338

DECONVOLUTION OF KINETIC DECAYS AND FITTING OF MULTIPLE EXPONENTIAL DECAYCURVES BY THE LINEAR PREDICTION METHOD

J. TANG t ) and J.R . NORRIS 1,21"Chemistry Division, Argonne National Laboratory, Argonne, IL 60439, USA and°'Department of Chemistry, University of Chicago, Chicago, IL 60637, USA

Received 20 January 1988

A method based on linear prediction (LP) theory is employed m analyzing multiple exponential decay signals . A generalized LPmethod is also proposed to determine true system response in deconvolution problems when the convoluted signals have a decay timecomparable to the duration of excitation pulse. These LP methods linearize the least-squares fit and require no prior knowledge ofthe exact number of the decaying exponents .

1 . Introduction

The system response function of a large number ofphysical systems can be characterized by multiple ex-ponential decays . The signals y(t) obtained in time-re-solved experiments are convoluted signals of the systemresponse function g(t) and the effective excitation pulseprofile f(t) as [1]

J. (t)= f If(T)g(t - T) dT=ff(t - T)g(T) dT,

(1)u

where the effective pulse profile f(t) itself is actuallyalso a convolution of the real excitation pulse and theinstrument response function . Because it is the effectivepulse profile f(t) that is measured from experiments,wherever in the later discussion the excitation pulse willrefer to the effective pulse profile for brevity . Theproblem m deconvolution is to determine the systemresponse function g(t) from the above integral equationwith given y(t) and f(t) . If the excitation pulse is muchshorter than the characteristic time of the system re-sponse function and can be approximated by a Diracdelta-function, then the observed signal is the same asthe system response function except noise contamina-tion . In this case, a deconvolution procedure is notnecessary and the problem is reduced to the usual andmuch simpler multiple exponential curve fitting . How-ever, if the pulse width is comparable to the decay timeof the system, the true response function must be de-convoluted to order to remove the effects of the pulseprofile from the measured signals .A nonlinear least-squares fit is the most common

method in use to analyze composite damping transients .For example, the method often used m deconvolutionof kinetic decays such as the Grinvald-Steinberg itera-

0168-9002/88/$03.50 © Elsevier Science Publishers B.V.(North-Holland Physics Publishing Division)

Nuclear Instruments and Methods m Physics Research A273 (1988) 338-342North-Holland, Amsterdam

tive reconvolution method is also a nonlinear routine[2] . In general, these nonlinear approaches require aspecific model, an exact number of damping exponents,and a good guess of initial parameters for iteration .Otherwise, the minimization mechanism may be trappedinto other local minima . Methods using integral trans-forms are sometimes used to provide the necessaryinitial estimates of a nonlinear least-squares fit [3,4] .

Noniterative methods exist but they have been ap-plied only to single exponential or double exponentialtransients [5,6] . Prony's linear prediction (LP) theory[7], however, linearizes the least-squares fit and requiresno prior knowledge about the precise number of thedecaying components . This method can be employed todetermine multiple exponential (real or complex) decays[8,9] and has also been applied by us and others tomagnetic resonance spectroscopy to improve resolutionand sensitivity [10-131. Recently, the LP theory hasbeen generalized and extended by us to deconvolutekinetic decays from pulse excitation of a finite duration .A coherent presentation of the LP theory for bothfitting of multiple exponential decay curves and decon-volution of kinetic decays will be given in this paper.

2. Theory

The LP theory was first developed by Prony [7] inthe late eighteenth century and has not been appliedextensively m spectroscopy until dust recently . Thefundamental assumption of the LP theory is that thesignal is a superposition of a finite number (M) ofexponential functions, either exponentially dampedsinusoids or pure exponential decays . Since signals areusually sampled at a regular time interval of T, first let

us assume a noiseless discrete data sequence yn with Mexponents as given by

M

yn -akzkk=1M

_

ak exp( -n YkT- 1nWkT),

2k=1

where 0 < n < N- 1 and N is the total number of datapoints . Yk is the decay rate and co, is the oscillationfrequency . The effects of noise will be discussed later . Itcan be shown [14] that if the zk's in eq . (2) are roots ofthe following polynomial equation of order M,

M

where ak's are the LP coefficients . Because each datavalue is related to others in a linear fashion, eq . (4) isknown as the linear prediction relation with a LP filterlength M.

2.1 . Fitting of multiple exponential decay curves

J Tang, J.R. Norris / Deconvolution of kinetic decays

As discussed earlier, the LP relation in eq, (4) isrigorous for a noiseless signal of multiple exponents . Inpractical cases, however, a signal contains noise ofvarious extent . It is a good approximation to assumethat the LP relation still holds in the presence of noise,except that the LP filter length M should be increasedto accommodate the presence of noise . Eq. (4) is astandard linear equation and can be solved by a linearleast-squares procedure using a Householder decom-position (QRD) [11] or a singular value decomposition(SVD) [8,10] routine . The algorithms and Fortran pro-gram listings for both routines are available from LIN-PACK [15] . Either QRD or SVD allows one to de-termine the rank of a data matrix (i .e ., the number ofindependent equations) and to reject noise by excludingsmall eigenvalues usually associated with noise. Oncethe LP coefficients are determined from eq . (4), one canconstruct the polynomial eq . (3) and solve for its roots[8,10,11] . The roots of the polynomial equation aregenerally complex numbers as zk = exp( -YkT -'WkT)with real parts related to the decay rates and imaginaryparts related to the oscillation frequencies .

In the special case of kinetic decay, only the physicalroots corresponding to a damping exponential need tobe kept for further process . Other roots correspondingto an increasing exponential or an oscillation at highfrequency can be rejected . Once zk 's are solved, thefinal step is to determine the amplitude for each decaycomponent. Eq . (2) is a linear equation with M un-

knowns in ak and is usually overdetermined . The am-plitude can be solved by a standard linear least-squaresroutine .

The rooting of the polynomial eq . (3) with real LPcoefficients can be solved by using the routine from ref.[16] . For data with complex values the correspondingLP coefficients are also complex . In this case, one mayuse the ZCPOLY routine from IMSL [17] . If the orderof the polynomial equation exceeds 50 which is theupper limit of the ZCPOLY routine, we have developeda different approach by finding the eigenvalues of theso-called companion matrix [18] . One can show that theinverse of the roots of the polynomial eq . (3) are eigen-values of the following MxM matrix

By using routines from the EISPACK [19], one cansolve eq . (5) for its eigenvalues and determine the rootsof the corresponding polynomial .

2.2. Deconvolutton of kinetic decays

In the time-resolved experiments the observed signaly(t) is generally a convoluted response of the systemresponse function g(t) and the excitation pulse profilef(t) as shown in eq. (1) . For discretely sampled data,the convoluted integral can be expressed as a convo-luted sum of two sequences gn and fn ,

k

yk - Y_ fn gk -nn=0

If the system response function, g(t), is a multipleexponential decay or a nonexponential decay but can beapproximated by a multiple exponential decay in awhole observation period, then according to the LPtheory discussed earlier the following equation holds,

P

gn - - Y_ akgn+k ,k=1

and

yn = - F akyn+k - Y bkfn+k ,k=1 k=1

P

bn = - Y_ akgk-n'k=n

339

6

7

As in the LP method for curve fitting of multipleexponential decay, a larger filter length (P) should beused instead of M in order to improve temporal resolu-tion [10,11] . By substitution of the above LP relationinto eq . (6), we have derived the following generalizedLP relation between yn and fn ,

8

A(z)=1+ Y akz k, (3) 1 0 0 0k=1

-a ] . . .

i .e ., if A(zk)= 0 for k = 1, 2, - - - M, then-a2 0 1 0-a3 0 0 1 . . .M

yn - Y_ akyn+k, 0 < n < N-M- 1, (4) . . . . . . 0k=1 - a.,H-1 0 0 1

34 0

The above eq . (8) is a standard linear equation with2P unknowns, a t , a2 , .' -, ap , bi , 62, . --, and bp.Again these unknowns can be determined by a linearleast-squares procedure using the Householder decom-position [11,15] or the singular value decomposition[8,10,15] . They can also be solved by inverting thematrix of the corresponding normal equation if thematrix is nonsingular . Once the LP coefficients of a,n 's

are determined from the first linear least-squares proce-dure, a rooting procedure of the polynomial equation asin eq . (3) can be applied to calculate the decay rateconstants and the oscillation frequencies . Since the sys-tem response is generally a pure decaying function, onlya few roots (M) corresponding to zero frequency or lowfrequencies need to be kept for further processing . Allother extraneous roots (P - M) with exponentially in-creasing rate constants or high frequencies should berejected . Finally, the amplitude for each of the Mphysical roots, ak , can be determined by the secondlinear least-squares fit to

M

Yn - E Vnkak+k=1

wherenr

Vnk-f f( t) e-l'stnT -~> dt .

0

Since the excitation pulse profile is usually measured indiscrete points, the integration shown above can beevaluated by either the trapezoidal rule, Simpson's ruleor any other more accurate integration method .

3. Applications

9

In our test of the LP method for curve fitting anddeconvolution, we are primarily interested in signals ofmultiple exponential decays, although the method canalso be applied to exponentially damped sinusoidalsignals such as magnetic resonance free induction de-cays . In the first example, a simulated exponential de-cay signal with three components is generated with 200data points and a sampling time interval of one second.The data values are rounded to the second decimal andnoise is introduced primarily by the rounding errors .The theoretical values for each exponent and their fittedvalues by using the LPQRD method with various LPfilter lengths are listed in table 1. Based on our experi-ence, an optimal filter length is usually about twice theeffective decay time of the signal, i.e., it is about twicethe number of points where the signal drops below 1/eof the initial amplitude.We have also tested the generalized LP method for

deconvolution by analyzing simulated signals with vari-ous conditions for pulse shape, decay function andnoise level . Poissonian noise was added to the signal

J Tang, J. R Norris / Deconoolution ofkinetic decays

Table 1Fitting of triple exponential decay

according to the single photon statistics [1] . The noiselevel in each channel was chosen with a standard devia-tion equal to the square root of the number of photoncounts in that given channel . In the example shown mfig. 1, two exponential decay components were convo-luted with a simulated pulse of a functional form ast 2exp(-t/2 .5) x 200 data points are generated with onesecond for the sampling time . The amplitude of thepulse was adjusted so that the signal peak counts equal

Fig. 1 . (A) Simulated pulse profile of f(t) =t2 exp(-t/2 .5)with 1 s sampling time . (B) Convoluted kinetic decay, y(t), ofdouble exponents with g(t) =10exp(-t/12 .5)+exp(-t/50) .The pulse profile in the figure is normalized to the peak counts

of the kinetic decay curve for easier display.

Amplitude Decay timeIs]

Theoretical value 3.00 50002.00 20 .001 00 500

LP filter lengthP = 40 2.86 5095

210 21 .04104 5 11

P = 50 3 .02 49 .951.99 19 .80099 495

P =60 302 49.931.99 19.810.99 498

P =70 303 49871 .99 19.660.98 4.91

Table 2Parameter values for deconvolution of double exponential decay

to 5000 . The system response function is given byS(t) = aiexp(-t/ß,) + a2exp( - t/ß2)i with a, = 10.00,ß, = 12.50 s, a2 =1.00 and ß2 = 50.00 s. In this test aLP filter length was set at 50 and the rooting procedureof A(z) yields only two roots with zero frequencyindicating a double exponential decay. In fig. 2, thesimulated convoluted signal and the fitted curve by thegeneralized LP method are displayed with a semiloga-rithmic scale . We have analyzed five data sets of thesame peak counts but with various distribution of noise.The resultant fits for the kinetic parameters by the LPmethod and the nonlinear Grinvald-Steinberg iterativereconvolution method [2] are listed in table 2. Bothmethods have similar accuracy but in the Grinvald-Steinberg method, one has to assume two exponents forthe system response function .

4. Discussion

In this report, two distinct kinds of problem areaddressed : the fitting of multiple exponential decaysand the deconvolution of kinetic decays . The LP method

Fig. 2 . (A) Semi-logarithmic plot of the convoluted kineticdecay as indicated by the (lower) solid line . (B) Fitted curve bythe LP method as represented by the dotted curve (under thesolid curve) (C) Residuals of the fit as shown in the top curve .

J. Tang, J.R. Norris / Deconoolution of kinetic decays

f(t) = t 2 exp(- t/2.5), g(t) = al exp(- t/ß,)+ a2 exp(- t/#2), peak counts = 5000

Acknowledgement

References

[3][41[5][6][71[81

34 1

we have presented can be applied to both areas as wellas to systems with exponentially damped sinusoids.Unlike the common nonlinear least-squares approaches,the "a priori" assumption of the total number of com-ponents is not required in the LP method . The matrixeq . (4) of the LP relation or the generalized LP relationin eq . (8) can be solved by any linear least-squaresprocedure using a Householder decomposition (QRD)or a singular value decomposition (SVD) [15] . One mayalso use a Cholesky decomposition (CHD) to solve thecorresponding normal equation [15] . For improvedcomputational efficiency, we have used the QRD orCHD routine with a matrix pivoting procedure insteadof SVD. In most applications we have found them asreliable as SVD.

This work was supported by the US Department ofEnergy, Office of Basic Energy Sciences, Division ofChemical Sciences, under contract W-31-109-Eng-38.

J N.

Demas,

Excited

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Lifetime

Measurements(Academic Press, New York, 1983).

[2] A. Grmvald and I.Z . Steinberg, Anal Biochem. 59 (1974)583.S.W . Provencher, J. Chem . Phys . 64 (1976) 2772 .H.J . St6chmann, Nucl . Instr. and Meth. 150 (1978) 273.A.H. Kalantar, Nucl . Instr. and Meth . 215 (1983) 437T. Mukoyama, Nucl . Instr . and Meth . 179 (1981) 357.G.R.B . Prony, J. L'Ecole Polytechnique 1 (1795) 22 .R. Kumaresan and D.W . Tufts, IEEE Trans. ASSP-30(1982) 833.F. Villuendas and J. Pelayo, Nucl . Instr. and Meth . A253(1987) 288.

[10] H. Barkhuilsen, J. de Beer, W M.M.J . Bovee and D. vanOrmondt, J Magn . Reson. 61 (1985) 465.

[I1] J . Tang, C.P . Lin, M.K. Bowman and J.R . Norris, J.Magn . Reson. 62 (1985) 167.

[12] J. Tang and J R. Norris, J. Chem . Phys. 84 (1986) 5210 .[13] J . Tang and J.R . Norris, J. Magn. Reson. 69 (1986) 180.[14] S.M . Kay and S.L . Marple, Proc. IEEE 69 (1981) 1380

ai ß, Is] a2 92 [s]Theoretical values 10 .00 12 .50 1 .00 50 .00

LP method, filter length P= 50 10.01+0.06 12.44±0.20 1 .02±0.07 49.55±1.30

Grinvald-Steinberg iterative reconvolution method 10.03+0.09 12.47±0.14 0.99±0.06 50.40±131

342

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J. Tang, J.R Norris / Deconoolution ofkinetic decays

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