TELKOM
NIKA
, Vol. 11, No. 10, Octobe
r 2013, pp. 6
025 ~ 6
033
ISSN: 2302-4
046
6025
Re
cei
v
ed Ap
ril 21, 2013; Revi
sed
Jul
y
1
1
, 2013; Acce
pted Jul
y
22,
2013
Moiré Fringe Phase Difference Measurement Based on
Spectrum Zoom Technology
Li Chang*, Gai Cui
Schoo
l of Information Sci
enc
e and En
gi
neer
ing, She
n
y
a
ng
Univers
i
t
y
of
T
e
chn
o
lo
g
y
No.11
1
She
n
li
ao W
e
st Road,
Economic a
n
d
T
e
chnica
l Dev
e
lo
pment Z
o
n
e
,
Shen
yan
g
,
Lia
oni
ng Prov.
PR Chi
na. 11
0
870
*Corres
p
o
ndi
n
g
author, e-ma
i
l
: chang
lia
nli
@
163.com, sgd
j
i
ance
2
0
04@
16
3.com
A
b
st
r
a
ct
Grating dis
p
l
a
cement
meas
ure
m
e
n
t tech
niq
ue
is a
n
i
m
p
o
rtant
mea
n
s to achi
eve
precis
e
displ
a
ce
ment me
asur
e
m
ent of
na
no
meter
scale.
Mo
ir
é
fri
nge ph
ase differenc
e me
asur
ement
is
the b
a
si
s
of grati
n
g
dis
p
lace
ment
me
a
s
ure
m
e
n
t tech
niq
ue. T
r
a
d
itio
nal
Moir
é fri
n
g
e
su
bdiv
i
si
on
meth
od
b
a
sed
o
n
DF
T
algor
ith
m
has l
o
w
frequ
e
n
cy reso
luti
on
relativ
e
ly w
h
e
n
extractin
g
Mo
i
r
é fring
e
fu
nd
a
m
e
n
tal fr
equ
en
cy
spectral l
i
ne,
w
h
ich causes
frequency
an
d phas
e
mea
s
ure
m
e
n
t erro
rs of Moiré fring
e
fund
a
m
e
n
ta
l
freque
ncy. CZ
T
spectrum
z
o
om tec
hni
qu
e can i
m
pr
ov
e th
e freque
ncy re
soluti
on of freq
uency b
a
n
d
ne
a
r
Moiré fring
e
funda
mental freq
uency
a
nd get
funda
me
ntal frequ
ency spec
tr
um li
nes
mor
e
exactly, w
h
i
c
h
can solve the
problem
of lo
w frequency res
o
lution of D
FT. The paper studies the
relation of Moiré fringe
phas
e differen
c
e and grati
ng
displ
a
ce
ment, and a
naly
z
e
s
the w
o
rking pri
n
ciples of
DF
T
a
nd CZ
T
spectr
u
m
z
o
o
m
metho
d
.
DF
T
and CZ
T
spectrum
z
o
om
are us
ed for Moiré frin
g
e
spectru
m
a
n
a
lysis a
nd p
h
a
s
e
differenc
e calc
ulati
on resp
ect
i
vely. Si
mul
a
ti
on resu
lt
s sho
w
measur
eme
n
t error of Moiré fring
e
pha
se
differenc
e w
i
th
DF
T
algor
ith
m
is in
10
-3
degr
ee sca
le, w
h
il
e
me
asur
e
m
ent
error w
i
th CZ
T
spectru
m
z
o
o
m
alg
o
rith
m is
a
p
p
roxi
mat
e
0
de
gree. T
h
e co
mparis
on r
e
sults
show
CZ
T
sp
ectrum
z
o
o
m
meth
od
has
b
e
tter
perfor
m
a
n
ce
a
nd a
p
p
lica
b
il
ity, w
h
ich i
m
prov
es the
phas
e d
i
fference
me
as
ure
m
e
n
t accur
a
cy of Moir
é fri
n
g
e
effectively.
Ke
y
w
ords
: Mo
iré fring
e
, DF
T
,
CZ
T
spectrum
z
o
om, p
hase
differenc
e mea
s
ure
m
e
n
t.
Copy
right
©
2013 Un
ive
r
sita
s Ah
mad
Dah
l
an
. All rig
h
t
s r
ese
rved
.
1. Introduc
tion
Grating
di
spl
a
cem
ent m
e
asu
r
em
ent t
e
ch
niqu
e is wid
e
ly appl
ied in
the f
i
elds of
sup
e
rfini
s
hin
g
an
d numb
e
rical co
ntrol
.
With
th
e
developm
ent
of
sci
en
ce
and
te
chn
o
logy,
displ
a
cement
mea
s
ureme
n
t techni
que
has
entered
i
n
to nan
omet
er
scale me
a
s
ureme
n
t times.
As the found
ation of grati
ng displa
cem
ent me
a
s
ure
m
ent,
Moiré fringe subdivi
sion
te
chn
o
lo
gy
can imp
r
ove
displ
a
cement
measu
r
em
e
n
t resol
u
tion
and mea
s
u
r
e
m
ent accu
ra
cy, which is the
most effectiv
e way to achi
eve nanom
et
er scal
e gr
ati
ng displa
cem
ent mea
s
ure
m
ent. Moiré f
r
inge
sub
d
ivision
m
e
thod
s in
clud
e pha
se
su
b
d
ivisi
on
and
amplitude
su
bdivisio
n
. In 1982, M. Ta
kada
publi
s
hed
the
Moiré
fring
e
sub
d
ivision
m
e
thod
with Fo
urie
r tra
n
sfo
r
m [1], which
analyzed Moi
r
é
fringe
sig
nal
comp
one
nts i
n
freq
uen
cy-d
omain th
e fi
rst time an
d got
high
er
gratin
g di
spla
cem
e
nt
accuracy by improvin
g Moi
r
é fring
e
pha
se differen
c
e
measurement
accura
cy.
The p
ape
r a
nalyze
s
th
e
grating
di
spla
ceme
nt me
a
s
ureme
n
t pri
n
cipl
e ba
se
d
on M
o
iré
fringe ph
ase
differen
c
e an
d the mea
s
urement effe
ct
of Fourie
r tra
n
sform su
bdi
vision metho
d
,
then utilize
s
CZT (Chi
rp
Z Tran
sfo
r
m) spe
c
trum
zoom te
ch
nique to
cal
c
ulate th
e p
hase
differen
c
e of Moiré frin
ge fundam
ental freque
nc
y to a
m
eliorate the measurement
result.
2. Moiré Fringe Displac
e
m
ent Mea
s
u
r
ement Princ
i
ple
Grating di
spl
a
cem
ent se
n
s
or
con
s
i
s
ts
of light
source, main grati
ng, indicative grating
and p
hotoel
ectri
c
conve
r
sion
elem
ent
s, in
whi
c
h
main g
r
ating
and i
ndi
cati
ve gratin
g a
r
e
s
u
pe
r
i
mp
os
ed
w
i
th a
n
g
l
e
θ
in
sp
ace to
form the
g
r
ati
ng p
a
ir. T
he
grating
pitch i
s
in
mi
cro
n
scale
and
t
w
o gratings
lo
cate cl
osely, so opti
c
inte
rf
eren
ce
phe
nome
n
o
n
will
o
c
cur to ge
nerate M
o
iré
fringe when li
ght passe
s th
roug
h the gra
t
ing pair [2, 3].
Evaluation Warning : The document was created with Spire.PDF for Python.
ISSN: 23
02-4
046
TELKOM
NIKA
Vol. 11, No
. 10, Octobe
r 2013 : 602
5 –
6033
6026
Moiré frin
ge
has an a
m
plification
effe
ct to grating pitch
and can re
duce the
measurement
error
cau
s
e
d
by the no
n
-
unifo
rmity
of grating
pitch
,
which is im
protant fo
r th
e
reali
z
ation
of
pre
c
i
s
e
grati
ng di
spla
ce
m
ent mea
s
u
r
e
m
ent. The
di
stributio
n of
Moiré
fring
e
l
i
ght
intensity is a
s
soci
ated wit
h
imagin
g
po
sition an
d im
aging time. T
he law
of Moiré frin
ge lig
ht
intensity distri
bution at (x, y) of
time t is
s
hown as
Eq.1 [4, 5].
12
I(
x
,
y
,
t)
=
I
(
x
,y
,
t
)
+
I
(
x
,
y
,t)
c
o
s
φ
(x,
y
,
t
)
(1)
Whe
r
e, I(x,y,t) is Moiré frin
ge
light inten
s
ity distributio
n. I
1
(x,y,t) is the ba
ckgro
u
n
d
light
intensity. I
2
(x,y,t) is amplitude of light intensity variatio
n.
φ
(x,y,t) is
Moiré frin
ge p
hase.
Grating
displ
a
cem
ent
se
nso
r
ge
ne
rat
e
s ve
rtical
Moiré f
r
inge
s. Kno
w
n from the
cha
r
a
c
teri
stic of vertical
fri
nge
s, the lig
ht intens
ity i
s
con
s
tant i
n
y dire
ction, a
nd ju
st chan
ges
perio
dically in x direction, so Eq
.1 is equi
valent to Eq.2 [6].
12
I(
x
,t)
=
I
(
x
,t)
+
I
(
x
,
t
)
c
os
φ
(
x
,t)
(2)
Sampling Moi
r
é fringe lig
ht intensity in x
dire
ct
ion can extract the variation of fring
e
light
intensity. Sa
mple the M
o
iré frin
ge lig
ht intens
ity o
f
position x
with hig
h
-spe
ed ph
otoele
c
tric
conve
r
si
on
e
l
ement. Whe
n
the
sam
p
li
ng inte
rval
∆
t is sho
r
t en
ough to
en
sure th
e p
h
a
s
e
d
i
ffe
r
e
nc
e o
f
a
d
j
ac
en
t samp
lin
g fr
in
ge le
ss th
a
n
π
/2, the p
h
a
s
e differen
c
e
∆φ
i
of
adja
c
en
t
sampli
ng frin
ge ca
n be cal
c
ulate
d
with
Eq.3.
ii
i
-
1
∆φ
=
φ
(x
,
t
)-
φ
(x
,
t
)
(3)
Whe
r
e,
φ
(x
,t
i
) and
φ
(x
,t
i-1
) are Moi
r
é frin
ge pha
se of time t
i
and t
i-1
.
Whe
n
Moiré fringe lig
ht intensity is sam
p
led N time
s from time t
0
to t, the Moiré fringe
pha
se va
riati
on
∆φ
rel
a
tive to the ini
t
ial mea
s
ure
m
ent time t
0
can
be fig
u
r
ed
out with
the
accumul
a
tion
of all phase
differen
c
e
∆φ
i
.
N
i
i=
1
∆φ
=
∆φ
(4)
Grating m
o
ve
ment lead
s to Moiré frin
g
e
pha
se cha
nging, so the
corre
s
po
ndin
g
grating
displ
a
cement
can be calcu
l
ated with the
relati
on bet
ween grating di
spla
cem
ent a
nd Moiré frin
g
e
phas
e
differenc
e in Eq.5 [7].
∆φ
∆
x=
d
2
π
(5)
Whe
r
e,
∆
x is
grating m
o
ve
ment displ
a
cement. d is grating pitch.
The rel
a
tion
of grating di
splacement a
nd Moiré frin
ge pha
se
differen
c
e i
s
not
only in
quantity, but also in moving dire
ction.
From t
he a
nalysi
s
abov
e, utilizing in
terfere
n
ce Moiré
fringe a
nd
sa
mpling
real
-time light inten
s
ity of Moiré f
r
inge
ca
n cal
c
ulate th
e Mo
iré frin
ge p
h
a
s
e
differen
c
e a
n
d
gratin
g mo
vement displ
a
cem
ent. Th
erefo
r
e, the
cal
c
ulatio
n a
c
cura
cy of M
o
iré
fringe ph
ase differen
c
e is
essential to g
r
ating di
spla
cement mea
s
u
r
eme
n
t.
3. Moiré Fringe Phase
Differen
ce Me
a
s
uremen
t M
e
thod
s
Moiré
frin
ge
p
hase diffe
ren
c
e
mea
s
u
r
em
ent in
clud
es
digital m
e
tho
d
an
d
analo
g
method,
of which di
gital method usually has mo
re meas
ure
m
ent advantag
es, so utilize digital spe
c
trum
analysi
s
tech
nique to
calculate the Moiré fringe
p
h
a
s
e differen
c
e.
Traditio
nal DFT algo
rithm
and
CZT
sp
ect
r
u
m
zoom
alg
o
rithm
are
a
nalyze
d
in
Moiré
fring
e
pha
se
diffe
ren
c
e
cal
c
ul
ation
respec
tively as
follows
.
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
2302-4
046
Moiré F
r
ing
e
Phase
Differe
nce Me
asure
m
ent Based on Spect
r
um
Zoom
Techn
o
log
y
(Li Cha
ng)
6027
3.1. Discre
t
e
Fourier Tra
n
sform Me
th
od
DFT alg
o
rith
m is widely applie
d in digital signal p
r
ocessin
g
field, which
can
analyze
time-dom
ain
sign
als in f
r
e
quen
cy-d
oma
i
n and a
c
hi
e
v
e the sign
al
spe
c
tru
m
di
scretization[8
,
9].
Whe
n
the sa
mpling of pho
toelectri
c
con
v
ersio
n
el
em
ent to Moiré fringe lig
ht intensity meets
the
spatial
sampli
ng the
o
re
m, the di
strib
u
tio
n
la
w of
Mo
iré fring
e
lig
ht i
n
tensity in
Eq
.2 is
equival
e
nt
to Eq.6.
01
0
I(
x
,t)
=
I
(
x
,t)
+
I
c
o
s
(
2
π
fx
+
φ
(
x
,t)
)
(6)
Whe
r
e, f
0
is fundam
ental freque
ncy of Moiré frin
ge lig
ht intensity variation in x direction.
Moiré frin
ge light intensity is peri
odi
c si
gnal, and pe
riodic si
gnal
can be expan
ded into
Fouri
e
r seri
es as Eq.7.
0n
n
0
n
n
n=
1
n
=
1
f(
x
)
=a
+
(
a
c
osnx
+
b
sin
n
x
)
=a
+
A
sin(nx
+
φ
)
(7)
Whe
r
e, a
0
is
c
o
ns
tant term. A
n
is the amplitude of e
a
ch h
a
rm
oni
c wave.
φ
n
is the initial
pha
se of ea
ch harm
oni
c wave.
Eq.7 sho
w
s a perio
dic function
co
nsi
s
ts
of co
nstant term
and ha
rmo
n
ic wave
comp
ositio
ns.
According
to
the
sam
p
lin
g Moi
r
é
fring
e
data
in
a
Moiré
frin
ge
perio
d, the
fri
nge
light inten
s
ity paramete
r
s
of a
0
, A
n
and
φ
n
ca
n b
e
calcul
ated. Fo
r Moi
r
é f
r
ing
e
light i
n
ten
s
ity
sign
al, the ph
ase
of funda
mental fre
q
u
ency i
s
a
s
sociated with
gra
t
ing displa
ce
ment, so
DF
T of
Moiré f
r
inge
li
ght inten
s
ity data is
proce
s
sed to
extra
c
t the fun
d
a
m
ental fre
q
u
ency
spe
c
tral
line
.
The DF
T re
sults of Moiré f
r
inge
seq
uen
ce are co
mpl
e
x numbe
rs,
whi
c
h are sh
own a
s
Eq.8.
2
π
N-
1
-j
(
)
n
k
N
n=
0
X
(
k)
=
x
(
n
)
e
=
R
e
[
X
(
k)
]
+
jI
m
[
X
(
k)
]
(8)
Whe
r
e, x(n) i
s
N-point finite-len
gth se
q
uen
ce
of Moi
r
é fring
e
data
.
Re[X(k)] is the real
part of Moi
r
é
fringe frequ
e
n
cy. Im[X(k)] i
s
the im
agi
na
ry part of M
o
i
r
é frin
ge fre
q
uen
cy. k is
se
rial
numbe
r of sp
ectral lin
es.
The amplitu
d
e
spe
c
trum A
k
and pha
se
spectrum
φ
k
of Moiré frin
ge l
i
ght intensity can b
e
obtaine
d with
Eq.9.
，
22
kk
Im
[X
(
k
)
]
A=
R
e
[
X
(
k
)
]
+
I
m
[
X
(
k
)
]
φ
=a
r
c
t
a
n
Re
[
X
(k
)]
(9)
Moiré f
r
ing
e
fundame
n
tal
freque
ncy
sp
ectral
line
h
a
s the
big
g
e
s
t po
we
r
sp
ectru
m
.
Extracting th
e funda
ment
al freq
uen
cy
spe
c
tral
line
can
cal
c
ul
ate
the Moi
r
é fri
nge fun
dam
e
n
tal
freque
ncy ph
ase of
the sa
mpling
time. Moiré
f
r
i
nge
fundam
ental f
r
equ
en
cy pha
se diffe
ren
c
e
of
sampli
ng inte
rval
∆
t
i
can b
e
cal
c
ulate
d
with Eq.10.
ii
-
1
ii
i
-
1
ii
-
1
Im
[X
(
k
)
]
Im
[X
(
k
)
]
∆φ
=
φ
-
φ
=
a
r
c
tan
(
)
-
ar
c
t
an(
)
Re
[
X
(k
)]
Re
[
X
(k
)]
(10
)
Ho
wever, th
e
accu
ra
cy of DFT f
r
equ
e
n
cy an
al
ysi
s
is rest
rain
e
d
by inhe
re
n
t
barri
er
effect. DFT freque
ncy reso
lution is
re
cip
r
ocal wi
th
the
length of
sa
mpling d
a
ta. If the frequ
en
cy
resolution n
e
eds to imp
r
ov
e, the sampli
ng time also
need
s to be i
n
crea
sed to g
e
t more samp
ling
data, which
will lead to
a substa
ntial i
n
crease
of system comput
ation quantity. Therefore,
the
measurement
effect of
DF
T algo
rithm i
n
Moi
r
é
fri
n
g
e
ph
ase diff
eren
ce
me
asurem
ent i
s
n
o
t
outstan
ding.
And CZ
T sp
ectru
m
zoom
techni
que i
s
applie
d in
Moiré fri
nge
pha
se differe
nce
measurement
to improve the defect
s
of DFT alg
o
rith
m.
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TELKOM
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Vol. 11, No
. 10, Octobe
r 2013 : 602
5 –
6033
6028
3.2. Chirp Z Trans
f
orm Spectr
u
m Zoo
m Method
Spectrum zo
om tech
niqu
e
is an im
port
ant mean
s in
sign
al analy
s
is,
whi
c
h
ca
n analy
z
e
the freq
uen
cy stru
cture
o
f
a sel
e
cte
d
frequ
en
cy
region
with
specifie
d an
d
sufficie
n
tly high
freque
ncy re
solutio
n
[10]. Spectrum zo
om
techni
q
u
e
mainly
conta
i
ns Z
oom
FF
T algo
rithm
a
nd
CZT al
gorith
m
. CZT
spe
c
trum
zo
om
algorith
m
takes d
e
n
s
e val
ues i
n
refin
e
m
ent freq
uen
cy
band, incre
a
s
e
s
the amo
unt of freque
ncy analys
i
s
points, and
calcul
ates the
amplitude a
n
d
pha
se inform
ation of corre
s
po
ndin
g
freq
uen
cy points.
CZT spe
c
tru
m
zoom alg
o
r
ithm can b
e
use
d
to
calcu
l
ate the z-tra
n
sform of a fixed path
in z-plan
e. If the fixed pat
h is a
ci
rcula
r
arc
o
n
the
unit circle, in
cre
a
si
ng the
output poi
nts to
make
it more
than in
put p
o
ints, the
sp
ecified f
r
eq
ue
ncy ba
nd
ca
n be
su
bdivid
ed. For N-poi
nts
seq
uen
ce x(n
)
, if the sampling data z
k
lo
cate on the u
n
it circl
e
with equal inte
rval
of N-point
s, the
z-tran
sform i
s
DFT in thi
s
case,
so DFT
can b
e
re
ga
rded a
s
a
spe
c
ial case of z-tran
sfo
r
m. The
sampli
ng poi
nt z
k
of comm
on path in z-p
l
ane can be e
x
presse
d as
Eq.11.
-k
k
z=
A
W
(11
)
Whe
r
e,
k=0,1,…,M-1. M i
s
compl
e
x spectr
um a
nal
ysis p
o
ints.
A and
W a
r
e
ran
dom
compl
e
x num
bers, whi
c
h can
be expressed a
s
Eq.12.
，
00
j
θ
-j
φ
00
A
=A
e
W
=
W
e
(12
)
So the sampli
ng data z
k
ca
n be expre
ssed as Eq.13.
00
0
0
j
θ
jk
φ
j(
θ
+k
φ
)
-k
-k
k0
0
0
0
z
=
A
e
We
=
A
We
(13
)
Whe
r
e, A
0
is the vector
radi
us len
g
th of initial sampli
n
g
point z
0
.
θ
0
is the pha
se
angle of
initial sampli
ng poi
nt z
0
.
φ
0
is the a
n
g
le differen
c
e of adj
acen
t sampli
ng p
o
ints.
W
0
is
the
extensio
nal p
r
opo
rtion of sampling
spi
r
a
l
. k=0,1,2…
M
-
1.
Figure 1 sh
o
w
s the
co
rre
spondi
ng curv
e of samplin
g
point z
k
in z-plane [11].
Figure 1. Curve of CZT sa
mpling poi
nt z
k
in z-plan
e
The sa
mplin
g
data z
k
ha
s the followi
ng chara
c
te
risti
c
s.
(1) Wh
en
A
0
<1, the
samp
ling spi
r
al lo
cates in
side th
e unit ci
rcl
e
, while
whe
n
A
0
>
1
, the
sampli
ng spiral locate
s out
side the u
n
it circle.
(2) Whe
n
W
0
<1, the sa
m
p
ling spiral
rotates in
side,
while when
W
0
>
1
, the
sampling
s
p
iral rotates
outs
i
de.
(3) Wh
en
A
0
=W
0
=1, the
CZT p
a
th is
a arc of the
unit circle, th
e amou
nt of pro
c
e
ssi
ng
data M can b
e
different fro
m
the input seque
nce poin
t
s N.
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TELKOM
NIKA
ISSN:
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046
Moiré F
r
ing
e
Phase
Differe
nce Me
asure
m
ent Based on Spect
r
um
Zoom
Techn
o
log
y
(Li Cha
ng)
6029
CZT of
D mu
ltiples i
s
divid
i
ng the o
r
igin
al frequ
en
cy resolution i
n
to D pa
rts
eq
ually to
get more
spe
c
tral lin
es. T
he clo
s
e
r
the
spe
c
tral lin
e
s
locate to the spe
c
tral pe
ak, the sm
all
e
r
errors frequ
e
n
cy
cal
c
ulatio
n re
sult
ha
s,
and th
e p
h
a
s
e an
d p
h
a
s
e
differen
c
e
cal
c
ulatio
n resul
t
s
have
smalle
r
error too. So
CZT
sp
ect
r
u
m
zoom
ca
n
also
be
rega
rded
as
a
co
rrection
meth
o
d
,
whi
c
h a
c
hiev
es ph
ase co
rrection by freq
uen
cy corre
c
t
i
on.
If the calcul
ation erro
r of DFT algo
rithm
is
ca
used by
barrier
effect, CZT alg
o
rith
m can
redu
ce the m
easure
m
ent error an
d imp
r
ove the
mea
s
ureme
n
t accura
cy with the pro
c
e
ssi
ng of
freque
ncy
su
bdivisio
n
. But if severe spe
c
tral i
n
terfe
r
e
n
ce
phe
nome
non h
app
en
s, CZT
algo
rith
m
can’t
sepa
rat
e
the interfe
r
ing freq
uen
cy compo
nent
and elimin
ate the effect
of measure
m
ent
error, be
cau
s
e
CZT sp
ect
r
um zoom ju
st enlarge
s a part of frequ
ency ban
d o
n
the freque
ncy
axis. In this case, th
e me
a
s
ureme
n
t error
can
be
red
u
ce
d by a
ddi
ng the
origi
n
a
l
sam
p
ling
po
ints
to improve th
e origin
al freq
uen
cy resolut
i
on. The z-tra
n
sform of z
k
is
:
N-1
N
-1
-n
-
n
n
k
kk
n=
0
n
=
0
X
(
z
)
=
x
(n)z
=
x
(
n
)
A
W
(14
)
Known from the Bluestei
n
equality
of Eq.15, Eq.16 ca
n be ded
uced
.
22
2
1
nk
=
[
n
+
k
-
(k
-
n
)
]
2
(15
)
22
2
2
2
2
2
2
n
(
k-
n
)
k
k
n
(
k-
n
)
k
N-
1
N
-
1
N
-
1
--
-n
-
n
22
2
2
2
2
2
k
n=
0
n
=
0
n=
0
k
2
X
(
z
)
=
x
(
n
)
A
W
W
W
=
W
[
x
(
n)
A
W
]
W
=
W
g(
n)
h(
k
-
n)
=W
[
g
(n
)*
h
(
n
)
]
(16
)
Whe
r
e,
，
22
nn
-
-n
22
g
(
n
)
=x
(n
)A
W
h
(n)=W
(17
)
In Eq.16, z-transfo
rm of z
k
can b
e
calculated with li
near
co
nvolut
ion of g(n) a
nd h(n),
then multiplie
d by
2
k
2
W
. The z-tran
sform
cal
c
ulation struct
ure is
sho
w
n i
n
Figure 2.
Figure 2. Cal
c
ulatio
n st
ru
cture of z-t
r
an
sform
Therefore, for the fi
xed parameters of A
0
,
θ
0
,
φ
0
and
W
0
, z-tran
sform of z
0
, z
1
,…, z
k
ca
n
be pro
c
e
s
sed
in z-pla
ne wit
h
Eq.12 and
Eq.16.
4. Simulation Anal
y
s
is
Moiré frin
ge signal seque
n
c
e x(n
)
in Eq.2 can b
e
de
scrib
ed a
s
Eq.18.
h(n
)
X(z
n
)
2
n
-n
2
AW
x(
n)
g(n
)
2
k
2
W
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ISSN: 23
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046
TELKOM
NIKA
Vol. 11, No
. 10, Octobe
r 2013 : 602
5 –
6033
6030
s
2
π
nf
x
(
n
)
=
A
+Bc
o
s(
+
φ
)
,
n=
0,1,.
..,N
-
1
f
(18
)
Whe
r
e, A i
s
dire
ct
com
pone
nt. B is sig
nal va
ria
t
ion amplitu
d
e
. f is M
o
iré fring
e
freque
ncy. f
s
is Moi
r
é frin
g
e
sam
p
ling f
r
eque
ncy.
φ
i
s
Moi
r
é fringe initial phase. N is
sampl
i
ng
s
i
ze [12].
Whe
n
A=1, B=3, f=10
0Hz, f
s
=1000
Hz, N=10
00,
φ
=0
°, the Moiré frin
ge se
que
nce x(n) is:
100
x(
n
)
=
1
+
3
c
o
s
(
2
π
n)
1000
(19
)
Figure 3 sh
o
w
s the x(n) waveform of time-d
omain
when
n
[
1,50]
.
0
5
10
15
20
25
30
35
40
45
50
-2
0
2
4
S
a
m
p
l
i
ng N
u
m
ber
Am
p
l
i
t
u
d
e
T
i
me
D
o
ma
i
n
o
f
x
(
n
)
Figure 3. x(n) waveform of
time-dom
ain
Utilize tra
d
itio
nal DFT alg
o
r
ithm to analyze t
he spe
c
trum of Moiré fringe
seq
uen
ce x(n
)
in Eq.18. The
spe
c
trog
ram
betwe
en 95
Hz and 1
0
5
H
z i
s
sh
own in Figure 4.
95
96
97
98
99
100
10
1
102
103
104
105
0
100
200
300
400
500
600
700
800
x
(
n) S
pec
t
r
og
r
a
m
w
i
t
h
D
F
T
A
l
gor
i
t
hm
F
r
equenc
y
/
H
z
Am
p
l
i
t
u
d
e
Figure 4. Spectrum a
nalysi
s
of x(n) with
traditional
DF
T algorithm
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
2302-4
046
Moiré F
r
ing
e
Phase
Differe
nce Me
asure
m
ent Based on Spect
r
um
Zoom
Techn
o
log
y
(Li Cha
ng)
6031
In Figure 4,
the freq
uen
cy resolution
of x(n)
sp
ect
r
um a
nalysi
s
with tra
d
itio
nal DFT
algorith
m
is
2Hz. The
sp
ectral
line
s
fi
gure
d
out
do
n’t locate
at
the freq
uen
cy point of 10
0Hz
exactly. The
spe
c
tral
line
whi
c
h
ha
s th
e big
g
e
s
t po
wer
spe
c
tru
m
is Moi
r
é f
r
inge fu
ndam
ental
freque
ncy
sp
ectral
line,
so the
mea
s
ured
Moi
r
é f
r
inge
fund
a
m
ental frequ
ency i
s
99.6
H
z.
Ho
wever, the
theoretical M
o
iré frin
ge fu
ndame
n
ta
l freque
ncy is 1
00Hz in Eq.1
8, traditional
DFT
algorith
m
cau
s
e
s
0.4
H
z fre
quen
cy me
asurem
ent e
rro
r of Moi
r
é
fri
nge fu
ndam
e
n
tal freq
uen
cy,
whi
c
h
will caused phase
difference m
easurement
error
of Moiré fringe
fundamental frequency
further.
Utilize CZT spectrum zoo
m
algorith
m
to
su
b
d
ivide t
he freq
uen
cy
band of x(n
)
betwee
n
95Hz an
d 10
5Hz. The pro
c
e
ssi
ng
procedure is de
scribed b
e
lo
w.
(1)
Determine
the subdivi
si
on frequ
en
cy
band a
nd the
quantity of output data.
(2) Ma
ke the
CZT path o
n
the unit circle, det
ermin
e
the positio
ns
of the starting point
and the en
din
g
point, and set the angle d
i
fference of adjacent sam
p
ling point.
(3)
Cal
c
ulate
the CZT of the sampli
ng p
o
ints on the p
a
th.
(4) Extra
c
t the spe
c
tral lin
e of x(n) fund
amental fre
q
u
ency.
The dist
ributi
on of x(n) spe
c
tral line
s
bet
we
e
n
95
Hz a
nd 105
Hz is shown in Figu
re 5.
95
96
97
98
99
10
0
10
1
10
2
10
3
10
4
10
5
0
50
0
10
00
15
00
F
r
eq
ue
nc
y
/
Hz
F
i
g
u
r
e
a
:
x(
n
)
Sp
e
c
tr
u
m
Z
o
o
m
o
f
C
Z
T
w
i
th
1
2
8
m
u
l
t
i
p
l
e
s
Am
p
l
i
t
u
d
e
95
96
97
98
99
10
0
10
1
10
2
10
3
10
4
10
5
0
50
0
10
00
15
00
F
r
eq
ue
nc
y
/
Hz
F
i
g
u
r
e
b
:
x(
n
)
Sp
e
c
tr
u
m
Z
o
o
m
o
f
C
Z
T
w
i
th
2
5
6
m
u
l
t
i
p
l
e
s
Am
p
l
i
t
u
d
e
Figure 5. Spectrum a
nalysi
s
re
sult
s of
x(n) with CZT spectrum zoo
m
algorithm
In Figure 5,
Figure a a
nd Figure b
are x(n) fre
quen
cy sub
d
i
vision re
sult
s of 128
multiples a
n
d
256 multiple
s with CZ
T sp
ectru
m
zo
om algorith
m
. The freque
ncy resol
u
tion of x(n)
spe
c
tru
m
a
n
a
lysis are 0.
078
Hz and
0.039
Hz re
spectively, wh
ich
have m
o
re
spe
c
tral li
nes
betwe
en 9
5
Hz an
d 10
5Hz
than DFT alg
o
rithm. Th
e x(
n)
sp
ect
r
al li
ne of fre
quen
cy 100
Hz can
be
extracted
exa
c
tly. And increasi
ng
the
su
bdivisio
n
multiples
can im
p
r
ove the fre
q
uen
cy re
solut
i
on
furthermore. From the
con
t
rast of Figu
re 4 and Fi
g
u
re 5, it can be
con
c
lu
ded th
at CZT spect
r
um
zoom
alg
o
rith
m ha
s p
r
omi
nent
spe
c
tru
m
analy
s
is ef
fect for Moi
r
é
fringe
fund
a
m
ental frequ
ency
spe
c
tral lin
e.
Whe
n
the M
o
iré fri
nge
ini
t
ial pha
se
φ
of x(n) i
n
Eq.
18 increases 1° each tim
e
, utilize
DFT alg
o
rith
m and CZT
algorithm resp
ectively to measure t
he pha
se dif
f
eren
ce of e
a
ch
variation
of
φ
. 0-1
0
° i
s
ch
o
s
en
a
s
th
e M
o
iré
fring
e
p
h
ase
vari
ation
regio
n
,
which
ca
n
obtain
1
0
grou
ps of ph
ase differen
c
e measurem
ent re
sults.
T
he mea
s
u
r
e
m
ent re
sults
with two alg
o
r
ithms
are sho
w
n in
Table 1.
Evaluation Warning : The document was created with Spire.PDF for Python.
ISSN: 23
02-4
046
TELKOM
NIKA
Vol. 11, No
. 10, Octobe
r 2013 : 602
5 –
6033
6032
Table 1. Moiré Fring
e
Pha
s
e Diffe
ren
c
e
Measu
r
em
en
t Result
s of x(n) With
DFT
Algorithm an
d
CZT Algo
rith
m
Grou
p Numbe
r
DFT Algorithm
(°
)
CZT Algorithm
(°
)
1
0.9966
1.0000
2 0.9968
1.0000
3 0.9968
1.0000
4 0.9970
1.0000
5 0.9970
1.0000
6 0.9972
1.0000
7 0.9972
1.0000
8 0.9974
1.0000
9 0.9975
1.0000
10 0.9976
1.0000
Table
1
sh
ows the
Moi
r
é f
r
inge
pha
se
di
ffer
ence m
e
a
s
ureme
n
t result of CZT
al
gorithm
is mo
re
accu
rate tha
n
that
of DF
T alg
o
rithm.
Figure
6 sho
w
s th
e
measurement
error curve
s
of
Moiré frin
ge p
hase differen
c
e mea
s
u
r
em
ent with DFT
algorith
m
and
CZT algo
rith
m.
1
2
3
4
5
6
7
8
9
10
-4
-3
-2
-1
0
1
2
x 1
0
-3
C
u
rv
e of
M
e
a
s
ur
em
ent
E
r
ror
G
r
o
up N
u
m
ber
M
eas
u
r
em
ent
E
r
ror (°)
DF
T
CZ
T
Figure 6. Erro
r cu
rves of M
o
iré frin
ge ph
ase
differen
c
e measureme
n
t with DFT a
l
gorithm a
nd
CZT alg
o
rith
m
In Figu
re
6, the ave
r
ag
e p
hase differen
c
e
me
a
s
u
r
em
ent erro
r of
DFT is -0.002
9
deg
ree,
while the
me
asu
r
em
ent error
of CZT
sp
ectru
m
zoom
algorithm i
s
approximate
0 deg
ree,
whi
c
h
prove
s
the su
perio
rity of CZT spe
c
trum zoom te
chni
q
ue.
Comp
ared wi
th traditional
DFT al
gorith
m
applie
d in
Moiré fri
nge f
undam
ental f
r
equ
en
cy
pha
se diffe
re
nce
cal
c
ul
ation, CZT
spe
c
trum
zo
om t
e
ch
niqu
e can
achi
eve p
r
e
c
ise Moi
r
é fri
nge
pha
se
difference me
asu
r
eme
n
t with
out mo
dulati
on o
r
filteri
ng p
r
o
c
e
ssi
ng. Be
side
s,
the
freque
ncy
re
solutio
n
of CZT sp
ect
r
um
analysi
s
i
s
adju
s
ted exp
ediently. Co
n
s
eq
uently CZT
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
2302-4
046
Moiré F
r
ing
e
Phase
Differe
nce Me
asure
m
ent Based on Spect
r
um
Zoom
Techn
o
log
y
(Li Cha
ng)
6033
spe
c
tru
m
zo
o
m
techniq
ue
can redu
ce
Moiré fr
in
ge
pha
se differe
nce me
asure
m
ent error, which
is ben
efit for grating di
spl
a
ceme
nt mea
s
urem
ent.
5. Conclusio
n
Moiré frin
ge fundam
ental freque
ncy pha
se diffe
re
nce measurement
effect can affect the
accuracy of grating
di
spla
ceme
nt
me
asurem
en
t.
Tra
d
itional DFT algorith
m
can
analy
z
e
M
o
i
r
é
fringe fun
d
a
m
ental fre
q
u
ency to o
b
tain the pha
se
of Moiré fri
n
g
e
first ha
rmo
n
ic
wave. Ho
wever,
due to the i
n
fluen
ce of
freque
ncy
re
solutio
n
in spectrum a
n
a
l
ysis, the ph
ase
differen
c
e
measurement
accuracy i
s
relatively low.
The
ap
plication of CZT sp
ectru
m
zoom
techni
que
ca
n
solve thi
s
pro
b
lem
effectively. CZT
spe
c
trum
zoom
i
m
prove
s
th
e
freque
ncy
re
solution
of Moi
r
é
fringe
sp
ect
r
u
m
analy
s
is by
frequ
en
cy su
bdivisio
n
, whi
c
h
ca
n o
b
tain
exact
sp
ect
r
al line
of Moi
r
é
fringe
funda
mental frequ
ency fo
r
pha
se
and
ph
ase differen
c
e
cal
c
ulatio
n.
Simulation
re
sult
s
sho
w
th
e
pha
se
differe
nce
mea
s
u
r
em
e
n
t accu
ra
cy o
f
Moiré
frin
ge
funda
mental
freq
uen
cy
wi
th
CZT alg
o
rith
m is highe
r than that of traditional
DF
T algorith
m
. CZ
T spe
c
trum
zoom metho
d
can
achi
eve the
function of
Moiré fri
n
g
e
sp
ectr
um
corre
c
tion, redu
ce fre
q
u
ency a
nd p
hase
measurement
errors, an
d a
c
hieve
high
-p
reci
sio
n
ph
ase differe
nce
measurement
of Moiré
frin
ge
fundame
n
tal
freque
ncy, which
ha
s a
certai
n si
gnif
i
can
c
e fo
r p
r
eci
s
e
gratin
g displa
cem
ent
measurement
.
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