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Thisistocertifythatthc

dissertation entitled

I . ABSORPTION- CORRECTED FLUORESCENCE
THROUGH FIBER OPTICS

II. DESIGN AND ANALYSIS OF NONRECURSIVE
DIGITAL FILTERS
presented by

Thomas Patrick Doherty

has been accepted towards fulfillment
of the requirements for

Ph . D . Chemis try

‘degreein

 

 

afi?fi%;4.

 

Major professor

Datew

MS U is an Afl‘innatiw Action/Equal Opportunity Institution

 

 

 

PLACE IN RETURN BOX to remove thls checkout from your record.
TO AVOID FINES return on or More due duo.

 

P_________________———-————-—
DATE DUE DATE DUE DATE DUE

 

 

 

 

 

 

 

 

 

I I I
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_, MSU Is An Allirmdive Action/Emmi Opportunity Inditution #

 

7 _—_—___

 

 

I. ABSORPTION-CORRECTED FLUORESCENCE THROUGH FIBER OPTICS
II. DESIGN AND ANALYSIS OF NONRECURSIVE DIGITAL FILTERS

By
Thomas Patrick Doherty

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY

Department of Chemistry

1990

9001944

ABSTRACT

I. ABSORPTION-CORRECTED FLUORESCENCE THROUGH FIBER OPTICS
II. DESIGN AND ANALYSIS OF NONRECURSIVE DIGITAL FILTERS

m

Thomas Patrick Doherty

I

Several instrumental and mathematical methods have been developed that correct
fluorescence measurements for the inner filter effect. Almost all are designed for use
with 1 cm cuvettes, and are thus incompatible with flowing sample streams. A fiber optic
fluorometer was constructed that addresses this problem. The instrument simultaneously
measures the fluorescence at the front and rear of a cylindrical flowcell, as well as the
transmittance of the excitation beam. From the absorbance at the excitation wavelength
and the ratio of the fluorescence at the front of the cell to that at the rear, the absorbance
at the emission wavelength is estimated. The computer that controls the instrument uses
these two absorbances to calculate what the fluorescence measurement would be in the

absence of inner filter effects. The corrections are useful to a total absorbance of 2.5.

The equation that is used to calculate the corrected fluorescence contains an
optical transfer function that describes the propagation of light through a fiber optic
bundle. This transfer function was originally derived empirically, based on the measured
fluorescence of a series of solutions of known absorbance. An expression is derived from
first principles that matches the shape of the empirical function under certain conditions.
The expression could prove useful for modifying the correction equation should the

refractive index of the sample, or the wavelength of excitation or emission be changed.

II

The Savitzky-Golay filter familiar to many chemists is one example of a
nonrecursive digital filter. Techniques for examining the frequency response of a
nonrecursive filter are described, and are used to study three types of filters, including
sliding window filters that can be used at the extremes of a data set. These are shown to
have undesirable gain and phase properties that could lead to the erroneous
interpretation of data, especially if they are applied repeatedly. Techniques for the
design of a filter with a given frequency response are also described. Two types of filters
are designed: a bandpass filter that removes correlated noise, and IOWpaSS filters for use

when the sampling rate of a digitized signal is changed.

To Paul and Joan, who know I can
and

to Linda, who is my inspiration

ACKNOWLEDGMENTS

None of the work described in this dissertation would have been possible without
the support and guidance of my research advisor, Stan Crouch. I am perhaps most
indebted to him for something which is not embodied in this text: the knowledge I gained
because I was given the freedom to pursue any interesting idea, whether related to the
projects described here or not. It is this "education” that made coming to Michigan State

University truly worthwhile to me.
That, and some memorable Lions games.

I would also like to thank the other members of my committee: James Dye, Jack
Holland, and especially my second reader, Chris Enke. Every time I spoke with him I was
filled with a renewed sense of excitement at the many ways we can use computer software
to get more information from our spectroscopic data. My list of mentors would not be
complete without those who helped me decide to go to graduate school: John
Zimmerman, Professor of Chemistry at Wabash College, and Garry Buettner, formerly at
Wabash.

Due to the size of the Crouch group during my stay, it is not possible to thank all
the members of my research group who helped me at State. But you know who you are.
I thank all of you for advice and help in the lab, and friendship inside it and out.
However, several members of the research group must be mentioned separately because

their work directly affected what is described herein.

Pete Wentzell was my chief collaborator in the digital filter work. Without his
focus and clarity it would have remained merely a hobby and never would have made it
into this dissertation. I am also gateful to Cheryl Stults and Eric Erickson for providing

other interesting applications to try out my filters on.

Pete’s counterpart on the fluorescence side was Mark Victor. We argued
incessantly over the finer points of the model presented in Chapter 2, which ultimately
led to geat improvements. Mark also provided plenty of experimental help: be refined
the design of the stepper motor controller and built the final version, and designed the
fiber Optic interface system for the PTR monochromators. My thanks also go out to
Deak Watters and Russ Geyer in the machine shop who built the components of the

interface.

Max Hineman served as my main opties, math and quantum chemistry resource,
and was also a geat roommate for three years. The members of the lunch bunch: Pete,
Max, Dean Peterson, Kim Ratanathanawongs, and Bob Harfmann, provided something
much more valuable and lasting than research papers: memories. Ditto for all those with
whom I picnicked, camped, skied, and played softball and volleyball. For reasons not
beyond my control, I am able to thank the expediter of this dissertation, Larry Bowman,

with whom I also had many interesting discussions about fluorescence.

I owe an immeasurable debt to my parents, Joan and Paul. They always knew I
could do it, even when I didn’t believe it. Finally, to the person to whom I owe the most,

my wife Linda: thank you. Ifyou hadn’t come to Michigan State, I’d still be there.

vi

TABLE OF CONTENTS

 

List of Figures _ X:
List of Tables -xvi

 

I. Absorption-Corrected Fluorescence Through Fiber Optics

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Chapter 1 : Introduction to Absorption-Corrected Fluorescence ...................................... 1
Review _ - ........................ 4
Mathematical Corrections .......... ..... 4

Instrumental Corrections .................... 5

Applications _ - - - ................ 6

Miscellaneous Phenomena Related to the Inner Filter Effect ................ 7

Absorption Correction For Flowing Sample Streams - -- - ........... 8
References----- - ............................................... 10
Chapter 2 : A Fluorescence Intensity Expression - 12
Review .......... 12

Fiber Optics ................................... 13

A Fluorescence Intensity Expression - - -- - - ............. 15
Summary of Previous Work - - - -- 17
Comparison of Theoretin and Experimental Models _ .......... 18
Conclusions --22
References ......... 24
Chapter 3 : Experiments in Absorption-Corrected Fluorescence - .......... 25
Theory .......................................... 25
Experimental Details .................................... 27

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Apparatus 27
The Effect of Primary and Secondary Absorbance on the
Fluorescence Ratio ...... 31
Introduction - 31
Experimental Details 33
Reagents - 33
Procedure 33
Results and Discussion 35
A Test of the Correction Procedure - ................. 50
Introduction- - -_ ............................. 50
Experimental Details - . ................. 52
Reagents .................... 52
Procedure -- .... 52
Results and Discussion - ......... 52
Real-Time Performance 56
References 64
Chapter 4 : Perspectives, Conclusions, and Suggestions for Further Study .................... 66
Limitations of the Current Instrument and Suggestions for
Improvement ....66
Comparison of the Front Surface / Rear Surface Fluorometer and Cell
Shift Instruments - -72
Suggestions for Further Study -73
Modifications of the Current Instrument 73
Other Experiments in Absorption-Corrected Fluorescence .................. 76
References 82

 

viii

11. Design and Analysis of Nonrecursive Digital Filters

 

Chapter 5: Introduction to Digital Filters

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

-- 84

Analog Filters 84
Digital Filters 86
Linearity and Time-Invariance 87
Advantages of Digital Filters 90
A Brief History of Convolution Filters in Analytical Chemisry 92
Savitzky-Golay Filters 92

Other Symmetric Filters 95
Asymmetric Filters 96
Conclusions 96
References 97
Chapter 6 : Frequency Responses of Digital Filters 99
Examples 103
Savitzky-Golay Filters - 103

Digital Up/ Down Integration 104

Sliding Window Filters 112
Performance of the Sliding Window Filters 1 18
Conclusions 125
References -- 126
Chapter 7 : Design of Digital Filters 127
Theory 127
Examples 128
Low-Pass Filters 128
Differentiators 132

Why Design Filters? -136

 

Examples
Correlated Noise Removal

Decimation

 

 

Interpolation

Conclusions

 

References

137
137
141
145
151
153

Figure 2-1.

Figure 2-2.

Figure 2-3.

Figure 2-4.

Figure 3-1.

Figure 3-2.

Figure 3-3.

Figure 3-4.

Figure 3-5.

Figure 3-6.

Figure 3-7.

LIST OF FIGURES

Optical fiber acceptance cone

Overlap region of light emitted from and collected by two
adjacent optical fibers: (a) schematic diagram, (b) head-on

View ..

Comparison of experimental and theoretical transfer
functions.

............... 16

.............. 19

 

Effect of changes in the parameters used in the theoretical
model: (a) change in numerical aperture (1 - 4 = numerical
aperature of .1, .05, .025, .0125), (b) change in fiber radius

 

(1 - 4 = fiber radius of .1, .3, .9, 2.7 mm). --

Cylindrical cell model. - ..................................................

 

 

Diagram of fluorescence spectrometer. ............

Diagram of sample cell: (a) cutaway view, (b) view showing
fiber optic windows.

 

............... 23

............... 26

............... 28

............... 30

Output of the four detectors for 500 11M quinine sulfate (08),
0 to 910 11M bromocresol purple (BP): (3) reference channel,

(b) transmittance channel, (c) front surface fluorescence

channel, (d) rear surface fluorescence channel. ..........................

Fluorescence ratio versus absorbance at 438 nm. 1= 250 um
quinine sulfate (OS), 2= 500 uM 08, 3= 750 14M 08, 4=
1000 11M OS, 5- 1250 MM 08, 6-- 1500 uM QS.

 

Parameter 0 versus absorbance at 353 nm.

 

Parameter b versus absorbance at 353 nm. - ......

 

............... 34

............... 36

............... 38

Figure 3-8.

Figure 3-9.

Figure 3-10.

Figure 3-11.

Figure 3-12.

Figure 3- 13.

Figure 3-14.

Figure 3-15.

Figure 3-16.

Figure 3-17.

Absorbance at 438 nm (1 - calculated, 2 - measured) versus
sodium fluorescein (SF) concentration: (a) 250 um quinine
sulfate (OS), (b) 500 um OS

 

Absorbance at 438 nm (1 a calculated, 2 = measured) versus
sodium fluorescein (SF) concentration: (a) 750 11M quinine

 

sulfate (OS), (b) 1000 um OS

Absorbance at 438 nm (1 = calculated, 2 = measured) versus
sodium fluorescein (SF) concentration: (a) 1250 11M quinine
sulfate (OS), (b) 1500 I‘M OS

 

Front surface fluorescence intensity ( 1 = measured, 2 =
corrected using calculated absorbance at 438 nm) versus
sodium fluorescein concentration: (a) 250 um quinine

sulfate (OS), (b) 500 um OS

 

Front surface fluorescence intensity (1 = measured, 2 =
corrected using calculated absorbance at 438 nm) versus
sodium fluorescein concentration: (a) 750 um quinine

 

sulfate (OS), (b) 1000 nM OS

Front surface fluorescence intensity (1 8 measured, 2 a
corrected using calculated absorbance at 438 nm) versus
sodium fluorescein concentration: (a) 1250 11M quinine
sulfate (OS), (b) 1500 14M QS

 

Absorption spectrum of 0.909 11M bromocresol purple (BP)
in 0.1 E HClO4.

 

Absorbance at 438 nm (1 = calculated, 2 = measured) versus
bromocresol purple (BP) concentration: (a) 250 11M quinine
sulfate (OS), (b) 500 um OS

 

Absorbance at 438 nm (1 = calculated, 2 - measured) versus
bromocresol purple (BP) concentration: (a) 750 um quinine
sulfate (OS), (b) 1000 11M OS

41

---42

43

46

47

48

51

53

- ,_54

 

Fluorescence intensity (1 :- uncorrected, 2 = corrected using
calculated absorbance at 438 nm, 3 - corrected using
measured absorbance at 438 nm) versus bromocresol purple
(BP) concentration: (a) 250 11M quinine sulfate (OS), (b) 500
14M 05

 

--57

Figure 3-18.

Figure 3-19.

Figure 3-20.

Figure 4-1.

Figure 4-2.

Figure 4-3.

Figure 4-4.

Figure 5-1.

Figure 6-1.

Figure 6-2.

Figure 6-3.

Figure 6-4.

Fluorescence intensity (1 = uncorrected, 2 - corrected using
calculated absorbance at 438 nm, 3 - corrected using
measured absorbance at 438 nm) versus bromocresol purple
(BP) concentration: (a) 750 11M quinine sulfate (OS), (b)
1000 11M OS.

 

 

 

 

filters in place.

 

 

 

 

 

passes.

 

58
Real-time results: (a) measured primary absorbance, (b)
estimated secondary absorbance 60
Real-time results: (a) front surface fluorescence, (b)
corrected front surface fluorescence. - 61
Time profile of rear surface fluorescence measured while
sample cell is rinsed with 0.1 E HClO4: (a) 0 to 4 minutes,
(b) 0 to 25 minutes ...... -...68
Front surface fluorescence blank: (3) filters removed, (b)
............. 70
Calculated ratio of front surface fluorescence measurement
from 2 cm flow cell to front surface fluorescence
measurement from 0.5 cm flow cell. ............. 78
Flowing bubble system: (a) diagram showing time-varying
pathlength, (b) calculated front surface fluorescence signal
(1 a total absorbance negligible, 2 8 total absorbance of 2.0).
80
Illustration of the Nyquist theorem. ............ 88
Frequency responses of 11 point Savitzky-Golay filters: (3)
linear, (b) quadratic, (c) quartic 105
Frequency responses of quadratic Savitzky-Golay filters: (a)
7 point, (b) 9 point, (c) 11 point. 106
Frequency responses of multiple passes of a quadratic 11
point Savitzky-Golay filter: (a) 1 pass, (b) 2 passes, (c) 3
- 107
Frequency response of 11 point quadratic first derivative

Savitzky-Golay filter.

 

108 -

Figure 6-5.

Figure 6-6.

Figure 6-7.

Figure 6-8.

Figure 6-9.

Figure 6-10.

Figure 6-11.

Figure 6- 12.

Figure 7-1.

Figure 7-2.

Figure 7-3.

Figure 7-4.

Figure 7-5.

Figure 7-6.

Normalized frequency responses of up/ down integration

 

 

 

 

 

 

 

 

 

 

 

with rc=o.1.

filtered

 

 

filters: (a) 11 point, (b) 21 point, (c) 101 point. ....... 111
Frequency responses of quadratic initial point filters: (a) 9
point, (b) 11 point, (c) 13 point. - 114
Phase response of quadratic initial point filters: (a) 9 point,
(b) 11 point, c) 13 point. 116
Frequency response of 11 point initial point filters: (a)
linear, (b) quadratic, (c) cubic. 117
Frequency response of quadratic 11 point filters: (a) minus
first point, (b) initial point, (c) third point. -119
Effect of sliding window filter on noise-free sine wave. .. 121
Effect of sliding window filter on noise - _ ....... 123
Effect of initial slope filter on st0pped-flow data: (a)
stopped-flow transient, (b) calculated derivative. 124
Frequency responses of unwindowed 11 point filters: (a) fc
= 0.1 times sampling frequency, (b) fc = 0.2 times sampling
frequency, (c) fc= 0.3 times sampling frequency. 131
Frequency responses of the windowed versions of the filters
shown in Figure 7-1. ............ 133
Frequency response of windowed 21 point derivative filter

135
Peak obtained on flow injection analyzer. (a) unfiltered, (b)

138
Frequency responses of (a) low-pass filter, (b) high-pass
filter, (c) notch filter 140
Decimation by a factor of two. ............ 142

 

xiv

Figure 7-7.

Figure 7-8.

Figure 7-9.

Figure 7-10.

Figure 7-11.

Time-of-flight mass spectrum of xenon: (a) original
spectrum, (b) decimated by 10, no filtering.

 

(3) Frequency response of decimation filter. (b) Filtered
time-of-flight mass spectrum of xenon; filtered, then
decimated by 10

Interpolation by a factor of two

Interpolated Gaussian profile: (a) before filtering, (b)
results of Savitzky-Golay interpolation filter, (c) results of
windowed ideal low-pass interpolation filter.

 

Frequency responses of decimation filters: (a) windowed
ideal low-pass filter, fc - 0.477, (b) 21 point eighth order

 

Savitzky-Golay filter.

144

146

148

- 149

150

Table 3-1.

Table 3-2.

Table 3-3.

Table 3-4.

Table 3-5.

Table 3-6.

LIST OF TABLES

Comparison of measured and calculated secondary
absorbance

 

 

values

 

absorbance

 

 

44
Comparison of linear regression results for measured and
calculated secondary absorbance 44
Comparison of uncorrected and corrected fluorescence

49
Comparison of measured and calculated secondary

55
Comparison of linear regression results for measured and
calculated secondary absorbance 55
Comparison of uncorrected and corrected fluorescence

...... 59

values

 

CHAPTER 1 : INTRODUCTION TO ABSORPTION-CORRECTED
FLUORESCENCE

Fluorescence techniques are used in quantitative analysis when their sensitivity,
selectivity, and wide dynamic range are justified. A fluorescence assay is typically 10-1000
times more sensitive than an absorption assay. Selectivity is provided in part by the
ability of the analyst to select the excitation and emission wavelengths and in part by the
fact that many compounds do not fluoresce with high efi'iciency (1). Measurement of the
lifetime of a fluorophore or the polarization of its emission can also be used to enhance

selectivity.

In most quantitative applications it is assumed that the observed fluorescence
intensity is proportional to the concentration of the fluorophore. In fact, the fluorescence
intensity expression involves many terms associated with the fluorophore, the
characteristics of the sample matrix and the geometry of the spectrometer (2). Although
the geometry of the spectrometer is usually constant, changes in the nature of the sample
from one measurement to the next can cause this assumption of linearity to be incorrect.

The intensity of the total fluorescence emitted is assumed to be given by:
FocIo(1 - 10 *1”)

where F is the fluorescence intensity, I0 is the incident monochromatic radiant power, 6 is
the absorptivity of the fluorophore, c is the concentration, and b is the pathlength, so that
she - A, the absorbance of the fluorophore. In the limit of low absorbance this equation
can be approximated by

Forloebc

2
which is the basis of the assumption of linearity between the observed intensity and
concentration, although it should be stressed that the expression does not apply to a

typical measurement made in a spectrometer.

The simplest example in which the obsen/ed fluorescence is not proportional to
the fluorophore concentration occurs when the fluorophore absorbs significant amounts
of radiation. This is often what limits the upper end of the linear dynamic range of a
fluorescence calibration curve. Samples containing such high concentrations of
fluorophore are typically diluted until the concentration lies within the linear range of the
calibration curve (3). This is not a general approach for several reasons. In high
throughput, automatic assays, either all of the samples must be diluted, possibly causing
the concentrations of certain samples to fall below the detection limit, or none are
diluted, in which case the concentration of other samples may be erroneously estimated.
Also, dilution may upset the position of an equilibrium, or it may not be possible, as in

the case of in situ measurements.

In other cases the absorbance may be high at the excitation wavelength (primary
absorbance) or the emission wavelength (secondary absorbance) due to other
components of the sample matrix. This phenomenon is known as the 111' per filter effect.
If this occurs, estimates of the concentration of a fluorophore may be in error if the
matrix of the unknown sample absorbs a different amount of radiation than the matrix of
the sample used to develop the calibration curve. In this case, a calibration curve is often
generated using the sample itself via standard additions, but once again this is not a
general solution. Standard additions are usually not possible with high throughput
automated assays or in situ measurements. Furthermore, if the fluorophore is
contributing to the inner filter effect, standard additions will not produce a linear

calibration curve.

3

In other cases that fluorescence detection is used, the concentration of the
fluorophore remains constant, and some species is added to the solution to change the
emission intensity by some other means, such as a change in the equilibrium between
fluorescent and non-fluorescent forms of the fluorophore, or a change in the quantum
efiiciency due to quenching. The former technique can be used to determine the binding
of substrates to enzymes. The latter technique is particularly popular for biochemical
applications, since the extent of quenching is a function of polarity of the environment of
the fluorescent site, the accessibility of the fluorophore to the quencher, and the
proximity of acceptors of energy transfer (4). There are also some assays, particularly
immunoassays, that are based on fluorescence quenching. In both cases, the change in
fluorescence intensity is assumed to be due to the phenomenon in question and not to

changes in primary or secondary absorbance.

Reaction-rate methods sometimes employ fluorescence detection of a reactant or
product. If the primary or secondary absorbance of the solution should change with time,
or the reaction should start or end in the non-linear region of the calibration curve for the
fluorophore, the estimate of the concentration of the fluorophore, and hence the reaction
rate, could be in error. One example is the measurement of reduced nicotinamide

adenine dinucleotide (NADH) in enzyme immunoassays (5).

In some cases inner filter effects are unavoidable. For this reason, many
researchers have proposed mathematical corrections of observed fluorescence values that
require measurement of the absorbance of the sample at the excitation and emission
wavelengths. Others have constructed spectrofluorometers that automatically produce
absorption-corrected fluorescence spectra. The discussion above indicates that there is

clearly a need for ways to make absorption-corrected fluorescence measurements.

4
REVIEW

There is a long history of work in absorption-corrected fluorescence at Michigan
State University, including the work of Holland et at (6-8), Christmann et at. (9-12),
Adamsons et at. (13-16), and Ratzlaff et al. (17,18). Several of these researchers have
written thorough reviews of absorption correction, and another good review has recently

appeared (19). It is the purpose of this section to update those reviews.
Mathematical Corrections

Wiechelman, in addition to reviewing the field, has proposed an empirical
mathematical correction for absorption (19). The difference between the fluorescence in

the presence and absence of absorbers was fit by an equation of the form:
AF = a(1 - e‘bA)

where A is the sum of the primary and secondary absorbances measured on a
spectrophotometer. Values of a and b were determined for several absorbers of
biochemical interest added to several different proteins containing the fluorescent
tryptophan residue. The parameters a and b were found to depend on the absorbing
wecies, but not on the proteins. This study was interesting because the difference in the

two fluorescence values, rather than the ratio, was fit by a curve.

Batke has derived an expression for correction that can be used with fluorometers
of side excitation, bottom emission design (20). The equation has been used to study the
concentration dependence of the quantum efi'iciency for fluorophores that associate at
high concentration. The concentration of the fluorophore was varied over four orders of
magnitude, and at the highest concentrations, the fluorescence values had to be corrected

for absorption by the fluorophore.

5

Van Geel et al. have developed a general expression for the observed intensity in
molecular fluorescence spectrometry (2). They have divided the areas of the sample into
the prefilter region (that part of the excitation beam that does not overlap with the
emission beam), the post filter region (that part of the emission beam that does not
overlap with the excitation beam), and the primary absorption/reabsorption region
(where the two beams overlap). The expression assumes that the light is collimated and
under those conditions is identical to the equation used by Ratzlaff (17) for a prefilter
and post-filter length of zero.

Instrumental Corrections

Lutz has reported a simple cell shift instrument (21). This instrument shifts the
cell along the diagonal of the cuvette. For the first measurement, both the excitation and
emission pathlengths are short; for the second measurement the pathlengths are long.

This method has the advantage of a simple expression for correction:
ch F1(F1/F2)a

where I“l is the short pathlength measurement, F2 is the long pathlength measurement,
and a is given by 5/01 - 17), where ’1 is the short pathlength and 12 is the long pathlength.
The worst case deviation of the absorption-corrected values was 3% with a primary or

secondary absorbance as high as 2.0.

Street has investigated the use of a mirrored sample cell compartment to enhance
the observed fluorescence signal, and to diagnose and correct inner filter effects (22-25).
Measurements are made with and without the mirrors covered. The ratio of the signal
obtained with the mirrors blocked to the difference between the signals is related to the
measured primary and secondary absorbance by a quadratic polynomial. Under

conditions of collimated excitation and point source/collimated emission, the equation

6
reduces to a straight line with a zero intercept. The author suggests the construction of a
working curve for each chemical system, making this method much less general than the
cell shift method.

Applications

Many of the methods of correcting for inner filter effects continue to see use in
applications of fluorescence spectroscopy. The equation of Demas and Adamson (26),
originally derived to correct for errors caused when quenchers are added to a solution
containing a fluorophore, has continued to be the most popular mathematical method. It
was used successfully by Haga et al., who studied the quenching of Ru(bipyrizine)32+ by
amines in acetonitrile (27). Gamache et al. had less success with this method (28). They
found that the equation could not adequately correct their data in a study of the
quenching of Cu(2,9-diphenyl-1,10-phenanthrocene)2+ by various Cr(III) complexes.
The workers had to resort to the more complex and less precise lifetime method to
determine the quenching constants for these systems. This underscores the need for
more theoretically sound methods of correction, such as the cell-shift method. This
equation has also been used by Marcus et al. to correct data for inner filter effects (29).

They studied the binding of benzo(a)pyrene to cytochrome P-450.

Blewitt et al. used the equation of Kinka and Faulkner (30) to study the quenching
of the fluorescence of the tryptophan residues in dyptheria toxin at low pH, which was
ultimately correlated to the tertiary structure of the protein (31). Dithiothreitol was
added to the solution to reduce disulfide linkages; corrections were required to account

for its absorbance at the excitation wavelength.

The cell shift instrument built by Novak (32) has been used to correct for inner
filter errors in the study of the quenching of the chlorophyll fluorescence by herbicide

ureas (33). The corrections suggested for use with the side/bottom fluorometer by Batke

7
(20) have been used by Vas and Batke in studies of the binding of 3-phosphoglycerate to
3-phosphoglycerate kinase using ANS as both the competitive binder and fluorescent
indicator (34).

Schroeder et 0!. continue to report applications of the instrument based on the
design of Holland (35). The latest applications involve membrane studies using a number
of membrane-bound fluorophores. Dehydroergosterol has been used to study the fluidity
of cancer cell membranes (36) and the effect of the fluidity on the action of squalene
carrier protein (37). Cholestatrienol was used in a similar study (38). The fluorophore
1,6- diphenylhexa-1,3,5-triene (DPH) has been used to study the effects of cationic and
anionic anesthetics on the fluidity of the leaflets of plasma cell membrane and on the
action of several membrane-bound enzymes (39), and on the uptake of 7-aminobutyric

acid (GABA) by the GABA receptor (40).
Miscellaneous Phenomena Related to the Inner Filter Efi‘ect

O’Neil and Schulman have discovered a simple means of determining the
quantum efficiency of a fluorophore by the relative method, using highly concentrated
solutions and right angle geometry (41). They found that, for a pure fluorophore, the
product of the absorptivity of a compound and the concentration at which its maximum
fluorescence occurs (where the calibration curve bends over) is dependent only on the
geometric properties of the fluorometer used. The quantum efi'iciency of the unknown
can be calculated by:

‘0 g 08 (Fu/Fs)

where O“ is the quantum efficiency of the unknown, (as is the quantum efficiency of the
standard, Fu is the maximum fluorescence intensity of the unknown, and F s is the

maximum fluorescence intensity of the standard.

8
Chu et al. have put the inner filter effect to good use in the design of a universal
fluorescence detector for HPLC (42,43). The mobile phase contains a small amount of a
fluorescent compound. As a compound is eluted, the steady-state fluorescence signal
either decreases (if the compound does not fluoresce) or increases (if the compound
fluoresces). In the case of a non-fluorescent eluent, one of the mechanisms responsible

for the decrease in the fluorescence is the inner filter effect (2).
ABSORPTION-CORRECTION FOR FLOWING SAMPLE STREAMS

Most work in absorption-corrected fluorescence to date has involved instruments
that use conventional one centimeter square cuvettes. While this the is most versatile
method of making fluorescence measurements, it does have some drawbacks. For
example, none of the instrumental corrections developed are compatible with flowing
sample streams. This is unfortunate, because many techniques in which fluorescence
measurements are made on a flowing stream - including continuous flow analysis, flow
injection analysis, high performance liquid chromatography and stopped-flow kinetics -

could benefit from absorption correction.

In order to fill this need, an instrument was designed by Ratzlafi to make primary
absorption-corrected fluorescence measurements in real time (17,18). The sample cell is
a cylindrical flow celL and light is conducted to and from the cell by a bifurcated fiber
optic bundle. The instrument has three detectors, and simultaneously monitors source
intensity, front surface fluorescence intensity and transmittance. A mathematical model
was developed that corrects the observed intensity based on the value of the primary
absorbance.

The model developed by Ratzlaff, which is detailed in Chapter 2 of this
dissertation, contains a term in which the primary and secondary absorbance are added

together. By switching the excitation and transmittance monochromators to the emission

9
wavelength, and measuring the absorbance at this wavelength, it was possible to correct
the fluorescence emission for secondary absorbance errors as well. However, the
instrument was now no longer able to obtain the corrections in real time. Furthermore,
to operate in near real time, it would have been necessary to slew the monochromators
rapidly between the excitation and emission wavelengths, and either change the setting of
the fluorescence monochromator or gate the aperture to the emission photomultiplier
tube to protect it from the scattered excitation beam. The wavelength setting imprecision

was thought to be a major source of error in the corrected values.

This inability to correct for secondary absorption errors in real time was felt to be
a major shortcoming of Ratzlaff’s instrument. The next three chapters describe an
instrument and methodology that allow one to make fluorescence measurements
corrected for both primary and secondary absorption errors in real time on a flowing
sample stream. First, a fluorescence intensity expression that relates the measured
fluorescence to the sample’s absorbance at the excitation and emission wavelengths is
derived. Then the instrumentation and methodology are described, and experimental
results are presented. Finally, the limitations of the current instrument are described.
Some suggestions for its improvement are given, as well as suggestions for other ways of

making absorption-corrected fluorescence measurements on a flowing sample stream.

990.495»

10.
11.
l2.

13.
14.
15.

16.

17.
18.

19.

10

CHAPTER 1 REFERENCES

Schulman, S.G., ed., Molecular Luminescence Spectroscopy, John Wiley and
Sons: New York, 1985.

Van Geel, F.; Voightman, E.; Winefordner, J.D. Appl. Spectrosc. 1984, 38, 228.

Mielenz, K.D., ed., Optical Radiation Measurements, Volume 3: Measurement of
Photoluminescence, Academic Press: New York, 1982.

Lakowitz, J.R. Principles of Fluorescence Spectroscopy, Plenum Press: New York,
1983.

Harkonen, M.; Adlercreutz, H. Trends Anal Chem. 1983, 2, 176.

Holland, J.F.; Teets, R.E.; Timnick, A. Anal. Chem. 1973, 45, 145.

Holland, J.F.; Teets, R.E.; Kelly, P.M.; Timnick, A. Anal. Chem 1977, 49, 706.
Holland, J.F. Ph. D. Dissertation, Michigan State University, 1971.

Christmann, D.R.; Crouch, S.R.; Holland, J.F.; Timnick, A. Anal Chem. 1980,
52, 291.

Christmann, D.R.; Crouch, S.R.; Timnick, A. Anal. Chem 1981, 53, 276.
Christmann, D.R.; Crouch, S.R.; Timnick, A. Anal. Chem. 1981, 53, 2040.

Christmann, D.R Ph.D. Dissertation, Michigan State University, East Lansing,
MI, 1980.

Adamsons, K.; Timnick, A.; Holland, J.F.; Sell, J .E. Anal. Chem. 1982, 54, 2186.
Adamsons, K.; Sell, J.E.; Holland, J.F.; Timnick, A. Am. Lab. 1984, 16, 16.

Adamsons, K. M.S. Dissertation, Michigan State University, East Lansing, MI,
1982.

Adamsons, K. Ph.D. Dissertation, Michigan State University, East Lansing, MI,
1985.

Ratzlaff, E.H.; Harfmann, R.G.; Crouch, S.R. Anal. Chem. 1984, 56, 342.

Ratzlaff, E.H. Ph.D. Dissertation, Michigan State University, East Lansing, MI,
1982. .

Wiechelman, K. Am. Lab. 1986, 18, 49.

20.
21.

i3

26.
27.

28.

29.
30.
31.
32.
33.

35.
36.

37.
38.

39.
40.
41.
42.
43.

1 1
Batke, J. Anal. Biochem 1982, 121, 123.
Lutz, H-P.; Luisi, P.L. Helv. Chim Acta 1983, 66, 1929.
Street, K.W.; Singh, A. Anal. Lett. 1985, 18, 529.
Street, K.W. Analyst (London) 1985, 110, 1169.
Street, K.W. Analyst (London) 1987, 112, 167.
Street, KW. Analyst (London) 1987, 112, 921.
Demas, J.N.; Adamson, AW. 1. Am Chem. Soc. 1973, 95, 5159.

Haga, M.; Dodsworth, E.S.; Eryavek, (3.; Seymour,P.; Lever, A.B.P. Inorg.
(Diem 1985,24, 1901.

Gamache, R.E. Jr.; Rader, R.A.; McMillan, DR. 1. Am. Chem Soc. 1985, 107,
1141.

Marcus, C.B.;T\1rner, C.R.; Jefcoate, C.R. Biochemistry 1984, 24, 5115.
Kinka, G.W.; Faulkner, L.R. J. Am. Chem. Soc. 1976, 98, 3897.

Blewitt, M.G.; Chung, L.A.; London, E. Biochemistry 1984, 24, 5458.
Novak, A. Collect. Czech. Chem Commun. 1978, 43, 4869.

Kaplanov, N. Photosynthesis 1985, I9, 221.

Vas, M.; Batke, J. Eur. J. Biochem 1984, 139, 115.

' Schroeder, F.; Holland, J.F.; Vagelos, as. J. Biol Chem 1976, 251, 6739.

Fischer, R.T.; Cowlen, M.S.; Dempsey, M.E.; Schroeder, F. Biochemistry 1985,
24, 3322.

Schroeder, F.; Dempsey, M.E.; Fischer, R.T. J. Biol Chem 1985, 260, 2904.

Kier, A.B.; Sweet, W.D.; Cowlen, M.S.; Schroeder, F. Biochim. Biophys. Acta
1986, 861 , 287.

Sweet, W.D.; Schroeder, F. Biochem. J. 1986, 239, 301.

Roberts, E.; Liron, 2.; Wong, E.; Schroeder, F. Era). Neurol. 1985, 88, 13.
O’Neal, J .S.; Schulman, S.G. Anal. Lett. 1986, 19, 495.

Su, S.Y.; Jurgensen, A.; Bolton, D.; Winefordner, J.D. Anal Lett. 1981, 14A, 1.
Su, S.Y.; Lai, E.P.; Winefordner, J .D. Anal. Lett. 1982, 15A, 439.

CHAPTER 2: A FLUORESCENCE INTENSITY EXPRESSION

In previous work in absorption-corrected fluorescence, the excitation and
emission beams were collimated to simplify the fluorescence intensity expression and
make it integrable, and to satisfy one of the requirements of Beer’s law (1). However,
light that exits a fiber or bundle of fibers is not collimated, so the expressions derived in
previous studies are not applicable to this study. In this chapter an expression for the
optical transfer function of a pair of fibers is derived, and is compared to a semiempirical
transfer function that is integrable and forms the basis of the correction equation used in

Chapter 3.
Review

Mitchell and coworkers were among the first to use optical fibers to measure
molecular fluorescence in solution. They found that a front surface fluorescence
measurement with a bifurcated bundle offered superior immunity to inner filter effects
compared to measurements made with conventional 90° fluorometers (2). In designing a
flowcell for their filter fluorometer, they studied the effects of the focal length of the
excitation optics and the cell length on the observed fluorescence signal (3). Recently, the
emphasis has shifted from bundles of fibers to single fibers for fluorescence
measurements (4). Although much of the work involves the use of fluorescent reagents
immobilized on fibers, there have been several reports of measurements of the
fluorescence of discrete volumes of solution with single fibers (5-8), so the optical

properties of fibers are still a concern.

There have also been reports of the use of a pair of fibers to measure Raman
scattering in solution (9,10). These researchers have investigated ways of maximizing the

observed Raman scattering signal by using fibers with different diameters and numerical

12

13
apertures (defined below) (11,12) and by varying the angle formed by the fiber pair (12).
In both studies mentioned above, the authors started with equations describing the
optical properties of fibers and developed expressions for the observed signal that were
numerically integrated. The model described below, although similar to those mentioned
above, was derived independently of them and is analytically integrable.

Fiber Optics

An optical fiber consists of a core and a cladding with different indices of
refraction. As shown in Figure 2-1, light that enters the fiber such that it strikes the fiber
face at an angle of less than 0 undergoes total internal reflection and propagates along the
fiber. If the angle of incidence is greater than a, the ray enters the cladding and leaves the
fiber. Based on this simple model, it would be expected that the light leaving the end of
an optical fiber is contained in a cone whose walls make an angle 0 with the vector normal
to the fiber face. Furthermore, any rays not wholly contained within this same cone

would not propagate along the fiber if incident on the fiber face.

The sine of the angle defined by the wall of the cone and the vector normal to the
fiber face is called the numerical aperture (n.a.) of the fiber, and is a function of the
wavelength of light, and the refractive indices of the core, cladding, and external medium.

Numerical apertures are usually reported for an external medium of air.

Consider two fibers placed side by side, both of core radius r, and cladding
thickness c, so that they have a total radius of R = r + c. At any distance 2 from the face
of the fibers, the light emitted by one fiber forms a circle of radius B(2) 8 r + z - t, where t
a tan (sin'1 na. ). The base of the acceptance cone of the other fiber is a circle with the
same diameter, with its center located a distance of 2R away from the center of the other
circle. Those two circles will not overlap until they are both of radius R, or in other

words, until

14

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15
r + c - r + z - t, or z = c / t. When they do begin to overlap, the overlap region is roughly
elliptical, as shown in Figure 2-2. This region is symmetric about both the x and y axes; its
area is given by 4 times the area of either circle evaluated from its extreme x value to
where it intersects with the other circle. Adhering to the conventions of Figure 2-2, and
using the equation for the circle on the left, the area of the overlap region is given by:

B s)
area(z)=4 [Bz(z)-xz]1/2dz
R

=2[(n/2-sin-1fimafin-Rmhzynzwz1 (2-1)
Asz _. co, area(z) —- 1122, as expected.
A Fluorescence Intensity Expression

Assume that these fibers are immersed in a homogeneous medium with a linear
absorption coefficient (absorbance/pathlength) at the excitation wavelength and emission
wavelength of ‘ex and ‘em respectively. The medium contains a fluorescent compound
with a linear absorption coefficient of Ecxf and a quantum efficiency of Q. In a plane
located a distance 2 from the surface of the excitation fiber, the power at any point can be
approximated by Pd = POIO"‘°"Z / 82(2), where P0 is the radiative power at the face of
the excitation fiber. The fluorescence emitted by an infinitely small volume element at
this point is PdQecxf dx dy dz. The emission is assumed to be isotropic. A certain solid
angle of the emitted light is collected by the emission fiber; it is approximated by r2 /
«(R2 + 22), where (R2 + 2:2)1/2 is the distance from the center of the overlap area to the
center of the emission fiber. Before reaching the emission fiber, the emission is
attenuated by absorption by a factor of 10““2. The power that enters the emission fiber

is then given by:

16

 

 

 

 

cladding core overlap

rodiu s=r+ zt

‘3") (b)

 

x=0

formula of circle:
F(B' (2)-X’)"’

 

Figure 2-2. Overlap region of light emitted from and collected by two adjacent
optical fibers: (a) schematic diagram, (b) head-on View.

17

2
-8 z -8 z 1‘
P Tm} as: 10 ex 10 "n —2———-d:cdydz
f 0 x B (z)1r(R2+zz)

 

b
.. f ’(eex ‘l’ 8cm) z r2 dz- (1
-130 Q Ens/10 [f—B‘I'Yz) 110224-22) dy z

where b is the length of the cell. The double integral consists only of geometric quantities
which relate to how much excitation radiation is reaching a point in the cell or how much
light emitted from a point is collected by the emission fiber. Since none of the variables
in the integral depend cm: or y, Equation 2-1 for the overlap area derived in the previous
section can be combined with the equation above. The result, after converting to natural

exponentials is:

b 2
_ f -20303 (8e: + £2111): \ 1‘
Pf “Po Q 8e: 3]- e [area(z, 132(2) n(R2+zz) ] dz

 

(2-2)
Hereafter, this equation will be called the theoretical model.
Summary of Previous Work

If we compare Equation 2-2 to a similar equation derived by Ratzlaff (13), it is
seen that the terms in brackets following the exponential term are identical to Ratzlaff’s
transfer function, T(z), for which he proposed no theoretical expression. Instead, he
assumed that there is an exponential rise of the transfer function near the face of the
bundle, mostly due to the increase in overlap area of the cones of excitation and emission.
A mathematical simulation showed that there is an erponential decrease in the transfer
function at greater distances, due to the drop ofi in irradiated power and the decreased
solid angle of collection. Based on those assumptions, the following transfer function was

proposed:

T(z)=G(e‘Hz-e'KZ) (2-3)

18
Hereafter, this will be called the experimental model.

This transfer function has the important advantage that it is easily integrated after

substitution into equation 2-2 for the term in brackets, to yield the following fluorescence

 

 

intensity equation:
_ f 1 _ -(H+2.303(sex+cem))b
“)0 Q a" G[ H+2.303(£,,+ 8m) (1 e )
(2-4)
1 ( l_e-(x+2.303(e,,+ em))b)]
K+2.303(£,,+£,m)

The values of H and K were determined experimentally by measuring the front surface
fluorescence and primary absorbance of a series of solutions of quinine sulfate (OS) (14).
The resulting values were regressed on equation 2-4, with ‘ex equal to ‘exf and Gem: 0.

The values reported by Ratzlaff are H = 1.108 :t 0.052, K = 9.60 :t 0.33.
Comparison of Theoretical and Experimental Models

The theoretical and experimental transfer functions are plotted in Figure 2-3, with
the ordinate for the experimental transfer function on the bottom and the ordinate for
the theoretical model on the top. Both curves are normalized to a maximum value of 1.0.
Parameters for the theoretical model were chosen which approximate those of the

equipment used in the experiments described in Chapter 3:
r = core diameter = 116 um
c = cladding thickness = 17 um
na. 8 0.128

The value of the numerical aperture is not that of the fibers; it is the numerical aperture

of the monochromators. The numerical aperture of an optical system is equal to the

z

o

E

z

E.’

a:

11’

i

0.0

Figure 2-3.

19

 

 

 

 

 

* 1 ' 1 ' 1 ' 1 '
0.2 0.4 0.0 0.0 1.0

DISTANCE FROM FIBER FACE (cm)

Comparison of experimental and theoretical transfer functions.

20
smallest numerical aperture of all the components of the system. The monochromators
are f/3; the numerical aperture is related to the f/no. by na. = (2-f/no.)'l, which gives a
na. of 0.167 for the monochromators, compared to ca. 0.3 for the fibers. Since the ends
of the fibers are immersed in aqueous solution with a refractive index of ca. 1.3, the

effective na. of the system is 0.167 / 1.3 = 0.128.

It can be seen that the theoretical model provides a close match to the shape of
the experimentally determined transfer function. However, the difference in scale
required to represent each on the same graph is disconcerting. The theoretical model
indicates that the transfer function is zero until 2 > c / t, and also indicates that it peaks

much closer to the fiber face.

The first difference is easy to explain. It is assumed in the theoretical model that
the optical power in the cladding is zero, but this is not true. If an electric field model,
rather than the simple ray tracing model, is used to describe the propagation of light
through an optical fiber, it is found that there is an ‘evanescent field’ in the cladding that
is nonzero. This evanescent field produces an overlap area between the two fibers

infinitely close to the face, as has been demonstrated by light scattering experiments (2).

The difference between the locations of the peak maxima of the two transfer
functions is more difficult to explain. The difference is probably a result of the omission
of two factors in the theoretical model: (1) the fact that a fiber optic bundle was used in
the experiments, whereas a fiber pair was employed in the model and (2) the effect of the
cell walls was not considered. With numerical apertures near those used in the model,
bundling would not shift the location of the peak maximum. This is because the
maximum occurs at a distance less than that required to cause overlap between an

excitation fiber and an emission fiber located at a distance of 2R. In a randomized

21
bundle, an emission fiber could be found at any distance from an excitation fiber, but this

probabilistic effect is diffith to incorporate into the model.

The presence of cell walls has two possible effects. If the walls are totally non-
reflective, the overlap of the area between the two fibers is limited to 1R 3, where R c is
the cell radius. This would cause a more rapid drop off of the transfer function at
distances greater than where the top of the overlap area first contacts the cell wall;
however, this would not effect the location of the transfer function maximum. If the walls
did reflect some energy, which is likely, it would cause the incident power in the overlap
area to be greater than estimated in Equation 2-2. Also, a larger solid angle of emission
would be collected. This increase in power would be caused by the light which otherwise
would have been lost reflecting off the cell wall. However, if large amounts of light were
reflected off the walls, the absorption pathlength assumptions used to obtain the

integrated fluorescence expression would not be correct.

Two assumptions that were made in developing the theoretical model may also
have contributed to the difference in the locations of the transfer function maxima. It
was assumed that the power is constant everywhere in a plane at a given distance from
the fiber face. A better approximation is a two dimensional Gaussian power distribution
with a full width at half maximum defined by the numerical aperture (3). The other
approximation that was made is that each point in the overlap plane is the same distance
from the emission fiber as the center of the overlap area. Both assumptions were made
so that the transfer function could be integrated analytically in the xy plane; without these
assumptions numerical integration is required. Plaza et al. (12) compared the results of
numerical integration both with and without the simplifying assumptions and found that

the assumptions had little effect on the results.

22
Without abandoning the model derived here, there are two factors that could be
included that would cause the maximum of the fiber transfer function to move farther out
into the cell: an increase the estimate Of the fiber diameter and a decrease in the estimate
of the numerical aperture. The effects of these two changes are shown in Figure 2-4. The
first of these could be considered as a way of taking the bundling of the fibers into
account. The second could be considered as a simple means to account for reflections

from the cell walls.
Conclusions

The model described in this chapter has given the empirical transfer function of
Ratzlaff (13) a more solid theoretical foundation. Because Ratzlaff’s transfer function
has proven itself experimentally, and because the newly proposed equation is so unwieldy,
the corrections in this dissertation are based on Equation 2-4, which was originally
derived by Ratzlaff. The coefficients in this expression have already been proven
experimentally to be independent of wavelength over a short range. This is expected,
because the numerical aperture of the optical system is determined by the
monochromators (wavelength-independent numerical apertures) rather than the fibers
(wavelength-dependent numerical apertures). How much the coefiicients depend on the
refractive index of the solution remains to be seen (15), but the dependence is expected
tobe large. This is unfortunate, because it means that a new set of coefficients will have
to be determined experimentally for each solvent in which fluorescence is measured,
unless an equation can be found that relates the coefficients to the refractive index of the

solution. Perhaps such an equation could be derived from the model presented here.

TRANSFER FUNCTION

TRANSFER FUNCTION

Figure 2-4.

1.0 T

00"
O

00‘ -

0.2 _

1.0

0.4

0.2

 

” , \\

 

 

 

 

 

fl 1 ' l ' j ' T ' I
0.0 0.2 0.4 0.0 0.0 1 .0

DISTANCE FROM FIBER FACE (cm)

. ' (b)

 

 

 

 

fi ' U I I | I I s —'
0.0 ' 0.2 0.4 0.0 0.0 1.0

ousrmc: FROM FIBER FACE (cm)

Efi'ect of changes in the parameters used in the theoretical model:
(a) change in numerical aperture (1 - 4 = numerical a rature of
.1, .05, .025, .0125), (b) change in fiber radius (1 - 4 -- ber radius
of .1, .3, .9, 2.7 mm).

9919933!"

9°

10.
11.
12.

13.

14.
15.

24

CHAPTER 2 REFERENCES

Christmann, D.R.; Crouch, S.R.; Holland, J .F.; Timnick, A. Anal. Chem 1980,
52, 291.

Mitchell, D.G.; Garden, J .S.; Aldous, KM. Anal. Chem 1976, 48, 2275.
Smith, R.M.; Jackson, K.W.; Aldous, K.M. Anal Chem 1977, 49, 2051.
Wolfbeis, O.S. Fresnius Z. Anal. Chem 1986, 325, 387.

Vurek, G.G. Anal Chem 1982, 54, 840.

Sepaniak, Ml; Tromberg, BJ., Eastham, J.F. Clin. Chem. 1983, 29, 1678.

Renault, G.; Raynal, E.; Sinet, M.; Muffat-Joly, M.; Berthier, J .; Cornillault, J .;
Godard, B.; Pocidalo, J. Am J. Physiol 1984, 246, H491.

Chudyk, W.A.; Carrabba, M.M.; Kenny, J.E. Anal. Chem 1985, 57, 1237.
McCreery, R.L.; Fleischmann, M.; Hendra, P. Anal. Chem 1983, 55, 148.
Dao, N.O.; Plaza, P. Analusis 1986, 14, 119.

Schwab, S.D.; McCreey, R.L. Anal Chem 1984, 56, 2199.

Plaza, P.; Dao, N.O.; Jouan, M.; Fevrier, H.; Saisse, H. Appl Opt. 1986, 25,
3448.

Ratzlaff, E.H. Ph.D. Dissertation, Michigan State University, East Lansing, MI,
1982, Equation 3-4, p. 44.

Ratzlaff, E.H.; Harfmann, R.G.; Crouch, S.R. Anal Chem 1984, 56, 342.
Victor, M.A.; Ph.D. research in progress, Michigan State University, 1987.

CHAPTER 3: EXPERIMENTS IN ABSORPTION-CORRECTED
FLUORESCENCE
In this chapter the instrumentation and methodology for making fluorescence
measurements that are corrected for both primary and secondary inner filter effects are
described. The system is an extension of a previous system described by Ratzlaff (1,2)

that corrects only primary inner filter errors in real time.
THEORY

An intuitive description of the phenomena underlying the correction method that
was used in this work is explained below. Assume that monochromatic light enters a
cylinder at the front of the cylinder as shown in Figure 3-1. The cylinder is assumed to
contain a homogeneous distribution of a fluorescent material. If the light entering the
cell is collimated, and is not attenuated significantly by absorption, then the right angle
fluorescence measured at either end of the cell would be the same. However, if some of
the light is absoer by the medium in the cell, the rear of the cell is less efficiently
excited than the front. Since fluorescence is emitted isotropically, the front of the cell
would present a larger solid angle for collection of emission than the rear, and the ratio of
the fluorescence measured at the front to that measured at the rear would be less than

one.

Now imagine that the sample cylinder does not absorb the incident radiation, but
does absorb the emitted radiation. In this case the fluorescence signals at both ends will
decrease by the same magnitude. This is because the cylinder is uniformly illuminated, so
the average plane of excitation is in the center of the cell, and the pathlength to either
side is the same. Finally, consider the case where there is absorption at both the
excitation and emission wavelengths. As before, the front is more efficiently illuminated

than the rear, and on the basis of the solid angle of collection, the fluorescence signal at

25

26

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27
the front is expected to be larger. Another factor is that, on the average, the emitted
radiation will travel farther to exit the rear of the cell than the front, so the pathlength for
absorption for most of the radiation exiting the from of the cell is shorter (x in Figure 3-1)
than that for the radiation exiting the rear of the cell (b-x in Figure 3-1). The following
statements summarize the results of this model:

1) As the primary absorbance increases, while the secondary absorbance remains
constant, the ratio of the front/ rear surface fluorescence increases.

2) If the secondary absorbance increases while the primary absorbance remains
constant, the ratio Of the front /rear surface fluorescence increases only if the
primary absorbance is nonzero, and the greater the primary absorbance, the
greater the rate of increase of the ratio.

Even if the incident light were not collimated, one would expect that the ratio of the
fluorescence measurements made at each end of a cylindrical sample cell illuminated only
at one end would depend on the primary and secondary absorbance of the solution the in
the cylinder. If the primary absorbance were known, it might be possible to calculate the
secondary absorbance based on appropriate curve fitting to the ratio of the front surface
fluorescence measurement to the rear surface fluorescence measurement. The values of
the primary and secondary absorbance could then be used to correct the front surface
fluorescence measurement for inner filter effects. The goal of this project was to test this

hypothesis.
Experimental Details
APPARATUS

A diagram of the instrument designed to test the correction procedure is shown in
Figure 3-2. Excitation is provided by a 150 W Xe arc lamp bulb (3) and power supply (4).
The light is focused by a quartz condensing lens (f/2.5) through a colored glass bandpass

.efioEebeon—m 853.23: we Bahama .N-m 053m

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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29
filter (center wavelength =360 nm, bandpass=70 nm) and onto the entrance slit of a small
(f/3, focal length = 100 mm) monochromator (5). At the exit slit of the monochromator
one leg of a bifurcated fiber optic bundle (6) is positioned. This bundle consists Of
approximately 130 plastic clad silica (PCS) fibers with a core diameter of 116 um and an
outer diameter of 150 pm. The bundle is slit shaped at the monochromator end, with a
length of ca. 3.5 mm and a width of ca. 0.5 mm, and is circular at the common end, with a
diameter of 2 mm. The common end of the bundles fits into a threaded nylon nut and is
screwed tightly into a black delrin block which serves as the sample cell, and is shown in
Figure 3-3 and described later. The other leg of the bundle is terminated at the entrance
slit of a small (f/3, focal length = 74 mm) Fastie-Ebert monochromator (7). The
wavelength drive of the monochromator is controlled by a stepper motor (8), and a
stepper motor drive circuit that was designed and built locally. A photomultiplier tube
(9) in a black delrin housing is located at the exit slit. The tube is at an angle Of 17° to the
monochromator axis, which was found to give the maximum signal. Between the slit and

the photomultiplier tube is a colored glass longpass filter (cutoff = 410 nm).

The other end of the sample cell is terminated by a similar bifurcated fiber optic
bundle (10). This bundle consists of approximately 60 PCS fibers with a core diameter of
210 um and an outer diameter of 230 pm. One of its legs is terminated at a small
monochromator identical to the excitation monochromator. A PIN photodiode (11) is
located at the exit slit. The other leg of this bundle is terminated at the entrance slit of a
monochromator/photomultiplier tube combination identical to the one previously
described. A single fiber leads from the exit slit of the excitation monochromator to an
integrated diode detector/amplifier (12). All of the optical components and the sample
cell are housed in a light-tight box.

Each photomultiplier tube is powered by a high voltage supply (13). The output

current of the tube is converted to a voltage by an electrometer (14). The photodiode

30

. 531-3)

(0)

 

 

 

 

 

 

 

 

flow
excitation in out transmittance
F.__ _:_|
bifurcated
fiber
Optic
front surface emission bundle rear surface emission

(b)

Figure 3-3. Diagram of sample cell: (a) cutaway view, (b) view showing fiber
optic windows.

31
current is converted to a voltage by an electrometer built locally (15). The voltage output
of each of the four amplifiers (reference, transmittance, front fluorescence and rear
fluorescence) is digitized by a microcomputer (16) equipped with an analog input / output
board. This board contains four synchronously clocked dual slope A/D converters
Operating at 30 Hz. Programs were written in FORTRAN to collect data, control the
stepper motors, and calculate fluorescence values. Data are transmitted to a

minicomputer (17) via RS-232 serial link for storage and plotting.

The sample cell, shown in Figure 3-3 was constructed out of a block of black
Delrin (18). The center section is 1 cm in length and 2 mm in diameter for a total volume
of 31.4 11L. The top is threaded to accept standard cheminert fittings. The ends of the
sample cell are formed by the common ends Of the two bifurcated fiber Optic bundles. A

peristaltic pump (19) is used to pump the solution to be measured through the cell.

For all the experiments described in this chapter the photomultiplier tubes were
operated at -700 V. The PMT electrometers had a conversion factor of 108 V/A and a
time constant of 300 ms. The tubes were operated at room temperature, and had a dark
current of ca. 12 pA at -700 V. The transmittance photodiode electrometer had a
conversion factor of 1011 V/A and a time constant of 150 ms. The Xe arc lamp was
operated at a current of 6.2 A at 20 V. For the fluorescence measurements, the
excitation and transmittance monochromators were set to 353 nm, and the emission

monochromators were set to 438 nm.
The Effect of Primary and Secondary Absorbance on the Fluorescence Ratio
INTRODUCTION

It was desired to see the individual effects of primary and secondary absorbance

on the ratio of fluorescence values measured with the instrument described above. In

32
order to test secondary absorbance alone, it was necessary to find a compound which has
an absorptivity minimum near the excitation maximum of quinine sulfate (OS) (350 nm),
an absorptivity maximum near the emission maximum of OS (450 nm), chemical stability
at pH - 1 and high photostability. It was also required that the compound does not
quench the fluorescence of OS. The location of a compound that satisfies these
requirements was difficult. The Am” and ‘max for dyes are routinely reported, but not
the Am and ‘min' Furthermore, few values are reported at pH - 1, and many dyes
change Am” at low pH, as ionic groups become protonated. Indicator dyes for pH
titrations seemed like a natural choice, since their color, and hence approximate Am” ,
could be found at pH= 1 (20), and they are guaranteed to be stable at this pH. However,

among 20 indicators tested, none has an absorptivity at 415 run that is more than 5 times

as great as that at 350 nm.

Cobalt (II) salts have absorptivities that differ by >100 times between 350 and
450 nm at pH 8 1, but their small absorptivities mean that very high concentrations (0.2
M) are required. At such high concentrations they were found to quench OS
fluorescence. Another compound that was considered was N ,N-diethylparanitrosoaniline.
Its synthesis was attempted, but the synthesis required the use of nitrite ion, which has a
high absorptivity at 350 nm and is difficult to remove by recrystallization. Other
compounds can be added to oxidized the excess nitrite to nitrate, but they absorb at 350
nm. The compound that was found to be most useful is sodium fluorescein. Although its

absorptivity at 428 nm (A is only ca. 15 times that at 345 nm (1min), it is stable at low

max)

pH and its use as a secondary absorber is well documented (15). Although sodium

fluorescein is fluorescent, its quantum yield is quite low at low pH.

33

' EXPERIMENTAL DETAILS

Reagents

All stock solutions and dilutions were made 0.1 H in HClO4, which was prepared
by diluting 9.0 mL of 72% HCIO4 (21) to 1.00 liter with distilled water. All solid reagents
were used without purification. Solutions containing quinine sulfate (OS) (22) were
made from a stock solution Of 1.00 mM OS. Solutions containing sodium fluorescein

(SF) (23) were made from a stock solution Of 8.00 mM SF.

A series of solutions of each combination of 0, 250, 500, 750, 1000, 1250 and 1500
11L of OS stock solution and 0, 125, 250, 375, 500, 625, 750, 875, and 1000 11L of SF stock
solution was made and diluted to 10 mL in volumetric flasks with 0.1 N, HClO4.

Procedure

All measurements were made with solution flowing continuously through the
sample cell at a rate Of 0.8 mL min'l. For each channel, 30 analog-tO-digital conversions
were averaged and stored to create approximately a 1 second time constant. A blank
signal was collected with 0.1 H HCIO4 flowing through the sample cell for 2 minutes.
Then, the series of solutions containing 0 11L OS and 0 to 1000 11L SF was flawed through
the cell, allowing 2 minutes for each solution, followed by 5 minutes of 0.1 H HClO4
blank. The source was then covered and the dark signal for each channel recorded. This
procedure was repeated for each concentration of OS. The output of the four detectors

for a typical experiment is shown in Figure 3-4.

For each channel, the middle 80 seconds of the 120 second measurement for each
solution were averaged together to produce one value. From this value, the dark signal
for that channel was subtracted. Finally, the fluorescence channels were adjusted for

source fluctuations and blank corrected by dividing by the reference channel

34

 

 

 

 

 

 

 

 

0.1 -
(a)
. 0.0 - L—
1.2 _,
(b)
0.0 -
8
> 5.0 -
(0)
0.0 - ——i
5.0 q
(d)
0’0 T 1 ‘ I l
0 s 10 15 20 25

m: (mums)

Figure 34. Output of the four detectors for 500 14M qUinine sulfate (OS), 0 to
910 um bromocresol purple (BP): (a) reference channel, (b)
transmittance channel, (c) front surface fluorescence channel, d
rear surface fluorescence channel.

35
measurement collected during the blank period, and subtracting the blank signal. For
each concentration of OS and SF, the blank that was subtracted was the fluorescence
measurement for the corresponding solution containing no OS but the same amount of
SF. This was necessary because the blank signal was found to decrease with increasing SF
concentration, indicating the large magnitude of stray light. The stray light problem will
be discussed in Chapter 4.

The transmittance signal was corrected for source fluctuations by dividing by the
value of the reference channel measurement for that time period, then multiplying times
the reference value for the blank signal. Transmittance was then calculated as (corrected
transmittance) / (blank transmittance), and absorbance as the negative base 10 logarithm
of the transmittance. The absorbance of each solution at 353 nm and at 438 nm was

measured on a separate spectrophotometer (24).
RESULTS AND DISCUSSION

The ratio Of front surface fluorescence to rear surface fluorescence for each Of the
42 solutions containing OS is shown in Figure 3-5. As predicted, the ratio changes little
with secondary absorbance when the primary absorbance is low (symbol= 1) and changes
greatly when the primary absorbance is high (symbol=6). It can also be seen that the
ratio changes little with primary absorbance when the secondary absorbance is low (left
side of plot) but changes notably when the secondary absorbance is high (right side of

plot).

In order to use the fluorescence ratio as a way to estimate the secondary
absorbance, a mathematical relationship between the ratio and the primary and
secondary absorbance was sought. This was done by assuming the primary absorbance to
be constant for each set of data in Figure 3-5, and fitting each data set to an equation that

gives the ratio as a function of the secondary absorbance for that value of the primary

36

 

 

 

 

[SF]. W
0 10 20 30 40 50 60 70 so
13.0 1 1 1 1 1 1 1
s

7.0 -
g 5.0 -
<
1:
11.1 5.0 -
o
z
I.1..I
o
K} 4.0 -
a:
o
E

3.0 -

20 - ./

g M
‘-° n l l 1 l 1 1 1 1
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6
ABSORBANCE AT 438 nM

Figure 3-5. Fluorescence ratio versus absorbance at 438 nm. 1 = 250 um

1n1nesulfate (,QS 2= 500nm OS, 3= 750 #1511 OS, 4: 1000 um
S, 5.. 1250uMQ,6=1500uMOS.

37
absorbance. A simplex fitting program written by Peter Wentzell based on the algorithm
of Nedler and Mead (25) was used to determine the best fitting exponential equations,
while a least squares polynomial fitting program (26) was used to determine the best
fitting polynomial equations.

In choosing an equation to fit to the data in Figure 3-5, two criteria were
important: 1) the equation used to fit the data should be monotonic over the interval
considered here, since the theory predicts that sort of behavior and 2) the parameters
generated by the fit should themselves be monotonic with respect to the primary
absorbance to reflect what is seen in Figure 3-4, and to produce a reasonable fit of the

few resulting estimates of the parameters by an equation.

Two models with three parameters were tried: ratio = a(A”)2+bA”+c, and ratio
= aebA"+c, where A" is the secondary absorbance and a, b, and c are the variable
parameters. Both functions were found to give parameters which were not monotonic.
Since a two parameter polynomial fit is a straight line, and that would Obviously not work
well, a fit of that equation was not attempted. Instead, two different two parameter
exponential models were tried: ratio = aebA", and ratio 2 ea" + b. The former was
found to be a better fit, and gave monotonic parameters with respect to the primary
absorbance. These best fit equations are the lines drawn in Figure 3-5, and the

agreement is good.

The parameters a and b produced by the model are shown in Figures 3-6 and 3-7.
These, too must be fit to an equation to allow interpolation between experimental points.
Parameter a appears to rise exponentially, so both of the two parameter exponential
models were tried; neither seemed to produce a reasonable fit. However the three
parameter exponential model, a a few, + h did produce reasonable results, with f =
0.00386 , g = 2.02 and h = 1.20 . The second parameter, b in Figure 3-7, did not appear

PARAMETER a

1.45

.1.40

1 .35

1 .30

1.25

1.20

38

 

 

 

 

0.0 0.4 0.0 1.2 1.0 . 2.0
ABSORBANCE AT 353 nm

Figure 3-6. Parameter a versus absorbance at 353 nm.

PARAMETER b

1 .25 -

0.75 -*

0.50 -

0.25 -

39

 

 

 

0.00
0.0

Figure 3.7.

I I I l I I
0.4 0.8 1.2

ABSORBANCE AT 353 nm

Parameter 6 versus absorbance at 353 nm.

1.6

2.0

40
to have any particular shape to it, so it was fit by a straight line b = jA' + k, where j =
0.551 and k = 0.144.

It is possible to reverse the procedure and estimate the secondary absorbance of

each solution. The algorithm is as follows:

1) With the measured values Of the primary absorbance and f, g, and h, calculate
a.

2) With the measured value of the primary absorbance and j and k, calculate b.

3) With the measured value of the front/ rear surface fluorescence ratio R, and a
and b, calculate the secondary absorbance asA" = log (R / a) / b.

The estimates of the secondary absorbances of these solutions calculated by the
procedure above are shown in Figures 3-8, 3-9 and 3-10 and are summarized in Tables 3-1
and 3-2. In these figures and tables, the estimated secondary absorbance is compared
with the value measured on a separate spectrophotometer (20). The residuals in Table 3-
1 are the differences between the measured and estimated secondary absorbances. The
variance of linear regression in Table 3-2 is the root-mean-square of the differences
between the calculated or measured secondary absorbances and the line that best fits the

points.

, The results show that the estimated secondary absorbance is not as linear with
concentration as the measured secondary absorbance, as would be expected. The average
difference between the estimated and measured values is ca. 0.03 absorbance units. At all
concentrations of OS, the relative residual is largest at low concentrations of SF, and the
estimates are always lower than the measured absorbances. The consequences of this will

be evident when the corrected fluorescence values are shown.

41

(a)

ABSORBANCE AT 438 nm

 

 

 

1.6 -‘
(b)
1.4 -
1.2 '-
1.0 ~

0.0 -‘

ABSORBANCE AT 438 nm

0.4 -

0.2 '-

 

o.0 -1

 

 

[5"]. IN

Figure 3-8. Absorbance at 438 nm (1 I: calculated, 2 8 measured) versus
sodium fluorescein SF) concentration: (a) 250 uM Quinine sulfate

(05). (b) 500 14M 0 .

42

A
m
V

ABSORBANCE AT 438 nm

 

 

 

 

 

(b)
E
C
,9,
i’
’2
la)
O
i
8
9

I I I I I I I I

0 10 20 30 40 50 00 70 00

[SF]. W
Figure 3-9. Absorbance at 438 nm (1 2 calculated, 2 = measured) versus

sodium fluorescein (SF) concentration: (a) 750 um quinine sulfate

43

A
33

ABSORBANCE AT 438 nm

 

 

(b)

Aesonamcs AT 438 nm

 

 

 

[5F]. I‘M

Figure 3-10. Absorbance at 438 nm (1 = calculated, 2 = measured) versus
sodium fluorescein (SF) concentration: (a) 1250 pm Quinine
sulfate (OS), (b) 1500 14M OS.

TABLE 3-1. COMPARISON OF MEASURED AND CALCULATED
SECONDARY ABSORBAN CE

IOS|,10'6M AVE. RESIDUAL MAX. RESIDUAL MIN. RESIDUAL

250 0.018 0.04 0.005
500 0.024 0.07 0.002
750 0.045 0.07 0.003
1000 0.040 0.09 0.02
1250 0.027 0.04 0.008
1500 0.022 0.1 0.001
TABLE 3.2. COMPARISON OF LINEAR REGRESSION RESULTS FOR
MEASURED AND CALCULATED SECONDARY
ABSORBANCE
VARIANCE OF R
REGRESSION (x 10 )
MEASURED CALCULATED
SECONDARY SECONDARY
[OS]I 10'6M ABSORBANCE ABSORBANCE
250 7.0 16
500 6.5 19
750 9.5 23
1000 9.2 20
1250 3.7 14

1500 6.5 21

45
The values of the secOndary absorbance, along with the measured values of the
primary absorbance, can now be used to correct the measured front surface fluorescence
values for absorption effects. The equation was derived by Ratzlaff (24); its derivation,
and a model which justifies the use of a formerly empirical transfer function, is presented

in Chapter 2. The equation is:
F: = F” - C

where Fc is the corrected fluorescence, F” is the measured front surface fluorescence and

C is a correction factor which is given by:

C = f,.,,°/fm

where fm" and fm are equations which give the theoretical observed fluorescence signal in

the absence and presence of absorption effects, respectively. The value offIn is given by:

 

f = l (1_e-(1.1os+2.303(c,,+5mm)
m 1.1oa+2.303(a,,+£,m)

l

- -(0.s+2.303(e,,+€.m))b)
9.6+2.303(8,,+8,m)

(l-e
where ‘ex and ‘em are the linear absorption coefficients (absorbance/pathlength) for
excitation and emission respectively, and b is the pathlength, which was experimentally
determined to be 1.00 cm. The value offm‘ is given by the equation above with ‘ex and
rem equal to zero. The results of the correction procedure are shown in Figures 3-1 1, 3-
12 and 3-13 and are summarized in Table 3-3. The range in Table 3-3 is the difference

between the smallest and largest values in the data set.

The corrected fluorescence values follow the same trend as the estimated
secondary absorbance. For each concentration of OS, the second corrected fluorescence

value is always lower than the first, which has no SF. This is due to the systematically low

46
TOTAL ABSORBANCE (353 nm + 438 nm)

0.50 0.75 1 .00 1 .25 1 .50 1.75 2.00
.0 l J J l J l l

.W
(a) ”j '

 

 

 

 

a...
3
3.4
8
8
8
8
3
8

[Sf]. 1N

TOTAL ABSORBANCE (353 nm 4 438 nm)

0.75 1.00 1.25 1.50 1.75 200 2.25
J J l I l

I 1
Ni: -¥ P ‘5 4N

 

 

(b). -

surmooscuaaanmx
8
1 I .

 

 

 

o
3
8
8
8
3.1
g-
3.

[SF]. m

Figure 3-11. Front surface fluorescence intens312~)( 1= measured, 2 = corrected
using calculated absorbance at 4 11m) versus sodium fluorescein
concentration: (a) 250 pM quinine sulfate (OS), (b) 500 um OS.

47
TOTAL Aesoaamcs (355 nm + 430 nm)

1.00 1.25 1.50 1.75 2.00 2.25 2.50
l l l l J l l

(a) 150: m

 

PMTANOMCURRENT.M
8
l

 

 

 

 

 

 

 

 

 

 

30 Cl
° 1 I 1 I U I l 1 I
0 10 20 30 40 50 50 70 50
[57]. W
TOTAL ABSORBANCE (353 nm 4' 458 nm)
1.25 1.50 1.75 2.00 2.25 2.50 2.75 5.00
240 1 1 1 1 J 1 1 1
(b) ‘ \ = -.- - -
zoo -1 . . N
E d
g 150 "
120 -
5 '° '
‘0 -
o I I r I 1 1 I I r
0 10 20 50 40 50 .0 70 .0

[SF]. u“

Figure 3-12. Front surface fluorescence intensitgy (1 = measured, 2 = corrected
using calculated absorbance at 43 nm) versus sodium fluorescein
concentration: (a) 750 MA quinine sulfate (OS), (b) 1000 AM OS.

(a)

. 11A

PMTANODE

(b)

Figure 3.13.

, 11A

'PMTANODE

48
101111 1135012310105 (553 nm 4» 453 nm)
1.50 1.75 2.00 2.25 2.50 2.75 5.00 5.25

 

 

 

 

 

 

m J I I I J J I I
240 '-
200 '1
100 "I
.1
120 -'
m .1

2 .\.\‘\.\¥

‘0 q

' A
1

° 1 1 1 I 1 r 1 r r

10 20 50 4O 50 50 70 00

[SF]. W

TOTAL ABSORBANCE (555 nm + 458 nm)

1.50 1.75 2.00 2.25 2.50 2.75 5.00 5.25 5.50

 

 

 

 

 

 

1 1 1 1 L 1 1 1 1
500 _ N
250 '-
«1
200 -1
150 .1
100 -
w - \
° 1 1 1 1 1 1 1 1 1
0 10 20 50 40 50 00 70 00
isri. uM

Front surface fluorescence intensi (1 = measured, 2 = corrected
using calculated absorbance at 43 nm) versus sodium fluorescein
concentration: (a) 1250 um quinine sulfate (OS), (b) 1500 11M 08.

49

 

TABLE 3-3.
FLUORESCENCE VALUES
6 REL.
[OS], 10' M “ MEAN ST. DEV. ST. DEV. RANGE
250, U 21.0 9.6 0.46
C 56.2 0.91 0.016
500, U 32.2. 13.4 0.42
C 105 1.8 0.017
750, U 39.1 14.6 0.37
C 158 2.5 0.016
1000, U 45.3 16.6 0.37
C 209 7.6 0.036
1250, U 47.4 15.2 0.32
C 263 2.6 0.010
1500, U 49.1 14.8 0.30
C 312 6.5 0.021

“ U = uncorrected, C = corrected
ST. DEV. = standard deviation
REL. ST. DEV. = relative standard deviation

29
2.6

41
5.6

44
8.0

51
27

46
8.8

45
24

COMPARISON OF UNCORRECTED AND CORRECTED

REL.

RANGE

1.4
0.046

1.3
0.053

1.1
0.050

1.1
0.13
0.97
0.033

0.91
0.078

50
estimate of the secondary absorbance for low concentrations of SF that was described
earlier. Furthermore, the corrected fluorescence values peak near the middle of the
concentration range, and drop Off again at high concentrations of SF. It may be that the
correction equation is no longer usable at high absorbances. However, these data Show
that the correction procedure is useful up to an absorbance of at least 3.25. Note that the
relative standard deviation and relative range of the corrected values, as shown in Table
3-3, are 10 to 30 times lower than the uncorrected values. The corrected values have an

average relative standard deviation of 1.9% and an average range of 6.5%.
A Test of the Correction Procedure
INTRODUCTION

It is not surprising that parameters generated from fitting a curve to a set of
experimental points could then be used to generate a set of curves which fit that same set
well. In order for a reasonable test of the correction procedure to be made, it must work
well when used on another set Of data generated under a different set of conditions. In
order to accomplish this, a compound was sought which is stable at pH= 1, is not
fluorescent and does not quench the fluorescence of the OS. Furthermore, it is important
that the compound chosen have absorptivities at 345 and 438 run that are nearly equal, so
that both primary and secondary absorbances could be varied at roughly the same rate.
Alizarin red was tried first, but it quenched the fluorescence of OS. This was ultimately
traced to the presence Of NaCl in the unpurified indicator. However, even after
recrystallization from ethanol the alizarin red still quenched the OS fluorescence. The
next choice, bromocresol purple (BP), was found to be nearly ideal. Its only drawback
was its surfactant character, which made pipetting difficult. Figure 3-14 is a spectrum of
909 11M BP in 0.113 HClO4.

51

 

 

 

 

 

secs 2 3 5 CE 2&3 68885 2.. 83 Co 5.58% 5:98? EA Esme
2:5 Ikozmmm><3
com
com 4 » 0.0
nan mme
i/
V
8
S
O
H
8
V
N
3
3
ON

52

WERIMENT’AL DETAILS
Reagents

Stock solutions of I-IClO4 and OS were used as described in the previous section.
Solutions containing bromocresol purple (BP) (20) were made from a stock solution of
9.09mM BP. A series of solutions consisting of each combination of 0, 250, 500, 750 and
1000 14L OS stock solution and 0, 100, 200,..., 900 11L BP stock solution was made and
diluted to 10 mL in a volumetric flask with 0.1 H HClO4.

Procedure

The procedure for measurement of these solutions was the same as that in the
previous section. In addition, the absorbance of these solutions was measured at 438 nm
by changing the setting of the excitation monochromator until the largest scattering signal
was measured at the fluorescence channel, then the transmittance monochromator setting
was varied until the largest transmittance Signal was measured. After these adjustments

were made the absorbance was determined.
RESULTS AND DISCUSSION

Figures 3-15 and 3-16 Show the estimated secondary absorbance for each of the 40
solutions which contained OS. The results are summarized in Tables 3-4 and 3-5. These
secondary absorbances were estimated with the parameters found for SF in the previous
section and the measured values of the primary absorbance Obtained by this instrument.
Except for the series of solutions containing 250 11M OS, the values of the average
residuals are about the same as those found in the previous section. It is apparent that
the ability of this method to predict the secondary absorbance of the solutions is not
reliable at small primary absorbances, as was predicted. In fact, an absorbance of less

than zero is estimated in several cases. The results shown in Figure 3-15a are particularly

A
m
v

ABSORBANCE AT 458 nm

(b)

ABSORBANCE AT 458 nm

1.60 -'

1.40 -'

1.20 -‘

1.00 -'

(LOO -

(L00 '-

€L40 '-

IL20 -

(L00 '-

53

 

 

 

-{L20

1.60 1
1.40 -
1.20 -
1.00 -
(L00 -
(L00 -
(L40 '-
IL20 -

(L00 -‘

 

I I I I I I I _ I I I
100 200 500 400 500 600 700 .000 000

[3"]- IN

 

 

-4L20

Figure 3.15.

1' I I I I I I . I I
100 200 500 400 500 000 700 000 000

[9"]- W

fibsorbancel at 4318e 11mP 1 = calculated, 2 (=) néggsured) versus
romocreso Sconcentration: a 11M uinine
sulfate (OS), $311000 amp 0

1.50 -'

1.40 '1

A
33

1.20 -'

1.00 -'

0.00 -'

0.00 -'

0.40-I ‘

ABSORBANCE AT 458 nm

0.20 -‘

 

0.00 -1

 

 

"”0 I 1 1 I I I 1 1 T I
0 100 200 300 400 500 500 700 500 000

[SP]. W

1.50 -
(b) 1.40 -
1.20 -
1.00 4
0.50 «-
0.50 '-

0.40 -'

ABSORBANCE AT 458 nm

0.20 -

0.00 -'

 

 

 

Figure 3-16. Absorbance at 4318c nrr1P 1 = calculated, 2 = measured) versus
bromocresol concentration: (a) 750 uM quinine

sulfate (OS), )000(B 11M OS

55

TABLE 34. COMPARISON OF MEASURED AND CALCULATED
SECONDARY ABSORBAN CE

IQSI: 10'6M AVE. RESIDUAL MAX. RESIDUAL MIN. RESIDUAL

 

 

250 0.072 02 0.004
500 0.046 0.1 0.004
750 0.026 0.06 0.001
1000 0.013 0.04 0.002
TABLE 35. COMPARISON OF LINEAR REGRESSION RESULTS FOR
MEASURED AND CALCULATED SECONDARY
ABSORBANCE
VARIANCE OF R
REGRESSION (x 10 )
MEASURED CALCULATED
6 SECONDARY SECONDARY
[081,10 M ABSORBANCE ABSORBANCE
250 2.8 58
500 1.7 37
750 33 26

1000 1.4 44

56
discouraging. However, at high primary absorbances the agreement is respectable, as it is

in at least one region of each Of the four data sets.

With the measured value of the primary absorbance, and the estimated value of
the secondary absorbance, the fluorescence values were corrected for inner filter effects
as described in the previous section. The results of the correction are shown in Figures 3-
17 and 3-18, where these values are compared to the corrected values Obtained with the
measured primary and secondary absorbances. The results are also summarized in Table
3-6.

The corrected values obtained with the measured secondary absorbances, those
labelled M in Table 3-6, are as good as the results Obtained with SF in the previous
section. The average relative standard deviation for the four data sets is 1.9%, and the
average relative range is 5.3%. Unfortunately, the corrected fluorescence values obtained
with the estimated secondary absorbances, those labelled C in Table 3-6, are not as good,
particularly with the series of solutions that contained 250 pM OS. For the corrected
fluorescence values Obtained with the estimated secondary absorbances, the average
relative Standard deviation is 3.3% and the average relative range is 10%, which is about
half as good as the values obtained with the measured secondary absorbances. For these

solutions, the corrections appear to be useful to a total absorbance of 2.5.
Real-time Performance

Figures 3- 19 and 3-20 show the real-time performance of the instrument in
determining the absorption-corrected front surface fluorescence of a series of solutions
containing 500 14M OS and 0 to 909 um BP. Figure 19a Shows the measured primary
absorbance. The spikes in this trace mark the occurrence of a bubble flowing through the
cell. The bubble is introduced when the sample tube is moved from one sample container

to another, and serves as a marker between solutions. The difference in refractive indices

(a)

. M

(b)

www.m'

Figure 3-17.

Furnaces.

57
- 10m. Aesosamcz (353 nm + 458 nm)

 

 

 

 

 

 

 

 

0.25 0.50 0.75 1 .00 1 .25 1 .50 I .75 2.00 2.25 2.50

a I L I I I 4 I J I
‘ r, 7“:

4° -1

q
a .1
a .1
10 ‘-
° I I I I 1 1 I I I

100 200 500 400 500 000 700 000 000
[8?]. W
TOTAL ABSORBANCE (555 11111 + 458 nm)

0.50 0.75 1 .00 1 .25 1 .50 1.75 2.00 2.25 2.50 2.75
00 I I I I J I I I I I
w u
04 -1
a -
a -
10 '-
° I I I I I m I I —T

0 100 200 500 400 500 000 7” 000 000
[SP]. uM

Fluorescence intensity (1 s uncorrected, 2 -= corrected usin
calculated absorbance at 438 nm, 3 = corrected using measure)
absorbance at 438 nm) versus bromocresol urp le
concentration: (a) 250 11M quinine sulfate (OS), (b) 1100M O.

58

TOTAL ABSORBANCE (555 nm + 458 nm)

0.75 1 .00 1.25 1.50 1 .75 2.00 2.25 2.50 2.75 5.00
(a) I .0 I I I I I I I I I J

W

 

PMTANWECURRENT.IIA
8
In

 

 

 

 

 

 

 

20 -
J
° I i I I l I I I I
0 100 200 500 400 500 000 700 000 000
[3"]- IN
TOTAL ABSORBANCE (555 nm 4' 458 nm)
(b) 1 .00 I .25 1 .50 1 .75 2.00 2.25 2.50 2.75 5.00 5.25
1 1 1 I I I I 1 1 1
I” -' M
150 "'
E .
E 120 -
g .0 .-
5 '° -
a -
° I I I I I I I I I
0 I” 2W 5W 4W 0” 000 100 000 ”0
[BP], nu

Figure 3-18. Fluorescence intensity (1 - uncorrected, 2 1- corrected usin
calculated absorbance at 438 nm, 3 = corrected using measure
absorbance at 438 nm) versus bromocresol p 1e éBP)
concentration: (a) 750 11M quinine sulfate (OS), (b) 1 14M S.

59

 

TABLE 3-6.
FLUORESCENCE VALUES
6 REL.
[OS], 10' M * MEAN ST. DEV. ST. DEV. RANGE

250, U 16.2 8.1 0.50

C 44.9 2.1 0.046

M 46.6 0.91 0.019
500, U 25.4 12 0.47

C 87.6 2.1 0.024

M 88.1 0.88 0.0099

750, U 31.0 13 0.42

C 130 3.0 0.023

M 130 1.6 0.012
1000, U 36.9 15 0.41

C 170 6.5 0.038

M 170 6.1 0.036

f U = uncorrected, C = corrected using the calculated 2 absorbance, and M = corrected

using the measured 2 absorbance.
ST. DEV. = standard deviation
REL ST. DEV. = relative standard deviation

25
7.0
2.5

37
5.8
2.7

40
8.0
4.0

46
18
17

COMPARISON OF UNCORRECTED AND CORRECTED

REL.

RANGE

1.6
0.16
0.053

1.5
0.067
0.031

1.3
0.061
0.031

1.2
0.11
0.098

 

 

 

 

 

 

 

 

8 0.0 .—
i
3 0.5 -
g 6.. -
a. 0.2 -1 (a)
0.0 -——i ' .L__
g ..
a 1.2 -
g 0.9 -
é 1 . (b)
‘I’IME (MINUTES) 20 25

Figure 3-19. Real-time results: (a) measured primary absorbance, (b)
estimated secondary absorbance.

61

 

 

 

 

 

 

 

 

 

 

so-
,. (a)
8 1°-
8
g..-
5
5...
...—J “
(b)
8 ..
8
:40-
5
05-20-
J
o l I I. I LI—
_ 0 5 10 15 20 25

TIME (MINUTES)

Figure 3-20. Real-time results: (a) front surface fluorescence, (b) corrected
front surface fluorescence. -

62
between the aqueous sample and the air in the bubble causes most of the excitation beam
to be reflected at the air/ solution interface, and thus only a small fraction of the beam

reaches the other end of the cell. This has the same effect as a high sample absorbance.

Figure 3-19b shows the estimated secondary absorbance, based on the measured
primary absorbance and the ratio of the measured front surface fluorescence to the
measured rear surface fluorescence (not shown). The spikes in this trace occur for the
reason given above, but they are usually in the opposite direction as the spikes in the
primary absorbance trace. For a given value of the fluorescence ratio, the higher the
primary absorbance, the lower the estimated secondary absorbance, which explains why
they should go in opposite directions. The large spike near 15 minutes is due to incorrect
placement of the sample tube. This solution was the run twice as long as the others.

Note the concentration linearity Of the estimated secondary absorbance.

Figure 3-20a shows the measured front surface fluorescence, and Figure 3-20b
shows the corrected front surface fluorescence. These figures reflect what was shown in
Table 3-6: the corrected values have a much lower standard deviation, and the range of
highest value to lowest value is smaller. The microcomputer that controls the instrument
(16) is capable of displaying a corrected fluorescence value at the rate of 5 Hz, which is

fast enough to accurately characterize continuous flow analysis (CFA) and flow injection

analysis (FIA) peaks.

The measured front surface fluorescence, shown in Figure 3-20a, is the least noisy
of the four traces. The other three traces rely on the transmittance measurement, either
directly, as in the case of the primary absorbance (Figure 3- 19a), or indirectly via a
calculation, as in the case of the estimated secondary absorbance (Figure 3-19b) and the
corrected front surface fluorescence (Figure 3-20b). The end result is that the corrected

front surface fluorescence (Figure 3-20b) is noisier than the measured front surface

63
fluorescence (Figure 3-20a), primarily due to the uncertainty in the transmittance

measurement. This weak link in the system will be addressed in the next chapter.

99°89???“

10.
11.
12.

13.
14.
15.

16.
17.
18.
19.
20.
21.

CHAPTER 3 REFERENCES

Ratzlaff, E,H.; Harfmann, R.G.; Crouch, S.R. Anal Chem. 1984, 56, 342.

Ratzlafi', E.H. Ph.D. Dissertation, Michigan State University, East Lansing, MI,
1982.

The Second Source, Duarte, CA.

Model M303, PRA International, Inc., Oak Ridge, TN.

Model H10, Instruments SA, Metuchen, NJ.

Maxlight Optical Waveguides, Inc., Phoenix, AZ.

Model MC1-02, PTR Optics Corp., Waltham, MA.

Model A83210-M2, North American Philips Control Corp, Cheshire, CT.
Model R1527, Hammamatsu Corp., Middlesex, NJ.

Highlight Fiber Optics, Caldwell, ID.

Model S780-5BO, Hammamatsu Corp., Middlesex, NJ.

Patton, C.J. Ph.D. Dissertation, Michigan State University, East Lansing, MI,
1982.

Model EU42A, Heath Corp., Benton Harbor, MI.
Model 427, Keithley Instruments Inc., Cleveland, OH.

Christmann, D.R., Ph.D. Dissertation, Michigan State University, East Lansing,
MI, 1980.

System 9000, IBM Instruments Inc., Danbury, CT.

LSI-ll, Digital Equipment Corp., Maynard, MA.

Cadillac Plastic and Chemical Co., Detroit, MI.

Ismatec Model IP- 12, Cole-Parmer Instrument Company, Chicago, IL.
Bishop, E., ed. Indicators, Pergamon Press: New York, 1972.
Malincrodt, Inc., Paris, KY.

N
9‘

65
Aldrich Chemical Co., Inc., Milwalkee, WI.
Eastman Kodak Company, Rochester, NY.
Model Lambda 5, Perkin Elmer Corp., Norwalk, CT.
Nedler, J.A.; Mead, R. ComputerJ. 1965, 7, 308.

Johnson, K.J. Numerical Methods in Chemistry, Marcel Dekker, Inc.: New York,
1980.

CHAPTER 4: PERSPECTIVES, CONCLUSIONS, AND SUGGESTIONS FOR
FURTHER STUDY

The instrument described in the previous chapter allows one to correct
fluorescence measurements for inner filter efi'ects in real time. The performance of the
instrument lies somewhere between standard fluorescence Spectrometers that use
mathematical corrections of fluorescence measurements (low end) and the current state
of the art in cell shift technology (1). In this chapter some basic limitations of the
instrument are discussed, then some comparisons of the new instrument to the cell shift
instrument are made. Finally, some suggestions for future work are proposed.

LIMITATIONS OF THE CURRENT INSTRUMENT AND SUGGESTIONS FOR
IMPROVEMENT

One of the major limitations of the instrument thus far appears to be the ability to
measure the primary absorbance. A photodiode was chosen as the transducer of the
transmitted light to simplify the system by not requiring a high voltage power supply, but
a photomultiplier tube (PMT) would have been a better choice. Not only are
photomultipliers much more sensitive to ultraviolet light (UV) than UV enhanced
photodiodes, but most PMT’s have peak sensitivity in the blue region of the spectrum.
Photodiodes Show maximum sensitivity in the red, and thus are quite susceptible to stray
light in the infrared. Due to the low UV sensitivity of the photodiode, a conversion factor
of 1011 V/A was required to measure its current, which made this measurement

extremely susceptible to noise.

Since the front surface/ rear surface fluorometer incorporates a flow cell, the flow
characteristics of the system are of importance. Because Ratzlaff reported that his
instrument had poor wash characteristics (2), the wash time of the instrument described

in this thesis was tested. A solution Of 100 11M OS was passed through the cell, followed

67

by a four-second bubble, followed by 0.1 N HClO4. The time profile of the rear surface
fluorescence measurement after the bubble passed through the cell is shown in Figure 4-
1. Figure 4-1a shows that the half-life of cleaning the cell is roughly 1.25 minutes. Figure
4-1b shows that the measured fluorescence settles to within 20% of the steady-state blank
value of 0.35 nA after about 20 minutes, which would make it difficult to exploit this
instrument for the high throughput methods Of flow injection analysis (FIA) and air
segmented continuous flow analysis (CFA). The wash characteristics of the instrument
could be improved with a cell that is made out of quartz instead of Delrin, and is
surrounded with some black material to satisfy the non-reflective cell wall requirements
of the model described in Chapter 2. Quartz has the added advantage of being more
inert chemically than Delrin.

Unfortunately, use of a quartz flowcell is expected to cause two problems. The
first is that some reflection is bound to occur between the solution/glass interface and the
glass wall/black surface interface. The second is that each end would now consist of a
fiber/air/glass/solution interface rather that a fiber/ solution interface, which would
increase the amount of reflected light, as well as increase the stray light reaching the
PMT’S. The reflections could be minimized by flooding the region between the fiber
bundle and the glass wall of the cell with a liquid with a refractive index nearly equal to

that of quartz.

Although absorption-correction techniques can extend the upper end of the
dynamic range of fluorescence measurements, the lower end is still important; after all,
sensitivity is one of the reasons for choosing fluorescence spectrometry instead of
another method. With a S / N criterion of 2, the detection limit of the front surface/ rear
surface fluorometer was found to be 0.01 11M quinine sulfate (OS) , which is comparable
to what is obtained by the cell shift instrument (3). The greatest source of noise was, as in

Ratzlafi’s instrument, the source flicker noise superimposed on the high blank signal.

PMT ANODE CURRENT. nA

PMTANODECURRENT,IIA

Figure 4-1.

 

 

 

I I I
1 2 5 4

TIME (MINUTES)

(9)

 

TIME (umurss)

Time profile of rear surface fluorescence measured while sample
cell is rinsed with 0.1 N HClO4: (a) 0 to 4 minutes, (b) 0 to 25
minutes.

69

Figure 4-2 shows the front surface fluorescence Spectrum of the 0.1 N HClO4
blank, both with (Figure 4-2a) and without (Figure 4-2b) the bandpass filter in front of
the excitation monochromator and the longpass filter in front of the PMT. Without the
filters, most of the blank signal is due to the tail of the excitation peak at 353 nm. There
does appear to be some additional contribution to the tail near 410 nm, 470 nm, and 540
nm. The peak near 540 nm is probably a reentry peak due to specular reflection of the
scattered 353 nm excitation beam inside the emission monochromator. The peak near
410 nm is probably a Raman band of water. The peak near 470 nm can be seen more
clearly in the spectrum obtained with the filters in place. It is the spectrum of the Xenon
arc lamp that was used for excitation. There does not appear to be any fluorescence

contributing to the blank signal.

Further evidence that the large fluorescence blank signal was not due to adsorbed
OS, as was previously suggested by Ratzlaff, was Obtained by measuring the fluorescence
blank Signal from 1 M NaCl in water and 1 M NaOH in water. Neither blank signal was
lower than that due to the 0.1 H HClO4. NaCl is a strong quencher of OS fluorescence
(4), and the quantum efficiency of OS is low at high pH, so both solutions would be
expected to suppress any contribution from adsorbed OS to the blank Signal. It is still
possible that there was some contribution to the fluorescence blank from the fluorescence
of impurities in the quartz fibers, in the epoxy that holds the fibers together, or in the
Delrin flow cell, but this is considered unlikely. It would be possible to distinguish
fluorescence from stray light if the lifetime of the blank could be determined.

It is not difficult to understand why the blank signal at the rear of the cell should
be so large: the source is pointed directly at it. It is more difficult to understand why the
signal is so large at the front end of the cell. Part of it is undoubtedly due to reflections
from the rear of the cell, but only about 0.2% of the incident light is expected to be

reflected at a quartz/water interface. Thus, the stray light at the front is expected to be at

70

 

0.5 -
(a)

3.
8
8
5
5
O.

0.0 v I I I ' l ' I

400 450 500 550
WAVELENGTH (nm)
0.02 '-
(b)

E
E
8
3
O
5
’é

0.00 -- - . 1 . I . r

 

400 450 500 550
WAVELENGTH (nm)

Figure 4-2. Frorlit surface fluorescence blank: (2) filters removed, (b) filters
in p ace.

71
most 0.002 times that at the rear. This assumption neglects the light reflected by the
epoxy between the fibers. It is also possible that a portion of the stainless steel ferrule
that surrounds the rear fiber bundle is aposed to the excitation beam, which would
dramatically increase the amount Of light that is reflected back to the front from the rear
of the cell. Broken fibers in the common end Of the bundle also contribute to stray light
in the front surface measurement. Light that leaves a broken excitation fiber in the
common leg of the bundle can enter a broken emission fiber. This problem of high blank
signal is both a technological and a theoretical concern. Although it would be possible to
remove more stray light with better monochromators and better filters, and thus reduce
the blank Signal, the blank could never be made as low as that Obtained from a 90°

fluorescence measurement in a cuvette.

The predominant source of noise is the source flicker noise, which manifests itself
both within the blank and the fluorescence Signal Source flicker noise and drift is
normally minimized by dividing the fluorescence signal by the reference Signal, as was
done in these experiments. Examination of the reference, transmittance and fluorescence
signals revealed that the reference Signal could not be correlated with the others over
short time periods, but only over periods of minutes or more. In fact, the reference
channel Signal would Often fall when the fluorescence rose and the opposite also held
true. This could have been due in part to the different time constants of the amplifiers
used, but the major cause is more likely the spatial characteristics of the source flicker
noise from the arc lamp. The reference signal is produced by a photodiode connected to
a single fiber placed in the vicinity of the exit slit Of the excitation monochromator. Thus
only a small part of the lamp image that formed the excitation beam is sampled. In a
typical fluorescence spectrometer a fraction of the entire excitation beam is sampled, so

spatial noise is averaged out by the reference detector (5).

72

Two modifications to improve the quality of reference correction are suggested.
The first is to replace the single reference fiber with a bundle of fibers that are randomly
positioned along the exit slit. The second solution is to split off a fraction of the source
before the entrance slit of the excitation monochromator. This is not normally done
because the lamp power at the excitation wavelength is not always correlated with the
total power of the lamp. However, the chromatic noise of the arc might be less severe
than its spatial noise.
COMPARISON OF THE FRONT SURFACE/REAR SURFACE FLUOROMETER AND

CELL SHIFT INSTRUMENTS

The front surface/ rear surface fluorometer has some advantages over instruments
that use the cell Shift method. The efficiency of the transport and collection of light in
this instrument is better that that Of the cell shift instrument, due to the collimation
requirement of the latter (2). Also, the use of fiber Optics allows greater flexibility of the
optical alignment. For example, the only critical optical axis in the instrument is the
lamp/lens/ excitation monochromator axis; any other component of the system may be
placed in essentially any position. The use of fiber optics results in good immunity to
vibration of the sample cell, which would be important in stopped-flow kinetics
measurements and process control applications. Unlike the cell shift instrument, the new
instrument is compatible with flowing streams and capable of real-time analysis. The
time response of the new instrument for the measurement and display of corrected
fluorescence data is limited by the speed at which the calculations can be made, rather
than by any mechanical movement. If real time display of corrected fluorescence is not

required, the time response of the system is unlimited.

Unfortunately, there are also disadvantages to the front surface/rear surface
fluorometer. The ability of this instrument to correct for inner filter effects is not as good

as the cell-shift instrument. The correction equation for the new system contains an

73
Optical transfer function that was Obtained by curve fitting. Thus, it is not currently
known under what conditions these corrections would be expected to work. As explained
in Chapter 2, the transfer function is expected to change based on changes in the
numerical aperture of the optical system, as well as the refractive index of the solution
being measured. Although studies are currently underway to determine the wavelength
dependence of the transfer function experimentally, and the efi'ect of the refractive index

(6), these are not currently known.

As a general instrument for fluorescence spectroscopy, the new design is less
versatile than the cell shift instrument. Nephelometry (90’ scattering) is not possible, and
turbidirnetric measurements (0’ scattering) would have to be interpreted semiempirically,
as the fluorescence was. Measurements that require polarized light are not possible,
since the multirnode fibers convert linearly polarized light to elliptically polarized light
(7). The cell Shift method provides the operator with two different estimates of the
primary and secondary absorbance, whereas this instrument does not. Differences in the
absorbance estimates indicate that a compound is adsorbing to the cell walls or that the

sample is inhomogeneous (1). Those errors would go undiagnosed with this instrument.
SUGGESTIONS FOR FURTHER STUDY
Modifications of the Current Instrument

In spite of the fact that demand by the communications industry has helped to
lower the price of monochromators, an instrument which uses four monochromators is
unlikely to be considered practical by many people. Furthermore, the dynamic range was
limited on the lower end by stray light in the monochromators, and additional filters were
required to achieve reasonable performance. The use of bandpass filters instead of
monochromators is therefore encouraged. Interference filters, in combination with

shortpass filters (to remove higher orders) would be most useful. The greatest obstacle to

74
the use Of filters would be their wider spectral bandpass, which could cause deviations
from Beer’s law at high absorbances. A study of the effectiveness of absorption
correction with respect to excitation and emission bandpass would prove useful in any
case. If wider bandpasses can be used than were used in these studies, lower detection

limits could be achieved, providing the stray light could be reduced.

The use of a laser as an excitation source would provide many advantages.
Recently, several low cost integrated dye laser systems, which could replace the arc
lamp / excitation monochromator combination have been introduced (8). The bandwidth
of a laser is quite narrow, and its use would eliminate stray light from the excitation
optics. The high power of a laser would lower the detection limit of the instrument, if the
high stray light of the emission optics could be lessened. Laser sources can be focused to
diffraction limited spot diameters, which makes them ideal for introducing light into a
single Optical fiber. Thus the fiber bundles in the current instrument could be replaced
with single fiber/beam splitter combinations. This would allow the construction of much
smaller flow cells. A 600 um fiber would produce a 2.8 11L flowcell; a 200 um fiber would
produce a 0.3 11L flowcell (9). Furthermore, the excitation/emission cone overlap
problem described in Chapter 2 would not exist; the transfer function for a single fiber
system would probably be monotonic. The disadvantage of a single fiber would be that,
since the same fiber would conduct light to the cell and from the cell, reflections at each

fiber face would contribute directly to the stray light.

Continuous wave (CW) lasers, although more expensive than the pulsed lasers
mentioned above, could provide an advantage that would justify the extra expense. With
a CW laser as the excitation source, the powerful technique of phase fluorometry (10)
could be combined with absorption correction. In phase fluorometry, the excitation
source is sinusoidally modulated at a high frequency, typically several tens of megahertz.
Most optical modulators that operate at such high frequencies require a highly collimated

75
light source, which is why a laser is necessary. The fluorescence emission intensity will
then oscillate at the modulation frequency, but the AC portion of the emission will be
phase shifted relative to the excitation beam, and demodulated as well. Either the phase
shift or the demodulation factor can be used to calculate the lifetime of the fluorophore.

One reason that phase fluorometry is preferred over pulsed lifetime methods is
that phase sensitive detection (lock-in amplification) can be used to great advantage. The

phase shift 8 between the excitation and emission beams is given by:
6 = tan'1 (or)

where w is the modulation frequency and r is the lifetime of the fluorophore. Any
contribution to the emission intensity caused by elastic or inelastic scattering appears the
same as a contribution from a fluorophore with an infinitely short lifetime. Thus
scattered light is never phase shifted relative to the excitation beam. It is possible to
modulate the excitation beam at a frequency such that the fluorescence emission is phase
shifted by 45° relative to the excitation beam. Phase sensitive detection can then used to
discriminate fluorescence from inelastic scattering (11) and Raman scattering (12). The
former is especially relevant to this dissertation. It represents a way to minimize the stray
light due to the scattered excitation beam, which was the major cause of the large

fluorescence blank reported earlier in this chapter.

Another advantage phase fluorometry has over pulsed lifetime methods is that it
is possible to determine the lifetime of a fluorophore in less than one second by phase
fluorometry. This has applications in both qualitative and quantitative analysis. For
example, it is possible to determine the lifetime of a fluorescent eluite as it elutes from a
column (13). Knowledge of the lifetime aids in the identification of fluorophores with

similar spectra but different lifetimes.

76

However, the opportunities that rapid lifetime determinations Offer for
quantitative analysis are greater. By dividing the steady-state fluorescence emission
intensity by the lifetime of the fluorophore, one obtains a quantity that is independent of
the quantum efficiency, and thus independent of the concentration of any species that
quenches the fluorophore (14,15). If this Operation were combined with the primary and
secondary absorption correction scheme presented in this dissertation, it would be
possible, on a one-second time scale, to make fluorescence measurements that are

independent of the concentration of any absorbers or quenchers.

The final suggestion for modification of the current instrument is the use of an
inert, white flowcell. Although a new model might be required, a white flowcell would
increase the Optical throughput in somewhat the same way that mirrored cuvettes do
when they are used with a conventional fluorometer (16). Due to difficulty in its
machining, Teflon would probably be a poor choice; a better one would be ultrahigh
molecular weight polyethylene, which is nearly as chemically inert (17).

Other Experiments in Absorption-Corrected Fluorescence

The instrument described in this dissertation was designed to satisfy the
requirements of flowing stream compatibility and real time data acquisition. If the
second requirement is relaxed, there are some modifications that could simplify the
instrument and correction algorithm. One such modification was suggested earlier: the
excitation and transmittance monochromators could be slewed rapidly between the
«citation and emission wavelengths, while the front surface fluorescence PMT is
protected by a shutter when the «citation monochromator is set to the emission
wavelength. The advantage to this change would be the increased accuracy, relative to
the present method, with which the secondary absorbance value would be known. The

uncertainty in the value Of the secondary absorbance would not depend on the value of

77
the primary absorbance. Furthermore, the second fluorescence channel would not be
required. The disadvantages Of this modification would include the sacrifice of real-time
performance, the imprecision of the wavelength setting, and the additional need for
slewing motors for the two monochromators. Another way to measure the secondary
absorbance accurately would be to add a second flow cell, either sequential or parallel,

with its own continuum source (ie. a tungsten bulb), monochromator, and detector.

All of the techniques mentioned thus far have been based on three separate
measurements - either two fluorescence and one absorbance or two absorbance and one
fluorescence - because three values are being determined: corrected fluorescence,
primary absorbance and secondary absorbance. Recall that in the equation used to
correct for inner filter effects, the 2 absorbance values are added together everywhere
they appear. In other words, all that is needed is the total absorbance (primary 4-
secondary) in order to correct for inner filter effects. If the individual absorbance values
are not required, two other configurations can be suggested. In the first, two flow cells of
different lengths would be used. If the sum of the absorbances were small, the signal
from the long flow cell would be greater than that from the Short flow cell, because the
the «citation beam would reach the end of the cell, and the light emitted from there
would reach the front Of the cell without attenuation. In the case of high primary or
secondary absorbance, or both, the signals from the two cells would be similar, because
the solution at the rear of the long cell would contribute little to the measured front
surface fluorescence signal. The ratio Of the two signals would be indicative of the total
absorbance. Figure 4-3 shows how the ratio of the two signals is expected to change as a
function of the total absorbance based on the model used in the «perirnental section.
Unlike the ratio Of front / rear fluorescence, the ratio of the long cell fluorescence to the

short cell fluorescence can be «pressed analytically and solved «actly. The stray light

78

 

 

 

2.5 -
O -
p 2.0
E
1.1.!
o I
5
Q
In
Iii
g 1.5 -
1:!
1.0 -
I . I v I ' I ' I
0.0 1.0 2.0 5.0 4.0
TOTAL ABSORBANCE
Figure 4-3. Calculated ratio of front surface fluorescence measurement from 2

cm flow cell to front surface fluorescence measurement from 0.5
cm flow cell.

79
problem would be expected to be less severe with this instrument as there is no

solution / glass interface directly Opposite the «citation window.

The method proposed above relies on one value of the ratio of the signal
measured with two different pathlengths. If the ratio of signals from many cells of
different lengths could be obtained, a more reliable corrected fluorescence measurement
would result. One way to achieve this would be with a single channel, front surface
fluorescence instrument. A variable pathlength would be Obtained by flowing a bubble
through the flow cell, or a segment of some liquid which is immiscible with the sample
solution. The liquid would also be required to have a much different refractive ind«
than the sample. As the bubble or liquid entered the flow cell, the solution to be
measured would follow it, resulting in a time-dependent sample pathlength. The
interface between the bubble or the liquid and the sample would effectively function as
the rear of the cell, because the difference in the refractive indices would guarantee that
light would be scattered once it reached that point, and the curved meniscus between the
two solutions would mean that most of the radiation would scatter to the walls, rather
than back into the solution. Figure 4-4a illustrates how this concept could be
implemented. The front surface fluorescence signals expected for two solutions that
produce the same signal when they fill the flow cell completely are shown in Figure 4-4b.
These signals would be indistinguishable by a single, steady-state measurement. Note
that the rise time of the sample with a high total absorbance is less than the rise time Of
the sample with low absorbance. It might be possible to derive an equation that relates
the rise time of the front surface fluorescence signal to the total absorbance. Of course,
with this method, real-time response is sacrificed, and it would only be compatible with
sample streams that could be segmented by gas or liquid. The advantage is that the
system would require only one source/ detector and would thus be very simple to

implement. It is expected that a low noise syringe pump would be required, because with

80

f
in IOW out (a)
excitation ‘ I

    

‘—\

 

 

bubble
bifurcated
fiber
optic

front surface emission bundle

1.0 -
g 0.0 -
‘8?
E. 0.0 -
0
iii
8 (b)
d 0.4 -
9
ii
8 0.2 -

0.0 '-

I ' I T l U I U I u I
0.0 0.2 0.4 0.0 0.0 1 .0
LENGTH or SAMPLE SEGMENT (cm)
Figure 4-4. Flowing bubble tem: (a) diagram showing time- ' g

pathlength, (b) culated front surface fluorescence Si 1 =
total absorbance negligible, 2 -= total absorbance of 2.0).21131

81

a peristaltic pump the position of the interface between the sample and the bubble would

oscillate.

When this project was initiated most researchers using fiber optics for in situ
fluorescence measurements used bundles, like the ones that were used in the project.
Early workers found that, because of the Often complicated nature Of the sample matrix,
the best one could hope to do was to report changes in fluorescence relative to some
initial value (18). In many cases it was not clear whether changes were due to changes in
the concentration of the fluorophore or changes in the concentration of some absorber in
the matrix. Most recent in situ measurements have involved the use of a single fiber for
the combined «citation and collection of emission (19,20). The smaller diameter Of a
single fiber allows it to reach places inaccessible tO a bundle; a single fiber is also much
less expensive than a bundle. In measurements made in vivo, the matrix has been Shown
to cause significant inner filter errors. In «perirnents involving the in vivo monitoring Of
NADH, an equation was derived that allows absorption correction based on the
measurements of reflectance from surrounding tissue (20). With other in situ

measurements, however, no surface esdsts for this type of correction (21).

The switch to single fibers has made it possible to suggest a way of correcting in
situ measurements for inner filter errors. In this case, as in any other, the most reliable
method is to make two or more measurements with different pathlengths of «citation
and emission. This could be done by forming lenses on the ends of the fibers that are
placed in situ (22,23). Two measurements could be made: one with a fiber with short
focal length and one with a fiber with along focal length. The latter measurement would
be affected more by absorption by the concommitants in the matrix than the former. The
ratio of the two signals could be related to the sum of the primary and secondary

absorbance as it was with the two pathlength system described above.

10.

11.
12.
13.

14.
15.
16.
17.
18.
19.
20.

21.

82
CHAPTER 4 REFERENCES

Adamsons, K.; Sell, J.E.; Holland, J .F.; Timnick, A. Am Lab. 1984, 16, 16.
Ratzlaff, E.H.; Harfmann, R.G.; Crouch, S.R. Anal Chem. 1984, 56, 342.

Christmann, D.R. Ph.D. Dissertation, Michigan State University, East Lansing,
MI, 1980.

Miller, J .N., ed., Standards in Fluorescence Spectrometry, Chapman and Hall:
New York, 1981.

Model LS-5 Operator’s Manual, Perkin Elmer Corp., Danbury, CT.
Victor, M.A.; Ph.D. research in progress, Michigan State University, 1987.

Culshaw, B. Optical Fibre Sensing and Signal Processing, Peter Peregrinus, Ltd.:
New York, 1984.

Laser Science Inc., Cambridge, MA
Vurek, G.G. Anal Chem 1982, 54, 840.

Lakowicz, J.R. Principles of Fluorescence Spectroscopy, Plenum Press: New York,
1983.

Nithipatikom, K.; McGown, L.B. Appl Spectrosc. 1987, 41, 1080.
Demas, J.N.; Keller, R.A. Anal Chem 1985, 57, 538.

Brushwyler, K.R.; Bright, F.V.; Hieftje, G.M. Abstract #104, FACSS XIV,
Detroit, MI, 1987.

Hieftje, G.M.; Haugen, G.R. Anal Chim. Acta 1981, 123, 255.

Demas, J .N.; Jones, W.M.; Keller, R.A. Anal Chem 1986, 58, 538.

Street, K.W.; Singh, A. Anal Lett. 1985, 18, 529.

Cadillac Plastic and Chemical Co., Detroit, MI.

Chance, B.; Legallais, V.; Sarge, J .; Graham, N. Anal Biochem 1975, 66, 498.
Sepaniak, MJ.; Tromberg, B.J., Eastham, J .F. Clin. Chem 1983, 29, 1678.

Renault, G.; Raynal, E.; Sinet, M.; Muffat-Joly, M.; Berthier, J.; Cornillault, J .;
Pocidalo, J. Adv. Exp. Med Bio., 1986, 121, 229.

Chudyk, W.A.; Carrabba, M.M.; Kenny, J.E. Anal Chem 1985, 57, 1237.

83
22. Lee, KA; Barnes, Es. Appl Opt. 1985, 24, 3134.
23. Russo, V.; Righini, G.C.; Sottini, S.; Triganri, S. Appl Opt. 1984, 23, 3277.

CHAPTER 5: INTRODUCTION TO DIGITAL FILTERS

Most «perirnental scientists are somewhat familiar with the process of filtering
analog electrical signals. At the very least, one learns how to set the rise time on a
preamplifier or chart recorder to be short enough so as not to distort the signal, and long
enough to reject high frequency noise. However, many scientists are not aware that any
filtering Operation that can be done by analog circuits can be done by arithmetic
operations on a digitized version of the signal. The arithmetic versions, which are known
as digital filters, have certain advantages compared to the widely used analog filters.
Conversely, many scientists are not aware that some of the arithmetic operations they
perform on data, such as smoothing and averaging, are filtering Operations. In many
cases, realizing when an arithmetic Operation is a filtering process allows one to achieve
better results more quickly than would have been obtained through many hours of trial

and error.

In this chapter some of the general characteristics of filters are discussed. These
concepts will shed some light on the techniques used to analyze digital filters, which are
«plained in Chapter 6, and those used to design digital filters, which are «plained in
Chapter 7. This chapter is concluded by a survey of the use of digital filters in analytical

chemistry.
ANALOG FILTERS

Digital filters have much in common with analog filters. An analog filter can be
characterized by its impulse response, which is the output of the filter when an infinitely

narrow pulse, known as the delta function, is applied to its input. From the impulse

85
response, the output of the filter for any input waveform can be deduced. The output is

given by the convolution of the impulse response and the input waveform:

y(t)=fh(7)x(t-T)d7 (5-1)
where x(t) is the input, y(t) is the output, and h(r) is the impulse response of the. filter.

The FOurier transform of the impulse response of a filter is given by :

X(6I)= :1:(t)e‘""dt

where i is (-1)1/2. In this dissertation, i will be used rather than the electrical engineering
convention j. X(w) is called the transfer function of the filter, and is a function of
frequency. The transfer function is generally more useful than the impulse response for
characterizing a filter . The reason for this stems from one of the theorems of Fourier
transforms, which states that convolution in the time domain is equivalent to
multiplication in the frequency domain (1). In other words, the Fourier transform of the
output of a filter is given by the Fourier transform of the input times the transfer function
of the filter. Since this multiplication process is much easier to visualize than the
convolution process, filters are usually characterized by their frequency domain
characteristics. The transfer function is usually converted from its compl« form, A(x) +
iB(x), to the magnitude/phase representation, where the magnitude, or gain is given by
[A2(x) +Bz(x) ]1/2, and the phase is given by tan“1 [B(x) / A(x)]. In this form, the transfer
function is Often called the frequency response of the filter. The magnitude part of the
frequency response is the gain the filter has for a sinusoid of a given frequency. The
phase part Of the frequency response is the amount by which the phase of a sinusoid of a
given frequency is shifted upon passing through the filter.

86
DIGITAL FILTERS

All of the characteristics of analog filters listed above also pertain to digital filters,
but in a slightly different form. Digital filters are characterized by an impulse response.
The effect they have on any signal can be predicted by the discrete convolution function:

ya: 2 ckx‘n-k (5'2)
kz-aa

where the x" values are the input data, the y" values are the output data, and the ck values

constitute the impulse response of the filter; that is, the output when the input is:

xn=1,n=0
xn=0,n+0

The impulse response of a digital filter is synonymous with the terms convolution fitnction
and filter coeflicients. This is because Equation 5-2 is more than a way of predicting what
the output of the filter will be; it is also the way the output is Obtained. Unlike analog
filtering, which requires the use Of discrete circuit elements such as amplifiers, capacitors,
resistors and inductors to achieve the convolution described in Equation 5-1, digital
filtering only requires that one do the calculation of Equation 5-2 with digital multipliers

and accumulators.

There is also a discrete version of the Fourier transform that holds for a finite
number Of data points N. It is given by:
N-t .
n=0
As with analog filters, the Fourier transform of the impulse response of a digital filter
gives its transfer function. Unfortunately it is in a discrete form, like the impulse
response itself. In Chapter 6 a method for determining the form of the continuous

transfer function of a digital filter is given.

87
Note that the discrete Fourier transform only involves a summation from 0 to N-l
rather than from -00 to co. This is one Of the many results of the Nyquist theorem, also
known as the sampling theorem. This theorem states that a continuous signal which has a
Fourier transform that is nonzero above fs / 2 is not accurately represented if digitized
and sampled at a frequency of f8. Any components of the signal at frequencies greater
than fs / 2 are folded back and interpreted as having a frequency between 0 and f8 / 2.

The sampling theorem is difficult to «plain but easy to demonstrate. Figure 5-1a
shows what happens when three sinusoids of frequency 10, 30 and 50 Hz are sampled at a
rate of 40 Hz. Note that all three sine waves take on the same value at each sampling
point. Only the 10 Hz sine wave has been sampled correctly. Figure 5-1b Shows how 30
Hz and 50 Hz fold back into the region between 0 and 20 Hz, and appear
indistinguishable from 10 Hz. The frequency fs / 2 is called the Nyquist frequency or
folding fiequency. Where analog and digital filter theory differ, the difference is most

likely due to some consequence of the sampling theorem.

It should be noted that Equation 5-2 is a Simplified version of the general digital
filtering formula:

yn=k=2mckxn-k "I' E djy‘n-j

j=-ea
If d]- = 0 for all j, as in Equation 5-2, the filter is called a nonrecursive filter. If one or
more d j are not equal to zero, the filter is called a recursive filter. In this chapter and

those that follow only nonrecursive filters will be considered.
LINEARITY AND TIME INVARIAN CE

The relationships between the impulse response of a filter, its Fourier transform,

and the Fourier transforms of the input and output signals only hold for linear, time-

 
VIII I

 

 

 

 

 

 

e l (b)
l 1..
50 Hz
._ I
50 Hz
l
I _ T _ I
DO 10 Hz 20 Hz

Figure 5-1. Illustration of the Nyquist theorem.

89
invariant (LTI) systems. In Order for a system to be an LTI system, it must obey the
following three criteria:

1. If inputA results in output A’, and input B results in output B’, inputA + B must
result in output A’ + B'.

2. If inputA results in outputA’, input c-A must result in output c-A', where c is a
constant.

3. If inputA(t) results in outputA’(t), inputA(t + s) must result in outputA'(t + s),
where s is a constant.

An analog filter is a LTI system when the input is kept within a range of
amplitudes and frequencies. A digital filter is an LTI system if the filter coefficients do
not change with time. If they do, the filter is no longer an LTI system, and the results of
filtering can vary in unpredictable ways, even when similar signals are filtered. When a
filter is not linear or time-invariant, it must be studied by observation of its output when
the signal of interest is the input. Chapter 6 contains a study of a particular non-linear,

time-variant filter.

LTI systems are capable of performing the common filtering Operations of low-
pass, high-pass, bandpass and bandstop (notch) filtering. They are also capable of
performing differentiation and integration, which are LTI Operations. It was noted
earlier that when a sinusoidal signal passes through a filter, a sinusoid of the same
frequency emerges, but in general the amplitude and phase of the output is not the same
as the input. This is one consequence of the fact that sinusoids are the eigenfunctions Of

filter operations. They are also the eigenfunctions Of differentiation:
d/dt eiw‘ - be“

where eiw‘ 8 cos wt 4» Sin at (the Euler relation). The output is simply the input
multiplied by a constant. Therefore, a differentiator is nothing more than a filter with a

transfer function of i0. Since the transfer function rises linearly with frequency, a

90
differentiator is a high-pass filter. This explains the common observation that
differentiation degrades the signal to noise (S/N) ratio of most signals; the gain of a

differentiator is highest in the region of the spectrum where the S /N ratio is often lowest.

Sinusoids are also the eigenfunctions of integration:

f eimt 61:48“

W

An integrator is a filter with a transfer function of 1 / iw, which makes it a low-pass filter.
Integration commonly increases the S/N ratio of a signal. Because an integrator has

infinite gain at w = 0, it cannot be implemented by a non-recursive digital filter.
ADVANTAGES OF DIGITAL FILTERS

Although digital and analog filters have much in common, there are many
advantages to the use of digital filters. One of the greatest is that a digital filter need not
be causal. A causal filter is one that has an impulse response that is zero for r > 0. In
other words, a causal filter does not respond to the delta function until the pulse actually
reaches the input terminal. All electronic filters are causal; they are sometimes called

realizable filters.

In contrast, digital filters are often used on stored, digitized versions of a Signal.
In these cases the value of the signal for future time is available; that is, the value of It”)
, x“) , etc. can be used in filtering x". Digital filters are usually designed with
coefi‘icients that are symmetric about k a 0. These symmetric filters have the important
property that the phase shift is zero throughout the entire frequency range. All electronic

filters cause a frequency-dependent phase shift, and are therefore more likely to distort a

signal.

91

Electronic filters must be implemented with electronic components such as
resistors, capacitors, inductors and operational amplifiers. To change the frequency
response of an electronic filter, one must change the values of these components.
Furthermore, the values of these components can change from the nominal values due to
changes in temperature and humidity, aging, and other factors. The frequency response
of a digital filter is determined by its coefficients. Changing the frequency response of a
digital filter merely requires a change in the coefficients, which can be done by a
computer program. Furthermore, the. coefficients, and thus the frequency response, are

independent of changes in the environment.

Perhaps the greatest advantage digital filters ofl'er the «perirnental scientist is
that they operate on a copy of the original signal rather than the signal itself. If a Signal is
judged to have been filtered incorrectly by a digital filter it may be filtered again and
again until the desired result is achieved. If a signal is filtered incorrectly by an analog
filter, the «perirnent must be repeated and the signal regenerated.

One should not get the impression that analog electronic filters are no longer
required. At the very least an antialiasing filter is needed immediately before the
digitizer. An antialiasing filter is a low-pass filter with a cutoff frequency near fs / 2 that
prevents the aliasing phenomenon described earlier in this chapter. Furthermore, low-
pass filters are sometimes needed to keep the analog signal within the dynamic range of
the digitizer. However, most other filtering operations that were once done by analog

filters can now be done by digital filters, with better results and more versatility.

92
A BRIEF HISTORY OF CONVOLUTION FILTERS IN ANALYTICAL CHEMISTRY

Savitzky-Golay Filters

The history of the use of digital filters in analytical chemistry begins with the work
Of Savitzky and Golay (2) in 1962. They published several tables of filter coefficients,
which now bear their names, that can be used to smooth (low-pass filter) data or calculate
the first through fourth derivative of a data set. The coefficients are derived on the
assumption that the data can be described by a polynomial Of a certain degree. Savitzky
and Golay showed that convolution of the data with the coefficients is equivalent to fitting
a polynomial to a range of points, and replacing the center point of the range with the
calculated value of the polynomial at that point. Although mathematically equivalent to

curve fitting, the convolution process is much simpler and faster.

Unfortunately, several Of the coefficients they published are in error; these were
later corrected by Steiner et al (3). Further corrections were given by Madden (4), who
also provided an equation that allows one to calculate the coefi‘icients of filters with

widths greater than 25, which is the limit of the tables published by Savitzky and Golay.

Within a few years there were hundreds of reported applications of Savitzky-
Golay (SG) filters. Out of concern that the filters were used improperly in some cases,
Willson and Edwards (5) and Enke and Nieman (6) thoroughly studied the effects of SG
filters on the signal-to-noise (S/N) ratio of Gaussian and Lorentzian peaks, and on the
distortion SG filters cause to these peaks. The former is an «cellent tutorial on virtually
all aspects Of the handling of digitized data, and is recommended to anyone with an
interest in signal processing. The concept of the smoothing ratio was introduced in the
latter article. The smoothing ratio is the ratio of the full width at half maximum
(FWHM) of the convolution function to the FWHM of the spectral peak. Although the

recommendations of the two groups sometimes differ - Enke and Nieman recommend a

93
smoothing ratio of 2 for maximum S/N enhancement, whereas Willson and Edwards
recommend a ratio of 1.2 - they are still widely quoted and heeded by those who use 56
filters. Both of these studies were confined to smoothing filters; a similar «amination of

derivative filters was done by O’Haver and Begley (7).

The studies described above were all empirical, in that S/N ratios and degrees of
distortion were determined by the actual use of $6 filters on calculated peak profiles,
both with and without added noise. Recent work has focused on the effects that
convolution has on the equations that describe these peak profiles, and this work has
produced results that are more generally useful (8-12). Gans and Gill (8) derived an
«pression that shows that the sum of the squares of the residuals (SSR) measured
between a filtered data set and its unfiltered version is composed of a signal distortion
term and a noise term. They also showed «perirnentally that a plot of the SSR versus the
number Of coefficients in an SG filter has an inflection point at some number of
coefficients N,- , where the greatest contributor to the SSR ceases to be the noise and
becomes the distortion. They propose that the distortion caused by the filter is
minimized, and the noise removal is maximized, by an SG filter Of N,- points.

Bromba and Ziegler (9) reported some of the properties «hibited by SG filters.
These properties include, for an SG filter of order m, conservation of all polynomials of
degree m +1 or less and conservation of the first m +1 moments of the data set. Based on
these properties, they questioned the experimental results of Enke and Nieman, who
found that the area (zeroth moment) of a Gaussian peak is changed by some SG filters.
They reiterated that the factor by which the variance of white noise is reduced by any
nonrecursive filter is given by the sum of the squares of the coefficients of that filter
regardless of the distribution of the noise, which once again contradicts the «perirnental
results of Enke and Nieman. They also stated that, for SG smoothing filters, the noise

reduction factor is also given by the square of the zeroth filter coefficient. Gans and Gill

94
(8) showed that the noise reduction factor is given by the square of the nth coefficient for
a nth derivative filter. The paper by Bromba and Ziegler also contains the frequency
response plots of some SG filters, and the authors point out that the tangency of the gain
curve at zero frequency is of order n for an ntll order SG filter. A more comprehensive

collection of frequency response plots was published by Betty and Horlick (24).

Ziegler (10) compared SG filters to analog RC filters. He recommended that
those who practice slow-scan spectrometry replace their analog RC filters with digital SG
filters. This paper also contains several useful plots of the calculated attenuation of
Gaussian and Lorentzian peaks, and the increase in the linewidths Of these peaks, as a
function of the rise time (similar to Enke and Nieman’s smoothing ratio) of SG filters of

orders zero through six.

Bromba and Ziegler (11) provided a mathematical definition of a typical spectral
lineshape. This definition encompasses Gaussian and Lorentzian peaks, as well as many
Others. They tabulated the time domain and frequency domain characteristics that a
smoothing filter must have in order to conserve the symmetry, positivicity, position,
height, width, and range of values (maximum - minimum) of a spectrometric lineshape.

They also showed which classes of filters have one or more of these characteristics.

Perhaps the most impressive summary of filter performance to date was done by
de Blasi et al (12). They showed that the deformation caused by a filter can be estimated
at any point in a signal. The higher order terms of a Taylor series «pansion of the signal
are calculated; these are multiplied by the corresponding moments of a continuous time
version of the impulse response of the filter to yield the deformation. As one might guess
from this description of the method, it is complicated and time consuming, but powerful.
This paper also includes a plot that gives the amount by which a Cauchy - Lorenz peak is

deformed by linear and quadratic SG filters of various widths.

95
Other Symmetric Filters

Bromba and Ziegler presented a recursive implementation of 86 filters (13).
They later found that these filters were sometimes unstable due tO roundoff errors, a
common problem with recursive filters, and derived a new class of filters that can be
implemented recursively with little susceptibility to roundoff errors (14). The authors
also «tended the filter design to include filters that are capable of resolution
enhancement (15). Biermann and Ziegler (16) published plots that allow one to select
the parameters of the filter based on the desired height, S/N ratio, and resolution of the

output.

Gianelli and Altamura showed that SG filters can be replaced by rectangular
weighing functions in which the coefficients take on only two nonzero values (17). These
filters can be implemented with only two multiplications and n additions to achieve the
same effect as an SG filter with a width of n. They also showed, through the analysis
method of de Blasi et al, that these simpler filters cause only slightly more distortion than

an Optimum filter (18).

Kaiser and Reed (19) published equations that allow one to design low-pass and
derivative filters. These filters are similar to those described in Chapter 7 of this
dissertation. Three types Of filters are described: those with ripples in both the passband
and the stopband, those with ripples only in the stopband, and monotone filters with no
ripples in the frequency response. For the first two types, the designer specifies the
desired cutoff frequency and the maximum allowable magnitude of the ripple. For the
third type the designer specifies the cutoff frequency and the width of the transition from
the passband to the stopband. In all three cases, the specification determines the number
of coefficients in the filter. Marchand and Mamet have discussed the advantages of the

monotone filters for some applications (20).

96
Asymmetric Filters

In all the work reviewed thus far, the coefi'icients of the filter have been
symmetric. A symmetric filter of n points cannot be used on the first or last (n / 2) - 1
points in a data set, so these points are often discarded, unless one is willing to decrease
the number of coefiicients in the filter that is used at the extremes (6,21). In an attempt
to work around this disadvantage, Proctor and Sherwood (22) and Leach et al (23)
designed SG filters in which the polynomial that is fit to a region of the data is used to
estimate any point within the region, rather than just the center point. These filters are

discussed in greater detail in Chapter 6.
CONCLUSIONS

From the literature cited above, it is obvious that an understanding of the
frequency response of a filter is necessary if the filter is to be applied properly. However,
the literature contains no «planation of how one Obtains the frequency response of an
arbitrary nonrecursive filter. New filter designs are Often reported without disclosure of
their frequency domain characteristics. Furthermore, most of the work on filters in
analytical chemistry has involved studies of the effects of S6 filters on spectral lineshapes.
There is little mention of how one would design a filter to remove correlated noise or do

any of the other operations filters might be called upon to do besides low-pass filtering.

The engineering literature contains a vast amount of information about the design
and analysis Of digital filters, but much of it is difficult to understand without an electrical
engineering background. The n«t two chapters contain an «planation of how to
calculate and interpret the frequency domain characteristics of nonrecursive digital

filters, and how to design a filter that matches a given frequency domain specification.

109399???)

hit-5H
Ni"?

13.
14.
15.
16.
17.
18.
19.
20.
21.

97

CHAPTER 5 REFERENCES

Bracewell, R. The Fourier TIonsfonn and itsApplications, McGraw-Hill: New
York, 1965.

Savitsky, A.; Golay, MJ.E. Anal Grern. 1964, 36, 1627.

Steiner, J .; Termonia, Y.; Deltour, J. Anal Chem 1972, 44, 1906.
Madden, H.H. Anal Chem 1978, 50, 1383.

Willson, P.D.; Edwards, T.H. Appl Spectroscop. Rev. 1976, 12, 1.
Enke, C.G.; Nieman, T.A. Anal Chem 1976, 48, 705A.

O’Haver, T.C.; Begley, T. Anal Chem 1981, 53, 1876.

Gans, P.; Gill, J .B. Appl Spectrosc. 1983, 37, 515.

Bromba, M.U.A.; Ziegler, H. Anal Chem 1981, 53, 1583.
Ziegler, H. Appl Spectrosc. 1981, 58, 687.

Bromba, M.UA.; Ziegler, H. Anal Chem 1983, 55, 648.

de Blasi, M.; Gianelli, G.; Papoff, P.; Rotunno, T. Ann. Chim (Rome) 1975, 65,
183.

Bromba, M.U.A.; Ziegler, H. Anal Chem 1979, 51, 1760.
Bromba, M.U.A.; Ziegler, H. Anal Chem 1983, 55, 1299.
Bromba, M.U.A.; Ziegler, H. Anal Chem 1984, 56, 2052.
Biermann, G; Ziegler, H. Anal Chem 1986, 58, 536.
Gianelli, G.; Altamura, 0. Rev. Sci. Instrum 1976, 47, 27.
Gianelli, G.; Altamura, 0. Rev. Sci. Instrum 1976, 47, 32.
Kaiser, J .F.; Reed, W.A. Rev. Sci. Instrum 1977, 48, 1447.
Marchand, P.; Marmet, L Rev. Sci. Instrum 1983, 54, 1034.
Khan, A. Anal Chem 1987, 59, 654.

Proctor, A.; Sherwood, P.M.A. Anal Chem 1980, 52, 2315.

98
23. Leach, R.A.; Carter, C.A.; Harris, J.M. Anal Chem 1984, 56, 2304.
24. Betty, K.; Horlick, G. Anal Chem 1977, 49, 351.

CHAPTER 6 : FREQUENCY RESPONSES OF DIGITAL FILTERS

The use of digital filters, including Savitzky-Golay (SG) filters, in analytical
chemistry has largely been a matter of trial and error. This is due in part to the inability
of the analyst to specify the desired result mathematically, and in part to a lack of
understanding of the fundamentals of digital filters. A filter is normally characterized by
its frequency response. The frequency response of a filter is a measure of the gain of the
filter for a sinusoid of a given frequency, and the degree to which the phase of the
sinusoid is shifted. A filter can be characterized further by its noise equivalent
bandwidth, which is a measure of the change in the variance of uncorrelated (white) noise
that is input to the filter. These concepts apply to digital filters as well as to analog filters.
In this chapter, techniques for the analysis of digital filters are «plained; appropriate

applications are illustrated by some examples from research at Michigan State University.

As «plained in Chapter 5, convolution with a time-invariant convolution function

is a linear, time-invariant (LTI) operation. The equation for this Operation is:
00
yn=k2mckxn-k (6-1)
where the ck values constitute the convolution function. In order to determine the
frequency response of a filter, we need to perform this convolution on the function xn =
em”, which is the eigenfunction of all LTI systems. The result is:
y“: 2 ch et'o (ft-k) - = 810112 eke-50k
k=- k=-°°

which demonstrates that cm" is indeed the eigenfunction of a digital filter. The transfer

function of the filter is obtained by dividing the output by the input eiw", and is given by:

99

100

2 eke-10k = 2 ck(coskt.I-isinko) : A+Bi (6-2)
kz—ss 53"“

This is the polar form of the transfer function. It is converted to the frequency response

through the following transformation:

' k
Gain: r——A2+B2 = V (kg-“chem k0) + (kg-”chain or)

 

kz-ae

Phase=arctan(B/A)=arctan (——.-,

 

i 0,, sin lco ) (6'3)

cpcos km

in...
When a filter has symmetric coefficients (ck - -ck), the sine terms in each summation
above equal zero, since the sine is an Odd function. In these cases, which include the
derivative SG filters of even order (smoothing, 2nd derivative, 4th derivative, etc.), the
equations are:

Gain: .4 = 2 01,008 km

b..-” (6-4)

Phase=11rct.an(0/A)=0o
The fact that the symmetric filters cause zero phase Shift throughout the spectrum is
important. It results in the characteristic that symmetric filters will not cause a shift in

the location of the maximum value of a Single symmetric peak.

When a filter has antisymmetric coefficients (ck -= -c_k), the cosine terms in
equation 6-3 sum to equal zero, since the cosine is an even function. The equations for

the frequency response are then:

101

Gain=B=2 ck sin kw
k=-¢5
, (6-5)
Phase=arctan(B/0)=90
These equations apply to odd-order-derivative Savitzky-Golay filters (1St derivative, 3"d

derivative, etc.).

The frequency responses of filters that calculate the first or higher derivatives can
be normalized to make them easier to interpret. The gain calculated by equation 6-4 or
6-5 is divided by the true value of the derivative, which for the nth derivative is (iw)".
Where the normalized frequency response is one, the derivative is accurate. Where it is
greater than one or less than one, the derivative is overestimated or underestimated,
respectively. Note that the normalized frequency response is undefined at w=0; it is

normally set to one.

The other way a digital filter can be characterized is by its noise equivalent

bandwidth (NEB). The variance of a signal is related to its power spectral density P(w)
by:

02: gluon.) (6.6)

where the power spectral density (PSD) is the square of the transfer function.
Furthermore, the power spectral density of the output of the filter P'(w) is related to the

power spectral density of the input P(w), by:
P.(or=|ri(n)|2 10(6) (6'7)

where H ( or) is the transfer function of the filter. Combining equations 6-6 and 6-7 we

obtain:

102

062 = zirtfP (0M0: 21'le(61)|2 PM do) (6-8)

For white noise the power spectral density is constant, so it can be placed outside
the integral. Furthermore, we can substitute equation 6-2 into equation 6-8 as the

transfer function to get:

 

—:°2—- 2117 f 0,,(008 Ice-i sinktd)

k:-c
The square of the transfer function, which is what remains inside the integral, is
composed of three kinds of terms: cos mw cos no, sin mw sin no and cos no) sin no. The
only terms that evaluate to a nonzero integral over the interval (1r, -1r) are the terms of the

form cos2 kw and sin2 kw. Thus,

~11 k=-oo
and
2 co
0 2
T00 = 2 Cr: (6-9)

Therefore, the variance of white noise that is filtered by a digital filter is reduced by a

factor equal to the sum of the squares of the coefficients of the filter.

These two ways of characterizing a filter are in a sense complementary. The
frequency response of a filter helps one to determine if its use will severely distort the
underlying waveform of a signal, be it a spectral peak, an exponential decay, or any other
waveform. If the gain of the filter is low in a region of the spectrum where the power
spectral density of the waveform is high, distortion is likely to result. However, the

distortion of a waveform does not always destroy the information that the waveform

103
contains. For «ample, it was stated earlier that a symmetric filter, no matter what its
frequency response, will not shift the location of the center of a Single symmetric peak.
The frequency response also indicates if the filter will be able to attenuate correlated
noise, which is usually confined to a narrow region of the spectrum.

However, the frequency response does not completely characterize a filter. For
«ample, if one wishes to avoid distorting a waveform, one could filter it with a digital
filter that has a gain of one at all frequencies. While this would indeed avoid distortion, it
would not produce the desired efl'ect of filtering. This filter, known as an all-pass filter,

has coefficients given by:
Ck = 1, k = 0
ck -= 0, k 11 0

which results in a noise amplification of 1.0 as calculated via equation 6-9. Thus, this
filter does not distort the waveform, nor does it remove noise; in fact, it does nothing, as
one would «pect. Although this an extreme «ample, it illustrates that tradeoff between
noise reduction and distortion that must be faced by anyone who uses a filter. Greater
noise reduction (smaller NEB) can only be achieved through reduction of the integral of
the transfer function, which is achieved by lowering the cutoff frequency of the filter,

which results in greater distortion of the underlying waveform.
EXAMPLES
Savitzky-Golay (SG) Filters

Savitzky-Golay filters are symmetric or antisymmetric, depending on the order of
the derivative being calculated, so that equations 64 (even order derivative) and 6-5 (odd

order derivative) can be used to calculate the frequency response. Aside from the order

104
of the derivative, there are three other variables that can be specified to produce the
desired frequency response: the order of the polynomial that is used to fit the data
(linear, quadratic, cubic, etc.), the number of coefficients in the filter, and the number of
passes Of the data through the filter. The effect that each of these variables has on the

frequency response will be «amined below.

Figure 6-1 shows the frequency responses of linear, quadratic and quartic 1 1 point
SG filters. It is Obvious that increasing the order of the polynomial causes the cutoff
frequency to move to higher frequencies. Higher order filters also «hibit greater
tangency at w=0. Figure 6-2 Shows the frequency response of 7, 11 and 13 point
quadratic filters. For a given order, an increase in the number of points in the filter
causes the cutoff frequency to move to a lower frequency, and increases the number of

ripples in the stopband.

Figure 6-3 Shows the frequency responses Of 1, 2, and 3 passes of an 11 point
quadratic 86 filter. With each pass of the filter the cutoff frequency decreases, the
transition band widens, and the stopband ripples decrease in magnitude. These
frequency response plots are readily calculated by taking advantage of the equivalence of
convolution in the time domain and multiplication in the frequency domain. If a Single
pass of a filter has a frequency response of H(w), then the effect of 11 passes of the filter
has a frequency response of [H(w)]". Figure 6-4 shows the frequency response of an 11
point linear differentiator, and the normalized frequency response of the same filter. The

advantage of the normalized representation is easily seen.

Digital Up/Down Integration

It is shown in Chapter 5 that a differentiator is a high pass filter. When
differentiation is done by analog circuits, the output is often extremely noisy. For this

reason, an electronic circuit was designed that incorporates the noise-rejection

105

1.0-

0.0 0.5
FREQUENCY

 

 

 

Figure 6-1. Frequency responses Of.1 1 point Savitzky-Golay filters: (a) linear, '
(b) quadratlc, (c) quartic.

1.0-

106

(a)

.1"

 

 

Figure 6-2.

 

0.5
FREQUENCY

Frequen re uses of quadratic Savitzky-Golay filters. (a) 7
point.(b (139 9059139 (9)11 point-

-

107

1.0 -
(a)

GAIN

(b

0.0- (C) ‘ A 4.
~ v

0.0 0.5
FREQUENCY

 

 

 

Figure 6—3. Freqmen responses Of multi le asses of a quadratic 11 point
Saviltzkyzolay filter: (a) 1 pas}; (g) 2 passes, (c) 3 passes.

108

1.0-1

 

 

0.0 0.5
FREQUENCY

Figure 64. Frequency response of 11 point quadratic first derivative Savitzky-
Golay filter.

109
characteristics of integration to estimate the derivative (1). The method is known as
up / down integration. The principle of the method is that the derivative of a signal may
be estimated by the slope of a line drawn between two points equidistant from the point
at which the derivative is desired. If the actual measured value of the Signal at these two
points is used, the slope of the line is bound to be influenced by random errors. A less
noisy estimate of the signal at each endpoint of the line can be made if the signal is
integrated over some interval surrounding this point, and the integral is divided by the

width of the interval. The algorithm is as follows:
1. Integrate the signal at time tl for duration At and divide by At .
2. Integrate the Signal at time t2 for duration At and divide by At .
3. Subtract the integral at time t1 from the integral at time t2.
4. Divide by (t1 - t2) to estimate the derivative.

Since the up/down method has proven itself in several inplementations for stopped-flow
kinetics (1,10,11) this seemed like a natural place to start as a way to calculate the

derivative for digitized stopped-flow data (2).

In these experiments, the integral was calculated via Simpson’s rule, yn = y,” +
0.5(xn +1 + x"). Although this is a recursive formula for the derivative, it can be recast
into a nonrecursive form, because yo = 0, and the duration of the integral is finite. A

simple «ample will demonstrate this.

Let us suppose that the point at which we derive the integral is at the center of a
set 9 points, with the center point known as x0. Simpson’s rule gives the area of the

region beyond the center point as :

110

area2=[ 0.5(x1-1-xo) + 0.5(x2+x1) + 0.5(x3-1-x2) + 0.5(x4+x3)] Ax
:-=(051c0+1t1 +x2+1t3 + 0.51:4)Ax
The area of the region before the center point is given by:
area1-[ 0.5(x_1+x0) + 0.5(x_2+x_1) + 0.5(x_3+x_7) + 0.5(x_4+x_3) ] Ax
=(0.5!0 +x_l +x_2 +x_3 + 0&4)”
The difference in the two areas is:

areaz-area1=( 0.5x4 + x3 + x2 + x1 -x_1 -x_2 -x_3 - 0.51:4)

This quantity must be divided by 4Ax to convert the difference in areas to a difference in
mean values, then again by 4A1: to convert the difference in mean values to the dam of
the line connecting those mean values. The final result for the estimate of the derivative

is
x’o=( 0.34 ‘I‘ X3 "I' x2 '1" x1 'x_1 “x.2‘x-3' 0.5x_4)/16Ax

Note that previous values of the integral y" are not represented in this equation; thus the
recursive formula for the up/down integration has been recast in a non-recursive form,
which can be represented as follows:

xn=( l )ickxn-h

m2 Ax kz—aa

 

where ck=0.5 for k=-m, ck-l fory-m<k<0, ck-O for ksO, char-1 for 0<k<m, elf-0.5

for ksm. Other equations for estimating the integral would give different results.

The normalized frequency response of 11, 21, and 101 point up/down integration
filters is shown in Figure 6-5. These filters have several features in common with

Savitzky-Golay filters, including the ripples in the stopband. This is not surprising,

GAIN

Figure 6-5.

 

 

 

111

 

0.5

Normalized frequency responses of up/down integration filters:
(a) 11point, (b) 21 point, (e) 101 point.

112
because they are equivalent to Savitzky-Golay linear derivative filters with the end terms
changed from 1.0 to 0.5. Hamming (3) and Nevins and Pardue (4) have pointed out the
beneficial aspects of modifying smoothing filters in this way.

The advantage of having recognized this procedure as a non-recursive filter is that
the frequency response can now be used to determine whether these filters will distort the
signal. When these filters were used on «ponential decays, the estimate of the derivative
was found to be independent of the cutofi' frequency of the filters in the absence of noise.
This was later discovered to be a remarkable property of an «ponential decay, that
adding together any number of points in a «ponential decay produces a new Signal with a
new preexponential factor, but the same decay constant. In the stopped-flow experiments
mentioned previously, the largest practical filters were used, up to 101 points, and these
were found not to cause significant systematic errors. Another advantage of having
recognized this procedure as a nonrecursive filter is that the sum of the squares of the
coefficients of the filter can be used to estimate the variance of the derivative estimate if
the variance of the input signal is known, and if the noise in the signal can be assumed to
be uncorrelated. In the case of stopped-flow data collected in our laboratory, the white

noise assumption appears to be valid.
Sliding Window Filters

One of the major disadvantages of the use of symmetric SG filters is that a filter
with 2N + 1 coefficients cannot be used on the first N or last N points in a data set.
These points are often discarded because they cannot be filtered. In some applications
this does not present a problem; for «ample, when a spectrum that has large stretches of
baseline at the beginning and end is filtered. However, if the beginning or end of a signal

contains important information it cannot be discarded.

1 13

A solution to this dilemma was provided by Proctor and Sherwood (5) and again
by Leach et al. (6). Proctor and Sherwood modified the equation of Savitzky and Golay
so that the filter could estimate any point that was fit by the polynomial. They called their
modified filters extended sliding window filters. Leach et al. considered the case of
estimating the endpoint of a data set or the first derivative at that point, and included not
only the equations but several tables of coefficients as well. Unfortunately, the tables
contained some errors that were later corrected by Baedecker (7). They called their
filters initial point and initial slope filters. For the remainder of this chapter, these filters
will be called N‘h point filters, where the value of N is relative to the first filter coefficient.
Thus a fourth point filter is one that, regardless of its length, may be used to estimate the

fourth point in a data set.

The modified Savitzky-Golay filters seemed like the ideal tool for calculating the
initial rate of a reaction monitored by a stopped-flow mixing system. However, the
performance of the initial slope filters is poor compared to the symmetric Savitzky-Golay
filters that can be used in the center of the data set (2). It was presumed that the reason
for their poor performance could be found in the frequency response of these filters, and
in their noise equivalent bandwidth. Therefore, the characteristics of these filters were

investigated (8).

The design of these modified 86 filters requires the specification of three
variables: the order of the polynomial, the number of points, and the number Of the point
being fit, specified relative to the first filter coefficient (first, second, etc.). Figure 6-6
shows the frequency responses of 9, 11, and 13 point quadratic initial point filters. There
are two major differences between these filters and their symmetric counterparts. The
first is that the gain of these filters is greater than one at some frequencies. This means
that certain frequencies in the signal will be amplified rather than attenuated. This is

expected to cause distortion of some waveforms. In addition, these filters would

114

 

 

\
(a)
1.0 -
8
0‘0 Q
. . °I'\’\ .
0.0 l
0.0 0.5
FREQUENCY
Figure 6—6. $100!!! 3:81:83“ p311: quadratic initial point filters. (a) 9 point,

115
increase the variance of any signal in which the majority of the noise is contained in

theband Of frequencies for which the gain is greater than one.

The second major difference between these filters and symmetric filters is that the
gain does not go to zero beyond the cutoff frequency. While this is «pected to decrease
the amount of distortion these filters cause, it also means that these filters are not as

good at rejecting while noise as the symmetric filters.

Figure 6—6 shows that several trends in the frequency response occur as the
number of coefficients in the filter is increased. The gain curve reaches a higher
maximum value, while the frequency at which the maximum occurs and the cutoff
frequency are decreased. The number Of ripples in the stopband increases, while their
amplitude decreases. Finally, an increase in the number of coefficients causes the filter
to have a lower average gain in the stopband; that is, the frequency response of the filter

approaches zero at higher frequencies.

Since these filters are not symmetric, the phase Shift they induce is a function of
frequency. Figure 6-7 shows how the phase response depends on the number of
coefficients. Note that, regardless of the number of coefficients, all filters have nearly
zero phase shift in the passband of the filter. However, as the gain of the filter begins to
rise, the phase Shift Of the filter begins to deviate from zero. The phase shift peaks at the
cutoff frequency of the filter, then begins to oscillate and converge in much the same way
as the gain. The effect this would have on the distortion of a signal is probably compl«;
however, for slowly changing signals the phase shift is nearly zero, and little distortion is
«pected. As the number of coefficients increases, the maximum phase shift increases, as

well as the number of ripples.

Figure 6-8 shows the frequency responses of 11 point linear, quadratic and cubic
initial point filters. AS the order of the polynomial increases, the frequency at which the

PHASE SHIFT. DEGREES

116

 

 

Figure 6-7.

FREQUENCY

Phase response of quadratic initial point filters: (a) 9 point, (b) -

11 point, c) 13 point.

117

_ I) -  
, ¢Q..0

GAIN

 

 

0.0 0.5
FREQUENCY

Figure 6-8. Freqlgncy response of 11 point initial point filters: (a) linear, (b)
qua tic, (c) cubic. -

118
largest gain is found becomes higher, while the maximum gain first increases, then
decreases. Furthermore, increasing the order causes the stopband to settle to higher
values. As for the symmetric Savitzky-Golay filters, higher order means greater tangency
of the transfer function at «1:0 both in the gain and in the phase shift, which is not Shown.

The other major efi'ect on the phase is that the higher order filters have a lower average

phase shift.

The last variable associated with these filters is the number Of the point that is
being estimated. Figure 6-9 shows the frequency response of minus-first-point, initial-
point, and third-point quadratic 11 point filters. The first of these filters extrapolates to 2
points before the beginning of the filter coefficients. Obviously, the farther the estimated
point is from the center Of the filter coefficients, the higher the maximum gain in the
frequency response, and the higher the value to which the stopband gain settles. Based

on Figure 6-9, one should not use these filters to extrapolate data.
Performance of the Sliding Window Filter

Proctor and Sherwood originally derived these filters so that they could smooth
photoelectron spectra. In order to filter their spectra with several passes of an SG filter,
they filtered the first point with an initial point filter, then the second point with a second
point filter, and so on, until they reached the first point that could be filtered with a
symmetric filter. They continued to use the symmetric filter until they neared the end of
the data set, when they again used asymmetric filters. They ended by smoothing the last
point in the data set with a final point filter that is the mirror image Of the initial point

filter.

When used this way, the coefficients of the filter change with time near the
beginning and end Of the data set. Consequently, the filter is no longer linear or time

invariant. Filters that are not LTI systems cannot be described completely by their

119

(a)

(b)

(e)

 

 

3,0 ..
2.0 d
3
1.0 -
0.0
0.0
Figure 6-9.

0.5
FREQUENCY

Frequen res rise of uadratic 11 point filters: (a) minus first
point. (beginitlglopoint, c) third point.

120
frequency response; they must be tested in order to determine what effect they will have
on a signal.

In order to separate the efi'ects that the sliding least squares filters have on the
correlated and uncorrelated parts of a signal, the filters were tested on a sine wave and on
random noise. The effect of the filter on a sine wave is shown in Figure 6-10. The period
of the sine wave is 10 points, so its frequency is 0.1 times the sampling frequency.
Quadratic 11 point filters were used throughout the data set. The gain of the initial point
filter for f = 0.1 is ca. 1.4, as seen in Figure 66, while the gain for the symmetric filter is ca.
0.7, as can be seen in Figure 6-1.

Figure 6-10 shows the effect of 1, 5, and 20 passes of the «tended sliding window
filter. In the data set that was smoothed one time, the 5 points at each end were the only
ones filtered by an asymmetric filter. Obviously, these are also the only points that no
longer fall on the Sine wave. In fact, the filter has forced these points into nearly a
straight line on the left. Notice that on the right side of the plot, the last data point is
lower than any point in the original sine wave; this is a consequence of the gain of greater
than one that the initial point and final point filters possess at this frequency. In the
center of the data set, the points still constitute a sine wave of the same frequency and
phase, as one would predict from the transfer function of the symmetric SG filter. The

amplitude is ca. 0.7 times the original amplitude.

The efi'ects of the sliding window filter extend much farther into the data set that
has been filtered 5 times. This is to be «pected, because some of the first 5 points from
the first pass are used to calculate the first 10 points of the second pass, and so on. On
this basis, one would predict that the first and last 25 points of this data set would fail to
fall on the sine wave. While this is true, the innermost points in the first and last set of 25

are distorted so little that it is not noticeable. However, the first minimum and the last

121

 

 

_ Raw Data

_1 Pass‘

_ 5 Passes

__ 20 Passes

l l l 4 l I I I
I I 5 I 1 I I I
d as 00 u co to on on -
o o a o o o o e a e o e O
C b o e o o e e o e e a e e e e
A
'5 o O o o o o a o o o a o o 1
z . on o. .0 to a. on u on -
c
3
>4 " ..
L I 00 o. o oo o. o.
D .0. o e o o e e a
.- a
fin! - O O o o a O o o
n o O o o o e o e o
I- n u o. e. on u o. o.
s. -. -
. 4
o e o.
‘0 - . . .0 O 0'... .0” 0’... a”. ’0’... .
a a"... 00° '00. '00. on ’0' '00. as u.
v
3, .. .—
E '. l
< e
- 0. 0'...
C”.
d 4 b
I I l J I I I l
I I I 1 I r I I

 

 

 

Figure 6-10.

(arbitrary units)

Effect of sliding window filter on noise-free sine wave.

 

122
maximum are shifted in time (or equivalently, phase shifted) relative to the first
minimum and last maximum in the original sine wave. The center of the data set still has
the same frequency and phase as the original sine wave, and has an amplitude that is ca.

0.16 times the original amplitude, which is expected, since (0.7)2 a 0.16.

In the data set that has been filtered 20 times, the edge effects extend even
further into the data set. In the center of the data set, the sine wave has nearly
disappeared. It is expected to have an amplitude of (0.7)20 which is ca. 8 x 10 ‘4. It is
apparent that the data deviate from this small sine wave as far as 25 points from either
end. Most important is the fact that the value at the ends have changed little from the

values they took after one pass of the filter.

Figure 6-11 shows the effect that the sliding windows have on pure noise. Again
the 11 point quadratic filter was used. The points are drawn from a Gaussian distribution
of random noise with a mean of zero and unit variance (9). The line with the large
oscillations results from smoothing the data 20 times, while the smoother line results
from filtering the data 100 times. Since the mean of the noise is zero, an infinite number
of passes of the symmetric filter is expected to reduce the data to a straight line with a
slope of zero. Near the center of the data set that is clearly happening, but once again,
because of the large gains associated with the initial and final point filters, the end values
do not change much from the unfiltered values. Since the data are uncorrelated, the
phase shift of the asymmetric filters does not contribute to the edge effects in this data

set.

Figure 6—12 shows the performance of 2 different 21 point quadratic first
derivative filters: initial and symmetric. Figure 6-l2a is a plot of absorbance versus time
obtained with an automated stopped-flow spectrophotometer (12). Figure 6-12b shows

the derivative calculated by the two filters mentioned above. Since the symmetric filter

Amplitude

1%

 

 

 

 

 

_2 -- q
r I n I l l l
-3. I I ' l ' I ' 1
O 20 4O 60. 80. 100

Point Number

Figure 6-11. Effect of sliding window filter on noise.

 

124

 

   

 

 

 

 

 

0 5 ' I 1 ‘ I
0.4 --
A
D .,
<
o 0.3 - Raw Data
0
C
O
f 0.2 u u
0
a d
.D
<
0.1 - -
(a)
o 4 : : : .
1|- (b) 4
0.3 -" . . u
’3 lmtlal Point
O
a 1
\
D
5 o 2 -- -
‘0
'0
\
<
.0 0.1 -- u
Symmetric
1’ -
. J . I I
0.0 v ' I I I I I
o. 1 2. 3. 4.

Time (see)

Figure 6-12. Effect of initial slope filter on stopped-flow data: (a) stopped-flow
transient, (b) calculated derivative. -

125
cannot calculate the derivative at the first 10 points in the data set, and the initial point
filter cannot calculate the derivative at the last 20 points, these points are not shown. The
variance of the derivative calculated by the initial point filter is obviously much greater
than the variance of the derivative calculated by the symmetric filter. Furthermore, the
phase shift introduced by the initial point filter can be seen in Figure 6-12b. Several
minima and maxima in the lower peak are also present in the upper peak, but they occur
earlier in the upper peak due to the positive phase shift of the initial point filter. Of
course, it is not suggested that an initial point filter be used when the symmetric filter can
be used. This example merely shows that, when an initial point filter is used, one should
not assume that the variance of the resulting estimate is the same as the variance of an

estimate produced by the symmetric filter.
CONCLUSIONS

The initial point filters provide a good example of why it is necessary that one
have a knowledge of the frequency response of a digital filter one intends to use. Many
people probably assumed, as we did at first, that because these filters are derived using
the same assumptions used to derive the symmetric SG filters, they will perform the
same. Analysis of the initial point filters shows that under some conditions they do not
act like filters at all. It is hoped that this chapter has demonstrated the ease with which
frequency response plots can be generated, and how useful they can be to users of

nonrecursive filters.

i"

PWNP‘S"?

10.
11.
12.

126

CHAPTER 6 REFERENCES

Cordos, E.M.; Crouch, S.R.; Malmstadt, H.V. Anal. Chem 1968, 40, 1812.
Wentzell, P.D.; Crouch, S.R. Anal. Chem 1986, 58, 2855.

Hamming, R.W. Digital Filters, 2nd ed., Prentice-Hall: Englewood Cliffs, NJ,
1983.

Nevins, TA; Pardue, H.L.Anal. Chem 1984, 56, 2249.

Proctor, A.; Sherwood, P.M.A. Anal. Chem 1980, 52, 2315.

Leach, R.A.; Carter, 0A.; Harris, J.M.Anal. Chem. 1984, 56, 2304.
Baedeker, PA. Anal Chem 1985, 57, 1477.

Wentzell, P.D.; Doherty, T.P.; Crouch, S.R. Anal. Chem 1987, 59, 367.
Wentzell, P.D.; Crouch, S.R. J. Chem Ed, submitted for publication.
Ingle, J.D., Jr.; Crouch, 8.12. Anal. Chem 1972, 42, 1055.

Holler, FJ.; private communication.

Crouch, S.R.; Holler, Fl; Notz, P.K.; Beckwith, P.M. Appl. Spectrosc. Rev. 1977,
13, 165.

 

CHAPTER 7 : DESIGN OF DIGITAL FILTERS

The preceding chapter detailed the procedure used to determine the frequency
response of any nonrecursive digital filter. In many cases the Savitzky-Golay (SG) filters,
which were the focal point of the previous chapter, are adequate for any low-pass filtering
or differentiation that may be required. At other times, a filter is required that more
accurately mimics the ideal low-pass filter; that is, one that has a gain of exactly one for a
range of frequencies, and a gain of zero for the remainder. In this chapter, techniques for
the design of a filter with a given frequency response are demonstrated, and several
examples of their use are shown. These filters are sometimes known as windowed ideal
low-pass (WILP) filters. Much of the material in this chapter is based on two excellent
and readable textbooks on the subject (1,2).

THEORY

It has already been stated that, because of the sampling process, a digital filter is
characterized by a transfer function that is periodic, with a period equal to the sampling
period. Because the transfer function is periodic and continuous, it has a Fourier
expansion. In other words, the continuous transfer function can be described by a
discrete sum of weighted sines and cosines. This Fourier expansion, which is specified in
the time domain, is equivalent to the frequency domain transfer function in the sense that
convolution in the time domain is equivalent to multiplication in the frequency domain.
Therefore, the Fourier expansion is the set of filter coefficients that must be convolved

with the data to obtain the desired filtering. The Fourier expansion of a function H ( w) is

given by:

127

 

128

11(0): g9- +k21ak cos ka+bksin lea

where the “k and bk are the time domain coefficients of the expansion. We wish to turn
this equation around; namely, given the frequency response, H(w) we want the values of

“k and bk' They are determined by the following set of equations:

“In: 711' f H(0)cos lea) do; (7-2)
bk :71!” f H(w)sinlcm do; (7-3)

The results are converted to functions of the sampling frequency through the relation, to

g 21f,
EXAMPLES
Low-pass Filters
If a low-pass filter is required, the frequency specification is:
H(w) = l, wc< w < a)c
H(w) = 0, w > wc,w < -wc

where we is the cutoff frequency of the filter. Note that this is an even function, while the
sine is odd. The product of the two, which is the function to be integrated in equation 7-3
to obtain the coefficients b k’ is therefore odd, and its integral over a region symmetric

about u = 0 is zero. Thus, the value of all the coefficients bk is zero.

To evaluate the coefficients of a k we need to solve the equation:

,: — “-1.

 

129

”c

n .
ak=7lffH(a)coslcm do = 711' coslca d0
-II

'"c

=1|:.1_sinlcm]me = 23inlcan = 28inlcfc
7r k _0 Fit- ° 75—1?

o

7-4
(0

”c
_ 1 .. 2 _
aa-fl'fdm "' Tn - 4f¢
"we
If one wishes, for example, a filter that passes all frequencies lower than one tenth of the

sampling frequency, then fc - 0.1, and the coefficients are given by:
Ck = 0.4, k = 0
ck = 2-(sin 0.21rk)/1rk, k If 0

Two points must be made here. First, the equations above hold for all k.
Remember that, as shown in equation 7-1, the Fourier expansion of a frequency response
involves a summation to infinity over k. If one wishes to design a filter that has a
frequency response that matches the specification exactly, the filter must have an infinite
number of coefficients. Obviously, the use of such a filter would require an infinite
number of multiplications and additions, as well as the loss of an infinite number of data
ponits at each end of the data set. Therefore, one is forced to evaluate the coefficients
only out to some practical value of k (and -k). The minimum number of coefficients one
will use in the actual filter depends on the cutoff frequency and the desired attenuation
rate in the transition band. The maximum depends on the rate at which output from the
filter is desired and the number of data points that one can afford to lose at each end,

since both depend linearly on the number of coefi'icients in the filter.

The second important point is that this design equation does not guarantee that

the output of the filter is equal to the input when the input is constant. Normally one

 

130
wishes the output and input to be equal under those conditions, since it guarantees that
the DC component of a waveform will not be distorted. Therefore, to avoid DC

distortion, after the coefficients are calculated, they should be normalized so that:

2a=1

Figure 7-1 shows an 11 point filter with cutoff frequencies at 0.1, 0.2, and 0.3 times
the sampling frequency. In all three curves, the gain starts at one, but soon begins to
oscillate. Unlike the SG filters, these filters contain ripples in both the passband and the
stopband. These ripples occur because the transfer function contains a discontinuity,
where the gain goes from one to zero. In most cases such a sharp cutoff is not needed. A
better choice would be to have a transition region of finite duration, and have the transfer
function decrease linearly from one to zero in that region. While it would be possible to
find the Fourier expansion of such a frequency specification with equation 7-3, there is an

easier way to achieve the same result.

If the discontinuous transfer function were convolved with a narrow rectangular
function in the frequency domain, the result would be exactly what we want: a transfer
function with a transition band where the gain decreases linearly. Because of the
equivalence of convolution in the frequency domain and multiplication in the time
domain, this same result can be obtained if the filter coefficients are multiplied by the

Fourier expansion of the small rectangular function, which is given by:
ak = N~sin(1rk / N)/1rk
do = 1

This process is known as windowing the coefficients. There are many windowing
prowdures, and all of them offer a compromise between the width and the shape of the

transition region in the transfer function, and the height of the ripples in the transfer

I 'i-

 

131

 

 

 

 

1.0 - E
(b) (0
§
l
0.0 0.5
FREQUENCY
Figure 7-1. Frequency responses of unwindowed 11 point filters: (a fc = 0.1

times sampling frequen , ) f = 0.2 tnmes sampling equency,
(c) fe = 0.3 times sampling eqniency.

132
function (1,3). Figure 7-2 shows the windowed versions of the same filters shown earlier
in Figure 7-1. Each of the filters now has a much wider transition region (less sharp
cutoff) than the filters in Figure 7-1, but the ripples in both the passband and the
stopband have been geatly reduced. Because of the geater accuracy of the windowed
filters near DC, they would be preferred in most applications.

Differentiators

As was stated in Chapter 5, a perfect differentiator behaves as a high pass filter.
The signals that are analyzed by chemists often have power spectral densities (PSD) that
are concentrated at lower frequencies. The PSD at high frequencies is due mostly to the
noise in the signal. Therefore, a perfect differentiator is seldom called for; such a filter
has its highest gain for frequencies where there is little signal power density. A useful
differentiator must have a cutoff frequency beyond which its gain is close to zero. The

derivative of the eigenfunction of an LTT system is given by:
d/dn = cw" = imam
and the transfer function (output/input) is thus:
imam NW = in»

This is easily extended to any derivative; the transfer function of the N‘11 derivative is
(11.2)”. This transfer function can be substituted into Equations 7-2 and 7-3 to obtain the
filter coefficients. If the derivative is odd (1“, 3", etc.), the transfer function is odd. The
product of the transfer function and the cosine, which is even, is then odd. Thus, a k is
zero for all k. If the derivative is even, (2“, 4th, etc), bk is zero for all 1:. For a symmetric
filter it is always true that either all the “I; are zero or all the bi: are zero, and the

coefficients of the filter are the ones that are not equal to zero.

133

 

 

 

1.0 "
(a) (b) (C)

2
5

0.0 - 4‘.— A—“:. e.._

0.0 0.5
FREQUENCY

Figure ‘7-2.

Frequen? responses of the windowed versions of the filters shown
in Figure -1

 

134

The results are, for the first derivative:

_ gi ( sin 21Tlctc _ cos 21mg, )
blc' 11 k2 k

 

7-5
50: 0
and for the second derivative:
_ 2 2
at: 131!!— sin 21:ka - 47% cos 21w,
’63 ’52 7-6
“0: 0

These filters are normally windowed in the same way the low-pass filters are. However,
there is a different set of normalization criteria associated with derivative filters. We are
no longer concerned about the effect of constant input, since the derivative of a constant
is zero. Instead, we are concerned about what is obtained when a polynomial of degee m
is differentiated. An accurate differentiator would give the result that if xn = n'",

a'"/(dn)"' x" = m!. Therefore, derivative filter coefficients are normalized so that
z (ck k”) = M!
For example, a first derivative filter requires that

Z (ckk) =1

Figure 7-3 shows the frequency response of a windowed 21 point first derivative
filter with cutoff frequency of 0.1 times the sampling frequency. The curve with a gain of
one at w=0 is the normalized frequency response, as described in Chapter 6. Aside from
the lack of ripples in the stopband, it looks very much like the differentiator shown in
Figure 6-4.

135

 

 

 

 

1.0 '-
normalized
i
0.0 l “I.
0.0 0.5

FREQUENCY

Figure 7-3. fregnllency response of windowed 21 point derivative filter with
c: 0 O

136
WHY DESIGN FILTERS?

We have seen that $6 filters are characterized by large ripples in the stopband,
whereas windowed ideal low-pass (WILP) filters have much smaller ripples that are
distributed throughout the transfer function. This difference endows the SG filters with
greater accuracy near DC, which means they would cause less distortion that the WILP
filters, although the extra distortion caused by WILP filters can be made arbitrarily small.
However, the presence of the ripples in the stopband could cause the SG filters to
perform worse than the WILP filters on data in which the noise is not white, but
concentrated in certain areas of the spectrum. In this case the noise equivalent
bandwidth (NEB) of the filter is irrelevant; what is important is the gain of the filter at
those frequencies where the noise power is geat. One cannot treat 86 filters as general
low-pass filters, because one cannot be sure that the gain of the filter is approximately

zero for the high frequencies.

Because WILP filters do not have large ripples anywhere in the spectrum, they
are easily converted to other types of filters, including high-pass, bandpass and bandstop
(notch) filters. While these filters could be designed by calculating the Fourier expansion
of the desired frequency response, it is simple to obtain them from converted low-pass
filters. For example, high-pass filters are useful for analyzing a rapidly changing signal
that is superimposed on a slowly changing backgound (4). To be converted to a high-
pass filter, a low-pass filter must be changed so that the gain is one everywhere it was
zero, and zero everywhere it was one. This can be done by subtracting the low-pass
transfer function from a transfer function that is equal to one at all frequencies. Recall
that this is an all-pass filter, which has the filter coefficients c0 8 1. One of the theories
of Fourier transforms states that the Fourier expansion of the sum of two functions is
given by the sum of the Fourier expansions of the individual functions. All one need do is

reverse the sign of each coefficient except the central one and subtract the central

137
coefficient from one to obtain a high-pass filter with the same cutoff frequency as the

original low-pass filter.

Bandpass and notch filters can also be designed in this way, although it is a trial
and error process. A notch filter will serve as an example. Here, the gain is required to
be zero for all frequencies located within a distance Af of fc , the center frequency of the
stopband. A filter having approximately this frequency can be made by the addition of a
low-pass filter with a cutoff frequency of fc - Af and a high-pass filter with a cutoff
frequency of j"c + Af . The part that makes this a trial and error process is that this

design algorithm does not guarantee what the value of the gain of the filter will be at fc.
EXAMPLES
Correlated Noise Removal

Figure 7-4a shows a peak that was obtained with a flow injection analysis (FIA)
(8). The height of the peak is a measure of the absorbance of the solution, which
consisted of a plug of dye injected into the carrier stream. These data were collected as
part of an experiment to determine the effect of temperature on dispersion in FIA (5).
The sinusoidal component in the peak is pump noise, although this was not immediately
apparent. The sampling frequency for these experiments was 1.43 Hz, and the sinusoid
has a period of approximately 16 points, which gives a frequency of 1.43 / 16 = 0.09 Hz
for the sinusoid. This frequency is much to low to be ascribed to the rollers in the pump,
which for this experiment revolved at approximately 1.5 Hz. However, due to aliasing, a
signal at 1.52 Hz actually shows up at 0.09 Hz. Since the sinusoid is composed of
approximately 16 points per cycle, it has a frequency that is approximately 1/ 16 (0.0625)
times the sampling frequency. To remove this noise with minimum distortion of the FIA

peak, a bandstop filter with fc = 0.0625 is required.

138

(a)

ABSORBANCE -)

(b)

TIME ->

Figure 7-4. gleakcgbtained on flow injection analyzer: (a) unfiltered, (b)
ter ‘

139

In this case a compromise must be made because the data set consists of just 100
points. Although the peak is near the center of the data set, and an estimate of its height
is all that is required, one cannot afford to lose more that about 15 points at each end.
Figure 7-5 shows the frequency response of the filter that was designed to remove the
correlated noise, and the high-pass and low-pass filters that were combined to produce
the bandstop filter. Note that the gain of the filter is not zero at fc. To obtain a lower
gain at fc, one would have to use more filter coefficients, which would cause the loss of
more data at each end of the data set, or specify a lower fc for the low-pass filter and a
higher fc for the high-pass filter, which would cause the bandstop region to be wider and
the peak to be more distorted. The filter that was used here was found to be a good

compromise between distortion and correlated noise rejection.

Figure 7-4b shows the FIA peak after it has been filtered by the bandstop filter.
The peak height has been decreased by about 4 times the amplitude of the correlated
noise. The ratio of FWHM to the peak height has increased by about 25%. Both of
these factors indicate that the peak has been distorted somewhat. However, examination
of the baseline in Figure 7-4a and 7-4b shows that the amplitude of the correlated noise
has been decreased dramatically.

The distortion caused by the bandstop filter would not be of geat concern in
typical FIA applications. The shape of an FIA peak is largely independent of the
concentration of the analyte in a typical assay where the reaction goes to completion.
Since the peaks produced by different analyte concentrations are related by a scaling
factor, and the bandstop filter is an LTT system, the outputs of the filter are also related
by a scaling factor. However, the heights of the filtered versions would be eXpected to be
more linear with concentration than the unfiltered versions, since the correlated noise

source has random phase relative to the peak maximum.

 

140

 

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141
The bandstop filter is expected to perform better than baseline subtraction
because the filter works well independently of the phase of the correlated noise and its
amplitude. Its use requires no human invention, and a new filter could be designed

automatically to cope with changes in either the pump speed or sampling rate.
Decimation

There are other cases when 86 filters are inadequate due to the ripples they
possess in the stopband. In this section it will be shown that changing the number of data

points in a set requires the use of filters that are flat throughout the frequency spectrum

(6).

Decreasing the size of a data set by retaining only a fraction of the original signal
is known in the field of electrical engineering as decimation. Although this term literally
means retention of every tenth point in a data set, retention of every N‘11 point is called
decimation by a factor of N. Decimation is employed when the signal is sampled too

quickly, producing a quantity of data that is unmanageable.

Decimation by a factor of N is exactly analogous to having sampled the original
data at l/Nth the original sampling frequency. The process of decimation produces a new
Nyquist frequency that is 1/N times the original Nyquist frequency. Any components of
the signal or noise that existed at frequencies geater than the new Nyquist frequency will

be folded back into the spectrum.

The process is illustrated in Figure 7-6 for decimation by a factor of two. Note
that the process of throwing out every second point causes the frequency axis to be cut in
half, and the power spectral density of those components in the upper half of the
spectrum folds back and adds to the power spectral density (PSD) of the components in

the lower half of the spectrum. In many cases this process causes only insignificant

 

142

remove every noise folds
other point back

—” _/

 

 

 

 

filter

remove every
other point

#

amplitude

 

 

 

 

l l
frequency

Figure 7-6. Decimation by a factor of two. ‘

143
changes in the spectrum and appearance of the data. However, there may be cases where
the original sigial had significant power spectral density at the highest frequencies in the
spectrum. This is more likely as the degee of decimation increases, because then the

fraction of the old spectrum that folds into the new spectrum increases.

Even if significant amounts of noise become aliased to lower frequencies,
subsequent filtering can often be used to remove it. However, if the frequency of the
noise is such that it folds back into a region of the spectrum where the underlying
waveform has high power spectral density, any attempts to remove that noise will result in
distortion of the waveform. For these reasons it is essential that the signal be filtered
mm decimation occurs. Figure 7-6 demonstrates that, if the proper filter is used, the
aliasing of noise and signal components can be prevented. Just as the last step before
sampling should be an analog antialiasing filter, the last step before decimation should be
a digital antialiasing filter.

Figure 7-7a shows a portion of the mass spectrum of Xe obtained with the CVC
Model 2000 time-of-flight (TOF) mass spectrometer. These data were collected as part
of a study of the effects of suboptimal time lag focusing on TOF data (9). On this system,
sampling is performed by a boxcar integator. The output of the boxcar integator
contains a correlated noise component that is of unknown origin, but is believed to
originate within the spectrometer. This data set consists of 4064 points, and represents a
typical oversampling situation: the signal was sampled rapidly so that nothing would be
missed, but the high sampling rate resulted in a nearly unmanageable amount of data.

If only every 10th point is retained, the spectrum in Figure 7-7b results. Note the
presence of a sinusoidal component. This component does not appear to be in the
original spectrum. Closer examination reveals that this component is the aliased version

of a high frequency sinusoid that is the cause of the 'fuzziness" of the signal in Figure

144

 

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of xenon: (a) original spectrum, (b)

tering.

t mass
10, no

Time—of-fli
decimated

Figure 7-7.

145
7-7a. The sinusoid now has a period that is approximately equal to the FWHM of the
peaks. If a bandstop filter were used to remove the sinusoid after the decimation, it

would significantly distort the shapes of the two peaks.

Since we plan to decimate the data set by a factor of 10, it should be filtered first
by a low-pass filter with a cutoff frequency of one tenth the Nyquist frequency, or 0.05
times the sampling frequency. This is a large data set, with large stretches of baseline at
each end, so a filter with a large number of coefficients may be used in order to obtain a
sharp cutofl'. Figure 7~8a shows the frequency response of the 41 point filter that was
used, and Figure 7-8b shows the result of filtering followed by decimation. The spectrum
in Figure 7-8b looks to be much closer to the average of the fuzzy spectrum in Figure 7-7a
than does the spectrum in Figure 7-7b. Furthermore, the FWHM of the peaks'in Figure
7-8b is the same as in Figure 7-7a, because the peaks’ power spectral densities were small
in the upper 90% of the frequency range.

Interpolation

This same analysis of the effect of changes in the sampling rate on the PSD of a
signal can be used to provide a multipurpose method for interpolation. Interpolation can
be viewed as increasing the effective sampling rate of a system. We interpolate a data set
by a factor of N by adding N - 1 new data points between the original sampled data

points.

Interpolation schemes difi'er by how the magnitudes of the new data are
determined. Consider the effect of placing a data point with a value of zero between each
point of original data. Since the data still represent the same time duration, but there are
twice as many data points, the sampling rate has been increased by a factor of 2. The
PSD of the signal now extends to a Nyquist frequency that is twice as geat as the original

Nyquist frequency. Also, the PSD of the new signal in the upper half of the frequency

146

 

 

 

 

1.0 "
(a)
z
<
(9
0.0
°'° FREQUENCY °""
(b)
T
E
U)
2
E
E
F “1
FLIGHT TIME —.

Figure 7-8. - a) Frequency response of decimation filter. (b Filtered time-of-
flight mass spectrum of xenon; filtered, then decn)mated by 10.

147
spectrum is the mirror image of the PSD in the lower half of the spectrum, as shown in
Figure 7-9. Although this is not intuitively obvious, consider what would happen if every
second point were removed. The result would be the original waveform, with its original
PSD. If the upper half of the PSD of the interpolated signal were anything other than the
mirror image of the lower half, this process of interpolation followed by decimation would

result in a different PSD than the original waveform had.

Obviously, placing alternating zeros in a data set is not what is desired in the
process of interpolation by two. What we want is for the PSD of the signal that results
from the interpolation process to be equal to the PSD of the signal that would have been
obtained if the analog signal had been sampled at twice the sampling frequency. We
presume that the original sampling frequency was adequate for representing the signal;
that is, the PSD of the analog signal was nearly zero at frequencies higher than half the
sampling frequency. If this is true, the upper half of the PSD of the interpolated signal
should be nearly zero. Therefore, all we need to do is filter the signal that has the
alternating zeros with a low-pass filter that has a cutoff frequency fc of 0.25 (half the
Nyquist frequency) to obtain an interpolated version of the original waveform. This

process is shown in Figure 7-9.

Figure 7-10 shows the results of interpolation by a factor of two applied to a
calculated Gaussian profile with a FWHM of approximately 9 points. Figure 7-10a shows
the zero-filled data set. Figure 7-10b shows what happens when a 21 point eighth order
SG filter is used to filter the zero-filled data set. Figure 7-10c shows the results obtained
with a WILP filter that has a cutoff frequency of 0.477. The frequency responses of the
two filters are shown in Figure 7-11. The SG filter is clearly inadequate for this purpose.
The ripples in the stopband of the 86 filter let through some high frequency components
that show up as sinusoidal components in the interpolated version of the Gaussian

profile. One can see from examining Figure 7-9 that inserting the alternating zeros into

148

 

 

‘ ladd zeros

 

amplitude

 

jfflter

 

 

 

 

I
frequency

Figure 7-9. Interpolation by a factor of two.

149

 

 

 

 

 

.. °.' (a)

M‘s-aooooooooooooooe.o°w

Figure 7-10. Interpolated Gaussian ofile: (a) before filtering, (b) results of
Savitzky-Golay interpo ation filter, (c) results of windowed ideal
low-pass interpolation filter.

150

 

1.0

(b) (8)

GAIN

\ A /

 

 

FREQUENCY

Figure 7-11. Frequency responses of decimation filters: a3. windowed ideal
low-pass filter, f = 0.477, (b) 21 point e' order Savitzky-

C

Golay filter.

151
the data set causes the PSD to be as geat at the Nyquist frequency as it is at DC. In the
previous chapter it was shown that the SG filters all have gains that are nonzero at the
Nyquist frequency. Therefore any 86 filter would be expected to perform poorly as an

interpolation filter.

As an interpolation scheme, the filtering method described above has some
advantages over other methods. The filtering method makes no assumptions about what
the form of the data is; it only assumes that the PSD of the data is insignificant above the
original Nyquist frequency. In this way it is similar to the Fourier "zoom transform"
approach, where one pads the Fourier transform of a signal with zeros to double its
length, and then inverse transforms to produce the interpolated data (7). This contrasts
with polynomial interpolation, where it is assumed that the data can be described by a
polynomial of low degee. Choice of the proper degee for the polynomial is often a
difficult task.

The convolution method is much faster to implement than polynomial
interpolation, especially if digital signal microprocessors are used, because it only
requires multiplication and addition. The convolution method is more versatile than the
zoom transform. Through combinations of interpolation and decimation any rational
amount of interpolation can be done, such as interpolation by 4 and decimation by 3 to
increase the effective sampling rate by 4/3. The zoom transform is limited to
interpolation and decimation by powers of two unless the much slower discrete Fourier

transform is used in place of the fast Fourier transform.
CONCLUSIONS

In this chapter it has been shown that Savitzky-Golay filters are not adequate for
all filtering purposes. The removal of correlated noise, decimation and interpolation can

be done better by filters that are designed specifically for those tasks. The design of a

152
filter is not a complicated process; simple filters can be designed with the aid of
calculator. While not everyone who uses filters should feel compelled to design them as

well, it is hoped that this chapter has shown that a little extra effort can sometimes pay

off.

95"?!”

>3

153

CHAPTER 7 REFERENCES

Hamming, R.W. Digital Filters, 2nd ed., Prentice-Hall: Englewood Cliffs, NJ,
1983.

Williams, C.S. Designing Digital Filters, Prentice-Hall: Englewood Cliffs, NJ,
1986.

Ramirez, R.W. The FFT, Prentice-Hall: Englewood Cliffs, NJ, 1985.
Marchand, P.; Marmet, L. Rev. Sci. Instrum I977, 48, 512.
Stults, C.L.M.; Wade, A.P.; Crouch, S.R. Anal. Chim Acta 1987, 192, 301.

Crochiere, R.E.; Rabiner, LR. Multimte Digital Signal Processing, Prentice-Hall:
Englewood Cliffs, NJ, 1983.

Francl, T.J.; Hunter, R.L.; McIver, R.T., Jr.;Anal. Chem 1983, 55, 2094.
Patton, CJ. Ph. D. Dissertation, Michigan State University, 1982.

Erickson E.; private communiation.