Wireless Network Design: Optimization Models and Solution ...

324 Nitin Salodkar and Abhay Karandikar
In this chapter, we consider ergodic capacity (also termed as throughput capacity
or expected capacity in the literature). This notion of capacity measures the rates
achievable in the long run averaged over the channel variations. Here, the channel is
assumed to vary sufficiently fast, yet the variations are slow enough such that ‘reasonably’
long codes can be transmitted. We first derive an expression for the ergodic
capacity of a single user point-to-point link under an average power constraint and
then extend the notion to multiuser scenario.
14.2.1 Point-to-Point Capacity with Perfect Transmitter CSI
Consider a single user wireless channel depicted in Figure 14.1. We assume transmitter
has perfect knowledge of CSI. Let Pn denote the transmission power in slot n.
Let X1 = x1,...,XM = xM be a given realization of channel states. We assume that
the transmitter has an average power constraint of ¯P.
Bits
Fig. 14.1 Point-to-point transmission model
X
Feedback path with zero delay
The restriction on the average power expenditure makes the problem of achieving
capacity to be a power allocation problem. Specifically, the problem is to determine
the transmission power in each slot that maximizes the information rate (and
hence achieves capacity), while keeping the average power expenditure below the
prescribed limit. The problem can be stated as:
max
P 1,...,PM
1
M
M �
∑ log
n=1
1 + PnXn
N0
�
, (14.3)
subject to,
M 1
M ∑ Pn = ¯P. (14.4)
n=1
This is a constrained optimization problem and can be solved using standard Lagrangian
relaxation [8]. Let x + denote max(0,x). A solution to the optimization
problem stated in (14.3) and (14.4) is a policy that determines the optimal power in
nth slot to be: