wice) { [ ]1 a n = n sin(n) n = 4 n { [ n cos(n) ] n sin(n)d 1 cos(n)d. The inegral gives zero so a n = 4 cos n. n Now wrie ou he final form of he Fourier Series. We have f() = cos n n { 4 cos() 1 9 cos(3)+... cos(n) = cos() HELM (VERSION 1: March 18, 4): Workbook Level 3.: Represening Periodic Funcions by Fourier Series
20 Exercises For each of he following periodic signals skech he given funcion over a few periods find he rigonomeric Fourier coefficiens wrie ou he firs few erms of he Fourier Series. 1 <</ 1. f() = f( +) =f() / << square wave. f() = 1 <<1 f( +)=f() 1 T/ << 3. f() = f( + T )=f() square wave 1 <<T/ << 4. f() = f( +) =f() <<, T/ << 5. f() = A sin half wave recifier T, <<T/ HELM (VERSION 1: March 18, 4): Workbook Level 3.: Represening Periodic Funcions by Fourier Series
21 Answers { cos cos 3 3 cos { sin + sin + sin sin sin cos 4 { sin ω+ 13 sin 3 ω+ 15 sin 5ω+... { cos cos 3 cos where ω =/T { cos + cos + {( 4 cos ) sin ( sin ) sin 3 4 sin A + A sin ω A { cos ω (1)(3) + cos 4ω (3)(5) HELM (VERSION 1: March 18, 4): Workbook Level 3.: Represening Periodic Funcions by Fourier Series
Chapter 7. Response of FirstOrder RL and RC Circuits
Chaper 7. esponse of FirsOrder L and C Circuis 7.1. The Naural esponse of an L Circui 7.2. The Naural esponse of an C Circui 7.3. The ep esponse of L and C Circuis 7.4. A General oluion for ep and Naural
More information17 Laplace transform. Solving linear ODE with piecewise continuous right hand sides
7 Laplace ransform. Solving linear ODE wih piecewise coninuous righ hand sides In his lecure I will show how o apply he Laplace ransform o he ODE Ly = f wih piecewise coninuous f. Definiion. A funcion
More information9. Capacitor and Resistor Circuits
ElecronicsLab9.nb 1 9. Capacior and Resisor Circuis Inroducion hus far we have consider resisors in various combinaions wih a power supply or baery which provide a consan volage source or direc curren
More informationDifferential Equations and Linear Superposition
Differenial Equaions and Linear Superposiion Basic Idea: Provide soluion in closed form Like Inegraion, no general soluions in closed form Order of equaion: highes derivaive in equaion e.g. dy d dy 2 y
More informationThe Transport Equation
The Transpor Equaion Consider a fluid, flowing wih velociy, V, in a hin sraigh ube whose cross secion will be denoed by A. Suppose he fluid conains a conaminan whose concenraion a posiion a ime will be
More informationMathematics in Pharmacokinetics What and Why (A second attempt to make it clearer)
Mahemaics in Pharmacokineics Wha and Why (A second aemp o make i clearer) We have used equaions for concenraion () as a funcion of ime (). We will coninue o use hese equaions since he plasma concenraions
More information4.2 Trigonometric Functions; The Unit Circle
4. Trigonomeric Funcions; The Uni Circle Secion 4. Noes Page A uni circle is a circle cenered a he origin wih a radius of. Is equaion is as shown in he drawing below. Here he leer represens an angle measure.
More informationCHARGE AND DISCHARGE OF A CAPACITOR
REFERENCES RC Circuis: Elecrical Insrumens: Mos Inroducory Physics exs (e.g. A. Halliday and Resnick, Physics ; M. Sernheim and J. Kane, General Physics.) This Laboraory Manual: Commonly Used Insrumens:
More informationcooking trajectory boiling water B (t) microwave 0 2 4 6 8 101214161820 time t (mins)
Alligaor egg wih calculus We have a large alligaor egg jus ou of he fridge (1 ) which we need o hea o 9. Now here are wo accepable mehods for heaing alligaor eggs, one is o immerse hem in boiling waer
More informationAP Calculus BC 2010 Scoring Guidelines
AP Calculus BC Scoring Guidelines The College Board The College Board is a noforprofi membership associaion whose mission is o connec sudens o college success and opporuniy. Founded in, he College Board
More informationPROFIT TEST MODELLING IN LIFE ASSURANCE USING SPREADSHEETS PART ONE
Profi Tes Modelling in Life Assurance Using Spreadshees PROFIT TEST MODELLING IN LIFE ASSURANCE USING SPREADSHEETS PART ONE Erik Alm Peer Millingon 2004 Profi Tes Modelling in Life Assurance Using Spreadshees
More informationInductance and Transient Circuits
Chaper H Inducance and Transien Circuis Blinn College  Physics 2426  Terry Honan As a consequence of Faraday's law a changing curren hrough one coil induces an EMF in anoher coil; his is known as muual
More informationAP Calculus AB 2013 Scoring Guidelines
AP Calculus AB 1 Scoring Guidelines The College Board The College Board is a missiondriven noforprofi organizaion ha connecs sudens o college success and opporuniy. Founded in 19, he College Board was
More informationMTH6121 Introduction to Mathematical Finance Lesson 5
26 MTH6121 Inroducion o Mahemaical Finance Lesson 5 Conens 2.3 Brownian moion wih drif........................... 27 2.4 Geomeric Brownian moion........................... 28 2.5 Convergence of random
More informationCircuit Types. () i( t) ( )
Circui Types DC Circuis Idenifying feaures: o Consan inpus: he volages of independen volage sources and currens of independen curren sources are all consan. o The circui does no conain any swiches. All
More informationFullwave rectification, bulk capacitor calculations Chris Basso January 2009
ullwave recificaion, bulk capacior calculaions Chris Basso January 9 This shor paper shows how o calculae he bulk capacior value based on ripple specificaions and evaluae he rms curren ha crosses i. oal
More informationA Mathematical Description of MOSFET Behavior
10/19/004 A Mahemaical Descripion of MOSFET Behavior.doc 1/8 A Mahemaical Descripion of MOSFET Behavior Q: We ve learned an awful lo abou enhancemen MOSFETs, bu we sill have ye o esablished a mahemaical
More informationLectures # 5 and 6: The Prime Number Theorem.
Lecures # 5 and 6: The Prime Number Theorem Noah Snyder July 8, 22 Riemann s Argumen Riemann used his analyically coninued ζfuncion o skech an argumen which would give an acual formula for π( and sugges
More informationChapter 2 Problems. s = d t up. = 40km / hr d t down. 60km / hr. d t total. + t down. = t up. = 40km / hr + d. 60km / hr + 40km / hr
Chaper 2 Problems 2.2 A car ravels up a hill a a consan speed of 40km/h and reurns down he hill a a consan speed of 60 km/h. Calculae he average speed for he rip. This problem is a bi more suble han i
More informationDuration and Convexity ( ) 20 = Bond B has a maturity of 5 years and also has a required rate of return of 10%. Its price is $613.
Graduae School of Business Adminisraion Universiy of Virginia UVAF38 Duraion and Convexiy he price of a bond is a funcion of he promised paymens and he marke required rae of reurn. Since he promised
More informationFourier Series & The Fourier Transform
Fourier Series & The Fourier Transform Wha is he Fourier Transform? Fourier Cosine Series for even funcions and Sine Series for odd funcions The coninuous limi: he Fourier ransform (and is inverse) The
More information5.8 Resonance 231. The study of vibrating mechanical systems ends here with the theory of pure and practical resonance.
5.8 Resonance 231 5.8 Resonance The sudy of vibraing mechanical sysems ends here wih he heory of pure and pracical resonance. Pure Resonance The noion of pure resonance in he differenial equaion (1) ()
More informationRC (ResistorCapacitor) Circuits. AP Physics C
(ResisorCapacior Circuis AP Physics C Circui Iniial Condiions An circui is one where you have a capacior and resisor in he same circui. Suppose we have he following circui: Iniially, he capacior is UNCHARGED
More informationDifferential Equations. Solving for Impulse Response. Linear systems are often described using differential equations.
Differenial Equaions Linear sysems are ofen described using differenial equaions. For example: d 2 y d 2 + 5dy + 6y f() d where f() is he inpu o he sysem and y() is he oupu. We know how o solve for y given
More information3 RungeKutta Methods
3 RungeKua Mehods In conras o he mulisep mehods of he previous secion, RungeKua mehods are singlesep mehods however, muliple sages per sep. They are moivaed by he dependence of he Taylor mehods on he
More informationAppendix A: Area. 1 Find the radius of a circle that has circumference 12 inches.
Appendi A: Area workedou s o OddNumbered Eercises Do no read hese workedou s before aemping o do he eercises ourself. Oherwise ou ma mimic he echniques shown here wihou undersanding he ideas. Bes wa
More informationThe Torsion of Thin, Open Sections
EM 424: Torsion of hin secions 26 The Torsion of Thin, Open Secions The resuls we obained for he orsion of a hin recangle can also be used be used, wih some qualificaions, for oher hin open secions such
More informationChapter 2: Principles of steadystate converter analysis
Chaper 2 Principles of SeadySae Converer Analysis 2.1. Inroducion 2.2. Inducor volsecond balance, capacior charge balance, and he small ripple approximaion 2.3. Boos converer example 2.4. Cuk converer
More informationA Curriculum Module for AP Calculus BC Curriculum Module
Vecors: A Curriculum Module for AP Calculus BC 00 Curriculum Module The College Board The College Board is a noforprofi membership associaion whose mission is o connec sudens o college success and opporuniy.
More informationFourier Series and Fourier Transform
Fourier Series and Fourier ransform Complex exponenials Complex version of Fourier Series ime Shifing, Magniude, Phase Fourier ransform Copyrigh 2007 by M.H. Perro All righs reserved. 6.082 Spring 2007
More informationRandom Walk in 1D. 3 possible paths x vs n. 5 For our random walk, we assume the probabilities p,q do not depend on time (n)  stationary
Random Walk in D Random walks appear in many cones: diffusion is a random walk process undersanding buffering, waiing imes, queuing more generally he heory of sochasic processes gambling choosing he bes
More informationChapter 2 Problems. 3600s = 25m / s d = s t = 25m / s 0.5s = 12.5m. Δx = x(4) x(0) =12m 0m =12m
Chaper 2 Problems 2.1 During a hard sneeze, your eyes migh shu for 0.5s. If you are driving a car a 90km/h during such a sneeze, how far does he car move during ha ime s = 90km 1000m h 1km 1h 3600s = 25m
More information1 HALFLIFE EQUATIONS
R.L. Hanna Page HALFLIFE EQUATIONS The basic equaion ; he saring poin ; : wrien for ime: x / where fracion of original maerial and / number of halflives, and / log / o calculae he age (# ears): age (halflife)
More informationSignal Rectification
9/3/25 Signal Recificaion.doc / Signal Recificaion n imporan applicaion of juncion diodes is signal recificaion. here are wo ypes of signal recifiers, halfwae and fullwae. Le s firs consider he ideal
More informationCapacitors and inductors
Capaciors and inducors We coninue wih our analysis of linear circuis by inroducing wo new passive and linear elemens: he capacior and he inducor. All he mehods developed so far for he analysis of linear
More informationEquation for a line. Synthetic Impulse Response 0.5 0.5. 0 5 10 15 20 25 Time (sec) x(t) m
Fundamenals of Signals Overview Definiion Examples Energy and power Signal ransformaions Periodic signals Symmery Exponenial & sinusoidal signals Basis funcions Equaion for a line x() m x() =m( ) You will
More information1. y 5y + 6y = 2e t Solution: Characteristic equation is r 2 5r +6 = 0, therefore r 1 = 2, r 2 = 3, and y 1 (t) = e 2t,
Homework6 Soluions.7 In Problem hrough 4 use he mehod of variaion of parameers o find a paricular soluion of he given differenial equaion. Then check your answer by using he mehod of undeermined coeffiens..
More informationPermutations and Combinations
Permuaions and Combinaions Combinaorics Copyrigh Sandards 006, Tes  ANSWERS Barry Mabillard. 0 www.mah0s.com 1. Deermine he middle erm in he expansion of ( a b) To ge he kvalue for he middle erm, divide
More information4 Convolution. Recommended Problems. x2[n] 1 2[n]
4 Convoluion Recommended Problems P4.1 This problem is a simple example of he use of superposiion. Suppose ha a discreeime linear sysem has oupus y[n] for he given inpus x[n] as shown in Figure P4.11.
More informationPresent Value Methodology
Presen Value Mehodology Econ 422 Invesmen, Capial & Finance Universiy of Washingon Eric Zivo Las updaed: April 11, 2010 Presen Value Concep Wealh in Fisher Model: W = Y 0 + Y 1 /(1+r) The consumer/producer
More informationModule 4. Singlephase AC circuits. Version 2 EE IIT, Kharagpur
Module 4 Singlephase A circuis ersion EE T, Kharagpur esson 5 Soluion of urren in A Series and Parallel ircuis ersion EE T, Kharagpur n he las lesson, wo poins were described:. How o solve for he impedance,
More informationRC, RL and RLC circuits
Name Dae Time o Complee h m Parner Course/ Secion / Grade RC, RL and RLC circuis Inroducion In his experimen we will invesigae he behavior of circuis conaining combinaions of resisors, capaciors, and inducors.
More informationSection 5.1 The Unit Circle
Secion 5.1 The Uni Circle The Uni Circle EXAMPLE: Show ha he poin, ) is on he uni circle. Soluion: We need o show ha his poin saisfies he equaion of he uni circle, ha is, x +y 1. Since ) ) + 9 + 9 1 P
More informationRC Circuit and Time Constant
ircui and Time onsan 8M Objec: Apparaus: To invesigae he volages across he resisor and capacior in a resisorcapacior circui ( circui) as he capacior charges and discharges. We also wish o deermine he
More informationSecond Order Linear Differential Equations
Second Order Linear Differenial Equaions Second order linear equaions wih consan coefficiens; Fundamenal soluions; Wronskian; Exisence and Uniqueness of soluions; he characerisic equaion; soluions of homogeneous
More informationChapter 4: Exponential and Logarithmic Functions
Chaper 4: Eponenial and Logarihmic Funcions Secion 4.1 Eponenial Funcions... 15 Secion 4. Graphs of Eponenial Funcions... 3 Secion 4.3 Logarihmic Funcions... 4 Secion 4.4 Logarihmic Properies... 53 Secion
More information2.5 Life tables, force of mortality and standard life insurance products
Soluions 5 BS4a Acuarial Science Oford MT 212 33 2.5 Life ables, force of moraliy and sandard life insurance producs 1. (i) n m q represens he probabiliy of deah of a life currenly aged beween ages + n
More informationSolution of a differential equation of the second order by the method of NIGAM
Tire : Résoluion d'une équaion différenielle du second[...] Dae : 16/02/2011 Page : 1/6 Soluion of a differenial equaion of he second order by he mehod of NIGAM Summarized: We presen in his documen, a
More informationChapter 2 Kinematics in One Dimension
Chaper Kinemaics in One Dimension Chaper DESCRIBING MOTION:KINEMATICS IN ONE DIMENSION PREVIEW Kinemaics is he sudy of how hings moe how far (disance and displacemen), how fas (speed and elociy), and how
More informationName: Algebra II Review for Quiz #13 Exponential and Logarithmic Functions including Modeling
Name: Algebra II Review for Quiz #13 Exponenial and Logarihmic Funcions including Modeling TOPICS: Solving Exponenial Equaions (The Mehod of Common Bases) Solving Exponenial Equaions (Using Logarihms)
More informationOn the degrees of irreducible factors of higher order Bernoulli polynomials
ACTA ARITHMETICA LXII.4 (1992 On he degrees of irreducible facors of higher order Bernoulli polynomials by Arnold Adelberg (Grinnell, Ia. 1. Inroducion. In his paper, we generalize he curren resuls on
More informationKeldysh Formalism: Nonequilibrium Green s Function
Keldysh Formalism: Nonequilibrium Green s Funcion Jinshan Wu Deparmen of Physics & Asronomy, Universiy of Briish Columbia, Vancouver, B.C. Canada, V6T 1Z1 (Daed: November 28, 2005) A review of Nonequilibrium
More informationWhy Did the Demand for Cash Decrease Recently in Korea?
Why Did he Demand for Cash Decrease Recenly in Korea? Byoung Hark Yoo Bank of Korea 26. 5 Absrac We explores why cash demand have decreased recenly in Korea. The raio of cash o consumpion fell o 4.7% in
More informationThe option pricing framework
Chaper 2 The opion pricing framework The opion markes based on swap raes or he LIBOR have become he larges fixed income markes, and caps (floors) and swapions are he mos imporan derivaives wihin hese markes.
More informationSection 7.1 Angles and Their Measure
Secion 7.1 Angles and Their Measure Greek Leers Commonly Used in Trigonomery Quadran II Quadran III Quadran I Quadran IV α = alpha β = bea θ = hea δ = dela ω = omega γ = gamma DEGREES The angle formed
More informationAP Calculus AB 2010 Scoring Guidelines
AP Calculus AB 1 Scoring Guidelines The College Board The College Board is a noforprofi membership associaion whose mission is o connec sudens o college success and opporuniy. Founded in 1, he College
More informationRevisions to Nonfarm Payroll Employment: 1964 to 2011
Revisions o Nonfarm Payroll Employmen: 1964 o 2011 Tom Sark December 2011 Summary Over recen monhs, he Bureau of Labor Saisics (BLS) has revised upward is iniial esimaes of he monhly change in nonfarm
More informationRotational Inertia of a Point Mass
Roaional Ineria of a Poin Mass Saddleback College Physics Deparmen, adaped from PASCO Scienific PURPOSE The purpose of his experimen is o find he roaional ineria of a poin experimenally and o verify ha
More informationEntropy: From the Boltzmann equation to the Maxwell Boltzmann distribution
Enropy: From he Bolzmann equaion o he Maxwell Bolzmann disribuion A formula o relae enropy o probabiliy Ofen i is a lo more useful o hink abou enropy in erms of he probabiliy wih which differen saes are
More informationTwo Compartment Body Model and V d Terms by Jeff Stark
Two Comparmen Body Model and V d Terms by Jeff Sark In a onecomparmen model, we make wo imporan assumpions: (1) Linear pharmacokineics  By his, we mean ha eliminaion is firs order and ha pharmacokineic
More informationGraduate Macro Theory II: Notes on Neoclassical Growth Model
Graduae Macro Theory II: Noes on Neoclassical Growh Model Eric Sims Universiy of Nore Dame Spring 2011 1 Basic Neoclassical Growh Model The economy is populaed by a large number of infiniely lived agens.
More informationHedging with Forwards and Futures
Hedging wih orwards and uures Hedging in mos cases is sraighforward. You plan o buy 10,000 barrels of oil in six monhs and you wish o eliminae he price risk. If you ake he buyside of a forward/fuures
More informationA Note on Using the Svensson procedure to estimate the risk free rate in corporate valuation
A Noe on Using he Svensson procedure o esimae he risk free rae in corporae valuaion By Sven Arnold, Alexander Lahmann and Bernhard Schwezler Ocober 2011 1. The risk free ineres rae in corporae valuaion
More informationCointegration: The Engle and Granger approach
Coinegraion: The Engle and Granger approach Inroducion Generally one would find mos of he economic variables o be nonsaionary I(1) variables. Hence, any equilibrium heories ha involve hese variables require
More informationVariance Swap. by Fabrice Douglas Rouah
Variance wap by Fabrice Douglas Rouah www.frouah.com www.volopa.com In his Noe we presen a deailed derivaion of he fair value of variance ha is used in pricing a variance swap. We describe he approach
More informationSignal Processing and Linear Systems I
Sanford Universiy Summer 214215 Signal Processing and Linear Sysems I Lecure 5: Time Domain Analysis of Coninuous Time Sysems June 3, 215 EE12A:Signal Processing and Linear Sysems I; Summer 1415, Gibbons
More informationChapter 8: Regression with Lagged Explanatory Variables
Chaper 8: Regression wih Lagged Explanaory Variables Time series daa: Y for =1,..,T End goal: Regression model relaing a dependen variable o explanaory variables. Wih ime series new issues arise: 1. One
More informationChabot College Physics Lab RC Circuits Scott Hildreth
Chabo College Physics Lab Circuis Sco Hildreh Goals: Coninue o advance your undersanding of circuis, measuring resisances, currens, and volages across muliple componens. Exend your skills in making breadboard
More informationTHE PRESSURE DERIVATIVE
Tom Aage Jelmer NTNU Dearmen of Peroleum Engineering and Alied Geohysics THE PRESSURE DERIVATIVE The ressure derivaive has imoran diagnosic roeries. I is also imoran for making ye curve analysis more reliable.
More informationVoltage level shifting
rek Applicaion Noe Number 1 r. Maciej A. Noras Absrac A brief descripion of volage shifing circuis. 1 Inroducion In applicaions requiring a unipolar A volage signal, he signal may be delivered from a bipolar
More informationSteps for D.C Analysis of MOSFET Circuits
10/22/2004 Seps for DC Analysis of MOSFET Circuis.doc 1/7 Seps for D.C Analysis of MOSFET Circuis To analyze MOSFET circui wih D.C. sources, we mus follow hese five seps: 1. ASSUME an operaing mode 2.
More informationAcceleration Lab Teacher s Guide
Acceleraion Lab Teacher s Guide Objecives:. Use graphs of disance vs. ime and velociy vs. ime o find acceleraion of a oy car.. Observe he relaionship beween he angle of an inclined plane and he acceleraion
More informationPricing FixedIncome Derivaives wih he ForwardRisk Adjused Measure Jesper Lund Deparmen of Finance he Aarhus School of Business DK8 Aarhus V, Denmark Email: jel@hha.dk Homepage: www.hha.dk/~jel/ Firs
More information11/6/2013. Chapter 14: Dynamic ADAS. Introduction. Introduction. Keeping track of time. The model s elements
Inroducion Chaper 14: Dynamic DS dynamic model of aggregae and aggregae supply gives us more insigh ino how he economy works in he shor run. I is a simplified version of a DSGE model, used in cuingedge
More informationEconomics Honors Exam 2008 Solutions Question 5
Economics Honors Exam 2008 Soluions Quesion 5 (a) (2 poins) Oupu can be decomposed as Y = C + I + G. And we can solve for i by subsiuing in equaions given in he quesion, Y = C + I + G = c 0 + c Y D + I
More informationMOTION ALONG A STRAIGHT LINE
Chaper 2: MOTION ALONG A STRAIGHT LINE 1 A paricle moes along he ais from i o f Of he following alues of he iniial and final coordinaes, which resuls in he displacemen wih he larges magniude? A i =4m,
More informationPart 1: White Noise and Moving Average Models
Chaper 3: Forecasing From Time Series Models Par 1: Whie Noise and Moving Average Models Saionariy In his chaper, we sudy models for saionary ime series. A ime series is saionary if is underlying saisical
More informationNewton s Laws of Motion
Newon s Laws of Moion MS4414 Theoreical Mechanics Firs Law velociy. In he absence of exernal forces, a body moves in a sraigh line wih consan F = 0 = v = cons. Khan Academy Newon I. Second Law body. The
More informationA Probability Density Function for Google s stocks
A Probabiliy Densiy Funcion for Google s socks V.Dorobanu Physics Deparmen, Poliehnica Universiy of Timisoara, Romania Absrac. I is an approach o inroduce he Fokker Planck equaion as an ineresing naural
More informationMultiprocessor SystemsonChips
Par of: Muliprocessor SysemsonChips Edied by: Ahmed Amine Jerraya and Wayne Wolf Morgan Kaufmann Publishers, 2005 2 Modeling Shared Resources Conex swiching implies overhead. On a processing elemen,
More informationMorningstar Investor Return
Morningsar Invesor Reurn Morningsar Mehodology Paper Augus 31, 2010 2010 Morningsar, Inc. All righs reserved. The informaion in his documen is he propery of Morningsar, Inc. Reproducion or ranscripion
More informationAP Calculus AB 2007 Scoring Guidelines
AP Calculus AB 7 Scoring Guidelines The College Board: Connecing Sudens o College Success The College Board is a noforprofi membership associaion whose mission is o connec sudens o college success and
More informationTorsion of Closed Thin Wall (CTW) Sections
9 orsion of losed hin Wall (W) Secions 9 1 Lecure 9: ORSION OF LOSED HIN WALL (W) SEIONS ALE OF ONENS Page 9.1 Inroducion..................... 9 3 9.2 losed W Secions.................. 9 3 9.3 Examples......................
More informationUsefulness of the Forward Curve in Forecasting Oil Prices
Usefulness of he Forward Curve in Forecasing Oil Prices Akira Yanagisawa Leader Energy Demand, Supply and Forecas Analysis Group The Energy Daa and Modelling Cener Summary When people analyse oil prices,
More informationAnalogue and Digital Signal Processing. First Term Third Year CS Engineering By Dr Mukhtiar Ali Unar
Analogue and Digial Signal Processing Firs Term Third Year CS Engineering By Dr Mukhiar Ali Unar Recommended Books Haykin S. and Van Veen B.; Signals and Sysems, John Wiley& Sons Inc. ISBN: 073807 Ifeachor
More information6.5. Modelling Exercises. Introduction. Prerequisites. Learning Outcomes
Modelling Exercises 6.5 Inroducion This Secion provides examples and asks employing exponenial funcions and logarihmic funcions, such as growh and decay models which are imporan hroughou science and engineering.
More informationNotes for Signals and Systems Version 1.0
Noes for Signals and Sysems Version.0 Wilson J. Rugh These noes were developed for use in 50.4, Signals and Sysems, Deparmen of Elecrical and Compuer Engineering, Johns Hopkins Universiy, over he period
More informationMotion Along a Straight Line
Moion Along a Sraigh Line On Sepember 6, 993, Dave Munday, a diesel mechanic by rade, wen over he Canadian edge of Niagara Falls for he second ime, freely falling 48 m o he waer (and rocks) below. On his
More informationWorking Paper On the timing option in a futures contract. SSE/EFI Working Paper Series in Economics and Finance, No. 619
econsor www.econsor.eu Der OpenAccessPublikaionsserver der ZBW LeibnizInformaionszenrum Wirschaf The Open Access Publicaion Server of he ZBW Leibniz Informaion Cenre for Economics Biagini, Francesca;
More informationAnswer, Key Homework 2 David McIntyre 45123 Mar 25, 2004 1
Answer, Key Homework 2 Daid McInyre 4123 Mar 2, 2004 1 This prinou should hae 1 quesions. Muliplechoice quesions may coninue on he ne column or page find all choices before making your selecion. The
More informationANALYTIC PROOF OF THE PRIME NUMBER THEOREM
ANALYTIC PROOF OF THE PRIME NUMBER THEOREM RYAN SMITH, YUAN TIAN Conens Arihmeical Funcions Equivalen Forms of he Prime Number Theorem 3 3 The Relaionshi Beween Two Asymoic Relaions 6 4 Dirichle Series
More informationarxiv:math/0111328v1 [math.co] 30 Nov 2001
arxiv:mah/038v [mahco 30 Nov 00 EVALUATIONS OF SOME DETERMINANTS OF MATRICES RELATED TO THE PASCAL TRIANGLE C Kraenhaler Insiu für Mahemaik der Universiä Wien, Srudlhofgasse 4, A090 Wien, Ausria email:
More informationSuggested Reading. Signals and Systems 42
4 Convoluion In Lecure 3 we inroduced and defined a variey of sysem properies o which we will make frequen reference hroughou he course. Of paricular imporance are he properies of lineariy and ime invariance,
More informationEstablishing Prefabricated Wood IJoist Composite EI
July 008 Esablishing Prefabricaed Wood I Composie EI INTRODUCTION Composie (glued/nailed) floors are common in boh residenial and commercial consrucion, and have been successfully designed by Prefabricaed
More informationTechnical Appendix to Risk, Return, and Dividends
Technical Appendix o Risk, Reurn, and Dividends Andrew Ang Columbia Universiy and NBER Jun Liu UC San Diego This Version: 28 Augus, 2006 Columbia Business School, 3022 Broadway 805 Uris, New York NY 10027,
More informationFollow links Class Use and other Permissions. For more information, send to:
COPYRIGHT NOTICE: David A. Kendrick, P. Ruben Mercado, and Hans M. Amman: Compuaional Economics is published by Princeon Universiy Press and copyrighed, 2006, by Princeon Universiy Press. All righs reserved.
More informationPrincipal components of stock market dynamics. Methodology and applications in brief (to be updated ) Andrei Bouzaev, bouzaev@ya.
Principal componens of sock marke dynamics Mehodology and applicaions in brief o be updaed Andrei Bouzaev, bouzaev@ya.ru Why principal componens are needed Objecives undersand he evidence of more han one
More informationnonlocal conditions.
ISSN 17493889 prin, 17493897 online Inernaional Journal of Nonlinear Science Vol.11211 No.1,pp.39 Boundary Value Problem for Some Fracional Inegrodifferenial Equaions wih Nonlocal Condiions Mohammed
More informationReturn Calculation of U.S. Treasury Constant Maturity Indices
Reurn Calculaion of US Treasur Consan Mauri Indices Morningsar Mehodolog Paper Sepeber 30 008 008 Morningsar Inc All righs reserved The inforaion in his docuen is he proper of Morningsar Inc Reproducion
More informationGoRA. For more information on genetics and on Rheumatoid Arthritis: Genetics of Rheumatoid Arthritis. Published work referred to in the results:
For more informaion on geneics and on Rheumaoid Arhriis: Published work referred o in he resuls: The geneics revoluion and he assaul on rheumaoid arhriis. A review by Michael Seldin, Crisopher Amos, Ryk
More information