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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 THE EFFECTS ON SNR AND  AND  RESOLUTION  By Judy P i h s i e n B.Sc,  Lee  Dalhousie University,  1984  A THESIS SUBMITTED I N PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE  in 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 a s c o n f o r m i n g to the required  standard  THE UNIVERSITY OF B R I T I S H COLUMBIA J u n e 1986 ©  Judy P i h s i e n  Lee  In p r e s e n t i n g requirements  this thesis f o r an  of  British  it  freely available  Columbia, I agree that for reference  understood for  by  that  h i s or  be  her  copying or  f i n a n c i a l gain  shall  DE-6  (3/81)  J~~WAg  2.0,  lt$l>  Library  s h a l l make  and  study.  I  publication be  the  of  further this  Columbia  thesis  head o f  this  my  It is thesis  a l l o w e d w i t h o u t my  C-Kg./^; s/ry  The U n i v e r s i t y o f B r i t i s h 1956 Main Mall V a n c o u v e r , Canada V6T 1Y3 Date  University  representatives.  not  the  the  g r a n t e d by  permission.  Department of  the  f o r extensive copying of  s c h o l a r l y p u r p o s e s may  department or  f u l f i l m e n t of  advanced degree at  agree that permission for  in partial  written  ABSTRACT  The  problem  of  transform  spectrum  performed  by  function. on  at  specific  the  for  the  3-term  m a g n i t u d e mode.  and  absorption the  apodization.  This  is  a  of  functions  the  e f f e c t s of window  window i s shown t o be h a l f of  mode.  efficient  window  S e l e c t i o n of  are  resolved, with a  The 10%  m a g n i t u d e s p e c t r a do  not  absorption criterion  d i s p l a y a simple  components.  increasing  Filler mode;  i i  and  the the  expense  except  spectra  are  pattern, by  and the  of  examined.  SNRs  These f i n d i n g s l e n d  various applications.  E0.20,  for well  for resolution.  a phase dependence; however t h e s e a r e e x p l a i n e d dispersion  an  these f a c t o r s are a l s o  the  is  recommended  o f t e n e l i m i n a t e d at the  apodized  v a l l e y as  shape  K a i s e r - B e s s e l work f o r  recommended windows show s u f f i c i e n t  Noest-Kort.  For  Norton-Beer F3,  B l a c k m a n - H a r r i s and Sidelobes  a  and  effective  symmetrical  f o r the a b s o r p t i o n  and/or r e s o l u t i o n , t h e r e f o r e of t h e  the  r e q u i r e d dynamic range.  t h e N o e s t - K o r t and  Kaiser-Bessel are  Hamming,  and  Fourier  dynamic r a n g e s f o r both the a b s o r p t i o n  m a g n i t u d e mode,  dynamic range,  the  a  t i m e - d o m a i n f u n c t i o n by  A symmetrical  window i s b a s e d on  All  study  by  a peak i n  damped t i m e - d o m a i n s i g n a l s i s made by e x a m i n i n g t h e r e s u l t i n g  better  SNR  surrounding  alleviated  m u l t i p l y i n g the  m a g n i t u d e modes.  and  is  A systematic  lineshapes  for  sidelobes  The  a l s o show absorption  themselves  to  TABLE OF CONTENTS  Chapter  Page  1 . Introduction 2. A p o d i z a t i o n  1 of A b s o r p t i o n  10  2.2 P r o c e d u r e  18 Mode R e s u l t s a n d D i s c u s s i o n s  25  2.4 M a g n i t u d e Mode R e s u l t s a n d 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  Signal-To-Noise  Ratio  3.1 I n t r o d u c t i o n  54  3.2 P r 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 a n d D i s c u s s i o n s  57  3.4 M a g n i t u d e Mode R e s u l t s a n d 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 o f A p o d i z e d  Spectra  4.1 I n t r o d u c t i o n  66  4.2 P r o c e d u r e  69  4.3 A b s o r p t i o n  5.  Spectra  2.1 I n t r o d u c t i o n  2.3 A b s o r p t i o n  3.  and Magnitude  Mode R e s u l t s a n d D i s c u s s i o n s  72  4.4 M a g n i t u d e Mode R e s u l t s a n d 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  Summary  1  References  1  iii  01  06  Appendix A  1 1 1  Appendix B  1 1 4  Appendix C  1 1 6  Appendix D  •  iv  1  20  LIST  OF T A B L E S  Table  Page  I.  Window f u n c t i o n s f o r a b s o r p t i o n  26  II.  Apodized 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  27  III.  Window f u n c t i o n s f o r magnitude  40  IV.  Apodized magnitude s p e c t r a h a l f w i d t h s  41  V.  R e l a t i v e SNRs : a b s o r p t i o n windows  58  VI.  R e l a t i v e SNRs : magnitude windows  62  VII.  Absorption  l i n e s h a p e v a l l e y s due to the  K a i s e r - B e s s e l window VIII. Absorption  75  l i n e s h a p e v a l l e y s due to the  3-term Blackman-Harris window  v  75  L I S T OF  FIGURES  Figure  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  2.  Unapodized  absorption  lineshapes  due  13  to  various  dampings 3.  Unapodized  15 magnitude  lineshapes  due  to  various  dampings 4.  16  Window t y p e s a n d r e s u l t i n g a b s o r p t i o n a n d  magnitude  lineshapes 5.  21  C o m p a r i s o n o f s u i t a b l e and u n f a v o r a b l e  absorption  lineshapes 6.  Absorption  31 l i n e s h a p e s due t o recommended windows  and  moderate damping 7.  Absorption  34  l i n e s h a p e s due t o recommended windows a n d  h i g h damping 8.  Absorption  35  l i n e s h a p e s due t o recommended windows  and  complete damping 9.  Comparison  of  36 s u i t a b l e and  unfavorable  magnitude  lineshapes  44  10. M a g n i t u d e l i n e s h a p e s due t o recommended  windows  and  moderate damping  .....46  11. M a g n i t u d e l i n e s h a p e s due t o recommended complete damping  windows and 47  vi  12. M a g n i t u d e l i n e s h a p e s Kaiser-Bessel 13.  Relative  resulting  signal-to-noise  r a t i o s due  16. V a l l e y h e i g h t s  as a f u n c t i o n  63  of spacing  and  apodized absorption of spacing  damping  peaks and  due t o t h e K a i s e r - B e s s e l  two p e a k s s e p a r a t e d by t h r e e  73  damping  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  19. M a g n i t u d e l i n e s h a p e for  magnitude  68  as a f u n c t i o n  18. M a g n i t u d e l i n e s h a p e for  to  of damping  two p e a k s  two  absorption 59  r a t i o s due  15. V a l l e y between  two K a i s e r - B e s s e l  to  of damping  signal-to-noise  17. V a l l e y h e i g h t s  i n the 49  windows a s a f u n c t i o n  of  a  window  windows a s a f u n c t i o n 14. R e l a t i v e  from v a r y i n g  of  peaks window  spacings  79  due t o t h e K a i s e r - B e s s e l  window  two p e a k s s e p a r a t e d by two s p a c i n g s  20. M a g n i t u d e l i n e s h a p e  80  due t o t h e K a i s e r - B e s s e l  window  f o r two p e a k s s e p a r a t e d by one s p a c i n g 21.  Valley heights of  23.  and  damping  of spacing  apodized  as a f u n c t i o n  magnitude of spacing  82 and  damping  peaks and  a s a' f u n c t i o n  of spacing  two H a n n i n g a p o d i z e d m a g n i t u d e p e a k s  vii  and  84  damping  two 4 - t e r m B l a c k m a n - H a r r i s a p o d i z e d m a g n i t u d e p e a k s  24. V a l l e y h e i g h t s of  as a f u n c t i o n  two K a i s e r - B e s s e l  Valley heights of  of spacing  81  two u n a p o d i z e d m a g n i t u d e p e a k s  22. V a l l e y h e i g h t s of  as a f u n c t i o n  76  85  damping 87  25. L i n e s h a p e s o f a s i n g l e peak a p o d i z e d by t h e 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 o f two p e a k s s e p a r a t e d by two s p a c i n g s  and 91  a p o d i z e d by t h e K a i s e r - B e s s e l window 27. L i n e s h a p e s o f two p e a k s s p a r a t e d by one  spacing  and  a p o d i z e d by t h e K a i s e r - B e s s e l window  92  28. M a g n i t u d e l i n e s h a p e s f r o m s i g n a l w i t h no p h a s e s 29. M a g n i t u d e l i n e s h a p e s f r o m s i g n a l where one  component  h a s no p h a s e , and t h e o t h e r h a s a p h a s e of T T / 2 30. M a g n i t u d e l i n e s h a p e s f r o m s i g n a l where one has no p h a s e , and t h e o t h e r has a p h a s e o f  viii  96  97  compoment ir  98  ACKNOWLEDGEMENT  I  would l i k e  t o take  t h i s o p p o r t u n i t y t o thank Dr.  Comisarow f o r h i s g u i d a n c e and e n t h u s i a s m i n t h i s for his discussions c l a r i f y i n g this  field.  spectra  I  to test  discussions  also  wish  his  research,  t o thank  and f o r p r o o f r e a d i n g ;  Mark A a r s t o l  t h i s manuscript,  all  of h i s help.  f o r running  and f o r a l l  and Greg S t a t t e r  of the for  his  T h a n k s a l s o go t o D r . M. W. B l a d e s a n d  group f o r the use o f t h e i r microcomputer f o r the  of  and  t h e many s u b t l e t i e s e n c o u n t e r e d i n  apodization functions,  explanations of formulas.  M e l B.  and a s p e c i a l  ix  processing  t h a n k y o u t o Zane W a l k e r f o r  CHAPTER 1 INTRODUCTION  Fourier analysis. analyze  i s a powerful technique f o r  With t h i s mathematical procedure, a  complicated  components, resulting This  transformation  time-domain  however a f r e q u e n t l y  signal  i t i s possible for  i t s frequency  feature  p r o b l e m c a n be a l l e v i a t e d by a p o d i z a t i o n ,  of s i d e l o b e s .  but t h i s  a f f e c t s t h e s i g n a l - t o - n o i s e r a t i o and t h e r e s o l u t i o n .  resulting  examination  and  Given  a w a v e f o r m composed o f a sum o f s i n u s o i d s ,  identify  were s t r u c k  their  simultaneously  different  with  to separate  out  One e x a m p l e w o u l d be an  but with d i f f e r e n t f o r c e ,  intensities.  i t w o u l d be  the d i f f e r e n t  Another  example  notes  would  be  and an  w a v e f o r m whose f r e q u e n c y c o m p o n e n t s a r e i d e n t i f i e d i n  spectrum.  components  each  I f , f o r e x a m p l e , two d i f f e r e n t p i a n o s t r i n g s  f o r t h e human e a r t o d i s c e r n  electrical a  i t i s possible  each i n d i v i d u a l component.  a c o u s t i c waveform.  possible  sidelobes  defined.  i t s own f r e q u e n c y a n d a m p l i t u d e , and  of  to  e f f e c t s some b a s i c c o n c e p t s a b o u t F o u r i e r a n a l y s i s be  removal  Prior  the  will  the  i n turn  and  apodization  of  to  encountered problem i s that the  s p e c t r u m c a n have t h e u n d e s i r a b l e  detailed  spectral  A n a l y s i s of a time-domain s i g n a l f o r i t s frequency can  be p e r f o r m e d by a p r o c e d u r e known a s  a  Fourier  transform (FT). Fourier  transform  t e c h n i q u e s encompass a wide  1  variety  of  applications.  The c o n c e p t  into  heat  The  relationship  common  conduction  [1,2],  between  applications  systems  itself  to  was c o n c e i v e d d u r i n g  a n d i s now u s e d Fourier  mechanical  i n many  transformation and  electrical  Other  a p p l i c a t i o n s c a n be f o u n d  as m e d i c i n e ,  a r c h i t e c t u r e and geology  employs  FT i s o p t i c s  the  holography In  and o p t i c a l  chemistry,  studies  [7,8],  spectroscopy magnetic atomic  their  [9,10].  resonance  have  (NMR),  [4].  Another  been a p p l i e d  include  in  detail fields  field  which  diverse  as  to electrochemical is  in  infrared  ioncyclotron  emission or o p t i c a l  the  [6],  predominance  These  and  i n such v a r i e d  w i t h a p p l i c a t i o n as  spectroscopy  FTs  but  [5],  areas.  engineering  a s w e l l a s o t h e r s y s t e m s h a v e been d i s c u s s e d  [1,2,3].  research  the area  of  ( I R ) , nuclear  resonance  emission spectroscopies.  (ICR),  and  There  are  e x t e n s i v e r e v i e w s on F T - I R [ 1 1 , 1 2 , 1 3 ] ,  FT-NMR [ 1 4 , 1 5 ] , a n d F T - I C R  [16,17,18,19],  emission  a n d a l s o work on a t o m i c  Although commercially The  first  FT-NMR  One  commercial  factor  in  spectroscopy  t h e p a s t few d e c a d e s .  was p r o d u c e d  was m a n u f a c t u r e d  some y e a r s  the enticement  i n 1958.  in  1964, t h e  later  [ 9 ] , and  i n 1981.  of the m u l t i p l e x ,  detailed  until  i t was n o t a p p l i e d  FT-IR i n s t r u m e n t  F T - I C R was s o l d  realization  time  t o chemical  spectrometer  the f i r s t  first  t h e FT h a s e x i s t e d s i n c e 1 8 2 2 ,  [20,21].  of t h e use o f  FTs  or F e l l g e t t advantage.  was  the  This  was  I t c a n . be a p p r o a c h e d i n t e r m s o f  advantage or a s i g n a l - t o - n o i s e r a t i o  2  (SNR) a d v a n t a g e o f  a an  FT  spectrometer  be  the  comparison  spectrometer. containing method SNR.  over a c o n v e n t i o n a l spectrometer.  N  would Thus  If  a  the  scanning FT  need a t o t a l  spectrometer  spectrometer  s p e c t r a l elements  the  conventional  of  t i m e T,  b a s i c a d v a n t a g e o f an  one i s s p e e d .  T,  a  signal scanning the  FT s p e c t r o m e t e r  a scanning spectrometer  times.  same  over  In a  length  a in of  examines N d i f f e r e n t  spectral  I n t h e same l e n g t h o f t i m e , a FT s p e c t r o m e t e r  examines  time-domain data p o i n t s which c o n t a i n s p e c t r a l size  of a s i g n a l  that  i t i s observed,  time,  a  FT  An a l t e r n a t i v e v i e w c a n be made  time,  i s directly  element  information.  The  p r o p o r t i o n a l t o the l e n g t h of  time  a n d random n o i s e i s p r o p o r t i o n a l  root of the s i g n a l .  spectral  an  t i m e o f NT i n o r d e r t o g e t  o f t h e SNR due t o e q u a l s a m p l i n g  square  to  acquires  fora total  terms  elements.  A model c o u l d  the  For the scanning spectrometer,  i s o n l y examined f o r a f r a c t i o n of  that i s the signal  to  i s proportional  t o T/N.  each  the  total  Thus t h e 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 ' , w h i c h makes t h e r a t i o o f t h e s i g n a l t o 1 / 2  the  noise  spectrometer total  equal  to  T/N ' .  examines  time,  T,  proportional  to T  each  giving 1 > / 2  ,  On  the  other  hand,  the  component f o r a t i m e e q u a l  a  signal  proportional  a n d a SNR o f T ^ . 1  2  to  T h i s SNR i s N  to T,  FT the  noise times  1 / / 2  b e t t e r t h a n t h a t f r o m a c o n v e n t i o n a l m e t h o d , b u t t h e r e i s o n l y an a d v a n t a g e when N i s g r e a t e r t h a n o n e . Another spectroscopy  factor has  which been  aided  the  in  the  progress  in  3  development computing.  of  FT  Before  d e s c r i b i n g the computer breakthrough, necessary.  Basically,  r e l a t i n g a general  a  FT  function  can  i n the  be  with the  thought of as  time domain,  frequency-domain d e s c r i p t i o n , G ( f ) . is  some background on FTs  One  a  g(t),  is  formula with  its  method of d e p i c t i n g  this  formula  (1)  This  is  a  continuous  a n a l y t i c a l l y by  function,  integration.  and  G(f)  However,  can  be  obtained  i n experimental p r a c t i c e  the a n a l y t i c a l e x p r e s s i o n  f o r the time-domain f u n c t i o n , g ( t ) , can  be  i s not  q u i t e c o m p l i c a t e d and  an a n a l y t i c a l t r a n s f o r m can still  possible  d e s c r i b i n g the  to  not  compute  always e a s i l y a t t a i n a b l e and be  a  r e a d i l y performed.  FT.  If  the  Yet  numerical  time-domain f u n c t i o n are o b t a i n a b l e ,  that  so  i t is values  is i f a  value i s known f o r the time-domain f u n c t i o n at d i s c r e t e i n t e r v a l s in  time,  t . n  performed.  then  An  continuous FT,  expression  Eq.  1 , can  G(f >k  where  N  i s the  a discrete Fourier  I  similar  be given  transform  to  that  f o r the DFT  n  k  t o t a l number of  the t r a n s f o r m s t i l l  can  describing  required  4  the  (2)  n  d i s c r e t e p o i n t s sampled.  extensive  be  by  g(t )exp(-i2*f t )  d i g i t a l data i s i d e a l f o r computer a n a l y s i s , of  (DFT)  but  the  computing  The  computation time.  The  turning the  p o i n t came w i t h t h e c r e a t i o n of an e f f i c i e n t  Cooley-Tukey  Fourier  algorithm  transform  (FFT),  about FFTs [ 2 2 ] , emergence of  which  i s a l s o known  t o compute DFTs.  o f t h e FFT  i n 1965  was  fast  been w r i t t e n  algorithms  a great a i d  the  [23].  i n the  The  computation  FTs. Fourier  calculated the  transform q u i c k l y and  s p e c t r a of l a r g e d a t a efficiently.  t r a n s f o r m a t i o n o f d a t a do  sampled  at  spectrum  occurs  discrete  which  Erroneous  exist.  The  i n t e r v a l s to y i e l d  can  the sampling  be  However a l l o f t h e s e  be  DFT  discrete,  analysis,  equally  l e n g t h T.  the  spaced  points,  then  height [24],  of  intervals  be  from  is  frequency the  peaks.  aliasing  which  Sidelobes  l e n g t h of the d a t a  is  can set.  collected  f o r an a c q u i s i t i o n  frequency 1/T  between  time  at of  spectrum w i l l c o n s i s t  b e t w e e n them. two  i t s c o r r e c t h e i g h t would not  errors, can  a peak f e l l  be  alleviated.  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 value  discrete  time-domain data  In the t r a n s f o r m , the  maximum  a  rate i s insufficient.  problems can  now  time-domain s i g n a l  formulated  finite  In  can  c o u l d have e r r o r s i n the h e i g h t s of  frequencies  when  sets  However some p r o b l e m s w i t h  f o r m a r o u n d a peak b e c a u s e o f t h e  of  as  Much h a s  i n c l u d i n g d e t a i l s of new  algorithm,  discrete  the  frequency  detected.  Peak  sometimes r e f e r r e d t o as the p i c k e t fence  effect  r e m e d i e d by e x t e n d i n g  the time-domain data set  zeros,  which i s  called zero-filling  points  between the p r e v i o u s d i s c r e t e  5  be  If  [25,26]. frequency  This  with  incorporates  points,  and  an  infinite  number  frequency obtained In two  Thus  in a discrete order  time  per  cycle  are  c o r r e c t peak  sampled  required  i t s a m p l i t u d e and  the  continuous  heights  properly,  to  can  The  t w i c e the  i s the N y q u i s t sampling theorem.  be  least  sufficient  sampling  rate in  maximum  frequency.  I f i t i s not  satisfied,  i t s e l f or i s a l i a s e d  selecting a sufficient  at  obtain  frequency.  d o m a i n must be a t l e a s t  a v o i d e d by  yield  frequency l i n e s h a p e .  t h e s p e c t r u m f o l d s b a c k on be  the  f o r a w a v e f o r m t o be  i n f o r m a t i o n on  This  z e r o - f i l l i n g s would  lineshape.  points  the  of  sampling  [22]. rate  This  in  the  can time  domain. S i d e l o b e s around peaks can on  a few FT p a i r s .  domain,  its  frequency  time-domain f u n c t i o n Obviously, infinite it  If a signal  transform  domain;  sampling  sine  an  i s a delta of  i s equal to zero a f t e r  time  is  w i t h some k n o w l e d g e  is infinitely infinitely  f o r example,  l e n g t h of t i m e .  function,  be u n d e r s t o o d  narrow  t h e FT o f an function  long i n the peak  infinite,  the a c q u i s i t i o n  t i m e , T.  and  Thus a t r u n c a t e d t i m e s i g n a l  origin  for  When a  an  boxcar in  the  i t yields  leads to a  a  frequency  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 , definition  domain.  i s truncated; that i s ,  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,  f u n c t i o n w i t h a peak s u r r o u n d e d  the  constant  a s i g n a l c a n n o t be u n d e r t a k e 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 t i m e T,  function.  in  i n the frequency  Thus t h e s i g n a l  time  of t h i s term have  6  been  and  the  discussed [27].  Apodization  can  be  p e r f o r m e d by  multiplying  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 There  is  a  multitude  windowing. function window  In  sampled  this  by  apodization.  terminology  work,  There  time-domain are  function  numerous  apodization  weighting  function,  and t i m e - d o m a i n  Fourier  negative  The lobes  Sometimes lobes,  in  this  work,  the  nor  literature  is  literature  a  l e n g t h o f t i m e T.  A  the  terms  for this,  work, t h e t e r m s k i r t  a  smoothing  which  is  smooth.  of  window  function,  a  a frequency-domain  function,  a  is  weightfunction.  peak i n  shape a t the base of a i t  purpose  for  discrete  spectrum  and  discrete  term s i d e l o b e r e f e r s t o both the surrounding  on  function r e f e r s to  T h i s h a s been r e f e r r e d t o a s a  frequency-response  transform.  the  function,  transformation,  i s produced.  spectrum,  transformation.  for  other  including  lineshape  time-domain  i s the f u n c t i o n d e f i n e d over a time T which the  transformation  in  a time-domain  function  After  to Fourier  from a time of 0 t o a t o t a l  function  multiplied  of  the  the  peak i s n o t  There are c o l o r f u l  such as wings,  wiggles  r e f e r s to the general  and  shape  measured Fourier  positive  frequency of terms feet.  or  and  domain. distinct in  the  In  this  a t the base of a  peak. There  is  literature, thorough literature  extensive  ranging  analyses  d i s c u s s i o n o f window f u n c t i o n s  from  applications  of dozens  were e x c l u d e d  windows.  from t h i s  7  study.  of Many  single windows  in  windows from  the to the  T h e s e i n c l u d e d windows  w h i c h had p a r a m e t e r s  requiring  differing  [27,28,29];  the  conditions  frequency  application  domain  adjustments,  and t h o s e w h i c h were a p p l i e d  [30,31,32],  procedures  often empirical, for  besides  and  had  windowing  various  i n . the  to  other  time-domain  [33,34,35] . The  windows chosen  f o r t h i s study are those which are  implemented  by m u l t i p l i c a t i o n  complex  empirical  or  windows  deals  presents the r e s u l t s this  i n t h e t i m e domain w i t h o u t need f o r  adjustments  only  with  .  Most of. t h e l i t e r a t u r e  undamped  in either study  time-domain  Thus  signals  f o r b o t h t h e a b s o r p t i o n and m a g n i t u d e  systems  deals with both  undamped modes.  absorption and  damped  The  signals  h o w e v e r t h e m o d e l s c a n be a p p l i e d  C h a p t e r 2,  t h e e f f e c t s o f windows a r e e x a m i n e d  f o r the a b s o r p t i o n and magnitude signals  lineshapes resulting  of v a r i o u s dampings.  windows  and  magnitude  lineshapes i s discussed.  recommended  the  as  behavior  a result  of  the  Aside  to  real  narrow  signal-to-noise ratio  f u n c t i o n can a f f e c t  damping of t h e time-domain  resulting  t h e SNR  time-  shapes  absorption  of and  window f u n c t i o n s a r e they  eliminate  lineshape.  a good (SNR).  lineshape  should  Application  of a s p e c t r u m .  s i g n a l a l s o changes  8  in detail  from  different  Specific  f r o m h a v i n g low s i d e l o b e s ,  h a v e an a d e q u a t e  Two  of t h e e x t e n t t o w h i c h  sidelobes while retaining a f a i r l y  window  or  and  s u c h a s F T - I C R and FT-NMR.  In  domain  on  signals,  t h e power, magnitude  mode.  have been s y n t h e s i z e d ,  easily  of  Altering  t h e SNR.  a the  Chapter  3  examines  the absorption  and m a g n i t u d e SNRs due  window f u n c t i o n s a n d d i f f e r e n t amounts An  important  resolution. lineshapes  aspect  of  C h a p t e r 4 examines a p o d i z e d with  two  closely  spaced  t o t h e amount o f d a m p i n g  peaks.  9  different  damping.  frequency-domain  f u n c t i o n s a r e a p p l i e d , and t h e r e s u l t s with respect  of  to  absorption peaks.  spectra  is  and magnitude  Various  window  f o r two p e a k s a r e e x a m i n e d and t h e s p a c i n g  between  the  2  CHAPTER APODIZATION  2.1  Fourier  composite  of  such  a  transform  signal  AND  MAGNITUDE  where  T  K  the  is  is  the  SPECTRA  d e t a i l s  ICR  have  been  on  Eq.  3  cos(2fff t)  This  s u f f i c i e n t l y resulting  at T  Thus  continuous  a  or  by  sampling  at  a  rate  values  is  the  time,  T,  is  S  of  signal  functions.  An  is  example  if  and  the  data  The  continuous N  by  the  FTs  is  zero  have  if  been  is  discrete signal,  values.  signal.  The  rate,  for  enough,  a  the  continuous  y i e l d  the  t o t a l  a  quasi-  signal  form  The  S,  been  time-domain  time-domain with  FT-  performed  the can  to  sampled  f i l l e d  signal  [14],  have  the  it  and  FT-NMR  relation  resemble  sampling  time,  lineshapes  and  numerical  10  in  to  spectroscopy,  w i l l  time-domain  determined  equation  points;  (3)  relaxation  applicable  time-domain  spectrum.  discrete  is  the  resulting  Fourier  storing  is  this  lineshape  the  0<t<T  Analytical  the  discrete  frequency  in  discrete  frequency  lineshape.  on  For  r  equation  [36].  d e t a i l s  long  sum  time,  variables  described  sampled  a  time-domain  e x p ( - t / r )  D  [36,37],  is  obtained  the  and  calculated signal  = K  acquisition  of  of  the  is  amplitude.  and  spectroscopy,  consisting  function  g(t)  3,  ABSORPTION  Introduction In  a  OF  of  array  is Eq.  of  N  acquisition  and  the  total  number o f d a t a  p o i n t s , N, a c c o r d i n g t o  T = (N-1)/S.  A  numerical  frequency  Fourier transform  spectrum  (4)  i s performed t o y i e l d  w i t h t h e d i s t a n c e between  the  a  discrete  points,  the  s p a c i n g , g i v e n by  Af =  Recall imaginary  that  a DFT,  part.  The  Eq.  real  1/T.  2,  part,  (5)  c o n t a i n s both a r e a l also  transform, g i v e s t h e absorption spectrum. sine  transform,  lineshape squares mode  as  the  The i m a g i n a r y  y i e l d s the d i s p e r s i o n spectrum.  results  an  cosine  part, or  The m a g n i t u d e  f r o m t a k i n g t h e s q u a r e r o o t o f t h e sum o f t h e  o f t h e a b s o r p t i o n and d i s p e r s i o n  of  known  and  presentation  lineshapes.  i s power w h i c h i s m e r e l y  A  the  final  magnitude  squared. The  a b s o r p t i o n mode i s u s e d f o r s p e c t r a l  high  r e s o l u t i o n FT-NMR.  ICR,  and  also  representation  The m a g n i t u d e mode i s u t i l i z e d  f o r c e r t a i n a p p l i c a t i o n s o f NMR  been b e c a u s e o f what a p p e a r s t o be c o n t a m i n a t i o n component. it  has  discrete  This takes  been  nature  by a  of sampling  are actually  and n e g a t i v e  1 1  the  dispersion  result  absorption  The  but has not  t h e form o f n e g a t i v e components,  shown t h a t t h e s e  f o r FT-  [14,38,39].  a b s o r p t i o n mode c a n i n p r i n c i p l e be u s e d f o r F T - I C R ,  in  however of the  intensities,  and  that  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  absorption  mode t o F T - I C R [ 4 0 ] .  For engineering  open  the  purposes,  the  power mode i s u s e d . Chemical involve  applications,  such as FT-ICR  damped t i m e - d o m a i n s i g n a l s .  time-domain s i g n a l  i s r e p r e s e n t e d by t h e r a t i o o f  showing v a r i o u s r a t i o s i n F i g .  the  signal  i s undamped.  by  T/T  moderate,  A partially  and  high  domain  signal  creates  that  bottom  signal  reference  to  calculated  with  a  totally T/r  consistent  i n the factor  of  different  signal  damping [ 3 6 , 4 1 ] , held constant  f i x e d and o n l y  t o low,  virtually  signal,  totally  r  :  a  time-  amplitude. this  thus  value future  corresponds  Some f i g u r e s  a n d i n o t h e r s T was v a r i e d . held  A  to  one  h a v e been shown  however t h e y  for  were  i n some f i g u r e s  not  r  Here, the a c q u i s i t i o n  i s varied  3  i s represented  i t i s evident  damped  damped  =3.0.  amounts  is  1,  totally  different  T,  i n Eq.  T/ r = 3.0 w h i c h d e p i c t s  in Fig.  often  T/ r = 0.0,  h a s d e c a y e d t o 5% o f i t s o r i g i n a l  an e s s e n t i a l l y  varied,  When  damped s i g n a l  amounts of damping.  signal corresponds t o  the  1.  T/ r  1.0, a n d 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  damped  From  FT-NMR,  The amount o f d a m p i n g o f a  with plots  = 0.5,  and  was time,  t o produce s i g n a l s  of  dampings.  Damping o f t h e t i m e - d o m a i n s i g n a l h a s a m a r k e d e f f e c t on t h e frequency spectrum. lineshapes  due  a b s o r p t i o n mode,  to  T h i s i s a p p a r e n t by c o m p a r i n g t h e f r e q u e n c y d i f f e r e n t amounts o f d a m p i n g  for  F i g . 2, a n d t h e m a g n i t u d e mode, F i g .  12  both 3.  the These  T / t = 0.0  T/t=1.0  0.0  T / t =  3.0  0.0  Figure  1.  Time-domain f u n c t i o n s with v a r i o u s dampings. Top : undamped, middle : moderately damped, bottom : v i r t u a l l y completely damped.  13  lineshapes different the  were c a l c u l a t e d f r o m t h e amounts o f d a m p i n g .  o r i g i n s are  offset  3 with  scale i s linear, has  i n u n i t s of Eq.  _  Q  show p o o r  sidelobes.  positive  magnitude  The  value.  and  negative  lineshapes,  lineshapes,  and  Fig.  are  a r e a l s o p l a g u e d by  central  particularly  for  produce s p e c t r a  with  lineshapes,  sidelobes 3,  The  5.  characteristics,  absorption  and  i t s baseline  t o i t s r e s p e c t i v e damping  undamped t i m e - d o m a i n s i g n a l s w h i c h  very high large  is f f »  spectra  totally  o f Eq.  s c a l e i s s c a l e d as t h e d i s t a n c e away f r o m t h e  maximum, t h a t The  amplitude  such t h a t each l i n e s h a p e  at the dash c o r r e s p o n d i n g frequency  The  transformation  at  low  Fig.  2,  have  dampings.  The  wider than the sidelobes.  absorption  The  lineshapes  h a v e s m o o t h e r y e t w i d e s k i r t s w h i c h e x t e n d f o r many s p a c i n g s , 5,  from  t h e c e n t r a l maximum f o r h i g h e r  the a b s o r p t i o n the  magnitude l i n e s h a p e s .  need f o r a p o d i z a t i o n .  sidelobes  as they  an a d j a c e n t  peak  of  strong  Of  w h i c h emerge f o r low  undesirable of  and  low  damping v a l u e s  can  peak.  These l i n e s h a p e s  considerable disturbance damping  values.  i n t e r f e r e w i t h and This  distort  both reveal  are  the  Sidelobes  are  the  i s particularly detrimental  i n t e n s i t y must be d e t e c t e d  for  Eq.  intensity when  a  i n close proximity to  a  peak.  From  Figs.  considerably dependent lineshape  2 and  with  upon  the  3,  i t i s noted t h a t the  d i f f e r e n t amounts of damping, T/r  ratio.  Since  the  lineshapes and  thus  are  frequency-domain  i s a f u n c t i o n of d a m p i n g t h e e f f e c t o f a window c a n  14  vary  be  ABSORPTION  FREQUENCY (Af) Figure  2.  Unapodized dampings.  absorption  15  lineshapes  due  to  various  o  MAGNITUDE  I  -4  I  I  I  -2  I  I  0  I  I  2  I  I  due  to  4  I  FREQUENCY (Af) Figure  3.  Unapodized dampings.  magnitude  lineshapes  16  various  expected  t o vary with  The in  damping.  e f f e c t s of a f i n i t e  the  shape  sampling  of t h e s k i r t s  interval are clearly  in Figs.  2 and  3.  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, windowing whereby prior  in  the time  domain.  a time-domain s i g n a l  This  of  i s a mathematical  process  i s m u l t i p l i e d by a window  function  to Fourier transformation.  study,  have  come f r o m e n g i n e e r i n g  and used i n  literature;  mode r e s u l t s were n o t commented u p o n .  Also,  Window results;  functions however,  have  been  these  a p p l i c a t i o n s r a t h e r than  applied  studies  theoretical.  cases,  d e t a i l s o f t h e amount o f i m p r o v e m e n t ,  These  windows  were a d j u s t e d ,  frequency  detail  signal.  often  absorption experimental results,  apodization,  b u t no q u a n t i t a t i v e r e s u l t s ,  exact  specific  engineering  Thus q u a l i t a t i v e no  responses.  this  absorption  yielding  were  improvement o f windowing compared t o  shown f o r s p e c i f i c  thus these  s t u d i e s d e a l t o n l y w i t h an undamped t i m e - d o m a i n  in  method  a p o d i z a t i o n , i s by  Many o f t h e windows p r e v i o u s l y m e n t i o n e d ,  the  A  seen  such  were g i v e n  often empirically,  to  were as  [28,42]. suit  O t h e r windows h a v e been  the  examined  f o r t h e a b s o r p t i o n mode f o r s u c h d e t a i l s a s t h e  width  of t h e l i n e s h a p e p r o d u c e d a n d t h e s i z e o f t h e s i d e l o b e s .  However  the  earlier  criteria  studies. [43],  and  Of  for acceptable importance,  windows were d i f f e r e n t  f o r example,  the width at half height  sidelobes, although  recorded,  [44].  in  were t h e c o m p u t i n g The h e i g h t s  were n o t a g r e a t c o n c e r n .  17  time  of the Like the  engineering  studies,  cases.  this  In  functions It  apparent  for  that a detailed  time-domain  study s y s t e m a t i c a l l y  time-domain  signals  determined in  the primary reason  the  detail  of  of  different  is  different  lacking, dampings.  e x a m i n e s t h e e f f e c t o f many windows on  of v a r i o u s  dampings.  Their  efficacy  a c c o r d i n g to the extent that s i d e l o b e s are  a b s o r p t i o n and magnitude l i n e s h a p e s . below,  window  sidelobes.  examination  signals  undamped  for applying  on b o t h t h e a b s o r p t i o n and m a g n i t u d e modes  particularly This  study,  i s f o r t h e r e d u c t i o n of t h e s i z e of t h e is  windows  these works a l s o d e a l t o n l y w i t h  As  is  eliminated  described  t h e g e n e r a l shape o f window f u n c t i o n s f o r use  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  f r o m t h e g e n e r a l shape  in in used  i n m a g n i t u d e mode s p e c t r o s c o p y .  2.2  Procedure In the p r o c e s s of a p o d i z a t i o n ,  is  m u l t i p l i e d by a window f u n c t i o n ,  then F o u r i e r For  this  signal.  was  The  Eq.  time,  T,  was  s i g n a l can i n t h e KHz  and  scale.  last  the  The  For example,  is  spectrum. time-domain  from t = 0 t o the  set to 1 sec.  total  frequency, f , Q  parameters,  i n FT-ICR, the t i m e -  i n t h e o r d e r o f msec a n d  the  range, thus the r e s u l t s here would  18  g(t),  t h i s product  frequency  used t o r e p r e s e n t  w h i c h was  signal,  T h i s model c o r r e s p o n d s t o a c t u a l  e x c e p t on a d i f f e r e n t  c o u l d be  3  l e n g t h of the s i g n a l l a s t e d  s e t t o 32 Hz.  domain  w(t),  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  study,  acquisition  a time-domain  frequencies appear  the  same e x c e p t f o r t h e s c a l e . An  array  domain  signal,  calculated to  o f 129 d a t a p o i n t s was c a l c u l a t e d Eq.  3,  and  another  f o r a window f u n c t i o n .  produce  function,  set  of  for 129  the values  was  T h e s e were m u l t i p l i e d t o g e t h e r  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 g(t)w(t).  time-  T h i s time-domain  time-domain  d a t a was t h e n  zero-filled  [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. DFT was p e r f o r m e d on t h e z e r o - f i l l e d ,  windowed t i m e - d o m a i n  A  array  u s i n g t h e r o u t i n e POLFT2 on t h e UBC-MTS s y s t e m , a n d t h e r e s u l t i n g real  and  i m a g i n a r y v a l u e s were u s e d t o c a l c u l a t e  the  apodized  absorption  and magnitude  s p e c t r a a s shown i n A p p e n d i x  A.  This  calculation  was p e r f o r m e d  f o r a range of v a l u e s  for  each  window.  The  values  for  T/ r ,  r e p r e s e n t i n g 0.0, a n d t o 0.5, The Filler  window [43],  examinations spectra. chosen  work,  Filler's were  and  Norton  windows  and  Beer  [44]  windows w h i c h p r o d u c e d  windows.  used  results dampings,  adaptation  actually  performed  detailed  for  absorption  the lowest sidelobes  were  f o r the absorption-mode from Norton and  Beer's  portions  of  f o r a b s o r p t i o n a n d one f o r m a g n i t u d e  f r o m N o e s t and K o r t ' s work [ 4 5 ] .  were  sources.  by t a k i n g o n l y t h e p o s i t i v e  A window  0.001  various  f o r t h e undamped c a s e  t h e windows were t a k e n d i r e c t l y  and  were s e t t o  1.0, 2.0, a n d 3.0.  f r o m t h e s e w o r k s t o be i m p l e m e n t e d  studies;  3,  f u n c t i o n s were o b t a i n e d f r o m  on  The  i n Eq.  for r  examined  for  cases  In t h e i r due  to  work,  different  h o w e v e r t h e i r p u r p o s e was t o i l l u s t r a t e windows  19  the  which  produced  l o w peak h e i g h t e r r o r s .  rectangle, in  [46,47,48,49,50],  known windows a r e c o m p i l e d  remaining Aside  windows were a d a p t e d  from  symmetrical  i n H a r r i s ' s study,  t h i s extensive report [51], the  remaining  studies  was  necessary  The w i n d o w s d e s c r i b e d i n t h e s e  windows, s u i t e d  f o r engineering  i n t h e power mode w h i c h  m a g n i t u d e mode,  less  thus the  i n shape w i t h r e s p e c t t o t h e m i d p o i n t  symmetrical  implemented  along with  a n d H a r r i s ' s windows  the absorption studies.  These  the  results.  Adaptation of F i l l e r ' s  are  from  These,  t h e work o f N o e s t a n d K o r t ,  d i s p l a y e d o n l y undamped  for  such as  t r i a n g l e , Hamming a n d H a n n i n g a r e d e s c r i b e d i n d e t a i l  numerous s o u r c e s  commonly  Common w i n d o w s ,  i sdirectly  in  works time.  purposes, are  related  t o the  and so they a r e a l s o a p p l i c a b l e t o t h e magnitude  mode. F i g u r e 4A shows one s u c h e x a m p l e o f a s y m m e t r i c a l the  3-term  Blackman-Harris  corresponding adapted time domain from range.  When  function,  window.  symmetrical  functions,  the  lineshapes  large  typically  maxima. with  magnitude-type lobes  shows  the  i n the  t = 0 t o t = T, a n d a r e s e t t o z e r o o u t s i d e  a p p l i e d and t h e r e s u l t s p r e s e n t e d  calculated  4B  T h e s e windows b o t h e x t e n d  are  auxiliary  and F i g .  function,  are  o r magnitude-type windows, i n the  absorption  show l a r g e n e g a t i v e  Figure  4C shows t h e  T/ r = 1.0,  resulting  3-term B l a c k m a n - H a r r i s  mode,  intensities  absorption  positive  and  lineshape,  from windowing w i t h  window o f  undesirable since the  20  this  F i g . 4A. ones  can  the The be  -1  0  I  raxoutxcv (at)  Figure  4.  *  -  4  -  1  0  2  mourner (*<)  Window t y p e s a n d r e s u l t i n g a b s o r p t i o n a n d magnitude lineshapes. A : M a g n i t u d e - t y p e window. B : A b s o r p 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 : M a g n i t u d e l i n e s h a p e due t o window i n A. F : M a g n i t u d e l i n e s h a p e due t o window i n B. 21  mistaken as s p e c t r a l l i n e s , and t h e n e g a t i v e totally the  conceal  l a s t h a l f of  an a c t u a l p e a k . a symmetrical  However windows c o n s i s t i n g window g i v e a b s o r p t i o n  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 are  defined  ones c a n d i m i n i s h o r  intensities.  lineshapes  T h e s e windows  such that t h e c e n t r a l p o i n t of a symmetrical  window  is  now  l o c a t e d a t t = 0 a n d t h e e n d o f t h e window i s a t t =  and  will  be r e f e r r e d t o a s a b s o r p t i o n - t y p e  Fig.  4B shows t h e window f o r a b s o r p t i o n  3-term  Blackman-Harris  absorption  spectrum  absorption-type  window  of  resulting  window  of  from  Fig.  4B  windows.  4A.  from t h e  The  apodized  t h e windowing i s given  T,  F o r example,  studies derived  Fig.  of  in  with  the  Fig.  4D.  C o m p a r i s o n o f F i g . 4C w i t h F i g . 4D shows t h e o b v i o u s  improvement.  Although  height  the  the lineshape  absorption-type  better 2.  i s w i d e r a t 5 0 % o f t h e peak  window,  i t i s f r e e from l a r g e l o b e s and i s  than t h e unapodized l i n e s h a p e  Thus t h e window shape i n F i g .  studies.  lineshapes  inFig.  shown i n t h e c e n t e r  of F i g .  4B i s u s e d f o r t h e a b s o r p t i o n  Comparison of the apodized  unapodized  with  lineshape  i n F i g . 4D t o t h e  2 shows t h e i m p r o v e m e n t  due  to  windowing. It these  i s known i n FT-NMR t h a t windows o f t h e s h a p e  a b s o r p t i o n mode s t u d i e s ,  the absorption-type  be a p p l i e d t o i m p r o v e t h e SNR [ 1 5 ] , increasing the  exponential  Windows  windows,  of  lineshapes, that  the  shape  22  can  Windows w i t h t h e s h a p e o f an  [15,52] a r e u t i l i z e d f o r improvement  r e s o l u t i o n of a b s o r p t i o n  lineshapes.  used f o r  used  of  i s t o make n a r r o w e r f o r magnitude  mode  studies,  those  that are symmetrical,  a b s o r p t i o n mode s t u d i e s [ 4 2 ] . than  h a v e a l s o been  These c r e a t e a narrower  t h e a b s o r p t i o n - t y p e windows used i n t h i s  by c o m p a r i n g F i g s .  4C a n d 4D;  lobes are unacceptable  a s shown i n  as i n F i g .  Comparison of F i g .  produce  H o w e v e r , when an  shown i n F i g .  a d a p t e d window shape i s n o t n e c e s s a r y  studies.  The windows  4B, i s a p p l i e d t h e 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 , the  peaks.  F i g . 4A,  n a r r o w m a g n i t u d e l i n e s h a p e s a s seen i n F i g . 4E. a b s o r p t i o n t y p e window,  negative  the absorption-type  f o r t h e m a g n i t u d e mode.  f o r the magnitude s t u d i e s ,  lineshape  large  i n t e r f e r e with adjacent  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 ,  used  for  s t u d y , a s was shown  however t h e e n s u i n g  as they  window i s n o t v e r y e f f e c t i v e  used  4F.  Thus  f o r t h e m a g n i t u d e mode  4E w i t h t h e c e n t e r c u r v e  in Fig. 3  shows t h e e f f e c t o f a p o d i z a t i o n . Besides windows were  the  w i t h both  applied  3-term B l a c k m a n - H a r r i s  window,  t h e a b s o r p t i o n type and magnitude type  f o r the absorption  and  magnitude  windows i n c l u d e d t h e 4-term B l a c k m a n - H a r r i s , Hanning  a=2,  and t r i a n g l e .  resulted  for  produced  a  result,  due  extremely windows  these  very to  smooth the  lineshapes,  the  If  absorption  corresponding as i n  produced  shapes  modes.  Gaussian,  other  These Hamming,  The same t r e n d s a s d e s c r i b e d a b o v e  windows.  large lobes, which  various  an  lineshape,  corresponding  slightly  For  window  absorption  window  had  absorption-type  improved  magnitude-type  23  the  magnitude — type  F i g . 4C.  only  absorption-type  window  absorption produced  absorption  lineshapes  lineshape  due  to  window was smooth lineshape positive  due  of  window,  lobes.  absorption-type  and q u i t e narrow,  lobe  a greatly  improved l i n e s h a p e  window.  window  and  an  F i g . 4C.  had  The a b s o r p t i o n - t y p e  d i d not  window,  cause such  extremely T h i s was a  corresponding  t r i a n g l e window d i d  extreme  sidelobe  was a b o u t 4 0 % o f t h e peak  for  about  and t h e magnitude-type  was  absorption  shown  inFig.  1% o f t h e peak  sidelobes. height,  height.  mode s t u d i e s a r e t h o s e 4B,  of a d j a c e n t  which gives t h e l e a s t peaks.  The l i n e s h a p e s  w i n d o w s were e x a m i n e d  of t h e i r peak h e i g h t s . position  advantageous  positive  the  negative  The w i n d o w s  with a general  being  the spectra  chosen  shape  as mode  i s t o produce  interference i n the resulting  f o rt h e i r widths  at specific  window d e p e n d e d  The  choice  fractions  f o r t h e s i z e and  sidelobes with the absolute  included.  detection  from t h e a p p l i c a t i o n  They were a l s o e x a m i n e d  of t h e i r highest  sidelobes  The  a s shown i n F i g . 4A.  The p r i n c i p l e p u r p o s e o f a p o d i z i n g lineshape  and  3-  triangle  a n d t h e windows u s e d f o r t h e m a g n i t u d e  studies are symmetrical,  a  due t o t h e a b s o r p t i o n - t y p e  shape due t o t h e  sidelobe  the  Blackman-Harris  shape o f t h i s  6 0 % o f t h e peak h e i g h t ,  Blackman-Harris  window  of  3-term  absorption  improve t h e l i n e s h a p e a s w e l l a s d i d t h e a b s o r p t i o n - t y p e  term  a  The  F i g . 4D, b u t t h e a b s o r p t i o n  magnitude-type  and a s e v e r e l y d i s t o r t e d  magnitude-type not  smaller  t h a t was 10% o f t h e peak h e i g h t  large negative case  the  to  lobe  with  of  values the  upon t h e window w h i c h p r o d u c e d  24  of most the  narrowest  lineshape  required.  0.1%  sidelobes  below  largest tolerable  sidelobe  was  of  t h e peak h e i g h t  w o u l d be t h e window  allow  with,  a t m o s t , a 0.1% e r r o r i n peak h e i g h t .  range  were r e q u i r e d ,  a peak t h e s i z e o f t h e h i g h e s t  a l s o been t a b u l a t e d , 10,000:1.  peak h e i g h t of  damping  controlled, T/ T v a l u e s  2.3  The w i d t h s  1000:1,  the  the at  peak t o be  This  detected  I f a smaller  dynamic  halfwidths  have  f o r a l a r g e dynamic range  time-domain s i g n a l  up  a t 50% a n d 10% o f t h e  can  not  The amount  be  precisely  i s desired. i s apodized,  i n Table I . section,  w(t),  i t s peak i n t e n s i t y  t h e p e a k s h a v e been  Mode R e s u l t s a n d  windows,  previous  then  t h u s a window w i t h good r e s u l t s o v e r a w i d e r a n g e o f  Absorption  listed  highest  choice.  f o r a complete p i c t u r e of the l i n e s h a p e s . of  range  lineshape  of  these  were a l s o r e c o r d e d  a l l of the i l l u s t r a t i o n s ,  The  are  f o r example 100:1, as have t h o s e  When a l i n e s h a p e For  dynamic  which produced the narrowest s i d e l o b e - f r e e  would  to  the  For example, i f the r a t i o of the s i z e of the  peak t o t h e window  with  normalized.  Discussion  used  f o r the absorption  studies  They were i m p l e m e n t e d a s d e s c r i b e d a n d t h e r e s u l t s o f t h e windows  a s s e m b l e d i n T a b l e I I . The a p o d i z e d  according  to  t h e i r widths  heights.  The  desired  narrowest lineshape;  decreases.  that  at specific  spectra  which  i t has t h e l o w e s t  25  the  for absorption are  evaluated  f r a c t i o n s of t h e i r  window i s t h e one is,  in  are  produces halfwidth  peak the value  TABLE I. Window functions, w(t), for absorption. Window  A n a l y t i c a l form,  0<t<T  Rectangle  1 .0  Blackman-Harris, 3-term  . 42323+. 49755cos(*t/T) + .07922cos(2irt/T)  Blackman-Harris, 4-term  .35875+.48829cos(»t/T)+.14128cos(2»t/T) + .01 l68cos(3irt/T)  Filler  DO.24  [cos(#t/2T) + .24cos(3*t/2T)]/1 .24  Filler  E0.13  [ 1 + 1.13cos(*t/T) + .l3cos(2*t/T)]/2.26  Filler  E0.20  [1+1,2cos(*t/T)+.2cos(2*t/T)]/2.4  Gaussian Hamming  2 exp[-0.5(3.5t/T) ] .54+.46cos(wt/T)  Hanning Kaiser-Bessel  a cos  (»t/2T)  where a=2,4 2 0.5  I.[3.5ir{l .0-(t/T) }  0  where I.fx] i s a zero-order modified  Kaiser-Bessel, 4-term  Noest-Kort*  ]/I [3.5*]  Bessel function .40243+.49804cos(*t/T) + .09831 cos(2*t/T) +.00l22cos(3»t/T) AC where A=.260+.520cos(*t) + .2l9cos(2«-t)  Norton-Beer F3 Triangle  2 - ( t / r ) /(41n2)] 2 2 24 .045335+.554883(1-(t/T) ) +.399782(1-(t/T) ) 1.0-t/T C=.5exp[t/r  *The equation f o r r «1.25 from Ref.[45] i s not employed as i t produces poorer r e s u l t s than those depicted in t h i s work.  26  TABLE I I .  Apodized  absorption  spectra  halfwidths.  Halfwidth 50% T/T  Window  0.29  Rectangle  at  10%  1%  0.1%  o f Peak  Height  .3) (13 . 2 ) ( > 3 2  (1  Blackman-Harris,  3-term  0.56  0  Blackman-Harris,  4-term  0.66  1 .2  .97  0.01%  ) (>32  )  (13.1 )  1  .3  1 .4  1  .6  1 .8  1.9  ( 2 . 8)  (5.8)  Filler  E0.13  0.55  0  Filler  E0.20  0.58  1 .0  1  .3  1 .4  (4.8)  Gaussian  0.65  1 .2  1  .7  2.  1  (8.2)  Hanning, a=2  0.50  0  (  .97)  (2.  9)  (6.8)  H a n n i n g , a=4.  0.64  1. 1  1  .4  (1 .  5)  (2.5)  Kaiser-Bessel  0.64  1. 1  1  .5  1 .7  1.8  0.59  1 .0  1  .3  1 .5  (2.9)  0.94  1  .3  1  .4  0.48  0  .81  1 .0  Kaiser-Bessel,  4 term  Noest-Kort Norton-Beer  F3  0.32  Rectangle  (1  .94  .81  1 .2  3-term  0.61  1 . 1  2 .0  Blackman-Harris,  4-term  0.66  1 .2  2 .2  DO.24  0.55  0  Filler  E0.13  0.58  Filler  E0.20  0.62  27  (12.  .3) ( 1 0 . 2 ) ( > 3 2  Blackman-Harris,  Filler  (19. 6)(>32 ) 2)(>32 ) ) (>32  )  ( 5 . 5) ( 2 4 . 1 ) 6. 2  (20.2)  1 .8  ( 5 . 7) ( 2 0 . 6 )  1 .0  2 .0  (5. 8)  (18.0)  1. 1  2 .0  5. 8  (18.7)  .97  TABLE I I c o n t d . Gaussian  0. 69  1 ,3 .  2.3  Hamming  0. 49  0.,85  (2.3)  6.3  (23.1)  (15.2)(>32  )  Hanning,  a=2  0. 53  0.,91  (1.9)  (5.1)  (16.1)  Hanning,  a=4  0. 68  1 .2 .  2.2  6.0  19.6  0. 64  1 .. 1  2.1  6.0  19.7  0. 63  1 ., 1  2.0  5.8  (19.2)  0. 94  1 .3 ,  1.5  (18.6)(>32  )  0. 52  0,.94  1.7  (10.3) (>32  )  Triangle  0. 48  0,.84  (2.8)  (10.5)(>32  )  Rectangle  0. 35  0,.55  (8.2)(>32  Kaiser-Bessel Kaiser-Bessel,  4--term  Noest-Kort Norton-Beer  F3  ) (>32  Blackman-Harris,  3- t e r m  0. 61  1 ., 1  2.8  (8.4)  Blackman-Harris,  4- t e r m  0. 70  1 .3 ,  3.0  9.1  )  (32.0) >32  Filler  E0.13  0. 63  1 .2  2.8  8.3  30.1  Filler  E0.20  0. 66  1 .2  2.8  8.5  31.4  Hanning,  a=2  0. 57  1 .0  2.7  7.9  26.0  Hanning,  a=4  0. 72  1 .3  3.0  8.8  >32  0. 72  1 .3  3.0  8.9  >32  0. 68  1 .3  2.9  8.6  (>32  )  0. 94  1 .3  1.5  (13.7)(>32  )  0. 56  1. 1  2.5  (11.2)(>32  )  0. 42  (1 .3)  3- t e r m  0. 74  1 .6  4.3  13.3  (>32  B l a c k m a n - H a r r i s, 4-•term  0, 84  1 .7  4.5  14.0  >32  Kaiser-Bessel Kaiser-Bessel,  4--term  Noest-Kort Norton-Beer  F3  Rectangle Blackman-Harris,  28  (6.2)(>32  ) (>32  ) )  TABLE I I . c o n t d . Filler  E0.13  0.72  1.5  4.2  13.1  >32  Filler  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  Kaiser-Bessel  0.81  1.7  4.5  13.8  >32  K a i s e r - B e s s e l , 4-- t e r m  0.77  1.6  4.4  13.5  Noest-Kort  0.94  1 .4  1 .7  (4.8)(>32 )  0.65  1.4  4.2  (13.3)(>32 )  0.52  1.5  (5.4)  (28.1)(>32 )  Norton-Beer 3.0  F3  Rectangle  (>32 )  Blackman-Harris,  3-term  0.84  1.9  5.7  18.1  >32  Blackman-Harris,  4-term  0.93  2.1  5.9  18.9  >32  Filler  DO.24  0.78  1.8  5.6  17.6  >32  Filler  E0.13  0.81  1.9  5.6  17.8  >32  Filler  E0.20  0.84  1.9  5.7  18.1  >32  Gaussian  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  Kaiser-Bessel  0.91  2.0  5.9  18.7  >32  0.87  2.0  5.8  18.3  0.96  1.5  2.0  2.3  0.75  1 .8  5.5  0.72  2.0  6.5  Kaiser-Bessel,  4- t e r m  Noest-Kort Norton-Beer  F3  Triangle  (  ) denote s i d e l o b e s  a t s p e c i f i e d peak  29  (>32 ) ( 8 . 7)  (17.3)(>32 ) 20.8  height.  (>32 )  in  Table  I I f o r the  u n i t s of  1/T,  Eq.  required conditions.  5, away f r o m t h e c e n t r a l maximum.  same s c a l e a s u s e d i n t h e window, every  the  of  from the w i d t h  difficult  if  Table  line,  This  in  i s the  i n d i c a t e d a poor calculated  for  i t s a c t u a l shape i s a l s o  D e t e c t i o n of an a d j a c e n t , t h e p r i n c i p l e peak had  II  Kaiser-Bessel  windows  5 which presents  of the v a l u e s below  10  -4  sidelobes,  i n a narrow envelope.  both  Hanning  T h i s can  lineshapes  be  even  For  if  example,  a=2,  produce l i n e s h a p e s of  r = 0.5.  the  s m a l l peak would  large  i t i s apparent t h a t the  e q u i v a l e n t w i d t h when T/ Fig.  of t h e  l o b e s were c o n t a i n e d  from  When t r e n d s  are  value.  importance.  these  figures.  h a l f w i d t h s f o r t h a t window were not  damping Aside  These v a l u e s  and  the  approximately  a l s o be o b s e r v e d  in  in a logarithmic scale.  All  • , i n c l u d i n g negative  s i d e l o b e s , h a v e been  -4 set  to  the  rectangle  case  10  f o r the  (which  absorption half  has  corresponds  to the  lineshapes are  t h e peak h e i g h t .  parenthesis  lineshapes  symmetrical,  Note t h a t the  many s i d e l o b e s ,  of an a d j a c e n t  the  Fig.  to  unapodized 2).  o n l y the h i g h  Since frequency  to the Hanning  a=2  l a r g e s t o f w h i c h i s a l m o s t 2%  of  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  peak,  i n Table  particularly  a s m a l l one.  The  II i n d i c a t e sidelobes higher  T h u s t h e numbers i n t h e  of  are enclosed  II  l i n e s h a p e due  f o r the  in  l i n e s h a p e due  peak h e i g h t s . Table  The  window i s shown f o r r e f e r e n c e  i s shown.  window  logarithmic figures.  1%,  i n parentheses 30  0.1%  and  f o r the  numbers  than  the  in  given  0.01%  columns  case  of  the  ABSORPTION  FREQUENCY (Af) F i g u r e 5.  Comparison of s u i t a b l e and u n f a v o r a b l e lineshapes.  absorpt ion  Hanning  a=2 window when T/ r = 0.5.  Bessel  window  dynamic  produces  range  a lineshape  examined,  thus  In c o n t r a s t , without  none  of  the  Kaiser-  sidelobes  i n the  i t s values  are  in  parentheses. The  presence of sidelobes  Table  I I .  these  values  widths  w h i c h a r e n o t i n p a r e n t h e s e s i n d i c a t e a window w h i c h  be  used  Since  i s q u i t e common f o r t h e windows i n  windows w i t h t h i s p r o p e r t y  h a v e been e n c l o s e d  should  n o t be u s e d ,  i n parentheses.  A l l of t h e could  w i t h t h e 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 b e l o w  required  range.  Details  sidelobes are l i s t e d Since  which  these  satisfactory  can  windows.  produce be  lines  eliminated  Windows  which  These  produce  with from  sidelobes the  create  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 t h e Hamming, triangle.  of t h e  i n A p p e n d i x B.  windows  unfavorable,  of t h e h e i g h t s and p o s i t i o n s  the  quest  for  lineshapes  with  H a n n i n g a=2,  sidelobes higher  are  than  and t h e  1% o f t h e  peak  c e r t a i n ones produce t h e  best  height. From  the remaining  results.  The  choice  windows,  o f t h e b e s t window i s d e p e n d e n t  r e q u i r e d c o n d i t i o n s o f damping and dynamic r a n g e . windows E0.20  upon  Some e f f e c t i v e  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 t h e N o e s t - K o r t , and  linewidths Depending  Kaiser-Bessel. according  on  the  to  amount  r e q u i r e d , c e r t a i n of these  These  windows  t h e summary o f of  damping  a l lyield  results  and  the  Filler narrow  i n Table I I . dynamic  windows a r e more a p p l i c a b l e .  32  the  range  If  sufficient, signals II,  of any  lowest  difference and  the  i t is clear  the  Noest-Kort  i s t h e most  amount o f d a m p i n g .  damping except  halfwidths i s negligible. peak  widths  increase.  In c o n t r a s t ,  Noest-Kort  window r e m a i n p r a c t i c a l l y  f o r a l l dampings.  o f most o f  the widths  For  T/r  fairly  s e e n by c o m p a r i n g t h e v a l u e s  illustrated  windows f o r c o m p a r i s o n .  amount  of  Table  II.  increases from F i g s .  The  general  the p e c u l i a r q u a l i t y  damping  increase in width. For  is  increased  This  8.  o t h e r windows :  This  II,  is  and  is also  rectangle  to the  Noest-  of becoming narrower as while  window,  i s the  retaining This  l i n e s h a p e due  the  i s most o b s e r v a b l e  the Noest-Kort  i t is  height.  other  the  lineshapes  i n the 0 . 1 % column i n  a s t h e amount o f  the h a l f w i d t h s a c t u a l l y decrease which 7 and  in  the peak  larger,  6 t o 8 w i t h t h e K a i s e r - B e s s e l and  in Figs.  K o r t window has  i n Table  to  1% o f t h e  1 and  v a l u e s of  1% of t h e peak  n a r r o w peak down t o  spectra  o f t h e l i n e s h a p e s due at  the  damping  frequency  constant  are  halfwidth  t h e amount o f  the  Table  However  t h a t t h i s window i s f a r s u p e r i o r t o t h e o t h e r s  readily  the  As  for  window  f o r t h e undamped c a s e b e t w e e n t h e N o e s t - K o r t  the  a  r = 0.0.  T/  is  window  1% c o l u m n i n  From t h e  increases,  clear  efficient  t h a t the h a l f w i d t h s f o r the Noest-Kort  f o r every  the other  height  1% o f t h e peak h e i g h t  a modest d y n a m i c r a n g e t o o n l y  is  damping  noticeable  r e v e r s e of the b e h a v i o r  due  to  w i t h t h e K a i s e r - B e s s e l a s an e x a m p l e ,  note  l i n e s h a p e w i d e n s f r o m a m o d e r a t e amount o f d a m p i n g ,  Fig.  how  the  6,  t o a h i g h amount o f d a m p i n g ,  33  Fig.  7.  Thus t h e  Noest-Kort  ABSORPTION  FREQUENCY Figure  6.  (Af)  Absorption lineshapes due to recommended windows moderate damping.  and  ABSORPTION  0.0  4.0  8.0  12.0  16.0  FREQUENCY Figure  7.  Absorption lineshapes h i g h damping.  20.0  24.0  28.0  32.0  (Af) due t o recommended windows and  ABSORPTION  to  FREQUENCY ( A f ) Figure  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.  outperforms  the  other  windows,  especially  for  highly  damped  signals. The in  N o e s t - K o r t has a v e r y s i m p l e p a r a m e t e r  which,  T a b l e I , matches the damping of t h e time-domain  as  shown  signal.  If i t  i s n o t d e s i r a b l e t o a d j u s t a window a t a l l , a n o t h e r window h a v e t o be c h o s e n consistently  f o r t h i s dynamic range.  narrow  as e f f e c t i v e  lineshapes.  range  If 0.1%  the  i s required,  1% c a s e .  v a l u e i n t h e 0.1%  narrow.  At  T/ r  windows b o t h p r o d u c e  peak h e i g h t .  For  T/r  the best performance, Kort  only marginally  better  to  T a k i n g t h e window p r o d u c i n g  column of T a b l e I I as  = 0.5,  and  and  E0.20  the F i l l e r  2.0,  for a totally  best  F o r t h e undamped  the narrowest  = 1.0  the  E0.20 a n d  a=2  damped s i g n a l  are  Kaiser-  w i d t h s a t 0.1%  the Hanning  case,  results  of  the  provides  the  Noest-  i s e a s i l y the best c h o i c e . I n s t e a d o f r e c o m m e n d i n g a d i f f e r e n t window f o r e a c h  c o m b i n a t i o n of damping and made i n o r d e r t o l i m i t of  than  f o r e x a m p l e down  3 - t e r m B l a c k m a n - H a r r i s and F i l l e r  identically Bessel  from T a b l e I I , i t i s not  y i e l d s an a b u n d a n c e o f c h o i c e s . the  has  a p r e c i s e c h o i c e f o r t h e b e s t window i s  c u t as f o r t h e  smallest  both  F3  E 0 . 1 3 , however i t d o e s s u f f i c e f o r t h e  dynamic range  o f t h e peak h e i g h t ,  window  Norton-Beer  of dampings.  a larger  not as c l e a r  seen  a s t h e N o e s t - K o r t and  o t h e r s , s u c h as t h e F i l l e r entire  As  The  would  dynamic range,  s l i g h t compromises are  t h e number o f c h o i c e s .  T a b l e I I r e v e a l s t h a t some w i n d o w s ,  37  specific  Closer examination  w h i l e not the v e r y  best  for  a p a r t i c u l a r case,  the  best c h o i c e s  produce h a l f w i d t h s which are very c l o s e to  listed.  Two  windows y i e l d e q u a l l y  results.  Both the F i l l e r  E0.20 and  similarly  smooth l i n e s h a p e s down t o 0.1%  are  relatively  The  K a i s e r - B e s s e l r e s u l t can  6,  but  Figs.  narrow, p a r t i c u l a r l y be  becomes l e s s e f f e c t i v e 7 and  identical  8.  The  to those  shows t h e = 3.0,  of t h e peak h e i g h t  f o r the  s e e n t o be v e r y e f f e c t i v e  in Fig.  f o r h i g h e r dampings as to the F i l l e r  been i l l u s t r a t e d .  i s only a  lower  which  values.  f o r the K a i s e r - B e s s e l ,  s i m i l a r i t y of t h e s e  there  K a i s e r - B e s s e l windows p r o d u c e  damping  r e s u l t s due  t h e K a i s e r - B e s s e l has  acceptable  two  shown  E0.20 a r e  thus  virtually  for c l a r i t y  Examination  in  only  of Table  w i n d o w s , f o r e x a m p l e when  II T/r  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 down  to  t h e K a i s e r - B e s s e l window p r o d u c e s a 0.01%  o f t h e peak h e i g h t  This  i s shown i n F i g .  the  0.01%  5,  and  column i n T a b l e  can  II.  f o r both  When of  i n c r e a s i n g l y wider  T/ r the  = 1.0,  6,  l i n e s h a p e does not  frequency  range  constant,  thus  Kaiser-Bessel notably  Fig.  examined.  a l s o be o b s e r v e d  A smooth l i n e s h a p e  by  reach  w i t h the  h i g h e r , F i g s . 7 and 0.01%  is effective  for a  large  f o r s i g n a l s w i t h low amounts o f d a m p i n g ,  38  increases. 8,  is  the  skirt  in  the  virtually  peaks.  dynamic T/r  from  lineshape  o f t h e peak h e i g h t  However i t s l i n e s h a p e  0.5.  examining  a s t h e amount o f d a m p i n g and  and  results  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 window  lineshape  T/ r = 0.0  t h i s window f o r a l l of t h e d a m p i n g s s t u d i e d , becoming  smooth  = 0.0  The  range, and  0.5.  2.4  M a g n i t u d e Mode R e s u l t s and The  windows  presented shape and  t  magnitude  similar  are h a l f of these  to those are  those  and  the  obtained  studies,  w i t h p a r e n t h e s e s and the  sidelobes are  a r e not  presented  than  reference windows,  such  the exact  lineshapes values  the a b s o r p t i o n  windows  mode.  f o r the  f o r a l l of the damping  the  The  spectra should With the  magnitude  lineshapes,  IV.  s i z e s and  As  for  the  indicated  positions  R e s u l t s of p o o r e r  of  windows  values. be  narrow w i t h  sidelobes  r e c t a n g l e window a s  T a b l e I V shows  triangle,  are  windows i n T a b l e  with s i d e l o b e s are  f o u n d i n A p p e n d i x B.  no a p o d i z a t i o n , as  these however  resulting  a desired limit. for  for  windows,  i s d i s p l a y e d i n Table  P r e f e r a b l y , apodized lower  expressions  while  f o r the a b s o r p t i o n  h a v e been e x a m i n e d f o r t h e  absorption  in  tapering to zero at t = 0  f o r a good window f o r t h e m a g n i t u d e mode  III  data  are  windows i s s y m m e t r i c a l  f o r the a b s o r p t i o n  symmetrical  spectra  shapes.  criteria  same as  Each of t h e s e  Note t h a t the a n a l y t i c a l  windows  The  the magnitude  i t s maximum i n t h e c e n t e r ,  = T.  windows a r e  the  used f o r a p o d i z i n g  i n Table I I I .  with  Discussion  that  well  o n l y produce s l i g h t l y  spectra  compared t o the performance of o t h e r ,  complex  windows.  more  the known  improved  analytically  D e p e n d i n g upon t h e amount o f d a m p i n g and  the  d y n a m i c r a n g e r e q u i r e d , c e r t a i n o f t h e windows a r e p r e f e r r e d o v e r  39  TABLE I I I .  Window functions, w(t), for magnitude.  Window  A n a l y t i c a l form,  0<t<T  Rectangle  1 .0  Blackman-Harris, 3-term  . 42323-. 49755cos(2trt/T) + .07922cos(4»t/T)  Blackman-Harris, 4-term  .35875-.48829cos(2*t/T)+.14128cos(4»t/T) -.01168cos(6*t/T) 2  Gaussian Hamming  exp[-0.5(3.5(2t-T)/T) ] •54-.46cos(2»t/T) a  Hanning Kaiser-Bessel  sin  Ot/T)  where a=2,4  Kaiser-Bessel, 4-term  as f o r absorption, but s u b s t i t u t i n g 2t-T for t .40243-.49804cos(2*t/T) + .09831 cos(4»t/T) -.00122cos(6*t/T)  Noest-Kort M C  where M=.430+.843cos(»t) -.239cos(2«t)+.043cos(5*t)-.602U Osame as for absorption U=1-4t for 0<t<l/4 =3-4t f o r 3/4<t< 1 Triangle  «0 elsewhere 1- |2t-T|/T  40  (Table I)  TABLE IV.  Apodized magnitude spectra  halfwidths.  Halfwidth 50% T/r  0.0  0.5  Window  10%  at  1%  0.1%  0.01%  of Peak Height  (2.7) (31.2)(>32 ) (>32 )  Rectangle  0.59  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  K a i s e r - B e s s e l , 4 term  1.2  2.1  2.7  3.0  (5.7)  Noest-Kort  1.2  1 .6  Rectangle  0.60  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  Gaussian  1.3  2.4  3.3  4.1  Hamming  0.90  1.5  1.9 (>32 ) (>32 )  Hanning, a=2  1.0  1.6  (2.8)  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  K a i s e r - B e s s e l , 4 term  1.2  2.1  2.7  3.0  (5.7)  Noest-Kort  1.2  1 .6  Triangle  0.88  1.5  41  (22.10(>32 ) (>32 )  (2.7) (31.4)(>32 ) (>32 )  3.9 (20.4)  (6.6) (13.5)  (22.1)(>32 ) (>32 ) (5.3) (19.1)(>32 )  TABLE IV.  contd. 0.62  Rectangle  (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  K a i s e r - B e s s e l , 4 term  1.2  2.1  2.7  3.1  (5.8)  Rectangle  0.69  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  Gaussian  1.3  2.4  3.4  4.0 (>32 )  Hamming  0.92  1.6  Hanning, a=2  1.0  1 .7  3.6  Kaiser-Bessel  1.3  2.3  3.0  3.5  4.4  K a i s e r - B e s s e l , 4 term  1 .2  2.1  2.8  3.4  (5.9)  Rectangle  0.84  (3.8)(>32 ) (>32 ) (>32 )  4.6  (5.6)(>32 ) (>32 ) (7.3) (14.7)  (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  Gaussian  1.3  2.4  3.4  4.0 (>32 )  Hamming  0.94  1.7  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  K a i s e r - B e s s e l , 4 term  1.2  2.1  2.9  3.6  24.4  Triangle  0.94  1.7  (9.5)(>32 ) (>32 )  (7.1) (23.2)(>32 )  ( ) denote sidelobes at s p e c i f i e d peak h e i g h t .  42  5.0  others. Some c o n t r a s t i n g m a g n i t u d e a l o g a r i t h m i c amplitude  scale,  lineshapes  are symmetrical  f  the high  ,  only  r e s u l t s a r e shown i n F i g . and low d a m p i n g .  with respect  frequency  Comparison of the l i n e s h a p e s  half  Since  frequency,  of the s p e c t r a a r e  displayed.  i n F i g . 9 r e v e a l s the consequence of  a=2  window i s a commonly e m p l o y e d window.  an  improvement  rectangle  result,  lineshape. above  The  apodization,  t h e peak h e i g h t . very  While i t i s d e f i n i t e l y as  represented  by  lineshape  when c o m p a r e d  to the  Hanning  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 The  lineshape  as i t has a s i d e l o b e l a r g e r  window i s  = 0.5  sidelobes The  :  t h e n t h e H a n n i n g a=2,  t h e r e c t a n g l e window c r e a t e s t h e h i g h e s t  sidelobes.  reverse  order  height.  i s observed f o r the widths  at half  i n t h e 50% c o l u m n o f T a b l e I V .  the  n a r r o w e s t peak a t 50% o f t h e peak h e i g h t ,  a=2,  and t h e n t h e K a i s e r - B e s s e l .  43  its are  The  the  Kaiser-Bessel  finally  visible  1% o f  coulumns.  b r o a d e n s a t 50% o f t h e peak h e i g h t . sidelobes,  a=2  t h a n 2% o f  f o r t h i s window a t T/ r  9 a l s o shows an e f f e c t o f r e d u c i n g  window p r o d u c e s t h e l o w e s t  below  due t o t h e H a n n i n g a=2  i n p a r e n t h e s e s f o r t h e 1%, 0.1% and 0.01%  also  the  K a i s e r - B e s s e l window d o e s b r o a d e n t h e  peak h e i g h t , and so t h e v a l u e s  lineshape  Hanning  smoother  acceptable  Figure  The  t h e K a i s e r - B e s s e l window p r o d u c e s a  1% o f t h e peak h e i g h t ,  window, however  not  no  magnitude  t o the center  a p o o r window c o n t r a s t e d w i t h a recommended window.  over  9 with  The  This i s  rectangle  then the  and  has  Hanning  MAGNITUDE  32.0  FREQUENCY F i g u r e 9.  Comparison of 1ineshapes.  (Af)  s u i t a b l e and  unfavorable  magnitude  If the  smoothing  Hamming  undamped  window  to  dampings,  down t o o n l y 1% o f t h e peak h e i g h t i s d e s i r e d , gives the narrowest  moderately  damped  apodized  signals.  t h e o t h e r windows r e s u l t  For  spectra  this  range  i s e v i d e n t i n t h e 1% c o l u m n o f T a b l e I V .  also  from  moderate damping. the  Hamming  signals, of  Note,  for this  the  lineshapes  of  i n F i g . 11 a n d by t h e p a r e n t h e s e s f o r t h e  window i n T a b l e I V ,  the  3-term  that  for highly  F o r l a r g e dampings and a dynamic Blackman-Harris  T h i s i s t r u e f o r T/ r  column of T a b l e I V , The r e s u l t s  damped  and i n F i g .  window  they a r e adequate  = 2.0 a n d 3.0 a s s e e n  i n t h e 1%  i l l u s t r a t e s the l a t t e r  f o r t h e 3 - t e r m B l a c k m a n - H a r r i s window a r e n o t  lineshapes.  not s u f f i c e ,  especially  1% i s  window  produces  Table  window h a s t h e n a r r o w e s t  IV,  narrow  i t i s readily visible  lineshapes,  t o 0.1% o f t h e peak h e i g h t .  smooth,  needed,  for heavily  I f a d y n a m i c r a n g e o f 1000:1 i s n e c e s s a r y ,  From  however  a s shown i n F i g s . 10.  window w i l l  Blackman-Harris  of  narrowest  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  signals.  1%  the  11 w h i c h  has  range  a s good a s t h o s e f o r t h e Hamming window a t l o w d a m p i n g s ,  Hamming  due  t h e Hamming no l o n g e r r e d u c e s t h e l i n e s h a p e t o b e l o w  results.  case.  shows  As t h e d a m p i n g i n c r e a s e s , t h e l i n e s h a p e due t o  t h e peak h e i g h t .  100:1,  F i g . 10 w h i c h  This i s  window becomes much w i d e r t h a n t h e o t h e r s a t 1%  t h e peak h e i g h t . values  of  i n s p e c t r a which a r e a t l e a s t  30% w i d e r w h i c h apparent  for  damped  t h e 3-term magnitude that  this  f r e e from s i d e l o b e s ,  down  T h i s o c c u r s f o r e v e r y damping  45  the  except  MAGNITUDE T/t=1.0 KAISER-BESSEL 3-TERM  BLACKMAN-HARRIS, HAMMING  '  0.0  Figure  4.0  10.  8.0  1  12.0 16.0 FREQUENCY  r  20.0 (A f )  <'  \ l  "  \' '' \ '  1  * , '  4  /1  24.0  M a g n i t u d e l i n e s h a p e s due t o recommended w i n d o w s moderate damping.  and  MAGNITUDE  FREQUENCY Figure  11.  Magnitude lineshapes complete damping.  (Af)  due to recommended windows and  T/ T  = 3.0,  Thus  the  entire  however i t i s o n l y  5% w i d e r t h a n t h e  3 - t e r m 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  range of dampings s t u d i e d .  Its results  f o r m o d e r a t e d a m p i n g i n F i g . 10, and For of  value by  an  produce  the  examined.  to  a r e w i t h i n 6%  the K a i s e r - B e s s e l being  value.  Figures  10,  a smooth l i n e s h a p e I t can sidelobes  be are  and  11  noted from both F i g s . reduced,  smallest half height  half  height,  as  seen  l i n e w i d t h s at  of each o t h e r  10 and  with  a l l but  those T/ r  one  11 t h a t t h e more  the wider the w i d t h  also  at  results,  a t 50%  Hamming window p r o d u c e s t h e h i g h e s t  the o p p o s i t e  damping  has  f o r a l a r g e dynamic range f o r a l l dampings.  The  are  .01%  show t h a t t h e K a i s e r - B e s s e l window  height. the  every  Their  narrower f o r  11.  4-term Blackman-  for  l i n e w i d t h s i n T a b l e IV.  of t h e peak h e i g h t  due  halfwidth values  the  illustrated  smoothing to  b o t h t h e K a i s e r - B e s s e l and lowest  case.  for  h i g h damping i n F i g .  T h e s e windows p r o d u c e s i m i l a r  comparing t h e i r  0.01%  are  even g r e a t e r d y n a m i c r a n g e , s u c h a s  t h e peak h e i g h t ,  Harris  narrowest  w i t h the while  width.  lowest  the  The  of the  s i d e l o b e s and  peak  sidelobes  Kaiser-Bessel  but  results  the widest  3-term B l a c k m a n - H a r r i s  the  width  results  are  intermediate. The  Kaiser-Bessel  altered.  For  dynamic  range.  changes. being  this  has  a  a  was  s e t t o 3.5  study,  This  When a. = 1.5,  a l m o s t 2%  window  value the  can  parameter  be a l t e r e d ,  s i d e l o b e s are  of t h e peak h e i g h t , b u t  48  which  can  to give but  the  a  large  lineshape  l a r g e w i t h the  the width  be  at half  highest height  Figure  12.  Magnitude l i n e s h a p e s r e s u l t i n g K a i s e r - B e s s e l window.  from v a r y i n g  a  i n the  i s q u i t e n a r r o w as  shown i n F i g .  increased,  the  half  increases.  height  2.5  s i z e of  the  12.  N o t e t h a t as  sidelobes decreases,  the  but  a value  is  the width  at  Conclusion The  purpose  appropriate dampings  for  window  of  the  this  study  function  f o r m Eq.  is  f o r use  3.  to with  of a window i s t o l i m i t  skirt  peak i n t h e d e t e c t i o n o f an  one  examination the  of the e x t e n s i v e  frequency  considerably  lineshapes  in their  characteristic  was  signals  In c h e m i c a l  p r i n c i p l e value of  determine  the  literature due  to  on  different extent  into  i n t e r f e r e n c e from  the  adjacent  different  of  the  damping  of the  most  An that  windows  varied  A further  general  t h a t windows w h i c h p r o d u c e d t h e  the  peak.  w i n d o w i n g showed  r e d u c t i o n of s i d e l o b e s .  emerge as b e i n g  various the  lowest  skirts  i t s height.  c o n s i d e r a t i o n b o t h d a m p i n g and  windows  of  most  spectroscopies  t e n d e d t o c a u s e t h e peak t o be w i d e r a t h a l f o f Taking  the  dynamic  range,  effective.  time-domain s i g n a l  can  The 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 g o o d window s h o u l d p e r f o r m w e l l for  a range of damping One  range,  consideration the  sidelobe. is  to  values.  ratio  A general  c h o o s e one  interest.  of  Opting  when c h o o s i n g the  a window  is  the  dynamic  l a r g e s t peak h e i g h t  to  the  highest  principle  for selecting a particular  which s u f f i c e s  just  f o r the dynamic  window  range  of  f o r a window w i t h a g r e a t e r d y n a m i c r a n g e i s o f  50  no a d v a n t a g e as The  other  damping If  i t i n c r e a s e s the w i d t h  p o i n t t o be  of t h e  not,  it  taken  time-domain f u n c t i o n ,  i s best  to  a  large  size while  range  dampings.  this  eliminate  figures,  it  as is  with  apodization  of a b s o r p t i o n  the  Fig.  lineshape, F i g . The  9,  studies,  case.  but  for  tables However  the  sidelobes  s u c h as  not  t h e H a n n i n g and t h e windows  in Figs.  windows  are  5 and  a  lineshapes.  Gaussian,  9.  From of  greater  in  these  magnitude  effect  than  A clear illustration reduces  a  do  recommended  apodization  has  for  comparable.  the  only g r a d u a l l y reduces the  is  magnitude absorption  i s t h e one  which provides  the  narrowest  f r e e f r o m s i d e l o b e s f o r e a c h g i v e n d a m p i n g and  for  narrowing  for  5.  range c o m b i n a t i o n . Table II,  others  window w h i c h s h a r p l y  window o f c h o i c e  lineshape  these  these  Kaiser-Bessel  sidelobes,  Most  apparent that  lineshapes  of  be c o n t r o l l e d .  r e t a i n i n g a narrow l i n e s h a p e  illustrated also  height.  damped s i g n a l s .  s i d e l o b e s as w e l l as  chapter,  can  t h e c r i t e r i a of e l i m i n a t i n g  specified  Commonly e m p l o y e d w i n d o w s , not  i f this  t o s e l e c t a window w h i c h s u f f i c e s  windows f u l f i l l  of  o f t h e peak  i n t o c o n s i d e r a t i o n i s t h e amount  r a n g e f r o m undamped t o t o t a l l y A few  a t 50%  T h a t i s , i t has  absorption a specified  the  studies, s e t of  halfwidth value  o r T a b l e IV, conditions.  i n d i c a t e t h a t the best  for  in  magnitude  Examination  of  window v a r i e s f r o m c a s e  to  t h e r e a r e o f t e n a few  p r o p e r t i e s f o r a given case.  51  lowest  dynamic  windows w i t h v e r y This  fact  similar  reduces  the  number o f windows w h i c h need t o be recommended. D e p e n d i n g on t h e c o n d i t i o n s , d i f f e r e n t for  the  absorption  criterion,  mode.  Taking  dynamic  a l l dampings  as  the  t o 1% o f t h e peak h e i g h t ,  0.1% o f t h e peak h e i g h t , are e f f i c i e n t  both the F i l l e r  f o r a l l dampings,  works w e l l  a n d i f no a d j u s t m e n t s F o r a dynamic  range t o  E0.20 a n d K a i s e r - B e s s e l but a r e best  for  dampings.  The F i l l e r  E0.20 h a s a s i m p l e r a n a l y t i c a l  listed  Table I .  The K a i s e r - B e s s e l window w o r k s f o r a  large  in  dynamic  dampings  of  effectively any  down t o 0.01% o f  0.5 a n d l e s s .  apodized  damping  0.01%  range,  the  Thus a b s o r p t i o n  peak  window,  The a n a l y t i c a l  expressions  as  very  height,  for  can  be  f o r s i g n a l s of  a n d down t o  0.1%  f o r undamped o r s l i g h t l y damped s i g n a l s w i t h t h e  B e s s e l window.  lower  formula,  lineshapes  down t o 1% o f t h e peak h e i g h t  with the Noest-Kort  main  some w i t h  The N o e s t - K o r t  are p r e f e r r e d , the Norton-BeerF3 s u f f i c e s .  windows  range  t h e f o l l o w i n g r e c o m m e n d a t i o n s c a n be made,  c o n s t r a i n t s on t h e damping c o n d i t i o n s . for  windows a r e e f f e c t i v e  f o r these  and  Kaiser-  windows a r e  found i n Table I . I n t h e m a g n i t u d e mode, i s p r e f e r r e d f o r a dynamic Harris  window  10,000:1. T/ r not  for  when  r a n g e o f 100:1,  1000:1,  i s e i t h e r 2.0 o r 3.0,  results  and t h e  This pattern also holds  readily observable. i n Table IV,  T / r = 0.0, t h e Hamming window t h e 3-term  Kaiser-Bessel  Blackmanwindow  f o r T/ r = 0.5 a n d 1.0.  for When  a p a t t e r n f o r the narrowest case i s  However upon c l o s e r e x a m i n a t i o n  there are other  52  windows w i t h v a l u e s  of  the  almost  as  small  as the n a r r o w e s t w i d t h s .  100:1, t h e Hamming window i s b e s t less.  For  t h e same d y n a m i c  Thus f o r a d y n a m i c f o r a damping  range,  of  range  T/ r  and a h i g h l y damped  = 1.0  of or  signal,  the  3 - t e r m 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  for  low  dampings,  Hamming.  the  dynamic  r a n g e i s 100:1,  Blackman-Harris one  to  but not as e f f e c t i v e l y  and t h e damping  window i s t h e c h o i c e .  use f o r a g r e a t e r  amount o f d a m p i n g . Bessel  suffices  as t h e  dynamic  For a dynamic  f o r a l l dampings.  53  range,  i s unknown, This  If  the 3-term-  window i s a l s o  1000:1,  and  for  the any  range of 10,000:1, t h e K a i s e r -  CHAPTER  3  SIGNAL-TO-NOISE  3.1  Introduct ion The  obtained random  frequency  lineshapes  from s i m u l a t e d noise.  the  In r e a l i t y ,  transformed  lineshapes  spectrum.  which  and  N o r t o n and  The  only  Beer  Apodization  alter  necessarily  be  2 do  the  s i g n a l content equal  exponential  not  show SNRs.  (SNR)  of  frequency Frequency  n o i s e by F i l l e r  [43]  due  [53]. to  the  to d i f f e r  f r o m t h a t of  the  a window f u n c t i o n t o t h e and  the n o i s e  proportions.  It  will  i n c r e a s e the  have d i f f e r e n t  has  SNR  e f f e c t s on  t o t h e windows u s e d s h o u l d  time  content, been  t h a t a p p l y i n g a window w i t h t h e  of course  SNRs due  contain  noted that the  sidelobe behavior  c a u s e t h e SNR  Applying  decreasing  the  1 and  case.  studies  windows w i l l  contain  noise.  absorption  so  I t should  were  [ 4 4 ] h a v e been m i s t a k e n t o d e p i c t SNRs  will  in  chapters  signal-to-noise ratio  h a v e been d e r i v e d w i t h o u t  of t h e window, n o t  will  the  t h i n g t h a t i s shown i s t h e  unapodized  previous  a time-domain s i g n a l would  shown i n C h a p t e r s  lineshapes  i n the  time-domain s i g n a l s which d i d not  random n o i s e w h i c h w o u l d l i m i t  FT  RATIO  be  data  but  not  known  for  shape  [52].  of  a  Different  the  signal,  found  to  and  ensure  t h a t t h e w i n d o w s recommended h a v e a d e q u a t e SNRs. The  absolute  amount o f n o i s e  SNR  of the  frequency  s p e c t r u m d e p e n d s upon  i n the time-domain s i g n a l .  54  Thus t h e SNRs  the  before  and  after  a p o d i z a t i o n a r e a f u n c t i o n of t h e a b s o l u t e  time-domain compare  the  reference that  signal,  Eq.  relative  The  SNR  include  such  [41].  lineshapes  to  where  of the unapodized  the  lineshape,  window. h a v e been s e t f o r t h t o  represent  however many a r e n o t a p p l i c a b l e SNR  expressions  requiring  and t h e c o n d i t i o n s of  as the t e m p e r a t u r e ,  A l l of these  the  meaningful  about d i m e n s i o n s of the equipment [ 5 4 ] ,  of t h e FT-ICR c e l l ,  [55],  i t i s more  o f windowed  equations  o f FT l i n e s h a p e s ,  equations  Hence  i s t h e SNR  i s from the r e c t a n g l e  information size  SNRs  f o r comparison  Several different the  3.  SNR o f  equations  them a r e a p p l i c a b l e s i n c e t h e y  specific such as t h e  the  experiment  a n d a l s o more g e n e r a l  are similar  i n form,  here.  equations  b u t none  d e a l o n l y w i t h an unwindowed  of  time-  domain f u n c t i o n . General  trends  h a v e been shown f o r t h e r e l a t i o n s h i p b e t w e e n  t h e amount o f d a m p i n g o f t h e t i m e s i g n a l a n d t h e s i z e o f t h e For  the  absorption  mode,  i t i s known t h a t  d a m p i n g i n c r e a s e s , t h e SNR a l s o i n c r e a s e s SNR  decreases  with  Apparently  conflicting  as  [52],  i n c r e a s e d damping f o r  the  the  amount  Conversely, magnitude  SNR. of the mode  [54].  been  shown  damping the  [36,41].  increases,  results  These  patterns  w o r k s show t h a t a s t h e  t h e SNRs o f a b s o r p t i o n  SNRs o f m a g n i t u d e s p e c t r a  conflict  f o r SNR v s T/ r  increase.  a r i s e s from the d e f i n i t i o n  55  amount  spectra decrease However  of the problem  the :  have of and  apparent these  were  done u n d e r a d i f f e r e n t c o n d i t i o n . results  a r e due  T,  changed  was  T/ r .  and  3.2  not  situation.  r  held constant  values.  This  that i s ,  time-domain s i g n a l  ratio  and  i s an  of c h a n g e s i n T/ r  is altered;  signal,  which changed the  s t u d y , T was  Mention  contradictory  l e n g t h of t h e t i m e  held constant  the d u r a t i o n of the  r  was  altogether  in this  the  work  relaxation  changes.  Procedure formula  be  signal,  t o c a l c u l a t e t h e SNR  general Eq.  function.  3, This  enough t o a p p l y but  lineshapes  relative  SNR.  windowing specific  has  also  equation  apodized  and  r  t o t h e c a s e where  The must  with  t h u s c h a n g i n g t h e T/ r  different refer  t o s t u d i e s where t h e  R e c a l l that in t h i s  altered,  These seemingly  A  include  a parameter  general  formula  for a total  : for  T h i s has  s i g n a l e n v e l o p e has l e n g t h of time  s t u d y , and  T=1  the equation  lineshapes  simulated  time-domain  for  s h o u l d a l s o compare t h e  been d e r i v e d [ 5 6 ] .  c o n d i t i o n s of t h i s  to the  to the unapodized  c a s e where t h e  lasts  of t h e a p o d i z e d  a  of  the  i t should portray  the  a  SNR  window  relative  SNR  been t a i l o r e d the [45].  shape These  for  for  the  exp(-t/r ) are  the  is  (6)  where w ( t )  i s t h e window f u n c t i o n , and  56  t h e t e r m SNR  r e f e r s to  the  relative for  SNR.  The v a l u e s  a b s o r p t i o n , and Table The  SNRs  integration SNR d a t a  3.3  were c a l c u l a t e d on t h e UBC-MTS s y s t e m  at  increases.  Thus  the  SNR.  the v a l u e s  holds.  namely  i n Table  A l l o f t h e windows have which has a  T/ r  > 1.0.  similar notably case. case  SNR  t h e t r i a n g l e has t h e  and t h e 4-term B l a c k m a n - H a r r i s  has t h e lowest  w h i c h c a n be s e e n i n F i g .  13.  large difference, case, case. can  almost but  This be  18%,  clarity,  the remaining  windows  There i s a  b e t w e e n t h e s e two w i n d o w s f o r t h e  l e s s than general  next  For  between t h e 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 .  dampings  If o r  At h i g h e r dampings a l l of  s e l e c t e d windows have been p l o t t e d ,  damped  SNR  h i g h e r than t h e unapodized  t h e i r values are very c l o s e ,  the Noest-Kort  undamped  The  converge.  Although  fall  f o r absorption  from E q . 6 w i t h t h e e x p r e s s i o n s  windows h a v e a SNR  h i g h dampings,  only  the  resulting  I t s v a l u e drops as low a 55% of t h e unapodized  remaining  to  The  t h e SNR  general trend  a r e d i s p l a y e d i n T a b l e V.  highest  with  Mode R e s u l t s a n d D i s c u s s i o n  r e s u l t s w i t h the exception of the Noest-Kort  The  I  was t h e n p l o t t e d a g a i n s t T/ r .  values, calculated  lower  from Table  f o r magnitude.  t h e amount o f d a m p i n g i n c r e a s e s ,  windows  w(t),  II  p e r f o r m e d by t h e s u b r o u t i n e DCADRE.  Absorption As  f o r w ( t ) were c a l c u l a t e d  a 1% d i f f e r e n c e f o r t h e trend  seen i n F i g .  13,  57  of convergence and exact  at  details  totally greater can  be  TABLE V.  Relative  siqnal-to-noise  ratios  Window  : A b s o r p t i o n windows.  T/T 0.0  0.5  1 .0  2.0  3 .0  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 - t e r m 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 - t e r m 0.71  0.81  0.91  1.10  1 .27  Filler  DO.24  0.81  0.89  0.97  1.13  1 .26  Filler  E0.13  0.78  0.87  0.96  1.12  1 .26  Filler  E0.20  0.76  0.85  0.94  1.12  1 .27  a=2  0.82  0.90  0.98  1.13  1 .25  0.72  0.82  0.92  1.10  1 .27  0.75  0.84  0.94  1.11  1 .27  Noest-Kort  0.55  0.65  0.76  0.91  1 .18  N o r t o n - B e e r F3  0.83  0.91  0.99  1.13  1 .25  Triangle  0.87  0.94  1.01  1.14  1 .25  Rectangle  Hanning  Kaiser-Bessel Kaiser-Bessel  4-term  58  o = o = ^= += x = o =  BIockman-Harris, F i l l e r E0.13 Filler E0.20 Kai s e r - B e s s e I Noest-Kort Triangle  4-term  CN  d. 0.0  1 0.5  1 1.0  1 1.5  V Figure  13.  1 2.5  to  absorption  1  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 a s a f u n c t i o n o f d a m p i n g .  59  1 2.0  1 3.0  obtained  from  i n c r e a s e s as to  Table  For  of  whose h i g h e s t  The  SNR  the  windows,  greatest  the  increase  SNR  is  due  shown i s more t h a n t w i c e i t s  value.  The  p a t t e r n of to  signal  a higher amplitude  has  recall Fig.  the  increased  attributed  1;  highest  with  of  and  noise The  taper  damped  time-domain  level  i s constant  has  little  windows used f o r a b s o r p t i o n  to small amplitudes  i s very  small.  so t h e window w i l l the  sampling  the  SNRs  effect  a t t = T,  higher  towards t = T  where  In c o n t r a s t , the n o i s e  Thus t h e SNR  than  1.0  are  as  in  such  where i t i s s t r o n g e s t , n e a r t  decrease the n o i s e content  time.  T,  throughout  Thus m u l t i p l y i n g a damped t i m e - d o m a i n f u n c t i o n by  and  strength  A  be  s i g n a l at t = 0 than at t =  a window r e t a i n s t h e s i g n a l c o n t e n t 0,  i n c r e a s e d d a m p i n g can  s h a p e o f t h e window.  period.  a t t = 0,  4B.  SNR  however, the  the e n t i r e sampling  Fig.  each  the damping i n c r e a s e s .  the Noest-Kort  lowest  V.  the  signal  i s constant,  and  t o w a r d s t h e end  of  increases.  This  i s noted  by  for signals with a  large  amount  of  damping. There SNR.  If  narrow produce  is the  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 d e f i n i t i o n of  lineshape,  then  the  relatively  dynamic  as  meaning  a  w i n d o w s recommended i n  Chapter  1  l i n e s w i t h good r e s o l u t i o n .  produced narrow l i n e s h a p e s a  r e s o l u t i o n i s taken  low  r a n g e and  SNR. has  The  Kaiser-Bessel  f o r a l a r g e dynamic range,  The  Filler  a higher  and  SNR.  60  E0.20 was  but  good f o r a  w h i l e the F i l l e r  window i t has smaller  E0.13  only  worked  f o r a l i m i t e d dynamic range,  the t h r e e . Kort  yet  and  Kaiser-Bessel,  show a l o s s of SNR  resolution,  a=2.  i s c l e a r at lower  This pattern a drastically  should  3.4  be  Magnitude  exact  in F i g . from  are  14.  Table  unapodized Table  VI  the case;  however,  The  a t low  SNR  t o t a l l y damped SNR  Hanning  Noest-Kort  dampings,  and  shown  6 and  In every i n damping. increased are  the  This  follows  SNRR the  damping. lower  than  the  Examination  of  decreases  by  g i v e n window  i s o n l y a b o u t 25%  windows  case,  range i s s m a l l .  f o r any  the  For  any  l e s s than the  window, undamped  value. The  triangle  yields  Blackman-Harris  has  v a l u e s a l m o s t as  low.  and  to  some s e l e c t e d w i n d o w s a r e  o n l y a s m a l l amount as t h e d a m p i n g i s i n c r e a s e d . the  compared  is required.  t o windows  the  shows t h a t t h e SNR  Noest-  The  increase  due  of  pattern.  decreases with  values  the  t r i a n g l e and  r e s u l t s show a c o n s i s t e n t  w i t h an  SNR  Discussion  I I I were u s e d f o r w ( t ) .  of  when  T h e s e were c a l c u l a t e d w i t h Eq.  t r e n d t h a t SNR  All  i f a l a r g e SNR  i n T a b l e V I , and  decreases s l i g h t l y general  s u c h as  dampings.  particularly  Mode R e s u l t s and  m a g n i t u d e SNR  values  s u c h as t h e  r e d u c e d SNR,  used w i t h c a u t i o n  The  the h i g h e s t  T h u s windows w i t h good r e s o l u t i o n ,  windows w i t h p o o r e r  has  has  resolution.  the  The  the  lowest  highest values  SNRs,  with  the  and  the  4-term  Kaiser-Bessel's  T h i s d e m o n s t r a t e s a t r a d e o f f between t r i a n g l e window p r o d u c e d p o o r l y  61  SNR  resolved  TABLE V I .  Relative  siqnal-to-noise  ratios  : Magnitude  windows.  T/r  Window  Rectangle  0.0  0.5  1.0  2.0  3.0  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  Kaiser-Bessel  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  Triangle  0,.87  0,.86  0,.85  0,.80  0,.73  62  CO  d  • = o = A = += x =  CM  d  o.o Figure  14.  0.5  n— 1.0  B l a c k m a n - H a r r i s , 3BI a c k m a n - H a r r i s, 4 Hamming Kai ser-Besse I Triangle  i  1.5  Relative s i g n a l - t o - n o i s e r a t i o s due windows a s a f u n c t i o n o f d a m p i n g .  63  2.0  to  term term  2.5  magnitude  3.0  l i n e s h a p e s a t 1% o f t h e peak h e i g h t , y e t i t h a s t h e b e s t the o t h e r hand, good  a recommended  resolution  generality  has  window,  the  with  SNR.  this  :  However  t h e Hamming  w i t h good r e s o l u t i o n a t 1% o f i t s l i n e s h a p e ' s peak h a s t h e SNR v a l u e s H a r r i s window, h e i g h t , h a s SNR  3.5  second t o the best;  height,  i n the higher  SNRs  as  f u n c t i o n of  for  both  also  and t h e 3-term Blackmanpeak  range.  Conclusion The  pattern  lineshapes,  a  the  different  w i t h the former  window  spread  value  a t h i g h dampings.  small  provide  absorption  over  a  distinct  and  magnitude  i n c r e a s i n g and t h e l a t t e r  decreasing  T h i s p a t t e r n i s due t o t h e two  s h a p e s u s e d f o r t h e two modes.  SNRs  reference  damping  apodized  as t h e t h e damping i s i n c r e a s e d .  the  window,  w i t h good r e s o l u t i o n t o 0.1% o f l i n e s h a p e ' s values  On  Kaiser-Bessel,  c l o s e to the poorest  does not h o l d f o r a l l c a s e s  SNR.  a wide range and r i s e  above  The m a g n i t u d e SNRs  f o r the unapodized case,  The  absorption  the  unapodized  a r e a l l below  and t h e v a l u e s  cover  1.0, a  range. A  judged  t r a d e o f f b e t w e e n SNR a n d r e s o l u t i o n , where r e s o l u t i o n i s by  observed  a t 1% o f t h e p e a k h e i g h t ,  f o r the absorption  Noest-Kort absorption magnitude  the l i n e width  window  produced  lineshapes results,  results,  the  but  the  also  clearly  but not f o r magnitude. narrowest,  the  lowest  3-term B l a c k m a n - H a r r i s  64  is  best SNRs. window  The  resolved, For  the  created  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 For  the absorption  windows,  a pattern  w i n d o w s w h i c h were d e r i v e d  i s observed.  Blackman-Harris  and  Gaussian  remaining  windows  absorption  a n d m a g n i t u d e modes.  the v a l u e  descend  f o r i t s SNR  magnitude c o u n t e r p a r t T  = 1000  are  the  i s t h e b e s t , and t h e 3 -  are same  In fact  i n the absorption at  T/r  = 0.0.  o f e a c h window  the  worst,  order  for  thus only  and  the  both  the  f o r any d e r i v e d  window,  mode i s i d e n t i c a l t o i t s E x a m i n a t i o n of Eq. 6 w i t h  shows t h a t b o t h p o r t i o n s o f t h e e q u a t i o n  reduce t o 1 . 0 , area  in  from magnitude  The SNR q u a l i t y  i s t h e same f o r b o t h modes : t h e t r i a n g l e term  SNRs.  with  thisterm  the terms c o n t a i n i n g w ( t ) remain.  o f a window a n d i t s d e r i v e d equal.  •65  form a r e e q u a l ,  hence the  The SNRs  CHAPTER RESOLUTION  4. 1  OF  4  APODIZED  SPECTRA  Introduction The  windows  chapters  were  which  chosen  h a v e been recommended i n for  their ability  to  the  create  a  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 o f t h e peak h e i g h t . definition height; the  of r e s o l u t i o n  height  were  or r e s o l u t i o n ,  a t 1%,  o f more i n t e r e s t .  t h e r e a r e two p e a k s ,  at  half  o f t h e recommended w i n d o w s , 0.1% a n d 0.01% o f t h e  Thus t h e w i d t h a t  range r e q u i r e d i s used t o d e f i n e t h e r e s o l u t i o n If  narrow  The common  o f a s i n g l e peak i s t h e w i d t h  however, f o r the a p p l i c a t i o n  widths,  previous  the  peak  dynamic  of a s i n g l e  peak.  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 m e a s u r i n g t h e w i d t h o f a p e a k , t h e h e i g h t b e t w e e n t h e p e a k s i s measured. Window lineshapes.  functions However  have there  r e d u c t i o n a n d peak w i d t h . wider  the  windows,  peak  been a p p l i e d  at half  is  a trade  to  improve  spectral  off  between  sidelobe  The s m o o t h e r a l i n e s h a p e h e i g h t becomes.  For  i s made, t h e  the  recommended  t h e w i d t h a t 50% o f t h e peak h e i g h t h a s been s a c r i f i c e d  f o r n a r r o w e r l i n e s h a p e s a t a n d b e l o w 1% o f t h e peak h e i g h t . the lost  usual  idea of r e s o l u t i o n ,  for  improved r e s o l u t i o n  that defined at half at lower  fractions  height, of  the  Thus is peak  height. Resolution  has  been f o u n d t o be r e l a t e d  66  to the  amount  of  damping  : a s t h e damping  should  decrease  The  widths  increases, the r e s o l u t i o n  f o r both a b s o r p t i o n  o f t h e m a g n i t u d e l i n e s h a p e s do n o t c h a n g e a s  For a s i g n a l w i t h a t o t a l  h i g h l y damped u n a p o d i z e d than  the  totally case.  These  i s 1.6 t i m e s  h a v e been c a l c u l a t e d  damped l i n e w i d t h s  wider  at  FT l i n e s h a p e s  the  height w i l l  be p o o r e r  although  the  resolution  of Table  that resolution  at a specified  i n the previous chapters  d e f i n i t i o n of r e s o l u t i o n  b e t w e e n them [ 5 7 , 5 8 ] .  illustrated  totally  I I and  will  of a s i n g l e  peak,  increases. peak  signals.  spaced peaks.  The e f f e c t o f  be e x a m i n e d .  be u s e d f o r two p e a k s .  A new  One m e t h o d  o f two p e a k s i s t o m e a s u r e t h e v a l l e y  The r a t i o o f t h e h e i g h t  15.  increase  f r a c t i o n of the  o f a peak i s known a s t h e s i z e  in Fig.  Table  deal with single peaks,  spaced peaks w i l l  defining the resolution  height  undamped  l i n e w i d t h s do  f o r more h i g h l y damped  a p o d i z a t i o n on two c l o s e l y  the  the  the i n c r e a s e s f o r magnitude a r e  h o w e v e r s p e c t r a may c o n t a i n c l o s e l y  to  than  a n y f r a c t i o n o f t h e peak h e i g h t , w o r s e n s a s d a m p i n g  The r e s u l t s  wider  absorption,  f r o m t h e undamped a n d  [36]. Observation  This pattern confirms  Therefore,  of  o f T = 1, t h e  a t h a l f h e i g h t l i s t e d a s p a r t o f a s t u d y on t h e  i n c r e a s e d damping,  slight.  line  i s 1.4 t i m e s  For  IV r e v e a l t h a t t h e u n a p o d i z e d and a p o d i z e d with  time  magnitude l i n e width  damped l i n e s h a p e  rapidly  for absorption  acquisition  undamped m a g n i t u d e l i n e w i d t h .  continuous  height  [ 5 2 ] , and magnitude [ 5 4 ] .  w i t h c h a n g i n g a m o u n t s o f d a m p i n g a s do t h o s e widths.  at half  between t h e peaks the  valley,  A v a l l e y o f 10% i s c o n v e n t i o n a l  i n mass  67  of  Figure  15.  V a l l e y between two p e a k s . T o p : smooth Bottom: l i n e s h a p e with sidelobes.  68  lineshape.  spectrometry,  and w i l l  be t h e c r i t e r i o n  f o r t h e r e s o l u t i o n o f two  peaks. As  m e n t i o n e d i n C h a p t e r 2,  apodization  many s t u d i e s have been done  f o r a s i n g l e peak,  d e a l t w i t h two a p o d i z e d  but only  peaks.  In h i s study,  s p a c e d p e a k s were 5.5 b i n s a p a r t , frequency  spacing,  undamped  magnitude  absorption  and  signals w i l l This Chapter  mode  h i s study In  lineshapes  of the study abilities  containing  conformation dampings  Yet  the  dealt  and  spectra  to  two  following  f r o m damped  checks  windows  t o r e s o l v e two  peaks w i l l  be  peak s e p a r a t i o n s .  and  f r o m undamped t o t o t a l l y  damped.  fora  for  their  variety  Both magnitude and  of  absorption  signals  Their v a l l e y s w i l l  in  Apodized  ranging  be m e a s u r e d  peaks t h a t a r e separated  by 0.5 up t o 6.0 s p a c i n g s ,  see  the l i m i t  peaks w i l l  4.2  undamped  recommended  for  t o which these  with  spacings.  examined  be e x a m i n e d f o r t i m e - d o m a i n  to a  sections,  peaks.  t h e 10% r e s o l u t i o n c r i t e r i o n  will  closely  only  be e x a m i n e d , a n d f o r p e a k s o f even c l o s e r  fortheir  lineshapes  t h e most  [51]  where a b i n i s e q u i v a l e n t  lineshapes.  magnitude  portion 2  E q . 5.  t h e one by H a r r i s  on  E q . 5, t o  be r e s o l v a b l e .  Procedure The  contained  lineshapes  i n C h a p t e r 2,  o n l y a s i n g l e peak.  s i g n a l was s i m u l a t e d  from t h e t r a n s f o r m s  Retaining this  f o r two f r e q u e n c i e s by  69  of Eq.  3,  form, a time-domain  f(t)  O  0  1  1  B o t h p e a k s were s e t t o have e q u a l a m p l i t u d e , relaxation time  of  time.  The a c q u i s i t i o n  The  frequency, windows. in  f  1  and  =511.5 at  The b a s i c f r e q u e n c y  order  lineshape  f o r an  = 512 H z . at  frequency  undamped  a n d 0.33 f o r  The o t h e r  intervals  of  for certain  was i n c r e a s e d f r o m  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 which  t = 0 a total  signal.  Hz t o 506 Hz  every half  and e q u a l  1  damped s i g n a l ,  r e f e r e n c e peak was s e t t o f from  = K ,  Q  t i m e was s e t f r o m  1.0 a n d 0.5 f o r a p a r t i a l l y  an e s s e n t i a l l y c o m p l e t e l y damped  varied  K  T = 1 s e c , a n d r was s e t t o 1000.0  signal,  (7)  = (K c o s ( 2 f f f t ) + K , c o s ( 2 j r f , t ) ) e x p ( - t / r ) .  32 Hz t o 512 Hz frequency  The i n t e r f e r e n c e was  due t o t h e n a t u r e o f t h e FT o f t h e t i m e - d o m a i n s i g n a l . domain  function  positive  of  frequency  t h e form domain  i n Eq.  and  also  every  magnitude  the negative  a l t e r e d t h e peak h e i g h t s .  peak  A  3 has a l i n e s h a p e a  mirror  image  timei n the at the  corresponding negative frequencies.  T h i s h a s been d e s c r i b e d f o r  a  of the negative  single  extend  peak  into  the positive  lineshape. negligible difference  [40,52],  For  the  The s k i r t  r e g i o n and o v e r l a p w i t h  results  i n Chapter  2,  the  this  f o r one p e a k . When t h e r e a r e two p e a k s ,  image  frequencies.  is  f a r enough from  interference.  At higher  When a peak i s a t 512 H z ,  70  is  a considerable  frequencies, the positive  t h e n e g a t i v e one t o r e c e i v e  positive  effect  i n t h e peak h e i g h t s c a n o c c u r when t h e p e a k s  low  can  any  are of lineshape  noticeable  t h e c o n t r i b u t i o n due t o  the t a i l A This  o f t h e peak a t -512  Hz  i s l e s s than  0.3%  f a c t o r w h i c h h a s been o m i t t e d f r o m E q . phase  refers  t o a phase  i n the time-domain  t a k e s t h e f o r m o f a c o n s t a n t a d d e d t o t h e 2nf t h a t phase  affects absorption  lineshapes  are  purely  d i s p e r s i o n component, absorption  spectra.  absorption  a  peak  since  component.  zero.  when t h e  contaminating However ,  For magnitude,  on t h e shape  of a  phases  single  The  one  frequency the  worst case f o r the  phase  apodized  of both peaks  equaled  o f b o t h p e a k s h a v e been s e t t o z e r o t h u s  s i g n a l has  for  i n Chapter  t h e s i g n a l c o n t a i n s two c o m p o n e n t s ,  l i n e s h a p e o c c u r r e d when t h e p h a s e  The  used  I t i s known  best case, any  which  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  i t h a s no e f f e c t When  time-domain is  t e r m was  [14,52].  t h e f r e q u e n c y outcome.  magnitude  signal  term.  without  phase.  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 c a n be p h a s e a d j u s t e d  phase  affects  The  the  o c c u r s when t h e r e i s no p h a s e .  to e l i m i n a t e the i n t e r f e r e n c e 2,  7 is  t h e form i n Eq.  7,  and  p r e s e n t i n g the v a l l e y s of the  so t h e w o r s t  apodized  the case  magnitude  results. The w(t), The  time-domain  from  E q . 7 was  function,  then z e r o - f i l l e d  16,385 p o i n t s .  by a window,  resulting  f ( t ) w ( t ) was  magnitude.  sampled  f o r 2049  three times to a complete data set  A discrete Fourier  t h e s e p o i n t s i n t h e same manner The  multiplied  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  windowed time-domain  data p o i n t s , of  function,  t r a n s f o r m was  as d e s c r i b e d  l i n e s h a p e s were e x a m i n e d  .71  performed  i n Chapter  on  2.  f o r the s i z e of  their  valleys.  For  between  the  height  smooth l i n e s h a p e s , peaks  of the  was  valley  the  considered  as  f o r the  shown i n F i g .  s m a l l s i d e l o b e s between the peaks; the h i g h e s t as  s i d e l o b e was  illustrated  in  window,  the  sizes  against  T/ r  and  shown and  4.3  the  peak.  Narrow,  10%  particularly valleys  smooth  Kaiser-Bessel  lineshapes  criterion  low  was  and  of  valley  For  then  each plotted  These p l o t s  to  two  are  peaks for  e v e n when t h e  satisfied  dampings and  widely  and  one  peaks  summarize  the p r e v a i l i n g  the  patterns,  examination  This general  t r e n d i s seen i n F i g .  VII.  this  table  results  and of  i t is clear  72  10%  The  resulting  from  separations. are  in  A  a  three  This gives a  general  valley  provided  when  the  in  criterion.  w i t h the d e t a i l s  that  cases,  peaks.  d e t a i l s are  the  16,  many  spaced  w i t h peaks of a range of  a s d i s p l a y e d i n F i g . 16.  for closer  for  lineshapes  plot  From  15.  l i n e s h a p e as  resulted  dimensional  VII  Fig.  f o r the a b s o r p t i o n  to  Table  size  s i z e of the  window  way  of  the  had  Discussion  convenient  idea  the  sections.  examined f o r a b s o r p t i o n  v a r i o u s dampings,  cases,  between the peaks.  following  of  together.  valley for  were  the  results  were b r o u g h t c l o s e The  i n the  of  similar  i n these  were g a t h e r e d  Mode R e s u l t s and  Application revealed  spacing  point  Some l i n e s h a p e s  used to determine the  of the v a l l e y s  lowest  calculation  15.  t h e b o t t o m d i a g r a m of  discussed  Absorption  s i z e of the  i n Table  peaks  are  KAISER-BESSEL  Fiaure Figure  16 16.  V a l l e y h e i g h t s a s a f u n c t i o n o f s p a c i n g and d a m p i n g vaxxey ^ ^ . ^ d by t h e K a i s e r - B e s s e l p  e  a  k  window,  73  s  a  p  o  d  i  z  e  separated  by  6.0  all  dampings.  At  and  i n c l u d i n g a d a m p i n g of T/ * to T/r  spacings, 5.0  true  up  true  for a totally  there  spacings,  = 1.0;  and  = 2.0.  undamped c a s e .  f o r t h e undamped c a s e .  1%  T/r  = 0.0  this criterion For  The  t o and  valley  for  is satisfied  3 spacings  up  to  this  is  this  peaks are e x t r e m e l y  In f a c t ,  f o r p e a k s up  10%  4 spacings,  f o r a s c l o s e as  resolved for  i s l e s s than a  well  the v a l l e y s are  as c l o s e  as  4  is  below  spacings  apart. A low  systematic  p a t t e r n i s observed  v a l l e y s for widely  where t h e  spaced peaks,  and  lineshapes  gradually  increasing  valley  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  peaks  being  apart. that  totally  Also,  unresolved  as damping i n c r e a s e s ,  is,  as T / r  This  exact  increases, pattern  Blackman-Harris  window  exact  in  values,  compared These also  to  those  was  also  was.applied  Table  be  only  VIII,  resolution  observed as o b s e r v e d  show v e r y  identically  similar  results  explained  by  the  with one  the space  decreases;  increase. when  the  in Fig.  similar  for  3-term 17.  when  Table  VII.  two  peaks,  f o r a s i n g l e peak. nature  of  The  results  f o r t h e K a i s e r - B e s s e l window i n  showed r e l a t i v e l y can  the  are  the v a l l e y h e i g h t s  windows b e h a v e v i r t u a l l y  similarity  when t h e y  have  the  and This  FT  for  absorpt ion. As in are  the peaks are  set c l o s e r together,  the space between the p e a k s . added,  the  height  skirts  overlap  When t h e h e i g h t s o f t h e  of the v a l l e y  74  their  is  obtained.  skirts  This  is  TABLE V I I .  Absorption  lineshape  valleys  : Kaiser-Bessel  CM)  1_0  2A)  3^0  1.0  100  100  100  100  2.0  34  53  68  81  14  27  40  13  21  1  3.0  1.9  4.0  0.00  5.0  0.01  s  3.1  7.4  6.0  0.01  s  2.0  4.9  5.4  s denotes  TABLE V I I I .  Absorption 3-term  window.  13 8.6  sidelobes.  lineshape  valleys :  B l a c k m a n - H a r r i s window.  V i f  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  6.0  0.05 s  1.8  4.4  s denotes  sidelobes.  75  12 7.9  BLACKMAN-HARRIS,  Figure  17.  3-TERM  V a l l e y h e i g h t s a s 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 p e a k s a p o d i z e d by t h e 3 - t e r m B l a c k m a n - H a r r i s window.  76  confirmed the  by c l o s e e x a m i n a t i o n  linearity  function,  of  Fourier  h^t),  has  of T a b l e I I . transforms.  t h e FT H ^ f ) ,  This  If  and  f o l l o w s due  there  h ( t ) has  is  a  time  t h e FT  2  to  H (f); 2  then h^t) where  <  + h (t) <  > H^f)  2  + K (f)  > denotes the F o u r i e r t r a n s f o r m  f u n c t i o n c o u l d be composed o f a r e a l and these  parts  are merely a d d i t i v e .  an  In the  case  this  t h o u g h t o f as p u r e l y t h e a d d i t i o n o f  transforms  of t h e  are  additive,  simply  a p p l i e d t o two the peaks can  4.4  time s i g n a l s . the  peaks.  By  transform  of  frequency  part,  thus  absorption  i s examined  frequency  the a b s o r p t i o n  lineshapes peak c a n  examining Table I I , the h e i g h t s  be c a l c u l a t e d .  lineshapes,  Discussion  with  the  The [59]  claim  magnitude r e s u l t s o b t a i n e d that  additive.  A simple  not  f o r the magnitude s t u d i e s .  found  steadily  as  the  like  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  t h e v a l l e y s w o u l d be e x p e c t e d . agree  be  between  i t i s assumed t h a t m a g n i t u d e l i n e s h a p e s a r e a d d i t i v e  absorption  and  the  recommended windows f o r one  M a g n i t u d e Mode R e s u l t s and If  not  Since  The  imaginary  only  be  r e a l p a r t of the  pair.  lineshapes, can  the  (8)  2  pattern resembling  spacing  is  lineshapes the a b s o r p t i o n Rather  decreased,  the  than  are  of do  merely  results  is  increasing  valley  heights  fluctuate. Apodization  is  effective  i n smoothing the  77  spectral  skirt  associated with  f r e q u e n c y - d o m a i n l i n e s h a p e s , however  r e s u l t s ensued.  When f - f , > 6, t h e c r i t e r i o n o 1  is  However f o r p e a k s o f c l o s e r  satisfied.  pattern  i s not observed.  consequences undamped The  of  signal  magnitude  altering  has  lineshape  as i n F i g .  peaks,  as  19 a n d 20 i l l u s t r a t e t h e  Kaiser-Bessel  18.  increases,  seen i n F i g .  Kaiser-Bessel  spacings,  f -f^,  When  18.  window  Note t h a t t h e unapodized  However,  peaks.  lineshape  a r e three  spacings  when t h e y a r e b r o u g h t  closer  T h u s , f o r t h e undamped  case,  the v a l l e y s increase to a c e r t a i n  21 w h i c h sums up t h e v a l l e y s  f o r many  lineshapes.  apart  i s a p p l i e d t o two p e a k s the s k i r t  which  peak. are 6  between t h e peaks i s l e s s than  two p e a k s a r e s e p a r a t e d  by 3  spacings,  the  valley  them d i p s t o l e s s t h a n 0.1% f o r t h e undamped c a s e a s i n R e t a i n i n g t h i s amount o f d a m p i n g a n d b r i n g i n g t h e p e a k s  to only 2 spacings  apart causes the apodized  lineshapes  as  (the height  i s 100%) a s i n  19.  only  is  window,  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 o f one  the  between  f o r an  l i n e , and t h e u n a p o d i z e d , r e c t a n g l e  and dampings f o r t h e u n a p o d i z e d  The  2%.  simple  i n F i g s . 19 a n d 2 0 , t h e r e a r e no s i d e l o b e s b e t w e e n t h e  peak s e p a r a t i o n  spacings  a  b e t w e e n two p e a k s  the  hence t h e v a l l e y d e c r e a s e s .  extent  Fig.  due t o  i s t h e clashed l i n e .  together,  When  the spacing  spacing,  s i d e l o b e s b e t w e e n t h e two p e a k s when t h e y  apart,  as  18,  f o r t h e 10% v a l l e y  p l o t t e d on t h e same s c a l e a s t h e s i n g l e  d e p i c t e d by t h e s o l i d lineshape  Figures  unsystematic  one peak  However  of the v a l l e y  i f t h e peaks a r e s e t even c l o s e r , 78  to  t o appear  1  Fig.  spacing  f, =509, f =512 0  VO  502.0  504.5  Figure  18.  507.0  509.5  512.0  514.5  517.0  519.5  522.0  M a g n i t u d e l i n e s h a p e due t o t h e K a i s e r - B e s s e l f o r two p e a k s s e p a r a t e d by t h r e e s p a c i n g s .  window  Figure  19.  M a g n i t u d e l i n e s h a p e due t o t h e K a i s e r - B e s s e l window f o r two p e a k s s e p a r a t e d by two s p a c i n g s .  f, =511, f =512 0  oo  502.0  504.5  F i g u r e 20.  519.5  M a g n i t u d e l i n e s h a p e due t o t h e K a i s e r - B e s s e l window f o r two p e a k s s e p a r a t e d by one s p a c i n g .  522.0  RECTANGLE  apart,  the apodized  lineshape c l e a r l y  by a v a l l e y o f l e s s t h a n It  shows two p e a k s  0.1% a s i n F i g . 20.  i s a p p a r e n t t h a t t h e 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  t h e peak s e p a r a t i o n c h a n g e s .  t o t h e K a i s e r - B e s s e l window s u m m a r i z e d  The similar  with  to those were  the  pattern  two p e a k s  in Fig.  22.  for  Blackman-Harris  valley occurred.  F i g . 21.  4-term  window t o  Again  separations  if  a  similar as  d i f f e r e n t p a t t e r n a s s e e n by  not  produce  seen i n Table  IV.  However,  h e i g h t s due t o t h e K a i s e r - B e s s e l a n d t h e  chapters,  resolve closely  single-  t o t h e K a i s e r - B e s s e l ' s a s d i d t h e 4-  identical.  the 3-term B l a c k m a n - H a r r i s  heights  The  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  window,  Bessel are p r a c t i c a l l y  previous  see  very  the heights f l u c t u a t e d ,  K a i s e r - B e s s e l window d i d  Blackman-Harris  valley  the  i s observable.  peaked l i n e s h a p e s as s i m i l a r term  as  peak.  c a l c u l a t e d f o r two p e a k s o f d i f f e r e n t  comparing t h i s w i t h  The  increase  window p r o d u c e d l i n e s h a p e s  shown i n F i g . 2 3 , b u t w i t h a t o t a l l y  damping  I t i s clear  by t h e K a i s e r - B e s s e l window f o r one  4-term  due  decreases.  4-term B l a c k m a n - H a r r i s  valleys  for  with the r e s u l t s  the h e i g h t of the v a l l e y s does not simply  s p a c i n g between  as  The v a l l e y s were e x a m i n e d f o r p e a k s  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 d a m p i n g s  that  separated  4-term  A similar pattern i s  window.  T h i s was  t o be an e f f e c t i v e window,  Kaiserobserved  shown,  i n the  but i t does  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 .  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 ,  83  the  and  not The no  KAISER-BESSEL  22.  V a l l e y h e i g h t s a s a f u n c t i o n o f s p a c i n g and d a m p i n g o f two m a g n i t u d e p e a k s a p o d i z e d by t h e K a i s e r - B e s s e l window.  84  BLACKMAN-HARRIS,  F i g u r e 23.  4-TERM  V a l l e y h e i g h t s a s a f u n c t i o n o f s p a c i n g and damping o f two m a g n i t u d e p e a k s a p o d i z e d by t h e 4 - t e r m 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 t h e amount of d a m p i n g can The or  H a n n i n g window was  the  4-term B l a c k m a n - H a r r i s  Windowing  with  sidelobes.  the  lineshapes  have  t o as c l o s e as  increase Fig.  for  24.  lineshape  3.0  1.5  when a p p l i e d t o  a  single  a lineshape  spacings.  spacings  low  for  a  s m o o t h , but  c o u l d be m i s t a k e n  for other  Unlike  the a b s o r p t i o n  valley  heights  f r o m one  for  two  c o n t a i n s many a u x i l i a r y  results,  i s observable  for  magnitude.  overview.  Appendix  in this  Exact section,  peaks.  The  maxima w h i c h  values  damping heights  There i s a l s o  p l o t s h a v e been p r e s e n t e d  a few  in  problem  The  was  discussed  shown  no p a t t e r n b e t w e e n  window t o t h e n e x t .  general  separated  peaks.  spacing value to the next.  only  for  valleys consistently  c o n s i s t e n t p a t t e r n f r o m one observed,  the  valleys  f o r a l l dampings as  s i n g l e peak p e r s i s t s  many  peaks,  When p e a k s a r e the  peak.  with  i s a more c o n s i s t e n t p a t t e r n , h o w e v e r t h e  i s not  fluctuate  consistently  spacings,  decreasing  This  encountered  and  as e f f e c t i v e a s t h e K a i s e r - B e s s e l  H a n n i n g window l e f t  s e p a r a t i o n s g r e a t e r than 3.0  detected.  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  resulting  by  not  be  Since  no  no  pattern  to  give  f o r the v a l l e y s of the  a  windows  w i t h d e t a i l s about s i d e l o b e s ,  are  in  C.  The additive  reason  behavior  mathematically. absorption  t h a t t h e m a g n i t u d e l i n e s h a p e s do  Let  spectrum,  as  shown h(t)  by  absorption  represent  a time  not can  signal,  D i s the d i s p e r s i o n spectrum,  86  show  simple  be  shown  A and  is  the  M i s the  H A N N I N G  Figure  24.  V a l l e y h e i g h t s a s a f u n c t i o n o f s p a c i n g and damping of two m a g n i t u d e p e a k s a p o d i z e d by t h e H a n n i n g window.  87  magnitude spectrum i n the frequency domain. h^t),  the  transform produces A ,  D ,  1  s i g n a l , h , gives A , D , 2  2  h(t)  and M .  2  = hjU)  For a time  and  1  .  signal,  A second time  If the time s i g n a l i s  2  + h (t)  (9)  2  then the transform y i e l d s the frequency f u n c t i o n s  A = A  1  + A  2  (10)  D = D  1  + D  2  (11)  as e x p l a i n e d If  i n the p r e v i o u s s e c t i o n .  magnitude  were  merely  additive,  l i n e s h a p e f o r two time s i g n a l s would  M = M  + M  1  = (A,  Calculation  of  the  magnitude  be  (12)  2  + Dj )  2  then  2  1  7  + (A  2  + D  2 2  2 2  )  1 / 2  .  (13)  the a c t u a l magnitude lineshape due to  two  time  s i g n a l s g i v e s the f o l l o w i n g  M = (A  2  = ((A  + D ) 2  1  + A )  = (A^ +  2  (14)  1 / 2  2  + (D  + A  2 2  + D ) ) 2  1  1  /  2  2  + D  88  2 2  + 2A^  + 2B D ) ^ . ]  2  }  2  2  (15)  This  i s not e q u a l  not  merely  t o Eq.  additive.  13,  t h e r e f o r e magnitude l i n e s h a p e s are  The  magnitude l i n e s h a p e  M +M , b u t , by r e a r r a n g e m e n t o f E q . 1  2  the r e l a t i o n  i n Eq.  is  not  simply  15 w i t h t h e i m p l e m e n t a t i o n  of  14, t h e m a g n i t u d e l i n e s h a p e w i t h two p e a k s i s  actually  M = (Mj  2  + M  + 2k k  2 2  }  + 2D D )  2  1  When t h e p e a k s a r e f a r a p a r t , between spaced,  the peaks are very the  portion  p o s i t i v e values  for M  small.  between n  and  create  .  (16)  the values  for M ,  A  n  and  n  D  n  However when p e a k s a r e c l o s e l y  t h e p e a k s may  be made up o f  and p o s s i b l y l a r g e n e g a t i v e  and e i t h e r p o s i t i v e o r n e g a t i v e together  1 / / 2  2  values  a high v a l l e y ,  for A .  values  large for  These c o u l d  n  o r c a n c e l a n d make  a  D, n  add low  valley. 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  25, 26, a n d 27. The then  a peak  closer  to  dot-dash A  examination, the  Hz,  Fig.  by t h e s o l i d  line,  single  i s s e t a t 512 H z , F i g . 2 5 , a n d  i s b r o u g h t i n t o 510 H z , 511  represented  r e f e r e n c e peak  The  line,  Fig.  26,  and  magnitude  the a b s o r p t i o n  then  lineshape  t o be composed  can  be  even  lineshapes  are  l i n e s h a p e by  the  a n d t h e d i s p e r s i o n l i n e s h a p e by t h e d a s h e d  magnitude  absorption  27.  in Figs.  seen,  with  line. close  of the square root of the squares of  l i n e s h a p e and t h e d i s p e r s i o n l i n e s h a p e s  •89  in Fig.  to  =512  in cr  o b  in d i  o T  MAGNITUDE  if> 7  Figure  25.  506.0  —I 508.0  —i— 510.0  L i n e s h a p e s of a s i n g l e B e s s e l window.  90  —I  512.0  f  peak a p o d i z e d  —  ABSORPTION  —  DISPERSION  —I  516.0  514.0  by  the  Kaiser-  518.0  f, =510,  f. =512  in  MAGNITUDE — ABSORPTION - - DISPERSION 506.0  508.0  510.0  512.0  f Figure  26.  —i 514.0  1  516.0  Lineshapes of two peaks s e p a r a t e d by two s p a c i n g s apodized by t h e K a i s e r - B e s s e l window.  91  518.0  and  f, =511, in  92  f. =512  25.  A t a d i f f e r e n c e of  totally Fig.  unresolved,  26,  i s not  together, and  but  be  at  512  26.  to a combination When  The  reason of  and  can  the  be  another  then  added t o g e t h e r ,  The  same c a n  the  lineshape  2  for  But  dispersion  t o two  peaks  single  peak.  25  be  t h i s can  i s placed Hz  the  and  result.  n o t be  done  window s h o u l d  type  Hanning  a=2  not  be  totally  as  the  4-term  of the magnitude v a l l e y s f o r unexpected.  The  magnitude-  window p r o d u c e d an a b s o r p t i o n s h a p e w i t h  distortions,  predictable behavior,  and  the  Fig.  the 4-term B l a c k m a n - H a r r i s , point  l i n e s h a p e s were n o t i c e d  m a g n i t u d e - t y p e windows s u c h Thus t h e b e h a v i o r  this  Another  the  lineshapes.  Blackman-Harris.  severe  to  lineshapes.  in Fig.  i n F i g . 27 w i l l  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 Chapter  well  s h a p e i s p l a c e d a t 511  be done f o r d i s p e r s i o n .  f o r the magnitude  in  and  even  very  s e e n t o be due  l i n e s h a p e as  identical  absorption  suddenly  absorption  added  brought  t o the l i n e s h a p e of a  i f an a b s o r p t i o n  25,  of t h e  the peaks are  are  lineshape,  Fig.  d i s p e r s i o n l i n e s h a p e s due  attributed directly  and  seen t h a t t h i s  t o the a d d i t i o n of the magnitude  absorption  Hz  be  the magnitude l i n e s h a p e s are  not  For example,  the magnitude l i n e s h a p e s  magnitude l i n e s h a p e s ,  combinations  s p e c t r a , and The  two  I t can  lineshapes.  Fig.  complicated  can  just  together,  resolved,  F i g . 26.  i s i n s t e a d due  dispersion  closer  2 spacings,  r e s u l t i n g m a g n i t u d e v a l l e y s had 24, Fig.  t h a n t h o s e due  more  t o windows s u c h  as  i s observable  in  23.  about magnitude l i n e s h a p e s  93  less  Fig.  27.  fall  on t h e c o r r e c t f r e q u e n c i e s ,  This  The maximum v a l u e s  i s directly  due  they  should,  do  not  b u t have been s h i f t e d o u t w a r d s .  t o the a d d i t i o n of  d i s p e r s i o n components. where  of t h e magnitude l i n e s h a p e s  the  absorption  Yet note that the a b s o r p t i o n and t h e d i s p e r s i o n l i n e s h a p e s  and  peaks  are  fall  properly  placed. A  single  magnitude-mode unlike  the  However  does  not  examinations  excellent  different  valleys,  Blackman-Harris  however  their  shape o f t h e l i n e The  peaks,  for single 6  spaced peaks.  spacings,  windows  both  yield  lineshape.  a It  seen  The H a n n i n g window, on t h e o t h e r  p a t t e r n o f peak h e i g h t s , however t h e  time  shift,  i s well  has  known  no  that  or a time s h i f t ;  effect  on  mode becomes d e p e n d e n t  on p h a s e .  This  signal, magnitude mode  is  h o w e v e r , when a p h a s e i s the  magnitude  time-domain i s not merely  b u t h a s become a new c o m p l i c a t e d  94  a  t h e magnitude  o n l y one o f two t i m e - d o m a i n c o m p o n e n t s  i n time,  by  i s n o t smooth.  independent of phase, to  of  fluctuating  a d d i t i o n o f a p h a s e component t o a t i m e - d o m a i n  causes  peaks.  The K a i s e r - B e s s e l  p a t t e r n s a r e not i d e n t i c a l as  produces a systematic  shifted  spaced  for  g e n e r a l i t y c a n n o t be r e a c h e d a b o u t t h e e f f e c t  c o m p a r i n g F i g . 22 w i t h F i g . 2 3 .  added  obtained  results  o v e r unwindowed s i g n a l s .  w i n d o w s on two c l o s e l y  4-term  which  improved  o f two v e r y c l o s e l y  improvements  i s an i m p r o v e m e n t  A concise  hand,  yield  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  windowing  and  window  signal  whose  transform  i s no l o n g e r t h e same a s t h a t due t o a s i g n a l  without  phases. If will  a phase i s added t o h ^ t ) ,  have  then the magnitude  a f o r m d e s c r i b e d by E q .  components  will  differ  16;  however  t h e A,  from those without phase.  terms a r e unchanged.  An i l l u s t r a t i o n  lineshape  The  and remaining  of t h e e f f e c t s of phase i s  shown f o r an undamped t i m e - d o m a i n s i g n a l a p o d i z e d by t h e  Kaiser-  B e s s e l window where t h e r e i s one peak a t f = 5 1 2 Hz a n d a n o t h e r a t Q  ^ = 5 0 8 Hz + p h a s e .  The p h a s e h a s been s e t t o 0 i n F i g . 28,  in  ir  Fig.  29,  increases,  and  the  in  Fig.  30.  Note t h a t  the  phase  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 ,  t h a t t h e w o r s t c a s e o c c u r s f o r p h a s e = 0, the  as  ir /2  Fig.  28.  and  In c o n t r a s t ,  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  increased  with  the v a l l e y  b e i n g l o w e s t when t h e p h a s e e q u a l s  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 p h a s e a d d e d t o one o f two domain  components;  relative  however,  phase d i f f e r e n c e .  time-domain  signal,  this  phenomenon  +  where 8,  2  each w i t h a d i f f e r e n t phase,  + M  + 2((A A  2 2  1  2  h_(t).  +  2  The  1  1  i s t h e phase of h ^ t ) ,  component,  for  and 9  the  Eq.  16  D ^ J c o s ^ - B ^ 2  2  1 / 2  (17)  i s t h e phase of t h e second  d e r i v a t i o n of t h i s  95  then  a  expression  (A D - A D ) s i n ( 6 - e ) ) ) 2  true  time-  I f t h e r e a r e two c o m p o n e n t s i n  i n c o r p o r a t e s t h e p h a s e w i t h t h e new  M = (M,  is  0.  formula i s found  in  PHASE=0  Figure  28.  Magnitude  lineshapes  96  from s i g n a l  with  phase = 0.  PHASE=TT/2  F ig au ur re e Fi  29 2*.  M agnitude wagn^  l i n^e s h aFp e^s Q  97  f^ rom s i g n a l O  T  H  E  R  H  A  where one component ,  S  A  P  H  A  S  E  O  F  /  2  >  PHASE=n  KAISER-BESSEL RECTANGLE i  502.0504.0  i  506.0  i  508.0  1 510.0  1 512.0  1 514.0  1 516.0  1 518.0  1 520.0  1 522.0  f 30.  M a g n i t u d e l i n e s h a p e s f r o m s i g n a l where one compoment h a s a p h a s e o f 0, a n d t h e o t h e r h a s a p h a s e o f it.  98  A p p e n d i x D. there  Note t h a t Eq.  17 r e t u r n s  Conclusion The  smooth  Kaiser-Bessel absorption  dynamic has  window,  lineshape  which  for  was  found  to  a s i n g l e peak  range and v a r i e t y of dampings,  create  over  a  less  are separated than  together,  continue  to  satisfy In  absorption  lineshapes,  recommended f o r a s i n g l e peak w i l l The  together,  peaks  is  closer  up  the  to  3  absorption  m e r e l y by a d d i t i o n o f  therefore  the  windows  are  brought  i n an e a s i l y p r e d i c t a b l e  pattern.  close  together,  The m a g n i t u d e l i n e s h a p e s  regular pattern magnitude  dispersion  brought  when two p e a k s a r e  they  are  poorly  b u t when t h e y a r e s e t e v e n c l o s e r t h e y s u d d e n l y become  resolved.  magnitude  them  a l s o r e s o l v e two p e a k s .  however  they a r e not r e s o l v e d  resolved, well  the  K a i s e r - B e s s e l window was a l s o f o u n d 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 ,  the  general,  SNR,  If  set  this criterion  due t o two p e a k s c a n be o b t a i n e d  single  As  t h e v a l l e y between As t h e p e a k s a r e  f o r an undamped s i g n a l .  lineshape the  10% f o r a l l d a m p i n g s . they  spacings  by 6 s p a c i n g s ,  a  large  and t o have a f a i r  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 p e a k s .  peaks  the  16 when  i s no d i f f e r e n c e i n p h a s e .  4.5  No  t o t h e form of Eq.  i s observable,  but the f l u c t u a t i n g behavior  v a l l e y s c a n be e x p l a i n e d  components.  a r e not simply a d d i t i v e .  by  the  T h e s e two c o m p o n e n t s  s p e c t r a a r e d e p e n d e n t on p h a s e .  99  absorption  also  explain  I f there  is a  of and why  phase  difference lineshape  between  t h e two c o m p o n e n t s ,  o f t h e two p e a k s i s b e t t e r  lower v a l l e y both components  than a lineshape have no  the apodized  resolved,  that  magnitude  i s , i t has a  due t o a t i m e - d o m a i n s i g n a l where  phase.  100  CHAPTER 5 SUMMARY  Truncated time-domain s i g n a l s , yield  lineshapes  sidelobes,  was  differing shape  with sidelobes.  transformation,  Apodization,  elimination  of  p e r f o r m e d by m u l t i p l y i n g t i m e - d o m a i n f u n c t i o n s of  d e g r e e s o f d a m p i n g by v a r i o u s window  of t h e window f u n c t i o n s was  studies;  however,  this  type  sidelobes  in  the  consisting  of  h a l f of t h e  equal  upon F o u r i e r  symmetrical  The  f o r the  magnitude  o f window p r o d u c e d l a r g e  negative  absorption  t i m e p e r i o d was  functions.  lineshape.  symmetrical  f o u n d t o be  An  shape  adapted  extended  satisfactory  f o r the  shape over  an  absorption  studies. Specific  conditions  damping  of  the  dynamic  range  window,  with  the of  various  windows  For  the  spectrum.  the  required The  peak h e i g h t .  closely  the a b s o r p t i o n  as  from  range  s e l e c t i o n of  sidelobes  From t h i s being  different  101  at  as  a an  specific  systematic  search,  effective.  s p a c e d p e a k s c o u l d be  mode,  minimum  dynamic  were e x a m i n e d t o e n s u r e t h a t t h e y  t h a t two  the  were e x a m i n e d f o r t h e window w h i c h  peak f r e e  emerged  include  t o a v a r i e t y of dampings t a k e n  lineshapes  narrowest  fractions  SNRs, and  practice  c h o s e n as t h e m a i n f a c t o r f o r t h e  The  recommendations  in  s i g n a l and  frequency  applicability  added f a c t o r . yielded  time-domain of the  r e q u i r e m e n t was  encountered  had  These adequate  resolved.  windows were recommended  for  specific  dynamic  required,  the  dampings;  however,  coincide single  ranges.  Noest-Kort  with  this  the  window  I f a dynamic  due  dynamic  rather  both  damping  of the  time-domain  slightly  1000:1,  yielded  sidelobes  10,000:1  was n e e d e d ,  to  level  Kaiser-Bessel lineshape  the F i l l e r  f o r a l l dampings,  r a n g e o f 100:1,  of a b s o r p t i o n  dampings.  increase  relative with  SNRs f o r t h e a b s o r p t i o n  102  were  range  of  signals.  The  though r a t h e r  showed  a  wide,  damping.  more  marked  lineshapes.  For  recommended  lineshapes  a  for  due t o t h e  sufficed  The K a i s e r - B e s s e l window e l i m i n a t e d  i n c r e a s e d damping.  windows  for a l l  r e q u i r e m e n t s f o r a dynamic  f o r a l l d a m p i n g s down t o a d y n a m i c r a n g e o f The  These  damped  while wider,  T h i s window a l s o s a t i s f i e d  r a n g e o f 1000:1.  The  a  r e d u c e d s i d e l o b e s down  t h e Hamming window was  B l a c k m a n - H a r r i s window,  For  f o r s i g n a l s of low  magnitude l i n e s h a p e s  than a p o d i z a t i o n  than  Kaiser-Bessel  I f a dynamic  window d i d p r o d u c e a s m o o t h ,  of  and  a the  but the l i n e s h a p e s  none o f t h e windows  undamped t o m o d e r a t e l y damped s i g n a l s . 3-term  If  lineshapes  results.  for moderately to t o t a l l y  Apodization  dynamic  signal.  wider  E0.20  satisfactory  a t 0.01% o f t h e peak h e i g h t  improvement  for a l l  f o r a l l dampings.  w i d e f o r h i g h l y damped s i g n a l s .  this  was  window h a s a p a r a m e t e r w h i c h i s s e t t o  t o t h e N o e s t - K o r t window,  eliminated  100:1  w h i c h 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 ,  range of  windows  of  window e l i m i n a t e d s i d e l o b e s  Norton-Beer F 3 w i n d o w s u f f i c e d , with those  range  sidelobes  10,000:1. windows  were  found  The v a l u e s r a n g e d f r o m  to  about  half  the  u n a p o d i z e d SNR  unapodized  SNR  f o r the  f o r h i g h l y damped  y i e l d e d narrow l i n e s h a p e s , SNRs  compared  recommended. except  to The  the  undamped c a s e t o g r e a t e r  the  those  cases. listed  ratios  for  Most  than  windows  which  above, a l s o produced  windows  which  were  which  has  rather small values  low not  recommended w i n d o w s r e t a i n e d a d e q u a t e  Noest-Kort  the  SNRs,  for  low  a l l o f t h e SNRs were l o w e r  than  dampings. For those  the magnitude windows,  due  t o the unapodized case f o r a l l  decreased  a  small  recommended than the  windows a l l had  two and  as  damping  sufficient  SNRs,  two  may  peaks,  be  d e f i n e d as  the h e i g h t of the  with a height  of  between t h e s e  10%  skirts  p e a k s was  for a single absorption  two  closely  spaced peaks.  two  peaks  c o u l d be  three  peaks  between  often being  higher  was  the  peaks  simply  of  the  valley  satisfactory. skirts  t h e a d d i t i o n of Thus the  t h e amount o f  (1/T).  As  10% the  As  overlapped the  windows  peak were a l s o e f f e c t i v e  resolution  for  damping,  valley  for  spacing  between  decreased.  As  as  the  r e s o l u t i o n worsened.  the magnitude l i n e s h a p e s , the  their  D e p e n d i n g on  spacings,  decreased, the  being  r e s o l v e d w i t h a t most a  frequency  damping i n c r e a s e d For  The  f o r t h e a b s o r p t i o n mode.  recommended  c l o s e as  SNRs  increased.  the h e i g h t  p e a k s were b r o u g h t c l o s e t o g e t h e r ,  sizes  the  was  The  unrecommended w i n d o w s .  Resolution between  amount  dampings.  were n o t  the h e i g h t s  f o u n d t o be a  103  of  simple  the  valleys  function  of  frequency  spacing  observed  since  windows  The  f u n c t i o n of absorption  spacing  t h e r e was a the  peaks  smaller  additive. valley  for  and d i s p e r s i o n components also  of t h e magnitude  explain  of magnitude s p e c t r a .  six  b u t i n no c o n s i s t e n t  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  components  The'  the  as  of t h e lineshape.  surprising  A p a t t e r n was  a  phase  observed  when  p h a s e d i f f e r e n c e between two t i m e - d o m a i n c o m p o n e n t s . the  apodized  became more r e s o l v e d up t o a p h a s e d i f f e r e n c e o f  ir , a n d  are various  The windows  to  absorption  be  increased,  in resolution.  a p p l i c a t i o n s f o rthe  recommended f o r a b s o r p t i o n  effective  absorption  c o u l d be  this  applied window  t h e windows  negative  nature  of sampling  i s solved;  mode c o u l d be i m p l e m e n t e d i n F T - I C R .  of a phase dependence o f magnitude s p e c t r a b o t h FT-ICR a n d FT-NMR. in  in  i f a single  Since  i n eliminating  results  were  sidelobes  of  s p e c t r a , t h e p r o b l e m o f a p p a r e n t p h a s e d i s t o r t i o n s due  the d i s c r e t e  shifts  was n o t  of the v a l l e y h e i g h t s  p r e f e r r e d o v e r a d j u s t i n g windows.  shown  to  10%  FT-NMR i n p l a c e o f t h e e m p i r i c a l windows  were  pattern  not  spacings,  r e l a t i v e p h a s e d i f f e r e n c e was  There  for  are  with a  behavior  the unapodized peaks d e c r e a s e d  study.  A clear  lineshapes resolved  fluctuating  two  dependency  As  peaks  and f o r v a r i o u s  pattern.  These  o f damping.  magnitude  produced  spacings,  nor  the  The o b s e r v a t i o n  frequency  importance, p a r t i c u l a r l y  p o s i t i o n s of  t o FT-ICR.  104  therefore  the  The r e c o g n i t i o n  i s of s i g n i f i c a n c e t o  t h a t t h e r e c a n be s l i g h t apodized  Recommendations  peaks  i s of  i n this  work  have  seen  windows  p r a c t i c a l application with  promising  recommended f o r m a g n i t u d e h a v e been a p p l i e d  t o FT-ICR e x p e r i m e n t s ,  improvement  rigorous effects  over better  the unapodized  spectra,  than the e x i s t i n g  testing i s s t i l l of a p o d i z a t i o n  The  successfully  a n d t h e 3 - t e r m B l a c k m a n - H a r r i s window  found t o have a s u f f i c i e n t dynamic range.  sidelobes  results.  required,  T h e s e r e s u l t s were an and  apodizing  eliminated function.  105  the More  p a r t i c u l a r l y t o examine the  on t h e r e s o l u t i o n o f v e r y c l o s e l y  peaks.  was  spaced  REFERENCES 1.  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«LARGE-L0W+1 DIV-1.ODO/128.ODO DT»DIV TAU=2.0D0 Pl=3.14159265358979D0 TW0PI«6.28318530717958D0 WO=32.ODO*TW0PI  1 1 1  59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 11 1 112 113 1 14 1 15 116  C  C C  THE ALPHA PARAMETER A=3.50O TBW=PI*A X*10.000  IN THE KAISER-BESSEL WINDOW IS "A"  CALCULATE THE DENOMINATOR OF THE KAISER-BESSEL WINDOW. CC»1.ODO CX*1.000 DO 5 K*1.20 CC=K*CC CD=(TBW/2.0D0)**K CY«=(CD/CC)**2D0 CX-CX+CY CONTINUE i5 C WRITE THE HEADING OF THE TABLE WRITE(6.80)CX 80 F0RMAT('THE DENOMINATOR IS '.F9.4) WRITE(6.85)A 85 FORMAT('FOR A « '.F3.1) C -CALCULATE Y, THE TIME DOMAIN SIGNAL; T RANGES FROM 0 TO 1 DO 10 1=1.129 T=(1-1)*DT C CALCULATE THE KAISER-BESSEL WINDOW. NOTE: TO DO OTHER WIDOWS, C INSERT THEIR FORM FROM TABLE I I I HERE. B=TBW*(DABS(1.O00-(2.OD0*DABS(T-O.5DO))**2DO)**O.5D0) C=1.ODO T0T=1.ODO 00 7 K» 1,20 C-K*C 0=(B/2.0D0)**K SUM=(D/C)**2D0 TOT-TOT+SUM 7 CONTINUE WINDOW(I)-TOT/CX WINDOW!" 129)=0.0D0 C Y(I)=X*DEXP(-T/TAU)«DCOS(WO*T)*(WINDOW(I)) 10 CONTINUE C ZERO FILL TO A TOTAL OF M POINTS DO 15 I-130.M Y(I)=0.0D0 15 CONTINUE C -CALCULATE FT OF Y & STORE IN DATA AS DESCRIBED IN UBC FOURT' CALL P0LFT2(Y.DATA.M.DT) C -CALCULATE F-FO. THE FREQUENCY, AND STORE IN W(I) C FCOR CORRECTS THE FREQUENCY 0 RANGE FROM 0 TO MAXF. FCOR=DFLOAT(MAXF)/(DFLOAT(NPTS)-1.ODO) DO 20 1*1,NPTS W(I)«(DFLOAT(I)+32.ODO*TW0PI-WO-1.0D0)*FC0R 20 CONTINUE C -CALCULATE THE MAGNITUDE LINESHAPE AND STORE IN OMAG(J) DO 30 I'LOW.LARGE U«I-L0W+1 DATAR(J)=DREAL(DATA(I)) DATAI(d)-DIMAG(DATA(I)) . DMAG(J)*(((DATAR(d))**2)+((DATAI(d))**2))**0.5D0 30 CONTINUE C -CALCLATE RELATIVE INTENSITY OF THE MAGNITUDE LINESHAPE  112  117 1 18 1 19 120 121 122 123 124  50  RM=DMAG(1) DO 5 0 I - 1 . N P T S RELM(I)=(DMAG(I))/RM CONTINUE  C -PRINT T I T L E • WRITE(6,100) C . . . W R I T E T H E NAME  O F T H E WINDOW  100  FORMAT('WINDOW  125 126 127  120  WRITEC6,120) F0RMAT('MAGNITUDE  128  122  WRITEC6.122)M FORMAT('POINTS  129 130 131 132 133 134 135 136 137  AFTER  "WINDOW  -"  - KAISER-BESSEL') FT')  TRANSFORMED  :'.I5)  TTAU=1.ODO/TAU WRITE(6.125)TTAU 125  FORMAT('T/TAU=',F5.3)  WRITE (6.150) 150 FORMAT(' ') I S I N U N I T S O F 1/T C -PRINT OUTPUT : W ( I ) , THE FREOUENCY DMAG I S THE ABSOLUTE MAGNITUDE I N T E N S I T Y C C RELM I S THE NORMALIZED MAGNITUDE I N T E N S I T Y WRITE(6,200) FORMATC W-WO [X2PIJ INTENSITY WRITE(6.300)(W(I).DMAG(I).RELM(I),I=1.200)  138 139  200  140 141 142  300 FORMAT(1X,F15.5,F15.5,F15.5) STOP END  11 3  REL.INT  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  sidelobe  Window  : Absorption  windows.  T/r  Rectangle  0.0  0.5  1 .0  2^0  3^  21% 0.72  1 3% 0.75  9.7% 1 .2  1 1% 1 .2  5.6% 2.2  0.12% 8.1  0.07% 16.0  -  B l a c k m a n - H a r r i s 3-term  0.05% 1 .8  0.14% 5.1  B l a c k m a n - H a r r i s 4-term  0.01% 2.2  0.01% 19.0  -  -  0.70% 2.4  X X  X X  X X  0.55% 2.6  -  -  -  -  -  -  X X  Filler  DO.24  Filler  E0.13  0.33% 1 .8  Filler  E0.20  0.09% 1 .8 0.02% 2.8  Gaussian  X X  Hamming Hanning  a=2  2.6% 1 .2  Hanning  a=4  0.44% 1 .7  0.08% 7.2  0.07% J 1. 1  -  1.1% 2.2  X X  X X  1 .7% 1 .7  X X  X X  -  -  -  -  0.04% 9.1  0.05% 13.0  0.03% 29.9  Noest-Kort  0.91% 1.7  0.77% 1 .7  0.19% 2.1  0.19% 2.1  N o r t o n - B e e r F3  0.36% 2.3  0.89% 2.2  0.82% 3.2  0.67% 5.1  4-term  -  X X  0.04% 1.7  Kaiser-Bessel  0.59% 7.1  X X  0.01% 2.1  Kaiser-Bessel  -  -  114  0.03% 2.7 X X  APPENDIX B c o n t d . Triangle  Height  X X  and p o s i t i o n  X X  X X  X X  of h i g h e s t s i d e l o b e s : Magnitude  Window  0.0 22% 1 .4  Rectangle  0.5 22% 1 .4  T/r  0.08% 24.3  windows.  1.0  2.0  3.0  24% 1 .4  29% 1 .3  1 3% 4.3  B l a c k m a n - H a r r i s 3 - t e r m 0.04% 3.6  0.04% 3.7  0.04% 3.6  0.04% 4.4  0.05% 10.4  B l a c k m a n - H a r r i s 4-term  0.01% 4.5  0.01% 4.5  0.01% 4.5  0.00% 7.3  0.00% 16.8  Gaussian  0.02% 5.9  0.02% 6.0  0.03% 5.6  0.06% 4.7  0.08% 4.8  Hamming  0.76% 4.4  0.80% 4.4  0.87% 4.4  1 .2% 4.4  1 .6% 4.4  0.53% 4.3  0.02% 14.3  Hanning  a=2  X X  2.7% 2.3  X X  Hanning  a=4  X X  0.46% 3.3  X X  X X  -  -  -  —  —  0.01% 3.8  0.01% 4.3  0.00% 4.9  K a i s e r - B e s s e l 4-term  0.05% 3.3  0.05% 3.3  0.02% 4.5  Noest-Kort  9.7% 2.2  9.4% 2.2  X X  X X  X X  4.6% 2.8  X X  X X  X X  Kaiser-Bessel  Triangle  Upper Lower  X X  0.03% 4.5  0.01% 7.4  value = height of s i d e l o b e (% o f peak h e i g h t ) v a l u e = p o s i t i o n o f s i d e l o b e (1/T f r o m c e n t r a l maximum) = no s i d e l o b e s i n r a n g e e x a m i n e d X = no v a l u e c a l c u l a t e d  115  APPENDIX C Magnitude  lineshape  valleys  V 1 f  : R e c t a n g l e window.  T  (KO  A  1_;_C)  2M)  1.0  0.07  36  2.0  0.11  4.7  3.0  5.2s  1.7  4.1  4.0  9.8s  12s  4.9  3_^0  65  84  11  20 7.6 11  5.0  12s  14s  4.1  7.6  6.0  14 s  16 s  2.2  4.9  s denotes  Magnitude  lineshape  ._ _ 1.0 2.0 3.0 4.0 5.0 6.0  valleys  CM) 0.07 100 0.02 19  sidelobes.  : 3-term  Blackman-Harris.  KO  2_;J)  3_;_0  18  33  43  100  100  100  23  46  68  s  21  s  29 s  41  0.04  11  s  12  21  10s  s denotes  11s  sidelobes.  1 16  3.6  8.6  Magnitude lineshape  valleys  : Kaiser-Bessel  0.0 0.5  100 0.02  1 .0  1.0  window.  2.0  3.0  100  100  100  22  43  61  1 .5  100  100  97  90  2.0  100  100  100  100  2.5  73  83  90  94  16  32  46  3.0  0.02  3.5  42  27  15  4.0  34  33  29  22  4.5  14  18  21  21  1.2  12  5.0  0.00  4.6  8.6  5.5  3.4  1 .2  1 .5  4.2  6.0  1.9  1 .5  0.55  0.88  11 7  Magnitude  lineshape  valleys  0.0 0.5 1 .0 1 .5 2.0 2.5 3.0  100 0.07 100 0.11 77 5.2  : 4-term B l a c k m a n - H a r r i s  1.0  window.  2.0  3.0  100  100  100  36  65  84  100  99  94  18  34  92  96  1 1  20  4.9 86 4.7  3.5  42  31  19  4.0  39  38  34  28  4.5  17  22  25  26  10  14  5.4  5.0  0.01  5.3  5.5  5.1  2.4  0.94  4.4  6.0  3.2  2.8  1 .7  0.10  118  Magnitude  lineshape v a l l e y s  0.0 0.5 1.0  : H a n n i n g a = 2 window.  1.0  100 0.03  2.0  3.0  100  100  100  26  50  70  1.5  96  86  71  52  2.0  98  97  95  89  2.5  45  59  69  76  16  32  45  11  23  3.0  0.02  3.5  8.8  0.08  4.0  0.00  1.5  4.5  3.5  5.0  1.2s  1.1  2.8  5.4  5.5  2.6s  2.7s  1.1  3.2  6.0  2.0  0.23  1.0  2.3  s  s  s denotes  3.3  5.9 s  sidelobes.  119  5.0  13 s  8.5  APPENDIX D  Derivation  A  o f E q . 17.  time-domain  dispersion  signal transforms to y i e l d  component i n t h e f r e q u e n c y  h(t)  A  an a b s o r p t i o n  and  domain  > A - iD.  s h i f t e d time-domain  signal  i s related to the unshifted  signal  by  h(t-t ) Q  =  (exp(-i2*ft ))(A-iD) Q  = Acos(2ffft ) 0  - Dsin(2ffft ) 0  - i (Dcos(2jrft  The  term 2 j r f t  shift general  in  Q  absorption  ) + A s i n ( 2 f f f t > ). o  c a n be r e p l a c e d by 8  frequency u n i t s .  case  o  where  h  n  Q  which  Substituting  represents  this  i s s h i f t e d by a t i m e  (A1 )  the  i n t o Eq.  A1,  t  have  R  will  time the an  l i n e s h a p e of t h e form  A. = A cos6„ - D sin6„ t n n n n  (A2)  n  and  the dispersion  D  tn  t  lineshape w i l l  = A sin&V + D c o s 8 . n n n n  120  be o f t h e f o r m  (A3)  F o r a s i g n a l c o n s i s t i n g o f two c o m p o n e n t s , shifts each  o f 9. a n d 0 \  z  O  respectively,  t a k e t h e form of Eq.  have t h e f o r m o f E q . A  time s h i f t  change,  but  expression  not  A2;  and h , w i t h  t h e r e s u l t i n g A. a n d A t>i t  a n d t h e r e s u l t i n g D.  0  causes the A the  M  n  and D  n  n  components.  f o r M where t h e r e  are substituted  time-shifted  components In order  by A ,  A  , D.  is  y i e l d s Eq.  17 a s t h e e x p r e s s i o n  different  and D  of Eq. to  phase  will  expanded,  16  to  obtain  an  and  use of  2  and  where  the  The  trigonometric  some  A ,  respectively  c o m p o n e n t s h a v e been d e s c r i b e d a b o v e .  time-domain  and D  i s a r e l a t i v e phase d i f f e r e n c e ,  expression  a  will 2  A3.  s u b s t i t u t i o n s a r e made i n E q . 16 : t h e c o m p o n e n t s A,, D  phase  2  resulting identities  f o r a m a g n i t u d e l i n e s h a p e due t o  s i g n a l c o n s i s t i n g o f two c o m p o n e n t s angle.  121  each  with  a  Publications M. B. C o m i s a r o w a n d J . L e e , '"Phase D i s t o r t i o n s " i n Absorption-Mode F o u r i e r Transform 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 ( 1 9 8 5 ) . J . P. L e e a n d M. B. C o m i s a r o w , ' A d v a n t a g e o u s 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 Spectroscopy', A p p l . S p e c t r o s c . ( i n p r e s s ). J . P. L e e a n d M. B. C o m i s a r o w , ' A d v a n t a g e o u s A p o d i z a t i o n Functions f o rAbsorption-Mode F o u r i e r Transform Spectroscopy', 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. L e e a n d M. B. C o m i s a r o w , 'The P h a s e D e p e n d e n c e o f M a g n i t u d e S p e c t r a ' , Chem. P h y s . L e t . ( t o be s u b m i t t e d ) . J . P. L e e a n d M. B. C o m i s a r o w , 'Anomolous I n t e n s i t i e s a n d the Phase Dependence o f A p o d i z e d Magnitude S p e c t r a ' , ( manuscript i n p r e p a r a t i o n ).  

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