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Homework Problem Set 3
Submission deadline: Nov. 16, 23h59
1. Following Problem 3. of Homework2, and the large-scale fading is log-normally distributed
with µp,dB = 0 dB and σp,dB = 7 dB. (Hint: when having large-scale fading, the total power
reduction can be viewed as P L(d) + X in dB-scale, where X is a normal distribution.)
(a) (5pt) How large is the outage probability if only the large-scale fading is considered (ignore
the small-scale fading)?
(b) (5pt) How large is the outage probability if both the small-scale and large-scale fading
are considered?
2. Following Problem 4. of Homework2, and now the narrow-band link is subjected to both
large-scale log-normal fading, with µp,dB = −1 dB and σp,dB = 5 dB, and small-scale fading.
(a) (5pt) We aim to achieve 97% boundary coverage for the system, meaning it should operate
effectively at 97% of the locations at the maximum distance dmax . Calculate the fading
margin needed as a consequence of large-scale fading to fulfill this requirement.
(b) (10pt) By adding two fading margins obtained in the case with only consider small-scale
fading and only consider large-scale fading, we can get a total fading margin, denoted as
Mtotal . Furthermore, with given µp,dB and σp,dB , we can also compute the fading margin
Mk,dB for a Suzuki distribution relative to the mean-dB value of the shadowing so that
the outage probability is smaller than 3%. Then, what is the difference between these two
values |Mtotal − Mk,dB |? (Hint: Mtotal should be relative to mean power for calculation.)
3. Consider the following modulated signal formulation:

cos [(2πfc t + bm fsep t)] , 0 < t ≤ Ts
gm (t) =

0, otherwise
where bm ∈ {−3, −1, 1, 3}. Please answer the following questions.
(a) (3pt) What kind of modulation is this? Why?
(b) (2pt) How many bits can be transmitted for a modulated signal?
(1)
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(c) (5pt) Suppose we want orthogonality between modulated signal with different bm . Then,
what is the minimum feasible value for fsep ?
4. We want to compare between the 16-QAM and 16-APSK (refer to Fig.1). Suppose the distance
between the adjacent constellation points in 16-QAM is 2d. Then, please answer the following
questions.
Fig. 1
(a) (3pt) What is the average signal energy of 16-QAM in terms of d?
(b) (10pt) Suppose the energy per bit of both 16-QAM and 16-APSK is EB . Please give the
mathematical expressions for both 16-QAM and 16-APSK. Suppose we have an ideal power
amplifier, which modulation can have a better resistance to the Gaussian noise (i.e. better
bit-error-rate)? Why?
(c) (4pt) What is the corresponding peak-to-average-power ratio (PAPR) for 16-QAM and 16APSK?
(d) (4pt) Suppose the power amplifier is non-ideal and we want to operate at the linear region
for all symbols. Please comment the impact of the non-ideal power amplifier on 16-QAM
and 16-APSK.
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5. Consider three functions:

T

 1, 0 < t ≤
3 ;
x(t) =

T
−2,
<t≤T
3

T

−1, 0 < t ≤



3


T
2T
y(t) =
1,
<t≤

3
3




2T
−1,
<t≤T
3

6t
T

− 1, 0 < t ≤

T
3 ;
z(t) =

−3t
T

+ 2,
<t≤T
T
3
Please answer the following questions. (Note: To simplify the derivation process, you
(2)
(3)
(4)
can
consider the special case of T = 3.)
(a) (12pt) Use Gram–Schmidt orthogonalization to find a set of expansion functions ϕ1 (t),
ϕ2 (t), and ϕ3 (t) corresponding to x(t), y(t), and z(t).
(b) (9pt) Find points in the signal space diagram for x(t), y(t) and z(t), and then calculate
their correlation coefficient in pair, i.e., ρx,y , ρx,z and ρy,z ?
6. We consider a system using raised cosine pulses with α = 0.7 and T = 1. Suppose both user
1 and user 2 transmit their signals to the BS with power being 20 dBm. The pathloss model is
as follows: before the breaking point distance, i.e., d < dbreak , it follows the Friis’ law; after
the breaking point distance, i.e., d > dbreak , it follows the pathloss factor n = 3.2. The breaking
point distance is dbreak = 5 m. Please answer the following questions.
GN (f, α, T ) =







1−α
2T
1−α
1+α
< |f | <
2T
2T
1, 0 < |f | ≤
T
π
0.5(1
−
sin(
(|2πf
|
−
))),

2α
T





0, otherwise
(5)
(a) (8pt) Suppose that both user 1 and user 2 are 250 m away from the BS. What is the
minimum separation between the carrier frequencies of user 1 and user 2 such that the SIR
is at least 30 dB.
(b) (4pt) Following (a), now, user 2 moves toward the BS such that the distance between user
2 and the BS is 150 m. Then, what is the SIR of user 1 and user 2, respectively?
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(c) (8pt) Following (b), we now want the SIR of both user 1 and user 2 to be at least 30 dB.
Then, what is the minimum separation between the carrier frequencies of two users?
(d) (3pt) Following (c), instead of changing the separation between the carrier frequencies, what
we can do to let the SIR of both user 1 and user 2 to be at least 30 dB?