Power of Random Processes 1/40
|
|
- Camilla Doris Cain
- 6 years ago
- Views:
Transcription
1 Power of Random Processes 40
2 Power of a Random Process Recall : For deerminisic signals insananeous power is For a random signal, is a random variable for each ime. hus here is no single # o associae wih insananeous power. o ge he Epeced Insananeous Power i.e., on average we compue he saisical ensemble average of : 40
3 Power of a Random Process Ofen drop he Average erminology his is also called he mean square value of he process Avg. Power of : P X E{ } In general, can depend on ime. Bu we ll see i doesn for WSS & saionary processes 340
4 Relaionship of Power o ACF Recall: R X, +τ E { +τ } Clearly, seing τ 0 makes his equal o P X P X R X, If he Process is WSS or SS R X, R X 0 For WSS or SS P X R X 0 R X τ P X Power of process τ 440
5 Power vs. Variance Recall Variance: σ R E { [ ] } 0 for WSS P P X σ + Power & Variance are Equal for Zero mean Process AC Power DC Power Noe: If he WSS process is Zero Mean, hen: PX σ 540
6 640 Power Specral Densiy of a Random Process Recall: PSD for Deerminisic Signal : where lim j d e X X S
7 Power Specral Densiy of a Random Process For a random Process: each realizaion sample funcion of process has differen F and herefore a differen PSD. We again rely on averaging o give he Epeced PSD or Average PSD Bu Usually jus call i PSD. 740
8 Define PSD for WSS RP We define PSD of WSS process o be : S lim E X <Compare his o PSD for Deerminisic Signal> his definiion isn very useful for analysis so we seek an alernaive form he Wiener-Khinchine heorem provides his alernaive!!! 840
9 Weiner- Khinchine heorem Le be a WSS process w ACF R X τ and w PSD S X as defined in hen R X τ and S X form a F pair : S X F{ R X τ } or Equivalenly R X τ S X 940
10 040 Proof of WK heorem By definiion : d e X j For Real : d d e d e d e X X X j j j
11 40 Proof of WK heoremcon d hus : Move E {.} inside inegrals : lim d d e R S j lim d d e E S j
12 Proof of WK heoremcon d We were almos here BU we have one-oo-many inegrals. For convenience define: φ - R X - e-j- S lim φ d d 40
13 Proof of WK heoremcon d Change of Variables Change of Aes Insead of inegraing Row-by-Row as in we inegrae Diagonal-by-Diagonal. Le: τ τ + λ λ + λ τ 340
14 Proof of WK heoremcon d τ λ + 440
15 540 Proof of WK heoremcon d J λ τ λ τ Use Jacobian resul for -D change of variables From Calculus III
16 Proof of WK heoremcon d S lim φ τ dτdλ???? Q: As τ ranges over - τ how does λ range? J A: For each τ, λ mus be resriced o say inside original square see Figure on ne Char 640
17 Proof of WK heoremcon d τ τ 6 τ τ τ 4 λ + Noe: φ Doesn Change In he Direcion Of + Ais 740
18 Proof of WK heoremcon d τ τ + - τ λ τ τ λ + λ + - τ - τ When τ > 0 we need From Aes Figure: λ - τ - λ - + τ Works ou similarly for τ < 0 840
19 940 Proof of WK heoremcon d So τ λ τ φ τ τ d d S lim + - τ { } lim τ R d d I τ τ φ τ τ τ φ <End of Proof>
20 Some Properies of PSD & ACF he PSD is an even funcion of for a real process proof : since each sample funcion is real valued hen we know ha is even. is even: even even even So is S X w his is clear from 040
21 Some Prop. of PSD & ACF S X is real-valued and 0. proof : again from since w is realvalued & 0, so is S X w 3 R X τ is an even funcion of τ R τ E E R { { σ τ σ τ } τ } Also follows from IF{ Real & Even } Real & Even <Propery of he F> + 40
22 Graphical View of Prop #3 For WSS, ACF does no depend on absolue ime only relaive ime - τ +τ τ τ Doesn maer if you look forward by τ or look backward by τ 40
23 Compuing Power from PSD From i s name Power Specral Densiy we know wha o epec : P So le s Prove his!!!! S d π Well his par is no obvious!! 340
24 440 Compuing Power from PSD We Know: { } I π τ τ d e S S R j P X R X 0 Q.E.D.! 0 π π d S d e S P j X
25 Unis of PSD Funcion P S d π S Was Hz [ Was Hz ] 540
26 Using Symmery of S X w P 0 S d π Double o accoun for he - 0 par of inegral Inegrae only from 0 640
27 PSD for D Processes No much changes mosly, jus use DF insead of CF!! S X Ω DF { R X [m] } P S π Periodic in Ω wih period π π π Ω dω Need only look a -π Ω<π 740
28 Big Picure of PSD & ACF Narrow ACF Broad PSD Less Correlaed Sample-o-sample Process ehibis Rapid flucuaions i.e. High Frequencies have Large power conen Broad ACF Narrow PSD More Correlaed Sample-o-sample Process ehibis Slow flucuaions i.e. High Frequencies have Small power conen <<See Big Picure: Filered RP Chars in V-3 RP Eamples >> 840
29 Whie Noise he erm Whie Noise refers o a WSS process whose PSD is fla over all frequencies C- Whie Noise S X Convenion o use his form i.e. w division by N S X N Whie Noise Has Broades Possible PSD 940
30 C- Whie Noise NOE : C- whie noise has infinie Power : N d Can really eis in pracice bu sill a very useful Model for Analysis of Pracical Scenarios 3040
31 C- Whie Noise Q : wha is he ACF of C- whie Noise? A: ake he IF of he fla PSD : R τ I { N } R X τ N δ τ Dela funcion! Narrowes ACF Area N τ & are uncorrelaed for any Also. P X R X 0 N δ0 Infinie Power.. I Checks! 340
32 PSD is: S X Ω N Ω bu focus on Ω [-π,π] ACF is: R X [m] IDF {N } N δ [m] Dela sequence D- Whie Noise S X Ω N -π π R X [m] N Ω m Broades Possible PSD Narrowes ACF [k ] & [k ] are uncorrelaed for any k k 340
33 D- Whie Noise Noe: P P R π [0 ] π π N N dω was N was D- Whie Noise has Finie Power unlike C- Whie Noise 3340
34 Eamples of PSD Eample : BANDLIMIED WHIE NOISE his looks like whie noise wihin some bandwidh bu is PSD is zero ouside ha bandwidh hence he name. hus: S N rec 4πB -πb S X N πb And P B π N π πb d N π πb + πb NB 3440
35 Eample: BL Whie Noise he ACF using F pair for rec and sinc is: R X τ N B sinc πbτ Again we see P X N B, since R X 0 N B R X τ N B -3 B - B - B B B 3 B τ Noe: For τ > B & + τ are Approimaely Uncorrelaed 3540
36 Eample # of PSD Eample : SINUSOID WIH RANDOM PHASE A cos C + θ We eamined his RP before: R X τ A cos C τ So using F Pair for a Cosine gives he PSD: S X πa [δ + C + δ C ] Area πa S X Area πa - c c 3640
37 Eample # of PSD Noe : Can ge P X in wo ways:.. R π 0 S A d A <Sifing Propery> P X A 3740
38 Eample #3 of PSD Eample 3: FILERED D- RANDOM PROCESS < See Also: Class Noes on Filered RPs > [k] D- Filer y[k] [k] + [k +] Zero mean R X [m] σ δ[m] Inpu ACF Whie noise S X Ω σ² Ω Inpu PSD S X Ω σ² Ω 3840
39 Eample #3 of PSD For his case we showed earlier ha for his filer oupu he ACF is : R Y [m] σ² { δ[m] + δ[m-] + δ[m+] } So he Oupu PSD is: S Y Ω σ² [ + e -jω + e -jω ] σ² [cosω + ] Use he resul for DF of δ[m] and also ime-shif propery cos Ω By Euler 3940
40 Eample #3 of PSD S Y Ω σ² [cosω + ] Replicas S y Ω 4σ² Replicas -π -π π π Ω Remember: Limi View o [-π,π] General Idea Filer Shapes Inpu PSD: Here i suppresses High Frequency power 4040
Chapter 4 The Fourier Series and Fourier Transform
Represenaion of Signals in Terms of Frequency Componens Chaper 4 The Fourier Series and Fourier Transform Consider he CT signal defined by x () = Acos( ω + θ ), = The frequencies `presen in he signal are
More informationGuest Lectures for Dr. MacFarlane s EE3350 Part Deux
Gues Lecures for Dr. MacFarlane s EE3350 Par Deux Michael Plane Mon., 08-30-2010 Wrie name in corner. Poin ou his is a review, so I will go faser. Remind hem o go lisen o online lecure abou geing an A
More informationEECE 301 Signals & Systems Prof. Mark Fowler
EECE 3 Signals & Sysems Prof. Mark Fowler Noe Se # Wha are Coninuous-Time Signals??? /6 Coninuous-Time Signal Coninuous Time (C-T) Signal: A C-T signal is defined on he coninuum of ime values. Tha is:
More informationSensors, Signals and Noise
Sensors, Signals and Noise COURSE OUTLINE Inroducion Signals and Noise: 1) Descripion Filering Sensors and associaed elecronics rv 2017/02/08 1 Noise Descripion Noise Waveforms and Samples Saisics of Noise
More informationLinear Response Theory: The connection between QFT and experiments
Phys540.nb 39 3 Linear Response Theory: The connecion beween QFT and experimens 3.1. Basic conceps and ideas Q: How do we measure he conduciviy of a meal? A: we firs inroduce a weak elecric field E, and
More informationChapter 3: Signal Transmission and Filtering. A. Bruce Carlson Paul B. Crilly 2010 The McGraw-Hill Companies
Communicaion Sysems, 5e Chaper 3: Signal Transmission and Filering A. Bruce Carlson Paul B. Crilly 00 The McGraw-Hill Companies Chaper 3: Signal Transmission and Filering Response of LTI sysems Signal
More informationEE 224 Signals and Systems I Complex numbers sinusodal signals Complex exponentials e jωt phasor addition
EE 224 Signals and Sysems I Complex numbers sinusodal signals Complex exponenials e jω phasor addiion 1/28 Complex Numbers Recangular Polar y z r z θ x Good for addiion/subracion Good for muliplicaion/division
More information7 The Itô/Stratonovich dilemma
7 The Iô/Sraonovich dilemma The dilemma: wha does he idealizaion of dela-funcion-correlaed noise mean? ẋ = f(x) + g(x)η() η()η( ) = κδ( ). (1) Previously, we argued by a limiing procedure: aking noise
More informationEECE 301 Signals & Systems Prof. Mark Fowler
EECE 3 Signals & Sysems Prof. Mark Fowler Noe Se #2 Wha are Coninuous-Time Signals??? Reading Assignmen: Secion. of Kamen and Heck /22 Course Flow Diagram The arrows here show concepual flow beween ideas.
More informationCHAPTER 2 Signals And Spectra
CHAPER Signals And Specra Properies of Signals and Noise In communicaion sysems he received waveform is usually caegorized ino he desired par conaining he informaion, and he undesired par. he desired par
More information6.302 Feedback Systems Recitation 4: Complex Variables and the s-plane Prof. Joel L. Dawson
Number 1 quesion: Why deal wih imaginary and complex numbers a all? One answer is ha, as an analyical echnique, hey make our lives easier. Consider passing a cosine hrough an LTI filer wih impulse response
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 informationLecture Notes 2. The Hilbert Space Approach to Time Series
Time Series Seven N. Durlauf Universiy of Wisconsin. Basic ideas Lecure Noes. The Hilber Space Approach o Time Series The Hilber space framework provides a very powerful language for discussing he relaionship
More informationContinuous Time Linear Time Invariant (LTI) Systems. Dr. Ali Hussein Muqaibel. Introduction
/9/ Coninuous Time Linear Time Invarian (LTI) Sysems Why LTI? Inroducion Many physical sysems. Easy o solve mahemaically Available informaion abou analysis and design. We can apply superposiion LTI Sysem
More information2.160 System Identification, Estimation, and Learning. Lecture Notes No. 8. March 6, 2006
2.160 Sysem Idenificaion, Esimaion, and Learning Lecure Noes No. 8 March 6, 2006 4.9 Eended Kalman Filer In many pracical problems, he process dynamics are nonlinear. w Process Dynamics v y u Model (Linearized)
More informationA Bayesian Approach to Spectral Analysis
Chirped Signals A Bayesian Approach o Specral Analysis Chirped signals are oscillaing signals wih ime variable frequencies, usually wih a linear variaion of frequency wih ime. E.g. f() = A cos(ω + α 2
More informationES.1803 Topic 22 Notes Jeremy Orloff
ES.83 Topic Noes Jeremy Orloff Fourier series inroducion: coninued. Goals. Be able o compue he Fourier coefficiens of even or odd periodic funcion using he simplified formulas.. Be able o wrie and graph
More informationSignals and Systems Linear Time-Invariant (LTI) Systems
Signals and Sysems Linear Time-Invarian (LTI) Sysems Chang-Su Kim Discree-Time LTI Sysems Represening Signals in Terms of Impulses Sifing propery 0 x[ n] x[ k] [ n k] k x[ 2] [ n 2] x[ 1] [ n1] x[0] [
More information6.003 Homework #8 Solutions
6.003 Homework #8 Soluions Problems. Fourier Series Deermine he Fourier series coefficiens a k for x () shown below. x ()= x ( + 0) 0 a 0 = 0 a k = e /0 sin(/0) for k 0 a k = π x()e k d = 0 0 π e 0 k d
More informationLecture 2: Optics / C2: Quantum Information and Laser Science
Lecure : Opics / C: Quanum Informaion and Laser Science Ocober 9, 8 1 Fourier analysis This branch of analysis is exremely useful in dealing wih linear sysems (e.g. Maxwell s equaions for he mos par),
More informationDiebold, Chapter 7. Francis X. Diebold, Elements of Forecasting, 4th Edition (Mason, Ohio: Cengage Learning, 2006). Chapter 7. Characterizing Cycles
Diebold, Chaper 7 Francis X. Diebold, Elemens of Forecasing, 4h Ediion (Mason, Ohio: Cengage Learning, 006). Chaper 7. Characerizing Cycles Afer compleing his reading you should be able o: Define covariance
More information2 int T. is the Fourier transform of f(t) which is the inverse Fourier transform of f. i t e
PHYS67 Class 3 ourier Transforms In he limi T, he ourier series becomes an inegral ( nt f in T ce f n f f e d, has been replaced by ) where i f e d is he ourier ransform of f() which is he inverse ourier
More informationLaplace Transforms. Examples. Is this equation differential? y 2 2y + 1 = 0, y 2 2y + 1 = 0, (y ) 2 2y + 1 = cos x,
Laplace Transforms Definiion. An ordinary differenial equaion is an equaion ha conains one or several derivaives of an unknown funcion which we call y and which we wan o deermine from he equaion. The equaion
More information23.2. Representing Periodic Functions by Fourier Series. Introduction. Prerequisites. Learning Outcomes
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 informationDEPARTMENT OF ELECTRICAL AND ELECTRONIC ENGINEERING EXAMINATIONS 2008
[E5] IMPERIAL COLLEGE LONDON DEPARTMENT OF ELECTRICAL AND ELECTRONIC ENGINEERING EXAMINATIONS 008 EEE/ISE PART II MEng BEng and ACGI SIGNALS AND LINEAR SYSTEMS Time allowed: :00 hours There are FOUR quesions
More information23.5. Half-Range Series. Introduction. Prerequisites. Learning Outcomes
Half-Range Series 2.5 Inroducion In his Secion we address he following problem: Can we find a Fourier series expansion of a funcion defined over a finie inerval? Of course we recognise ha such a funcion
More informationEECE 301 Signals & Systems Prof. Mark Fowler
EECE 31 Signals & Sysems Prof. Mark Fowler Noe Se #1 C-T Sysems: Convoluion Represenaion Reading Assignmen: Secion 2.6 of Kamen and Heck 1/11 Course Flow Diagram The arrows here show concepual flow beween
More informationLaplace transfom: t-translation rule , Haynes Miller and Jeremy Orloff
Laplace ransfom: -ranslaion rule 8.03, Haynes Miller and Jeremy Orloff Inroducory example Consider he sysem ẋ + 3x = f(, where f is he inpu and x he response. We know is uni impulse response is 0 for
More information6.003: Signal Processing
6.003: Signal Processing Coninuous-Time Fourier Transform Definiion Examples Properies Relaion o Fourier Series Sepember 5, 08 Quiz Thursday, Ocober 4, from 3pm o 5pm. No lecure on Ocober 4. The exam is
More information3.1.3 INTRODUCTION TO DYNAMIC OPTIMIZATION: DISCRETE TIME PROBLEMS. A. The Hamiltonian and First-Order Conditions in a Finite Time Horizon
3..3 INRODUCION O DYNAMIC OPIMIZAION: DISCREE IME PROBLEMS A. he Hamilonian and Firs-Order Condiions in a Finie ime Horizon Define a new funcion, he Hamilonian funcion, H. H he change in he oal value of
More informationZürich. ETH Master Course: L Autonomous Mobile Robots Localization II
Roland Siegwar Margaria Chli Paul Furgale Marco Huer Marin Rufli Davide Scaramuzza ETH Maser Course: 151-0854-00L Auonomous Mobile Robos Localizaion II ACT and SEE For all do, (predicion updae / ACT),
More informationCharacteristics of Linear System
Characerisics o Linear Sysem h g h : Impulse response F G : Frequency ranser uncion Represenaion o Sysem in ime an requency. Low-pass iler g h G F he requency ranser uncion is he Fourier ransorm o he impulse
More informationMath 2142 Exam 1 Review Problems. x 2 + f (0) 3! for the 3rd Taylor polynomial at x = 0. To calculate the various quantities:
Mah 4 Eam Review Problems Problem. Calculae he 3rd Taylor polynomial for arcsin a =. Soluion. Le f() = arcsin. For his problem, we use he formula f() + f () + f ()! + f () 3! for he 3rd Taylor polynomial
More informationContinuous Time. Time-Domain System Analysis. Impulse Response. Impulse Response. Impulse Response. Impulse Response. ( t) + b 0.
Time-Domain Sysem Analysis Coninuous Time. J. Robers - All Righs Reserved. Edied by Dr. Rober Akl 1. J. Robers - All Righs Reserved. Edied by Dr. Rober Akl 2 Le a sysem be described by a 2 y ( ) + a 1
More information28. Narrowband Noise Representation
Narrowband Noise Represenaion on Mac 8. Narrowband Noise Represenaion In mos communicaion sysems, we are ofen dealing wih band-pass filering of signals. Wideband noise will be shaped ino bandlimied noise.
More information5. Stochastic processes (1)
Lec05.pp S-38.45 - Inroducion o Teleraffic Theory Spring 2005 Conens Basic conceps Poisson process 2 Sochasic processes () Consider some quaniy in a eleraffic (or any) sysem I ypically evolves in ime randomly
More informationEE 313 Linear Signals & Systems (Fall 2018) Solution Set for Homework #8 on Continuous-Time Signals & Systems
EE 33 Linear Signals & Sysems (Fall 08) Soluion Se for Homework #8 on Coninuous-Time Signals & Sysems By: Mr. Houshang Salimian & Prof. Brian L. Evans Here are several useful properies of he Dirac dela
More informationRepresenting a Signal. Continuous-Time Fourier Methods. Linearity and Superposition. Real and Complex Sinusoids. Jean Baptiste Joseph Fourier
Represening a Signal Coninuous-ime ourier Mehods he convoluion mehod for finding he response of a sysem o an exciaion aes advanage of he lineariy and imeinvariance of he sysem and represens he exciaion
More information4/9/2012. Signals and Systems KX5BQY EE235. Today s menu. System properties
Signals and Sysems hp://www.youube.com/v/iv6fo KX5BQY EE35 oday s menu Good weeend? Sysem properies iy Superposiion! Sysem properies iy: A Sysem is if i mees he following wo crieria: If { x( )} y( ) and
More informationon the interval (x + 1) 0! x < ", where x represents feet from the first fence post. How many square feet of fence had to be painted?
Calculus II MAT 46 Improper Inegrals A mahemaician asked a fence painer o complee he unique ask of paining one side of a fence whose face could be described by he funcion y f (x on he inerval (x + x
More informationMATH 31B: MIDTERM 2 REVIEW. x 2 e x2 2x dx = 1. ue u du 2. x 2 e x2 e x2] + C 2. dx = x ln(x) 2 2. ln x dx = x ln x x + C. 2, or dx = 2u du.
MATH 3B: MIDTERM REVIEW JOE HUGHES. Inegraion by Pars. Evaluae 3 e. Soluion: Firs make he subsiuion u =. Then =, hence 3 e = e = ue u Now inegrae by pars o ge ue u = ue u e u + C and subsiue he definiion
More informationSpectral Analysis of Random Processes
Spectral Analysis of Random Processes Spectral Analysis of Random Processes Generally, all properties of a random process should be defined by averaging over the ensemble of realizations. Generally, all
More informationReceivers, Antennas, and Signals. Professor David H. Staelin Fall 2001 Slide 1
Receivers, Anennas, and Signals Professor David H. Saelin 6.66 Fall 00 Slide A Subjec Conen A Human Processor Transducer Radio Opical, Infrared Acousic, oher Elecromagneic Environmen B C Human Processor
More informationSystem Processes input signal (excitation) and produces output signal (response)
Signal A funcion of ime Sysem Processes inpu signal (exciaion) and produces oupu signal (response) Exciaion Inpu Sysem Oupu Response 1. Types of signals 2. Going from analog o digial world 3. An example
More informationTwo Coupled Oscillators / Normal Modes
Lecure 3 Phys 3750 Two Coupled Oscillaors / Normal Modes Overview and Moivaion: Today we ake a small, bu significan, sep owards wave moion. We will no ye observe waves, bu his sep is imporan in is own
More informationψ ( t) = c n ( t ) n
p. 31 PERTURBATION THEORY Given a Hamilonian H ( ) = H + V( ) where we know he eigenkes for H H n = En n we ofen wan o calculae changes in he ampliudes of n induced by V( ) : where ψ ( ) = c n ( ) n n
More informationADDITIONAL PROBLEMS (a) Find the Fourier transform of the half-cosine pulse shown in Fig. 2.40(a). Additional Problems 91
ddiional Problems 9 n inverse relaionship exiss beween he ime-domain and freuency-domain descripions of a signal. Whenever an operaion is performed on he waveform of a signal in he ime domain, a corresponding
More informationChapter 4. Truncation Errors
Chaper 4. Truncaion Errors and he Taylor Series Truncaion Errors and he Taylor Series Non-elemenary funcions such as rigonomeric, eponenial, and ohers are epressed in an approimae fashion using Taylor
More informationME 452 Fourier Series and Fourier Transform
ME 452 Fourier Series and Fourier ransform Fourier series From Joseph Fourier in 87 as a resul of his sudy on he flow of hea. If f() is almos any periodic funcion i can be wrien as an infinie sum of sines
More informationψ ( t) = c n ( t) t n ( )ψ( ) t ku t,t 0 ψ I V kn
MIT Deparmen of Chemisry 5.74, Spring 4: Inroducory Quanum Mechanics II p. 33 Insrucor: Prof. Andrei Tokmakoff PERTURBATION THEORY Given a Hamilonian H ( ) = H + V ( ) where we know he eigenkes for H H
More informationSOLUTIONS TO ECE 3084
SOLUTIONS TO ECE 384 PROBLEM 2.. For each sysem below, specify wheher or no i is: (i) memoryless; (ii) causal; (iii) inverible; (iv) linear; (v) ime invarian; Explain your reasoning. If he propery is no
More informationF This leads to an unstable mode which is not observable at the output thus cannot be controlled by feeding back.
Lecure 8 Las ime: Semi-free configuraion design This is equivalen o: Noe ns, ener he sysem a he same place. is fixed. We design C (and perhaps B. We mus sabilize if i is given as unsable. Cs ( H( s = +
More informationAn random variable is a quantity that assumes different values with certain probabilities.
Probabiliy The probabiliy PrA) of an even A is a number in [, ] ha represens how likely A is o occur. The larger he value of PrA), he more likely he even is o occur. PrA) means he even mus occur. PrA)
More information( ) = Q 0. ( ) R = R dq. ( t) = I t
ircuis onceps The addiion of a simple capacior o a circui of resisors allows wo relaed phenomena o occur The observaion ha he ime-dependence of a complex waveform is alered by he circui is referred o as
More informationDYNAMIC ECONOMETRIC MODELS vol NICHOLAS COPERNICUS UNIVERSITY - TORUŃ Józef Stawicki and Joanna Górka Nicholas Copernicus University
DYNAMIC ECONOMETRIC MODELS vol.. - NICHOLAS COPERNICUS UNIVERSITY - TORUŃ 996 Józef Sawicki and Joanna Górka Nicholas Copernicus Universiy ARMA represenaion for a sum of auoregressive processes In he ime
More informationThe complex Fourier series has an important limiting form when the period approaches infinity, i.e., T 0. 0 since it is proportional to 1/L, but
Fourier Transforms The complex Fourier series has an imporan limiing form when he period approaches infiniy, i.e., T or L. Suppose ha in his limi () k = nπ L remains large (ranging from o ) and (2) c n
More information6.003 Homework #9 Solutions
6.003 Homework #9 Soluions Problems. Fourier varieies a. Deermine he Fourier series coefficiens of he following signal, which is periodic in 0. x () 0 3 0 a 0 5 a k a k 0 πk j3 e 0 e j πk 0 jπk πk e 0
More informationPhysics 127b: Statistical Mechanics. Fokker-Planck Equation. Time Evolution
Physics 7b: Saisical Mechanics Fokker-Planck Equaion The Langevin equaion approach o he evoluion of he velociy disribuion for he Brownian paricle migh leave you uncomforable. A more formal reamen of his
More informationChapter 7: Solving Trig Equations
Haberman MTH Secion I: The Trigonomeric Funcions Chaper 7: Solving Trig Equaions Le s sar by solving a couple of equaions ha involve he sine funcion EXAMPLE a: Solve he equaion sin( ) The inverse funcions
More informationRandom Processes 1/24
Random Processes 1/24 Random Process Oher Names : Random Signal Sochasic Process A Random Process is an exension of he concep of a Random variable (RV) Simples View : A Random Process is a RV ha is a Funcion
More informationSolutions from Chapter 9.1 and 9.2
Soluions from Chaper 9 and 92 Secion 9 Problem # This basically boils down o an exercise in he chain rule from calculus We are looking for soluions of he form: u( x) = f( k x c) where k x R 3 and k is
More informationR t. C t P t. + u t. C t = αp t + βr t + v t. + β + w t
Exercise 7 C P = α + β R P + u C = αp + βr + v (a) (b) C R = α P R + β + w (c) Assumpions abou he disurbances u, v, w : Classical assumions on he disurbance of one of he equaions, eg. on (b): E(v v s P,
More informationLinear Time-invariant systems, Convolution, and Cross-correlation
Linear Time-invarian sysems, Convoluion, and Cross-correlaion (1) Linear Time-invarian (LTI) sysem A sysem akes in an inpu funcion and reurns an oupu funcion. x() T y() Inpu Sysem Oupu y() = T[x()] An
More informationChapter 2. First Order Scalar Equations
Chaper. Firs Order Scalar Equaions We sar our sudy of differenial equaions in he same way he pioneers in his field did. We show paricular echniques o solve paricular ypes of firs order differenial equaions.
More information1/8 1/31/2011 ( ) ( ) Amplifiers lecture. out. Jim Stiles. Dept. of o EECS
1/31/2011 Amplifiers lecure 1/8 Amplifiers An ideal amplifier is a wo-por circui ha akes an pu signal v and reproduces i exacly a is oupu, only wih a larger magniude! ( ) i ( ) v + ( ) A I v + ou ( ) (
More informationKEY. Math 334 Midterm I Fall 2008 sections 001 and 003 Instructor: Scott Glasgow
1 KEY Mah 4 Miderm I Fall 8 secions 1 and Insrucor: Sco Glasgow Please do NOT wrie on his eam. No credi will be given for such work. Raher wrie in a blue book, or on our own paper, preferabl engineering
More informationOrdinary dierential equations
Chaper 5 Ordinary dierenial equaions Conens 5.1 Iniial value problem........................... 31 5. Forward Euler's mehod......................... 3 5.3 Runge-Kua mehods.......................... 36
More information2.3 SCHRÖDINGER AND HEISENBERG REPRESENTATIONS
Andrei Tokmakoff, MIT Deparmen of Chemisry, 2/22/2007 2-17 2.3 SCHRÖDINGER AND HEISENBERG REPRESENTATIONS The mahemaical formulaion of he dynamics of a quanum sysem is no unique. So far we have described
More informationUnit Root Time Series. Univariate random walk
Uni Roo ime Series Univariae random walk Consider he regression y y where ~ iid N 0, he leas squares esimae of is: ˆ yy y y yy Now wha if = If y y hen le y 0 =0 so ha y j j If ~ iid N 0, hen y ~ N 0, he
More information6.003 Homework #9 Solutions
6.00 Homework #9 Soluions Problems. Fourier varieies a. Deermine he Fourier series coefficiens of he following signal, which is periodic in 0. x () 0 0 a 0 5 a k sin πk 5 sin πk 5 πk for k 0 a k 0 πk j
More informationR =, C = 1, and f ( t ) = 1 for 1 second from t = 0 to t = 1. The initial charge on the capacitor is q (0) = 0. We have already solved this problem.
Theoreical Physics Prof. Ruiz, UNC Asheville, docorphys on YouTube Chaper U Noes. Green's Funcions R, C 1, and f ( ) 1 for 1 second from o 1. The iniial charge on he capacior is q (). We have already solved
More informationLinear Circuit Elements
1/25/2011 inear ircui Elemens.doc 1/6 inear ircui Elemens Mos microwave devices can be described or modeled in erms of he hree sandard circui elemens: 1. ESISTANE () 2. INDUTANE () 3. APAITANE () For he
More informationBernoulli numbers. Francesco Chiatti, Matteo Pintonello. December 5, 2016
UNIVERSITÁ DEGLI STUDI DI PADOVA, DIPARTIMENTO DI MATEMATICA TULLIO LEVI-CIVITA Bernoulli numbers Francesco Chiai, Maeo Pinonello December 5, 206 During las lessons we have proved he Las Ferma Theorem
More informationTraveling Waves. Chapter Introduction
Chaper 4 Traveling Waves 4.1 Inroducion To dae, we have considered oscillaions, i.e., periodic, ofen harmonic, variaions of a physical characerisic of a sysem. The sysem a one ime is indisinguishable from
More informationOscillation of an Euler Cauchy Dynamic Equation S. Huff, G. Olumolode, N. Pennington, and A. Peterson
PROCEEDINGS OF THE FOURTH INTERNATIONAL CONFERENCE ON DYNAMICAL SYSTEMS AND DIFFERENTIAL EQUATIONS May 4 7, 00, Wilmingon, NC, USA pp 0 Oscillaion of an Euler Cauchy Dynamic Equaion S Huff, G Olumolode,
More informationExplaining Total Factor Productivity. Ulrich Kohli University of Geneva December 2015
Explaining Toal Facor Produciviy Ulrich Kohli Universiy of Geneva December 2015 Needed: A Theory of Toal Facor Produciviy Edward C. Presco (1998) 2 1. Inroducion Toal Facor Produciviy (TFP) has become
More informationStochastic Structural Dynamics. Lecture-6
Sochasic Srucural Dynamics Lecure-6 Random processes- Dr C S Manohar Deparmen of Civil Engineering Professor of Srucural Engineering Indian Insiue of Science Bangalore 560 0 India manohar@civil.iisc.erne.in
More informationIntegration Over Manifolds with Variable Coordinate Density
Inegraion Over Manifolds wih Variable Coordinae Densiy Absrac Chrisopher A. Lafore clafore@gmail.com In his paper, he inegraion of a funcion over a curved manifold is examined in he case where he curvaure
More informationOBJECTIVES OF TIME SERIES ANALYSIS
OBJECTIVES OF TIME SERIES ANALYSIS Undersanding he dynamic or imedependen srucure of he observaions of a single series (univariae analysis) Forecasing of fuure observaions Asceraining he leading, lagging
More informationDelhi Noida Bhopal Hyderabad Jaipur Lucknow Indore Pune Bhubaneswar Kolkata Patna Web: Ph:
Serial : 0. ND_NW_EE_Signal & Sysems_4068 Delhi Noida Bhopal Hyderabad Jaipur Lucknow Indore Pune Bhubaneswar Kolkaa Pana Web: E-mail: info@madeeasy.in Ph: 0-4546 CLASS TEST 08-9 ELECTRICAL ENGINEERING
More informationGround Rules. PC1221 Fundamentals of Physics I. Kinematics. Position. Lectures 3 and 4 Motion in One Dimension. A/Prof Tay Seng Chuan
Ground Rules PC11 Fundamenals of Physics I Lecures 3 and 4 Moion in One Dimension A/Prof Tay Seng Chuan 1 Swich off your handphone and pager Swich off your lapop compuer and keep i No alking while lecure
More information- The whole joint distribution is independent of the date at which it is measured and depends only on the lag.
Saionary Processes Sricly saionary - The whole join disribuion is indeenden of he dae a which i is measured and deends only on he lag. - E y ) is a finie consan. ( - V y ) is a finie consan. ( ( y, y s
More information9/9/99 (T.F. Weiss) Signals and systems This subject deals with mathematical methods used to describe signals and to analyze and synthesize systems.
9/9/99 (T.F. Weiss) Lecure #: Inroducion o signals Moivaion: To describe signals, boh man-made and naurally occurring. Ouline: Classificaion ofsignals Building-block signals complex exponenials, impulses
More informationSMT 2014 Calculus Test Solutions February 15, 2014 = 3 5 = 15.
SMT Calculus Tes Soluions February 5,. Le f() = and le g() =. Compue f ()g (). Answer: 5 Soluion: We noe ha f () = and g () = 6. Then f ()g () =. Plugging in = we ge f ()g () = 6 = 3 5 = 5.. There is a
More informationHamilton- J acobi Equation: Explicit Formulas In this lecture we try to apply the method of characteristics to the Hamilton-Jacobi equation: u t
M ah 5 2 7 Fall 2 0 0 9 L ecure 1 0 O c. 7, 2 0 0 9 Hamilon- J acobi Equaion: Explici Formulas In his lecure we ry o apply he mehod of characerisics o he Hamilon-Jacobi equaion: u + H D u, x = 0 in R n
More informationProblem Set 5. Graduate Macro II, Spring 2017 The University of Notre Dame Professor Sims
Problem Se 5 Graduae Macro II, Spring 2017 The Universiy of Nore Dame Professor Sims Insrucions: You may consul wih oher members of he class, bu please make sure o urn in your own work. Where applicable,
More informationIntroduction to AC Power, RMS RMS. ECE 2210 AC Power p1. Use RMS in power calculations. AC Power P =? DC Power P =. V I = R =. I 2 R. V p.
ECE MS I DC Power P I = Inroducion o AC Power, MS I AC Power P =? A Solp //9, // // correced p4 '4 v( ) = p cos( ω ) v( ) p( ) Couldn' we define an "effecive" volage ha would allow us o use he same relaionships
More informationThe Quantum Theory of Atoms and Molecules: The Schrodinger equation. Hilary Term 2008 Dr Grant Ritchie
e Quanum eory of Aoms and Molecules: e Scrodinger equaion Hilary erm 008 Dr Gran Ricie An equaion for maer waves? De Broglie posulaed a every paricles as an associaed wave of waveleng: / p Wave naure of
More informationElements of Stochastic Processes Lecture II Hamid R. Rabiee
Sochasic Processes Elemens of Sochasic Processes Lecure II Hamid R. Rabiee Overview Reading Assignmen Chaper 9 of exbook Furher Resources MIT Open Course Ware S. Karlin and H. M. Taylor, A Firs Course
More informationAn Introduction to Malliavin calculus and its applications
An Inroducion o Malliavin calculus and is applicaions Lecure 5: Smoohness of he densiy and Hörmander s heorem David Nualar Deparmen of Mahemaics Kansas Universiy Universiy of Wyoming Summer School 214
More information1 Review of Zero-Sum Games
COS 5: heoreical Machine Learning Lecurer: Rob Schapire Lecure #23 Scribe: Eugene Brevdo April 30, 2008 Review of Zero-Sum Games Las ime we inroduced a mahemaical model for wo player zero-sum games. Any
More informationMath 106: Review for Final Exam, Part II. (x x 0 ) 2 = !
Mah 6: Review for Final Exam, Par II. Use a second-degree Taylor polynomial o esimae 8. We choose f(x) x and x 7 because 7 is he perfec cube closes o 8. f(x) x / f(7) f (x) x / f (7) x / 7 / 7 f (x) 9
More informationt is a basis for the solution space to this system, then the matrix having these solutions as columns, t x 1 t, x 2 t,... x n t x 2 t...
Mah 228- Fri Mar 24 5.6 Marix exponenials and linear sysems: The analogy beween firs order sysems of linear differenial equaions (Chaper 5) and scalar linear differenial equaions (Chaper ) is much sronger
More information12: AUTOREGRESSIVE AND MOVING AVERAGE PROCESSES IN DISCRETE TIME. Σ j =
1: AUTOREGRESSIVE AND MOVING AVERAGE PROCESSES IN DISCRETE TIME Moving Averages Recall ha a whie noise process is a series { } = having variance σ. The whie noise process has specral densiy f (λ) = of
More informationEE 315 Notes. Gürdal Arslan CLASS 1. (Sections ) What is a signal?
EE 35 Noes Gürdal Arslan CLASS (Secions.-.2) Wha is a signal? In his class, a signal is some funcion of ime and i represens how some physical quaniy changes over some window of ime. Examples: velociy of
More informationSignals and Systems Prof. Brian L. Evans Dept. of Electrical and Computer Engineering The University of Texas at Austin
EE 345S Real-Time Digial Signal Processing Lab Spring 26 Signals and Sysems Prof. Brian L. Evans Dep. of Elecrical and Compuer Engineering The Universiy of Texas a Ausin Review Signals As Funcions of Time
More informationEECE 301 Signals & Systems Prof. Mark Fowler
EECE 31 Signals & Sysems Prof. Mar Fowler Noe Se #1 C-T Signals: Circuis wih Periodic Sources 1/1 Solving Circuis wih Periodic Sources FS maes i easy o find he response of an RLC circui o a periodic source!
More informationLecture 4. Goals: Be able to determine bandwidth of digital signals. Be able to convert a signal from baseband to passband and back IV-1
Lecure 4 Goals: Be able o deermine bandwidh o digial signals Be able o conver a signal rom baseband o passband and back IV-1 Bandwidh o Digial Daa Signals A digial daa signal is modeled as a random process
More informationLecture #8 Redfield theory of NMR relaxation
Lecure #8 Redfield heory of NMR relaxaion Topics The ineracion frame of reference Perurbaion heory The Maser Equaion Handous and Reading assignmens van de Ven, Chapers 6.2. Kowalewski, Chaper 4. Abragam
More information