6.02 Fall 2012, Quiz 2
Page 1 of 17
Name:
DEPARTMENT OF EECS
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
6.02: Digital Communication Systems, Fall 2012
Quiz II
November 13, 2012
“
×
” your section
Section
Time
Recitation Instructor
TA
D
1
1011
Victor Zue
Ruben Madrigal
D
2
1112
Victor Zue
Cassandra Xia
D
3
121
Jacob White
Kyu Seob Kim
D
4
12
Yury Polyanskiy
ShaoLun Huang
D
5
23
Yury Polyanskiy
Rui Hu
D
6
34
Jacob White
Eduardo Sverdlin Lisker
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(mostly short!) in VI sections, and
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in this quiz booklet.
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120 minutes
.
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6.02 Fall 2012, Quiz 2
Page 2 of 17
Bring in The Noise
A particular digital communication scheme sends signals over a channel that behaves essentially as a linear,
timeinvariant (LTI) system at baseband. The binary source at the transmitter generates the symbols “0” and
“1” with equal probability. The characteristics of the channel and the timing at the receiver are such that the
receiver is able to obtain
M
≥
1
good samples in each bit slot. The good samples in a bit slot corresponding
to a “0” are all of the form
y
[
n
] =
V
0
+
w
[
n
]
while in a bit slot corresponding to a “1” they are all of the form
y
[
n
] =
V
1
+
w
[
n
]
.
Here
w
[
n
]
denotes a noise term that is a Gaussian random variable of mean 0 and variance
σ
2
, and is
independent across samples, i.e., the samples at the receiver are perturbed by additive white Gaussian noise.
For onoff signaling,
V
0
= 0
and
V
1
=
V
. The linearity of the channel then ensures that for bipolar signaling
V
0
=
−
V
and
V
1
=
V
.
The receiver decides on whether a “0” or “1” was sent by comparing the
average
value of the
M
samples in
any bit slot with a threshold voltage
V
th
. If the average is below
V
th
, the receiver decides a “0” was sent; if
the average is above
V
th
, the receiver decides a “1” was sent.
Suppose we are using
bipolar
signaling and
M
= 1
, i.e., only a single sample is taken in each bit slot.
Then you’ve seen that the probability of the receiver making an error in deciding whether a “0” or “1” was
sent in any particular bit slot, i.e., the bit error rate (BER), is minimized by choosing
V
th
= 0
, with the
corresponding BER being given by
V
BER
bipol
= 0
.
5
erfc
√
.
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 Fall '12
 GeorgeVerghese
 Frequency, unit sample response

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