Task1_Part2_SignalProcessing

Task1-2: Signal Processing II

Announcement: February 06, 2019
1st due date: February 20, 2019 (Wednesday) before 11:59 pm
2nd due date: February 23, 2019 (Saturday) before 11:59 pm

The main goal of this task is to understand the fundamental of data acquisition system and signal processing. You must study the following two tutorials (signal_processing_V4.mlx, data_aquisition_V3.mlx) to solve the corresponding problems. Please answer all sub-questions in each problem. You should write your own code to solve these questions.

Problem 1: Convolution (10 points)

(a) Compute an analytic y(t) which is the convolution of f(t) and g(t):

(b) Write a code to numerically compute y(t) and plot y(t). Please use for-loop and do not use conv. You can randomly assign the values of A and a.

(c) Write a code to numerically compute y(t) and plot y(t). Please use conv. You can randomly assign the values of A and a.

Problem 2: Convolution Theorem (10 points)

(a) Proof the convolution theorem and explain the meaning of these relationships in your words.

(b) Compute a Fourier transform of the triangular function in both analytic and numeric ways (Note that this function is not a periodic):

(c) Please explain the result in (b) using your answers for Problem 1.

Problem 3: Discrete Fourier Transform 1 (10 points)

(a) What is the meaning of the following relationship in the lecture slide? Please explain it.

(b) What is the meaning of the following relationship in the lecture slide? Please explain it.

(c) What is the difference between these two functions in the lecture slide? :

  and   ,

Problem 4: Discrete Fourier Transform 3 (20 points) - Use FFT

     where a = 2, b = 2, c= 6, f₁ = 3, and f₂ = 6

     where a = 0.3, b = 10, c= 3, f₁ = 5, and f₂ = 8

(a) z1 and z2 are discrete signals, which are obtained by digitizing y1(t) and y2(t) with a sampling rate of 50 Hz and collecting them for 5 seconds, respectively. Please plot z1 and z2 in the time domain (include a proper time axis).

(b) Perform the discrete Fourier transform of z1 and z2, and plot your graphs in the frequency domain (include a proper frequency axis). Plot only positive frequency signals.

(c) Please compare the shape of the frequency curves of z1 and z2. Which frequency curve is thinner (more narrow)? For example, compare the frequency curve at f₁ in both graphs. Which one is thinner? Please explain your answer. What makes the difference?

Problem 5: Discrete Fourier Transform 2 (30 points) - Use FFT

where A1 = 3, A2 = 10, and A3 = 5.

(a) y1 is a discrete signal, which is obtained by digitizing y(t) with a sampling rate of 500 Hz for 3 seconds. Please plot y1 in the time domain (include a proper time axis).

(b) Perform the discrete Fourier transform of y1 and plot your graph in the frequency domain (include a proper frequency axis). Plot only a positive frequency signal.

(c) y2 is a discrete signal, which is obtained by digitizing y(t) with a sampling rate of 100 Hz for 3 seconds. Please plot y2 in the time domain (include a proper time axis).

(d) Perform the discrete Fourier transform of y2 and plot your graph in the frequency domain (include a proper frequency axis). Plot only a positive frequency signal.

(e) If you digitize a longer-duration signal (let's say 20 seconds) with a sampling rate of 100 Hz, can you measure and extract all frequencies contained in the original signal, y(t)? Please explain your answer.

(f) If you digitize the signal with a sampling rate of 105 Hz for 3 seconds, can you measure and extract all frequencies contained in the original signal, y(t)? please explain your answer.

(g) If you digitize the signal with a sampling rate of 251 Hz for 3 seconds, can you measure and extract all frequencies contained in the original signal, y(t)? please explain your answer.

Problem 6: Frequency Analysis (10 points)

Two sinusoidal accelerations are measured using an accelerometer in a smartphone. Each of the waves is stored at vib_data1.mat and vib_data2.mat.

(a) Load vib_data1.mat and plot the acceleration signal in a z direction (zvib) using the corresponding time info (time). What is the frequency of this wave?

(b) Load vib_data2.mat and plot the acceleration signal in a z direction (zvib) using the corresponding time info (time). What is the frequency of this wave?

Submission of Your Work

• You should turn in a report and codes to [email protected]. When you send your email, please cc your email for future reference.
• A subject of your email must be the format of "Task1@2_Your name _ Degree _ ID"
  • Your name: first five letter your first + last name. The first letter is uppercase and the rest of them are lowercase (i.g Chul Min Yeum -> Chulmi, Juan Park -> Juanp)
  • Degree: pick your degree among BA, ME, MA, and PH (BA=Undergraduate, ME=MEng, MA=MASc, and PH=PhD)
  • ID: your school ID
  • Please do not include any other text except this subject line.
• For writing equations, I recommend the use of latex equations editors introduced in the Markdown tutorial and inserting equation links. However, I also accept for attaching an image of your handwritten equation (but not recommend).
• Your report includes your answers and styled codes for questions in the problems.
• You have to submit a single code file that includes the codes for all problems.
• The formats of the folder and files are
  • Folder name: Task1@2_Your name _ Degree _ ID (same with the subject of your email)
  • File names: "Code_Your name _ Degree _ ID.m or .py" for codes and "Rept_Your name _ Degree _ ID.md and .pdf".
• The report must be written with Markdown script (GFM) and all other formats like docx or pptx are not permitted.
• You should also include a report in pdf that must be converted from your report in Markdown.
• Please review the general submission instruction in the course syllabus.
• When you violate these submission guideline, your report will be returned and must be resubmitted.

For example, Juan Park is using MATLAB to complete the Task1-2. Juan needs to submit his report and codes to [email protected] with an attachment of Task1@2_Juanp_BA_000000.zip. In the folder, there are at least three files:

• Code_Juanp_BA_000000.m
• Rept_Juanp_BA_000000.md
• Rept_Juanp_BA_000000.pdf

• You may need to include all figures used for writing your report.

Note

• Please post a question if you need to help understand the problem and/or tutorials.
• You are permitted to discuss the task with your colleague.
• Your grade depends on the completeness and clarity of your work.
• You should include clear and concise comments in your codes.