ESE 471 Spring 2020: Exam 1
NAME:
• Solve each problem completely, showing each step. Each successful step shown can be
awarded partial credit. For example, write down the definition or formula before
doing the math.
• No need to show work on multiple choice or true/false questions, as there is no partial
credit for such problems.
• Feel free to use extra sheets to answer any question. Include these extra sheets with the
exam booklet and label each sheet with your name.
• For clarity, copy your final answer to the space after the word “Answer:” for each part
of each problem. (Proofs and diagrams do not need to be copied.)
• You may leave your expression in terms of constants, such as Q(2.5) or e−2 – this is not
a test of your calculator. Full credit will be given if the expression is correct.
• You have 80 minutes for this exam.
• You may have with you one note sheet, up to two sides of a 8.5 x 11 inch sheet of paper.
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# Score Out of
1 25
2,3 16
4 20
5 21
6 18
Total 100
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Problems
1. (5 points each) True/False Questions. Circle either True or False.
(a) True or False: The pulse shape with Fourier transform of its autocorrelation function,
Rp(f), for 0 < b < 1, shown below, meets the Nyquist Filtering theorem condition
for zero inter-symbol interference.
1
0.5
1+a
2Ts
1−a
2Ts
−1+a2Ts −1−a2Ts
R(f)
f
(b) True or False: In 8-ary PSK, all symbols are equally distant from the origin.
(c) True or False: A waveform has infinite energy.
(d) True or False: For a pulse shape p(t) which meets the Nyquist sampling theorem,
and has symbol period Ts, the two waveforms φ0(t) = p(t) and φ1(t) = p(t− Ts/2)
are always orthogonal.
(e) True or False: An OFDM system with K subcarriers has 8K orthogonal waveforms.
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2. (8 points) Short answer (Answer in two sentences or less): What is the major problem
with using a rectangular pulse for the pulse shape p(t) in a wireless communication
system?
3. (8 points) Consider an 8-QAM modulation with the following constellation diagram:
A
−A
A
−A
2A
2A2A
φ0
φ1
What is the average symbol energy?
Answer:
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4. (20 total points) A pulse shape p(t) is formed by convolving the following two functions
s(t) and g(t):
s(t) = exp
®
−|t|
σ
´
(1)
g(t) =
®
1, −T < t < T
0, o.w.
(2)
where σ and T are positive constants. What is the Fourier transform of the pulse shape
P (jω)? (You may instead provide P (f) if you prefer.)
Answer:
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5. (21 points) In this problem, consider binary 1-D detection when noise is not Gaussian.
Instead, noise W has the following distribution,
fW (w) =
1
pi(1 + w2)
(3)
Let X be the received signal, and let the transmitted signal be either a0 = 0 or a1 = 1.
The two hypotheses at the receiver are:
H0 : X = 0 +W
H1 : X = 1 +W,
Assume equally likely symbols. Starting from the optimal (lowest probability of error) de-
tection formula for arbitrary distributions, prove that the optimal detector has a threshold
of 0.5.
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6. (18 total points) Consider the functions:
φ0(t) =
® »
1
2 , −1 ≤ t ≤ 1
0, o.w.
φ1(t) =
®
t
»
3
2 , −1 ≤ t ≤ 1
0, o.w.
(a) (12 points) Prove that φ0(t) and φ1(t) are orthogonal.
(b) (6 points) Now, assume that I give you a waveform φ2(t) that is orthogonal to φ1(t).
True or False: φ2(t) must be orthogonal to φ0(t).
Answer:
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