If we integrate the given modulating signal, m(t), we arrive at the following FM signal:
|
|
- Luke Greene
- 5 years ago
- Views:
Transcription
1
2 Part b If w intgrat th givn odulating signal, (, w arriv at th following signal: ( Acos( πf t + β sin( πf W can us anothr trigonotric idntity hr. ( Acos( β sin( πf cos( πf Asin( β sin( πf sin( πf Now, sinc β sin ( πf is a sall, slowly ti varying signal (rlativ to f w can trat it as our s( signal. Using this notation, th abov Hilbrt transfor proprtis, and so trigonotric idntitis w arriv at H { ( } ˆ ( Acos( β sin( πf sin( πf + Asin( β sin( πf cos( πf A + A Asin + A sin Asin + A sin [ sin( β sin( πf πf + sin( β sin( πf + πf ] [ sin( πf t β sin( πf + sin( πf t + β sin( πf ] ( πf t + β sin( πf ( β sin( πf πf + sin( πf t β sin( πf ( πf t + β sin( πf W thus arriv at th final solution: ( β sin( πf πf sin( β sin( πf πf ( Asin( πf t + β sin( πf
3 Probl Part a This is idntical to th intgral in probl.44, part b: ( A ( f t + β sin( πwt π, β cos ha W Part b f ha Part c This was covrd in class. Th first BPF liinats any nois in th spctru. Th output of th hard liitr loos li a variabl frquncy squar wav. W can writ th output of th hard liitr as follows: HL sin( nπ jnθ ( t ( c c n n ( πf t + β sin( πwt and, n θ nπ Rcall that copl ponntials can b pandd using Eulr s forula: j cos + j sin j cos j sin Thus, if w valuat our Fourir su fro inus infinity to plus infinity, all of th jsin trs cancl whil all of th cosin trs add. W ar lft with HL sin( nπ jnθ ( t ( c n n 4 cos π n nπ cos 4 3π ( nθ ( ( πf t + β sin( πwt cos( 6πf t + β sin( πwt + L All of th high frquncy trs ar liinatd whn th hard-liitd signal passs through th scond BPF. Also not that th aiu aplitud of th hard liitr is. Thus, du to th nonlinar natur of th hard liitr non of our trs can hav an aplitud largr than. W ar lft with
4 ( cos( πf t + β sin( πwt Passing through th diffrntiator rsults in π d dt π sin ( ( sin( πf t + β sin( πwt ( πf + β πw cos( πwt ( πf t + β sin( πwt ( f + βw cos( πwt This loos siilar to our AM signal, with th HF part insid th sin tr and th right hand factor rprsnting th odulating signal. Thus, aftr w pass th abov signal through th nvlop dtctor w arriv at v nv ( f + β W cos( πwt, βw f Th final stp is to pass this signal through th DC blocing capacitor. W thus arriv at th final output signal: ( f cos( πwt
5 Part d AM ( A[ + s( ] cos( πf, s( B cos( πwt W travl through th hard liitr. This ti, du to th natur of AM, our signal is truly priodic and loos as follows: X HL ( - T /f t W can s that now our signal coing out of th hard liitr, HL, is truly priodic and loos li a squar wav with aplitud. Th scond BPF following th hard liitr liinats all frquncis abov f. If you rcall, a squar wav can b writtn as a su of sinusoids of incrasing frquncis. Sinc f is th fundantal of this squar wav it is allowd to pass whil all of th raining HF trs ar liinatd. Thus our squar wav is soothd and th raining signal is ( cos( πf W pass this signal through th diffrntiator. π d dt π ( ( πf sin( πf f sin( πf This signal is in turn passd through th nvlop dtctor producing th following signal. ( f v nv
6 Th DC blocing capacitor liinats this signal to produc th final rsult. ( Part AM Cas, SSB ( cos( πf Th first BPF has no ffct sinc it is cntrd at our carrir frquncy, f. Multiplying by th cosin squars our cosin tr. W can us a trigonotric proprty. ( π f [ + cos( π ( f ] cos Th final LPF liinats th HF cosin tr. W ar lft with a constant. ( AM Cas, DSB Th only diffrnc hr is th BW of th first BPF. Sinc our signal is siply a spi in th frquncy doain th diffrnt BW has no ffct on th signal. Thus w obtain th sa answr as w did in th SSB cas. (
7 Cas, SSB ( cos( πf t + β sin( πwt W now fro our class nots that this signal can b writtn as { } jπf t jβ sin( πwt ( R And fro hr w now that thin of our incoing signal as jβ sin( πwt can b writtn as a Fourir Sris. Thus, w can ( J ( β cos( π ( f + nw n n Again, onc w pass this signal through th first BPF it liinats th HF trs whr n, laving us with ( J ( β cos( π ( f W BPF + Not that sinc our BPF only tnds to th right of our cntr frquncy w only pass th tr f + W and w liinat th tr f W. W ultiply b th carrir cosin and us a trigonotric proprty. BPF ( cos( πf J ( β cos( π ( f + W cos( πf J ( β cos( π ( f + W + J ( β cos( πwt Our final LPF liinats th HF tr and w ar lft with th solution. ( J ( β cos( πwt
8 Cas, DSB ( cos( πf t + β sin( πwt For this final cas w can again prss our input signal using th Fourir Sris prssion. ( J ( β cos( π ( f + nw n n W pass this signal through th BPF. Howvr, this ti notic that th filtr tnds lft by W as wll as right. So now w not only pass th low frquncy tr f + W but w also pass th low frquncy tr f W. W arriv at BPF ( J ( β cos( π ( f + W J ( β cos( π ( f W Not that th Bssl function is an odd function, i J ( β J( β th inus sign in th iddl of our BPF ( tr. W ultiply by our carrir cosin and apply th sa trigonotric proprty. BPF. This accounts for ( cos( πf J ( β cos( π ( f + W cos( πf J ( β cos( π ( f W cos( πf J ( β [ cos( π ( f + W + cos( πwt ] J ( β [ cos( π ( f W + cos( πwt ] Th LPF liinats th HF trs as usual and w ar lft with th following. Not that cosin is also an odd function. ( J ( β cos( πwt J ( β cos( πwt J ( β cos( πwt J ( β cos( πwt ( t
10. The Discrete-Time Fourier Transform (DTFT)
Th Discrt-Tim Fourir Transform (DTFT Dfinition of th discrt-tim Fourir transform Th Fourir rprsntation of signals plays an important rol in both continuous and discrt signal procssing In this sction w
More informationINTEGRATION BY PARTS
Mathmatics Rvision Guids Intgration by Parts Pag of 7 MK HOME TUITION Mathmatics Rvision Guids Lvl: AS / A Lvl AQA : C Edcl: C OCR: C OCR MEI: C INTEGRATION BY PARTS Vrsion : Dat: --5 Eampls - 6 ar copyrightd
More informationFourier Transforms and the Wave Equation. Key Mathematics: More Fourier transform theory, especially as applied to solving the wave equation.
Lur 7 Fourir Transforms and th Wav Euation Ovrviw and Motivation: W first discuss a fw faturs of th Fourir transform (FT), and thn w solv th initial-valu problm for th wav uation using th Fourir transform
More informationProbability and Stochastic Processes: A Friendly Introduction for Electrical and Computer Engineers Roy D. Yates and David J.
Probability and Stochastic Procsss: A Frindly Introduction for Elctrical and Computr Enginrs Roy D. Yats and David J. Goodman Problm Solutions : Yats and Goodman,4.3. 4.3.4 4.3. 4.4. 4.4.4 4.4.6 4.. 4..7
More informationCalculus Revision A2 Level
alculus Rvision A Lvl Tabl of drivativs a n sin cos tan d an sc n cos sin Fro AS * NB sc cos sc cos hain rul othrwis known as th function of a function or coposit rul. d d Eapl (i) (ii) Obtain th drivativ
More informationMathematics. Complex Number rectangular form. Quadratic equation. Quadratic equation. Complex number Functions: sinusoids. Differentiation Integration
Mathmatics Compl numbr Functions: sinusoids Sin function, cosin function Diffrntiation Intgration Quadratic quation Quadratic quations: a b c 0 Solution: b b 4ac a Eampl: 1 0 a= b=- c=1 4 1 1or 1 1 Quadratic
More informationMath 120 Answers for Homework 14
Math 0 Answrs for Homwork. Substitutions u = du = d d = du a d = du = du = u du = u + C = u = arctany du = +y dy dy = + y du b arctany arctany dy = + y du = + y + y arctany du = u du = u + C = arctan y
More informationProblem Set 4 Solutions Distributed: February 26, 2016 Due: March 4, 2016
Probl St 4 Solutions Distributd: Fbruary 6, 06 Du: March 4, 06 McQuarri Probls 5-9 Ovrlay th two plots using Excl or Mathatica. S additional conts blow. Th final rsult of Exapl 5-3 dfins th forc constant
More information4 x 4, and. where x is Town Square
Accumulation and Population Dnsity E. A city locatd along a straight highway has a population whos dnsity can b approimatd by th function p 5 4 th distanc from th town squar, masurd in mils, whr 4 4, and
More informationANALYSIS IN THE FREQUENCY DOMAIN
ANALYSIS IN THE FREQUENCY DOMAIN SPECTRAL DENSITY Dfinition Th spctral dnsit of a S.S.P. t also calld th spctrum of t is dfind as: + { γ }. jτ γ τ F τ τ In othr words, of th covarianc function. is dfind
More information1 General boundary conditions in diffusion
Gnral boundary conditions in diffusion Πρόβλημα 4.8 : Δίνεται μονοδιάτατη πλάκα πάχους, που το ένα άκρο της κρατιέται ε θερμοκραία T t και το άλλο ε θερμοκραία T 2 t. Αν η αρχική θερμοκραία της πλάκας
More informationSolution: APPM 1360 Final (150 pts) Spring (60 pts total) The following parts are not related, justify your answers:
APPM 6 Final 5 pts) Spring 4. 6 pts total) Th following parts ar not rlatd, justify your answrs: a) Considr th curv rprsntd by th paramtric quations, t and y t + for t. i) 6 pts) Writ down th corrsponding
More informationIntro to QM due: February 8, 2019 Problem Set 12
Intro to QM du: Fbruary 8, 9 Prob St Prob : Us [ x i, p j ] i δ ij to vrify that th anguar ontu oprators L i jk ɛ ijk x j p k satisfy th coutation rations [ L i, L j ] i k ɛ ijk Lk, [ L i, x j ] i k ɛ
More informationQuasi-Classical States of the Simple Harmonic Oscillator
Quasi-Classical Stats of th Simpl Harmonic Oscillator (Draft Vrsion) Introduction: Why Look for Eignstats of th Annihilation Oprator? Excpt for th ground stat, th corrspondnc btwn th quantum nrgy ignstats
More informationAndre Schneider P621
ndr Schnidr P61 Probl St #03 Novbr 6, 009 1 Srdnicki 10.3 Vrtx for L 1 = gχϕ ϕ. Th vrtx factor is ig. ϕ ig χ ϕ igur 1: ynan diagra for L 1 = gχϕ ϕ. Srdnicki 11.1 a) Dcay rat for th raction ig igur : ynan
More information6.1 Integration by Parts and Present Value. Copyright Cengage Learning. All rights reserved.
6.1 Intgration by Parts and Prsnt Valu Copyright Cngag Larning. All rights rsrvd. Warm-Up: Find f () 1. F() = ln(+1). F() = 3 3. F() =. F() = ln ( 1) 5. F() = 6. F() = - Objctivs, Day #1 Studnts will b
More informationDifferentiation of Exponential Functions
Calculus Modul C Diffrntiation of Eponntial Functions Copyright This publication Th Northrn Albrta Institut of Tchnology 007. All Rights Rsrvd. LAST REVISED March, 009 Introduction to Diffrntiation of
More informationIntegration by Parts
Intgration by Parts Intgration by parts is a tchniqu primarily for valuating intgrals whos intgrand is th product of two functions whr substitution dosn t work. For ampl, sin d or d. Th rul is: u ( ) v'(
More information2008 AP Calculus BC Multiple Choice Exam
008 AP Multipl Choic Eam Nam 008 AP Calculus BC Multipl Choic Eam Sction No Calculator Activ AP Calculus 008 BC Multipl Choic. At tim t 0, a particl moving in th -plan is th acclration vctor of th particl
More information( ) = ( ) ( ) ( ) ( ) τ τ. This is a more complete version of the solutions for assignment 2 courtesy of the course TA
This is a mor complt vrsion o th solutions or assignmnt courtsy o th cours TA ) Find th Fourir transorms o th signals shown in Figur (a-b). -a) -b) From igur -a, w hav: g t 4 0 t = t 0 othrwis = j G g
More informationCalculus II (MAC )
Calculus II (MAC232-2) Tst 2 (25/6/25) Nam (PRINT): Plas show your work. An answr with no work rcivs no crdit. You may us th back of a pag if you nd mor spac for a problm. You may not us any calculators.
More informationPrelim Examination 2011 / 2012 (Assessing Units 1 & 2) MATHEMATICS. Advanced Higher Grade. Time allowed - 2 hours
Prlim Eamination / (Assssing Units & ) MATHEMATICS Advancd Highr Grad Tim allowd - hours Rad Carfull. Calculators ma b usd in this papr.. Candidats should answr all qustions. Full crdit will onl b givn
More informationClassical Magnetic Dipole
Lctur 18 1 Classical Magntic Dipol In gnral, a particl of mass m and charg q (not ncssarily a point charg), w hav q g L m whr g is calld th gyromagntic ratio, which accounts for th ffcts of non-point charg
More information2. Background Material
S. Blair Sptmbr 3, 003 4. Background Matrial Th rst of this cours dals with th gnration, modulation, propagation, and ction of optical radiation. As such, bic background in lctromagntics and optics nds
More informationLegendre Wavelets for Systems of Fredholm Integral Equations of the Second Kind
World Applid Scincs Journal 9 (9): 8-, ISSN 88-495 IDOSI Publications, Lgndr Wavlts for Systs of Frdhol Intgral Equations of th Scond Kind a,b tb (t)= a, a,b a R, a. J. Biazar and H. Ebrahii Dpartnt of
More informationEngineering Differential Equations Practice Final Exam Solutions Fall 2011
9.6 Enginring Diffrntial Equation Practic Final Exam Solution Fall 0 Problm. (0 pt.) Solv th following initial valu problm: x y = xy, y() = 4. Thi i a linar d.. bcau y and y appar only to th firt powr.
More informationDSP-First, 2/e. LECTURE # CH2-3 Complex Exponentials & Complex Numbers TLH MODIFIED. Aug , JH McClellan & RW Schafer
DSP-First, / TLH MODIFIED LECTURE # CH-3 Complx Exponntials & Complx Numbrs Aug 016 1 READING ASSIGNMENTS This Lctur: Chaptr, Scts. -3 to -5 Appndix A: Complx Numbrs Complx Exponntials Aug 016 LECTURE
More informationThe graph of y = x (or y = ) consists of two branches, As x 0, y + ; as x 0, y +. x = 0 is the
Copyright itutcom 005 Fr download & print from wwwitutcom Do not rproduc by othr mans Functions and graphs Powr functions Th graph of n y, for n Q (st of rational numbrs) y is a straight lin through th
More informationEEO 401 Digital Signal Processing Prof. Mark Fowler
EEO 401 Digital Signal Procssing Prof. Mark Fowlr Dtails of th ot St #19 Rading Assignmnt: Sct. 7.1.2, 7.1.3, & 7.2 of Proakis & Manolakis Dfinition of th So Givn signal data points x[n] for n = 0,, -1
More informationAddition of angular momentum
Addition of angular momntum April, 0 Oftn w nd to combin diffrnt sourcs of angular momntum to charactriz th total angular momntum of a systm, or to divid th total angular momntum into parts to valuat th
More information1997 AP Calculus AB: Section I, Part A
997 AP Calculus AB: Sction I, Part A 50 Minuts No Calculator Not: Unlss othrwis spcifid, th domain of a function f is assumd to b th st of all ral numbrs for which f () is a ral numbr.. (4 6 ) d= 4 6 6
More informationThe Frequency Response of a Quarter-Wave Matching Network
4/1/29 Th Frquncy Rsons o a Quartr 1/9 Th Frquncy Rsons o a Quartr-Wav Matchg Ntwork Q: You hav onc aga rovidd us with conusg and rhas uslss ormation. Th quartr-wav matchg ntwork has an xact SFG o: a Τ
More informationTP A.31 The physics of squirt
thnial proof TP A. Th physis of squirt supporting: Th Illustratd Prinipls of Pool and Billiards http://illiards.olostat.du y David G. Aliator, PhD, PE ("Dr. Dav") thnial proof originally postd: 8//7 last
More informationSeptember 23, Honors Chem Atomic structure.notebook. Atomic Structure
Atomic Structur Topics covrd Atomic structur Subatomic particls Atomic numbr Mass numbr Charg Cations Anions Isotops Avrag atomic mass Practic qustions atomic structur Sp 27 8:16 PM 1 Powr Standards/ Larning
More information2F1120 Spektrala transformer för Media Solutions to Steiglitz, Chapter 1
F110 Spktrala transformr för Mdia Solutions to Stiglitz, Chaptr 1 Prfac This documnt contains solutions to slctd problms from Kn Stiglitz s book: A Digital Signal Procssing Primr publishd by Addison-Wsly.
More informationCollisions between electrons and ions
DRAFT 1 Collisions btwn lctrons and ions Flix I. Parra Rudolf Pirls Cntr for Thortical Physics, Unirsity of Oxford, Oxford OX1 NP, UK This rsion is of 8 May 217 1. Introduction Th Fokkr-Planck collision
More informationIntroduction to the Fourier transform. Computer Vision & Digital Image Processing. The Fourier transform (continued) The Fourier transform (continued)
Introduction to th Fourir transform Computr Vision & Digital Imag Procssing Fourir Transform Lt f(x) b a continuous function of a ral variabl x Th Fourir transform of f(x), dnotd by I {f(x)} is givn by:
More informationSinusoidal Response Notes
ECE 30 Sinusoidal Rspons Nots For BIBO Systms AStolp /29/3 Th sinusoidal rspons of a systm is th output whn th input is a sinusoidal (which starts at tim 0) Systm Sinusoidal Rspons stp input H( s) output
More informationCOMPUTER GENERATED HOLOGRAMS Optical Sciences 627 W.J. Dallas (Monday, April 04, 2005, 8:35 AM) PART I: CHAPTER TWO COMB MATH.
C:\Dallas\0_Courss\03A_OpSci_67\0 Cgh_Book\0_athmaticalPrliminaris\0_0 Combath.doc of 8 COPUTER GENERATED HOLOGRAS Optical Scincs 67 W.J. Dallas (onday, April 04, 005, 8:35 A) PART I: CHAPTER TWO COB ATH
More informationOptics and Non-Linear Optics I Non-linear Optics Tutorial Sheet November 2007
Optics and Non-Linar Optics I - 007 Non-linar Optics Tutorial Sht Novmbr 007 1. An altrnativ xponntial notion somtims usd in NLO is to writ Acos (") # 1 ( Ai" + A * $i" ). By using this notation and substituting
More informationREADING ASSIGNMENTS. Signal Processing First. Problem Solving Skills LECTURE OBJECTIVES. x(t) = cos(αt 2 ) Fourier Series ANALYSIS.
Signal Procssing First Lctur 5 Priodic Signals, Harmonics & im-varying Sinusoids READING ASSIGNMENS his Lctur: Chaptr 3, Sctions 3- and 3-3 Chaptr 3, Sctions 3-7 and 3-8 Nxt Lctur: Fourir Sris ANALYSIS
More informationThe Matrix Exponential
Th Matrix Exponntial (with xrciss) by D. Klain Vrsion 207.0.05 Corrctions and commnts ar wlcom. Th Matrix Exponntial For ach n n complx matrix A, dfin th xponntial of A to b th matrix A A k I + A + k!
More informationa 1and x is any real number.
Calcls Nots Eponnts an Logarithms Eponntial Fnction: Has th form y a, whr a 0, a an is any ral nmbr. Graph y, Graph y ln y y Th Natral Bas (Elr s nmbr): An irrational nmbr, symboliz by th lttr, appars
More information1973 AP Calculus AB: Section I
97 AP Calculus AB: Sction I 9 Minuts No Calculator Not: In this amination, ln dnots th natural logarithm of (that is, logarithm to th bas ).. ( ) d= + C 6 + C + C + C + C. If f ( ) = + + + and ( ), g=
More informationHydrogen Atom and One Electron Ions
Hydrogn Atom and On Elctron Ions Th Schrödingr quation for this two-body problm starts out th sam as th gnral two-body Schrödingr quation. First w sparat out th motion of th cntr of mass. Th intrnal potntial
More informationPhysics. X m (cm)
Entranc xa 006-007 Physics Duration: hours I- [ pts] An oscillator A chanical oscillator (C) is ford of a solid (S), of ass, attachd to th xtrity A of a horizontal spring of stiffnss (constant) = 80 N/
More informationDifferential Equations
Prfac Hr ar m onlin nots for m diffrntial quations cours that I tach hr at Lamar Univrsit. Dspit th fact that ths ar m class nots, th should b accssibl to anon wanting to larn how to solv diffrntial quations
More informationAddition of angular momentum
Addition of angular momntum April, 07 Oftn w nd to combin diffrnt sourcs of angular momntum to charactriz th total angular momntum of a systm, or to divid th total angular momntum into parts to valuat
More informationThe Matrix Exponential
Th Matrix Exponntial (with xrciss) by Dan Klain Vrsion 28928 Corrctions and commnts ar wlcom Th Matrix Exponntial For ach n n complx matrix A, dfin th xponntial of A to b th matrix () A A k I + A + k!
More informationdx equation it is called a second order differential equation.
TOPI Diffrntial quations Mthods of thir intgration oncption of diffrntial quations An quation which spcifis a rlationship btwn a function, its argumnt and its drivativs of th first, scond, tc ordr is calld
More informationDIFFERENTIAL EQUATION
MD DIFFERENTIAL EQUATION Sllabus : Ordinar diffrntial quations, thir ordr and dgr. Formation of diffrntial quations. Solution of diffrntial quations b th mthod of sparation of variabls, solution of homognous
More informationare given in the table below. t (hours)
CALCULUS WORKSHEET ON INTEGRATION WITH DATA Work th following on notbook papr. Giv dcimal answrs corrct to thr dcimal placs.. A tank contains gallons of oil at tim t = hours. Oil is bing pumpd into th
More informationReview of Exponentials and Logarithms - Classwork
Rviw of Eponntials and Logarithms - Classwork In our stud of calculus, w hav amind drivativs and intgrals of polnomial prssions, rational prssions, and trignomtric prssions. What w hav not amind ar ponntial
More informationA Propagating Wave Packet Group Velocity Dispersion
Lctur 8 Phys 375 A Propagating Wav Packt Group Vlocity Disprsion Ovrviw and Motivation: In th last lctur w lookd at a localizd solution t) to th 1D fr-particl Schrödingr quation (SE) that corrsponds to
More informationAs the matrix of operator B is Hermitian so its eigenvalues must be real. It only remains to diagonalize the minor M 11 of matrix B.
7636S ADVANCED QUANTUM MECHANICS Solutions Spring. Considr a thr dimnsional kt spac. If a crtain st of orthonormal kts, say, and 3 ar usd as th bas kts, thn th oprators A and B ar rprsntd by a b A a and
More informationNEW APPLICATIONS OF THE ABEL-LIOUVILLE FORMULA
NE APPLICATIONS OF THE ABEL-LIOUVILLE FORMULA Mirca I CÎRNU Ph Dp o Mathmatics III Faculty o Applid Scincs Univrsity Polithnica o Bucharst Cirnumirca @yahoocom Abstract In a rcnt papr [] 5 th indinit intgrals
More informationLinear-Phase FIR Transfer Functions. Functions. Functions. Functions. Functions. Functions. Let
It is impossibl to dsign an IIR transfr function with an xact linar-phas It is always possibl to dsign an FIR transfr function with an xact linar-phas rspons W now dvlop th forms of th linarphas FIR transfr
More informationAnnounce. ECE 2026 Summer LECTURE OBJECTIVES READING. LECTURE #3 Complex View of Sinusoids May 21, Complex Number Review
ECE 06 Summr 018 Announc HW1 du at bginning of your rcitation tomorrow Look at HW bfor rcitation Lab 1 is Thursday: Com prpard! Offic hours hav bn postd: LECTURE #3 Complx Viw of Sinusoids May 1, 018 READIG
More informationPHA 5127 Answers Homework 2 Fall 2001
PH 5127 nswrs Homwork 2 Fall 2001 OK, bfor you rad th answrs, many of you spnt a lot of tim on this homwork. Plas, nxt tim if you hav qustions plas com talk/ask us. Thr is no nd to suffr (wll a littl suffring
More informationSearch sequence databases 3 10/25/2016
Sarch squnc databass 3 10/25/2016 Etrm valu distribution Ø Suppos X is a random variabl with probability dnsity function p(, w sampl a larg numbr S of indpndnt valus of X from this distribution for an
More informationCHAPTER 5. Section 5-1
SECTION 5-9 CHAPTER 5 Sction 5-. An ponntial function is a function whr th variabl appars in an ponnt.. If b >, th function is an incrasing function. If < b
More informationTotal Wave Function. e i. Wave function above sample is a plane wave: //incident beam
Total Wav Function Wav function abov sampl is a plan wav: r i kr //incidnt bam Wav function blow sampl is a collction of diffractd bams (and ): r i k r //transmittd bams k ks W nd to know th valus of th.
More informationChapter two Functions
Chaptr two Functions -- Eponntial Logarithm functions Eponntial functions If a is a positiv numbr is an numbr, w dfin th ponntial function as = a with domain - < < ang > Th proprtis of th ponntial functions
More informationMaxwellian Collisions
Maxwllian Collisions Maxwll ralizd arly on that th particular typ of collision in which th cross-sction varis at Q rs 1/g offrs drastic siplifications. Intrstingly, this bhavior is physically corrct for
More informationEinstein Equations for Tetrad Fields
Apiron, Vol 13, No, Octobr 006 6 Einstin Equations for Ttrad Filds Ali Rıza ŞAHİN, R T L Istanbul (Turky) Evry mtric tnsor can b xprssd by th innr product of ttrad filds W prov that Einstin quations for
More informationElectron energy in crystal potential
Elctron nry in crystal potntial r r p c mc mc mc Expand: r r r mc mc mc r r p c mc mc mc r pc m c mc p m m m m r E E m m m r p E m r nr nr whr: E V mc E m c Wav quation Hamiltonian: Tim-Indpndnt Schrodinr
More informationNARAYANA I I T / P M T A C A D E M Y. C o m m o n P r a c t i c e T e s t 1 6 XII STD BATCHES [CF] Date: PHYSIS HEMISTRY MTHEMTIS
. (D). (A). (D). (D) 5. (B) 6. (A) 7. (A) 8. (A) 9. (B). (A). (D). (B). (B). (C) 5. (D) NARAYANA I I T / P M T A C A D E M Y C o m m o n P r a c t i c T s t 6 XII STD BATCHES [CF] Dat: 8.8.6 ANSWER PHYSIS
More informationMATH 319, WEEK 15: The Fundamental Matrix, Non-Homogeneous Systems of Differential Equations
MATH 39, WEEK 5: Th Fundamntal Matrix, Non-Homognous Systms of Diffrntial Equations Fundamntal Matrics Considr th problm of dtrmining th particular solution for an nsmbl of initial conditions For instanc,
More informationu x v x dx u x v x v x u x dx d u x v x u x v x dx u x v x dx Integration by Parts Formula
7. Intgration by Parts Each drivativ formula givs ris to a corrsponding intgral formula, as w v sn many tims. Th drivativ product rul yilds a vry usful intgration tchniqu calld intgration by parts. Starting
More informationMSLC Math 151 WI09 Exam 2 Review Solutions
Eam Rviw Solutions. Comput th following rivativs using th iffrntiation ruls: a.) cot cot cot csc cot cos 5 cos 5 cos 5 cos 5 sin 5 5 b.) c.) sin( ) sin( ) y sin( ) ln( y) ln( ) ln( y) sin( ) ln( ) y y
More informationWeek 3: Connected Subgraphs
Wk 3: Connctd Subgraphs Sptmbr 19, 2016 1 Connctd Graphs Path, Distanc: A path from a vrtx x to a vrtx y in a graph G is rfrrd to an xy-path. Lt X, Y V (G). An (X, Y )-path is an xy-path with x X and y
More informationMor Tutorial at www.dumblittldoctor.com Work th problms without a calculator, but us a calculator to chck rsults. And try diffrntiating your answrs in part III as a usful chck. I. Applications of Intgration
More information1997 AP Calculus AB: Section I, Part A
997 AP Calculus AB: Sction I, Part A 50 Minuts No Calculator Not: Unlss othrwis spcifid, th domain of a function f is assumd to b th st of all ral numbrs x for which f (x) is a ral numbr.. (4x 6 x) dx=
More information3 2x. 3x 2. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB
Math B Intgration Rviw (Solutions) Do ths intgrals. Solutions ar postd at th wbsit blow. If you hav troubl with thm, sk hlp immdiatly! () 8 d () 5 d () d () sin d (5) d (6) cos d (7) d www.clas.ucsb.du/staff/vinc
More information6. The Interaction of Light and Matter
6. Th Intraction of Light and Mattr - Th intraction of light and mattr is what maks lif intrsting. - Light causs mattr to vibrat. Mattr in turn mits light, which intrfrs with th original light. - Excitd
More informationEE140 Introduction to Communication Systems Lecture 2
EE40 Introduction to Communication Systms Lctur 2 Instructor: Prof. Xiliang Luo ShanghaiTch Univrsity, Spring 208 Architctur of a Digital Communication Systm Transmittr Sourc A/D convrtr Sourc ncodr Channl
More informationCS 491 G Combinatorial Optimization
CS 49 G Cobinatorial Optiization Lctur Nots Junhui Jia. Maiu Flow Probls Now lt us iscuss or tails on aiu low probls. Thor. A asibl low is aiu i an only i thr is no -augnting path. Proo: Lt P = A asibl
More informationThe van der Waals interaction 1 D. E. Soper 2 University of Oregon 20 April 2012
Th van dr Waals intraction D. E. Sopr 2 Univrsity of Orgon 20 pril 202 Th van dr Waals intraction is discussd in Chaptr 5 of J. J. Sakurai, Modrn Quantum Mchanics. Hr I tak a look at it in a littl mor
More informationMA1506 Tutorial 2 Solutions. Question 1. (1a) 1 ) y x. e x. 1 exp (in general, Integrating factor is. ye dx. So ) (1b) e e. e c.
MA56 utorial Solutions Qustion a Intgrating fator is ln p p in gnral, multipl b p So b ln p p sin his kin is all a Brnoulli quation -- st Sin w fin Y, Y Y, Y Y p Qustion Dfin v / hn our quation is v μ
More information5. B To determine all the holes and asymptotes of the equation: y = bdc dced f gbd
1. First you chck th domain of g x. For this function, x cannot qual zro. Thn w find th D domain of f g x D 3; D 3 0; x Q x x 1 3, x 0 2. Any cosin graph is going to b symmtric with th y-axis as long as
More informationChapter 1. Chapter 10. Chapter 2. Chapter 11. Chapter 3. Chapter 12. Chapter 4. Chapter 13. Chapter 5. Chapter 14. Chapter 6. Chapter 7.
Chaptr Binomial Epansion Chaptr 0 Furthr Probability Chaptr Limits and Drivativs Chaptr Discrt Random Variabls Chaptr Diffrntiation Chaptr Discrt Probability Distributions Chaptr Applications of Diffrntiation
More informationThomas Whitham Sixth Form
Thomas Whitham Sith Form Pur Mathmatics Unit C Algbra Trigonomtr Gomtr Calculus Vctor gomtr Pag Algbra Molus functions graphs, quations an inqualitis Graph of f () Draw f () an rflct an part of th curv
More informationLecture 2: Discrete-Time Signals & Systems. Reza Mohammadkhani, Digital Signal Processing, 2015 University of Kurdistan eng.uok.ac.
Lctur 2: Discrt-Tim Signals & Systms Rza Mohammadkhani, Digital Signal Procssing, 2015 Univrsity of Kurdistan ng.uok.ac.ir/mohammadkhani 1 Signal Dfinition and Exampls 2 Signal: any physical quantity that
More information( ) Differential Equations. Unit-7. Exact Differential Equations: M d x + N d y = 0. Verify the condition
Diffrntial Equations Unit-7 Eat Diffrntial Equations: M d N d 0 Vrif th ondition M N Thn intgrat M d with rspt to as if wr onstants, thn intgrat th trms in N d whih do not ontain trms in and quat sum of
More informationIntroduction to Condensed Matter Physics
Introduction to Condnsd Mattr Physics pcific hat M.P. Vaughan Ovrviw Ovrviw of spcific hat Hat capacity Dulong-Ptit Law Einstin modl Dby modl h Hat Capacity Hat capacity h hat capacity of a systm hld at
More informationperm4 A cnt 0 for for if A i 1 A i cnt cnt 1 cnt i j. j k. k l. i k. j l. i l
h 4D, 4th Rank, Antisytric nsor and th 4D Equivalnt to th Cross Product or Mor Fun with nsors!!! Richard R Shiffan Digital Graphics Assoc 8 Dunkirk Av LA, Ca 95 rrs@isidu his docunt dscribs th four dinsional
More information10. Limits involving infinity
. Limits involving infinity It is known from th it ruls for fundamntal arithmtic oprations (+,-,, ) that if two functions hav finit its at a (finit or infinit) point, that is, thy ar convrgnt, th it of
More informationVTU NOTES QUESTION PAPERS NEWS RESULTS FORUMS
Diffrntial Equations Unit-7 Eat Diffrntial Equations: M d N d 0 Vrif th ondition M N Thn intgrat M d with rspt to as if wr onstants, thn intgrat th trms in N d whih do not ontain trms in and quat sum of
More informationA 1 A 2. a) Find the wavelength of the radio waves. Since c = f, then = c/f = (3x10 8 m/s) / (30x10 6 Hz) = 10m.
1. Young s doubl-slit xprint undrlis th instrunt landing syst at ost airports and is usd to guid aircraft to saf landings whn th visibility is poor. Suppos that a pilot is trying to align hr plan with
More informationThings I Should Know Before I Get to Calculus Class
Things I Should Know Bfor I Gt to Calculus Class Quadratic Formula = b± b 4ac a sin + cos = + tan = sc + cot = csc sin( ± y ) = sin cos y ± cos sin y cos( + y ) = cos cos y sin sin y cos( y ) = cos cos
More informationExercise 1. Sketch the graph of the following function. (x 2
Writtn tst: Fbruary 9th, 06 Exrcis. Sktch th graph of th following function fx = x + x, spcifying: domain, possibl asymptots, monotonicity, continuity, local and global maxima or minima, and non-drivability
More informationSAFE HANDS & IIT-ian's PACE EDT-15 (JEE) SOLUTIONS
It is not possibl to find flu through biggr loop dirctly So w will find cofficint of mutual inductanc btwn two loops and thn find th flu through biggr loop Also rmmbr M = M ( ) ( ) EDT- (JEE) SOLUTIONS
More informationTwo New Sliding DTFT Algorithms for Phase Difference Measurement Based on a New Kind of Windows
.78/sr--8 MEASUEMET SCIECE EVIEW, Volu, o. 6, Two w Sliding DTFT Algoriths for Phas Diffrnc Masurnt Basd on a w Kind of Windows Yaqing Tu, Ting ao Shn,, Haitao Zhang, Ming Li Dpartnt of Inforation Enginring,
More informationVII. Quantum Entanglement
VII. Quantum Entanglmnt Quantum ntanglmnt is a uniqu stat of quantum suprposition. It has bn studid mainly from a scintific intrst as an vidnc of quantum mchanics. Rcntly, it is also bing studid as a basic
More informationThomas Whitham Sixth Form
Thomas Whitham Sith Form Pur Mathmatics Cor rvision gui Pag Algbra Moulus functions graphs, quations an inqualitis Graph of f () Draw f () an rflct an part of th curv blow th ais in th ais. f () f () f
More informationExam 1. It is important that you clearly show your work and mark the final answer clearly, closed book, closed notes, no calculator.
Exam N a m : _ S O L U T I O N P U I D : I n s t r u c t i o n s : It is important that you clarly show your work and mark th final answr clarly, closd book, closd nots, no calculator. T i m : h o u r
More informationPROOF OF FIRST STANDARD FORM OF NONELEMENTARY FUNCTIONS
Intrnational Journal Of Advanc Rsarch In Scinc And Enginring http://www.ijars.com IJARSE, Vol. No., Issu No., Fbruary, 013 ISSN-319-8354(E) PROOF OF FIRST STANDARD FORM OF NONELEMENTARY FUNCTIONS 1 Dharmndra
More information1 1 1 p q p q. 2ln x x. in simplest form. in simplest form in terms of x and h.
NAME SUMMER ASSIGNMENT DUE SEPTEMBER 5 (FIRST DAY OF SCHOOL) AP CALC AB Dirctions: Answr all of th following qustions on a sparat sht of papr. All work must b shown. You will b tstd on this matrial somtim
More informationComplex representation of continuous-time periodic signals
. Complx rprsntation of continuous-tim priodic signals Eulr s quation jwt cost jsint his is th famous Eulr s quation. Brtrand Russll and Richard Fynman both gav this quation plntiful prais with words such
More information1 Input-Output Stability
Inut-Outut Stability Inut-outut stability analysis allows us to analyz th stability of a givn syst without knowing th intrnal stat x of th syst. Bfor going forward, w hav to introduc so inut-outut athatical
More information