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Distributed Space-Time Coded Protocols for
Exploiting Cooperative Diversity in Wireless Networks
J. Nicholas Laneman
Department of Electrical Engineering
University of Notre Dame
Notre Dame, IN USA
Email: jnl@nd.edu
Gregory W. Wornell
Research Laboratory of Electronics
Massachusetts Institute of Technology
Cambridge, MA USA
Email: gww@allegro.mit.edu
Abstract— We develop and analyze space-time coded cooper-
ative diversity protocols for combating multipath fading across
multiple protocol layers in a wireless network. The protocols ex-
ploit spatial diversity available among a collection of distributed
terminals that relay messages for one another in such a manner
that the destination terminal can average the fading, even though
it is unknown a priori which terminals will be involved. In partic-
ular, a source initiates transmission to its destination, and many
relays potentially receive the transmission. Those terminals that
can fully decode the transmission utilize a space-time code to co-
operatively relay to the destination. We demonstrate that these
protocols achieve full spatial diversity in the number of cooper-
ating terminals, not just the number of decoding relays, and can
be used effectively for higher spectral efficiencies than repetition-
based schemes. We discuss issues related to space-time code de-
sign for these protocols, emphasizing codes that readily allow for
appealing distributed versions.
I. INTRODUCTION
In wireless networks, signal fading arising from multipath
propagation is a particularly severe form of interference that
can be mitigated through the use of diversity—transmission of
redundant signals over essentially independent channel realiza-
tions in conjunction with suitable receiver combining to aver-
age the channel effects. Space, or multi-antenna, diversity tech-
niques are particularly attractive as they can be readily com-
bined with other forms of diversity, e.g., time and frequency
diversity, and still offer dramatic performance gains when other
forms of diversity are unavailable. In contrast to the more con-
ventional forms of single-user space diversity with physical ar-
rays, this work builds upon the classical relay channel model
[1] and examines the problem of creating and exploiting space
diversity using a collection of distributed antennas belonging to
multiple terminals, each with its own information to transmit.
We refer to this form of space diversity as cooperative diver-
sity (cf. user cooperation diversity of [2]) because the terminals
share their antennas and other resources to create a “virtual ar-
ray” through distributed transmission and signal processing.
Cooperative diversity between two cooperating terminals is
examined in [3], [4], and a variety of repetition-based proto-
J. Nicholas Laneman was with the Research Laboratory of Electronics, MIT,
Cambridge, MA. This work has been supported in part by Hewlett-Packard un-
der the MIT/HP Alliance, by ARL Federated Labs under Cooperative Agree-
ment No. DAAD19-01-2-0011, and by NSF under Grant No. CCR-9979363.
cols are developed and analyzed. For example, a relay either
amplifies what it receives, or fully decodes, re-encodes, and re-
peats the source message. It is shown in [4] that these sim-
ple protocols can be extended to more than two terminals to
provide full spatial diversity: if m is the number of cooper-
ating terminals, each with a single transmit antenna, system
performance can behave as if each terminal employs m trans-
mit antennas. For example, the outage probability performance
of repetition-based cooperative diversity decays asymptotically
proportional to 1/SNR
m(1−mR
norm
)
, where SNR corresponds to
the average signal-to-noise ratio (SNR) between terminals, and
0 < R
norm
< 1/m corresponds to a suitably-normalized spec-
tral efficiency of the protocol [4]. In this context, full diversity
refers to the fact that, as R
norm
→ 0, the outage probability
decays as 1/SNR
m
. By contrast, the outage probability perfor-
mance of non-cooperative transmission decays asymptotically
as 1/SNR
(1−R
norm
)
, where 0 < R
norm
< 1 is allowed, and as
1/SNR as R
norm
→ 0. Thus, while the outage probability per-
formance of cooperative diversity can decay faster, it does so
only for small R
norm
, in particular, for R
norm
< 1/(m + 1).
Of course, there are more general forms of decode-and-
forward transmission, just as there are more general forms of
space-time codes. Indeed, we will see in this paper that, once
we introduce a few variations on the decode-and-forwardtheme
laid out in [4], the vast array of space-time coding literature can
be brought to bear in the context of cooperative diversity, lead-
ing to a class of protocols that we call space-time coded coop-
erative diversity. Essentially, our new protocols consist of the
following: all relays that can decode the original transmission
re-transmit in the same subchannel using a suitably designed
space-time code. Fig. 1 illustrates the two phases of the proto-
col.
Space-time coded cooperative diversity leads to schemes
whose outage probability performance decays asymptotically
proportional to roughly 1/SNR
m(1−2R
norm
)
. Thus, they (a)
achieve full spatial diversity order m as R
norm
→ 0, (b) have
larger diversity order than repetition-based algorithms for all
R
norm
, and (c) are preferable to non-cooperativetransmission if
R
norm
< (m − 1)/(2m − 1). Moreover, we will see that these
protocols may be readily implemented in a distributed fashion,
because theyonly require the relays to estimate the SNR of their
received signals, decode them if the SNR is sufficiently high, re-
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Phase I Phase II
Fig. 1. Illustration of the two-phases of space-time coded cooperative diversity
protocols. In the first phase, the source broadcasts to the destination as well as
potential relays. Decoding relays are shaded. In the second phase, the decoding
relays use a space-time code to transmit to the destination.
encode with the appropriate waveform from a space-time code,
and re-transmit in the same subchannel.
In broader context, cooperative diversity can be viewed as a
form of network coding, in this case designed to exploit spa-
tial diversity in the network. There is a growing body of work
focused on network coding for enhancing performance of wire-
less and other communication networks [5], [6], [7].
II. SYSTEM MODEL
In our model for the wireless channel in Fig. 1, narrowband
transmissions suffer the effects of frequency nonselective fad-
ing and additive noise. Our analysis in Section III focuses on
the case of slow fading, and measures performance by outage
probability, to isolate the benefits of space diversity. While our
protocols can be naturally extended to the kinds of wideband
and highly mobile scenarios in which frequency- and time-
selective fading, respectively, are encountered, the potential im-
pact of our protocols becomes less substantial as other forms of
diversity can be exploited in the system.
A. Medium Access
As in many current wireless networks, we divide the avail-
able bandwidth into orthogonal channels and allocate these
channels to the transmitting terminals. The medium-access
control (MAC) sublayer typically performs this function. For
example, the MAC in many cellular networks seeks to allocate
orthogonal channels, e.g., frequency-division, time-division, or
code-division, to the terminals in a cell for communicating to
the basestation of that cell. As anotherexample, the MAC in the
IEEE 802.11 wireless LAN standard uses similar structures for
LANs controlled by an access point, or a distributedcontention-
resolution/collision avoidance algorithm which facilitates ran-
dom time-division.
For our cooperative diversity protocols described in Sec-
tion III, transmitting terminals must also process their received
signals; however, current limitations in radio implementation
preclude the terminals from transmitting and receiving at the
same time in the same frequencyband. Because of severesignal
attenuation over the wireless channel, and insufficient electrical
isolation between the transmit and receive circuitry, a terminal’s
PSfrag replacements
Frequency
Time
{1} Transmits
D(1) Transmit
{2} Transmits
D(2) Transmit
{m} Transmits D(n) Transmit
.
.
.
.
.
.
Phase I
Phase II
Fig. 2. Example channel allocations across frequency and time. M =
{1, 2, . . . , m} denotes the set of cooperating terminals in the network. For
source s, D(s) denotes the set of decoding relays participating in a space-time
code during the second phase.
transmitted signal drowns out the signals of other terminals at
its receiver input. Thus, we further divide each channel into
orthogonal subchannels. Fig. 2 illustrates an example channel
allocation satisfying these constraints.
B. Equivalent Channel Models
Under the above orthogonality constraints, we can now con-
veniently, and without loss of generality, characterize our chan-
nel models. Let M = {1, 2, . . . , m} be the set of cooperating
terminals in the network. Due to the symmetry of the chan-
nel allocations, we focus on the message of the source s ∈ M
in transmitting to its destination d(s), potentially using termi-
nals M − {s} as relays. Thus there are m cooperating termi-
nals communicating to d(s). We utilize a baseband-equivalent,
discrete-time channel model for the continuous-time channel,
and we consider N consecutive uses of the channel, where N
is a large integer.
During the first phase, each potential relay r ∈ M − {s}
receives
y
r
[n] = a
s,r
x
s
[n] + z
r
[n] , (1)
for, say, n = 1, . . . , N/2, where x
s
[n] is the source transmitted
signal and y
r
[n] is the received signal at r. If the SNR is suffi-
ciently large for r to decode this transmission, then r serves as
a decoding relay for the source s, so that r ∈ D(s). We charac-
terize the set D(s) more specifically in Section III, but for now
it is sufficient to define it qualitatively and bear in mind that it
is a random set.
The destination receives signals during both phases. During
the first phase, we model the received signal at d(s) as
y
d(s)
[n] = a
s,d(s)
x
s
[n] + z
d(s)
[n] , (2)
for n = 1, . . . , N/2. During the second phase, we model the
received signal at d(s) as
y
d(s)
=
X
r∈D(s)
a
r,d(s)
x
r
[n] + z
d(s)
[n] , (3)
for n = N/2 + 1, . . . , N, where x
r
[n] is the transmitted signal
of relay r. It is during this second phase that the decoding relays
TO APPEAR IN IEEE GLOBECOM 2002, TAIPEI, TAIWAN. 3
employ an appropriately designed space-time code, allowing
d(s) to separate, weight, and combine the signals even though
they are transmitted in the same subchannel.
In (1)-(2), a
i,j
captures the effects of path-loss, shadowing,
and frequency nonselective fading, and z
j
[n] captures the ef-
fects of receiver noise and other forms of interference in the
system. We consider the scenario in which the fading coeffi-
cients are known to, i.e., accurately measured by, the appropri-
ate receivers, but not fully known to (or not exploited by) the
transmitters. Statistically, we model a
i,j
as zero-mean, inde-
pendent, circularly-symmetric complex Gaussian random vari-
ables with variances 1/λ
i,j
, so that the magnitudes |a
i,j
| are
Rayleigh distributed (|a
i,j
|
2
are exponentially distributed with
parameter λ
i,j
) and the phases ]a
i,j
are uniformly distributed
on [0, 2π). Furthermore, we model z
j
[n] as zero-mean mutually
independent, circularly-symmetric, complex Gaussian random
sequences with variance N
0
.
C. Parameterizations
As in [3], [4], two important parameters of the system are the
transmit signal-to-noise ratio SNR and the spectral efficiency R.
We now define these parameters in terms of standard param-
eters in the continuous-time channel. For a continuous-time
channel with bandwidth W Hz available for transmission, the
discrete-time model contains W two-dimensional symbols per
second (2D/s).
If the transmitting terminals have an average power con-
straint in the continuous-time channel model of P
c
Joules/s, we
see that this translates into a discrete-time power constraint of
P = 2P
c
/W Joules/2D since each terminal transmits in a frac-
tion 1/2 of the available degrees of freedom (cf. Fig. 1). Thus,
the channel model is parameterized by the SNR random vari-
ables SNR |a
i,j
|
2
, where
SNR =
2P
c
N
0
W
=
P
N
0
(4)
is the SNR without fading.
In addition to SNR, transmission schemes are further param-
eterized by the spectral efficiency R b/s/Hz attempted by the
transmitting terminals. Note that throughout the paper R is the
transmission rate normalized by the number of degrees of free-
dom utilized by each terminal, not by the total number of de-
grees of freedom in the channel.
Nominally, one could parameterize the system by the pair
(SNR, R); however, our results lend more insight, and are sub-
stantially more compact, when we parameterize the system by
(SNR, R
norm
), where
1
R
norm
= R/ log(1 + SNR) . (5)
III. SPACE-TIME CODED COOPERATIVE DIVERSITY
We now develop and analyze a decode-and-forward based
class of cooperative diversity protocols that we call space-time
1
Unless otherwise indicated, logarithms in this paper are taken to base 2.
coded cooperative diversity. As we alluded in Section I, such
protocols consist of the source broadcasting its transmission to
its destination and potential relays. Potential relays that can
decode the transmission become decoding relays and partici-
pate in the second phase of the protocol. Although the set of
decoding relays D(s) is a random set, we will see that proto-
cols of this form offer full spatial diversity in the number of
cooperating terminals, not just the number of decoding relays
participating in the second phase. Interestingly, potential relays
that cannot decode contribute as much to the performance of
the protocol as the decoding relays.
A. Mutual Information and Outage Probability
Since the channel average mutual information I is a func-
tion of, e.g., the coding scheme, the rule for including potential
relays into the decoding set D(s), and the fading coefficients
of the channel, it too is a random variable. The event I < R
that this mutual information random variable falls below some
fixed spectral efficiency R is referred to as an outage event, be-
cause reliable communication is not possible for realizations in
this event. The probability of an outage event, Pr [I < R], is
referred to as the outage probability of the channel.
Since D(s) is a random set, we first use the total probability
law to write
Pr [I < R] =
X
D(s)
Pr [D(s)] Pr [I < R|D(s)] , (6)
and we examine each term in the summation.
1) Outage Conditional on Decoding Set: Conditioned on
D(s) being the decoding set, the mutual information between
s and d(s) for random codebooks generated i.i.d. circularly-
symmetric, complex Gaussian at the source and all potential
relays can be shown to be
1
2
log(1 + SNR |a
s,d(s)
|
2
) +
1
2
log(1 + SNR
X
r∈D(s)
|a
r,d(s)
|
2
) ,
(7)
the sum of the mutual informations for two “parallel” chan-
nels, one from the source to the destination, and one
from the set of decoding relays to the destination. Thus
Pr [I < R|D(s)] involves |D(s)| + 1 independent fading coef-
ficients, so we might expect it to decay asymptotically propor-
tional to 1/SNR
|D(s)|+1
. Indeed, while we leave out the details
due to space considerations, [4] develops the high SNR approx-
imation
2
Pr [I < R|D(s)] ∼
2
2R
− 1
SNR
|D(s)|+1
× λ
s,d(s)
Y
r∈D(s)
λ
r,d(s)
× A
|D(s)|
(2
2R
− 1) , (8)
2
The approximation f(SNR) ∼ g(SNR) is in the sense of f (SNR)/g(SNR) →
1 as SNR → ∞.
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where
A
n
(t) =
1
(n − 1)!
Z
1
0
w
(n−1)
(1 − w)
(1 + tw)
dw , (9)
for n > 0, and A
0
(t) = 1. Note that we have expressed (8)
in such a way that the first term captures the dependence upon
SNR and the second term captures the dependence upon {λ
i,j
}.
2) Decoding Set Probability: Next, we consider the term
Pr [D(s)], the probability of a particular decoding set. As one
rule for selecting from the potential relays, we can require that a
potential relay fully decode the source message in order to par-
ticipate in the second phase. Indeed, full decoding is required
in order for the mutual information expression (7) to be correct;
however, nothing prevents us from imposing additional restric-
tions on the members of the set D(s). For example, we might
require that a potential relay fully decode and see a realized
SNR some factor larger than its average.
Since the realized mutual information between s and r for
i.i.d. complex Gaussian codebooks is given by
1
2
log
1 + SNR |a
s,r
|
2
,
we have under this rule
Pr [r ∈ D(s)] = Pr
|a
s,r
|
2
> (2
2R
− 1)/SNR
= exp[−λ
s,r
(2
2R
− 1)/SNR] .
Moreover, since each potential relay makes this decision inde-
pendently, and the fading coefficients are independentunder our
model, we have
Pr [D(s)] =
Y
r∈D(s)
exp[−λ
s,r
(2
2R
− 1)/SNR]
×
Y
r6∈D(s)
(1 − exp[−λ
s,r
(2
2R
− 1)/SNR])
∼
2
2R
− 1
SNR
m−|D(s)|−1
×
Y
r6∈D(s)
λ
s,r
. (10)
Note that, any selection means through which Pr [r ∈ D(s)] ∼
1 and (1 − Pr [r ∈ D(s)]) ∝ 1/SNR, for SNR large, indepen-
dently for each r, will result in similar behavior for Pr [D(s)].
Combining (8) and (10) in to (6), we obtain
Pr [I < R] ∼
2
2R
− 1
SNR
m
×
X
D(s)
λ
s,d(s)
Y
r∈D(s)
λ
r,d(s)
Y
r6∈D(s)
λ
s,r
× A
|D(s)|
(2
2R
− 1) . (11)
B. Convenient Bounds
While the approximation given in (11) is quite general and
can be numerically evaluated to determine performance, it is
not very convenient for further analysis. There are two factors
contributingto its complexity: dependenceupon {λ
i,j
}, and the
unwieldy closed-form expression for A
n
(t) as n grows. In this
section, we developed upper and lower bounds for (11) that we
exploit in the sequel.
Our objective is simplify the summation in (11). To this end,
we note that for a given decoding set D(s), either r ∈ D(s), in
which case λ
r,d(s)
appears in the correspondingterm in (11), or
r 6∈ D(s), in which case λ
s,d(s)
appears in the corresponding
term in (11). We therefore define
λ
r
= min{λ
r,d(s)
, λ
s,r
} ,
λ
r
= max{λ
r,d(s)
, λ
s,r
} , (12)
and λ
s
= λ
s
= λ
s,d(s)
. Then the product dependent upon
{λ
i,j
} is bounded by
λ
m
≤ λ
s,d(s)
Y
r∈D(s)
λ
r,d(s)
Y
r6∈D(s)
λ
s,r
≤
λ
m
, (13)
where λ is the geometric mean of the λ
i
and
λ is the geometric
mean of the λ
i
, for i ∈ M. We note that the upper and lower
bounds in (13) are independent of D(s). We also note that the
bounds in (13) coincide, i.e., λ = λ, if (though not only if)
λ
i
= λ
i
for all i ∈ M. Viewing λ
i,j
as a measure of distance
between terminals i and j, the class of network geometries in
two dimensions that satisfy this condition are those in which
all the relays lie with arbitrary spacing along the perpendicular
bisector between the source and destination. A more general
study of the effects of such network geometry on performance
is beyond the scope of this paper.
To avoid dealing with (9), we exploit the bounds
1
(n + 1)!(1 + t)
≤ A
n
(t) ≤
1
n!
. (14)
Combining (13) and (14) into (11), we arrive at the following
simplified bounds for outage probability
Pr [I < R] ≥
2
2R
− 1
SNR/λ
m
2
−2R
X
D(s)
1
(|D(s)| + 1)!
(15)
Pr [I < R] ≤
2
2R
− 1
SNR/λ
m
X
D(s)
1
|D(s)|!
. (16)
C. Diversity-Multiplexing Tradeoff
An interesting tradeoff between diversity and multiplex-
ing arises when we parameterize ours results in terms of
(SNR, R
norm
), with R
norm
given in (5). Specifically, when we
approximate Pr [I < R]
.
= SNR
−∆(R
norm
)
, in the sense of equal-
ity to first-order in the exponent, i.e.,
∆(R
norm
) = lim
SNR→∞
−
log(Pr [I < R])
log(SNR)
, (17)
we find that increasing R
norm
reduces ∆. This tradeoff was
originally observed in the context of multi-antenna systems [8],
TO APPEAR IN IEEE GLOBECOM 2002, TAIPEI, TAIWAN. 5
Non−Cooperative
Repetition
Space−Time, Upper Bound
Space−Time, Lower Bound
PSfrag replacements
∆(R
norm
)
R
norm
m
1
1
2
1
m
1
m+1
m−1
2m−1
m−1
2m−3
m
2(m−1)
Fig. 3. Diversity order ∆(R
norm
) for non-cooperative transmission (red),
repetition-coded cooperative diversity (green), and space-time coded coopera-
tive diversity (blue, bounds from (18)). As R
norm
→ 0, all cooperative di-
versity protocols provide full spatial diversity order m, the number of cooper-
ating terminals. Relative to direct transmission, space-time coded cooperative
diversity can be effectively utilized for a much broader range of R
norm
than
repetition-coded cooperative diversity, especially as m becomes large.
so it is not surprising that it also arise in the context of cooper-
ative diversity [4].
Utilizing our lower and upper bounds (15)-(16) in (17) yields
upper and lower bounds, respectively, on the diversity order
m(1 − 2R
norm
) ≤ ∆(R
norm
) ≤ m
1 −
m − 1
m
2R
norm
(18)
Fig. 3 compares these bounds, along with the corresponding
tradeoffs for non-cooperative transmission, ∆(R
norm
) = 1 −
R
norm
, and repetition-based cooperative diversity ∆(R
norm
) =
m(1 − mR
norm
). Clearly, space-time coded cooperative di-
versity offers larger diversity order than repetition-based algo-
rithms and can be effectively utilized for higher spectral effi-
ciencies than repetition-based schemes.
IV. PRACTICAL ISSUES
A. Space-Time Code Design
The outage analysis in Section III relies on a random cod-
ing argument, and demonstrates that full spatial diversity can
be achieved using such a rich set of codes. In practice, one may
wonder whether or not there exist space-time codes for which
the number of participating antennas is not known a priori and
yet full diversity can be achieved. More specifically, if we de-
sign a space-time code for a maximum of N transmit antennas,
but only a randomly selected subset of n of those antennas ac-
tually transmit, can the space-time code offer diversity n? It
turns out that the class of space-time block codes based upon
orthogonal designs have this property [9]. Essentially, these
codes have orthogonal waveforms emitted from each antenna,
corresponding to columns in a code matrix. Absence of an an-
tenna corresponds to deletion of a column in the matrix, but
the columns remain orthogonal, allowing the code to maintain
its diversity benefits. Thus space-time coded cooperative diver-
sity protocols may be readily deployed in practice using these
codes.
B. Distributed Implementation
Given a suitably designed space-time code, space-time coded
cooperative diversity reduces to a simple, distributed network
protocol. When each terminal transmits its message, the other
terminals receive and potentially decode, requiring only an
SNR measurement. If a relay can decode, it transmits the in-
formation in the second phase using its column from the space-
time code matrix. Because the destination receiver can measure
the fading, it can determinewhich relays are involved in the sec-
ond phase and adapt its decoding rule appropriately. Although
certainly the terminals could exchange more information in or-
der to adapt power to the network geometry, for example, such
overhead is not required in order to obtain full diversity.
One of the key challenges to implementing such a protocol
could be block and symbol synchronization of the cooperating
terminals. Such synchronization might be obtained through pe-
riodic transmission of known synchronization prefixes, as pro-
posed in current wireless LAN standards. A detailed study of
issues involved with synchronization is beyond the scope of the
present paper.
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