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Capacity of Fading Broadcast Channels with Transmitter Ordering CSI
Capacity of Fading Broadcast Channels with
Transmitter Ordering CSI
Rajiv Agarwal, Yifan Liang, Andrea Goldsmith
Dept. of Electrical Engineering
Stanford University
{rajivag, y, andrea}@wsl.stanford.edu
Abstract
We derive the ergodic capacity region for a fading broadcast channel when
the transmitter knows only the channel ordering information of the users and the
receiver knows its transmitter-receiver channel gain and the ordering information
at any time instant. We show that superposition coding based on the ordering
information achieves the ergodic capacity region. Moreover, we prove that as the
number of users goes to innity, the capacity region with just transmitter ordering
information is the same as the capacity region with perfect channel knowledge at
the transmitter. Numerical results are provided for a Rayleigh fading channel.
1
Introduction
The time-varying nature of the underlying channel is one of the most signicant challenges
in designing wireless communication systems. Dynamic allocation of power, bandwidth,
and rate can result in signicant performance improvements over constant resource alloca-
tion strategies. Practical systems are beginning to incorporate more and more elements of
adaptation in order to eectively utilize the time-varying channels found in most wireless
systems. One important scenario for multiuser wireless communications is a broadcast
channel where a single transmitter sends independent information to many receivers,
for example in the downlink of a cellular system. The capacity of a fading broadcast
channel with perfect channel state information (CSI) at the transmitter along with the
capacity-achieving transmission strategy of adaptive resource allocation was found in [1].
However, knowing the channel perfectly at the transmitter requires 1) perfect channel
measurements at the receiver and 2) a perfect feedback link to send this channel infor-
mation to the transmitter. In certain settings, one or both of these requirements may be
impractical to implement. Thus, in this work, we investigate the broadcast channel with
only partial CSI at the transmitter.
Specically, in this work we assume that at any time instant the transmitter knows the
ordering of the users in terms of their instantaneous channel gains, but not the channel
gains themselves. Finding the capacity region for this case, besides its theoretical impor-
tance, has much practical signicance as well due to the amount of feedback information
required to provide the transmitter with perfect CSI. In particular, if the distribution of
the channel gain is such that |||| is large with non-negligible probability
1
(or innite for
1
The notation ||x|| for a random variable x denotes the number of possible dierent values that x
can take. continuous fading distributions), then in order to provide perfect CSI at the transmitter
the number of bits that has to be fed back for each user can be very large (or innite).
In this case, feeding back ordering information requires only log
2
(K!) K log
2
(K)
bits, where K is the number of users. Based on channel ordering information, we derive
the ergodic capacity region for the fading BC in Section 3. Obviously these rates will
not be as high as those when the channel is known perfectly. However, we will see that
the capacity region with only channel ordering information is quite close to the capacity
region assuming perfect transmitter CSI. Thus, it is practical and near optimal to feed
back ordering information to the transmitter.
The case of limited feedback has been investigated in other works. In [2], the authors
studied the diversity performance of an MIMO system under limited feedback. In an
unpublished work [3], Chakraborty investigated the capacity region for fading broadcast
channels with partial transmitter CSI, including the case of transmitter channel ordering
information. A comprehensive reference on the eect of imperfect CSI on channel capacity
can be found in [4].
In this paper, we nd the ergodic capacity region for the fading broadcast channel
when the transmitter has the ordering information only and the receivers know their local
channel (dened as the transmitter-receiver gain) and the ordering information. We also
prove that for i.i.d. continuously fading channels, ordering information is asymptotically
optimal, i.e. as the number of users in the system grows, the capacity region with trans-
mitter ordering information only, equals that when the transmitter knows the channel
perfectly.
The rest of the paper is organized as follows. In Section 2, we describe the channel
model for the fading broadcast channel under consideration. In Section 3, we derive the
ergodic capacity region with ordering information only at the transmitter for the two
user case. In Section 4, we prove the asymptotic optimality of the ordering information
in the limit of a large number of users. In Section 5, we give numerical results showing
the capacity improvement over no channel knowledge by knowing just one bit of ordering
information
2
. Concluding remarks and ideas for future work are given in 6.
2
The Fading Broadcast Channel
We consider a fading broadcast channel with a single transmitter communicating inde-
pendent information to 2 users. All the results can be easily extended to more than two
users. The signal source X[i] is composed of 2 independent information sources, where
i represents the time index. We assume a signal bandwidth of B. The time-varying
channel gain of the path to user j is denoted by h
j
[i], which remains constant during the
ith channel use. Each receiver has additive Gaussian noise with power spectrum density
N
0
. The received signal of user j is thus
Y
j
[i] = h
j
[i]X[i] + w
j
[i],
j = 1, 2
(1)
where w
j
[i] is white Gaussian noise with power N
0
B. For simplicity, we assume B = 1
Hz throughout this paper. We also dene the channel power gain
j
[i] = |h
j
[i]|
2
, where
the distribution of h
j
[i] induces a distribution on
j
[i].
We are interested in the case of imperfect channel knowledge. Specically, the fading
state vector [i] = (
1
[i],
2
[i]) is unknown to the transmitter, however, the ordering is
2
For two users the number of bits to convey the ordering information is log
2
(2!) = 1. known, i.e. at time instant i the transmitter knows whether
1
[i] >
2
[i] or
1
[i]
2
[i].
We denote this information at the transmitter as e[i]. For the two user case, e[i] can take
one of 2 possible values for any i i.e.
e[i] =
0 , if
1
[i] >
2
[i]
1 , if
1
[i]
2
[i].
(2)
Since e[i] is known for all i, the transmitter can therefore vary the power of the
signal transmitted to each user P
j
[i] as a function of e[i] subject to an average power
constraint P . The two receivers have knowledge of their respective
j
[i]s and also the
ordering information. Actually, the transmitter adapts its transmission scheme based
on the available CSI, and this adaptation strategy is made known to the receiver. The
receiver knows the transmitter CSI since this CSI is a function of the receiver CSI.
3
Ergodic Capacity Region
The ergodic capacity region is dened as the set of all long-term average rates achievable
in a fading channel with arbitrarily small probability of error. In this section, we derive
the ergodic capacity region assuming a continuous fading process
3
.
For the two user case, with the channel ordering information at the transmitter, the
overall channel can be decomposed into two component channels (e[i] = 0 or 1) at any
time instant i. We assume that the channel associated with each user is stationary and
ergodic so we can drop the time index i. Each component channel is a degraded BC,
whose capacity region is known [6]. In the rst component channel, the information
received by user 2 is always a degraded version of that received by user 1, while in the
second component channel the relationship is reversed.
This can be considered as a generalization of the probabilistic broadcast channel de-
ned in [1]. In this channel model each component channel is a degraded AWGN BC.
However in our model, each component channel although degraded, still experiences
fading. The transmitter will generate 2 codebooks corresponding to each component
channel, choose one of them according to the ordering information at any time instant
and rely on the ergodicity of the channel to achieve the long-term average rate. Unlike
the transmission scheme for perfect CSI that uses a dierent codebook for each joint
fading state [1], this transmission scheme using only 2 codebooks is much simpler.
Although the component channels are degraded, the capacity is known only in terms
of a mutual information expression with a maximization over the input distribution
because the component channels experience fading and the optimal input distribution
depends on the underlying fading process. The optimal input distribution is not known
in closed-form even for simple fading processes like i.i.d. Rayleigh fading for both the
users with dierent means [7]. Hence, in the discussion that follows, we will restrict to
Gaussian input and derive an achievable region in closed form to get some insights. The
encoding scheme used to prove achievability, as described later in this Section, leads to
the ergodic capacity region when Gau