Inter
national
J
our
nal
of
Electrical
and
Computer
Engineering
(IJECE)
V
ol.
8,
No.
3,
June
2018,
pp.
1692
–
1700
ISSN:
2088-8708
1692
I
ns
t
it
u
t
e
o
f
A
d
v
a
nce
d
Eng
ine
e
r
i
ng
a
nd
S
cie
nce
w
w
w
.
i
a
e
s
j
o
u
r
n
a
l
.
c
o
m
P
erf
ormance
Analysis
of
Differ
ential
Beamf
orming
in
Decentralized
Netw
orks
Samer
Alabed
Department
of
Electrical
Engineering,
American
Uni
v
ersity
of
the
Middle
East,
K
uw
ait.
Article
Inf
o
Article
history:
Recei
v
ed
Sep
16,
2017
Re
vised
Feb
18,
2018
Accepted
Mar
13,
2018
K
eyw
ord:
Distrib
uted
beamforming
T
w
o-w
ay
relay
netw
orks
Distrib
uted
space-time
coding
Dif
ferential
space-time
coding
Cooperati
v
e
communications
ABSTRA
CT
This
paper
proposes
and
analyzes
a
no
v
el
dif
ferential
distrib
uted
beamforming
strate
gy
for
decentralized
tw
o-w
ay
relay
netw
orks.
In
our
strate
gy
,
the
phases
of
the
recei
v
ed
signals
at
all
relays
are
synchronized
without
requiring
channel
feedback
or
training
symbols.
Bit
error
rate
(BER)
e
xpressions
of
the
propose
d
strate
gy
are
pro
vided
for
coherent
and
dif-
ferential
M-PSK
modulation.
Upper
bounds,
lo
wer
bounds,
and
simple
approximations
of
the
BER
are
also
deri
v
ed.
Based
on
the
theoretical
and
simulated
BER
perform
ance,
the
proposed
strate
gy
of
fers
a
high
system
performance
and
lo
w
decoding
com
ple
xity
and
de-
lay
without
requiring
channel
state
information
at
an
y
transmitting
or
recei
ving
antenna.
Furthermore,
the
simple
approximation
of
the
BER
upper
bound
sho
ws
that
the
proposed
strate
gy
enjo
ys
the
full
di
v
ersity
g
ain
which
i
s
equal
to
the
number
of
transmitting
anten-
nas.
Copyright
c
2018
Institute
of
Advanced
Engineering
and
Science
.
All
rights
r
eserved.
Corresponding
A
uthor:
Name:
Samer
Alabed
Af
filiation:
Assistant
professor
Address:
Department
of
Electrical
Engineering,
American
Uni
v
ersity
of
the
Middle
East,
Block
3,
Building
1,
Eg
aila,
K
uw
ait.
Phone:
+965
2225
1400
Ext.:
1790
Email:
samer
.al-abed@aum.edu.kw
,
samer
.alabed@nt.tu-darmstadt.de
1.
INTR
ODUCTION
Depending
on
the
a
v
ailability
of
channel
state
information
(CSI)
on
the
relay
nodes,
se
v
eral
strate
gies
for
wireless
sensor
netw
orks
ha
v
e
been
recently
suggested
to
generalize
and
impro
v
e
cooperati
v
e
di
v
ersity
strate
gies
[1–15].
Some
strate
gies
are
based
on
unrealistic
assumptions
such
as
perf
ect
CSI
a
v
ailable
at
all
nodes
[15].
These
popular
strate
gies,
such
as
distrib
uted
beamforming
strate
gy
utilized
in
wi
reless
sensor
netw
orks,
e
xploit
perfect
CSI
to
coherently
process
the
relay
signals.
These
strate
gies
enjo
y
a
high
system
performance
and
lo
w
decoding
comple
xity
and
delay
.
Other
strate
gies,
such
as
the
distrib
uted
space-time
coding
(DS
TC),
consider
the
case
of
perfect
or
partial
CSI
a
v
ailable
at
the
recei
ving
antenna
only
[6–8].
Recent
strate
gies
for
wireless
sensor
netw
orks,
such
as
dif
ferential
distrib
uted
space-time
coding
(Dif
f-
DSTC)
strate
gies
[6–14],
ha
v
e
been
recently
designed
based
on
more
practical
assumption
of
no
CSI
required
to
decode
the
information
symbols.
Therefore,
these
strate
gies
do
not
associated
wit
h
an
y
o
v
erhead
i
n
v
olv
ed
in
chan-
nel
estimation.
Ho
we
v
er
,
Dif
f-DSTC
strate
gies
suf
fer
from
lo
w
system
performance
in
terms
of
bit
error
rate
and
a
comparably
high
decoding
comple
xity
and
latenc
y
.
In
[16],
another
dif
ferential
approaches
based
on
beamform-
ing
strate
gies,
i.e.,
dif
ferential
recei
v
e
beamforming
strate
gies,
ha
v
e
been
suggested
which
do
not
need
CSI
at
an
y
transmitting
or
recei
ving
antenna
to
decode
the
information
symbols
by
combining
the
dif
ferential
di
v
ersity
strat-
e
gy
and
the
recei
v
e
beamforming
strate
gy
.
Ho
we
v
er
,
these
strate
gies
require
(
R
+
1)
time
slots
to
transmit
each
information
symbol
where
R
is
the
number
of
relay
nodes.
Therefore,
the
y
suf
fer
from
lo
w
spectral
ef
ficienc
y
for
a
lar
ge
number
of
relay
nodes,
i.e.,
their
symbol
rate
is
lo
wer
than
that
of
the
con
v
entional
transmit
beamforming.
In
this
paper
,
we
propose
and
analyze
a
no
v
el
decentralized
beamforming-based
noncoherent
relaying
strate
gy
.
Our
strate
gy
can
be
vie
wed
as
a
non-tri
vial
combination
of
tw
o
multiple
antenna
strate
gies,
i.e.,
the
dif
ferential
di
v
ersity
strate
gy
and
the
distrib
uted
beamforming
strate
gy
,
in
which
the
benefits
of
both
strate
gies
[9,
10]
are
retained.
J
ournal
Homepage:
http://iaescor
e
.com/journals/inde
x.php/IJECE
I
ns
t
it
u
t
e
o
f
A
d
v
a
nce
d
Eng
ine
e
r
i
ng
a
nd
S
cie
nce
w
w
w
.
i
a
e
s
j
o
u
r
n
a
l
.
c
o
m
,
DOI:
10.11591/ijece.v8i3.pp1692-1700
Evaluation Warning : The document was created with Spire.PDF for Python.
IJECE
ISSN:
2088-8708
1693
r
e
p
l
a
c
e
m
e
n
t
s
f
1
f
2
f
R
f
0
g
1
g
2
g
R
T
1
T
2
R
1
R
2
R
R
Figure
1.
TWRN
with
R
+
2
nodes.
2.
WIRELESS
RELA
Y
NETW
ORK
MODEL
Let
us
consider
a
half-duple
x
tw
o-w
ay
relay
netw
ork
consisting
of
tw
o
single-antenna
terminals
T
1
and
T
2
that
e
xchange
their
information
via
R
single-antenna
relays,
i.e.,
R
1
;
:
:
:
;
R
R
as
sho
wn
in
Fig.
1.
W
e
assumed
that
the
channels
are
reciprocal
for
the
transmission
from
terminal
T
1
to
terminal
T
2
and
vice
v
ersa.
W
e
further
consider
the
e
xtended
block
f
ading
channel
model,
for
which
the
channels
do
not
change
within
tw
o
consecuti
v
e
transmission
blocks
and
then
change
randomly
outside
this
time
interv
al.
Further
,
we
consider
the
case
of
no
CSI
a
v
ailable
at
an
y
transmitting
or
recei
ving
antenna
in
the
whole
netw
ork
and
the
nodes
T
1
,
T
2
,
R
1
;
:
:
:
;
R
R
ha
v
e
limited
a
v
erage
transmit
po
wers
P
T
1
,
P
T
2
,
P
1
;
:
:
:
;
P
R
,
respecti
v
ely
.
Throughout
this
paper
,
jj
,
]
,
(
)
,
kk
,
and
E
fg
denote
the
a
bsolute
v
alue,
the
ar
gument
of
a
comple
x
number
,
the
comple
x
conjug
ate,
the
Frobenious
norm,
and
the
statistical
e
xpectation,
respecti
v
ely
.
3.
THE
OPTIMAL
DECENTRALIZED
COHERENT
BEAMFORMING
WITH
PERFECT
CSI
Let
us
consider
the
case
where
all
channel
coef
ficients
and
the
statistics
of
the
noise
processes
at
the
relays
and
terminals
are
perfectly
kno
wn
at
an
y
time.
In
this
part,
the
encoded
symbols
are
identical
with
the
corresponding
information
symbols,
such
that
x
(
k
)
T
t
=
s
(
k
)
T
t
(1)
where
s
(
k
)
T
t
denotes
the
information
symbol
of
the
k
th
block
transmitted
by
terminal
T
t
and
t
=
1
;
2
.
During
the
first
and
second
time
slot
of
the
k
th
transmission
block,
the
r
th
relay
recei
v
es
from
T
1
and
T
2
the
follo
wing
signals
y
(
k
)
R
1
;r
=
p
P
T
1
f
(
k
)
r
x
(
k
)
T
1
+
n
(
k
)
R
1
;r
(2)
y
(
k
)
R
2
;r
=
p
P
T
2
g
(
k
)
r
x
(
k
)
T
2
+
n
(
k
)
R
2
;r
(3)
where
n
(
k
)
R
t;r
denotes
the
noise
signal
at
the
r
th
relay
in
the
t
th
time
slot
of
the
k
th
transmission
block
and
f
(
k
)
r
and
g
(
k
)
r
stand
for
the
channel
from
T
1
to
the
r
th
relay
and
from
T
2
to
the
r
th
relay
,
respecti
v
ely
,
in
the
k
th
block.
Note
that
the
noise
is
modeled
as
an
independent
and
identically
distrib
uted
Gaussian
random
v
ariable
with
zero
mean
and
v
ariance
2
n
.
The
relay
weights
its
recei
v
ed
signal
y
(
k
)
R
1
r
and
y
(
k
)
R
2
;r
defined
in
(2)
and
(3),
by
a
beamforming
scaling
f
actor
,
w
3
and
w
4
,
respecti
v
ely
,
to
form
a
beam
steering
to
w
ards
the
destination
terminal.
The
r
th
relay
transmits
then
the
resulting
signal
t
(
k
)
R
3
=
w
3
y
(
k
)
R
1
;r
;
t
(
k
)
R
4
=
w
4
y
(
k
)
R
2
;r
(4)
in
the
third
and
forth
time
slot
of
the
k
th
transmission
block.
In
[10,
20],
it
has
been
sho
wn
that
the
optimum
amplify
and
forw
ard
(AF)
beamforming
f
actors
are
gi
v
en
by
w
3
=
c
3
g
r
f
r
;
w
4
=
c
4
f
r
g
r
;
(5)
where
c
3
=
q
P
R
(
2
+
P
T
1
j
f
r
j
2
)
j
f
r
j
2
j
g
r
j
2
and
c
4
=
q
P
R
(
2
+
P
T
2
j
g
r
j
2
)
j
g
r
j
2
j
f
r
j
2
.
Note
that
the
optimum
relay
beamforming
f
actors
w
3
and
w
4
require
perfect
kno
wledge
of
the
instantaneous
channel
coef
ficients
f
r
and
g
r
.
Note
that
channel
coef
ficients
can
be
estimated
by
sending
pilot
symbols
through
the
communications
between
the
terminals,
ho
we
v
er
,
this
es
timation
process
results
in
a
reduced
throughput
and
additional
o
v
erhead.
Another
dra
wback
of
this
process
is
that
more
pilot
symbols
are
required
if
the
channel
changes
rapidly
.
P
erformance
Analysis
of
Dif
fer
ential
Beamforming
in
Decentr
alized
Networks(Samer
Alabed)
Evaluation Warning : The document was created with Spire.PDF for Python.
1694
ISSN:
2088-8708
4.
THE
DECENTRALIZED
NON-COHERENT
BEAMFORMING
Let
us
consider
the
case
of
no
CSI
a
v
ailable
at
an
y
transmitting
or
recei
ving
antenna.
T
o
f
acilitate
coherent
processing
at
the
relays
and
the
destinations,
T
t
transmits
in
the
first
time
slot
of
the
k
th
block
the
dif
ferentially
encoded
symbol
x
(
k
)
T
t
to
the
relays,
gi
v
en
by
x
(
k
)
T
t
=
x
(
k
1)
T
t
s
(
k
)
T
t
:
(6)
In
the
first
tw
o
time
slots
of
the
k
th
transmission
block,
the
recei
v
ed
signals
at
the
r
th
relay
are
gi
v
en
by
y
(
k
)
R
1
;r
=
p
P
T
1
f
(
k
)
r
x
(
k
)
T
1
+
n
(
k
)
R
1
;r
;
(7)
y
(
k
)
R
2
;r
=
p
P
T
2
g
(
k
)
r
x
(
k
)
T
2
+
n
(
k
)
R
2
;r
(8)
where
n
(
k
)
R
1
;r
and
n
(
k
)
R
2
;r
denote
the
noise
at
the
r
th
relay
in
the
first
and
second
time
slot
of
the
k
th
transmission
block
and
f
(
k
)
r
and
g
(
k
)
r
stand
for
the
channel
from
T
1
to
the
r
th
relay
and
from
T
2
to
the
r
th
relay
,
respecti
v
ely
,
in
the
k
th
block.
Similar
to
(4),
in
the
third
and
fourth
time
slot
of
the
k
th
block,
the
r
th
relay
weights
its
recei
v
ed
signals
y
(
k
)
R
1
;r
and
y
(
k
)
R
2
;r
by
weighting
f
actors
w
3
and
w
4
and
transmits
the
resulting
signals
t
(
k
)
R
3
;r
=
w
(
k
)
R
3
;r
y
(
k
)
R
1
;r
;
t
(
k
)
R
4
;r
=
w
(
k
)
R
4
;r
y
(
k
)
R
2
;r
;
(9)
respecti
v
ely
.
In
the
follo
wing,
let
us
consider
the
block
f
ading
model,
i.e.,
f
(
k
1)
r
=
f
(
k
)
r
and
g
(
k
1)
r
=
g
(
k
)
r
and
consider
also
the
signals
recei
v
ed
at
T
2
.
The
recei
v
ed
signal
at
T
1
can
be
reco
v
ered
correspondingly
.
During
the
third
time
slot,
the
recei
v
ed
signal
at
T
2
is
gi
v
en
by
y
(
k
)
T
2
=
R
X
r
=1
g
r
t
(
k
)
R
3
;r
=
g
r
w
(
k
)
R
3
;r
y
(
k
)
R
1
;r
:
(10)
From
(5),
the
optimal
v
alue
of
w
R
3
;r
,
r
=
1
;
2
;
;
R
,
whi
ch
leads
to
a
coherent
superposition
of
the
signals
from
all
relays
and
maximizes
the
signal
to
noise
ratio
(SNR)
at
both
terminals,
is
e
xpressed
as
w
R
3
;r
=
c
R
3
;r
g
r
f
r
=
^
c
R
3
;r
e
j
(
k
)
R
1
;r
(11)
where
c
R
3
;r
=
q
P
r
(
2
+
P
T
1
j
f
r
j
2
)
j
f
r
j
2
j
g
r
j
2
and
^
c
R
3
;r
=
q
P
r
(
2
+
P
T
1
j
f
r
j
2
)
are
scaling
f
actors
to
maintain
the
po
wer
constraint
and
(
k
)
R
1
;r
=
f
r
g
r
j
f
r
jj
g
r
j
=
]
f
(
k
)
r
+
]
g
(
k
)
r
.
In
the
proposed
strate
gy
,
it
is
assumed
that
t
here
is
no
CSI
a
v
ailable
at
an
y
transmitting
or
recei
ving
antenna.
Where
as,
to
ensure
coherent
reception,
the
phase
rotation
(
k
)
R
t;r
applied
at
the
r
th
relay
in
the
k
th
block
must
be
as
close
as
possible
to
its
optimal
v
alue
opt
R
t;r
defined
in
(11).
In
the
follo
wing,
we
proposed
an
ef
ficient
encoding
strate
gy
performed
at
the
relays
in
which
the
phases
of
their
recei
v
ed
signals
are
adjus
ted
and
a
beam
steering
to
w
ards
the
destination
terminal
is
formed
without
requiring
CSI.
Making
use
of
the
recei
v
ed
signals
at
each
relay
from
both
terminals,
(
k
)
R
1
;r
can
be
e
xpressed
as
(
k
)
R
1
;r
=
]
y
(
k
1)
R
1
;r
+
]
y
(
k
)
R
2
;r
:
(12)
F
or
suf
ficiently
lar
ge
SNR
and
from
(7)
and
(8),
]
y
(
k
)
R
1
;r
]
f
(
k
)
r
+
]
x
(
k
)
R
1
;r
and
]
y
(
k
)
R
2
;r
]
g
(
k
)
r
+
]
x
(
k
)
R
2
;r
.
In
this
case,
equation
(12)
can
be
e
xpressed
as
^
(
k
)
R
1
;r
]
f
(
k
1)
r
+
]
g
(
k
)
r
]
x
(
k
1)
T
1
+
]
x
(
k
)
T
2
(13)
where
the
phase
term
]
x
(
k
1)
T
1
+
]
x
(
k
)
T
2
in
(13)
is
a
constant.
F
or
high
SNR,
the
ef
fect
of
the
noise
on
the
phase-
shift
becomes
insignificant
and
therefore,
^
(
k
)
R
1
;r
(
k
)
R
1
;r
.
Making
use
of
the
e
xtended
block
f
ading
assumption,
i.e.,
the
channels
do
not
change
o
v
er
tw
o
consecuti
v
e
transmission
blocks,
we
can
use
f
r
and
g
r
to
represent
f
(
k
)
r
and
g
(
k
)
r
,
respecti
v
ely
,
where,
g
(
k
)
r
=
g
(
k
1)
r
=
g
r
,
f
(
k
)
r
=
f
(
k
1)
r
=
f
r
.
F
or
suf
ficiently
lar
ge
S
NR,
equation
(10)
can
be
e
xpressed
as
y
(
k
)
T
2
R
X
r
=1
R
1
;r
j
f
(
k
)
r
jj
g
(
k
)
r
j
e
j
]
x
(
k
)
T
2
s
(
k
)
T
1
+
w
(
k
)
T
2
(14)
IJECE
V
ol.
8,
No.
3,
June
2018:
1692
–
1700
Evaluation Warning : The document was created with Spire.PDF for Python.
IJECE
ISSN:
2088-8708
1695
where
w
(
k
)
T
2
=
R
X
r
=1
R
1
;r
g
(
k
)
r
e
(
j
(
k
)
R
1
;r
)
n
(
k
)
R
1
;r
+
n
(
k
)
T
2
:
(15)
In
the
case
that
the
direct
link
between
the
tw
o
terminals,
T
1
and
T
2
,
is
a
v
aila
ble,
the
recei
v
ed
signal
at
terminal
T
2
during
the
first
transmission
phase
is
gi
v
en
by
y
(
k
)
T
2
;
dl
=
p
P
T
1
f
0
x
(
k
)
T
1
+
n
(
k
)
T
2
(16)
where
n
(
k
)
T
2
denotes
the
the
recei
v
er
noise
of
the
k
th
block
at
terminal
T
2
and
x
(
k
)
T
1
is
defined
in
(6).
Making
use
of
the
recei
v
ed
signal
y
(
k
)
T
2
;
dl
defined
in
(16),
the
decoder
at
terminal
T
2
can
be
e
xpressed
as
arg
m
in
s
(
k
)
8
<
:
y
(
k
)
T
2
e
(
j
]
x
(
k
)
T
2
)
y
(
k
1)
T
2
e
(
j
]
x
(
k
1)
T
2
)
s
(
k
)
2
+
y
(
k
)
T
2
;
dl
y
(
k
1)
T
2
;
dl
s
(
k
)
2
9
=
;
:
(17)
5.
BER
PERFORMANCE
AN
AL
YSIS
5.1.
Non-coher
ent
technique
In
this
section,
the
theoret
ical
performance
of
the
proposed
di
f
ferential
strate
gy
in
terms
of
BER
based
on
the
assumptions
in
Sec.
2.
and
assuming
no
CSI
a
v
ailable
at
an
y
transmitting
or
recei
ving
antenna
is
introduced.
The
general
conditional
BER
e
xpression
of
coherent
and
non-coherent
techniques
for
(
R
+
1)
independent
f
ading
paths
between
the
communicating
terminals
can
be
approximated
as
[16,
18,
19]
P
b
(
)
1
2
2(
R
+1)
Z
f
(
)
exp(
(
)
)
d
(18)
where
f
(
)=
b
2
2
(
)
R
+1
X
r
=1
2
R
+
1
R
R
R
+2
cos
R
(
+
2
)
(
R
+1
R
+1
)
cos
(
R
+
1)(
+
2
)
;
(19)
(
)=
b
2
(1
+
2
sin(
)
+
2
)
2
;
(20)
=
a=b
is
constant,
the
v
alues
of
a
and
b
depend
on
the
modulation
order
,
e.g.,
for
4-PSK,
a
=
p
2
p
2
and
b
=
p
2
+
p
2
[18],
and
=
s
+
R
P
r
=1
r
where
s
and
r
denote
the
instantaneous
SNR
for
the
direct
link
between
T
1
and
T
2
and
the
link
between
T
1
and
T
2
via
the
r
th
relay
,
respecti
v
ely
,
such
that
r
=
P
T
1
P
T
2
P
r
j
f
r
j
2
j
g
r
j
2
2
n
(2
P
T
2
P
r
j
g
r
j
2
+
P
T
1
P
r
j
f
r
j
2
+
P
T
1
P
T
2
2
f
r
+
P
T
2
2
n
)
(21)
s
=
P
T
1
j
f
0
j
2
2
2
n
:
(22)
The
a
v
erage
BER
is
obtai
ned
by
a
v
eraging
the
conditional
BER
e
xpression
P
b
(
)
gi
v
en
in
(18)
o
v
er
the
random
v
ariables
using
the
moment
generation
function
method
[16,
18],
such
that
P
b
1
2
2(
R
+1)
Z
f
(
)
M
s
(
)
R
Y
r
=1
M
r
(
)
d
(23)
where
M
i
(
)
denotes
the
moment
generation
function
of
the
instantaneous
SNR
i
,
i
2
f
s;
1
;
;
R
g
.
M
r
(
)
can
be
obtained
through
inte
gration
o
v
er
tw
o
e
xponential
random
v
ariables
j
f
r
j
2
and
j
g
r
j
2
,
such
that
M
r
(
)
=
1
1
+
K
f
1
+
A
(
)
Z
1
0
exp(
u=
2
g
r
)
u
+
R
r
(
)
du
!
(24)
P
erformance
Analysis
of
Dif
fer
ential
Beamforming
in
Decentr
alized
Networks(Samer
Alabed)
Evaluation Warning : The document was created with Spire.PDF for Python.
1696
ISSN:
2088-8708
where
A
(
)=
K
f
2
P
T
2
(
K
f
+
1)
P
T
1
P
T
2
2
f
r
+
P
T
1
P
r
2
f
r
+
P
T
2
2
n
P
r
1
2
g
r
;
(25)
R
r
(
)
=
P
T
1
P
T
2
2
f
r
+
P
T
1
P
r
2
f
r
+
P
T
2
2
n
2
P
r
P
T
2
(1
+
K
f
)
;
(26)
K
f
=
K
d
f
;
K
d
f
=
(
)
P
T
1
2
f
r
2
2
n
:
(27)
Similar
to
(24),
to
obtain
M
s
(
)
,
we
inte
grate
o
v
er
the
e
xponential
random
v
ariable
j
f
0
j
2
to
get
M
s
(
)
=
1
1
+
(
)
P
T
1
2
f
r
2
2
n
:
(28)
Substituting
=
=
2
into
(20),
(
)
can
be
bounded
as
(
)
b
2
(1
+
)
2
=
2
.
Hence,
the
BER
is
upper
bounded
by
P
b
1
2
2(
R
+1)
Z
f
(
)
B
(
)
R
Y
r
=1
1
1
+
K
f
1
+
A
(
)
Z
1
0
exp(
u=
2
g
r
)
u
+
R
r
;min
(
)
du
!
d
(29)
where
B
(
)
=
1
1
+
K
0
;
(30)
K
0
=
K
d
0
;
(31)
K
d
0
=
(
)
P
T
1
2
f
0
2
2
n
;
(32)
R
r
;min
(
)
=
P
T
1
P
T
2
2
f
r
+
P
T
1
P
r
2
f
r
+
P
T
2
2
n
2
P
r
P
T
2
1
+
P
T
1
2
f
r
b
2
(1
+
)
2
4
2
n
!
1
:
(33)
In
the
case
that
the
SNR
is
lar
ge,
hence
K
f
>>
1
and
K
0
>>
1
,
we
can
approximate
the
terms
1
=
(1
+
K
f
)
in
(29)
by
1
=K
f
and
1
=
(1
+
K
0
)
in
(31)
by
1
=K
0
.
F
or
the
sak
e
of
simplicity
,
let
us
further
assume
that
f
0
=
f
1
=
=
f
R
=
f
.
In
this
case,
the
simple
approximation
of
the
BER
upper
bound
can
be
e
xpressed
as
P
b
2
2
n
P
T
1
2
f
!
(
R
+1)
Z
max
(
)
(34)
where
Z
max
(
)
=
1
2
2(
R
+1)
Z
f
(
)
(
)
(
R
+1)
1
+
A
(
)
Z
1
0
exp(
u=
2
g
r
)
u
+
R
r
;min
(
)
du
!
d
:
(35)
Similar
to
(29),
(
)
can
be
bounded
by
substituting
=
=
2
into
(20)
where
(
)
(
b
2
(1
)
2
=
2)
.
Hence,
the
BER
is
lo
wer
bounded
by
P
b
1
2
2(
R
+1)
Z
f
(
)
B
(
)
R
Y
r
=1
1
1
+
K
f
1
+
A
(
)
Z
1
0
exp(
u=
2
g
r
)
u
+
R
r
;max
(
)
du
!
d
(36)
where
R
r
;max
(
)
=
P
T
1
P
T
2
2
f
r
+
P
T
1
P
r
2
f
r
+
P
T
2
2
n
2
P
r
P
T
2
1
+
P
T
1
2
f
r
b
2
(1
)
2
4
2
n
!
1
:
(37)
IJECE
V
ol.
8,
No.
3,
June
2018:
1692
–
1700
Evaluation Warning : The document was created with Spire.PDF for Python.
IJECE
ISSN:
2088-8708
1697
F
or
suf
ficiently
lar
ge
SNR
and
using
t
he
same
approximations
and
assumptions
as
in
(34),
we
obtain
the
simple
approximation
of
the
BER
lo
wer
bound
as
P
b
2
2
n
P
T
1
2
f
!
(
R
+1)
Z
min
(
)
(38)
where
Z
min
(
)
=
1
2
2(
R
+1)
Z
f
(
)
(
)
(
R
+1)
1
+
A
(
)
Z
1
0
exp(
u=
2
g
r
)
u
+
R
r
;max
(
)
du
!
d
:
(39)
From
(34)
and
(38),
(
R
+
1)
and
Z
(
)
denote
the
di
v
ersity
g
ain
and
the
coding
g
ain,
respecti
v
ely
,
where
it
can
be
observ
ed
that
the
full
di
v
ersity
order
(
R
+
1)
can
be
achie
v
ed
when
R
relay
nodes
are
used
in
the
netw
ork.
5.2.
Coher
ent
technique
In
this
section,
the
theoretical
performance
of
the
proposed
strate
gy
in
terms
of
BER
using
the
same
assumptions
as
in
Sec.
2.
and
assuming
perfect
CSI
a
v
ailable
at
all
transmitting
and
recei
ving
antenna
is
proposed.
Similar
to
Sec.
5.1.,
the
general
conditional
BER
e
xpression
P
b
(
)
is
gi
v
en
in
(18).
F
or
the
coherent
technique,
c
=
c
s
+
R
P
r
=1
c
r
with
c
r
=
P
T
1
P
r
j
f
r
j
2
j
g
r
j
2
2
n
(
P
r
j
g
r
j
2
+
P
T
1
2
f
r
+
2
n
)
;
(40)
and
c
s
=
2
s
where
s
is
defined
in
(22).
After
a
v
eraging
the
conditional
BER
e
xpression
P
b
(
)
defined
i
n
(18)
o
v
er
the
Rayleigh
distrib
uted
random
v
ariables
similar
as
in
Sec.
5.1.,
the
a
v
erage
BER
(
P
b
)
can
be
approximated
as
in
(23).
Similar
to
Sec.
5.1.,
M
s
(
)
,
obtained
by
a
v
eraging
o
v
er
the
e
xponential
random
v
ariable
j
f
0
j
2
,
is
e
xpressed
in
(28)
and
M
r
(
)
,
obtained
by
first
a
v
eraging
o
v
er
the
e
xponential
random
v
ariables
j
f
r
j
2
and
then
a
v
eraging
o
v
er
the
e
xponential
random
v
ariables
j
g
r
j
2
,
is
e
xpressed
in
(24)
where
K
f
=
2
K
d
f
,
K
d
f
is
defined
in
(27),
A
(
)
=
K
f
(
K
f
+
2)
P
T
1
2
f
r
+
2
n
P
r
1
2
g
r
;
(41)
and
R
r
(
)
=
P
T
1
2
f
r
+
2
n
P
r
(1
+
K
f
)
:
(42)
Similarly
as
in
Sec.
5.1.,
the
upper
bound
BER,
obtained
by
substituting
=
=
2
into
(20),
is
defined
in
(29)
where
K
0
=
2
K
d
0
,
K
d
0
is
defined
in
(32),
and
R
r
;min
(
)
=
P
T
1
2
f
r
+
2
n
P
r
1
+
P
T
1
2
f
r
b
2
(1
+
)
2
2
2
n
!
1
:
(43)
F
or
suf
ficiently
lar
ge
SNR
and
using
the
same
approximations
and
assumptions
as
in
(34),
the
simple
approximation
of
the
BER
upper
bound
can
be
e
xpressed
as
P
b
2
n
P
T
1
2
f
0
!
(
R
+1)
Z
max
(
)
(44)
where
Z
max
(
)
is
defined
in
(35).
The
lo
wer
bound
BER,
obtained
by
substituting
=
=
2
into
(20),
is
defined
in
(29)
where
K
0
=
2
K
d
0
,
K
d
0
and
A
(
)
are
defined
in
(32)
and
(41),
respecti
v
ely
,
and
R
r
;max
(
)
=
P
T
1
2
f
r
+
2
n
P
r
1
+
P
T
1
2
f
r
b
2
(1
)
2
2
2
n
!
1
:
(45)
P
erformance
Analysis
of
Dif
fer
ential
Beamforming
in
Decentr
alized
Networks(Samer
Alabed)
Evaluation Warning : The document was created with Spire.PDF for Python.
1698
ISSN:
2088-8708
In
the
case
that
the
SNR
is
lar
ge
and
using
the
same
approximations
and
assumptions
as
in
(34),
the
simple
approx-
imation
of
the
BER
lo
wer
bound
can
be
e
xpressed
as
P
b
2
n
P
T
1
2
f
0
!
(
R
+1)
Z
min
(
)
(46)
where
Z
min
(
)
is
defined
in
(39).
Similar
to
Sec.
5.1.
and
from
(44)
and
(46),
it
can
also
be
observ
ed
that
we
can
achie
v
e
the
di
v
ersity
order
of
R
+
1
when
R
relay
nodes
are
used
in
the
netw
ork.
6.
RESUL
TS
AND
AN
AL
YSIS
In
our
simulations,
we
assume
a
relay
netw
ork
with
independent
flat
Rayleigh
f
ading
channels.
In
Fig.
2,
we
compare
the
proposed
non-coherent
strate
gy
with
the
four
-phase
coherent
and
non-coherent
DSTC
strate
gy
[12,
13]
us
ing
the
Alamouti
[19]
and
the
random
unitary
code,
the
optimal
coherent
distrib
uted
transmit
beamforming
strate
gy
,
the
strate
gy
proposed
in
[17],
and
the
tw
o-phase
Dif
f-DSTC
strate
gy
for
TWRNs
[14].
T
o
f
airly
compare
the
performance
of
all
strate
gies,
the
same
total
transmitted
po
wer
(
P
T
=
P
T
1
+
P
T
2
+
R
P
r
=1
P
r
,
where
P
T
1
=
P
T
2
=
R
P
r
=1
P
r
,
P
1
=
P
2
=
=
P
R
)
and
bit
rate
are
used.
0
5
10
15
20
25
30
35
10
−2
10
−1
10
0
B
E
R
S
N
R
(
d
B
)
D
i
f
.
4
-
p
h
a
s
e
s
c
h
e
m
e
(
A
l
a
m
o
u
t
i
)
[
8
]
C
o
h
.
4
-
p
h
a
s
e
s
c
h
e
m
e
(
A
l
a
m
o
u
t
i
)
[
9
]
G
e
n
e
r
a
l
r
a
n
k
b
e
a
m
f
o
r
m
i
n
g
[
1
3
]
D
i
f
.
4
-
p
h
a
s
e
s
c
h
e
m
e
(
r
a
n
d
o
m
)
[
8
]
C
o
h
.
4
-
p
h
a
s
e
s
c
h
e
m
e
(
r
a
n
d
o
m
)
[
9
]
P
r
o
p
o
s
e
d
D
i
f
.
2
-
p
h
a
s
e
s
c
h
e
m
e
(
A
l
a
m
o
u
t
i
)
[
1
0
]
B
e
a
m
f
o
r
m
i
n
g
w
i
t
h
p
e
r
f
e
c
t
C
S
I
Figure
2.
BER
v
ersus
SNR
for
dif
ferent
coherent
and
dif
ferential
strate
gies
with
R
=
2
and
a
rate
of
0
:
5
bpcu.
It
is
observ
ed
that
the
proposed
strate
gy
outperforms
the
s
tate-of-the-art
t
w
o-
and
four
-phase
strate
gies
and
the
dif
fer
ence
between
the
coherent
and
non-coherent
distrib
uted
strate
gy
for
the
random
unitary
and
Alamouti
scheme
is
around
3
dB.
Similarly
,
there
e
xists
a
3
dB
mar
gin
between
the
proposed
non-coherent
strate
gy
and
the
distrib
uted
beamforming
strate
gy
.
This
means
that
the
proposed
strate
gy
achie
v
es
full
di
v
ersi
ty
g
ain
with
coding
g
ain
only
3
dB
less
than
that
of
a
system
which
requires
full
CSI
at
all
transmitting
and
recei
ving
trans
cei
v
ers.
The
3
dB
loss
in
performance
can
be
e
xplained
by
the
absence
of
CSI
at
all
transmitting
and
recei
ving
antennas.
In
Fig.
3,
we
consider
a
relay
netw
ork
with
one
relay
when
there
is
a
direct
link
between
the
communicating
terminals.
In
this
figure,
we
sho
w
the
theoretical
and
simulated
BER
performance
of
the
proposed
s
trate
gy
at
T
2
v
ersus
the
transmitted
SNR
with
and
without
CSI
and
using
4-PSK
modulation.
From
Fig.
3,
it
is
observ
ed
that
the
simulated
BER
per
formance
of
our
proposed
strate
gy
with
and
without
CSI
is
v
ery
close
to
the
theoretical
BER
performance
obtained
from
the
e
xpressions
deri
v
e
in
Sec.
5..
IJECE
V
ol.
8,
No.
3,
June
2018:
1692
–
1700
Evaluation Warning : The document was created with Spire.PDF for Python.
IJECE
ISSN:
2088-8708
1699
0
5
10
15
20
25
30
10
−5
10
−4
10
−3
10
−2
10
−1
10
0
B
E
R
S
N
R
(
d
B
)
C
o
h
e
r
e
n
t
4
-
P
S
K
-
s
i
m
u
l
a
t
i
o
n
N
o
n
-
c
o
h
e
r
e
n
t
4
-
P
S
K
-
s
i
m
u
l
a
t
i
o
n
B
E
R
e
x
a
c
t
-
a
n
a
l
y
t
i
c
a
l
-
c
o
h
e
r
e
n
t
(
1
8
)
B
E
R
e
x
a
c
t
-
a
n
a
l
y
t
i
c
a
l
-
n
o
n
-
c
o
h
e
r
e
n
t
(
2
3
)
B
E
R
l
o
w
e
r
b
o
u
n
d
-
a
n
a
l
y
t
i
c
a
l
-
c
o
h
e
r
e
n
t
(
3
6
)
B
E
R
u
p
p
e
r
b
o
u
n
d
-
a
n
a
l
y
t
i
c
a
l
-
c
o
h
e
r
e
n
t
(
2
9
)
S
i
m
p
l
e
B
E
R
l
o
w
e
r
b
o
u
n
d
-
a
n
a
l
y
t
i
c
a
l
-
c
o
h
e
r
e
n
t
(
3
8
)
S
i
m
p
l
e
B
E
R
u
p
p
e
r
b
o
u
n
d
-
a
n
a
l
y
t
i
c
a
l
-
c
o
h
e
r
e
n
t
(
3
4
)
B
E
R
l
o
w
e
r
b
o
u
n
d
-
a
n
a
l
y
t
i
c
a
l
-
n
o
n
-
c
o
h
e
r
e
n
t
(
3
6
)
B
E
R
u
p
p
e
r
b
o
u
n
d
-
a
n
a
l
y
t
i
c
a
l
-
n
o
n
-
c
o
h
e
r
e
n
t
(
2
9
)
S
i
m
p
l
e
B
E
R
l
o
w
e
r
b
o
u
n
d
-
a
n
a
l
y
t
i
c
a
l
-
n
o
n
-
c
o
h
e
r
e
n
t
(
3
8
)
S
i
m
p
l
e
B
E
R
u
p
p
e
r
b
o
u
n
d
-
a
n
a
l
y
t
i
c
a
l
-
n
o
n
-
c
o
h
e
r
e
n
t
(
3
4
)
Figure
3.
BER
v
ersus
SNR
using
4-PSK.
7.
CONCLUSION
In
this
paper
,
we
propose
and
analyze
a
no
v
el
dif
ferential
distrib
uted
transmit
beamforming
strate
gy
for
decentralized
tw
o-w
ay
sensor
netw
orks.
In
our
proposed
strate
gy
,
the
phases
of
the
recei
v
ed
signals
at
all
relay
nodes
are
synchronized
without
requiring
channel
feedback
or
training
symbols
and
with
symbol
rate
equi
v
alent
to
that
of
the
con
v
entional
transmit
beamforming
strate
gy
.
BER
e
xpressions
of
the
proposed
strat
e
gy
are
pro
vided
for
coherent
and
dif
ferential
M-PSK
modulation.
Upper
bounds,
lo
wer
bounds,
and
simple
approximations
of
the
BER
are
also
deri
v
ed.
The
simple
approximation
of
the
BER
upper
bound
sho
ws
that
the
proposed
strate
gy
enjo
ys
the
full
di
v
ersity
g
ain
which
is
equal
to
the
number
of
transmitting
antennas.
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BIOGRAPHY
OF
A
UTHOR
Samer
Alabed
joined
American
Uni
v
ersity
of
the
Middle
East
as
an
assistant
professor
of
electri-
cal
and
computer
engineering
in
2015.
He
w
as
a
researcher
in
the
communication
s
ystems
group
at
Darmstadt
Uni
v
ersity
of
T
echnology
,
Darmstadt,
Ge
rman
y
from
2008
to
2015.
He
recei
v
ed
his
PhD
de
gree
in
electrical
engineering
and
information
technology
with
great
honor
(”magna
cum
laude”),
from
Darmstadt
Uni
v
ersity
of
T
echnology
,
Darms
tadt,
German
y
and
his
Bachelor
and
Master
de
gree
with
grea
t
honor
.
During
the
last
13
years,
he
has
w
ork
ed
as
an
assistant
professor
,
(post-doctoral)
researcher
,
and
lecturer
in
s
e
v
eral
uni
v
ersities
in
German
y
and
Middle
East
where
he
has
taught
more
than
50
courses
in
Electrical,
Electronic,
Communication,
and
Computer
Engi-
neering
and
supervised
tens
of
master
theses
and
se
v
eral
PhD
students.
Dr
.
Alabed
recei
v
ed
se
v
eral
a
w
ards
from
IEE,
IEEE,
D
AAD
...
etc.,
where
the
last
one
w
as
the
best
paper
a
w
ard
from
the
International
IEEE
WSA
in
March,
2015.
Dr
.
Alabed
has
w
ork
ed
as
a
researcher
in
se
v
eral
uni
v
er
-
sities
and
companies
and
w
as
in
vited
to
man
y
conferences
and
w
orkshops
in
Europe,
US,
and
North
Africa.
The
main
idea
of
his
research
is
to
de
v
elop
adv
anced
DSP
algorithms
in
the
area
of
wireless
communication
systems
a
nd
netw
orks
including
(Massi
v
e)
MIMO
systems,
distrib
uted
systems,
co-operati
v
e
communications,
relay
netw
orks,
space-time
block
and
trellis
coding,
dif
ferential
and
blind
mul
ti-antenna
techniques,
MIMO
channel
estimation,
MIMO
decoders,
channel
coding
and
modulation
techniques,
distrib
uted
communication
systems,
tw
o-w
ay
relaying,
baseband
communi-
cations,
multi-carrier
transmission
(OFDM),
modeling
of
wireless
channel
characteristics,
adapti
v
e
beamforming,
se
nsor
array
processing,
transcei
v
er
design,
multi-user
and
multi-carrier
wireless
communication
systems,
con
v
e
x
optimization
algorithms
for
signal
processing
communications,
channel
equalization,
and
other
kinds
of
distortion
and
interference
mitig
ation.
Further
info
on
his
homepage:
http://drsameralabed.wixsite.com/samer
IJECE
V
ol.
8,
No.
3,
June
2018:
1692
–
1700
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