Description
In this lab, we will investigate two important Fourier Transform applications: filtering and
modulation. We will apply both of these to a real-world problem of creating and decoding a
message using Morse Code. This will be a two-week lab.
Lab 6 Turn-in Checklist
• 5 pre-lab exercises
• 4 lab assignments with 4 TA check-offs
• 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
sections.
In the first week of the lab, you should do Pre-lab parts 1-3, and assignments 1-3.
Pre-lab (Part I)
Read the Lab 6 Background document, then complete the following exercises. Only the first
three questions of the pre-lab are due in week 1, but you will get further with the lab if you do
more than that.
1. Filtering.
a. Suppose a real-valued LTI filter has the frequency response: �(��) = ‘
‘()*
i. Find the magnitude response |H(jw)|. Draw a rough sketch of |H(jw)| vs. w and
label the axes using � = 200.
ii. Based on your sketch in (ii), classify the filter type.
iii. What should the filter coefficients b and a be for the function signal.lsim?
b. Consider the input signal x1(t) = cos(100t). Is this input in the pass band of the filter?
Find the response y1(t) of the filter to this input.
c. If you were to implement x1(t) on the computer with fs=4000 Hz for 500ms, how
long would the time vector be?
2. Modulation, in general, is an important part of all communication systems. In short,
modulation is a process that transforms a low frequency signal into a signal of higher
frequency. This is useful because low frequency signals cannot travel long distance over
the air, but higher frequency signals can.
a. Consider the modulation system below, with input X(jw) provided on the left:
EE 235 Lab 4
Using Fourier Transform properties, find the expression for Y(jw) in terms of X(jw).
Then, using the plot for X(jw) above, sketch Y(jw) vs. w. What is the carrier
frequency for this system?
b. For the signal X(jw) above, what is the smallest carrier frequency that you could use
without causing the shifted signals to overlap in frequency?
3. Demodulation is used to undo the modulation operation. It is performed using a twostage process, as shown below.
The first stage requires another multiplication operation using the same sinusoidal signal
that was used for modulation. The second stage involves a low pass filter to recover the
signal x(t). The purpose of these two stages is to recover the original X(jw), which is
centered on w = 0.
a. Suppose y(t) = x(t)cos(200t), where X(jw) and Y(jw) are the same as in part 2.
Using Fourier Transform properties, find an expression for Z(jw) in terms of Y(jw)
and sketch Z(jw) vs. w. You should notice that your plot for Z(jw) contains two
parts: the original signal X(jw) centered around w = 0 with amplitude scaled by ½,
and other copies of X(jw) centered at higher frequencies
b. The low-pass filter we will use to recover X(jw) has the following LCCDE relating
input z(t) to output xr(t):
(240)
�0�2(�)
��0 + (3 × 100)
�8�2(�)
��8 + (2.2 × 10:)
�;�2(�)
��; + (10<)
��2(�)
��
+ (2 × 10=)�2(�) = (2 × 10=)�(�)
EE 235 Lab 4
i. Find the filter’s frequency response�(��) = ?2()*)
@()*) .
ii. What should the filter coefficients b and a be for the function lsim?
iii. What is the DC gain |H(j0)| of this filter?
4. The Morse Code was an early means of communicating information digitally. We use it
for communicating messages in this lab.
a. The Morse Code, which uses two symbols, a dot and a dash, so we will need two
signals to represent them. As we did in lab 4, we will choose one signal to be the
reverse of the other.
�ABCD(�) = 50��GHIJ�(�) �AMJ(�) = −�ABCD(�)
What should A be if you want the area of the signal to be 1? Using a definition of
bandwidth as the frequency where the magnitude is reduced by a factor of √2 , find
the bandwidth of these signals. Will this bandwidth work with a carrier frequency of
wc=200? If not, how would you change the exponent?
b. Given the International Morse Code table below:
A · – H · · · · O – – – V · · · –
B – · · · I · · P · – – · W · – –
C – · –
· J · – – – Q – – · – X – · · –
D – · · K – · – R · – · Y – · –
–
E · L · – · · S · · · Z – – · ·
F · · – · M – – T –
G – – · N – · U · · –
Letting dah be the array for �ABCD(�) and dit be the array for �AMJ(�), give the
Python code that you would use to create a time signal xm that communicates the
letter ‘X’ as a sequence of dots and dashes.
c. To decode a message with a sequence of dots and dashes, we need to compare the
received signal in each time slot to the two possible signals and find the closest
match. One way to find the closest match is to measure the sum of squared errors at
each time sample. Write code that computes this quantity for a given signal using the
numpy.sum function.
5. Frequency-division multiplexing is a method for sending multiple messages together at
the same time. Assuming that each message is limited to a low frequency bandwidth, the
different messages are modulated to different, non-overlapping positions in frequency.
EE 235 Lab 4
The individual messages can be recovered by demodulating using a carrier matched to
the appropriate position in frequency.
Consider a signal y(t) of the form
�(�) = �1(�)���(100�) + �2(�)���(200�) + �3(�)���(400�)
where m1(t), m2(t), and m3(t) each correspond to a single letter of the alphabet which
has been encoded using International Morse Code. This signal consists of a sum of three
different modulated signals.
In the last exercise, you learned how to recover a single signal x(t) from y(t) =
x(t)cos(200t). That same process can actually be used to individually recover m1(t),
m2(t), and m3(t) using three separate systems, as shown below:
a. What three carrier signals are needed for each system shown? In other words, what
frequency cosine will give m1(t)? With m2(t)? With m3(t)?
b. The signals that you transmit for any given letter can have different lengths (some
letters have one symbol but others have up to four symbols), but you can’t determine
the length until after you demodulate. If a letter has less than four symbols, then the
ending slots of the original signal will be zero valued, but after recovery there may
some non-zero (but low energy) values due to non-ideal filtering. Specify a test for
determining when the message ends, i.e. when a slot has zero value.
EE 235 Lab 4
Lab Assignments
This lab has 4 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 signal
from scipy. As always, you should add comments to your code for clarity.
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: Filtering
In this Assignment, we’ll implement and analyze a continuous-time filter using the Python scipy
signal package. Write your code in a new cell for Assignment 1, inside your Lab 6 notebook.
A. Using fs=4000 Hz, create a time vector t for the range [0,500ms). Use t to create the
signal x1 corresponding to x1(t) from part 1 of the prelab.
B. Define a filter for a first-order system using the parameters you found in the pre-lab, part
1. Using the signal.freqresp function, find the frequency response of the system and plot
the magnitude |H(jw)| and phase in side-by-side (1×2) plots. For the magnitude, create the
plot with overlayed version of standard magnitude and magnitude with a dB scale
(20 logHX |�(��)| , specifically with the left axis corresponding to the linear amplitude
and the right to dB. Label the frequency axis using frequency in rad/s.
C. Using the signal.lsim function, find the output y1 that is the response of the system to x1.
Create a new plot, and plot x1 and y1 together on the same plot. Confirm that the
resulting signal matches what you expect from the pre-lab in terms of changes in
amplitude and phase (associated with using a non-ideal filter).
Assignment Check-Off #1 of 4: Demonstrate this Assignment to the lab TA .
Report discussion:
If you put two identical ideal low-pass filters in series, then the overall response will be
unchanged. Comment on what will happen if you put two identical versions of the above filter in
series. Do you expect the amplitude of the response to x1 to be greater or smaller than for a
single filter? (Note that you can test this by letting the output y1 be an input to the same filter.)
Assignment 2: Amplitude Modulation
Modulation, an important part of all communication systems, is a process which transforms a
signal of low frequency to a signal of higher frequency. This is useful because low frequency
EE 235 Lab 4
signals cannot travel longer distances over the air, but high frequency signals can. In this
exercise, we implement and analyze amplitude modulation in software.
Use a new cell for Assignment 2. Comment your code and title/label your plots.
A. First create two signals to communicate.
a. Using sampling frequency of fs=4000 Hz, create time signals x1 and x0
corresponding to
�H(�) = 50��GHIJ�(�) �X(�) = −�H(�)
using a time vector defined over [0,0.5).
b. Using 2×1 subplots, plot the time signal for x1 and the magnitude of its Fourier
transform. For the Fourier transform, use Nfft = 8192 samples, and in the plot use
rad/s and limit the frequency axis to the range [-100,100).
B. Next create and modulate the message signal.
a. Construct a message signal xm for communicating the bit sequence [1 0 1 0] by
concatenating x1 and x0.
b. Create the carrier signal: c(t)=cos(200t).
c. Finally, implement the modulated signal: y(t) = xm(t)c(t)
C. Using 2×1 subplots, plot the time signals xm and y.
D. Using 3×1 subplots, plot the magnitude of the Fourier transforms for xm, c and y. Again
use Nfft = 8192 samples, and a frequency axis of rad/s in the plot. For xm, limit the
frequency axis to the range [-150,150), but for the other two plots you should use the
range [-500,500). Verify that the Fourier transforms correctly put the carrier frequency at
wC = 200 rad/s.
Assignment Check-Off #2 of 4: Demonstrate this assignment to the lab TA
Report discussion:
In this Assignment, we analyzed a system with input-output relationship y(t) = x(t)cos(200t). A
student in class claims this system is LTI. Explain why they are incorrect.
Assignment 3: Amplitude Demodulation
Now we’ll learn to use (ideal) demodulation, to undo the modulation operation performed in 2.0.
In a communication system, once a signal is transmitted to the intended receiver, the receiver
will use demodulation to that transformation and recover the original signal, as in part 3 of the
pre-lab. The first stage is a multiplication operation using the same carrier that was used for
modulation. The second stage is a filter used to fully recover the signal x(t). The purpose of these
two stages is to recover the original X(jw), which is centered on w = 0.
EE 235 Lab 4
Use a new cell for Assignment 3. Comment your code and title/label your plots.
A. Using the carrier from assignment 2, create the signal z(t)=y(t)c(t). Using 2×1 subplots,
plot the time signal z and the magnitude of its Fourier transform. For the Fourier
transform, use Nfft = 8192 samples, and in the plot use rad/s and limit the frequency axis
to the range [-1000,1000).
B. Using the scipy signal package, create a continuous-time filter using the coefficients that
you found in part 3 of the pre-lab. Find the frequency response of the filter using the
signal.freqresp function. Using a single plot with parallel axes, plot the magnitude (dB
scale) and phase of the system frequency response, with magnitude on the left axis and
phase on the right axis. Use frequency range [-300,300) in rad/s.
C. Use the signal.lsim function to filter z(t) with the filter you designed, giving the
recovered signal xr(t). Using 2×1 subplots, plot the time signal xr and the magnitude of
its Fourier transform. For the Fourier transform, use Nfft = 8192 samples, and in the plot
use rad/s and limit the frequency axis to the range [-150,150) rad/s. Because the filter is
not ideal, you should notice some differences in the recovered and original signals.
Assignment Check-Off #3 of 4: Demonstrate this assignment to the lab TA
Report discussion:
A student decides to use a shorter signal of the form �H(�) = ���GHIXJ�(�), thinking that the
message can be communicated in a tenth of the time. They claim to see the exact same graph for
xr(t) as they saw in their pre-lab. Explain why the student is incorrect, and why this solution will
not work.
Assignment 4: Decoding a Morse Code Message
We will decode a Morse Code signal of a three letter word, using what we learned in the
previous three exercises. This Assignment is a modification from an exercise in Computer
Explorations in Signals and Systems using Matlab by John R. Buck, Michael M. Daniel, and
Andrew C. Singer.
You are given a signal y(t) which contains a simple message from Agent 007. When loading the
file, you magically transform into Agent 008, the code-breaking sleuth. The last words of the
aging Agent 007 were “The future of technology lies in…” at which point Agent 007 saved the
remaining message to a csv file and decided to return to school and give up the next four years to
pursue the American dream of getting a PhD.
Your job now is to decipher the message encoded
in y(t) and decode Agent 007’s final words. The signal was obtained by adding (not
concatenating) three 2 second message signals each modulated with carrier frequencies of 100,
200 and 400 rad/sample, and digitized with a sampling frequency of 4kHz.
A. Using a new, separate cell, write a function to decode the sequence of dots and dashes in
a received message of fixed length 2 s with 0.5 s time slots for each symbol. Be mindful
EE 235 Lab 4
of the fact that not all letters have the same number of Morse code characters. If there are
fewer than 4 symbols, the remaining length is padded with zeros, so you will have to
have a check to see if the time-slice in question is empty or a Morse code signal. You
should use your work from pre-lab parts 4 and 5. Test your function by creating the
signal for ‘X’, and decoding the original and a modulated-demodulated recovered
version.
B. Start another new cell. Load the data file ‘message_modulated.csv’ which is available
from the course web site. The file contains the signal y(t) with three messages combined
using frequency division multiplexing.
C. Write a loop that:a. Assigns the demodulating frequencies to extract the respective message signal from
the composite signal, using your work from the pre-lab part 5 and the knowledge
that the carrier signals are y1(t)=cos(100t), y2(t)=cos(200t), y3(t)=cos(400t);
b. Demodulates the signal and then filters it as was done in the previous parts and uses
the function from part (A) to recover the i
th recovered message and output str; and
c. Prints out the “Message mi(t) is: str” with the values filled in for i and str
Assignment Check-Off #4 of 4: Demonstrate this assignment to the lab TA
Report discussion:
In this lab, we matched the received and candidate signals using a minimum squared error
criterion. Explain why the matched filter from lab 4 might be more effective.
Use the Morse Code table provided in pre-lab part (4) to decode the letter from each signal plot.
Complete Agent 007’s final words: “The future of technology lies in ________.”
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 Lab6-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.
Closing Note
This quarter you have decoded innumerable messages, and learned how to perform convolution,
convert signals from the time domain to the frequency domain, and filter signals. What an
EE 235 Lab 4
accomplishment! Unfortunately, all good things must come to an end.
It was a pleasure working
with you, Agent 008, but it is now time for a new group of EE students to follow in your
footsteps and take on the EE235 challenge. Good luck and keep in touch!
