Cooperative Diversity for Wireless Fading Channels

without Channel State Information

Deqiang Chen and J. Nicholas Laneman
Department of Electrical Engineering
University of Notre Dame
Notre Dame, IN 46556
Email: {dchen2, jlaneman}@nd.edu

Abstract— Relaying and cooperative diversity allow multiple Relay 1

wireless radios to effectively share their antennas and create a
virtual antenna array, thereby leveraging the spatial diversity
benefits of multiple-input, multiple-output (MIMO) antenna sys- Source Dest.
tems. This paper examines the benefits of cooperative diversity for
scenarios in which the receivers cannot exploit accurate channel Relay 2
state information (CSI). In particular, noncoherent demodulation
is explored for two classes of relay processing, namely, detect- ..
.
and-forward and amplify-and-forward. A complete maximum
likelihood (ML) framework for noncoherent demodulation is de- Relay N
veloped for detect-and-forward, and is shown to naturally extend
the corresponding framework for coherent demodulation. By
contrast, the intractability of ML demodulation for noncoherent Fig. 1. Communication between a source and destination with multiple
amplify-and-forward is demonstrated, suggesting a disconnect relays.
from the well-developed framework for coherent demodulation.
Simulation results exhibit the diversity benefits of the detect-and-
forward algorithms.
(CRC) code to screen for block decision errors at the relays
I. I NTRODUCTION [6]. Another possibility is to explicitly take the effects of
Cooperative diversity allows a collection of radios to relay relay decision errors into account in the destination decoding
signals for each other and effectively create a virtual antenna algorithm [7], [8]. Our expectation is that some combination
array for combating multipath fading in wireless channels. of both approaches is required in practical systems.
Fig. 1 depicts a simple example of such a communication With this motivation, we turn to the study of modulation and
scenario. Many studies of cooperative diversity focus on an demodulation for cooperative wireless systems. Some algo-
information-theoretic perspective, employing either Shannon rithms for coherent demodulation are introduced and partially
capacity (see [1], [2] and references therein) or outage ca- analyzed in [7]. In particular, relay processing takes two simple
pacity (see [3]–[5] and references therein) as performance forms: detect-and-forward, in which the relay demodulates and
measures for ergodic or non-ergodic channel environments, remodulates the signal transmitted by the source, and amplify-
respectively. In real networks, especially those such as sensor and-forward, in which the relay simply amplifies its received
networks with delay-constrained applications and complexity- signal. Similar schemes are also considered in [9]. Further
constrained radios, practical codes with finite blocklength must analysis of coherent amplify-and-forward is developed [10]–
be considered. [12].
To obtain the best performance from practical decoding A majority of the work on cooperative diversity has fo-
algorithms such as maximum-likelihood decoding or iterative cused on scenarios in which the receivers, and perhaps the
decoding, it becomes necessary to take into account the effects transmitters, obtain channel state information (CSI) in the
of relay processing when implementing the destination decod- form of accurate estimates of the fading coefficients. The
ing algorithm. In the extreme case, one can consider uncoded receivers can utilize available CSI for coherent reception, and
transmissions and study modulation and demodulation for the transmitters can utilize available CSI for power control and
various kinds of relay processing. The processing elements coherent beamforming. This paper summarizes some results
and performance analysis obtained for uncoded symbols can for coherent demodulation with CSI at the receivers only
then be extended to the analysis of coded systems as is done [7], and extends the framework to noncoherent demodulation
for classical channel models. without CSI at the transmitters or receivers. The case of binary
If the relays perform some sort of detection and/or decoding, transmission is analyzed in more depth in [8]. We note that,
the constraint of limited blocklength requires the cooperative even when CSI is available at the receivers, noncoherent mod-
protocol to cope with relay decision errors. One way of dealing ulation and demodulation may be required for low-complexity,
with the issue is to employ an outer cyclic-redundancy check low-power hardware implementations.
 II. S YSTEM M ODEL B. Maximum-Likelihood (ML) Demodulation
For simplicity of exposition, we focus on the communica- In the sequel, we consider modulation and demodulation,
tion model depicted by Fig. 1, with one source denoted s, one with an emphasis wherever possible on maximum-likelihood
destination denoted d, and one or more relays denoted r = (ML) demodulation. We develop our results for general M -
1, 2, . . . , N . The relays must satisfy a half-duplex constraint, ary signaling, i.e., transmit signal take values xm , for m =
i.e., they cannot transmit and receive simultaneously in any 1, 2, . . . , M with equal probability. To provide specific simu-
given frequency band. To study the demodulation issues in lation results, we specialize the results to the case of M -ary
their simplest setting, we further constrain the radios to relay orthogonal signaling, so that the symbol duration K = M ,
orthogonally, e.g., in time or in frequency. Optimization within and the signal vectors take values
this model, in the form of power and bandwidth allocation, √
as well as extensions to more general models, are certainly xm = Eim , m = 1, 2, . . . , M , (5)
important but beyond the scope of this paper. As we will see, where im is a unit vector with 1 as its m-th element and 0 as all
even these simple models generate quite challenging detection its other elements. If the fading captured by our model is flat
problems. across frequency, then the orthogonal signal set corresponds
A. Channel Model to frequency-shift keying (FSK). If the fading captured by
our model flat is across time, then the orthogonal signal set
In the scenario described above, a baseband-equivalent, corresponds to pulse position modulation (PPM).
discrete-time channel model is as follows.1 We consider an Because the fading coefficients are modeled as being mutu-
input “symbol” (modulation symbol, channel codeword, and ally independent, and the relays do not interact, the destination
so forth) as a block of K complex channel uses, and col- received signals yd,s and yd,r , r = 1, 2, . . . , N , are condi-
lect time varying signals into vectors, so that, for example, tionally independent given xs . It is therefore natural, when
x = [x[1] x[2] . . . x[K]]T . To isolate the benefits of spatial considering ML detection at the destination, to define log-
diversity, our model captures the effects of block fading, either likelihood ratios in order to simplify analysis of the problem.
in time or in frequency. For a given symbol, the source Specifically, let
transmission xs is received by the destination as
pyd,s |xs (y|xm )
yd,s = ad,s xs + zd,s (1) `sm (y) := ln (6)
pyd,s |xs (y|x1 )
and by the relays as be the log likelihood ratio (LLR) for yd,s , the vector received
yr,s = ar,s xs + zr,s , r = 1, 2, . . . , N . (2) at the destination from the source, given the source trans-
mits symbol xm . Note that, although normalization by the
After processing their received signals, the relays transmit likelihood given the first symbol x1 is not necessary, it is
signals xr , r = 1, 2, . . . , N , to the destination over the convenient for obtaining simplified LLRs. For example, we
channels have `s1 (y) = 0. Similarly, let
yd,r = ad,r xr + zd,r , r = 1, 2, . . . , N . (3) pyd,r |xs (y|xm )
`rm (y) := ln (7)
We denote the average energy per symbol for the source pyd,r |xs (y|x1 )
and relays as Es := E[x†s xs ] and Er := E[x†r xr ], for r = be the LLR for yd,r , the vector received at the destination from
1, 2, . . . , N , respectively. the relay, given the source transmits symbol xm . Again, we
In (1)–(3), ai,j captures the effects of narrowband fading, have `r1 (y) = 0, for r = 1, 2, . . . , N . With these definitions,
and zi,j captures the effects of additive noise and other the destination ML decision rule can be compactly written as
interference in the system. We model ai,j as being mutually
N
independent zero-mean, complex Gaussian random variables X
with variances σa2i,j , and zi,j as mutually independent, zero- m̂ = arg max `sm (yd,s ) + `rm (yd,r ) . (8)
m=1,2,...,M
r=1
mean, complex Gaussian random vectors with covariance
matrices Ni IK , where IK is the K × K identify matrix. With this high-level structure in place for the destination
Furthermore, the fading coefficients ai,j , noise vectors zi,j , receiver, we now specialize (6) and (7) to various types of re-
and transmitted signals xs and xr are all modeled as mutually lay processing, including detect-and-forward and amplify-and-
independent. Among other possible parameterizations, we de- forward, as well as various types of demodulation, including
fine the average received signal-to-noise ratio (SNR) between coherent and noncoherent.
transmitter j and receiver i as
III. D ETECT- AND -F ORWARD
σa2i,j Ej We begin our discussion with detect-and-forward processing
SNRi,j = . (4)
Ni at the relays. First, we draw some conclusions about the
1 San serif fonts denote random variables, e.g., x, and serif fonts denote
general structure of the ML receiver for detect-and-forward.
deterministic quantities, e.g., x. Vectors are denoted in the appropriate bold We point out how the results specialize to the more well-
face, so that x denotes a random vector, and x denotes a deterministic vector. known cases of noncoherent spatial diversity with receive
antenna arrays. We then specialize the results to the coherent length K = K 0 L into L symbols of length K 0 , and apply
and noncoherent cases, respectively. the detect-and-forward strategy to these shorter symbols. The
exact tradeoffs involved, as well as code designs for detect-
A. General Formulation and Observations and-forward relaying, represent an interesting area for further
Detect-and-forward proceeds as follows. The source trans- research.
mits√ signal xs .√The relays make decision errors, so that Another caveat is that, for M -ary signal sets, the relay
xr / Er 6= xs / Es with nonzero probability. Although it is r
transition probabilities Pm 0 ,m can be difficult to obtain without

possible to find the ML detector, its nonlinear form makes resorting to bounds or Monte Carlo integration. This issue
detailed analysis of the probability of error quite difficult. can be partially alleviated when the signal sets exhibit some
For a given channel model, (6) is relatively straightforward kind of symmetry such are geometrically uniformity [14].
to compute. What remains is to determine (7). To this end, let Phase-shift keying (PSK) and orthogonal signal sets such
pyd,r |xr (y|xm0 ) as FSK and PPM can have this property, depending upon
`ˆrm0 (y) := ln (9) the channel model; general quadrature amplitude modulation
pyd,r |xr (y|x1 ) (QAM) constellations often are not geometrically uniform.
be the LLR for yd,r , the vector received at the destination from A final caveat is that, because of the non-linearities in (11),
the relay, given the relay transmits symbol xm0 . As before, analyzing the performance of the detector (8), for example, in
`ˆr1 (y) = 0, r = 1, 2, . . . , N . To write an expression for (7) terms of symbol-error rate, is quite challenging. Some progress
in terms of (9), we need to know the conditional probability can be made in the case of symmetric binary transmissions,
law for each relay transmit symbol given the source transmit by observing that (11) reduces in this case to `r1 (y) = 0 and
symbol. In particular, let "
ˆr (y))
#
 r + (1 −  r ) exp(` 2
r `r2 (y) = ln , (12)
Pm 0 ,m := Pr[xr = xm0 |xs = xm ] (10) (1 − r ) + r exp(`ˆr (y))
2
be the transition probabilities that capture the effects of relay where r = P1,2 2
= P2,1r
is the relay decision error probability.
decisions. Then we can readily show that, for each m = ˆ
That is, `2 (y) = fr (`r2 (y)), where
r
1, 2, . . . , M , 
" PM
r ˆr
#
t T r , t > Tr
0 =1 Pm0 ,m exp(`m0 (y))
  
m r + (1 − r )e
`rm (y) = ln PM . (11) fr (t) = ln ≈ t, −Tr ≤ t ≤ Tr ,
P r 0 exp(`ˆr 0 (y)) (1 − r ) + r et 
m =10 m ,1 m −Tr , t < −Tr

Suppose it were possible for the relays to avoid making (13)
decision errors, i.e., xr /Er = xs /Es with probability one for and Tr = ln [(1 − r )/r ]. The sigmoidal behavior of fr (t)
r = 1, 2, . . . , N . Then Pm0 ,m = δm0 ,m so that (11) reduces suggests the piecewise linear approximation in (13). This is
to `rm (y) = `ˆrm (y). This fictional scenario is equivalent to re- appealing because it eliminates the logarithm and exponenti-
moving the noisy, faded paths between the source and relays in ations in (12), is amenable to analysis in some cases [7], [8],
Fig. 1 and replacing them with noise-free paths. If we maintain and provides tight approximations to performance.
orthogonal relaying, the detection problem becomes equivalent
B. Coherent Demodulation
to one for receive antenna diversity, possibly with different
branch SNRs [13]. In fact, the corresponding receive antenna For coherent demodulation, we assume the destination re-
diversity problem provides a convenient way of developing a ceiver can obtain accurate estimates of the realizations of
lower bound on performance. However, as we might expect, the fading coefficients ad,s and ad,r , for r = 1, 2, . . . , N .
this lower bound is only tight for scenarios in which the paths Given ad,s and xs , yd,s is conditionally complex Gaussian
between the source and relays are very strong. with density
For more general channel conditions, (8), in combination CN (y; ad,s xs , Nd IK ) , (14)
with (9)–(11), tell us that it is possible for the destination to
where CN(y; µ, Σ) denotes a K-dimensional complex Gaus-
explicitly take into account the effects of relay decision errors.
sian probability density function with mean vector µ and
However, a caveat is that computational complexity may limit
covariance matrix Σ. Thus, the LLR (6) simplifies to2
the use of this approach to signal sets of moderate size. To see
this, we observe from (8) and (11) that we require O(M 2 N ) ∗
2Re{ad,s (xm − x1 )† y} + |ad,s |2 (x†1 x1 − x†m xm )
multiplications and additions, O(M 2 N ) exponentiations, and `sm (y) = .
Nd
O(M N ) logarithms to find the ML decision at the destination. (15)
Although the number of relays N may be reasonably small, M We recognize the first term of (15) as the appropriate matched-
can become quite large for coded modulations. For example, filter operation for our problem. Note that the relays employ
a binary linear block code of rate R and blocklength K
2 There is a slight abuse of notation here. The LLR in (6) involves a
has M = 2RK codewords. One alternative for reducing
ratio of likelihoods without conditioning on receiver CSI, but (15) assumes
the computational burden, but presumably at the expense of conditioning on receiver CSI. The appropriate interpretation should be clear
degraded performance, would be to break a codeword of from the context.
similar demodulators to form their decisions, with substitution IV. A MPLIFY- AND -F ORWARD
of the appropriate fading realizations and noise variances.
For completeness, we include some discussion of demodula-
Similarly, given ad,r and xr , yd,r is conditionally complex tion for amplify-and-forward relay processing. Under coherent
Gaussian with density reception, amplify-and-forward has received considerable at-
tention. By contrast, we have found no results for noncoherent
CN (y; ad,r xr , Nd IK ) , (16) amplify-and-forward. As we will see in this section, there is
likely a good reason for this: it is impossible to obtain the ML
Thus the LLR (9) simplifies to detector without resorting to numeric integration. Moreover,
simple linear combiners inspired by optimal combining and
∗
2Re{ad,r (xm0 − x1 )† y} + |ad,r |2 (x†1 x1 − x†m0 xm0 ) equal-gain combining (EGC) for noncoherent receive arrays
`ˆm0 (y) =
r
.
Nd [13] appear to perform worse than direct transmission.
(17)
Under amplify-and-forward, each relay scales its received
Although (17) is a linear operation, substitution into (11) to
signal, i.e.,
form `rm (y) in general results in a nonlinear operation on the
received signal. xr = βr yr,s ,

where βr is the scaling factor at relay r. To satisfy an average

C. Noncoherent Demodulation output energy constraint per symbol, several constraints can
be imposed at the relay, e.g.,
For noncoherent demodulation, the destination receiver can-
not obtain accurate estimates of the realizations of the fading Er
βr2 ≤ (22)
coefficients ad,s and ad,r , for r = 1, 2, . . . , N . Given xs , the ||yr,s ||2
signal yd,s received by the destination directly from the source Er
is conditionally complex Gaussian with density βr2 ≤ (23)
|as,r |2 Es + K · Nr
Er
βr2 ≤ 2
 
CN y; 0, σa2d,s xs x†s + Nd IK . (18) . (24)
σas,r Es + K · Nr

Thus, the LLR (6) simplifies to The first constraint, (22), ensures that the relay output energy
  per symbol is no larger than Er with probability one. This
σ 2 † † power constraint is suitable for both coherent and noncoherent
ad,s † xm xm x1 x1
`sm (y) = y  − y scenarios, but yields models that are very difficult to analyze.
Nd2 σa2 † σa2 †
The second constraint, (23), uses receive CSI from the source-
1+ Nd xm xm
d,s
1+ Nd,s
d
x x
1 1
relay link to ensure that an average output energy per symbol is
σa2d,s x†1 x1 +Nd
" #
+ ln (19) maintained for each realization of ar,s . This power constraint
σa2d,s x†m xm +Nd is suitable for full or partially coherent scenarios in which
each relay obtains accurate receiver CSI for at least its source-
We recognize the first term of (19) as a generalized energy relay fading magnitude. The third and final constraint, (24),
detector. Again, note that the relays utilize similar LLRs in only ensures that an average output energy per symbol is
making their decisions, with substitution of the appropriate maintained, but allows for the instantaneous output power
fading and noise variances. to be much larger than the average. This power constraint
Similarly, given xr , yd,r is conditionally complex Gaussian is also suitable for both coherent and noncoherent scenarios.
with density It is particularly convenient for purposes of analysis because
  the relay scaling factor βr is a constant instead of a random
CN y; 0, σa2d,r xr x†r + Nd IK . (20) variable, as in (22) and (23).
In the following sections, we summarize the existing results
Thus, the LLR (9) simplifies to for coherent amplify-and-forward and discuss how challenging
  it is to extend these ideas to noncoherent amplify-and-forward.
σ 2 † † The key differences lie in the density of the signal received at
x m m0 x x x
1 1
`ˆrm0 (y) =
ad,r † 0
y  − y the destination through a relay path, i.e.,
Nd2 σa2 † σa2
1+ Nd xm0 xm0
d,r
1+ Nd x†1 x1
d,r

yd,r = ad,r βr (ar,s xs + zr,s ) + zd,r . (25)

σa2d,r x†1 x1 +Nd
" #
+ ln (21) A. Coherent Demodulation
σa2d,r x†m0 xm0 +Nd
For coherent demodulation under (23), we assume the desti-
Although (21) is a quadratic form in y, substitution into (11) nation receiver can obtain accurate estimates of the realizations
to form `rm (y) in general results in a non-quadratic form in of the fading coefficients ad,s , ar,s , ad,r , and the relay scaling
y. factors βr , for r = 1, 2, . . . , N . The signal yd,s is identical to
the case of coherent detect-and-forward, so that its density is the form (29). However, this essentially makes the ML detector
given by (14) and the LLR (6) again reduces to (15). unsuitable for practical implementation.
Given all the CSI along with xs , and under the power con- Since ML detection is too complex for analysis and imple-
straints (23) or (24), the received signals yd,r , r = 1, 2, . . . , N , mentation, we could consider suboptimal diversity combiners
are conditionally independent and complex Gaussian with inspired by (21). Specifically, one can consider
densities  
2 † †
σaeff †  xm xm x1 x1
CN y; ad,r βr ar,s xs , (|ad,r |2 βr2 Nr + Nd )IK , `rm (y) = 2 y − y


(26) Neff σa2 † σa2 †
1+ Neff xm xm
eff
1+ Neff
eff
x 1 x1
respectively. Thus, the LLR (7) simplifies to " #
σa2eff x†1 x1 +Neff
+ ln , (30)
2Re{g ∗ (xm − x1 )† y} + |g |2 (x†1 x1 − x†m xm ) σa2eff x†m xm +Neff
`sm (y) = ,
|ad,r |2 βr Nr + Nd
(27) where the effective fading and noise variances are
where g = ad,r βr ar,s . Unlike coherent detect-and-forward, σa2eff := σa2d,r βr2 σa2r,s (31)
the ML demodulator for coherent amplify-and-forward is
equivalent to a linear operation on the received signals. For Neff := σa2d,r βr2 Nr + Nd . (32)
the Gaussian noise channel model, performance conditioned Of course, employing (30) in (8) sacrifices optimality of
on the receive CSI is relatively straightforward to obtain; the demodulator. More concerning, however, is the fact that
however, because (27) involves nonlinear operations on the numerical investigation suggests the demodulator based upon
fading coefficients, the main challenge for coherent amplify- (30) appears to perform worse than the case of no relays, i.e.,
and-forward is in averaging performance over the densities direct transmission from the source to the destination. Similar
for the fading coefficients. Some progress in this direction has empirical conclusions have been obtained for demodulators
been obtained in [10]–[12]. based upon EGC.
B. Noncoherent Demodulation V. S IMULATIONS
Under noncoherent demodulation, neither the amplitude nor This section provides empirical simulation results for non-
the phase of the effective fading coefficients are known to coherent, detect-and-forward cooperative diversity with up to
the appropriate relay or destination receivers. Thus, only the three relays. We specialize the signal schemes and demodu-
power constraints (22) and (24) are allowed. We focus on (24) lation algorithms to the case of M -ary orthogonal signaling
throughout this section to simplify the discussion. (cf. (5)) to illustrate the results. In particular, this choice is
The signal yd,s is identical to the case of noncoherent detect- convenient because the signals are geometrically uniform [14],
and-forward, so that its density is given by (18) and the LLR and because a closed form expression exists for the symbol
(6) reduces to (19). error rate for these signals transmitted over noncoherent fading
None of signals yd,r received through the relays (cf. (25)) channels [13]. Thus, (10) is readily computable without having
are conditionally Gaussian given only the transmitted signal. to resort to extensive simulations.
To see this, we can condition on the transmitted signal xs and The simulation conditions follow the same lines as in [7],
the fading coefficient ad,r to obtain a conditionally complex [8]. Specifically, the coordinates of the whole communication
Gaussian random vector with zero-mean and covariance matrix network are normalized by the distance ld,s between the source
† and destination transceivers. Without loss of generality, the
E[yd,r yd,r |xs , ad,r ] source is assumed to be located at (0, 0), and the destination
= |ad,r |2 βr2 (σa2r,s xs x†s + Nr IK ) + Nd IK (28) located at (1, 0). For simplicity of exposition, the relays are
assumed to be located at (l, 0), 0 < l < 1. The fading
For Rayleigh fading, |ad,r |2 is exponentially distributed with variances σa2i,j are assigned using a path-loss model of the
parameter λ = σa−2d,r
. Thus, the conditional density of yd,r −α
form σa2i,j ∝ li,j , where li,j is the distance from node i to
given only xs is the average of a complex Gaussian vector node j, and α is the path-loss exponent, chosen as α = 4 for
with zero-mean and covariance matrix (28) over an exponential our results. The total network energy per transmitted symbol
density, is also normalized to unity. Specifically, we set Es = 1 for
pyd,r |xs (y|x) direct transmission; for cooperative diversity transmission, we
Z ∞ assign equal energy among the source and relays, so that
†
= CN(y; 0, E[yd,r yd,r |x, a])λe−λa da . (29) Es = Er = 1/(N + 1), r = 1, 2, . . . , N . We stress that this
0 power allocation need not be optimal in general.
As far as we know, there is no closed form solution to the Fig. 2 displays simulation results for the symbol error
integral (29), even for the case of a single dimension, i.e., rate (SER) of M -ary orthogonal signaling, with M = 4,
K = 1 so that yd,r is a scalar, or for orthogonal signal sets and N = 0, 1, 2, 3 relays. Diversity benefits of cooperative
(5). In principle, the LLR (7) can be implemented as the transmission appear as faster decay of the SER with SNR.
logarithm of the ratio of two numerically computed integrals of Similar observations about diversity gains can also be made for
 0
10
N=0 focusing on the case of noncoherent demodulation without
N=1 receiver CSI. For amplify-and-forward relay processing, little
N=2
−1
N=3 insights have been obtained in the noncoherent case. In the
10
coherent case, amplify-and-forward is convenient because the
receiver processing is linear, and there is only extra additive
−2 noise in the transmitted signal. On the other hand, diversity
PE

10
benefits of coherent amplify-and-forward are only available
with orthogonal relaying for the single source-destination
10
−3
model that we consider. Detect-and-forward relay processing
is convenient because it integrates better with existing net-
work protocol stacks and allows for more bandwidth efficient
−4
10 operation via non-orthogonal relaying. On the other hand,
0 5 10 15 20 25
SNR (dB)
taking into account the effects of relay demodulation errors
becomes intractable for moderate to large blocklengths. Both
Fig. 2. 4-FSK with noncoherent detect-and-forward and N = 0, 1, 2, 3 approaches produce thorny detection problems: either the ML
relays. Relays located in the middle.
detection rule is impossible to find or intractable to implement,
and analysis of bit-error probability is quite involved. Much
0
10
l = 0.1
more insight is needed for a more complete understanding of
−1
l = 0.5 these demodulation problems.
10 l = 0.9

ACKNOWLEDGMENT
−2
10
This work has been supported in part by the State of Indiana
PE

10
−3 through the Twenty-First Century Research and Technology
Fund and by NSF through grant ECS03-29766.
−4
10
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−5
10 [1] A. Sendonaris, E. Erkip, and B. Aazhang, “User Cooperation Diversity,
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