Digital Signal Processing Reference
In-Depth Information
q
(
t
)
x
(
t
)
x
(
t
)
0
F
(
f
)
0
H
(
f
)
channel
G
(
f
)
0
(a)
x
(
t
)
1
F
(
f
)
1
G
(
f
)
1
F
(
f
)
0
G
(
f
)
0
and
f
(b)
0
B
/2
−
B
/2
F
(
f
)
1
G
(
f
)
1
and
f
−
B
−
B
/2
0
B
/2
B
N
/ 2
lowpass noise
(c)
q
(
t
)
0
0
H
(
f
)
0
x
(
t
)
0
0
B
/2
−
B
/2
lowpass channel
q
(
t
)
1
bandpass noise
N
/ 2
1
H
(
f
)
1
bandpass channel
x
(
t
)
1
0
B
/2
B
Figure 6.5
. (a) A bank of filters at the transmitter and at the receiver, to split
the channel into subbands, (b) frequency responses of the filters, and (c) equivalent
parallel pair of channels.
Thus, when power is optimally allocated, most of it goes to subband 1,
and nearly nothing goes to subband 0. Note that there is a significant
improvement in capacity owing to band spitting, and further improvement
owing to power allocation.
6.6 Circularly symmetric complex random vectors
When random vectors arising in digital communication systems are complex,
they often satisfy a property called circular symmetry. We now describe this
property, which plays a crucial role in the derivation of channel capacity for the
complex case. Let
x
=
x
re
+
j
x
im
(6
.
31)
be a complex random vector with autocorrelation
R
xx
=
E
[
xx
†
]=
P
+
j
Q
(6
.
32)
Search WWH ::
Custom Search