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ELE315: Random Signals I
The last topic in our review of your previous course is the concept of expectation or expected value for functions of a random variable. The definition of what we mean by expected value is fairly straightforward. Essentially if a random variable X, has a probability density function pX(x), then the function g(X) has the expected value,
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This integral calculation should pose no more difficulty than any other integral you come across. The hard part, is to find and develop some appreciation for what the result of the calculation means. For now, the main results we need to understand are,
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We will focus more on these and other measures during the course, but within the context of random processes rather than random variables. The web-notes for this topic collect together the main ideas and results that I think you should be familiar with and come from the recommended reading in chapter 4 of the textbook. In general, you should know how to calculate the mean and variance of a random variable with a specified probability density function. The ideas and results associated with multiple random variables are also treated quite extensively in chapter 4 and will become important for us later in the course so we'll review them at that time. Also, the mathematics of calculating pdfs for functions of a random variable (this is what happens when a random signal is processed by a physical system) are described at the end of chapter 4 and provides most of the ammunition needed to solve the end of chapter problems.
To provide us with some motivation for what we'll be looking at in the coming weeks, I have constructed a MATLAB simulation for problems 4.22 and 4.5M. Copy the files model2.mdl and anal2.m from the ELE315 subdirectory on the ftp site. We'll spend some time in class looking at the results of the simulation.
Review
the recommended textbook sections in Chapter 4.Solve the following problems from chapters 3 and 4 of the textbook. Hand-in your solutions no later than 6:00 p.m. on Wednesday, September 23rd, 1998.
Chapter 3: 3.3, 3.6, 3.9, 3.12
Chapter 4: 4.2, 4.4, 4.6, 4.8
Read sections 5.1 and 5.2 of the textbook through to Ex. 5.11 on page 147.
The simulation model
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Sample Results
Input and output signals
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Input and output histograms
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Output histogram showing effects of removing the
two peaks caused by the saturation effects
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MATLAB script file for processing the simulation data
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% % ELE315; Random Processes I % % % Assignment 2. Due date Sept. 23rd., 1998 % % Mean and Variance calculations % % Transformations of a random variable % clc;close all; % Plot the simulation input and output time series figure subplot(211);plot(tout,x) subplot(212);plot(tout,y) % Plot the simulation input and output time series (truncated) figure subplot(211);plot(tout(1:200),x(1:200));grid subplot(212);plot(tout(1:200),y(1:200));grid % Compute the input and output data statistics mean(x) var(x) mean(y) var(y) % Plot the input and output histogram of the data figure subplot(211);hist(x,100) subplot(212);hist(y,100) % remove the two "impulse" functions from py(y) figure subplot(211);hist(y,100) [N,X]=hist(y,100); subplot(212);bar(X(2:99),N(2:99)) |
