Indonesian
J
our
nal
of
Electrical
Engineering
and
Computer
Science
V
ol.
23,
No.
1,
July
2021,
pp.
285
292
ISSN:
2502-4752,
DOI:
10.11591/ijeecs.v23i1.pp285-292
r
285
An
impr
o
v
ement
to
h
ybrid
beamf
orming
pr
ecoding
scheme
f
or
mmW
a
v
e
massi
v
e
MIMO
systems
based
on
channel
matrix
Zahra
Amirifar,
J
amshid
Abouei
Department
of
Electrical
Engineering,
Y
azd
Uni
v
ersity
,
Y
azd,
Iran
Article
Inf
o
Article
history:
Recei
v
ed
Mar
24,
2021
Re
vised
Jun
10,
2021
Accepted
Jun
18,
2021
K
eyw
ords:
5G
netw
ork
Channel
matrix
Hybrid
beamforming
Massi
v
e
MIMO
Optimization
Precoding
ABSTRA
CT
The
massi
v
e
multiple-input
multiple-output
(MIMO)
technology
has
been
applied
in
ne
w
generation
wireless
systems
due
to
gro
wing
demand
for
reliability
and
high
data
rate.
Hybrid
beamforming
architectures
in
both
recei
v
er
and
transmitter
,
including
analog
and
digital
precoders,
play
a
significant
role
in
5G
communication
netw
orks
and
ha
v
e
recently
attracted
a
lot
of
attention.
In
this
paper
,
we
propose
a
simple
and
ef
fecti
v
e
beamforming
precoder
approach
for
mmW
a
v
e
massi
v
e
MIMO
systems.
W
e
first
solv
e
an
optimization
problem
by
a
simplification
subject,
and
in
the
second
step,
we
use
the
co
v
ariance
channel
matrix
f
C
k
=
C
o
v
(
H
k
)
and
B
k
=
H
k
H
H
k
instead
of
chan-
nel
matrix
H
k
.
Si
mulation
results
v
erify
that
the
proposed
scheme
can
enjo
y
a
higher
sum
rate
and
ener
gy
ef
ficienc
y
than
pre
vious
methods
such
as
spatially
sparse
method,
analog
method,
and
con
v
entional
h
ybrid
method
e
v
en
with
inaccurate
Channel
State
Information
(CSI).
Percentage
dif
ference
of
the
achie
v
able
rate
of
C
k
=
C
o
v
(
H
k
)
and
B
k
=
H
k
H
H
k
schemes
compared
to
con
v
entional
methods
are
2
:
51%
and
48
:
94%,
re-
specti
v
ely
.
This
is
an
open
access
article
under
the
CC
BY
-SA
license
.
Corresponding
A
uthor:
Jamshid
Abouei
Department
of
Electrical
Engineering
Y
azd
Uni
v
ersity
Saf
aie,
Y
azd,
Iran
Email:
abouei@yazd.ac.ir
1.
INTR
ODUCTION
An
increase
in
the
number
of
antenna
arrays
has
been
recognized
as
the
most
important
f
actor
in
wireless
communication
due
to
achie
ving
higher
spectral
and
ener
gy
ef
ficiencies,
higher
data
rates,
and
the
capability
of
interference
mitig
ation
[1].
F
or
such
a
technology
,
kno
wn
as
massi
v
e
multiple
-input
multiple-
output
(MIMO)
[2]-[4],
man
y
number
of
antennas
can
be
le
v
eraged
in
di
f
ferent
structures.
Among
the
most
popular
approaches,
spatial
multiple
xing,
spatial
di
v
ersity
,
and
beamforming
are
promising
candidates
for
the
ne
xt
generation
of
wireless
communications
[5].
These
technologies
are
certain
to
play
a
significant
role
in
increasing
t
he
netw
ork’
s
throughput.
In
recent
years,
beamforming
algorithms
ha
v
e
been
widely
used
in
v
ari-
ous
MIMO
communications
applications,
in
particular
in
mmW
a
v
e
systems
and
5G
cellular
netw
orks
[6].
This
technique
significantly
increases
the
array
g
ain,
pro
vides
additional
radio
link
mar
gin
that
reduces
propag
ation
path
loss,
and
mitig
ates
the
co-channel
interference.
Beamforming
in
mmW
a
v
e
systems
of
fers
enormous
po-
tentials
where
highly
directional
adapti
v
e
antennas
ca
n
be
b
uilt
in
v
ery
small
form
f
actors
to
transmit
signals
for
maximal
signal-to-noise
ratio
(SNR).
Despite
its
astounding
adv
antages,
other
issues
such
as
cost,
po
wer
consumption,
comple
xity
in
the
corresponding
optimization
algorithms,
channel
estimation,
and
modeling
pose
J
ournal
homepage:
http://ijeecs.iaescor
e
.com
Evaluation Warning : The document was created with Spire.PDF for Python.
286
r
ISSN:
2502-4752
significant
challenges
to
the
beamforming
technique
in
massi
v
e
MIMO
systems
[7],
[8].
In
an
analog
beam-
former
,
the
antenna
array
is
connected
to
a
single
radio
frequenc
y
(RF)
chain
including
amplifier
and
analog
to
digital
con
v
erter
(ADC).
Besides,
its
phase
shifters
are
cheaper
and
ha
v
e
a
lo
wer
po
wer
consumption
com-
pared
to
RF
chains.
Digital
beamformer
,
on
the
other
hand,
is
the
optimal
scheme
based
on
the
singular
v
alue
decomposition
(SVD)
of
the
channel
matrix
H,
where
a
single
phase
array
antenna
is
connected
to
a
single
RF
chain.
The
optimal
precoder
and
combiner
in
transmitter/recei
v
er
sides
are
set
according
to
the
right
and
left
singular
v
ectors,
respecti
v
ely
.
Ho
we
v
er
,
this
scheme
is
not
feasible
in
ener
gy-constrained
massi
v
e
MIMO
netw
orks.
There
are
some
specific
challenges
in
deplo
ying
analog
and
digital
beamforming
techniques
in
mas-
si
v
e
MIMO
systems
such
as
theoretical
analysis
with
practical
constraints,
po
wer
consumption
of
RF
chains,
pilot
contamination
in
the
uplink,
channel
estimation,
ef
ficient
channel
feedback
mechanism,
and
emplo
ying
lo
w-comple
xity
signal
detection
algorit
hms
[9].
Ho
we
v
er
,
massi
v
e
MIMO
systems
contain
a
lar
ge
number
of
antennas,
hence,
one
should
a
v
oid
using
one
RF
chain
at
each
antenna
due
to
the
cost,
po
wer
composition,
and
comple
xity
in
v
olv
ed.
On
one
hand,
an
essential
option
in
achie
ving
the
aforementioned
benefits
is
to
emplo
y
massi
v
e
MIMO
technique.
On
the
other
hand,
in
the
traditional
MIMO
architecture,
each
antenna
requires
to
be
connected
to
one
RF
chain.
Hence,
the
fully
digital
beamforming
solution
will
lead
to
high
comple
xity
and
more
ener
gy
consumption
in
massi
v
e
MIMO
scenarios,
especially
in
mmW
a
v
e
frequenc
y
band
[10],
[11].
T
o
solv
e
this
problem,
v
arious
h
ybrid
analog
and
digital
beamforming
structures
ha
v
e
been
proposed
in
the
literature
[12]-[15].
A
k
e
y
f
actor
in
v
olv
ed
with
such
a
h
ybrid
beamforming
is
the
design
of
small-size
digital
signal
processor
in
baseband
and
lar
ge-size
analog
phase
shifters
in
RF
band
to
increase
the
antenna
array
g
ain.
Therefore,
h
ybrid
beamforming
can
enjo
y
a
lo
wer
number
of
RF
chains
and
higher
ener
gy
ef
ficienc
y
rather
than
full
digital
beamforming
without
ob
vious
performance
loss
[16].
There
are
tw
o
h
ybrid
beamforming
ar
-
chitectures
namely
fully-connected
and
sub-connected
structures,
where
each
RF
chain
is
connected
to
either
all
antennas
or
a
set
of
selected
antennas,
respecti
v
ely
.
The
fully-connected
h
ybrid
beamforming
architecture
is
not
practical,
since
it
requires
a
huge
number
of
phase
shifters
which
increase
the
computational
comple
x-
ity
,
cost,
and
the
ener
gy
consumption
[17].
In
contrast,
the
sub-connected
h
ybrid
beamforming
architecture
displays
a
significant
reduction
in
the
number
of
phase
shifters.
Another
challenge
that
should
be
addressed
for
the
analysis
and
e
v
aluation
of
h
ybrid
beamforming
with
sub-connected
architecture
is
the
a
v
ailability
of
the
channel
state
information
(CSI)
in
transmitter
and/or
recei
v
er
.
In
the
MIM
O
system,
the
CSI
is
g
athered
in
the
form
of
a
matrix
which
is
kno
wn
as
channel
matrix
H.
From
the
information
theoretic
points
of
vie
w
,
matrix
H
has
been
recognized
as
an
inte
gral
part
of
MIMO
systems,
because
an
y
channel
estimation
method
(e.g.,
minimum
mean
square
error
(MMSE)),
and
precoding
and
postcoding
matrices
are
optimized
based
on
this
matrix.
The
full
CSI,
when
used
appropriately
achie
v
es
the
highest
performance
of
an
y
MIMO
s
y
s
tem.
Ho
we
v
er
,
a
MIMO
system
is
more
preferable
that
k
eeps
its
per
-
formance
and
be
non-sensiti
v
e
under
the
imperfect
estimation
of
channel
matrix
H
[19].
Note
that
in
practical
situations,
perfect
CSI
at
the
tr
ansmitter
is
not
perfectly
a
v
ailable
because
of
feedback
error
,
channel
estima-
tion
error
,
the
number
of
channel
parameters,
and
time
v
arying
channel,
while
the
high
capaci
ty
rises
with
the
number
of
paths
in
the
channel
[20].
T
o
tackle
this
problem,
one
should
emplo
y
the
MIMO
channel
co
v
ari-
ance
matrix
est
imation
which
e
v
aluates
ho
w
much
the
v
ariables
change
together
.
In
this
re
g
ard,
we
propose
an
ef
fecti
v
e
optimization
solution
for
sub-connected
h
ybri
d
beamforming
in
mmW
a
v
e
massi
v
e
MIMO
systems
based
on
the
co
v
ariance
of
channel
matrix
C
k
=
C
o
v
(
H
k
)
and
B
k
=
H
k
H
H
k
instead
of
the
channel
matrix
H
k
.
The
proposed
beamformi
ng
scheme
displays
a
higher
data
rate,
simpler
,
and
achie
v
es
the
best
po
s
sible
ener
gy
ef
ficienc
y
.
Simulation
results
sho
w
that
our
proposed
structure
achie
v
es
an
e
v
en
higher
data
rate
when
the
number
of
recei
v
ed
antennas
is
doubled.
Moreo
v
e
r
,
we
compute
the
ener
gy
ef
ficienc
y
of
the
v
arious
pre-
coding
schemes.
In
addition,
we
apply
the
impact
of
imperfect
CSI
on
the
mas
si
v
e
MIMO
system
performance
to
sho
w
that
our
proposed
method
has
a
special
adv
ance
which
is
non-sensiti
v
e
to
imperfect
CSI.
The
rest
of
the
paper
is
or
g
anized
as
follo
ws:
describes
the
system
model
including
channel
matrix,
the
recei
v
ed
signal
v
ector
,
the
achie
v
able
rates,
and
optimization
problem
in
section
2
and
section
3.
specifies
the
proposed
schem
es
and
describes
well-kno
wn
precoders
such
as
spatially
sparse
and
analog
methods.
The
simulation
results,
in
terms
of
the
ener
gy
ef
ficienc
y
of
the
proposed
scenario
compared
with
the
aforementioned
precoders
are
pro
vided
in
section
4.
Finally
,
we
conclude
the
paper
in
section
5.
2.
THE
PR
OPOSED
METHOD
In
this
w
ork,
we
consider
the
h
ybrid
beamforming
with
sub-connected
architecture
the
sase
station
(BS)
is
equipped
with
M
T
antennas
connected
to
N
T
RF
chains
to
serv
e
K
users,
where
N
T
<
M
T
.
In
the
Indonesian
J
Elec
Eng
&
Comp
Sci,
V
ol.
23,
No.
1,
July
2021
:
285
–
292
Evaluation Warning : The document was created with Spire.PDF for Python.
Indonesian
J
Elec
Eng
&
Comp
Sci
ISSN:
2502-4752
r
287
h
ybrid
beamforming
scenario,
the
signal
is
then
processed
digitally
by
digital
precoder
in
baseband
and
is
then
processed
by
analog
precoder
in
RF
domain.
Accordingly
,
in
the
recei
v
er
,
the
recei
v
ed
signal
is
processed
by
analog
combiner
and
then
modified
by
digital
combiner
.
Finally
,
the
recei
v
ed
v
ector
at
user
k
is
gi
v
en
by
[10].
y
k
=
W
(
k
)
H
BB
W
(
k
)
H
RF
H
k
F
RF
F
(
k
)
BB
s
k
+
W
(
k
)
H
BB
W
(
k
)
H
RF
n
k
(1)
Where
s
k
is
the
symbol
v
ector
of
r
k
data
streams
with
the
total
number
of
data
streams
r
=
å
K
k
=
1
r
k
,
W
(
k
)
BB
2
C
N
R
k
r
k
and
W
(
k
)
RF
2
C
M
R
k
N
R
k
denote
the
baseband
decoder
and
the
RF
combiner
,
respecti
v
ely
.
The
number
of
recei
v
e
antennas
and
the
number
of
antennas
connected
to
one
RF
chain
are
denoted
by
M
R
k
and
N
R
k
,
respecti
v
ely
.
In
addition,
H
k
2
C
M
R
k
M
T
is
the
channel
matrix
between
BS
and
user
k
.
Similarly
,
F
RF
2
C
M
T
N
T
and
F
(
k
)
BB
2
C
N
T
r
represent
the
analog
RF
and
digital
baseband
precoders,
respecti
v
ely
.
Finally
,
n
k
N
(
0
;
s
2
I
M
R
k
)
is
comple
x
additi
v
e
white
g
aussian
noise
(A
WGN)
v
ector
with
the
independent
and
i
dentical
distrib
ution
(i.i.d)
with
zero
mean
and
v
ariance
s
2
.
In
this
paper
,
we
adopt
the
same
mmW
a
v
e
channel
model
as
in
[10,
11,
14,
15],
where
it
is
assumed
that
the
channel
matrix
is
sum
of
the
scatters
in
mmW
a
v
e
propag
ation
en
vironment
for
narro
wband
do
wnlink
MIMO
system,
i.e.,
H
=
b
å
L
l
=
1
a
l
L
l
(
j
r
l
)
L
l
(
j
t
l
)
a
r
(
j
r
l
)
a
H
t
(
j
t
l
)
(2)
Where
b
is
a
normalization
f
actor
,
L
is
the
number
of
multipath,
a
l
is
the
g
ain
of
l
t
h
path,
L
l
(
j
t
l
)
and
L
l
(
j
r
l
)
denote
the
transmit
and
recei
v
e
antenna
array
g
ains
[17],
a
r
and
a
t
represent
the
array
response
v
ectors.
In
addition,
j
t
and
j
r
are
angle
of
departure
(AoD)
and
angle
of
arri
v
al
(AoA),
respecti
v
ely
.
Assuming
that
the
BS
and
each
user
are
equipped
with
linear
array
(LA),
the
array
response
v
ector
with
N
elements
can
be
e
xpressed
as
[21]:
a
L
A
(
j
)
=
1
N
h
1
;
e
j
2
p
l
d
sin
j
;
:
::
;
e
j
(
N
1
)
2
p
l
d
sin
j
i
T
(3)
where
l
is
the
w
a
v
elength
and
d
is
the
antenna
spacing.
Additionally
,
mmW
a
v
e
channel
H
follo
ws
the
sparse-
scattering
model
because
of
the
limited
number
of
scatters
in
the
en
vironment
[1].
The
performance
of
our
proposed
scheme
is
e
v
aluated
in
terms
of
the
achie
v
able
rate
of
user
k
and
is
compared
that
with
other
precoding
methods.
Hence,
R
k
can
be
formulated
as
[11]:
R
k
=
log
2
j
I
M
R
k
+
r
N
T
s
2
H
k
F
RF
F
(
k
)
BB
F
(
k
)
H
BB
F
H
RF
H
H
k
j
(4)
where
r
denotes
the
transmitter
po
wer
b
udget.
In
addition,
we
present
the
ener
gy
ef
ficienc
y
of
the
aforemen-
tioned
schemes
defined
as
[22]:
g
=
R
k
P
t
o
t
al
=
R
k
P
t
+
N
T
P
RF
+
N
PS
P
PS
(
bps/Hz/W
)
(5)
where
P
t
o
t
al
is
the
tot
al
ener
gy
consumption,
N
PS
is
the
number
of
required
phase
shifters,
P
t
,
P
RF
,
P
PS
are
the
transmitted
ener
gy
,
ener
gy
consumed
by
RF
chains
and
the
ener
gy
consumed
by
phase
shifters,
respecti
v
ely
.
3.
RESEARCH
METHOD
In
this
section,
we
propose
a
simple
and
ef
fecti
v
e
approach
which
achie
v
es
better
performance
than
traditional
precoders.
Therefore,
we
aim
to
maximize
the
objecti
v
e
function
in
massi
v
e
MIMO
sys
tems,
which
can
be
e
xpressed
as
[23]:
max
F
RF
;
W
(
k
)
RF
;
8
k
K
å
k
=
1
W
(
k
)
H
RF
H
k
F
RF
2
F
s.t.
F
H
RF
F
RF
=
I
N
T
;
W
(
k
)
H
RF
W
(
k
)
RF
=
I
N
R
k
;
8
k
(6)
Using
the
Frobenius
norm
[25],
the
objecti
v
e
function
in
(6)
can
be
re
written
as:
max
F
RF
t
r
ace
0
B
@
F
H
RF
K
å
k
=
1
(
H
H
k
W
(
k
)
RF
W
(
k
)
H
RF
H
k
)
|
{z
}
F
RF
P
1
C
A
s.t.
F
H
RF
F
RF
=
I
N
T
(7)
An
Impr
o
vement
to
Hybrid
Beamforming
Pr
ecoding
Sc
heme
for
mmW
ave
Massive
...
(Zahr
a
Amirifar)
Evaluation Warning : The document was created with Spire.PDF for Python.
288
r
ISSN:
2502-4752
for
gi
v
en
W
(
k
)
RF
;
8
k
,
and
max
W
(
k
)
RF
;
8
k
t
r
ace
å
K
k
=
1
W
(
k
)
H
RF
(
H
k
F
RF
F
H
RF
H
H
k
)
|
{z
}
W
(
k
)
RF
Q
k
!
s.t.
W
(
k
)
H
RF
W
(
k
)
RF
=
I
N
R
k
;
8
k
(8)
for
gi
v
en
F
RF
.
The
solutions
of
objecti
v
e
functions
in
(7)
and
(8)
are
the
same.
Lik
ed
by
the
generalized
lo
w
rank
approximation
of
matrices
(GLRAM)
algorithm
in
[23],
these
equations
are
solv
ed
by
an
iterati
v
e
procedure.
In
this
paper
,
we
propose
a
simpler
solution
where
we
first
simplify
(7)
and
(8)
and
then
apply
the
SVD
method
[23]
on
the
channel
matrix
to
compute
optimal
F
RF
and
W
(
k
)
RF
.
In
the
proposed
algorithm,
the
optimal
precoder
and
combiner
are
set
according
to
the
right
and
left
singular
v
ectors,
respecti
v
ely
.
The
SVD
of
matrix
H
k
with
rank
r
can
be
e
xpressed
as
[24]:
H
k
=
U
k
S
V
H
k
(9)
where
U
k
and
V
k
are
M
R
k
r
and
M
T
r
dimensional
unitary
matrices
(
i
:
e
:
;
U
H
k
U
k
=
I
r
and
V
H
k
V
k
=
I
r
),
respec-
ti
v
ely
,
and
S
denotes
r
r
diagonal
mat
rix
with
d
1
;
:
::
;
d
r
eigen
v
alues.
Thus,
the
optimal
precoding
matrix
can
be
e
xpressed
as
[4]:
W
(
H
)
RF
;
o
p
t
=
V
k
(10)
As
a
result,
the
matrix
H
k
can
be
re
written
as:
H
k
=
r
å
i
=
1
d
i
u
i
;
k
v
H
i
;
k
(11)
where
u
i
;
k
and
v
i
;
k
are
the
column
v
ectors
of
matrices
U
k
and
V
k
,
re
specti
v
ely
.
In
addition,
the
matrix
B
k
=
H
k
H
H
k
with
dimension
M
R
k
M
R
k
can
be
decomposed
as:
H
k
H
H
k
=
D
k
L
D
H
k
(12)
where
D
k
is
M
R
k
M
R
k
dimensional
modal
m
atrices
(
D
H
k
D
k
=
I
M
R
k
),
and
L
d
eno
t
es
M
R
k
M
R
k
diagonal
matrix
with
h
1
;
:
::
;
h
M
R
k
eigen
v
alues.
As
a
result,
it
is
easy
to
sho
w
that
h
i
=
(
d
2
i
i
=
1
;
2
;
:
::
;
r
0
i
=
r
+
1
;
::
:
;
M
R
k
(13)
Note
that
for
quality
of
service
(QoS)
reasons,
the
number
of
RF
chains
(
N
T
)
ought
not
to
be
too
small
during
the
peak
time
when
compared
with
the
number
of
antennas
(
M
T
).
Therefore,
for
the
object
i
v
e
functions
(7)
and
(8),
we
ha
v
e
W
(
k
)
RF
W
(
k
)
H
RF
'
N
T
M
T
I
M
R
k
and
F
(
k
)
RF
F
(
k
)
H
RF
'
N
T
M
T
I
M
T
,
respecti
v
ely
.
Hence,
without
an
y
loss
of
generality
,
we
choose
these
matrix
substitution
approaches
for
simplicity
of
problem.
Therefore,
problems
(7)
and
(8)
can
be
re
written
as:
max
F
RF
t
r
ace
0
B
@
F
H
RF
K
å
k
=
1
H
H
k
H
k
|
{z
}
F
RF
P
1
C
A
s.t.
F
H
RF
F
RF
=
I
N
T
(14)
and
max
W
(
k
)
RF
;
8
k
t
r
ace
å
K
k
=
1
W
(
k
)
H
RF
(
H
k
H
H
k
)
|
{z
}
W
(
k
)
RF
Q
k
!
s.t.
W
(
k
)
H
RF
f
W
(
k
)
RF
=
I
N
R
k
;
8
k
(15)
In
the
second
step,
problems
(14)
and
(15)
can
be
solv
ed
via
the
SVD
method
to
find
F
RF
and
W
(
k
)
RF
including
the
N
T
lar
gest
eigen
v
ectors
of
P
and
the
N
R
k
lar
gest
eigen
v
alues
of
Q
k
,
respecti
v
ely
.
Indonesian
J
Elec
Eng
&
Comp
Sci,
V
ol.
23,
No.
1,
July
2021
:
285
–
292
Evaluation Warning : The document was created with Spire.PDF for Python.
Indonesian
J
Elec
Eng
&
Comp
Sci
ISSN:
2502-4752
r
289
Recalling
the
f
act
that
the
transmitter
of
user
k
sends
pilot
w
a
v
eforms
to
estimate
the
channel
responses
matrix
H
k
,
in
the
proposed
precoder
scheme
for
the
underlying
mmW
a
v
e
massi
v
e
MIMO
system,
we
calculate
the
co
v
ariance
of
this
channel
matrix,
denoted
by
C
k
,
and
use
this
matrix
instead
of
H
k
in
all
precoder
methods.
Simulation
results
in
Section
4.
v
erify
that
our
proposed
structure
can
achie
v
e
higher
rate
and
ener
gy-ef
ficienc
y
when
compared
with
the
case
when
only
matrix
H
k
is
used.
In
practice,
a
channel
matrix
H
k
is
described
as
(2).
Hence,
the
co
v
ariance
of
such
a
matrix
is
calcu-
lated
using
the
follo
wing
formula
C
k
=
C
o
v
(
H
k
)
=
E
h
(
H
k
E
(
H
k
)
)
(
H
k
E
(
H
k
)
)
T
i
(16)
Thus,
the
formula
(4)
can
be
written
as
ˆ
R
k
=
log
2
j
I
M
R
k
+
r
N
T
s
2
B
k
F
RF
F
(
k
)
BB
F
(
k
)
H
BB
F
H
RF
B
k
j
(17)
and,
˜
R
k
=
log
2
j
I
M
T
+
r
N
T
s
2
C
k
F
RF
F
(
k
)
BB
F
(
k
)
H
BB
F
H
RF
C
k
j
(18)
In
this
paper
,
we
compare
our
proposed
algorithm
with
the
con
v
entional
structure
based
on
analog
precoder
,
con
v
entional
h
ybrid
precoder
,
and
spatially
sparse
precoder
[11].
4.
RESUL
TS
AND
DISCUSSION
In
this
section,
we
pro
vide
some
simulation
results
in
terms
of
the
achie
v
able
rate
and
the
ener
gy
ef
ficienc
y
.
The
performance
e
v
aluation
of
the
proposed
precoding
is
also
e
xamined
in
detail
and
compared
that
with
v
arious
well-kno
wn
recently
w
orks
such
as
analog
precoding,
spatially
sparse
precoding,
and
h
ybrid
precoding.
Furthermore,
we
assume
that
both
transmit
and
recei
v
e
antennas
correspond
to
a
typical
uniform
linear
array
configuration
with
spacing
d
=
l
=
2
.
According
to
the
rayleigh
f
ading
channel
model,
the
afore-
mentioned
precoding
schemes
can
be
applied
to
the
mmW
a
v
e
system
with
a
fe
w
number
of
ef
fecti
v
e
multipath
components.
Hence,
for
the
channel
model
in
(2),
we
define
L
=
3
[9],
and
also
carrier
frequenc
y
28
GHz
[12],
AoDs
and
AoAs
uniform
distrib
ution
within
the
interv
al
p
6
;
p
6
and
[
p
;
p
]
,
respecti
v
ely
,
because
of
using
omnidirectional
antennas
by
users.
In
addition,
we
consider
SNR
equal
to
r
=
s
2
;
M
R
k
=
8
;
16
;
M
T
=
64
;
N
T
=
16
(number
of
RF
chains)
and
N
R
k
=
M
T
=
N
T
.
In
this
paper
,
we
set
P
RF
=
250
m
W
[18],
P
PS
=
1
m
W
[12],
and
P
t
=
1
W
[26].
In
the
first
step
of
our
simulation,
we
consider
the
perfect
CSI
conditions,
where
the
deri
v
ed
achie
v
able
rate
e
xpressions
in
(4),
(17),
and
(18)
v
ersus
SNR
are
simulated
in
Figure
1
for
all
precoding
schemes
including
our
proposed
algorithm
and
for
M
R
k
=
8
,
M
T
=
64,
N
T
=
16,
and
L
=
3.
It
is
seen
that
the
proposed
precoding
and
analog
precoding
ha
v
e
the
highest
and
the
lo
west
achie
v
able
rate,
respecti
v
ely
.
After
that,
spatially
sparse
precoding
and
h
ybrid
precoding
achie
v
e
higher
rates.
When
we
apply
matrices
B
k
and
C
k
,
all
precoders
with
using
these
matrices
display
m
o
r
e
achie
v
able
rate
compared
to
the
case
of
using
matrix
H
k
.
Note
that
spatially
sparse
precoding
is
a
strong
method
and
has
higher
achie
v
able
rate
than
h
ybrid
and
analog
schemes,
since
its
channel
matrix
includes
the
limited
paths
which
is
suitable
for
sparse
en
vironments.
Ho
we
v
er
,
as
sho
wn
in
Figure
1,
the
spatially
sparse
precoder
has
less
achie
v
able
rate
than
our
proposed
precoding.
Using
the
f
act
that
the
accurac
y
of
CSI
is
af
fected
by
the
number
of
elements
of
channel
matrix
and
in
spi
te
of
spatially
sparse
precoding,
we
increase
the
elements
of
channel
matrix
mathematically
,
and
without
adding
an
y
antennas
or
RF
chains.
T
o
sho
w
the
superiority
of
our
proposed
scheme
in
increasing
the
achie
v
able
rate,
let
us
consider
the
precoders
in
T
able
1.
F
or
instance,
for
h
ybrid
precoder
case,
the
achie
v
abl
e
rates
are
25.21,
26.22,
and
43.03
in
“
hybr
id
pr
ecod
ing
”,
“
hybr
id
C
k
pr
ecod
ing
”,
and
“
hybr
id
B
k
pr
ecod
ing
”
le
gends,
respecti
v
ely
,
where
the
percentage
dif
ferences
are
3.92%
and
52.22%
compared
to
“
hybr
id
pr
ecod
ing
”.
An
Impr
o
vement
to
Hybrid
Beamforming
Pr
ecoding
Sc
heme
for
mmW
ave
Massive
...
(Zahr
a
Amirifar)
Evaluation Warning : The document was created with Spire.PDF for Python.
290
r
ISSN:
2502-4752
T
able
1.
Comparison
achie
v
able
rate
of
all
precoders
at
SNR=10
d
B
Precoding
Schemes
Achie
v
able
rate
Percentage
Dif
ference
pr
o
posed
me
t
hod
27.44
0%
anal
o
g
16.98
0%
s
pa
t
ial
l
y
s
par
se
26.69
0%
hybr
id
25.21
0%
pr
o
posed
B
k
me
t
hod
45.22
48.94%
anal
o
g
B
k
34.7
68.57%
s
pa
t
ial
l
y
s
par
se
B
k
44.48
49.99%
hybr
id
B
k
43.03
52.22%
pr
o
posed
C
k
me
t
hod
28.14
2.51%
anal
o
g
C
k
18.73
9.80%
s
pa
t
ial
l
y
s
par
se
C
k
27.45
2.80%
hybr
id
C
k
26.22
3.92%
T
o
summarize
the
abo
v
e
ar
guments,
we
can
claim
that
our
proposed
precoding
scheme
based
on
the
B
k
matrix
and
the
co
v
ariance
matrix
C
k
instead
of
channel
matri
x
H
k
achie
v
es
t
he
highest
achi
e
v
able
rate
wi
th
a
slight
increase
in
comple
xity
.
This
issue
will
be
decreased
when
the
number
of
recei
v
ed
antennas
is
lar
ger
(e.g,.
M
R
k
=
16
;
M
T
=
64
).
Figure
2
illustrates
this
matter
,
where
it
is
observ
ed
that
the
proposed
precoding
still
has
a
higher
achie
v
able
rate
than
other
precoder
schemes.
The
abo
v
e
results
are
achie
v
ed
under
the
perfect
CSI,
while
in
the
real
communication
systems,
the
conditions
are
not
al
w
ays
perfect.
W
e
apply
the
impact
of
imperfect
CSI
on
the
massi
v
e
MIMO
system
performance
which
can
be
modeled
as
[27]:
b
H
k
=
a
H
k
+
p
1
a
2
E
(19)
where
b
H
is
the
act
ual
channel
matrix,
E
is
the
error
matrix,
a
2
[
0
;
1
]
is
scalar
and
can
be
considered
as
the
CSI
accurac
y
,
which
means
that
lo
wer
a
is
quite
poor
accurac
y
of
CSI
and
closer
to
1
means
closer
perfect
CSI.
From
Figure
3,
it
is
observ
ed
that
when
a
is
0.9,
the
performance
of
the
proposed
precoding
scheme
measurably
close
to
perfect
CSI
conditions
(
a
=
1).
Hence,
we
can
conclude
that
the
proposed
precoding
is
not
sensiti
v
e
to
the
CSI
accurac
y
e
v
en
when
a
is
0.5.
-20
-15
-10
-5
0
5
10
15
20
SNR (dB)
0
10
20
30
40
50
60
Achievable Rate
proposed method
analog method
spatially sparse precoding
hybrid precoding
proposed
B
k
method
analog
B
k
method
spatially sparse
B
k
precoding
hybrid
B
k
precoding
proposed
C
k
method
analog
C
k
method
spatially sparse
C
k
precoding
hybrid
C
k
precoding
Figure
1.
Achie
v
able
rate
comparison
v
ersus
SNR
for
M
R
k
=
8
;
M
T
=
64
;
N
T
=
16
;
L
=
3
mmW
a
v
e
MIMO
system
with
using
H
k
,
B
k
,
and
C
k
matrices
-20
-15
-10
-5
0
5
10
15
20
SNR (dB)
0
10
20
30
40
50
60
70
Achievable Rate
proposed method
analog method
spatially sparse precoding
hybrid precoding
proposed
B
k
method
analog
B
k
method
spatially sparse
B
k
precoding
hybrid
B
k
precoding
Figure
2.
Achie
v
able
rate
comparison
v
ersus
SNR
for
M
R
k
=
16
;
M
T
=
64
;
N
T
=
16
;
L
=
3
mmW
a
v
e
MIMO
system
with
using
H
k
and
B
k
matrices
The
ener
gy
ef
ficienc
y
of
the
aforementioned
schemes
defined
in
(5),
v
ersus
the
number
of
RF
chains
N
T
is
e
v
aluated
in
Figure
4.
W
e
can
find
from
Figure
4
that
analog
precoding
based
on
H
k
has
the
lo
west
ener
gy
Indonesian
J
Elec
Eng
&
Comp
Sci,
V
ol.
23,
No.
1,
July
2021
:
285
–
292
Evaluation Warning : The document was created with Spire.PDF for Python.
Indonesian
J
Elec
Eng
&
Comp
Sci
ISSN:
2502-4752
r
291
ef
ficienc
y
because
of
using
analog
elements.
In
addition,
it
is
clear
that
our
proposed
precoding
based
on
B
k
has
the
highest
ener
gy
ef
ficienc
y
,
especially
when
N
T
is
lo
wer
than
30.
-20
-15
-10
-5
0
5
10
15
20
SNR (dB)
10
15
20
25
30
35
40
45
50
55
Achievable Rate
proposed
B
k
method, alpha = 1
proposed
B
k
method, alpha = 0.9
proposed
B
k
method, alpha = 0.7
proposed
B
k
method, alpha = 0.5
Figure
3.
Impact
of
imperfect
CSI
on
achie
v
able
rate
comparison
v
ersus
SNR
for
M
R
k
=
16
;
M
T
=
64
;
N
T
=
16
;
L
=
3
mmW
a
v
e
MIMO
system
with
using
B
k
matrix
0
10
20
30
40
50
60
70
Number of N
T
(RF chains)
0
500
1000
1500
2000
2500
3000
3500
4000
4500
Energy Efficiency
proposed method
analog method
spatially sparse precoding
hybrid precoding
proposed
B
k
method
analog
B
k
method
spatially sparse
B
k
precoding
hybrid
B
k
precoding
8
10
12
14
800
900
1000
1100
1200
1300
1400
Figure
4.
Ener
gy
ef
ficienc
y
comparison
for
M
R
k
=
8
;
M
T
=
64
;
N
T
=
16
;
L
=
3
mmW
a
v
e
MIMO
system
using
H
k
and
B
k
matrices
5.
CONCLUSION
This
paper
presented
an
impro
v
ement
to
h
ybrid
beamforming
precoding
scheme
suitable
for
mmW
a
v
e
massi
v
e
MIMO
system
based
on
the
matrices
C
k
=
C
o
v
(
H
k
)
and
B
k
=
H
k
H
H
k
instead
of
the
channel
matrix
H
k
.
As
result,
by
applying
a
fe
w
more
comple
xity
,
a
higher
achie
v
able
rate
can
be
achie
v
ed,
while
this
a
bit
more
comple
xity
will
be
decreased
when
M
T
is
lar
ger
,
because
this
comple
xity
is
about
the
dimension
of
the
matrix.
Our
proposed
precoding
is
also
suitable
when
the
system
dimension
is
lar
ge.
Simulation
results
sho
wed
that
the
achie
v
able
rate
of
the
proposed
precoding,
spatially
sparse,
analog
method
and
h
ybrid
precoders
wit
h
using
C
k
and
B
k
is
higher
than
when
these
algorithm
s
use
only
the
channel
matrix
H
k
.
Furthermore,
the
ener
gy
ef
ficienc
y
of
the
proposed
precoding
w
as
higher
than
other
precoders.
In
addition,
we
e
v
aluated
the
impact
of
inaccurac
y
of
the
CSI
and
it
w
as
sho
wn
that
the
proposed
precoding
is
not
sensiti
v
e
to
CSI
accurac
y
.
Thus,
our
proposed
precoding
has
a
better
performance
in
achie
ving
a
high
throughput.
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