IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 22, NO. 6, AUGUST 2004 1099
Fading Relay Channels: Performance Limits and
Space–Time Signal Design
Rohit U. Nabar, Member, IEEE, Helmut Bölcskei, Senior Member, IEEE, and
Felix W. Kneubühler, Student Member, IEEE
Abstract—Cooperative diversity is a transmission technique,
where multiple terminals pool their resources to form a virtual
antenna array that realizes spatial diversity gain in a distributed
fashion. In this paper, we examine the basic building block of
cooperative diversity systems, a simple fading relay channel where
the source, destination, and relay terminals are each equipped
with single antenna transceivers. We consider three different
time-division multiple-access-based cooperative protocols that
vary the degree of broadcasting and receive collision. The relay
terminal operates in either the amplify-and-forward (AF) or
decode-and-forward (DF) modes. For each protocol, we study the
ergodic and outage capacity behavior (assuming Gaussian code
books) under the AF and DF modes of relaying. We analyze the
spatial diversity performance of the various protocols and find
that full spatial diversity (second-order in this case) is achieved
by certain protocols provided that appropriate power control is
employed. Our analysis unifies previous results reported in the
literature and establishes the superiority (both from a capacity, as
well as a diversity point-of-view) of a new protocol proposed in
this paper. The second part of the paper is devoted to (distributed)
space–time code design for fading relay channels operating in the
AF mode. We show that the corresponding code design criteria
consist of the traditional rank and determinant criteria for the case
of colocated antennas, as well as appropriate power control rules.
Consequently space–time codes designed for the case of colocated
multiantenna channels can be used to realize cooperative diversity
provided that appropriate power control is employed.
I. INTRODUCTION
T
RANSMISSION over wireless channels suffers from
random fluctuations in signal level known as fading and
from cochannel interference. Diversity is a powerful technique
to mitigate fading and improve robustness to interference. In
classical diversity techniques, the data signal is conveyed to
the receiver over multiple (ideally) independently fading signal
paths (in time/frequency/space). Appropriate combining at the
receiver realizes diversity gain, thereby improving link relia-
bility. Spatial or antenna diversity techniques are particularly
attractive since they provide diversity gain without incurring an
expenditure of transmission time or bandwidth. Signal design
for multiantenna systems with colocated antennas (also known
as space–time coding) aimed at extracting spatial diversity gain
has been studied extensively in the literature [1]–[4].
A new way of realizing spatial diversity gain (in a distributed
fashion) has recently been introduced in [5]–[8] under the name
of
user cooperation diversity or cooperative diversity. Here,
Manuscript received July 15, 2003; revised February 1, 2004.
The authors are with the Communication Technology Laboratory, Swiss
Federal Institute of Technology (ETH), Zürich CH-8092, Switzerland (e-mail:
nabar@nari.ee.ethz.ch; boelcskei@nari.ee.ethz.ch; fwk@nari.ee.ethz.ch).
Digital Object Identifier 10.1109/JSAC.2004.830922
Fig. 1. Schematic of fading relay channel.
multiple terminals (sensors) in a network cooperate to form a
virtual antenna array realizing spatial diversity in a distributed
fashion. In [9], it has been demonstrated that uplink capacity can
be increased via user cooperation diversity. A variety of coop-
eration protocols for channels with a single relay terminal have
been studied and analyzed in [10]–[13]. In [14], it is shown that
for channels with multiple relays, cooperative diversity with ap-
propriately designed codes realizes full spatial diversity gain.
We note that many cooperative diversity schemes can be cast
into the framework of network coding [15]–[17]. Finally, we
refer to [18], [19] for fundamental results on nonfading relay
channels and to [20] and [21] for recent results on scaling laws
in large (relay) networks.
Contributions and relation to previous work. The first part of
this paper is devoted to the information-theoretic performance
limits of three different time-division multiple-access (TDMA)-
based transmission protocols for the single relay channel shown
in Fig. 1. The protocols we consider implement varying de-
grees of broadcasting and receive collision in the network.
1
In
each of the protocols, the relay terminal is allowed to either am-
plify-and-forward (AF) or decode-and-forward (DF) the signal
received from the source terminal. The second part of the paper
deals with (distributed) space–time code design for the fading
relay channel operating in the AF mode. Our detailed contribu-
tions in relation to previous work reported in [5]–[14] are sum-
marized as follows.
• We establish a unified framework for the results on
fading relay channels reported in [5]–[14], propose a
new protocol which is superior to existing protocols for
the single-relay fading channel, and put the performance
gains achievable in the distributed multiantenna case into
1
The degree of broadcasting is determined by the number of nodes listening
to a broadcasted message.
0733-8716/04$20.00 © 2004 IEEE
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1100 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 22, NO. 6, AUGUST 2004
the context of traditional multiple-input–multiple-output
(MIMO) gains.
• Assuming a Gaussian codebook, we derive closed form
expressions for the mutual information associated with
each of the protocols analyzed. Based on these results,
we compare the performance of the different protocols
in terms of achievable rates and establish the superiority
of protocols implementing maximum degrees of broad-
casting and receive collision.
• Based on an outage capacity analysis, we investigate the
diversity performance of the proposed protocols. In par-
ticular, we find that full spatial diversity is achieved by
certain protocols provided that appropriate power control
is employed.
• For an AF single-relay fading channel, we derive the
design criteria for (distributed) space–time codes. Our
results indicate that optimal space–time code design in
the single-relay case consists of satisfying the classical
rank and determinant criteria for colocated antennas [2],
as well as appropriate power control rules between the
terminals. It is shown that the power control rule arising
in the context of (distributed) space–time code design
is equivalent to the power control rule obtained through
an outage capacity analysis. Finally, we note that the
differences between [14] and the space–time code design
problem considered in this paper will be explained in
greater detail in Section V.
Organization of the paper. The rest of this paper is orga-
nized as follows. Section II describes AF and DF single-relay
channels and introduces the three different TDMA-based pro-
tocols, as well as the corresponding channel and signal models.
Sections III and IV provide an information-theoretic compar-
ison of the different protocols for the AF and DF cases, re-
spectively. Section V deals with (distributed) space–time signal
design for AF single-relay fading channels. We conclude in
Section VI.
Notation. The superscripts
, and stand for transposi-
tion, conjugate transposition and element-wise conjugation, re-
spectively. denotes the expectation operator, is the
identity matrix, stands for an all zeros matrix of appropriate
dimensions, and
is the Euclidean norm of the vector .A
circularly symmetric complex Gaussian random variable is a
random variable , where and
are independent identically distributed (i.i.d.) .
II. P
ROTOCOL DESCRIPTIONS AND CHANNEL
AND
SIGNAL MODELS
A. General Setup and Protocol Descriptions
Consider the fading relay channel shown in Fig. 1. Data is
to be transmitted from the source terminal S to the destination
terminal D with the assistance of the relay terminal R. All
terminals are equipped with single antenna transmitters and
receivers. Throughout this paper, we assume that a terminal
cannot transmit and receive simultaneously. The relay terminal
assists in communication with the destination terminal by either
amplifying-and-forwarding (AF) or decoding-and-forwarding
(DF) the received signal. In the AF mode, the relay terminal
TABLE I
T
HREE DIFFERENT
TDMA-BASED PROTOCOLS. S, R,
AND,DS
TAND FOR THE
SOURCE,RELAY,
AND DESTINATION
TERMINALS,R
ESPECTIVELY.
SIGNIFIES COMMUNICATION
BETWEEN TERMINALS
A AND
B
simply amplifies and retransmits the signal received from the
source terminal (the signal received at the relay terminal is
corrupted by fading and additive noise). No demodulation or
decoding of the received signal is performed in this case. In
the DF mode, the signal received from the source terminal is
demodulated and decoded before retransmission. The signal
models associated with the AF and DF transmission modes
are discussed in greater detail in Section II-B. We note that
in practice the AF mode when compared with the DF mode
requires significantly lower implementation complexity at the
relay terminal.
For each of the two forwarding modes (AF and DF) we shall
next describe three different cooperative protocols, which im-
plement varying degrees of broadcasting and receive collision in
the network. The degree of broadcasting is given by the number
of nodes simultaneously (i.e., in the same time slot) listening to
the source node (i.e., 2 if both R and D listen, 1 if only R or D
listens). Furthermore, receive collision is said to be maximum if
the destination node receives information simultaneously from
both S and R.
Protocol I: The source terminal communicates with the relay
and destination terminals during the first time slot. In the second
time slot, both the relay and source terminals communicate with
the destination terminal. This protocol realizes maximum de-
grees of broadcasting and receive collision.
Protocol II: In this protocol, the source terminal communi-
cates with the relay and destination terminals over the first time
slot. In the second time slot, only the relay terminal communi-
cates with the destination terminal. This protocol realizes a max-
imum degree of broadcasting and exhibits no receive collision.
Protocol III: The third protocol is identical to Protocol I apart
from the fact that the destination terminal chooses not to receive
the direct
2
signal during the first time slot for reasons
that will be motivated later in this section. This protocol does
not implement broadcasting but realizes receive collision.
The protocols are summarized in Table I. Protocols II and III
were first proposed in [8] and [22], respectively. Protocol I ap-
pears to be new. Note that while the signal conveyed to the relay
and destination terminals over the two time slots is the same
under Protocol II, Protocols I and III can potentially convey dif-
ferent signals to the relay and destination terminals. This fact
will be exploited in Section V in the context of (distributed)
space–time code design for fading relay channels.
Additional comments on the three protocols described above
are in order. The conditions and setup for Protocol I are self-
evident. Protocol II is logical in a scenario where the source ter-
minal engages in data reception from another terminal in the
network over the second time slot thereby rendering it unable
2
signifies the link between terminals A and B.
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NABAR et al.:FADINGRELAYCHANNELS:PERFORMANCELIMITSANDSPACE–TIME SIGNAL DESIGN 1101
to transmit. Similarly, for Protocol III the destination terminal
may be engaged in data transmission to another terminal during
the first time slot. Hence, the transmitted signal is received only
at the relay terminal and buffered for subsequent forwarding.
We assume that the source terminal expends the same amount
of power over the two time slots. In Protocol II, the source ter-
minal is silent over the second time slot, which implies that this
protocol is more efficient than Protocols I and III in terms of
battery life.
B. Channel and Signal Models
Throughout this paper, we assume frequency-flat fading, no
channel knowledge in the transmitters, perfect channel state in-
formation in the receivers and perfect synchronization. Perfect
channel state information in the receivers implies that the
channel is known to the relay terminal, while the individual
, and channels are known to the
destination terminal. Depending on the relaying mode (AF or
DF), knowledge of a specific individual channel gain may not
be required at the relay/destination terminal. Such a relaxation
of the assumption on channel knowledge will be highlighted in
the corresponding discussion. The assumption on synchroniza-
tion is most critical since synchronization becomes increasingly
challenging in larger networks. Protocols II and III are essen-
tially derivatives of Protocol I. We shall, therefore, first provide
the input-output relation for Protocol I for both the AF and DF
modes and then specialize to Protocols II and III.
Input–output relation for Protocol I in the AF mode. The
signals transmitted by the source terminal during the first and
second time slots are denoted as
and , respectively.
In the following, we consider symbol-by-symbol transmission
so that the time index
can be dropped and we simply write
and for the symbols transmitted in the first and second time
slots, respectively. We assume that
and
for . The data symbols may be chosen from a com-
plex-valued finite constellation such as quadrature amplitude
modulation (QAM) or from a Gaussian codebook. The signal
received at the destination terminal in the first time slot is given
by
(1)
where
is the average signal energy received at the desti-
nation terminal over one symbol period through the
link (having accounted for path loss and shadowing between
the source and destination terminals),
is the random,
3
com-
plex-valued, unit-power channel gain between source and desti-
nation terminals and
is additive white noise.
The signal received at the relay terminal during the first time slot
is given by
(2)
where
is the average signal energy over one symbol period
received at the relay terminal (having accounted for path loss
and shadowing between the source and relay terminals),
is
the random, complex-valued, unit-power channel gain between
3
Unless specified otherwise, we do not make any assumptions on the precise
distribution of the channel gains.
the source and relay terminals and is addi-
tive white noise. Note that in general
due to dif-
ferences in path loss and shadowing between the and
links.
The relay terminal normalizes the received signal by a factor
of
(so that the average energy is unity) and re-
transmits the signal during the second time slot. The destination
terminal receives a superposition of the relay transmission and
the source transmission during the second time slot according
to
(3)
where
is the average signal energy over one symbol pe-
riod received at the destination terminal through the
link (having accounted for path loss and shadowing between the
relay and destination terminals),
is the random, complex-
valued, unit-power channel gain between the relay and destina-
tion terminals and
is additive white noise.
We note that (3) contains the additional assumption of constant
and over the two time slots. Using
, we can rewrite (3) as
(4)
where the effective noise term
with
. Finally, we
assume that the receiver normalizes
by a factor
4
. This normalization does
not alter the signal-to-noise ratio (SNR) but simplifies the
ensuing presentation. The effective input–output relation for
Protocol I in the AF mode can now be summarized as
5
(5)
where
is the received signal vector, is
the effective 2
2 channel matrix given by
(6)
is the transmitted signal vector, and (when
conditioned on the channel
) is circularly symmetric com-
plex Gaussian noise with
and
. We shall make use of the fact that conditioned on
is Gaussian when calculating the mutual information for the
AF-based protocols in Section III.
Input–output relation for Protocol I in the DF mode. In the
DF mode, still assuming Protocol I, the signal received at the
destination terminal during the first time slot is identical to that
for the AF mode and is, hence, given by (1). The signal received
at the relay terminal is given by (2). Unlike the AF mode, the
relay terminal now demodulates and decodes the signal received
during the first time slot. Assuming that the signal is decoded
correctly and retransmitted, we obtain
4
Recall that we assumed perfect channel state information in the receiver.
5
The subscript 1 in reflects the fact that we are dealing with Protocol I.
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1102 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 22, NO. 6, AUGUST 2004
The effective input-output relation in the DF mode for Protocol
I can be summarized as
(7)
where
is the received signal vector, is the
effective 2
2 channel matrix given by
(8)
is the transmitted signal vector, and is additive
white Gaussian noise with
and .
From (8) it is clear that knowledge of
is not required at the
destination terminal in the DF mode.
Input-output relation for Protocols II and III. The corre-
sponding input–output relations for Protocols II and III in the
AF and DF modes may be derived from (5) and (7), respec-
tively. For Protocol II, the received signal for either forwarding
mode can be written as
(9)
where
denotes the first column of (chosen appropriately
from (6) or (8) depending on the transmission mode) and
(conditioned on in the AF mode) is the 2 1 additive
white complex Gaussian noise vector with
and
. Similarly, the signal received at the
destination terminal under Protocol III (the received signal is
scalar in this case) satisfies
(10)
where
is the second row of (chosen appropriately from (6)
or (8) depending on the transmission mode) and
(conditioned
on
in the AF mode) is scalar additive white noise.
Note that the different protocols convert the spatially dis-
tributed antenna system into effective single-input–multiple-
output (SIMO) (with Protocol II), multiple-input–single-output
(MISO) (with Protocol III), and MIMO (with Protocol I)
channels allowing the fundamental gains of multiple-antenna
systems such as diversity gain, array gain and interference
canceling gain to be exploited in a distributed fashion. We
emphasize that multiplexing gain (i.e., a linear increase in
achievable rate with the number of antennas in MIMO channels
[23]–[26]) is conspicuously absent, since time is expended
to create a virtual MIMO channel thereby negating any mul-
tiplexing gain. Further, note that the general structure and
statistics of the effective channels created by the different pro-
tocols are different from the classical i.i.d. circularly symmetric
complex Gaussian behavior widely used in the MIMO literature
[2], [24], [25].
III. I
NFORMATION-THEORETIC PERFORMANCE OF
PROTOCOLS IN THE AF MODE
In this section, we analyze the information-theoretic perfor-
mance of the three different AF-based protocols introduced in
Section II.
A. Mutual Information of AF-Based Protocols
In the following, we employ an ergodic block-fading channel
model (with independent blocks) and assume an i.i.d. Gaussian
codebook with covariance matrix
. More-
over, we assume that the destination terminal has perfect knowl-
edge of
, and . The mutual information for Proto-
cols I-III is obtained from (5), (9), and (10) as
6
bps/Hz
(11)
where , and the factor accounts
for the fact that information is conveyed to the destination ter-
minal over two time slots. If coding is performed over an infi-
nite number of independent channel realizations, the capacity of
each of the three protocols,
, is given by the
ergodic capacity
with the expectation carried
out with respect to the random channel. We emphasize that
is the capacity of the single-relay fading channel in conjunction
with Protocol j. If coding is performed only within one block
the Shannon capacity is zero. In this case, we resort to the
%
outage capacity [27], [28],
, defined as
% (12)
or equivalently, the rate
is guaranteed to be supported
for
% of the channel realizations. In the following,
we compare the different protocols in the AF mode both from a
capacity (ergodic and outage) and a diversity point-of-view.
B. Comparison From a Capacity Point-of-View
We begin with a comparison of Protocols I and II. Note that
, where is the mutual information be-
tween the vectors
and as defined in (5). Applying the chain
rule for mutual information [29], we have
where is the mutual information between and
, while is the conditional mutual informa-
tion between
and given . It is easy to verify that
, where is defined in (9). Noting
that
it then follows that
Since it follows immediately that ,
which shows that the achievable rate for Protocol I is higher than
that for Protocol II. We have, therefore, shown the intuitive re-
sult that the information rate is reduced if the source terminal
does not transmit to the destination terminal in the second time
slot. We note, however, that the superiority of Protocol I comes
at the cost of increased receiver complexity which is due to the
fact that in the second time slot the destination terminal receives
the superposition of the signals from source and relay termi-
nals whereas Protocol II is collision-free in the second time slot.
This result establishes the importance of receive collision for
6
Recall that the noise is conditionally (on the channel) Gaussian.
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NABAR et al.:FADINGRELAYCHANNELS:PERFORMANCELIMITSANDSPACE–TIME SIGNAL DESIGN 1103
achieving high throughput. In the context of multiaccess fading
channels, a similar observation has been made by Gallager in
[30].
We shall next compare Protocols II and III and start by noting
that for
and defined in (9) and (10), respectively, we have
(since ), and consequently, .
We can, therefore, summarize our results as follows:
(13)
establishing the superiority of Protocol I over the other two pro-
tocols in terms of achievable rate. We emphasize that the or-
dering in (13) applies to ergodic and outage capacities for all
three protocols.
7
Note that
implies . The
factor
may be viewed as a noise amplification factor. In order
to have
, we need , which
is the case if the
link is good (i.e., ) and
much stronger than the
link. Physically, this may occur
when the source terminal is located very close to the relay ter-
minal resulting in high SNR for the
link. On the other
hand, if
the noise amplification
will be substantial and the performance of Protocol III will de-
teriorate significantly compared with Protocol II. The reason for
this is intuitively clear. In Protocol II, the destination terminal
receives the source transmission over the first time slot without
any added amplified noise from the relay terminal, whereas in
Protocol III the information transmitted in the first time slot ar-
rives at the destination terminal through the noise-amplifying
relay link. Hence, Protocol II is expected to outperform Protocol
III if the noise amplification is large. Due to the assumption of
i.i.d. (across time slots) codebooks, the information transmitted
in the second time slot of Protocol III on the
link is inde-
pendent of the corrupted signal transmitted in the first time slot
and can, therefore, not compensate for the poor relay link.
Finally, we shall interpret the ordering in (13) in terms of
traditional MIMO gains. From (11), we can see that the price
to be paid for cooperative transmission over two time slots
is a reduction in spectral efficiency (compared with a MIMO
system with colocated antennas) accounted for by the factor
in front of the log term. As evidenced by (11), Protocol I
is the only protocol that can realize a multiplexing gain in the
classical sense and, hence, recover (to a certain extent) from
this 50% loss in spectral efficiency. We note, however, that the
effective channel is not i.i.d. complex Gaussian as is the case
in traditional MIMO systems. This implies that in general we
may not recover fully from the loss in spectral efficiency. The
corresponding difference in performance can be attributed to
the fact that we are dealing with a distributed system where
the individual terminals have to cooperate through noisy links.
A more detailed quantitative discussion of this performance
difference is in many cases possible but seems beyond the scope
of this paper. Protocols II and III do not provide multiplexing
gain, which explains their inferior performance when compared
with Protocol I. Finally, the fact that Protocol II is superior
to Protocol III can be attributed to the fact that Protocol II
corresponds to a SIMO system realizing array gain, whereas
Protocol III corresponds to a MISO system devoid of array
7
Recall that the source terminal was assumed to expend the same amount of
power over the two time slots. Allowing a flexible allocation of transmit power
across the two time slots can lead to an ordering different from (13).
gain (recall that we assumed perfect channel knowledge in
the receivers and no channel knowledge in the transmitters).
Maximizing the degree of broadcasting and receive collision
(as is done in Protocol I) will in general result in a higher
number of degrees-of-freedom (and, hence, higher achievable
rates in the degrees-of-freedom limited case) reflected by the
creation of an effective MIMO channel.
C. Diversity Performance
We shall next analyze and compare the different protocols
from a diversity point-of-view. Following the approach in [31]
and [8], we shall interpret the outage probability at a certain
transmission rate as the packet-error rate (PER). The diversity
order is then given by the magnitude of the slope of the PER as
a function of SNR (on a log-log scale). To be more precise, we
define the diversity order for transmission rate
as
(14)
where
denotes the PER or outage probability
at transmission rate
as a function of SNR. Equivalently, a
scheme achieving diversity order
at rate has an error
probability that behaves as
at high
SNR. In the remainder of this subsection, we assume that
the channel gains
and are independent ,
which corresponds to Rayleigh fading on these two links.
Furthermore, we take the channel between the relay terminal
and the destination terminal to be additive white Gaussian
noise (AWGN) (i.e.,
). We note that the latter as-
sumption is conceptual and simplifies the performance analysis
significantly. The general case seems difficult to deal with
analytically. Physically, this assumption could correspond to
a scenario where the destination and relay terminals are static
and have line-of-sight connection, while the source terminal is
moving.
We start by investigating Protocol III and noting that
can
be lower-bounded as
(15)
where
(16)
and
under the simplifying
assumption,
, made above. It follows that the outage
probability at transmission rate
can be upper-bounded
according to
Recalling that and are independent Rayleigh dis-
tributed and using the approximation
for sufficiently large, we obtain
(17)
which using (14) shows that second-order diversity is achieved
in the effective SNR
. We emphasize, however, that the di-
versity performance being determined by the effective SNR
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