Digital Signal Processing Reference
In-Depth Information
reconstruction
error
bits power
q
(
n
)
0
s
(
n
)
0
x
(
n
)
0
x
(
n
)
1
y
(
n
)
0
y
(
n
)
1
s
(
n
) +
e
(
n
)
0
=
s
(
n
)
0
s
(
n
)
1
0
(
b
,
P
)
0 0
(
b
,
P
)
1
s
(
n
)
1
s
(
n
) +
e
(
n
)
1
1
=
F
H
G
s
(
n
)
M
−
1
q
(
n
)
J
−
1
s
(
n
)
M
−
1
x
(
n
)
P
y
(
n
)
J
−
1
−
1
s
(
n
)
s
(
n
)
channel
noise
equalizer
precoder
channel
Figure 14.2
. The transceiver with all signals shown in component form.
Note here that the approximate nature of Eq. (14.4) comes about for a number
of reasons:
1.
Low error assumption
. As we saw in Sec. 2.3.2, the probability of error for
the QAM case is actually given by
P
PAM
−P
PAM
,
2
where
P
PAM
is the probability of error for an appropriately defined PAM
constellation. Equation (14.2) makes the approximation that
0
.
This is certainly an excellent approximation for any practically acceptable
value of
P
PAM
.
P
PAM
≈
2
−b
k
/
2
) has been replaced
with unity. This approximation is valid only when
b
k
is reasonably large.
Under this assumption, the further approximation (2
b
k
2.
High bit rate assumption
.
The quantity (1
−
1) = 2
b
k
−
is also
reasonable, so that Eq. (14.4) is often replaced with
2
Q
−
1
P
e
(
k
)
4
σ
e
k
2
b
k
3
P
k
≈
.
(14
.
5)
This equation expresses the average power required for the
k
th user to operate
at the probability of error
P
e
(
k
)witha
b
k
-bit QAM constellation, when the
noise variance at the
k
th output of
G
(input of
k
th detector) is
σ
e
k
.Sincethe
error probability
P
e
(
k
) is often regarded as the
quality of service
(QoS), the
specification of the set of numbers
{P
e
(
k
)
}
is equivalent to the specification of
QoS requirements.
DMT versus OFDM systems.
In traditional OFDM systems the signals
s
k
(
n
)are
obtained from a scalar symbol stream
s
(
n
) by the process of
blocking
.A lthe
signals
s
k
(
n
) then have identical power, that is,
P
k
=
P
0
for all
k.
In such systems
the error
P
e
(
k
) is fixed for all
k,
and no bit allocation is performed. While OFDM
systems do not have bit allocation, discrete multitone, or
DMT
, systems used in
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