# Practice Quiz 1

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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 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
- 2) Problem 2: Impulsive Images
- 3) Problem 3: Inverse DTFT
- 4) Problem 4: Number Transforms
- 5) Problem 5: Continuous-Time Transforms
- 6) Problem 6: Transformational

## 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

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

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

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

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

## 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:

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

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

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.

## 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?

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

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

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

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

## 5) Problem 5: Continuous-Time Transforms

Consider the following three CT signals.

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?

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

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

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

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

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

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].

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