3062 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 50, NO.

12, DECEMBER 2004

Cooperative Diversity in Wireless Networks:

Efficient Protocols and Outage Behavior
J. Nicholas Laneman, Member, IEEE, David N. C. Tse, Member, IEEE, and Gregory W. Wornell, Fellow, IEEE

Abstract—We develop and analyze low-complexity cooperative

diversity protocols that combat fading induced by multipath
propagation in wireless networks. The underlying techniques
exploit space diversity available through cooperating terminals’
relaying signals for one another. We outline several strategies
employed by the cooperating radios, including fixed relaying
schemes such as amplify-and-forward and decode-and-forward,
selection relaying schemes that adapt based upon channel mea-
surements between the cooperating terminals, and incremental Fig. 1. Illustration of radio signal paths in an example wireless network
relaying schemes that adapt based upon limited feedback from the with terminals T and T transmitting information to terminals T and T ,
destination terminal. We develop performance characterizations respectively.
in terms of outage events and associated outage probabilities,
which measure robustness of the transmissions to fading, focusing
on the high signal-to-noise ratio (SNR) regime. Except for fixed
decode-and-forward, all of our cooperative diversity protocols tiple-antenna, diversity techniques are particularly attractive as
are efficient in the sense that they achieve full diversity (i.e., they can be readily combined with other forms of diversity, e.g.,
second-order diversity in the case of two terminals), and, more- time and frequency diversity, and still offer dramatic perfor-
over, are close to optimum (within 1.5 dB) in certain regimes. Thus, mance gains when other forms of diversity are unavailable. In
using distributed antennas, we can provide the powerful benefits
of space diversity without need for physical arrays, though at a loss contrast to the more conventional forms of space diversity with
of spectral efficiency due to half-duplex operation and possibly at physical arrays [2]–[4], this work builds upon the classical relay
the cost of additional receive hardware. Applicable to any wireless channel model [5] and examines the problem of creating and
setting, including cellular or ad hoc networks—wherever space exploiting space diversity using a collection of distributed an-
constraints preclude the use of physical arrays—the performance
characterizations reveal that large power or energy savings result tennas belonging to multiple terminals, each with its own in-
from the use of these protocols. formation to transmit. We refer to this form of space diversity
Index Terms—Diversity techniques, fading channels, outage as cooperative diversity (cf., user cooperation diversity of [6])
probability, relay channel, user cooperation, wireless networks. because the terminals share their antennas and other resources
to create a “virtual array” through distributed transmission and
signal processing.
I. INTRODUCTION

I N wireless networks, signal fading arising from multipath

propagation is a particularly severe channel impairment that
can be mitigated through the use of diversity [1]. Space, or mul-
A. Motivating Example
To illustrate the main concepts, consider the example wire-
less network in Fig. 1, in which terminals and transmit
Manuscript received January 22, 2002; revised June 10, 2004. The work of to terminals and , respectively. This example might cor-
J. N. Laneman and G. W. Wornell was supported in part by ARL Federated respond to a snapshot of a wireless network in which a higher
Labs under Cooperative Agreement DAAD19-01-2-0011, and by the National
Science Foundation under Grant CCR-9979363. The work of J. N. Laneman level network protocol has allocated bandwidth to two terminals
was also supported in part by the State of Indiana through the 21st Century Re- for transmission to their intended destinations or next hops. For
search and Technology Fund, and by the National Science Foundation under example, in the context of a cellular network, and might
Grant ECS03-29766. The work of D. N. C. Tse was supported in part by the
National Science Foundation under Grant ANI-9872764. The material in this correspond to handsets and might correspond to the
paper was presented in part at the 38th Annual Allerton Conference on Com- base station [7]. As another example, in the context of a wire-
munications, Control and Computing, Monticello, IL, October 2000, and at the less local-area network (LAN), the case might corre-
IEEE International Symposium on Information Theory, Washington, DC, June
2001. spond to an ad hoc configuration among the terminals, while the
J. N. Laneman was with the Department of Electrical Engineering and Com- case might correspond to an infrastructure configura-
puter Science, Massachusetts Institute of Technology (MIT), Cambridge. He is tion, with serving as an access point [8]. The broadcast na-
now with the Department of Electrical Engineering, University of Notre Dame,
Notre Dame, IN 46556 USA (e-mail: jnl@nd.edu). ture of the wireless medium is the key property that allows for
D. N. C. Tse is with the Department of Electrical Engineering and Com- cooperative diversity among the transmitting terminals: trans-
puter Science, University of California, Berkeley, CA 94720 USA (e-mail: mitted signals can, in principle, be received and processed by
dtse@eecs.berkeley.edu).
G. W. Wornell is with the Department of Electrical Engineering and Computer any of a number of terminals. Thus, instead of transmitting in-
Science, Massachusetts Institute of Technology (MIT), Cambridge, MA 02139 dependently to their intended destinations, and can listen
USA (e-mail: gww@mit.edu). to each other’s transmissions and jointly communicate their in-
Communicated by L. Tassiulas, Associate Editor for Communication Net-
works. formation. Although these extra observations of the transmitted
Digital Object Identifier 10.1109/TIT.2004.838089 signals are available for free (except, possibly, for the cost of
0018-9448/04$20.00 © 2004 IEEE
LANEMAN et al.: COOPERATIVE DIVERSITY IN WIRELESS NETWORKS 3063

additional receive hardware) wireless network protocols often Loosely speaking, cooperation yields highest achievable rates
ignore or discard them. when the source-relay channel quality is very high, and obser-
In the most general case, and can pool their resources, vation yields highest achievable rates when the relay-destination
such as power and bandwidth, to cooperatively transmit their channel quality is very high. Various extensions to the case of
information to their respective destinations, corresponding to a multiple relays have appeared in the work of Schein and Gal-
wireless multiple-access channel with relaying for , and lager [18], [19], Gupta and Kumar [20], [21], Gastpar et al.
to a wireless interference channel with relaying for . [22]–[24], and Reznik et al. [25]. For channels with multiple
At one extreme, corresponding to a wireless relay channel, the information sources, Kramer and Wijngaarden [26] consider a
transmitting terminals can focus all their resources on transmit- multiple-access channel in which the sources communicate to a
ting the information of ; in this case, acts as the “source” single destination and share a single relay.
of the information, and serves as a “relay.” Such an approach 2) Multiple-Access Channels With Generalized Feed-
might provide diversity in a wireless setting because, even if the back: Work by King [27], Carleial [28], and Willems et al.
fading is severe between and , the information might be [29]–[32] examines multiple-access channels with generalized
successfully transmitted through . Similarly, and can feedback. Here, the generalized feedback allows the sources
focus their resources on transmitting the information of , cor- to essentially act as relays for one another. This model relates
responding to another wireless relay channel. most closely to the wireless channels we have in mind. The
constructions in [28]–[30] can be viewed as two-terminal gen-
B. Related Work eralizations of the cooperation scheme in [5]; the construction
Relay channels and their extensions form the basis for our [27] may be viewed as a two-terminal generalization of the
study of cooperative diversity. This section summarizes some observation scheme in [5]. Sendonaris et al. introduce multipath
of the relevant literature in this area. Because relaying and fading into the model of [28], [30], calling their approaches for
cooperative diversity essentially create a virtual antenna array, this system model user cooperation diversity [6], [33], [34]. For
work on multiple-antenna systems, or multiple-input, multiple- ergodic fading, they illustrate that the adapted coding scheme
output (MIMO) systems, is of course relevant, as are different of [30] enlarges the achievable rate region.
ways of characterizing fundamental performance limits in
wireless channels, in particular outage probability for noner- C. Summary of Results
godic settings. Throughout the rest of the paper, we assume
that the reader is familiar with these latter areas, and refer the We now highlight the results of the present paper, many of
interested reader to [2]–[4], [9], [10], and references therein, for which were initially reported in [35], [36], and recently ex-
an introduction to the relevant concepts from multiple-antenna tended in [37]. This paper develops low-complexity cooperative
systems and to [11] for an introduction to outage capacity for diversity protocols that explicitly take into account certain im-
fading channels. plementation constraints in the cooperating radios. Specifically,
1) Relay Channels: The classical relay channel models a while previous work on relay and cooperative channels allows
class of three-terminal communication channels originally ex- the terminals to transmit and receive simultaneously, i.e., full-
amined by van der Meulen [12], [13]. Cover and El Gamal [5] duplex, we constrain them to employ half-duplex transmission.
treat certain discrete memoryless and additive white Gaussian Furthermore, although previous work employs channel state in-
noise relay channels, and they determine channel capacity for formation (CSI) at the transmitters in order to exploit coherent
the class of physically degraded1 relay channels. More gener- transmission, we utilize CSI at the receivers only. Finally, al-
ally, they develop lower bounds on capacity, i.e., achievable though previous work focuses primarily on ergodic settings and
rates, via three structurally different random coding schemes: characterizes performance via Shannon capacity or capacity re-
gions, we focus on nonergodic or delay-constrained scenarios
• facilitation [5, Theorem 2], in which the relay does not and characterize performance by outage probability [11].
actively help the source, but rather, facilitates the source We outline several cooperative protocols and demonstrate
transmission by inducing as little interference as possible; their robustness to fairly general channel conditions. In addition
• cooperation [5, Theorem 1], in which the relay fully de- to direct transmission, we examine fixed relaying protocols
codes the source message and retransmits, jointly with the in which the relay either amplifies what it receives, or fully
source, a bin index (in the sense of Slepian–Wolf coding decodes, re-encodes, and retransmits the source message. We
[14], [15]) of the previous source message; call these options amplify-and-forward and decode-and-for-
ward, respectively. Obviously, these approaches are inspired
• observation2 [5, Theorem 6], in which the relay encodes a by the observation [5], [18], [28] and cooperation [5], [6],
quantized version of its received signal, using ideas from [30] schemes, respectively, but we intentionally limit the com-
source coding with side information [14], [16], [17]. plexity of our protocols for ease of potential implementation.
1At a high level, degradedness means that the destination receives a corrupted Furthermore, our analysis suggests that cooperating radios may
version of what the relay receives, all conditioned on the relay transmit signal. also employ threshold tests on the measured channel quality
While this class is mathematically convenient, none of the wireless channels between them, to obtain adaptive protocols, called selection
found in practice are well modeled by this class.
2The names facilitation and cooperation were introduced in [5], but Cover
relaying, that choose the strategy with best performance. In
and El Gamal did not give a name to their third approach. We use the name addition, adaptive protocols based upon limited feedback from
observation throughout the paper for convenience. the destination terminal, called incremental relaying, are also
3064 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 50, NO. 12, DECEMBER 2004

Fig. 2. Example time-division channel allocations for (a) direct transmission with interference, (b) orthogonal direct transmission, and (c) orthogonal cooperative
diversity. We focus on orthogonal transmissions of the form (b) and (c) throughout the paper.

developed. Selection and incremental relaying protocols rep- case of slow fading, to capture scenarios in which delay con-
resent new directions for relay and cooperative transmission, straints are on the order of the channel coherence time, and mea-
building upon existing ideas. sures performance by outage probability, to isolate the benefits
For scenarios in which CSI is unavailable to the transmitters, of space diversity. While our cooperative protocols can be nat-
even full-duplex cooperation cannot improve the sum capacity urally extended to the kinds of wide-band and highly mobile
for ergodic fading [38]. Consequently, we focus on delay-lim- scenarios in which frequency- and time-selective fading, respec-
ited or nonergodic scenarios, and evaluate performance of our tively, are encountered, the potential impact of our protocols be-
protocols in terms of outage probability [11]. We show analyt- comes less substantial when other forms of diversity can be ex-
ically that, except for fixed decode-and-forward, each of our ploited in the system.
cooperative protocols achieves full diversity, i.e., outage prob-
ability decays proportional to , where is signal-to- A. Medium Access
noise ratio (SNR) of the channel, whereas it decays proportional As in many current wireless networks, such as cellular and
to without cooperation. At fixed low rates, amplify-and- wireless LANs, we divide the available bandwidth into orthog-
forward and selection decode-and-forward are at most 1.5 dB onal channels and allocate these channels to the transmitting
from optimal and offer large power or energy savings over di- terminals, allowing our protocols to be readily integrated into
rect transmission. For sufficiently high rates, direct transmis- existing networks. As a convenient by-product of this choice,
sion becomes preferable to fixed and selection relaying, because we are able to treat the multiple-access (single receiver) and in-
these protocols repeat information all the time. Incremental re- terference (multiple receivers) cases described in Section I-A
laying exploits limited feedback to overcome this bandwidth in- simultaneously, as a pair of relay channels with signaling be-
efficiency by repeating only rarely. More broadly, the relative tween the transmitters. Furthermore, removing the interference
attractiveness of the various schemes can depend upon the net- between the terminals at the destination radio(s) substantially
work architecture and implementation considerations. simplifies the receiver algorithms and the outage analysis for
purposes of exposition.
D. Outline For all of our cooperative protocols, transmitting terminals
must also process their received signals; however, current limi-
An outline of the remainder of the paper is as follows. Sec- tations in radio implementation preclude the terminals from full-
tion II describes our system model for the wireless networks duplex operation, i.e., transmitting and receiving at the same
under consideration. Section III outlines fixed, selection, and time in the same frequency band. Because of severe attenuation
incremental relaying protocols at a high level. Section IV char- over the wireless channel, and insufficient electrical isolation
acterizes the outage behavior of the various protocols in terms between the transmit and receive circuitry, a terminal’s trans-
of outage events and outage probabilities, using several results mitted signal drowns out the signals of other terminals at its re-
for exponential random variables developed in Appendix I. Sec- ceiver input.3 Thus, to ensure half-duplex operation, we further
tion V compares the results from a number of perspectives, and divide each channel into orthogonal subchannels. Fig. 2 illus-
Section VI offers some concluding remarks. trates our channel allocation for an example time-division ap-
proach with two terminals.
We expect that some level of synchronization between the ter-
II. SYSTEM MODEL minals is required for cooperative diversity to be effective. As
In our model for the wireless channel in Fig. 1, narrow-band suggested by Fig. 2 and the modeling discussion to follow, we
transmissions suffer the effects of frequency nonselective fading 3Typically, a terminal’s transmit signal is 100–150 dB above its received
and additive noise. Our analysis in Section IV focuses on the signal.
LANEMAN et al.: COOPERATIVE DIVERSITY IN WIRELESS NETWORKS 3065

consider the scenario in which the terminals are block, carrier, complex Gaussian random variables with variances . Fur-
and symbol synchronous. Given some form of network block thermore, we model as zero-mean mutually independent,
synchronization, carrier and symbol synchronization for the net- circularly symmetric, complex Gaussian random sequences
work can build upon the same between the individual transmit- with variance .
ters and receivers. Exactly how this synchronization is achieved,
and the effects of small synchronization errors on performance, C. Parameterizations
is beyond the scope of this paper. Two important parameters of the system are the SNR without
fading and the spectral efficiency. We now define these param-
B. Equivalent Channel Models eters in terms of standard parameters in the continuous-time
Under the above orthogonality constraints, we can now channel. For a continuous-time channel with bandwidth
conveniently, and without loss of generality, characterize our hertz available for transmission, the discrete-time model con-
channel models using a time-division notation; frequency-di- tains two-dimensional symbols per second (2D/s).
vision counterparts to this model are straightforward. Due to If the transmitting terminals have an average power constraint
the symmetry of the channel allocations, we focus on the mes- in the continuous-time channel model of joules per second,
sage of the “source” terminal , which potentially employs we see that this translates into a discrete-time power constraint
terminal as a “relay,” in transmitting to the “destination” of J/2D since each terminal transmits in half of the
terminal , where and . We utilize available degrees of freedom, under both direct transmission and
a baseband-equivalent, discrete-time channel model for the cooperative diversity. Thus, the channel model is parameterized
continuous-time channel, and we consider consecutive uses by the SNR random variables , where
of the channel, where is large.
For direct transmission, our baseline for comparison, we (5)
model the channel as
is the common SNR without fading. Throughout our analysis,
(1) we vary , and allow for different (relative) received SNRs
through appropriate choice of the fading variances. As we will
for, say, , where is the source transmitted
see, increasing the source-relay SNR proportionally to increases
signal, and is the destination received signal. The other ter-
in the source-destination SNR leads to the full diversity benefits
minal transmits for as depicted in Fig. 2(b).
of the cooperative protocols.
Thus, in the baseline system, each terminal utilizes only half of
In addition to SNR, transmission schemes are further param-
the available degrees of freedom of the channel.
eterized by the rate bits per second, or spectral efficiency
For cooperative diversity, we model the channel during the
first half of the block as b/s/Hz (6)
(2) attempted by the transmitting terminals. Note that (6) is the rate
(3) normalized by the number of degrees of freedom utilized by
each terminal, not by the total number of degrees of freedom
for, say, , where is the source transmitted in the channel.
signal and and are the relay and destination received Nominally, one could parameterize the system by the pair
signals, respectively. For the second half of the block, we model ; however, our results lend more insight, and are sub-
the received signal as stantially more compact, when we parameterize the system by
either of the pairs or , where4
(4)

for , where is the relay transmitted R

(7)
signal and is the destination received signal. A similar
setup is employed in the second half of the block, with the roles
For a complex additive white Gaussian noise (AWGN)
of the source and relay reversed, as depicted in Fig. 2(c). Note
channel with bandwidth and SNR given by ,
that, while again half the degrees of freedom are allocated to
is the SNR normalized by the minimum SNR
each source terminal for transmission to its destination, only a
required to achieve spectral efficiency [39]. Similarly,
quarter of the degrees of freedom are available for communica-
is the spectral efficiency normalized by the max-
tion to its relay.
imum achievable spectral efficiency, i.e., channel capacity [9],
In (1)–(4), captures the effects of path-loss, shadowing,
[10]. In this sense, parameterizations given by
and frequency nonselective fading, and captures the
and are duals of one another. For our setting with
effects of receiver noise and other forms of interference in the
fading, as we will see, the two parameterizations yield tradeoffs
system, where and . We consider the
between different aspects of system performance: results under
scenario in which the fading coefficients are known to, i.e.,
exhibit a tradeoff between the normalized SNR
accurately measured by, the appropriate receivers, but not fully
gain and spectral efficiency of a protocol, while results under
known to, or not exploited by, the transmitters. Statistically, we
model as zero-mean, independent, circularly symmetric 4Unless otherwise indicated, logarithms in this paper are taken to base 2.
3066 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 50, NO. 12, DECEMBER 2004

exhibit a tradeoff between the diversity order and by first appropriately combining the signals from the two sub-
normalized spectral efficiency of a protocol. The latter tradeoff blocks using one of a variety of combining techniques; in the
has also been called the diversity-multiplexing tradeoff in [9], sequel, we focus on a suitably designed matched filter, or max-
[10]. imum-ratio combiner.
Note that, although we have parameterized the transmit 2) Decode-and-Forward: For decode-and-forward trans-
powers and noise levels to be symmetric throughout the net- mission, the appropriate channel model is again (2)–(4). The
work for purposes of exposition, asymmetries in average SNR source terminal transmits its information as , say, for
and path loss can be lumped into the fading variances . . During this interval, the relay processes
Furthermore, while the tools are powerful enough to consider by decoding an estimate of the source transmitted signal.
general rate pairs , we consider the equal rate point, i.e., Under a repetition-coded scheme, the relay transmits the
, for purposes of exposition. signal

III. COOPERATIVE DIVERSITY PROTOCOLS

In this section, we describe a variety of low-complexity co- for .
operative diversity protocols that can be utilized in the network Decoding at the relay can take on a variety of forms.
of Fig. 1, including fixed, selection, and incremental relaying. For example, the relay might fully decode, i.e., estimate
These protocols employ different types of processing by the without error, the entire source codeword, or it might em-
relay terminals, as well as different types of combining at the ploy symbol-by-symbol decoding and allow the destination
destination terminals. For fixed relaying, we allow the relays to to perform full decoding. These options allow for trading off
either amplify their received signals subject to their power con- performance and complexity at the relay terminal. Note that
straint, or to decode, re-encode, and retransmit the messages. we focus on full decoding in the sequel; symbol-by-symbol
Among many possible adaptive strategies, selection relaying decoding of binary transmissions has been treated from an
builds upon fixed relaying by allowing transmitting terminals to uncoded perspective in [40]. Again, the destination can employ
select a suitable cooperative (or noncooperative) action based a variety of combining techniques; we focus in the sequel on a
upon the measured SNR between them. Incremental relaying suitably modified matched filter.
improves upon the spectral efficiency of both fixed and selec-
tion relaying by exploiting limited feedback from the destina- B. Selection Relaying
tion and relaying only when necessary. As we might expect, and the analysis in Section IV confirms,
In any of these cases, the radios may employ repetition or fixed decode-and-forward is limited by direct transmission be-
more powerful codes. We focus on repetition coding throughout tween the source and relay. However, since the fading coeffi-
the sequel, for its low implementation complexity and ease of cients are known to the appropriate receivers, can be mea-
exposition. Destination radios can appropriately combine their sured to high accuracy by the cooperating terminals; thus, they
received signals by exploiting control information in the pro- can adapt their transmission format according to the realized
tocol headers. value of .
This observation suggests the following class of selection re-
A. Fixed Relaying laying algorithms. If the measured falls below a certain
1) Amplify-and-Forward: For amplify-and-forward trans- threshold, the source simply continues its transmission to the
mission, the appropriate channel model is (2)–(4). The destination, in the form of repetition or more powerful codes. If
source terminal transmits its information as , say, for the measured lies above the threshold, the relay forwards
. During this interval, the relay processes , what it received from the source, using either amplify-and-for-
and relays the information by transmitting ward or decode-and-forward, in an attempt to achieve diversity
gain.
(8) Informally speaking, selection relaying of this form should
offer diversity because, in either case, two of the fading coef-
for . To remain within its power con- ficients must be small in order for the information to be lost.
straint (with high probability), an amplifying relay must use gain Specifically, if is small, then must also be small
for the information to be lost when the source continues its
transmission. Similarly, if is large, then both and
must be small for the information to be lost when the
(9) relay employs amplify-and-forward or decode-and-forward. We
formalize this notion when we consider outage performance of
where we allow the amplifier gain to depend upon the fading selection relaying in Section IV.
coefficient between the source and relay, which the relay
estimates to high accuracy. This scheme can be viewed as rep- C. Incremental Relaying
etition coding from two separate transmitters, except that the As we will see, fixed and selection relaying can make ineffi-
relay transmitter amplifies its own receiver noise. The destina- cient use of the degrees of freedom of the channel, especially for
tion can decode its received signal for high rates, because the relays repeat all the time. In this section,
LANEMAN et al.: COOPERATIVE DIVERSITY IN WIRELESS NETWORKS 3067

we describe incremental relaying protocols that exploit limited as a function of the fading coefficient . The outage event for
feedback from the destination terminal, e.g., a single bit indi- spectral efficiency is given by and is equivalent to the
cating the success or failure of the direct transmission, that we event
will see can dramatically improve spectral efficiency over fixed R
and selection relaying. These incremental relaying protocols can (11)
be viewed as extensions of incremental redundancy, or hybrid
automatic-repeat-request (ARQ), to the relay context. In ARQ, For Rayleigh fading, i.e., exponentially distributed
the source retransmits if the destination provides a negative ac- with parameter , the outage probability satisfies6
knowledgment via feedback; in incremental relaying, the relay R
retransmits in an attempt to exploit spatial diversity.
As one example, consider the following protocol utilizing
feedback and amplify-and-forward transmission. We nominally R
allocate the channels according to Fig. 2(b). First, the source
transmits its information to the destination at spectral efficiency R
. The destination indicates success or failure by broadcasting
a single bit of feedback to the source and relay, which we as-
sume is detected reliably by at least the relay.5 If the source-des-
where we have utilized the result of Fact 1 in Appendix I with
tination SNR is sufficiently high, the feedback indicates success R
, , and .
of the direct transmission, and the relay does nothing. If the
source-destination SNR is not sufficiently high for successful B. Fixed Relaying
direct transmission, the feedback requests that the relay am-
1) Amplify-and-Forward: The amplify-and-forward pro-
plify-and-forward what it received from the source. In the latter
tocol produces an equivalent one-input, two-output complex
case, the destination tries to combine the two transmissions.
Gaussian noise channel with different noise levels in the out-
As we will see, protocols of this form make more efficient use
puts. As explained in detail in Appendix II, the maximum
of the degrees of freedom of the channel, because they repeat
average mutual information between the input and the two
only rarely. Incremental decode-and-forward is also possible;
outputs, achieved by i.i.d. complex Gaussian inputs, is given
however, its analysis is more involved, and its performance is
by
slightly worse than the above protocol.

IV. OUTAGE BEHAVIOR

(12)
In this section, we characterize performance of the protocols as a function of the fading coefficients, where
of Section III in terms of outage events and outage probabilities
[11]. To facilitate their comparison in the sequel, we also de- (13)
rive high-SNR approximations of the outage probabilities using
results from Appendix I. For fixed fading realizations, the ef- We note that the amplifier gain does not appear in (12), be-
fective channel models induced by the protocols are variants of cause the constraint (9) is met with equality.
well-known channels with AWGN. As a function of the fading The outage event for spectral efficiency is given by
coefficients viewed as random variables, the mutual information and is equivalent to the event
for a protocol is a random variable denoted by ; in turn, for a R
target rate , denotes the outage event, and de- (14)
notes the outage probability.
For Rayleigh fading, i.e., independent and exponen-
A. Direct Transmission tially distributed with parameters , analytic calculation of
To establish baseline performance under direct transmission, the outage probability becomes involved, but we can approxi-
the source terminal transmits over the channel (1). The max- mate its high-SNR behavior as
imum average mutual information between input and output in
this case, achieved by independent and identically distributed R
( i.i.d.) zero-mean, circularly symmetric complex Gaussian in-
puts, is given by

(10) where we have utilized the result of Claim 1 in Appendix I, with

5Such an assumption is reasonable if the destination encodes the feedback

bit with a very-low-rate code. Even if the relay cannot reliably decode, useful
R
protocols can be developed and analyzed. For example, a conservative protocol
might have the relay amplify-and-forward what it receives from the source in all
cases except when the destination reliably receives the direct transmission and 6As we develop more formally in Appendix I, the approximation f (SNR) 
the relay reliably decodes the feedback bit. g (SNR) for SNR large is in the sense of f (SNR)=g (SNR) ! 1 as SNR ! 1.
3068 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 50, NO. 12, DECEMBER 2004

2) Decode-and-Forward: To analyze decode-and-forward The behavior in (18) indicates that fixed decode-and-
transmission, we examine a particular decoding structure at forward does not offer diversity gains for large , because re-
the relay. Specifically, we require the relay to fully decode the quiring the relay to fully decode the source information limits
source message; examination of symbol-by-symbol decoding the performance of decode-and-forward to that of direct trans-
at the relay becomes involved because it depends upon the par- mission between the source and relay.
ticular coding and modulation choices. The maximum average
mutual information for repetition-coded decode-and-forward C. Selection Relaying
can be readily shown to be
To overcome the shortcomings of decode-and-forward trans-
mission, we described selection relaying corresponding to adap-
tive versions of amplify-and-forward and decode-and-forward,
(15) both of which fall back to direct transmission if the relay cannot
decode. We cannot conclude whether or not these protocols
as a function of the fading random variables. The first term in are optimal, because the capacities of general relay and related
(15) represents the maximum rate at which the relay can reliably channels are long-standing open problems; however, as we will
decode the source message, while the second term in (15) rep- see, selection decode-and-forward enables the cooperating ter-
resents the maximum rate at which the destination can reliably minals to exploit full spatial diversity and overcome the limita-
decode the source message given repeated transmissions from tions of fixed decode-and-forward.
the source and destination. Requiring both the relay and desti- As an example analysis, we determine the performance of
nation to decode the entire codeword without error results in the selection decode-and-forward. Its mutual information is some-
minimum of the two mutual informations in (15). We note that what involved to write down in general; however, in the case
such forms are typical of relay channels with full decoding at of repetition coding at the relay, using (10) and (15), it can be
the relay [5]. readily shown to be
The outage event for spectral efficiency is given by
and is equivalent to the event
R
(16) (19)
where R . This threshold is motivated
by our discussion of direct transmission, and is analogous to
For Rayleigh fading, the outage probability for repetition-
(11). The first case in (19) corresponds to the relay’s not being
coded decode-and-forward can be computed according to
able to decode and the source’s repeating its transmission; here,
the maximum average mutual information is that of repetition
coding from the source to the destination, hence the extra factor
of in the SNR. The second case in (19) corresponds to the
relay’s ability to decode and repeat the source transmission;
here, the maximum average mutual information is that of repe-
tition coding from the source and relay to the destination.
(17)
The outage event for spectral efficiency is given by
R and is equivalent to the event
where . Although we may readily com-
pute a closed-form expression for (17), for compactness we ex-
amine the large behavior of (17) by computing the limit

(20)

The first (resp., second) event of the union in (20) corresponds to

the first (resp., second) case in (19). We observe that adapting to
the realized fading coefficient ensures that the protocol performs
no worse than direct transmission, except for the fact that it po-
tentially suffers the bandwidth inefficiency of repetition coding.
Because the events in the union of (20) are mutually exclu-
sive, the outage probability becomes a sum

as , using the results of Facts 1 and 2 in Appendix I.

Thus, we conclude that
R
(18)
(21)
LANEMAN et al.: COOPERATIVE DIVERSITY IN WIRELESS NETWORKS 3069

and we may readily compute a closed-form expression for (21). The outage event is equivalent to the event
For comparison to our other protocols, we examine the large R
behavior of (21) by computing the limit (25)

For exponentially distributed with parameters , the

outage probability satisfies

R
(26)

where we have applied the result of Fact 2 in Appendix I.

2) Orthogonal Transmit Diversity Bound: The transmit di-
versity bound (26) does not take into account the half-duplex
constraint. To capture this effect, we constrain the transmit di-
(22) versity scheme to be orthogonal.
When the source and relay can cooperate perfectly, an equiv-
as , using the results of Facts 1 and 2 of Appendix I. alent model to (23), incorporating the relay orthogonality con-
Thus, we conclude that the large performance of selection straint, consists of parallel channels
decode-and-forward is identical to that of fixed amplify-and- (27)
forward. (28)
Analysis of more general selection relaying becomes in-
volved because there are additional degrees of freedom in This pair of parallel channels is utilized half as many times as the
choosing the thresholds for switching between the various op- corresponding direct transmission channel, so the source must
tions such as direct, amplify-and-forward, and decode-and-for- transmit at twice the spectral efficiency in order to achieve the
ward. These issues represent a potentially useful direction for same spectral efficiency as direct transmission.
future research, but a detailed analysis of such protocols is For each fading realization, the maximum average mutual in-
beyond the scope of this paper. formation can be obtained using independent complex Gaussian
inputs. Allocating a fraction of the power to , and the re-
D. Bounds for Cooperative Diversity maining fraction of the power to , the average
mutual information is given by
We now develop performance limits for fixed and selection
relaying. If we suppose that the source and relay know each (29)
other’s messages a priori, then instead of direct transmis-
sion, each would benefit from using a space–time code for The outage event is equivalent to the outage region
two transmit antennas. In this sense, the outage probability R
(30)
of conventional transmit diversity [2]–[4] represents an opti-
mistic lower bound on the outage probability of cooperative As in the case of amplify-and-forward, analytical calculation
diversity. The following sections develop two such bounds: of the outage probability becomes involved; however, we can
an unconstrained transmit diversity bound and an orthogonal approximate its high-SNR behavior for Rayleigh fading as
transmit diversity bound that takes into account the half-duplex
constraint. R
1) Transmit Diversity Bound: To utilize a space–time code (31)
for each terminal, we allocate the channel as in Fig. 2(b). Both
terminals transmit in all the degrees of freedom of the channel, using the result of Claim 2 in Appendix I, with
so their transmitted power is J/2D, half that of direct trans-
mission. The spectral efficiency for each terminal remains .
For transmit diversity, we model the channel as
R R

(23)
Clearly, (31) is minimized for , yielding
R
for, say, . As developed in Appendix III, an (32)
optimal signaling strategy, in terms of minimizing outage prob-
ability in the large regime, is to encode information using so that i.i.d. complex Gaussian inputs again minimize outage
i.i.d. complex Gaussian, each with power . probability for large . Note that for , (32) converges
Using this result, the maximum average mutual information as to (26), the transmit diversity bound without orthogonality con-
a function of the fading coefficients is given by straints. Thus, the orthogonality constraint has little effect for
small , but induces a loss in proportional to
(24)
3070 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 50, NO. 12, DECEMBER 2004

with respect to the unconstrained transmit diversity bound for along with their associated performance limits, is beyond the
large . scope of this paper.

E. Incremental Relaying V. DISCUSSION

Outage analysis of incremental relaying is complicated by its In this section, we compare the outage results developed in
variable-rate nature. Specifically, the protocols operate at spec- Section IV in various regimes. To partition the discussion, and
tral efficiency when the source-destination transmission is make it clear in which context certain observations hold, we
successful, and spectral efficiency when the relay repeats divide the exposition into two sections. Section V-A considers
the source transmission. Thus, we examine outage probability general asymmetric networks in which the fading variances
as a function of and the expected spectral efficiency . may be distinct. The primary observations of this section are
For incremental amplify-and-forward, the outage probability comparing the performance of the cooperative protocols to the
as a function of and is given by transmit diversity bound and examining how spectral efficiency
and network asymmetry affect that comparison. Section V-B
focuses on the special case of symmetric networks in which
the fading variances are identical, e.g., , without
loss of generality. Focusing on this special case allows us to
substantially simplify the exposition for comparison of per-
formance under different parameterizations, e.g.,
(33) and .
where and are given by (10) and (12), respectively, A. Asymmetric Networks
R
As indicated by the results in Section IV, for fixed rates,
and is given in (13). The second equality follows from the simple protocols such as fixed amplify-and-forward, selection
fact that the intersection of the direct and amplify-and-forward decode-and-forward, and incremental amplify-and-forward
outage events is exactly the amplify-and-forward outage event each achieve full (i.e., second-order) diversity: their outage
at half the rate. Furthermore, the expected spectral efficiency probability decays proportional to (cf. (15), (22), and
can be computed as (35)). We now compare these protocols to the transmit diversity
R R
bound, discuss the impacts of spectral efficiency and network
geometry on performance, and examine their outage events.
1) Comparison to Transmit Diversity Bound: Equating per-
R R
formance of amplify-and-forward (15) or selection decode-and-
forward (22) to the transmit diversity bound (26), we can de-
R termine the relative SNR losses of cooperative diversity. In the
SNR (34) low-spectral-efficiency regime, the protocols without feedback
are within a factor of
where the second equality follows from substituting standard
exponential results for . R
An important question is the value of to employ in (33) for a R
given expected spectral efficiency . A fixed value of can arise
from several possible , depending upon the value of ; thus, in SNR from the transmit diversity bound, suggesting that the
we see that the pre-image SNR can contain several points. powerful benefits of multiple-antenna systems can indeed be
We define a function SNR SNR to capture a useful obtained without the need for physical arrays. For statistically
mapping from to ; for a given value of , it seems clear from symmetric networks, e.g., , the loss is only or
the outage expression (33) that we want the smallest possible. 1.5 dB; more generally, the loss decreases as the source-relay
For fair comparison to protocols without feedback, we char- path improves relative to the relay-destination path.
acterize a modified outage expression in the large-SNR regime. For larger spectral efficiencies, fixed and selection relaying
Specifically, we compare outage of fixed and selection relaying lose an additional 3 dB per transmitted bit per second per hertz
protocols to the modified outage SNR . For large (bit/s/Hz) with respect to the transmit diversity bound. This ad-
, we have ditional loss is due to two factors: the half-duplex constraint and
the repetition-coded nature of the protocols. As suggested by
R
Fig. 3, of the two, repetition coding appears to be the more sig-
SNR
nificant source of inefficiency in our protocols. In Fig. 3, the
(35) SNR loss of orthogonal transmit diversity with respect to uncon-
where we have combined the results of Claims 1 and 3 in Ap- strained transmit diversity is intended to indicate the cost of the
pendix I. half-duplex constraint, and the loss of our cooperative diversity
Bounds for incremental relaying can be obtained by suitably protocols with respect to the transmit diversity bound indicates
normalizing the results developed in Section IV-D; however, we the cost of both imposing the half-duplex constraint and em-
stress that treating protocols that exploit more general feedback, ploying repetition-like codes. The figure suggests that, although
LANEMAN et al.: COOPERATIVE DIVERSITY IN WIRELESS NETWORKS 3071

Fig. 3. SNR loss for cooperative diversity protocols (solid) and orthogonal transmit diversity bound (dashed) relative to the (unconstrained) transmit diversity
bound.

the half-duplex constraint contributes, “repetition” in the form 3) Effects of Geometry: To isolate the effect of network ge-
of amplification or repetition coding is the major cause of SNR ometry on performance, we compare the high-SNR behavior
loss for high rates. By contrast, incremental amplify-and-for- of direct transmission (12) with that of incremental amplify-
ward overcomes these additional losses by repeating only when and-forward (35). Comparison with fixed and selection relaying
necessary. is similar, except for the additional impact of SNR loss with
2) Outage Events: It is interesting that amplify-and-forward increasing spectral efficiency. Using a common model for the
and selection decode-and-forward have the same high-SNR path-loss (fading variances), we set , where
performance, especially considering the different shapes isthe distance between terminals and , and is the path-loss
of their outage events (cf. (14), (20)), which are shown exponent [7]. Under this model, comparing (12) with (35), as-
in the low-spectral-efficiency regime in Fig. 4. When the suming both approximations are good for the of interest,
relay can fully decode the source message and repeat it, we prefer incremental amplify-and-forward whenever
i.e., , the outage event for selection de-
code-and-forward is a strict subset of the outage event of (36)
amplify-and-forward, with amplify-and-forward approaching
that of selection decode-and-forward as . On the Thus, incremental amplify-and-forward is useful whenever the
other hand, when the relay cannot fully decode the source relay lies within a certain normalized ellipse having the source
message and the source repeats, i.e., , the and destination as its foci, with the size of the ellipse increasing
outage event of amplify-and-forward is neither a subset nor a in . What is most interesting about the structure of this
superset of the outage event for selection decode-and-forward. “utilization region” for incremental amplify-and-forward is that
Apparently, averaging over the Rayleigh-fading coefficients it is symmetric with respect to the source and destination. By
eliminates the differences between amplify-and-forward and comparison, a certain circle about only the source gives the uti-
selection decode-and-forward, at least in the high-SNR regime. lization region for fixed decode-and-forward.
3072 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 50, NO. 12, DECEMBER 2004

Fig. 4. Outage event boundaries for amplify-and-forward (solid) and selection decode-and-forward (dashed and dash-dotted) as functions of the realized fading
coefficientja j between the cooperating terminals. Outage events are to the left and below the respective outage event boundaries. Successively lower solid curves
correspond to amplify-and-forward with increasing values of ja j . The dashed curve corresponds to the outage event for selection decode-and-forward when the
relay can fully decode and the relay repeats, i.e., SNR ja j  2, while the dash-dotted curve corresponds to the outage event of selection decode-and-forward
when the relay cannot fully decode and the source repeats, i.e., SNR ja j < 2. Note that the dash-dotted curve also corresponds to the outage event for
direct transmission.

Utilization regions of the form (36) may be useful in devel- The results in Appendix I are all general enough to allow this
oping higher layer network protocols that select between direct particular parameterization. To demonstrate their application,
transmission and cooperative diversity using one of a number of we consider amplify-and-forward. The outage event under this
potential relays. Such algorithms and their performance repre- alternative parameterization is given by
sent an interesting area of further research, and a key ingredient R
for fully incorporating cooperative diversity into wireless net-
works.
For , the outage probability is approximately
B. Symmetric Networks
We now specialize all of our results to the case of statistically R
symmetric networks, e.g., without loss of generality. R
We develop the results, summarized in Table I, under the two
parameterizations and , respectively. where we have utilized the results of Claim 1 in Appendix I with
1) Results Under Different Parameterizations: Parameter-
izing the outage results from Section IV in terms of
is straightforward because remains fixed; we simply substitute R
R to obtain the results listed in the second
column of Table I. Parameterizing the outage results from Sec- The other results listed in the third column of Table I can be
tion IV in terms of is a bit more involved because obtained in similar fashion using the appropriate results from
increases with . Appendix I.
LANEMAN et al.: COOPERATIVE DIVERSITY IN WIRELESS NETWORKS 3073

TABLE I
SUMMARY OF OUTAGE PROBABILITY APPROXIMATIONS FOR STATISTICALLY SYMMETRIC NETWORKS

Fig. 5. Outage probabilities versus SNR , small R regime, for statistically symmetric networks, i.e.,  = 1. The outage probability curve for
amplify-and-forward was obtained via Monte Carlo simulation, while the other curves are computed from analytical expressions. Solid curves correspond to
exact outage probabilities, while dash-dotted curves correspond to the high-SNR approximations from Table I. The dashed curve corresponds to the transmit
diversity bounds in this low spectral efficiency regime.

2) Fixed Systems: Fig. 5 shows outage probabilities for ward with respective to the transmit diversity bound is also ap-
the various protocols as functions of in the small, fixed parent. The curves for fixed and selection relaying shift to the
regime. Both exact and high-SNR approximations are dis- right by 3 dB for each additional bit/s/Hz of spectral efficiency in
played, demonstrating the wide range of SNRs over which the the high regime. By contrast, the performance of incremental
high-SNR approximations are useful. The diversity gains of our amplify-and-forward is unchanged at high SNR for increasing
protocols appear as steeper slopes in Fig. 5, from a factor of . Note that, at outage probabilities on the order of , coop-
decrease in outage probability for each additional 10 dB of erative diversity achieves large energy savings over direct trans-
SNR in the case of direct transmission, to a factor of de- mission—on the order of 12–15 dB.
crease in outage probability for each additional 10 dB of SNR 3) Fixed Families of Systems: Another way to ex-
in the case of cooperative diversity. The relative loss of 1.5 amine the high spectral efficiency regime as SNR becomes
dB for fixed amplify-and-forward and selection decode-and-for- large is to allow to grow with increasing . In particular,
3074 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 50, NO. 12, DECEMBER 2004

the choice of is a natural one: for it into account, there may be additional power costs of relays op-
slower growth, the outage results essentially behave like fixed erating instead of powering down. Despite these costs, our anal-
systems for sufficiently large , while for faster growth, the ysis demonstrates significant performance enhancements, par-
outage probabilities all tend to . These observations motivate ticularly in the low-spectral-efficiency regime (up to roughly 1
our parameterization in terms of . bit/s/Hz) often found in practice. Like other forms of diversity,
Parameterizing performance in terms of leads to these performance enhancements take the form of decreased
interesting tradeoffs between the diversity order and normalized transmit power for the same reliability, increased reliability for
spectral efficiency of a protocol. Because these tradeoffs arise the same transmit power, or some combination of the two.
naturally in the context of multiple-antenna systems [9], [10], it The observations in Section V-B suggest that, among other
is not surprising that they show up in the context of cooperative issues, a key area of further research is exploring cooperative
diversity. Diversity order can be viewed as the power to which diversity protocols in the high-spectral-efficiency regime. It re-
is raised in our outage expressions in the third column of mains unclear at this point whether our simple protocols are
Table I. To be precise, we can define diversity order as close to optimal in this regime, among all possible cooperative
diversity protocols, yet our results indicate that direct transmis-
(37) sion eventually becomes preferable. Useful work in this area
SNR would develop tighter lower bounds on performance, which is
akin to developing tighter converses for the relay channel [5],
Larger implies more robustness to fading (faster
or demonstrating other protocols that are more efficient for high
decay in the outage probability with increasing SNR),
spectral efficiencies. Some of our own work in this direction ap-
but generally decreases with increasing .
pears in [37].
For example, the diversity order of amplify-and-forward is
More broadly, there are a number of channel circumstances in
; thus, its maximum diversity
addition to those considered here that warrant further investiga-
order is achieved as , and maximum normal-
tion. In particular, for scenarios in which the transmitters obtain
ized spectral efficiency is achieved as . Fig. 6
accurate knowledge of the channel realizations, via feedback or
compares the tradeoffs for direct transmission and cooperative
other means, beamforming and power and bandwidth allocation
diversity. As we might expect from our previous discussion,
become possible. These options allow the cooperating terminals
among the protocols developed in this paper, incremental am-
to adapt to their specific channel conditions and geometry and
plify-and-forward yields the highest for each ;
select appropriate coding schemes for various regimes. Again,
this curve also corresponds to the transmit diversity bound
better understanding of the relay channel will continue to yield
in the high-SNR regime. What is most interesting about the
insight on these problems.
results in Fig. 6 is the sharp transition at between
We note that we have focused on the case of a pair of terminals
our preference for amplify-and-forward (as well as selection
cooperating; extension to more than two terminals is straightfor-
decode-and-forward) for and our preference for
ward except for the fact that comparatively more options arise.
direct transmission for .
For example, in the case of three cooperating terminals, one of
the relays might amplify-and-forward the information, while the
VI. CONCLUDING REMARKS AND FUTURE DIRECTIONS other relay might decode-and-forward the information, or vice
We develop in this paper a variety of low-complexity, coop- versa. Moreover, as the number of terminals forming a network
erative protocols that enable a pair of wireless terminals, each grows, higher layer protocols for organizing terminals into co-
with a single antenna, to fully exploit spatial diversity in the operating groups become increasingly important. Some prelim-
channel. These protocols blend different fixed relaying modes, inary work in this direction is reported in [38]. Finally, because
specifically amplify-and-forward and decode-and-forward, cooperative diversity is inherently a network problem, it could
with strategies based upon adapting to CSI between cooper- be fruitful to take into account additional higher layer network
ating source terminals (selection relaying) as well as exploiting issues such as queuing of bursty data, link layer retransmissions,
limited feedback from the destination terminal (incremental and routing.
relaying). For delay-limited and nonergodic environments, we
analyze the outage probability performance, in many cases APPENDIX I
exactly, and in all cases using accurate, high-SNR approxima- ASYMPTOTIC CDF APPROXIMATIONS
tions. To keep the presentation in the main part of the paper concise,
There are costs associated with our cooperative protocols. For we collect in this appendix several results for the limiting be-
one thing, cooperation with half-duplex operation requires twice havior of the cumulative distribution function (CDF) of certain
the bandwidth of direct transmission for a given rate, and leads combinations of exponential random variables. All our results
to larger effective SNR losses for increasing spectral efficiency. are of the form
Furthermore, depending upon the application, additional receive
hardware may be required in order for the sources to relay for (38)
one another. Although this may not be the case in emerging ad
hoc or multihop cellular networks, it would be the case in the up- where is a parameter of interest; is the CDF of a
link of current cellular systems that employ frequency-division certain random variable that can, in general, depend upon
duplexing. Finally, although our analysis has not explicitly taken ; and are two (continuous) functions; and and
LANEMAN et al.: COOPERATIVE DIVERSITY IN WIRELESS NETWORKS 3075

Fig. 6. Diversity order 1(R ) in (37) versus R for direct transmission and cooperative diversity.

are constants. Among other things, for example, (38) implies Claim 1: Let , , and be independent exponential
the approximation is accurate for close random variables with parameters , , and , respectively.
to . Let as in (13). Let be positive, and
Fact 1: Let be an exponential random variable with param- let be continuous with and
eter . Then, for a function continuous about and as . Then
satisfying as
(39)
(43)
Fact 2: Let , where and are independent ex- Moreover, if a function is continuous about and
ponential random variables with parameters and , respec- satisfies as , then
tively. Then the CDF

(44)
(40)
The following lemma will be useful in the proof of Claim 1.
satisfies
Lemma 1: Let be positive, and let ,
(41) where and are independent exponential random variables
with parameters and , respectively. Let be
Moreover, if a function is continuous about and continuous with and as .
satisfies as , then Then the probability satisfies

(42) (45)
3076 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 50, NO. 12, DECEMBER 2004

Proof of Lemma 1: We begin with a lower bound

(52)

where remains finite for any as

.
Combining (49), (51), and (52), we have
(46)
so, utilizing Fact 1
(53)
(47)
and furthermore
To prove the other direction, let be a fixed constant

since and, by assumption,

and as .
The constants are arbitrary. In particular, can be
chosen arbitrarily large, and arbitrarily close to . Hence,
(54)

(48) Combining (47) with (54), the lemma is proved.

But Proof of Claim 1:

(49)

which takes care of the first term of (48). To bound the second
term of (48), let be another fixed constant, and note that

(55)
where in the second equality we have used the change of vari-
ables . But by Lemma 1 with and
, the quantity in brackets approaches as
, so we expect
(50)

where the first term in the bound of (50) follows from the fact
that
(56)
To fully verify (56), we must utilize the lower and upper bounds
is nonincreasing in , and the second term in the bound of (50) developed in Lemma 1.
follows from the fact that . Using the lower bound (47), (55) satisfies
Now, the first term of (50) satisfies

(51)

and, by a change of variable , the second term of

(50) satisfies
(57)
where the first equality results from the Dominated Conver-
gence Theorem [41] after noting that the integrand is both
bounded by and converges to the function .
LANEMAN et al.: COOPERATIVE DIVERSITY IN WIRELESS NETWORKS 3077

Using the upper bound (54), (55) satisfies where the last equality follows from the change of variables
.
To upper-bound (63), we use the identities for
all and for all , so that (63) becomes

(58)

where the last equality results from the fact and the
fact that
whence

(64)

To lower-bound (63), we use the concavity of , i.e.,

remains finite for all even as . for any ,
Again, the constants are arbitrary. In particular,
can be chosen arbitrarily large, and arbitrarily close to . for all
Hence,
and the identity for all , so that (63) becomes
(59)

Combining (57) and (59) completes the proof.

Claim 2: Let and be independent exponential random
variables with parameters and , respectively. Let be pos-
itive and let be continuous with as .
Define

(60)

Then

(61)

Moreover, if is continuous about with as

, then

(62)

Proof: First, we write CDF in the form

Thus,

(65)

Since the bounds in (64) and (65) are equal, the claim is
proved.
Claim 3: Suppose pointwise as , and
(63) that is monotone increasing in for each . Let
3078 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 50, NO. 12, DECEMBER 2004

be such that , pointwise as , exists an such that . The

and is monotone decreasing in for each . Define result follows from , where the first
. Then inequality follows from the definition of and the second
inequality follows from the fact that .
(66)
APPENDIX II
Proof: Since for all , we have , and AMPLIFY-AND-FORWARD MUTUAL INFORMATION
consequently because is monotone
increasing. Thus, For completeness, in this appendix we compute the maximum
average mutual information for amplify-and-forward transmis-
(67) sion (12). The result borrows substantially from the vector re-
sults in [2], [3], aside from taking into account the amplifier
The upper bound is a bit more involved. Fix . Lemma 2 power constraint in the relay as well as simplifying manipula-
shows that for each there exists such that tions.
We write the equivalent channel (2)–(4), with relay pro-
cessing (8), in vector form as
for all such that . Then we have

Thus,

and since can be made arbitrarily small

(68) where the source signal has power constraint , and
relay amplifier has constraint
Combining (67) with (68), we obtain the desired result.
The following Lemma is used in the proof of the upper bound (69)
of Claim 3.
Lemma 2: Let be such that , and the noise has covariance .
pointwise as , and is monotone decreasing in Note that we determine the mutual information for arbitrary
for each . Define . For each and transmit powers, relay amplification, and noise levels, even
any , there exists such that though we utilize the result only for the symmetric case. Since
the channel is memoryless, the average mutual information
satisfies
for all such that .
Proof: Fix and , and select such that
. with equality for zero-mean, circularly symmetric complex
Because point-wise as , for each Gaussian [2], [3]. Noting that
and any , there exists a such that
all
Moreover, since is monotone decreasing in , if is
sufficient for convergence at , then it is sufficient for conver-
gence at all . Thus, for any and there exists we have
a such that
all
Throughout the rest of the proof, we only consider and (70)
such that .
Consider the interval , and note that It is apparent that (70) is increasing in , so the amplifier power
implies . Since , we have . constraint (69) should be active, yielding, after substitutions and
Also, since by the above construction, we algebraic manipulations
have . By continuity, assumes all
intermediate values between and on the
interval [42, Theorem 4.23]; in particular, there with given by (13).
LANEMAN et al.: COOPERATIVE DIVERSITY IN WIRELESS NETWORKS 3079

APPENDIX III which, using Fact 2, decays proportional to for

INPUT DISTRIBUTIONS FOR TRANSMIT DIVERSITY BOUND large if . Thus, minimizing the outage probability
In this appendix, we derive the input distributions that min- for large is equivalent to maximizing
imize outage probability for transmit diversity schemes in the
high-SNR regime. Our derivation is a slight extension of the re- (74)
sults in [2], [3] dealing with asymmetric fading variances.
An equivalent channel model for the two-antenna case can be such that . Clearly, (74) is maximized for
summarized as and . Thus, zero-mean,
i.i.d. complex Gaussian inputs minimize the outage probability
(71) in the high-SNR regime.

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