EE544 Homework Assignment # 4
1. (Proakis problem 2.18) Use the Chernoff bound to show that
Q
(
x
)
≤
e

x
2
/
2
where
Q
(
x
) is defined
by
Q
(
x
) =
1
√
2
π
Z
∞
x
e

t
2
/
2
dt,
x
≥
0
2. A real bandpass (BP) signal
σ
(
t
) is filtered by a real linear timeinvariant (LTI) BP filter
g
(
t
).
Let the output signal, which is
real
, be
r
(
t
). Define
σ
l
(
t
),
g
l
(
t
), and
r
l
(
t
) be the complex envelopes
of
σ
(
t
),
g
(
t
), and
r
(
t
), respectively, with respect to the angular frequency
ω
c
1. Also define the
real and imaginary components of
σ
l
(
t
),
g
l
(
t
), and
r
l
(
t
) in the following way:
σ
l
(
t
)
=
σ
I
(
t
) +
j σ
Q
(
t
)
,
g
l
(
t
)
=
g
I
(
t
) +
j g
Q
(
t
)
,
r
l
(
t
)
=
r
I
(
t
) +
j r
Q
(
t
)
.
(a) Draw the system block diagram with the labels
σ
(
t
)
, g
(
t
) and
r
(
t
).
(b) Write the real signals
σ
(
t
),
g
(
t
), and
r
(
t
) in terms of
σ
l
(
t
),
g
l
(
t
),
r
l
(
t
), exp(
jω
c
t
), and the
operator Re
{·}
.
(c) Write the convolution operation
r
(
t
) =
σ
(
t
)
*
g
(
t
) in terms of real signals
σ
I
(
t
)
, σ
Q
(
t
)
, g
I
(
t
),
g
Q
(
t
)
,
cos(
ω
c
t
) and sin(
ω
c
t
).
(d) Write the convolution operation
r
l
(
t
) =
σ
l
(
t
)
*
1
2
g
l
(
t
) of two complex signals in terms of their
real and imaginary components.
(e) Based on (c) and (d), show that
r
(
t
)
=
Re
‰
σ
l
(
t
)
*
1
2
g
l
(
t
)
¶
exp(
jω
c
t
)
,
Re
{
r
l
(
t
) exp(
jω
c
t
)
}
.
(f) Draw the complex equivalent system block diagram with the labels
σ
l
(
t
)
,
1
2
g
l
(
t
) and
r
l
(
t
).
Note that, for the impulse response of the complex equivalent system, we should use onehalf
of the complex envelope of the real impulse response.
Now, suppose
σ
(
t
)
=
v
(
t
) cos(
ω
c
t
+
θ
) and
g
(
t
)
=
2
h
(
t
) cos(
ω
c
t
+
φ
)
,
where
v
(
t
) and
h
(
t
) are real lowpass signals.
1