Virginia Tech
The Bradley Department of Electrical and Computer Engineering ECE 5605 Stochastic Signals and Systems Fall 08 Problem Set 9
Problem 1. Let X(t) and Y (t) be independent, WSS random processes with zero means and the same covariance function C
Virginia Tech
The Bradley Department of Electrical and Computer Engineering ECE 5605 Stochastic Signals and Systems Fall 08 Problem Set 8
Problem 1. Let Un be a sequence of i.i.d. zero-mean, unit-variance Gaussian random variables. A "low-pass filter" tak
Virginia Tech
The Bradley Department of Electrical and Computer Engineering ECE 5605 Stochastic Signals and Systems Fall 08 Problem Set 7
Problem 1. Let be a random variable uniformly distributed on [0, 2). Consider the process X(t) = cos2 (t + ). Find th
Virginia Tech
The Bradley Department of Electrical and Computer Engineering ECE 5605 Stochastic Signals and Systems Fall 08 Problem Set 5
Problem 1. The random variables X and Y have joint density fX,Y (x, y) = Evaluate the probability P [X Y ]. e-x 0 x <
Virginia Tech
The Bradley Department of Electrical and Computer Engineering ECE 5605 Stochastic Signals and Systems Fall 08 Problem Set 4
Problem 1. The random variable X is Gaussian with mean zero and variance 2 . Find E[X|X > 0] and V AR[X|X > 0]. Probl
Virginia Tech
The Bradley Department of Electrical and Computer Engineering ECE 5605 Stochastic Signals and Systems Fall 08 Problem Set 3
Problem 1. (a) Given that the events A1 , . . . , An form a partition of S, and using the total probability theorem,
Virginia Tech
The Bradley Department of Electrical and Computer Engineering ECE 5605 Stochastic Signals and Systems Fall 08 Problem Set 2
Problem 1. Suppose that a point is selected at random from inside the unit circle. Let Y be the distance of the point
Virginia Tech
The Bradley Department of Electrical and Computer Engineering ECE 5605 Stochastic Signals and Systems Fall 08 Problem Set 1
Problem 1. A random experiment has sample space S = cfw_a, b, c. Suppose that P [cfw_a, c] = 5/8 and P [cfw_b, c] = 7
Stochastic Signals and Systems
Markov Chains Virginia Tech
Fall 2008
Markov Chains: Introduction
Markov processes represent the simplest generalization of independent processes by permitting the outcome at any instant to depend only on the outcome that pr
Stochastic Signals and Systems
Analysis and Processing of Random Signals
Virginia Tech
Fall 2008
Ergodicity
Until now we have generally assumed that a statistical
description of the random process is available. Of course, this is seldom true in practice;
Stochastic Signals and Systems
Analysis and Processing of Random Signals
Virginia Tech
Fall 2008
Linear Systems
In this lecture we look at transformations of random processes.
Linear. Let X1 (t) and X2 (t) be two arbitrary time functions
and let a1 and a
Stochastic Signals and Systems
Analysis and Processing of Random Signals
Virginia Tech
Fall 2008
Power Spectral Density
For deterministic processes (signals), Fourier series and
Fourier transform represent the frequency components as a "frequency spectru
Stochastic Signals and Systems
Random Processes Virginia Tech
Fall 2008
Introduction
If we view a system as acting on an input random process
to produce an output random process, we find that we need to develop a new "calculus" for random processes.
In
Stochastic Signals and Systems
Random Processes Virginia Tech
Fall 2008
Properties Autocorrelation Func. Stationary Processes
In this lecture we treat random processes that are jointly stationary and of second order, that is, E X (t)2 < . Some important p
Stochastic Signals and Systems
Random Processes Virginia Tech
Fall 2008
Stationary Random Processes
We say a random process is stationary when its statistics do not change with time.
An observation of the process in the time interval (t0 , t1 )
exhibits
Stochastic Signals and Systems
Random Processes Virginia Tech
Fall 2008
Random Telegraph Signal
Consider a random process X (t) that starts at the value
X (0) = 1 with equal probability. At random times t1 , t2 , . . . thereafter it changes sign. The "sw
Stochastic Signals and Systems
Random Processes Virginia Tech
Fall 2008
Poisson Process
Events occur at random instants of time at an average rate
of events per second. Let N(t) be the number of event occurrences in the time interval [0, t]. N(t) is a no
Stochastic Signals and Systems
Random Processes Virginia Tech
Fall 2008
Autoregressive Random Processes
A first order autoregressive (AR) process Yn has the form
Yn = Yn-1 + Xn where Xn are iid random variables.
is a measure of statistical dependence of
Stochastic Signals and Systems
Random Processes Virginia Tech
Fall 2008
Gaussian Random Variables
A random process X (t) is a Gaussian random process if
the samples X (t1 ), X (t2 ), . . . , X (tk ) are jointly Gaussian random variables for all k , and a
Stochastic Signals and Systems
Random Processes Virginia Tech
Fall 2008
Mean and Correlation Functions
As in the case of rvs, the moment functions play an
important role in practical applications. These functions, however, only partially describe a rando
Stochastic Signals and Systems
Random Processes Virginia Tech
Fall 2008
Random Processes: Example
In certain random experiments, the outcome is a function of time, position, angle, or some other parameter.
Statistical multipath channel model: a multipath
Stochastic Signals and Systems
Multiple Random Variables
Virginia Tech
Fall 2008
PDF of Sums of Independent RVs
Let X1 , X2 , . . . , Xn be n independent random variables.
We're interested in finding the pdf of S = X1 + X2 + . . . + Xn
If n = 2, the pdf