EE 359: Wireless Communications  Fall 2005
SECOND EXAM
This exam is open book and notes, and calculators and the textbook are needed. Graded exams and solutions
will be available within the next week. Good luck!
Problem 1 (25 points): Short Answer.
(a) Name one advantage and one disadvantage of having a DSSS system whose spreading codes
have an autocorrelation approximately equal to a delta function?
(b) Consider a MIMO system with 10 transmit antennas and 6 receive antennas with matrix
H
describing the antenna gains from each transmit to each receive antenna. If the SVD of
H
has
4 nonzero singular values, then how many singular values are zero and what is the rank of
H
?
(c) Why are precoding and frequency equalization ineﬀective techniques for treating subcarrier
fading in OFDM systems?
(d) Consider an adaptive system with instantaneous SNR at the transmitter ˆ
γ
that is not necessarily
equal to the true SNR
γ
. Assume ˆ
γ
has the same distribution as
γ
,so
p
(ˆ
γ
)=
p
(
γ
). For the
variablerate variablepower MQAM scheme described in Section 9.3 of the text, suppose that
the transmit power and rate is adapted relative to ˆ
γ
instead of
γ
. Will the average transmitted
data rate and power be larger, smaller, or the same as under perfect channel estimates (ˆ
γ
=
γ
),
and why?
Problem 2 (25 points): MIMOOFDM Systems
MIMO techniques can be applied to systems with ISI as well as ﬂat fading. In this problem we illustrate
how to combine MIMO and OFDM. Consider a 2x2 MIMO system, where the channel from transmit
antenna
i
to receive antenna
j
,
i, j
=1
,
2, has discretetime FIR
h
ij
[
n
]=
α
ij
δ
[
n
]+
β
ij
δ
[
n

1], assuming
a sampling time equal the symbol time
T
s
. Assume a cyclic preﬁx of length equal to the delay spread is
appended on the symbol sequence transmitted over each antenna. Also assume that received symbols in a
given block that are aﬀected by ISI in a previous block are discarded.
(a) What is the delay spread associated with the MIMO channel, and how many OFDM subchannels
are needed to insure a ﬂatfading MIMO system?
(b) Let
x
1
[
i
]and
x
2
[
i
] denote, respectively, the symbol transmitted on the 1st and 2nd transmit
antenna at time
i
. Similarly, let
y
1
[
i
]and
y
2
[
i
] denote, respectively, the symbol received on the
1st and 2nd receive antenna at time
i
, and let
n
1
[
i
]and
n
2
[
i
] denote, respectively, the noise
sample on the 1st and 2nd receive antenna at time
i
. Assuming 5 OFDM subchannels, write
the inputoutput relationship of your MIMOOFDM system in matrix form
y
=
Hx
+
n
,where
y
[
i
]=[
y
1
[
i
]
y
1
[
i

1]
... y
1
[
i

4]
y
2
[
i
]
... y
2
[
i

4]]
T
and
x
[
i
]=[
x
1
[
i
]
x
1
[
i

1]
... x
1
[
i

4]
x
2
[
i
]
... x
2
[
i

4]]
T
,and
n
[
i
]=[
n
1
[
i
]
n
1
[
i

1]
... n
1
[
i

4]
n
2
[
i
]
... n
2
[
i

4]]
T
.No
t
e
that this is a generalization of the matrix representation of OFDM given in (12.24).
(c) Suppose the matrix
H
obtained in part (d) has nonzero singular values
σ
1
=1.5,
σ
2
=1.2, and
σ
3
=.8. Assuming
ρ
=10dBand
B
= 1 Hz, ﬁnd the capacity of the MIMOOFDM system
under optimal power and rate adaptation
and
under beamforming.