Abstract-In this paper, we investigate the resource allocation in a differentially modulated relay network. In addition to the energy optimization, we also consider location optimization to minimize the average symbol error rate (SER). The closedform solution is derived for the single-relay case, and formulas allowing numerical search are provided for multiple-relay cases. Analytical and simulated comparisons confirm that the optimized systems provide considerable improvement over the unoptimized systems, and that the minimum SER can be achieved via the joint energy-location optimization. Index Terms-Differential phase shift keying, resource management, relays, cooperative system

Senior Member, IEEE Liuqing Yang

Student Member, IEEE Woong Cho

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IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 56, NO. 4, APRIL 2008 531
Optimum Resource Allocation for
Relay Networks with Differential Modulation
Woong Cho, Student Member, IEEE, and Liuqing Yang, Senior Member, IEEE
Abstract—In this paper, we investigate the resource allocation
in a differentially modulated relay network. In addition to the
energy optimization, we also consider location optimization to
minimize the average symbol error rate (SER). The closed-
form solution is derived for the single-relay case, and formulas
allowing numerical search are provided for multiple-relay cases.
Analytical and simulated comparisons confirm that the optimized
systems provide considerable improvement over the unoptimized
systems, and that the minimum SER can be achieved via the joint
energy-location optimization.
Index Terms—Differential phase shift keying, resource man-
agement, relays, cooperative system.
I. INTRODUCTION
RELAY networks rely on virtual antenna arrays to pro-vide diversity gains without imposing antenna packag-
ing limitations [8]. To reduce communication overhead and
transceiver complexity, cooperative diversity schemes relying
on noncoherent or differential modulations have recently been
introduced to obviate the need for channel state information
(CSI) [3], [10], [11].
Optimum resource allocation emerges as an important di-
rection for the improvement of the performance and energy
efficiency in relay networks. In the context of coherent relay
networks with full CSI, the optimum power allocation was
developed in [6] for Gaussian parallel relays, the optimum
allocation of energy and bandwidth in multihop links is
considered in [4], and more recently, the opportunistic relay
selection was introduced in [2]. In this paper, we consider relay
systems with differential modulation that does not require any
CSI. Instead of focusing only on the energy allocation, we
formulate the relay location optimization as a parallel problem
to the energy allocation optimization. The resource allocation
will be optimized based on an upper bound of the overall
symbol error rate (SER) of a relay network employing the
decode-and-forward (DF) protocol. For single-relay scenario,
we derive a closed-form optimum solution. For multi-relay
cases where the relays are placed together, we provide formu-
las allowing numerical searches for the optimum solution. We
will also show that, though the optimizations can be carried
out independently, the joint energy and location optimization
gives the optimum SER performance.
Paper approved by E. Erkip, the Editor for Cooperation Diversity of IEEE
Communication Society. Manuscript received February 21, 2006; revised
December 7, 2006 and April 10, 2007. Part of the results in this paper
was presented at the IEEE Military Communications Conference, Washington,
DC, Oct. 23-25, 2006, and the IEEE International Conference on Acoustics,
Speech, and Signal Processing, Honolulu, Hawaii, Apr. 15-20, 2007. This
work is supported in part by the Office of Naval Research under grants
N00014-07-1-0868 and N00014-07-1-0193.
The authors are with the Department of Electrical and Computer Engi-
neering, University of Florida, Gainesville, FL 32611, USA (e-mail: woong-
cho@ufl.edu, lqyang@ece.ufl.edu).
Digital Object Identifier 10.1109/TCOMM.2008.060104.
II. SYSTEM MODEL
Consider a network with one source node s, L relay
nodes {rk}Lk=1, and one destination node d. The DF relaying
protocol is considered, in which the relay nodes de-modulate
the signal from the source, then re-modulate and forward to
the destination. We assume that obstacles disable a direct
transmission resulting in no direct link between the source
node and the destination node.
With the nth phase-shift keying (PSK) modulated symbol
being sn = ej2πcn/M , cn ∈ {0, . . . ,M−1}, the corresponding
transmitted signal from the source is xsn = x
s
n−1sn with initial
value xs0 = 1. Each symbol duration is partitioned into two
segments. During the first segment, the source broadcast the
symbol via a common channel to all relays, and the received
signal at the kth relay is given by
yrk,sn =
√
Eshrk,sn xsn + zrkn , k = 1, 2, . . . , L , (1)
During the second segment, the relays form estimates ŝrkn ,
differentially modulate them as xrkn , and forward to the
destination through their distinct channels. The received signal
at the destination corresponding to each relay node is given
by
yd,rkn =
√Erkhd,rkn xrkn + zdn, k = 1, 2, . . . , L. (2)
In (1) and (2), Ei is the energy per symbol at node i, the
fading coefficient hi,jn and noise z
i
n are zero-mean complex
Gaussian with variance σ2hi,j , ∀i, j ∈ {s, rk, d}, and N0,
respectively. Accordingly, the received instantaneous signal-
to-noise ratio (SNR) between the transmitter j and the receiver
i is γi,j = (|hi,jn |2Ej)/N0, and the average received SNR is
γ̄i,j = (σ2hi,jEj)/N0.
Given that the channel is slowly time-varying and the fad-
ing coefficient remains nearly invariant over two consecutive
symbols, the conditional distribution of the received signal
yn is complex Gaussian with mean yn−1sn and variance
2N0. Hence, the log likelihood function (LLF) of yn is
li,jm (yn) ={yny∗n−1e−j2πm/M}, where i, j ∈ {s, rk, d} and
m ∈ {0, 1, . . . , M − 1}. Then, the differential demodulator at
the kth relay and the destination nodes are given by
ŝrkn =e
j2πm′/M:m′=arg max
m
lrk,sm (y
rk,s
n )
=arg max
m
{(yrk,sn )∗yrk,sn−1ej2πm/M}
ŝdn =e
j2πm′/M:m′=arg max
m
L∑
k=1
ld,rkm (y
d,rk
n )
=arg max
m
L∑
k=1
{(yd,rkn )∗yd,rkn−1ej2πm/M} .
(3)
With no CSI assumed at either the relays or the destination
node, the decision rule in (3) turns out to be the differential
0090-6778/08$25.00 c© 2008 IEEE
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532 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 56, NO. 4, APRIL 2008
detection with postdetection equal gain combining (EGC) [9,
Ch. 9].
III. ERROR PERFORMANCE ANALYSIS
Let us denote the average SER at the kth relay as Pe,rk . For
differential M -ary PSK (DMPSK) signaling, the s − rk link
SER Pe,rk can be obtained as in [9, Ch. 8]. At the destination,
provided that the symbol sn is correctly demodulated at all
relays, the conditional SER Pe,d can be obtained using the
results in [7, Appendix C]. Based on these, we formulate an
upper bound on the average SER Pe at the destination as
follows:
Proposition 1 With any given Pe,rk and Pe,d, an upper bound
on Pe can be found as :
Pe ≤ P̄e = 1 −
L∏
k=1
(1 − Pe,rk)(1 − Pe,d). (4)
This is an upper bound since the cases where the rk − d
link error corrects the s − rk link error are ignored. Fig. 1
plots the SER bound as a function of both γ̄d,rk and γ̄rk,s
when L = 2, where we assume that all rk −d and s−rk links
have the same SNR; i.e., γ̄d,rk = γ̄d,r and γ̄rk,s = γ̄r,s, ∀k.
Notice that the surface flattens along the γ̄d,r axis, but keeps
descending along the γ̄r,s axis. These observations suggest
that the overall error performance of the DF based cooperative
system depends more on the s−r link than the r−d link. Such
unbalanced effects of the relay links confirm that optimum
resource allocation is critical in achieving the optimum error
performance.
IV. OPTIMUM RESOURCE ALLOCATION
In this section, we will investigate the effects of energy
allocation and relay location on the SER performance. For
analytical tractability, we consider a network topology where
all relay nodes are located at the same distance from the source
and the destination nodes; that is, Ds,rk = Ds,r and Drk,d =
Dr,d, ∀k. Even with such a topology, resource optimization
among relays is possible as in [2]. However, in the absence
of CSI, it is reasonable to assign equal energies to all relays
Erk = Er, ∀k.
To carry out the optimization, we will also exploit the rela-
tionship between the variance of the channel fading coefficient
and the inter-node distance: σ2hi,j = C ·D−νj,i , i, j ∈ {s, r, d},
where ν is the path loss exponent of the wireless channel and
C is a constant which we henceforth set to 1 without loss of
generality.
A. Energy Allocation Optimization
Problem Statement 1 For any given source, relay and des-
tination node locations (Ds,r and Dr,d), and the total energy
per symbol E , determine the optimum energy allocation Eos
and Eor which minimize P̄e in Eq. (4) while satisfying the total
energy constraint : Es + LEr = E .
Clearly, the energy constraint is equivalent to the SNR
constraint: ρ = ρs + Lρr, where ρ := E/N0, ρs := Es/N0,
and ρr := Er/N0. To gain some insight, we consider a single-
relay setup with DBPSK and establish the following result:
10
10
10
10
10
10
−4
−5
−3
−2
−1
0
γ̄d,r (dB)γ̄r,s (dB)
Fig. 1. SER bound versus γ̄d,r and γ̄r,s (L = 2, M = 2, γ̄dk,r = γ̄d,r
and γ̄rk,s = γ̄r,s,∀k).
Proposition 2 For a single-relay setup with L = 1, at given
s− r and r − d distances Ds,r and Dr,d, and under the total
energy constraint, the optimum energy allocation ρos should
satisfy Eq. (5)1 and correspondingly, ρor = ρ − ρos.
Although (5) is an exact solution for any E and N0 values
and s − r and r − d distances, its complex form does not
provide much insight. Under the high SNR assumption, we
neglect the constant terms in (5) and obtain an approximate
solution for the optimum energy allocation
ρos =
D
−ν/2
r,d
D
−ν/2
s,r + D
−ν/2
r,d
· ρ ⇔ Eos =
D
−ν/2
r,d
D
−ν/2
s,r + D
−ν/2
r,d
· E . (6)
Interestingly, this solution coincides with the optimum power
allocation obtained by minimizing the outage probability [5,
(8)] and by minimizing the error probability bound of the
coherent (de-)modulation [1, (8)].
From (6), it readily follows that the optimum energy allo-
cation ratio between the source and the relay nodes satisfies
Eos
Eor
=
(
Ds,r
Dr,d
)ν/2
. (7)
Eq. (7) reveals explicitly that the optimum energy allocation
heavily hinges upon the inter-node distances. In addition, the
path loss exponent of the wireless channel, ν , also affects the
optimum energy allocation.
B. Relay Location Optimization
Problem Statement 2 For any given transmit energies at the
source and relay nodes (Es and Er), and the path loss exponent
ν of the wireless channel, determine the optimal location of the
relays, Dos,r, which minimizes P̄e in Eq. (4) while satisfying
0 < Dos,r < Ds,d .
Considering the single-relay setup with DBPSK and apply-
ing the high-SNR approximation, we establish the following
result:
1This solution can be readily obtained using the Lagrange multiplier. The
detailed proof is omitted due to the space limit.
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CHO and YANG: OPTIMUM RESOURCE ALLOCATION FOR RELAY NETWORKS WITH DIFFERENTIAL MODULATION 533
ρos=
√√√√2D−2νr,d + D−νs,r D−νr,d (6D−νr,dρ + 5) + 2D−2νs,r (2D−2νr,d ρ2 + 3D−νr,dρ + 1)
4D−νs,r D−νr,d (D
−ν
s,r − D−νr,d )2
− 2D
−ν
r,dρ + 3
2(D−νs,r − D−νr,d )
, (5)
0 0.2 0.4 0.6 0.8 1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
D
s,r
ρ s
/ρ
 
 
Exact: ρ
s
o/ρ
Approximation: ρ
s
o/ρ
Simulated: ρ
s
o/ρ
Numerical search: D
s,r
o
Approximation: D
s,r
o
Simulated: D
s,r
o
D
s,r
o , ν=4
ρ
s
o/ρ, ν=4
ρ
s
o/ρ and D
s,r
o , ν=2
ρ
s
o/ρ, ν=1.5
D
s,r
o , ν=1.5
Fig. 2. Optimum energy allocation and relay location (ρ = 10dB,
L = 1, ν = 1.5, 2, 4).
Proposition 3 For a single-relay setup with L = 1 and the
s−d distance Ds,d, and given the prescribed transmit energy
Es and Er, the optimum location of the relay is
Dos,r =
ρ
1/(ν−1)
s
ρ
1/(ν−1)
s + ρ
1/(ν−1)
r
· Ds,d , (8)
and accordingly, Dor,d = Ds,d − Dos,r .
Alternatively, (8) can be represented as
Dos,r
Dor,d
=
(
ρs
ρr
)1/(ν−1)
=
(Es
Er
)1/(ν−1)
. (9)
Eq. (9) bears a very similar form as its counterpart for the
optimum energy allocation in (7). In fact, when the path loss
exponent ν = 2, they are identical. However, for general ν
values, (7) and (9) are quite different. Such a discrepancy is
actually very reasonable, because (7) and (9) result from two
distinct optimization problems.
C. Discussions and Extensions
The optimum energy allocation and relay location are
depicted in Fig. 2 when ρ = 10dB and L = 1, where lines
with circle mark represent the optimum energy allocation and
lines with plus mark represent the optimum relay location,
respectively. As mentioned before, these curves overlap when
ν = 2. For all cases, Fig. 2 shows that the approximated
values are nearly identical to the exact values and the Monte
Carlo simulations. For general L values, the path loss exponent
ν renders it impossible to derive the closed-form optimum
solution, even with the high SNR approximation. In such
cases, one can resort to the numerical search using the SER
bound in Proposition 1.
0 0.2 0.4 0.6 0.8 1
10
−3
10
−2
10
−1
10
0
D
s,r
A
ve
ra
ge
 S
E
R
 
 
L = 1
L = 2
L = 3
L = 4
ρ
s
=ρ
r
: Unoptimized
ρ
s
=ρ
s
o: Optimized
Fig. 3. SER comparison between relay systems with and without
energy optimization (ρ = 10dB, ν = 4).
0 0.2 0.4 0.6 0.8 1
10
−3
10
−2
10
−1
10
0
ρ
s
/ρ
A
ve
ra
ge
 S
E
R
 
 
L = 1
L = 2
L = 3
L = 4Ds,r=Dr,d: Unoptimized
Ds,r =Ds,r
o : Optimized
Fig. 4. SER comparison between relay systems with and without
location optimization (ρ = 10dB, ν = 4).
It is worth stressing that, the direct link is ignored in our
considerations. The relay location optimization is independent
of the direct link, because the relay link(s) and the direct
link are parallel for any given energy distribution. The energy
allocation, however, can be considerably affected by the in-
clusion of the direct link, especially when the relay is close to
the source node. In the extreme case, where Ds,r approaches
zero and SNR is sufficiently high, uniform energy allocation
becomes optimum.
As aforementioned, the energy and location optimizations
can be performed individually. However, the minimum SER
can only be achieved through joint energy and location
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534 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 56, NO. 4, APRIL 2008
optimization. For L = 1, Eqs. (7) and (9) suggest that,
when ν = 2, any relay location is equally good in terms of
minimizing SER, as long as the energy allocation satisfies (7);
whereas, for other ν values, it is always optimum to place
the relay at the midpoint of the s − d link. For general L
values, the joint optimization can be carried out in an iterative
manner: i) find the optimum relay location, and ii) optimize
the energy distribution based on the updated location. Our
simulations show that the SER function is generally convex,
which ensures convergence. In addition, we will see that the
uniform energy allocation is a very good starting point for the
iterative optimization.
V. SIMULATIONS
Figs. 3 and 4 depict the SER of the relay system with and
without energy and location optimizations, respectively. The
system parameters are: ρ = 10dB, L = (1, 2, 3, 4), Ds,d = 1
and ν = 4. In the system without energy optimizations, a
uniform energy allocation ρs = ρr = ρ/(L+1) is considered.
In the system without location optimization, the relays located
at the midpoint of the source-destination link are considered.
From these figures, we observe that the optimized system
universally outperforms the unoptimized system with the same
total energy.
Notice that the minima of the energy-optimized SER curves
almost coincide with those of the unoptimized system. This
implies that (near-)optimum SER can be achieved even with
the uniform energy allocation across the source and relay
nodes, provided that the relay location is carefully selected.
However, the minima of the location-optimized SER curves
are generally far from those of the upoptimized system (see
Fig. 4). These indicate that placing the relay nodes at the
midpoint cannot achieve the minimum SER even with careful
energy allocation. In addition, these observations suggest that
the uniform energy allocation is an ideal starting point for the
iterative energy-location optimization.
VI. CONCLUSIONS
In this paper, we investigated the optimum energy distribu-
tion and optimum location of relays in a system employing
differential demodulation. Our simulations confirm that both
the energy and location optimizations provide considerable
SER advantages. Without optimization, the system with more
relays may at times underperform the system with less or
no relays. We also showed that the minimum SER can be
achieved by the joint energy-location optimization, and that
the location optimization is generally more critical than the
energy optimization.
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Optimum Resource Allocation for Relay Networks with Differential Modulation

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Optimum Resource Allocation for Relay Networks with Differential Modulation

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