EE3723 : Digital Communications
|
|
- Thomasina Lane
- 6 years ago
- Views:
Transcription
1 EE373 : Digial Communicaions Week 6-7: Deecion Error Probabiliy Signal Space Orhogonal Signal Space MAJU-Digial Comm.-Week-6-7
2 Deecion Mached filer reduces he received signal o a single variable zt, afer which he deecion of symbol is carried ou The concep of maximum likelihood deecor is based on Saisical Decision Theory I allows us o formulae he decision rule ha operaes on he daa opimize he deecion crierion z T MAJU-Digial Comm.-Week-6-7 H > < H γ
3 Deecion of Binary Signal in Gaussian Noise The oupu of he filered sampled a T is a Gaussian random process MAJU-Digial Comm.-Week-6-7 3
4 Hence Baye s Decision Crierion and Maximum Likelihood Deecor z H > a + a < H γ where z is he minimum error crierion and γ is opimum hreshold For anipodal signal, s - s a - a z H > < H MAJU-Digial Comm.-Week-6-7 4
5 Error will occur if s is sen s is received P H s P e s s is sen s is received P H s P e s Probabiliy of Error P e s p z s dz γ P e s p z s dz γ The oal probabiliy of error is he sum of he errors P P e, s P e s P s + P e s P s B i i P H s P s + P H s P s MAJU-Digial Comm.-Week-6-7 5
6 If signals are equally probable P P H s P s + P H s P s B + [ P H s P H s ] by Symmery PB [ P H s + P H s ] P H s Numerically, P B is he area under he ail of eiher of he condiional disribuions pz s or pz s and is given by: P P H s dz p z s d z B γ γ z a ex p γ σ π σ dz MAJU-Digial Comm.-Week-6-7 6
7 P B z a exp γ σ π σ u z a σ u a a exp du σ π The above equaion canno be evaluaed in closed form Qfuncion Hence, a a dz PB Q equaion B σ Q z exp z π z.8 MAJU-Digial Comm.-Week-6-7 7
8 Recall: Error probabiliy for binary signals P B a a Q σ equaion B Where we have replaced a by a..8 To minimize P B, we need o maximize: a a σ We have Therefore, or a a σ a a Ed Ed σ N / N a a a a E E d d σ σ N N MAJU-Digial Comm.-Week-6-7 8
9 [ ] [ ] [ ] [ ] + T T T T d s s d s d s d s s E 3.63 N E Q P d B The probabiliy of bi error is given by: MAJU-Digial Comm.-Week [ ] [ ] [ ] + s s d s d s
10 The probabiliy of bi error for anipodal signals: The probabiliy of bi error for orhogonal signals: N E Q P b B E MAJU-Digial Comm.-Week-6-7 The probabiliy of bi error for unipolar signals: N E Q P b B N E Q P b B
11 Error probabiliy for binary signals Table for compuing of Q-Funcions MAJU-Digial Comm.-Week-6-7
12 Relaion Beween SNR S/N and E b /N In analog communicaion he figure of meri used is he average signal power o average noise power raion or SNR. In he previous few slides we have used he erm E b /N in he bi error calculaions. How are he wo relaed? E b can be wrien as ST b and N is N/W. So we have: Eb STb S W N where σ N N / W N Rb Thus E b /N can be hough of as normalized SNR. Makes more sense when we have muli-level signaling. Reading: Page 7 and 8. MAJU-Digial Comm.-Week-6-7
13 Bipolar signals require a facor of increase in energy compared o orhogonal signals Since log 3 db, we say ha bipolar signaling offers a 3 db beer performance han orhogonal MAJU-Digial Comm.-Week-6-7 3
14 Comparing BER Performance For P P E b / B, orhogonal B, anipodal N db 9.x 7.8x 4 For he same received signal o noise raio, anipodal provides lower bi error rae han orhogonal MAJU-Digial Comm.-Week-6-7 4
15 Problem: Evaluaing Error Performance Consider a Binary Communicaion Sysem ha receives equally likely signals s and s plus AWGN see he following figure. Assume ha he receiving filer is a Mached Filer MF, and ha he noise Power Specral Densiy N is equal o - Wa/Hz. Use he values of received signal volage and ime shown on figure o compue he Bi Error Probabiliy. s millivols 3 µs 3 µs s millivols MAJU-Digial Comm.-Week-6-7 5
16 Problem: Error Performance based Designing Consider ha NRZ binary pulses are ransmied along a communicaion cable ha aenuaes he signal power by 3 db from ransmier o receiver. The pulses are coherenly deeced a he receiver, and he daa rae is 56 kbis/s. Assume Gaussian noise wih N -6 Was/Herz. Wha is he minimum amoun of Power needed a he ransmier in order o mainain a bi-error probabiliy of Pe -3? MAJU-Digial Comm.-Week-6-7 6
17 Signals vs vecors Represenaion of a vecor by basis vecors Orhogonaliy of vecors Orhogonaliy of signals MAJU-Digial Comm.-Week-6-7 7
18 Wha is a signal space? Signal space Vecor represenaions of signals in an N-dimensional orhogonal space Why do we need a signal space? I is a means o conver signals o vecors and vice versa. I is a means o calculae signals energy and Euclidean disances beween signals. Why are we ineresed in Euclidean disances beween signals? For deecion purposes: The received signal is ransformed o a received vecors. The signal which has he minimum disance o he received signal is esimaed as he ransmied signal. MAJU-Digial Comm.-Week-6-7 8
19 Orhogonal signal space N-dimensional orhogonal signal space is characerized by N linearly independen funcions { ψ j } N called basis funcions. j The basis funcions mus saisfy he orhogonaliy condiion. ρ ji T < ψ i, ψ j > ψ ψ * K i j K d T j, i,..., N Where ρ ij i i j j is he correlaion coefficien. K is he normalizing consan which makes he signal space Orhonormal. MAJU-Digial Comm.-Week-6-7 9
20 Example of an orhonormal basis Example: -dimensional orhonormal signal space, / sin / cos > < < < d T T T T T T T ψ ψ ψ ψ π ψ π ψ ψ ψ MAJU-Digial Comm.-Week-6-7 Example: -dimensional orhonornal signal space, > < d ψ ψ ψ ψ ψ ψ T ψ T ψ ψ
21 Signal space Any arbirary finie se of waveforms where each member of he se is of duraion T, can be expressed as a linear combinaion of N orhonogal waveforms where. s i { ψ } N j j N N M { } M i i a ijψ j i,..., M j N M s where a ij < s, ψ > s ψ i j T i * j d j,..., N i,..., M T s i ai, ai,..., a in Vecor represenaion of waveform E i N j a ij Waveform energy Parseval s heorem MAJU-Digial Comm.-Week-6-7
22 Signal space a T * ij si ψ j d Waveform o vecor conversion i N s a ψ j Vecor o waveform conversion ij j s i ψ ψ N T T a i a in a ai M a in s m s m a ai M a in a i a in ψ ψ N s i s m ai, ai,..., a in MAJU-Digial Comm.-Week-6-7
23 Basis Funcions: An example A se of 8 orhogonal of basis funcions Wha signals can we form wih his se of basis funcions? MAJU-Digial Comm.-Week-6-7 3
24 Basis Funcions: An example Linear combinaion of basis funcions φ [n] + φ [n] + /3 φ 3 [n] + φ 4 [n] + /5 φ 5 [n] + φ 6 [n] + /7 φ 7 [n] + φ 8 [n] Waveforms in he span of basis funcions φ [n] + φ [n] + /9 φ 3 [n] + φ 4 [n] + /5 φ 5 [n] + φ 6 [n] + /49 φ 7 [n] + φ 8 [n] φ [n] + / φ [n] + /3 φ 3 [n] + /4 φ 4 [n] + /5 φ 5 [n] + /6 φ 6 [n] + /7 φ 7 [n] + /8 φ 8 [n] MAJU-Digial Comm.-Week-6-7 4
25 Represenaion of a signal in signal space MAJU-Digial Comm.-Week-6-7 5
26 Example: Baseband Anipodal Signals MAJU-Digial Comm.-Week-6-7 6
27 Example: BPSK MAJU-Digial Comm.-Week-6-7 7
28 Example QPSK MAJU-Digial Comm.-Week-6-7 8
29 Synhesis Equaion Modulaion MAJU-Digial Comm.-Week-6-7 9
30 Example: Baseband Anipodal Signals MAJU-Digial Comm.-Week-6-7 3
31 Example: BPSK MAJU-Digial Comm.-Week-6-7 3
32 Correlaion Measure of similariy beween wo signals c n + g z d. E E g z Cross correlaion ψ gz Auocorrelaion + τ g z + τ d. + ψ g τ g g + τ d. MAJU-Digial Comm.-Week-6-7 3
33 Analysis Equaion Deecion MAJU-Digial Comm.-Week
34 Correlaion Deecor MAJU-Digial Comm.-Week
35 Correlaion Deecor: Examples MAJU-Digial Comm.-Week
36 Correlaion Deecor Example: QPSK MAJU-Digial Comm.-Week
Block Diagram of a DCS in 411
Informaion source Forma A/D From oher sources Pulse modu. Muliplex Bandpass modu. X M h: channel impulse response m i g i s i Digial inpu Digial oupu iming and synchronizaion Digial baseband/ bandpass
More informationSolutions to the Exam Digital Communications I given on the 11th of June = 111 and g 2. c 2
Soluions o he Exam Digial Communicaions I given on he 11h of June 2007 Quesion 1 (14p) a) (2p) If X and Y are independen Gaussian variables, hen E [ XY ]=0 always. (Answer wih RUE or FALSE) ANSWER: False.
More informationUNIVERSITY OF CALIFORNIA College of Engineering Department of Electrical Engineering and Computer Sciences EECS 121 FINAL EXAM
Name: UNIVERSIY OF CALIFORNIA College of Engineering Deparmen of Elecrical Engineering and Compuer Sciences Professor David se EECS 121 FINAL EXAM 21 May 1997, 5:00-8:00 p.m. Please wrie answers on blank
More informationA First Course in Digital Communications
A Firs Course in Digial Communicaions Ha H. Nguyen and E. Shwedyk February 9 A Firs Course in Digial Communicaions /58 Block Diagram of Binary Communicaion Sysems m { b k } bk = s b = s k m ˆ { bˆ } k
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 informationLecture 8. Digital Communications Part III. Digital Demodulation
Lecure 8. Digial Communicaions Par III. Digial Demodulaion Binary Deecion M-ary Deecion Lin Dai (Ciy Universiy of Hong Kong) EE38 Principles of Communicaions Lecure 8 Analog Signal Source SOURCE A-D Conversion
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 informationEE456 Digital Communications
EE456 Digial Communicaions Professor Ha Nguyen Sepember 6 EE456 Digial Communicaions Inroducion o Basic Digial Passband Modulaion Baseband ransmission is conduced a low frequencies. Passband ransmission
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 informationDemodulation of Digitally Modulated Signals
Addiional maerial for TSKS1 Digial Communicaion and TSKS2 Telecommunicaion Demodulaion of Digially Modulaed Signals Mikael Olofsson Insiuionen för sysemeknik Linköpings universie, 581 83 Linköping November
More informationAnswers to Exercises in Chapter 7 - Correlation Functions
M J Robers - //8 Answers o Exercises in Chaper 7 - Correlaion Funcions 7- (from Papoulis and Pillai) The random variable C is uniform in he inerval (,T ) Find R, ()= u( C), ()= C (Use R (, )= R,, < or
More informationMultiphase Shift Keying (MPSK) Lecture 8. Constellation. Decision Regions. s i. 2 T cos 2π f c t iφ 0 t As iφ 1 t. t As. A c i.
π fc uliphase Shif Keying (PSK) Goals Lecure 8 Be able o analyze PSK modualion s i Ac i Ac Pcos π f c cos π f c iφ As iφ π i p p As i sin π f c p Be able o analyze QA modualion Be able o quanify he radeoff
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 informationImpacts of both Tx and Rx IQ Imbalances on OFDM Systems - Analytical Approach
mpacs of boh Tx and Rx Q mbalances on OFD Sysems - Analyical Approach Hassan Zareian, Vahid Tabaaba Vakili ran Universiy of Science and Technology UST, Tehran, ran Faculy of he slamic Republic of ran Broadcasing
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 informationOutline Chapter 2: Signals and Systems
Ouline Chaper 2: Signals and Sysems Signals Basics abou Signal Descripion Fourier Transform Harmonic Decomposiion of Periodic Waveforms (Fourier Analysis) Definiion and Properies of Fourier Transform Imporan
More informationProblem Formulation in Communication Systems
Problem Formulaion in Communicaion Sysems Sooyong Choi School of Elecrical and Elecronic Engineering Yonsei Universiy Inroducion Problem formulaion in communicaion sysems Simple daa ransmission sysem :
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 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 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 informationLecture 2 April 04, 2018
Sas 300C: Theory of Saisics Spring 208 Lecure 2 April 04, 208 Prof. Emmanuel Candes Scribe: Paulo Orensein; edied by Sephen Baes, XY Han Ouline Agenda: Global esing. Needle in a Haysack Problem 2. Threshold
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 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 informationSpeaker Adaptation Techniques For Continuous Speech Using Medium and Small Adaptation Data Sets. Constantinos Boulis
Speaker Adapaion Techniques For Coninuous Speech Using Medium and Small Adapaion Daa Ses Consaninos Boulis Ouline of he Presenaion Inroducion o he speaker adapaion problem Maximum Likelihood Sochasic Transformaions
More information14 Autoregressive Moving Average Models
14 Auoregressive Moving Average Models In his chaper an imporan parameric family of saionary ime series is inroduced, he family of he auoregressive moving average, or ARMA, processes. For a large class
More informationStochastic Signals and Systems
Sochasic Signals and Sysems Conens 1. Probabiliy Theory. Sochasic Processes 3. Parameer Esimaion 4. Signal Deecion 5. Specrum Analysis 6. Opimal Filering Chaper 6 / Sochasic Signals and Sysems / Prof.
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 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 informationst semester. Kei Sakaguchi
0 s semeser MIMO Communicaion Sysems #5: MIMO Channel Capaciy Kei Sakaguchi ee ac May 7, 0 Schedule ( s half Dae Tex Conens # Apr. A-, B- Inroducion # Apr. 9 B-5, B-6 Fundamenals
More informationVehicle Arrival Models : Headway
Chaper 12 Vehicle Arrival Models : Headway 12.1 Inroducion Modelling arrival of vehicle a secion of road is an imporan sep in raffic flow modelling. I has imporan applicaion in raffic flow simulaion where
More informationEELE Lecture 8 Example of Fourier Series for a Triangle from the Fourier Transform. Homework password is: 14445
EELE445-4 Lecure 8 Eample o Fourier Series or a riangle rom he Fourier ransorm Homework password is: 4445 3 4 EELE445-4 Lecure 8 LI Sysems and Filers 5 LI Sysem 6 3 Linear ime-invarian Sysem Deiniion o
More informationChapter One Fourier Series and Fourier Transform
Chaper One I. Fourier Series Represenaion of Periodic Signals -Trigonomeric Fourier Series: The rigonomeric Fourier series represenaion of a periodic signal x() x( + T0 ) wih fundamenal period T0 is given
More informationLicenciatura de ADE y Licenciatura conjunta Derecho y ADE. Hoja de ejercicios 2 PARTE A
Licenciaura de ADE y Licenciaura conjuna Derecho y ADE Hoja de ejercicios PARTE A 1. Consider he following models Δy = 0.8 + ε (1 + 0.8L) Δ 1 y = ε where ε and ε are independen whie noise processes. In
More informationLAPLACE TRANSFORM AND TRANSFER FUNCTION
CHBE320 LECTURE V LAPLACE TRANSFORM AND TRANSFER FUNCTION Professor Dae Ryook Yang Spring 2018 Dep. of Chemical and Biological Engineering 5-1 Road Map of he Lecure V Laplace Transform and Transfer funcions
More information20. Applications of the Genetic-Drift Model
0. Applicaions of he Geneic-Drif Model 1) Deermining he probabiliy of forming any paricular combinaion of genoypes in he nex generaion: Example: If he parenal allele frequencies are p 0 = 0.35 and q 0
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 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 informationENERGY SEPARATION AND DEMODULATION OF CPM SIGNALS
ENERGY SEPARATION AND DEMODULATION OF CPM SIGNALS Balu Sanhanam SPCOM Laboraory Deparmen of E.E.C.E. Universiy of New Meico Moivaion CPM has power/bandwidh efficiency and used in he wireless sysem infrasrucure.
More informationVoltage/current relationship Stored Energy. RL / RC circuits Steady State / Transient response Natural / Step response
Review Capaciors/Inducors Volage/curren relaionship Sored Energy s Order Circuis RL / RC circuis Seady Sae / Transien response Naural / Sep response EE4 Summer 5: Lecure 5 Insrucor: Ocavian Florescu Lecure
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 informationODEs II, Lecture 1: Homogeneous Linear Systems - I. Mike Raugh 1. March 8, 2004
ODEs II, Lecure : Homogeneous Linear Sysems - I Mike Raugh March 8, 4 Inroducion. In he firs lecure we discussed a sysem of linear ODEs for modeling he excreion of lead from he human body, saw how o ransform
More informationProperties of Autocorrelated Processes Economics 30331
Properies of Auocorrelaed Processes Economics 3033 Bill Evans Fall 05 Suppose we have ime series daa series labeled as where =,,3, T (he final period) Some examples are he dail closing price of he S&500,
More informationTransform Techniques. Moment Generating Function
Transform Techniques A convenien way of finding he momens of a random variable is he momen generaing funcion (MGF). Oher ransform echniques are characerisic funcion, z-ransform, and Laplace ransform. Momen
More informationEE 435 Lecture 42. Phased Locked Loops and VCOs
EE 435 Lecure 42 d Locked Loops and VCOs Basis PLL Archiecure Loop Filer (LF) Volage Conrolled Oscillaor (VCO) Frequency Divider N Applicaions include: Frequency Demodulaion Frequency Synhesis Clock Synchronizaion
More informationIII-A. Fourier Series Expansion
Summer 28 Signals & Sysems S.F. Hsieh III-A. Fourier Series Expansion Inroducion. Divide and conquer Signals can be decomposed as linear combinaions of: (a) shifed impulses: (sifing propery) Why? x() x()δ(
More informationGeneralized Least Squares
Generalized Leas Squares Augus 006 1 Modified Model Original assumpions: 1 Specificaion: y = Xβ + ε (1) Eε =0 3 EX 0 ε =0 4 Eεε 0 = σ I In his secion, we consider relaxing assumpion (4) Insead, assume
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 informationThen. 1 The eigenvalues of A are inside R = n i=1 R i. 2 Union of any k circles not intersecting the other (n k)
Ger sgorin Circle Chaper 9 Approimaing Eigenvalues Per-Olof Persson persson@berkeley.edu Deparmen of Mahemaics Universiy of California, Berkeley Mah 128B Numerical Analysis (Ger sgorin Circle) Le A be
More informationChapter 2. Models, Censoring, and Likelihood for Failure-Time Data
Chaper 2 Models, Censoring, and Likelihood for Failure-Time Daa William Q. Meeker and Luis A. Escobar Iowa Sae Universiy and Louisiana Sae Universiy Copyrigh 1998-2008 W. Q. Meeker and L. A. Escobar. Based
More informationEELE Lecture 3,4 EE445 - Outcomes. Physically Realizable Waveforms. EELE445 Montana State University. In this lecture you:
EELE445 Monana Sae Universiy Lecure 3,4 EE445 - Oucomes EELE445-4 Lecure 3,4 Poer, Energy, ime average operaor secion. In his lecure you: be able o use he ime average operaor [] for finie ime duraion signals
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 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 informationHomework Solution Set # 3. Thursday, September 22, Textbook: Claude Cohen Tannoudji, Bernard Diu and Franck Lalo, Second Volume Complement G X
Deparmen of Physics Quanum Mechanics II, 570 Temple Universiy Insrucor: Z.-E. Meziani Homework Soluion Se # 3 Thursday, Sepember, 06 Texbook: Claude Cohen Tannoudji, Bernard Diu and Franck Lalo, Second
More informationRetrieval Models. Boolean and Vector Space Retrieval Models. Common Preprocessing Steps. Boolean Model. Boolean Retrieval Model
1 Boolean and Vecor Space Rerieval Models Many slides in his secion are adaped from Prof. Joydeep Ghosh (UT ECE) who in urn adaped hem from Prof. Dik Lee (Univ. of Science and Tech, Hong Kong) Rerieval
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 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 informationMacroeconomic Theory Ph.D. Qualifying Examination Fall 2005 ANSWER EACH PART IN A SEPARATE BLUE BOOK. PART ONE: ANSWER IN BOOK 1 WEIGHT 1/3
Macroeconomic Theory Ph.D. Qualifying Examinaion Fall 2005 Comprehensive Examinaion UCLA Dep. of Economics You have 4 hours o complee he exam. There are hree pars o he exam. Answer all pars. Each par has
More information10. State Space Methods
. Sae Space Mehods. Inroducion Sae space modelling was briefly inroduced in chaper. Here more coverage is provided of sae space mehods before some of heir uses in conrol sysem design are covered in he
More informationExcel-Based Solution Method For The Optimal Policy Of The Hadley And Whittin s Exact Model With Arma Demand
Excel-Based Soluion Mehod For The Opimal Policy Of The Hadley And Whiin s Exac Model Wih Arma Demand Kal Nami School of Business and Economics Winson Salem Sae Universiy Winson Salem, NC 27110 Phone: (336)750-2338
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 informationJoint Transmitter-Reciever Optimization for Multiple Input Multiple Output (MIMO) Systems
Join Transmier-Reciever Opimizaion for Muliple Inpu Muliple Oupu (MIMO Sysems eun Chul WANG and wang Bo (Ed LEE School of Elecrical Engineering, Seoul Naional Universiy, OREA Absrac Muliple ransmi (Tx
More informationEE123 Digital Signal Processing
Discree Transforms (Finie) EE3 Digial Signal Processing Lecure 9 DFT is only one ou of a LARGE class of ransforms Used for: Analysis Comression Denoising Deecion Recogniion Aroximaion (Sarse) Sarse reresenaion
More informationAugmented Reality II - Kalman Filters - Gudrun Klinker May 25, 2004
Augmened Realiy II Kalman Filers Gudrun Klinker May 25, 2004 Ouline Moivaion Discree Kalman Filer Modeled Process Compuing Model Parameers Algorihm Exended Kalman Filer Kalman Filer for Sensor Fusion Lieraure
More informationMath 10B: Mock Mid II. April 13, 2016
Name: Soluions Mah 10B: Mock Mid II April 13, 016 1. ( poins) Sae, wih jusificaion, wheher he following saemens are rue or false. (a) If a 3 3 marix A saisfies A 3 A = 0, hen i canno be inverible. True.
More informationStationary Time Series
3-Jul-3 Time Series Analysis Assoc. Prof. Dr. Sevap Kesel July 03 Saionary Time Series Sricly saionary process: If he oin dis. of is he same as he oin dis. of ( X,... X n) ( X h,... X nh) Weakly Saionary
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 informationcs/ee 143 Communication Networks
cs/ee 143 Communicaion Neworks Chaper 3 Eherne Tex: Walrand & Parakh, 2010 Seven Low CMS, EE, Calech Warning These noes are no self-conained, probably no undersandable, unless you also were in he lecure
More informationSystem of Linear Differential Equations
Sysem of Linear Differenial Equaions In "Ordinary Differenial Equaions" we've learned how o solve a differenial equaion for a variable, such as: y'k5$e K2$x =0 solve DE yx = K 5 2 ek2 x C_C1 2$y''C7$y
More informationGMM - Generalized Method of Moments
GMM - Generalized Mehod of Momens Conens GMM esimaion, shor inroducion 2 GMM inuiion: Maching momens 2 3 General overview of GMM esimaion. 3 3. Weighing marix...........................................
More informationThe Arcsine Distribution
The Arcsine Disribuion Chris H. Rycrof Ocober 6, 006 A common heme of he class has been ha he saisics of single walker are ofen very differen from hose of an ensemble of walkers. On he firs homework, we
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 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 informationApplication of Speed Transform to the diagnosis of a roller bearing in variable speed
Applicaion of Speed Transform o he diagnosis of a roller bearing in variable speed Julien Roussel 1, Michel Hariopoulos 1, Edgard Sekko 1, Cécile Capdessus 1 and Jérôme Anoni 1 PRISME laboraory 1 rue de
More informationChapter 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 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 information5. Response of Linear Time-Invariant Systems to Random Inputs
Sysem: 5. Response of inear ime-invarian Sysems o Random Inpus 5.. Discree-ime linear ime-invarian (IV) sysems 5... Discree-ime IV sysem IV sysem xn ( ) yn ( ) [ xn ( )] Inpu Signal Sysem S Oupu Signal
More informationMathematical Theory and Modeling ISSN (Paper) ISSN (Online) Vol 3, No.3, 2013
Mahemaical Theory and Modeling ISSN -580 (Paper) ISSN 5-05 (Online) Vol, No., 0 www.iise.org The ffec of Inverse Transformaion on he Uni Mean and Consan Variance Assumpions of a Muliplicaive rror Model
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 informationModule 2 F c i k c s la l w a s o s f dif di fusi s o i n
Module Fick s laws of diffusion Fick s laws of diffusion and hin film soluion Adolf Fick (1855) proposed: d J α d d d J (mole/m s) flu (m /s) diffusion coefficien and (mole/m 3 ) concenraion of ions, aoms
More informationψ(t) = V x (0)V x (t)
.93 Home Work Se No. (Professor Sow-Hsin Chen Spring Term 5. Due March 7, 5. This problem concerns calculaions of analyical expressions for he self-inermediae scaering funcion (ISF of he es paricle in
More information1 Evaluating Chromatograms
3 1 Evaluaing Chromaograms Hans-Joachim Kuss and Daniel Sauffer Chromaography is, in principle, a diluion process. In HPLC analysis, on dissolving he subsances o be analyzed in an eluen and hen injecing
More informationTHE DISCRETE WAVELET TRANSFORM
. 4 THE DISCRETE WAVELET TRANSFORM 4 1 Chaper 4: THE DISCRETE WAVELET TRANSFORM 4 2 4.1 INTRODUCTION TO DISCRETE WAVELET THEORY The bes way o inroduce waveles is hrough heir comparison o Fourier ransforms,
More informationNavneet Saini, Mayank Goyal, Vishal Bansal (2013); Term Project AML310; Indian Institute of Technology Delhi
Creep in Viscoelasic Subsances Numerical mehods o calculae he coefficiens of he Prony equaion using creep es daa and Herediary Inegrals Mehod Navnee Saini, Mayank Goyal, Vishal Bansal (23); Term Projec
More informationTwo Popular Bayesian Estimators: Particle and Kalman Filters. McGill COMP 765 Sept 14 th, 2017
Two Popular Bayesian Esimaors: Paricle and Kalman Filers McGill COMP 765 Sep 14 h, 2017 1 1 1, dx x Bel x u x P x z P Recall: Bayes Filers,,,,,,, 1 1 1 1 u z u x P u z u x z P Bayes z = observaion u =
More informationCHERNOFF DISTANCE AND AFFINITY FOR TRUNCATED DISTRIBUTIONS *
haper 5 HERNOFF DISTANE AND AFFINITY FOR TRUNATED DISTRIBUTIONS * 5. Inroducion In he case of disribuions ha saisfy he regulariy condiions, he ramer- Rao inequaliy holds and he maximum likelihood esimaor
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 informationL1, L2, N1 N2. + Vout. C out. Figure 2.1.1: Flyback converter
page 11 Flyback converer The Flyback converer belongs o he primary swiched converer family, which means here is isolaion beween in and oupu. Flyback converers are used in nearly all mains supplied elecronic
More informationInnova Junior College H2 Mathematics JC2 Preliminary Examinations Paper 2 Solutions 0 (*)
Soluion 3 x 4x3 x 3 x 0 4x3 x 4x3 x 4x3 x 4x3 x x 3x 3 4x3 x Innova Junior College H Mahemaics JC Preliminary Examinaions Paper Soluions 3x 3 4x 3x 0 4x 3 4x 3 0 (*) 0 0 + + + - 3 3 4 3 3 3 3 Hence x or
More information(10) (a) Derive and plot the spectrum of y. Discuss how the seasonality in the process is evident in spectrum.
January 01 Final Exam Quesions: Mark W. Wason (Poins/Minues are given in Parenheses) (15) 1. Suppose ha y follows he saionary AR(1) process y = y 1 +, where = 0.5 and ~ iid(0,1). Le x = (y + y 1 )/. (11)
More information4. Electric field lines with respect to equipotential surfaces are
Pre-es Quasi-saic elecromagneism. The field produced by primary charge Q and by an uncharged conducing plane disanced from Q by disance d is equal o he field produced wihou conducing plane by wo following
More informationProposal of atomic clock in motion: Time in moving clock
Proposal of aomic clock in moion: Time in moving clock Masanori Sao Honda Elecronics Co., d., 0 Oyamazuka, Oiwa-cho, Toyohashi, ichi 441-3193, Japan E-mail: msao@honda-el.co.jp bsrac: The ime in an aomic
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 informationThe electromagnetic interference in case of onboard navy ships computers - a new approach
The elecromagneic inerference in case of onboard navy ships compuers - a new approach Prof. dr. ing. Alexandru SOTIR Naval Academy Mircea cel Bărân, Fulgerului Sree, Consanţa, soiralexandru@yahoo.com Absrac.
More informationNature Neuroscience: doi: /nn Supplementary Figure 1. Spike-count autocorrelations in time.
Supplemenary Figure 1 Spike-coun auocorrelaions in ime. Normalized auocorrelaion marices are shown for each area in a daase. The marix shows he mean correlaion of he spike coun in each ime bin wih he spike
More informationB Signals and Systems I Solutions to Midterm Test 2. xt ()
34-33B Signals and Sysems I Soluions o Miderm es 34-33B Signals and Sysems I Soluions o Miderm es ednesday Marh 7, 7:PM-9:PM Examiner: Prof. Benoi Boule Deparmen of Elerial and Compuer Engineering MGill
More informationENGI 9420 Engineering Analysis Assignment 2 Solutions
ENGI 940 Engineering Analysis Assignmen Soluions 0 Fall [Second order ODEs, Laplace ransforms; Secions.0-.09]. Use Laplace ransforms o solve he iniial value problem [0] dy y, y( 0) 4 d + [This was Quesion
More informationJoint Optimization of Rate Allocation and BLAST Ordering to Minimize Outage Probability
Join Opimizaion of Rae Allocaion and BLAST Ordering o Minimize Ouage Probabiliy Arumugam Kannan, Badri Varadarajan and John R. Barry School of Elecrical and Compuer Engineering Georgia Insiue of Technology,
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 informationSTATE-SPACE MODELLING. A mass balance across the tank gives:
B. Lennox and N.F. Thornhill, 9, Sae Space Modelling, IChemE Process Managemen and Conrol Subjec Group Newsleer STE-SPACE MODELLING Inroducion: Over he pas decade or so here has been an ever increasing
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 information