Practice Quiz 2
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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 handwritten 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).
For all subparts requiring symbolic expressions, you should enter your answers using Python syntax. Your answers may contain any of the following:
omega
represents \omegaOMEGA
represents \Omegasin
andcos
represent \sin(\cdot) and \cos(\cdot), respectivelysqrt
represents the square root functione
,j
, andpi
represent e, j, and \pi, respectively
Table of Contents
1) Peaks and Valleys
Consider a family of signals that can be described by the following equation (where \alpha and m are parameters):
An example of such a function is shown below, for \alpha = 0.9 and m = 3.
Consider also the following graphs, each of which shows the magnitude of the DTFT of such a signal when
2) DFTs
The left column below shows six discretetime signals for 0\le n\le31. The right column shows plots of the magnitudes of six DFTs computed for N=32. For each discretetime signal in the left column below, find the matching DFT magnitude (one of plots AF).
3) Trigonometric Fourier Series
Part 1
We would like to represent the following CT signal
using a trigonometric Fourier series of the following form:
Determine the following parameters of the trigonometric representation:
T = ~
M =~
Nonzero c_k values:
Nonzero d_k values:
Part 2
Let f[n] represent a discretetime signal whose Fourier series coefficients F[k] are periodic in N=5, i.e., F[k]=F[k+5] for all integers k. The following plots show the magnitude and angle of F[k] over one period:
We wish to find the coefficients of a trigonometric representation for f[n] with the following form:
Determine the following parameters of the trigonometric representation:
N = ~
M =~
Nonzero c_k values:
Nonzero d_k values:
4) Signal Facts
Suppose we are given the following facts about a signal x[\cdot] (and its Fourier series coefficients X[\cdot]):

x[\cdot] is periodic and realvalued.

\displaystyle \max\left(x[n]\right) = 5

X[k6] = X[k] for all k

X[2] = 0

y[n] = x[n]  1 is a antisymmetric function of n.

\displaystyle \frac{1}{6}\sum_{n=2}^{3}(1)^nx[n] = 0

{\rm Im}(X[1]) > 0
Answer the following questions about this signal. If there is not enough
information to solve any of the questions, enter {\tt NEI} (for not enough
information'') in those boxes.
N =~
x[n] =~
5) DTFT Matching
Let x_0[n] = \delta[n+1] + \delta[n] + \delta[n1], and let x_1[\cdot] be a scaled and periodicallyextended version of x_0[\cdot], with repetitions every N_0 samples:
As an example of the general shape of this function, here is an example with N_0=17 (though you should not assume that N_0=17 throughout the problem):
Also let x_2[n] = B\cos(\Omega_0 n) for some value \Omega_0, and let x_3[n] = x_1[n]+x_2[n].
Also consider the following plots, each of which shows the (purely real) DTFT of some function.
Which of the graphs (16) corresponds to X_3(\Omega), the DTFT of x_3[n] = x_1[n]+x_2[n]? And what are the values of A, N_0, B, and \Omega_0? Enter a single number in each box below.