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.
ESE 471 Spring 2020 2
# Score Out of
1 25
2,3 16
4 20
5 21
6 18
Total 100
ESE 471 Spring 2020 3
ESE 471 Spring 2020 4
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 subc
arriers has 8K orthogonal waveforms.
ESE 471 Spring 2020 5
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:
ESE 471 Spring 2020 6
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: ESE 471 Spring 2020 7 Blank work page ESE 471 Spring 2020 8 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. ESE 471 Spring 2020 9 Blank work page ESE 471 Spring 2020 10 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: ESE 471 Spring 2020 11 Blank work page