a straight line connecting the points C
1
(P
1
) and C
2
(P
2
), where
C
i
(P
i
) =
max
{
ki
:
k
k
ki
=P
i
} k
k
C
k
(
k
i
).
(14.26)
Fixed frequency division partitions the total bandwidth B into nonoverlapping segments B
1
and B
2
,
which are then allocated to the respective transmitters. Since the bandwidths are separate, the users
are independent, and they can allocate their time-varying power independently, subject only to the total
power constraint P
i
. The fixed frequency division rate region (R
1
, R
2
) thus satisfies
R
i
max
ki
k
C
k
(
k
i
, B
i
),
(14.27)
where
C
k
(
k
i
, B
i
) = B
i
log 1 +
k
i
nB
i
,
(14.28)
and the
n
i
s satisfy the power constraint
k
k
k
i
= P
i
.
It can be shown [?] that fixed frequency division dominates time division. and superposition coding
dominates both. Thus, as for the broadcast channel, the relative performance of the different spectrum
sharing techniques is the same in AWGN and in fading, although the shape of the capacity region is
different.
14.5
Random Access
Given a channelization scheme, each user can be assigned a different channel for some period of time.
However, most data users do not require continuous transmission, so dedicated channel assignment can
be extremely inefficient. Moreover, most systems have many more total users (active plus idle users) than
channels, so at any given time channels can only be allocated to users that need them. Random access
strategies are used is such systems to assign channels to the active users.
Random access techniques were pioneered by Abramson with the Aloha protocol [7]. In the ALOHA
random access protocol, packets are buffered at each terminal and transmitted over a common channel
to a common hub or base station. In unslotted, or "pure" Aloha, no control is imposed on the channel to
synchronize transmission from the various users, and therefore the start times of packets from different
users in the network can be modeled as a Poisson point process. Should two users "collide," they both
wait a random amount of time before retransmitting. The goal, of course, is to prevent the users from
colliding once again when they retransmit. Under these circumstances packets from different users will
be transmitted with a high probability of success if there is a light to moderate amount of traffic on
the network. As the traffic on the network increases the probability of a collision between packets from
different users increases.
In slotted Aloha, the users are further constrained by a requirement that they only begin transmitting
at the start of a time slot. The use of such time slots increases the maximum possible throughput of
the channel [8], but also introduces the need for synchronization of all nodes in the network, which can
entail significant overhead. Even in a slotted system, collisions occur whenever two or more users attempt
transmission in the same slot. Error control coding can result in correct detection of a packet even after
a collision, but if the error correction is insufficient then the packet must be retransmitted, resulting in
a complete waste of the energy consumed in the original transmission. A study on design optimization
between error correction and retransmission is described in [9].
311