Databases Reference
In-Depth Information
(a)
Generate a random binary sequence with
P
(
0
)
=
0
.
8, and compare your simulation
results with the analytical results.
(b)
Generate a binary first-order Markov sequence with
P
(
|
)
=
.
(
|
)
=
0
0
0
9, and
P
1
1
.
9. Encode it using your program. Discuss and comment on your results.
4.
Show that
0
∞
∞
H
(
X
d
|
Y
d
)
=−
f
X
|
Y
(
x
i
|
y
j
)
f
Y
(
y
j
)
log
f
X
|
Y
(
x
i
|
y
j
)
−
log
(99)
j
=−∞
i
=−∞
5.
For two random variables
X
and
Y
, show that
H
(
X
|
Y
)
H
(
X
)
with equality if
X
is independent of
Y
.
Hint: E
(Jensen's inequality).
6.
Given two random variables
X
and
Y
, show that
I
[
log
(
f
(
x
))
]
log
{
E
[
f
(
x
)
]}
(
X
;
Y
)
=
I
(
Y
;
X
)
.
7.
For a binary source with
P
(
0
)
=
p
,
P
(
X
=
0
|
Y
=
1
)
=
P
(
X
=
1
|
Y
=
0
)
=
D
, and
distortion measure
d
(
x
i
,
y
j
)
=
x
i
⊕
y
j
show that
I
(
X
;
Y
)
=
H
b
(
p
)
−
H
b
(
D
)
(100)
2
8.
Find the autocorrelation function in terms of the model coefficients and
σ
for
(a)
an AR(1) process,
(b)
an MA(1) process, and
(c)
an AR(2) process.
