- Email: [email protected]
The effect of apodisation and finite resolution on Fourier transform infrared and Raman spectra
SPECTROCHIMICA ACTA PART
A
SpectrochimicaActa Part A 53 (1997)2245-2252
ELSEVIER
The effect of apodisation and finite resolution on Fourier transform infrared and Raman spectra Stewart F. Parker ap*,Philip B. Tooke b b British
a ISIS Facility, Rutherford Appleton Laboratory, Petroleum Research Centre, Chertsey Road,
Chilton, Didcot, Sunbury-on-Thames,
Oxon OX11 Middlesex
OQX, TW16
UK 7LN,
UK
Received9 July 1997; accepted11 July 1997
Abstract
The interferograms measured on Fourier transform infrared (FTIR) and Raman (FT-Raman) spectrometers are routinely apodised before the Fourier transform is carried out. The effects of hnite resolution and different apodisation functions on synthetic bands of known Lorentz-Gauss character have been investigated. (The apodisation functions chosen were those commonly available on commercial FTIR and FT-Raman spectrometers). The effects are dramatic, the ratio of resolution/(true bandwidth) required to obtain less than 10% distortion (as judged by the peak height and width) in the band varied from 1 for boxcar to less than 0.2 for Norton-Beer (strong). Increasing Gaussian character of the band results in initially (i.e. at high resolution) the band being less altered by resolution and apodisation than a pure Lorentzian band. As the resolution is degraded, the band is more strongly affected. These results are explained in terms of the effect of truncation and distortion of the Fourier transform of the bandshape. A comparison of calculated and experimental results gave good agreement for the 1216 cm - ’ band of chloroform in deuterochloroform. The experimental results also showed an increasing degree of Gaussian character as the resolution became comparable with the full width at half height of the band. 0 1997 Elsevier Science B.V. Keywords:
FT-Raman; Infrared; Apodisation; Resolution
1. Introduction The effect of finite resolution on infrared spectra is of critical importance for quantitative analysis. In the case of dispersive spectrometers there has been a considerable body of work on the subject (see Ref. [l] for a review). In contrast, the subject has attracted very little attention with regard to Fourier transform (FT) spectrometers, * Correspondingauthor.
both infrared (FTIR) and FT-Raman. With FT spectrometers, apart from the effects of resolution, there is an additional factor to consider. has been Typically, after the interferogram recorded, it is multiplied by an apodisation function before the Fourier transform is carried out. (The apodisation function is applied to reduce the sidelobes that result from truncation of the interferogram [2]). Mathematically, multiplication in the Fourier domain is equivalent to convolution in the frequency domain. Thus, the observed spec-
1386-1425/97/$17.000 1997 ElsevierScienceB.V. All rights reserved. PIIS1386-1425(97)00163-7
71-16
,S.F. Pm&r.
P.B.
Tookr
Spemocitimica
trum is the convolution of the true spectrum and the Fourier transform of the apodisation function. Furthermore, because the experimentally obtained interferogram is of finite length, the net result is the same as the product of a boxcar apodisation function with an interferogram of infinite length. Thus. any real spectrum has been apodised with a boxcar function even in the apparent absence of apodisation. For quantitative analysis it is essential to know under what conditions accurate data may be obtained. It is important to be able to accurately measure the intensity (expressed as either the band height or the area) and the bandshape of a peak. The latter is particularly vital for any subsequent curve fitting that may be carried out. Two factors are particularly important: whether the spectrometer is operating in a regime where the response is linear (i.e. whether the Beer-Lambert law is obeyed) and how the instrumental parameters affect the spectrum. Non-linearity can result from concentration dependent interactions (e.g. hydrogen-bonding in solution) or can be intrinsic to the detector and its electronics. This is usually manifested at high absorbance values in FTIR spectrometers. A modern FTIR spectrometer should be linear up to at least 1.5 absorbance units (30% transmission) and values of 2 (1% transmission) are not unexpected. In the present work we assume that the spectrometer is operating in the linear regime and it is the instrumental factors that are the focus of the present study. Apart from a study by Griffiths [3] on the effect of resolution and apodisation (boxcar and triangular functions only were considered) on a Lorentzian lineshape, there have been no studies of the effects of these important variables on infrared spectra. In previous publications [4-61 we have discussed the effects of resolution and apodisation with particular reference to FT-Raman spectra. In the present work (which is appliboth FTIR and FT-Raman cable to spectroscopies) we describe the methodology in detail and we have carried out a systematic study of the effects of both resolution and apodisation function on synthetic peaks of mixed Lorentzian and Gaussian character and verified the results experimentally. Commercial FTIR and FT-Ra-
Actu
Ptrrt
A 53 (1997)
121.5
2.52
man spectrometers usually offer a range of apodisation functions as part of the operating system; the apodisation functions investigated here are those commonly employed on a variety of instruments.
2. Experimental The flowchart for the program used for the computer studies is shown in Fig. 1. Single Lorentzian and Gaussian peaks, each with a full width at half-height (FWHH) of 20.25 cm- I. were generated and stored separately. The arrays thus obtained correspond to data measured at 1 cm-’ resolution. The program then generates and stores a series of files for each apodisation function and increasing Gaussian character (in steps of 10%) as a function of the spectral resolution. Infrared spectra were recorded using a Nicolet 60SX FTIR spectrometer in the range 0.25-32
Fig. 1. Flow chart peaks with different
for generation of mixed resolutions and apodisation
Lorentz/Gauss functions.
S.F. Table 1 Mathematical the Fourier Box: Trap: Tri: Happ: NBW:
NBM:
NBS:
Parker,
P.B.
Tooke
/Spectrochimica
Acta
Part
A 53 (1997)
2245-2252
2141
3. Results and discussion forms domain
of the apodisation
functions
F(x) = 1 F(x) = 1 F(x) = ?-2x/L F(x) = 1 -x/L F(x) = 0.54+0.46 cos(n.x/L) F(x) = 0.384093 -0.087577[1 -(x/Ly] +0.703484[1 -(x/L)‘]’ F(x) = 0.152442-0.136176[1 +0.983734[1 -(x/Ly] F’(x) = 0.045335+0.554883[1 +0.399782[1 -(x/L)~]’
Note that the functions where L is the length resolution.
as applied
in
O<.x
-(x/L)‘]
-(x/L)~]’
are defined on the interval 0 2x 5 L, of the interferogram for the chosen
The effect of resolution on the bandshape of a single, pure Lorentzian line that has been apodised with a boxcar apodisation function (i.e. the least possible apodisation that can be applied in any real situation) is shown in Fig. 3. Figs. 4 and 5 show the effect on the peak height ratio (ratio of observed height to true height) and peak width ratio (ratio of observed width to true width) as a function of resolution and apodisation. It was stated in the introduction, that the purpose of apodisation is to reduce, and preferably eliminate, the sidelobes that may occur from truncation of the interferogram. Inspection of a series of figures (not shown) similar to Fig. 3, shows that the success of the apodisation functions in this regard is variable. In order of decreasing effectiveness they are: Norton-Beer
cm-’ resolution, each successive measurement differing by a factor of 2. A deuterated triglycine sulphate detector was used to ensure good linearity. The sample was CDCl, (Goss 99.8% D) in a 0.2 mm KBr cell. Prior to the Fourier transform being carried out, both interferograms (1000 scans each background and sample) were apodised and then zero filled using one of boxcar, triangular or Happ-Genzel apodisation functions. The degree of zero filling was chosen such that the number of transformed data points was the same as that of the 0.25 cm ~ ’ file so that each spectrum had the same number of data points irrespective of the resolution used. This operation ensured that the band width and height of the peaks in the experimental spectra could be measured to the same degree of accuracy in each case. The apodisation functions investigated were; boxcar (Box), trapezoidal (Trap), triangular (Tri), Happ-Genzel (Happ), Norton-Beer weak (NBW), Norton-Beer medium (NBM) and Norton-Beer strong (NBS). The functions as they are applied in the Fourier domain are defined and displayed in Table 1. The coefficients are those commonly employed by the instrument manufacturers.
(strong) z Norton-Beer
x Happ-Genzel
> Norton-Beer
(medium) (weak)
x triangular > trapezoidal > boxcar
Trapezoidal
Triangular
Happ-Genzel
0
Norton-Beer
(weak)
Norton-Beer
(medium)
Norton-Beer
(strong) Data
Fig. 2. Functional forms of applied in the Fourier domain.
points
the
L
apodisation
functions
as
S.F.
Parker.
P.B.
To&e
~‘Sprctrochi~~zica
Actor
Purt
A 53 (I9971
0
0.5
1X-~
252
1.0
1.5
2.0
Resolution/band
2.5
3.0
3.5
width
Fig. 4. The effect of resolution and choice of apodisation function on the peak height ratio of a Lorentzian peak. I 100
I 200 Data
I 300
I 400
500
points
Fig. 3. The effect of finite resolution and boxcar apodisation on the lineshape of a 100% Lorentzian line (FWHH = 20.25 cm ‘).
Since the sidelobes are caused by the step in the interferogram that is introduced by truncation, it might be expected that apodisation functions that result in the interferogram smoothly dying away to zero will cause the least degree of sidelobes. However, consideration of Fig. 2 shows that this is not the case; thus Norton-Beer (weak) and triangular are about equally effective, yet the former introduces a sizeable step whereas the latter does not. Clearly, there are additional factors that are important. If a 10% distortion is defined as an acceptable level then the resolution at which this occurs is shown in Table 2. (The table gives values for a 50% Gaussian lineshape, the trend is the same irrespective of Gaussian content and the numerical values are similar). We have also investigated the effects of an increasing Gaussian contribution to the lineshape. All of the apodisation functions behave similarly,
thus the discussion will be limited to the NortonBeer (strong) function since this shows the largest effects. The effect on the relative height is shown in Fig. 6 for the extremes: 100% Lorentzian (dashed line) and 100% Gaussian (solid line). Peaks of mixed Lorentz/Gauss character lie between these extremes.
+ Norton-Beer l Norton-Beer
0
0.5
1.0
(medium) (strong)
1.5
Resolution/band
2.0
2.5
3.0
3.5
width
Fig. 5. The effect of resolution and choice of apodisation function on the peak width ratio of a Lorentzian peak.
S.F.
Parker,
P.B.
Tooke/Spectrochimica
Acta
Part
A 53 (1997)
2245-2252
2249
Table 2 Table of resolution and relative resolution at which a 50% Gaussian band undergoes a 10% distortion as judged by peak height and width AFN”
Box Trap Ham Tri NBW NBM NBS
Height
Width
Resolutionb
Relative resolution’
Resolution
Relative resolution
32.0 23.1 12.0 7.2 5.2 4.2 2.8
1.58 1.17 0.59 0.36 0.29 0.21 0.14
24.3 11.6 10.1 1.1 6.9 4.8 3.2
1.20 0.87 0.50 0.38 0.34 0.24 0.16
I AFN, apodisation function. ’ Resolution in cm ’ ’ Relative resolution:resolution/(true
FWHH).
From Fig. 6, two regimes need to be distinguished. The first is where the FWHH is much smaller than the resolution and the second is where the FWHH is comparable with or larger than the resolution. In the first regime as the Gaussian character increases the bandshape is degraded less. Inspection of the magnitude of the transform of the bandshape, Fig. 7, shows that the transform of a Gaussian band decays to zero very much faster than that of a Lorentzian band. Degrading the resolution in the frequency domain is equivalent to truncating the transform in the Fourier domain. Thus a band that decays quickly in the Fourier domain will be less affected by truncation than a more slowly decaying band and hence the bandshape will be affected more slowly as the resolution is degraded. In the second regime, at some resolution, which is dependent on the apodisation function, there is a smooth crossover, such that the 100% Lorentzian band is least affected and the 100% Gaussian band the most affected. In order of decreasing resolution at which the crossover occurs the functions are: boxcar > trapezoidal > Happ - Genzel > triangular > Norton - Beer (weak) > Norton-Beer (medium) > Norton-Beer (strong). The effect is quite marked; the resolution at which crossover occurs is N 50 cm- ’ for box(strong). car and N 10 cm-’ for Norton-Beer The apodisation functions that have the lowest crossover values are those that have a significant
effect close to the origin (in the Fourier domain). As the resolution is degraded, the apodisation function has a progressively greater effect closer to the origin and hence will affect lineshapes that have relatively larger values closer to the origin increasingly rapidly. This is precisely the case as the Gaussian content of a lineshape is increased. In order to check the predictions made by the computer study we have measured the height and FWHH of the 1216 cm- ’ band of residual chloroform in deuterochloroform as a function of resolution and apodisation function. This band was chosen because it is reasonably isolated and symmetrical. The FWHH is 5.69 cm- ’ when measured at 0.25 cm-’ resolution, this value was taken as the true FWHH. Curve fitting the peak showed that the lineshape was 20% Gaussian. Figs. 8 and 9 compare the calculated values for peak height ratio and FWHH ratio for a 20% Gaussian peak with those obtained experimentally. It can be seen that the agreement is generally excellent. We have also investigated from the experimental spectra how the lineshape changes with resolution and apodisation. The results are shown in Table 3. Irrespective of the apodisation function, the lineshape becomes increasingly Gaussian as the resolution becomes comparable to the bandwidth. This must reflect truncation of the interferogram and is thus true of all apodisation functions. In addition, apodisation causes the
S.F.
Parker,
P.B.
To&e
Spectrochirnica
Acfu
Part
A 53 (1997)
.X45-
2252
4 0.8
1.0
1.1 Resolutionbandwidth
1.2
Fig. 6. The effect of hnite resolution and Norton-Beer (strong) apodisation character. Dashed line: 100% Lorentzian, solid line: 100% Gaussian.
bandshape to become Gaussian much earlier (in terms of resolution/(true FWHH)) than its absence. Table 3 suggests that the degree to which this occurs is apodisation function dependent and may be a by-product of the degree of efficiency to which sidelobes are suppressed by it.
1500 Data
% Gauss
= 0
% Gauss
= 50
2000
2500
3000
1.4
on the relative height of peaks of varying Gaussian
One area that has not been addressed in this study is the effect of the apodisation function on the noise level present in the spectrum. A smoothing function works by attenuating the high frequency components in the Fourier domain and all of the apodisation functions considered here have
1.1 1
Resolution
1000
1.3
o Boxcar Triangular
A
Resolution /bandwidth
points
Fig. 7. Fourier transforms of lineshapes of different Gaussian character.
Fig. 8. Comparison (for a 20% Gaussian peak) of predicted and experimental results of the relative height. (Open symbols are calculated points, filled symbols are experimental points).
S.F. Parker, P.B. Tooke /Spectrochimica Acta Part A 53 (1997) 2245-2252
0 Boxcar A Triangular 0 Happ-Genzel
5 z
2.5
s ‘g
2.0
2 1.5
Ov5015 . . . . . . . Resolution
3.5
/ bandwidth
Fig. 9. Comparison (for a 20% Gaussian peak) of predicted and experimental results of the relative width. (Open symbols are calculated points, filled symbols are experimental points).
this property (see Fig. 2), thus all of them act as smoothing functions to a greater or lesser degree. Thus by an appropriate choice of apodisation function it is possible to increase the signal-tonoise ratio in the spectrum over that obtained by boxcar alone. For condensed phase spectra bandwidths are typically > 10 cm- i and spectra are routinely run at 4 cm - ’ resolution hence the relative resolution is 0.4 < and from Figs. 4 and 5 it is clear that boxcar and trapezoidal provide the best results. It is interesting to speculate that the smoothing aspect of apodisation may be the reaTable 3 Table of how the Gaussian content of the 1216 cm-r band of deuterochloroform changes with resolution and apodisation Resolution
0.25 0.5 1
2 4 8 16
Relative resolution
0.04 0.09 0.18 0.35 0.70 1.41 2.81
% Gaussian content Box
Tri
Ham
17 17 19 24 26 69 -
17 17 21 28 38 76 100
17 17 23 32 49 88 100
2251
son why some form of apodisation is the default option on all commercial FTIR and FT-Raman spectrometers! The authors are unaware of any systematic study of the effectiveness of apodisation functions on signal-to-noise ratio, this is clearly an area worth investigating since it provides a simple (and cheap) way of improving the performance of FT spectrometers. Finally, as has been noted previously [3], it is not possible to recover the true lineshape from an experimental spectrum to a degree that is better than that obtained from using boxcar alone. This is because a more accurate approximation to the lineshape requires the presence of higher frequency components in the interferogram, i.e. higher resolution and these are not present in a low resolution interferogram.
4. Conclusions
The work described here has demonstrated the dramatic effect the choice of apodisation function can have on spectra obtained using an FTIR or FT-Raman spectrometer. This is particularly marked when the resolution is comparable to the FWHH of the band in question. The results indicate that in circumstances where accurate bandshape information is important, boxcar apodisation is the function of choice. The major drawback of this function is the sidelobes that may be generated; these can obscure a neighbouring weak band. In situations where this is unacceptable or unavoidable (e.g. the spectrometer is already at its maximum resolution or the increased time required to record spectra at higher resolution is unacceptable), apodisation offers a solution. Similar effects to those described here as the result of the interplay of resolution and choice of apodisation function can be observed by an incorrect selection of the size of the Jacquinot stop for a particular resolution. The purpose of the Jacquinot stop is to control the beam divergence through the spectrometer. If the divergence becomes large then there is a path difference between on- and off-axis rays in the spectrometer. These interfere and produce what is known as
2’51
S.F. Parker.
P.B.
Tooke
, Spectrochirnica
‘self-apodisation’ [2]. The effect was illustrated for an FT-Raman spectrum in [4]. In practice the size of the Jacquinot stop is usually under software control and is set automatically for the chosen resolution. It is difficult to rank the functions investigated overall, because there is a large degree of subjectivity involved in what is considered acceptable in terms of band distortion and sidelobes. (In addition, some FT instruments only offer a very limited choice of apodisation functions.) Each of the functions has its merits and demerits, the most vital consideration is that the experimenter should be aware that their choice of operating conditions can have a very large effect on the spectra they observe. Perhaps caveat emptor should be the first item that appears on the menu for the instrument’s operating system!
Actu
Purt
A 53 (1997)
.?245- 2252
Acknowledgements
Permission to publish has been given by the British Petroleum Company.
References [l] K.S. Seshadri. R.N. Jones. Spectrochim. Acta Part A 19A (1963) 1013. [2] P.R. Griffiths, J.A. de Haseth, Fourier Transform Infrared Spectroscopy, Wiley, New York. 1986. [3] R.J. Anderson, P.R. Griffiths, Anal. Chem. 47 (1975) 2339. [4] S.F. Parker, V. Patel, P.B. Tooke, K.P.J. Williams, Spectrochim. Acta Part A 47A (1991) 1171-l 178. [5] P.B. Tooke, V. Patel, K.P.J. Williams, S.F. Parker, 8th Int. Conf. on Fourier Transform Spectroscopy, Proc. SPIE 1575 (1992) 248-249. [6] S.F. Parker, Spectrochim. Acta Part A SOA (1994) 1841L 1856.
A
SpectrochimicaActa Part A 53 (1997)2245-2252
ELSEVIER
The effect of apodisation and finite resolution on Fourier transform infrared and Raman spectra Stewart F. Parker ap*,Philip B. Tooke b b British
a ISIS Facility, Rutherford Appleton Laboratory, Petroleum Research Centre, Chertsey Road,
Chilton, Didcot, Sunbury-on-Thames,
Oxon OX11 Middlesex
OQX, TW16
UK 7LN,
UK
Received9 July 1997; accepted11 July 1997
Abstract
The interferograms measured on Fourier transform infrared (FTIR) and Raman (FT-Raman) spectrometers are routinely apodised before the Fourier transform is carried out. The effects of hnite resolution and different apodisation functions on synthetic bands of known Lorentz-Gauss character have been investigated. (The apodisation functions chosen were those commonly available on commercial FTIR and FT-Raman spectrometers). The effects are dramatic, the ratio of resolution/(true bandwidth) required to obtain less than 10% distortion (as judged by the peak height and width) in the band varied from 1 for boxcar to less than 0.2 for Norton-Beer (strong). Increasing Gaussian character of the band results in initially (i.e. at high resolution) the band being less altered by resolution and apodisation than a pure Lorentzian band. As the resolution is degraded, the band is more strongly affected. These results are explained in terms of the effect of truncation and distortion of the Fourier transform of the bandshape. A comparison of calculated and experimental results gave good agreement for the 1216 cm - ’ band of chloroform in deuterochloroform. The experimental results also showed an increasing degree of Gaussian character as the resolution became comparable with the full width at half height of the band. 0 1997 Elsevier Science B.V. Keywords:
FT-Raman; Infrared; Apodisation; Resolution
1. Introduction The effect of finite resolution on infrared spectra is of critical importance for quantitative analysis. In the case of dispersive spectrometers there has been a considerable body of work on the subject (see Ref. [l] for a review). In contrast, the subject has attracted very little attention with regard to Fourier transform (FT) spectrometers, * Correspondingauthor.
both infrared (FTIR) and FT-Raman. With FT spectrometers, apart from the effects of resolution, there is an additional factor to consider. has been Typically, after the interferogram recorded, it is multiplied by an apodisation function before the Fourier transform is carried out. (The apodisation function is applied to reduce the sidelobes that result from truncation of the interferogram [2]). Mathematically, multiplication in the Fourier domain is equivalent to convolution in the frequency domain. Thus, the observed spec-
1386-1425/97/$17.000 1997 ElsevierScienceB.V. All rights reserved. PIIS1386-1425(97)00163-7
71-16
,S.F. Pm&r.
P.B.
Tookr
Spemocitimica
trum is the convolution of the true spectrum and the Fourier transform of the apodisation function. Furthermore, because the experimentally obtained interferogram is of finite length, the net result is the same as the product of a boxcar apodisation function with an interferogram of infinite length. Thus. any real spectrum has been apodised with a boxcar function even in the apparent absence of apodisation. For quantitative analysis it is essential to know under what conditions accurate data may be obtained. It is important to be able to accurately measure the intensity (expressed as either the band height or the area) and the bandshape of a peak. The latter is particularly vital for any subsequent curve fitting that may be carried out. Two factors are particularly important: whether the spectrometer is operating in a regime where the response is linear (i.e. whether the Beer-Lambert law is obeyed) and how the instrumental parameters affect the spectrum. Non-linearity can result from concentration dependent interactions (e.g. hydrogen-bonding in solution) or can be intrinsic to the detector and its electronics. This is usually manifested at high absorbance values in FTIR spectrometers. A modern FTIR spectrometer should be linear up to at least 1.5 absorbance units (30% transmission) and values of 2 (1% transmission) are not unexpected. In the present work we assume that the spectrometer is operating in the linear regime and it is the instrumental factors that are the focus of the present study. Apart from a study by Griffiths [3] on the effect of resolution and apodisation (boxcar and triangular functions only were considered) on a Lorentzian lineshape, there have been no studies of the effects of these important variables on infrared spectra. In previous publications [4-61 we have discussed the effects of resolution and apodisation with particular reference to FT-Raman spectra. In the present work (which is appliboth FTIR and FT-Raman cable to spectroscopies) we describe the methodology in detail and we have carried out a systematic study of the effects of both resolution and apodisation function on synthetic peaks of mixed Lorentzian and Gaussian character and verified the results experimentally. Commercial FTIR and FT-Ra-
Actu
Ptrrt
A 53 (1997)
121.5
2.52
man spectrometers usually offer a range of apodisation functions as part of the operating system; the apodisation functions investigated here are those commonly employed on a variety of instruments.
2. Experimental The flowchart for the program used for the computer studies is shown in Fig. 1. Single Lorentzian and Gaussian peaks, each with a full width at half-height (FWHH) of 20.25 cm- I. were generated and stored separately. The arrays thus obtained correspond to data measured at 1 cm-’ resolution. The program then generates and stores a series of files for each apodisation function and increasing Gaussian character (in steps of 10%) as a function of the spectral resolution. Infrared spectra were recorded using a Nicolet 60SX FTIR spectrometer in the range 0.25-32
Fig. 1. Flow chart peaks with different
for generation of mixed resolutions and apodisation
Lorentz/Gauss functions.
S.F. Table 1 Mathematical the Fourier Box: Trap: Tri: Happ: NBW:
NBM:
NBS:
Parker,
P.B.
Tooke
/Spectrochimica
Acta
Part
A 53 (1997)
2245-2252
2141
3. Results and discussion forms domain
of the apodisation
functions
F(x) = 1 F(x) = 1 F(x) = ?-2x/L F(x) = 1 -x/L F(x) = 0.54+0.46 cos(n.x/L) F(x) = 0.384093 -0.087577[1 -(x/Ly] +0.703484[1 -(x/L)‘]’ F(x) = 0.152442-0.136176[1 +0.983734[1 -(x/Ly] F’(x) = 0.045335+0.554883[1 +0.399782[1 -(x/L)~]’
Note that the functions where L is the length resolution.
as applied
in
O<.x
-(x/L)‘]
-(x/L)~]’
are defined on the interval 0 2x 5 L, of the interferogram for the chosen
The effect of resolution on the bandshape of a single, pure Lorentzian line that has been apodised with a boxcar apodisation function (i.e. the least possible apodisation that can be applied in any real situation) is shown in Fig. 3. Figs. 4 and 5 show the effect on the peak height ratio (ratio of observed height to true height) and peak width ratio (ratio of observed width to true width) as a function of resolution and apodisation. It was stated in the introduction, that the purpose of apodisation is to reduce, and preferably eliminate, the sidelobes that may occur from truncation of the interferogram. Inspection of a series of figures (not shown) similar to Fig. 3, shows that the success of the apodisation functions in this regard is variable. In order of decreasing effectiveness they are: Norton-Beer
cm-’ resolution, each successive measurement differing by a factor of 2. A deuterated triglycine sulphate detector was used to ensure good linearity. The sample was CDCl, (Goss 99.8% D) in a 0.2 mm KBr cell. Prior to the Fourier transform being carried out, both interferograms (1000 scans each background and sample) were apodised and then zero filled using one of boxcar, triangular or Happ-Genzel apodisation functions. The degree of zero filling was chosen such that the number of transformed data points was the same as that of the 0.25 cm ~ ’ file so that each spectrum had the same number of data points irrespective of the resolution used. This operation ensured that the band width and height of the peaks in the experimental spectra could be measured to the same degree of accuracy in each case. The apodisation functions investigated were; boxcar (Box), trapezoidal (Trap), triangular (Tri), Happ-Genzel (Happ), Norton-Beer weak (NBW), Norton-Beer medium (NBM) and Norton-Beer strong (NBS). The functions as they are applied in the Fourier domain are defined and displayed in Table 1. The coefficients are those commonly employed by the instrument manufacturers.
(strong) z Norton-Beer
x Happ-Genzel
> Norton-Beer
(medium) (weak)
x triangular > trapezoidal > boxcar
Trapezoidal
Triangular
Happ-Genzel
0
Norton-Beer
(weak)
Norton-Beer
(medium)
Norton-Beer
(strong) Data
Fig. 2. Functional forms of applied in the Fourier domain.
points
the
L
apodisation
functions
as
S.F.
Parker.
P.B.
To&e
~‘Sprctrochi~~zica
Actor
Purt
A 53 (I9971
0
0.5
1X-~
252
1.0
1.5
2.0
Resolution/band
2.5
3.0
3.5
width
Fig. 4. The effect of resolution and choice of apodisation function on the peak height ratio of a Lorentzian peak. I 100
I 200 Data
I 300
I 400
500
points
Fig. 3. The effect of finite resolution and boxcar apodisation on the lineshape of a 100% Lorentzian line (FWHH = 20.25 cm ‘).
Since the sidelobes are caused by the step in the interferogram that is introduced by truncation, it might be expected that apodisation functions that result in the interferogram smoothly dying away to zero will cause the least degree of sidelobes. However, consideration of Fig. 2 shows that this is not the case; thus Norton-Beer (weak) and triangular are about equally effective, yet the former introduces a sizeable step whereas the latter does not. Clearly, there are additional factors that are important. If a 10% distortion is defined as an acceptable level then the resolution at which this occurs is shown in Table 2. (The table gives values for a 50% Gaussian lineshape, the trend is the same irrespective of Gaussian content and the numerical values are similar). We have also investigated the effects of an increasing Gaussian contribution to the lineshape. All of the apodisation functions behave similarly,
thus the discussion will be limited to the NortonBeer (strong) function since this shows the largest effects. The effect on the relative height is shown in Fig. 6 for the extremes: 100% Lorentzian (dashed line) and 100% Gaussian (solid line). Peaks of mixed Lorentz/Gauss character lie between these extremes.
+ Norton-Beer l Norton-Beer
0
0.5
1.0
(medium) (strong)
1.5
Resolution/band
2.0
2.5
3.0
3.5
width
Fig. 5. The effect of resolution and choice of apodisation function on the peak width ratio of a Lorentzian peak.
S.F.
Parker,
P.B.
Tooke/Spectrochimica
Acta
Part
A 53 (1997)
2245-2252
2249
Table 2 Table of resolution and relative resolution at which a 50% Gaussian band undergoes a 10% distortion as judged by peak height and width AFN”
Box Trap Ham Tri NBW NBM NBS
Height
Width
Resolutionb
Relative resolution’
Resolution
Relative resolution
32.0 23.1 12.0 7.2 5.2 4.2 2.8
1.58 1.17 0.59 0.36 0.29 0.21 0.14
24.3 11.6 10.1 1.1 6.9 4.8 3.2
1.20 0.87 0.50 0.38 0.34 0.24 0.16
I AFN, apodisation function. ’ Resolution in cm ’ ’ Relative resolution:resolution/(true
FWHH).
From Fig. 6, two regimes need to be distinguished. The first is where the FWHH is much smaller than the resolution and the second is where the FWHH is comparable with or larger than the resolution. In the first regime as the Gaussian character increases the bandshape is degraded less. Inspection of the magnitude of the transform of the bandshape, Fig. 7, shows that the transform of a Gaussian band decays to zero very much faster than that of a Lorentzian band. Degrading the resolution in the frequency domain is equivalent to truncating the transform in the Fourier domain. Thus a band that decays quickly in the Fourier domain will be less affected by truncation than a more slowly decaying band and hence the bandshape will be affected more slowly as the resolution is degraded. In the second regime, at some resolution, which is dependent on the apodisation function, there is a smooth crossover, such that the 100% Lorentzian band is least affected and the 100% Gaussian band the most affected. In order of decreasing resolution at which the crossover occurs the functions are: boxcar > trapezoidal > Happ - Genzel > triangular > Norton - Beer (weak) > Norton-Beer (medium) > Norton-Beer (strong). The effect is quite marked; the resolution at which crossover occurs is N 50 cm- ’ for box(strong). car and N 10 cm-’ for Norton-Beer The apodisation functions that have the lowest crossover values are those that have a significant
effect close to the origin (in the Fourier domain). As the resolution is degraded, the apodisation function has a progressively greater effect closer to the origin and hence will affect lineshapes that have relatively larger values closer to the origin increasingly rapidly. This is precisely the case as the Gaussian content of a lineshape is increased. In order to check the predictions made by the computer study we have measured the height and FWHH of the 1216 cm- ’ band of residual chloroform in deuterochloroform as a function of resolution and apodisation function. This band was chosen because it is reasonably isolated and symmetrical. The FWHH is 5.69 cm- ’ when measured at 0.25 cm-’ resolution, this value was taken as the true FWHH. Curve fitting the peak showed that the lineshape was 20% Gaussian. Figs. 8 and 9 compare the calculated values for peak height ratio and FWHH ratio for a 20% Gaussian peak with those obtained experimentally. It can be seen that the agreement is generally excellent. We have also investigated from the experimental spectra how the lineshape changes with resolution and apodisation. The results are shown in Table 3. Irrespective of the apodisation function, the lineshape becomes increasingly Gaussian as the resolution becomes comparable to the bandwidth. This must reflect truncation of the interferogram and is thus true of all apodisation functions. In addition, apodisation causes the
S.F.
Parker,
P.B.
To&e
Spectrochirnica
Acfu
Part
A 53 (1997)
.X45-
2252
4 0.8
1.0
1.1 Resolutionbandwidth
1.2
Fig. 6. The effect of hnite resolution and Norton-Beer (strong) apodisation character. Dashed line: 100% Lorentzian, solid line: 100% Gaussian.
bandshape to become Gaussian much earlier (in terms of resolution/(true FWHH)) than its absence. Table 3 suggests that the degree to which this occurs is apodisation function dependent and may be a by-product of the degree of efficiency to which sidelobes are suppressed by it.
1500 Data
% Gauss
= 0
% Gauss
= 50
2000
2500
3000
1.4
on the relative height of peaks of varying Gaussian
One area that has not been addressed in this study is the effect of the apodisation function on the noise level present in the spectrum. A smoothing function works by attenuating the high frequency components in the Fourier domain and all of the apodisation functions considered here have
1.1 1
Resolution
1000
1.3
o Boxcar Triangular
A
Resolution /bandwidth
points
Fig. 7. Fourier transforms of lineshapes of different Gaussian character.
Fig. 8. Comparison (for a 20% Gaussian peak) of predicted and experimental results of the relative height. (Open symbols are calculated points, filled symbols are experimental points).
S.F. Parker, P.B. Tooke /Spectrochimica Acta Part A 53 (1997) 2245-2252
0 Boxcar A Triangular 0 Happ-Genzel
5 z
2.5
s ‘g
2.0
2 1.5
Ov5015 . . . . . . . Resolution
3.5
/ bandwidth
Fig. 9. Comparison (for a 20% Gaussian peak) of predicted and experimental results of the relative width. (Open symbols are calculated points, filled symbols are experimental points).
this property (see Fig. 2), thus all of them act as smoothing functions to a greater or lesser degree. Thus by an appropriate choice of apodisation function it is possible to increase the signal-tonoise ratio in the spectrum over that obtained by boxcar alone. For condensed phase spectra bandwidths are typically > 10 cm- i and spectra are routinely run at 4 cm - ’ resolution hence the relative resolution is 0.4 < and from Figs. 4 and 5 it is clear that boxcar and trapezoidal provide the best results. It is interesting to speculate that the smoothing aspect of apodisation may be the reaTable 3 Table of how the Gaussian content of the 1216 cm-r band of deuterochloroform changes with resolution and apodisation Resolution
0.25 0.5 1
2 4 8 16
Relative resolution
0.04 0.09 0.18 0.35 0.70 1.41 2.81
% Gaussian content Box
Tri
Ham
17 17 19 24 26 69 -
17 17 21 28 38 76 100
17 17 23 32 49 88 100
2251
son why some form of apodisation is the default option on all commercial FTIR and FT-Raman spectrometers! The authors are unaware of any systematic study of the effectiveness of apodisation functions on signal-to-noise ratio, this is clearly an area worth investigating since it provides a simple (and cheap) way of improving the performance of FT spectrometers. Finally, as has been noted previously [3], it is not possible to recover the true lineshape from an experimental spectrum to a degree that is better than that obtained from using boxcar alone. This is because a more accurate approximation to the lineshape requires the presence of higher frequency components in the interferogram, i.e. higher resolution and these are not present in a low resolution interferogram.
4. Conclusions
The work described here has demonstrated the dramatic effect the choice of apodisation function can have on spectra obtained using an FTIR or FT-Raman spectrometer. This is particularly marked when the resolution is comparable to the FWHH of the band in question. The results indicate that in circumstances where accurate bandshape information is important, boxcar apodisation is the function of choice. The major drawback of this function is the sidelobes that may be generated; these can obscure a neighbouring weak band. In situations where this is unacceptable or unavoidable (e.g. the spectrometer is already at its maximum resolution or the increased time required to record spectra at higher resolution is unacceptable), apodisation offers a solution. Similar effects to those described here as the result of the interplay of resolution and choice of apodisation function can be observed by an incorrect selection of the size of the Jacquinot stop for a particular resolution. The purpose of the Jacquinot stop is to control the beam divergence through the spectrometer. If the divergence becomes large then there is a path difference between on- and off-axis rays in the spectrometer. These interfere and produce what is known as
2’51
S.F. Parker.
P.B.
Tooke
, Spectrochirnica
‘self-apodisation’ [2]. The effect was illustrated for an FT-Raman spectrum in [4]. In practice the size of the Jacquinot stop is usually under software control and is set automatically for the chosen resolution. It is difficult to rank the functions investigated overall, because there is a large degree of subjectivity involved in what is considered acceptable in terms of band distortion and sidelobes. (In addition, some FT instruments only offer a very limited choice of apodisation functions.) Each of the functions has its merits and demerits, the most vital consideration is that the experimenter should be aware that their choice of operating conditions can have a very large effect on the spectra they observe. Perhaps caveat emptor should be the first item that appears on the menu for the instrument’s operating system!
Actu
Purt
A 53 (1997)
.?245- 2252
Acknowledgements
Permission to publish has been given by the British Petroleum Company.
References [l] K.S. Seshadri. R.N. Jones. Spectrochim. Acta Part A 19A (1963) 1013. [2] P.R. Griffiths, J.A. de Haseth, Fourier Transform Infrared Spectroscopy, Wiley, New York. 1986. [3] R.J. Anderson, P.R. Griffiths, Anal. Chem. 47 (1975) 2339. [4] S.F. Parker, V. Patel, P.B. Tooke, K.P.J. Williams, Spectrochim. Acta Part A 47A (1991) 1171-l 178. [5] P.B. Tooke, V. Patel, K.P.J. Williams, S.F. Parker, 8th Int. Conf. on Fourier Transform Spectroscopy, Proc. SPIE 1575 (1992) 248-249. [6] S.F. Parker, Spectrochim. Acta Part A SOA (1994) 1841L 1856.