# Practice Quiz 1

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You will have *one hour and fifty minutes* to complete this quiz. During the
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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
- 2) Find the Magnitude
- 3) Angular Trends
- 4) More than Meets the Eye
- 5) Dome, Sweet Dome

## 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.

X_1(\omega) =~

#### Part 2

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

and illustrated below.

X_2(\omega)=~

#### Part 3

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

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.

Corresponding Graph:

Corresponding Graph:

Corresponding Graph:

Corresponding Graph:

Corresponding Graph:

Corresponding Graph:

Corresponding Graph:

## 4) More than Meets the Eye

#### Part 1

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

X_1(\omega) =~

#### Part 2

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

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:

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

`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.

Matching graph: