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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 proce
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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-vari
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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 [
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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,
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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 an
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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 pa
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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 ins
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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,
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 outco
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
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
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 rep
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
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
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 o
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 ti
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 even
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 rando
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 )
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. Th
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 param
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