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Accuracy limits of reconstruction in sampling digital interferometers
Volume 78, number 5,6
OPTICS COMMUNICATIONS
15 September 1990
Accuracy limits of reconstruction in sampling digital interferometers Peter Jani
CentralResearchInstitutefor Physics,H-1525 Budapest, P.O.Box 49, Hungary Received 20 March 1990; revised manuscript received 18 June 1990
The results of error analysis in sampling theory are extended for the determination of accuracy limits in sampling digital interferometers. An estimate is given for the truncation, round-off, jitter and frequency-aliasing sources of error of the reconstructed functions of motion. Conditions of measurement parameters are defined for the case when the accuracy of the reconstructed part of motion is equal to the resolution of the conventional interferometer.
1. Introduction The inherent advantage o f sampling the d a t a prov i d e d by a digital interferometer is to use the same set o f d a t a for the reconstruction o f displacement, velocity, power spectrum, etc. It is the purpose o f this p a p e r to extend the results o f error analysis in sampiing theory for the d e t e r m i n a t i o n o f accuracy limits o f the a b o v e functions. The present p a p e r does not deal with the other sources o f error e.g. detector non-linearity, frequency stability o f the light source, changes o f the refractive index o f air, etc. to m e n t i o n but a few, which affect the accuracy o f all digital interferometers including the sampling ones. Several schemes exist for the reconstruction o f the function f ( t ) from its samples values f ( n T ) . Here we deal with the reconstruction in the f u n d a m e n t a l Shannon series only, which does not necessarily lead to the best error bounds, sin oJ( t - n T ) f(t)=_~
f(nT)
og(t-nT)
(1)
The m a i n sources o f error in any reconstructed signal are the t r u n c a t i o n error, the r o u n d - o f f error, the j i t t e r error a n d the frequency-aliasing error [ 1 ]. Following the results o f T h o m a s et al. [ 2 ] a n d Papoulis [ 3 ], well established b o u n d s can be written for the above errors in the case when the reconstruction is
p e r f o r m e d for a section o f the signal that is intercepted by the sampling. F o r the truncation error one can write leN(t) I <
41f(t)m~x I
rt2Nq
,
(2)
where sin to(t--nT)
N
e u ( t ) = f ( t ) - - _ ~U f ( n T )
to(t-nT)
(3)
is the error o f the reconstructed signal due to the finite n u m b e r 2 N o f sampling points, f ( n T ) are the sample values and q= 1-r, ro9~/o9 is the ratio between the highest allowed v i b r a t i o n a l frequency ~o~ o f m o t i o n a n d the sampling frequency ~o. It is shown in i n f o r m a t i o n theory that though different in origin the r o u n d - o f f error and the j i t t e r error can be h a n d l e d in the same way [ 3]. The first is caused by the fact that in any d a t a processing system the analogue signals are substituted by a suitable integer with an error c o m p a r a b l e to the resolution o f the A / D converter. The second is caused by the fact, that the samples are taken at points different from those m a r k e d for s a m p l i n g due to the finite precision o f the system clock. The b o u n d s for the r o u n d - o f f and j i t t e r error can be written as follows: I~Rj(t) I ~< ( o ) I E / T c )
1/2 ,
(4)
where
0030-4018/90/$03.50 © 1990 - Elsevier Science Publishers B.V. ( North-Holland )
309
Volume 78, number 5,6
OPTICS COMMUNICATIONS
eRj(t) = f ( t ) -- ~ f ' (nT) -oo
sin co( t - n T ) co( t - n T )
(5)
is the difference between the true signal and the one reconstructed with the false sampling values f ' (n T), co~ is the frequency band limit of the error,
15 September 1990
v(iT) = m , 2 / ( k T ) ,
(10)
where v(iT) is the average velocity at time iT referred to the sampling time T. The displacement belonging to two samples i, j of the record is
s(iT)-sOT)=
(2/k) ~ m,.
(11)
I I
E= i
~(t)
dt<~,
(6)
--oo
is the energy of the error. For the frequency aliasing error of the reconstruction one can write: lea(t)b ~<(B/2zc) Isin o)tl ,
L tinT)
-de
sin co( t - n T ) o)( t - nT)
(8)
o~ /i
B = J IF(~o,) I do),,
2) ( m s - m s ) / ( i - j ) .
(12)
For the reconstruction of the velocity function v(t) e.g., one uses the v(iT) sampled values v ( t ) = ~ v(iT) _oo
is the difference between the true signalf(t) bound at a higher frequency than 0)/2 and the one reconstructed with the samples taken at sampling frequency m, and (9)
(o
where F(oJ) is the Fourier transform o f f ( t ) .
2. Sampling the interferometric data
Here we consider a Michelson type interferometer, fig. 1. The movement of one of the arms of this interferometer relative to the other is represented by a pulse train, each pulse signifying a 2 / k displacement of the moving arm, where 2 is the wavelength of light and k is an integer. For each pulse a sign corresponding to the direction of movement is assigned. Now we sum up and store the pulses arriving during each sample time. By retaining the mutual correspondence between physical time and sample number i we obtain a set of data {ms} representing a record of the movement. So m, is an integer which equals to the sum of pulses arrived during the ith sample time. The physical interpretation of the sample value m, is obvious: 310
a[(i-j)T]=2/(kT
(7)
where q(t) =fit)-
The acceleration belonging to two samples i, j of the record is
sin o ) ( t - - i T ) co(t-iT)
(13)
The same set of data {m,} can be used for the computation of the quantity
v2( iT) = ( 2 / k T ) 2 rn 2 ,
(14)
providing thus the sampling values for the reconstruction of the v2(t) function. Performing the Fourier transformation of the v2(t) function one obtains the power density spectrum of the motion. Thus we obtain that the proposed digital sampling interferometer retains all the properties and measurement facilities of the conventional metrology interferometers and at the same time provides the possibility of measuring other functions of motion such as velocity, acceleration, power density spectrum, etc.
3. Errors of reconstruction
While it is straightforward to compute from (2) the truncation error for any of the functions of motion the computation of the round-off-jitter error requires further considerations. Using the equality [ 3 ]
E= i ~,(t)dr= Z T4,(nT"),
(15)
where in case of displacement measurement eR,(nT) = f ( n T ) - - f ' ( n T ) = (2/k) Amn,
(16)
and - 1 < Arnn < 1, for a motion that does not con-
Volume 78, number 5,6
OPTICS COMMUNICATIONS
15 September 1990
/////H/{/
BS
I
M
L
I
-
+
D
I
t
'~,
ill.1/
k
[
\ \ ,,\\
/ /,
"
Fig. 1. A symbolic lay-out of the digital interferometer. At equidistant points of phase-difference, 2n x2/k (where x is the wave vector, 2 is the wave length of light, k is an integer) a pulse is formed giving rise to a pulse train during motion. tain any non-zero velocity c o m p o n e n t outside the time interval [ - N T , NT] one obtains
E = ~ T ( 2 / k ) 2 AmZ~ < 2 N T ()t/k) 2 .
(17)
the cycle t i m e o f the highest vibration. F o r at least how m a n y we obtain from the c o n d i t i o n that the t r u n c a t i o n error should not exceed the ()t/k) resolution o f the interferometer either.
--oo
]~N(t) I ~< ()t/k)
Substituting this value in ( 4 ) we have I~Rj(t) I ~< (2k) (2 N T / T I ) 1/2 ,
(18)
where T~ = n/COl is the cycle time o f the highest vibration o f the motion. So the condition for the roundo f f - j i t t e r error not to exceed the ()t/k) resolution o f the interferometer is
2 NT<~ T t .
( 19 )
This implies in itself that no aliasing type error can occur since several samples should be taken during
--
4lf(t)max I 4lf(t)max I l~2Nq - l~2N(l_N/2) ,
(2o)
where we used the equality c o n d i t i o n in (19). Resolving ( 2 0 ) for N we have If(t)m.x I N>~0.4 - -
(Mk)
(21)
The o p t i m u m sampling rate can be defined so that 2 N samples are taken during the T~ cycle time, where 311
Volume 78, number 5,6
OPTICS COMMUNICATIONS
N is found from ( 2 1 ) a n d f ( t ) m a x is the m a x i m u m value o f the physical q u a n t i t y to be reconstructed re-ferred to T~ cycle time.
4. Summary We p r o p o s e d a scheme o f a sampling digital interferometer. It was shown that all interesting functions o f m o t i o n can be reconstructed from a single set o f data. The results o f error analysis were e x t e n d e d to this
312
15 September 1990
measuring principle when the reconstruction is perf o r m e d in the f u n d a m e n t a l Shannon series. The o p t i m u m value o f sampling rate was defined, ( 1 9 ) a n d (21 ), for the case when the error o f the reconstructed displacement does not exceed the resolution o f the conventional interferometer.
References [ 1] A.J. Jerri, Proc. IEEE, Vol. 65, No. 11 (1977). [2] J.B. Thomas and B. Liu, IEEE Int. Cony. Rec. (USA) 12 (1964) 269. [ 3 ] A. Papoulis, Proc. IEEE, Vol. 54, No. 7 (1966).
OPTICS COMMUNICATIONS
15 September 1990
Accuracy limits of reconstruction in sampling digital interferometers Peter Jani
CentralResearchInstitutefor Physics,H-1525 Budapest, P.O.Box 49, Hungary Received 20 March 1990; revised manuscript received 18 June 1990
The results of error analysis in sampling theory are extended for the determination of accuracy limits in sampling digital interferometers. An estimate is given for the truncation, round-off, jitter and frequency-aliasing sources of error of the reconstructed functions of motion. Conditions of measurement parameters are defined for the case when the accuracy of the reconstructed part of motion is equal to the resolution of the conventional interferometer.
1. Introduction The inherent advantage o f sampling the d a t a prov i d e d by a digital interferometer is to use the same set o f d a t a for the reconstruction o f displacement, velocity, power spectrum, etc. It is the purpose o f this p a p e r to extend the results o f error analysis in sampiing theory for the d e t e r m i n a t i o n o f accuracy limits o f the a b o v e functions. The present p a p e r does not deal with the other sources o f error e.g. detector non-linearity, frequency stability o f the light source, changes o f the refractive index o f air, etc. to m e n t i o n but a few, which affect the accuracy o f all digital interferometers including the sampling ones. Several schemes exist for the reconstruction o f the function f ( t ) from its samples values f ( n T ) . Here we deal with the reconstruction in the f u n d a m e n t a l Shannon series only, which does not necessarily lead to the best error bounds, sin oJ( t - n T ) f(t)=_~
f(nT)
og(t-nT)
(1)
The m a i n sources o f error in any reconstructed signal are the t r u n c a t i o n error, the r o u n d - o f f error, the j i t t e r error a n d the frequency-aliasing error [ 1 ]. Following the results o f T h o m a s et al. [ 2 ] a n d Papoulis [ 3 ], well established b o u n d s can be written for the above errors in the case when the reconstruction is
p e r f o r m e d for a section o f the signal that is intercepted by the sampling. F o r the truncation error one can write leN(t) I <
41f(t)m~x I
rt2Nq
,
(2)
where sin to(t--nT)
N
e u ( t ) = f ( t ) - - _ ~U f ( n T )
to(t-nT)
(3)
is the error o f the reconstructed signal due to the finite n u m b e r 2 N o f sampling points, f ( n T ) are the sample values and q= 1-r, ro9~/o9 is the ratio between the highest allowed v i b r a t i o n a l frequency ~o~ o f m o t i o n a n d the sampling frequency ~o. It is shown in i n f o r m a t i o n theory that though different in origin the r o u n d - o f f error and the j i t t e r error can be h a n d l e d in the same way [ 3]. The first is caused by the fact that in any d a t a processing system the analogue signals are substituted by a suitable integer with an error c o m p a r a b l e to the resolution o f the A / D converter. The second is caused by the fact, that the samples are taken at points different from those m a r k e d for s a m p l i n g due to the finite precision o f the system clock. The b o u n d s for the r o u n d - o f f and j i t t e r error can be written as follows: I~Rj(t) I ~< ( o ) I E / T c )
1/2 ,
(4)
where
0030-4018/90/$03.50 © 1990 - Elsevier Science Publishers B.V. ( North-Holland )
309
Volume 78, number 5,6
OPTICS COMMUNICATIONS
eRj(t) = f ( t ) -- ~ f ' (nT) -oo
sin co( t - n T ) co( t - n T )
(5)
is the difference between the true signal and the one reconstructed with the false sampling values f ' (n T), co~ is the frequency band limit of the error,
15 September 1990
v(iT) = m , 2 / ( k T ) ,
(10)
where v(iT) is the average velocity at time iT referred to the sampling time T. The displacement belonging to two samples i, j of the record is
s(iT)-sOT)=
(2/k) ~ m,.
(11)
I I
E= i
~(t)
dt<~,
(6)
--oo
is the energy of the error. For the frequency aliasing error of the reconstruction one can write: lea(t)b ~<(B/2zc) Isin o)tl ,
L tinT)
-de
sin co( t - n T ) o)( t - nT)
(8)
o~ /i
B = J IF(~o,) I do),,
2) ( m s - m s ) / ( i - j ) .
(12)
For the reconstruction of the velocity function v(t) e.g., one uses the v(iT) sampled values v ( t ) = ~ v(iT) _oo
is the difference between the true signalf(t) bound at a higher frequency than 0)/2 and the one reconstructed with the samples taken at sampling frequency m, and (9)
(o
where F(oJ) is the Fourier transform o f f ( t ) .
2. Sampling the interferometric data
Here we consider a Michelson type interferometer, fig. 1. The movement of one of the arms of this interferometer relative to the other is represented by a pulse train, each pulse signifying a 2 / k displacement of the moving arm, where 2 is the wavelength of light and k is an integer. For each pulse a sign corresponding to the direction of movement is assigned. Now we sum up and store the pulses arriving during each sample time. By retaining the mutual correspondence between physical time and sample number i we obtain a set of data {ms} representing a record of the movement. So m, is an integer which equals to the sum of pulses arrived during the ith sample time. The physical interpretation of the sample value m, is obvious: 310
a[(i-j)T]=2/(kT
(7)
where q(t) =fit)-
The acceleration belonging to two samples i, j of the record is
sin o ) ( t - - i T ) co(t-iT)
(13)
The same set of data {m,} can be used for the computation of the quantity
v2( iT) = ( 2 / k T ) 2 rn 2 ,
(14)
providing thus the sampling values for the reconstruction of the v2(t) function. Performing the Fourier transformation of the v2(t) function one obtains the power density spectrum of the motion. Thus we obtain that the proposed digital sampling interferometer retains all the properties and measurement facilities of the conventional metrology interferometers and at the same time provides the possibility of measuring other functions of motion such as velocity, acceleration, power density spectrum, etc.
3. Errors of reconstruction
While it is straightforward to compute from (2) the truncation error for any of the functions of motion the computation of the round-off-jitter error requires further considerations. Using the equality [ 3 ]
E= i ~,(t)dr= Z T4,(nT"),
(15)
where in case of displacement measurement eR,(nT) = f ( n T ) - - f ' ( n T ) = (2/k) Amn,
(16)
and - 1 < Arnn < 1, for a motion that does not con-
Volume 78, number 5,6
OPTICS COMMUNICATIONS
15 September 1990
/////H/{/
BS
I
M
L
I
-
+
D
I
t
'~,
ill.1/
k
[
\ \ ,,\\
/ /,
"
Fig. 1. A symbolic lay-out of the digital interferometer. At equidistant points of phase-difference, 2n x2/k (where x is the wave vector, 2 is the wave length of light, k is an integer) a pulse is formed giving rise to a pulse train during motion. tain any non-zero velocity c o m p o n e n t outside the time interval [ - N T , NT] one obtains
E = ~ T ( 2 / k ) 2 AmZ~ < 2 N T ()t/k) 2 .
(17)
the cycle t i m e o f the highest vibration. F o r at least how m a n y we obtain from the c o n d i t i o n that the t r u n c a t i o n error should not exceed the ()t/k) resolution o f the interferometer either.
--oo
]~N(t) I ~< ()t/k)
Substituting this value in ( 4 ) we have I~Rj(t) I ~< (2k) (2 N T / T I ) 1/2 ,
(18)
where T~ = n/COl is the cycle time o f the highest vibration o f the motion. So the condition for the roundo f f - j i t t e r error not to exceed the ()t/k) resolution o f the interferometer is
2 NT<~ T t .
( 19 )
This implies in itself that no aliasing type error can occur since several samples should be taken during
--
4lf(t)max I 4lf(t)max I l~2Nq - l~2N(l_N/2) ,
(2o)
where we used the equality c o n d i t i o n in (19). Resolving ( 2 0 ) for N we have If(t)m.x I N>~0.4 - -
(Mk)
(21)
The o p t i m u m sampling rate can be defined so that 2 N samples are taken during the T~ cycle time, where 311
Volume 78, number 5,6
OPTICS COMMUNICATIONS
N is found from ( 2 1 ) a n d f ( t ) m a x is the m a x i m u m value o f the physical q u a n t i t y to be reconstructed re-ferred to T~ cycle time.
4. Summary We p r o p o s e d a scheme o f a sampling digital interferometer. It was shown that all interesting functions o f m o t i o n can be reconstructed from a single set o f data. The results o f error analysis were e x t e n d e d to this
312
15 September 1990
measuring principle when the reconstruction is perf o r m e d in the f u n d a m e n t a l Shannon series. The o p t i m u m value o f sampling rate was defined, ( 1 9 ) a n d (21 ), for the case when the error o f the reconstructed displacement does not exceed the resolution o f the conventional interferometer.
References [ 1] A.J. Jerri, Proc. IEEE, Vol. 65, No. 11 (1977). [2] J.B. Thomas and B. Liu, IEEE Int. Cony. Rec. (USA) 12 (1964) 269. [ 3 ] A. Papoulis, Proc. IEEE, Vol. 54, No. 7 (1966).