Boston University
Department of Electrical and Computer Engineering
ENG EC515 Digital Communication (
Fall 2009
)
Problem set 7
Assigned:
Wed 3 Dec
Due:
Fri 11 Dec
Suggested reading:
Classnotes, handouts, and Ch. 7 textbook
O
ffi
ce hours:
M, W 4:00–6:00pm PHO438
Problem 7.1
Problem 7.1
Problem 7.2
Show that raised cosines (Page 167 of Handout) are Nyquist pulses.
Problem 7.3
A real BPSK waveform
x
BPS K
(
t
)
=
∑
n

1
l
=
0
a
l
s
(
t

lT
),
a
l
∈ {
A
,
+
A
}
is transmitted over a
channel that introduces ISI described by the impulseresponse
h
(
t
)
=
1
+
0
.
25
δ
(
t

T
). The received signal
is described by
y
(
t
)
=
(
x
BPS K
?
h
)(
t
)
+
Z
(
t
) where
Z
(
t
) is a real zeromean AWGN processes with psd
σ
2
=
N
0
/
2. The symbols
a
l
are all independent and equally likely.
For such a channel we know that the optimum receiver can base it’s decisions on samples of
u
(
t
) taken at
time instants that are multiples of
T
where
u
(
t
)
=
y
(
t
)
?
s
*
(

t
). By design, the pulse
s
(
t
)
?
s
*
(

t
) is Nyquist
for the symbol rate 1
/
T
. Hence, the equivalent discretetime channel model reduces to:
u
l
=
a
l
+
0
.
25
a
l

1
+
Z
l
where the
Z
l
’s are real, i.i.d., Normal random variables with zeromean and variance
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 Fall '09
 VenkateshSaligrama
 Probability theory, Nyquist rate, equivalent discretetime channel, discretetime channel model

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