LITERATURE 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11.

CITED

I. K i r s c h e n b a u m , Heavy W a t e r [Russian translation], IL, Moscow (1953). K . F . L a b a r r and F. Galld, Usp. Khim., 40, 654 (1971). V . K . Miloslavskii and N. S. T r o i t s k i i , in: The S p e c t r o s c o p y of Solids [in Russian], Vol. 4, Nauka, L e n i n g r a d (1969). F . G . Shach, P h y s . Rev., 4 6 , 3 4 5 (1934). L . R . I n g e r s o l l and D. H. L i e h e n b e r g , J. Opt. Soe. A m e r . , 48, No. 5, 339 (1958). I . I . Ushakov, Izv. Vyssh. Uchebn. Zaved., P r i b o r s t r . , 10, No. 8, 86 (1967). I . I . Ushakov, P r i b . Tekh. Eksp., No. 5, 155 (1969). H. L a r z u l , M. F. Gelebart, and A. J o h a n n i n - G i l l e s , Compt. Rend. Acad. Sci. P a r i s , 261, No. 6, 4701 (1965). J . W . Johns, Canad. J. P h y s . , 4 1 , 2 0 9 (1963). G. Dupouy and Ch. F e r t , Compt. Rend., 208, 1298 (1939). S . I . Sorokin, G. P. Kal'yanov, and V. A. Z'abelin, Zh. Fiz. Khim., 47, No. 9, 2440 (1973). r

INTERPOLATION TRANSFORM E.

USING

IN FOURIER

M. Sharov

and

THE

FAST

FOURIER

SPECTROSCOPY V. A . M a m o n t o v

UDC 535.853,4

The role of a c c u r a t e r e g i s t r a t i o n of the position of lines relative to the w a v e - n u m b e r scale is l a r g e . F o r e x a m p l e , in m o l e c u l a r s p e c t r o s c o p y the wave n u m b e r s of the position of a b s o r p t i o n lines contain information about the s t r u c t u r a l p a r a m e t e r s of the m o l e c u l e s being investigated. The r e c o r d i n g and p r o c e s s i n g of i n f o r m a tion in c o n t e m p o r a r y F o u r i e r s p e c t r o s c o p y a r e c a r r i e d out only with d i s c r e t e m e t h o d s , which, owing to t h e i r well-known a d v a n t a g e s , have b e c o m e standard. But an e r r o r of one-half a d i s c r e t i z a t i o n step m a y p r o v e to be i n a d m i s s i b l y l a r g e in a p p r a i s i n g m e a s u r e m e n t and information p r o c e s s i n g r e s u l t s . V a r i o u s interpolation methods a r e employed to reduce this e r r o r . An interpolation method using the f a s t F o u r i e r t r a n s f o r m (FFT) [1] with application to F o u r i e r s p e c t r o s copy p r o b l e m s is d i s c u s s e d in this p a p e r . Interpolation both in the frequency and the time domain i s cons i d e r e d . The l a t t e r is i m p o r t a n t in solving i n t e r f e r o m e t r i c p r o b l e m s by F o u r i e r s p e c t r o s c o p y m e t h o d s . P r i o r to the a p p e a r a n c e of the F F T , interpolation in F o u r i e r s p e c t r o s c o p y was a c c o m p l i s h e d e i t h e r by m e a n s of the d i r e c t calculation of the s p e c t r o g r a m using a finer step, which resulted in c o l o s s a l expenditures of machine time, o r by using K o t e l ' n i k o v ' s t h e o r e m Z I (X) =

I~ (x)

sin (2g~rrnaxX - - n n ) (2~O'maxX- - nn) '

which can be written f o r the f r e q u e n c y domain in the f o r m +| Z sin (2mrL - - nn) S (~r) =

S,~ (~)

( 2 ~ u r L - - n~x)

(i)

(2) '

w h e r e x, a and L, a m a x a r e the c u r r e n t and m a x i m u m v a l u e s , r e s p e c t i v e l y , of the path d i f f e r e n c e and the wave n u m b e r s . Equation (2) can be r e p r e s e n t e d in d i s c r e t e f o r m f o r a finite n u m b e r N of readings as Nil

Sh = Z

,=0

Sn

sin (rtk/d - - n~) (~xk/d - - n u )

(3)

T r a n s l a t e d f r o m Zhurnal Prikladnoi Spektroskopii, Vol. 24, No. 6, pp. 1059-1063, June, 1976. Original a r t i c l e submitted April 7, 1975. This material is protected by copyright registered in the name of Plenum Publishing Corporation, 227 West t 7th Street, New York, N.Y. 10011. No part I of this publication may be reprodueed, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission of the publisher. A copy of this article is available from the publisher for $Z50.

I 755

f 9

a

0

N-Z/N-3 ""~'N-I

i

-L

zL

~

9

%_

+L

M~z

NI~IN-3~/N-: Wlz

Fig. 1. The shaping of a large data set for interpolation in the case ami n = 0: a) the original large data set with N = 8 and b) the large data set p r e p a r e d for the F F T . where d is the specified n u m b e r of points per resolvable s p e c t r a l interval, and the p a r a m e t e r k can take on the M values from kl to k2. In o r d e r to shorten the calculations, one can combine the interpolation based on Eq. (3) with apodization [2], which is accomplished by the r e p l a c e m e n t of the function sin (2~rgL-mr)/(27rgL-a~) with a rapidly decaying apodized i n s t r u m e n t a l function, defined consequently in the smalle st interval. Eq. (3) is a convolution for which the d i r e c t computation time is proportional to the product M x N. I f M ~ log2N, then it is m o r e efficient to p e r f o r m the convolution operation on the basis of the inverse convolution theorem F [I (x) A (x)] = F [I (x)l ~ F [A (x)],

(4)

where F is the F F T o p e r a t o r and the * sign is the convolution o p e r a t o r . Upon c o m p a r i s o n of Eqs. (3) and (4), we can write Sn = F [i(x)], where I(x) is an i n t e r f e r o g r a m r e c o r d e d from - L to + L with d i s c r e t i z a t i o n step h = 2 L / N and sin 0rk/d-alr)/(lrk/d-n~r) = F[A(x)], w h e r e A(x) is d e s c r i b e d by the conditions A(x)=

1 for - - d L ~ x ~ d L , 0 for x ~ - - d L H x ~ d L .

T h e r e f o r e , in o r d e r to determine S k = F [I(x)A(x)], it is n e c e s s a r y to calculate the product I(x)A(x), which is e s s e n t i a l l y the a s s i g n m e n t of (d-1)N/2 null "readings" to the right and the left of I(x), and to c a r r y out the F F T . This operation of f o r m a l l y expanding the domain in which the d i s c r e t e function is specified, o r a s s i g n ing null readings, is determined by the p r o p e r t i e s of the d i s c r e t e F o u r i e r t r a n s f o r m (DFT) [9] and the c h a r a c t e r i s t i c s of the standard F F T p r o g r a m s . The standard F F T p r o g r a m s which have been developed for d o m e s tic c o m p u t e r s (BI~SM-4, BI~SM-6, a n d o t h e r s ) require the r e p r e s e n t a t i o n of a large data set amounting to N = 2~

(5)

readings, where y is an integer. The n e c e s s i t y in this case of completing with null readings the n u m b e r of readings of an i n t e r f e r o g r a m obtained in an experiment, which does not satisfy the condition (5), and the use of this operation for partial interpolation has been communicated e a r l i e r [3]. An interpolation method utilizing the d i r e c t and inverse F F T has been briefly illustrated in the paper [4], but it is inadequate for the practical application of the method. The procedure for the application of interpolation utilizing the F F T is modified depending on the actual position of the s p e c t r a l range Ag on the w a v e - n u m b e r axis (Aa = a m a x - g m i n , where groin is the minimum wave n u m b e r in the spectral signal). In the case of the F F T the calculated position Ag of the i n t e r f e r o g r a m always lies in the region 0 - l / h , where h is the d i s c r e t i z a t i o n step (1/h = 2 ~ a ) . The multiplier m in the exp r e s s i o n m ~ a = area x is always an integer upon the c o r r e c t choice of h. Two c a s e s are thereby distinguished. 1. m = 1, i.e., gmin = 0 and ~ a = amax. One point is n e c e s s a r y per resolvable spectral interval in the uninterpolated s p e c t r o g r a m Sn. Let us pose the problem of obtaining d points. The i n t e r f e r o g r a m illustrated in Fig. l a is shaped p r i o r to the F F T just as is shown in Fig. lb. The numbering of the readings In for N = 8 (Fig. lb) exhibits the rule for the shaping of the l a r g e data set of an i n t e r f e r o g r a m p r i o r to the F F T ( d - 1 ) N null readings are fitted into the middle of the sequence In. The new sequence with N' = dN readings should satisfy the condition (5); t h e r e f o r e , one selects d to be equal to an integral power of two (d = 21, 22, 23,. .. ). There will be d points per resolvable spectral interval in the s p e c t r o g r a m Sn a f t e r the F F T . The interpolation of an i n t e r f e r n g r a m is c a r r i e d out in the following m a n n e r . After the F F T of an int e r f e r o g r a m I n having N readings a s p e c t r o g r a m Sn located in the region 0 - 1 / t l is obtained: the actual spectrum

'/56

a.

I

1/2;~

"//Tz

0

fm-l)/21i

ml21z

~fnin

C~rnax

b ]

I

m

;flZtz lib

0

{m-l)/2/z

m/2iz .

6rain Omnx C

[7

~min

~max

Fig. 2. The shaping of a large data set for i n t e r polation in the case in which ~min ~ 0 and m ~ a = amax: i n t e r p r e t a t i o n of the s p e c t r u m for odd (a) and even m (b); c) the large data set p r e p a r e d f o r FFT. (N/2 points) in the region 0 - 1 / 2 h and its reflection (N/2 points) in the region 1 / 2 h - I / h . Null readings totaling ( d - 1 ) N points a r e fitted into the middle of the sequence Sn, and a f t e r an inverse F F T of the new sequence an i n t e r f e r o g r a m Ik is obtained in which 2d points o c c u r p e r cycle of the component of m a x i m u m frequency. An even function is illustrated in Fig. 1. The overwhelming m a j o r i t y of F o u r i e r s p e c t r o m e t e r s (including all the F o u r i e r s p e c t r o m e t e r s with continuous scanning) have a s y m m e t r i c a l i n t e r f e r o g r a m s whose null reading is located in the best case in the s t e a d y - s t a t e phase region [5], and the null delay (path difference), which is the same for all frequency components, is not realized. In this case a phase c o r r e c t i o n is n e c e s s a r y , which a c c o r d i n g to the method of [5] can follow t h e interpolation, in the c o u r s e of which we obtain interpolated values of the sine and cosine t r a n s f o r m s . In the case of a large d it m a y be m o r e efficient to c a r r y out the phase c o r r e c t i o n of the i n t e r f e r o g r a m f i r s t [6] and then the interpolation. Due to the change in the size of the large data set after a s s i g n m e n t of nulls f r o m N to N' values, it is n e c e s s a r y to multiply by the coefficient K = (N'/N) 1/2 a f t e r interpolation of the value of the function (the s p e c t r o g r a m o r the i n t e r f e r o g r a m ) . Since N' = 2T', and N = 27, then K = 2 (T'-Y)/2 = d 1/2. 2. m >-=2 (In = 2, 3, 4 . . . . ), i.e., ~min ~ 0 and ama x = m~,z. If m is odd, the actual calculated value of the s p e c t r u m will be a r r a n g e d just as in the f i r s t case; if m is even, the reflection of the s p e c t r u m is located in the region 0 - 1 / 2 h, and the actual s p e c t r u m is located in the region 1 / 2 h - 1 / h . It is n e c e s s a r y to take this situation into account in the interpretation of the s p e c t r u m a f t e r the F F T of an i n t e r f e r o g r a m (Fig. 2a, b}. Interpolation of the s p e c t r o g r a m is c a r r i e d out as in the f i r s t case. The interpolation of the i n t e r f e r o g r a m is accomplished by the a s s i g n m e n t of (m-1)N/2 nulls to the right and the left of the sequence Sn obtained by m e a n s of the F F T of In, if the task of obtaining no less than two points per cycle of the m a x i m u m frequency is set. If the n u m b e r of readings obtained does not satisfy the condition (5) (this will be the c a s e when m ~ 2 k, where k is an integer), the n u m b e r of additional nulls n e c e s s a r y to satisfy the given condition is fitted into the middle of the sequence Sn. If we would like to obtain no less than 2d points per cycle of the component of maximum frequency, it is n e c e s s a r y to fit ( d - 1 ) m N nulls into the middle of the sequence Sn (Fig. 2c) in addition to the ( m - 1 ) N / 2 nulls indicated above, which were assigned at the boundaries of this sequence. The additional nulls f o r the satisfaction of condition (5) are also fitted into the middle of the sequence Sn. Since according to T this condition m is always just an integer, then in this case d can be f r a c t i o n a l : dm = 2)' - T . The fact that the p r e s e n t interpolation method does not require any additional p r o g r a m m i n g besides the shaping of large data sets is an attractive feature. Since the method in question r e q u i r e s a c o m p u t e r with an operational m e m o r y device (OMD) of large capacity, it cannot replace the d i r e c t interpolation method based on Kotel'nikov's theorem, which can be applied m o r e efficiently when it is n e c e s s a r y to interpolate n a r r o w ranges of functions of r e l a t i v e l y large g e n e r a l extent. When the c a p a c i t y of the OMD is insufficient to c a r r y out i n t e r polation by the method d i s c u s s e d and external d i r e c t - a c c e s s m e m o r y devices (disk m e m o r y d e v i c e s , magnetic

757

d r u m s , and so on) are components of the c o m p u t e r s y s t e m , a procedure f o r p e r f o r m i n g the F F T [7] can be used which p e r m i t s p r o c e s s i n g a large data set in p a r t s . The formulation of the standard p r o g r a m s according to the method of [8], which p e r m i t s eliminating operations with the null readings such that the time of the calculations b e c o m e s proportional to M log2N , can give a large saving of the time and OMD storage of the c o m p u t e r . LITERATURE

CITED

J. W. Cooley and J. W. Tukey, Math. Comput., 1 9 , 2 9 7 (1965). A. S. F i l l e r , J. Opt. Soc. A m e r . , 54, 762 (1964). M. L. F o r m a n , J . Opt. Soc. A m e r . , 56, 978 (1966). B. Gold and C. M. Rader, Digital P r o c e s s i n g of Signals, McGraw-Hill (1969). L. Mertz, T r a n s f o r m a t i o n s in Optics, Wiley (1965). M. L. F o r m a n , W. H. Steel, and G. A. Vanasse, J. Opt. Soc. A m e r . , 56, 59 (1966). J. Connes, "Computing p r o b l e m s in F o u r i e r s p e c t r o s c o p y , " in: International Conference on F o u r i e r Spectroscopy, Aspen (1970). C. M. Randall, Appl. Opt., 6, No. 8 , 1432 (1967). G. D. Bergland, Zarubezh. Radioelektron., No. 3, 52 (1971).

1.

2. 3. 4. 5. 6. 7. 8o

9.

INTERFERENCE OF

THE

THE

PATH

MODULATION DIFFERENCE

ZERO

VALUE

P. F. A. A.

Parshin, Bytsenko,

BY P E R I O D I C RELATIVE

VARIATION

TO

V. M. A r k h i p o v , and A. P. Kiselev

UDC 535.853.4

A g e n e r a l r e p r e s e n t a t i o n of the t r a n s m i s s i o n s p e c t r u m of an interference modulator has been obtained in [1] in the case when the path difference is varied periodically in a sawtooth m a n n e r . In this paper we c o n s i d e r different special c a s e s that are useful in practice. F o r the c a s e of the t r a n s m i s s i o n s p e c t r u m when 70 = 0 we have T

F (n, 0, ~, T; co) = L (n - - l, 2(oT) 2 f G (t) cos (r + %) dt 0 T

+ L (n; 2(oT)4 cos (2no)T - - (o"0 .f G (t) cos (oldt 0

+ L (n; 2~T) 2..i" G (t) cos (~t + %) dt + 2 S G (t) cos (~l + %) dt + 2

6 (t) cos (~t + %) dt,

(1)

where % = --o)'~ + 2no)T, q).z= 0, % = co-r-- (n + I) oT,

% = - - a)1:,

(2)

% = --~o'~ + 4n(oT. When ~- = 0 the e x p r e s s i o n b e c o m e s T

F (n, 0, 0, T; co) = L (4n; oT) 2 ./' G (t) cos (~t - - ~T) dr.

(3)

0

Translated f r o m Zhurnal Prikladnoi Spektroskopii, Vol. 24, No. 6, pp. 1064-1069, June, 1976. Original a r t i c l e submitted J a n u a r y 27, 1975; revision submitted F e b r u a r y 3, 1976. This material is protected by copyright registered ~n the name of Plenum Publishing Corporation, 227 West 17th Street, New York, iV. Y. 10011. No part I of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, [ microfilming, recording or otherwise, without written permission of the publisher. A copy of this article is available from the publisher for $ 7.50. I

758