TLT-5200/5206 COMMUNICATION THEORY, Exercise 3, Fall TLT-5200/5206 COMMUNICATION THEORY, Exercise 3, Fall Problem 1.
|
|
- Damian Gibbs
- 5 years ago
- Views:
Transcription
1 TLT-5/56 COMMUNICATION THEORY, Exercise 3, Fall Problem. The "random walk" was modelled as a random sequence [ n] where W[i] are binary i.i.d. random variables with P[W[i] = s] = p (orward step with probability p) P[W[i] = s] = p (backward step with probability p) n i W[ i] To illustrate, one possible sample path (realization) could look something like 3s s s -s -s [n] ut keep in mind that the above igure is just an example (one realization). We are really talking about statistical characterizations here. n TLT-5/56 COMMUNICATION THEORY, Exercise 3, Fall In order to the sequence to be wide-sense stationary (WSS), the autocovariance (or autocorrelation) unction should be shit invariant and the mean unction should be constant (see Carlson 4 th Ed. p ). This means also that the variance K (m,m) o the sequence should be constant in time as well. Now K but K ( m, m) ( m l, m l) ( m) m(s) p( p) ( m l) ( m l)(s) p( p) ( m) meaning that the variance o this random sequence is increasing in time. Thus, the process is not WSS. Furthermore, the mean unction is ( n) ns( p ) which is also time-dependent (in general) implying non-stationarity. Thus, we conclude that the sequence is NOT WSS! This signal model can also be easily examined using Matlab. In the ollowing, we generate independent realizations (o length samples) with s = and p = /. So according to the theory, the mean and variance are in this case given by [ n] E [ n] ns(p) n ( n) E [ n] [ n] E [ n] min( n, n)() (/4) n These results can now be veriied by calculating the corresponding ensemble averages over the generated independent realizations. The results are illustrated in general in the ollowing two igures. / 6 / 6
2 TLT-5/56 COMMUNICATION THEORY, Exercise 3, Fall Three Independent Realizations Time Index 5 Ensemble Mean and Variance (Estimates) TLT-5/56 COMMUNICATION THEORY, Exercise 3, Fall Problem. Oscillators are ubiquitous in physical systems, especially electronic and optical ones. For example, in radio requency communication systems they are used or requency translation o inormation signals and or channel selection. Oscillators are also present in digital electronic systems which require a time reerence, i.e., a clock signal, in order to synchronize operations. Noise is o major concern in oscillators, because introducing even small noise into an oscillator leads to dramatic changes in its requency spectrum and timing properties. This phenomenon is known as phase noise or timing jitter. A perect oscillator would have localized tones at discrete requencies (i.e., harmonics), but any corrupting noise spreads these perect tones, resulting in high power levels at neighbouring requencies. This eect is the major contributor to undesired phenomena such as interchannel intererence, leading to increased bit error rates (ER s) in RF communication systems. Another maniestation o the same phenomenon, jitter, is important in clocked and sampled data systems. Uncertainties in switching instants caused by noise lead to synchronization problems. Characterizing how noise aects oscillators is thereore crucial or practical applications. 5 variance In our case, we study the eects o a small phase noise present in a j( t( t)) complex oscillator signal, xt () Ae. In ideal case, () t, and we have only a discrete impulse at requency in the spectral density unction G (). Let s start by rewriting the given signal x(t) as j( t( t)) j t j () t j t x() t Ae Ae e Ae v() t. 5 mean It is important to note that v(t) is a random process and thereore x(t) is also a random process! First we study is the x(t) a WSS process and we start by taking the expected value (or ensemble average) with t held ixed at arbitrary value, as Time Index 3 / 6 4 / 6
3 TLT-5/56 COMMUNICATION THEORY, Exercise 3, Fall x t E x t E Ae v t Ae E[()] v t () [ ()] [ j t ()] j t ased on the small noise phase assumption ( () t rad), we can approximate v(t) as vt e cos t jsin t j ( t), j () t ( ) ( ( )) ( ( )) and the expected value simpliies to x t Ae E j t Ae. () j t [ ()] j t We notice that the expected value depends on time, and thereore x(t) is not a WSS process! Even though, x(t) is not a WSS process, we can still calculate the autocorrelation o the random process. When we investigate the relationship between two random variables, we use autocorrelation unction which is deined or complex random unctions as R ( t, t ) E[ x( t ) x*( t )]. This unction measures the relatedness or dependence between two random processes. Now that we have the required tools, let s start with the derivation. R ( t, t ) E[ x( t ) x*( t )] [ ( ) *( )] j t j( t) E Ae v t Ae v t j ( tt) Ae Evt [()*( v t)]. TLT-5/56 COMMUNICATION THEORY, Exercise 3, Fall Recalling the small noise phase assumption ( () t rad), the expected value E[v(t )v*(t )] simpliies to E[ v( t) v*( t)] E[( j ( t))( j ( t))] E[ ( t ) ( t )]. E[ ( t)] I we now assume that the random variable () t is WSS, we are only interested in the time dierence between the unctions and thereore we mark t =t and t =t-τ. Thus, the autocorrelation unction o x(t) can be rewritten as j ( ) { E[ ( ) ( )]} j R A e t t Ae ( R( ) ) j j Ae Ae R(). We notice that the dependency orm time t disappears and that the autocorrelation unction depends only in the time dierence τ. Now, in the autocorrelation unction we have the original spectral impulse at and in addition we have requency shited version o the autocorrelation unction o () t around requency. Now it is easy to deine the Fourier transorm o R x (τ), which represents the spectral density unction G x (). G( ) F{ R ( )} A ( ) A G ( ). ecause we assumed that () t is a realization rom WSS process, there exists a Fourier transorm pair R ( ) G ( ). 5 / 6 6 / 6
4 TLT-5/56 COMMUNICATION THEORY, Exercise 3, Fall In the spectral density unction it is even clearer that the power spectrum o the phase noise is shited around requency. The idea is even urther illustrated below. This holds when the phase noise is small. TLT-5/56 COMMUNICATION THEORY, Exercise 3, Fall Problem 3. The variance o the thermal noise voltage v(t) at the open-circuit terminals o a resistor R at temperature T is given by (Carlson 4 th Ed. p.37) G ( ) Gx ( ) vt () kt v R where 3h k = oltzmann constant =.38* -3 h = Planc constant = 6.6* -34 In the book/course notes, the spectral density unction or the noise process is given as G ( ) x Rh G V ( ) h / kt e So, we immediately note that the process is not exactly white (uncorrelated) because the spectral density unction varies with requency. We could calculate the autocorrelation unction or the process by inverse Fourier transorm o G V (). This autocorrelation unction at delay tells us the value o the covariance or t i - t j =, since the mean is zero (when the mean is zero, autocovariance = autocorrelation). We can calculate the total power o the process directly with the help o the irst ormula or v or by integrating G V () rom -ininity to ininity. We do that with Maple and get 7 / 6 8 / 6
5 TLT-5/56 COMMUNICATION THEORY, Exercise 3, Fall > k:=.37*^(-3);r:=;t:=9;h:=6.6*^(-34); - k :=.37 R := T := 9-33 h :=.66 > G:=*R*h*abs()/(exp(h*abs()/(k*T))-): > total_power_:=eval(*(pi*k*t)^*r/(3*h)); -5 total_power_ := > total_power_:=int(g,=-ininity...ininity); -5 total_power_ := To get the desired approximation or G V () we proceed as ollows. First we can write G V () as Rh ( ) h GV Rh h / kt e kt h! kt using the terms up to nd order o the series expansion or e x o the orm e x x! x... TLT-5/56 COMMUNICATION THEORY, Exercise 3, Fall Using only the terms up to the nd order is justiied under the given assumption that h kt. The given orm can be simpliied to G V ( ) h Rh kt h! kt h RkT kt h kt Rh h kt Then we use the trick that (+x) - -x when x is small and we get (use x = h /kt) the desired result h G V ( ) RkT kt The approximation and true density are given in the ollowing Figure. The power in some requency range is given by the integral o G V () over the desired requencies. The integration is easy using the approximation and can be carried out by pen and paper. For example or = - P G ( ) d V symmetry h G V ( ) d 4RkT 4kT Remember, however, that the result is valid only i kt/h. 9 / 6 / 6
6 TLT-5/56 COMMUNICATION THEORY, Exercise 3, Fall TLT-5/56 COMMUNICATION THEORY, Exercise 3, Fall 9 x 9 8. x 6 True Approximation Spectral Density Power True Approximation Frequency [Hz] Figure : Approximative and true power spectral densities (notice the logarithmic scaling o the requency axis).. Maximum Frequency (andwidth) [Hz] Figure : The power in the requency range < using both approximative and true densities. We can also calculate the wanted noise power in the requency range < GHz with Maple, using both the accurate and approximative densities: > G:=*R*k*T*(-h*abs()/(*k*T)): > power_:=int(g,=-^exponent...^exponent): > power_:=int(g,=-^exponent...^exponent): > eval(subs(exponent=9,power_)); > eval(subs(exponent=9,power_)); Clearly, the approximation is good. In general, the power in the requency range is illustrated in the ollowing Figure using both the approximative and true densities. Finally it was asked how big a portion o the total power is within the requency range < GHz, so >raction:=*eval(subs(exponent=9,power_))/total_power_; raction := meaning that only.% o the total noise power is within the requency range < GHz - so 99.9% o the total noise power is outside that range at higher requencies ( GHz and above)! o sounds somewhat strange (?) but this is indeed the case!! o this simply relects the act that we are really dealing with an extremely wideband (close to white) noise process here!! / 6 / 6
7 TLT-5/56 COMMUNICATION THEORY, Exercise 3, Fall Problem 4. When white noise with spectral density / is iltered with H(), the spectral density o the ilter output signal is / H(). This being the case, the power N o the output signal is given by N H ( ) d H ( ) d Noise equivalent bandwidth N is now deined as (see Carlson 4 th Ed. p.378) N g H ( ) d where g = max H(). This means that N is the bandwidth o the ideal ilter having the same maximum power gain and output power as H(). utterworth ilters are characterized with (n = ilter order and = 3d bandwidth) H ( ) / n meaning that g = or all utterworth ilters. We can then use the deinition to get N g H ( ) d / n d TLT-5/56 COMMUNICATION THEORY, Exercise 3, Fall To choose the easy way out, we can check some table o standard integrals (e.g. Carlson 4 th Ed., p. 786) and note that / k dx sin( / k) x k, k > Now, use the substitution x=/ and dx=d/. N nsin( / n) sin( / n) / n d n / x and in the limit we get lim lim n sin( / n) / n N n n / n dx sin( / n) because sin(x)/x as x (Hint: Prove with L Hospitals rule). Results are easy to check with Maple: > assume(k>);int(/(+x^k),x=..ininity); > N:=/(sin(Pi/(*n))/(Pi/(*n))); Pi Pi sin(----) k k Pi N := / Pi sin(/ ----) n n > limit(n,n=ininity); 3 / 6 4 / 6
8 TLT-5/56 COMMUNICATION THEORY, Exercise 3, Fall TLT-5/56 COMMUNICATION THEORY, Exercise 3, Fall Noise Equivalent andwidth N Filter Order n Figure 3: Noise equivalent bandwidth N as a unction o the ilter order n or a utterworth ilter with a 3d bandwidth o Hz. H() n = n = 4 n = Frequency [Hz] Figure 4: Squared amplitude responses or three utterworth ilters with 3d bandwidth = Hz..9 3 d bandwidth = Hz In this case the, noise equivalent bandwidth approaches the 3d bandwidth quite rapidly when the ilter order is increased. This means that the utterworth ilters become closer and closer to "ideal" ilters in the average output power sense when the ilter order is increased. With some other ilter structures, this behaviour can be dierent (slower/aster). H() Noise equivalent bandwidth The ollowing igures illustrate the concept o noise equivalent bandwidth urther Frequency [Hz] Figure 5: Squared amplitude response or a utterworth ilter o order n = with 3 d bandwidth = Hz, the noise equivalent bandwidth is also in the igure. 5 / 6 6 / 6
ENSC327 Communications Systems 2: Fourier Representations. School of Engineering Science Simon Fraser University
ENSC37 Communications Systems : Fourier Representations School o Engineering Science Simon Fraser University Outline Chap..5: Signal Classiications Fourier Transorm Dirac Delta Function Unit Impulse Fourier
More informationAdditional exercises in Stationary Stochastic Processes
Mathematical Statistics, Centre or Mathematical Sciences Lund University Additional exercises 8 * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *
More informationFigure 3.1 Effect on frequency spectrum of increasing period T 0. Consider the amplitude spectrum of a periodic waveform as shown in Figure 3.2.
3. Fourier ransorm From Fourier Series to Fourier ransorm [, 2] In communication systems, we oten deal with non-periodic signals. An extension o the time-requency relationship to a non-periodic signal
More informationAnalog Communication (10EC53)
UNIT-1: RANDOM PROCESS: Random variables: Several random variables. Statistical averages: Function o Random variables, moments, Mean, Correlation and Covariance unction: Principles o autocorrelation unction,
More informationReferences Ideal Nyquist Channel and Raised Cosine Spectrum Chapter 4.5, 4.11, S. Haykin, Communication Systems, Wiley.
Baseand Data Transmission III Reerences Ideal yquist Channel and Raised Cosine Spectrum Chapter 4.5, 4., S. Haykin, Communication Systems, iley. Equalization Chapter 9., F. G. Stremler, Communication Systems,
More informationMixed Signal IC Design Notes set 6: Mathematics of Electrical Noise
ECE45C /8C notes, M. odwell, copyrighted 007 Mied Signal IC Design Notes set 6: Mathematics o Electrical Noise Mark odwell University o Caliornia, Santa Barbara rodwell@ece.ucsb.edu 805-893-344, 805-893-36
More informationObjectives. By the time the student is finished with this section of the workbook, he/she should be able
FUNCTIONS Quadratic Functions......8 Absolute Value Functions.....48 Translations o Functions..57 Radical Functions...61 Eponential Functions...7 Logarithmic Functions......8 Cubic Functions......91 Piece-Wise
More informationIntroduction to Analog And Digital Communications
Introduction to Analog And Digital Communications Second Edition Simon Haykin, Michael Moher Chapter Fourier Representation o Signals and Systems.1 The Fourier Transorm. Properties o the Fourier Transorm.3
More informationRATIONAL FUNCTIONS. Finding Asymptotes..347 The Domain Finding Intercepts Graphing Rational Functions
RATIONAL FUNCTIONS Finding Asymptotes..347 The Domain....350 Finding Intercepts.....35 Graphing Rational Functions... 35 345 Objectives The ollowing is a list o objectives or this section o the workbook.
More informationELEG 3143 Probability & Stochastic Process Ch. 4 Multiple Random Variables
Department o Electrical Engineering University o Arkansas ELEG 3143 Probability & Stochastic Process Ch. 4 Multiple Random Variables Dr. Jingxian Wu wuj@uark.edu OUTLINE 2 Two discrete random variables
More information2. SPECTRAL ANALYSIS APPLIED TO STOCHASTIC PROCESSES
2. SPECTRAL ANALYSIS APPLIED TO STOCHASTIC PROCESSES 2.0 THEOREM OF WIENER- KHINTCHINE An important technique in the study of deterministic signals consists in using harmonic functions to gain the spectral
More informationSignals & Linear Systems Analysis Chapter 2&3, Part II
Signals & Linear Systems Analysis Chapter &3, Part II Dr. Yun Q. Shi Dept o Electrical & Computer Engr. New Jersey Institute o echnology Email: shi@njit.edu et used or the course:
More information9.1 The Square Root Function
Section 9.1 The Square Root Function 869 9.1 The Square Root Function In this section we turn our attention to the square root unction, the unction deined b the equation () =. (1) We begin the section
More informationA Fourier Transform Model in Excel #1
A Fourier Transorm Model in Ecel # -This is a tutorial about the implementation o a Fourier transorm in Ecel. This irst part goes over adjustments in the general Fourier transorm ormula to be applicable
More informationStochastic Processes
Elements of Lecture II Hamid R. Rabiee with thanks to Ali Jalali Overview Reading Assignment Chapter 9 of textbook Further Resources MIT Open Course Ware S. Karlin and H. M. Taylor, A First Course in Stochastic
More information2 Frequency-Domain Analysis
2 requency-domain Analysis Electrical engineers live in the two worlds, so to speak, o time and requency. requency-domain analysis is an extremely valuable tool to the communications engineer, more so
More informationChapter 6. Random Processes
Chapter 6 Random Processes Random Process A random process is a time-varying function that assigns the outcome of a random experiment to each time instant: X(t). For a fixed (sample path): a random process
More informationLecture 13: Applications of Fourier transforms (Recipes, Chapter 13)
Lecture 13: Applications o Fourier transorms (Recipes, Chapter 13 There are many applications o FT, some o which involve the convolution theorem (Recipes 13.1: The convolution o h(t and r(t is deined by
More informationIMPROVED NOISE CANCELLATION IN DISCRETE COSINE TRANSFORM DOMAIN USING ADAPTIVE BLOCK LMS FILTER
SANJAY KUMAR GUPTA* et al. ISSN: 50 3676 [IJESAT] INTERNATIONAL JOURNAL OF ENGINEERING SCIENCE & ADVANCED TECHNOLOGY Volume-, Issue-3, 498 50 IMPROVED NOISE CANCELLATION IN DISCRETE COSINE TRANSFORM DOMAIN
More informationNotes on Wavelets- Sandra Chapman (MPAGS: Time series analysis) # $ ( ) = G f. y t
Wavelets Recall: we can choose! t ) as basis on which we expand, ie: ) = y t ) = G! t ) y t! may be orthogonal chosen or appropriate properties. This is equivalent to the transorm: ) = G y t )!,t )d 2
More informationAnalog Computing Technique
Analog Computing Technique by obert Paz Chapter Programming Principles and Techniques. Analog Computers and Simulation An analog computer can be used to solve various types o problems. It solves them in
More informationSystems & Signals 315
1 / 15 Systems & Signals 315 Lecture 13: Signals and Linear Systems Dr. Herman A. Engelbrecht Stellenbosch University Dept. E & E Engineering 2 Maart 2009 Outline 2 / 15 1 Signal Transmission through a
More informationProbability and Statistics for Final Year Engineering Students
Probability and Statistics for Final Year Engineering Students By Yoni Nazarathy, Last Updated: May 24, 2011. Lecture 6p: Spectral Density, Passing Random Processes through LTI Systems, Filtering Terms
More informationResearch Article. Spectral Properties of Chaotic Signals Generated by the Bernoulli Map
Jestr Journal o Engineering Science and Technology Review 8 () (05) -6 Special Issue on Synchronization and Control o Chaos: Theory, Methods and Applications Research Article JOURNAL OF Engineering Science
More informationConference Article. Spectral Properties of Chaotic Signals Generated by the Bernoulli Map
Jestr Journal o Engineering Science and Technology Review 8 () (05) -6 Special Issue on Synchronization and Control o Chaos: Theory, Methods and Applications Conerence Article JOURNAL OF Engineering Science
More informationStochastic Processes. M. Sami Fadali Professor of Electrical Engineering University of Nevada, Reno
Stochastic Processes M. Sami Fadali Professor of Electrical Engineering University of Nevada, Reno 1 Outline Stochastic (random) processes. Autocorrelation. Crosscorrelation. Spectral density function.
More informationLongitudinal Waves. Reading: Chapter 17, Sections 17-7 to Sources of Musical Sound. Pipe. Closed end: node Open end: antinode
Longitudinal Waes Reading: Chapter 7, Sections 7-7 to 7-0 Sources o Musical Sound Pipe Closed end: node Open end: antinode Standing wae pattern: Fundamental or irst harmonic: nodes at the ends, antinode
More information( x) f = where P and Q are polynomials.
9.8 Graphing Rational Functions Lets begin with a deinition. Deinition: Rational Function A rational unction is a unction o the orm ( ) ( ) ( ) P where P and Q are polynomials. Q An eample o a simple rational
More informationDefinition: Let f(x) be a function of one variable with continuous derivatives of all orders at a the point x 0, then the series.
2.4 Local properties o unctions o several variables In this section we will learn how to address three kinds o problems which are o great importance in the ield o applied mathematics: how to obtain the
More information2A1H Time-Frequency Analysis II
2AH Time-Frequency Analysis II Bugs/queries to david.murray@eng.ox.ac.uk HT 209 For any corrections see the course page DW Murray at www.robots.ox.ac.uk/ dwm/courses/2tf. (a) A signal g(t) with period
More informationTwo-step self-tuning phase-shifting interferometry
Two-step sel-tuning phase-shiting intererometry J. Vargas, 1,* J. Antonio Quiroga, T. Belenguer, 1 M. Servín, 3 J. C. Estrada 3 1 Laboratorio de Instrumentación Espacial, Instituto Nacional de Técnica
More informationScattering of Solitons of Modified KdV Equation with Self-consistent Sources
Commun. Theor. Phys. Beijing, China 49 8 pp. 89 84 c Chinese Physical Society Vol. 49, No. 4, April 5, 8 Scattering o Solitons o Modiied KdV Equation with Sel-consistent Sources ZHANG Da-Jun and WU Hua
More informationCHAPTER 8 ANALYSIS OF AVERAGE SQUARED DIFFERENCE SURFACES
CAPTER 8 ANALYSS O AVERAGE SQUARED DERENCE SURACES n Chapters 5, 6, and 7, the Spectral it algorithm was used to estimate both scatterer size and total attenuation rom the backscattered waveorms by minimizing
More information10. Joint Moments and Joint Characteristic Functions
10. Joint Moments and Joint Characteristic Functions Following section 6, in this section we shall introduce various parameters to compactly represent the inormation contained in the joint p.d. o two r.vs.
More informationIn many diverse fields physical data is collected or analysed as Fourier components.
1. Fourier Methods In many diverse ields physical data is collected or analysed as Fourier components. In this section we briely discuss the mathematics o Fourier series and Fourier transorms. 1. Fourier
More informationLeast-Squares Spectral Analysis Theory Summary
Least-Squares Spectral Analysis Theory Summary Reerence: Mtamakaya, J. D. (2012). Assessment o Atmospheric Pressure Loading on the International GNSS REPRO1 Solutions Periodic Signatures. Ph.D. dissertation,
More informationQuadratic Functions. The graph of the function shifts right 3. The graph of the function shifts left 3.
Quadratic Functions The translation o a unction is simpl the shiting o a unction. In this section, or the most part, we will be graphing various unctions b means o shiting the parent unction. We will go
More informationTSKS01 Digital Communication Lecture 1
TSKS01 Digital Communication Lecture 1 Introduction, Repetition, and Noise Modeling Emil Björnson Department of Electrical Engineering (ISY) Division of Communication Systems Emil Björnson Course Director
More informationRandom Error Analysis of Inertial Sensors output Based on Allan Variance Shaochen Li1, a, Xiaojing Du2,b and Junyi Zhai3,c
International Conerence on Civil, Transportation and Environment (ICCTE 06) Random Error Analysis o Inertial Sensors output Based on Allan Variance Shaochen Li, a, Xiaojing Du, and Junyi Zhai3,c School
More information8.4 Inverse Functions
Section 8. Inverse Functions 803 8. Inverse Functions As we saw in the last section, in order to solve application problems involving eponential unctions, we will need to be able to solve eponential equations
More informationA Systematic Approach to Frequency Compensation of the Voltage Loop in Boost PFC Pre- regulators.
A Systematic Approach to Frequency Compensation o the Voltage Loop in oost PFC Pre- regulators. Claudio Adragna, STMicroelectronics, Italy Abstract Venable s -actor method is a systematic procedure that
More informationELEG 3143 Probability & Stochastic Process Ch. 6 Stochastic Process
Department of Electrical Engineering University of Arkansas ELEG 3143 Probability & Stochastic Process Ch. 6 Stochastic Process Dr. Jingxian Wu wuj@uark.edu OUTLINE 2 Definition of stochastic process (random
More informationParametrization of the Local Scattering Function Estimator for Vehicular-to-Vehicular Channels
Parametrization o the Local Scattering Function Estimator or Vehicular-to-Vehicular Channels Laura Bernadó, Thomas Zemen, Alexander Paier, ohan Karedal and Bernard H Fleury Forschungszentrum Teleommuniation
More informationThu June 16 Lecture Notes: Lattice Exercises I
Thu June 6 ecture Notes: attice Exercises I T. Satogata: June USPAS Accelerator Physics Most o these notes ollow the treatment in the class text, Conte and MacKay, Chapter 6 on attice Exercises. The portions
More informationENSC327 Communications Systems 19: Random Processes. Jie Liang School of Engineering Science Simon Fraser University
ENSC327 Communications Systems 19: Random Processes Jie Liang School of Engineering Science Simon Fraser University 1 Outline Random processes Stationary random processes Autocorrelation of random processes
More informationDigital Image Processing. Lecture 6 (Enhancement) Bu-Ali Sina University Computer Engineering Dep. Fall 2009
Digital Image Processing Lecture 6 (Enhancement) Bu-Ali Sina University Computer Engineering Dep. Fall 009 Outline Image Enhancement in Spatial Domain Spatial Filtering Smoothing Filters Median Filter
More informationCurve Sketching. The process of curve sketching can be performed in the following steps:
Curve Sketching So ar you have learned how to ind st and nd derivatives o unctions and use these derivatives to determine where a unction is:. Increasing/decreasing. Relative extrema 3. Concavity 4. Points
More informationSIO 211B, Rudnick. We start with a definition of the Fourier transform! ĝ f of a time series! ( )
SIO B, Rudnick! XVIII.Wavelets The goal o a wavelet transorm is a description o a time series that is both requency and time selective. The wavelet transorm can be contrasted with the well-known and very
More information2A1H Time-Frequency Analysis II Bugs/queries to HT 2011 For hints and answers visit dwm/courses/2tf
Time-Frequency Analysis II (HT 20) 2AH 2AH Time-Frequency Analysis II Bugs/queries to david.murray@eng.ox.ac.uk HT 20 For hints and answers visit www.robots.ox.ac.uk/ dwm/courses/2tf David Murray. A periodic
More informationA UNIFIED FRAMEWORK FOR MULTICHANNEL FAST QRD-LS ADAPTIVE FILTERS BASED ON BACKWARD PREDICTION ERRORS
A UNIFIED FRAMEWORK FOR MULTICHANNEL FAST QRD-LS ADAPTIVE FILTERS BASED ON BACKWARD PREDICTION ERRORS César A Medina S,José A Apolinário Jr y, and Marcio G Siqueira IME Department o Electrical Engineering
More informationSignals and Spectra - Review
Signals and Spectra - Review SIGNALS DETERMINISTIC No uncertainty w.r.t. the value of a signal at any time Modeled by mathematical epressions RANDOM some degree of uncertainty before the signal occurs
More informationDouble-slit interference of biphotons generated in spontaneous parametric downconversion from a thick crystal
INSTITUTE OF PHYSICS PUBLISHING JOURNAL OF OPTICS B: QUANTUM AND SEMICLASSICAL OPTICS J. Opt. B: Quantum Semiclass. Opt. 3 (2001 S50 S54 www.iop.org/journals/ob PII: S1464-4266(0115159-1 Double-slit intererence
More informationECE 2100 Lecture notes Wed, 1/22/03
HW #4, due, /24 Ch : 34, 37, 43 ECE 0 Lecture notes Wed, /22/03 Exercises: 2., 2.2, 2.4, 2.5 Stu or hints etc., see lecture notes or, /7 Problem Sessions: W, :50-2:40 am, WBB 22 (tall brick geology building),
More informationChapter 2. Basic concepts of probability. Summary. 2.1 Axiomatic foundation of probability theory
Chapter Basic concepts o probability Demetris Koutsoyiannis Department o Water Resources and Environmental Engineering aculty o Civil Engineering, National Technical University o Athens, Greece Summary
More informationMassachusetts Institute of Technology
Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science 6.011: Introduction to Communication, Control and Signal Processing QUIZ, April 1, 010 QUESTION BOOKLET Your
More informationX-ray Diffraction. Interaction of Waves Reciprocal Lattice and Diffraction X-ray Scattering by Atoms The Integrated Intensity
X-ray Diraction Interaction o Waves Reciprocal Lattice and Diraction X-ray Scattering by Atoms The Integrated Intensity Basic Principles o Interaction o Waves Periodic waves characteristic: Frequency :
More informationProbabilistic Model of Error in Fixed-Point Arithmetic Gaussian Pyramid
Probabilistic Model o Error in Fixed-Point Arithmetic Gaussian Pyramid Antoine Méler John A. Ruiz-Hernandez James L. Crowley INRIA Grenoble - Rhône-Alpes 655 avenue de l Europe 38 334 Saint Ismier Cedex
More information3. Several Random Variables
. Several Random Variables. Two Random Variables. Conditional Probabilit--Revisited. Statistical Independence.4 Correlation between Random Variables. Densit unction o the Sum o Two Random Variables. Probabilit
More informationThe concept of limit
Roberto s Notes on Dierential Calculus Chapter 1: Limits and continuity Section 1 The concept o limit What you need to know already: All basic concepts about unctions. What you can learn here: What limits
More informationSupplement To: Search for Tensor, Vector, and Scalar Polarizations in the Stochastic Gravitational-Wave Background
Supplement To: Search or Tensor, Vector, and Scalar Polarizations in the Stochastic GravitationalWave Background B. P. Abbott et al. (LIGO Scientiic Collaboration & Virgo Collaboration) This documents
More informationEDGES AND CONTOURS(1)
KOM31 Image Processing in Industrial Sstems Dr Muharrem Mercimek 1 EDGES AND CONTOURS1) KOM31 Image Processing in Industrial Sstems Some o the contents are adopted rom R. C. Gonzalez, R. E. Woods, Digital
More informationFig 1: Stationary and Non Stationary Time Series
Module 23 Independence and Stationarity Objective: To introduce the concepts of Statistical Independence, Stationarity and its types w.r.to random processes. This module also presents the concept of Ergodicity.
More informationChapter 4 Imaging. Lecture 21. d (110) Chem 793, Fall 2011, L. Ma
Chapter 4 Imaging Lecture 21 d (110) Imaging Imaging in the TEM Diraction Contrast in TEM Image HRTEM (High Resolution Transmission Electron Microscopy) Imaging or phase contrast imaging STEM imaging a
More informationNew Functions from Old Functions
.3 New Functions rom Old Functions In this section we start with the basic unctions we discussed in Section. and obtain new unctions b shiting, stretching, and relecting their graphs. We also show how
More informationFundamentals of Noise
Fundamentals of Noise V.Vasudevan, Department of Electrical Engineering, Indian Institute of Technology Madras Noise in resistors Random voltage fluctuations across a resistor Mean square value in a frequency
More informationAn Alternative Poincaré Section for Steady-State Responses and Bifurcations of a Duffing-Van der Pol Oscillator
An Alternative Poincaré Section or Steady-State Responses and Biurcations o a Duing-Van der Pol Oscillator Jang-Der Jeng, Yuan Kang *, Yeon-Pun Chang Department o Mechanical Engineering, National United
More informationStochastic Processes. A stochastic process is a function of two variables:
Stochastic Processes Stochastic: from Greek stochastikos, proceeding by guesswork, literally, skillful in aiming. A stochastic process is simply a collection of random variables labelled by some parameter:
More informationNumerical Methods - Lecture 2. Numerical Methods. Lecture 2. Analysis of errors in numerical methods
Numerical Methods - Lecture 1 Numerical Methods Lecture. Analysis o errors in numerical methods Numerical Methods - Lecture Why represent numbers in loating point ormat? Eample 1. How a number 56.78 can
More informationMath Review and Lessons in Calculus
Math Review and Lessons in Calculus Agenda Rules o Eponents Functions Inverses Limits Calculus Rules o Eponents 0 Zero Eponent Rule a * b ab Product Rule * 3 5 a / b a-b Quotient Rule 5 / 3 -a / a Negative
More informationPhysics 5153 Classical Mechanics. Solution by Quadrature-1
October 14, 003 11:47:49 1 Introduction Physics 5153 Classical Mechanics Solution by Quadrature In the previous lectures, we have reduced the number o eective degrees o reedom that are needed to solve
More informationarxiv: v1 [gr-qc] 18 Feb 2009 Detecting the Cosmological Stochastic Background of Gravitational Waves with FastICA
1 arxiv:0902.3144v1 [gr-qc] 18 Feb 2009 Detecting the Cosmological Stochastic Background o Gravitational Waves with FastICA Luca Izzo 1,2,3, Salvatore Capozziello 1 and MariaFelicia De Laurentis 1 1 Dipartimento
More informationSec 3.1. lim and lim e 0. Exponential Functions. f x 9, write the equation of the graph that results from: A. Limit Rules
Sec 3. Eponential Functions A. Limit Rules. r lim a a r. I a, then lim a and lim a 0 3. I 0 a, then lim a 0 and lim a 4. lim e 0 5. e lim and lim e 0 Eamples:. Starting with the graph o a.) Shiting 9 units
More informationEECE 301 Signals & Systems Prof. Mark Fowler
EECE 3 Signals & Systems Pro. Mark Fowler Discussion #9 Illustrating the Errors in DFT Processing DFT or Sonar Processing Example # Illustrating The Errors in DFT Processing Illustrating the Errors in
More informationEstimation and detection of a periodic signal
Estimation and detection o a periodic signal Daniel Aronsson, Erik Björnemo, Mathias Johansson Signals and Systems Group, Uppsala University, Sweden, e-mail: Daniel.Aronsson,Erik.Bjornemo,Mathias.Johansson}@Angstrom.uu.se
More informationReview D: Potential Energy and the Conservation of Mechanical Energy
MSSCHUSETTS INSTITUTE OF TECHNOLOGY Department o Physics 8. Spring 4 Review D: Potential Energy and the Conservation o Mechanical Energy D.1 Conservative and Non-conservative Force... D.1.1 Introduction...
More informationPhiladelphia University Faculty of Engineering Communication and Electronics Engineering
Module: Electronics II Module Number: 6503 Philadelphia University Faculty o Engineering Communication and Electronics Engineering Ampliier Circuits-II BJT and FET Frequency Response Characteristics: -
More informationCHAPTER 2 RANDOM PROCESSES IN DISCRETE TIME
CHAPTER 2 RANDOM PROCESSES IN DISCRETE TIME Shri Mata Vaishno Devi University, (SMVDU), 2013 Page 13 CHAPTER 2 RANDOM PROCESSES IN DISCRETE TIME When characterizing or modeling a random variable, estimates
More informationECE-340, Spring 2015 Review Questions
ECE-340, Spring 2015 Review Questions 1. Suppose that there are two categories of eggs: large eggs and small eggs, occurring with probabilities 0.7 and 0.3, respectively. For a large egg, the probabilities
More information14 - Gaussian Stochastic Processes
14-1 Gaussian Stochastic Processes S. Lall, Stanford 211.2.24.1 14 - Gaussian Stochastic Processes Linear systems driven by IID noise Evolution of mean and covariance Example: mass-spring system Steady-state
More informationRoberto s Notes on Differential Calculus Chapter 8: Graphical analysis Section 1. Extreme points
Roberto s Notes on Dierential Calculus Chapter 8: Graphical analysis Section 1 Extreme points What you need to know already: How to solve basic algebraic and trigonometric equations. All basic techniques
More informationThe achievable limits of operational modal analysis. * Siu-Kui Au 1)
The achievable limits o operational modal analysis * Siu-Kui Au 1) 1) Center or Engineering Dynamics and Institute or Risk and Uncertainty, University o Liverpool, Liverpool L69 3GH, United Kingdom 1)
More informationFeasibility of a Multi-Pass Thomson Scattering System with Confocal Spherical Mirrors
Plasma and Fusion Research: Letters Volume 5, 044 200) Feasibility o a Multi-Pass Thomson Scattering System with Conocal Spherical Mirrors Junichi HIRATSUKA, Akira EJIRI, Yuichi TAKASE and Takashi YAMAGUCHI
More information1. Interference condition. 2. Dispersion A B. As shown in Figure 1, the path difference between interfering rays AB and A B is a(sin
asic equations or astronomical spectroscopy with a diraction grating Jeremy Allington-Smith, University o Durham, 3 Feb 000 (Copyright Jeremy Allington-Smith, 000). Intererence condition As shown in Figure,
More informationNumerical Solution of Ordinary Differential Equations in Fluctuationlessness Theorem Perspective
Numerical Solution o Ordinary Dierential Equations in Fluctuationlessness Theorem Perspective NEJLA ALTAY Bahçeşehir University Faculty o Arts and Sciences Beşiktaş, İstanbul TÜRKİYE TURKEY METİN DEMİRALP
More information( 1) ( 2) ( 1) nan integer, since the potential is no longer simple harmonic.
. Anharmonic Oscillators Michael Fowler Landau (para 8) considers a simple harmonic oscillator with added small potential energy terms mα + mβ. We ll simpliy slightly by dropping the term, to give an equation
More information13. Power Spectrum. For a deterministic signal x(t), the spectrum is well defined: If represents its Fourier transform, i.e., if.
For a deterministic signal x(t), the spectrum is well defined: If represents its Fourier transform, i.e., if jt X ( ) = xte ( ) dt, (3-) then X ( ) represents its energy spectrum. his follows from Parseval
More informationSOUND. Responses to Questions
SOUND Responses to Questions. Sound exhibits several phenomena that give evidence that it is a wave. ntererence is a wave phenomenon, and sound produces intererence (such as beats). Diraction is a wave
More informationDiscrete-Time Fourier Transform (DTFT)
Connexions module: m047 Discrete-Time Fourier Transorm DTFT) Don Johnson This work is produced by The Connexions Project and licensed under the Creative Commons Attribution License Abstract Discussion
More informationLecture - 30 Stationary Processes
Probability and Random Variables Prof. M. Chakraborty Department of Electronics and Electrical Communication Engineering Indian Institute of Technology, Kharagpur Lecture - 30 Stationary Processes So,
More information1. Definition: Order Statistics of a sample.
AMS570 Order Statistics 1. Deinition: Order Statistics o a sample. Let X1, X2,, be a random sample rom a population with p.d.. (x). Then, 2. p.d.. s or W.L.O.G.(W thout Loss o Ge er l ty), let s ssu e
More informationStanding Waves If the same type of waves move through a common region and their frequencies, f, are the same then so are their wavelengths, λ.
Standing Waves I the same type o waves move through a common region and their requencies,, are the same then so are their wavelengths,. This ollows rom: v=. Since the waves move through a common region,
More informationLab 3: The FFT and Digital Filtering. Slides prepared by: Chun-Te (Randy) Chu
Lab 3: The FFT and Digital Filtering Slides prepared by: Chun-Te (Randy) Chu Lab 3: The FFT and Digital Filtering Assignment 1 Assignment 2 Assignment 3 Assignment 4 Assignment 5 What you will learn in
More informationSTAT 801: Mathematical Statistics. Hypothesis Testing
STAT 801: Mathematical Statistics Hypothesis Testing Hypothesis testing: a statistical problem where you must choose, on the basis o data X, between two alternatives. We ormalize this as the problem o
More informationPhysics 107 TUTORIAL ASSIGNMENT #7
Physics 07 TUTORIL SSIGNMENT #7 Cutnell & Johnson, 7 th edition Chapter 6: Problems 5, 65, 79, 93 Chapter 7: Problems 7,, 9, 37, 48 Chapter 6 5 Suppose that sound is emitted uniormly in all directions
More informationAPPENDIX 1 ERROR ESTIMATION
1 APPENDIX 1 ERROR ESTIMATION Measurements are always subject to some uncertainties no matter how modern and expensive equipment is used or how careully the measurements are perormed These uncertainties
More information( ) ( ) ( ) + ( ) Ä ( ) Langmuir Adsorption Isotherms. dt k : rates of ad/desorption, N : totally number of adsorption sites.
Langmuir Adsorption sotherms... 1 Summarizing Remarks... Langmuir Adsorption sotherms Assumptions: o Adsorp at most one monolayer o Surace is uniorm o No interaction between adsorbates All o these assumptions
More information9.3 Graphing Functions by Plotting Points, The Domain and Range of Functions
9. Graphing Functions by Plotting Points, The Domain and Range o Functions Now that we have a basic idea o what unctions are and how to deal with them, we would like to start talking about the graph o
More informationIntrinsic Small-Signal Equivalent Circuit of GaAs MESFET s
Intrinsic Small-Signal Equivalent Circuit o GaAs MESFET s M KAMECHE *, M FEHAM M MELIANI, N BENAHMED, S DALI * National Centre o Space Techniques, Algeria Telecom Laboratory, University o Tlemcen, Algeria
More informationStatistical signal processing
Statistical signal processing Short overview of the fundamentals Outline Random variables Random processes Stationarity Ergodicity Spectral analysis Random variable and processes Intuition: A random variable
More informationSyllabus Objective: 2.9 The student will sketch the graph of a polynomial, radical, or rational function.
Precalculus Notes: Unit Polynomial Functions Syllabus Objective:.9 The student will sketch the graph o a polynomial, radical, or rational unction. Polynomial Function: a unction that can be written in
More information