Inter
national
J
our
nal
of
Electrical
and
Computer
Engineering
(IJECE)
V
ol.
8,
No.
3,
June
2018,
pp.
1583
–
1595
ISSN:
2088-8708
1583
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
T
ranscei
v
er
Design
f
or
MIMO
Systems
with
Indi
vidual
T
ransmit
P
o
wer
Constraints
Raja
Muthalagu
Department
of
EEE,
BITS,
Pilani,
Dubai
Campus,
Dubai,
U
AE-
345055.
Article
Inf
o
Article
history:
Recei
v
ed:
Aug
17,
2017
Re
vised:
Feb
21,
2018
Accepted:
Mar
13,
2018
K
eyw
ord:
MIMO
indi
vidual
transmit
po
wer
constraint
(ITPC)
channel
state
information
(CSI)
ABSTRA
CT
This
paper
in
v
estig
ate
the
transcei
v
er
design
for
single-user
mul
tiple-input
multiple-
output
system
(SU-MIMO).
Joint
transcei
v
er
design
with
an
improper
modulation
is
de
v
eloped
based
on
the
minimum
total
mean-squared
error
(TMSE)
criterion
under
tw
o
dif
ferent
cases.
One
is
equal
po
wer
all
ocation
(EP
A)
and
other
is
the
po
wer
con-
straint
that
jointly
meets
both
EP
A
and
total
transmit
po
wer
constraint
(TTPC)
(i.e
ITPC).
T
ranscei
v
er
is
designed
based
on
the
ass
umption
that
both
the
perfect
and
im-
perfect
channel
state
information
(CSI)
is
a
v
ailable
at
both
the
transmitter
and
recei
v
er
.
The
simulation
results
sho
w
the
performance
impro
v
ement
of
the
proposed
w
ork
o
v
er
con
v
entional
w
ork
in
terms
of
bit
error
rate
(BER).
Copyright
c
2018
Institute
of
Advanced
Engineering
and
Science
.
All
rights
r
eserved.
Corresponding
A
uthor:
Raja
Muthalagu
Department
of
EEE,
BITS,
Pilani,
Dubai
Campus
Dubai,
U
AE
+971-562249227
raja.sanjee
v
e@gmail.com
1.
INTR
ODUCTION
1.1.
Backgr
ound
Multiple-input
multiple-output
(MIMO)
systems
are
widely
used
to
substantially
increase
the
spect
ral
ef
ficienc
y
of
wireless
c
hannels.
Ho
we
v
er
,
the
benefits
of
multi-user
MIMO
highly
depend
on
the
type
of
channel
state
information
(CSI)
at
both
ends
and
on
the
le
v
el
of
accurac
y
of
t
his
information.
Practical
high
data
rates
wireless
systems
can
only
ha
v
e
imperfect
CSI
at
the
recei
v
er(CSIR),
i.e.,
an
estimate
of
the
channel
based
on
training
sequence.
Multiple-input
multiple-output
(MIMO)
systems
are
widely
used
to
substantially
increase
the
spect
ral
ef
ficienc
y
of
wireless.
The
spectral
ef
ficienc
y
of
the
MIMO
systems
is
increased
linearly
with
the
increase
in
the
number
of
transmit
and
recei
v
e
antennas.
Ho
we
v
er
,
the
ef
ficienc
y
of
MIMO
system
highly
rely
on
whether
channel
state
information
(CSI)
at
both
ends
or
not.
In
practice,
getting
the
perfect
CSI
is
impractical
because
of
the
dynamic
nature
of
the
channel
and
the
channel
estimation
errors.
Thusly
,
it
is
important
to
outline
a
system
suf
ficiently
enough
to
imperfect
CSIT
and/or
CSIR.
An
MIMO
systems
can
be
sub-di
vided
into
three
fundament
al
classifications,
spatial
di
v
ersity
[1,
2,
3,
4],
spatial
multiple
xing
[5,
6,
7]
and
beamforming
[8,
9,
10].
In
single-user
multiple-input
multiple-output(SU-MIMO)
system,
the
di
v
ersity
can
be
got
through
the
utilization
of
space-time
codes
[11,
12].
T
o
accomplis
h
full
di
v
ersity
,
t
he
transmit
beamforming
with
recei
v
e
combining
w
as
one
of
the
least
dif
ficult
methodologies.
T
o
enable
spatial
multiple
xing
in
SU-MIMO
systems,
the
appropriat
e
transmit
precoding
design
or
joint
precoder
-decoder
designs
were
proposed
under
a
v
ariety
of
system
objecti
v
es
and
dif
ferent
CSI
assumptions.
Another
beamforming
method
utilizing
singular
v
alue
decomposition
(SVD)
for
closed
loop
SU-MIMO
systems
with
a
con
v
olution
encoder
and
modulation
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.pp1583-1595
Evaluation Warning : The document was created with Spire.PDF for Python.
1584
ISSN:
2088-8708
techniques,
for
e
xample,
M
-quadrature
amplitude
modulation
(
M
-QAM)
and
M
-phase
shift
k
e
ying
(
M
-PSK)
o
v
er
the
Rayleigh
f
ading
ha
v
e
been
proposed
in
our
past
w
orks.
As
f
ar
as
spectral
ef
fecti
v
eness,
a
SU-MIMO
system
ought
to
be
intended
to
approach
the
capacity
of
the
channel
[13,
14,
15].
In
the
light
of
this
perception,
a
frequenc
y-selecti
v
e
MIMO
channel
can
be
managed
by
taking
a
multicarrier
approach,
which
is
a
well-kno
wn
capacity
lossless
the
struct
ure
and
permits
us
to
treat
e
v
ery
carrier
a
flat
M
IMO
channel.
A
capacity-
achie
ving
design
manages
that
t
he
channel
matrix
at
e
v
ery
carrier
must
be
diagonalized,
and
afterw
ard,
a
w
ater
-filling
po
wer
distrib
ution
must
be
utilized
on
the
spatial
subchannels
of
all
carriers.
Note
that
this
obliges
CSI
a
v
ailable
at
both
the
recei
v
er
and
transmitter
.
As
design
criteria,
dif
ferent
perform
ance
measures
are
considered,
for
e
xample,
W
eighted
Minimum
MSE
[16,
17],
TMSE
[18],
least
BER
[19].
From
signal
processing
point
of
vie
w
,
so
as
to
minimize
the
information
estimation
error
from
the
recei
v
ed
signal,
TMSE
is
a
critical
metric
for
transcei
v
er
design
and
has
been
embraced
in
SU-MIMO
systems.A
joint
transcei
v
er
design
for
a
SU-MIMO
frame
w
orks,
utilizing
an
MSE
paradigm
is
presented
in
[16].
A
no
v
el
optimization
method
is
proposed
to
s
olv
e
the
probabilistic
MSE
constrained
multiuser
multiple-input
single-output
(MU-MISO)
transcei
v
er
design
problem
[20].
1.2.
The
Pr
oblem
All
the
schemes
that
are
introduced
in
the
abo
v
e
w
orks
is
general
and
addressed
a
fe
w
opti
mization
criteria
lik
e
e
xtreme
data
rate,
least
BER,
and
MMSE.
The
issue
of
designing
an
optimum
linear
transcei
v
er
for
a
SU-MIMO
channel,
possibly
with
delay
spread,
utilizing
a
weighted
MMSE
paradigm
subject
to
a
transmit
po
wer
constraint
is
composed
in
[16].
These
studies
assume
that
the
perfect
CSI
w
as
a
v
ailable
at
the
transmitter
side.
Ho
we
v
er
,
in
practical
communication
s
y
s
tems,
the
propag
ation
en
vironment
may
be
more
challenging,
and
the
recei
v
er
and
transmitter
can
not
ha
v
e
a
perfect
kno
wledge
of
the
CSI.
The
imperfect
CSI
may
emer
ge
from
an
assortment
of
sources,
for
e
xample,
outdated
channel
estimat
es,
error
in
channel
estimation,
quantiza-
tion
of
the
channel
estimate
in
the
feedback
channel
and
so
forth
[21].
T
o
obtain
a
rob
ust
communications
system,
the
MIMO
systems
design
with
imperfect
CSI
is
an
im-
portant
issue
to
in
v
estig
ate.
The
optimal
precoding
strate
gies
in
SU-MIMO
systems
were
proposed
under
the
assumption
that
imperfect
CSI
is
a
v
ailable
at
the
transmitter
,
and
perfect
CSI
is
a
v
ailable
at
the
t
ransmitter
[22].
The
rob
ust
joint
precoder
and
decoder
design
to
reduce
the
TMSE
with
imperfect
CSI
at
both
the
transmitter
and
recei
v
er
of
SU-MIMO
systems
were
proposed
in
[23,
24,
25].
A
no
v
el
precoding
techniques
to
enhance
the
performance
of
the
do
wnlink
in
MU-MIMO
system
w
as
studied
with
improper
constellation
[26].
Precoding
designed
in
[26]
is
more
appropriate
for
a
MIMO
system
with
i
mproper
signal
constellation.
MMSE
and
modified
zero-forcing
(ZF)
precoder
designs
are
demonstrated
to
accomplish
an
unri
v
aled
performance
than
the
routine
linear
and
non-linear
precoders.
Both
instances
of
imperfect
and
perfect
CSI
are
considered,
where
the
imperfect
CSI
case
considers
the
correlation
data
and
channel
mean.
A
joint
precoder
and
decoder
design
under
the
minimum
TMSE
measure
produced
e
xceptional
BER
performance
for
,
proper
constellation
techniques,
e.g.,
M
-PSK
and
M
-QAM
[27,
28].
Then
ag
ain,
when
ap-
plying
the
same
outline
to
the
improper
constellation
techniques,
e.g.,
M
-ASK
and
BPSK,
the
performance
corrupts
fundam
entally
.
The
minimum
TMSE
des
ign
for
SU-MIMO
system
with
improper
modulation
tech-
niques
w
as
proposed
in
[26]
and
indicated
to
gi
v
e
a
predominant
performance
in
terms
of
BER
than
the
tradi-
tional
design
in
[27].
The
opti
mum
joint
precoder
and
decoder
designs
for
the
SU-MIMO
frame
w
orks
which
utilize
improper
constellation
strate
gies,
either
under
the
imperfect
or
perfect
CSI
w
as
proposed
in
[29,
30,
31].
In
both
instances
of
imperfect
and
perfect
CSI,
a
minimum
TMSE
measure
is
created
and
used
to
de
v
elop
an
iterati
v
e
design
technique
for
the
optimum
precoding
and
decoding
matrices
[29,
30,
31].
1.3.
The
Pr
oposed
Solution
In
all
of
these
designs
only
the
TMSE
measure
is
considered.
The
TMSE
measure
leads
to
wide
po
wer
v
ariations
across
the
transmit
antennas
and
poses
s
e
v
er
e
constraint
on
the
po
wer
amplifier
design.
Ho
we
v
er
,
to
the
best
of
our
kno
wledge,
no
attention
has
been
paid
to
either
the
ITPC
or
EP
A
based
joint
SU-MIMO
transcei
v
er
design
which
emplo
y
improper
modulation
techniques,
either
under
the
perfect
CSI
or
imperfect
CSI
assumption.
T
o
fill
the
g
ap,
this
paper
shall
address
the
problem
of
designing
jointly
optimum
SU-MIMO
transcei
v
er
under
improper
modulation
that
minimize
the
sum
of
symbol
estimation
error
subject
to
EP
A
and
IJECE
V
ol.
8,
No.
3,
June
2018:
1583
–
1595
Evaluation Warning : The document was created with Spire.PDF for Python.
IJECE
ISSN:
2088-8708
1585
ITPC(i.e
jointly
optimize
the
TTPC
and
EP
A).
It
assumes
the
perfect
and
imperfect
CSI
with
correlation
infor
-
mation
is
a
v
ailable
at
both
the
transmitter
and
recei
v
er
.
An
iterati
v
e
design
procedure
is
de
v
eloped
to
find
the
optimum
precoding
and
decoding
matrices.
The
rest
of
the
paper
is
or
g
anized
as
follo
ws.
The
system
model,
po
wer
constraint
and
problem
for
-
mulation
are
presented
in
Section
2.
The
proposed
optimum
joint
precoder
and
decoder
design
for
SU-MIMO
under
imperfect
CSI
is
presented
in
Section
3.
The
si
mulation
results
for
the
proposed
system
is
presented
in
Section
4.
Finally
conclusions
are
gi
v
en
in
Section
5.
Notations
:
Throughout
this
paper
,
(
)
T
denotes
matrix
transpose,
upper
(lo
wer)
case
boldf
ace
letters
are
for
matrices
(v
ectors),
(
)
H
stands
for
matrix
conjug
ate
transpose,
E
(
)
is
e
xpectation,
(
)
means
matrix
conjug
ate,
k
k
is
Euclidian
norm,
I
N
is
an
N
N
identity
matrix
and
T
r(
)
is
the
trace
operation.
2.
SYSTEM
MODEL,
PO
WER
CONSTRAINT
AND
PR
OBLEM
FORMULA
TION
2.1.
System
Model
A
general
SU-MIMO
system
model
consist
of
M
T
transmit
and
M
R
recei
v
e
antennas.
The
input
bit
streams
are
modulated
by
some
improper
modulation
te
chniques
to
generate
symbol
streams.
The
symbol
streams
to
be
sent
are
denoted
by
a
B
1
v
ector
s
=
[
S
1
;
:
:
:
;
S
B
]
T
,
where
B
is
the
number
of
data
streams
(i.e)
B
=
r
ank
(
H
)
min
(
M
R
;
M
T
)
,
where
H
is
M
R
M
T
channel
matrix
with
its
(
i;
j
)
th
element
h
i
;
j
denoting
the
channel
response
from
the
i
th
transmit
antenna
and
j
th
recei
v
e
antenna.
The
modulated
symbols
are
passed
through
the
precodi
ng
matrix
U
of
size
M
T
B
to
produce
a
M
T
1
precoded
v
ector
x
=
Us
.
The
precoding
matrix
with
the
comple
x
components
adds
redundanc
y
to
the
modulated
symbol
to
enhance
the
MIMO
system
performance.
The
pre
coded
v
ector
is
passed
through
the
MIMO
channel
through
N
T
antennas.
The
data
symbols
are
assumed
to
be
uncorrelated
and
ha
v
e
zero
mean
and
unit
ener
gy
,
i.e.,
E
[
ss
H
]
=
I
B
.
At
the
recei
v
er
end,
recei
v
ed
signal
at
recei
ving
antennas
are
processed
by
the
linear
decoder
matrix
V
of
size
B
M
R
.
F
or
a
MIMO
channel
without
an
y
delay-spread,
the
M
R
1
recei
v
ed
signal
v
ector
is
defined
as
y
=
Hx
+
n
(1)
y
is
fed
to
the
decoder
V
.
Then
the
resultant
v
ector
is:
^
s
=
VHUs
+
Vn
(2)
where
the
M
R
1
v
ector
n
represents
spatially
and
temporally
additi
v
e
white
Gaussian
noise
(A
WGN)
of
zero
mean
and
v
ariance
2
n
.
2.2.
P
o
wer
Constraint
The
con
v
entional
joint
precoder
and
decoder
design
are
based
on
the
follo
wing
TTPC
[29]:
E
[
k
x
k
2
]
=
E
[
k
Us
k
2
]
=
T
r(
UU
H
)
=
P
:
(3)
where
P
is
the
total
t
ransmitted
po
wer
from
all
the
antennas
at
the
tr
ansmitter
.
Most
o
f
the
precoding
or
joint
precoding
and
decoding
design
for
the
MIMO
systems
is
studied
with
TTPC
across
all
antennas.
Here,
we
consider
the
more
realistic
ITPC.
The
p
-norm
concept
is
a
multitasking
algorithm,
and
the
dif
ferent
po
wer
allocation
can
be
obtaine
d
by
changing
the
v
alue
of
p
.
In
linear
algebra
theory
,
the
P-norm
is
gi
v
en
by
[32,
33]
k
x
k
p
:=
B
X
i
=1
j
x
i
j
p
!
1
=p
f
or
p
1
(4)
1.
F
or
p
=
1
,
k
x
k
1
:=
P
B
i
=1
j
x
i
j
1
.
This
is
1-norm
and
it
is
si
mply
the
sum
of
the
absolute
v
alue
of
x
i
.
So
this
referes
to
TTPC
if
x
i
denotes
the
po
wer
in
each
antenna.
2.
F
or
p
=
1
,
k
x
k
1
:=
max(
x
1
;
:::;
x
M
)
.
In
linear
algebra
theory
,
this
infinity
norm
is
a
special
case
of
uniform
norm,
so
this
refers
to
equal
po
wer
allocation
(EP
A).
T
r
ansceiver
Design
for
MIMO
Systems
with
Individual
T
r
ansmit
...
(Raja
Muthala
gu)
Evaluation Warning : The document was created with Spire.PDF for Python.
1586
ISSN:
2088-8708
3.
F
or
1
<
p
<
1
,
the
p
-norm
constraint
is
formulated
as
an
optimization
problem
and
can
satisfy
both
the
TTPC
and
EP
A
with
an
appropriate
v
alue
for
p
,
so
this
refers
to
indi
vidual
transmi
t
po
wer
constraint
(ITPC).
2.3.
Pr
oblem
F
ormulation
The
optimum
joint
precoder
and
decoder
for
SU-MIMO
systems
which
emplo
ying
a
proper
modula-
tion
techniques
(e.g.,
M
-PSK,
M
-ASK
for
which
E
[
ss
T
]
=
0
)
i
s
deri
v
ed
by
minimizing
the
TMSE
under
the
TTPC
specified
by
(
??
).
The
TMSE
matrix
is
calculated
as
e
=
E
[
k
^
s
s
k
2
]
=
E
[
k
(
VHUs
+
Vn
)
s
k
2
]
(5)
This
TMSE
criterion
e
xpressed
in
(5)
is
optimum
for
the
SU-MIMO
systems
with
proper
modulations.
In
an
y
case,
with
improper
modulation
techniques
(for
which
E
[
ss
T
]
6
=
0
)
considered
in
this
w
ork
,
the
TMSE
criterion
for
SU-MIMO
systems
design
e
xpressed
by
(5)
is
not
optimum.
Since
the
tr
aditional
methodology
e
xpressed
by
(5)
yields
a
comple
x-est
eemed
filter
output.
But,
only
the
real
part
of
t
his
output
is
rele
v
ant
for
the
decision
in
an
MIMO
system
with
improper
constellations
[30].
In
this
w
ork,
the
MIMO
design
under
TTPC
in
[31]
is
e
xtended
to
both
the
EP
A
and
ITPC.
By
considering
the
improper
constellations,
the
error
v
ector
is
e
xpressed
as
follo
ws:
e
=
^
s
s
(6)
where
^
s
=
<
(
VHUs
+
Vn
)
.
W
atch
that
the
estimation
of
the
recei
v
ed
signal
^
s
is
changed
from
the
con
v
en-
tional
design
e
xpressed
in
(5).
Thusly
,
the
M
SE
criterion
with
respect
to
only
the
real
part
of
the
recei
v
ed
signal
with
TTPC
will
result
in
a
better
design.
W
ith
the
ne
wly
defined
error
v
ector
,
the
TMSE
can
be
computed
as
follo
ws:
E
[
k
e
k
2
]
=
E
[
k<
(
VHUs
+
Vn
)
s
k
2
]
=
E
[
k
(
VHUs
+
V
H
U
s
)
=
2
+
(
Vn
+
V
n
)
=
2
s
k
2
]
(7)
=
T
r
f
E
f
[0
:
5(
VHUs
+
V
H
U
s
)
+0
:
5(
Vn
+
V
n
)
s
]
0
:
5(
s
H
U
H
H
H
V
H
+
s
T
U
T
H
T
V
T
)
+0
:
5(
n
H
V
H
+
n
T
V
T
)
s
H
(8)
we
consider
the
follo
wing
assumptions
on
the
statistics
of
the
data,
noise
and
channel
(i.e.
E
[
n
]
=
E
[
nn
T
]
=
E
[
n
n
H
]
=
0
,
E
[
nn
H
]
=
2
n
I
N
T
and
E
[
ss
H
]
=
E
[
ss
T
]
=
I
B
).
By
using
those
assumption
and
after
some
manipulation
(8)
can
be
simplified
to
E
[
k
e
k
2
]
=
T
r
n
0
:
25
VHUU
H
H
H
V
H
+
VHUU
T
H
T
V
T
+
V
H
U
U
H
H
H
V
H
+
V
H
U
U
T
H
T
V
T
0
:
5(
VHU
+
V
H
U
+
U
H
H
H
V
H
+
U
T
H
T
V
T
)
+
I
B
+0
:
25
2
n
(
VV
H
+
V
V
T
)
o
(9)
The
goal
is
to
find
an
optimum
U
and
V
which
minimize
E
[
k
e
k
2
]
subject
to
the
TTPC,
total
transmit
po
wer
(T
r(
UU
H
)
and
the
transm
it
po
wer
constraint
that
jointly
optimize
the
TTPC
and
EP
A
(i.e
ITPC).
IJECE
V
ol.
8,
No.
3,
June
2018:
1583
–
1595
Evaluation Warning : The document was created with Spire.PDF for Python.
IJECE
ISSN:
2088-8708
1587
Mathematically
,
it
is
defined
as
min
U
;
V
E
[
k<
(
VHUs
+
Vn
)
s
k
2
]
s
:
t
:
(T
r(
UU
H
)
p
)
1
=p
P
T
:
(10)
where
P
T
is
a
constant.
If
p
=
1
,
then
the
P
T
will
be
equal
to
TTPC
(
),
p
=
1
corresponds
to
an
EP
A
(
)
where
=
=B
.
And
for
p
in
the
interv
al
1
<
p
<
1
,
P
T
=
1
,
p
-norm
condition
is
suf
ficient
to
meet
both
the
EP
A
and
a
TTPC
with
upper
bound
.
The
can
be
chosen
between
the
interv
al
of
[
=B
;
].
Here,
we
form
the
Lagrangian
to
find
the
optimum
solution
to
optimization
problem
e
xpressed
in
(10)
=
E
[
k<
(
VHUs
+
Vn
)
s
k
2
]
+
(T
r(
UU
H
)
p
)
1
=p
P
T
(11)
where
is
the
Lagrange
multiplier
.
The
optimum
v
alues
for
precoder
and
decoder
matrix
are
found
by
follo
w-
ing
the
same
procedure
as
in
[31].
3.
PR
OPOSED
MA
THOS:
OPTIMUM
JOINT
PRECODER
AND
DECODER
DESIGN
FOR
SU-
MIMO
WITH
IMPERFECT
CSI
This
section
proposes
a
design
of
the
optimum
linear
precoder
and
decoder
for
SU-MIMO
system
emplo
ying
improper
modulations
based
on
the
po
wer
constraint
t
hat
jointly
optimize
TTPC
and
EP
A.
Here
it
consider
the
imperfect
CSI
along
with
the
transmit
and
recei
v
e
correlation
is
a
v
ailable
at
both
the
end.
W
e
use
the
channel
model
in
our
pre
vious
w
ork
[31],
that
is
H
=
R
1
=
2
R
H
w
R
1
=
2
T
(12)
where
H
w
is
a
spatially
white
matrix
whose
entries
are
independent
and
identically
distrib
uted
(i.i.d.)
N
c
(0
;
1)
.
The
matrices
R
R
and
R
T
represent
the
normalized
recei
v
e
and
transmit
correlations,
respecti
v
ely
.
Both
trans-
mit
and
recei
v
e
correlations
are
assumed
to
be
full-rank
and
kno
wn
to
both
the
recei
v
er
and
the
transmitter
.
The
orthogonal
training
method
[29]
is
performed
to
estimate
the
channel
error
.
It
can
be
described
as
follo
ws
H
=
^
H
+
E
(13)
where
^
H
=
R
1
=
2
R
^
H
w
R
1
=
2
T
is
the
estimated
o
v
erall
channel
matrix,
^
H
w
is
the
MMSE
estimation
of
H
w
,
E
=
R
1
=
2
e
;
R
E
w
R
1
=
2
T
is
the
channel
estimation
error
matrix,
R
1
=
2
e
;
R
=
[
I
M
R
+
2
ce
R
1
R
]
1
is
the
ef
fect
of
the
recei
v
e
correlation
on
the
channel
estimation
error
,
2
ce
is
the
quality
of
the
channel
estimate
and
the
e
ntries
of
E
w
are
i.i.d.
N
c
(0
;
2
ce
)
.
By
modeling
the
true
channel
as
in
(13)
under
the
MMSE
channel
estimation,
the
TMSE
function
for
joint
tr
ansceiver
design
can
be
e
v
aluated
for
improper
modulation
as
follo
ws:
E
[
k
e
k
2
]
=
E
[
k
^
s
s
k
2
]
=
E
[
k<
(
V
(
^
H
+
E
)
Us
+
Vn
)
s
k
2
]
=
E
[
k
(
V
(
^
H
+
E
)
Us
+
V
(
^
H
+
E
)
U
s
)
=
2
+(
Vn
+
V
n
)
=
2
s
k
2
]
=
T
r
n
E
nh
0
:
5(
V
(
^
H
+
E
)
Us
+
V
(
^
H
+
E
)
U
s
)+
0
0
:
5(
Vn
+
V
n
)
s
]
h
0
:
5(
s
H
U
H
(
^
H
+
E
)
H
V
H
+
s
T
U
T
(
^
H
+
E
)
T
V
T
)
+0
:
5(
n
H
V
H
+
n
T
V
T
)
s
H
(14)
T
r
ansceiver
Design
for
MIMO
Systems
with
Individual
T
r
ansmit
...
(Raja
Muthala
gu)
Evaluation Warning : The document was created with Spire.PDF for Python.
1588
ISSN:
2088-8708
Substituting
E
=
R
1
=
2
e
;
R
E
w
R
1
=
2
T
in
(14)
and
after
taking
e
xpect
ation
with
respect
to
s
,
E
w
,
and
n
,
(14)
be-
comes
1
:
E
[
k
e
k
2
]
=
T
r
n
0
:
25
V
^
HUU
H
^
H
H
V
H
+
0
:
25
V
^
HUU
T
^
H
T
V
T
0
:
5
V
^
HF
+
0
:
25
VR
e
;
R
V
H
T
r(
R
T
UU
H
)
2
ce
+0
:
25
VV
H
2
n
+
0
:
25
V
^
H
U
U
H
^
H
H
V
H
+0
:
25
V
^
H
U
U
T
^
H
T
V
T
0
:
5
V
^
H
U
+0
:
25
V
R
e
;
R
V
T
f
T
r(
R
T
UU
H
)
g
2
ce
+
0
:
25
V
V
T
2
n
0
:
5
U
H
^
H
H
V
H
0
:
5
U
T
^
H
T
V
T
+
I
B
o
(15)
By
substituti
ng
(15)
in
(11)
and
taking
the
deri
v
ati
v
es
of
with
respect
to
V
and
U
,
it
can
be
sho
wn
that
the
associated
Karush-K
uhn-T
uck
er
(KKT)
conditions
can
be
obtained
and
gi
v
en
in
the
follo
wing.:
First,
the
v
alue
of
@
@
V
can
be
found
by
using
the
c
yclic
property
of
the
trace
function.
Setting
@
@
V
=
0
and
taking
the
comple
x
conjug
ates
of
both
sides
gi
v
es
V
(
^
HUU
H
^
H
H
+
R
e
;
R
2
ce
T
r(
R
T
UU
H
))
+
V
^
H
U
U
H
^
H
H
+
2
n
V
=
2
U
H
^
H
H
(16)
Similarly
,
setting
@
@
U
=
0
.
Ag
ain,
taking
the
comple
x
conjug
ates
of
both
sides
has
gi
v
es
(
^
H
H
V
H
V
^
H
+
R
T
2
ce
T
r(
R
e
;
R
V
H
V
))
U
+
^
H
H
V
H
V
^
H
U
+
2
(T
r(
UU
H
)
p
)
1
=p
=
2
^
H
H
V
H
(17)
Ne
xt,
by
post-multiplying
both
sides
of
(16)
by
V
H
one
obtains
(
V
(
^
HUU
H
^
H
H
+
R
e
;
R
2
ce
T
r(
R
T
UU
H
))
+
V
^
H
U
U
H
^
H
H
+
2
n
V
)
V
H
=
2
V
H
U
H
^
H
H
(18)
Lik
e
wise,
pre-multiplying
both
sides
of
(17)
by
U
H
produces
((
^
H
H
V
H
V
^
H
+
R
T
2
ce
T
r(
R
e
;
R
V
H
V
))
U
+
^
H
H
V
H
V
^
H
U
+
2
(T
r(
UU
H
)
p
)
1
=p
)
U
H
=
2
^
H
H
V
H
U
H
(19)
It
then
follo
ws
from
(18)
and
(19)
that:
(
V
(
^
HUU
H
^
H
H
+
R
e
;
R
2
ce
T
r(
R
T
UU
H
))
+
V
^
H
U
U
H
^
H
H
+
2
n
V
)
V
H
=
((
^
H
H
V
H
V
^
H
+
R
T
2
ce
T
r(
R
e
;
R
V
H
V
))
U
+
^
H
H
V
H
V
^
H
U
+
2
(T
r(
UU
H
)
p
)
1
=p
)
U
H
(20)
1
In
performing
the
e
xpectation,
the
follo
wing
results
are
used:
E
[
E
w
]
=
E
[
E
H
w
]
=
0
,
E
[
E
w
AE
H
w
]
=
2
ce
tr(
A
)
I
N
and
E
[
E
w
AE
T
w
]
=
0
.
IJECE
V
ol.
8,
No.
3,
June
2018:
1583
–
1595
Evaluation Warning : The document was created with Spire.PDF for Python.
IJECE
ISSN:
2088-8708
1589
Then,
by
taking
the
traces
of
both
sides
of
(20)
one
has:
=
2
n
2
P
T
T
r(
VV
H
)
(21)
As
in
[31],
the
optimum
solution
for
precoding
and
decoding
matrix
are
obtained
by
using
the
e
xplicit
relationship
between
(16),
(17)
and
(21).
An
iterati
v
e
algorithm
is
used
to
find
the
solutions.
The
optimum
decoding
matrix
is
V
Re
V
Im
=
C
Re
C
Im
AB
A
Im
+
B
Im
B
Im
A
Im
AC
1
(22)
where
AB
=
A
Re
+
B
Re
+
2
n
I
N
R
,
AC
=
A
Re
+
B
Re
2
n
I
N
R
V
=
V
Re
+
j
V
Im
,
HUU
H
H
H
=
A
Re
+
j
A
Im
,
H
U
U
H
H
H
=
B
Re
+
j
B
Im
and
2
F
H
H
H
=
C
Re
+
C
Im
.
Similarly
,
the
optimum
precoding
matrix
is
U
Re
U
Im
=
AD
Q
Im
P
Im
P
Im
+
Q
Im
AE
1
R
Re
R
Im
(23)
where
AD
=
P
Re
+
Q
Re
+
2
I
N
T
,
AE
=
P
Re
+
Q
Re
2
I
N
T
U
=
U
Re
+
j
U
Im
,
H
H
V
H
GH
=
P
Re
+
j
P
Im
,
H
H
V
H
V
H
=
Q
Re
+
j
Q
Im
and
2
H
H
V
H
=
R
Re
+
R
Im
Based
on
the
abo
v
e
e
xpressions,
the
optimum
precoder
and
decoder
can
be
solv
ed
by
an
iteration
procedure
as
outlined
in
follo
wing
algorithm:
1.
Initialize
U
,
and
U
by
setting
the
B
B
upper
sub-matrix
of
U
a
scaled
identity
matrix
(which
satisfies
the
po
wer
constraint
with
equality),
while
all
the
other
remaining
entries
of
U
are
zero.
2.
Find
the
v
alue
of
V
using
(22).
3.
Find
the
v
alue
of
using
(21).
4.
Find
U
using
(23).
5.
If
(T
r(
UU
H
)
p
)
1
=p
P
T
for
1
<
p
<
1
,
scale
U
such
that
T
r
n
UU
H
o
=
P
T
,
else
go
to
the
ne
xt
step.
6.
If
T
r
U
i
U
i
1
U
i
U
i
1
H
p
1
=p
<
10
4
,
then
terminate,
else
go
to
Step
2.
4.
RESUL
TS
AND
DISCUSSION
This
section
presents
the
perfor
mance
of
the
proposed
transcei
v
er
design
for
SU-MIMO
system
em-
plo
ying
im
proper
modulation,
EP
A
and
ITPC.
The
Matlab
simulation
has
been
used
for
modelling
the
proposed
SU-MIMO
system
and
channel.
The
performance
of
the
proposed
system
o
v
er
the
imperfect
CSI
is
measured
in
terms
of
BER.
The
BPSK,
4-ASK
and
QPSK
are
applied
to
modulate
the
data.
T
o
illustrate
the
performance
impro
v
ement
of
the
proposed
system,
the
BER
performance
of
the
SU-MIMO
system
with
improper
modu-
lation
under
TTPC
[31]
is
compared
with
the
proposed
SU-MIMO
system
with
improper
modulation
under
both
ITPC
and
EP
A.
The
simulation
results
are
a
v
eraged
o
v
er
at
least
10,000
channel
realizations.
In
all
the
simulation
results
reported
in
this
section,
the
number
of
parallel
date
streams
are
set
as
B
=
4
and
the
number
of
transmit
and
recei
v
er
antennas
are
fix
ed
as
M
T
=
M
R
=
4
.
The
transmit
correlation
matric
is
defi
ned
as
R
T
(
i;
j
)
=
j
i
j
j
T
for
i;
j
=
1
;
2
;
:
:
:
;
M
T
,
where
recei
v
e
correlation
metric
is
defined
as
R
R
(
i;
j
)
=
j
i
j
j
R
for
i;
j
=
1
;
2
;
:
:
:
;
M
R
.
T
r
ansceiver
Design
for
MIMO
Systems
with
Individual
T
r
ansmit
...
(Raja
Muthala
gu)
Evaluation Warning : The document was created with Spire.PDF for Python.
1590
ISSN:
2088-8708
S
N
R
=
P
T
/
σ
2
n
(
d
B
)
0
2
4
6
8
10
12
14
16
B
E
R
10
-4
10
-3
10
-2
10
-1
10
0
p=inf, Proposed-EPA
p=4.84, Proposed-ITPC
p=2.69, Proposed-ITPC
p=1.7, Proposed-ITPC
p=1, Conventional-TTPC
Figure
1.
Performance
comparison
of
the
con
v
entional
transcei
v
er
and
proposed
transcei
v
er
for
BPSK
modu-
lations
and
perfect
CSI.
M
T
=
M
R
=
4
,
B
=
4
,
2
ce
=
0
,
T
=
R
=
0
:
0
.
S
N
R
=
P
T
/
σ
2
n
(
d
B
)
0
2
4
6
8
10
12
14
B
E
R
10
-4
10
-3
10
-2
10
-1
10
0
p=inf, Proposed-EPA
p=4.84, Proposed-ITPC
p=2.69, Proposed-ITPC
p=1.7, Proposed-ITPC
p=1, Conventional-TTPC
Figure
2.
Performance
comparison
of
the
con
v
entional
transcei
v
er
and
proposed
transcei
v
er
for
4-ASK
modu-
lations
and
perfect
CSI.
M
T
=
M
R
=
4
,
B
=
4
,
2
ce
=
0
,
T
=
R
=
0
:
0
.
The
SNR
for
all
the
simulation
results
in
this
paper
is
defined
as
SNR
=
P
T
2
n
and
the
training
phase
SNR
is
defined
as
SNR
tr
=
P
tr
2
n
=
26
:
016
dB.
The
number
of
iteration
required
to
con
v
er
ge
the
optimum
v
alue
precoder
decoder
may
v
ary
between
6
to
9
iterations
and
it
is
mainly
based
on
the
SNR
and
channel
condition.
F
or
the
v
alue
of
p
=
1
corresponds
to
the
con
v
entional
TTPC,
where
the
p
=
1
corresponds
to
the
proposed
EP
A.
F
or
case
p
between
0
and
1
corresponds
to
the
practical
solution
that
sati
sfies
ITPC.
F
or
the
case
of
ITPC,
three
dif
f
erent
v
alues
f
o
r
and
are
cons
idered
based
on
the
p
v
alue.
Note
that,
with
p
=
f
1
:
7
;
2
:
69
;
4
:
84
g
,
one
has
=
f
5
:
2
;
2
:
8
;
1
:
5
g
and
=
f
9
:
8
;
6
:
7
;
4
:
5
g
.
First,
Fig.
1
sho
ws
t
h
e
performance
comparisons
of
the
con
v
entional
TTPC
based
linear
SU-MIMO
transcei
v
er
design
for
improper
modulation
in
[31]
with
that
of
the
proposed
ITPC
and
EP
A
based
linear
SU-
MIMO
transcei
v
er
design
for
improper
modulation.
The
BPSK
modulation
is
applied
to
modulate
the
data,
and
it
assumes
the
perfect
CSI
is
a
v
ailable
at
both
the
transmitter
and
recei
v
er
.
The
main
purpose
of
this
simulation
to
sho
w
the
performance
in
terms
of
BER
of
the
proposed
ITPC
and
EP
A
based
linear
SU-MIMO
transcei
v
er
design
for
improper
modulation.
As
can
be
seen
from
the
Fig.
1,
the
proposed
EP
A
based
SU-
MIMO
system
system
leads
to
a
SNR
performance
de
gradation
of
about
4
dB
for
BER
10
3
when
compared
to
the
con
v
entional
TTPC
base
SU-MIMO
system.
F
or
p=1.7,
2.69
and
4.84
of
the
proposed
ITPC
based
IJECE
V
ol.
8,
No.
3,
June
2018:
1583
–
1595
Evaluation Warning : The document was created with Spire.PDF for Python.
IJECE
ISSN:
2088-8708
1591
S
N
R
=
P
T
/
σ
2
n
(
d
B
)
0
2
4
6
8
10
12
14
B
E
R
10
-4
10
-3
10
-2
10
-1
10
0
p=inf, Proposed-EPA
p=4.84, Proposed-ITPC
p=2.69, Proposed-ITPC
p=1.7, Proposed-ITPC
p=1, Conventional-TTPC
Figure
3.
Performance
comparison
of
the
con
v
entional
transcei
v
er
and
proposed
transcei
v
er
for
OQPSK
mod-
ulations
and
perfect
CSI.
M
T
=
M
R
=
4
,
B
=
4
,
2
ce
=
0
,
T
=
R
=
0
:
0
.
S
N
R
=
P
T
/
σ
2
n
(
d
B
)
0
5
10
15
20
B
E
R
10
-4
10
-3
10
-2
10
-1
10
0
p=inf, Proposed-EPA
p=4.84, Proposed-ITPC
p=2.69, Proposed-ITPC
p=1.7, Proposed-ITPC
p=1, Conventional-TTPC
Figure
4.
Performance
comparison
of
the
con
v
entional
transcei
v
er
and
proposed
transcei
v
er
for
for
BPSK
and
imperfect
CSI.
M
T
=
M
R
=
4
,
B
=
3
or
B
=
4
,
2
ce
=
0
:
015
,
T
=
R
=
0
:
5
.
SU-MIMO
system,
1dB,
2dB
and
3
dB
SNR
de
gradation
is
observ
ed,
correspondingly
when
compared
to
the
con
v
entional
TTPC
based
linear
SU-MIMO
system.
This
comparison
sho
ws
the
con
v
entional
TTPC
based
SU-MIMO
system
achie
v
es
a
little
superior
performance
than
the
proposed
methods.
Ho
we
v
er
,
in
practice,
EP
A
and
ITPC
based
SU-MIMO
system
is
more
suitable
as
the
po
wer
at
each
transmit
antenna
is
l
imited
indi
vidually
by
the
linearity
of
the
po
wer
amplifier
.
In
that
w
ay
all
the
four
types
of
proposed
methods
are
prepared
to
design
a
SU-MIMO
transcei
v
er
rather
than
con
v
entional
TTPC
based
SU-MIMO
system.
Note
also
that,
since
perfect
CSI
is
a
v
ailable
at
both
the
transmitter
and
recei
v
er
ends,
the
performance
curv
es
impro
v
e
e
xponentially
with
SNR
and
there
is
no
error
floor
in
all
performance
curv
es.
Fig.
2
and
Fig.
3
also
sho
ws
the
similar
type
of
performance
comparisons
as
in
Fig.
1
b
ut
for
the
case
of
4-ASK
and
OQPSK,
correspondingly
under
perfect
CSI.
Ag
ain,
the
performance
de
gradation
of
our
proposed
design
o
v
er
the
con
v
entional
design
is
clearly
observ
ed
from
Fig.
2
and
Fig.
3.
Fig.
4,
Fig.
5
and
Fig.
6
sho
ws
performance
comparisons
of
the
con
v
entional
TTPC
based
linear
SU-
MIMO
transcei
v
er
design
for
improper
modulation
in
[31]
with
that
of
the
proposed
ITPC
and
EP
A
based
linear
SU-MIMO
transcei
v
er
design
for
BPSK,
4-ASK
and
OQPSK,
respecti
v
ely
b
ut
for
the
case
of
imperfect
CSI.
As
mentioned
before
the
MIMO
system
design
tak
es
into
account
the
one-dimensional
property
of
improper
T
r
ansceiver
Design
for
MIMO
Systems
with
Individual
T
r
ansmit
...
(Raja
Muthala
gu)
Evaluation Warning : The document was created with Spire.PDF for Python.
1592
ISSN:
2088-8708
S
N
R
=
P
T
/
σ
2
n
(
d
B
)
0
2
4
6
8
10
12
14
B
E
R
10
-4
10
-3
10
-2
10
-1
10
0
p=inf, Proposed-EPA
p=4.84, Proposed-ITPC
p=2.69, Proposed-ITPC
p=1.7, Proposed-ITPC
p=1, Conventional-TTPC
Figure
5.
Performance
comparison
of
the
con
v
entional
transcei
v
er
and
proposed
transcei
v
er
for
for
4-ASK
and
imperfect
CSI.
M
T
=
M
R
=
4
,
B
=
3
or
B
=
4
,
2
ce
=
0
:
015
,
T
=
R
=
0
:
5
.
S
N
R
=
P
T
/
σ
2
n
(
d
B
)
0
2
4
6
8
10
12
14
B
E
R
10
-4
10
-3
10
-2
10
-1
10
0
p=inf, Proposed-EPA
p=4.84, Proposed-ITPC
p=2.69, Proposed-ITPC
p=1.7, Proposed-ITPC
p=1, Conventional-TTPC
Figure
6.
Performance
comparison
of
the
con
v
entional
transcei
v
er
and
proposed
transcei
v
er
for
for
OQPSK
and
imperfect
CSI.
M
T
=
M
R
=
4
,
B
=
3
or
B
=
4
,
2
ce
=
0
:
015
,
T
=
R
=
0
:
5
.
modulations.
As
can
be
seen
from
the
figure,
the
proposed
joint
linear
transcei
v
er
lea
ds
to
a
little
performance
de
gradation,
especially
for
EP
A
based
SU-MIMO
system
with
BPSK
modulation
(an
SNR
de
gradation
of
about
3
dB
is
observ
ed
for
BER
of
10
3
).
F
or
the
case
of
imperfect
CSI
follo
wing
v
alues
are
assumed
f
o
r
correlation
T
=
R
=
0
:
5
.
Note
that,
with
SNR
tr
=
P
tr
2
n
=
26
:
016
dB
and
T
=
0
:
5
,
one
has
2
ce
=
0
:
015
.
W
e
also
pres
ented
the
performance
comparison
for
BPSK,
4-ASK
and
OQPSK
for
proposed
ITPC
based
SU-MIMO
under
both
the
perfect
and
imperfect
CSI
which
are
illustrated
in
Fig.
7.
F
or
v
alue
of
p
=
4
:
84
,
one
has
=
1
;
2
and
=
4
:
5
.
Results
of
Fig.
7
sho
w
the
ef
fect
of
CSI
on
proposed
design
in
terms
of
the
BER.
It
is
observ
ed
the
f
act
that
the
proposed
design
for
BPSK,
4-ASK
and
OQPSK
has
much
better
BER
performance
in
perfect
CSI,
and
the
channel
estimation
errors
cause
a
lar
ge
performance
de
gradation
on
the
BER.
Fig.
8
e
xamines
the
ef
fect
of
channel
correlations
on
the
proposed
system
BER
performance
under
imperfect
CSI.
F
or
this
figure,
OQPSK
modulation
is
emplo
yed
with
the
number
of
data
streams
B
=
4
.
V
arious
sets
of
transmit/recei
v
e
correlations
considered
are
f
T
=
0
:
9
;
R
=
0
:
9
g
;
f
T
=
0
:
9
;
R
=
0
:
5
g
;
f
T
=
0
:
5
;
T
=
0
:
9
g
;
and
f
T
=
0
:
5
;
T
=
0
:
5
g
.
W
ith
p
=
1
:
7
,
one
has
=
5
:
2
and
=
9
:
8
.
In
general,
Fig.
8
sho
ws
that
higher
v
alues
of
the
transmit
and
recei
v
e
correlations
cause
bigger
performance
losses.
IJECE
V
ol.
8,
No.
3,
June
2018:
1583
–
1595
Evaluation Warning : The document was created with Spire.PDF for Python.