Digital Signal Processing Reference
In-Depth Information
Problems
6.1.
Sketch a plot of the capacity
C
in Eq. (6.6) as a function of
B
for 0
≤ B ≤
∞
,forfixed
p
0
/N
0
.
6.2.
Note that
p
0
/
N
0
B
)
is dimensionless. For a channel with bandwidth
B
= 100 Hz what is the
capacity if the ratio
p
0
/
N
0
in Eq. (6.6) should be specified in hertz so that
p
0
/
(
N
0
is 100 Hz? If the bandwidth is increased to 1
MHz (with
p
0
/
N
0
at 100 Hz) then what is the capacity?
6.3.
In Problem 6.2 is it possible to increase the bandwidth so that the capacity
is twice its value at the bandwidth of 1 MHz?
If so, what is the new
bandwidth?
6.4.
For
B
=1MHzand
p
0
/
N
0
= 100 Hz, let
C
1
be the value of the capacity
calculated from Eq. (6.6).
1. Suppose we want to increase the power
p
0
(with all other quantities
fixed) so that the capacity is doubled. What is the new ratio
p
0
/
N
0
?
2. Suppose we want to increase the power
p
0
(with all other quantities
fixed) so that the capacity is ten times higher than
C
1
.
What is the
new ratio
p
0
/
N
0
?
6.5.
Does the capacity
in Eq. (6.6) have any convexity property? That is, is
it a convex (or a concave) function of
B
? Justify your answer. (A review
of convex functions can be found in Sec. 21.2.)
C
P
e
=10
−
7
the
SNR gap from capacity is 9.76 dB. What is the SNR gap with
P
e
=10
−
5
?
How about with
P
e
=10
−
9
?
6.6.
In Sec. 6.3 we showed that for PAM with error probability
6.7.
Consider again Ex. 6.1, where we had
C
unsplit
>
1
.
Is there any
upper bound on this ratio? If not, find an example of
C
split
/
N
0
and
N
1
such that
this ratio is arbitrarily large, say 10
.
0
.
6.8.
Compute the mutual information (6.80) for the following channel:
H
=
10
1
,
assuming
C
xx
=
I
and
σ
q
=1
.
Plot this as a function of
>
0. How large
can you make this if you are free to choose
?
6.9.
Consider the problem of maximizing the mutual information (6.95) sub-
ject to the power constraint (6.96). Assume
P
=3
,σ
q
=1
,
and channel
singular values
σ
h,k
such that
σ
h,
0
=1
,σ
h,
1
=0
.
1
,
and
σ
h,
2
=0
.
01
.
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