Description
Convolution (II)
In the lab we will continue our study of convolution to learn how it can be used to measure the
similarity between two signals, which is useful in decoding transmitted messages in digital
communications. This lab will also be used to review functions and formally introduce loops and
decision statements.
Lab 4 Turn-in Checklist
• 3 pre-lab exercises
• 2 lab assignment check-offs with TA
• Lab report, submitted as a Jupyter notebook following the format provided in earlier labs
Note: All assignments except the prelab should be completed in groups of 2-3 people.
The pre-lab exercises are to be completed individually and submitted via Canvas before lab
section.
Pre-lab
Read the Lab 4 Background document, then complete the following exercises.
Let s1(t) = u(t) – u(t – 1) and s0(t) = -s1(t). Which of the plots below is the correct plot for y1(t)
= s1(t) s1(t) ∗ s1(t) ? Find the peak value and critical time points for y1(t), including start time, peak
time and end time. Repeat for y0(t) = s1(t) s0(t) ∗ s1(t) .
In digital communications, we can use can convolve a received signal r(t) separately with s1(t)
and then with s0(t) to determine which signal was transmitted.
EE 235 Lab 4
After convolving the received signal with a signal template, you can find the correlation by
extracting the signal value at its midpoint. We will refer to the correlation values as y1_corr
and y0_corr, for the correlation with s1 and s0, respectively.
Write a Python decision
statement (if/else) to meet the following conditions: Let s be 0, unless the variable y1_corr is
greater than the variable y0_corr – in that case, assign the value 1 to s.
Write a function decode for deciding which of two signals was sent. Your function should take
the following inputs: a signal r, the sampling rate fs, the 0-signal s0, and the 1-signal s1. It
should return the decoded symbol, which is either 0 or 1.
In the function, you implement the
correlation using convolution (don’t forget to apply the amplitude scaling factor) and then
extracting the peak value at the midpoint of the resulting signals. The decoded symbol will be
decided based on the correlation measurements y0_corr (between r and s0) and y1_corr
(between r and s1). If y1_corr exceeds y0_corr, return 1.
Otherwise, return 0. For this lab, you
can return 0/1 either as a number or as a string, but it is a little more convenient to use strings.
Lab Assignments
This lab has 3 assignments. Each should be given a separate code cell in your Notebook, and
each should be associated with a markdown cell with subtitle and discussion. As in previous labs,
your notebook should start with a markdown title and overview cell, which should be followed
by an import cell that has the import statements for all assignments. In this lab, you will need to
use most of the import commands that you have used in earlier labs, plus “import csv”. As
always, you should add comments to your code for clariy.
You will again be working with concepts from previous labs, so you may want to refer back to
the background files for those labs or the py_ref document.
Assignment 1: Matched Filter with Ideal Signals
In this Assignment, we’ll use a received signal that has no noise and convolve it with itself and
with the template for a different signal. Our signals involve unit pulses, and they will serve as the
basis for the digital communications problem we will explore later in this lab. Write your code in
a new cell for Assignment 1, inside your Lab 4 notebook.
A. Implement the signals s1(t) = u(t) – u(t-1) and s0(t) = -s1(t) with fs=1000 and 0 < t < 1.
B. Use numpy.convolve() to compute the convolution y1(t) = s1(t) s1(t) ∗ s1(t) and y0(t) = s1(t)
∗ s1(t) s0(t). Don’t forget to apply the amplitude scaling factor in implementing convolution.
C. Compute the time samples vector t_y for y1 and y0 (they should be the same length). On
a 2×1 subplot, plot y1 vs. t_y and y0 vs. t_y. Title and label each subplot. Verify that the
plots match your pre-lab calculations.
D. To get the correlation measurement y1_corr, extract the value of y1 at the index which
corresponds to when time t equals the width of input s1 in seconds, as in pre-lab. Print
y1_corr. Repeat to get y2_corr. Compare your correlation measurements to the peak
EE 235 Lab 4
values you found in pre-lab problem 1. (Note: We are using discrete-time signals while
pre-lab used continuous-time ones, so correlation measurements may not exactly match.)
Assignment Check-Off #1 of 3: Demonstrate this Assignment to the lab TA
Report discussion:
In this Assignment, we assumed signals s1(t) and s0(t) were both box
signals, but with opposite (+/-) amplitudes. Suppose instead that s0(t)=v(t)-v(t-0.5), where
v(t)=u(t)-u(t-0.5). Using your understanding of correlation, do you expect the correlation
measurement between s1(t) and s0(t) to be greater or less than the correlation measurement was
when both signals were box signals? Which s0(t) would be more useful for communications?
Assignment 2: Signal Decoding with a Matched Filter
As always, start a new cell in your Jupyter notebook for this assignment. In this assignment, you
will use a random noise generator. Note that each time you call it, the function will use a
different random seed, so if you run the cell multiple times without changing your code, you will
get different results for the plots. Don’t forget to title and label all plots.
A. Using a separate cell, implement the decode function that you wrote in the pre-lab.
B. Using the definitions of fs, and the s1 and s0 signals from Assignment 1, create a
received signal r(t) by adding signal s1(t) with a random noise signal n(t).
a. We create a noise vector of the same length as s1, and standard deviation 10 with:
n = np.random.normal(0,3,len(s1))
b. Create received signal vector r using r = s1 + n, a second received signal vector
r0 = s0 + n.
c. Concatenate the two signals s1 and s0 in sequence and repeat for r and r0. Create
a time samples vector and plot them in one plot with the clean signal overlaying
the noisy version. The plot should look roughly like:
EE 235 Lab 4
C. Now compute the correlation measurements between r and s1, and r and s0:
a. Compute yr1 = r s1 ∗ s1(t) , and yr0 = r ∗ s1(t) s0. Create t_yr, the time samples vector
associated with yr1 and yr0.
b. In Plot 2, plot yr1 vs. t_yr and yr0 vs. t_yr in different colors. Provide a legend.
The plot should look roughly as follows.
c. Compute yr1_corr as the correlation between r and s1, by sampling yr1 at the
correct time. Print yr1_corr. Repeat for y0_corr, the correlation between r and
x0, by sampling yr0.
D. Test decode, by decoding the 1-signal s1 and the 0-signal s0. If the function works as
expected, passing the input signal r=s1+n should produce the symbol 1, while passing
r=s0+n should produce the symbol 0.
Assignment Check-Off #2 of 3: Demonstrate this Assignment to the lab TA
Report discussion:
In a real communication system, having shorter signals means that you can
communicate information faster. How would the correlation functions yr1 and yr0 change if the
signal was only 20 ms long?
Assignment 3: Decipher Received Message
In this exercise, we apply what we learned in previous exercises to a digital communication
system. A primary goal of a digital communication system is to decipher received signals that
have been distorted with noise during transmission. As we’ve seen, we can use correlation
measurements to determine whether the received signal is 0 or 1 at a given time.
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EE 235 Lab 4
You will decipher a secret message by decoding a bit sequence from the received message r(t),
and translating the bits to ASCII characters.
Start a new cell for Assignment 3. You will use the decode function from the previous
assignment. Since the message was created with a higher noise level and sampling rate, you will
need to rebuild s0 and s1 using fs=8000.
A. The received message r(t) has been reshaped as an array with N indices, one for each
binary signal and saved as a CSV file. Download the file ‘receivedmsg.csv’ from the
class web page. Using the guidelines in the Lab 4 background document, read in the
CSV file containing the points from the received message and call it rm. Each row of the
CSV file corresponds to one of the message’s bits worth of data (a 1 second time slice),
transmitted and received as ri(t), i=1, …, N.
B. To decode the message, you will need to loop over each row, and use decode to
determine if the data for the given time slice constitutes a 1 or 0 signal, then save the
appropriate value. If your decode function produces a binary output, you can store it in
an array. If it produces a string, then you can just concatenate each new bit to the string of
previous results.
C. Print the sequence of bits.
Report discussion:
Provide the decoded message in your lab report. You can copy the
result and use this link: http://www.binaryhexconverter.com/binary-to-ascii-text-converter to
translate the bits to ASCII symbols.
Team Report
When you’ve tested and cleaned up all your code (remember, you should only submit code for
the Assignments, each in their own cell), go to ‘File’ then ‘Download as’, then select ‘.ipynb’. The
file you download is a Notebook that your TA will be able to open and grade for you, once you
submit it on Canvas.
Remember, only one notebook per team! Make sure that your notebook is
titled Lab4-XYZ.ipynb, where XYZ are the initials of the lab partners. You may want to also
download the file as pdf to have a nicer documentation of your records.
Submit the .ipynb file via Canvas.
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