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Practice Quiz 1

The questions below are due on Monday March 29, 2021; 10:00:00 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 practice 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).

Table of Contents

1) Fourier Transforms

For all subparts of this question, you should enter your answers as Python expressions. 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, j, and pi represent e, j, and \pi, respectively

Part 1

The continuous-time signal x_1(t) is defined by the following plot

and is zero outside the indicated range of t.

Find X_1(\omega), the Fourier Transform of x_1(t).
X_1(\omega) =~

Part 2

Determine the Fourier transform of x_2(t) given by the following expression

x_2(t) = \begin{cases} e^{|t|}&\text{if}~-1\lt t\lt1\\ 0&\text{otherwise} \end{cases}

and illustrated below.

Enter a closed-form expression for the Fourier transform in the box below, in terms of sines and cosines.
X_2(\omega)=~

Part 3

Consider a DT signal x_3[\cdot] described by the following diagram:

Determine a closed-form expression for the Fourier transform of x_3[\cdot].
X_3(\Omega) =~

2) Find the Magnitude

Consider the following plots of the magnitudes of the Fourier series coefficients for four discrete-time signals, that are each periodic in N=20.

For each of the following signals, determine which if any of the previous plots shows the magnitude of its Fourier series coefficients.

Matching Panel:

Matching Panel:

Matching Panel:

Matching Panel:

3) Angular Trends

Consider the following plots:

 

For each of the expressions below, indicate the letter of the corresponding plot from above.

\angle \left(e^{-jx}\right)

Corresponding Graph:

\angle \left(1 + 0.8e^{jx}\right)

Corresponding Graph:

\angle \left({1+0.4e^{jx}\over 2+0.8e^{jx}}\right)

Corresponding Graph:

\angle \left(1 + e^{jx}\right)

Corresponding Graph:

\angle \left(1+0.8e^{j2x}\right)

Corresponding Graph:

\angle \left(0.9e^{jx}+0.8e^{-jx}\right)

Corresponding Graph:

\angle \left({1\over 1+0.8e^{jx}}\right)

Corresponding Graph:

4) More than Meets the Eye

Part 1

Consider the function x_1 described by the following expression and plot:

x_1(t) = 3te^{-2|t|}

Determine the Fourier transform X_1(\omega) of this signal. Express your answer in closed form.
X_1(\omega) =~

Part 2

Now consider a signal x_2 whose Fourier transform X_2 is given by the following expression:

X_2(\omega) = 3\omega e^{-2|\omega|}

Determine a closed-form expression for x_2(t).
x_2(t) =~

Part 3

Assume that a function x_3 has a Fourier transform given by X_3.

Let y_3 be defined in terms of x_3, as follows:

y_3(t) = \dot{x}_3\big(3(t+5)\big)

where \dot{x}_3(t) is the time derivative of x_3(t).

Find Y_3(\omega) in terms of X_3. Use X_3 to represent X_3.
Y_3(\omega) =~

5) Dome, Sweet Dome

Ben Bitdiddle created a signal x_0[n] representing the MIT dome, but he only saved the DTFS coefficients X_0[k] (and not the original signal). However, he knew that one period of the original signal (which is periodic in N=51) looked like this:

Ben tried several different methods of recovering the original image based on X_0[k], by applying the DTFS synthesis equation to the following sets of coefficients.

For each set of Fourier coefficients described below (X_A through X_I), determine the corresponding signal from the 24 options shown below (x_1 through x_{24}). Assume that all 24 of those signals are purely real and are periodic in N=51.

If the required signal would be complex-valued, choose the complex option. Otherwise, write the name of the signal from the following page.

X_A[k] = \text{Re}\left(X_0[k]\right)

Matching graph:

X_B[k] = \text{Im}\left(X_0[k]\right)

X_C[k] = j\text{Im}\left(X_0[k]\right)

X_D[k] = \begin{cases} 0 & \text{if}~k=0\\ X_0[k] & \text{otherwise} \end{cases}

X_E[k] = \begin{cases} 0 &\text{if}~k=25\\ X_0[k] & \text{otherwise} \end{cases}

X_F[k] = X_0[k] + 1/51

X_G[k] = e^{j\pi}X_0[k]

X_H[k] = \begin{cases} X_0[0] & \text{if}~k=0\\ e^{j\pi}X_0[k] & \text{otherwise} \end{cases}

X_I[k] = |X_0[k]|e^{j\left(-\angle X_0[k]\right)}

Ben's Graphs: