Cooperative Multiplexing: Toward Higher Spectral Efficiency in Multi-antenna Relay Networks

Abstract

Previous work on cooperative communications has concentrated primarily on the diversity benefits of such techniques. This paper, instead, considers the multiplexing benefits of cooperative communications. First, a new interpretation on the fundamental tradeoff between the transmission rate and outage probability in multi-antenna relay networks is given. It follows that multiplexing gains can be obtained at any finite SNR, in full-duplex multi-antenna relay networks. Thus relaying can offer not only stronger link reliability, but also higher spectral efficiency. Specifically, the decode-and-forward protocol is applied and networks that have one source, one destination, and multiple relays are considered. A receive power gain at the relays, which captures the network large scale fading characteristics, is also considered. It is shown that this power gain can significantly affect the system diversity-multiplexing tradeoff for any finite SNR value. Several relaying protocols are proposed and are shown to offer nearly the same outage probability as if the transmit antennas at the source and the relay(s) were co-located, given certain SNR and receive power gains at the relays. Thus a higher multiplexing gain than that of the direct link can be obtained if the destination has more antennas than the source. Much of the analysis in the paper is valid for arbitrary channel fading statistics. These results point to a view of relay networks as a means for providing higher spectral efficiency, rather than only link reliability.

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arXiv:0903.2471v1 [cs.IT] 13 Mar 2009
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Cooperative Multiplexing: Toward Higher
Spectral Efficiency in Multi-antenna Relay
Networks
Yijia (Richard) Fan, Chao Wang, H. Vincent Poor, John S. Thompson
Abstract
Previous work on cooperative communications has concentrated primarily on the diversity benefits
of such techniques. This paper, instead, considers the multiplexing benefits of cooperative communica-
tions. First, a new interpretation on the fundamental tradeoff between the transmission rate and outage
probability in multi-antenna relay networks is given. It follows that multiplexing gains can be obtained
at any finite SNR, in full-duplex multi-antenna relay networks. Thus relaying can offer not only stronger
link reliability, but also higher spectral efficiency.
Specifically, the decode-and-forward protocol is applied and networks that have one source, one
destination, and multiple relays are considered. A receive power gain at the relays, which captures
the network large scale fading characteristics, is also considered. It is shown that this power gain can
significantly affect the system diversity-multiplexing tradeoff for any finite SNR value. Several relaying
protocols are proposed and are shown to offer nearly the same outage probability as if the transmit
antennas at the source and the relay(s) were co-located, given certain SNR and receive power gains at
the relays. Thus a higher multiplexing gain than that of the direct link can be obtained if the destination
has more antennas than the source.
Y. Fan and H. V. Poor are with the Department of Electrical Engineering, Princeton University, Princeton, NJ, 08544, USA.
(e-mail: yijiafan@princeton.edu, poor@princeton.edu)
C. Wang and J. S. Thompson are with the Institute for Digital Communications, University of Edinburgh, Edinburgh, EH9
3JL, UK. (e-mail: chao.wang@ed.ac.uk, john.thompson@ed.ac.uk)
This research was supported in part by the U.S. National Science Foundation under Grants ANI-03-38807 and CNS-06-25637.
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Much of the analysis in the paper is valid for arbitrary channel fading statistics. These results point
to a view of relay networks as a means for providing higher spectral efficiency, rather than only link
reliability.
I. INTRODUCTION
A. Background
Before describing the contributions of the paper, we first review some basic results from two
areas of research on which they are based: Multiple-input Multiple-output (MIMO) communi-
cations and cooperative communications.
1) MIMO communications: MIMO systems have been extensively studied during the last
decade. It is well known that a MIMO system has two advantages over single-input single-
output (SISO) systems, namely multiplexing gain and diversity gain. The diversity gain can
improve the link reliability, while the multiplexing gain enhances the spectral efficiency. It has
been revealed that the tradeoff between diversity and multiplexing gain is a key characteristic
of MIMO systems. In [3], the diversity-multiplexing tradeoff (DMT) has been defined assuming
that the signal-to-noise ratio (SNR) is infinitely high:
Definition 1 (Infinite-SNR DMT [3]): Consider a family of Gaussian codes Cη operating at
SNR η and having rates Rη. Assuming sufficiently long codewords, the multiplexing gain and
diversity order are defined as
r
∆= lim
η→∞
Rη
log2η, and d
∆= − lim
η→∞
log2Pout(Rη)
log2η
,
(1)
where Pout(Rη) is the outage probability corresponding to the transmission rate R.
Given any multiplexing gain r, the diversity gain d in this scenario describes the rate of decay
of the outage probability when the SNR tends to infinity. Graphically, d is approximately equal
to the negative slope of the log-log plot of the outage probability versus SNR when the SNR
tends to infinity. It is well known that, for Rayleigh fading, the infinite-SNR DMT for an
Mr×Mtpoint-to-point MIMO system (i.e., a system with Mttransmit antennas and Mrreceive
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antennas) is a piece-wise linear function connecting the points (k,(Mt− k)(Mr− k)), k =
0,··· ,min(Mt,Mr).
More recently, the analysis of DMT has been extended to the finite SNR scenario [4].
Specifically, the definition is given as follows.
Definition 2 (Finite-SNR DMT [4]): The finite-SNR multiplexing gain r and diversity gain d
are defined as
r =
R
log(1 + gη), and d(r,η) = −∂ lnPout(r,η)
∂ lnη
(2)
where g denotes an array gain achieved at low SNR, and Pout(r,η) is the outage probability at
rate R = r log2(1 + gη).
Here the diversity gain quantifies the negative slope of the log-log ratio of outage probability
to SNR for any value of SNR, instead of the asymptotic slope for infinite SNR. Obviously, the
finite SNR DMT approaches the infinite-SNR DMT when η → +∞, but the finite-SNR DMT
definition is clearly more general, and also more important in terms of practical implications,
as most of the SNR operating points encountered in wireless local area networks (WLANs) and
cellular networks are in the range of −10dB to 25dB.
Generally speaking, link reliability is usually the primary consideration in wireless commu-
nications, and thus diversity gain is often of paramount importance. However, once a sufficient
link reliability is established, i.e., once the diversity gain increases to a sufficient level, higher
spectral efficiency (i.e., multiplexing gain) then rises in importance. In this sense, multiplexing
gain is the more important advantage of MIMO systems in terms of enabling higher data rate
transmission.
2) Cooperative communications: Cooperative communications is a more recent concept that
combines the benefits of MIMO systems with relay technologies. In a relay network where
the nodes are equipped with either single or multiple antennas, cooperative communications
allow the nodes to help each other forward (relay) all messages to the destination, rather than
transmitting only their own messages. As the antennas at the transmitters in such network are
distributed, the network thus forms a “distributed MIMO” system. A question naturally arises:
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How does the performance of a distributed MIMO system compare with that of a point-to-point
MIMO system in terms of DMT?
Instead of looking at both multiplexing and diversity behavior simultaneously, most previous
work in this area emphasizes primarily the diversity benefits of relay networks (e.g., [1], [2]).
The term, cooperative diversity, thus has been quoted extensively. The reason for ignoring the
multiplexing gain of such networks is primarily because, unlike a point-to-point MIMO link, in
a relay network, a multiplexing gain higher than the direct link (i.e., the source to destination
link) is difficult to obtain due to the additional receive and transmission time slots the relays
require. Several schemes have been proposed to improve the multiplexing gain of half-duplex
relay networks [5], [15]–[17]. But still, no multiplexing gain higher than that of the direct link
can be obtained in terms of infinite-SNR DMT. In fact, it can be shown [6] that, for a full-duplex
network consisting of one source, one destination, the infinite-SNR DMT d(r) of the network,
at least under Rayleigh fading, is upper bounded by
d(r) ≤ min(dS,RD,dSR,D),
(3)
where dS,RDis the infinite-SNR DMT when the receive antennas at the relay(s) and the des-
tination are co-located, and dSR,Dis the infinite-SNR DMT when the transmit antennas at the
source and the relay(s) are co-located. Clearly, the maximal r of the network is always the same
as that of the direct link.1
This result is not encouraging. Since the infinite-SNR case is an extreme case of the finite-
SNR case, one might conjecture that the same conclusion should apply for the finite-SNR case.
However, in this paper, we will show that this conjecture is not true.
B. Contributions of the paper
In this paper, a new interpretation on the fundamental tradeoff between the transmission rate
and outage probability in multi-antenna relay networks is given. It follows that, under any
1Note that the same conclusion is also conjectured in [6] even if the nodes are clustered, i.e., when the channel between either
the source and the relay or the relay and the destination is an additive white Gaussian noise (AWGN) channel.
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fading statistics, a higher multiplexing gain than that of the direct link may be obtained by
using relaying, for any finite SNR value, in multi-antenna relay networks. Thus, cooperative
communications can offer not only stronger link reliability, but also higher spectral efficiency as
well.
Specifically, we consider a network that has one source, one destination, and multiple relays.
We apply the decode-and-forward protocol, where the relays decode, re-encode and forward the
message of the source to the destination. The signal received by the relay has average linear
power gain ϕ over that received by the destination, where ϕ satisfies 0 < ϕ ≤ +∞. This power
gain ϕ results from the different distances (path losses) between the nodes, and captures the
large scale fading characteristics2of the network. The gain ϕ is an important factor affecting
the multi-node network performance. However, its importance has not been considered fully in
previous work. In this paper, we show that the DMT of the network is, in fact, significantly
affected by the value of ϕ. We summarize the content of the paper as follows.
• For single relay networks, a full-duplex relaying scheme in which the relay uses a fixed set
of antennas to re-transmit the message is introduced. Also, an adaptive relaying protocol
is proposed based on the principle that the relay transmits only if it decodes the message
correctly. This protocol is shown to offer considerable insight into the fundamentals of
system outage and DMT performance. Specifically, it is shown that when the SNR is finite,
the network DMT performance is mainly determined by the relationship between SNR and
network fading characteristics including both ϕ (i.e., large scale fading) and small scale
fading coefficients (e.g., Rayleigh fading). The network can offer an outage probability
PSR,D as if the antennas at the source and the relay are co-located, given a sufficiently
large ϕ. Thus a multiplexing gain higher than that of the direct transmission can be obtained
once the destination is equipped with more antennas than the source. It is shown that the
higher the SNR becomes, the less likely that PSR,Dcan be obtained for a fixed value of
ϕ. However, when the SNR is below a certain threshold, which is determined primarily
2We will also discuss the impact of lognormal shadowing effects in Section VIII.
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