Internati
o
nal
Journal of Ele
c
trical
and Computer
Engineering
(IJE
CE)
V
o
l.4
,
No
.2
,
Ap
r
il 2
014
, pp
. 29
5
~
30
2
I
S
SN
: 208
8-8
7
0
8
2
95
Jo
urn
a
l
h
o
me
pa
ge
: h
ttp
://iaesjo
u
r
na
l.com/
o
n
lin
e/ind
e
x.ph
p
/
IJECE
Analysis of Multiple-Bit S
h
ift-Left Operations on Complex
Numbers in (
−
1+j)-Base Binary Form
at
Ta
riq J
a
m
il* a
n
d
Usman Ali*
*
*Department of Electr
ical and
C
o
mputer Engin
e
ering,
Sultan
Qaboos University
, OMAN
**Laborator
y
of
Signals and
S
y
stems,
Unive
r
sity
of Pa
ris, FRAN
CE
Article Info
A
B
STRAC
T
Article histo
r
y:
Received Dec 10, 2013
Rev
i
sed
Feb 7, 20
14
Accepte
d
Mar 2, 2014
Complex numbers find var
i
ous applications
in
the field of
engineering
.
To
avoid excessive delay
s
in prod
uction
of results
obtained b
y
im
plementing
divide-
a
nd-conq
uer
technique in dealing
with ar
ithmetic oper
a
tion
s
involving
this ty
pe of numbers in today
’
s com
puter s
y
stems, Complex Binar
y
Number
S
y
ste
m
with base
(
−
1+j) has been proposed in scientific liter
a
ture. In this
paper, we h
a
ve investigat
ed the
effects of bit-wis
e
shift lef
t
operations (from
1-8 bits) on the complex binar
y
r
e
presentation
of complex nu
mbers and
anal
yz
ed
thes
e
r
e
s
u
lts
us
ing m
a
t
h
em
atic
al
equa
ti
ons
.
Keyword:
Co
m
p
lex
nu
mb
er
C
o
m
p
l
e
x bi
na
r
y
Bin
a
ry nu
m
b
er
Mu
ltip
le sh
ift
Sh
ift left
Copyright ©
201
4 Institut
e
o
f
Ad
vanced
Engin
eer
ing and S
c
i
e
nce.
All rights re
se
rve
d
.
Co
rresp
ond
i
ng
Autho
r
:
Tariq Jam
i
l,
Depa
rt
m
e
nt
of
El
ect
ri
cal
and
C
o
m
put
er E
ngi
neeri
n
g
,
C
o
l
l
e
ge of En
g
i
neeri
n
g – P.
O.
B
o
x 33
,
Sultan Qa
bo
os
Uni
v
ersity
, AlKh
o
d
12
3,
M
u
scat,
OM
AN
Em
a
il: tj
a
m
i
l
@
squ
.
edu
.
o
m
1.
INTRODUCTION
C
o
m
p
l
e
x num
bers fi
nd
vari
ous a
ppl
i
cat
i
o
ns i
n
t
h
e fi
el
d of e
ngi
neer
i
ng. I
n
t
o
day
’
s com
put
er
syste
m
s, ad
d
itio
n of t
w
o co
m
p
lex
nu
m
b
ers
and
inv
o
l
v
e
s two
i
n
d
i
v
i
du
al ad
d
ition
s
,
on
e for
the real pa
rts
and one for the im
aginary parts
. Sim
i
la
rly, th
e su
b
t
ractio
n
o
f
th
ese two
com
p
l
e
x num
bers i
n
v
o
l
v
e
s
t
w
o i
n
di
vi
dual
subt
ract
i
o
n
s
,
one
f
o
r t
h
e
re
al
part
s
and one f
o
r
t
h
e
im
aginary pa
rts
. Mu
ltip
licatio
n
i
n
vo
lv
es fo
ur ind
i
v
i
du
al m
u
ltip
lica
tio
n
s
,
,
,
, one
subt
ract
i
o
n
, an
d on
e
add
itio
n
. A
n
d
f
i
nal
l
y
, di
vi
si
o
n
i
n
vol
ves
si
x i
n
di
vi
d
u
a
l
m
u
l
tip
licatio
n
s
,
,
,
,
,
, two add
itio
ns
and
, one
s
u
bt
ract
i
o
n
, and
th
en
t
w
o
i
n
di
vi
dual
di
vi
si
o
n
s
a
n
d
. Th
ese
sub
-
o
p
e
ration
s
wit
h
in
an
op
eration
d
ealing
with co
m
p
lex
num
bers i
n
c
r
e
a
se t
h
e del
a
y
i
n
t
h
e cal
cul
a
t
i
on o
f
o
v
eral
l
resul
t
of t
h
e
ope
rat
i
o
n an
d hence
deg
r
a
d
e
s
t
h
e
per
f
o
r
m
a
nce of t
h
e com
put
er
sy
st
em
. The
que
st
t
o
fi
nd a
bet
t
e
r conci
s
e
m
e
t
hod f
o
r re
prese
n
t
i
n
g co
m
p
l
e
x
num
bers
has y
i
el
ded t
h
e
f
o
r
m
ul
at
i
on o
f
a
uni
que
n
u
m
b
er sy
st
em
referr
ed t
o
as C
o
m
p
l
e
x B
i
nary
N
u
m
b
er
Syste
m
(CBNS) [1
]. Ex
ten
s
iv
e d
e
tails o
f
th
e arith
m
e
ti
c al
gori
t
h
m
s
t
o
be fol
l
o
wed
i
n
C
B
N
S fo
r bi
na
ry
rep
r
ese
n
t
a
t
i
ons
and o
p
erat
i
o
n
s
of com
p
l
e
x num
bers can
be f
o
und
in
[
1
,2
,3
] an
d
th
e har
d
w
a
r
e
d
e
sign
s and
i
m
p
l
e
m
en
tatio
n
s
of th
e arithmetic
circu
its
b
a
sed
o
n
th
is
num
ber sy
stem
can be fou
n
d
in [4
,5
,6
,7]
.
In this
p
a
p
e
r,
we h
a
ve p
r
esen
ted
th
e effects o
f
m
u
ltip
le-b
it (fro
m
1
b
it to
8
b
its) sh
ift-left op
eratio
n
s
on
a com
p
lex
num
ber
rep
r
es
ent
e
d i
n
C
B
N
S
.
Thi
s
p
a
pe
r i
s
or
ga
ni
zed as
f
o
l
l
o
w
s
:
I
n
sect
i
on
2,
we’l
l
pr
esent
ba
si
c i
n
f
o
rm
at
i
on ab
ou
t
C
B
N
S an
d
h
o
w to
rep
r
esen
t an
in
teg
e
r-on
l
y co
m
p
lex
nu
m
b
er in
to
th
i
s
n
e
w
nu
m
b
er syste
m
. Th
en
we’ll tak
e
th
e
CBNS
rep
r
ese
n
t
a
t
i
o
n
of com
p
l
e
x num
bers an
d,
i
n
sect
i
on 3,
gi
ve a com
p
rehen
s
i
v
e anal
y
s
i
s
of t
h
e effect
s o
f
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN
:
2
088
-87
08
I
J
ECE
Vo
l. 4
,
N
o
. 2
,
Ap
r
il 20
14
:
295
–
3
02
29
6
m
u
l
tip
le-b
it shift left o
p
e
ratio
n
s
on
th
e com
p
lex
n
u
m
b
e
rs. Co
n
c
l
u
sion
is p
r
esen
ted
in sectio
n
4
,
which
is
followe
d
by ac
knowledgem
ents and re
fere
nc
es.
2.
-
B
A
S
E C
O
MPLEX
BINARY
N
U
M
BER SY
STEM
M
a
t
h
em
at
i
cally
, t
h
e val
u
e o
f
an
n-
bi
t
bi
na
r
y
num
ber wi
t
h
base
1
can
b
e
written
in
t
h
e fo
rm
of a
powe
r s
e
ries as
1
+
1
+
1
+
… +
1
+
1
+
1
where
the
co-effecients
,
,
,…,
,
,
are
bi
na
ry
i
n
nat
u
re (
0
or
1
)
a
n
d
bel
o
n
g
t
o
com
p
l
e
x
b
i
nary
num
ber s
y
st
em
. Usi
n
g
t
h
e c
o
n
v
e
r
si
o
n
al
go
ri
t
h
m
s
gi
ven i
n
[
1
]
,
we ar
e abl
e
to obtain
a
1
b
i
nary
rep
r
ese
n
t
a
t
i
on
of
any
gi
ve
n c
o
m
p
l
e
x n
u
m
b
er (
w
he
t
h
er i
t
i
s
m
a
de f
r
om
i
n
t
e
gers
,
fract
i
ons
,
or
fl
oat
i
n
g
p
o
i
n
t
num
bers
) i
n
C
o
m
p
l
e
x
B
i
nary
N
u
m
b
er Sy
st
em
(C
B
N
S)
.
For e
x
am
ple, as shown in
[1],
2012
111000
0000
001
1100
000
1000
0
2012
1100
0000
000
0110
111
0100
00
2012
100000
0000
001
0000
110
000
2012
111010
0000
001
1101
000
1110
000
2012
2012
111010
0000
0011
101
0001
110
0000
3.
MULTIPLE
-
BIT SHIFT
-
LEFT OPERATIONS IN CB
NS
To a
n
alyze the
effects
of shi
f
t-
left operati
o
ns on a
c
o
m
p
le
x
num
ber re
presented in CBNS
form
at, a
co
m
p
u
t
er pro
g
ram
in
C++ la
n
g
u
a
g
e
was written
wh
ich
a
llo
wed
for au
to
-v
ariation
s
in
m
a
g
n
itud
e
and
sig
n
of
bot
h
real
a
n
d
im
agi
n
ary
c
o
m
ponent
s
o
f
a
com
p
l
e
x
num
ber
i
n
a l
i
n
ea
r
fas
h
i
o
n,
an
d
t
h
en
decom
pos
ed t
h
e
com
p
l
e
x bi
nar
y
num
ber aft
e
r
t
h
e shi
f
t
-
l
e
ft
ope
rat
i
o
n i
n
t
o
i
t
s
real
and i
m
agi
n
a
r
y
com
ponent
s
.
The l
e
n
g
t
h
o
f
th
e o
r
ig
i
n
al b
i
n
a
ry b
it array was li
m
i
ted
to
8
0
0
b
its a
nd
0s we
re pad
d
e
d
o
n
t
h
e l
e
ft
-si
d
e of t
h
e
bi
nar
y
dat
a
wh
en
t
h
e
g
i
v
e
n
co
m
p
lex
num
b
e
r requ
ired less th
an m
a
x
i
m
u
m
allo
wab
l
e
b
its fo
r rep
r
esen
tatio
n in CBNS
fo
rm
at.
To
b
e
tter
u
n
d
e
rstand
th
ese restrictio
n
s
, let’s
co
nsid
er th
e
follo
wing
co
m
p
lex
n
u
m
b
e
r:
Ori
g
i
n
al
c
o
m
p
lex num
ber rep
r
esent
e
d
i
n
C
B
NS bef
o
re pad
d
i
n
g:
90
90
1101
00
01
00
01
00
0
Padded com
p
lex
bina
ry array
suc
h
that
t
h
e total size o
f
t
h
e array is
80
0 b
its.
90
90
0
…
011
01
00
01
00
01
00
0
Sh
ifting
t
h
is b
i
n
a
ry array
b
y
1
-
b
it to
th
e left
will yield
0
…
01101
0001
000
1000
0
en
su
ring
t
h
at to
tal array-size rem
a
in
s 8
00 b
its. Th
is was do
n
e
b
y
rem
ovi
ng
o
n
e
0
fr
om
t
h
e l
e
ft
-
s
i
d
e a
n
d
i
n
se
rt
i
n
g
o
n
e
0
o
n
t
h
e ri
g
h
t
-
si
de
of t
h
e
num
ber.
Si
m
ilarly, sh
iftin
g
of th
e
orig
in
al
b
i
n
a
ry
array
b
y
2,3
,
4
,
5
,
6
,
7
,
8
-
b
its to
th
e left will yield
respectively:
0
…
01101
0001
000
00
0
…
01101
0001
000
000
0
…
01101
0001
000
0000
0
…
01101
0001
000
0000
0
0
…
01101
0001
000
0000
00
0
…
01101
0001
000
0000
000
0
…
01101
0001
000
0000
000
0
Table 1 prese
n
ts an ove
r
all summary
of the e
ffect
on t
h
e signs
of t
h
e com
p
lex num
b
ers,
represe
n
ted
in
CBNS
fo
rm
at,
b
ecau
s
e o
f
m
u
l
tip
le-b
it
sh
i
f
t-left o
p
e
ration
s
(1
to
8
b
its).
Shi
f
t
-
l
e
ft
o
p
er
at
i
ons
on c
o
m
p
l
e
x
bi
na
ry
n
u
m
bers affect
n
o
t
o
n
l
y
t
h
e si
gns
o
f
t
h
e
gi
v
e
n com
p
l
e
x
num
bers
(as s
h
own i
n
Ta
ble 1) but also ha
ve
im
pact
on t
h
e
magnitude
s of
the com
p
lex
num
b
ers according t
o
vari
ous m
a
t
h
em
at
i
cal
rel
a
ti
onshi
ps.
To
fi
n
d
o
u
t
t
h
e e
ffec
t
s of s
h
i
f
t
-
l
e
ft
ope
rat
i
o
ns o
n
t
h
e m
a
gni
t
udes
of t
h
e
com
p
l
e
x num
bers
, we va
ri
ed t
h
e
m
a
gni
t
u
de of real
an
d
im
agi
n
ary
com
ponent
s o
f
t
h
e ori
g
i
n
al
co
m
p
l
e
x
num
bers in a linear fas
h
ion (Figure 1). T
h
e com
p
lex
nu
m
b
ers o
b
t
ained
after sh
i
f
t-left o
p
e
ratio
ns were
anal
y
zed
by
o
b
t
ai
ni
ng m
a
t
h
em
at
i
cal
equat
i
ons
desc
ri
bi
n
g
t
h
ei
r
be
havi
or,
as gi
ven
i
n
Fi
g
u
res
.
2-
9.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
ECE
I
S
SN
:
208
8-8
7
0
8
Ana
l
ysis o
f
Multip
le-Bit S
h
ift-Left Op
era
tions on
C
o
mp
lex
Nu
mb
ers (Ta
r
i
q
Ja
mil)
29
7
To f
u
l
l
y
un
der
s
t
a
nd t
h
e
vari
a
t
i
ons i
n
t
h
e si
gn a
nd m
a
gni
t
ude
of t
h
e c
o
m
p
l
e
x num
bers bef
o
re a
n
d
after th
e sh
ift-l
e
ft op
erati
o
n,
we
use
d
M
i
cr
o
s
oft
Excel
t
o
dr
aw
gra
p
hs as
s
h
o
w
n i
n
t
h
e
Fi
gu
res.
1
0
-
1
7
.
Tabl
e 1.
Effect on
si
g
n
s
of com
p
lex
n
u
m
b
e
rs in
CB
NS
format after sh
i
f
t-left op
eration
s
Befo
re Shi
ft-L
eft
After
Shi
ft-Le
f
t
by 1-bit
After
Shi
ft-Le
f
t
by 2-bits
After
Shi
ft-Le
f
t
by 3-bits
After
Shi
ft-Le
f
t
by 4-bits
Real
I
m
aginar
y
Real
I
m
aginar
y
Real
I
m
aginar
y
Real
I
m
aginar
y
Real
I
m
aginar
y
+
0
–
+
0
–
+
+
–
0
–
0
+
–
0
+
–
–
+
0
0
+
–
–
+
0
–
+
0
–
0
–
+
+
–
0
+
–
0
+
+
+
–
0
+
–
0
+
–
–
+
–
0
+
–
–
+
0
–
+
–
+
0
–
+
+
–
0
+
–
–
–
+
0
–
+
0
–
+
+
Before Shift-Left
After Shift-Left
by
5-
bits
After Shift-Left
by
6-
bits
After
Shi
ft-Le
f
t
by 7-bits
After
Shi
ft-Le
f
t
by 8-bits
Real
I
m
aginar
y
Real
I
m
aginar
y
Real
I
m
aginar
y
Real
I
m
aginar
y
Real
I
m
aginar
y
+
0
+
–
+
+
–
–
+
–
–
0
–
+
–
–
+
+
–
+
0
+
+
+
–
+
+
–
+
+
0
–
–
–
+
–
–
+
–
–
+
+
+
–
–
+
–
–
+
+
+
–
–
–
+
+
–
+
+
–
–
+
+
+
–
–
+
–
–
+
–
–
–
+
+
–
+
+
–
–
Fi
gu
re
1.
B
e
f
o
r
e
shi
f
t
-
l
e
ft
Fig
u
re
2
.
After sh
ift
-
left b
y
1-b
it
Fig
u
re3
.
After sh
ift-left b
y
2
-
b
its
Fig
u
re
4
.
After sh
ift
-
left b
y
3-b
its
Fig
u
re
5
.
After sh
ift
-
left b
y
4-b
its
Fig
u
re
6
.
After sh
ift
-
left b
y
5-b
its)
Fig
u
re
7
.
After sh
ift
-
left b
y
6-b
its
Fig
u
re
8
.
After sh
ift
-
left b
y
7-b
its
Fig
u
re
9
.
After sh
ift
-
left b
y
8-b
its
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN
:
2
088
-87
08
I
J
ECE
Vo
l. 4
,
N
o
. 2
,
Ap
r
il 20
14
:
295
–
3
02
29
8
Fig
u
re
10
. Effects o
f
sh
ift-left
o
p
e
ration
o
n
sig
n
and
m
a
g
n
i
t
u
d
e
of a po
sitiv
e
real-on
l
y com
p
lex
n
u
m
b
e
r
(1
-8
bits)
Fi
gu
re
1
1
. E
ffe
ct
s of
s
h
i
f
t
-
l
e
ft
ope
rat
i
o
n
on
si
gn
an
d m
a
gni
t
ude
o
f
a
ne
gat
i
v
e
real
-o
nl
y
c
o
m
p
l
e
x num
ber
(1
-8
bits)
Fig
u
re
12
. Effects o
f
sh
ift-left
o
p
e
ration
o
n
sig
n
and
m
a
g
n
i
t
u
d
e
of a po
sitiv
e im
ag
in
ary-on
ly co
m
p
lex
num
ber (1
-
8
bi
t
s
)
Fi
gu
re
1
3
. E
ffe
ct
s of
s
h
i
f
t
-
l
e
ft
ope
rat
i
o
n
on
si
gn
an
d m
a
gni
t
ude
o
f
a
ne
gat
i
v
e i
m
agi
n
ary
-
o
n
l
y
com
p
l
e
x
num
ber (1
-
8
bi
t
s
)
-2
5
-2
0
-1
5
-1
0
-5
0
5
10
15
20
25
1
4
7
10
13
16
19
R
eal
Im
a
g
R
eal
Sh
i
f
t
Lef
t
1-
b
i
t
I
m
ag S
h
i
f
t
L
e
f
t
1-
bi
t
-5
0
-4
0
-3
0
-2
0
-1
0
0
10
20
30
14
7
1
0
1
3
1
6
1
9
R
eal
Im
a
g
R
eal
S
h
i
f
t
Lef
t
2
-
bi
t
I
m
ag S
h
i
f
t
Lef
t
2
-
bi
t
0
5
10
15
20
25
30
35
40
1
3
5
7
9
1
11
3
1
5
1
71
9
R
eal
Im
a
g
R
eal
S
h
i
f
t
Lef
t
3-
bit
I
m
ag S
h
i
f
t
Lef
t
3-
bit
-
100
-8
0
-6
0
-4
0
-2
0
0
20
40
13
57
9
1
1
1
3
1
5
1
7
1
9
Re
a
l
Im
a
g
R
e
a
l
S
h
i
f
t L
e
ft 4
-
bi
t
I
m
ag S
h
i
f
t
L
e
f
t
4-
bi
t
-1
0
0
-8
0
-6
0
-4
0
-2
0
0
20
40
60
80
10
0
1
4
7
1
01
3
1
61
9
Re
a
l
Im
a
g
Re
a
l
S
h
i
f
t
Le
f
t
5-
b
i
t
Im
a
g
S
h
i
f
t
Le
f
t
5-
b
i
t
0
20
40
60
80
100
120
140
160
1
4
7
1
01
3
1
61
9
Re
a
l
Im
a
g
Re
a
l
S
h
i
f
t
Lef
t
6-
b
i
t
Im
a
g
S
h
i
f
t
Lef
t
6-
b
i
t
-
180
-
160
-
140
-
120
-
100
-8
0
-6
0
-4
0
-2
0
0
20
40
14
7
1
0
1
3
1
6
1
9
Re
a
l
Im
a
g
Re
a
l
S
h
i
f
t
L
e
f
t
7-
bi
t
Im
a
g
S
h
i
f
t
L
e
f
t
7-
bi
t
0
50
10
0
15
0
20
0
25
0
30
0
35
0
13
57
9
1
1
1
3
1
5
1
7
1
9
Re
a
l
Im
a
g
Re
a
l
S
h
i
f
t
Le
f
t
8-bi
t
Im
a
g
S
h
i
f
t
Le
f
t
8-bi
t
-2
5
-2
0
-1
5
-1
0
-5
0
5
10
15
20
25
1
4
7
10
13
16
19
R
eal
Im
a
g
R
eal
S
h
i
f
t
Lef
t
1-
bi
t
I
m
ag S
h
i
f
t
Lef
t
1-
bi
t
-3
0
-2
0
-1
0
0
10
20
30
40
50
13
57
9
1
1
1
3
1
5
1
7
1
9
Re
a
l
Im
a
g
Re
a
l
S
h
i
f
t
Lef
t
2
-
bit
Im
a
g
S
h
i
f
t
Lef
t
2
-
bit
-40
-35
-30
-25
-20
-15
-10
-5
0
13
57
9
1
1
1
3
1
5
1
7
1
9
R
eal
Im
a
g
R
eal
S
h
i
f
t
L
e
f
t
3-
bi
t
I
m
a
g
S
h
i
f
t
L
e
ft
3-
bi
t
-4
0
-2
0
0
20
40
60
80
10
0
13579
1
1
1
3
1
5
1
7
1
9
Re
a
l
Im
a
g
Re
a
l
S
h
i
f
t L
e
ft
4-
bi
t
I
m
a
g
S
h
i
f
t L
e
ft
4-
bi
t
-
100
-8
0
-6
0
-4
0
-2
0
0
20
40
60
80
100
1
3
5
7
9
1
1
1
31
51
7
1
9
Re
a
l
Im
a
g
Re
a
l
S
h
i
f
t
Le
f
t
5-
b
i
t
I
m
ag S
h
ift
Le
f
t
5-
b
i
t
-1
6
0
-1
4
0
-1
2
0
-1
0
0
-8
0
-6
0
-4
0
-2
0
0
1
4
7
1
0
13
16
19
Re
a
l
Im
a
g
Re
a
l
S
h
i
f
t
Le
f
t
6-
bi
t
Im
a
g
S
h
i
f
t
Le
f
t
6-
bi
t
-5
0
0
50
10
0
15
0
20
0
1
4
7
1
01
3
1
61
9
Re
a
l
Im
a
g
Re
a
l
S
h
i
f
t
Le
f
t
7
-
bi
t
Im
a
g
S
h
i
f
t
Le
f
t
7
-
bi
t
-
350
-
300
-
250
-
200
-
150
-
100
-5
0
0
50
1
3
5
7
9
1
1
1
31
51
7
1
9
Re
a
l
Im
a
g
Re
a
l
S
h
i
f
t
Le
f
t
8-
b
i
t
I
m
ag S
h
ift
Le
f
t
8-
b
i
t
-2
5
-2
0
-1
5
-1
0
-5
0
5
10
15
20
25
1
4
7
10
13
1
6
19
Re
a
l
Im
a
g
Re
a
l
S
h
i
f
t
Le
f
t
1
-
bit
I
m
ag S
h
ift
Le
f
t
1
-
bit
0
5
10
15
20
25
30
35
40
1
3
5
7
9
11
13
15
17
19
Re
a
l
Im
a
g
R
e
a
l
S
h
i
ft L
e
f
t
2-
bi
t
I
m
ag S
h
i
f
t
Lef
t
2-
bi
t
-5
0
-4
0
-3
0
-2
0
-1
0
0
10
20
30
40
50
1
4
7
10
13
16
19
Re
a
l
Im
a
g
R
e
a
l
S
h
i
f
t
L
e
f
t
3-
bi
t
I
m
a
g
S
h
i
f
t
L
e
ft
3-
bi
t
-1
0
0
-8
0
-6
0
-4
0
-2
0
0
20
40
1
4
7
1
01
31
6
1
9
Re
a
l
Im
a
g
Re
a
l
S
h
i
f
t
L
e
f
t
4-
bi
t
Im
a
g
S
h
i
f
t
L
e
f
t
4-
bi
t
0
10
20
30
40
50
60
70
80
1
4
7
10
1
3
16
19
R
eal
Im
a
g
R
eal
S
h
i
f
t
Le
ft 5-
b
i
t
Im
a
g
S
h
i
f
t
Le
ft 5-
b
i
t
-
200
-
150
-
100
-5
0
0
50
1
4
7
1
01
3
1
61
9
R
eal
Im
a
g
R
eal
S
h
i
f
t
Lef
t
6-
bi
t
I
m
ag S
h
i
f
t
Lef
t
6-
bi
t
-
200
-
150
-
100
-5
0
0
50
100
150
200
1
4
7
10
13
16
1
9
R
eal
Im
a
g
R
eal
S
h
i
f
t
Left
7-
bi
t
Im
a
g
S
h
i
f
t
Left
7-
bi
t
0
50
100
150
200
250
300
350
14
7
1
0
1
3
1
6
1
9
R
eal
Im
a
g
R
eal S
h
ift
L
e
f
t
8-
bit
I
m
ag S
h
ift
L
e
f
t
8-
bit
-2
5
-2
0
-1
5
-1
0
-5
0
5
10
15
20
25
1
4
7
1
01
3
1
61
9
Re
a
l
Im
a
g
Re
a
l
S
h
i
f
t
Lef
t
1-
bi
t
I
m
ag S
h
i
f
t
Lef
t
1-
bi
t
-4
0
-3
5
-3
0
-2
5
-2
0
-1
5
-1
0
-5
0
1
3
5
7
9
1
1
13
15
17
1
9
Re
a
l
Im
a
g
Re
a
l
S
h
i
f
t
Lef
t
2
-
bi
t
Im
a
g
S
h
i
f
t
Lef
t
2
-
bi
t
-5
0
-4
0
-3
0
-2
0
-1
0
0
10
20
30
40
50
1
4
7
1
01
31
61
9
R
eal
Im
a
g
R
eal
S
h
i
f
t
Lef
t
3-
b
i
t
I
m
ag
S
h
i
f
t
Lef
t
3-
b
i
t
-4
0
-2
0
0
20
40
60
80
100
1
4
7
10
13
16
19
Re
a
l
Im
a
g
Re
a
l
S
h
i
f
t
Lef
t
4-
bi
t
Im
a
g
S
h
i
f
t
Lef
t
4-
bi
t
-8
0
-7
0
-6
0
-5
0
-4
0
-3
0
-2
0
-1
0
0
1
3
5
7
9
11
13
15
17
19
R
eal
Im
a
g
R
eal
S
h
i
f
t
Lef
t
5-
bit
I
m
ag S
h
if
t
Lef
t
5-
bit
-4
0
-2
0
0
20
40
60
80
100
120
140
160
180
1
4
7
1
01
31
6
1
9
Re
a
l
Im
a
g
Re
a
l
S
h
i
f
t
Lef
t
6-
bi
t
I
m
ag S
h
i
f
t
Lef
t
6-
bi
t
-2
0
0
-1
5
0
-1
0
0
-5
0
0
50
10
0
15
0
20
0
1
4
7
1
01
31
61
9
Re
a
l
Im
a
g
Re
a
l
S
h
i
f
t
L
e
f
t
7-
bi
t
Im
a
g
S
h
i
f
t
L
e
f
t
7-
bi
t
-3
5
0
-3
0
0
-2
5
0
-2
0
0
-1
5
0
-1
0
0
-5
0
0
13579
1
1
1
3
1
5
1
7
1
9
Re
a
l
Im
a
g
Re
a
l
S
h
i
f
t
Le
f
t
8-
b
i
t
I
m
ag
S
h
if
t
Le
f
t
8-
b
i
t
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
ECE
I
S
SN
:
208
8-8
7
0
8
Ana
l
ysis o
f
Multip
le-Bit S
h
ift-Left Op
era
tions on
C
o
mp
lex
Nu
mb
ers (Ta
r
i
q
Ja
mil)
29
9
Fi
gu
re
1
4
. E
ffe
ct
s of
s
h
i
f
t
-
l
e
ft
ope
rat
i
o
n
on
si
gn
an
d m
a
gni
t
ude
o
f
a
+R
eal
+Im
a
gi
nary
co
m
p
l
e
x num
ber
(1
-8
bits)
Fi
gu
re
1
5
. E
ffe
ct
s of
s
h
i
f
t
-
l
e
ft
ope
rat
i
o
n
on
si
gn
an
d m
a
gni
t
ude
o
f
a
+R
eal
–Im
a
gi
nary
c
o
m
p
l
e
x num
ber
(1
-8
bits)
Fi
gu
re
1
6
. E
ffe
ct
s of
s
h
i
f
t
-
l
e
ft
ope
rat
i
o
n
on
si
gn
an
d m
a
gni
t
ude
o
f
a
–R
eal
+Im
a
gi
nary
co
m
p
l
e
x num
ber
(1
-8
bits)
Fi
gu
re
1
7
. E
ffe
ct
s of
s
h
i
f
t
-
l
e
ft
ope
rat
i
o
n
on
si
gn
an
d m
a
gni
t
ude
o
f
a
–R
eal
–Im
a
gi
nary
c
o
m
p
l
e
x num
ber
(1
-8
bits)
Aft
e
r
anal
y
z
i
n
g Fi
gu
res.
1
-
17
, we
are ab
le t
o
ob
tain
th
e ch
aracteris
t
i
c
eq
ua
t
i
ons
desc
ri
bi
n
g
c
o
m
p
l
e
x
num
bers i
n
C
B
NS
fo
rm
at
aft
e
r s
h
i
f
t
-
l
e
ft
o
p
er
at
i
ons.
The
s
e e
quat
i
o
ns
are
gi
ven
i
n
Tabl
e
2.
-5
0
-4
0
-3
0
-2
0
-1
0
0
10
20
30
1
3
5
7
9
1
1
13
15
17
19
R
eal
Im
a
g
R
eal S
h
if
t
L
e
ft
1-
bit
Im
a
g
S
h
i
f
t
L
e
ft
1-
bit
-5
0
-4
0
-3
0
-2
0
-1
0
0
10
20
30
40
50
1
4
7
1
01
3
1
61
9
R
eal
Im
a
g
R
eal
S
h
i
f
t
Lef
t
2-
bi
t
I
m
ag S
h
i
f
t
Lef
t
2-
bi
t
0
10
20
30
40
50
60
70
80
1
3
5
7
9
1
11
3
1
51
7
1
9
Re
a
l
Im
a
g
Re
a
l
S
h
i
ft L
e
ft
3-
bi
t
I
m
a
g
S
h
i
ft L
e
ft
3-
bi
t
-1
0
0
-8
0
-6
0
-4
0
-2
0
0
20
40
1
3
5
7
9
1
1
1
31
51
7
1
9
Re
a
l
Im
a
g
R
e
a
l
S
h
i
f
t
L
e
ft
4
-
bi
t
I
m
a
g
S
h
i
f
t L
e
ft
4
-
bi
t
0
20
40
60
80
10
0
12
0
14
0
16
0
1
3
5
7
9
1
11
3
1
51
7
1
9
Re
a
l
Im
a
g
Re
a
l
S
h
i
f
t
Le
f
t
5-
bi
t
Im
a
g
S
h
i
f
t
Le
f
t
5-
bi
t
-
200
-
150
-
100
-5
0
0
50
100
150
200
14
7
1
0
1
3
1
6
1
9
R
eal
Im
a
g
R
eal S
h
ift
Left
6-
bit
Im
a
g
S
h
i
f
t
Left
6-
bit
-3
5
0
-3
0
0
-2
5
0
-2
0
0
-1
5
0
-1
0
0
-5
0
0
50
1
4
7
1
01
31
61
9
Re
a
l
Im
a
g
R
e
a
l
S
h
i
ft
L
e
f
t
7-
bi
t
I
m
a
g
S
h
i
ft L
e
ft
7-
bi
t
0
50
100
150
200
250
300
350
1
4
7
10
13
16
19
Re
a
l
Im
a
g
R
e
a
l
S
h
i
f
t
Lef
t
8-
bi
t
Im
a
g
S
h
i
f
t
Le
f
t
8
-
bi
t
-3
0
-2
0
-1
0
0
10
20
30
40
50
1
4
7
1
0
13
16
1
9
Re
a
l
Im
a
g
Re
a
l
S
h
i
f
t
Le
f
t
1-
bi
t
I
m
ag S
h
i
f
t
Le
f
t
1-
bi
t
-5
0
-4
0
-3
0
-2
0
-1
0
0
10
20
30
1
3
5
7
9
11
13
15
1
7
19
Re
a
l
Im
a
g
R
e
a
l
S
h
i
f
t L
e
ft
2
-
bi
t
I
m
a
g
S
h
i
ft
L
e
ft 2
-
bi
t
-4
0
-2
0
0
20
40
60
80
100
1
4
7
1
0
13
16
19
Re
a
l
Im
a
g
Re
a
l
S
h
i
ft L
e
ft
3-
bi
t
I
m
a
g
S
h
i
ft L
e
ft
3-
bi
t
-1
0
0
-8
0
-6
0
-4
0
-2
0
0
20
40
60
80
10
0
1
4
7
1
01
3
1
61
9
R
eal
Im
a
g
R
eal
S
h
i
f
t
Le
f
t
4-
b
i
t
I
m
ag Sh
i
f
t
Le
f
t
4-
b
i
t
-
180
-
160
-
140
-
120
-
100
-8
0
-6
0
-4
0
-2
0
0
20
40
1
4
7
1
01
3
1
61
9
R
eal
Im
a
g
R
eal
S
h
i
f
t
Lef
t
5-
b
i
t
I
m
ag S
h
i
f
t
Lef
t
5-
b
i
t
-5
0
0
50
10
0
15
0
20
0
14
7
1
0
1
3
1
6
1
9
Re
a
l
Im
a
g
Re
a
l
S
h
i
f
t
Lef
t
6-
bi
t
I
m
ag S
h
i
f
t
Lef
t
6-
bi
t
-350
-300
-250
-200
-150
-100
-5
0
0
50
1
4
7
1
01
3
1
61
9
Re
a
l
Im
a
g
Re
a
l
S
h
i
f
t
L
e
f
t
7-bi
t
I
m
ag S
h
i
f
t
L
e
f
t
7-bi
t
y
=
-
1
6
.
01
4x
+
0
.
1
-40
0
-30
0
-20
0
-10
0
0
10
0
20
0
30
0
40
0
1
4
7
1
0
1
31
61
9
Re
a
l
Im
a
g
Re
a
l
S
h
i
f
t
L
e
f
t
8-
bi
t
I
m
ag S
h
i
f
t
L
e
f
t
8-
bi
t
-5
0
-4
0
-3
0
-2
0
-1
0
0
10
20
30
1
3
5
7
9
1
11
3
1
5
1
71
9
Re
a
l
Im
a
g
R
e
a
l
S
h
i
ft L
e
ft
1-
b
i
t
I
m
a
g
S
h
i
ft L
e
ft
1-
b
i
t
-3
0
-2
0
-1
0
0
10
20
30
40
50
14
7
1
0
1
3
1
6
1
9
R
eal
Im
a
g
R
eal
Sh
i
f
t
L
e
ft
2
-
b
i
t
Im
a
g
S
h
i
f
t
L
e
ft
2
-
b
i
t
-
100
-8
0
-6
0
-4
0
-2
0
0
20
40
1
4
7
10
13
16
19
R
eal
Im
a
g
R
eal
S
h
i
f
t
Lef
t
3
-
bi
t
I
m
ag Sh
i
f
t
Lef
t
3
-
bi
t
-1
0
0
-8
0
-6
0
-4
0
-2
0
0
20
40
60
80
10
0
13579
1
1
1
3
1
5
1
7
1
9
Re
a
l
Im
a
g
R
e
a
l
S
h
i
ft L
e
ft
4
-
bi
t
I
m
a
g
S
h
i
f
t L
e
ft
4-
bi
t
-5
0
0
50
10
0
15
0
20
0
1
4
7
1
01
3
1
61
9
R
eal
Im
a
g
R
eal
S
h
i
f
t
Lef
t
5-
b
i
t
I
m
ag S
h
i
f
t
Lef
t
5-
b
i
t
-
200
-
150
-
100
-5
0
0
50
1
3
5
7
9
11
13
15
17
1
9
R
eal
Im
a
g
R
eal
S
h
ift
Left
6-
bi
t
I
m
ag S
h
if
t
Left
6-
bi
t
-5
0
0
50
100
150
200
250
300
350
1
4
7
1
01
31
6
1
9
R
eal
Im
a
g
R
eal
S
h
i
f
t
Le
f
t
7-
bi
t
I
m
ag S
h
i
f
t
Le
f
t
7-
bi
t
-4
0
0
-3
0
0
-2
0
0
-1
0
0
0
10
0
20
0
30
0
40
0
1
4
7
1
01
31
6
1
9
Re
a
l
Im
a
g
R
e
a
l
S
h
i
f
t L
e
f
t
8-
bit
I
m
ag S
h
i
f
t
L
e
f
t
8-
bi
t
-3
0
-2
0
-1
0
0
10
20
30
40
50
13579
1
1
1
3
1
5
1
7
1
9
R
eal
Im
a
g
R
eal
S
h
i
f
t
Lef
t
1-
b
i
t
I
m
a
g
S
h
i
f
t
L
e
ft
1-
b
i
t
-5
0
-4
0
-3
0
-2
0
-1
0
0
10
20
30
40
50
1
4
7
1
0
13
16
19
Re
a
l
Im
a
g
R
e
al
S
h
i
f
t
Lef
t
2-
bi
t
I
m
ag S
h
i
f
t
Lef
t
2-
bi
t
-8
0
-7
0
-6
0
-5
0
-4
0
-3
0
-2
0
-1
0
0
Re
a
l
Im
a
g
R
e
al
S
h
i
f
t
Le
f
t
3-
bi
t
I
m
a
g
S
h
i
ft
L
e
ft
3-
b
i
t
-4
0
-2
0
0
20
40
60
80
10
0
1
4
7
1
01
3
1
61
9
Re
a
l
Im
a
g
R
e
al
S
h
i
f
t
Lef
t
4-
bi
t
I
m
ag
S
h
i
f
t
Lef
t
4-
bi
t
-
180
-
160
-
140
-
120
-
100
-8
0
-6
0
-4
0
-2
0
0
20
40
1
4
7
1
0
1
31
61
9
R
eal
Im
a
g
R
eal
S
h
if
t
Le
f
t
5-
bi
t
I
m
ag S
h
if
t
Le
f
t
5-
bi
t
-
200
-
150
-
100
-5
0
0
50
100
150
200
14
7
1
0
1
3
1
6
1
9
R
eal
Im
a
g
R
eal
S
h
i
f
t
Lef
t
6-
bi
t
Im
a
g
S
h
i
f
t
Lef
t
6-
bi
t
-50
0
50
100
150
200
250
300
350
Re
a
l
Ima
g
Re
a
l
S
h
i
f
t
Lef
t
7
-
bi
t
I
m
ag S
h
i
f
t
Lef
t
7
-
bi
t
-3
5
0
-3
0
0
-2
5
0
-2
0
0
-1
5
0
-1
0
0
-5
0
0
1
4
7
1
01
31
61
9
Re
a
l
Im
a
g
Re
a
l
S
h
i
f
t
Lef
t
8-
bi
t
Im
a
g
S
h
i
f
t
Lef
t
8-
bi
t
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN
:
2
088
-87
08
I
J
ECE
Vo
l. 4
,
N
o
. 2
,
Ap
r
il 20
14
:
295
–
3
02
30
0
Ta
ble 2
.
C
h
ara
c
t
e
ri
st
i
c
equat
i
ons
d
e
scri
bi
n
g
com
p
l
e
x n
u
m
b
ers in CBNS form
at after sh
ift
-
left op
eratio
ns
Befo
re Shi
ft-L
eft
After
Shi
ft-Le
f
t
by 1-bit
After
Shi
ft-Le
f
t
by 2-bits
Real
old
Im
aginary
old
Real
new
Im
aginary
new
Real
new
Im
aginary
new
+
0
Real
old
+Real
old
0
2Real
old
–
0
Real
old
+Real
old
0
2Real
old
0
+
Im
ag
old
Im
ag
old
+2Im
ag
old
0
0
–
Im
ag
old
Im
ag
old
+2Im
ag
old
0
+
+
2Real
old
0 +2Real
old
2I
m
a
g
old
+
–
0
2I
m
a
g
old
2Real
old
+2Im
ag
old
–
+
0
2I
m
a
g
old
2Real
old
+2Im
ag
old
–
–
2Real
old
0 +2Real
old
2I
m
a
g
old
Befo
re Shi
ft-L
eft
After
Shi
ft-Le
f
t
by 3-bits
After
Shi
ft-Le
f
t
by 4-bits
Real
old
Im
aginary
old
Real
new
Im
aginary
new
Real
new
Im
aginary
new
+
0
+2Real
old
+2Real
old
4Real
old
0
–
0
+2Real
old
+2Real
old
4Real
old
0
0
+
2I
m
a
g
old
+2Im
ag
old
0
4I
m
a
g
old
0
–
2I
m
a
g
old
+2Im
ag
old
0
4I
m
a
g
old
+
+
0 +4
I
m
ag
old
4Real
old
4I
m
a
g
old
+
–
+4Real
old
0
4Real
old
4I
m
a
g
old
–
+
+4Real
old
0
4Real
old
4I
m
a
g
old
–
–
+4Real
old
0
4Real
old
4I
m
a
g
old
Before Shift-Left
After Shift-Left
by
5-
bits
After Shift-Left
by
6-
bits
Real
old
Im
aginary
old
Real
new
Im
aginary
new
Real
new
Im
aginary
new
+
0
+4Real
old
+1
4Real
old
+¾
2
+8Real
old
+¼
–
0
4Real
old
+4Real
old
0
8Real
old
0
+
+4Im
ag
old
¼
+4Im
ag
old
+
⅓
8I
m
a
g
old
½
0
–
4I
m
a
g
old
+
⅓
4I
m
a
g
old
¼
+8Im
ag
old
¼
+½
+
+
+8Real
old
¼
+½
8Real
old
¼
+8Im
ag
old
¾
+
–
0
8I
m
a
g
old
+8Real
old
+8Im
ag
old
¼
–
+
2
+8Im
ag
old
+¼
8Real
old
+1½
8I
m
a
g
old
2
–
–
8Real
old
½
+8Real
old
+½
8I
m
a
g
old
+
⅓
Befo
re Shi
ft-L
eft
After
Shi
ft-Le
f
t
by 7-bits
After
Shi
ft-Le
f
t
by 8-bits
Real
old
Im
aginary
old
Real
new
Im
aginary
new
Real
new
Im
aginary
new
+
0
8Real
old
+1
½
8Real
old
2
+16R
eal
old
+½
+3½
–
0
+8Real
old
+8Real
old
¼
16Real
old
+½
+¼
0
+
+8Im
ag
old
+½
8I
m
a
g
old
+
⅓
1
+16Im
a
g
old
0
–
8I
m
a
g
old
¼
+8Im
ag
old
¾
+1
16
I
m
ag
old
+
⅓
+
+
+1
16
I
m
ag
old
+
⅓
+16R
eal
old
½
+16Im
a
g
old
+½
+
–
16Real
old
+
½
+¼
+16R
eal
old
½
16
I
m
ag
old
–
+
+16R
eal
old
+
½
+3½
16Real
old
4¼
+16Im
a
g
old
3
–
–
+16R
eal
old
1
16Real
old
+¾
16
I
m
ag
old
1¼
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
ECE
I
S
SN
:
208
8-8
7
0
8
Ana
l
ysis o
f
Multip
le-Bit S
h
ift-Left Op
era
tions on
C
o
mp
lex
Nu
mb
ers (Ta
r
i
q
Ja
mil)
30
1
4.
CO
NCL
USI
O
N
The i
m
port
a
nt
rol
e
o
f
com
p
l
e
x
num
bers
i
n
al
l
ty
pes of en
gi
ne
eri
ng a
ppl
i
cat
i
o
n
s
cann
o
t
be
un
de
rst
a
t
e
d. T
o
im
pro
v
e t
h
e
per
f
o
r
m
a
nce of ari
t
h
m
e
t
i
c
op
erat
i
ons i
n
v
o
l
v
i
ng t
h
ese t
y
pe
of
num
bers. C
B
NS
p
r
ov
id
es a v
i
ab
le altern
ative to
represen
t th
ese n
u
m
be
rs in a concis
e form
at with the expectation
of
su
bstan
tial enhan
cem
en
t in
the sp
eed
o
f
arit
h
m
et
ic o
p
e
ratio
n
s
d
ealin
g wi
th
th
ese typ
e
s
o
f
nu
m
b
ers.
In th
is
pape
r,
we ha
ve
l
ook
ed i
n
det
a
i
l
on h
o
w s
h
i
f
t
-
l
e
ft
o
p
erat
i
o
ns
of 1
-
8 bi
t
s
o
n
a com
p
l
e
x nu
m
b
er represe
n
t
e
d i
n
CBNS a
ffect t
h
e si
gns
and m
a
gnitudes
of these num
b
ers.
ACKNOWLE
DGE
M
ENTS
The
w
o
r
k
pres
ent
e
d i
n
t
h
i
s
pape
r
has
bee
n
t
h
e
res
u
l
t
of
a resea
r
c
h
gr
ant
:
I
G
/
E
N
G
/
E
C
E
D/
0
6
/
0
2
pr
o
v
i
d
e
d
by
Sul
t
a
n Qa
bo
os
Uni
v
ersi
t
y
(O
m
a
n) an
d we g
r
at
eful
l
y
ack
no
wl
ed
ge t
h
e enc
o
u
r
a
g
em
ent
rende
r
e
d
to
u
s
fo
r th
is p
r
oj
ect. Prelimin
ary resu
lts related
to
sing
le-b
it sh
ift-left op
eration
s
o
n
co
m
p
lex
nu
m
b
ers
rep
r
ese
n
t
e
d i
n
C
B
N
S ha
ve ap
peare
d
p
r
ev
i
o
usly in
th
e Pro
c
eed
ing
s
of th
e Canadia
n
Conference
on Elec
trical
an
d Co
m
p
u
t
er Eng
i
n
e
er
ing 20
05
, an
d (u
p-
to 4-b
its sh
if
t-
lef
t
op
er
atio
n
s
) in th
e Pro
ceed
i
ng
s of
the
In
tern
ation
a
l Co
nferen
ce on
Co
m
p
u
t
er an
d
Co
mm
u
n
i
catio
n
En
g
i
n
e
ering
200
6. Th
e
p
o
s
itiv
e
feed
b
a
ck
receive
d from revie
w
ers of these
pre
v
ious publications
prom
pted us
to
e
nga
ge i
n
m
o
re thorough a
nd
ext
e
n
d
ed a
n
al
y
s
i
s
(up t
o
8
-
bi
t
s
) of s
h
i
f
t
-
l
e
ft
ope
rat
i
o
n
s
i
nvol
vi
n
g
co
m
p
l
e
x bi
nary
num
bers an
d
we are
th
ank
f
u
l
to th
ese rev
i
ewers
for th
eir v
a
l
u
ab
le in
pu
t.
REFERE
NC
ES
[1]
T
.
Ja
mi
l,
Compl
ex
Binar
y
Number
Sys
t
em
, Sprin
g
er-Verlag
,
G
e
r
m
an
y
,
2012.
[2]
T. Jam
il, N. H
o
lm
es, D. Blest
,
Towards im
plem
enta
tion of a binar
y
number
sy
stem for complex numbers,
Proceed
ings of
t
h
e IE
EE
Southe
astcon
, 2000
, pp
. 268-274
.
[3]
T. J
a
m
il,
The
co
m
p
lex binar
y
nu
m
b
er s
y
s
t
em
: Ba
s
i
c ari
t
hm
etic m
a
de s
i
m
p
le,
IEE
E
Pot
e
ntia
ls
20(5), 2002, pp
. 39-
41.
[4]
T.
Jamil,
B.
Arafeh,
A.
AlHabsi,
Hardware implemen
tation and performance ev
aluati
on of complex binar
y
adder
designs,
Proceedings of th
e 7th
World
Multiconference on Sy
stemics, Cyb
e
rnetic
s, and Informatics
, 2, 2003, pp.
68-
73.
[5]
T.
Jamil, A.
Abdulghani, A.
AlMaashari, D
e
sign of
a n
i
bb
le-size
subtr
acto
r
for (
−
1+j)-bas
e complex binar
y
numbers,
WSEAS Transactions o
n
Circuits
and S
y
stems
3(5), 200
4, pp
. 1067-107
2.
[6]
T. Jam
i
l
,
A. A
bdulghani
, A. A
l
Maashari
, Desi
gn
and im
plem
entation
of a n
i
b
b
le-size m
u
ltipl
i
e
r for (
−
1
+
j)-b
ase
complex bin
a
r
y
numbers,
WSEAS Transactions o
n
Circuits
and S
y
stems
4(11), 20
05, pp
. 1539-15
44.
[7]
T. Jamil, S. AlA
b
ri, Design of
a
divi
der
cir
c
uit fo
r complex bin
a
r
y
numbers,
Proceedings of the
W
o
rld Congress on
Engineering and
Computer Scien
c
e
, II. 2010, pp.
832-837.
BIOGRAP
HI
ES
OF AUTH
ORS
Dr. Tariq Jam
il is a facult
y m
e
m
b
er in the Depa
rtment of Electrical a
nd Compu
t
er Engineering
at Sultan Qaboos University
(S
QU,
Oman) wh
ere he
teaches and does research in the ar
eas of
com
puter ar
chit
ectur
e, p
a
ral
l
e
l
proces
s
i
ng,
co
m
puter arithm
e
t
i
c,
and cr
yp
tog
r
aph
y
. Be
fore
joining th
e facu
lty
at SQU in
y
e
ar 2000, he h
a
d
been a lecturer
at the University
of New Sout
h
W
a
les, S
y
dn
e
y
(Australia) and
the Universit
y
o
f
Tas
m
ania, L
a
u
n
ces
ton (Aus
tral
ia). Dr. J
a
m
i
l
holds a B
.
Sc.
(Honors) degree in
electr
i
ca
l
engineering fro
m the NWFP
University
o
f
Engineering and
Technolog
y
(Pakistan) and M.S.
/Ph.D. degr
ees in computer engineer
ing from
the Florida Institute of Techno
log
y
(USA). He
has authored th
r
ee books, holds
an Australian
Innovation
Patent on Com
p
lex
Binar
y
Associat
iv
e Dataflow Pr
ocessor, and
has written
over
fort
y
res
earch p
a
pers
in refere
ed
internat
iona
l co
nferenc
e
s
and journals
. He has
been a rec
i
pien
t
of resear
ch grants from the Australian Resear
ch
Council
and SQU. In 1996, he
was awarded th
e
IEEE Com
put
e
r
Societ
y
(
USA)/Upsilon Pi Ep
silon Honor Socie
t
y
Award
for Academ
ic
Excellen
c
e and
has also serv
ed
as a d
i
stinguished
speaker
in
the
IEEE Computer
Society
(USA)
Distinguished V
i
sitors Program
(DVP). He is a
se
nior m
e
m
b
er of
IEEE (USA),
a
m
e
m
b
er of the
IE
T
(UK), a Chart
e
re
d E
ngi
ne
e
r
(
UK), and
a r
e
gis
t
ered
Professional
Engineer (Pak
istan).
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN
:
2
088
-87
08
I
J
ECE
Vo
l. 4
,
N
o
. 2
,
Ap
r
il 20
14
:
295
–
3
02
30
2
Usman Ali received his
PhD degree in Comp
uter S
c
ienc
e
an
d Netw
orking f
r
om Telecom
P
a
ris
T
ech
, P
a
ris
(F
rance) and M
.
S
c
. degr
ee in
Signal Processing from the Laborator
y
of Signals
and S
y
stem
s (LS
S
)
, S
U
P
ELEC–Universit
y P
a
ri
s 11, Gif-sur-
y
v
e
tt
e (F
rance)
. He holds a B.S
c
.
degree in electr
onic engin
eerin
g from
the GIK In
stitute (Pakis
tan). He is curr
entl
y
a Product
Engine
er a
t
Int
e
lligen
t Im
aging
S
y
stem
s Inc.
, E
d
m
onton (Canad
a), wher
e he
do
es resear
ch and
development in
communication a
nd signal processing for transpo
r
tation industr
y
.
From 2004
to
2006, he worked as lecturer
at the COMSATS Un
iversity
(Pakistan). He w
a
s an internee
at
Texas Instrum
e
n
t
s, Munich, (Ger
m
a
n
y
) in 2007
a
nd at Fraunhofer
Institute, E
r
lang
en (Germ
a
n
y
)
in 2009. He h
a
s
authored
sever
a
l publications in
refereed
intern
ational conferences and journ
a
ls.
Dr. Ali is
als
o
a recipi
ent of M
i
cros
oft Res
earch
European P
h
D
s
c
holars
h
ip and is
a regis
t
ere
d
Professional En
gineer (Pakistan
)
. He was in
th
e organizing co
mmittee of IEEE ICEIS 2006
(Pakistan).
Evaluation Warning : The document was created with Spire.PDF for Python.