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Quiz 1

The questions below are due on Wednesday March 24, 2021; 09:25:00 PM.
 
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You will have one hour and fifty minutes to complete this quiz. During the quiz, you may reference one sheet (front and back) of hand-written or printed notes, but you may not make use of other outside resources, including other pages on the Internet. You may use as much scratch paper as you like during the quiz, and your scratch work will not be collected.

This quiz consists of 5 questions. Please make sure to submit your answer to each question (you may submit as many times as you like during the quiz).

For all subparts requiring symbolic expressions, you should enter your answers using Python syntax. Your answers may contain any of the following:

  • omega represents \omega
  • OMEGA represents \Omega
  • sin and cos represent \sin(\cdot) and \cos(\cdot), respectively
  • sqrt represents the square root function
  • e, 1j, and pi represent e, j, and \pi, respectively
  • Re and Im represent {\rm Re}(\cdot) and {\rm Im}(\cdot), respectively

Certain questions may allow additional variables; if this is the case, it will be mentioned in that problem.

Table of Contents

1) Sinusoids

Part a

Consider the function x_1[n] = A_1\cos\left(\Omega_1 n + \phi_1\right), which is plotted below:
 

Based on this equation and the graph, estimate the values of A_1, \Omega_1, and \phi_1, and enter your answers as Python expressions in the boxes below.

A_1 \approx~

\Omega_1 \approx~

\phi_1 \approx~

 

Part b

Consider the function \displaystyle x_2[n] = a_2e^{j\Omega_2 n} + a_2^*e^{-j\Omega_2 n}, which is plotted below:
 

Based on this equation and the graph, estimate the values of |a_2|, \angle a_2, and \Omega_2, and enter your answers as Python expressions in the boxes below.

What is the magnitude of a_2?
|a_2| \approx~

What is the phase of a_2?
\angle a_2 \approx~

What is the value \Omega_2?
\Omega_2 \approx~

 

Part c

Consider the function \displaystyle x_3[n] = c_3\cos(\Omega_3 n) + d_3\sin(\Omega_3 n), which is plotted below:
 

Based on this equation and the graph, estimate the values of c_3, d_3, and \Omega_3, and enter your answers as Python expressions in the boxes below.

c_3 \approx~

d_3 \approx~

\Omega_3 \approx~

2) Fourier Transforms

Consider the CT signal x(\cdot) defined by the following plot (you can assume that x(t) = 0 for all t not shown in the plot below):

This function has a Fourier transform given by X(\cdot). A portion of the magnitude of the Fourier transform is shown below:

Determine a simple expression for b in terms of the values in x(\cdot).
When entering your answer, you may use t_1 to refer to t_1, t_2 to refer to t_2, and/or a to refer to a.
b =~
Determine a simple expression for \omega_1 in terms of the values in x(\cdot).
When entering your answer, you may use t_1 to refer to t_1, t_2 to refer to t_2, and/or a to refer to a.
\omega_1 =~

Which of the plots below best represents the angle of the Fourier transform?
Best match for \angle\left\{X(\omega)\right\}:~

In your chosen graph, what is the value of c?
When entering your answer, you may use t_1 to refer to t_1, t_2 to refer to t_2, and/or a to refer to a.
Notice also that c's meaning depends on the particular graph you chose.
c =~

 

Phase Graphs:

3) Related Transforms

For this problem, we will consider the following signal f[n], which is known to be zero outside of the range of n plotted below:

Assume that f[\cdot] has a DTFT given by F(\cdot), such that \displaystyle F(\Omega) = \sum_{n=-\infty}^\infty x[n] e^{-j\Omega n}

For each of the related signals below (each of which is directly related to f[\cdot], determine its Fourier transform in terms of F(\cdot) (do not solve for the exact transform). Use F as a Python function in your answers below.

 
F_2(\Omega) =~

 
F_3(\Omega) =~

 
F_4(\Omega) =~

 
F_5(\Omega) =~

Note that f_6[n] is nonzero over a wider range of n, and that for all n that are divisible by three, we have the following relationship:

f_6[n] = f_6[n-1] = f_6[n+1] = f[n/3].

F_6(\Omega) =~

4) DFT

In this problem, we will consider a CT signal consisting of two distinct notes, one an "C" at 523Hz, and another an "A" at 1760Hz, each represented by a pure cosine.

Imagine sampling this signal at a rate f_s = 2000 samples per second to produce a DT signal, and then computing the DFT of that signal with some value N, where N is a power of 2. Plotting the DFT magnitudes gives us the following graph for some range of k values (not necessarily the whole range of k values). Note also that we have plotted this as a line graph for clarity even though X[k] is actually a function of discrete k):

Under this analysis, the "C" note (523Hz) corresponds to peaks centered at k=\pm 268. Given this information, answer the following questions:

What value of N was used in this calculation?
N=~

The "A" note (1760Hz) will manifest as peaks at \pm some k value, where k is in the range [0, N/2]. At what value of k will this peak occur?
Enter your answer as a Python expression (not necessarily simplified); and note that you can use Python's round function to round to the nearest integer; you do not need to calculate the exact result by hand.

Which of the following statements best describes the relationship of these signals to the graph above?
The "C" note caused the peaks at \pm k_1, and the "A" note caused the peaks at \pm k_2.
The "C" note caused the peaks at \pm k_2, and the "A" note caused the peaks at \pm k_1.

5) DTFS Components

In this question, we will consider 12 simple signals, each of which is periodic in N=6. For each of these signals, determine which of the graphs below (labeled A through U) best matches the magnitude, real part, and imaginary part of the signal's DTFS representation.

Note

Note that you may wish to sketch out these signals on scratch paper and solve on paper first, before entering any of your answers into the system, to avoid the need to scroll back and forth between the graphs.

Consider this function's DTFS, X_0[k], and answer the following questions about it:
 
Which graph best matches |X_0[k]|?
Which graph best matches {\rm Re}(X_0[k])?
Which graph best matches {\rm Im}(X_0[k])?
Consider this function's DTFS, X_1[k], and answer the following questions about it:
 
Which graph best matches |X_1[k]|?
Which graph best matches {\rm Re}(X_1[k])?
Which graph best matches {\rm Im}(X_1[k])?
Consider this function's DTFS, X_2[k], and answer the following questions about it:
 
Which graph best matches |X_2[k]|?
Which graph best matches {\rm Re}(X_2[k])?
Which graph best matches {\rm Im}(X_2[k])?
Consider this function's DTFS, X_3[k], and answer the following questions about it:
 
Which graph best matches |X_3[k]|?
Which graph best matches {\rm Re}(X_3[k])?
Which graph best matches {\rm Im}(X_3[k])?
Consider this function's DTFS, X_4[k], and answer the following questions about it:
 
Which graph best matches |X_4[k]|?
Which graph best matches {\rm Re}(X_4[k])?
Which graph best matches {\rm Im}(X_4[k])?
Consider this function's DTFS, X_5[k], and answer the following questions about it:
 
Which graph best matches |X_5[k]|?
Which graph best matches {\rm Re}(X_5[k])?
Which graph best matches {\rm Im}(X_5[k])?
Consider this function's DTFS, Y_0[k], and answer the following questions about it:
 
Which graph best matches |Y_0[k]|?
Which graph best matches {\rm Re}(Y_0[k])?
Which graph best matches {\rm Im}(Y_0[k])?
Consider this function's DTFS, Y_1[k], and answer the following questions about it:
 
Which graph best matches |Y_1[k]|?
Which graph best matches {\rm Re}(Y_1[k])?
Which graph best matches {\rm Im}(Y_1[k])?
Consider this function's DTFS, Y_2[k], and answer the following questions about it:
 
Which graph best matches |Y_2[k]|?
Which graph best matches {\rm Re}(Y_2[k])?
Which graph best matches {\rm Im}(Y_2[k])?
Consider this function's DTFS, Y_3[k], and answer the following questions about it:
 
Which graph best matches |Y_3[k]|?
Which graph best matches {\rm Re}(Y_3[k])?
Which graph best matches {\rm Im}(Y_3[k])?
Consider this function's DTFS, Y_4[k], and answer the following questions about it:
 
Which graph best matches |Y_4[k]|?
Which graph best matches {\rm Re}(Y_4[k])?
Which graph best matches {\rm Im}(Y_4[k])?
Consider this function's DTFS, Y_5[k], and answer the following questions about it:
 
Which graph best matches |Y_5[k]|?
Which graph best matches {\rm Re}(Y_5[k])?
Which graph best matches {\rm Im}(Y_5[k])?

 

Graphs for Question 5: