IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 58, NO. 7, SEPTEMBER 2009 3301
Cooperative Diversity for Intervehicular
Communication: Performance
Analysis and Optimization
Hacı
˙
Ilhan, Student Member, IEEE, Murat Uysal, Senior Member, IEEE,and
˙
Ibrahim Altunba¸s
Abstract—Although there has been a growing literature on
cooperative diversity, the current literature is mainly limited to
the Rayleigh fading channel model, which typically assumes a
wireless communication scenario with a stationary base station
antenna above rooftop level and a mobile station at street level. In
this paper, we investigate cooperative diversity for intervehicular
communication based on cascaded Nakagami fading. This channel
model provides a realistic description of an intervehicular channel
where two or more independent Nakagami fading processes are
assumed to be generated by independent groups of scatterers
around the two mobile terminals. We investigate the performance
of amplify-and-forward relaying for an intervehicular cooperative
scheme assisted by either a roadside access point or another
vehicle that acts as a relay. Our diversity analysis reveals that the
cooperative scheme is able to extract the full distributed spatial
diversity. We further formulate a power-allocation problem for
the considered scheme to optimize the power allocated to the
broadcasting and relaying phases. Performance gains up to 3 dB
are obtained through optimum power allocation, depending on the
relay location.
Index Terms—Cooperative diversity, fading channels, inter-
vehicular communication, relay-assisted transmission.
I. INTRODUCTION
I
NTERVEHICULAR communication is an integral part of
intelligent transportation systems (ITSs), which have been
receiving growing attention in recent years [1], [2]. The concept
of ITS has mainly been originated to advance transportation
safety and efficiency through dissemination of road and
traffic information, e.g., real-time updates regarding collisions,
incidents, congestion, surface, and weather conditions, and
Manuscript received May 5, 2008; revised November 5, 2008. First pub-
lished February 6, 2009; current version published August 14, 2009. This
paper was presented in part at the IEEE Vehicular Technology Conference
(VTC-Spring) [24], Singapore, May 2008, and the IEEE Vehicular Technology
Conference (VTC-Fall) [25], Calgary, AB, Canada, September 2008. The work
of H.
˙
Ilhan and
˙
I. Altunba¸s was supported by the Scientific and Technological
Research Council of Turkey under Project 107E022. The work of M. Uysal was
supported in part by the Natural Sciences and Engineering Research Council
under Collaborative Research and Development Grant CRDPJ348999-06. The
review of this paper was coordinated by Prof. W. Su.
H.
˙
Ilhan and
˙
I. Altunba¸s are with the Department of Electronics and Com-
munication Engineering, Istanbul Technical University, 34469 Istanbul, Turkey
(e-mail: ilhanh@itu.edu.tr; ibraltunbas@itu.edu.tr).
M. Uysal is with the Department of Electrical and Computer Engineering,
University of Waterloo, Waterloo, ON N2L3G1, Canada (e-mail: muysal@ece.
uwaterloo.ca).
Digital Object Identifier 10.1109/TVT.2009.2014685
coordination of vehicles at critical points such as blind cross-
ings and highway entries. A variety of broadband in-vehicle
applications (such as high-speed Internet access from within
the vehicular network, cooperative downloading, network gam-
ing among passengers of adjacent cars, and virtual meetings
among coworkers) is also envisioned [1]–[6] as a result of ever-
increasing dependence on the Internet.
Although there has been a growing literature on the network-
ing and application layers in vehicular networks, the relevant
literature on the physical-layer aspects is sparse. The main
challenge facing the deployment of vehicular ad hoc networks
(VANETs) indeed manifests itself as their main advantage, i.e.,
the lack of infrastructure. This makes cooperative diversity
(also known as user cooperation or cooperative communi-
cation) [7]–[9] an ideal physical-layer solution for VANETs.
Cooperative diversity exploits the broadcast nature of wireless
transmission, i.e., the cost-free possibility of the transmitted
signals being received by other than the destination node, and
thus, a source node can get help from other nodes by relaying
the information message to the destination node. The source
and its relays effectively form a virtual antenna array to exploit
spatial diversity advantages.
Cooperative diversity has extensively been investigated in
the literature [7]–[12]; however, the current results are mainly
limited to the Rayleigh fading channel model, which is com-
monly used to characterize the cellular radio systems. This
model typically assumes a wireless communication scenario
with a stationary base station antenna above rooftop level and a
mobile station at street level. On the other hand, in intervehicu-
lar communication systems, both the transmitter and receiver
antennas are in motion, and their antennas are relatively at
lower elevations, invalidating the Rayleigh fading assumption.
Various experimental results and theoretical analysis (see, e.g.,
[13], [14], and the references therein) demonstrate that the
Rayleigh channel model and the related second-order channel
statistics originally proposed for a base station-to-mobile link
fail to provide an accurate model for dynamic mobile-to-mobile
link. Instead, the cascaded (double) Rayleigh fading channel
model has been proposed [14], [15], which provides a realistic
description of an intervehicular channel where two Rayleigh
fading processes are assumed to be generated by independent
groups of scatterers around the two mobile terminals. A gen-
eralized channel model, i.e., the so-called N
∗
Nakagami, has
further been proposed in [16], which involves the product of N
Nakagami-m distributed random variables.
0018-9545/$26.00 © 2009 IEEE
3302 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 58, NO. 7, SEPTEMBER 2009
In this paper, we investigate cooperative diversity in the
context of vehicular communication. We consider a vehicle-to-
vehicle (V2V) scenario under two different scenarios. In the
first scenario, the source vehicle is assisted by another vehicle
in its vicinity. All underlying channels are modeled as cascaded
Nakagami fading (assuming N =2).
1
In the second scenario,
a roadside access point (AP) acts as a relay. Therefore, the
channel between source and destination vehicles is modeled
by cascaded Nakagami fading, whereas source-to-relay AP and
relay AP-to-destination channels are modeled by Nakagami
fading. We consider the receive diversity protocol of [7] with
amplify-and-forward relaying. We first obtain the diversity
order for these two relay-assisted vehicular scenarios through
the derivation of the pairwise error probability (PEP). Then,
building upon a union bound on the bit error rate (BER), we
formulate a power-allocation problem to determine how the
overall transmit power should be shared between broadcasting
and relaying phases for performance optimization.
The rest of this paper is organized as follows: In Section II,
we introduce the relay-assisted V2V transmission model. In
Section III, we derive the PEP expressions for two scenarios un-
der consideration and discuss the diversity order. In Section IV,
we present union bounds on the BER, which are used as ob-
jective functions for optimization, optimization procedure, and
results of optimization. In Section V, we provide Monte Carlo
simulation results to demonstrate t he error rate performance of
the relay-assisted V2V scheme for various relay locations and
compare the performance of equal power allocation (EPA) and
optimum power allocation (OPA) schemes. We finally conclude
in Section VI.
II. C
HANNEL AND TRANSMISSION MODEL
We consider a single-relay scenario in which source, relay,
and destination nodes operate in half-duplex mode and are
equipped with a single pair of transmit and receive antennas.
We study two different scenarios based on the relay type.
In the first scenario [see Fig. 1(a)], the source vehicle is
assisted by another vehicle, whereas in the second scenario
[see Fig. 1(b)], a roadside AP acts a relay. We assume an aggre-
gate channel model that takes into account both the long-term
path loss and short-term fading. This allows us to explicitly con-
sider the effects of the relay location in our transmission model.
In Fig. 1(a) and (b), d
SD
, d
SR
, and d
RD
are the distances
of source-to-destination (S → D), source-to-relay (S → R),
and relay-to-destination (R → D) links, respectively, and θ is
the angle between lines S → R and R → D. Assuming the
path loss between S → D to be unity, the relative gain of
S → R and R → D links are defined as G
SR
=(d
SD
/d
SR
)
v
and G
RD
=(d
SD
/d
RD
)
v
, respectively, where v is the path-
loss coefficient [12]. We further define the relative geometrical
gain μ = G
SR
/G
RD
(in decibels), which indicates the location
of the relay with respect to the source and destination. The
more negative this ratio (given in decibels) is, the more closely
1
For the sake of presentation, in the rest of this paper, we use the term
“Nakagami” instead of the actual term “Nakagami-m” in a similar manner
to [16].
Fig. 1. (a) Vehicle-assisted V2V communication. (b) AP-assisted V2V
communication.
the relay is placed to the destination node. On the other hand,
positive values of this ratio indicate that the relay is more close
to the source node. The particular case of μ =0dB means that
both source and destination nodes have the same distance to
the relay.
We assume that all underlying channels are quasi-static,
which is well justified for vehicular communication scenar-
ios in rush-hour traffic. In Fig. 1(a), α
SR
, α
RD
, and α
SD
represent S → R, R → D, and S → D links’ complex fad-
ing coefficients whose magnitudes h
SR
, h
RD
, and h
SD
,re-
spectively, follow the cascaded Nakagami distribution for the
vehicle-assisted scenario. These magnitudes are assumed to
be the product of statistically independent, but not necessarily
identically distributed, two Nakagami random variables [16].
Therefore, we have h
SD
= h
SD
1
h
SD
2
, h
SR
= h
SR
1
h
SR
2
, and
h
RD
= h
RD
1
h
RD
2
for S → D, S → R, and R → D links,
respectively, with the probability density function (pdf)
f
h
(h)=
2
hΓ(m
1
)Γ(m
2
)
G
2,0
0,2
m
1
m
2
h
2
Ω
1
Ω
2
−
m
1
,m
2
(1)
where the subscripts SD, SR, and RD are dropped for con-
venience. Here, G
2,0
0,2
(.| :) is the Meijer G-function, and Γ(.) is
the Gamma function [17]. In (1), m
l
, l =1, 2, is a parameter
describing the fading severity given by m
l
=Ω
2
l
/E[(h
2
l
−
Ω
l
)
2
] ≥ 1/2, with Ω
l
= E[h
2
l
] and E[.] denoting the expecta-
tion operator. Taking Ω
l
=1, one can normalize the power of
the fading process to unity. Furthermore, note that the pdf in
(1) reduces to the cascaded Rayleigh distribution when m
l
=1
[18]. In the AP-assisted scenario illustrated in Fig. 1(b), the
S → D link is still modeled as cascaded Nakagami; however,
S → R and R → D links are now subject to Nakagami fading.
This is justifiable considering that the relay node is an AP
˙
ILHAN et al.: COOPERATIVE DIVERSITY FOR INTERVEHICULAR COMMUNICATION 3303
elevated well above street level. Under this scenario, the pdf
for h
SR
and h
RD
is given by
f
h
(h)=
2m
m
Ω
m
Γ(m)
h
2m−1
exp
−
m
Ω
h
2
. (2)
Note that the Nakagami distribution encloses both the
Rayleigh and Rician distributions. For m>1,itcloselyap-
proximates the Rician distribution, which is used to model
fading channels with a line-of-sight (LOS) component. There is
a one-to-one mapping between the m parameter and the Rician
factor, which can be written as
√
m
2
− m/(m −
√
m
2
− m),
m ≥ 1 in terms of m [19].
The transmission model under consideration builds upon the
receive diversity cooperation protocol
2
[7], [12]. This protocol
effectively implements a single-input–multiple-output (SIMO)
scheme in a distributed fashion, realizing receive diversity
advantages. In the receive diversity protocol, the source node
broadcasts to the relay and destination nodes over the first
transmission phase. In the second transmission phase, only the
relay node communicates with the destination node. Therefore,
the signal transmitted both to the relay and destination nodes
over the two transmission phases is the same. Let x denote the
transmitted signal chosen from an M-ary phase-shift keying
(PSK) or M -ary quadratic-amplitude modulation (QAM) con-
stellation. Considering path-loss effects, the received signals at
the relay and destination are given as
r
R
=
2G
SR
KEh
SR
x + n
R
(3)
r
D1
=
√
2KEh
SD
x + n
D1
(4)
where n
R
and n
D
1
are the independent samples of zero-mean
complex Gaussian random variables with variance N
0
/2 per
dimension. Here, the total energy (to be used by both source and
relay terminals) is 2E during two time slots, yielding an average
power in proportion to E per time slot, i.e., assuming a unit
time duration. K is an optimization parameter that controls the
fraction of power reserved for the broadcasting phase. Setting
K =1/2 yields the EPA scheme.
The relay terminal normalizes the received signal r
R
by
a factor of
E[|r
R
|
2
]=
√
2G
SR
KE + N
0
and retransmits
the resulting signal during the second time slot. After proper
normalization, the received signal at the destination is therefore
given by [21]
r
D2
=
√
aEh
SR
h
RD
x + n
D2
(5)
where a =(2G
SR
K)/(A + h
2
RD
), with A =[1+
2G
SR
K(E/N
0
)]/2G
RD
(1 − K)(E/N
0
)], and n
D
2
is a
conditionally zero-mean complex Gaussian random variable
with variance N
0
/2 per dimension. Writing (4) and (5) in a
matrix notation, we have
[
r
D1
r
D2
]
T
r
=
h
SD
h
SR
h
RD
T
h
√
2KEx 0
0
√
aEx
X
+
n
D1
n
D2
T
n
.
(6)
2
This is referred as orthogonal amplify-and-forward relaying in [20].
III. PEP AND DIVERSITY GAIN ANALYSIS
In this section, we investigate t he diversity order for the
cooperative vehicular scheme under consideration through
the derivation of the PEP. The PEP is the building block for
the derivation of union bounds to the error rates, which will
later be used as an objective function for OPA. We assume
maximum-likelihood decoding with perfect knowledge of the
channel state information at the receiver. A Chernoff bound on
the conditional PEP is given by [22]
P (X,
ˆ
X|h) ≤ exp
−
d
2
(X,
ˆ
X|h)
4N
0
(7)
where the Euclidean distance (conditioned on fading channel
coefficients) between X and
ˆ
X is d
2
(X,
ˆ
X|h)=h(X −
ˆ
X)
(X −
ˆ
X)
H
h
H
Here, ()
H
denotes Hermitian transpose. Substi-
tuting d
2
(X,
ˆ
X|h) in (7), we have
P (X,
ˆ
X|h
SD
,h
SR
h
RD
)≤exp
−
β
2
2Kh
2
SD
+ ah
2
SR
h
2
RD
(8)
where β = |x − ˆx|
2
(E/N
0
)/2. Since the channel distributions
differ for two scenarios under consideration, we present them
separately in the following.
A. PEP for the Vehicle-Assisted Scenario
In this scenario, another vehicle in the vicinity of the source
vehicle acts as a relay. Therefore, all underlying channels
can be modeled as cascaded Nakagami fading. Let m
SD
1
,
m
SD
2
, m
SR
1
, m
SR
2
, m
RD
1
, and m
RD
2
denote the m pa-
rameters of the Nakagami random variables representing the
corresponding links. Let y
SD
, y
SR
, and y
RD
denote y
SD
=
h
2
SD
, y
SR
= h
2
SR
, and y
RD
= h
2
RD
, respectively, each with the
normalized pdf
f
y
(y)=
1
yΓ(m
1
)Γ(m
2
)
G
2,0
0,2
m
1
m
2
y|
−
m
1
,m
2
(9)
where the subscripts SD, SR and RD are dropped for conve-
nience. Averaging (8) with respect to y
SD
and using the closed-
form solution [ 17, eq. (713.1)]
∞
0
z
−ρ
exp(−γz)G
2,0
0,2
ηz|
−
b
1
,b
2
dz = γ
ρ−1
G
2,1
1,2
η
γ
ρ
b
1
,b
2
(10)
for the resulting expression, we have
P (X,
ˆ
X|y
SR
,y
RD
) ≤ exp
−
aβ
2
y
SR
y
RD
×
1
2
l=1
Γ(m
SD
l
)
G
2,1
1,2
m
SD
1
m
SD
2
Kβ
1
m
SD
1
,m
SD
2
. (11)
3304 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 58, NO. 7, SEPTEMBER 2009
Further averaging (11) over y
SR
and again using (10), we
obtain
P (X,
ˆ
X|y
RD
) ≤ G
2,1
1,2
m
SD
1
m
SD
2
Kβ
1
m
SD
1
,m
SD
2
×
1
2
l=1
Γ(m
SD
l
)Γ(m
SR
l
)
G
2,1
1,2
2m
SR
1
m
SR
2
aβy
RD
1
m
SR
1
,m
SR
2
.
(12)
Finally, averaging (12) over y
RD
, we obtain an upper bound
on the PEP as
P (X,
ˆ
X)
≤
1
2
l=1
Γ(m
SD
l
)Γ(m
SR
l
)Γ(m
RD
l
)
× G
2,1
1,2
m
SD
1
m
SD
2
Kβ
1
m
SD
1
,m
SD
2
×
∞
0
y
−1
RD
G
2,1
1,2
m
SR
1
m
SR
2
KG
SR
β
1+
A
y
RD
1
m
SR
1
,m
SR
2
× G
2,0
0,2
m
RD
1
m
RD
2
y
RD
|
−
m
RD
1
,m
RD
2
dy
RD
. (13)
To the best of our knowledge, a closed-form solution
for (13) is unfortunately not available for the general case,
yet this single integral can easily be numerically evaluated
through commercially available mathematics s oftware such as
MATLAB, Mathematica, or Maple. Furthermore, for some cer-
tain relay locations, (13) can further be simplified. For example,
consider the case when the relay is close to the destination
(i.e., μ 1). Under sufficiently high signal-to-noise ratios
(SNRs), it can be shown that (13) can be solved as
P (X,
ˆ
X) ≤G
−m
SR
1
SR
2
l=1
m
SR
l
m
SR
1
2
l=1
m
SD
l
m
SD
1
× U
m
SD
1
, 1+m
SD
1
− m
SD
2
,
m
SD
1
m
SD
2
Kβ
× U
m
SR
1
, 1+m
SR
1
− m
SR
2
,
m
SR
1
m
SR
2
KG
SR
β
× (Kβ)
−m
SD
1
−m
SR
1
(14)
where U (., ., .) is the hypergeometric U-function [17]. From
(14), one can check that this system achieves an asymptotical
diversity order of min(m
SD
1
,m
SD
2
) + min(m
SR
1
,m
SR
2
).
It can be noted that the derived PEP in (14) includes cas-
caded Rayleigh fading as a special case. Inserting m
SD
1
=
m
SD
2
= m
SR
1
= m
SR
2
= m
RD
1
= m
RD
2
=1 in (14), we
have U(1, 1,.) → 1, simplifying (14) to
P (X,
ˆ
X) ≤
4K
−2
G
SR
|x − ˆx|
4
E
N
0
−2
(15)
which yields an asymptotic diversity order of two.
Fig. 2. Effective diversity order for the V2V system over conventional and
cascaded Rayleigh and Nakagami (m =2)fading processes.
B. PEP for the AP-Assisted Scenario
In the AP-assisted case, the channel between source and
destination vehicles is modeled by cascaded Nakagami fading,
whereas source-to-relay AP and relay AP-to-destination chan-
nels are modeled by Nakagami fading. Averaging (11) with
respect to h
SR
and h
RD
, which are now Nakagami distributed,
we obtain
P (X,
ˆ
X) ≤
1
2
l=1
Γ(m
SD
l
)Γ(m
SR
l
)Γ(m
RD
l
)
× G
2,1
1,2
m
SD
1
m
SD
2
Kβ
1
m
SD
1
,m
SD
2
×
∞
0
y
−1
RD
G
1,1
1,1
m
SR
KG
SR
β
1+
A
y
RD
1
m
SR
× G
1,0
0,1
m
RD
y
RD
|
−
m
RD
dy
RD
. (16)
Under μ 1 and sufficiently high SNRs, we have
P (X,
ˆ
X) ≤(m
SR
)
m
SR
G
−m
SR
SR
2
l=1
m
SD
l
m
SD
1
× U
m
SD
1
, 1+m
SD
1
− m
SD
2
,
m
SD
1
m
SD
2
Kβ
× (Kβ)
−m
SD
1
−m
SR
. (17)
From (17), it can be shown that an asymptotic diversity
order of min(m
SD
1
,m
SD
2
)+m
SR
is available. Further note
that the derived PEP in (17) includes cascaded Rayleigh fad-
ing as a special case. Inserting m
SD
1
= m
SD
2
= m
SR
=1,
(17) simplifies to (15), which yields an asymptotic diversity
order of two.
To have a better understanding of how much diversity gain
is achievable in various SNR = E/N
0
ranges, we plot in
Fig. 2 the effective (instantaneous) diversity order [18], which
is simply the slope of the derived PEP as a function of the
average SNR, i.e., −log P (X,
ˆ
X)/ log(E/N
0
). We consider
the vehicle- and AP-assisted cases for both cascaded Nakagami
and cascaded Rayleigh fading under consideration.
˙
ILHAN et al.: COOPERATIVE DIVERSITY FOR INTERVEHICULAR COMMUNICATION 3305
As shown in Fig. 2, the asymptotical diversity orders for the
vehicle-assisted scheme are 4 and 2, respectively, for cascaded
Nakagami (m =2)and cascaded Rayleigh fading. This con-
firms our earlier observation on the diversity order given by
min(m
SD
1
,m
SD
2
) + min(m
SR
1
,m
SR
2
). For the AP-assisted
scheme, the diversity orders remain the same, which can be
confirmed through min(m
SD
1
,m
SD
2
)+m
SR
.
As benchmarks, we include the performance of maximal
ratio combining (MRC) with two colocated receive anten-
nas over Nakagami (m =2), cascaded Nakagami (m =2),
Rayleigh, and cascaded Rayleigh fading channels. It is obvious
in Fig. 2 that for the MRC scheme, the effective diversity
order converges to its asymptotical values of 4 and 2 over
conventional Nakagami and Rayleigh fading, respectively. Con-
vergence gets slower for the cascaded channels, and the asymp-
totical diversity order is observed for very high SNR values.
Another benchmark is the performance of the receive diver-
sity cooperative scheme under consideration over the Rayleigh
fading channel.
3
It is observed that the relaying link (i.e., the
cascaded nature of the channel over S → R → D link) further
slows down the convergence. The performance of our vehicle-
assisted scheme suffers from both the presence of cascaded
Nakagami (or Rayleigh) fading channels in the S → D link
and the cascaded structure of two Nakagami (or Rayleigh)
fading channels over the S → R → D link. Therefore, the
convergence of the diversity order to its asymptotical value
becomes the slowest.
C. Average PEP Over the Relay Location
In this section, we investigate the average PEP for the
vehicle-assisted scenario to take into account the vehicle relay’s
movement. Normalizing the distance between the source and
the destination to unity (i.e., d
SD
=1) and assuming v =2and
θ = π,wehaveμ = G
SR
/G
RD
= ((1 − d
SR
)/d
SR
)
2
, where
d
SR
is the distance between the s ource and the relay. We also
assume that d
SR
is modeled as a uniformly distributed random
variable. The pdf of d
SR
is therefore given as f(d
SR
)=1,
0 ≤ d
SR
≤ 1. The pdf of μ can be then calculated as
f(μ)=
1
2
√
μ(1 +
√
μ)
2
, 0 ≤ μ ≤∞. (18)
Rewriting (13) in terms of μ, we obtain
P (X,
ˆ
X)
≤
1
2
l=1
Γ(m
SD
l
)Γ(m
SR
l
)Γ(m
RD
l
)
×G
2,1
1,2
m
SD
1
m
SD
2
Kβ
1
m
SD
1
,m
SD
2
×
∞
0
y
−1
RD
G
2,1
1,2
m
SR
1
m
SR
2
K(1+
√
μ)
2
β
1+
B
y
RD
1
m
SR
1
,m
SR
2
×G
2,0
0,2
m
RD
1
m
RD
2
y
RD
|
−
m
RD
1
,m
RD
2
dy
RD
(19)
3
The corresponding PEP can be found in [21].
Fig. 3. BER versus K (μ = −30 dB, 4-PSK, θ = π,andv =2).
where
B =
[1 + 2(1 +
√
μ)
2
K(E/N
0
)]
[2(1 + 1/
√
μ)
2
(1 − K)(E/N
0
)]
.
The average PEP can then be calculated numerically by
P (X,
ˆ
X)=
∞
0
f(μ)P (X,
ˆ
X)dμ. (20)
IV. BER-O
PTIMIZED POW E R ALLOCATION
Although EPA guarantees full asymptotical diversity, only
partial diversity gains are exploited in the practical SNR ranges,
as illustrated in Fig. 2. To further improve the performance, we
aim to optimally allocate the power between broadcasting and
relaying phases. For optimization of the power allocation, we
consider the BER as our objective function. The derived PEP
expressions constitute the building block for the derivation of
BER bounds. Specifically, a union bound on the BER is given
by [23]
P
b
≤
1
n
X
p(X)
X=
ˆ
X
q(X,
ˆ
X)P (X,
ˆ
X) (21)
where n is the number of information bits per transmis-
sion, p(X) is the probability that codeword X is transmitted,
q(X,
ˆ
X) is the number of information bit errors in choosing
another codeword
ˆ
X instead of the transmitted codeword, and
P (X,
ˆ
X) is the corresponding PEP. The specific form of BER
bounds depends on the modulation scheme. Since PEPs are
dependent on the Euclidean distance, we introduce the nota-
tion f(Δ = |x − ˆx|
2
)ˆ=P (X,
ˆ
X) to explicitly demonstrate this
dependence. Union bounds on BER for binary PSK (BPSK),
4-PSK, 16-PSK, and 16-QAM can be then expressed as
P
b,BPSK
≤f(Δ = 4) (22)
P
b,4−PSK
≤f(Δ = 2) + f(Δ = 4) (23)
