Texas A&M, ECEN 455

Excerpt: ... ECEN 455 Digital Communications - Spring 2009 Homework 3 Date assigned: February 11, 2009 Date due: February 18, 2009 in class Suggested reading: Lecture Notes 1 and 2 1. Consider m independent random variables (sub-noises in the language of Lecture Note 2) n1 , n2 , . . . , nm . Show that these random variables are also uncorrelated, i.e., m m Var k=1 nk = k=1 Var(nk ). (1) 2. In this exercise, we study some properties of the Gaussian random variable. (a) Consider a Gaussian random variable n with mean 0 and variance 1. Such a Gaussian random variable is said to be standard. What is the probability density function of the random variable 2n? (b) Consider two independent, standard Gaussian random variables n1 and n2 . What is the probability density function of n1 + n2 ? (c) Re-answer the questions in (a) and (b) by using just the Central Limit Theorem. 3. In this exercise, we study some properties of the Gaussian Q() function. (a) Consider a (standard) Gaussian random variable n with mean 0 and varia ...

University of Illinois, Urbana Champaign, ECE 562

Excerpt: ... ECE 562 Fall 2008 Homework 1 Date Assigned: 29 August 2008. Date Due: 4 September 2006 in class. Suggested Reading: Lecture notes 1 and 2. 1. The lecture notes are only as good as how we make them look to be. To be sure, the instructor plays a key role in envisioning and developing them. But the students play an important role in keeping the notes alive and relevant by providing feedback. Through this question, we ask that you provide us with as many typos you can find in the lecture notes. A negligible fraction of the homework points will be reserved for the typos you can find, but enough to provide concrete incentive for those that need it. 2. In this exercise, we study some properties of the Gaussian random variable. (a) Consider a Gaussian random variable n with mean 0 and variance 1. Such a Gaussian random variable is said to be standard normal. What is the probability density function of the random variable 2n? (b) Consider two independent Gaussian random variables n1 and n2 . What is the probability ...

UCSD, ECE 244

Excerpt: ... Statistical Optics Lecture 2 Characteristic Functions Moments Transformation of RVs 1-D monotonic Sums of RVs Central Limit Theorem Bandpass Random Signals Joint Gaussian Random Variables Complex RVs Complex gaussian Statistical Optics Lecture 2 Winter Quarter 2009 ECE244a Statical Optics - Winter 2009 Lecture 2 1 Statistical Optics Lecture 2 Characteristic Function Characteristic Functions Moments Transformation of RVs 1-D monotonic The characteristic function is the Fourier transform of the pdf MU ( ) = pU ( u ) = 1 2 Sums of RVs Central Limit Theorem Bandpass Random Signals Joint Gaussian Random Variables Complex RVs Complex gaussian e j u pU (u )du MU ( )e j u d Considering g (u ) = e j u , then MU ( ) = E e j u ECE244a Statical Optics - Winter 2009 Lecture 2 2 Statistical Optics Lecture 2 Relationship between MU ( ) and moments of pdf Expand exponential in Taylor series e j u = then MU ( ) Characteristic Functions Moments Transform ...

Allan Hancock College, WEB 2226

Excerpt: ... ing the following steps: (i-a) Write down the log likelihood function (i-b) Differentiate the log likelihood function and set it to 0 (i-c) Solve the equation in part (i-b) to obtain an estimate of (ii) If there is a priori knowledge that the parameter is distributed according to ( ) = 1 2 e 2 , compute the MAP estimate of by doing the following steps: (ii-a) Write down the unnormalized PDF (the likelihood function multiplied by the prior PDF) then write down the log unnormalized PDF (the logarithm of the unnormalized PDF) (ii-b) Differentiate the log unnormalized PDF and set it to 0 (ii-c) Solve the equation in part (ii-b) to obtain an estimate of 3. (For this exercise you may want to have a look at Robs Notes on products of Gaussians on the ENGN2226 website) Suppose X and Y are jointly Gaussian random variables with the joint PDF XY (x, y) given by: 1 2( 1 XY (x, y) = e 21 2 (x1 )2 2 1 + (y2 )2 2 ) 2 , (i) Are X and Y independent? Justify y ...

Stanford, EE 179

Excerpt: ... + Sn(ffc)] Filtering a WSS Process Same PSD effect as for deterministic signals n(t) with PSD Sn(f) y(t) has PSD|H(f)|2Sn(f) H(f) Gaussian Processes X(t) is a Gaussian process if Yg = g (t ) X (t )dt is a Gaussian RV for any T and function g(t) Filtering a Gaussian process results in a 0 T Gaussian process Samples of a Gaussian process are jointly Gaussian random variables . independent. Uncorrelated samples of a Gaussian process are Examples of noise in Communication Systems Gaussian processes Filtering a Gaussian process yields a Gaussian process. Sampling a Gaussian process yields jointly Gaussian RVs If the autocorrelation at the sample times is zero, the RVs are independent. The signaltonoise power ratio (SNR) is obtained by integrating the PSD of the signal and integrating the PSD of the noise. In digital communications, the bit value is obtained by integrating the signal, and the probability of error by integrating Gaussian noise. Main Points Modulation lead ...

BYU, EE 485

Excerpt: ... Appendix A A Review of Statistics The principles of probability and statistics are used to describe the random nature of information and noise. Since an understanding of the randomness of both signals and noise is essential to understanding the performance of a communications system, we will review those principles here. This review is by no means exhaustive and focuses exclusively on Gaussian random variables . A.1 A.1.1 Gaussian Random Variables Density and Distribution Functions Let X be a Gaussian (or Normal) random variable. Then X has the probability density function 1 (x )2 exp fX (x) = 2 2 2 2 . (A.1) The probability density function of X fX (x) is used to compute the probability that the random variable X assumes a value between a and b: Pr {a < X < b} = b a fX (x)dx (A.2) for a b. The rst moment of X is the expected value which we call the mean and denote it using : E{X} = = . xfX (x)dx The second central moment of X is called the variance and is de ...

UConn, MATH 5160

Excerpt: ... Math5160 Final HW Fall 2008 All the homework has to be submitted by Friday December 12, any time either in my mailbox or under my oce door (a submission later than Friday may result in an incomplete grade). During the exam week, regular oce hours will be held on Monday and Friday at 2pm2:40pm, and on Wednesday noon1:00pm. If you use a theorem or a lemma, either give its name, or its number in the textbook or lecture notes (and dont forget to show that the assumptions hold). 1. (a) Show that if n=1 P(|Xn | > n) < then lim sup Xn /n 1 almost surely. n n (b) Deduce from (a) that if Xn have the same distribution and E|Xn | < then lim Xn /n = 0 almost surely. 1 2. (a) If n are iid Bernoulli random variables with P{ n =1} = P{ n =1} = 2 , give necessary and sucient conditions for non-random real coecients an so that the series an n n=1 converges almost surely. (b) If Xn are iid standard Gaussian random variables , which are also independent of al ...

Yale, BENG 449

Excerpt: ... Histogram of Gaussian Random Variables (10000 trials, n = 1000) 250 200 Observed Frequency, f(z) 150 100 50 0 -4 -3 -2 -1 0 z 1 2 3 4 ...

Yale, BENG 449

Excerpt: ... Histogram of Gaussian Random Variables (10000 trials, n = 1) 200 180 160 140 Observed Frequency, f(z) 120 100 80 60 40 20 0 -4 -3 -2 -1 0 z 1 2 3 4 ...

Yale, BENG 449

Excerpt: ... Histogram of Gaussian Random Variables (10000 trials, n = 10) 250 200 Observed Frequency, f(z) 150 100 50 0 -4 -3 -2 -1 0 z 1 2 3 4 ...

Yale, BENG 449

Excerpt: ... Histogram of Gaussian Random Variables (10000 trials, n = 5) 250 200 Observed Frequency, f(z) 150 100 50 0 -4 -3 -2 -1 0 z 1 2 3 4 ...

Berkeley, EE 121

Excerpt: ... EECS 121: Introduction to Digital Communication Systems Problem Set 6 Due Fri, March 14, 4pm at Thereses desk 1. In this exercise we study some properties of the Q() function dened as the following. (a) Consider a Gaussian random variable n with mea ...

University of Illinois, Urbana Champaign, EE 467

Excerpt: ... EE 467 Handout # 9 GAUSSIAN RANDOM VARIABLES The Gaussian Probability Density Function This is the most important pdf for this course. It also called a normal pdf. 2 1 - (x-m) fX (x) = e 22 . 2 Fall 1999 October 13, 1999 It can be shown this f X integrates to 1 (i.e., it is a valid pdf), and that the mean of the random variable X with the above pdf is m and the variance is 2 . The statement "X is Gaussian with mean m and variance 2 " is compactly written as "X N (m, 2 )." The cdf corresponding to the Gaussian pdf is given by x (u-m)2 1 e- 22 du. - 2 x FX (x) = - fX (u)du = This integral cannot be computed in closed-form, but if we make the change of variabe FX (x) = x-m v2 1 x-m e- 2 dv = , 2 u-m = v we get - where is the cdf of a N (0, 1) random variable, i.e., (x) = x - u2 1 e- 2 du. 2 Note that due to the symmetry of the Gaussian pdf, (-x) = 1 - (x). A closely related function to is the Q function which is defined by: Q(x) = 1 - (x) = Some end point properties of and Q are given be ...