# Engineering Department, UMass Boston XXXXXXXXXXSpring 2022 ENGIN 322: Probability and Random Processes Project #3: Correlation Functions and Spectral Density Project Description In this project, you...

Engineering Department, UMass Boston XXXXXXXXXXSpring 2022

ENGIN 322: Probability and Random Processes

Project #3: Co
elation Functions and Spectral Density
Project Description
In this project, you will observe the binary process in more detail. The binary process is a basic method for
transmitting data in a digital communication system. Your objectives are to determine/analyze the
spectral density of the process (i.e., of a transmitted signal) and use cross-co
elation to find a preamble
Part I
Create a Matlab script to generate 50,000 symbols of a random binary process with symbol rate of 1kHz
(i.e., ???? = XXXXXXXXXXand sample rate of 100kHz (i.e., 100 samples or repetitions for every symbol). Using this
signal, generate the autoco
elation function and power spectral density of the process using Matlab’s
xco
() and periodogram() functions. Be sure to label plots and set axis values appropriately.
Question 1: Relate your observations of the simulated signal to the theoretical autoco
elation function
and spectral density function for this process. Use the ACF determined in class (and in section 6-2 of the
textbook) in order to evaluate the spectral density for this function: ????(??) = ℱ{????(??)}
HINT: Use the Fourier Transform pairs and operations described in Appendix A (Tables A-4 and A-5)
Question 2: Use the equation for ????(??) determined in question 1 to determine the approximate frequency
where 90% of the average power is below. Rather than integrating, you can approximate this by summing
the values from the equation using a resolution of 1Hz. Compare your result to the simulated spectral
density in Matlab.
Part II
The file ‘part2_2022.mat’ has a saved a
ay that contains a simulated received signal. The transmitted
data signal has been delayed and noise has been added. Luckily, the transmitted data contains a known
preamble at the start of the transmission. The preamble was transmitted 4 times in a row directly prior
to starting the data transmission. PREAMBLE: [ XXXXXXXXXX XXXXXXXXXX XXXXXXXXXX]
• Use the cross-co
elation function to determine the start of the first preamble AND the start of the
data transmission. You can determine the start of the preamble by observing a plot of the cross-
co
elation function.
• Create a Matlab function that takes an a
ay of values as an input, determines the likely location
of the preamble, and returns this value. For example, if you input the a
ay from ‘part2.mat’, the
function should return the value for the start of the first preamble as determined above.
Question 3: When looking for the preamble, is it preferable to compare the observed signal with the
preamble or with the preamble repeated multiple times? Why?
Question 4: How does your output change when changing the order of the processes? In other words, if
the observed signal is defined as X and the preamble is defined as Y, how does ??????(??) compare to ??????(??)?
Part III
Question 5: Include a section (approximately 1 paragraph) in your report that discusses the relationship
etween this project and other real-world issues. Clearly label the section.
• Coding and Results: 30%
• Theoretical Analysis: 40%
• Written Report: 30%
Coding and Results: This portion of the project will be graded based on the implementation of code as
described for Part I and II above. Results should include a description of the observed outcomes and some
depiction of the results from your code.
Theoretical Analysis: This portion of the project will be graded based on your answers to the questions
above. Be sure to clearly indicate the answers AND REASONING for each of the questions within your
written report.
Written Report: The written report should be submitted on blackboard by midnight on May 11. The report
should be 3-5 pages including an overview of the project, expected outcomes, your analysis method,
esults, and observations. You may include any code as an appendix.
Project Description
Part I
Part II
Part III