Midterm 1 Exam - Requires Respondus LockDown
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! EE210, S22: Sample Midterm 1 Exam from F21
Instructions
All 12 questions are to be answered in the external document. On the screen, answer True if your
solution is to be included in the file to upload; answer False otherwise.
Start the solution of a question by listing the question number; for example: Question 2. End the
solution of a question with a solid horizontal line across the page.
All questions are displayed at once for time efficient access to any question. The first 4 questions are
time domain problems; the next 4 are frequency domain (DTFT) problems; the last 4 are complex
frequency domain (z-transform) problems. They can be answered in any order. Each question is
worth 3 points.
3 ptsQuestion 1
A discrete-time signal is as shown. Sketch and label carefully
the signal ; is the unit step
signal.
푥(푛)
(푛) = 푥(푛) [푢(푛 + 2) − 푢(푛 − 3)]푥1 푢(푛)
True
False
3 ptsQuestion 2
True
False
Determine if the signal is periodic. If
periodic, determine the period. If not, state why not.
푥(푛) = cos(휋푛/3) + sin(2휋푛/3)
3 ptsQuestion 3
True
False
The input and output of a given system are related by the
equation . i) Does this system satisfy the
homogeneity property of a linear system? ii) Does the system satisfy
the time invariance condition? Justify your answers. No credit without
justification.
푥(푛) 푦(푛)
푦(푛) = 푛 푥(푛 + 1) + 푥(푛)
3 ptsQuestion 4
True
False
Given the energy signal [2 3 - 1 1} at times 0 ≤ ≤ 3 and
zero otherwise, use time-domain calculations to determine
numerical values for i) only, where ' ' denotes
linear convolution and ii) is the
autocorrelation function of .
Do not waste time calculating convolution and autocorrelation at all
times.
푥(푛) = 푛
푥(푛) ∗ 푥(푛) at 푛 = 2 ∗
(푛) at 푛 = 2 only, where (푛)Γ푥푥 Γ푥푥
푥(푛)
3 ptsQuestion 5
True
False
Use the DTFT pair
together
with appropriate DTFT properties to write down in no more than few
lines the DTFT of each of the signals: i) , and ii)
푥(푛) = 푢(푛) , |푎| < 1 ↔ 푋( ) = 1/(1 − 푎 )푎푛 푒푗휔 푒−푗휔
(푛) = ( 푢(−푛)푥1 14 )−푛
(푛) = 푛 ( 푢(푛).푥2 14 )푛
3 ptsQuestion 6
True
False
Two LTI systems are connected in cascade as shown. Given
to be as shown and that , sketch the
frequency response of the full cascade system and label all
numerical values. In words, what will the output be if the input is
?
( )퐻1 푒푗휔 (푛) = 3ℎ2
sin( 푛)휋3
휋푛
퐻( )푒푗휔
푦(푛)
푥(푛) = cos(0.3휋푛) + cos(0.5휋푛)

3 ptsQuestion 7
True
False
Parseval's theorem states that a signal energy computed in the
time domain or the frequency domain should be equal, that is, if
then
. With the help of
Parseval's theorem, determine the total energy of the signal
and the fraction of that energy in the frequency band
퐸
푥(푛) ↔ 푋( ),푒푗휔
퐸 = |푥(푛) = |푋( 푑휔∑∞푛=−∞ |2 12휋 ∫
휋
−휋 푒
푗휔|2
푥(푛) = 3 sin( 푛)
휋
3
휋푛
−0.2휋 < 휔 < 0.2휋.

3 ptsQuestion 8
True
False
The input signal to an LTI system is white noise of
autocorrelation function . The system is an ideal LPF
of the real frequency response shown (
). Determine and sketch the power spectrum of the output
signal and use your sketch to calculate the total power of .
푥(푛)
(푛) = 3 훿(푛)푅푥푥
퐻(푓) 푓 = 퐹(퐻푧)/ (퐻푧)퐹푆
(푓)푆푦푦
푦(푛) 푦(푛)

3 ptsQuestion 9
The system function of some system has three poles located in the -
plane at .
i) If the system is known to be stable, what is the region of
convergence (ROC) of )? Briefly explain.
ii) Is there a ROC that would make both causal and stable? If
yes, list it. If no, state why not.
푧
푧 = −0.5, 0.7, 1.2
퐻(푧)
퐻(푧)
True
False

3 ptsQuestion 10
True
False
A causal LTI system has the known system function
. What is the difference equation relating
the output to the input State why this system is unstable.
퐻(푧) = +11−0.5푧−1
1
1+ 2푧−1
푦(푛) 푥(푛)?

3 ptsQuestion 11
True
False
The system function of an LTI system is known to be
. Determine the system's impulse
response h(n).
퐻(푧)
퐻(푧) = , |푧| > 1/41−푧−1
1− 14 푧−1

3 ptsQuestion 12
Not saved
True
False
The signals and ( -transform and ) are the input
and output to an LTI system of the causal and stable system
function , . If ,
determine and its ROC, hence determine .
푥(푛) 푦(푛) 푧 푋(푧) 푌 (푧)
퐻(푧) = 11− 12 푧−1
|푧| > 12 푥(푛) = 푢(−푛 − 1)
푌 (푧) 푦(푛)
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