Description
In this lab, we will learn how to transform signals from the time domain to the frequency domain
and try to identify the frequencies of ejωt that comprise a periodic signal.
The concepts we’ll focus on are finding the Fourier Series coefficients ak of x(t), identifying frequency components
of x(t), and understanding the relationship between Fourier Series coefficient index k and
frequency ω.
Lab 5 Turn-in Checklist
• 2 pre-lab exercises
• 3 lab assignments with 2 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
section.
Pre-lab
Read the Lab 5 Background document, then complete the following exercises.
1. Consider the sum of sinusoids d0(t) = sin(2π(941)t) + sin(2π(1336)t)
a. Find the Fourier Series representation of this signal, i.e. its Fourier series coefficients
ak, given that wo = 2π. (No need to prove wo = 2π.) For what values of k are the
coefficients non-zero?
b. Each Fourier series coefficient index k (with nonzero ak) corresponds to a frequency
ω = kwo in radians. The relationship between frequency in radians ω and frequency f
in Hertz (Hz) is ω=2πf. What are the frequencies of the two sinusoids in Hz?
c. If sample this sine wave with fs=8000 Hz, and then you take its Fourier transform
using an FFT with 4096 points, what is the frequency in Hz of the nearest FFT sample
to the true frequency? Find the value for each of the two sinusoids above.
2. The telephone touch-tone system uses signals composed of different frequencies to
indicate the digit pressed. This signaling scheme is called “dual-tone multi-frequency”
(DTMF) signaling. In DTMF signaling, each touch-tone on the keypad is represented as
the sum of two sinusoidal tones, where one tone is at a low frequency and the other is
high. DTMF keypad frequency pairs are given below:
High frequencies
Low frequencies
1209 Hz 1336 Hz 1477 Hz
697 Hz 1 2 3
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770 Hz 4 5 6
852 Hz 7 8 9
941 Hz 0
For example, to generate key ‘0’, we need the signal d0(t) above. To generate key ‘2’, we
need the signal d2(t) = sin(2π(697)t) + sin (2π(1336)t).
Write Python code to create a 2-D Numpy array tone_freqs, where each frequency pair is
ordered (lower frequency, higher frequency).
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. 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: Identifying Component Frequencies of a Signal
In this Assignment, we’ll create and analyze the signal d0(t) from the prelab. Write your code in
a new cell for Assignment 1, inside your Lab 5 notebook.
A. Using sampling frequency fs=8000, create a time samples vector t for 0 ≤ t < 0.25. Use
this vector to create d0(t) = sin(2π(941)t) + sin(2π(1336)t).
B. With the simpleaudio module, write a WAV file where d0 is played twice, with an 0.5-
second pause in between. You will need to amplify the signal to make it audible; try a
scaling factor of 50. Play the file to hear the telephone tones.
C. Find the Fourier Transform d0_ft of d0 using the numpy fft and fftshift functions. Create
a frequency vector f that corresponds the positive and negative frequency range. Create a
2×1 subplot, and plot d0_ft vs. f as Plot 1.
D. If this was an ideal continuous-time Fourier transform, then all d0_ft(i)=0 for i other than
the sinusoid frequencies. However, because of time and frequency sampling on the
computer, you can get non-zero values. Also, if this was a real communications problem,
the signal would have some noise. So, in order to automatically find the component
frequencies in the signal from the DFT, we’ll do an energy threshold test.
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a. Use the masking strategy described in the background document with a threshold of
900 to find the d0_ft array indices associated with FT peaks. Convert the resulting
indices to Hz.
b. Sinusoids should have symmetric positive and negative frequency complex
exponentials, but we are only interested in the positive frequencies. Print the values of
the positive frequencies.
E. As in lab 4, create a noise vector that is the same length as the signal, using a standard
deviation of 1
n = np.random.normal(0,1,len(d0))
and add it to d0 to create d0n. Compute the DFT as d0n_ft and plot d0n_ft vs. f as Plot 2
Assignment Check-Off #1 of 3: Demonstrate this Assignment to the lab TA by playing the
audio file and showing the FFT plots.
Report discussion:
To identify component frequencies, we used a threshold value of 900. Using your FFT
magnitude plot, explain why a threshold of 100 would not work. Would that threshold identify
more or fewer frequencies than the threshold of 900?
Comment on how the addition of noise changes the FFT plot.
What would the FFT look like if instead the signal had been
v(t) = 1 + sin(2π(941)t) + sin (2π(1336)t) ?
Assignment 2: Classifying Touch-Tone Telephone Signals
In this assignment, you will write a function to classify a tone signal by finding its component
frequencies, then test it. Use a separate cell for the function and the testing. in-line comments and
good form to keep your logic clear.
A. In a separate cell, define a function classify which takes inputs: tone signal x and
sampling rate fs. The output should be the digit associated with the signal.
a. Using pre-lab, define the tone frequency matrix tone_freqs. Remember to order the
pairs.
b. Use the procedure outlined in Assignment 1 to extract the component frequencies
from tone signal x, with N = 4096 fft samples and a threshold value of 750. Use the
variable names given below:
• xfft_abs as the shifted magnitude output of the FFT of x.
• index_tone as the indices in xfft_abs where the element of xfft_abs is greater
than a threshold value 900.
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• freq_tone as the array of frequencies in xfft_abs with spikes exceeding the
threshold value. freq_tone should be in Hz.
Recall that sinusoids should have symmetric positive and negative frequencies. Store
last positive frequencies of freq_tone in pos_freq.
c. Now that we have the two component frequencies of x, we can classify its touch tone
using your pre-lab code. Write a loop that does the following:
• Loop through the digits, and find the total squared error measurement digit_error
between the touch-tone frequency pair and pos_freq (this error measurement is
the sum of the two squared errors: (fL-fL(d))2+(fH-fH(d))2
, where fL is the
frequency of the low tone, fH is the frequency of the high tone and f*(d) are the
corresponding frequencies detected in the signal.
• Track the minimum error of the frequency pairs evaluated so far as min_e, with
the corresponding touch tone min_e_tone, by comparing the current total error to
our existing min_e and updating as needed.
• Return the touch-tone digit identified by min_e_tone.
B. Start a new cell. Using time sample vector t for the range 0 ≤ t < 0.25 and sampling rate
fs = 8000, test the classify function. Create the tone signals d4, d5, d8, and d9, and run
classify on each to verify that the correct digit is detected.
Assignment Check-Off #2 of 2: Demonstrate classify function test to the lab TA
Report discussion: If the signal had noise, as in assignment 1, what could you do to ensure high
detection accuracy?
Assignment 3: Decoding a Phone Number from Touch-Tone Signals
The ECE head TA wants to tell you about a San Diego restaurant, but would rather you decode
its phone number from a sound file. You can use your function from assignment 2. As always,
start a new cell for this assignment.
A. Load file phone.csv. The file has 11 time slices corresponding to 11 digits. Each
individual time-slice has 0.25 seconds of the DTMF signal followed by 0.25s of pause or
silence. The sample frequency used for creating the file was 8kHz.
B. We will be decoding the phone number in a loop, so we’ll need a buffer to store each
digit as we decode it. Declare a vector phone_num with 11 elements, each element -1.
C. Write a loop to fill in phone_num. For each element (digit), you will extract one of the
11 time slices (consisting of a tone signal and a silence signal), decode it, and print out
the digit. For each time slice, compute the corresponding start and end indices
(start_index, end_index) in x. Use these to extract the data samples for that time slice,
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storing them to signal. Classify the samples, and save your decoded tone number to the
appropriate index in phone_num.
D. Print the sequence of digits.
Report discussion: Provide the decoded message in your lab report. Optionally, if you are
ambitious, figure out the name of this well-know fast-food restaurant.
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 Lab5-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.
