Home / Problem Set 12 / Practice Quiz 1

Practice Quiz 1

The questions below are due on Monday May 17, 2021; 10:00:00 PM.
 
You are not logged in.

If you are a current student, please Log In for full access to the web site.
Note that this link will take you to an external site (https://shimmer.csail.mit.edu) to authenticate, and then you will be redirected back to this page.

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 6 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) Problem 1: Short Answers

Note: For these questions, you may use the constants j, e, and pi, as well as the cos and sin functions.

Part a. Find the fundamental period T_1 of the following signal

x_1(t)=2\cos\left({\pi\over3}t\right)\cos\left({\pi\over4}t\right)
and enter its value in the box provided.

T_1 =  

Part b. A trigonometric Fourier series of a periodic DT signal has the form

c_0+\sum_{k=1}^\infty \Big(c_k\cos(k\Omega_on) + d_k\sin(k\Omega_on)\Big)
where \Omega_o is 2\pi divided by the period (N) of the signal. How many non-zero coefficients are needed to represent
x_2[n]=3\sin(\Omega_on)-4\sin^3(\Omega_on)
in such an expansion?

Number of non-zero components:  

Part c. Let X_3(\omega) represent the Fourier transform of the following signal.

x_3(t)=\begin{cases} e^{-2(t-3)}& \text{if}~t\ge0\\ 0&\text{otherwise} \end{cases}
Find X_3(0).

X_3(0) =  

Part d. Let X_4[k] represent the DFT of the signal given below

x_4[n]=(\delta[n])^2+\delta[n]\delta[n-2]+2(\delta[n-5])^2
when N=10. Find X_4[2].

X_4[2] =  

Part e. The discrete cosine transform F_C[k] of a signal f[n] is defined by the following.

F_C[k] = {1\over N}\sum_{n=0}^{N-1} f[n]\cos\left({\pi k\over N}\Big(n{+}{1\over2}\Big)\right)
f[n] = F_C[0] + 2\sum_{k=1}^{N-1} F_C[k]\cos\left({\pi k\over N}\Big(n{+}{1\over2}\Big)\right)
The discrete cosine transform of
x_5[n]=A\cos\left({5\over8}\pi n+\phi\right)
was computed with N=8 and the result was found to have a single non-zero coefficient. The value of that coefficient was 2. Find k, \phi, and A.

k =  

\phi =  

A =  

2) Problem 2: Impulsive Images

Each part of this problem specifies a 2D signal f_i[r,c] as a function of row number r (increasing downward) and column number c (increasing to the right). Determine which of the panels at the bottom of this page shows the magnitude of F_i[k_r,k_c], which is the 2D DFT of f_i[r,c].

Part a. Let F_1[k_r,k_c] represent the 2D DFT of the following signal:

f_1[r,c]=2\delta[r]\delta[c]+\delta[r{+}2]\delta[c{-}1]+\delta[r{-}2]\delta[c{+}1]
Which of panels A–P (below) shows the magnitude of F_1[k_r,k_c]?

Panel  

Part b. Let F_2[k_r,k_c] represent the 2D DFT of the following signal:

f_2[r,c]= \Big(\delta[r{+}1]+2\delta[r]+\delta[r{-}1]\Big) \Big(\delta[c{-}1]-\delta[c{+}1]\Big)
Which of panels A–P (below) shows the magnitude of F_2[k_r,k_c]?

Panel  

Part c. Let F_3[k_r,k_c] represent the 2D DFT of the following signal:

f_3[r,c]=4\delta[r]\delta[c] +\delta[r]\delta[c{-}2] -\delta[r]\delta[c{+}2] +\delta[r{+}2]\delta[c] -\delta[r{-}2]\delta[c]
Which of panels A–P (below) shows the magnitude of F_3[k_r,k_c]?

Panel  

Each of the following panels shows the magnitude of the 2D DFT of a signal. For each panel, black represents the smallest magnitude in that panel (not necessarily 0), and white represents the largest magnitude in that panel.

3) Problem 3: Inverse DTFT

Let f[n] represent a discrete time signal whose DTFT is shown below.

Determine f[n] and enter its values for n=-3 through 3 in the boxes below.

f[-3] =  

f[-2] =  

f[-1] =  

f[0] =  

f[1] =  

f[2] =  

f[3] =  

4) Problem 4: Number Transforms

Images of the numbers 1 through 5 are shown below. Each image is 128x128 pixels.

Each of the following parts shows the magnitude of the 2D DFT of one of these digit images, where black represents the smallest magnitude in that 2D DFT (not necessarily 0), and white represents the largest magnitude in that 2D DFT.

For each part identify the corresponding digit image.

Part a. Which digit image was used to generate the following 2D DFT magnitude?

Digit  

Part b. Which digit image was used to generate the following 2D DFT magnitude?

Digit  

Part c. Which digit image was used to generate the following 2D DFT magnitude?

Digit  

Part d. Which digit image was used to generate the following 2D DFT magnitude?

Digit  

Part e. Which digit image was used to generate the following 2D DFT magnitude?

Digit  

5) Problem 5: Continuous-Time Transforms

Consider the following three CT signals.

f_1(t) = \begin{cases} a&\text{if}~|t|\lt b\\ 0&\text{otherwise} \end{cases}

f_2(t)=f_1(t)+c\delta(t)

f_3(t)={T\over2\pi}\sum_{m=-\infty}^\infty f_2(t+mT)

The plots below show the Fourier transforms for a number of different parameter values.

Part a. Which plot shows the Fourier transform of f_2(t) for a=3, b=1/2, and c=1?

Plot  

Part b. Which plot shows the Fourier transform of f_2(t) for a=3/2, b=1, and c=1?

Plot  

Part c. Which plot shows the Fourier transform of f_3(t) for a=3, b=1/2, c=1, and T=16?

Plot  

Part d. Which plot shows the Fourier transform of f_3(t) for a=3/2, b=1, c=1, and T=16?

Plot  

These panels show Fourier transforms. The horizontal scales are -4\pi to 4\pi for all panels. The vertical scales for panels A–H are 0 to 4 as indicated by numbers to the left of the plots. For panels I–L, the numbers to the left of the plots indicate the areas of the impulses.

6) Problem 6: Transformational

Let w[n] represent the following function

w[n]=\begin{cases} 1&\text{if}~0\le n\le5\\ 0&\text{otherwise} \end{cases}

and let W[k] represent the DFT of w[n] computed with analysis length N=8:

w[n]\quad{{{\rm DFT}\atop\displaystyle\Longleftrightarrow}\atop{\scriptstyle N=8}}\quad W[k]

This function is used to generate an output signal y[n] from an input signal x[n] as follows.

First, w[n] is multiplied times the input x[n] to get x_w[n]. Then the DFT of x_w[n] is multiplied times the DFT of w[n] to get the DFT of the output y[n].

x[n]w[n]\equiv x_w[n]\quad{{{\rm DFT}\atop\displaystyle\Longleftrightarrow}\atop{\scriptstyle N=8}}\quad X_w[k]

y[n]\quad{{{\rm DFT}\atop\displaystyle\Longleftrightarrow}\atop{\scriptstyle N=8}}\quad Y[k]\equiv X_w[k]W[k]

Determine y[n] when x[n]=n.

y[0] =  

y[1] =  

y[2] =  

y[3] =  

y[4] =  

y[5] =  

y[6] =  

y[7] =