4 Convolution. Recommended Problems. x2[n] 1 2[n]

Size: px
Start display at page:

Download "4 Convolution. Recommended Problems. x2[n] 1 2[n]"

Transcription

1 4 Convoluion Recommended Problems P4.1 This problem is a simple example of he use of superposiion. Suppose ha a discree-ime linear sysem has oupus y[n] for he given inpus x[n] as shown in Figure P Inpu x[n] Oupuy [n] III xn] yi[n] 0 0 n 0 n x2[n] 1 2[n] 0 0 n 0 0 n X 3 [n] y3[n].-. n -n Figure P4.1-1 Deermine he response y 4 [n] when he inpu is as shown in Figure P I x4[n] Figure P4.1-2 (a) Express x 4 [n] as a linear combinaion of x 1 [n], x 2 [n], and x 3 [n]. (b) Using he fac ha he sysem is linear, deermine y 4 [n], he response o x 4 [n]. (c) From he inpu-oupu pairs in Figure P4.1-1, deermine wheher he sysem is ime-invarian. P4-1

2 Signals and Sysems P4-2 P4.2 Deermine he discree-ime convoluion of x[n] and h[n] for he following wo cases. (a) x[n] h[n] Fiur 11J P (b) Figure P h[ni x [n ] X Figure P4.2-2 P4.3 Deermine he coninuous-ime convoluion of x() and h() for he following hree cases: (a) x() h () Figure P4.3-1

3 Convoluion / Problems P4-3 (b) x() h () 1-- -(- 1) u ( - 1) u( Figure P4.3-2 (c) x() h () F6(-2) -l Figure P4.3-3 P4.4 Consider a discree-ime, linear, shif-invarian sysem ha has uni sample response h[n] and inpu x[n]. (a) Skech he response of his sysem if x[n] = b[n - h[n] = (i)"u[n]. no], for some no > 0, and (b) Evaluae and skech he oupu of he sysem if h[n] = (I)"u[n] and x[n] = u[n]. (c) Consider reversing he role of he inpu and sysem response in par (b). Tha is, h[n] = u[n], x[n] = (I)"u[n] Evaluae he sysem oupu y[n] and skech. P4.5 (a) Using convoluion, deermine and skech he responses of a linear, ime-invarian sysem wih impulse response h() = e- 2 u() o each of he wo inpus x 1 (), x 2 () shown in Figures P4.5-1 and P Use yi() o denoe he response o x 1 () and use y 2 () o denoe he response o x 2 ().

4 Signals and Sysems P4-4 (i) X 1 () = u() 0 Figure P4.5-1 (ii) x 2 () Figure P4.5-2 (b) x 2 () can be expressed in erms of x,() as x 2 () = 2[x() - xi( - 3)] By aking advanage of he lineariy and ime-invariance properies, deermine how y 2 () can be expressed in erms of yi(). Verify your expression by evaluaing i wih yl() obained in par (a) and comparing i wih y 2 () obained in par (a). Opional Problems P4.6 Graphically deermine he coninuous-ime convoluion of h() and x() for he cases shown in Figures P4.6-1 and P4.6-2.

5 Convoluion / Problems P4-5 (a) h() x() Figure P4.6-1 (b) h () x() Figure P4.6-2 P4.7 Compue he convoluion y[n] = x[n] * h[n] when Assume ha a and # are no equal. x[n] =au[n], O < a< 1, h[n] =#"u[n], 0 < #< 1 P4.8 Suppose ha h() is as shown in Figure P4.8 and x() is an impulse rain, i.e., x() = ( of-kt) k= -o0

6 Signals and Sysems P4-6 (a) Skech x(). (b) Assuming T = 2, deermine and skech y() = x() * h(). P4.9 Deermine if each of he following saemens is rue in general. Provide proofs for hose ha you hink are rue and counerexamples for hose ha you hink are false. (a) x[n] *{h[ng[n]} = {x[n] *h[n]}g[n] (b) If y() = x() * h(), hen y(2) = 2x(2) * h(2). (c) If x() and h() are odd signals, hen y() = x() * h() is an even signal. (d) If y() = x() * h(), hen Ev{y()} = x() * Ev{h()} + Ev{x()} * h(). P4.10 Le 1 1 () and 2 2 () be wo periodic signals wih a common period To. I is no oo difficul o check ha he convoluion of 1 1 () and 2 () does no converge. However, i is someimes useful o consider a form of convoluion for such signals ha is referred o as periodicconvoluion.specifically, we define he periodic convoluion of 1 () and X 2 () as TO g() = T 1 (r)- 2 ( - r) dr = 1 ()* 2 () (P4.10-1) Noe ha we are inegraing over exacly one period. (a) Show ha q() is periodic wih period To. (b) Consider he signal a + T 0 Pa() 1(rF)2( - r) dr, = fa where a is an arbirary real number. Show ha 9() = Ya() Hin: Wrie a = kto - b, where 0 b < To. (c) Compue he periodic convoluion of he signals depiced in Figure P4.10-1, where To = 1.

7 Convoluion / Problems P4-7 e R2 () Figure P (d) Consider he signals x1[n] and x 2 [n] depiced in Figure P These signals are periodic wih period 6. Compue and skech heir periodic convoluion using No = 6. IT I '1 x, [n] I II... I-61 0II T II.. 2 1? 11 X2 [n] Figure P (e) Since hese signals are periodic wih period 6, hey are also periodic wih period 12. Compue he periodic convoluion of xi[n] and x2[n] using No = 12. P4.11 One imporan use of he concep of inverse sysems is o remove disorions of some ype. A good example is he problem of removing echoes from acousic signals. For example, if an audiorium has a percepible echo, hen an iniial acousic impulse is

8 Signals and Sysems P4-8 followed by aenuaed versions of he sound a regularly spaced inervals. Consequenly, a common model for his phenomenon is a linear, ime-invarian sysem wih an impulse response consising of a rain of impulses: h() = [ hkb(-kt) (P4.11-1) k=o Here he echoes occur T s apar, and hk represens he gain facor on he kh echo resuling from an iniial acousic impulse. (a) Suppose ha x() represens he original acousic signal (he music produced by an orchesra, for example) and ha y() = x() * h() is he acual signal ha is heard if no processing is done o remove he echoes. To remove he disorion inroduced by he echoes, assume ha a microphone is used o sense y() and ha he resuling signal is ransduced ino an elecrical signal. We will also use y() o denoe his signal, as i represens he elecrical equivalen of he acousic signal, and we can go from one o he oher via acousic-elecrical conversion sysems. The imporan poin o noe is ha he sysem wih impulse response given in eq. (P4.11-1) is inverible. Therefore, we can find an LTI sysem wih impulse response g() such ha y() *g() = x() and hus, by processing he elecrical signal y() in his fashion and hen convering back o an acousic signal, we can remove he roublesome echoes. The required impulse response g() is also an impulse rain: g() = ( k=o gkao-kt) Deermine he algebraic equaions ha he successive gk mus saisfy and solve for gi, g 2, and g 3 in erms of he hk. [Hin: You may find par (a) of Problem 3.16 of he ex (page 136) useful.] (b) Suppose ha ho = 1, hi = i, and hi = 0 for all i > 2. Wha is g() in his case? (c) A good model for he generaion of echoes is illusraed in Figure P4.11. Each successive echo represens a fedback version of y(), delayed by T s and scaled by a. Typically 0 < a < 1 because successive echoes are aenuaed. x() ± y() Delay T (i) Figure P4.11 Wha is he impulse response of his sysem? (Assume iniial res, i.e., y() = 0 for < 0 if x() = 0 for < 0.) (ii) Show ha he sysem is sable if 0 < a < 1 and unsable if a > 1. (iii) Wha is g() in his case? Consruc a realizaion of his inverse sysem using adders, coefficien mulipliers, and T-s delay elemens.

9 Convoluion / Problems P4-9 Alhough we have phrased his discussion in erms of coninuous-ime sysems because of he applicaion we are considering, he same general ideas hold in discree ime. Tha is, he LTI sysem wih impulse response h[n] = ( hks[n-kn] k=o is inverible and has as is inverse an LTI sysem wih impulse response g[n] = (g [nkn] k=o I is no difficul o check ha he gi saisfy he same algebraic equaions as in par (a). (d) Consider he discree-ime LTI sysem wih impulse response h[n] = ( S[n-kN] k=-m This sysem is no inverible. Find wo inpus ha produce he same oupu. P4.12 Our developmen of he convoluion sum represenaion for discree-ime LTI sysems was based on using he uni sample funcion as a building block for he represenaion of arbirary inpu signals. This represenaion, ogeher wih knowledge of he response o 5[n] and he propery of superposiion, allowed us o represen he sysem response o an arbirary inpu in erms of a convoluion. In his problem we consider he use of oher signals as building blocks for he consrucion of arbirary inpu signals. Consider he following se of signals: $[n] = (i)"u[n], #[n ] = [n - k], k = 0, 1, ±2 3,... (a) Show ha an arbirary signal can be represened in he form + 00 x[n] = ( ak4[n - k] k= by deermining an explici expression for he coefficien ak in erms of he values of he signal x[n]. [Hin:Wha is he represenaion for 6[n]?] (b) Le r[n] be he response of an LTI sysem o he inpu x[n] = #[n]. Find an expression for he response y[n] o an arbirary inpu x[n] in erms of r[n] and x[n]. (c) Show ha y[n] can be wrien as y[n] = 0[n] * x[n] * r[n] by finding he signal 0[n]. (d) Use he resul of par (c) o express he impulse response of he sysem in erms of r[n]. Also, show ha 0[n] *#[n] = b[n]

10 MIT OpenCourseWare hp://ocw.mi.edu Resource: Signals and Sysems Professor Alan V. Oppenheim The following may no correspond o a paricular course on MIT OpenCourseWare, bu has been provided by he auhor as an individual learning resource. For informaion abou ciing hese maerials or our Terms of Use, visi: hp://ocw.mi.edu/erms.

Suggested Reading. Signals and Systems 4-2

Suggested Reading. Signals and Systems 4-2 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 information

The Transport Equation

The 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 information

Differential Equations. Solving for Impulse Response. Linear systems are often described using differential equations.

Differential 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 information

Complex Fourier Series. Adding these identities, and then dividing by 2, or subtracting them, and then dividing by 2i, will show that

Complex Fourier Series. Adding these identities, and then dividing by 2, or subtracting them, and then dividing by 2i, will show that Mah 344 May 4, Complex Fourier Series Par I: Inroducion The Fourier series represenaion for a funcion f of period P, f) = a + a k coskω) + b k sinkω), ω = π/p, ) can be expressed more simply using complex

More information

Analogue and Digital Signal Processing. First Term Third Year CS Engineering By Dr Mukhtiar Ali Unar

Analogue 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: 0-7-380-7 Ifeachor

More information

Fourier series. Learning outcomes

Fourier series. Learning outcomes Fourier series 23 Conens. Periodic funcions 2. Represening ic funcions by Fourier Series 3. Even and odd funcions 4. Convergence 5. Half-range series 6. The complex form 7. Applicaion of Fourier series

More information

Fourier Series Solution of the Heat Equation

Fourier Series Solution of the Heat Equation Fourier Series Soluion of he Hea Equaion Physical Applicaion; he Hea Equaion In he early nineeenh cenury Joseph Fourier, a French scienis and mahemaician who had accompanied Napoleon on his Egypian campaign,

More information

23.3. Even and Odd Functions. Introduction. Prerequisites. Learning Outcomes

23.3. Even and Odd Functions. Introduction. Prerequisites. Learning Outcomes Even and Odd Funcions 3.3 Inroducion In his Secion we examine how o obain Fourier series of periodic funcions which are eiher even or odd. We show ha he Fourier series for such funcions is considerabl

More information

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

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

More information

23.3. Even and Odd Functions. Introduction. Prerequisites. Learning Outcomes

23.3. Even and Odd Functions. Introduction. Prerequisites. Learning Outcomes Even and Odd Funcions 23.3 Inroducion In his Secion we examine how o obain Fourier series of periodic funcions which are eiher even or odd. We show ha he Fourier series for such funcions is considerabl

More information

State Machines: Brief Introduction to Sequencers Prof. Andrew J. Mason, Michigan State University

State Machines: Brief Introduction to Sequencers Prof. Andrew J. Mason, Michigan State University Inroducion ae Machines: Brief Inroducion o equencers Prof. Andrew J. Mason, Michigan ae Universiy A sae machine models behavior defined by a finie number of saes (unique configuraions), ransiions beween

More information

cooking trajectory boiling water B (t) microwave 0 2 4 6 8 101214161820 time t (mins)

cooking 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 information

Chapter 7. Response of First-Order RL and RC Circuits

Chapter 7. Response of First-Order RL and RC Circuits Chaper 7. esponse of Firs-Order 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 information

Signal Processing and Linear Systems I

Signal Processing and Linear Systems I Sanford Universiy Summer 214-215 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 14-15, Gibbons

More information

Relative velocity in one dimension

Relative velocity in one dimension Connexions module: m13618 1 Relaive velociy in one dimension Sunil Kumar Singh This work is produced by The Connexions Projec and licensed under he Creaive Commons Aribuion License Absrac All quaniies

More information

Math 201 Lecture 12: Cauchy-Euler Equations

Math 201 Lecture 12: Cauchy-Euler Equations Mah 20 Lecure 2: Cauchy-Euler Equaions Feb., 202 Many examples here are aken from he exbook. The firs number in () refers o he problem number in he UA Cusom ediion, he second number in () refers o he problem

More information

Appendix A: Area. 1 Find the radius of a circle that has circumference 12 inches.

Appendix A: Area. 1 Find the radius of a circle that has circumference 12 inches. Appendi A: Area worked-ou s o Odd-Numbered Eercises Do no read hese worked-ou s before aemping o do he eercises ourself. Oherwise ou ma mimic he echniques shown here wihou undersanding he ideas. Bes wa

More information

HANDOUT 14. A.) Introduction: Many actions in life are reversible. * Examples: Simple One: a closed door can be opened and an open door can be closed.

HANDOUT 14. A.) Introduction: Many actions in life are reversible. * Examples: Simple One: a closed door can be opened and an open door can be closed. Inverse Funcions Reference Angles Inverse Trig Problems Trig Indeniies HANDOUT 4 INVERSE FUNCTIONS KEY POINTS A.) Inroducion: Many acions in life are reversible. * Examples: Simple One: a closed door can

More information

Graphing the Von Bertalanffy Growth Equation

Graphing the Von Bertalanffy Growth Equation file: d:\b173-2013\von_beralanffy.wpd dae: Sepember 23, 2013 Inroducion Graphing he Von Beralanffy Growh Equaion Previously, we calculaed regressions of TL on SL for fish size daa and ploed he daa and

More information

Understanding Sequential Circuit Timing

Understanding Sequential Circuit Timing ENGIN112: Inroducion o Elecrical and Compuer Engineering Fall 2003 Prof. Russell Tessier Undersanding Sequenial Circui Timing Perhaps he wo mos disinguishing characerisics of a compuer are is processor

More information

A Brief Introduction to the Consumption Based Asset Pricing Model (CCAPM)

A Brief Introduction to the Consumption Based Asset Pricing Model (CCAPM) A Brief Inroducion o he Consumpion Based Asse Pricing Model (CCAPM We have seen ha CAPM idenifies he risk of any securiy as he covariance beween he securiy's rae of reurn and he rae of reurn on he marke

More information

Equation for a line. Synthetic Impulse Response 0.5 0.5. 0 5 10 15 20 25 Time (sec) x(t) m

Equation 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 information

6.003 Homework #4 Solutions

6.003 Homework #4 Solutions 6.3 Homewk #4 Soluion Problem. Laplace Tranfm Deermine he Laplace ranfm (including he region of convergence) of each of he following ignal: a. x () = e 2(3) u( 3) X = e 3 2 ROC: Re() > 2 X () = x ()e d

More information

17 Laplace transform. Solving linear ODE with piecewise continuous right hand sides

17 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 information

DIFFERENTIAL EQUATIONS with TI-89 ABDUL HASSEN and JAY SCHIFFMAN. A. Direction Fields and Graphs of Differential Equations

DIFFERENTIAL EQUATIONS with TI-89 ABDUL HASSEN and JAY SCHIFFMAN. A. Direction Fields and Graphs of Differential Equations DIFFERENTIAL EQUATIONS wih TI-89 ABDUL HASSEN and JAY SCHIFFMAN We will assume ha he reader is familiar wih he calculaor s keyboard and he basic operaions. In paricular we have assumed ha he reader knows

More information

Chapter 2: Principles of steady-state converter analysis

Chapter 2: Principles of steady-state converter analysis Chaper 2 Principles of Seady-Sae Converer Analysis 2.1. Inroducion 2.2. Inducor vol-second balance, capacior charge balance, and he small ripple approximaion 2.3. Boos converer example 2.4. Cuk converer

More information

Cointegration: The Engle and Granger approach

Cointegration: The Engle and Granger approach Coinegraion: The Engle and Granger approach Inroducion Generally one would find mos of he economic variables o be non-saionary I(1) variables. Hence, any equilibrium heories ha involve hese variables require

More information

3 Runge-Kutta Methods

3 Runge-Kutta Methods 3 Runge-Kua Mehods In conras o he mulisep mehods of he previous secion, Runge-Kua mehods are single-sep mehods however, muliple sages per sep. They are moivaed by he dependence of he Taylor mehods on he

More information

Fourier Series & The Fourier Transform

Fourier 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 information

9. Capacitor and Resistor Circuits

9. 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 information

CHARGE AND DISCHARGE OF A CAPACITOR

CHARGE 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 information

RC, RL and RLC circuits

RC, 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 information

Differential Equations and Linear Superposition

Differential 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 information

Part 1: White Noise and Moving Average Models

Part 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 information

11/6/2013. Chapter 14: Dynamic AD-AS. Introduction. Introduction. Keeping track of time. The model s elements

11/6/2013. Chapter 14: Dynamic AD-AS. Introduction. Introduction. Keeping track of time. The model s elements Inroducion Chaper 14: Dynamic D-S 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 cuing-edge

More information

Mathematics in Pharmacokinetics What and Why (A second attempt to make it clearer)

Mathematics 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 information

Chapter 6. First Order PDEs. 6.1 Characteristics The Simplest Case. u(x,t) t=1 t=2. t=0. Suppose u(x, t) satisfies the PDE.

Chapter 6. First Order PDEs. 6.1 Characteristics The Simplest Case. u(x,t) t=1 t=2. t=0. Suppose u(x, t) satisfies the PDE. Chaper 6 Firs Order PDEs 6.1 Characerisics 6.1.1 The Simples Case Suppose u(, ) saisfies he PDE where b, c are consan. au + bu = 0 If a = 0, he PDE is rivial (i says ha u = 0 and so u = f(). If a = 0,

More information

Name: Algebra II Review for Quiz #13 Exponential and Logarithmic Functions including Modeling

Name: 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 information

Signal Rectification

Signal 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, half-wae and fullwae. Le s firs consider he ideal

More information

Economics 140A Hypothesis Testing in Regression Models

Economics 140A Hypothesis Testing in Regression Models Economics 140A Hypohesis Tesing in Regression Models While i is algebraically simple o work wih a populaion model wih a single varying regressor, mos populaion models have muliple varying regressors 1

More information

and Decay Functions f (t) = C(1± r) t / K, for t 0, where

and Decay Functions f (t) = C(1± r) t / K, for t 0, where MATH 116 Exponenial Growh and Decay Funcions Dr. Neal, Fall 2008 A funcion ha grows or decays exponenially has he form f () = C(1± r) / K, for 0, where C is he iniial amoun a ime 0: f (0) = C r is he rae

More information

4 Convolution. Solutions to Recommended Problems

4 Convolution. Solutions to Recommended Problems 4 Convolution Solutions to Recommended Problems S4.1 The given input in Figure S4.1-1 can be expressed as linear combinations of xi[n], x 2 [n], X 3 [n]. x,[ n] 0 2 Figure S4.1-1 (a) x 4[n] = 2x 1 [n]

More information

SOLID MECHANICS TUTORIAL GEAR SYSTEMS. This work covers elements of the syllabus for the Edexcel module 21722P HNC/D Mechanical Principles OUTCOME 3.

SOLID MECHANICS TUTORIAL GEAR SYSTEMS. This work covers elements of the syllabus for the Edexcel module 21722P HNC/D Mechanical Principles OUTCOME 3. SOLI MEHNIS TUTORIL GER SYSTEMS This work covers elemens of he syllabus for he Edexcel module 21722P HN/ Mechanical Principles OUTOME 3. On compleion of his shor uorial you should be able o do he following.

More information

Solution of a differential equation of the second order by the method of NIGAM

Solution 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 information

Circuit Types. () i( t) ( )

Circuit 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 information

Inductance and Transient Circuits

Inductance 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 information

3 Signals and Systems: Part II

3 Signals and Systems: Part II 3 Signals and Systems: Part II Recommended Problems P3.1 Sketch each of the following signals. (a) x[n] = b[n] + 3[n - 3] (b) x[n] = u[n] - u[n - 5] (c) x[n] = 6[n] + 1n + (i)2 [n - 2] + (i)ag[n - 3] (d)

More information

MTH6121 Introduction to Mathematical Finance Lesson 5

MTH6121 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 information

Chapter 15: Superposition and Interference of Waves

Chapter 15: Superposition and Interference of Waves Chaper 5: Superposiion and Inerference of Waves Real waves are rarely purely sinusoidal (harmonic, bu hey can be represened by superposiions of harmonic waves In his chaper we explore wha happens when

More information

FACULTY OF MATHEMATICAL STUDIES MATHEMATICS FOR PART I ENGINEERING. Lectures

FACULTY OF MATHEMATICAL STUDIES MATHEMATICS FOR PART I ENGINEERING. Lectures FACULY OF MAHEMAICAL SUDIES MAHEMAICS FOR PAR I ENGINEERING Lecures MODULE 3 FOURIER SERIES Periodic signals Whole-range Fourier series 3 Even and odd uncions Periodic signals Fourier series are used in

More information

Module 4. Single-phase AC circuits. Version 2 EE IIT, Kharagpur

Module 4. Single-phase AC circuits. Version 2 EE IIT, Kharagpur Module 4 Single-phase 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 information

Chabot College Physics Lab RC Circuits Scott Hildreth

Chabot 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 information

Acceleration Lab Teacher s Guide

Acceleration 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 information

2.6 Limits at Infinity, Horizontal Asymptotes Math 1271, TA: Amy DeCelles. 1. Overview. 2. Examples. Outline: 1. Definition of limits at infinity

2.6 Limits at Infinity, Horizontal Asymptotes Math 1271, TA: Amy DeCelles. 1. Overview. 2. Examples. Outline: 1. Definition of limits at infinity .6 Limis a Infiniy, Horizonal Asympoes Mah 7, TA: Amy DeCelles. Overview Ouline:. Definiion of is a infiniy. Definiion of horizonal asympoe 3. Theorem abou raional powers of. Infinie is a infiniy This

More information

Using RCtime to Measure Resistance

Using RCtime to Measure Resistance Basic Express Applicaion Noe Using RCime o Measure Resisance Inroducion One common use for I/O pins is o measure he analog value of a variable resisance. Alhough a buil-in ADC (Analog o Digial Converer)

More information

UMR EMC Laboratory UMR EMC Laboratory Technical Report: TR

UMR EMC Laboratory UMR EMC Laboratory Technical Report: TR UMR EMC Laboraory UMR EMC Laboraory Dep. of Elecrical & Compuer Engineering 870 Miner Circle Universiy of Missouri Rolla Rolla, MO 65409-0040 UMR EMC Laboraory Technical Repor: TR0-8-00 Effec of Delay

More information

Permutations and Combinations

Permutations 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 k-value for he middle erm, divide

More information

Capacitors and inductors

Capacitors 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 information

On the degrees of irreducible factors of higher order Bernoulli polynomials

On 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 information

AP Calculus AB 2007 Scoring Guidelines

AP Calculus AB 2007 Scoring Guidelines AP Calculus AB 7 Scoring Guidelines The College Board: Connecing Sudens o College Success The College Board is a no-for-profi membership associaion whose mission is o connec sudens o college success and

More information

Section 7.1 Angles and Their Measure

Section 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 information

INDEPENDENT MARGINALS OF OPERATOR LÉVY S PROBABILITY MEASURES ON FINITE DIMENSIONAL VECTOR SPACES

INDEPENDENT MARGINALS OF OPERATOR LÉVY S PROBABILITY MEASURES ON FINITE DIMENSIONAL VECTOR SPACES Journal of Applied Analysis 1, 1 (1995), pp. 39 45 INDEPENDENT MARGINALS OF OPERATOR LÉVY S PROBABILITY MEASURES ON FINITE DIMENSIONAL VECTOR SPACES A. LUCZAK Absrac. We find exponens of independen marginals

More information

Answer, Key Homework 2 David McIntyre 45123 Mar 25, 2004 1

Answer, Key Homework 2 David McIntyre 45123 Mar 25, 2004 1 Answer, Key Homework 2 Daid McInyre 4123 Mar 2, 2004 1 This prin-ou should hae 1 quesions. Muliple-choice quesions may coninue on he ne column or page find all choices before making your selecion. The

More information

YTM is positively related to default risk. YTM is positively related to liquidity risk. YTM is negatively related to special tax treatment.

YTM is positively related to default risk. YTM is positively related to liquidity risk. YTM is negatively related to special tax treatment. . Two quesions for oday. A. Why do bonds wih he same ime o mauriy have differen YTM s? B. Why do bonds wih differen imes o mauriy have differen YTM s? 2. To answer he firs quesion les look a he risk srucure

More information

Duration 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.

Duration 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 UVA-F-38 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 information

Entropy: From the Boltzmann equation to the Maxwell Boltzmann distribution

Entropy: 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 information

Graduate Macro Theory II: Notes on Neoclassical Growth Model

Graduate 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 information

INVESTIGATION OF THE INFLUENCE OF UNEMPLOYMENT ON ECONOMIC INDICATORS

INVESTIGATION OF THE INFLUENCE OF UNEMPLOYMENT ON ECONOMIC INDICATORS INVESTIGATION OF THE INFLUENCE OF UNEMPLOYMENT ON ECONOMIC INDICATORS Ilona Tregub, Olga Filina, Irina Kondakova Financial Universiy under he Governmen of he Russian Federaion 1. Phillips curve In economics,

More information

4.8 Exponential Growth and Decay; Newton s Law; Logistic Growth and Decay

4.8 Exponential Growth and Decay; Newton s Law; Logistic Growth and Decay 324 CHAPTER 4 Exponenial and Logarihmic Funcions 4.8 Exponenial Growh and Decay; Newon s Law; Logisic Growh and Decay OBJECTIVES 1 Find Equaions of Populaions Tha Obey he Law of Uninhibied Growh 2 Find

More information

RC Circuit and Time Constant

RC Circuit and Time Constant ircui and Time onsan 8M Objec: Apparaus: To invesigae he volages across he resisor and capacior in a resisor-capacior circui ( circui) as he capacior charges and discharges. We also wish o deermine he

More information

Chapter 4: Exponential and Logarithmic Functions

Chapter 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 information

Chapter 2 Kinematics in One Dimension

Chapter 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 information

Rotational Inertia of a Point Mass

Rotational 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 information

AP Calculus BC 2010 Scoring Guidelines

AP Calculus BC 2010 Scoring Guidelines AP Calculus BC Scoring Guidelines The College Board The College Board is a no-for-profi membership associaion whose mission is o connec sudens o college success and opporuniy. Founded in, he College Board

More information

4. International Parity Conditions

4. International Parity Conditions 4. Inernaional ariy ondiions 4.1 urchasing ower ariy he urchasing ower ariy ( heory is one of he early heories of exchange rae deerminaion. his heory is based on he concep ha he demand for a counry's currency

More information

Reporting to Management

Reporting to Management CHAPTER 31 Reporing o Managemen Inroducion The success or oherwise of any business underaking depends primarily on earning revenue ha would generae sufficien resources for sound growh. To achieve his objecive,

More information

Vector Autoregressions (VARs): Operational Perspectives

Vector Autoregressions (VARs): Operational Perspectives Vecor Auoregressions (VARs): Operaional Perspecives Primary Source: Sock, James H., and Mark W. Wason, Vecor Auoregressions, Journal of Economic Perspecives, Vol. 15 No. 4 (Fall 2001), 101-115. Macroeconomericians

More information

Basic Circuit Elements - Prof J R Lucas

Basic Circuit Elements - Prof J R Lucas Basic Circui Elemens - Prof J ucas An elecrical circui is an inerconnecion of elecrical circui elemens. These circui elemens can be caegorized ino wo ypes, namely acive elemens and passive elemens. Some

More information

RC (Resistor-Capacitor) Circuits. AP Physics C

RC (Resistor-Capacitor) Circuits. AP Physics C (Resisor-Capacior 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 information

Cannibalization and Product Life Cycle Management

Cannibalization and Product Life Cycle Management Middle-Eas Journal of Scienific Research 19 (8): 1080-1084, 2014 ISSN 1990-9233 IDOSI Publicaions, 2014 DOI: 10.5829/idosi.mejsr.2014.19.8.11868 Cannibalizaion and Produc Life Cycle Managemen Ali Farrukh

More information

Matrix Analysis of Networks

Matrix Analysis of Networks Marix Analysis of Neworks is edious o analyse large nework using normal equaions. is easier and more convenien o formulae large neworks in marix form. To have a nea form of soluion, i is necessary o know

More information

5.8 Resonance 231. The study of vibrating mechanical systems ends here with the theory of pure and practical resonance.

5.8 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 information

Physics 111 Fall 2007 Electric Currents and DC Circuits

Physics 111 Fall 2007 Electric Currents and DC Circuits Physics 111 Fall 007 Elecric Currens and DC Circuis 1 Wha is he average curren when all he sodium channels on a 100 µm pach of muscle membrane open ogeher for 1 ms? Assume a densiy of 0 sodium channels

More information

1. 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,

1. 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 information

Module 3. R-L & R-C Transients. Version 2 EE IIT, Kharagpur

Module 3. R-L & R-C Transients. Version 2 EE IIT, Kharagpur Module 3 - & -C Transiens esson 0 Sudy of DC ransiens in - and -C circuis Objecives Definiion of inducance and coninuiy condiion for inducors. To undersand he rise or fall of curren in a simple series

More information

EDEXCEL NATIONAL CERTIFICATE/DIPLOMA UNIT 67 - FURTHER ELECTRICAL PRINCIPLES NQF LEVEL 3 OUTCOME 2 TUTORIAL 1 - TRANSIENTS

EDEXCEL NATIONAL CERTIFICATE/DIPLOMA UNIT 67 - FURTHER ELECTRICAL PRINCIPLES NQF LEVEL 3 OUTCOME 2 TUTORIAL 1 - TRANSIENTS EDEXEL NAIONAL ERIFIAE/DIPLOMA UNI 67 - FURHER ELERIAL PRINIPLE NQF LEEL 3 OUOME 2 UORIAL 1 - RANIEN Uni conen 2 Undersand he ransien behaviour of resisor-capacior (R) and resisor-inducor (RL) D circuis

More information

Lectures # 5 and 6: The Prime Number Theorem.

Lectures # 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 ζ-funcion o skech an argumen which would give an acual formula for π( and sugges

More information

The naive method discussed in Lecture 1 uses the most recent observations to forecast future values. That is, Y ˆ t + 1

The naive method discussed in Lecture 1 uses the most recent observations to forecast future values. That is, Y ˆ t + 1 Business Condiions & Forecasing Exponenial Smoohing LECTURE 2 MOVING AVERAGES AND EXPONENTIAL SMOOTHING OVERVIEW This lecure inroduces ime-series smoohing forecasing mehods. Various models are discussed,

More information

Fourier Series Approximation of a Square Wave

Fourier Series Approximation of a Square Wave OpenSax-CNX module: m4 Fourier Series Approximaion of a Square Wave Don Johnson his work is produced by OpenSax-CNX and licensed under he Creaive Commons Aribuion License. Absrac Shows how o use Fourier

More information

Newton s Laws of Motion

Newton 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 information

4 Fourier series. y(t) = h(τ)x(t τ)dτ = h(τ)e jω(t τ) dτ = h(τ)e jωτ e jωt dτ. = h(τ)e jωτ dτ e jωt = H(ω)e jωt.

4 Fourier series. y(t) = h(τ)x(t τ)dτ = h(τ)e jω(t τ) dτ = h(τ)e jωτ e jωt dτ. = h(τ)e jωτ dτ e jωt = H(ω)e jωt. 4 Fourier series Any LI sysem is compleely deermined by is impulse response h(). his is he oupu of he sysem when he inpu is a Dirac dela funcion a he origin. In linear sysems heory we are usually more

More information

Lecture 2: Telegrapher Equations For Transmission Lines. Power Flow.

Lecture 2: Telegrapher Equations For Transmission Lines. Power Flow. Whies, EE 481 Lecure 2 Page 1 of 13 Lecure 2: Telegraher Equaions For Transmission Lines. Power Flow. Microsri is one mehod for making elecrical connecions in a microwae circui. I is consruced wih a ground

More information

Second Order Linear Differential Equations

Second 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 information

Economics Honors Exam 2008 Solutions Question 5

Economics 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 information

THE EQUATIONS OF THE IDEAL LATCHES

THE EQUATIONS OF THE IDEAL LATCHES THE EUATIONS OF THE IDEAL LATHES SERBAN E. VLAD Oradea iy Hall, iaa Unirii Nr., 4000, Oradea, Romania www.geociies.com/serban_e_lad, serban_e_lad@yahoo.com ABSTRAT We presen he eqaions ha model seeral

More information

A Curriculum Module for AP Calculus BC Curriculum Module

A 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 no-for-profi membership associaion whose mission is o connec sudens o college success and opporuniy.

More information

Foreign Exchange and Quantos

Foreign Exchange and Quantos IEOR E4707: Financial Engineering: Coninuous-Time Models Fall 2010 c 2010 by Marin Haugh Foreign Exchange and Quanos These noes consider foreign exchange markes and he pricing of derivaive securiies in

More information

Chapter 4. Properties of the Least Squares Estimators. Assumptions of the Simple Linear Regression Model. SR3. var(e t ) = σ 2 = var(y t )

Chapter 4. Properties of the Least Squares Estimators. Assumptions of the Simple Linear Regression Model. SR3. var(e t ) = σ 2 = var(y t ) Chaper 4 Properies of he Leas Squares Esimaors Assumpions of he Simple Linear Regression Model SR1. SR. y = β 1 + β x + e E(e ) = 0 E[y ] = β 1 + β x SR3. var(e ) = σ = var(y ) SR4. cov(e i, e j ) = cov(y

More information

Modeling Stock Price Dynamics with Fuzzy Opinion Networks

Modeling Stock Price Dynamics with Fuzzy Opinion Networks Modeling Sock Price Dynamics wih Fuzzy Opinion Neworks Li-Xin Wang Deparmen of Auomaion Science and Technology Xian Jiaoong Universiy, Xian, P.R. China Email: lxwang@mail.xju.edu.cn Key words: Sock price

More information

Use SeDuMi to Solve LP, SDP and SCOP Problems: Remarks and Examples*

Use SeDuMi to Solve LP, SDP and SCOP Problems: Remarks and Examples* Use SeDuMi o Solve LP, SDP and SCOP Problems: Remarks and Examples* * his file was prepared by Wu-Sheng Lu, Dep. of Elecrical and Compuer Engineering, Universiy of Vicoria, and i was revised on December,

More information