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UBC Theses and Dissertations

Apodization of absorption and magnitude mode fourier transform spectra and the effects on SNR and resolution Lee, Judy Pihsien 1986

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APODIZATION OF ABSORPTION AND MAGNITUDE MODE FOURIER TRANSFORM SPECTRA AND THE EFFECTS ON SNR AND RESOLUTION By Judy P i h s i e n Lee B . S c , D a l h o u s i e U n i v e r s i t y , 1984 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE i n THE FACULTY OF GRADUATE STUDIES ( Department of C h e m i s t r y ) We a c c e p t t h i s t h e s i s as c o n f o r m i n g t o the r e q u i r e d s t a n d a r d THE UNIVERSITY OF BRITISH June 1986 © Judy P i h s i e n Lee COLUMBIA In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t of the requirements f o r an advanced degree at the U n i v e r s i t y o f B r i t i s h Columbia, I agree t h a t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and study. I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e copying of t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the head o f my department or by h i s or her r e p r e s e n t a t i v e s . I t i s understood t h a t copying or p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l gain s h a l l not be allowed without my w r i t t e n p e r m i s s i o n . Department of C-Kg./^; s/ry  The U n i v e r s i t y of B r i t i s h Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 Date J~~WAg 2.0, lt$l> D E - 6 (3/81) A B S T R A C T The problem of s i d e l o b e s s u r r o u n d i n g a peak i n a F o u r i e r t r a n s f o r m spectrum i s a l l e v i a t e d by a p o d i z a t i o n . T h i s i s performed by m u l t i p l y i n g the time-domain f u n c t i o n by a window f u n c t i o n . A s y s t e m a t i c study of the e f f e c t s of window f u n c t i o n s on damped time-domain s i g n a l s i s made by e xamining the r e s u l t i n g l i n e s h a p e s a t s p e c i f i c dynamic ranges f o r both the a b s o r p t i o n and magnitude modes. A s y m m e t r i c a l window i s shown t o be e f f e c t i v e f o r the magnitude mode, and h a l f of the s y m m e t r i c a l shape i s b e t t e r f o r the a b s o r p t i o n mode. S e l e c t i o n of a recommended window i s based on the r e q u i r e d dynamic range. For an i n c r e a s i n g dynamic range, the N o e s t - K o r t and Norton-Beer F3, F i l l e r E0.20, and K a i s e r - B e s s e l a r e e f f i c i e n t f o r the a b s o r p t i o n mode; and the Hamming, 3-term B l a c k m a n - H a r r i s and K a i s e r - B e s s e l work f o r the magnitude mode. S i d e l o b e s a r e o f t e n e l i m i n a t e d a t the expense of SNR and/or r e s o l u t i o n , t h e r e f o r e t h e s e f a c t o r s a r e a l s o examined. A l l of the recommended windows show s u f f i c i e n t SNRs except f o r the N o e s t - K o r t . The a p o d i z e d a b s o r p t i o n s p e c t r a a r e w e l l r e s o l v e d , w i t h a 10% v a l l e y as the c r i t e r i o n f o r r e s o l u t i o n . The magnitude s p e c t r a do not d i s p l a y a s i m p l e p a t t e r n , and a l s o show a phase dependence; however t h e s e a r e e x p l a i n e d by the a b s o r p t i o n and d i s p e r s i o n components. These f i n d i n g s l e n d themselves t o v a r i o u s a p p l i c a t i o n s . i i TABLE OF CONTENTS Chapter Page 1 . I n t r o d u c t i o n 1 2. A p o d i z a t i o n of A b s o r p t i o n and Magnitude S p e c t r a 2.1 I n t r o d u c t i o n 10 2.2 Pr o c e d u r e 18 2.3 A b s o r p t i o n Mode R e s u l t s and D i s c u s s i o n s 25 2.4 Magnitude Mode R e s u l t s and D i s c u s s i o n s 39 2.5 C o n c l u s i o n 50 3. S i g n a l - T o - N o i s e R a t i o 3.1 I n t r o d u c t i o n 54 3.2 Pr o c e d u r e 56 3.3 A b s o r p t i o n Mode R e s u l t s and D i s c u s s i o n s 57 3.4 Magnitude Mode R e s u l t s and D i s c u s s i o n s 61 3.5 C o n c l u s i o n 64 4. R e s o l u t i o n of A p o d i z e d S p e c t r a 4.1 I n t r o d u c t i o n 66 4.2 Pr o c e d u r e 69 4.3 A b s o r p t i o n Mode R e s u l t s and D i s c u s s i o n s 72 4.4 Magnitude Mode R e s u l t s and D i s c u s s i o n s 77 4.5 C o n c l u s i o n 99 5. Summary 101 R e f e r e n c e s 1 06 i i i Appendix A 1 1 1 Appendix B 1 1 4 Appendix C 1 1 6 Appendix D • 1 20 i v L I S T OF TABLES Table Page I. Window functions for absorption 26 II. Apodized absorption spectra halfwidths 27 I I I . Window functions for magnitude 40 IV. Apodized magnitude spectra halfwidths 41 V. Relative SNRs : absorption windows 58 VI. Relative SNRs : magnitude windows 62 VII. Absorption lineshape valleys due to the Kaiser-Bessel window 75 VIII. Absorption lineshape valleys due to the 3-term Blackman-Harris window 75 v LIST OF FIGURES F i g u r e Page 1. Time-domain f u n c t i o n s w i t h v a r i o u s dampings 13 2. Unapodized a b s o r p t i o n l i n e s h a p e s due t o v a r i o u s dampings 15 3. Unapodized magnitude l i n e s h a p e s due t o v a r i o u s dampings 16 4. Window t y p e s and r e s u l t i n g a b s o r p t i o n and magnitude l i n e s h a p e s 21 5. Comparison of s u i t a b l e and u n f a v o r a b l e a b s o r p t i o n l i n e s h a p e s 31 6. A b s o r p t i o n l i n e s h a p e s due t o recommended windows and moderate damping 34 7. A b s o r p t i o n l i n e s h a p e s due t o recommended windows and h i g h damping 35 8. A b s o r p t i o n l i n e s h a p e s due t o recommended windows and complete damping 36 9. Comparison of s u i t a b l e and u n f a v o r a b l e magnitude l i n e s h a p e s 44 10. Magnitude l i n e s h a p e s due t o recommended windows and moderate damping .....46 11. Magnitude l i n e s h a p e s due t o recommended windows and complete damping 47 v i 12. Magnitude l i n e s h a p e s r e s u l t i n g from v a r y i n g a i n the K a i s e r - B e s s e l window 49 13. R e l a t i v e s i g n a l - t o - n o i s e r a t i o s due t o a b s o r p t i o n windows as a f u n c t i o n of damping 59 14. R e l a t i v e s i g n a l - t o - n o i s e r a t i o s due t o magnitude windows as a f u n c t i o n of damping 63 15. V a l l e y between two peaks 68 16. V a l l e y h e i g h t s as a f u n c t i o n of s p a c i n g and damping of two K a i s e r - B e s s e l a p o d i z e d a b s o r p t i o n peaks 73 17. V a l l e y h e i g h t s as a f u n c t i o n of s p a c i n g and damping of two 3-term B l a c k m a n - H a r r i s a p o d i z e d a b s o r p t i o n peaks 76 18. Magnitude l i n e s h a p e due t o the K a i s e r - B e s s e l window f o r two peaks s e p a r a t e d by t h r e e s p a c i n g s 79 19. Magnitude l i n e s h a p e due t o the K a i s e r - B e s s e l window f o r two peaks s e p a r a t e d by two s p a c i n g s 80 20. Magnitude l i n e s h a p e due t o the K a i s e r - B e s s e l window f o r two peaks s e p a r a t e d by one s p a c i n g 81 21. V a l l e y h e i g h t s as a f u n c t i o n of s p a c i n g and damping of two unapodized magnitude peaks 82 22. V a l l e y h e i g h t s as a f u n c t i o n of s p a c i n g and damping of two K a i s e r - B e s s e l a p o d i z e d magnitude peaks 84 23. V a l l e y h e i g h t s as a f u n c t i o n of s p a c i n g and damping of two 4-term B l a c k m a n - H a r r i s a p o d i z e d magnitude peaks 85 24. V a l l e y h e i g h t s as a' f u n c t i o n of s p a c i n g and damping of two Hanning a p o d i z e d magnitude peaks 87 v i i 25. L i n e s h a p e s of a s i n g l e peak a p o d i z e d by the K a i s e r - B e s s e l window 90 26. L i n e s h a p e s of two peaks s e p a r a t e d by two s p a c i n g s and a p o d i z e d by the K a i s e r - B e s s e l window 91 27. L i n e s h a p e s of two peaks s p a r a t e d by one s p a c i n g and a p o d i z e d by the K a i s e r - B e s s e l window 92 28. Magnitude l i n e s h a p e s from s i g n a l w i t h no phases 96 29. Magnitude l i n e s h a p e s from s i g n a l where one component has no phase, and the o t h e r has a phase of T T / 2 97 30. Magnitude l i n e s h a p e s from s i g n a l where one compoment has no phase, and the o t h e r has a phase of ir 98 v i i i ACKNOWLEDGEMENT I would l i k e t o tak e t h i s o p p o r t u n i t y t o thank Dr. Mel B. Comisarow f o r h i s g uidance and ent h u s i a s m i n t h i s r e s e a r c h , and f o r h i s d i s c u s s i o n s c l a r i f y i n g the many s u b t l e t i e s e n c o u n t e r e d i n t h i s f i e l d . I a l s o w i s h t o thank Mark A a r s t o l f o r r u n n i n g s p e c t r a t o t e s t a p o d i z a t i o n f u n c t i o n s , and f o r a l l of the d i s c u s s i o n s and f o r p r o o f r e a d i n g ; and Greg S t a t t e r f o r h i s e x p l a n a t i o n s of f o r m u l a s . Thanks a l s o go to Dr. M. W. Bl a d e s and h i s group f o r the use of t h e i r microcomputer f o r the p r o c e s s i n g of t h i s m a n u s c r i p t , and a s p e c i a l thank you t o Zane Walker f o r a l l of h i s h e l p . i x CHAPTER 1 INTRODUCTION F o u r i e r t r a n s f o r m a t i o n i s a p o w e r f u l t e c h n i q u e f o r s p e c t r a l a n a l y s i s . W i t h t h i s m a t h e m a t i c a l p r o c e d u r e , i t i s p o s s i b l e t o a n a l y z e a c o m p l i c a t e d time-domain s i g n a l f o r i t s fr e q u e n c y components, however a f r e q u e n t l y e n c o u n t e r e d problem i s t h a t the r e s u l t i n g spectrum can have the u n d e s i r a b l e f e a t u r e of s i d e l o b e s . T h i s problem can be a l l e v i a t e d by a p o d i z a t i o n , but t h i s i n t u r n a f f e c t s the s i g n a l - t o - n o i s e r a t i o and the r e s o l u t i o n . P r i o r t o d e t a i l e d e x a m i n a t i o n of the removal of s i d e l o b e s and the r e s u l t i n g e f f e c t s some b a s i c c o n c e p t s about F o u r i e r a n a l y s i s and a p o d i z a t i o n w i l l be d e f i n e d . G iven a waveform composed of a sum of s i n u s o i d s , each w i t h i t s own fr e q u e n c y and a m p l i t u d e , i t i s p o s s i b l e t o s e p a r a t e out and i d e n t i f y each i n d i v i d u a l component. One example would be an a c o u s t i c waveform. I f , f o r example, two d i f f e r e n t p i a n o s t r i n g s were s t r u c k s i m u l t a n e o u s l y but w i t h d i f f e r e n t f o r c e , i t would be p o s s i b l e f o r the human ear t o d i s c e r n the d i f f e r e n t n o t e s and t h e i r d i f f e r e n t i n t e n s i t i e s . Another example would be an e l e c t r i c a l waveform whose f r e q u e n c y components a r e i d e n t i f i e d i n a spectrum. A n a l y s i s of a time-domain s i g n a l f o r i t s f r e q u e n c y components can be performed by a p r o c e d u r e known as a F o u r i e r t r a n s f o r m ( F T ) . F o u r i e r t r a n s f o r m t e c h n i q u e s encompass a wide v a r i e t y of 1 a p p l i c a t i o n s . The concept i t s e l f was c o n c e i v e d d u r i n g r e s e a r c h i n t o heat c o n d u c t i o n [ 1 , 2 ] , and i s now used i n many a r e a s . The r e l a t i o n s h i p between F o u r i e r t r a n s f o r m a t i o n and the common a p p l i c a t i o n s t o m e c h a n i c a l and e l e c t r i c a l e n g i n e e r i n g systems as w e l l as o t h e r systems have been d i s c u s s e d i n d e t a i l [ 1 , 2 , 3 ] . Other a p p l i c a t i o n s can be found i n such v a r i e d f i e l d s as m e d i c i n e , a r c h i t e c t u r e and geology [ 4 ] . Another f i e l d which employs the FT i s o p t i c s [ 5 ] , w i t h a p p l i c a t i o n as d i v e r s e as holo g r a p h y and o p t i c a l s p e c t r o s c o p y [ 6 ] , In c h e m i s t r y , FTs have been a p p l i e d t o e l e c t r o c h e m i c a l s t u d i e s [ 7 , 8 ] , but t h e i r predominance i s i n the ar e a of s p e c t r o s c o p y [ 9 , 1 0 ] . These i n c l u d e i n f r a r e d ( I R ) , n u c l e a r magnetic resonance (NMR), i o n c y c l o t r o n resonance ( I C R ) , and atomic e m i s s i o n or o p t i c a l e m i s s i o n s p e c t r o s c o p i e s . There a re e x t e n s i v e r e v i e w s on FT-IR [11,12,13], FT-NMR [ 1 4 , 1 5 ] , and FT-ICR [16,17,18,19], and a l s o work on atomic e m i s s i o n [ 2 0 , 2 1 ] . A l t h o u g h the FT has e x i s t e d s i n c e 1822, i t was not a p p l i e d c o m m e r c i a l l y t o c h e m i c a l s p e c t r o s c o p y u n t i l t h e p a s t few decades. The f i r s t c ommercial FT-IR i n s t r u m e n t was produced i n 1964, the FT-NMR s p e c t r o m e t e r was manufactured some y e a r s l a t e r [ 9 ] , and the f i r s t FT-ICR was s o l d i n 1981. One f a c t o r i n the en t i c e m e n t of the use of FTs was the r e a l i z a t i o n of the m u l t i p l e x , or F e l l g e t t advantage. T h i s was f i r s t d e t a i l e d i n 1958. I t can . be approached i n terms of a time advantage or a s i g n a l - t o - n o i s e r a t i o (SNR) advantage of an 2 FT s p e c t r o m e t e r over a c o n v e n t i o n a l s p e c t r o m e t e r . A model c o u l d be the comparison of a s c a n n i n g s p e c t r o m e t e r t o an FT s p e c t r o m e t e r . I f the FT s p e c t r o m e t e r a c q u i r e s a s i g n a l c o n t a i n i n g N s p e c t r a l elements f o r a t o t a l time T, a s c a n n i n g method would need a t o t a l time of NT i n o r d e r t o get the same SNR. Thus the b a s i c advantage of an FT s p e c t r o m e t e r over a c o n v e n t i o n a l one i s speed. An a l t e r n a t i v e view can be made i n terms of the SNR due t o e q u a l s a m p l i n g t i m e s . In a l e n g t h of time , T, a s c a n n i n g s p e c t r o m e t e r examines N d i f f e r e n t s p e c t r a l e lements. In the same l e n g t h of t i m e , a FT s p e c t r o m e t e r examines time-domain d a t a p o i n t s which c o n t a i n s p e c t r a l i n f o r m a t i o n . The s i z e of a s i g n a l i s d i r e c t l y p r o p o r t i o n a l t o the l e n g t h of time t h a t i t i s o b s e r v e d , and random n o i s e i s p r o p o r t i o n a l t o the square r o o t of the s i g n a l . For the s c a n n i n g s p e c t r o m e t e r , each s p e c t r a l element i s o n l y examined f o r a f r a c t i o n of the t o t a l t i m e , t h a t i s the s i g n a l i s p r o p o r t i o n a l t o T/N. Thus the n o i s e 1 / 2 i s p r o p o r t i o n a l t o T/N ' , which makes the r a t i o of the s i g n a l t o 1 / 2 the n o i s e e q u a l t o T/N ' . On the o t h e r hand, the FT s p e c t r o m e t e r examines each component f o r a time e q u a l t o the t o t a l t i m e , T, g i v i n g a s i g n a l p r o p o r t i o n a l t o T, n o i s e p r o p o r t i o n a l t o T 1 > / 2, and a SNR of T 1^ 2. T h i s SNR i s N 1 / / 2 times b e t t e r than t h a t from a c o n v e n t i o n a l method, but t h e r e i s o n l y an advantage when N i s g r e a t e r than one. Another f a c t o r which a i d e d i n the development of FT s p e c t r o s c o p y has been the p r o g r e s s i n computing. B e f o r e 3 describing the computer breakthrough, some background on FTs i s necessary. B a s i c a l l y , a FT can be thought of as a formula rela t i n g a general function in the time domain, g ( t ) , with i t s frequency-domain description, G ( f ) . One method of depicting t h i s is with the formula This i s a continuous function, and G(f) can be obtained a n a l y t i c a l l y by integration. However, in experimental practice the a n a l y t i c a l expression for the time-domain function, g ( t ) , can be quite complicated and i s not always e a s i l y attainable and so an a n a l y t i c a l transform can not be readily performed. Yet i t i s s t i l l possible to compute a FT. If the numerical values describing the time-domain function are obtainable, that i s i f a value i s known for the time-domain function at discrete i n t e r v a l s in time, t . then a discrete Fourier transform (DFT) can be n performed. An expression s i m i l a r to that describing the continuous FT, Eq. 1 , can be given for the DFT by where N i s the t o t a l number of discrete points sampled. The d i g i t a l data i s ideal for computer analysis, but the computation of the transform s t i l l required extensive computing time. The ( 1 ) G(f k>- I g ( t n ) e x p ( - i 2 * f k t n ) ( 2 ) 4 t u r n i n g p o i n t came w i t h the c r e a t i o n of an e f f i c i e n t a l g o r i t h m , the Cooley-Tukey a l g o r i t h m which i s a l s o known as the f a s t F o u r i e r t r a n s f o r m ( F F T ) , t o compute DFTs. Much has been w r i t t e n about FFTs [ 2 2 ] , i n c l u d i n g d e t a i l s of new a l g o r i t h m s [ 2 3 ] . The emergence of the FFT i n 1965 was a g r e a t a i d i n the c o m p u t a t i o n of FTs. F o u r i e r t r a n s f o r m s p e c t r a of l a r g e d a t a s e t s can now be c a l c u l a t e d q u i c k l y and e f f i c i e n t l y . However some problems w i t h the t r a n s f o r m a t i o n of d a t a do e x i s t . The time-domain s i g n a l i s sampled at d i s c r e t e i n t e r v a l s t o y i e l d a d i s c r e t e f r e q u e n c y spectrum which c o u l d have e r r o r s i n the h e i g h t s of the peaks. Erroneous f r e q u e n c i e s can be f o r m u l a t e d from a l i a s i n g which o c c u r s when the s a m p l i n g r a t e i s i n s u f f i c i e n t . S i d e l o b e s can form around a peak because of t h e f i n i t e l e n g t h of the d a t a s e t . However a l l of t h e s e problems can be a l l e v i a t e d . In DFT a n a l y s i s , the time-domain d a t a i s c o l l e c t e d a t d i s c r e t e , e q u a l l y spaced i n t e r v a l s f o r an a c q u i s i t i o n time of l e n g t h T. In the t r a n s f o r m , t h e f r e q u e n c y spectrum w i l l c o n s i s t of d i s c r e t e p o i n t s w i t h a s p a c i n g of 1/T between them. I f the maximum v a l u e of a peak f e l l between two d i s c r e t e f r e q u e n c y p o i n t s , then i t s c o r r e c t h e i g h t would not be d e t e c t e d . Peak h e i g h t e r r o r s , sometimes r e f e r r e d t o as the p i c k e t fence e f f e c t [ 2 4 ] , can be remedied by e x t e n d i n g the time-domain d a t a s e t w i t h z e r o s , which i s c a l l e d z e r o - f i l l i n g [ 2 5 , 2 6 ] . T h i s i n c o r p o r a t e s p o i n t s between the p r e v i o u s d i s c r e t e f r e q u e n c y p o i n t s , and an 5 i n f i n i t e number of z e r o - f i l l i n g s would y i e l d the c o n t i n u o u s frequency l i n e s h a p e . Thus the c o r r e c t peak h e i g h t s can be o b t a i n e d i n a d i s c r e t e frequency l i n e s h a p e . In o r d e r f o r a waveform t o be sampled p r o p e r l y , a t l e a s t two p o i n t s per c y c l e a r e r e q u i r e d t o o b t a i n s u f f i c i e n t i n f o r m a t i o n on i t s a m p l i t u d e and f r e q u e n c y . The s a m p l i n g r a t e i n the time domain must be a t l e a s t t w i c e the maximum f r e q u e n c y . T h i s i s the N y q u i s t s a m p l i n g theorem. I f i t i s not s a t i s f i e d , the spectrum f o l d s back on i t s e l f or i s a l i a s e d [ 2 2 ] . T h i s can be a v o i d e d by s e l e c t i n g a s u f f i c i e n t s a m p l i n g r a t e i n the time domain. S i d e l o b e s around peaks can be u n d e r s t o o d w i t h some knowledge on a few FT p a i r s . I f a s i g n a l i s i n f i n i t e l y l o n g i n the time domain, i t s t r a n s f o r m i s an i n f i n i t e l y narrow peak i n the f r e q u e n c y domain; f o r example, the FT of an i n f i n i t e , c o n s t a n t time-domain f u n c t i o n i s a d e l t a f u n c t i o n i n the f r e q u e n c y domain. O b v i o u s l y , s a m p l i n g of a s i g n a l can not be u n d e r t a k e n f o r an i n f i n i t e l e n g t h of t i m e . Thus the s i g n a l i s t r u n c a t e d ; t h a t i s , i t i s e q u a l t o z e r o a f t e r the a c q u i s i t i o n t i m e , T. When a boxcar f u n c t i o n , a c o n s t a n t f u n c t i o n t r u n c a t e d a f t e r a time T, i n the time domain i s t r a n s f o r m e d t o the f r e q u e n c y domain, i t y i e l d s a s i n e f u n c t i o n . Thus a t r u n c a t e d time s i g n a l l e a d s t o a f r e q u e n c y f u n c t i o n w i t h a peak surrounded by s i d e l o b e s . E l i m i n a t i o n of the s i d e l o b e s i s c a l l e d a p o d i z a t i o n , and the d e f i n i t i o n and o r i g i n of t h i s term have been d i s c u s s e d [ 2 7 ] . 6 A p o d i z a t i o n can be performed by m u l t i p l y i n g the time-domain f u n c t i o n w i t h a window f u n c t i o n p r i o r t o F o u r i e r t r a n s f o r m a t i o n . There i s a m u l t i t u d e of t e r m i n o l o g y i n the l i t e r a t u r e on windowing. In t h i s work, a time-domain f u n c t i o n r e f e r s t o a f u n c t i o n sampled from a time of 0 t o a t o t a l l e n g t h of time T. A window f u n c t i o n i s the f u n c t i o n d e f i n e d over a time T which i s m u l t i p l i e d by the time-domain f u n c t i o n f o r the purpose of a p o d i z a t i o n . There are numerous o t h e r terms f o r a window f u n c t i o n i n c l u d i n g a p o d i z a t i o n f u n c t i o n , smoothing f u n c t i o n , w e i g h t i n g f u n c t i o n , and time-domain w e i g h t f u n c t i o n . A f t e r F o u r i e r t r a n s f o r m a t i o n , which i s a d i s c r e t e t r a n s f o r m a t i o n i n t h i s work, a frequency-domain spectrum or l i n e s h a p e i s produced. T h i s has been r e f e r r e d t o as a measured spectrum, f r e q u e n c y - r e s p o n s e f u n c t i o n , and d i s c r e t e F o u r i e r t r a n s f o r m . The term s i d e l o b e r e f e r s t o b o t h the p o s i t i v e and n e g a t i v e l o b e s s u r r o u n d i n g a peak i n t h e f r e q u e n c y domain. Sometimes the shape a t the base of a peak i s not of d i s t i n c t l o b e s , nor i s i t smooth. There a r e c o l o r f u l terms i n the l i t e r a t u r e f o r t h i s , such as wings, w i g g l e s and f e e t . In t h i s work, the term s k i r t r e f e r s t o the g e n e r a l shape a t the base of a peak. There i s e x t e n s i v e d i s c u s s i o n of window f u n c t i o n s i n the l i t e r a t u r e , r a n g i n g from a p p l i c a t i o n s of s i n g l e windows t o thorough a n a l y s e s of dozens windows. Many windows from the l i t e r a t u r e were e x c l u d e d from t h i s s t u d y . These i n c l u d e d windows 7 which had parameters r e q u i r i n g a d j u s t m e n t s , o f t e n e m p i r i c a l , f o r d i f f e r i n g c o n d i t i o n s [27,28,29]; and those which were a p p l i e d t o the f r e q u e n c y domain [30,31,32], and had v a r i o u s o t h e r a p p l i c a t i o n p r o c e d u r e s b e s i d e s windowing i n . the time-domain [33,34,35] . The windows chosen f o r t h i s s tudy a r e t h o s e which a r e e a s i l y implemented by m u l t i p l i c a t i o n i n the time domain w i t h o u t need f o r complex or e m p i r i c a l a d j u s t m e n t s . Most of. the l i t e r a t u r e on windows d e a l s o n l y w i t h undamped time-domain s i g n a l s , and p r e s e n t s the r e s u l t s i n e i t h e r the power, magnitude or a b s o r p t i o n mode. Thus t h i s s tudy d e a l s w i t h b o t h undamped and damped s i g n a l s f o r b o t h the a b s o r p t i o n and magnitude modes. The s i g n a l s have been s y n t h e s i z e d , however the models can be a p p l i e d t o r e a l systems such as FT-ICR and FT-NMR. In Chapter 2, the e f f e c t s of windows a r e examined i n d e t a i l f o r the a b s o r p t i o n and magnitude l i n e s h a p e s r e s u l t i n g from t i m e -domain s i g n a l s of v a r i o u s dampings. Two d i f f e r e n t shapes of windows and the b e h a v i o r of the r e s u l t i n g a b s o r p t i o n and magnitude l i n e s h a p e s i s d i s c u s s e d . S p e c i f i c window f u n c t i o n s are recommended as a r e s u l t of the e x t e n t t o which they e l i m i n a t e s i d e l o b e s w h i l e r e t a i n i n g a f a i r l y narrow l i n e s h a p e . A s i d e from h a v i n g low s i d e l o b e s , a good l i n e s h a p e s h o u l d have an adequate s i g n a l - t o - n o i s e r a t i o (SNR). A p p l i c a t i o n of a window f u n c t i o n can a f f e c t the SNR of a spectrum. A l t e r i n g the damping of the time-domain s i g n a l a l s o changes the SNR. Chapter 8 3 examines the a b s o r p t i o n and magnitude SNRs due t o d i f f e r e n t window f u n c t i o n s and d i f f e r e n t amounts of damping. An i m p o r t a n t a s p e c t of frequency-domain s p e c t r a i s r e s o l u t i o n . C h apter 4 examines a p o d i z e d a b s o r p t i o n and magnitude l i n e s h a p e s w i t h two c l o s e l y spaced peaks. V a r i o u s window f u n c t i o n s a r e a p p l i e d , and the r e s u l t s f o r two peaks a r e examined w i t h r e s p e c t t o the amount of damping and t h e s p a c i n g between the peaks. 9 C H A P T E R 2 A P O D I Z A T I O N O F A B S O R P T I O N A N D M A G N I T U D E S P E C T R A 2 . 1 I n t r o d u c t i o n I n F o u r i e r t r a n s f o r m s p e c t r o s c o p y , t h e t i m e - d o m a i n s i g n a l i s a c o m p o s i t e s i g n a l c o n s i s t i n g o f a s u m o f f u n c t i o n s . A n e x a m p l e o f s u c h a f u n c t i o n i s g ( t ) = K c o s ( 2 f f f D t ) e x p ( - t / r ) 0 < t < T ( 3 ) w h e r e T i s t h e a c q u i s i t i o n t i m e , r i s t h e r e l a x a t i o n t i m e , a n d K i s t h e a m p l i t u d e . T h i s e q u a t i o n i s a p p l i c a b l e t o F T - N M R [ 1 4 ] , a n d d e t a i l s o f t h e v a r i a b l e s i n t h i s e q u a t i o n i n r e l a t i o n t o F T -I C R h a v e b e e n d e s c r i b e d [ 3 6 ] . A n a l y t i c a l F T s h a v e b e e n p e r f o r m e d o n E q . 3 a n d d e t a i l s o n t h e r e s u l t i n g l i n e s h a p e s h a v e b e e n c a l c u l a t e d [ 3 6 , 3 7 ] , F o r F o u r i e r s p e c t r o s c o p y , t h e t i m e - d o m a i n s i g n a l i s s a m p l e d a t d i s c r e t e p o i n t s ; a n d i f i t i s s a m p l e d f o r a s u f f i c i e n t l y l o n g T o r i f t h e d a t a i s z e r o f i l l e d e n o u g h , t h e r e s u l t i n g f r e q u e n c y l i n e s h a p e w i l l r e s e m b l e t h e c o n t i n u o u s l i n e s h a p e . T h u s a d i s c r e t e t i m e - d o m a i n s i g n a l c a n y i e l d a q u a s i -c o n t i n u o u s f r e q u e n c y s p e c t r u m . T h e d i s c r e t e t i m e - d o m a i n s i g n a l i s o b t a i n e d b y s a m p l i n g t h e c o n t i n u o u s s i g n a l , w i t h t h e f o r m o f E q . 3 , a t a r a t e S a n d s t o r i n g N n u m e r i c a l v a l u e s . T h e a r r a y o f N v a l u e s i s t h e d i s c r e t e t i m e - d o m a i n s i g n a l . T h e t o t a l a c q u i s i t i o n t i m e , T , i s d e t e r m i n e d b y t h e s a m p l i n g r a t e , S , a n d t h e t o t a l 10 number of da t a p o i n t s , N, a c c o r d i n g t o T = (N-1)/S. (4) A n u m e r i c a l F o u r i e r t r a n s f o r m i s performed t o y i e l d a d i s c r e t e f r e q u e n c y spectrum w i t h the d i s t a n c e between the p o i n t s , the s p a c i n g , g i v e n by A f = 1/T. (5) R e c a l l t h a t a DFT, Eq. 2, c o n t a i n s both a r e a l and an ima g i n a r y p a r t . The r e a l p a r t , a l s o known as the c o s i n e t r a n s f o r m , g i v e s t h e a b s o r p t i o n spectrum. The i m a g i n a r y p a r t , or s i n e t r a n s f o r m , y i e l d s the d i s p e r s i o n spectrum. The magnitude l i n e s h a p e r e s u l t s from t a k i n g the square r o o t of the sum of the squares of the a b s o r p t i o n and d i s p e r s i o n l i n e s h a p e s . A f i n a l mode of p r e s e n t a t i o n i s power which i s merely the magnitude squared. The a b s o r p t i o n mode i s used f o r s p e c t r a l r e p r e s e n t a t i o n i n h i g h r e s o l u t i o n FT-NMR. The magnitude mode i s u t i l i z e d f o r FT-ICR, and a l s o f o r c e r t a i n a p p l i c a t i o n s of NMR [14,38,39]. The a b s o r p t i o n mode can i n p r i n c i p l e be used f o r FT-ICR, but has not been because of what appears t o be c o n t a m i n a t i o n by a d i s p e r s i o n component. T h i s t a k e s the form of n e g a t i v e components, however i t has been shown t h a t t h e s e a r e a c t u a l l y t he r e s u l t of the d i s c r e t e n a t u r e of s a m p l i n g and n e g a t i v e a b s o r p t i o n i n t e n s i t i e s , 1 1 and t h a t a p o d i z a t i o n c o u l d a l l e v i a t e t h i s problem and open the a b s o r p t i o n mode t o FT-ICR [ 4 0 ] . For e n g i n e e r i n g p u r p o s e s , the power mode i s used. C h e m i c a l a p p l i c a t i o n s , such as FT-ICR and FT-NMR, o f t e n i n v o l v e damped time-domain s i g n a l s . The amount of damping of a time-domain s i g n a l i s r e p r e s e n t e d by the r a t i o of T/ r i n Eq. 3 w i t h p l o t s showing v a r i o u s r a t i o s i n F i g . 1. When T/ r = 0.0, the s i g n a l i s undamped. A p a r t i a l l y damped s i g n a l i s r e p r e s e n t e d by T / T = 0.5, 1.0, and 2.0, c o r r e s p o n d i n g r e s p e c t i v e l y t o low, moderate, and h i g h amounts of damping. A v i r t u a l l y t o t a l l y damped s i g n a l c o r r e s p o n d s t o T/ r = 3.0 which d e p i c t s a t i m e -domain s i g n a l t h a t has decayed t o 5% of i t s o r i g i n a l a m p l i t u d e . From the bottom s i g n a l i n F i g . 1, i t i s e v i d e n t t h i s v a l u e c r e a t e s an e s s e n t i a l l y t o t a l l y damped s i g n a l , thus f u t u r e r e f e r e n c e t o a t o t a l l y damped s i g n a l c o r r e s p o n d s t o one c a l c u l a t e d w i t h T/r = 3 . 0 . Some f i g u r e s have been shown f o r d i f f e r e n t amounts of damping [ 3 6 , 4 1 ] , however they were not c o n s i s t e n t i n the f a c t o r h e l d c o n s t a n t : i n some f i g u r e s r was v a r i e d , and i n o t h e r s T was v a r i e d . Here, the a c q u i s i t i o n t i m e , T, i s h e l d f i x e d and o n l y r i s v a r i e d t o produce s i g n a l s of d i f f e r e n t dampings. Damping of the time-domain s i g n a l has a marked e f f e c t on the fr e q u e n c y spectrum. T h i s i s apparent by comparing the freq u e n c y l i n e s h a p e s due t o d i f f e r e n t amounts of damping f o r both the a b s o r p t i o n mode, F i g . 2, and the magnitude mode, F i g . 3. These 12 T / t = 0.0 T / t = 1 . 0 0.0 0 . 0 T / t = 3.0 Figure 1. Time-domain functions with various dampings. Top : undamped, middle : moderately damped, bottom : v i r t u a l l y completely damped. 13 l i n e s h a p e s were c a l c u l a t e d from the t r a n s f o r m a t i o n of Eq. 3 w i t h d i f f e r e n t amounts of damping. The a m p l i t u d e s c a l e i s l i n e a r , and the o r i g i n s a r e o f f s e t such t h a t each l i n e s h a p e has i t s b a s e l i n e a t the dash c o r r e s p o n d i n g t o i t s r e s p e c t i v e damping v a l u e . The f r e q u e n c y s c a l e i s s c a l e d as the d i s t a n c e away from the c e n t r a l maximum, t h a t i s f _f Q» i n u n i t s of Eq. 5. The s p e c t r a show poor c h a r a c t e r i s t i c s , p a r t i c u l a r l y f o r t o t a l l y undamped time-domain s i g n a l s which produce s p e c t r a w i t h v e r y h i g h s i d e l o b e s . The a b s o r p t i o n l i n e s h a p e s , F i g . 2, have l a r g e p o s i t i v e and n e g a t i v e s i d e l o b e s a t low dampings. The magnitude l i n e s h a p e s , F i g . 3, a r e wider than the a b s o r p t i o n l i n e s h a p e s , and a r e a l s o p l a g u e d by s i d e l o b e s . The l i n e s h a p e s have smoother ye t wide s k i r t s which e x t e n d f o r many s p a c i n g s , Eq. 5, from the c e n t r a l maximum f o r h i g h e r damping v a l u e s f o r b oth the a b s o r p t i o n and magnitude l i n e s h a p e s . These l i n e s h a p e s r e v e a l the need f o r a p o d i z a t i o n . Of c o n s i d e r a b l e d i s t u r b a n c e a r e the s i d e l o b e s which emerge f o r low damping v a l u e s . S i d e l o b e s a r e u n d e s i r a b l e as they can i n t e r f e r e w i t h and d i s t o r t the i n t e n s i t y of an a d j a c e n t peak. T h i s i s p a r t i c u l a r l y d e t r i m e n t a l when a peak of low i n t e n s i t y must be d e t e c t e d i n c l o s e p r o x i m i t y t o a s t r o n g peak. From F i g s . 2 and 3, i t i s n o t e d t h a t the l i n e s h a p e s v a r y c o n s i d e r a b l y w i t h d i f f e r e n t amounts of damping, and thus a r e dependent upon the T / r r a t i o . S i n c e the frequency-domain l i n e s h a p e i s a f u n c t i o n of damping the e f f e c t of a window can be 14 ABSORPTION FREQUENCY (Af) F i g u r e 2 . Unapodized a b s o r p t i o n l i n e s h a p e s due t o v a r i o u s dampings. 15 o MAGNITUDE I I I I I I I I I I I - 4 - 2 0 2 4 FREQUENCY (Af) F i g u r e 3 . Unapodized magnitude l i n e s h a p e s due t o v a r i o u s dampings. 16 expected t o v a r y w i t h damping. The e f f e c t s of a f i n i t e s a m p l i n g i n t e r v a l a r e c l e a r l y seen i n the shape of the s k i r t s i n F i g s . 2 and 3. A method of e l i m i n a t i n g unwanted s i d e l o b e s i n a spectrum, a p o d i z a t i o n , i s by windowing i n the time domain. T h i s i s a m a t h e m a t i c a l p r o c e s s whereby a time-domain s i g n a l i s m u l t i p l i e d by a window f u n c t i o n p r i o r t o F o u r i e r t r a n s f o r m a t i o n . Many of the windows p r e v i o u s l y mentioned, and used i n t h i s s t u d y , have come from e n g i n e e r i n g l i t e r a t u r e ; t h u s a b s o r p t i o n mode r e s u l t s were not commented upon. A l s o , t h e s e e n g i n e e r i n g s t u d i e s d e a l t o n l y w i t h an undamped time-domain s i g n a l . Window f u n c t i o n s have been a p p l i e d y i e l d i n g a b s o r p t i o n r e s u l t s ; however, t h e s e s t u d i e s were o f t e n e x p e r i m e n t a l a p p l i c a t i o n s r a t h e r than t h e o r e t i c a l . Thus q u a l i t a t i v e r e s u l t s , the improvement of windowing compared t o no a p o d i z a t i o n , were shown f o r s p e c i f i c c a s e s , but no q u a n t i t a t i v e r e s u l t s , such as ex a c t d e t a i l s of the amount of improvement, were g i v e n [ 2 8 , 4 2 ] . These windows were a d j u s t e d , o f t e n e m p i r i c a l l y , t o s u i t the s p e c i f i c f r e q u e n c y r e s p o n s e s . Other windows have been examined i n d e t a i l f o r the a b s o r p t i o n mode f o r such d e t a i l s as the w i d t h of the l i n e s h a p e produced and the s i z e of the s i d e l o b e s . However the c r i t e r i a f o r a c c e p t a b l e windows were d i f f e r e n t i n e a r l i e r s t u d i e s . Of imp o r t a n c e , f o r example, were the computing time [ 4 3 ] , and the w i d t h a t h a l f h e i g h t [ 4 4 ] . The h e i g h t s of the s i d e l o b e s , a l t h o u g h r e c o r d e d , were not a g r e a t c o n c e r n . L i k e the 1 7 e n g i n e e r i n g s t u d i e s , t h e s e works a l s o d e a l t o n l y w i t h undamped c a s e s . In t h i s s t u d y , the p r i m a r y reason f o r a p p l y i n g window f u n c t i o n s i s f o r the r e d u c t i o n of the s i z e of the s i d e l o b e s . I t i s apparent t h a t a d e t a i l e d e x a m i n a t i o n of d i f f e r e n t windows on b o t h the a b s o r p t i o n and magnitude modes i s l a c k i n g , p a r t i c u l a r l y f o r time-domain s i g n a l s of d i f f e r e n t dampings. T h i s study s y s t e m a t i c a l l y examines the e f f e c t of many windows on time-domain s i g n a l s of v a r i o u s dampings. T h e i r e f f i c a c y i s d e t e r m i n e d a c c o r d i n g t o the e x t e n t t h a t s i d e l o b e s a r e e l i m i n a t e d i n the a b s o r p t i o n and magnitude l i n e s h a p e s . As d e s c r i b e d i n d e t a i l below, the g e n e r a l shape of window f u n c t i o n s f o r use i n a b s o r p t i o n mode s p e c t r o s c o p y d i f f e r s from the g e n e r a l shape used i n magnitude mode s p e c t r o s c o p y . 2.2 P r o c e d u r e In the p r o c e s s of a p o d i z a t i o n , a time-domain s i g n a l , g ( t ) , i s m u l t i p l i e d by a window f u n c t i o n , w ( t ) , and t h i s p r o d u c t i s then F o u r i e r t r a n s f o r m e d t o y i e l d an a p o d i z e d f r e q u e n c y spectrum. For t h i s s t u d y , Eq. 3 was used t o r e p r e s e n t the time-domain s i g n a l . The l e n g t h of the s i g n a l l a s t e d from t = 0 t o the t o t a l a c q u i s i t i o n t i m e , T, which was s e t t o 1 s e c . The f r e q u e n c y , f Q , was s e t t o 32 Hz. T h i s model c o r r e s p o n d s t o a c t u a l p a r a m e t e r s , except on a d i f f e r e n t s c a l e . F or example, i n FT-ICR, the t i m e -domain s i g n a l can l a s t i n the o r d e r of msec and the f r e q u e n c i e s c o u l d be i n the KHz range, t h u s the r e s u l t s here would appear the 18 same except f o r the s c a l e . An a r r a y of 129 d a t a p o i n t s was c a l c u l a t e d f o r the t i m e -domain s i g n a l , Eq. 3, and a n o t h e r s e t of 129 v a l u e s was c a l c u l a t e d f o r a window f u n c t i o n . These were m u l t i p l i e d t o g e t h e r to produce an a r r a y r e p r e s e n t i n g a d i s c r e t e windowed time-domain f u n c t i o n , g ( t ) w ( t ) . T h i s time-domain d a t a was then z e r o - f i l l e d [25] f i v e t i m e s t o g i v e a q u a s i - c o n t i n u o u s f r e q u e n c y spectrum. A DFT was performed on the z e r o - f i l l e d , windowed time-domain a r r a y u s i n g the r o u t i n e POLFT2 on the UBC-MTS system, and t h e r e s u l t i n g r e a l and i m a g i n a r y v a l u e s were used t o c a l c u l a t e t h e a p o d i z e d a b s o r p t i o n and magnitude s p e c t r a as shown i n Appendix A. T h i s c a l c u l a t i o n was performed f o r a range of v a l u e s f o r r f o r each window. The v a l u e s f o r T/ r , i n Eq. 3, were s e t t o 0.001 r e p r e s e n t i n g 0.0, and t o 0.5, 1.0, 2.0, and 3.0. The window f u n c t i o n s were o b t a i n e d from v a r i o u s s o u r c e s . F i l l e r [ 4 3 ] , and Norto n and Beer [44] performed d e t a i l e d e x a m i n a t i o n s on windows f o r the undamped case f o r a b s o r p t i o n s p e c t r a . The windows which produced the low e s t s i d e l o b e s were chosen from t h e s e works t o be implemented f o r the absorption-mode s t u d i e s ; t h e windows were taken d i r e c t l y from Norton and Beer's work, and a d a p t a t i o n by t a k i n g o n l y the p o s i t i v e p o r t i o n s of F i l l e r ' s windows. A window f o r a b s o r p t i o n and one f o r magnitude were used from Noest and K o r t ' s work [ 4 5 ] . In t h e i r work, the r e s u l t s were a c t u a l l y examined f o r c a s e s due t o d i f f e r e n t dampings, however t h e i r purpose was t o i l l u s t r a t e windows which 19 produced low peak h e i g h t e r r o r s . Common windows, such as the r e c t a n g l e , t r i a n g l e , Hamming and Hanning are d e s c r i b e d i n d e t a i l i n numerous s o u r c e s [46,47,48,49,50], These, a l o n g w i t h l e s s commonly known windows a r e c o m p i l e d i n H a r r i s ' s s t u d y , thus the r e m a i n i n g windows were adapted from t h i s e x t e n s i v e r e p o r t [ 5 1 ] , A s i d e from the work of Noest and K o r t , the r e m a i n i n g s t u d i e s d i s p l a y e d o n l y undamped r e s u l t s . A d a p t a t i o n of F i l l e r ' s and H a r r i s ' s windows was n e c e s s a r y f o r the a b s o r p t i o n s t u d i e s . The windows d e s c r i b e d i n these works a r e s y m m e t r i c a l i n shape w i t h r e s p e c t t o the m i d p o i n t i n t i m e . These s y m m e t r i c a l windows, s u i t e d f o r e n g i n e e r i n g p u r p o s e s , a r e implemented i n the power mode which i s d i r e c t l y r e l a t e d t o the magnitude mode, and so they a r e a l s o a p p l i c a b l e t o the magnitude mode. F i g u r e 4A shows one such example of a s y m m e t r i c a l f u n c t i o n , the 3-term B l a c k m a n - H a r r i s f u n c t i o n , and F i g . 4B shows the c o r r e s p o n d i n g adapted window. These windows both e x t e n d i n the time domain from t = 0 t o t = T, and a r e s e t t o z e r o o u t s i d e t h i s range. When s y m m e t r i c a l f u n c t i o n s , or magnitude-type windows, ar e a p p l i e d and the r e s u l t s p r e s e n t e d i n the a b s o r p t i o n mode, the l i n e s h a p e s t y p i c a l l y show l a r g e n e g a t i v e i n t e n s i t i e s and a u x i l i a r y maxima. F i g u r e 4C shows the a b s o r p t i o n l i n e s h a p e , c a l c u l a t e d w i t h T/ r = 1.0, r e s u l t i n g from windowing w i t h the magnitude-type 3-term B l a c k m a n - H a r r i s window of F i g . 4A. The l a r g e l o b e s a r e u n d e s i r a b l e s i n c e the p o s i t i v e ones can be 20 -1 0 I * - 4 - 1 0 2 raxoutxcv (at) mourner (*<) F i g u r e 4. Window t y p e s and r e s u l t i n g a b s o r p t i o n and magnitude l i n e s h a p e s . A : Magnitude-type window. B : Absorp-t i o n - t y p e window. C : A b s o r p t i o n l i n e s h a p e due t o window i n A. D : A b s o r p t i o n l i n e s h a p e due t o window i n B. E : Magnitude l i n e s h a p e due t o window i n A. F : Magnitude l i n e s h a p e due t o window i n B. 21 m i s t a k e n as s p e c t r a l l i n e s , and the n e g a t i v e ones can d i m i n i s h or t o t a l l y c o n c e a l an a c t u a l peak. However windows c o n s i s t i n g of the l a s t h a l f of a s y m m e t r i c a l window g i v e a b s o r p t i o n l i n e s h a p e s f r e e of l a r g e s i d e l o b e s and n e g a t i v e i n t e n s i t i e s . These windows are d e f i n e d such t h a t the c e n t r a l p o i n t of a s y m m e t r i c a l window i s now l o c a t e d a t t = 0 and t h e end of the window i s a t t = T, and w i l l be r e f e r r e d t o as a b s o r p t i o n - t y p e windows. For example, F i g . 4B shows the window f o r a b s o r p t i o n s t u d i e s d e r i v e d from the 3-term B l a c k m a n - H a r r i s window of F i g . 4A. The a p o d i z e d a b s o r p t i o n spectrum r e s u l t i n g from the windowing w i t h the a b s o r p t i o n - t y p e window of F i g . 4B i s g i v e n i n F i g . 4D. Comparison of F i g . 4C w i t h F i g . 4D shows the o b v i o u s improvement. A l t h o u g h the l i n e s h a p e i s wider a t 50% of the peak h e i g h t w i t h the a b s o r p t i o n - t y p e window, i t i s f r e e from l a r g e l o b e s and i s b e t t e r than the unapodized l i n e s h a p e shown i n the c e n t e r of F i g . 2. Thus t h e window shape i n F i g . 4B i s used f o r the a b s o r p t i o n s t u d i e s . Comparison of the a p o d i z e d l i n e s h a p e i n F i g . 4D t o the unapodized l i n e s h a p e s i n F i g . 2 shows the improvement due t o windowing. I t i s known i n FT-NMR t h a t windows of t h e shape used f o r the s e a b s o r p t i o n mode s t u d i e s , the a b s o r p t i o n - t y p e windows, can be a p p l i e d t o improve the SNR [ 1 5 ] , Windows w i t h the shape of an i n c r e a s i n g e x p o n e n t i a l [15,52] a r e u t i l i z e d f o r improvement of the r e s o l u t i o n of a b s o r p t i o n l i n e s h a p e s , t h a t i s t o make narrower l i n e s h a p e s . Windows of the shape used f o r magnitude mode 22 s t u d i e s , t h o s e t h a t a r e s y m m e t r i c a l , have a l s o been used f o r a b s o r p t i o n mode s t u d i e s [ 4 2 ] . These c r e a t e a narrower l i n e s h a p e than the a b s o r p t i o n - t y p e windows used i n t h i s s t u d y , as was shown by comparing F i g s . 4C and 4D; however the e n s u i n g l a r g e n e g a t i v e l o b e s a r e u n a c c e p t a b l e as they i n t e r f e r e w i t h a d j a c e n t peaks. In c o n t r a s t t o the a b s o r p t i o n r e s u l t s , the a b s o r p t i o n - t y p e window i s not v e r y e f f e c t i v e f o r the magnitude mode. The windows used f o r the magnitude s t u d i e s , as shown i n F i g . 4A, produce narrow magnitude l i n e s h a p e s as seen i n F i g . 4E. However, when an a b s o r p t i o n type window, as i n F i g . 4B, i s a p p l i e d the r e s u l t i n g magnitude l i n e s h a p e has a h i g h e r s k i r t , shown i n F i g . 4F. Thus the adapted window shape i s not n e c e s s a r y f o r the magnitude mode s t u d i e s . Comparison of F i g . 4E w i t h the c e n t e r c u r v e i n F i g . 3 shows the e f f e c t of a p o d i z a t i o n . B e s i d e s the 3-term B l a c k m a n - H a r r i s window, v a r i o u s o t h e r windows w i t h both the a b s o r p t i o n type and magnitude type shapes were a p p l i e d f o r the a b s o r p t i o n and magnitude modes. These windows i n c l u d e d the 4-term B l a c k m a n - H a r r i s , G a u s s i a n , Hamming, Hanning a=2, and t r i a n g l e . The same t r e n d s as d e s c r i b e d above r e s u l t e d f o r t h e s e windows. I f an a b s o r p t i o n - t y p e window produced a v e r y smooth a b s o r p t i o n l i n e s h a p e , the a b s o r p t i o n r e s u l t , due t o the c o r r e s p o n d i n g magnitude — type window had e x t r e m e l y l a r g e l o b e s , as i n F i g . 4C. For a b s o r p t i o n - t y p e windows which produced o n l y s l i g h t l y improved a b s o r p t i o n l i n e s h a p e s , the c o r r e s p o n d i n g magnitude-type window produced 23 a b s o r p t i o n l i n e s h a p e s w i t h s m a l l e r l o b e s . The a b s o r p t i o n l i n e s h a p e due t o the a b s o r p t i o n - t y p e 3-term B l a c k m a n - H a r r i s window was smooth and q u i t e narrow, F i g . 4D, but the a b s o r p t i o n l i n e s h a p e due t o magnitude-type shape of t h i s window had a p o s i t i v e l o b e t h a t was 10% of the peak h e i g h t and an e x t r e m e l y l a r g e n e g a t i v e l o b e 60% of the peak h e i g h t , F i g . 4C. T h i s was a case of a g r e a t l y improved l i n e s h a p e due t o the a b s o r p t i o n - t y p e window, and a s e v e r e l y d i s t o r t e d shape due t o the c o r r e s p o n d i n g magnitude-type window. The a b s o r p t i o n - t y p e t r i a n g l e window d i d not improve the l i n e s h a p e as w e l l as d i d the a b s o r p t i o n - t y p e 3-term B l a c k m a n - H a r r i s window, and the magnitude-type t r i a n g l e window d i d not cause such extreme s i d e l o b e s . The p o s i t i v e s i d e l o b e was about 1% of the peak h e i g h t , and the n e g a t i v e s i d e l o b e was about 40% of the peak h e i g h t . The windows chosen f o r a b s o r p t i o n mode s t u d i e s a r e those w i t h a g e n e r a l shape as shown i n F i g . 4B, and the windows used f o r the magnitude mode s t u d i e s a r e s y m m e t r i c a l , as shown i n F i g . 4A. The p r i n c i p l e purpose of a p o d i z i n g the s p e c t r a i s t o produce a l i n e s h a p e which g i v e s t h e l e a s t i n t e r f e r e n c e i n t h e d e t e c t i o n of a d j a c e n t peaks. The l i n e s h a p e s r e s u l t i n g from the a p p l i c a t i o n of windows were examined f o r t h e i r w i d t h s a t s p e c i f i c f r a c t i o n s of t h e i r peak h e i g h t s . They were a l s o examined f o r the s i z e and p o s i t i o n of t h e i r h i g h e s t s i d e l o b e s w i t h the a b s o l u t e v a l u e s of the s i d e l o b e s b e i n g i n c l u d e d . The c h o i c e of the most advantageous window depended upon the window which produced the 24 narrowest l i n e s h a p e w i t h s i d e l o b e s below the dynamic range r e q u i r e d . For example, i f the r a t i o of the s i z e of the h i g h e s t peak t o the l a r g e s t t o l e r a b l e s i d e l o b e was 1000:1, then the window which produced the narrowest s i d e l o b e - f r e e l i n e s h a p e a t 0.1% of the peak h e i g h t would be the window of c h o i c e . T h i s would a l l o w a peak the s i z e of the h i g h e s t peak t o be d e t e c t e d w i t h , a t most, a 0.1% e r r o r i n peak h e i g h t . I f a s m a l l e r dynamic range were r e q u i r e d , f o r example 100:1, t h e s e h a l f w i d t h s have a l s o been t a b u l a t e d , as have t h o s e f o r a l a r g e dynamic range up t o 10,000:1. The w i d t h s were a l s o r e c o r d e d a t 50% and 10% of the peak h e i g h t f o r a complete p i c t u r e of the l i n e s h a p e s . The amount of damping of the time-domain s i g n a l can not be p r e c i s e l y c o n t r o l l e d , thus a window w i t h good r e s u l t s over a wide range of T/ T v a l u e s i s d e s i r e d . When a l i n e s h a p e i s a p o d i z e d , i t s peak i n t e n s i t y d e c r e a s e s . For a l l of the i l l u s t r a t i o n s , the peaks have been n o r m a l i z e d . 2.3 A b s o r p t i o n Mode R e s u l t s and D i s c u s s i o n The windows, w ( t ) , used f o r the a b s o r p t i o n s t u d i e s a r e l i s t e d i n T a b l e I . They were implemented as d e s c r i b e d i n the p r e v i o u s s e c t i o n , and the r e s u l t s of the windows f o r a b s o r p t i o n a r e assembled i n T a b l e I I . The a p o d i z e d s p e c t r a a r e e v a l u a t e d a c c o r d i n g t o t h e i r w i d t h s a t s p e c i f i c f r a c t i o n s of t h e i r peak h e i g h t s . The d e s i r e d window i s t h e one which produces the n a r r o w e s t l i n e s h a p e ; t h a t i s , i t has the l o w e s t h a l f w i d t h v a l u e 25 TABLE I. Window functions, w(t), for absorption. Window Analytical form, 0<t<T Rectangle Blackman-Harris, 3-term Blackman-Harris, 4-term F i l l e r DO.24 F i l l e r E0.13 F i l l e r E0.20 Gaussian Hamming Hanning Kaiser-Bessel Kaiser-Bessel, 4-term Noest-Kort* Norton-Beer F3 Triangle 1 .0 . 42323+. 49755cos(*t/T) + .07922cos(2irt/T) .35875+.48829cos(»t/T)+.14128cos(2»t/T) + .01 l68cos(3irt/T) [cos(#t/2T) + .24cos(3*t/2T)]/1 .24 [ 1 + 1.13cos(*t/T) + .l3cos(2*t/T)]/2.26 [1+1,2cos(*t/T)+.2cos(2*t/T)]/2.4 2 exp[-0.5(3.5t/T) ] .54+.46cos(wt/T) a cos (»t/2T) where a=2,4 2 0.5 I.[3.5ir{l .0-(t/T) } ]/I 0[3.5*] where I.fx] is a zero-order modified Bessel function .40243+.49804cos(*t/T) + .09831 cos(2*t/T) +.00l22cos(3»t/T) A C where A=.260+.520cos(*t) + .2l9cos(2«-t) 2 C=.5exp[t/r - ( t / r ) /(41n2)] 2 2 2 4 .045335+.554883(1-(t/T) ) +.399782(1-(t/T) ) 1.0-t/T *The equation for r «1.25 from Ref.[45] is not employed as i t produces poorer results than those depicted in this work. 26 TABLE I I . A p o d i z e d a b s o r p t i o n s p e c t r a h a l f w i d t h s . H a l f w i d t h a t 50% 10% 1% 0.1% 0.01% T/T Window of Peak H e i g h t R e c t a n g l e 0.29 (1 .3) (13 . 2 )(>32 ) (>32 ) B l a c k m a n - H a r r i s , 3-term 0.56 0 .97 1 .3 1 . 4 (13 . 1 ) B l a c k m a n - H a r r i s , 4-term 0.66 1 . 2 1 .6 1 . 8 1.9 F i l l e r E 0.13 0.55 0 .94 1 . 2 ( 2 . 8 ) ( 5 . 8 ) F i l l e r E 0 . 2 0 0.58 1 . 0 1 .3 1 . 4 ( 4 . 8 ) G a u s s i a n 0.65 1 . 2 1 .7 2 . 1 ( 8 . 2 ) Hanning, a=2 0.50 0 .81 ( .97) ( 2 . 9) ( 6 . 8 ) Hanning, a=4. 0.64 1 . 1 1 .4 (1 . 5) ( 2.5) K a i s e r - B e s s e l 0.64 1 . 1 1 .5 1 . 7 1 . 8 K a i s e r - B e s s e l , 4 term 0.59 1 . 0 1 .3 1 . 5 ( 2.9) N o e s t - K o r t 0.94 1 .3 1 .4 (19. 6)(>32 ) Norton-Beer F3 0.48 0 .81 1 . 0 ( 1 2 . 2)(>32 ) R e c t a n g l e 0.32 (1 .3) ( 1 0 . 2 )(>32 ) (>32 ) B l a c k m a n - H a r r i s , 3-term 0.61 1 . 1 2 . 0 ( 5 . 5) ( 2 4 . 1 ) B l a c k m a n - H a r r i s , 4-term 0.66 1 . 2 2 . 2 6. 2 ( 2 0 . 2 ) F i l l e r DO.24 0.55 0 .97 1 . 8 ( 5 . 7) ( 2 0 . 6 ) F i l l e r E 0.13 0.58 1 . 0 2 . 0 ( 5 . 8 ) ( 1 8 . 0 ) F i l l e r E 0 . 2 0 0.62 1 . 1 2 . 0 5. 8 ( 1 8.7) 27 TABLE II contd. Gaussian 0. 69 1 . ,3 Hamming 0. 49 0. ,85 Hanning, a=2 0. 53 0. ,91 Hanning, a=4 0. 68 1 . .2K a i s e r - B e s s e l 0. 64 1 . . 1 K a i s e r - B e s s e l , 4--term 0. 63 1 , . 1 Noest-Kort 0. 94 1 , .3Norton-Beer F3 0. 52 0, .94 T r i a n g l e 0. 48 0, .84 Rectangle 0. 35 0, .55 Blackman-Harris, 3- term 0. 61 1 , . 1 Blackman-Harris, 4- term 0. 70 1 , .3 F i l l e r E0.13 0. 63 1 .2 F i l l e r E0.20 0. 66 1 .2 Hanning, a=2 0. 57 1 .0 Hanning, a=4 0. 72 1 .3 K a i s e r - B e s s e l 0. 72 1 .3 K a i s e r - B e s s e l , 4--term 0. 68 1 .3 Noest-Kort 0. 94 1 .3 Norton-Beer F3 0. 56 1 . 1 Rectangle 0. 42 (1 .3) Blackman-Harris, 3- term 0. 74 1 .6 Blackman-Harri s, 4-•term 0, 84 1 .7 2.3 6.3 (23.1) (2.3) (15.2)(>32 ) (1.9) (5.1) (16.1) 2.2 6.0 19.6 2.1 6.0 19.7 2.0 5.8 (19.2) 1.5 (18.6)(>32 ) 1.7 (10.3) (>32 ) (2.8) (10.5)(>32 ) (8.2)(>32 ) (>32 ) 2.8 (8.4) (32.0) 3.0 9.1 >32 2.8 8.3 30.1 2.8 8.5 31.4 2.7 7.9 26.0 3.0 8.8 >32 3.0 8.9 >32 2.9 8.6 (>32 ) 1.5 (13.7)(>32 ) 2.5 (11.2)(>32 ) (6.2)(>32 ) (>32 ) 4.3 13.3 (>32 ) 4.5 14.0 >32 28 TABLE I I . c o n t d . F i l l e r E0.13 0.72 1.5 4.2 13.1 >32 F i l l e r E0.20 0.75 1.6 4.3 13.3 >32 Hanning, a=2 0.66 1 .4 4.0 12.5 >32 Hanning, a=4 0.81 1 .6 4.5 13.8 >32 K a i s e r - B e s s e l 0.81 1.7 4.5 13.8 >32 K a i s e r - B e s s e l , 4--term 0.77 1.6 4.4 13.5 (>32 ) N o e s t - K o r t 0.94 1 .4 1 .7 (4.8)(>32 ) Norton-Beer F3 0.65 1.4 4.2 (13.3)(>32 ) 3.0 R e c t a n g l e 0.52 1.5 (5.4) (28.1)(>32 ) B l a c k m a n - H a r r i s , 3-term 0.84 1.9 5.7 18.1 >32 B l a c k m a n - H a r r i s , 4-term 0.93 2.1 5.9 18.9 >32 F i l l e r DO.24 0.78 1.8 5.6 17.6 >32 F i l l e r E0.13 0.81 1.9 5.6 17.8 >32 F i l l e r E0.20 0.84 1.9 5.7 18.1 >32 G a u s s i a n 0.94 2.1 6.0 19.0 >32 Hamming 0.72 1.8 5.4 (18.1)(>32 ) Hanning, a=2 0.75 1 .8 5.4 17.2 >32 Hanning, a=4 0.91 2.0 5.9 18.6 >32 K a i s e r - B e s s e l 0.91 2.0 5.9 18.7 >32 K a i s e r - B e s s e l , 4 -term 0.87 2.0 5.8 18.3 (>32 ) N o e s t - K o r t 0.96 1.5 2.0 2.3 (8. 7) Norton-Beer F3 0.75 1 .8 5.5 (17.3)(>32 ) T r i a n g l e 0.72 2.0 6.5 20.8 (>32 ) ( ) denote s i d e l o b e s a t s p e c i f i e d peak h e i g h t . 29 i n T a b l e I I f o r the r e q u i r e d c o n d i t i o n s . These v a l u e s a r e i n u n i t s of 1/T, Eq. 5, away from the c e n t r a l maximum. T h i s i s the same s c a l e as used i n the f i g u r e s . When t r e n d s i n d i c a t e d a poor window, the h a l f w i d t h s f o r t h a t window were not c a l c u l a t e d f o r every damping v a l u e . A s i d e from the w i d t h of the l i n e , i t s a c t u a l shape i s a l s o of i m p o r t a n c e . D e t e c t i o n of an a d j a c e n t , s m a l l peak would be d i f f i c u l t i f the p r i n c i p l e peak had l a r g e s i d e l o b e s , even i f t h e s e l o b e s were c o n t a i n e d i n a narrow e n v e l o p e . For example, from T a b l e I I i t i s apparent t h a t the Hanning a=2, and the K a i s e r - B e s s e l windows b o t h produce l i n e s h a p e s of a p p r o x i m a t e l y e q u i v a l e n t w i d t h when T/ r = 0.5. T h i s can a l s o be o b s e r v e d i n F i g . 5 which p r e s e n t s the l i n e s h a p e s i n a l o g a r i t h m i c s c a l e . A l l -4 • of the v a l u e s below 10 , i n c l u d i n g n e g a t i v e s i d e l o b e s , have been -4 s e t t o 10 f o r the l o g a r i t h m i c f i g u r e s . The l i n e s h a p e due t o the r e c t a n g l e window i s shown f o r r e f e r e n c e f o r the unapodized case (which c o r r e s p o n d s t o the l i n e s h a p e s i n F i g . 2 ) . S i n c e a b s o r p t i o n l i n e s h a p e s a r e s y m m e t r i c a l , o n l y the h i g h f r e q u e n c y h a l f i s shown. Note t h a t the l i n e s h a p e due t o the Hanning a=2 window has many s i d e l o b e s , the l a r g e s t of which i s a l m o s t 2% of the peak h e i g h t . T h i s would i n t e r f e r e g r e a t l y w i t h the d e t e c t i o n of an a d j a c e n t peak, p a r t i c u l a r l y a s m a l l one. The numbers i n p a r e n t h e s i s i n T a b l e I I i n d i c a t e s i d e l o b e s h i g h e r than the g i v e n peak h e i g h t s . Thus the numbers i n the 1%, 0.1% and 0.01% columns of T a b l e I I a r e e n c l o s e d i n p a r e n t h e s e s f o r the case of the 30 A B S O R P T I O N FREQUENCY (Af) F i g u r e 5. Comparison of l i n e s h a p e s . s u i t a b l e and u n f a v o r a b l e a b s o r p t i o n Hanning a=2 window when T/ r = 0.5. In c o n t r a s t , the K a i s e r -B e s s e l window produces a l i n e s h a p e w i t h o u t s i d e l o b e s i n the dynamic range examined, thus none of i t s v a l u e s a r e i n p a r e n t h e s e s . The p r e s ence of s i d e l o b e s i s q u i t e common f o r the windows i n Ta b l e I I . S i n c e windows w i t h t h i s p r o p e r t y s h o u l d not be used, the s e v a l u e s have been e n c l o s e d i n p a r e n t h e s e s . A l l of the wi d t h s which a r e not i n p a r e n t h e s e s i n d i c a t e a window which c o u l d be used w i t h the a s s u r a n c e t h a t a u x i l i a r y maxima a r e below t h e r e q u i r e d range. D e t a i l s of the h e i g h t s and p o s i t i o n s of t h e s i d e l o b e s a r e l i s t e d i n Appendix B. S i n c e windows which produce l i n e s w i t h s i d e l o b e s a r e u n f a v o r a b l e , t h e s e can be e l i m i n a t e d from the quest f o r s a t i s f a c t o r y windows. Windows which c r e a t e l i n e s h a p e s w i t h n o t i c e a b l e s i d e l o b e s i n c l u d e the Hamming, Hanning a=2, and t h e t r i a n g l e . These produce s i d e l o b e s h i g h e r than 1% of the peak h e i g h t . From the r e m a i n i n g windows, c e r t a i n ones produce the b e s t r e s u l t s . The c h o i c e of the b e s t window i s dependent upon t h e r e q u i r e d c o n d i t i o n s of damping and dynamic range. Some e f f e c t i v e windows f o r a b s o r p t i o n mode a n a l y s i s a r e the N o e s t - K o r t , F i l l e r E0.20 and K a i s e r - B e s s e l . These windows a l l y i e l d narrow l i n e w i d t h s a c c o r d i n g t o the summary of r e s u l t s i n Ta b l e I I . Depending on the amount of damping and the dynamic range r e q u i r e d , c e r t a i n of th e s e windows a r e more a p p l i c a b l e . 32 I f a modest dynamic range t o o n l y 1% of the peak h e i g h t i s s u f f i c i e n t , the N o e s t - K o r t i s the most e f f i c i e n t window f o r s i g n a l s of any amount of damping. From the 1% column i n T a b l e I I , i t i s c l e a r t h a t the h a l f w i d t h s f o r the N o e s t - K o r t window are the l o w e s t f o r e v e r y damping except T/ r = 0 . 0 . However the d i f f e r e n c e f o r the undamped case between the N o e s t - K o r t h a l f w i d t h and the o t h e r h a l f w i d t h s i s n e g l i g i b l e . As the amount of damping i n c r e a s e s , the peak w i d t h s of most of the f r e q u e n c y s p e c t r a i n c r e a s e . In c o n t r a s t , the w i d t h s of the l i n e s h a p e s due t o the N o e s t - K o r t window remain p r a c t i c a l l y c o n s t a n t a t 1% of the peak h e i g h t f o r a l l dampings. For T / r v a l u e s of 1 and l a r g e r , i t i s c l e a r t h a t t h i s window i s f a r s u p e r i o r t o the o t h e r s i n r e t a i n i n g a f a i r l y narrow peak down t o 1% of the peak h e i g h t . T h i s i s r e a d i l y seen by comparing the v a l u e s i n T a b l e I I , and i s a l s o i l l u s t r a t e d i n F i g s . 6 t o 8 w i t h the K a i s e r - B e s s e l and r e c t a n g l e windows f o r c o m p a r i s o n . The g e n e r a l l i n e s h a p e due t o the N oest-K o r t window has the p e c u l i a r q u a l i t y of becoming narrower as the amount of damping i s i n c r e a s e d w h i l e the o t h e r l i n e s h a p e s i n c r e a s e i n w i d t h . T h i s i s most o b s e r v a b l e i n the 0 . 1 % column i n T a b l e I I . For the N o e s t - K o r t window, as the amount of damping i n c r e a s e s the h a l f w i d t h s a c t u a l l y d e c r e a s e which i s n o t i c e a b l e from F i g s . 7 and 8 . T h i s i s the r e v e r s e of the b e h a v i o r due t o the o t h e r windows : w i t h the K a i s e r - B e s s e l as an example, note how the l i n e s h a p e widens from a moderate amount of damping, F i g . 6 , t o a h i g h amount of damping, F i g . 7 . Thus the N o e s t - K o r t 33 A B S O R P T I O N FREQUENCY (A f ) Figure 6. Absorption lineshapes due to recommended windows and moderate damping. A B S O R P T I O N 0.0 4.0 8.0 12.0 16.0 20.0 24.0 28.0 32.0 FREQUENCY ( A f ) Figure 7. Absorption l i n e s h a p e s due to recommended windows and high damping. A B S O R P T I O N to FREQUENCY ( A f ) F i g u r e 8. A b s o r p t i o n l i n e s h a p e s due t o recommended windows complete damping. o u t p e r f o r m s the o t h e r windows, e s p e c i a l l y f o r h i g h l y damped s i g n a l s . The N o e s t - K o r t has a v e r y s i m p l e parameter w h i c h , as shown i n T a b l e I , matches the damping of the time-domain s i g n a l . I f i t i s not d e s i r a b l e t o a d j u s t a window at a l l , a n o t h e r window would have t o be chosen f o r t h i s dynamic range. The Norton-Beer F3 has c o n s i s t e n t l y narrow l i n e s h a p e s . As seen from T a b l e I I , i t i s not as e f f e c t i v e as the N o e s t - K o r t and o n l y m a r g i n a l l y b e t t e r than o t h e r s , such as the F i l l e r E0.13, however i t does s u f f i c e f o r the e n t i r e range of dampings. I f a l a r g e r dynamic range i s r e q u i r e d , f o r example down t o 0.1% of the peak h e i g h t , a p r e c i s e c h o i c e f o r the b e s t window i s not as c l e a r c u t as f o r the 1% c a s e . T a k i n g the window p r o d u c i n g the s m a l l e s t v a l u e i n the 0.1% column of T a b l e I I as the best window y i e l d s an abundance of c h o i c e s . For the undamped c a s e , b o t h the 3-term B l a c k m a n - H a r r i s and F i l l e r E0.20 r e s u l t s a r e i d e n t i c a l l y narrow. At T/ r = 0.5, the F i l l e r E0.20 and K a i s e r -B e s s e l windows b o t h produce the narrowest w i d t h s a t 0.1% of the peak h e i g h t . For T/r = 1.0 and 2.0, the Hanning a=2 p r o v i d e s the b e s t p e r f o r m a n c e , and f o r a t o t a l l y damped s i g n a l the Noest-K o r t i s e a s i l y the b e s t c h o i c e . I n s t e a d of recommending a d i f f e r e n t window f o r each s p e c i f i c c o m b i n a t i o n of damping and dynamic range, s l i g h t compromises are made i n o r d e r t o l i m i t the number of c h o i c e s . C l o s e r e x a m i n a t i o n of T a b l e I I r e v e a l s t h a t some windows, w h i l e not the v e r y b e s t 37 f o r a p a r t i c u l a r c a s e , produce h a l f w i d t h s which a r e v e r y c l o s e t o the best c h o i c e s l i s t e d . Two windows y i e l d e q u a l l y a c c e p t a b l e r e s u l t s . Both the F i l l e r E0.20 and K a i s e r - B e s s e l windows produce s i m i l a r l y smooth l i n e s h a p e s down t o 0.1% of the peak h e i g h t which are r e l a t i v e l y narrow, p a r t i c u l a r l y f o r the lower damping v a l u e s . The K a i s e r - B e s s e l r e s u l t can be seen t o be v e r y e f f e c t i v e i n F i g . 6, but becomes l e s s e f f e c t i v e f o r h i g h e r dampings as shown i n F i g s . 7 and 8. The r e s u l t s due t o the F i l l e r E0.20 a r e v i r t u a l l y i d e n t i c a l t o t h o s e f o r the K a i s e r - B e s s e l , thus f o r c l a r i t y o n l y the K a i s e r - B e s s e l has been i l l u s t r a t e d . E x a m i n a t i o n of T a b l e I I shows the s i m i l a r i t y of t h e s e two windows, f o r example when T / r = 3.0, t h e r e i s o n l y a 1% d i f f e r e n c e i n t h e i r h a l f w i d t h s a t 0.1% of t h e i r peak h e i g h t s . Only the K a i s e r - B e s s e l window produces a smooth l i n e s h a p e down t o 0.01% of the peak h e i g h t f o r both T/ r = 0.0 and 0.5. T h i s i s shown i n F i g . 5, and can a l s o be o b s e r v e d by e xamining the 0.01% column i n T a b l e I I . A smooth l i n e s h a p e r e s u l t s from t h i s window f o r a l l of the dampings s t u d i e d , w i t h the l i n e s h a p e becoming i n c r e a s i n g l y w i d e r as the amount of damping i n c r e a s e s . When T/ r = 1.0, F i g . 6, and h i g h e r , F i g s . 7 and 8, the s k i r t of the l i n e s h a p e does not r e a c h 0.01% of the peak h e i g h t i n the f r e q u e n c y range examined. However i t s l i n e s h a p e i s v i r t u a l l y c o n s t a n t , t h u s i t would not g r e a t l y d i s t o r t a d j a c e n t peaks. The K a i s e r - B e s s e l window i s e f f e c t i v e f o r a l a r g e dynamic range, n o t a b l y f o r s i g n a l s w i t h low amounts of damping, T/ r = 0.0 and 38 0.5. 2.4 Magnitude Mode R e s u l t s and D i s c u s s i o n The windows used f o r a p o d i z i n g the magnitude s p e c t r a are p r e s e n t e d i n T a b l e I I I . Each of t h e s e windows i s s y m m e t r i c a l i n shape w i t h i t s maximum i n the c e n t e r , t a p e r i n g t o z e r o a t t = 0 and t = T. Note t h a t the a n a l y t i c a l e x p r e s s i o n s f o r t h e s e windows a r e s i m i l a r t o those f o r the a b s o r p t i o n windows, however magnitude windows are s y m m e t r i c a l w h i l e the a b s o r p t i o n windows are h a l f of t h e s e shapes. The c r i t e r i a f o r a good window f o r the magnitude mode ar e the same as those f o r the a b s o r p t i o n mode. The windows i n T a b l e I I I have been examined f o r the r e s u l t i n g magnitude l i n e s h a p e s , and the d a t a o b t a i n e d i s d i s p l a y e d i n T a b l e IV. As f o r the a b s o r p t i o n s t u d i e s , the l i n e s h a p e s w i t h s i d e l o b e s a r e i n d i c a t e d w i t h p a r e n t h e s e s and e x a c t v a l u e s f o r the s i z e s and p o s i t i o n s of the s i d e l o b e s a r e found i n Appendix B. R e s u l t s of p o o r e r windows ar e not p r e s e n t e d f o r a l l of the damping v a l u e s . P r e f e r a b l y , a p o d i z e d s p e c t r a s h o u l d be narrow w i t h s i d e l o b e s lower than a d e s i r e d l i m i t . W i t h the r e c t a n g l e window as the r e f e r e n c e f o r no a p o d i z a t i o n , T a b l e IV shows t h a t w e l l known windows, such as the t r i a n g l e , o n l y produce s l i g h t l y improved s p e c t r a compared t o the performance of o t h e r , more a n a l y t i c a l l y complex windows. Depending upon the amount of damping and the dynamic range r e q u i r e d , c e r t a i n of t h e windows a r e p r e f e r r e d over 3 9 TABLE III. Window functions, w(t), for magnitude. Window Analytical form, 0<t<T Rectangle Blackman-Harris, 3-term Blackman-Harris, 4-term Gaussian Hamming Hanning Kaiser-Bessel Kaiser-Bessel, 4-term Noest-Kort Triangle 1 .0 . 42323-. 49755cos(2trt/T) + .07922cos(4»t/T) .35875-.48829cos(2*t/T)+.14128cos(4»t/T) -.01168cos(6*t/T) 2 exp[-0.5(3.5(2t-T)/T) ] •54-.46cos(2»t/T) a sin Ot/T) where a=2,4 as for absorption, but substituting 2t-T for t .40243-.49804cos(2*t/T) + .09831 cos(4»t/T) -.00122cos(6*t/T) M C where M=.430+.843cos(»t) -.239cos(2«t)+.043cos(5*t)-.602U Osame as for absorption (Table I) U=1-4t for 0<t<l/4 =3-4t for 3/4<t< 1 «0 elsewhere 1- |2t-T|/T 40 TABLE IV. Apodized magnitude spectra halfwidths. Halfwidth at 50% 10% 1% 0.1% 0.01% T/r Window of Peak Height 0.0 Rectangle 0.59 (2.7) (31.2)(>32 ) (>32 ) Blackman-Harris, 3 term 1.1 1.9 2.6 2.8 (>32 ) Blackman-Harris, 4 term 1.3 2.3 3.1 3.6 3.8 Gaussian 1.3 2.3 3.3 4.1 (18.5) Hamming 0.90 1.5 1.9 (>32 ) (>32 ) Kaiser-Bessel 1.3 2.2 3.0 3.4 3.6 Kaiser-Bessel, 4 term 1.2 2.1 2.7 3.0 (5.7) Noest-Kort 1.2 1 .6 (22.10(>32 ) (>32 ) 0.5 Rectangle 0.60 (2.7) (31.4)(>32 ) (>32 ) Blac kman-Harr i s, 3 term 1 . 1 1.9 2.6 2.9 (>32 ) Blackman-Harris, 4 term 1.3 2.3 3.1 3.6 3.9 Gaussian 1.3 2.4 3.3 4.1 (20.4) Hamming 0.90 1.5 1.9 (>32 ) (>32 ) Hanning, a=2 1.0 1.6 (2.8) (6.6) (13.5) Hanning, a=4 1.3 2.2 2.8 (3.8) (6.6) Kaiser-Bessel 1.3 2.2 3.0 3.4 3.8 Kaiser-Bessel, 4 term 1.2 2.1 2.7 3.0 (5.7) Noest-Kort 1.2 1 .6 (22.1)(>32 ) (>32 ) Triangle 0.88 1.5 (5.3) (19.1)(>32 ) 41 TABLE IV. contd. Rectangle 0.62 (2.7) (31.5)(>32 ) (>32 ) Blackman-Harris, 3 term 1 .1 2.0 2.6 3.0 (>32 ) Blackman-Harris, 4 term 1.3 2.3 3.1 3.6 3.9 Gaussian 1.3 2.4 3.3 4.0 (22.5) Hamming 0.91 1.5 2.0 (>32 ) (>32 ) Kaiser-Bessel 1.3 2.2 3.0 3.4 3.9 Kaiser-Bessel, 4 term 1.2 2.1 2.7 3.1 (5.8) Rectangle 0.69 (3.8)(>32 ) (>32 ) (>32 ) Blackman-Harris, 3 term 1 . 1 2.0 2.6 3.2 (>32 ) Blackman-Harris, 4 term 1.3 2.3 3.2 3.7 4.6 Gaussian 1.3 2.4 3.4 4.0 (>32 ) Hamming 0.92 1.6 (5.6)(>32 ) (>32 ) Hanning, a=2 1.0 1 .7 3.6 (7.3) (14.7) Kaiser-Bessel 1.3 2.3 3.0 3.5 4.4 Kaiser-Bessel, 4 term 1 .2 2.1 2.8 3.4 (5.9) Rectangle 0.84 (5.5)(>32 ) (>32 ) (>32 ) Blackman-Harris, 3 term 1.2 2.0 2.7 3.8 (>32 ) Blackman-Harris, 4 term 1.3 2.3 3.2 3.8 5.0 Gaussian 1.3 2.4 3.4 4.0 (>32 ) Hamming 0.94 1.7 (9.5)(>32 ) (>32 ) Hanning, a=2 1.0 1.9 3.9 8.0 (15.8) Hanning, a=4 1 .3 2.3 3.3 4.9 7.6 Kaiser-Bessel 1.3 2.3 3.1 3.7 5.1 Kaiser-Bessel, 4 term 1.2 2.1 2.9 3.6 24.4 Triangle 0.94 1.7 (7.1) (23.2)(>32 ) ( ) denote sidelobes at specified peak height. 42 o t h e r s . Some c o n t r a s t i n g magnitude r e s u l t s a r e shown i n F i g . 9 w i t h a l o g a r i t h m i c a m p l i t u d e s c a l e , and low damping. S i n c e magnitude l i n e s h a p e s a r e s y m m e t r i c a l w i t h r e s p e c t t o the c e n t e r f r e q u e n c y , f , o n l y the h i g h f r e q u e n c y h a l f of the s p e c t r a a r e d i s p l a y e d . Comparison of the l i n e s h a p e s i n F i g . 9 r e v e a l s the consequence of a poor window c o n t r a s t e d w i t h a recommended window. The Hanning a=2 window i s a commonly employed window. W h i l e i t i s d e f i n i t e l y an improvement over no a p o d i z a t i o n , as r e p r e s e n t e d by the r e c t a n g l e r e s u l t , the K a i s e r - B e s s e l window produces a smoother l i n e s h a p e . The K a i s e r - B e s s e l window does broaden the l i n e s h a p e above 1% of the peak h e i g h t , when compared t o the Hanning a=2 window, however i t d r a m a t i c a l l y narrows the l i n e s h a p e below 1% of the peak h e i g h t . The l i n e s h a p e due t o the Hanning a=2 window i s not v e r y a c c e p t a b l e as i t has a s i d e l o b e l a r g e r than 2% of i t s peak h e i g h t , and so the v a l u e s f o r t h i s window a t T/ r = 0.5 a r e i n p a r e n t h e s e s f o r the 1%, 0.1% and 0.01% coulumns. F i g u r e 9 a l s o shows an e f f e c t of r e d u c i n g s i d e l o b e s : t h e l i n e s h a p e broadens a t 50% of the peak h e i g h t . The K a i s e r - B e s s e l window produces the l o w e s t s i d e l o b e s , then the Hanning a=2, and f i n a l l y the r e c t a n g l e window c r e a t e s the h i g h e s t s i d e l o b e s . The r e v e r s e o r d e r i s ob s e r v e d f o r the w i d t h s a t h a l f h e i g h t . T h i s i s a l s o v i s i b l e i n the 50% column of T a b l e IV. The r e c t a n g l e has the n a r r o w e s t peak a t 50% of the peak h e i g h t , then the Hanning a=2, and then the K a i s e r - B e s s e l . 43 M A G N I T U D E 32.0 FREQUENCY (Af) F i g u r e 9. Comparison of s u i t a b l e and u n f a v o r a b l e magnitude 1ineshapes. I f smoothing down t o o n l y 1% of the peak h e i g h t i s d e s i r e d , the Hamming window g i v e s the nar r o w e s t a p o d i z e d s p e c t r a f o r undamped t o m o d e r a t e l y damped s i g n a l s . For t h i s range of dampings, the o t h e r windows r e s u l t i n s p e c t r a which a r e a t l e a s t 30% wider which i s e v i d e n t i n the 1% column of T a b l e IV. T h i s i s a l s o apparent from F i g . 10 which shows the l i n e s h a p e s due moderate damping. As the damping i n c r e a s e s , the l i n e s h a p e due t o the Hamming window becomes much wider than the o t h e r s a t 1% of the peak h e i g h t . Note, i n F i g . 11 and by the p a r e n t h e s e s f o r the v a l u e s f o r t h i s window i n T a b l e IV, t h a t f o r h i g h l y damped s i g n a l s , the Hamming no l o n g e r reduces the l i n e s h a p e t o below 1% of the peak h e i g h t . For l a r g e dampings and a dynamic range of 100:1, the 3-term B l a c k m a n - H a r r i s window has the narrowest r e s u l t s . T h i s i s t r u e f o r T/ r = 2.0 and 3.0 as seen i n the 1% column of T a b l e IV, and i n F i g . 11 which i l l u s t r a t e s the l a t t e r c a s e . The r e s u l t s f o r the 3-term B l a c k m a n - H a r r i s window a r e not as good as t h o s e f o r the Hamming window a t low dampings, however they a r e adequate as shown i n F i g s . 10. I f a l i n e s h a p e w i t h s i d e l o b e s lower than 1% i s needed, the Hamming window w i l l not s u f f i c e , e s p e c i a l l y f o r h e a v i l y damped s i g n a l s . I f a dynamic range of 1000:1 i s n e c e s s a r y , the 3-term B l a c k m a n - H a r r i s window produces smooth, narrow magnitude l i n e s h a p e s . From T a b l e IV, i t i s r e a d i l y v i s i b l e t h a t t h i s window has the narrowest l i n e s h a p e s , f r e e from s i d e l o b e s , down to 0.1% of the peak h e i g h t . T h i s o c c u r s f o r e v e r y damping except 45 M A G N I T U D E T / t=1 .0 KAISER-BESSEL BLACKMAN-HARRIS, 3-TERM HAMMING ' <' \ l " \' '' \' 1 * , ' 4 /1 0.0 4.0 8.0 1 r 12.0 16.0 20.0 FREQUENCY (A f ) 24.0 F i g u r e 10. Magnitude l i n e s h a p e s due t o recommended windows and moderate damping. M A G N I T U D E FREQUENCY (Af) Figure 11. Magnitude lineshapes due to recommended windows and complete damping. T/ T = 3.0, however i t i s o n l y 5% wider than the narrowest c a s e . Thus the 3-term B l a c k m a n - H a r r i s window i s a good c h o i c e f o r the e n t i r e range of dampings s t u d i e d . I t s r e s u l t s a r e i l l u s t r a t e d f o r moderate damping i n F i g . 10, and h i g h damping i n F i g . 11. For an even g r e a t e r dynamic range, such as smoothing t o .01% of the peak h e i g h t , b oth the K a i s e r - B e s s e l and 4-term Blackman-H a r r i s produce the l o w e s t h a l f w i d t h v a l u e s f o r every damping v a l u e examined. These windows produce s i m i l a r r e s u l t s , as seen by comparing t h e i r l i n e w i d t h s i n T a b l e IV. T h e i r l i n e w i d t h s a t 0.01% of the peak h e i g h t are w i t h i n 6% of each o t h e r w i t h those due t o the K a i s e r - B e s s e l b e i n g narrower f o r a l l but one T/ r v a l u e . F i g u r e s 10, and 11 show t h a t the K a i s e r - B e s s e l window has a smooth l i n e s h a p e f o r a l a r g e dynamic range f o r a l l dampings. I t can be n o t e d from both F i g s . 10 and 11 t h a t the more the s i d e l o b e s a r e reduced, the wider the w i d t h a t 50% of the peak h e i g h t . The Hamming window produces the h i g h e s t s i d e l o b e s but a l s o the s m a l l e s t h a l f h e i g h t w i d t h . The K a i s e r - B e s s e l r e s u l t s a r e the o p p o s i t e w i t h the l o w e s t s i d e l o b e s and the w i d e s t w i d t h a t h a l f h e i g h t , w h i l e the 3-term B l a c k m a n - H a r r i s r e s u l t s a r e i n t e r m e d i a t e . The K a i s e r - B e s s e l window has a parameter which can be a l t e r e d . For t h i s s t u d y , a was s e t t o 3.5 t o g i v e a l a r g e dynamic range. T h i s v a l u e can be a l t e r e d , but the l i n e s h a p e changes. When a. = 1.5, the s i d e l o b e s a r e l a r g e w i t h the h i g h e s t b e i n g almost 2% of the peak h e i g h t , but the w i d t h a t h a l f h e i g h t 48 F i g u r e 12 . Magnitude l i n e s h a p e s r e s u l t i n g from v a r y i n g a i n the K a i s e r - B e s s e l window. i s q u i t e narrow as shown i n F i g . 12. Note t h a t as the a v a l u e i s i n c r e a s e d , the s i z e of the s i d e l o b e s d e c r e a s e s , but the w i d t h a t h a l f h e i g h t i n c r e a s e s . 2.5 C o n c l u s i o n The purpose f o r t h i s study i s t o d e t e r m i n e the most a p p r o p r i a t e window f u n c t i o n f o r use w i t h s i g n a l s of v a r i o u s dampings of the form Eq. 3. In c h e m i c a l s p e c t r o s c o p i e s the p r i n c i p l e v a l u e of a window i s t o l i m i t the i n t e r f e r e n c e from th e s k i r t of one peak i n the d e t e c t i o n of an a d j a c e n t peak. An e x a m i n a t i o n of the e x t e n s i v e l i t e r a t u r e on windowing showed t h a t the f r e q u e n c y l i n e s h a p e s due t o d i f f e r e n t windows v a r i e d c o n s i d e r a b l y i n t h e i r r e d u c t i o n of s i d e l o b e s . A f u r t h e r g e n e r a l c h a r a c t e r i s t i c was t h a t windows which produced the l o w e s t s k i r t s tended t o cause the peak t o be w i d e r a t h a l f of i t s h e i g h t . T a k i n g i n t o c o n s i d e r a t i o n b o t h damping and dynamic rang e , d i f f e r e n t windows emerge as b e i n g the most e f f e c t i v e . The e x t e n t of the damping of the time-domain s i g n a l can not be p r e c i s e l y c o n t r o l l e d ; t h e r e f o r e a good window s h o u l d p e r f o r m w e l l f o r a range of damping v a l u e s . One c o n s i d e r a t i o n when c h o o s i n g a window i s the dynamic range, the r a t i o of the l a r g e s t peak h e i g h t t o t h e h i g h e s t s i d e l o b e . A g e n e r a l p r i n c i p l e f o r s e l e c t i n g a p a r t i c u l a r window i s t o choose one which s u f f i c e s j u s t f o r the dynamic range of i n t e r e s t . O p t i n g f o r a window w i t h a g r e a t e r dynamic range i s of 50 no advantage as i t i n c r e a s e s the w i d t h a t 50% of the peak h e i g h t . The o t h e r p o i n t t o be taken i n t o c o n s i d e r a t i o n i s the amount of damping of the time-domain f u n c t i o n , i f t h i s can be c o n t r o l l e d . I f n o t , i t i s b e s t t o s e l e c t a window which s u f f i c e s f o r the range from undamped t o t o t a l l y damped s i g n a l s . A few windows f u l f i l l the c r i t e r i a of e l i m i n a t i n g s i d e l o b e s t o a s p e c i f i e d s i z e w h i l e r e t a i n i n g a narrow l i n e s h a p e f o r a l a r g e range of dampings. Most o t h e r s a r e not comparable. Commonly employed windows, such as the Hanning and G a u s s i a n , do not e l i m i n a t e s i d e l o b e s as w e l l as the windows recommended i n t h i s c h a p t e r , as i l l u s t r a t e d i n F i g s . 5 and 9. From the s e f i g u r e s , i t i s a l s o apparent t h a t a p o d i z a t i o n of magnitude l i n e s h a p e s w i t h t h e s e windows has a g r e a t e r e f f e c t than a p o d i z a t i o n of a b s o r p t i o n l i n e s h a p e s . A c l e a r i l l u s t r a t i o n i s the K a i s e r - B e s s e l window which s h a r p l y reduces the magnitude s i d e l o b e s , F i g . 9, but o n l y g r a d u a l l y reduces the a b s o r p t i o n l i n e s h a p e , F i g . 5. The window of c h o i c e i s the one which p r o v i d e s the n a r r o w e s t l i n e s h a p e f r e e from s i d e l o b e s f o r each g i v e n damping and dynamic range c o m b i n a t i o n . That i s , i t has the l o w e s t h a l f w i d t h v a l u e i n T a b l e I I , f o r a b s o r p t i o n s t u d i e s , or T a b l e IV, f o r magnitude s t u d i e s , f o r a s p e c i f i e d s e t of c o n d i t i o n s . E x a m i n a t i o n of t h e s e t a b l e s i n d i c a t e t h a t the b e s t window v a r i e s from case t o c a s e . However t h e r e a r e o f t e n a few windows w i t h v e r y s i m i l a r n a r r o w i n g p r o p e r t i e s f o r a g i v e n c a s e . T h i s f a c t reduces the 51 number of windows which need t o be recommended. Depending on the c o n d i t i o n s , d i f f e r e n t windows are e f f e c t i v e f o r the a b s o r p t i o n mode. T a k i n g dynamic range as the main c r i t e r i o n , t he f o l l o w i n g recommendations can be made, some w i t h c o n s t r a i n t s on the damping c o n d i t i o n s . The N o e s t - K o r t works w e l l f o r a l l dampings t o 1% of the peak h e i g h t , and i f no a d j u s t m e n t s a r e p r e f e r r e d , the N o r t o n - B e e r F 3 s u f f i c e s . For a dynamic range t o 0.1% of the peak h e i g h t , b oth the F i l l e r E0.20 and K a i s e r - B e s s e l windows a r e e f f i c i e n t f o r a l l dampings, but a r e b e s t f o r lower dampings. The F i l l e r E0.20 has a s i m p l e r a n a l y t i c a l f o r m u l a , as l i s t e d i n Ta b l e I . The K a i s e r - B e s s e l window works f o r a v e r y l a r g e dynamic range, down t o 0.01% of the peak h e i g h t , f o r dampings of 0.5 and l e s s . Thus a b s o r p t i o n l i n e s h a p e s can be e f f e c t i v e l y a p o d i z e d down t o 1% of the peak h e i g h t f o r s i g n a l s of any damping w i t h the N o e s t - K o r t window, and down t o 0.1% and 0.01% f o r undamped or s l i g h t l y damped s i g n a l s w i t h the K a i s e r -B e s s e l window. The a n a l y t i c a l e x p r e s s i o n s f o r thes e windows a r e found i n T a b l e I . In the magnitude mode, when T / r = 0.0, the Hamming window i s p r e f e r r e d f o r a dynamic range of 100:1, the 3-term Blackman-H a r r i s window f o r 1000:1, and the K a i s e r - B e s s e l window f o r 10,000:1. T h i s p a t t e r n a l s o h o l d s f o r T/ r = 0.5 and 1.0. When T/ r i s e i t h e r 2.0 or 3.0, a p a t t e r n f o r the nar r o w e s t case i s not r e a d i l y o b s e r v a b l e . However upon c l o s e r e x a m i n a t i o n of the r e s u l t s i n T a b l e IV, t h e r e a r e o t h e r windows w i t h v a l u e s almost 52 as s m a l l as the n a r r o w e s t w i d t h s . Thus f o r a dynamic range of 100:1, the Hamming window i s b e s t f o r a damping of T/ r = 1.0 or l e s s . For the same dynamic range, and a h i g h l y damped s i g n a l , the 3-term B l a c k m a n - H a r r i s window i s b e t t e r . I t a l s o works f o r low dampings, but not as e f f e c t i v e l y as the Hamming. I f the dynamic range i s 100:1, and the damping i s unknown, the 3-term-B l a c k m a n - H a r r i s window i s the c h o i c e . T h i s window i s a l s o the one t o use f o r a g r e a t e r dynamic range, 1000:1, and f o r any amount of damping. For a dynamic range of 10,000:1, the K a i s e r -B e s s e l s u f f i c e s f o r a l l dampings. 53 CHAPTER 3 S I G N A L - T O - N O I S E R A T I O 3.1 I n t r o d u c t i o n The f r e q u e n c y l i n e s h a p e s i n the p r e v i o u s c h a p t e r s were o b t a i n e d from s i m u l a t e d time-domain s i g n a l s which d i d not c o n t a i n random n o i s e . In r e a l i t y , a time-domain s i g n a l would c o n t a i n random n o i s e which would l i m i t the s i g n a l - t o - n o i s e r a t i o (SNR) of the t r a n s f o r m e d spectrum. I t s h o u l d be noted t h a t the f r e q u e n c y l i n e s h a p e s shown i n C h a p t e r s 1 and 2 do not show SNRs. Frequency l i n e s h a p e s which have been d e r i v e d w i t h o u t n o i s e by F i l l e r [43] and N o r t o n and Beer [44] have been m i s t a k e n t o d e p i c t SNRs [ 5 3 ] . The o n l y t h i n g t h a t i s shown i s the s i d e l o b e b e h a v i o r due t o the FT of the window, not n o i s e . A p o d i z a t i o n w i l l cause the SNR t o d i f f e r from t h a t of the u n apodized c a s e . A p p l y i n g a window f u n c t i o n t o the time d a t a w i l l a l t e r the s i g n a l c o n t e n t and the n o i s e c o n t e n t , but not n e c e s s a r i l y i n e q u a l p r o p o r t i o n s . I t has been known f o r a b s o r p t i o n s t u d i e s t h a t a p p l y i n g a window w i t h the shape of a d e c r e a s i n g e x p o n e n t i a l w i l l i n c r e a s e the SNR [ 5 2 ] . D i f f e r e n t windows w i l l of c o u r s e have d i f f e r e n t e f f e c t s on the s i g n a l , and so the SNRs due t o the windows used s h o u l d be found t o ensure t h a t the windows recommended have adequate SNRs. The a b s o l u t e SNR of the f r e q u e n c y spectrum depends upon the amount of n o i s e i n the time-domain s i g n a l . Thus the SNRs b e f o r e 54 and a f t e r a p o d i z a t i o n a r e a f u n c t i o n of the a b s o l u t e SNR of the time-domain s i g n a l , Eq. 3. Hence i t i s more m e a n i n g f u l t o compare the r e l a t i v e SNRs of windowed l i n e s h a p e s where the r e f e r e n c e f o r comparison i s the SNR of the unapodized l i n e s h a p e , t h a t i s from the r e c t a n g l e window. S e v e r a l d i f f e r e n t e q u a t i o n s have been s e t f o r t h t o r e p r e s e n t the SNR of FT l i n e s h a p e s , however many a r e not a p p l i c a b l e h e r e . The e q u a t i o n s i n c l u d e SNR e x p r e s s i o n s r e q u i r i n g s p e c i f i c i n f o r m a t i o n about d i m e n s i o n s of the equipment [ 5 4 ] , such as the s i z e of the FT-ICR c e l l , and the c o n d i t i o n s of the experiment [ 5 5 ] , such as the t e m p e r a t u r e , and a l s o more g e n e r a l e q u a t i o n s [ 4 1 ] . A l l of the s e e q u a t i o n s a r e s i m i l a r i n form, but none of them a r e a p p l i c a b l e s i n c e they d e a l o n l y w i t h an unwindowed t i m e -domain f u n c t i o n . G e n e r a l t r e n d s have been shown f o r the r e l a t i o n s h i p between the amount of damping of the time s i g n a l and the s i z e of the SNR. For the a b s o r p t i o n mode, i t i s known t h a t as the amount of damping i n c r e a s e s , the SNR a l s o i n c r e a s e s [ 5 2 ] , C o n v e r s e l y , the SNR d e c r e a s e s w i t h i n c r e a s e d damping f o r the magnitude mode [ 5 4 ] . A p p a r e n t l y c o n f l i c t i n g r e s u l t s f o r SNR vs T/ r p a t t e r n s have been shown [ 3 6 , 4 1 ] . These works show t h a t as the amount of damping i n c r e a s e s , the SNRs of a b s o r p t i o n s p e c t r a d e c r e a s e and the SNRs of magnitude s p e c t r a i n c r e a s e . However the apparent c o n f l i c t a r i s e s from the d e f i n i t i o n of the problem : the s e were 55 done under a d i f f e r e n t c o n d i t i o n . These s e e m i n g l y c o n t r a d i c t o r y r e s u l t s a r e due t o s t u d i e s where the l e n g t h of the time s i g n a l , T, was changed w i t h r h e l d c o n s t a n t which changed the r a t i o T/ r . R e c a l l t h a t i n t h i s s t u d y , T was h e l d c o n s t a n t and r was a l t e r e d , thus c h a n g i n g the T/ r v a l u e s . T h i s i s an a l t o g e t h e r d i f f e r e n t s i t u a t i o n . M ention of changes i n T/ r i n t h i s work r e f e r t o the case where r i s a l t e r e d ; t h a t i s , the r e l a x a t i o n and not the d u r a t i o n of the time-domain s i g n a l changes. 3.2 P r o c e d u r e The f o r m u l a t o c a l c u l a t e the SNR of the a p o d i z e d l i n e s h a p e s must be g e n e r a l enough t o a p p l y t o the s i m u l a t e d time-domain s i g n a l , Eq. 3, but a l s o i n c l u d e a parameter f o r a window f u n c t i o n . T h i s e q u a t i o n s h o u l d a l s o compare the SNR of the a p o d i z e d l i n e s h a p e s t o the u n a p o d i z e d : i t s h o u l d p o r t r a y the r e l a t i v e SNR. A g e n e r a l f o r m u l a f o r a r e l a t i v e SNR f o r windowing has been d e r i v e d [ 5 6 ] . T h i s has been t a i l o r e d f o r t h e s p e c i f i c c a s e where the s i g n a l e n v elope has the shape e x p ( - t / r ) and l a s t s f o r a t o t a l l e n g t h of time T=1 [ 4 5 ] . These a r e t h e c o n d i t i o n s of t h i s s t u d y , and the e q u a t i o n i s where w(t) i s the window f u n c t i o n , and the term SNR r e f e r s to the (6) 56 r e l a t i v e SNR. The v a l u e s f o r w ( t ) were c a l c u l a t e d from T a b l e I f o r a b s o r p t i o n , and Table I I f o r magnitude. The SNRs were c a l c u l a t e d on the UBC-MTS system w i t h the i n t e g r a t i o n performed by the s u b r o u t i n e DCADRE. The r e s u l t i n g SNR d a t a was then p l o t t e d a g a i n s t T/ r . 3.3 A b s o r p t i o n Mode R e s u l t s and D i s c u s s i o n As the amount of damping i n c r e a s e s , t h e SNR f o r a b s o r p t i o n windows i n c r e a s e s . Thus the g e n e r a l t r e n d h o l d s . The SNR v a l u e s , c a l c u l a t e d from Eq. 6 w i t h the e x p r e s s i o n s i n T a b l e I f o r w ( t ) , a r e d i s p l a y e d i n T a b l e V. A l l of the windows have s i m i l a r r e s u l t s w i t h the e x c e p t i o n of the N o e s t - K o r t which has a n o t a b l y lower SNR. I t s v a l u e drops as low a 55% of the una p o d i z e d c a s e . The r e m a i n i n g windows have a SNR h i g h e r than the un a p o d i z e d case a t h i g h dampings, namely T/ r > 1.0. At h i g h e r dampings a l l of the v a l u e s c o n v e r g e . A l t h o u g h t h e i r v a l u e s a r e v e r y c l o s e , the t r i a n g l e has the h i g h e s t SNR and the 4-term B l a c k m a n - H a r r i s has the l o w e s t next t o t he N o e s t - K o r t which can be seen i n F i g . 13. For c l a r i t y , o n l y s e l e c t e d windows have been p l o t t e d , the r e m a i n i n g windows f a l l between the t r i a n g l e and 4-term B l a c k m a n - H a r r i s . There i s a l a r g e d i f f e r e n c e , almost 18%, between the s e two windows f o r the undamped c a s e , but l e s s than a 1% d i f f e r e n c e f o r the t o t a l l y damped c a s e . T h i s g e n e r a l t r e n d of convergence a t g r e a t e r dampings can be seen i n F i g . 13, and e x a c t d e t a i l s can be 57 TABLE V. R e l a t i v e s i q n a l - t o - n o i s e r a t i o s : A b s o r p t i o n windows. Window T / T 0.0 0.5 1 .0 2.0 3 .0 R e c t a n g l e 1 .00 1 .00 1 .00 1 .00 1 .00 B l a c k m a n - H a r r i s 3-term 0.77 0.88 0.95 1.12 1 .27 B l a c k m a n - H a r r i s 4-term 0.71 0.81 0.91 1.10 1 .27 F i l l e r DO.24 0.81 0.89 0.97 1.13 1 .26 F i l l e r E0.13 0.78 0.87 0.96 1.12 1 .26 F i l l e r E0.20 0.76 0.85 0.94 1.12 1 .27 Hanning a=2 0.82 0.90 0.98 1.13 1 .25 K a i s e r - B e s s e l 0.72 0.82 0.92 1.10 1 .27 K a i s e r - B e s s e l 4 -term 0.75 0.84 0.94 1.11 1 .27 N o e s t - K o r t 0.55 0.65 0.76 0.91 1 .18 Norton-Beer F3 0.83 0.91 0.99 1.13 1 .25 T r i a n g l e 0.87 0.94 1.01 1.14 1 .25 58 CN d. o = B I o c k m a n - H a r r i s , 4 - t e r m o = F i l ler E0.13 ^ = F i l ler E0.20 + = Kai s e r - B e s s e I x = N o e s t - K o r t o = T r i a n g l e 1 1 1 1 1 1 0.0 0.5 1.0 1.5 2.0 2.5 3.0 V 1 F i g u r e 13. R e l a t i v e s i g n a l - t o - n o i s e r a t i o s due t o a b s o r p t i o n windows as a f u n c t i o n of damping. 59 o b t a i n e d from T a b l e V. For each of the windows, the SNR i n c r e a s e s as the damping i n c r e a s e s . The g r e a t e s t i n c r e a s e i s due t o the N o e s t - K o r t whose h i g h e s t SNR shown i s more than t w i c e i t s l o w e s t v a l u e . The p a t t e r n of i n c r e a s e d SNR w i t h i n c r e a s e d damping can be a t t r i b u t e d t o the shape of t h e window. A damped time-domain s i g n a l has a h i g h e r a m p l i t u d e of s i g n a l a t t = 0 than a t t = T, r e c a l l F i g . 1; however, the n o i s e l e v e l i s c o n s t a n t throughout the e n t i r e s a m p l i n g p e r i o d . The windows used f o r a b s o r p t i o n a r e h i g h e s t a t t = 0, and t a p e r t o s m a l l a m p l i t u d e s a t t = T, as i n F i g . 4B. Thus m u l t i p l y i n g a damped time-domain f u n c t i o n by such a window r e t a i n s the s i g n a l c o n t e n t where i t i s s t r o n g e s t , near t 0, and has l i t t l e e f f e c t towards t = T where the s i g n a l s t r e n g t h i s v e r y s m a l l . In c o n t r a s t , the n o i s e i s c o n s t a n t , and so the window w i l l d e c r e a s e the n o i s e c o n t e n t towards the end of the s a m p l i n g t i m e . Thus the SNR i n c r e a s e s . T h i s i s noted by the SNRs h i g h e r than 1.0 f o r s i g n a l s w i t h a l a r g e amount of damping. There i s e v i d e n c e of the t r a d e o f f between r e s o l u t i o n and SNR. I f the d e f i n i t i o n of r e s o l u t i o n i s taken as meaning a narrow l i n e s h a p e , then the windows recommended i n Chapter 1 produce l i n e s w i t h good r e s o l u t i o n . The K a i s e r - B e s s e l window produced narrow l i n e s h a p e s f o r a l a r g e dynamic range, but i t has a r e l a t i v e l y low SNR. The F i l l e r E0.20 was good f o r a s m a l l e r dynamic range and has a h i g h e r SNR. w h i l e the F i l l e r E0.13 o n l y 60 worked f o r a l i m i t e d dynamic range, y e t has the h i g h e s t SNR of the t h r e e . Thus windows w i t h good r e s o l u t i o n , such as the N oest-K o r t and K a i s e r - B e s s e l , show a l o s s of SNR when compared t o windows w i t h p o o r e r r e s o l u t i o n , such as the t r i a n g l e and Hanning a=2. T h i s p a t t e r n i s c l e a r a t lower dampings. The N o e s t - K o r t has a d r a s t i c a l l y reduced SNR, p a r t i c u l a r l y a t low dampings, and s h o u l d be used w i t h c a u t i o n i f a l a r g e SNR i s r e q u i r e d . 3.4 Magnitude Mode R e s u l t s and D i s c u s s i o n The magnitude SNR r e s u l t s show a c o n s i s t e n t p a t t e r n . The e x a c t v a l u e s a r e i n T a b l e V I , and some s e l e c t e d windows a r e shown i n F i g . 14. These were c a l c u l a t e d w i t h Eq. 6 and t h e windows from T a b l e I I I were used f o r w ( t ) . In e v e r y c a s e , the SNRR de c r e a s e s s l i g h t l y w i t h an i n c r e a s e i n damping. T h i s f o l l o w s the g e n e r a l t r e n d t h a t SNR d e c r e a s e s w i t h i n c r e a s e d damping. A l l of the v a l u e s due t o windows a r e lower than t h e unapodized c a s e ; however, the range i s s m a l l . E x a m i n a t i o n of T a b l e VI shows t h a t t h e SNR f o r any g i v e n window d e c r e a s e s by o n l y a s m a l l amount as the damping i s i n c r e a s e d . For any window, the t o t a l l y damped SNR i s o n l y about 25% l e s s than the undamped SNR v a l u e . The t r i a n g l e y i e l d s the h i g h e s t SNRs, and the 4-term B l a c k m a n - H a r r i s has the l o w e s t v a l u e s w i t h t h e K a i s e r - B e s s e l ' s v a l u e s a l m o s t as low. T h i s demonstrates a t r a d e o f f between SNR and r e s o l u t i o n . The t r i a n g l e window produced p o o r l y r e s o l v e d 61 TABLE VI. R e l a t i v e s i q n a l - t o - n o i s e r a t i o s : Magnitude windows. Window T / r 0.0 0.5 1.0 2.0 3.0 Rectangle 1 . ,00 1 . .00 1 . .00 1 . .00 1 . ,00 Blackman-Harris 3-term 0. .77 0. .76 0, .74 0. .69 0. ,61 Blackman-Harris 4-term 0, .71 0. .70 0, .68 0. .63 0. ,54 Gaussian 0, .71 0. .71 0. .69 0. .63 0. ,55 Hamming 0, .86 0. .85 0, .84 0, .79 0. ,72 Hanning a=2 0, .82 0. .81 0, .80 0, .74 0. .67 Hanning a=4 0, .72 0. .71 0, .70 0. .64 0. .55 K a i s e r - B e s s e l 0, .72 0, .72 0, .70 0, .64 0. .56 K a i s e r - B e s s e l 4--term 0, .75 0, .74 0, .73 0, .67 0. .59 T r i a n g l e 0, .87 0, .86 0, .85 0, .80 0, .73 62 CO d CM d • = B l a c k m a n - H a r r i s , o = BI a c k m a n - H a r r i s, A = H a m m i n g + = Ka i s e r - B e s s e I x = T r i a n g l e 3 - t e r m 4 - t e r m o.o 0.5 n — 1.0 i 1.5 2.0 2.5 3.0 F i g u r e 14. R e l a t i v e s i g n a l - t o - n o i s e r a t i o s due windows as a f u n c t i o n of damping. to magnitude 63 l i n e s h a p e s a t 1% of the peak h e i g h t , y e t i t has the b e s t SNR. On the o t h e r hand, a recommended window, the K a i s e r - B e s s e l , w i t h good r e s o l u t i o n has c l o s e t o the p o o r e s t SNR. However t h i s g e n e r a l i t y does not h o l d f o r a l l c a s e s : the Hamming window, w i t h good r e s o l u t i o n a t 1% of i t s l i n e s h a p e ' s peak h e i g h t , a l s o has the SNR v a l u e s second t o the b e s t ; and the 3-term Blackman-H a r r i s window, w i t h good r e s o l u t i o n t o 0.1% of l i n e s h a p e ' s peak h e i g h t , has SNR v a l u e s i n the h i g h e r range. 3.5 C o n c l u s i o n The SNRs as a f u n c t i o n of damping p r o v i d e a d i s t i n c t p a t t e r n f o r both the a p o d i z e d a b s o r p t i o n and magnitude l i n e s h a p e s , w i t h the former i n c r e a s i n g and the l a t t e r d e c r e a s i n g as the the damping i s i n c r e a s e d . T h i s p a t t e r n i s due t o the two d i f f e r e n t window shapes used f o r the two modes. The a b s o r p t i o n SNRs sp r e a d over a wide range and r i s e above the un a p o d i z e d v a l u e a t h i g h dampings. The magnitude SNRs a r e a l l below 1.0, the r e f e r e n c e f o r the un a p o d i z e d c a s e , and the v a l u e s c o v e r a s m a l l range. A t r a d e o f f between SNR and r e s o l u t i o n , where r e s o l u t i o n i s judged by the l i n e w i d t h a t 1% of the peak h e i g h t , i s c l e a r l y o b s e r v e d f o r t h e a b s o r p t i o n r e s u l t s , but not f o r magnitude. The N o e s t - K o r t window produced the n a r r o w e s t , b e s t r e s o l v e d , a b s o r p t i o n l i n e s h a p e s but a l s o the l o w e s t SNRs. For the magnitude r e s u l t s , the 3-term B l a c k m a n - H a r r i s window c r e a t e d 64 narrow l i n e s h a p e s , y e t i t a l s o had r e l a t i v e l y h i g h SNRs. For the a b s o r p t i o n windows which were d e r i v e d from magnitude windows, a p a t t e r n i s o b s e r v e d . The SNR q u a l i t y of each window i s the same f o r both modes : the t r i a n g l e i s the b e s t , and the 3 -term B l a c k m a n - H a r r i s and G a u s s i a n a r e the w o r s t , and the r e m a i n i n g windows descend i n the same o r d e r f o r both the a b s o r p t i o n and magnitude modes. In f a c t f o r any d e r i v e d window, the v a l u e f o r i t s SNR i n the a b s o r p t i o n mode i s i d e n t i c a l t o i t s magnitude c o u n t e r p a r t a t T/ r = 0 . 0 . E x a m i n a t i o n of Eq. 6 w i t h T = 1 0 0 0 shows t h a t both p o r t i o n s of the e q u a t i o n w i t h t h i s t e r m reduce t o 1 . 0 , thus o n l y the terms c o n t a i n i n g w(t) remain. The ar e a of a window and i t s d e r i v e d form a r e e q u a l , hence the SNRs are e q u a l . •65 C H A P T E R 4 R E S O L U T I O N O F A P O D I Z E D S P E C T R A 4. 1 I n t r o d u c t i o n The windows which have been recommended i n the p r e v i o u s c h a p t e r s were chosen f o r t h e i r a b i l i t y t o c r e a t e a narrow l i n e s h a p e a t a s p e c i f i e d f r a c t i o n of the peak h e i g h t . The common d e f i n i t i o n of r e s o l u t i o n of a s i n g l e peak i s the w i d t h a t h a l f h e i g h t ; however, f o r the a p p l i c a t i o n of the recommended windows, the w i d t h s , or r e s o l u t i o n , a t 1%, 0.1% and 0.01% of the peak h e i g h t were of more i n t e r e s t . Thus the w i d t h a t the dynamic range r e q u i r e d i s used t o d e f i n e the r e s o l u t i o n of a s i n g l e peak. I f t h e r e a r e two peaks, a d i f f e r e n t d e f i n i t i o n i s used : i n s t e a d of measuring the w i d t h of a peak, the h e i g h t between the peaks i s measured. Window f u n c t i o n s have been a p p l i e d t o improve s p e c t r a l l i n e s h a p e s . However t h e r e i s a t r a d e o f f between s i d e l o b e r e d u c t i o n and peak w i d t h . The smoother a l i n e s h a p e i s made, the wider the peak a t h a l f h e i g h t becomes. For the recommended windows, the w i d t h a t 50% of the peak h e i g h t has been s a c r i f i c e d f o r narrower l i n e s h a p e s a t and below 1% of the peak h e i g h t . Thus the u s u a l i d e a of r e s o l u t i o n , t h a t d e f i n e d a t h a l f h e i g h t , i s l o s t f o r improved r e s o l u t i o n a t lower f r a c t i o n s of the peak h e i g h t . R e s o l u t i o n has been found t o be r e l a t e d t o the amount of 66 damping : as the damping i n c r e a s e s , the r e s o l u t i o n a t h a l f h e i g h t s h o u l d d e c r e a s e f o r both a b s o r p t i o n [ 5 2 ] , and magnitude [ 5 4 ] . The w i d t h s of the magnitude l i n e s h a p e s do not change as r a p i d l y w i t h c hanging amounts of damping as do those f o r a b s o r p t i o n l i n e w i d t h s . For a s i g n a l w i t h a t o t a l a c q u i s i t i o n time of T = 1, the h i g h l y damped unapodized magnitude l i n e w i d t h i s 1.4 times w i d e r than the undamped magnitude l i n e w i d t h . For a b s o r p t i o n , the t o t a l l y damped l i n e s h a p e i s 1.6 tim e s wider than the undamped c a s e . These have been c a l c u l a t e d from the undamped and t o t a l l y damped l i n e w i d t h s a t h a l f h e i g h t l i s t e d as p a r t of a s t u d y on the c o n t i n u o u s FT l i n e s h a p e s [ 3 6 ] . O b s e r v a t i o n of T a b l e I I and T a b l e IV r e v e a l t h a t the unapodized and a p o d i z e d l i n e w i d t h s do i n c r e a s e w i t h i n c r e a s e d damping, a l t h o u g h the i n c r e a s e s f o r magnitude a r e s l i g h t . T h i s p a t t e r n c o n f i r m s t h a t r e s o l u t i o n of a s i n g l e peak, a t any f r a c t i o n of the peak h e i g h t , worsens as damping i n c r e a s e s . T h e r e f o r e , the r e s o l u t i o n a t a s p e c i f i e d f r a c t i o n of the peak h e i g h t w i l l be p o o r e r f o r more h i g h l y damped s i g n a l s . The r e s u l t s i n the p r e v i o u s c h a p t e r s d e a l w i t h s i n g l e p e a ks, however s p e c t r a may c o n t a i n c l o s e l y spaced peaks. The e f f e c t of a p o d i z a t i o n on two c l o s e l y spaced peaks w i l l be examined. A new d e f i n i t i o n of r e s o l u t i o n w i l l be used f o r two peaks. One method of d e f i n i n g t h e r e s o l u t i o n of two peaks i s t o measure the v a l l e y between them [ 5 7 , 5 8 ] . The r a t i o of the h e i g h t between the peaks t o the h e i g h t of a peak i s known as the s i z e of the v a l l e y , i l l u s t r a t e d i n F i g . 15. A v a l l e y of 10% i s c o n v e n t i o n a l i n mass 67 F i g u r e 15. V a l l e y between two peaks. Top: smooth l i n e s h a p e . Bottom: l i n e s h a p e with s i d e l o b e s . 68 s p e c t r o m e t r y , and w i l l be the c r i t e r i o n f o r the r e s o l u t i o n of two peaks. As mentioned i n Chapter 2, many s t u d i e s have been done on a p o d i z a t i o n f o r a s i n g l e peak, but o n l y the one by H a r r i s [51] d e a l t w i t h two a p o d i z e d peaks. In h i s s t u d y , the most c l o s e l y spaced peaks were 5.5 b i n s a p a r t , where a b i n i s e q u i v a l e n t t o a freq u e n c y s p a c i n g , Eq. 5. Yet h i s study d e a l t o n l y w i t h undamped magnitude l i n e s h a p e s . In the f o l l o w i n g s e c t i o n s , a b s o r p t i o n and magnitude l i n e s h a p e s from damped and undamped s i g n a l s w i l l be examined, and f o r peaks of even c l o s e r s p a c i n g s . T h i s p o r t i o n of the study c h e c k s windows recommended i n Chapter 2 f o r t h e i r a b i l i t i e s t o r e s o l v e two peaks. A p o d i z e d l i n e s h a p e s c o n t a i n i n g two peaks w i l l be examined f o r t h e i r c o n f o r m a t i o n t o the 10% r e s o l u t i o n c r i t e r i o n f o r a v a r i e t y of dampings and peak s e p a r a t i o n s . Both magnitude and a b s o r p t i o n mode s p e c t r a w i l l be examined f o r time-domain s i g n a l s r a n g i n g from undamped t o t o t a l l y damped. T h e i r v a l l e y s w i l l be measured f o r peaks t h a t a r e s e p a r a t e d by 0.5 up t o 6.0 s p a c i n g s , Eq. 5, t o see the l i m i t t o which th e s e peaks w i l l be r e s o l v a b l e . 4.2 P r o c e d u r e The l i n e s h a p e s i n Chapter 2, from the t r a n s f o r m s of Eq. 3, c o n t a i n e d o n l y a s i n g l e peak. R e t a i n i n g t h i s form, a time-domain s i g n a l was s i m u l a t e d f o r two f r e q u e n c i e s by 69 f ( t ) = (K cos ( 2 f f f t ) + K,cos(2jrf , t ) ) e x p ( - t / r ). (7) O 0 1 1 Both peaks were s e t t o have e q u a l a m p l i t u d e , K Q = K 1, and e q u a l r e l a x a t i o n t i m e . The a c q u i s i t i o n time was s e t from t = 0 a t o t a l time of T = 1 s e c , and r was s e t t o 1000.0 f o r an undamped s i g n a l , 1.0 and 0.5 f o r a p a r t i a l l y damped s i g n a l , and 0.33 f o r an e s s e n t i a l l y c o m p l e t e l y damped s i g n a l . The r e f e r e n c e peak was s e t t o f = 512 Hz. The o t h e r peak v a r i e d from f 1 =511.5 Hz t o 506 Hz a t i n t e r v a l s of e v e r y f r e q u e n c y , and a t e v e r y h a l f f r e q u e n c y f o r c e r t a i n magnitude windows. The b a s i c f r e q u e n c y was i n c r e a s e d from 32 Hz t o 512 Hz i n o r d e r t o e l i m i n a t e i n t e r f e r e n c e s from the n e g a t i v e f r e q u e n c y l i n e s h a p e which a l t e r e d t he peak h e i g h t s . The i n t e r f e r e n c e was due t o the n a t u r e of the FT of the time-domain s i g n a l . A t i m e -domain f u n c t i o n of the form i n Eq. 3 has a l i n e s h a p e i n the p o s i t i v e f requency domain and a l s o a m i r r o r image a t the c o r r e s p o n d i n g n e g a t i v e f r e q u e n c i e s . T h i s has been d e s c r i b e d f o r a s i n g l e peak [ 4 0 , 5 2 ] , The s k i r t of the n e g a t i v e image can exte n d i n t o t he p o s i t i v e r e g i o n and o v e r l a p w i t h the p o s i t i v e l i n e s h a p e . For the r e s u l t s i n Chapter 2, t h i s e f f e c t i s n e g l i g i b l e f o r one peak. When t h e r e a r e two peaks, a c o n s i d e r a b l e d i f f e r e n c e i n the peak h e i g h t s can o c c u r when the peaks a re of low f r e q u e n c i e s . At h i g h e r f r e q u e n c i e s , the p o s i t i v e l i n e s h a p e i s f a r enough from the n e g a t i v e one t o r e c e i v e any n o t i c e a b l e i n t e r f e r e n c e . When a peak i s a t 512 Hz, the c o n t r i b u t i o n due t o 70 the t a i l of the peak a t -512 Hz i s l e s s than 0.3% A f a c t o r which has been o m i t t e d from Eq. 7 i s the phase. T h i s phase r e f e r s t o a phase i n the time-domain s i g n a l which t a k e s the form of a c o n s t a n t added t o the 2nf term. I t i s known t h a t phase a f f e c t s a b s o r p t i o n s p e c t r a . The b e s t c a s e , when the l i n e s h a p e s a r e p u r e l y a b s o r p t i o n w i t h o u t any c o n t a m i n a t i n g d i s p e r s i o n component, o c c u r s when t h e r e i s no phase. However , a b s o r p t i o n s p e c t r a c o n t a i n i n g c o n t a m i n a t i o n can be phase a d j u s t e d t o e l i m i n a t e the i n t e r f e r e n c e [ 1 4 , 5 2 ] . For magnitude, i n Chapter 2, a phase term was e x c l u d e d from the s i g n a l s g e n e r a t i n g one peak s i n c e i t has no e f f e c t on the shape of a s i n g l e f r e q u e n c y component. When the s i g n a l c o n t a i n s two components, the phase a f f e c t s the f r e q u e n c y outcome. The worst case f o r the a p o d i z e d magnitude l i n e s h a p e o c c u r r e d when the phase of both peaks e q u a l e d z e r o . The phases of b o t h peaks have been s e t t o z e r o t h u s the time-domain s i g n a l has the form i n Eq. 7, and so the worst case i s used f o r p r e s e n t i n g the v a l l e y s of the a p o d i z e d magnitude r e s u l t s . The time-domain f u n c t i o n , Eq. 7 was m u l t i p l i e d by a window, w ( t ) , from T a b l e I f o r a b s o r p t i o n and T a b l e I I I f o r magnitude. The windowed time-domain f u n c t i o n , f ( t ) w ( t ) was sampled f o r 2049 d a t a p o i n t s , then z e r o - f i l l e d t h r e e t i m e s t o a complete d a t a s e t of 16,385 p o i n t s . A d i s c r e t e F o u r i e r t r a n s f o r m was performed on t h e s e p o i n t s i n the same manner as d e s c r i b e d i n Chapter 2. The r e s u l t i n g l i n e s h a p e s were examined f o r the s i z e of t h e i r .71 v a l l e y s . For smooth l i n e s h a p e s , the s i z e of the l o w e s t p o i n t between the peaks was c o n s i d e r e d f o r the c a l c u l a t i o n of the h e i g h t of the v a l l e y as shown i n F i g . 15. Some l i n e s h a p e s had s m a l l s i d e l o b e s between the peaks; i n t h e s e c a s e s , the s i z e of the h i g h e s t s i d e l o b e was used t o d e t e r m i n e the s i z e of the v a l l e y as i l l u s t r a t e d i n the bottom diagram of F i g . 15. For each window, the s i z e s of the v a l l e y s were g a t h e r e d and then p l o t t e d a g a i n s t T/ r and the s p a c i n g between the peaks. These p l o t s a r e shown and d i s c u s s e d i n the f o l l o w i n g s e c t i o n s . 4.3 A b s o r p t i o n Mode R e s u l t s and D i s c u s s i o n A p p l i c a t i o n of the K a i s e r - B e s s e l window t o two peaks r e v e a l e d s i m i l a r r e s u l t s f o r the a b s o r p t i o n l i n e s h a p e as f o r one peak. Narrow, smooth l i n e s h a p e s r e s u l t e d even when the peaks were brought c l o s e t o g e t h e r . The 10% v a l l e y c r i t e r i o n was s a t i s f i e d f o r many c a s e s , p a r t i c u l a r l y f o r low dampings and w i d e l y spaced peaks. The v a l l e y s were examined f o r a b s o r p t i o n l i n e s h a p e s r e s u l t i n g from v a r i o u s dampings, and w i t h peaks of a range of s e p a r a t i o n s . A c o n v e n i e n t way t o summarize the r e s u l t s a r e i n a t h r e e d i m e n s i o n a l p l o t as d i s p l a y e d i n F i g . 16. T h i s g i v e s a g e n e r a l i d e a of the p r e v a i l i n g p a t t e r n s , and d e t a i l s a r e p r o v i d e d i n T a b l e V I I f o r c l o s e r e x a m i n a t i o n of the 10% v a l l e y c r i t e r i o n . T h i s g e n e r a l t r e n d i s seen i n F i g . 16, w i t h the d e t a i l s i n T a b l e V I I . From t h i s t a b l e i t i s c l e a r t h a t when the peaks a r e 72 K A I S E R - B E S S E L F i a u r e 16 V a l l e y h e i g h t s as a f u n c t i o n of s p a c i n g and damping F i g u r e 16. vaxxey ^ ^ . ^ p e a k s a p o d i z e d by the K a i s e r - B e s s e l window, 73 s e p a r a t e d by 6.0 s p a c i n g s , t h e r e i s l e s s than a 10% v a l l e y f o r a l l dampings. At 5.0 s p a c i n g s , t h i s c r i t e r i o n i s s a t i s f i e d up t o and i n c l u d i n g a damping of T/ * = 2.0. For 4 s p a c i n g s , t h i s i s t r u e up t o T / r = 1.0; and f o r as c l o s e as 3 s p a c i n g s t h i s i s t r u e f o r a t o t a l l y undamped c a s e . The peaks are e x t r e m e l y w e l l r e s o l v e d f o r the undamped c a s e . In f a c t , the v a l l e y s a r e below 1% f o r T / r = 0.0 f o r peaks up t o and as c l o s e as 4 s p a c i n g s a p a r t . A s y s t e m a t i c p a t t e r n i s o b s e r v e d where the l i n e s h a p e s have low v a l l e y s f o r w i d e l y spaced peaks, and g r a d u a l l y i n c r e a s i n g v a l l e y h e i g h t s as the peaks a r e brought c l o s e r t o g e t h e r w i t h the peaks b e i n g t o t a l l y u n r e s o l v e d when they a r e o n l y one space a p a r t . A l s o , as damping i n c r e a s e s , the r e s o l u t i o n d e c r e a s e s ; t h a t i s , as T / r i n c r e a s e s , the v a l l e y h e i g h t s i n c r e a s e . T h i s e x a c t p a t t e r n was a l s o o b s e r v e d when the 3-term B l a c k m a n - H a r r i s window w a s . a p p l i e d as o b s e r v e d i n F i g . 17. The e x a c t v a l u e s , i n T a b l e V I I I , show v e r y s i m i l a r r e s u l t s when compared t o t h o s e f o r the K a i s e r - B e s s e l window i n T a b l e V I I . These windows behave v i r t u a l l y i d e n t i c a l l y f o r two peaks, and a l s o showed r e l a t i v e l y s i m i l a r r e s u l t s f o r a s i n g l e peak. T h i s s i m i l a r i t y can be e x p l a i n e d by the n a t u r e of the FT f o r a b s o r p t i o n . As the peaks a r e s e t c l o s e r t o g e t h e r , t h e i r s k i r t s o v e r l a p i n the space between the peaks. When the h e i g h t s of the s k i r t s a r e added, the h e i g h t of the v a l l e y i s o b t a i n e d . T h i s i s 74 TABLE V I I . Absorption l i n e s h a p e v a l l e y s : K a i s e r - B e s s e l window. CM) 1_10 2A) 3^0 1.0 100 100 100 100 2.0 34 53 68 81 3.0 1.9 14 27 40 4.0 0.00 5.4 13 21 5.0 0.01 s 3.1 7.4 13 6.0 0.01 s 2.0 4.9 8.6 s denotes s i d e l o b e s . TABLE V I I I . Absorption l i n e s h a p e v a l l e y s : 3-term Blackman-Harris window. V f i T / r CL0 1.0 2.0 3_^ 0 1.0 100 100 100 100 2.0 19 39 58 72 3.0 0.01 9.9 22 34 4.0 0.05 s 4.5 11 18 5.0 0.04 s 2.7 6.5 12 6.0 0.05 s 1.8 4.4 7.9 s denotes s i d e l o b e s . 75 B L A C K M A N - H A R R I S , 3 - T E R M F i g u r e 17. V a l l e y h e i g h t s as a f u n c t i o n of s p a c i n g and damping of two a b s o r p t i o n peaks a p o d i z e d by the 3-term B l a c k m a n - H a r r i s window. 76 c o n f i r m e d by c l o s e e x a m i n a t i o n of T a b l e I I . T h i s f o l l o w s due t o the l i n e a r i t y of F o u r i e r t r a n s f o r m s . I f t h e r e i s a time f u n c t i o n , h ^ t ) , has the FT H ^ f ) , and h 2 ( t ) has the FT H 2 ( f ) ; then h ^ t ) + h 2 ( t ) < > H ^ f ) + K 2 ( f ) (8) where < > denotes the F o u r i e r t r a n s f o r m p a i r . The f r e q u e n c y f u n c t i o n c o u l d be composed of a r e a l and an i m a g i n a r y p a r t , thus the s e p a r t s a r e merely a d d i t i v e . In the case of a b s o r p t i o n l i n e s h a p e s , o n l y the r e a l p a r t of the t r a n s f o r m i s examined and t h i s can be thought of as p u r e l y the a d d i t i o n of the f r e q u e n c y t r a n s f o r m s of the time s i g n a l s . S i n c e the a b s o r p t i o n l i n e s h a p e s a r e s i m p l y a d d i t i v e , the recommended windows f o r one peak can be a p p l i e d t o two peaks. By e xamining T a b l e I I , the h e i g h t s between the peaks can be c a l c u l a t e d . 4 . 4 Magnitude Mode R e s u l t s and D i s c u s s i o n I f i t i s assumed t h a t magnitude l i n e s h a p e s a r e a d d i t i v e l i k e a b s o r p t i o n l i n e s h a p e s , then a s i m i l a r p a t t e r n f o r the h e i g h t s of the v a l l e y s would be e x p e c t e d . The magnitude r e s u l t s o b t a i n e d do not agree w i t h the c l a i m [59] t h a t l i n e s h a p e s a r e merely a d d i t i v e . A s i m p l e p a t t e r n r e s e m b l i n g the a b s o r p t i o n r e s u l t s i s not found f o r the magnitude s t u d i e s . Rather than i n c r e a s i n g s t e a d i l y as the s p a c i n g i s d e c r e a s e d , the v a l l e y h e i g h t s f l u c t u a t e . A p o d i z a t i o n i s e f f e c t i v e i n smoothing the s p e c t r a l s k i r t 77 a s s o c i a t e d w i t h frequency-domain l i n e s h a p e s , however u n s y s t e m a t i c r e s u l t s ensued. When f - f , > 6, the c r i t e r i o n f o r the 10% v a l l e y o 1 i s s a t i s f i e d . However f o r peaks of c l o s e r s p a c i n g , a s i m p l e p a t t e r n i s not ob s e r v e d . F i g u r e s 18, 19 and 20 i l l u s t r a t e the consequences of a l t e r i n g the s p a c i n g between two peaks f o r an undamped s i g n a l p l o t t e d on the same s c a l e as the s i n g l e peaks. The magnitude l i n e s h a p e due t o the K a i s e r - B e s s e l window i s d e p i c t e d by the s o l i d l i n e , and the u n a p o d i z e d , r e c t a n g l e window, l i n e s h a p e i s the clashed l i n e . Note t h a t the un a p o d i z e d l i n e s h a p e has s i d e l o b e s between the two peaks when they a r e t h r e e s p a c i n g s a p a r t , as i n F i g . 18. However, when they a r e brought c l o s e r t o g e t h e r , i n F i g s . 19 and 20, t h e r e a r e no s i d e l o b e s between the peaks, hence the v a l l e y d e c r e a s e s . Thus, f o r t h e undamped c a s e , as peak s e p a r a t i o n i n c r e a s e s , the v a l l e y s i n c r e a s e t o a c e r t a i n e x t e n t as seen i n F i g . 21 which sums up the v a l l e y s f o r many s p a c i n g s and dampings f o r the unapodized l i n e s h a p e s . The K a i s e r - B e s s e l window e l i m i n a t e d s i d e l o b e s of one peak. When the K a i s e r - B e s s e l i s a p p l i e d t o two peaks which a r e 6 s p a c i n g s , f - f ^ , a p a r t the s k i r t between the peaks i s l e s s than 2%. When two peaks a r e s e p a r a t e d by 3 s p a c i n g s , the v a l l e y between them d i p s t o l e s s than 0.1% f o r the undamped case as i n F i g . 18. R e t a i n i n g t h i s amount of damping and b r i n g i n g the peaks t o o n l y 2 s p a c i n g s a p a r t causes the a p o d i z e d l i n e s h a p e s t o appear as o n l y one peak (t h e h e i g h t of the v a l l e y i s 100%) as i n F i g . 19. However i f the peaks a r e s e t even c l o s e r , t o 1 s p a c i n g 78 f, =509, f0 =512 VO 502.0 504.5 507.0 509.5 512.0 514.5 517.0 519.5 522.0 F i g u r e 18. Magnitude l i n e s h a p e due t o the K a i s e r - B e s s e l window f o r two peaks s e p a r a t e d by t h r e e s p a c i n g s . F i g u r e 19. Magnitude l i n e s h a p e due t o the K a i s e r - B e s s e l window f o r two peaks s e p a r a t e d by two s p a c i n g s . f, =511, f0 =512 oo 502.0 504.5 519.5 522.0 F i g u r e 20. Magnitude l i n e s h a p e due t o the K a i s e r - B e s s e l window f o r two peaks s e p a r a t e d by one s p a c i n g . RECTANGLE a p a r t , the a p o d i z e d l i n e s h a p e c l e a r l y shows two peaks s e p a r a t e d by a v a l l e y of l e s s than 0.1% as i n F i g . 20. I t i s apparent t h a t the h e i g h t s of the v a l l e y s f l u c t u a t e as the peak s e p a r a t i o n changes. The v a l l e y s were examined f o r peaks a t many s e p a r a t i o n s and f o r v a r i o u s dampings w i t h the r e s u l t s due t o the K a i s e r - B e s s e l window summarized i n F i g . 22. I t i s c l e a r t h a t the h e i g h t of the v a l l e y s does not s i m p l y i n c r e a s e as the s p a c i n g between two peaks d e c r e a s e s . The 4-term B l a c k m a n - H a r r i s window produced l i n e s h a p e s v e r y s i m i l a r t o t h o s e by the K a i s e r - B e s s e l window f o r one peak. The v a l l e y s were c a l c u l a t e d f o r two peaks of d i f f e r e n t s e p a r a t i o n s w i t h the 4-term B l a c k m a n - H a r r i s window t o see i f a s i m i l a r p a t t e r n f o r v a l l e y o c c u r r e d . A g a i n the h e i g h t s f l u c t u a t e d , as shown i n F i g . 23, but w i t h a t o t a l l y d i f f e r e n t p a t t e r n as seen by comparing t h i s w i t h F i g . 21. No p a t t e r n f o r an a s s o c i a t i o n w i t h damping i s o b s e r v a b l e . The 4-term K a i s e r - B e s s e l window d i d not produce s i n g l e -peaked l i n e s h a p e s as s i m i l a r t o the K a i s e r - B e s s e l ' s as d i d the 4-term B l a c k m a n - H a r r i s window, seen i n T a b l e I V . However, the v a l l e y h e i g h t s due t o the K a i s e r - B e s s e l and the 4-term K a i s e r -B e s s e l a r e p r a c t i c a l l y i d e n t i c a l . A s i m i l a r p a t t e r n i s o b s e r v e d f o r the 3-term B l a c k m a n - H a r r i s window. T h i s was shown, i n the p r e v i o u s c h a p t e r s , t o be an e f f e c t i v e window, but i t does not r e s o l v e c l o s e l y spaced peaks w i t h any p r e d i c t a b l e p a t t e r n s . The h e i g h t s of t h e v a l l e y s f l u c t u a t e w i t h v a r y i n g s p a c i n g s , and no 83 K A I S E R - B E S S E L 22. V a l l e y h e i g h t s as a f u n c t i o n of s p a c i n g and damping of two magnitude peaks a p o d i z e d by the K a i s e r - B e s s e l window. 84 B L A C K M A N - H A R R I S , 4 - T E R M F i g u r e 23. V a l l e y h e i g h t s as a f u n c t i o n of s p a c i n g and damping of two magnitude peaks a p o d i z e d by the 4-term B l a c k m a n - H a r r i s window. 85 r e l a t i o n s h i p w i t h the amount of damping can be d e t e c t e d . The Hanning window was not as e f f e c t i v e as the K a i s e r - B e s s e l or the 4-term B l a c k m a n - H a r r i s when a p p l i e d t o a s i n g l e peak. Windowing w i t h the Hanning window l e f t a l i n e s h a p e w i t h many s i d e l o b e s . When a p p l i e d t o a s i g n a l c o n t a i n i n g two peaks, the r e s u l t i n g l i n e s h a p e s have c o n s i s t e n t l y low v a l l e y s f o r s e p a r a t i o n s g r e a t e r than 3.0 s p a c i n g s . When peaks a r e s e p a r a t e d by 3.0 t o as c l o s e as 1.5 s p a c i n g s , the v a l l e y s c o n s i s t e n t l y i n c r e a s e f o r d e c r e a s i n g s p a c i n g s f o r a l l dampings as shown i n F i g . 24. T h i s i s a more c o n s i s t e n t p a t t e r n , however the problem e n c o u n t e r e d f o r a s i n g l e peak p e r s i s t s f o r two peaks. The l i n e s h a p e i s not smooth, but c o n t a i n s many a u x i l i a r y maxima which c o u l d be m i s t a k e n f o r o t h e r peaks. U n l i k e the a b s o r p t i o n r e s u l t s , no p a t t e r n between damping and v a l l e y h e i g h t s i s o b s e r v a b l e f o r magnitude. The h e i g h t s f l u c t u a t e from one s p a c i n g v a l u e t o t h e n e x t . There i s a l s o no c o n s i s t e n t p a t t e r n from one window t o the n e x t . S i n c e no p a t t e r n was o b s e r v e d , o n l y a few p l o t s have been p r e s e n t e d t o g i v e a g e n e r a l o v e r v i e w . Exact v a l u e s f o r the v a l l e y s of the windows d i s c u s s e d i n t h i s s e c t i o n , w i t h d e t a i l s about s i d e l o b e s , a r e i n Appendix C. The reason t h a t the magnitude l i n e s h a p e s do not show s i m p l e a d d i t i v e b e h a v i o r as shown by a b s o r p t i o n can be shown m a t h e m a t i c a l l y . L e t h ( t ) r e p r e s e n t a time s i g n a l , A i s the a b s o r p t i o n spectrum, D i s the d i s p e r s i o n spectrum, and M i s the 86 H A N N I N G F i g u r e 24. V a l l e y h e i g h t s as a f u n c t i o n of spacing and damping of two magnitude peaks apodized by the Hanning window. 87 magnitude spectrum in the frequency domain. For a time signal, h ^ t ) , the transform produces A 1 , D 1 , and . A second time signal, h 2, gives A 2, D 2, and M2. If the time signal i s h ( t ) = h j U ) + h 2 ( t ) (9) then the transform yields the frequency functions A = A 1 + A 2 (10) D = D 1 + D 2 (11) as explained in the previous section. If magnitude were merely additive, then the magnitude lineshape for two time signals would be M = M1 + M 2 (12) = ( A , 2 + D j 2 ) 1 7 2 + ( A 2 2 + D 2 2 ) 1 / 2 . (13) Calculation of the actual magnitude lineshape due to two time signals gives the following M = ( A 2 + D 2 ) 1 / 2 (14) = ( ( A 1 + A 2 ) 2 + (D 1 + D 2 ) 2 ) 1 / 2 = ( A ^ + + A 2 2 + D 2 2 + 2 A ^ 2 + 2B}D2)]^2. (15) 88 T h i s i s not e q u a l t o Eq. 13, t h e r e f o r e magnitude l i n e s h a p e s a r e not merely a d d i t i v e . The magnitude l i n e s h a p e i s not s i m p l y M 1+M 2, but , by rearrangement of Eq. 15 w i t h the i m p l e m e n t a t i o n of the r e l a t i o n i n Eq. 14, the magnitude l i n e s h a p e w i t h two peaks i s a c t u a l l y M = ( M j 2 + M 2 2 + 2k}k2 + 2 D 1 D 2 ) 1 / / 2 . (16) When the peaks are f a r a p a r t , the v a l u e s f o r M n, A n and D n between the peaks a r e v e r y s m a l l . However when peaks a r e c l o s e l y spaced, the p o r t i o n between t h e peaks may be made up of l a r g e p o s i t i v e v a l u e s f o r M n and p o s s i b l y l a r g e n e g a t i v e v a l u e s f o r D n, and e i t h e r p o s i t i v e or n e g a t i v e v a l u e s f o r A n. These c o u l d add t o g e t h e r and c r e a t e a h i g h v a l l e y , or c a n c e l and make a low v a l l e y . The reason f o r the f l u c t u a t i n g v a l l e y s i s d e p i c t e d i n F i g s . 25, 26, and 27. The r e f e r e n c e peak i s s e t a t 512 Hz, F i g . 25, and then a peak i s brought i n t o 510 Hz, F i g . 26, and then even c l o s e r t o 511 Hz, F i g . 27. The magnitude l i n e s h a p e s a r e r e p r e s e n t e d by the s o l i d l i n e , the a b s o r p t i o n l i n e s h a p e by the d o t - d a s h l i n e , and the d i s p e r s i o n l i n e s h a p e by the dashed l i n e . A s i n g l e magnitude l i n e s h a p e can be seen, w i t h c l o s e e x a m i n a t i o n , t o be composed of the square r o o t of the squares of the a b s o r p t i o n l i n e s h a p e and the d i s p e r s i o n l i n e s h a p e s i n F i g . •89 to =512 in cr o b in d i o T if> 7 506.0 —I 508 .0 M A G N I T U D E — A B S O R P T I O N — D I S P E R S I O N — i — 510.0 —I 512.0 f —I 514.0 516.0 518.0 F i g u r e 25. Lineshapes of a s i n g l e peak apodized by the K a i s e r -B e s s e l window. 90 f, =510, f. =512 i n MAGNITUDE — ABSORPTION - - DISPERSION 506.0 508.0 510.0 512.0 f — i 1 514.0 516.0 518.0 F i g u r e 2 6 . L i n e s h a p e s o f t w o p e a k s s e p a r a t e d b y t w o s p a c i n g s a n d a p o d i z e d b y t h e K a i s e r - B e s s e l w i n d o w . 91 f, =511, f. =512 in 92 25. At a d i f f e r e n c e of 2 s p a c i n g s , the magnitude l i n e s h a p e s are t o t a l l y u n r e s o l v e d , F i g . 26. I t can be seen t h a t t h i s l i n e s h a p e , F i g . 26, i s not j u s t two magnitude l i n e s h a p e s , F i g . 25, added t o g e t h e r , but i s i n s t e a d due t o a c o m b i n a t i o n of the a b s o r p t i o n and d i s p e r s i o n l i n e s h a p e s . When the peaks a r e brought even c l o s e r t o g e t h e r , the magnitude l i n e s h a p e s are suddenly v e r y w e l l r e s o l v e d , F i g . 26. The reason can be seen t o be due t o the c o m p l i c a t e d c o m b i n a t i o n s of the a b s o r p t i o n and d i s p e r s i o n s p e c t r a , and not t o the a d d i t i o n of the magnitude l i n e s h a p e s . The a b s o r p t i o n and d i s p e r s i o n l i n e s h a p e s due t o two peaks can be a t t r i b u t e d d i r e c t l y t o the l i n e s h a p e of a s i n g l e peak. For example, i f an a b s o r p t i o n l i n e s h a p e as i n F i g . 25 i s p l a c e d a t 512 Hz and another i d e n t i c a l shape i s p l a c e d a t 511 Hz and then added t o g e t h e r , the l i n e s h a p e i n F i g . 27 w i l l be the r e s u l t . The same can be done f o r d i s p e r s i o n . But t h i s can not be done f o r the magnitude l i n e s h a p e s . R e c a l l t h a t the d i s t o r t e d a b s o r p t i o n l i n e s h a p e s were n o t i c e d i n Chapter 2 f o r magnitude-type windows such as the 4-term B l a c k m a n - H a r r i s . Thus the b e h a v i o r of the magnitude v a l l e y s f o r t h i s window s h o u l d not be t o t a l l y unexpected. The magnitude-type Hanning a=2 window produced an a b s o r p t i o n shape w i t h l e s s s e v e r e d i s t o r t i o n s , and the r e s u l t i n g magnitude v a l l e y s had more p r e d i c t a b l e b e h a v i o r , F i g . 24, than those due t o windows such as the 4-term B l a c k m a n - H a r r i s , F i g . 23. Another p o i n t about magnitude l i n e s h a p e s i s o b s e r v a b l e i n 9 3 F i g . 27. The maximum v a l u e s of the magnitude l i n e s h a p e s do not f a l l on the c o r r e c t f r e q u e n c i e s , but have been s h i f t e d outwards. T h i s i s d i r e c t l y due t o the a d d i t i o n of the a b s o r p t i o n and d i s p e r s i o n components. Yet note t h a t the a b s o r p t i o n peaks f a l l where they s h o u l d , and the d i s p e r s i o n l i n e s h a p e s a r e p r o p e r l y p l a c e d . A s i n g l e window does not y i e l d improved r e s u l t s f o r magnitude-mode e x a m i n a t i o n s of two v e r y c l o s e l y spaced peaks, u n l i k e the e x c e l l e n t improvements o b t a i n e d f o r s i n g l e peaks. However f o r peaks w i t h a s e p a r a t i o n of g r e a t e r than 6 s p a c i n g s , windowing i s an improvement over unwindowed s i g n a l s . A c o n c i s e g e n e r a l i t y can not be reached about the e f f e c t of d i f f e r e n t windows on two c l o s e l y spaced peaks. The K a i s e r - B e s s e l and 4-term B l a c k m a n - H a r r i s windows both y i e l d f l u c t u a t i n g v a l l e y s , however t h e i r p a t t e r n s a r e not i d e n t i c a l as seen by comparing F i g . 22 w i t h F i g . 23. The Hanning window, on the o t h e r hand, produces a s y s t e m a t i c p a t t e r n of peak h e i g h t s , however the shape of the l i n e i s not smooth. The a d d i t i o n of a phase component t o a time-domain s i g n a l , which causes a time s h i f t , has no e f f e c t on a magnitude l i n e s h a p e . I t i s w e l l known t h a t the magnitude mode i s independent of phase, or a time s h i f t ; however, when a phase i s added t o o n l y one of two time-domain components the magnitude mode becomes dependent on phase. T h i s time-domain i s not merely s h i f t e d i n t i m e , but has become a new c o m p l i c a t e d s i g n a l whose 94 t r a n s f o r m i s no l o n g e r the same as t h a t due t o a s i g n a l w i t h o u t phases. I f a phase i s added t o h ^ t ) , then the magnitude l i n e s h a p e w i l l have a form d e s c r i b e d by Eq. 16; however the A, and components w i l l d i f f e r from those w i t h o u t phase. The r e m a i n i n g terms a r e unchanged. An i l l u s t r a t i o n of the e f f e c t s of phase i s shown f o r an undamped time-domain s i g n a l a p o d i z e d by the K a i s e r -B e s s e l window where t h e r e i s one peak a t f Q=512 Hz and another a t ^=508 Hz + phase. The phase has been s e t t o 0 i n F i g . 28, ir /2 i n F i g . 29, and ir i n F i g . 30. Note t h a t as the phase i n c r e a s e s , t h e v a l l e y of the a p o d i z e d l i n e s h a p e d e c r e a s e s , and t h a t the worst c a s e o c c u r s f o r phase = 0, F i g . 28. In c o n t r a s t , the v a l l e y s of t h e u n a p o d i z e d l i n e s h a p e s i n c r e a s e as the phase i s i n c r e a s e d w i t h the v a l l e y b e i n g l o w e s t when the phase e q u a l s 0. The c a s e s a r e i l l u s t r a t e d w i t h a phase added t o one of two t i m e -domain components; however, t h i s phenomenon i s t r u e f o r a r e l a t i v e phase d i f f e r e n c e . I f t h e r e a r e two components i n the time-domain s i g n a l , each w i t h a d i f f e r e n t phase, then Eq. 16 i n c o r p o r a t e s the phase w i t h the new e x p r e s s i o n M = (M, 2 + M 2 2 + 2 ( ( A 1 A 2 + D ^ J c o s ^ - B ^ + (A D 2 - A 2 D 1 ) s i n ( 6 1 - e 2 ) ) ) 1 / 2 (17) where 8, i s t h e phase of h ^ t ) , and 9 2 i s the phase of the second component, h _ ( t ) . The d e r i v a t i o n of t h i s f o r m u l a i s found i n 95 PHASE=0 F i g u r e 28. Magnitude l i n e s h a p e s from s i g n a l with phase = 0. 96 P H A S E = T T / 2 F i a u r e 29 Magnitude l i n e s h a p e s from s i g n a l where one component F i g u r e 2*. wagn^ ^ Q F ^  ^ O T H E R H A S A P H A S E O F , / 2 > 97 PHASE=n KAISER-BESSEL RECTANGLE i i i 1 1 1 1 1 1 1 502.0504.0 506.0 508.0 510.0 512.0 514.0 516.0 518.0 520.0 522.0 f 30. Magnitude l i n e s h a p e s from s i g n a l where one compoment has a phase of 0, and the o t h e r has a phase of it. 98 Appendix D. Note t h a t Eq. 17 r e t u r n s t o the form of Eq. 16 when t h e r e i s no d i f f e r e n c e i n phase. 4.5 C o n c l u s i o n The K a i s e r - B e s s e l window, which was found t o c r e a t e a smooth a b s o r p t i o n l i n e s h a p e f o r a s i n g l e peak over a l a r g e dynamic range and v a r i e t y of dampings, and t o have a f a i r SNR, has been shown t o be e f f e c t i v e i n r e s o l v i n g two peaks. I f the peaks a r e s e p a r a t e d by 6 s p a c i n g s , the v a l l e y between them i s l e s s than 10% f o r a l l dampings. As the peaks a r e s e t c l o s e r t o g e t h e r , they c o n t i n u e t o s a t i s f y t h i s c r i t e r i o n up t o 3 s p a c i n g s f o r an undamped s i g n a l . I n g e n e r a l , the a b s o r p t i o n l i n e s h a p e due t o two peaks can be o b t a i n e d m erely by a d d i t i o n of the s i n g l e a b s o r p t i o n l i n e s h a p e s , t h e r e f o r e the windows recommended f o r a s i n g l e peak w i l l a l s o r e s o l v e two peaks. The K a i s e r - B e s s e l window was a l s o found t o be e f f i c i e n t f o r s i n g l e magnitude l i n e s h a p e s , however when two peaks a r e brought t o g e t h e r , they a r e not r e s o l v e d i n an e a s i l y p r e d i c t a b l e p a t t e r n . As the peaks a r e brought c l o s e t o g e t h e r , they a r e p o o r l y r e s o l v e d , but when they a r e s e t even c l o s e r they suddenly become w e l l r e s o l v e d . The magnitude l i n e s h a p e s a r e not s i m p l y a d d i t i v e . No r e g u l a r p a t t e r n i s o b s e r v a b l e , but the f l u c t u a t i n g b e h a v i o r of the magnitude v a l l e y s can be e x p l a i n e d by the a b s o r p t i o n and d i s p e r s i o n components. These two components a l s o e x p l a i n why magnitude s p e c t r a are dependent on phase. I f t h e r e i s a phase 99 d i f f e r e n c e between the two components, the a p o d i z e d magnitude l i n e s h a p e of the two peaks i s b e t t e r r e s o l v e d , t h a t i s , i t has a lower v a l l e y than a l i n e s h a p e due t o a time-domain s i g n a l where both components have no phase. 100 CHAPTER 5 SUMMARY T r u n c a t e d time-domain s i g n a l s , upon F o u r i e r t r a n s f o r m a t i o n , y i e l d l i n e s h a p e s w i t h s i d e l o b e s . A p o d i z a t i o n , e l i m i n a t i o n of s i d e l o b e s , was performed by m u l t i p l y i n g time-domain f u n c t i o n s of d i f f e r i n g d egrees of damping by v a r i o u s window f u n c t i o n s . The shape of the window f u n c t i o n s was s y m m e t r i c a l f o r the magnitude s t u d i e s ; however, t h i s t y pe of window produced l a r g e n e g a t i v e s i d e l o b e s i n the a b s o r p t i o n l i n e s h a p e . An adapted shape c o n s i s t i n g of h a l f of the s y m m e t r i c a l shape extended over an e q u a l time p e r i o d was found t o be s a t i s f a c t o r y f o r the a b s o r p t i o n s t u d i e s . S p e c i f i c c o n d i t i o n s e n c o u n t e r e d i n p r a c t i c e i n c l u d e the damping of the time-domain s i g n a l and the r e q u i r e d minimum dynamic range of the f r e q u e n c y spectrum. The dynamic range r e q u i r e m e n t was chosen as the main f a c t o r f o r the s e l e c t i o n of a window, w i t h a p p l i c a b i l i t y t o a v a r i e t y of dampings ta k e n as an added f a c t o r . The l i n e s h a p e s were examined f o r the window which y i e l d e d the n a r r o w e s t peak f r e e from s i d e l o b e s a t s p e c i f i c f r a c t i o n s of the peak h e i g h t . From t h i s s y s t e m a t i c s e a r c h , v a r i o u s windows emerged as b e i n g e f f e c t i v e . These recommendations were examined t o ensure t h a t they had adequate SNRs, and t h a t two c l o s e l y spaced peaks c o u l d be r e s o l v e d . For the a b s o r p t i o n mode, d i f f e r e n t windows were recommended 101 f o r s p e c i f i c dynamic ranges. I f a dynamic range of 100:1 was r e q u i r e d , the N o e s t - K o r t window e l i m i n a t e d s i d e l o b e s f o r a l l dampings; however, t h i s window has a parameter which i s s e t t o c o i n c i d e w i t h the damping of the time-domain s i g n a l . I f a s i n g l e window which r e q u i r e s no a d j u s t m e n t s was p r e f e r r e d , the Norton-Beer F 3 w i n d o w s u f f i c e d , w i t h s l i g h t l y w i d e r l i n e s h a p e s than those due t o the N o e s t - K o r t window, f o r a l l dampings. For a dynamic range of 1000:1, the F i l l e r E0.20 and K a i s e r - B e s s e l windows both y i e l d e d s a t i s f a c t o r y r e s u l t s . These windows e l i m i n a t e d s i d e l o b e s f o r a l l dampings, but the l i n e s h a p e s were r a t h e r wide f o r h i g h l y damped s i g n a l s . I f a dynamic range of 10,000:1 was needed, none of the windows reduced s i d e l o b e s down t o t h i s l e v e l f o r m o d e r a t e l y t o t o t a l l y damped s i g n a l s . The K a i s e r - B e s s e l window d i d produce a smooth, though r a t h e r wide, l i n e s h a p e a t 0.01% of the peak h e i g h t f o r s i g n a l s of low damping. A p o d i z a t i o n of magnitude l i n e s h a p e s showed a more marked improvement than a p o d i z a t i o n of a b s o r p t i o n l i n e s h a p e s . For a dynamic range of 100:1, the Hamming window was recommended f o r undamped t o m o d e r a t e l y damped s i g n a l s . The l i n e s h a p e s due t o the 3-term B l a c k m a n - H a r r i s window, w h i l e w i d e r , s u f f i c e d f o r a l l dampings. T h i s window a l s o s a t i s f i e d r e q u i r e m e n t s f o r a dynamic range of 1000:1. The K a i s e r - B e s s e l window e l i m i n a t e d s i d e l o b e s f o r a l l dampings down t o a dynamic range of 10,000:1. The r e l a t i v e SNRs f o r the a b s o r p t i o n windows were found t o i n c r e a s e w i t h i n c r e a s e d damping. The v a l u e s ranged from about 102 h a l f the unapodized SNR f o r the undamped case t o g r e a t e r than the u n apodized SNR f o r h i g h l y damped c a s e s . Most windows which y i e l d e d narrow l i n e s h a p e s , those l i s t e d above, a l s o produced low SNRs compared t o the r a t i o s f o r windows which were not recommended. The recommended windows r e t a i n e d adequate SNRs, except the N o e s t - K o r t which has r a t h e r s m a l l v a l u e s f o r low dampings. For the magnitude windows, a l l of the SNRs were lower than those due t o the unapodized case f o r a l l dampings. The SNRs d e c r e a s e d a s m a l l amount as damping was i n c r e a s e d . The recommended windows a l l had s u f f i c i e n t SNRs, o f t e n b e i n g h i g h e r than the unrecommended windows. R e s o l u t i o n may be d e f i n e d as the h e i g h t of the v a l l e y between two peaks, w i t h a h e i g h t of 10% b e i n g s a t i s f a c t o r y . As two peaks were brought c l o s e t o g e t h e r , t h e i r s k i r t s o v e r l a p p e d and the h e i g h t between t h e s e peaks was s i m p l y t h e a d d i t i o n of the s i z e s of the s k i r t s f o r the a b s o r p t i o n mode. Thus the windows recommended f o r a s i n g l e a b s o r p t i o n peak were a l s o e f f e c t i v e f o r two c l o s e l y spaced peaks. Depending on the amount of damping, two peaks c o u l d be r e s o l v e d w i t h a t most a 10% v a l l e y f o r as c l o s e as t h r e e f r e q u e n c y s p a c i n g s , ( 1 / T ) . As the s p a c i n g between the peaks was d e c r e a s e d , the r e s o l u t i o n d e c r e a s e d . As the damping i n c r e a s e d the r e s o l u t i o n worsened. For the magnitude l i n e s h a p e s , the h e i g h t s of the v a l l e y s between the peaks were not found t o be a s i m p l e f u n c t i o n of 103 f r e q u e n c y s p a c i n g nor of damping. A c l e a r p a t t e r n was not obs e r v e d s i n c e magnitude l i n e s h a p e s a r e not a d d i t i v e . The' windows produced peaks r e s o l v e d w i t h a 10% v a l l e y f o r s i x s p a c i n g s , and f o r v a r i o u s s m a l l e r s p a c i n g s , but i n no c o n s i s t e n t p a t t e r n . The f l u c t u a t i n g b e h a v i o r of the v a l l e y h e i g h t s as a f u n c t i o n of s p a c i n g was e x p l a i n e d by c l o s e e x a m i n a t i o n of the a b s o r p t i o n and d i s p e r s i o n components of the magnitude l i n e s h a p e . These two components a l s o e x p l a i n the s u r p r i s i n g phase dependency of magnitude s p e c t r a . A p a t t e r n was obser v e d when t h e r e was a phase d i f f e r e n c e between two time-domain components. As the r e l a t i v e phase d i f f e r e n c e was i n c r e a s e d , the a p o d i z e d peaks became more r e s o l v e d up t o a phase d i f f e r e n c e of ir , and the unapodized peaks d e c r e a s e d i n r e s o l u t i o n . There a r e v a r i o u s a p p l i c a t i o n s f o r the r e s u l t s i n t h i s s t u d y . The windows recommended f o r a b s o r p t i o n c o u l d be a p p l i e d f o r FT-NMR i n p l a c e of the e m p i r i c a l windows i f a s i n g l e window were p r e f e r r e d over a d j u s t i n g windows. S i n c e the windows were shown t o be e f f e c t i v e i n e l i m i n a t i n g n e g a t i v e s i d e l o b e s of a b s o r p t i o n s p e c t r a , the problem of apparent phase d i s t o r t i o n s due to the d i s c r e t e n a t u r e of sa m p l i n g i s s o l v e d ; t h e r e f o r e the a b s o r p t i o n mode c o u l d be implemented i n FT-ICR. The r e c o g n i t i o n of a phase dependence of magnitude s p e c t r a i s of s i g n i f i c a n c e t o both FT-ICR and FT-NMR. The o b s e r v a t i o n t h a t t h e r e can be s l i g h t s h i f t s i n the fr e q u e n c y p o s i t i o n s of a p o d i z e d peaks i s of im p o r t a n c e , p a r t i c u l a r l y t o FT-ICR. Recommendations i n t h i s work 104 have seen p r a c t i c a l a p p l i c a t i o n w i t h p r o m i s i n g r e s u l t s . The windows recommended f o r magnitude have been a p p l i e d s u c c e s s f u l l y t o FT-ICR e x p e r i m e n t s , and the 3-term B l a c k m a n - H a r r i s window was found t o have a s u f f i c i e n t dynamic range. These r e s u l t s were an improvement over the unapodized s p e c t r a , and e l i m i n a t e d the s i d e l o b e s b e t t e r than the e x i s t i n g a p o d i z i n g f u n c t i o n . More r i g o r o u s t e s t i n g i s s t i l l r e q u i r e d , p a r t i c u l a r l y t o examine the e f f e c t s of a p o d i z a t i o n on the r e s o l u t i o n of v e r y c l o s e l y spaced peaks. 1 0 5 REFERENCES 1. R. N. B r a c e w e l l , T h e F o u r i e r T r a n s f o r m a n d i t s A p p l i c a t i o n s ( M c G r a w - H i l l , New Y o r k , 1978 ) . 2. H. J . W e a v e r , A p p l i c a t i o n s o f D i s c r e t e a n d C o n t i n u o u s F o u r i e r A n a l y s i s ( W i l e y - I n t e r s c i e n c e , New Y o r k , 1983 ) . 3. D. 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Opt. 2J_, 1866 (1982). 57. R. C. Denney, A D i c t i o n a r y of S p e c t r o s c o p y ( M a c M i l l a n P r e s s , London, 1982 ). 109 58. R. W. K i s e r , I n t t r o d u c t i o n t o Mass S p e c t r o m e t r y and i t s A p p l i c a t i o n s ( Englewood C l i f f s , NJ, P r e n t i c e - H a l l , 1965 ). 59. S. L. M u l l e n and A. G. M a r s h a l l , A n a l . Chim. A c t a j_78, 17 (1985). 110 A P P E N D I X A 1 C THIS PROGRAM COMPUTES AND PRINTS OUT NUMERICAL VALUES FOR 2 C AN APODIZED MAGNITUDE LINESHAPE WITH THE WINDOW BEING 3 C THE KAISER-BESSEL WINDOW. 4 C SLIGHT VARIATIONS CAN BE MADE TO OBTAIN AN ABSORPTION SPECTRUM. 5 C THIS IS A FORTRAN PROGRAM FOR THE UBC-MTS G SYSTEM. 6 C THIS PROGRAM WAS USED TO OBTAIN DATA FOR TABLE IV AND APPENDIX B: 7 C AND WITH A FEW CHANGES. ADDING PLOTTING PARAMETERS. WAS USED 8 C TO FOR PLOTS SUCH AS FIGS. 9, 10 AND 11. 9 C. 10 C VARIABLES IN THIS PROGRAM 11 C 12 C A TIME DOMAIN SIGNAL."Y", IS CALCULATED OVER 128 POINTS. AND THEN 13 C ZERO FILLED ACCORDING TO "M"; AND HAS TIME,*T". RANGING FROM O TO 1 . 14 C AN FT IS PERFORMED ON *Y" YIELDING "DATA" WITH THE REAL PART (THE 15 C COSINE FT) IN "DATAR", THE IMAGINARY PART (SINE FT) IN "DATAI", AND 16 C THE RELATIVE INTENSITIES ARE CALCULATED. AND THAT OF "DMAG" IS IN RM 17 C THE VALUES "X". "TAU", AND *W0" ARE FROM THE EO. 3 WHERE WO " 2*PI*F. 18 C 19 C SETTING THE PARAMETERS 20 C "M" IS THE TOTAL NUMBER OF POINTS TO BE TRANSFORMED. 21 C "M" IS EQUAL TO A POWER OF TWO. PLUS ONE: MORE SPECIFICALLY. 22 C M=128*(2**N)+1 ...WHERE N IS THE NUMBER OF ZERO FILLINGS. 23 C IF N IS GREATER THAN 7. IT IS NECESSARY TO CHANGE THE SIZE OF 24 C DIM "Y" TO EQUAL "M" . AND DIM "DATA" TO EOUAL ((M-1)/2)+1. 25 C "MAXF" IS THE MAXIMUM FREQUENCY (IN HZ) DESIRED IN THE FT. 26 C FREQUENCY RANGES FROM M TO M+"MAXF" (THIS M IS NOT RELATED TO THE NUMBER 27 C OF POINTS TRANSFORMED. "M".) 28 C CHOOSE AN INTEGER VALUE LESS THAN OR EQUAL TO 32 FOR "MAXF". 29 C TO GET AN IDEA OF THE NUMBER OF POINTS THAT WILL BE PLOTTED. 30 C THE NUMBER OF POINTS PLOTTED."NPTS"=1+("M"-1)*MAXF/128. 31 C TO CHANGE THE NUMBER OF POINTS PLOTTED. "M" OR "MAXF" CAN BE CHANGED. 32 C "LOW" AND "LARGE" EXTRACT THE RANGE OF POINTS PLOTTED FROM "DATA" 33 C SUCH THAT THE INITIAL VALUE IS AT THE CENTRAL MAXIMUM. WO. 34 C AND THE LAST VALUE IS AT THE FREQUENCY MAXF. 35 C "TAU" CAN BE ALTERED TO SEE THE EFFECTS OF DIFFERENT T/TAU VALUES. 36 C 37 C SUMMARY OF PARAMETERS TO CHANGE 38 C M LINE 48 39 C MAXF LINE 49 40 C TAU LINE 55 41 C WINDOW FORM..LINES 83 TO 93 42 C WINDOW NAME..LINE 124 43 C -44 IMPLICIT REAL*8(A-H.O-Z) 45 DIMENSION Y(16385),DATAR(4097 ) ,W(4097),DATAI(4097).DMAG(4097),RELM(4097) 46 DIMENSION RELA(4097),WINDOW(129) 47 C0MPLEX*16 DATA(8193) 48 M=4097 49 MAXF=32DO 50 LOW=((M-1)/4)+1 51 LARGE*((MAXF+32D0)*(M-1D0)/128DO)+1D0 52 NPTS«LARGE-L0W+1 53 DIV-1.ODO/128.ODO 54 DT»DIV 55 TAU=2.0D0 56 Pl=3.14159265358979D0 57 TW0PI«6.28318530717958D0 58 WO=32.ODO*TW0PI 1 1 1 59 C THE ALPHA PARAMETER IN THE KAISER-BESSEL WINDOW IS "A" 60 A=3.50O 61 TBW=PI*A 62 X*10.000 63 C 64 C CALCULATE THE DENOMINATOR OF THE KAISER-BESSEL WINDOW. 65 CC»1.ODO 66 CX*1.000 67 DO 5 K*1.20 68 CC=K*CC 69 CD=(TBW/2.0D0)**K 70 CY«=(CD/CC)**2D0 71 CX-CX+CY 72 i 5 CONTINUE 73 C WRITE THE HEADING OF THE TABLE 74 WRITE(6.80)CX 75 80 F0RMAT('THE DENOMINATOR IS '.F9.4) 76 WRITE(6.85)A 77 85 FORMAT('FOR A « '.F3.1) 78 C -CALCULATE Y, THE TIME DOMAIN SIGNAL; T RANGES FROM 0 TO 1 79 DO 10 1=1.129 80 T=(1-1)*DT 81 C CALCULATE THE KAISER-BESSEL WINDOW. NOTE: TO DO OTHER WIDOWS, 82 C INSERT THEIR FORM FROM TABLE III HERE. 83 B=TBW*(DABS(1.O00-(2.OD0*DABS(T-O.5DO))**2DO)**O.5D0) 84 C=1.ODO 85 T0T=1.ODO 86 00 7 K» 1,20 87 C-K*C 88 0=(B/2.0D0)**K 89 SUM=(D/C)**2D0 90 TOT-TOT+SUM 91 7 CONTINUE 92 WINDOW(I)-TOT/CX 93 WINDOW!" 129)=0.0D0 94 C 95 Y(I)=X*DEXP(-T/TAU)«DCOS(WO*T)*(WINDOW(I)) 96 10 CONTINUE 97 C ZERO FILL TO A TOTAL OF M POINTS 98 DO 15 I-130.M 99 Y(I)=0.0D0 100 15 CONTINUE 101 C -CALCULATE FT OF Y & STORE IN DATA AS DESCRIBED IN UBC FOURT' 102 CALL P0LFT2(Y.DATA.M.DT) 103 C -CALCULATE F-FO. THE FREQUENCY, AND STORE IN W(I) 104 C FCOR CORRECTS THE FREQUENCY 0 RANGE FROM 0 TO MAXF. 105 FCOR=DFLOAT(MAXF)/(DFLOAT(NPTS)-1.ODO) 106 DO 20 1*1,NPTS 107 W(I)«(DFLOAT(I)+32.ODO*TW0PI-WO-1.0D0)*FC0R 108 20 CONTINUE 109 C -CALCULATE THE MAGNITUDE LINESHAPE AND STORE IN OMAG(J) 110 DO 30 I'LOW.LARGE 11 1 U«I-L0W+1 112 DATAR(J)=DREAL(DATA(I)) 113 DATAI(d)-DIMAG(DATA(I)) . 1 14 DMAG(J)*(((DATAR(d))**2)+((DATAI(d))**2))**0.5D0 1 15 30 CONTINUE 116 C -CALCLATE RELATIVE INTENSITY OF THE MAGNITUDE LINESHAPE 112 117 RM=DMAG(1) 1 18 DO 5 0 I - 1 . N P T S 1 19 R E L M ( I ) = ( D M A G ( I ) ) / R M 120 5 0 CONTINUE 121 C -PRINT T I T L E • 122 W R I T E ( 6 , 1 0 0 ) 123 C ...WRITE THE NAME OF THE WINDOW AFTER "WINDOW - " 124 100 FORMAT('WINDOW - K A I S E R - B E S S E L ' ) 125 W R I T E C 6 , 1 2 0 ) 126 120 F0RMAT('MAGNITUDE F T ' ) 127 W R I T E C 6 . 1 2 2 ) M 128 122 FORMAT('POINTS TRANSFORMED : ' . I 5 ) 129 TTAU=1.ODO/TAU 130 W R I T E ( 6 . 1 2 5 ) T T A U 131 125 F O R M A T ( ' T / T A U = ' , F 5 . 3 ) 132 WRITE ( 6 . 1 5 0 ) 133 150 FORMAT(' ') 134 C -PRINT OUTPUT : W ( I ) , THE FREOUENCY I S IN U N I T S OF 1/T 135 C DMAG IS THE ABSOLUTE MAGNITUDE I N T E N S I T Y 136 C RELM I S THE NORMALIZED MAGNITUDE I N T E N S I T Y 137 W R I T E ( 6 , 2 0 0 ) 138 2 0 0 F O R M A T C W-WO [ X 2 P I J I N T E N S I T Y R E L . I N T 139 W R I T E ( 6 . 3 0 0 ) ( W ( I ) . D M A G ( I ) . R E L M ( I ) , I = 1 . 2 0 0 ) 140 3 0 0 F O R M A T ( 1 X , F 1 5 . 5 , F 1 5 . 5 , F 1 5 . 5 ) 141 STOP 142 END 1 1 3 APPENDIX B H e i g h t and p o s i t i o n of h i g h e s t s i d e l o b e : A b s o r p t i o n windows. Window T / r 0.0 0.5 1 .0 2^0 3 ^ R e c t a n g l e 21% 0.72 1 3% 0.75 9.7% 1 .2 1 1% 1 .2 5.6% 2.2 B l a c k m a n - H a r r i s 3-term 0.05% 1 .8 0.14% 5.1 0.12% 8.1 0.07% 16.0 -B l a c k m a n - H a r r i s 4-term 0.01% 2.2 0.01% 19.0 - --F i l l e r DO.24 X X 0.70% 2.4 X X X X X X F i l l e r E0.13 0.33% 1 .8 0.55% 2.6 -- -F i l l e r E0.20 0.09% 1 .8 - - - -G a u s s i a n 0.02% 2.8 0.08% 7.2 0.07% J 1 . 1 - -Hamming X X 1.1% 2.2 X X X X 0.59% 7.1 Hanning a=2 2.6% 1 .2 1 .7% 1 .7 - --Hanning a=4 0.44% 1 .7 X X X X X X X X K a i s e r - B e s s e l 0.01% 2.1 - - - -K a i s e r - B e s s e l 4 -term 0.04% 1.7 0.04% 9.1 0.05% 13.0 0.03% 29.9 -N o e s t - K o r t 0.91% 1.7 0.77% 1 .7 0.19% 2.1 0.19% 2.1 0.03% 2.7 Norton-Beer F3 0.36% 2.3 0.89% 2.2 0.82% 3.2 0.67% 5.1 X X 114 APPENDIX B c o n t d . T r i a n g l e X X X X 0.08% X X X X 24.3 H e i g h t and p o s i t i o n of h i g h e s t s i d e l o b e s : Magnitude windows. Window T / r 0.0 0.5 1.0 2.0 3.0 R e c t a n g l e 22% 22% 24% 29% 1 3% 1 .4 1 .4 1 .4 1 .3 4.3 B l a c k m a n - H a r r i s 3-term 0.04% 0.04% 0.04% 0.04% 0.05% 3.6 3.7 3.6 4.4 10.4 B l a c k m a n - H a r r i s 4-term 0.01% 0.01% 0.01% 0.00% 0.00% 4.5 4.5 4.5 7.3 16.8 G a u s s i a n 0.02% 0.02% 0.03% 0.06% 0.08% 5.9 6.0 5.6 4.7 4.8 Hamming 0.76% 0.80% 0.87% 1 .2% 1 .6% 4.4 4.4 4.4 4.4 4.4 Hanning a=2 X 2.7% X 0.53% 0.02% X 2.3 X 4.3 14.3 Hanning a=4 X 0.46% X X -X 3.3 X X K a i s e r - B e s s e l 0.01% 0.01% 0.00% - -3.8 4.3 4.9 — — K a i s e r - B e s s e l 4 -term 0.05% 0.05% 0.02% 0.03% 0.01% 3.3 3.3 4.5 4.5 7.4 N o e s t - K o r t 9.7% 9.4% X X X 2.2 2.2 X X X T r i a n g l e X 4.6% X X X X 2.8 X X X Upper v a l u e = h e i g h t of s i d e l o b e (% of peak h e i g h t ) Lower v a l u e = p o s i t i o n of s i d e l o b e (1/T from c e n t r a l maximum) = no s i d e l o b e s i n range examined X = no v a l u e c a l c u l a t e d 115 APPENDIX C Magnitude l i n e s h a p e v a l l e y s : Rectangle window. V f 1 T A (KO 1_;_C) 2M) 3_^ 0 1.0 0.07 36 65 84 2.0 0.11 4.7 11 20 3.0 5 . 2 s 1.7 4.1 7.6 4.0 9 . 8 s 1 2 s 4.9 11 5.0 1 2 s 1 4 s 4.1 7.6 6.0 14 s 16 s 2.2 4.9 s denotes s i d e l o b e s . Magnitude l i n e s h a p e v a l l e y s : 3-term Blackman-Harris. ._ _ CM) K O 2_;J) 3_;_0 1.0 0.07 18 33 43 2.0 100 100 100 100 3.0 0.02 23 46 68 4.0 19 s 21 s 29 s 41 5.0 0.04 11 s 12 21 6.0 1 0 s 1 1 s 3.6 8.6 s denotes s i d e l o b e s . 1 16 Magnitude l i n e s h a p e v a l l e y s : K a i s e r - B e s s e l window.  0.0 1.0 2.0 3.0 0.5 100 100 100 100 1 .0 0.02 22 43 61 1 .5 100 100 97 90 2.0 100 100 100 100 2.5 73 83 90 94 3.0 0.02 16 32 46 3.5 42 27 15 1.2 4.0 34 33 29 22 4.5 14 18 21 21 5.0 0.00 4.6 8.6 12 5.5 3.4 1 .2 1 .5 4.2 6.0 1.9 1 .5 0.55 0.88 1 1 7 Magnitude l i n e s h a p e v a l l e y s : 4-term Blackman-Harris window. 0.0 1.0 2.0 3.0 0.5 100 100 100 100 1 .0 0.07 36 65 84 1 .5 100 100 99 94 2.0 0.11 4.9 18 34 2.5 77 86 92 96 3.0 5.2 4.7 1 1 20 3.5 42 31 19 5.4 4.0 39 38 34 28 4.5 17 22 25 26 5.0 0.01 5.3 10 14 5.5 5.1 2.4 0.94 4.4 6.0 3.2 2.8 1 .7 0.10 118 Magnitude l i n e s h a p e v a l l e y s : Hanning a = 2 window.  0.0 1.0 2.0 3.0 0.5 100 100 100 100 1.0 0.03 26 50 70 1.5 96 86 71 52 2.0 98 97 95 89 2.5 45 59 69 76 3.0 0.02 16 32 45 3.5 8.8 0.08 11 23 4.0 0.00 1.5 5.9 13 4.5 3.5 s 3.3 s 5.0 s 8.5 5.0 1.2s 1.1 2.8 5.4 5.5 2 . 6 s 2 . 7 s 1.1 3.2 6.0 2.0 s 0.23 1.0 2.3 s denotes s i d e l o b e s . 119 APPENDIX D D e r i v a t i o n of Eq. 17. A time-domain s i g n a l t r a n s f o r m s t o y i e l d an a b s o r p t i o n and d i s p e r s i o n component i n the fr e q u e n c y domain h ( t ) > A - i D . A s h i f t e d time-domain s i g n a l i s r e l a t e d t o t h e u n s h i f t e d s i g n a l by h ( t - t Q ) = ( e x p ( - i 2 * f t Q ) ) ( A - i D ) = A c o s ( 2 f f f t 0 ) - D s i n ( 2 f f f t 0 ) - i ( D c o s ( 2 j r f t ) + A s i n ( 2 f f f t > ). (A1 ) o o The term 2 j r f t Q can be r e p l a c e d by 8 Q which r e p r e s e n t s the time s h i f t i n f r e q u e n c y u n i t s . S u b s t i t u t i n g t h i s i n t o Eq. A1, the g e n e r a l case where h n i s s h i f t e d by a time t R w i l l have an a b s o r p t i o n l i n e s h a p e of the form A. = A cos6„ - D sin6„ (A2) t n n n n n and the d i s p e r s i o n l i n e s h a p e w i l l be of the form D t = A sin& V + D c o s 8 . (A3) tn n n n n 120 For a s i g n a l c o n s i s t i n g of two components, and h 2 , w i t h phase s h i f t s of 9. and 0 O r e s p e c t i v e l y , the r e s u l t i n g A. and A w i l l \ z t>i t 2 each t a k e the form of Eq. A2; and the r e s u l t i n g D. and D w i l l have the form of Eq. A3. A time s h i f t causes the A n and D n components of Eq. 16 t o change, but not the M n components. In o r d e r t o o b t a i n an e x p r e s s i o n f o r M where t h e r e i s a r e l a t i v e phase d i f f e r e n c e , some s u b s t i t u t i o n s a r e made i n Eq. 16 : the components A,, A 2, and D 0 a r e s u b s t i t u t e d by A , A , D. and D r e s p e c t i v e l y where the t i m e - s h i f t e d components have been d e s c r i b e d above. The r e s u l t i n g e x p r e s s i o n i s expanded, and use of t r i g o n o m e t r i c i d e n t i t i e s y i e l d s Eq. 17 as the e x p r e s s i o n f o r a magnitude l i n e s h a p e due t o a time-domain s i g n a l c o n s i s t i n g of two components each w i t h a d i f f e r e n t phase a n g l e . 121 P u b l i c a t i o n s M. B. Comisarow and J . Lee, '"Phase D i s t o r t i o n s " i n Absor p t i o n - M o d e F o u r i e r T r a n s f o r m Ion C y c l o t r o n Resonance S p e c t r a ' , A n a l . Chem. 57, 464 (1985). J . P. Lee and M. B. Comisarow, 'Advantageous A p o d i z a t i o n F u n c t i o n s f o r Magnitude-Mode F o u r i e r T r a n s f o r m S p e c t r o s c o p y ' , A p p l . S p e c t r o s c . ( i n p r e s s ). J . P. Lee and M. B. Comisarow, 'Advantageous A p o d i z a t i o n F u n c t i o n s f o r Absorption-Mode F o u r i e r T r a n s f o r m S p e c t r o s c o p y ' , A p p l . S p e c t r o s c . ( t o be s u b m i t t e d ). J . P. Lee and M. B. Comisarow, 'The Phase Dependence of Magnitude S p e c t r a ' , Chem. Phys. L e t . ( t o be s u b m i t t e d ). J . P. Lee and M. B. Comisarow, 'Anomolous I n t e n s i t i e s and the Phase Dependence of Ap o d i z e d Magnitude S p e c t r a ' , ( m a n u s c r i p t i n p r e p a r a t i o n ). 

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