ECE 650 Lecture #10 (was Part 1 & 2) D. van Alphen. D. van Alphen 1
|
|
- Adele Richard
- 6 years ago
- Views:
Transcription
1 ECE 650 Lecture #10 (was Part 1 & 2) D. van Alphen D. van Alphen 1
2 Lecture 10 Overview Part 1 Review of Lecture 9 Continuing: Systems with Random Inputs More about Poisson RV s Intro. to Poisson Processes More on Poisson Processes Introduction to Marov Processes D. van Alphen 2
3 Reviewing Lecture 9 0-mean RP V(t) is white if C V (t) = d(t) ( V(t i ) and V(t j ) are uncorrelated RV s if t i t j ) RP X(t) is normal (Gaussian) if RV s X(t 1 ),, X(t n ) are jointly normal for any n and any t 1, t 2,, t n RP X(t) is strict-sense stationary (SSS) if (all of) its statistical properties are invariant to shifts of the time origin. RP X(t) is wide-sense stationary (WSS) if E{X(t)} = h(t) = h, for all t (constant mean) R X (t 1, t 2 ) = R X (t) = E{ X(t + t) X(t) }, t = t 1 t 2 Note: E{ X 2 (t) } = R(0), the total average power of the RP D. van Alphen 3
4 Power Spectral Densities (PSD s) for WSS RP s Concept: The PSD, S X (w), of RP X(t) is a function which measures the distribution of power (on the average, of the RP) in the frequency domain. Wiener-Khinchin Theorem: For WSS processes, S X (w) = F { R X (t) } F R X (t) S X (w) (F : Fourier Transform) Integrate the PSD to find the total average power in the random process: P X 1 2 S ( w)dw X S X (f )df R X (0) E(X 2 (t)) D. van Alphen 4
5 White Noise, Re-visited White noise can be defined by either its autocorrelation function (delta-correlated) or its PSD (flat or constant for all w)): R X (t) (N 0 /2) t F S X (w) N 0 /2 w 1. E{ X 2 (t) } 1 2 S ( w)dw X R X (0) 2. E{X(t)} = (no constant term in R X (t)) 3. White noise has power, mean. D. van Alphen 5
6 Ergodic RP s A RP is ergodic if almost every member of the ensemble is typical of the ensemble, with the same statistical properties. For ergodic processes, time-averaging of a single sample function can be used to replace ensemble averaging. X(t) : notation for time-averaging of X(t) Ergodic stationary, but not conversely D. van Alphen 6
7 Continuing: Systems with Random Inputs Meaning: sample function X(t, z i ) yields output Y(t, z i ) Mean: E{Y(t)} = X(t) g Y(t) = g[x(t)] g(x) f (x; t)dx Autocorrelation: R Y (t 1, t 2 ) = E{ Y(t 1 ) Y(t 2 ) } x fix t; handle one RV at a time g(x1)g(x2) fx(x1,x2;t1,t2 )dx D. van Alphen 7
8 LTI Systems with Random Inputs X(t, x) = x(t) Inputting one sample function at a time h, H Y(t, x) = y(t) = x(t) * h(t) Goal: to describe the output RP, statistically Now consider LTI system with impulse response h(t), and frequency response H(f). From Lecture 9: S Y (f) = H(f) 2 S X (f) R YY (t) = R XX (t) * h(t) * h(-t) Also, we can find the mean of the output RP Y(t): h Y (t) = E[Y(t)] = E[h(t) * X(t, x)] = h(t) * h x (t) If the input RP is WSS, then h x (t) = h x, so h Y (t) = h X h(t t)dt h X H(0) D. van Alphen 8
9 Example Consider the RC LP Filter, with impulse response: h(t) = (1/t) exp(-t/t) u(t), where t = RC Describe the output RP if the input process is white Gaussian noise N(t), with PSD N 0 /2. Describe : give pdf, mean and Solution: R Y (t) or S Y (f). 1. Since the system is linear, the output RP is Gaussian From a FT Table: H(f) =, so H(f) 2 = 1 j2ft 1 (2ft) 3. N0 / 2 Hence, S Y (f) = 2 1 (2ft) Is this output white? 2 4. h Y = h X H(0) = 0 D. van Alphen 9
10 Summary: Systems with Random Inputs X(t) g, h, H Y(t) = g[x(t)] If the system is LTI, h Y (t) = E[Y(t)] = E[h(t) * X(t, x)] = h(t) * h x (t) If the system is LTI and X(t) is WSS S Y (f) = H(f) 2 S X (f) R YY (t) = R XX (t) * h(t) * h(-t) h Y = h X H(0), the input mean times the system dc gain D. van Alphen 10
11 Repeating: Lecture 1, p. 23 Poisson Random Variables (another ind of counting RV) RV X is Poisson with parameter a iff P X (m) Pr{ X m} m a m! e a m 0, 1, 2, 3 PDF setch for the case: a = 3 Pr{ X m} m! PX (0) Pr{ X 0} e e ! P X (1) Pr{ X 1} 1 3 e 1! 3 3e m e 3.. P X (m) (rounding) m ECE 650 D. van Alphen 11
12 More on Poisson RV s - Say X is Poisson with mean & var. a Introduced in Lect. 1, pp a a Repeating: PX (m) Pr{ X m} e m 0, 1, 2, m! a 1 e a Ratio of adjacent terms in the pdf: Pr{X 1} ( 1)! Pr{X } a a e a! Consider 3 cases for the relationship between mean a and : 1. < a Pr{X = -1} < Pr{X = } pdf increases with 2. > a Pr{X = -1} > Pr{X = } pdf decreases as increases 3. = a (only possible for integer values of a) P{X = -1} = Pr{X = } 2 equal (adjacent) values of pdf at X = -1 and X = (See previous page for example) ECE 650 D. van Alphen 12 m
13 Poisson Distributions, continued If a Poisson RV counts the number of occurrences of some event occurring with rate q, over a time period of t seconds, then the RV is Poisson with parameter a= qt: Pr{X } (qt)! e qt 2, Example: The number of phone calls to be handled in a cellular system during a particular hour of the day is a Poisson RV, with average rate 80 calls/minute. Let RV X count the number of calls to be handled in an interval of length 6 seconds. Pr{X } (qt)! e (qt), (80.1)! e 0, 1, (80.1) (8)! e (8) ECE 650 D. van Alphen 13
14 Poisson Approximation for Binomial Distribution (large n and small p (or n, p unnown but np nown)) If n is large, and p is small (so np is medium), and << n, then n n a a Pr( successes) p q e where a np! Note: a = np is the expected number of successes in n trials Example (Carol Ash): Consider a typist who maes (on the average) 3 mistaes per page. Find the probability that the typist maes 10 mistaes on p Note: p = prob. of error on typing a single character n = number of characters typed per page unnown But a = np = 3 = exp. number of errors per page n n 10 3 Pr(X 10) p q e 3 8.1x ! ECE 650 D. van Alphen 14
15 Queueing Theory: Poisson Arrivals As customers arrive (say to mae a ban transaction), write down the arrival time of each person Let a be the average # of customers arriving per hour Divide the hour into a large # (say n) of subintervals, so that at most 1 ( 0 or 1) person can arrive in a subinterval. Each subinterval is a Bernoulli trial, where success a customer arriving n, p unnown np = a, nown Use Poisson approximation for Binomial Poisson Arrivals: x 10:00 11:00 x x x x AM n AM ECE 650 D. van Alphen 15
16 Queueing Theory: Poisson Arrivals 10:00 AM Poisson Arrivals: x x x x x n 11:00 AM For independent arrivals, the # of arrivals in a time period is Poisson-distributed: Pr( arrivals in a time period) a! a where a: arrival rate (average # of arrivals/time period) e ECE 650 D. van Alphen 16
17 Example: Poisson Arrivals Suppose meteor bursts occur at a particular location in the upper atmosphere at a rate of 4 per minute. (a) Find the probability 2 bursts occur in the next minute. (b) Find the probability that at most 2 bursts occur in the next 3 minutes (a) Pr(2 bursts in 1 minute) e ! (b) Pr(at most 2 bursts in the next 3 minutes) = P(0, 1, or 2 bursts in the next 3 minutes). Note that for a period of 3 minutes, we have a = 12 bursts, on the average. Hence P(0, 1, or 2 bursts in the next 3 minutes) = e e e = x ! 1! 2! ECE 650 D. van Alphen 17
18 Poisson Points Experiment: Place n points at random on (-T/2, T/2). Find the probability that of the n points are in sub-interval (t 1, t 2 ), of length t a, as shown below: -T/2 t 1 t 2 T/2 t a Note: Each placement of a point is a Bernoulli trial, where success means that the point landed in (t 1, t 2 ), and failure meaning that it didn t. Thus, Pr{ pts. in t a } = Pr( successes) where p = (t 2 t 1 )/T = t a /T n p q n () ECE 650 D. van Alphen 18
19 Poisson Points, continued Now suppose that n >> 1 and t a << T ( n large, p = t a /T small) Poisson Approximation to Binomial Distribution O.K. Pr{ pts. in t a } = n p q n a! e a (nt a / T)! e (nt a / T) for near np = nt a /T. Now, let n, and T, with the constraint that n/t = a 1, a constant representing the density of points. Then Pr{ pts. in t a } ( a 1t a) a e 1t a! ( ) where a 1 = n/t, the density of points The # of Poisson Points in an Interval is a Poisson RV. ECE 650 D. van Alphen 19
20 Waiting Time for Poisson Points or Arrivals If events (e.g., arrivals) are Poisson distributed with a = n/t, then the waiting time W from a given event to the next event is exponentially distributed: f W (w) = a e -aw, w > 0 Proof: Suppose that an arrival occurs at time 0: 0 w t a = w F w (w) = Pr{W w} = Pr(at least 1 other event occurs at or before time w) = 1 Pr(0 events occur at or before time w) ( a w) 0 1 e 0! 1 e d f w (w) = F (w) ae aw w, dw w 0 ( aw) aw ECE 650 D. van Alphen 20
21 Poisson Points: Summary 1. Pr{ points in t a } at ) a! e ( at ) ( a, a n/ T (density of points) 2. If 2 intervals (t 1, t 2 ) and (t 3, t 4 ) are non-overlapping, then the events { a in (t 1, t 2 )} and { b in (t 3, t 4 )} are independent. 3. If events (e.g., arrivals) are Poisson distributed with a = n/t, then the waiting time W from a given event to the next event is exponentially distributed (with parameter a, mean 1/a): f w (w) = a e -aw, w > 0 ECE 650 D. van Alphen 21
22 Poisson Arrivals: Example Suppose that raindrops land on a weather-station collection grid at the rate of 100 drops/sec. Find the probability that the time interval between 2 successive drops is greater than 1 ms. Raindrop arrival Poisson points with a = 100 drops/sec. Waiting time W Exponential with a = 100 Pr{W >.001} = 1 Pr(W.001) = 1 F W (.001) =e -100(.001) = e -.1 =.905 ECE 650 D. van Alphen 22
23 Poisson RP s Consider a RP which counts the number of occurrences of some event in the time interval [0, t). Each occurrence of the event is said to be an arrival. The RP is a Poisson Process if it has the following properties: 1. (Independent Increments) The number of arrivals in any 2 non-overlapping intervals are independent of each other. 2. (Stationary Increments) The number of arrivals in an interval [t, t+t) depends only of the length of the interval, t. 3. (Distribution of Infinitesimal Increments) For an interval of infinitesimal length, [t, t+dt), the probability of a single arrival is proportional to Dt, and the probability of more than one arrival is negligible compared to Dt. D. van Alphen 23
24 Poisson Processes, continued X: counts arrivals on interval (0, t) If we let l be the constant of proportionality in property 3: i.e., Pr(1 arrival in [t, t + Dt) ) = l dt l: arrival rate It can be shown (see text, pp ) that Pr(X = 0 for interval (0, t) = P X (0; t) = e -lt u(t) ( lt) Pr(X = for interval (0, t)) = P X (; t) = e lt! u(t) Note: average # of arrivals in [0, t) is lt. PMF Function Note: process is not stationary in the mean. for RV at time t D. van Alphen 24
25 Poisson Processes, continued X: counts arrivals on interval (0, t) For a Poisson RP with arrival rate l, ( lt) Pr(X = for interval (0, t)) = P X (; t) = e lt u(t) (*)! Example (Miller & Childers, 8.37) Suppose that the arrival of calls at a switchboard is a Poisson process with rate l =.1 calls/min. Find the probability that the number of calls arriving in a 10- minute interval is less than (1) 1 From (*): Pr(X < 10) = e u(t) (since lt = 1) 0! In MATLAB: >> poisscdf(9,1) ans = D. van Alphen 25
26 Example, continued Since we are fixing time to t = 10, the Poisson Process (at t = 10) becomes a Poisson RV with parameter lt = 1, with PMF: PMF for Poisson RV (from process at t = 10) MATLAB Code: x = 0:12; poisspdf(x, 1) stem(x, ans) Checing prev. ans. >> x = 0:9; >> y = poisspdf(x,1); >> z = cumsum(y); >> z(10) ans = D. van Alphen 26
27 Poisson Process Sample Functions Now suppose that we want to setch a sample function from a (different) Poisson Process. Model the time between arrivals as exponential, mean 2: >> wt_time = exprnd(2,7,1); >> time_to_arrival = cumsum(wt_time); >> [wt_time time_to_arrival] ans = Arrival times D. van Alphen
28 Poisson Process Sample Function, Example Sample Function from Poisson Process MATLAB Code: >> x = [ ]; >> y = [0:7]; >> stairs(x, y) D. van Alphen 28
29 Marov Process Random process X(t) is a Marov process if, for any time points t 1 < t 2 < < t n < t n+1, the process satisfies the condition concerning the conditional pdf: f X (x n x n-1, x n-2,, x 1 ; t n,, t 1 ) = f X (x n x n-1 ; t n, t n-1 ) The Poisson counting process is an example of a Marov Process (due to the independent increments property). A discrete-valued Marov process satisfies the condition concerning the conditional PMF: 1 st order Marov P X (x n x n-1, x n-2,, x 1 ; t n,, t 1 ) = P X (x n x n-1 ; t n, t n-1 ) D. van Alphen 29
Stochastic 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 informationSRI VIDYA COLLEGE OF ENGINEERING AND TECHNOLOGY UNIT 3 RANDOM PROCESS TWO MARK QUESTIONS
UNIT 3 RANDOM PROCESS TWO MARK QUESTIONS 1. Define random process? The sample space composed of functions of time is called a random process. 2. Define Stationary process? If a random process is divided
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 informationfor valid PSD. PART B (Answer all five units, 5 X 10 = 50 Marks) UNIT I
Code: 15A04304 R15 B.Tech II Year I Semester (R15) Regular Examinations November/December 016 PROBABILITY THEY & STOCHASTIC PROCESSES (Electronics and Communication Engineering) Time: 3 hours Max. Marks:
More informationLecture Notes 7 Stationary Random Processes. Strict-Sense and Wide-Sense Stationarity. Autocorrelation Function of a Stationary Process
Lecture Notes 7 Stationary Random Processes Strict-Sense and Wide-Sense Stationarity Autocorrelation Function of a Stationary Process Power Spectral Density Continuity and Integration of Random Processes
More informationName of the Student: Problems on Discrete & Continuous R.Vs
Engineering Mathematics 05 SUBJECT NAME : Probability & Random Process SUBJECT CODE : MA6 MATERIAL NAME : University Questions MATERIAL CODE : JM08AM004 REGULATION : R008 UPDATED ON : Nov-Dec 04 (Scan
More informationName of the Student: Problems on Discrete & Continuous R.Vs
Engineering Mathematics 08 SUBJECT NAME : Probability & Random Processes SUBJECT CODE : MA645 MATERIAL NAME : University Questions REGULATION : R03 UPDATED ON : November 07 (Upto N/D 07 Q.P) (Scan the
More informationECE 450 Homework #3. 1. Given the joint density function f XY (x,y) = 0.5 1<x<2, 2<y< <x<4, 2<y<3 0 else
ECE 450 Homework #3 0. Consider the random variables X and Y, whose values are a function of the number showing when a single die is tossed, as show below: Exp. Outcome 1 3 4 5 6 X 3 3 4 4 Y 0 1 3 4 5
More information16.584: Random (Stochastic) Processes
1 16.584: Random (Stochastic) Processes X(t): X : RV : Continuous function of the independent variable t (time, space etc.) Random process : Collection of X(t, ζ) : Indexed on another independent variable
More informationUCSD ECE153 Handout #40 Prof. Young-Han Kim Thursday, May 29, Homework Set #8 Due: Thursday, June 5, 2011
UCSD ECE53 Handout #40 Prof. Young-Han Kim Thursday, May 9, 04 Homework Set #8 Due: Thursday, June 5, 0. Discrete-time Wiener process. Let Z n, n 0 be a discrete time white Gaussian noise (WGN) process,
More informationUCSD ECE250 Handout #27 Prof. Young-Han Kim Friday, June 8, Practice Final Examination (Winter 2017)
UCSD ECE250 Handout #27 Prof. Young-Han Kim Friday, June 8, 208 Practice Final Examination (Winter 207) There are 6 problems, each problem with multiple parts. Your answer should be as clear and readable
More informationECE Homework Set 3
ECE 450 1 Homework Set 3 0. Consider the random variables X and Y, whose values are a function of the number showing when a single die is tossed, as show below: Exp. Outcome 1 3 4 5 6 X 3 3 4 4 Y 0 1 3
More informationP 1.5 X 4.5 / X 2 and (iii) The smallest value of n for
DHANALAKSHMI COLLEGE OF ENEINEERING, CHENNAI DEPARTMENT OF ELECTRONICS AND COMMUNICATION ENGINEERING MA645 PROBABILITY AND RANDOM PROCESS UNIT I : RANDOM VARIABLES PART B (6 MARKS). A random variable X
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 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 informationChapter 6: Random Processes 1
Chapter 6: Random Processes 1 Yunghsiang S. Han Graduate Institute of Communication Engineering, National Taipei University Taiwan E-mail: yshan@mail.ntpu.edu.tw 1 Modified from the lecture notes by Prof.
More informationUCSD ECE 153 Handout #46 Prof. Young-Han Kim Thursday, June 5, Solutions to Homework Set #8 (Prepared by TA Fatemeh Arbabjolfaei)
UCSD ECE 53 Handout #46 Prof. Young-Han Kim Thursday, June 5, 04 Solutions to Homework Set #8 (Prepared by TA Fatemeh Arbabjolfaei). Discrete-time Wiener process. Let Z n, n 0 be a discrete time white
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 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 informationProblems on Discrete & Continuous R.Vs
013 SUBJECT NAME SUBJECT CODE MATERIAL NAME MATERIAL CODE : Probability & Random Process : MA 61 : University Questions : SKMA1004 Name of the Student: Branch: Unit I (Random Variables) Problems on Discrete
More informationProperties of the Autocorrelation Function
Properties of the Autocorrelation Function I The autocorrelation function of a (real-valued) random process satisfies the following properties: 1. R X (t, t) 0 2. R X (t, u) =R X (u, t) (symmetry) 3. R
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 informationName of the Student: Problems on Discrete & Continuous R.Vs
Engineering Mathematics 03 SUBJECT NAME : Probability & Random Process SUBJECT CODE : MA 6 MATERIAL NAME : Problem Material MATERIAL CODE : JM08AM008 (Scan the above QR code for the direct download of
More informationMarkov Chains. X(t) is a Markov Process if, for arbitrary times t 1 < t 2 <... < t k < t k+1. If X(t) is discrete-valued. If X(t) is continuous-valued
Markov Chains X(t) is a Markov Process if, for arbitrary times t 1 < t 2
More information2. (a) What is gaussian random variable? Develop an equation for guassian distribution
Code No: R059210401 Set No. 1 II B.Tech I Semester Supplementary Examinations, February 2007 PROBABILITY THEORY AND STOCHASTIC PROCESS ( Common to Electronics & Communication Engineering, Electronics &
More informationEAS 305 Random Processes Viewgraph 1 of 10. Random Processes
EAS 305 Random Processes Viewgraph 1 of 10 Definitions: Random Processes A random process is a family of random variables indexed by a parameter t T, where T is called the index set λ i Experiment outcome
More informationContinuous-time Markov Chains
Continuous-time Markov Chains Gonzalo Mateos Dept. of ECE and Goergen Institute for Data Science University of Rochester gmateosb@ece.rochester.edu http://www.ece.rochester.edu/~gmateosb/ October 23, 2017
More information7 The Waveform Channel
7 The Waveform Channel The waveform transmitted by the digital demodulator will be corrupted by the channel before it reaches the digital demodulator in the receiver. One important part of the channel
More informationExponential Distribution and Poisson Process
Exponential Distribution and Poisson Process Stochastic Processes - Lecture Notes Fatih Cavdur to accompany Introduction to Probability Models by Sheldon M. Ross Fall 215 Outline Introduction Exponential
More informationProbability and Statistics
Probability and Statistics 1 Contents some stochastic processes Stationary Stochastic Processes 2 4. Some Stochastic Processes 4.1 Bernoulli process 4.2 Binomial process 4.3 Sine wave process 4.4 Random-telegraph
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 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 informationName of the Student: Problems on Discrete & Continuous R.Vs
SUBJECT NAME : Probability & Random Processes SUBJECT CODE : MA645 MATERIAL NAME : Additional Problems MATERIAL CODE : JM08AM004 REGULATION : R03 UPDATED ON : March 05 (Scan the above QR code for the direct
More informationQuestion Paper Code : AEC11T03
Hall Ticket No Question Paper Code : AEC11T03 VARDHAMAN COLLEGE OF ENGINEERING (AUTONOMOUS) Affiliated to JNTUH, Hyderabad Four Year B Tech III Semester Tutorial Question Bank 2013-14 (Regulations: VCE-R11)
More informationThe exponential distribution and the Poisson process
The exponential distribution and the Poisson process 1-1 Exponential Distribution: Basic Facts PDF f(t) = { λe λt, t 0 0, t < 0 CDF Pr{T t) = 0 t λe λu du = 1 e λt (t 0) Mean E[T] = 1 λ Variance Var[T]
More informationSignals and Spectra (1A) Young Won Lim 11/26/12
Signals and Spectra (A) Copyright (c) 202 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version.2 or any later
More informationECE302 Spring 2006 Practice Final Exam Solution May 4, Name: Score: /100
ECE302 Spring 2006 Practice Final Exam Solution May 4, 2006 1 Name: Score: /100 You must show ALL of your work for full credit. This exam is open-book. Calculators may NOT be used. 1. As a function of
More informationECE 636: Systems identification
ECE 636: Systems identification Lectures 3 4 Random variables/signals (continued) Random/stochastic vectors Random signals and linear systems Random signals in the frequency domain υ ε x S z + y Experimental
More informationE&CE 358, Winter 2016: Solution #2. Prof. X. Shen
E&CE 358, Winter 16: Solution # Prof. X. Shen Email: xshen@bbcr.uwaterloo.ca Prof. X. Shen E&CE 358, Winter 16 ( 1:3-:5 PM: Solution # Problem 1 Problem 1 The signal g(t = e t, t T is corrupted by additive
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 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 informationCommunication Systems Lecture 21, 22. Dong In Kim School of Information & Comm. Eng. Sungkyunkwan University
Communication Systems Lecture 1, Dong In Kim School of Information & Comm. Eng. Sungkyunkwan University 1 Outline Linear Systems with WSS Inputs Noise White noise, Gaussian noise, White Gaussian noise
More informationECE6604 PERSONAL & MOBILE COMMUNICATIONS. Week 3. Flat Fading Channels Envelope Distribution Autocorrelation of a Random Process
1 ECE6604 PERSONAL & MOBILE COMMUNICATIONS Week 3 Flat Fading Channels Envelope Distribution Autocorrelation of a Random Process 2 Multipath-Fading Mechanism local scatterers mobile subscriber base station
More informationVALLIAMMAI ENGINEERING COLLEGE SRM Nagar, Kattankulathur 603 203. DEPARTMENT OF ELECTRONICS & COMMUNICATION ENGINEERING SUBJECT QUESTION BANK : MA6451 PROBABILITY AND RANDOM PROCESSES SEM / YEAR:IV / II
More informationIntroduction to Probability and Stochastic Processes I
Introduction to Probability and Stochastic Processes I Lecture 3 Henrik Vie Christensen vie@control.auc.dk Department of Control Engineering Institute of Electronic Systems Aalborg University Denmark Slides
More informationEE4601 Communication Systems
EE4601 Communication Systems Week 4 Ergodic Random Processes, Power Spectrum Linear Systems 0 c 2011, Georgia Institute of Technology (lect4 1) Ergodic Random Processes An ergodic random process is one
More informationPROBABILITY AND RANDOM PROCESSESS
PROBABILITY AND RANDOM PROCESSESS SOLUTIONS TO UNIVERSITY QUESTION PAPER YEAR : JUNE 2014 CODE NO : 6074 /M PREPARED BY: D.B.V.RAVISANKAR ASSOCIATE PROFESSOR IT DEPARTMENT MVSR ENGINEERING COLLEGE, NADERGUL
More informationFundamentals of Digital Commun. Ch. 4: Random Variables and Random Processes
Fundamentals of Digital Commun. Ch. 4: Random Variables and Random Processes Klaus Witrisal witrisal@tugraz.at Signal Processing and Speech Communication Laboratory www.spsc.tugraz.at Graz University of
More informationRandom Process. Random Process. Random Process. Introduction to Random Processes
Random Process A random variable is a function X(e) that maps the set of experiment outcomes to the set of numbers. A random process is a rule that maps every outcome e of an experiment to a function X(t,
More informationECE Lecture #10 Overview
ECE 450 - Lecture #0 Overview Introduction to Random Vectors CDF, PDF Mean Vector, Covariance Matrix Jointly Gaussian RV s: vector form of pdf Introduction to Random (or Stochastic) Processes Definitions
More informationG.PULLAIAH COLLEGE OF ENGINEERING & TECHNOLOGY DEPARTMENT OF ELECTRONICS & COMMUNICATION ENGINEERING PROBABILITY THEORY & STOCHASTIC PROCESSES
G.PULLAIAH COLLEGE OF ENGINEERING & TECHNOLOGY DEPARTMENT OF ELECTRONICS & COMMUNICATION ENGINEERING PROBABILITY THEORY & STOCHASTIC PROCESSES LECTURE NOTES ON PTSP (15A04304) B.TECH ECE II YEAR I SEMESTER
More informationChapter 3, 4 Random Variables ENCS Probability and Stochastic Processes. Concordia University
Chapter 3, 4 Random Variables ENCS6161 - Probability and Stochastic Processes Concordia University ENCS6161 p.1/47 The Notion of a Random Variable A random variable X is a function that assigns a real
More information= 4. e t/a dt (2) = 4ae t/a. = 4a a = 1 4. (4) + a 2 e +j2πft 2
ECE 341: Probability and Random Processes for Engineers, Spring 2012 Homework 13 - Last homework Name: Assigned: 04.18.2012 Due: 04.25.2012 Problem 1. Let X(t) be the input to a linear time-invariant filter.
More informationA Probability Primer. A random walk down a probabilistic path leading to some stochastic thoughts on chance events and uncertain outcomes.
A Probability Primer A random walk down a probabilistic path leading to some stochastic thoughts on chance events and uncertain outcomes. Are you holding all the cards?? Random Events A random event, E,
More informationRandom Processes Why we Care
Random Processes Why we Care I Random processes describe signals that change randomly over time. I Compare: deterministic signals can be described by a mathematical expression that describes the signal
More informationGaussian Basics Random Processes Filtering of Random Processes Signal Space Concepts
White Gaussian Noise I Definition: A (real-valued) random process X t is called white Gaussian Noise if I X t is Gaussian for each time instance t I Mean: m X (t) =0 for all t I Autocorrelation function:
More informationRelationship between probability set function and random variable - 2 -
2.0 Random Variables A rat is selected at random from a cage and its sex is determined. The set of possible outcomes is female and male. Thus outcome space is S = {female, male} = {F, M}. If we let X be
More informationEEM 409. Random Signals. Problem Set-2: (Power Spectral Density, LTI Systems with Random Inputs) Problem 1: Problem 2:
EEM 409 Random Signals Problem Set-2: (Power Spectral Density, LTI Systems with Random Inputs) Problem 1: Consider a random process of the form = + Problem 2: X(t) = b cos(2π t + ), where b is a constant,
More informationNorthwestern University Department of Electrical Engineering and Computer Science
Northwestern University Department of Electrical Engineering and Computer Science EECS 454: Modeling and Analysis of Communication Networks Spring 2008 Probability Review As discussed in Lecture 1, probability
More informationThings to remember when learning probability distributions:
SPECIAL DISTRIBUTIONS Some distributions are special because they are useful They include: Poisson, exponential, Normal (Gaussian), Gamma, geometric, negative binomial, Binomial and hypergeometric distributions
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 information3F1 Random Processes Examples Paper (for all 6 lectures)
3F Random Processes Examples Paper (for all 6 lectures). Three factories make the same electrical component. Factory A supplies half of the total number of components to the central depot, while factories
More informationLecture 13. Poisson Distribution. Text: A Course in Probability by Weiss 5.5. STAT 225 Introduction to Probability Models February 16, 2014
Lecture 13 Text: A Course in Probability by Weiss 5.5 STAT 225 Introduction to Probability Models February 16, 2014 Whitney Huang Purdue University 13.1 Agenda 1 2 3 13.2 Review So far, we have seen discrete
More informationLecture 4a: Continuous-Time Markov Chain Models
Lecture 4a: Continuous-Time Markov Chain Models Continuous-time Markov chains are stochastic processes whose time is continuous, t [0, ), but the random variables are discrete. Prominent examples of continuous-time
More informationA SIGNAL THEORETIC INTRODUCTION TO RANDOM PROCESSES
A SIGNAL THEORETIC INTRODUCTION TO RANDOM PROCESSES ROY M. HOWARD Department of Electrical Engineering & Computing Curtin University of Technology Perth, Australia WILEY CONTENTS Preface xiii 1 A Signal
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 informationECE 650 1/11. Homework Sets 1-3
ECE 650 1/11 Note to self: replace # 12, # 15 Homework Sets 1-3 HW Set 1: Review Assignment from Basic Probability 1. Suppose that the duration in minutes of a long-distance phone call is exponentially
More information3. Poisson Processes (12/09/12, see Adult and Baby Ross)
3. Poisson Processes (12/09/12, see Adult and Baby Ross) Exponential Distribution Poisson Processes Poisson and Exponential Relationship Generalizations 1 Exponential Distribution Definition: The continuous
More informationChapter 6 - Random Processes
EE385 Class Notes //04 John Stensby Chapter 6 - Random Processes Recall that a random variable X is a mapping between the sample space S and the extended real line R +. That is, X : S R +. A random process
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 informationThe Cooper Union Department of Electrical Engineering ECE111 Signal Processing & Systems Analysis Final May 4, 2012
The Cooper Union Department of Electrical Engineering ECE111 Signal Processing & Systems Analysis Final May 4, 2012 Time: 3 hours. Close book, closed notes. No calculators. Part I: ANSWER ALL PARTS. WRITE
More informationSystem Simulation Part II: Mathematical and Statistical Models Chapter 5: Statistical Models
System Simulation Part II: Mathematical and Statistical Models Chapter 5: Statistical Models Fatih Cavdur fatihcavdur@uludag.edu.tr March 20, 2012 Introduction Introduction The world of the model-builder
More informationSTAT/MATH 395 A - PROBABILITY II UW Winter Quarter Moment functions. x r p X (x) (1) E[X r ] = x r f X (x) dx (2) (x E[X]) r p X (x) (3)
STAT/MATH 395 A - PROBABILITY II UW Winter Quarter 07 Néhémy Lim Moment functions Moments of a random variable Definition.. Let X be a rrv on probability space (Ω, A, P). For a given r N, E[X r ], if it
More informationSTAT 430/510 Probability
STAT 430/510 Probability Hui Nie Lecture 16 June 24th, 2009 Review Sum of Independent Normal Random Variables Sum of Independent Poisson Random Variables Sum of Independent Binomial Random Variables Conditional
More informationStationary independent increments. 1. Random changes of the form X t+h X t fixed h > 0 are called increments of the process.
Stationary independent increments 1. Random changes of the form X t+h X t fixed h > 0 are called increments of the process. 2. If each set of increments, corresponding to non-overlapping collection of
More informationSimulating events: the Poisson process
Simulating events: te Poisson process p. 1/15 Simulating events: te Poisson process Micel Bierlaire micel.bierlaire@epfl.c Transport and Mobility Laboratory Simulating events: te Poisson process p. 2/15
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 information13.42 READING 6: SPECTRUM OF A RANDOM PROCESS 1. STATIONARY AND ERGODIC RANDOM PROCESSES
13.42 READING 6: SPECTRUM OF A RANDOM PROCESS SPRING 24 c A. H. TECHET & M.S. TRIANTAFYLLOU 1. STATIONARY AND ERGODIC RANDOM PROCESSES Given the random process y(ζ, t) we assume that the expected value
More informationSTOCHASTIC PROBABILITY THEORY PROCESSES. Universities Press. Y Mallikarjuna Reddy EDITION
PROBABILITY THEORY STOCHASTIC PROCESSES FOURTH EDITION Y Mallikarjuna Reddy Department of Electronics and Communication Engineering Vasireddy Venkatadri Institute of Technology, Guntur, A.R < Universities
More information3 Modeling Process Quality
3 Modeling Process Quality 3.1 Introduction Section 3.1 contains basic numerical and graphical methods. familiar with these methods. It is assumed the student is Goal: Review several discrete and continuous
More informationChapter 2 Random Processes
Chapter 2 Random Processes 21 Introduction We saw in Section 111 on page 10 that many systems are best studied using the concept of random variables where the outcome of a random experiment was associated
More informationTherefore the new Fourier coefficients are. Module 2 : Signals in Frequency Domain Problem Set 2. Problem 1
Module 2 : Signals in Frequency Domain Problem Set 2 Problem 1 Let be a periodic signal with fundamental period T and Fourier series coefficients. Derive the Fourier series coefficients of each of the
More informationStochastic Processes. Monday, November 14, 11
Stochastic Processes 1 Definition and Classification X(, t): stochastic process: X : T! R (, t) X(, t) where is a sample space and T is time. {X(, t) is a family of r.v. defined on {, A, P and indexed
More informationSpecial Discrete RV s. Then X = the number of successes is a binomial RV. X ~ Bin(n,p).
Sect 3.4: Binomial RV Special Discrete RV s 1. Assumptions and definition i. Experiment consists of n repeated trials ii. iii. iv. There are only two possible outcomes on each trial: success (S) or failure
More informationELEMENTS OF PROBABILITY THEORY
ELEMENTS OF PROBABILITY THEORY Elements of Probability Theory A collection of subsets of a set Ω is called a σ algebra if it contains Ω and is closed under the operations of taking complements and countable
More informationChapter 4 Random process. 4.1 Random process
Random processes - Chapter 4 Random process 1 Random processes Chapter 4 Random process 4.1 Random process 4.1 Random process Random processes - Chapter 4 Random process 2 Random process Random process,
More informationRecap. Probability, stochastic processes, Markov chains. ELEC-C7210 Modeling and analysis of communication networks
Recap Probability, stochastic processes, Markov chains ELEC-C7210 Modeling and analysis of communication networks 1 Recap: Probability theory important distributions Discrete distributions Geometric distribution
More informationDeterministic. Deterministic data are those can be described by an explicit mathematical relationship
Random data Deterministic Deterministic data are those can be described by an explicit mathematical relationship Deterministic x(t) =X cos r! k m t Non deterministic There is no way to predict an exact
More informationProbability Space. J. McNames Portland State University ECE 538/638 Stochastic Signals Ver
Stochastic Signals Overview Definitions Second order statistics Stationarity and ergodicity Random signal variability Power spectral density Linear systems with stationary inputs Random signal memory Correlation
More informationContinuous-Valued Probability Review
CS 6323 Continuous-Valued Probability Review Prof. Gregory Provan Department of Computer Science University College Cork 2 Overview Review of discrete distributions Continuous distributions 3 Discrete
More informationUCSD ECE250 Handout #24 Prof. Young-Han Kim Wednesday, June 6, Solutions to Exercise Set #7
UCSD ECE50 Handout #4 Prof Young-Han Kim Wednesday, June 6, 08 Solutions to Exercise Set #7 Polya s urn An urn initially has one red ball and one white ball Let X denote the name of the first ball drawn
More informationECE 313 Probability with Engineering Applications Fall 2000
Exponential random variables Exponential random variables arise in studies of waiting times, service times, etc X is called an exponential random variable with parameter λ if its pdf is given by f(u) =
More informationProbability Models in Electrical and Computer Engineering Mathematical models as tools in analysis and design Deterministic models Probability models
Probability Models in Electrical and Computer Engineering Mathematical models as tools in analysis and design Deterministic models Probability models Statistical regularity Properties of relative frequency
More informationOptimization and Simulation
Optimization and Simulation Simulating events: the Michel Bierlaire Transport and Mobility Laboratory School of Architecture, Civil and Environmental Engineering Ecole Polytechnique Fédérale de Lausanne
More informationCMPSCI 240: Reasoning Under Uncertainty
CMPSCI 240: Reasoning Under Uncertainty Lecture 5 Prof. Hanna Wallach wallach@cs.umass.edu February 7, 2012 Reminders Pick up a copy of B&T Check the course website: http://www.cs.umass.edu/ ~wallach/courses/s12/cmpsci240/
More informationMATH : EXAM 2 INFO/LOGISTICS/ADVICE
MATH 3342-004: EXAM 2 INFO/LOGISTICS/ADVICE INFO: WHEN: Friday (03/11) at 10:00am DURATION: 50 mins PROBLEM COUNT: Appropriate for a 50-min exam BONUS COUNT: At least one TOPICS CANDIDATE FOR THE EXAM:
More informationGaussian, Markov and stationary processes
Gaussian, Markov and stationary processes Gonzalo Mateos Dept. of ECE and Goergen Institute for Data Science University of Rochester gmateosb@ece.rochester.edu http://www.ece.rochester.edu/~gmateosb/ November
More informationBMIR Lecture Series on Probability and Statistics Fall, 2015 Uniform Distribution
Lecture #5 BMIR Lecture Series on Probability and Statistics Fall, 2015 Department of Biomedical Engineering and Environmental Sciences National Tsing Hua University s 5.1 Definition ( ) A continuous random
More informationMath/Stat 352 Lecture 8
Math/Stat 352 Lecture 8 Sections 4.3 and 4.4 Commonly Used Distributions: Poisson, hypergeometric, geometric, and negative binomial. 1 The Poisson Distribution Poisson random variable counts the number
More informationDiscrete Random Variables
CPSC 53 Systems Modeling and Simulation Discrete Random Variables Dr. Anirban Mahanti Department of Computer Science University of Calgary mahanti@cpsc.ucalgary.ca Random Variables A random variable is
More information