Digital Signal Processing Reference
In-Depth Information
Problems
2.1.
For a 2-bit QAM system suppose the SNR is such that the symbol error
probability is
P
e
has to be decreased to 10
−
6
,
then by how much should the SNR (in dB) be increased? Repeat for 6-bit
QAM.
P
e
=10
−
5
.
If the desired
2.2.
Repeat Problem 2.1 for 2-bit PAM and 6-bit PAM.
2.3.
Consider Fig. 2.7, which explains how decision errors arise due to additive
noise. In the figure the noise probability density function
f
E
(
e
) resembles
a Gaussian. Suppose instead that
f
E
(
e
) is a uniform pdf:
⎧
⎨
1
2
B
for
−
B<e<B
f
E
(
e
)=
⎩
0
otherwise.
1. What is the probability of error for an interior symbol, in terms of
A
and
B
?
2. Is it possible for this probability of error to be zero? If so, under what
conditions?
3. Find an example of a non-Gaussian noise pdf for which the probability
of error can never be zero no matter how small the (nonzero) noise
variance is.
2.4.
Consider a signal
x
(
t
)=
n
=
−∞
nT
)
.
This is of the same form
as the signals transmitted by digital communication systems (Sec. 2.1).
Assume that we know
f
(
t
)andweknowthat
x
(
t
) has this form.
s
(
n
)
f
(
t
−
1. Let
x
d
(
n
) be the uniformly sampled version
x
d
(
n
)=
x
(
nT
)
,
and let
f
d
(
n
)=
f
(
nT
)
.
Assume that
f
d
(
n
)has
z
-transform
F
d
(
z
) in some
appropriate region of convergence. Show that
s
(
n
) can be recovered
from the samples
x
d
(
n
)byusingafilteringoperationoftheform
∞
s
(
n
)=
g
(
k
)
x
d
(
n
−
k
)
.
k
=
−∞
How do you identify the impulse response
g
(
k
)?
2. What are the conditions on
F
d
(
z
)sothat
G
(
z
)=
k
g
(
k
)
z
−k
is
stable?
3. Given the samples
x
(
nT
)
,
explain how you can reconstruct the signal
x
(
t
) for all
t
?
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