J. Phys. Chem. 1984, 88, 5435-5441
band of 1-CNC as the temperature increases.
Conclusions Intramolecular hydrogen bonding between the N-H bond and the carbonyl group in 1-COOEC as well as the intermolecular double hydrogen bonding of the N-H-.N=Ctype between molecules of 1-CNC in CCl, solutions have been investigated by infrared spectroscopy. A distance of about 2.1 A between the N-H proton and the C=O group favors the occurrence of intramolecular hydrogen bonding in 1-COOEC. Nevertheless, the rotation of the carboethoxy group around the C,,-C bond causes this compound to exist in two configurations: the free N-H and the hydrogen-bonded N-H. These two configurations manifest themselves by the two well-separated bands in the N-H stretching frequency range of the infrared spectrum. From the temperature dependence of the intensities of these bands the intramolecular association constant K = 1.33 f 0.26 at 25 “C, the enthalpy change AH” = -7.40 f 0.22 kJ mol-’, and the entropy change AS” = -22.6 f 1.4 J mol-’ K-’ have been calculated for the intramolecular hydrogen bonding in 1-COOEC.
5435
1-CNC in nonpolar solvent is undergoing hydrogen-bonding self-association. The infrared spectrum of 1-CNC in carbon tetrachloride possesses two bands which have been well characterized as N-H band of the monomer and the double N-H band of the cyclic dimer. The relative intensities of these bands change with the concentration of 1-CNC. Moreover, the intensity of the N-H (dimer) band strongly depends on the variation of the temperature. The equilibrium constant of the dimer formation of 1-CNC in CC14 at 25 “C has been obtained as 92 f 16 mol-’ dm3. The enthalpy change and the entropy change for the complexation of 1-CNC in CCl, are -33.3 f 2.8 kJ mol-’ and -74 f 10 J mol-’ K-l, respectively. These results are in good agreement with the cyclic structure of the dimer discussed. Acknowledgment. The authors thank the Natural Sciences and Engineering Research Council of Canada and the ‘‘Ministzre de 1’Education du Qubbec” for financial assistance. Thanks are also due to Mr. Jean-Claude Bolduc for having taken many of the infrared spectra on the FTIR apparatus. Registry No. 1-(Carboethoxy)carbazole, 56995-05-2; l-cyanocarbazole, 83415-88-7.
Comparison of Fluorescence Lifetime Fitting Techniques Hans P. Good,* Alan J. Kallir, and Urs P. Wild Physical Chemistry Laboratory, Swiss Federal Institute of Technology, ETH-Zentrum. CH-8092 Zurich, Switzerland (Received: August 18, 1983: In Final Form: March 29, 1984)
The data obtained from fluorescence decay measurements using the time-correlated single-photon-countingtechnique may be analyzed with various fitting techniques. Each of the fitting procedures assumes a model function and determines the parameters therein. The mean values and the variances of the parameters are defined by the nature of the measuring process alone and should therefore be independent of the manner in which the data are analyzed. Methods of determining the variance in the parameters are discussed. The Fourier transform (FT) technique is compared with the iterative convolution (IC) method. We calculated error functions of these methods and verified them by simulation. In the limit of a large number of channels both techniques produce identical mean values and variances. The Fourier transform technique is appreciably quicker in execution time. The Fourier transform technique is therefore considered as the method of choice. Two derivatives of the Fourier transform technique, the square root and phase methods, are found to be good as rough first approximation techniques.
1. Introduction
Time-correlated single-photon decay spectroscopy is an experimental technique widely used in both the physical and biological sciences. The excited-state lifetimes may be determined with a time resolution better than 50 ps’v2 and therefore may be used to trace extremely fast deactivation paths. The first step of any data analysis is the construction of a hypothesis which involves an understanding of the physical and chemical processes. The second step requires the parameters contained in the hypothesis to be determined. The accept/reject criterium is based on statistical properties of the data and the model function. In this paper, we restrict ourselves to the consideration of single exponential decays. This simplifying assumption allows a proper comparison of the various fitting techniques. This does not imply that the methods compared here are only applicable to single (1) U.P. Wild, A. R. Holtzwarth, and H. P. Good, Reu. Sci. Instrum., 48, 1621 (1977). (2) H. P. Good, U. P. Wild, E. Haas, E. Fischer, E. P. Resenitz, and E. Lippert, Ber. Bunsenges. Phys. Chem., 86, 126 (1982).
0022-3654/84/2088-5435$01.50/0
exponential decays. In fact, the analysis of three exponentials is carried out routinely by the FT method. We deal here with the optimum choice of the fitting technique. Other deliberations treat the optimization of the TAC range3,, and the optimum pulse repetition rate (PRR).5 Only synthetic data are used in this study, since the parameters thereof are known, a priori. We previously presented an apparatus for the measurement of fluorescence lifetimes together with a new method of data analysis: the Fourier transform technique.’ A typical example is displayed in Figure 1. Recently OConnor et ale6have compared a “Fourier transform” technique with other methods of analysis. The Fourier transform technique they used is not to be confused with our approach, proposed in ref 1. Rather than minimizing the weighted least (3) P. Hall and B. K. Selinger, J . Phys. Chem., 85, 2941 (1981). (4) B. K. Selinger, C. M. Harris, and A. I. Kallir in ”Time Resolved Spectroscopy in Biochemistry and Biology”, R. B. Cundall and R. E. Dale, Ed., Plenum Press, New York, 1983. ( 5 ) H. P. Good, A. J. Kallir, and U. P. Wild, J . Lumin., in press. ( 6 ) D. V. OConnor, W. R. Ware, and J. C. Andre, J . Phvs. Chem.. 83, 1333 (1979).
0 1984 American Chemical Society
5436 The Journal of Physical Chemistry, Vol. 88, No. 22, 1984 PLOT NR.3 SUBSTRNCE:PYREN CONCENTRRT I ON :2E-5 SOLVENT:O.O3 ll SOS TEHPERRTURE:RllEIENT EXCl TRTION :314
RO:
0.60+-0.04 0.037+-0.003
rw:
i6.8+-1.3 NSEC 5.4*-0.5
NSEC
213.*-13. NSEC
EHISSION:393 REMRRKS:I4E-4
0.36+-0.05
fl CU++
TSHIFT:0.0000
NSEC
FLUORESCENCE
ORTR F I T
I
elements. The index (0 Ik IN - 1 ) is associated with the time t k = k / N in the normalized period [0,1]. From the theoretical standpoint one should consider the functions as sampled at the discrete points, t k . This allows proper application of the sampling theorem. With this theorem all frequency limited functions can be completely reconstructed from the data. However, from the experimental standpoint, the data point k of the array is not a sampled value but is the sum of the events accumulated in the interval [(k - l/z)/N, (k ‘/J/NI. Such an interval is called a channel and may be analytically represented as an integral over this interval. The number of counts observed in a channel is a random sample from the Poisson distribution of the expected number of counts for that channel. For a single channel this implies that the variance of the counts is where X is the mean number of counts for that channel. For large values of X ( 3 0 ) the Poisson distribution may be approximated by the Gaussian distribution. A distinction between the concepts of sampling and integration is only required if decays with a decay constant on the order of one channel are to be considered, a condition which the experimenter corrects by increasing the number of samples in the period to be used. The following conventions will be used in this paper. The discrete Fourier transform7 is required subsequently and is defined as 1 N-l F(k) = -Cf(t,) exp[-2aikt,] Nj=o
+
RAW D A T A
EXCITRTION
Good et al.
I
I
I
0
264.34 NSEC
1 N-l
E(k) =
M -6
M
-
k
Ce(tj)exp[-2?riktj] Nj=o
k = O...N - 1,
(3)
tj = O...j/N...N - 1 / N
In the Fourier space the deconvolution is given by ORTE:lS-JUL-82
NUMBER OF POINTS:512 NUHBER OF RUNSt141 PHOTOPHYSICRL CHEMISTRY ETH-ZUERICH OECDN
(4)
H = F/E
Figure 1. A multiexponential curve fit for the fluorescence of 20 X 10” M pyrene in 30 X M SDS with 1.4 X IO-’ M Cu2+.
The functions H , F, and E are complex valued.
squares of the real and imaginary parts of the Fourier transform they elected to determine the parameters by an unweighted quotient of the real and imaginary parts of the Fourier transform. Compare eq 40 of ref 6 and eq 18 of ref 1. The quotient used by OConnor et al. has no statistical foundation. Hence the results of the comparison, that the Fourier transform fitting technique is not optimal, do not apply to the original proposal. We have compared our Fourier transform technique with the iterative convolution (IC) method and obtained error functions of these methods and verified them by simulation. These show that the two techniques are statistically equivalent. Although the Fourier transform (FT) technique is conceptually more complex, it is appreciably quicker in computer execution, and therefore considered the method of choice.
We deal exclusively with a single exponential decay h(t),which may be written in the form
+ iHI(k)
H(k) = H,(k)
h(t) = a e x p [ - t / ~ ]
(5)
0 It Im
(6)
where a is the preexponential factor and r the decay parameter. Note, that the response function is not uniquely determined by the deconvolution. For instance the system response function n=+-
h’(t) =
a’exp[-(t n=-m
+n)/~]
0 It I 1
(7)
leads to the same system response as h(t). In the case of a 6 function e’(t) = 6 ( t ) it follows
2. Terminology and Notation The measured periodic decay curve, f(t), of the fluorescent system may be represented by a convolution integral of the periodic excitation pulse, e ( t ) , with the system response function, h(t):
At) = h(t)*e(t)
(1)
Suppose that the shape of one single excitation pulse can be characterized by e’(t). Then, if we normalize the repetition period to 1, e ( t ) will be given by n-+-
If the number of channels N is sufficiently large with respect to the decay parameter r , given in dimensionless units, H a n d respectively F, are given by 1 - 2nrik H=F=ar (9) 1 (2ark)’
+
3. Methods of Analysis
(2)
There are a variety of methods that may be utilized to analyze the collected data. These include iterative convolution (IC),*the
Equation 2 represents the periodic excitation function formed by the superposition of the nonperiodic function e’(t) shifted by all multiples of the normalized period. In order to perform any data analysis the functions must be represented by an array with N
(7) J. W. Cooky, P. A. W. Lenis, and P. D. Welch, IEEE Trans. Audio Electroacoust., Au/17 (2) (June 1969). (8) A. E. Knight and B. K. Selinger, Spectrochim. Acta, Part A , 27, 1223
e(t) =
n=-m
e’(t
+ n)
(1971).
Fluorescence Lifetime Fitting Techniques
The Journal of Physical Chemistry, Vol. 88, No. 22, 1984
method of moments (MM)9, the Fourier transform (FT),ls6 and modulating functions (MF)l0. However, only the IC and FT use all the information available and deliver a minimum variance in the lifetime estimates, Le., these methods smooth or suppress random occurrences in the best way. A detailed description of the FT program used is to be found elsewhere.” Two further methods, the square root (FT-SR) method and the phase (FT-P) method, are derived from the FT technique. These are to be considered as special cases, since their application is limited to specific circumstances. The expected and observed values for a function are denoted by the subscripts e and 0,respectively. The parameter vector is denoted by p. 3.1. Iterative Convolution ( I C ) . The I C method minimizes the sum of squared errors by iteratively approximating the parameters in the response function, h(t;p). This is a rather timeconsuming process, since the expected function, fe(t;p),has to be obtained by convoluting the excitation function with the hypothesized response function. This may be written as minimizing
where N- 1
g2Cf,(tt;P)) = fe(tk;P) = N-’ C e ( t k - t j ) h(tj;p) j=O
(1 1)
and p represents the set of parameters of the decay function. 3.2. Fourier Transform ( F T ) . Basically, the FT fitting procedure is a least-squares fit in the frequency domain, whereas the IC method is a least-squares fit in the time domain. The least-squares fit in Fourier space leads to the minimization of12 where the Fourier coefficient vector
is given by
Y
mization process. In practice the number of Fourier coefficients N C can be reduced for the computation of eq 12 and eq 14. This is necessary for eq 12 since the inverse of the covariance matrix CR is required. 3.3. The Square Root (FT-SR) and Phase (FT-P) Methods. Two special case methods may be derived from the FT method. These are the square root method (FT-SR) and the phase method (FT-P). Both rely upon the fact that, when the decay is single exponential and the statistical fluctuations in the data are ignored, the significant information resides in the first two Fourier coefficients. The lifetime estimates may be calculated by
Compared to a least-squares fit in Fourier space the FT-SR and FT-P methods use only two Fourier coefficients for the computation of T . This leads to a greater statistical error in T since there is no possibility of compensating the fluctuations in the Fourier coefficients. O’Connor et aL6 have used a similar method. Since a lifetime may be calculated from the real and imaginary part of every frequency the authors used a weighted expression for the calculation of the lifetime. They stated that this leads to better results than eq 15 and 16. However, this method must be biased, since the T’Sare strongly correlated, especially for small lifetimes. It is therefore not surprising that the method appeared to be worse that the conventional methods. 4.0. Analytical Comparison
All of the data analysis methods discussed in section 3, when applied to random data, provide parameter estimates. These parameter estimates p i are random variables and have a mean p i and a variance Ap?. Using straight forward statistical methods one may derive theoretical expressions for
H=
Ap; =
CR is the correlation matrix of the vector R. The superscript T stands for the transpose operation. In what follows we will denote by N C the number of the complex valued coefficients used. If the off-diagonal elements of the correlation matrix are set to zero1 the problem reduces to the minimization of NC
LsQ(P) = xwR(k)[He,R(k;p) - Ho,R(k)12 -k k=O
NC
C w d k ) [ H , , d k ~ )- H0,1(k)I2 (14)
k=O
The terms w R ( k ) and wI(k) are the weights of the Fourier coefficients. As previously defined H = FIE. The derivation of the matrix CR as well as wR and wI are contained in Appendix A. For example, He,R and He,Iare given by eq 9 for a single exponential decay. In comparison to the IC method, eq 10, the FT method, eq 12 and especially eq 14, require less computer time since the lengthy convolutions, eq 11, are no longer necessary during the mini(9) I. Isenberg, R. D. Dyson, and R. Hanson, Biophys. J., 13, 1090 (1973). (10) G. Striker in “Deconvolution-Reconvolution”,Nancy Symposium Proceedings, France, July 1982, M. Bouchy, Ed., 329. (11) H. P. Good, A. J. Kallir, and U.P. Wild, to be submitted for publication.
5437
~ @ j
- pi)’
(17)
These derivations are lengthy and are therefore not detailed here. In all cases, we linearized the nonlinear model function which depends on the parameters p. This linearization is certainly valid for all practical purposes. More generally, the covariance matrix C p of the parameter vector p, see ref 12, is given by cp
= (FC;lX)-l
(18)
where Cy is the covariance matrix of the experimental (or transformed) data, and X the matrix of the first partial derivatives of the model function. This technique is equally applicable to all methods discussed in section 4. The variances Ap? are the diagonal elements of Cp. 4.1. Iterative Convolution (IC). For a 6 pulse excitation and a model function h ( t ) = a exp[-t/~] it is shown in Appendix B that the normalized relative error in the parameter is given by
where N- 1
NT
= Cfo(tk) k=O
Le., the total number of counts accumulated. mi is the ith time moment, and N the number of channels. From this equation, with r = exp[-l/~], a number of useful relationships may be derived: (12) R. Deutsch, “Estimation Theory”, Prentice Hall, New York, 1965.
5438 The Journal of Physical Chemistry, Vol. 88, No. 22, 1984
Good et al.
3
i
1
0.01
7
0.1
1
-
Figure 2. Normalized tIC= ( A T / T ) N , I /for ~ different number of channels N: (1) N = 64; (2) N = 128; (3) N = 256; (4) N m.
=1
EIc(7+0,N+-) .$,-(T-L-,N-L~)
32
121/2~
0.1 1 10 T Figure 3. Normalized relative error t ( 7 ) = (Ar/7)NT1I2for different methods: (1) iterative convolution &c(T,N m ) and Fourier transform method [ ~ ( T , N C = 10); (2) square root method [FT-SR(~); (3) phase method t m + ( ~ ) . -+
(22)
(23)
In our derivation of eq 19 overlap is assumed. Overlap occurs if a photon due to the nth excitation pulse is not counted in the nth interval but a subsequent interval. Previous deliberation^,'^^*^^ for the limiting case of no overlap, have derived equations similar to eq 19. In order to obtain eq 20 the moments must be calculated analyti~ally.~ The proof of eq 21 starts by replacing the summations in the moments by integrals, whereas eq 22 and eq 23 are limiting cases of eq 21. In Figure 2 eq 20 is plotted for different number of channels. From this curve one may conclude that in the case N = 64, 7 should lie between 0.01 and 0.2. If the number of channels is increased, the lower limit is significantly reduced. 4.2. Fourier Transform (Fq.The covariance matrix Cyfor the Fourier coefficients may be found in Appendix A. The analytical Fourier transform H ( k ; a , ~which ) corresponds to the single exponential decay function h ( a , ~is) given by 1-2aik~ H(k;a,7) = a7 1 (2sk7)’
+
In this case the matrix X i s given by eq 25. Working with
I 0.1
10
1
Figure 4. Normalized relative error
7
~ ~ s R ( T )=
(A7/7)NT1/*for the Fourier transform square root method with a Gaussian excitation profile of width fwhm: (1) fwhm = 0.08; (2) fwhm = 0.05; (3) fwhm = 0.
4.3. Square Root (FT-SR) and Phase (FT-P) Methods. For the square root method the variance may be calculated from eq 15 and is
while that of the phase method is
where fl = ( 2 ~ 7 ) ~ . These are displayed in Figure 3. The derivation is contained and Ern+, which in Appendix C. It is not surprising that the only use part of the information available, are always greater than the EFT and tIc,which are based on the full data and are found to be equivalent. It is, however, at first glance surprising that in the limit 7 0 all four 5 functions approach almost the same value. For the Fourier transform square root method the simplifying assumption, that the excitation pulse is a 6 function, is relaxed in this section. If we assume that the excitation pulse +
x=
e’(t) =
-
4 In 2 exp[-42’ In 2/fwhmz] fwhmZ a1l2
can be approximated by a normalized Gaussian function with a full-width at half-maximum in dimensionless units (fwhm), then the computations above may be repeated, giving
continuous time functions and Fourier series this corresponds to the N m case treatment in section 4.1. In Figure 3 the function EFT (in curve 1) corresponds to .&(T&’--), eq 21, which coincides with EFT(7). In evaluating the function f m , the normalized standard error using the FT method, only the first 10 Fourier coefficients have been considered (NC = 10). ~~
~~
(13) Alan J. Kallir, Honours Thesis, Chemistry Department, Australian
National University, 1979.
where y = exp[fwhm2?r2/(2 In 2)]
(29)
The Journal of Physical Chemistry, Vol. 88, No. 22, 1984 5439
Fluorescence Lifetime Fitting Techniques TABLE I: Iterative Convolution (IC)
TABLE 11: Fourier Transform (FT), NC = 10 tm
€IC
simulation
7
1024; NT = lo* 1.1 1.1 1.9 2.9 3.7 7.4 18 30 39
1.o 1.1 1.9 2.9 3.6 7.0 17 28 35
N = 256; NT = IO6 1.o 1.1 1.o 1.2 1.1
1.o 1.o 1.o 1.o 1.o
N 0.1 0.2 0.5 0.8 1 2 5 8 10 0.01 0.02 0.03 0.04 0.05
theory
simulation
E
h-SR
512
1024
theory"
Diagonal Elements Set to Zero; N = 256, 512, 1024; NT = los 8.5 2.4 2.2 1.2 1.4 1.4 1.5 1.5 1.5 1.7 1.5 1.6 1.6 1.7 2.0 2.2 2.2 3.2 3.1 3.1 4.1 3.8 3.7 6.6 7.3 7.9
0.01 0.02 0.05 0.08 0.1 0.2 0.5 0.8 1.o 2.0
(30)
Working with broad excitation pulses leads to a significant decrease in the precision in the short lifetime region, see Figure 4.
Set to Zero: N = 1024, NT = IO5 1.1 1.1 2.0 2.9 3.7 7.7 32 78
1.o 1.o 1.o 1.o 1.o 1.1 1.9 2.9 3.5 7.0
Diagonal Elements Not
As required F*R-SR(fwhm+O)
256
7
0.1 0.2 0.5 0.8 1 2 8 20
'Diagonal elements not set to zero, N
5. Simulations The analytical results provide insight into the behavior of the normalized relative error functions for the simple cases. In this section we extend the domain of interest and verify the results obtained analytically using computer simulations. All the computer simulations described, vide infra, were performed on a DEC PDP 11/34. The synthetic data were generated by adding Gaussian "noise" to the decay curve. Each synthetic decay was fitted against the parent model, as if the data were truely experimental, providing a lifetime estimate. This process was repeated 200 times, in order to estimate mean value and variance of the lifetime. Under the assumption that the estimators are normally distributed this limits the error in A? to about f20%, with a confidence level of 95%. Since the true parameters of the synthesized data are known, a priori, the simulation provides a check for the analytical results and enables the bias and other numerical effects, caused by simplifying the fitting technique, to be evaluated. For a range of 7,NT, and N values the results obtained are displayed in Table I for the I C method. These data verify the analytical expressions obtained in section 4.1. The results for the FT-SR and FT-P methods are displayed in Tables I11 and IV, respectively. Except for tFT-sR(N= 256,NT=105,r=o.01) and EFT-~(N=256rNT=105,r=0.01) all variances lie within the theoretical limits of f20%. The bias in these cases is due to the fact that the analytical results were derived for N a. The FT-SR method has significantly less bias than the FT-P method. Table I1 shows the effect of the off-diagonal elements of the covariance matrix on tR. This corresponds to eq 14. The results clearly show that the bias is increased slightly by setting the the off-diagonal elements to zero. However, in practice systematic uncertainties are usually larger than the error introduced by the above-mentioned approximation. There is no difference between the theoretical results and the simulation if the full correlation matrix is used. The effect of finite excitation pulse width on the FT-SRmethod is found in Table V. Compared with theory (eq 28) the difference is found to be insignificant. For comparison calculations for the IC method are included.
-
6. Conclusion
The Fourier transform technique described has, in the limit, the same error function as the iterative convolution technique. These techniques share the same statistical basis. The FT method
-
1.0 1.1 1.9 2.9 3.5 7.0 28 70
m.
avoids the lengthy computations of the IC method. The number of channels, N, should be chosen as large as possible, and the whole correlation matrix used if the bias in the FT method is considered significant. For experiments with a &excitation pulse and a large number of channels the variance in the lifetime estimate is given by eq 21. This variance is due to the counting process. N o known fitting technique can produce less variance. Other fitting techniques may, at best, attain this value. If the excitation pulse is neglected, a "rule of thumb" may be constructed to estimate the statistical error in the lifetime estimate. AT/T = NT-'J~
T
5 0.3
= 3 . 5 i " ~ - ~ / ~ 7 > 0.3 (31) Though eq 31 cannot be easily extended to compute the variances in the lifetime estimates derived from decay curves contaminated by the excitation pulse, eq 27 may be used. The approximations contained in eq 27 do not severely restrict its applications to other fitting techniques. The principles outlined in this work may be applied to decay curves recorded with background noise and multiexponential decays. Acknowledgment. We thank the Swiss National Science Foundation for financial support. Appendix A
Suppose that e'(t) = S ( t ) , then Cfi = CF
-
H=
5440 The Journal of Physical Chemistry, Vol. 88, No. 22, 1984
TABLE V Influence of the Excitation Pulse, Square Root Method (FT-SR) with NT = lo6 and N = 256
TABLE III: Sauare Root Method (FT-SR) {FT-SR
simulation
T
N = 2048;NT = 1 .o 1 .o 0.9
0.01 0.05 0.1
theory“
EFT-SR
lo5
simulation theory fwhm = 0.05
T
1.1 1 .o 1 .o
N = 1024;NT =
“N-
Good et al.
lo5
0.01 0.05 0.1
1.1 1.0 0.9
1.1 1 .o 1 .o
0.01 0.05 0.1
N = 256;N~ = 105 2.2 1.1 1.1
1.1 1 .o 1 .o
0.01 0.02 0.03 0.04 0.05 0.06 0.07
2.4 1.4 1.3 1.1 1 .o 1.1 1.1
0.01
4.4 1.9 1.5 1.2 1.2 1.1 1.1
[IC theory
2.5 1.4 1.2 1.1 1.1 1.1 1.1
2.2 1.4 1.1 1.1
1.1
fwhm = 0.08 0.02 0.03 0.04 0.05 0.06 0.07
a.
TABLE I V Phase Method (FT-P)
4.8 1.9 1.41.1 1.2 1.1 1.1
3.4 1.8
CFT.P
simulation
7
2048;NT = 1 .o 0.9 1 .o
lo5
0.01 0.05 0.1
lo5
0.01 0.05 0.1
N = 1024;NT = 1.2 0.9 1 .o
0.01 0.05 0.1
N = 256;N~ = 105 1.9 1.1 1.2
N
If E ( k ) is not a random variable, then cov (HR(k),HR(j)), cov(HI(k),HI(j)), and cov(HR(k),HIu))may be calculated. It easily follows, for example, that
theory“ 1 .o 1 .o 1.2
a2(HR(k))= l/wR(k) = u2a2(FR(k)) 2uv cov (FR(k),FI(k)) v 2 ~ 2 ( F I ( k ) )
+
1 .o 1 .o 1.2
+
UZ(HI(k)) = l/w,(k) = u2a2(F1(k))+ 2uv COV (FR(k),FI(k))+ v 2 a 2 ( F ~ ( k (A.8) )) with
1 .o 1 .o 1.2
u=
ER(k)
V =
ER(k)2+
ER(k)2
+ EI(k)2
and
“+a.
2 N g 2 ( F ~ ( k )=) FR(O)+ F ~ ( 2 k ) 2Nn2(FI(k))= FR(O)- F ~ ( 2 k ) cov Cf(t,)f(ti)) = 0
=At,)
i #J
2N cov (FR(k),FI(k))= FI(2k)
i *j
which are derived in above.
one finds
2N
COV
Appendix B
+ FR(k+j)
2N cov(F~(k),FR(j))= FR(k-j)
(FI(~),FI(~)) = FR(k-j) - F ~ ( k + j )
(A.4)
+ F,(k+j)
2N cov ( F 1 ( k ) , F ~ ( j )=) FI(k-j)
a
At) = -exp[-t/r]
This means that the covariance matrix of the Fourier coefficients is easily deduced from the Fourier coefficients themselves. Note 03 that for a single exponential decay for N
-
F1(k) = 1
Suppose that e’(t) = 6(t) and h(t) = a exp[-t/r] then the periodic decayf(t) given in eq 1 becomes where r = exp[-l/r]
1- r
By definition
-27rkar2 (27rrk)2
+
Thus, for example 2N
COV
+
(FR(~ ) , F R1)) ( = FR(O) F R ( ~=) UT
+ 1 + ar(47rT)2 (A.6)
Now suppose that e’(t) #
6(t).
Therefore
a4 4
- E -
aa
afi = A [ 2, a?
a
+
41
1-r
$
Using the above definitions and eq 18 we find
(B.l)
The Journal of Physical Chemistry, Vol. 88, No. 22, 1984 5441
Fluorescence Lifetime Fitting Techniques Using
AT2 T N T
1 N-l
FR(k) = 6,
cy,
- Ef(rj) Nj=o
COS
=
m0
(B.17)
m2 - m12/mo From eq B.17 it easily follows
[27jk/"j
and fl are given by
It is convenient to make a change of variable X = l / N r , yielding
N-2-d2 In (mo)= m2 - m12 dx2 mo moz
(B.19)
Computing (d2/dx2) In (mo)by usiqg mo = 1 - exp[-Nx]/(l - exp[-x]) yields eq 20.
Appendix C if use is made of the above substitutions ( y l = -yc
a
=
p' = p ~ ~ -( r1) =
1 -r
Taking HR(O) = a7 and HR(1) = a7/(1 exp[-rj/
71
+ (277)')
gives
(B.7)
The matrix of the first partial derivatives may be written
C
-1
N- 1
mo = Cexp[-tj/7] j=o
(B.9)
(C.2)
l+P
N- 1
ml = E t j exp[-tj/r]
(B.lO)
j'0
N- 1
m2 = Ctjz exp[-tj/7]
(B.ll)
1'0
a
NT = mol-r
(B.12)
Equation B.3 may be rewritten in the form AT^ m0 T N T = cy' - P 2 / m o
(B.13)
with p = ( 2 7 ~ ) ~ . The covariance matrix Cyof the transformed experimental data is, according to Appendix A 1
1 l+P
[&
y2 1 t
I
(C.3)
-") 1 + 4p
from which the weighting matrix may be derived.
Insert
[
e' = Cr'i rj
+1:rl2
(B.14)
p' = Z r f tj
+1-r
'1
(B.15)
mo = C r Q in the right side of eq B.13 and verify
1-1
lfiL
(B.16)
I
I
-I
Using the definition NT = arN and substituting eq C2 and eq C4 into eq 18, one obtains eq 26.
