DEPARTMENT OF ELECTRICAL &ELECTRONICS ENGINEERING SIGNALS AND SYSTEMS. Assoc. Prof. Dr. Burak Kelleci. Spring 2018
|
|
- Leslie Chambers
- 5 years ago
- Views:
Transcription
1 DEPARTMENT OF ELECTRICAL &ELECTRONICS ENGINEERING SIGNALS AND SYSTEMS Aoc. Prof. Dr. Burak Kllci Spring 08
2 OUTLINE Th Laplac Tranform Rgion of convrgnc for Laplac ranform Invr Laplac ranform Gomric valuaion of h Fourir ranform Pol-zro plo Propri of Laplac ranform Linariy Tim hifing Shifing in h -domain Tim caling Conjugaion Convoluion Diffrniaion in h im domain Diffrniaion in -domain Ingraion in h im domain Iniial and final valu horm Signal and Sym - Wk 0
3 THE LAPLACE TRANSFORM A h bginning of Fourir Tranform, h rpon of a linar im-invarian ym wih impul rpon h() o a compl ponnial inpu wa analyzd. y whr H H For =jw hi ingral corrpond o Fourir Tranform of h() For gnral valu of compl variabl, i i rfrrd a h Laplac ranform of h() h d Signal and Sym - Wk 0 3
4 THE LAPLACE TRANSFORM Th Laplac ranform of a gnral ignal () i dfind a No ha i a compl variabl and can b wrin a =+jw, wih and w ar ral and imaginary par For convninc, h Laplac ranform will b dnod a L{()} and h ranform rlaionhip bwn () and () will b dnod a L d Signal and Sym - Wk 0 4
5 THE LAPLACE TRANSFORM Whn =jw, h Laplac ranform bcom Fourir ranform jw Th Laplac ranform hav alo a rlaionhip vn i no purly imaginary Th Laplac ranform can b inrprd a h Fourir ranform of () - Dpnding on h valu of hi ignal may b dcaying or growing in im. jw d jw jw d jw Signal and Sym - Wk 0 5 d
6 EAMPLE a u a a u d a a d a R a 0 0 No ha for R{}<-a h ingral will no convrg and h rul will b infiniy. Thrfor, hr i a rquirmn for in ordr o guaran h convrgnc. Signal and Sym - Wk 0 6
7 EAMPLE a u a a u d a 0 a d a R a 0 For convrgnc in hi ampl, R{+a}<0 or R{}<-a Comparing prviou ampl, h algbraic prion for h Laplac ranform ar idnical. Howvr, hy diffr bad on hir convrg rgion. Signal and Sym - Wk 0 7
8 THE LAPLACE TRANSFORM In gnral, h rang of valu for which h Laplac ingral convrg i rfrrd a h rgion of convrgnc (ROC). A convnin way o diplay h ROC i o how in h compl -plan. Th horizonal ai i R{} and vrical ai i Im{} For ampl h hadd ara ar h ROC of prviou ampl. Im Im -a R -a R Signal and Sym - Wk 0 8
9 EAMPLE 3 Th ROC i h rgion whr boh rm convrg. For hi ampl, h fir rm will convrg for R{}>- and cond rm will convrg for R{}>- Combining hi wo ROC, h ROC i R{}> d u d u d u u u u Signal and Sym - Wk 0 9
10 THE LAPLACE TRANSFORM A n in prviou ampl, h Laplac ranform i a raio of polynomial in h compl variabl, o ha whr N() and D() ar h numraor polynomial and dnominaor polynomial. Ecp a cal facor, h numraor and dnominaor polynomial can b pcifid by hir roo and marking h locaion of h roo provid a picorial way of dcribing h Laplac ranform. Each roo of dnominaor polynomial i indicad wih and ach roo of numraor polynomial i indicad wih o N D Signal and Sym - Wk 0 0
11 THE LAPLACE TRANSFORM For ampl, h roo of dnominaor and numraor polynomial of ampl 3 ar hown. Im - - ` -plan R Th roo of numraor polynomial ar commonly rfrrd o h zro of (), inc for h valu of, ()=0 Th roo of dnominaor polynomial ar commonly rfrrd o h pol of (), inc for h valu of, () i infini Th rprnaion of () uing i pol and zro in h - plan i rfrrd o a h pol-zro plo o () Signal and Sym - Wk 0
12 REGION OF CONVERGENCE FOR LAPLACE TRANSFORM Propry : Th ROC of () coni of rip paralll o h jw-ai in -plan Th ROC of () coni of h valu of =+jw for which h Fourir ranform of () - convrg Thi condiion dpnd only on, h ral par of Propry : For raional Laplac ranform, h ROC do no conain any pol. Sinc () i infini a a pol, h Laplac ranform will no convrg a a pol. Signal and Sym - Wk 0
13 REGION OF CONVERGENCE FOR LAPLACE TRANSFORM Propry 3: If () i a fini duraion ignal and aboluly ingrabl, hn ROC i h nir -plan. A fini duraion ignal i zro ouid an inrval of fini duraion. If () i aboluly ingrabl in hi fini inrval, muliplying i by dcaying or growing ponnial will alo b boundd. T T d T T d Signal and Sym - Wk 0 3
14 REGION OF CONVERGENCE FOR LAPLACE TRANSFORM Propry 4: If () i righ idd, and if h lin R{}= 0 i in h ROC, hn all valu of for which R{}> 0 will alo b in h ROC. A righ-idd ignal i a ignal for which ()=0 prior o om fini im T. If a poin i in h ROC, hn all h poin o h righ of (all poin wih largr ral par) ar in h ROC. For hi raon, h ROC in hi ca i commonly rfrrd o a a righ-half plan. d T d Signal and Sym - Wk 0 4
15 REGION OF CONVERGENCE FOR LAPLACE TRANSFORM Propry 5: If () i lf idd, and if h lin R{}= 0 in in h ROC, hn all valu of for which R{}< 0 will alo b in h ROC. A lf-idd ignal i a ignal for which ()=0 afr om fini im T. Th ROC i commonly rfrrd o a lf-half plan for hi ca. Signal and Sym - Wk 0 5
16 REGION OF CONVERGENCE FOR LAPLACE TRANSFORM Propry 6: If () i wo idd, and if h lin R{}= 0 i in h ROC, hn h ROC will coni of a rip in h -plan ha includ h lin R{}= 0 A wo-idd ignal i a ignal ha i of infini n for boh >0 and <0. Thi yp of ignal can b dividd ino righ-idd R () and lf-idd L ()ignal. Th Laplac ranform will convrg for valu of for which h ranform of boh R () and L () convrg. Signal and Sym - Wk 0 6
17 REGION OF CONVERGENCE FOR LAPLACE TRANSFORM Im Im r R L R Im r L R Signal and Sym - Wk 0 7
18 REGION OF CONVERGENCE FOR LAPLACE TRANSFORM Propry 7: If h Laplac ranform () of () i raional, hn i ROC i boundd by pol or nd o infiniy. In addiion, no pol of () ar conaind in h ROC. Propry 8: If h Laplac ranform () of () i raional, hn if () i righ idd, h ROC i h rgion in h -plan o h righ of h righmo pol. If () i lf idd, h ROC i h rgion in h -plan o h lf of h lfmo pol. Signal and Sym - Wk 0 8
19 EAMPLE Conidr h pol-zro parn of h following Laplac ranform Im Im R R Righ-idd ignal Im Im Lf-idd ignal R R Two-idd ignal Signal and Sym - Wk 0 9
20 THE INVERSE LAPLACE TRANSFORM Th Laplac ranform of a ignal () wih prd a =+jw and in h ROC Signal and Sym - Wk 0 0 w w w w w w w w d j d j d j j j j by muliplying boh id h invr Fourir Tranform Uing
21 THE INVERSE LAPLACE TRANSFORM () can b rcovrd from i Laplac ranform valuad for a of valu of in h ROC wih fid and w varying from - o. j Thi quaion a ha () can b rprnd a a wighd ingral of compl ponnial. j d Signal and Sym - Wk 0
22 EAMPLE L uing u u L L u R R R h parial- fracion panion L R Signal and Sym - Wk 0
23 GEOMETRIC EVALUATION OF THE FOURIER TRANSFORM FROM THE POLE-ZERO PLOT Th Fourir ranform of a ignal i h Laplac ranform valuad on h jwai. To dvlop a procdur o valua h Fourir ranform from Laplac ranform conidr a Laplac ranform wih a ingl zro [()=-a] L ay w wan o valua =. Th algbraic prion -a i h um of wo compl numbr and -a,which can b rprnd a vcor in h compl plan. Th vcor um of and a i a vcor from a o. Th lngh of hi vcor i h magniud of -a and h angl rlaiv o ral ai i h angl of hi compl numbr. If () ha a ingl pol [()=/(-a)], hn h rciprocal of h lngh of h vcor from pol i h magniud and h ngaiv of h angl i h ngaiv of h angl of h vcor wih h ral ai. Signal and Sym - Wk 0 3
24 EAMPLE Th, R Fourir Tranform i jw jw jw w jw an w jw Th lngh of hi vcor i h magniud Th ngaiv of h angl of h vcor i h angl. Signal and Sym - Wk 0 4
25 GEOMETRIC EVALUATION OF THE FOURIER TRANSFORM FROM THE POLE-ZERO PLOT Gnral raional Laplac ranform To valua () a = Th magniud of ( ) i h magniud of h cal facor M im h produc of lngh of h zro vcor dividd by h produc of h lngh of h pol vcor. Th angl of ( ) i h um of h angl of h zro vcor minu h um of h angl of h pol vcor. If h cal facor M i ngaiv, an addiional angl of would b includd. If () ha mulipl pol qual o ach ohr or mulipl zro qual o ach ohr (or boh), h lngh and angl of h vcor from ach of h pol or zro mu b includd. Signal and Sym - Wk 0 5 P j j R i i M P j j R i i P j j R i i M
26 EAMPLE Conidr h following all-pa ym Th pha of H H H h frquncy rpon jw inc jw inc an w a jw an w a Signal and Sym - Wk 0 6
27 PROPERTIES OF THE LAPLACE TRANSFORM Propry Signal Laplac Tranform ROC R R R Linariy a b a b Tim Shifing Shifing in h - Domain Tim Scaling a Diffrniaion in h Tim Domain A la R R R a Conjugaion Convoluion d Diffrniaion in h -Domain Ingraion in h Tim Domain if 0for 0 and if d 0 a Shifd vrion of R (i.., i in h ROCif 0 i in R) Scald ROC (i.., i in h ROC if / a i in R) R A la R R A la R d d d A la R R Iniial- and Final-Valu Thorm conain noimpul or highr - ordr ingularii a 0, hn 0for 0 and 0 lim lim ha a fini limi a, hn lim 0 R 0 Signal and Sym - Wk 0 7
28 THE INITIAL AND FINAL VALUE THEOREMS For h ignal ()=0, <0 and () conain no impul a h origin, i i poibl o calcula h iniial valu a approach zro from poiiv valu of. Thi horm i calld iniial-valu horm 0 lim If ()=0 for <0 and () ha a fini limi a, hn h final valu horm lim lim 0 Signal and Sym - Wk 0 8
29 EAMPLE Calcula h iniial-valu and final-valu for h following ignal lim lim lim lim lim Signal and Sym - Wk 0 9
30 SOME LAPLACE TRANSFORM PAIRS u Signal Tranform ROC All u R 0 u R 0 n R 0 u n n! n R 0 u n n! u R u R n u n! n R n R u n! n T T All cow 0u R 0 w0 inw 0u w0 R 0 w0 cow0 u R w0 inw 0 u w0 R w n n d un n d u u n im n n 0 All R 0 Signal and Sym - Wk 0 30
Chapter 5 The Laplace Transform. x(t) input y(t) output Dynamic System
EE 422G No: Chapr 5 Inrucor: Chung Chapr 5 Th Laplac Tranform 5- Inroducion () Sym analyi inpu oupu Dynamic Sym Linar Dynamic ym: A procor which proc h inpu ignal o produc h oupu dy ( n) ( n dy ( n) +
More informationInstructors Solution for Assignment 3 Chapter 3: Time Domain Analysis of LTIC Systems
Inrucor Soluion for Aignmn Chapr : Tim Domain Anali of LTIC Sm Problm i a 8 x x wih x u,, an Zro-inpu rpon of h m: Th characriic quaion of h LTIC m i i 8, which ha roo a ± j Th zro-inpu rpon i givn b zi
More informationChapter 12 Introduction To The Laplace Transform
Chapr Inroducion To Th aplac Tranorm Diniion o h aplac Tranorm - Th Sp & Impul uncion aplac Tranorm o pciic uncion 5 Opraional Tranorm Applying h aplac Tranorm 7 Invr Tranorm o Raional uncion 8 Pol and
More informationTransfer function and the Laplace transformation
Lab No PH-35 Porland Sa Univriy A. La Roa Tranfr funcion and h Laplac ranformaion. INTRODUTION. THE LAPLAE TRANSFORMATION L 3. TRANSFER FUNTIONS 4. ELETRIAL SYSTEMS Analyi of h hr baic paiv lmn R, and
More informationLaPlace Transform in Circuit Analysis
LaPlac Tranform in Circui Analyi Obciv: Calcula h Laplac ranform of common funcion uing h dfiniion and h Laplac ranform abl Laplac-ranform a circui, including componn wih non-zro iniial condiion. Analyz
More informationChapter 9 The Laplace Transform
Chapr 9 Th Laplac Tranform 熊红凯特聘教授 hp://min.ju.du.cn 电子工程系上海交通大学 7 Topic 9. DEFINATION OF THE LAPLACE TRANSFORM 9. THE REGION OF CONVERGENCE FOR LAPLACE THANSFORMS 9. PROPERTIES OF THE LAPLACE TRANSFORM
More informationREPETITION before the exam PART 2, Transform Methods. Laplace transforms: τ dτ. L1. Derive the formulas : L2. Find the Laplace transform F(s) if.
Tranform Mhod and Calculu of Svral Variabl H7, p Lcurr: Armin Halilovic KTH, Campu Haning E-mail: armin@dkh, wwwdkh/armin REPETITION bfor h am PART, Tranform Mhod Laplac ranform: L Driv h formula : a L[
More informationElementary Differential Equations and Boundary Value Problems
Elmnar Diffrnial Equaions and Boundar Valu Problms Boc. & DiPrima 9 h Ediion Chapr : Firs Ordr Diffrnial Equaions 00600 คณ ตศาสตร ว ศวกรรม สาขาว ชาว ศวกรรมคอมพ วเตอร ป การศ กษา /55 ผศ.ดร.อร ญญา ผศ.ดร.สมศ
More informationChapter 3: Fourier Representation of Signals and LTI Systems. Chih-Wei Liu
Chapr 3: Fourir Rprsnaion of Signals and LTI Sysms Chih-Wi Liu Oulin Inroducion Complx Sinusoids and Frquncy Rspons Fourir Rprsnaions for Four Classs of Signals Discr-im Priodic Signals Fourir Sris Coninuous-im
More informationBoyce/DiPrima 9 th ed, Ch 2.1: Linear Equations; Method of Integrating Factors
Boc/DiPrima 9 h d, Ch.: Linar Equaions; Mhod of Ingraing Facors Elmnar Diffrnial Equaions and Boundar Valu Problms, 9 h diion, b William E. Boc and Richard C. DiPrima, 009 b John Wil & Sons, Inc. A linar
More informationS.Y. B.Sc. (IT) : Sem. III. Applied Mathematics. Q.1 Attempt the following (any THREE) [15]
S.Y. B.Sc. (IT) : Sm. III Applid Mahmaics Tim : ½ Hrs.] Prlim Qusion Papr Soluion [Marks : 75 Q. Amp h following (an THREE) 3 6 Q.(a) Rduc h mari o normal form and find is rank whr A 3 3 5 3 3 3 6 Ans.:
More informationWhy Laplace transforms?
MAE4 Linar ircui Why Lalac ranform? Firordr R cc v v v KVL S R inananou for ach Subiu lmn rlaion v S Ordinary diffrnial quaion in rm of caacior volag Lalac ranform Solv Invr LT V u, v Ri, i A R V A _ v
More informationChap.3 Laplace Transform
Chap. aplac Tranorm Tranorm: An opraion ha ranorm a uncion ino anohr uncion i Dirniaion ranorm: ii x: d dx x x Ingraion ranorm: x: x dx x c Now, conidr a dind ingral k, d,ha ranorm ino a uncion o variabl
More informationLaplace Transforms recap for ccts
Lalac Tranform rca for cc Wha h big ida?. Loo a iniial condiion ron of cc du o caacior volag and inducor currn a im Mh or nodal analyi wih -domain imdanc rianc or admianc conducanc Soluion of ODE drivn
More informationPoisson process Markov process
E2200 Quuing hory and lraffic 2nd lcur oion proc Markov proc Vikoria Fodor KTH Laboraory for Communicaion nwork, School of Elcrical Enginring 1 Cour oulin Sochaic proc bhind quuing hory L2-L3 oion proc
More informationDISCRETE TIME FOURIER TRANSFORM (DTFT)
DISCRETE TIME FOURIER TRANSFORM (DTFT) Th dicrt-tim Fourir Tranform x x n xn n n Th Invr dicrt-tim Fourir Tranform (IDTFT) x n Not: ( ) i a complx valud continuou function = π f [rad/c] f i th digital
More informationLet s look again at the first order linear differential equation we are attempting to solve, in its standard form:
Th Ingraing Facor Mhod In h prvious xampls of simpl firs ordr ODEs, w found h soluions by algbraically spara h dpndn variabl- and h indpndn variabl- rms, and wri h wo sids of a givn quaion as drivaivs,
More informationBoyce/DiPrima 9 th ed, Ch 7.8: Repeated Eigenvalues
Boy/DiPrima 9 h d Ch 7.8: Rpad Eignvalus Elmnary Diffrnial Equaions and Boundary Valu Problms 9 h diion by William E. Boy and Rihard C. DiPrima 9 by John Wily & Sons In. W onsidr again a homognous sysm
More informationTHE LAPLACE TRANSFORM
THE LAPLACE TRANSFORM LEARNING GOALS Diniion Th ranorm map a ncion o im ino a ncion o a complx variabl Two imporan inglariy ncion Th ni p and h ni impl Tranorm pair Baic abl wih commonly d ranorm Propri
More informationCopyright 2012 Pearson Education, Inc. Publishing as Prentice Hall.
Chapr Rviw 0 6. ( a a ln a. This will qual a if an onl if ln a, or a. + k an (ln + c. Thrfor, a an valu of, whr h wo curvs inrsc, h wo angn lins will b prpnicular. 6. (a Sinc h lin passs hrough h origin
More informationLecture 1: Numerical Integration The Trapezoidal and Simpson s Rule
Lcur : Numrical ngraion Th Trapzoidal and Simpson s Rul A problm Th probabiliy of a normally disribud (man µ and sandard dviaion σ ) vn occurring bwn h valus a and b is B A P( a x b) d () π whr a µ b -
More informationwhere: u: input y: output x: state vector A, B, C, D are const matrices
Sa pac modl: linar: y or in om : Sa q : f, u Oupu q : y h, u u Du F Gu y H Ju whr: u: inpu y: oupu : a vcor,,, D ar con maric Eampl " $ & ' " $ & 'u y " & * * * * [ ],, D H D I " $ " & $ ' " & $ ' " &
More informationCSE 245: Computer Aided Circuit Simulation and Verification
CSE 45: Compur Aidd Circui Simulaion and Vrificaion Fall 4, Sp 8 Lcur : Dynamic Linar Sysm Oulin Tim Domain Analysis Sa Equaions RLC Nwork Analysis by Taylor Expansion Impuls Rspons in im domain Frquncy
More informationFinal Exam : Solutions
Comp : Algorihm and Daa Srucur Final Exam : Soluion. Rcuriv Algorihm. (a) To bgin ind h mdian o {x, x,... x n }. Sinc vry numbr xcp on in h inrval [0, n] appar xacly onc in h li, w hav ha h mdian mu b
More informationSample Final Exam (finals03) Covering Chapters 1-9 of Fundamentals of Signals & Systems
Sample Final Exam Covering Chaper 9 (final04) Sample Final Exam (final03) Covering Chaper 9 of Fundamenal of Signal & Syem Problem (0 mar) Conider he caual opamp circui iniially a re depiced below. I LI
More informationMidterm exam 2, April 7, 2009 (solutions)
Univrsiy of Pnnsylvania Dparmn of Mahmaics Mah 26 Honors Calculus II Spring Smsr 29 Prof Grassi, TA Ashr Aul Midrm xam 2, April 7, 29 (soluions) 1 Wri a basis for h spac of pairs (u, v) of smooh funcions
More informationWave Equation (2 Week)
Rfrnc Wav quaion ( Wk 6.5 Tim-armonic filds 7. Ovrviw 7. Plan Wavs in Losslss Mdia 7.3 Plan Wavs in Loss Mdia 7.5 Flow of lcromagnic Powr and h Poning Vcor 7.6 Normal Incidnc of Plan Wavs a Plan Boundaris
More informationAN INTRODUCTION TO FOURIER ANALYSIS PROF. VEDAT TAVSANOĞLU
A IRODUCIO O FOURIER AALYSIS PROF. VEDA AVSAOĞLU 994 A IRODUCIO O FOURIER AALYSIS ABLE OF COES. HE FOURIER SERIES ---------------------------------------------------------------------3.. Priodic Funcions-----------------------------------------------------------------------3..
More informationVeer Surendra Sai University of Technology, Burla. S u b j e c t : S i g n a l s a n d S y s t e m s - I S u b j e c t c o d e : B E E
Vr Surndra Sai Univriy of Tchnology, Burla Dparmn o f E l c r i c a l & E l c r o n i c E n g g S u b j c : S i g n a l a n d S y m - I S u b j c c o d : B E E - 6 0 5 B r a n c h m r : E E E 5 h m SYLLABUS
More informationLecture 26: Leapers and Creepers
Lcur 6: Lapr and Crpr Scrib: Grain Jon (and Marin Z. Bazan) Dparmn of Economic, MIT May, 005 Inroducion Thi lcur conidr h analyi of h non-parabl CTRW in which h diribuion of p iz and im bwn p ar dpndn.
More informationNikesh Bajaj. Fourier Analysis and Synthesis Tool. Guess.? Question??? History. Fourier Series. Fourier. Nikesh Bajaj
Guss.? ourir Analysis an Synhsis Tool Qusion??? niksh.473@lpu.co.in Digial Signal Procssing School of Elcronics an Communicaion Lovly Profssional Univrsiy Wha o you man by Transform? Wha is /Transform?
More informationSpring 2006 Process Dynamics, Operations, and Control Lesson 2: Mathematics Review
Spring 6 Procss Dynamics, Opraions, and Conrol.45 Lsson : Mahmaics Rviw. conx and dircion Imagin a sysm ha varis in im; w migh plo is oupu vs. im. A plo migh imply an quaion, and h quaion is usually an
More information2. The Laplace Transform
Th aac Tranorm Inroucion Th aac ranorm i a unamna an vry uu oo or uying many nginring robm To in h aac ranorm w conir a comx variab σ, whr σ i h ra ar an i h imaginary ar or ix vau o σ an w viw a a oin
More informationLecture 4: Laplace Transforms
Lur 4: Lapla Transforms Lapla and rlad ransformaions an b usd o solv diffrnial quaion and o rdu priodi nois in signals and imags. Basially, hy onvr h drivaiv opraions ino mulipliaion, diffrnial quaions
More informationEE 350 Signals and Systems Spring 2005 Sample Exam #2 - Solutions
EE 35 Signals an Sysms Spring 5 Sampl Exam # - Soluions. For h following signal x( cos( sin(3 - cos(5 - T, /T x( j j 3 j 3 j j 5 j 5 j a -, a a -, a a - ½, a 3 /j-j -j/, a -3 -/jj j/, a 5 -½, a -5 -½,
More information18.03SC Unit 3 Practice Exam and Solutions
Sudy Guide on Sep, Dela, Convoluion, Laplace You can hink of he ep funcion u() a any nice mooh funcion which i for < a and for > a, where a i a poiive number which i much maller han any ime cale we care
More informationInverse Fourier Transform. Properties of Continuous time Fourier Transform. Review. Linearity. Reading Assignment Oppenheim Sec pp.289.
Convrgnc of ourir Trnsform Rding Assignmn Oppnhim Sc 42 pp289 Propris of Coninuous im ourir Trnsform Rviw Rviw or coninuous-im priodic signl x, j x j d Invrs ourir Trnsform 2 j j x d ourir Trnsform Linriy
More informationEECE 301 Signals & Systems Prof. Mark Fowler
EECE 31 Signal & Syem Prof. Mark Fowler Noe Se #27 C-T Syem: Laplace Tranform Power Tool for yem analyi Reading Aignmen: Secion 6.1 6.3 of Kamen and Heck 1/18 Coure Flow Diagram The arrow here how concepual
More informationINTRODUCTION TO AUTOMATIC CONTROLS INDEX LAPLACE TRANSFORMS
adjoint...6 block diagram...4 clod loop ytm... 5, 0 E()...6 (t)...6 rror tady tat tracking...6 tracking...6...6 gloary... 0 impul function...3 input...5 invr Laplac tranform, INTRODUCTION TO AUTOMATIC
More information2.1. Differential Equations and Solutions #3, 4, 17, 20, 24, 35
MATH 5 PS # Summr 00.. Diffrnial Equaions and Soluions PS.# Show ha ()C #, 4, 7, 0, 4, 5 ( / ) is a gnral soluion of h diffrnial quaion. Us a compur or calculaor o skch h soluions for h givn valus of h
More informationMa/CS 6a Class 15: Flows and Bipartite Graphs
//206 Ma/CS 6a Cla : Flow and Bipari Graph By Adam Shffr Rmindr: Flow Nwork A flow nwork i a digraph G = V, E, oghr wih a ourc vrx V, a ink vrx V, and a capaciy funcion c: E N. Capaciy Sourc 7 a b c d
More informationCIVL 8/ D Boundary Value Problems - Quadrilateral Elements (Q4) 1/8
CIVL 8/7111 -D Boundar Vau Prom - Quadriara Emn (Q) 1/8 ISOPARAMERIC ELEMENS h inar rianguar mn and h iinar rcanguar mn hav vra imporan diadvanag. 1. Boh mn ar una o accura rprn curvd oundari, and. h provid
More informationThe transition:transversion rate ratio vs. the T-ratio.
PhyloMah Lcur 8 by Dan Vandrpool March, 00 opics of Discussion ransiion:ransvrsion ra raio Kappa vs. ransiion:ransvrsion raio raio alculaing h xpcd numbr of subsiuions using marix algbra Why h nral im
More informationfiziks Institute for NET/JRF, GATE, IIT JAM, JEST, TIFR and GRE in PHYSICAL SCIENCES MATEMATICAL PHYSICS SOLUTIONS are
MTEMTICL PHYSICS SOLUTIONS GTE- Q. Considr an ani-symmric nsor P ij wih indics i and j running from o 5. Th numbr of indpndn componns of h nsor is 9 6 ns: Soluion: Th numbr of indpndn componns of h nsor
More informationUNIT #5 EXPONENTIAL AND LOGARITHMIC FUNCTIONS
Answr Ky Nam: Da: UNIT # EXPONENTIAL AND LOGARITHMIC FUNCTIONS Par I Qusions. Th prssion is quivaln o () () 6 6 6. Th ponnial funcion y 6 could rwrin as y () y y 6 () y y (). Th prssion a is quivaln o
More informationApplied Statistics and Probability for Engineers, 6 th edition October 17, 2016
Applid Saisics and robabiliy for Enginrs, 6 h diion Ocobr 7, 6 CHATER Scion - -. a d. 679.. b. d. 88 c d d d. 987 d. 98 f d.. Thn, = ln. =. g d.. Thn, = ln.9 =.. -7. a., by symmry. b.. d...6. 7.. c...
More informationIntroduction to Fourier Transform
EE354 Signals and Sysms Inroducion o Fourir ransform Yao Wang Polychnic Univrsiy Som slids includd ar xracd from lcur prsnaions prpard y McClllan and Schafr Licns Info for SPFirs Slids his work rlasd undr
More informationwhereby we can express the phase by any one of the formulas cos ( 3 whereby we can express the phase by any one of the formulas
Third In-Class Exam Soluions Mah 6, Profssor David Lvrmor Tusday, 5 April 07 [0] Th vrical displacmn of an unforcd mass on a spring is givn by h 5 3 cos 3 sin a [] Is his sysm undampd, undr dampd, criically
More informationCharging of capacitor through inductor and resistor
cur 4&: R circui harging of capacior hrough inducor and rsisor us considr a capacior of capacianc is conncd o a D sourc of.m.f. E hrough a rsisr of rsisanc R, an inducor of inducanc and a y K in sris.
More informationDouble Slits in Space and Time
Doubl Slis in Sac an Tim Gorg Jons As has bn ror rcnly in h mia, a am l by Grhar Paulus has monsra an inrsing chniqu for ionizing argon aoms by using ulra-shor lasr ulss. Each lasr uls is ffcivly on an
More informationAR(1) Process. The first-order autoregressive process, AR(1) is. where e t is WN(0, σ 2 )
AR() Procss Th firs-ordr auorgrssiv procss, AR() is whr is WN(0, σ ) Condiional Man and Varianc of AR() Condiional man: Condiional varianc: ) ( ) ( Ω Ω E E ) var( ) ) ( var( ) var( σ Ω Ω Ω Ω E Auocovarianc
More informationLaplace Transform. National Chiao Tung University Chun-Jen Tsai 10/19/2011
plc Trnorm Nionl Chio Tung Univriy Chun-Jn Ti /9/ Trnorm o Funcion Som opror rnorm uncion ino nohr uncion: d Dirniion: x x, or Dx x dx x Indini Ingrion: x dx c Dini Ingrion: x dx 9 A uncion my hv nicr
More informationDE Dr. M. Sakalli
DE-0 Dr. M. Sakalli DE 55 M. Sakalli a n n 0 a Lh.: an Linar g Equaions Hr if g 0 homognous non-homognous ohrwis driving b a forc. You know h quaions blow alrad. A linar firs ordr ODE has h gnral form
More informationEE Control Systems LECTURE 2
Copyrigh F.L. Lewi 999 All righ reerved EE 434 - Conrol Syem LECTURE REVIEW OF LAPLACE TRANSFORM LAPLACE TRANSFORM The Laplace ranform i very ueful in analyi and deign for yem ha are linear and ime-invarian
More information46. Let y = ln r. Then dy = dr, and so. = [ sin (ln r) cos (ln r)
98 Scion 7.. L w. Thn dw d, so d dw w dw. sin d (sin w)( wdw) w sin w dw L u w dv sin w dw du dw v cos w w sin w dw w cos w + cos w dw w cos w+ sin w+ sin d wsin wdw w cos w+ sin w+ cos + sin +. L w +
More informationMicroscopic Flow Characteristics Time Headway - Distribution
CE57: Traffic Flow Thory Spring 20 Wk 2 Modling Hadway Disribuion Microscopic Flow Characrisics Tim Hadway - Disribuion Tim Hadway Dfiniion Tim Hadway vrsus Gap Ahmd Abdl-Rahim Civil Enginring Dparmn,
More informationNote 6 Frequency Response
No 6 Frqucy Rpo Dparm of Mchaical Egirig, Uivriy Of Sakachwa, 57 Campu Driv, Sakaoo, S S7N 59, Caada Dparm of Mchaical Egirig, Uivriy Of Sakachwa, 57 Campu Driv, Sakaoo, S S7N 59, Caada. alyical Exprio
More informationOn the Derivatives of Bessel and Modified Bessel Functions with Respect to the Order and the Argument
Inrnaional Rsarch Journal of Applid Basic Scincs 03 Aailabl onlin a wwwirjabscom ISSN 5-838X / Vol 4 (): 47-433 Scinc Eplorr Publicaions On h Driais of Bssl Modifid Bssl Funcions wih Rspc o h Ordr h Argumn
More informationCONTROL SYSTEMS. Chapter 3 Mathematical Modelling of Physical Systems-Laplace Transforms. Prof.Dr. Fatih Mehmet Botsalı
CONTROL SYSTEMS Chaper Mahemaical Modelling of Phyical Syem-Laplace Tranform Prof.Dr. Faih Mehme Boalı Definiion Tranform -- a mahemaical converion from one way of hinking o anoher o make a problem eaier
More informationInstitute of Actuaries of India
Insiu of Acuaris of India ubjc CT3 Probabiliy and Mahmaical aisics Novmbr Examinaions INDICATIVE OLUTION Pag of IAI CT3 Novmbr ol. a sampl man = 35 sampl sandard dviaion = 36.6 b for = uppr bound = 35+*36.6
More informationECE Connections: What do Roots of Unity have to do with OP-AMPs? Louis Scharf, Colorado State University PART 1: Why Complex?
ECE Conncion: Wha do Roo of Uni hav o do wih OP-AMP? Loui Scharf, Colorado Sa Univri PART : Wh Compl?. Curioi, M favori curioi i : π π ( ) 0.07... π π ECE Conncion: Colorado Sa Univri Ocobr 007 . Quion,
More information4.1 The Uniform Distribution Def n: A c.r.v. X has a continuous uniform distribution on [a, b] when its pdf is = 1 a x b
4. Th Uniform Disribuion Df n: A c.r.v. has a coninuous uniform disribuion on [a, b] whn is pdf is f x a x b b a Also, b + a b a µ E and V Ex4. Suppos, h lvl of unblivabiliy a any poin in a Transformrs
More informationChapter 9 - The Laplace Transform
Chaper 9 - The Laplace Tranform Selece Soluion. Skech he pole-zero plo an region of convergence (if i exi) for hee ignal. ω [] () 8 (a) x e u = 8 ROC σ ( ) 3 (b) x e co π u ω [] ( ) () (c) x e u e u ROC
More informationFourier. Continuous time. Review. with period T, x t. Inverse Fourier F Transform. x t. Transform. j t
Coninuous im ourir rnsform Rviw. or coninuous-im priodic signl x h ourir sris rprsnion is x x j, j 2 d wih priod, ourir rnsform Wh bou priodic signls? W willl considr n priodic signl s priodic signl wih
More informationEECE.3620 Signal and System I
EECE.360 Signal and Sysem I Hengyong Yu, PhD Associae Professor Deparmen of Elecrical and Compuer Engineering Universiy of Massachuses owell EECE.360 Signal and Sysem I Ch.9.4. Geomeric Evaluaion of he
More informationCHAPTER. Linear Systems of Differential Equations. 6.1 Theory of Linear DE Systems. ! Nullcline Sketching. Equilibrium (unstable) at (0, 0)
CHATER 6 inar Sysms of Diffrnial Equaions 6 Thory of inar DE Sysms! ullclin Skching = y = y y υ -nullclin Equilibrium (unsabl) a (, ) h nullclin y = υ nullclin = h-nullclin (S figur) = + y y = y Equilibrium
More informationBoyce/DiPrima/Meade 11 th ed, Ch 6.1: Definition of Laplace Transform
Boy/DiPrima/Mad h d, Ch 6.: Diniion o apla Tranorm Elmnary Dirnial Equaion and Boundary Valu Problm, h diion, by William E. Boy, Rihard C. DiPrima, and Doug Mad 7 by John Wily & Son, In. Many praial nginring
More informationChapter 7: Inverse-Response Systems
Chaper 7: Invere-Repone Syem Normal Syem Invere-Repone Syem Baic Sar ou in he wrong direcion End up in he original eady-ae gain value Two or more yem wih differen magniude and cale in parallel Main yem
More information(1) Then we could wave our hands over this and it would become:
MAT* K285 Spring 28 Anthony Bnoit 4/17/28 Wk 12: Laplac Tranform Rading: Kohlr & Johnon, Chaptr 5 to p. 35 HW: 5.1: 3, 7, 1*, 19 5.2: 1, 5*, 13*, 19, 45* 5.3: 1, 11*, 19 * Pla writ-up th problm natly and
More informationC From Faraday's Law, the induced voltage is, C The effect of electromagnetic induction in the coil itself is called selfinduction.
Inducors and Inducanc C For inducors, v() is proporional o h ra of chang of i(). Inducanc (con d) C Th proporionaliy consan is h inducanc, L, wih unis of Hnris. 1 Hnry = 1 Wb / A or 1 V sc / A. C L dpnds
More information2. Transfer function. Kanazawa University Microelectronics Research Lab. Akio Kitagawa
. ransfr funion Kanazawa Univrsiy Mirolronis Rsarh Lab. Akio Kiagawa . Wavforms in mix-signal iruis Configuraion of mix-signal sysm x Digial o Analog Analog o Digial Anialiasing Digial moohing Filr Prossor
More informationChapter 6. Laplace Transforms
6- Chaper 6. Laplace Tranform 6.4 Shor Impule. Dirac Dela Funcion. Parial Fracion 6.5 Convoluion. Inegral Equaion 6.6 Differeniaion and Inegraion of Tranform 6.7 Syem of ODE 6.4 Shor Impule. Dirac Dela
More informationEXERCISE - 01 CHECK YOUR GRASP
DIFFERENTIAL EQUATION EXERCISE - CHECK YOUR GRASP 7. m hn D() m m, D () m m. hn givn D () m m D D D + m m m m m m + m m m m + ( m ) (m ) (m ) (m + ) m,, Hnc numbr of valus of mn will b. n ( ) + c sinc
More informationPartial Fraction Expansion
Paial Facion Expanion Whn ying o find h inv Laplac anfom o inv z anfom i i hlpfl o b abl o bak a complicad aio of wo polynomial ino fom ha a on h Laplac Tanfom o z anfom abl. W will illa h ing Laplac anfom.
More informationGeneral Article Application of differential equation in L-R and C-R circuit analysis by classical method. Abstract
Applicaion of Diffrnial... Gnral Aricl Applicaion of diffrnial uaion in - and C- circui analysis by classical mhod. ajndra Prasad gmi curr, Dparmn of Mahmaics, P.N. Campus, Pokhara Email: rajndraprasadrgmi@yahoo.com
More information3(8 ) (8 x x ) 3x x (8 )
Scion - CHATER -. a d.. b. d.86 c d 8 d d.9997 f g 6. d. d. Thn, = ln. =. =.. d Thn, = ln.9 =.7 8 -. a d.6 6 6 6 6 8 8 8 b 9 d 6 6 6 8 c d.8 6 6 6 6 8 8 7 7 d 6 d.6 6 6 6 6 6 6 8 u u u u du.9 6 6 6 6 6
More informationFrequency Response. Lecture #12 Chapter 10. BME 310 Biomedical Computing - J.Schesser
Frquncy Rspns Lcur # Chapr BME 3 Bimdical Cmpuing - J.Schssr 99 Idal Filrs W wan sudy Hω funcins which prvid frquncy slciviy such as: Lw Pass High Pass Band Pass Hwvr, w will lk a idal filring, ha is,
More informationCPSC 211 Data Structures & Implementations (c) Texas A&M University [ 259] B-Trees
CPSC 211 Daa Srucurs & Implmnaions (c) Txas A&M Univrsiy [ 259] B-Trs Th AVL r and rd-black r allowd som variaion in h lnghs of h diffrn roo-o-laf pahs. An alrnaiv ida is o mak sur ha all roo-o-laf pahs
More information10. 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 informationSerial : 4LS1_A_EC_Signal & Systems_230918
Serial : LS_A_EC_Signal & Syem_8 CLASS TEST (GATE) Delhi oida Bhopal Hyderabad Jaipur Lucknow Indore Pune Bhubanewar Kolkaa Pana Web: E-mail: info@madeeay.in Ph: -56 CLASS TEST 8- ELECTROICS EGIEERIG Subjec
More informationExponential Sawtooth
ECPE 36 HOMEWORK 3: PROPERTIES OF THE FOURIER TRANSFORM SOLUTION. Exponenial Sawooh: The eaie way o do hi problem i o look a he Fourier ranform of a ingle exponenial funcion, () = exp( )u(). From he able
More informationLecture 1: Growth and decay of current in RL circuit. Growth of current in LR Circuit. D.K.Pandey
cur : Growh and dcay of currn in circui Growh of currn in Circui us considr an inducor of slf inducanc is conncd o a DC sourc of.m.f. E hrough a rsisr of rsisanc and a ky K in sris. Whn h ky K is swichd
More informationReview Lecture 5. The source-free R-C/R-L circuit Step response of an RC/RL circuit. The time constant = RC The final capacitor voltage v( )
Rviw Lcur 5 Firs-ordr circui Th sourc-fr R-C/R-L circui Sp rspons of an RC/RL circui v( ) v( ) [ v( 0) v( )] 0 Th i consan = RC Th final capacior volag v() Th iniial capacior volag v( 0 ) Volag/currn-division
More informationAlgorithmic Discrete Mathematics 6. Exercise Sheet
Algorihmic Dicree Mahemaic. Exercie Shee Deparmen of Mahemaic SS 0 PD Dr. Ulf Lorenz 7. and 8. Juni 0 Dipl.-Mah. David Meffer Verion of June, 0 Groupwork Exercie G (Heap-Sor) Ue Heap-Sor wih a min-heap
More informationSOLUTIONS. 1. Consider two continuous random variables X and Y with joint p.d.f. f ( x, y ) = = = 15. Stepanov Dalpiaz
STAT UIUC Pracic Problms #7 SOLUTIONS Spanov Dalpiaz Th following ar a numbr of pracic problms ha ma b hlpful for compling h homwor, and will lil b vr usful for suding for ams.. Considr wo coninuous random
More informationRelation between Fourier Series and Transform
EE 37-3 8 Ch. II: Inro. o Sinls Lcur 5 Dr. Wih Abu-Al-Su Rlion bwn ourir Sris n Trnsform Th ourir Trnsform T is riv from h finiion of h ourir Sris S. Consir, for xmpl, h prioic complx sinl To wih prio
More information6.8 Laplace Transform: General Formulas
48 HAP. 6 Laplace Tranform 6.8 Laplace Tranform: General Formula Formula Name, ommen Sec. F() l{ f ()} e f () d f () l {F()} Definiion of Tranform Invere Tranform 6. l{af () bg()} al{f ()} bl{g()} Lineariy
More information13.1 Circuit Elements in the s Domain Circuit Analysis in the s Domain The Transfer Function and Natural Response 13.
Chaper 3 The Laplace Tranform in Circui Analyi 3. Circui Elemen in he Domain 3.-3 Circui Analyi in he Domain 3.4-5 The Tranfer Funcion and Naural Repone 3.6 The Tranfer Funcion and he Convoluion Inegral
More informationChapter 2 The Derivative Business Calculus 99
Chapr Th Drivaiv Businss Calculus 99 Scion 5: Drivaivs of Formulas In his scion, w ll g h rivaiv ruls ha will l us fin formulas for rivaivs whn our funcion coms o us as a formula. This is a vry algbraic
More informationVoltage v(z) ~ E(z)D. We can actually get to this wave behavior by using circuit theory, w/o going into details of the EM fields!
Considr a pair of wirs idal wirs ngh >, say, infinily long olag along a cabl can vary! D olag v( E(D W can acually g o his wav bhavior by using circui hory, w/o going ino dails of h EM filds! Thr
More informationRevisiting what you have learned in Advanced Mathematical Analysis
Fourir sris Rvisiing wh you hv lrnd in Advncd Mhmicl Anlysis L f x b priodic funcion of priod nd is ingrbl ovr priod. f x cn b rprsnd by rigonomric sris, f x n cos nx bn sin nx n cos x b sin x cosx b whr
More informationTo become more mathematically correct, Circuit equations are Algebraic Differential equations. from KVL, KCL from the constitutive relationship
Laplace Tranform (Lin & DeCarlo: Ch 3) ENSC30 Elecric Circui II The Laplace ranform i an inegral ranformaion. I ranform: f ( ) F( ) ime variable complex variable From Euler > Lagrange > Laplace. Hence,
More informationResponse of LTI Systems to Complex Exponentials
3 Fourir sris coiuous-im Rspos of LI Sysms o Complx Expoials Ouli Cosidr a LI sysm wih h ui impuls rspos Suppos h ipu sigal is a complx xpoial s x s is a complx umbr, xz zis a complx umbr h or h h w will
More informationMixing Real-Time and Non-Real-Time. CSCE 990: Real-Time Systems. Steve Goddard.
CSCE 990: Ral-Tim Sym Mixing Ral-Tim and Non-Ral-Tim goddard@c.unl.du hp://www.c.unl.du/~goddard/cour/raltimsym 1 Ral-Tim Sym Mixd Job - 1 Mixing Ral-Tim and Non-Ral-Tim in Prioriy-Drivn Sym (Chapr 7 of
More information5. An object moving along an x-coordinate axis with its scale measured in meters has a velocity of 6t
AP CALCULUS FINAL UNIT WORKSHEETS ACCELERATION, VELOCTIY AND POSITION In problms -, drmin h posiion funcion, (), from h givn informaion.. v (), () = 5. v ()5, () = b g. a (), v() =, () = -. a (), v() =
More informationFourier Series: main points
BIOEN 3 Lcur 6 Fourir rasforms Novmbr 9, Fourir Sris: mai pois Ifii sum of sis, cosis, or boh + a a cos( + b si( All frqucis ar igr mulipls of a fudamal frqucy, o F.S. ca rprs ay priodic fucio ha w ca
More informationChapter 6 Differential Equations and Mathematical Modeling
6 Scion 6. hapr 6 Diffrnial Equaions and Mahmaical Modling Scion 6. Slop Filds and Eulr s Mhod (pp. ) Eploraion Sing h Slops. Sinc rprsns a lin wih a slop of, w should d pc o s inrvals wih no chang in.
More informationInterpolation and Pulse Shaping
EE345S Real-Time Digial Signal Proceing Lab Spring 2006 Inerpolaion and Pule Shaping Prof. Brian L. Evan Dep. of Elecrical and Compuer Engineering The Univeriy of Texa a Auin Lecure 7 Dicree-o-Coninuou
More information( ) ( ) if t = t. It must satisfy the identity. So, bulkiness of the unit impulse (hyper)function is equal to 1. The defining characteristic is
UNIT IMPULSE RESPONSE, UNIT STEP RESPONSE, STABILITY. Uni impulse funcion (Dirac dela funcion, dela funcion) rigorously defined is no sricly a funcion, bu disribuion (or measure), precise reamen requires
More information