Description
Time Transformations of Signals
In this lab, you will work through a series of exercises to perform transformations on continuous time signals, and get comfortable with relevant Python functions. Important concepts will
include transformations in time (time shift and time scale), as well as multiple transformations
using the order of operations.
Lab 2 Turn-in Checklist
• 4 pre-lab exercises
• 2 Assignment check-offs with TA
• Lab report, completed and submitted as a team
◦ Submit as a Jupyter notebook following the format of the template provided
◦ Submit the audio file recovered in assignment 3
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 2 Background document, then complete the following 4 exercises.
1. The envelope of an audio signal y(t) is approximated below:
a. Now consider the signal y(2t).
i. Sketch the signal y(2t). Label the height and the time points corresponding to t =
0, t = 0.5, t = 1.6 after the time scaling.
ii. If you played y(2t), would it sound like y(t) has higher frequencies or lower
frequencies?
b. Repeat (a) for the signal y(0.5t)
c. Sketch the signal y(t-1) and y(t+0.5), again labeling critical time points.
EE 235 Lab 1-1 Time Transformations of Signals
2. On a computer, we may have the constraint of keeping the time window fixed. If the time
window is constrained to be [0,3] sec, then which of the time transformations in part 1
will require you to throw away some of the transformed signal? If you were to implement
y(t)=x(2(t+1.5)) with a fixed time window, would it be better to scale first or shift first, or
does it not matter?
3. In lab, we will create a function that performs time scaling, and computes y(t) = x(at)
when sampled at rate fs. We will use the header:
def timescale(x, fs, a):
a. What are this function’s input parameters? Assume that it will return two variables y
and t, how will both be returned simultaneously? (Hint: it involves one return line and
commas)
b. How would you call this function if you wanted to time scale a signal vector y with
sampling rate fs, by a factor 2, and store the variables it returns to y1 and t_y1?
4. A signal y(t) was created from another signal x(t) by first scaling it by a factor of 3 and
then delaying it by 1. In other words, y(t)=x(3(t-1)). Write an equation to undo the
transformation and recover x(t) from y(t).
Download the template notebook for lab 2 (TemplateL2.ipynb) and rename it Lab2-XYZ, where
XYZ are the initials of the lab partners. Your Lab 2 notebook should be organized according to
this template.
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 Lab 1, 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.
For assignments 1 and 2, you will use the train signal from Lab 1. Make sure you have
“train32.wav” saved to your working directory. For assignment 3, your TA will assign you an
audio file to upload.
EE 235 Lab 1-1 Time Transformations of Signals
You will be working with arrays, audio files and plotting, so you may want to refer back to the
Lab 1 background discussion. Be sure to include the necessary import statements. Your input cell
should look like:
Assignment 1: Time Scaling Audio Signals
We will now implement and test a timescale function. This assignment will have four parts, A-D,
each of which should be indicated with comments, following the guidelines in the Lab 2
template.
A. Replicate the timescale function in Section 5 of the Lab 2 background document, and
save it in its own cell, as indicated in the Lab 2 template.
B. Write the necessary code to do the following. Feel free to use a loop for repeated steps:
a. Load “train32.wav” and save its signal as y and sampling rate as fs. Recall from lab
1 that this file only has one channel.
b. Compute the time samples t_y using y and fs. Refer to Lab 1 if needed.
C. Use the timescale function to obtain w(t)=y(2t) and v(t)=y(0.5t)
a. Create w(t) using a=2 and store the outputs of the timescale function as w and t_w.
b. Create v(t) using a=0.5 and store the outputs of the timescale function as v and t_v.
c. Play all three signals and verify that they sound different in the ways you predicted in
the prelab. Warning: the timescale function will give you a result with float format
and you need int16 for playing the audio file, so you will have to convert the format.
D. Load a new figure, and using a 3×1 subplot.
a. Plot the signal y vs. t_y on the 1st subplot. Adjust your x-axis to [0, 4], y-axis to
[-30000, 30000]. Label the axes and title your plot appropriately.
b. Repeat for the signal w vs. t_w on the 2nd subplot.
c. Repeat for the signal v vs. t_v on the 3rd subplot.
EE 235 Lab 1-1 Time Transformations of Signals
If you’ve implemented your code correctly, your figure should look something like:
Assignment Check-Off #1 of 2: Demonstrate this Assignment to the lab TA by playing the three
versions of the train signal.
Report discussion: What would happen if you used a time scaling factor of a=-1? Suppose a
student runs the figure command before every call to subplot. When you run your script, what
changes do you expect to see? How will the plots change?
Assignment 2: Time Shift Operation
We will now implement and test a timescale function. This assignment will have four parts, A-D,
each of which should be indicated with comments, following the guidelines in the Lab 2
template.
A. Write a function called timeshift that takes as input: a signal x, the sampling
frequency fs (in Hz), and a time shift t0 (in seconds). The function should implement
y(t) = x(t+t0) and produce as output the portion of the shifted signal starting at time 0.
Assume that the original signal has value zero outside the time window. Your function
should:
i. Find the integer shift n0 given t0 and fs.
ii. Use conditional control that tests whether the time shift is positive or negative
a. For a time delay, create y by concatenating a zero vector with the original
signal. (The output should be longer than the original signal.)
b. For a time advance, create y by copying the portion of the starting from n0
and then appending n0 zeroes at the end of the signal. (The output should be
the same length as the original signal.)
EE 235 Lab 1-1 Time Transformations of Signals
iii. Based on the length of the final signal and the sampling frequency, create a time
vector that corresponds to the output signal length, starting at 0.
iv. Return the new signal and the time vector.
Save the function it in its own cell, as indicated in the Lab 2 template.
B. Use the function to create y(t+0.5) and y(t-2). Plot the shifted signals with the
original in a 3×1 plot: y(t), y(t+0.5), and y(t-2). The x-axis should be between 0 and 4
for all three plots. Label axes and title the plot.
C. Play all three signals. For the signal that has been advanced, you should be able to
hear that part of the sound is missing, since we have not preserved those samples.
Assignment Check-Off #2 of 2: Demonstrate this Assignment by showing the plot to the lab TA
and playing the three signals.
Report discussion: There is a trivial case that you should ideally test for. If the shift is zero, then
the output is the original signal. If the shift is an advance bigger than the original signal, then the
output will be zero. Comment on whether your current implementation correctly handles these
cases and whether there a better implementation.
3.0 Recovering Popular TV/Movie Audio File
Mayday, mayday! The ECE head TA is in dire need of succor. His favorite audio files of popular
culture quotes that form the bulk of his personality have been tampered with by a mischievous
undergraduate! Please help restore his wit by undoing the evil student’s transformations in time
and recover the signals in the TA’s prized possession of audio files.
The professor is fairly certain that this was a harmless prank, and the student used a
transformation that would not be too difficult recover, with transformation factors of 1/2 or 2.
Therefore, the audio signal either went through a sequence of transformations to produce y(t) =
x(2t – 0.5) or y(t) = x(0.5t – 2). Your job is to implement the inverse transformation for both
cases using the functions you created in assignments 1 & 2, and choose the original based on
listening to the audio.
There are several audio files to fix. Your TA will assign each team a sound file to download; save
it to your workspace folder.
A. Using time scaling then time shift, implement transformations to recover x(t) for the two
options: a) y(t) = x(2t – 0.5), and b) y(t) = x(0.5t – 2). For the first case, call the output
signal ya and the corresponding time samples ta. For the second case, use yb and tb,
respectively.
B. Play both of the transformed signals, ya and yb. Determine which is most human like,
and from there determine which was the transformation actually performed by the
EE 235 Lab 1-1 Time Transformations of Signals
mischievous student: y(t) = x(2t – 0.5) or y(t) = x(0.5t – 2). Save the recovered
(corrected) signal.
C. Start a new figure. Plot the distorted audio in a 2×1 subplot figure window, after
computing the time sample t_y. Plot the recovered signal in the 2nd subplot of the figure.
Assignment Check-Off: No assignment check-offs for this part, but you can play the corrected
signal for the TA to get confirmation that you have it right.
Report discussion: Specify which transformation was used by the mischievous student, and
identify the movie or TV show your quote came from. Below are some hints and fun facts about
each quote.
S1: This quote comes from a popular science-fiction movie franchise directed by James Cameron that chronicles the
ongoing battle between humans and Skynet. The film series stars a former Governor of California. Five films have
been made so far with a sixth film is expected in 2019.
S2: This quote comes from the third entry in a popular movie franchise based on a series of British fantasy books
chronicling the life of a boy wizard and his friends. The director won a 2014 Academy Award for Best Director for
Gravity, which was the head TA’s favorite movie from 2013.
S3: This quote comes from a 2016 standalone spinoff film to a popular space opera film series. The movie ended up
being an unexpected prequel!
S4: This quote comes from a hugely popular HBO fantasy drama series. The head TA is a huge fan and waited 6+
hours in line to attend its San Diego Comic-Con program panel and was elated when HBO handed out free gift bags.
S5: This quote comes from an AMC post-apocalyptic horror drama TV series. At the 2013 San Diego Comic-Con,
the head TA and a group of friends started standing in line at 4am to gain entrance in to the show’s program panel
that started at 2pm!
S6: This quote comes from the 55th Disney animated feature film about an unlikely friendship between a rabbit and a
fox working to solve a crime. The movie is so good that its creators won a Golden Globe Award for best animated
film.
S7: This quote comes from an animated sequel to a classic fish tale from Pixar. It is voiced by a famous talk show
host and comedian recently honored by a Presidential Medal of Freedom.
S8: This quote comes from a web-slinging superhero movie trilogy that has been rebooted so many times that it is
hard to keep count. Regardless, the head TA is a huge fan and has plans to watch every movie this superhero is in.
S9: This quote comes from a television sitcom on CBS about a group of four socially awkward geeks (ahem,
intellectuals) living in Pasadena, CA. The word you hear is a regularly used catch phrase on the show that means
“fooled you!”
S10: This quote comes from a popular, female-led book-turned-movie set in the fictional country of “Panem.” The
head TA was too excited to see the film and only finished 75% of the book before seeing it. He reports being
surprised how many things the movie left out!
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
EE 235 Lab 1-1 Time Transformations of Signals
submit it on Canvas. Remember, only one notebook per team! Make sure that your notebook is
titled Lab2-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 via Canvas: i) the .ipynb file, and ii) the recovered audio file from assignment 3.
