29-06-2013, 02:51 PM
Capacity of wireless channels
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INTRODUCTION
In the previous two chapters, we studied specific techniques for communication
over wireless channels. In particular, Chapter 3 is centered on the
point-to-point communication scenario and there the focus is on diversity as
a way to mitigate the adverse effect of fading. Chapter 4 looks at cellular
wireless networks as a whole and introduces several multiple access and
interference management techniques.
The present chapter takes a more fundamental look at the problem of
communication over wireless fading channels. We ask: what is the optimal
performance achievable on a given channel and what are the techniques to
achieve such optimal performance? We focus on the point-to-point scenario in
this chapter and defer the multiuser case until Chapter 6. The material covered
in this chapter lays down the theoretical basis of the modern development in
wireless communication to be covered in the rest of the book.
The framework for studying performance limits in communication is information
theory. The basic measure of performance is the capacity of a channel:
the maximum rate of communication for which arbitrarily small error
probability can be achieved. Section 5.1 starts with the important example
of the AWGN (additive white Gaussian noise) channel and introduces
the notion of capacity through a heuristic argument. The AWGN channel
is then used as a building block to study the capacity of wireless
fading channels. Unlike the AWGN channel, there is no single definition
of capacity for fading channels that is applicable in all scenarios. Several
notions of capacity are developed, and together they form a systematic
study of performance limits of fading channels. The various capacity
measures allow us to see clearly the different types of resources available
in fading channels: power, diversity and degrees of freedom. We will see
how the diversity techniques studied in Chapter 3 fit into this big picture.
More importantly, the capacity results suggest an alternative technique,
opportunistic communication, which will be explored further in the later
chapters.
AWGN channel capacity
Information theory was invented by Claude Shannon in 1948 to characterize
the limits of reliable communication. Before Shannon, it was widely believed
that the only way to achieve reliable communication over a noisy channel,
i.e., to make the error probability as small as desired, was to reduce the data
rate (by, say, repetition coding). Shannon showed the surprising result that
this belief is incorrect: by more intelligent coding of the information, one
can in fact communicate at a strictly positive rate but at the same time with
as small an error probability as desired. However, there is a maximal rate,
called the capacity of the channel, for which this can be done: if one attempts
to communicate at rates above the channel capacity, then it is impossible to
drive the error probability to zero.
Packing spheres
Geometrically, repetition coding puts all the codewords (the M levels) in just
one dimension (Figure 5.1 provides an illustration; here, all the codewords
are on the same line). On the other hand, the signal space has a large number
of dimensions N. We have already seen in Chapter 3 that this is a very
inefficient way of packing codewords. To communicate more efficiently, the
codewords should be spread in all the N dimensions
Linear time-invariant Gaussian channels
We give three examples of channels which are closely related to the simple
AWGN channel and whose capacities can be easily computed. Moreover,
optimal codes for these channels can be constructed directly from an optimal
code for the basic AWGN channel. These channels are time-invariant, known
to both the transmitter and the receiver, and they form a bridge to the fading
channels which will be studied in the next section.
Waterfilling versus channel inversion
The capacity of the fading channel with full CSI (by using the waterfilling
power allocation) should be interpreted as a long-term average rate of
flow of information, averaged over the fluctuations of the channel. While
the waterfilling strategy increases the long-term throughput of the system
by transmitting when the channel is good, an important issue is the delay
entailed. In this regard, it is interesting to contrast the waterfilling power allocation
strategy with the channel inversion strategy. Compared to waterfilling,
channel inversion is much less power-efficient, as a huge amount of power is
consumed to invert the channel when it is bad. On the other hand, the rate of
flow of information is now the same in all fading states, and so the associated
delay is independent of the time-scale of channel variations. Thus, one
can view the channel inversion strategy as a delay-limited power allocation
strategy. Given an average power constraint, the maximum achievable rate by
this strategy can be thought of as a delay-limited capacity. For applications
with very tight delay constraints, this delay-limited capacity may be a more
appropriate measure of performance than the waterfilling capacity.