Algebraic Analysis of Noisy Exponential Decays

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JOIJRNAL OF MAGNETIC RESONANCE x,223-228 (1977)

Algebraic Analysisof Noisy ExponentialDecays

L. A. MCLACHLAN

Inst i tute of Nuclear Sciences, D.S .I .R., Pr ivate Bag, Lower Hutt , New Zealand

Received August 29,1976

A simple algebraic metho d o f determining the relaxation time of an expone ntial

decay with an unknown basel ine is presented. This method is sui table for on- linecompu ter analys is, is very stable, and is comparable in accuracy to other s impl i fied

wa ys of analyz ing exponent ial decay s which have been descr ibed in the last few years.

1 N T R O D U C T I O N

,4 common problem in pulsed nuclear magnetic resonance experiments is the ana-

lyz,ing of a noisy exponential decay to determine the time constant r. Such a signal is

described by

y(t) = A exp(-t/r) + B + v(t), PI

where A and B are constants and v(t) is a Gaussian random variable of rms amplitude

a. ,4t equispaced time intervals A, a digital signal averager makes Nsequential measure-ments of this signal, so it may be written

y(nA) = A exp(-nA/z) + B + u(nA), PI

where 0 < IZ < N. Linear or nonlinear least-squares-fitting techniques are then usually

used to extract r from this equation. Recently, however, two papers have appeared

describing much simpler methods of analysis which need only modest computing

facilities.

In the first of these (I), z is found from the difference between the logarithms of pairs

of points, with the baseline obtained by varying B until the variance of r is a minimum.

A subsequent paper (2) showed that by first grouping the points in threes and solving the

resulting simultaneous equations for B, the accuracy of the method proposed in the

first paper can be improved. It is shown in this paper that a further simplification is ob-

tained by grouping the data in four blocks and solving the resulting simultaneous equa-

tions for r. The expression obtained is in a form suitable for use with an on-line com-

puter, or even an ordinary scientific pocket calculator with limited memory capacity.

DATA ANALYSIS

The N measurements are split into four blocks and each block is summed to give

(1/4)N-1

&= z: y(nA)=A(1-Q)(l-R)-1++NB+(N/4)1’2VI,II=0

(l/Z)N-1

S,= 2 y(nA)=AQ(l-Q)(l-R)-1+~NB+(N/4)“2V2,n=(1/4)N

Copyright 0 1 977 by Academic Press, Inc. 22 3Al l rights of reproduction in any form reserved. ISSN 0022-2364Printed in Great Britain



224 L. A. McLACHLAN

(3/4)N-1

S, = 1 y(nA) = AQ’(1 - Q)(l - R)-’ + $NB i- (N/4)“2 V3,n=(l/Z)N

N-lS, = C y(nA) = AQ” (1 - Q) (1 - R)-’ + $NB + (N/4)“2 I’,, [31

n(3/4)N

where R = exp(-d/r), Q = exp(-$Nd/r), and V,, V,, V,, and V, are random variables

with rms values of a. Solving these four equations for z gives

7-l = 4(Nd)-‘ln[(S, - S,) (S, - S,)-l].

Using the approximation ln(1 + x) = x gives the fractional error E n r as

[41

E= 2aA-‘P(A/z) G(NA/z), PIwhere

F(x) = x-112 [l - exp(-x)], PI

G(x) = xm112 2{1 + exp($x)}]“2/[l - exp(-ix)]-’ [l - exp(-3x)]-‘. [71

The error in r depends upon two parameters, the normalized total measuring time

NA/z, and the normalized time between two sequential measurements A/z. Equations

[5], [6], and [7] show that, since F(A/z) is a monotonically decreasing function, NA

should be chosen so as to minimize G(NA/z). A minimum occurs in G when NA = 5.22,

which fortunately is broad so NA need not be set accurately. For instance, E s within 20 %

of its minimum value for NA in the range of 3.32 to 6.62. If at all possible, short sweep

times should be avoided since the error increases rapidly for small NA/z, being double

its minimum value for NA = 2.12. Conversely, long sweep times are insensitive toerrors in the choice of NA, requiring NA = 112 for a doubling in error.

At the minimum of G, the value of E s

E, = 3.07 Nli2aA-l [1 - exp(-52/N)]. [81

This monotonically decreasing expression reaches its familiar asymptotic form of

E, oc N-1’2 for N greater than about 30.

For N less than about 30, E, is smaller than the value given by its asymptotic form.

For some types of experiments, such as T1 measurements, a modest gain in accuracy

can thus be obtained by doing repeated measurements with a comparatively small N

and adding them, rather than doing one measurement with a very large N. Repeatedmeasurements, rather than a single measurement accupying the same time, also often

reduce the effects of slow drifts in the equipment.

The expression for E assumed ln(1 + x) = x in its derivation. This approximation is

valid provided S, - S, % (N/2a)‘/‘, which, at the E minimum, can be shown to be

equivalent to N1’2 B 40a/A by substituting in Eq. [3]. The latter inequality is easily

satisfied for A/a greater than about 40 and can be satisfied for much smaller values by

using an N of some hundreds. Unfortunately, this contradicts the desirability of a small

value of N, which, as mentioned in the preceding paragraph, sometimes occurs.

If an estimate of the accuracy of r is also required, then both a and A must be found

and substituted into Eq. [5] or Eq. [8]. If a is already known, then simply subtractingthe last value of y measured from the first wil l give a sufficiently accurate value of A for

calculating a. More commonly, a must be found from the root mean square difference



ALGEBRAIC ANALYSIS OF EXPONENTIALS 22 5

between the actual values of y and those calculated from an exponential decay using

the measured values of A, B, and r. Solving the simultaneous equations [3] gives

A= (~1-&)(&-&)(1-R)

(&+&-&-&)(I -Q>’and

B,T (s,s,-s,s,)N(S,+ s, - s, - S,)

[91

Along with the calculated value of r and the measured values ofy, these give a value of a

with a fractional error of order Neil2 which may then be used to obtain E.Unfortunately,

estimating the accuracy of a measurement of r requires better computing facilities than

those required for calculating r alone, since the N data points must now be stored while

A, B, and r are being calculated.At first sight, since there are only three unknowns in Eq. [I], it would seem best to

divide the data into three blocks and solve the resulting three equations for r. Indeed,

if this is done, the expression obtained,

z-l = 3(NA)-‘ln[(& - S,) (S, - S&i], [I11

is similar to Eq. [4]. There is, however, one significant difference between the two ex-

pressions. In Eq. [I 11, noise associated with S, appears in the numerator and denomi-

nator in such a way that its effect is maximized, but in Eq. [4] there is no correlation

between noise in the numerator and that in the denominator. Because of this difference

in noise correlation, when the detailed expressions for E are examined, it is found that

in a ll circumstances the error for Eq. [ 1 I] is at least 30 % larger than that for Eq. [4].Although there is a distinct advantage in dividing the data into four blocks for

calculating r, this does not apply to calculating A or B since in these cases correlations

exist between noise in the numerator and noise in the denominator in both [9] and [lo].

Detailed calculations show that for NA 5 2.5~, Eq. [IO] is slightly more accurate than

the corresponding equation for three data blocks, but in the asymptotic limit of NA = m,

it is about 13 “//, poorer in accuracy.

These equations have all been derived under the assumption that the noise voltages

at consecutive samplings are uncorrelated. This is true for Ti measurements, and is

usually true for spin-echo measurements, but it is not true for T2 measurements from

the free induction decay since the sampling theorem requires that the noise bandwidthbe narrow enough for at least two consecutive sampled noise voltages to be partially

correlated. To avoid signal distortion, however, the bandwidth must be wide enough

for the number of part ially correlated sequential noise voltages sampled to be much

less than +N. Periodic noise, such as mains frequency ripple, may also introduce corre-

lation between the noise voltages.

The effect of correlated noise voltages is to increase the rms value of the cumulative

noise voltage Vof Eq. [3]. Since the correlation extends over less than about N/40 con-

secutive samplings, the increase in V is small, usually less than 10 %. Furthermore, the

form of Eq. [4] is such that the net effect of correlations between V, and Vz, V2 and

I’,, and V, and V, is partially to cancel the increase in E expected from the increase inV alone. Indeed, in the case of periodic correlations, complete cancellation of the extra

correlated noise can sometimes occur. Thus, for almost all cases likely to be met in



226 L. A. McLACH LAN

practice, the assumption of uncorrelated noise is either exact, or an excellent

approximation.

DISCUSSION

A comparison of the various methods of analyzing exponential decays should always

include the three features : accuracy, computational complexity, and stabil ity against

noise. Of prime importance is the ability of a method to give a small value of s for a given

value of N. It is often also advantageous for a method to have its minimum value of E

occurring for a small value of Nd/r, since this keeps the total experimental time needed

for a given accuracy as short as possible. For some pulse sequences, the savings in time

can be considerable (3), but for other experimental situations the time saved may be a

minor consideration. Many workers have access to computers of sufficient power to

handle any of the least-squares methods commonly used, but for those less fortunate

the computational complexity becomes important. Neither must computational sim-

plicity bc obtained at the expense of length of running time, as occurs in some iterative

methods. Only the luckiest of experimenters always has signal-to-noise ratios of 100 or

more; yet many approaches to analyzing exponential decays are somewhat unstable in

the presence of noise and may be unreliable for signal-to-noise ratios lower than about

50 where they are most needed.

By fitting artif icial computer-generated noise exponential decays, it was shown that

the error in z was accurately given by Eq. [5] for A/a > 20. On the basis of this equation,

the accuracy of the method given in this paper was compared with that of the method of

Moore and Yalcin (I), and the modified method of Smith and Buckmaster (2). For ten

sampling points, A/a = 100, and an unknown baseline, Moore and Yalcin’s methodgives E, as 4.4 %, while both the present method and that of Smith and Buckmaster have

E, = 3.9 %. In the experimentally uncommon situation, at least for NMR, where the

baseline position is known to much greater accuracy than the rms noise level, Moore

and Yalcin’s method gives e, = 1.6 %.

Al l three methods have an optimum measuring time. Moore and Yalcin’s method has

an optimum time of 2.22, which is much shorter than the 5r of the present method and is

often shorter than the time required for Smith and Buckmaster’s method. This may

sometimes compensate for its poorer intrinsic accuracy. For the greatest accuracy,

Smith and Buckmaster’s approach requires as long a time as possible for accurately

measuring the baseline, and a much shorter measuring time in the region of r to 2.2~for actually evaluating the decay time.

The presence of an optimum measuring time means that prior knowledge of z to

within a factor of 2 is desirable for optimum use of any of these methods. A poor esti-

mate of z has less effect on the accuracy of the present method than on that of the other

two methods. Not only is the minimum broad, but it occurs at over twice the value of

Nd/r, so, for the same absolute error in the estimate of z, the fractional error is less than

half that of the other methods.

Of the many methods available for analyzing exponential decays, the one proposed

in this paper appears to be the simplest. It can readily be converted into an algorithm

for on-line computer analysis which uses the data points as they arrive, and then immedi-

ately discards them, storing only their sums. Because of the limited storage capacity

and trivial amount of mathematical manipulation needed, even a simple scientific



ALGEBRAIC ANALYSIS OF EXPONENTIALS 227

pocket calculator can be used. Prior manipulation of the data is unnecessary since the

abili ty to handle an arbitrary baseline means that it is immaterial whether y obeys Eq.

[I], or the form l-2 exp(-t/z) obtained from 180-90” spin-latt ice relaxation timemeasurements. As mentioned before, if an estimate of the accuracy of z is also required,

computational facil ities more elaborate than a pocket calculator are needed, but the

requirements are sti ll less than for most other methods.

Another advantage of the present method is its stabil ity in the presence of noise.

Many linear, or nonlinear, least-squares-fitting methods suffer from the disadvantage

that they are unstable with noise levels of only a few percent, even though they may be

very accurate at lower noise levels. This instabili ty always arises in some form or other

from a denominator involving the small difference of two numbers, both of which con-

tain noise. Probably all methods suffer from this defect to some extent, but in the pre-

sent method it is minimized both because the differences in the sums are comparatively

large numbers and because the noise is averaged by summing the individual points.

Inequalities similar to those in the earlier discussion on the validity of approximating

ln(1 + x) by x also govern the stability of the solution. The evaluation of computer-

generated data showed that the method was stable when a/A = 0.1 and was even stable

for a/A = 0.3, provided NA was in the region of 42 to 62. Only the most desperate ex-

perimenter would require stabil ity in higher noise levels than this. Experimentally, it

was found that even when the exponential had a superimposed sinusoidal component,

a.s found in Carr-Purcell-Gill-Meiboom experiments with pulse imperfections, the

rnethod remained stable for large amplitudes of the oscillation, provided the oscillatory

period was much less than NA/4.

There are two other effects which cause minor errors in some methods of analyzingclecay curves. Moore and Yalcin mentioned one of these: bias in r introduced by the

presence of noise. The present method, along with that of Moore and Yalcin, involves

taking the logarithm of the experimentally measured amplitude; this is the sum of the

true amplitude and a random noise voltage whose mean value is zero. Taking the loga-

rithm, however, is a nonlinear process so the mean value of the logarithmic function is

not the logarithm of the true amplitude, but is displaced from it by a small amount

which depends on the signal-to-noise ratio. This systematic error in amplitude becomes

a. systematic error in r whose magnitude depends to some extent on the method of

a.nalysis. Robinson (4) examined this problem for six different methods of analysis in

cases where the noise obeyed Poisson statistics. Using the same type of investigation,he found that for the present case of Gaussian statistics (5)

~~ = 2’ [ 1 + (a/A)’ (A/z’) Qw2 1 - Qm2) (1 + Q2)-l J, WI

where ~~ s the true value of r, and z’ is the experimentally determined value. A com-

parison of this expression with that for statistical errors (Eq. [5 J) shows that for all reas-

onable experimental situations the statistical error is much larger than the systematic

error. In the situation where repeated measurements are made ofthe same decay time,

a correction for the systematic error may, however, need to be made. Although no ex-

perimental examination of the bias in Eq. [4] was undertaken, it was noticed that with

computer-generated data, r tended to be about 5 y0 too high when a/A = 0.1. This trend

is in reasonable agreement with the 2 % error calculated from Eq. [12] for the same

parameters.



228 L. A. McLAC HLAN

The discrete nature of the analog-to-digital conversion process also limits the ac-

curacy with which r can be measured. If the discrete voltage step D is much less than the

rms noise, then the error in the baseline position is about &+D, so the error in z is in theorder of f$D/A. On the other hand, when a 2 D, the quantizing of the voltage becomes

less important, being partially or completely smoothed out by the averaging effect of

the random noise (6). Most commercial analog-to-digital converters have such good

resolution that the quantization effects can be ignored, but some fast analog-to-digital

converters have only 6-bit resolution and in this situation there may be no advantage

in having Nd > 42, even though the nominal optimum value is larger than this.

This method of analysis can be systematically extended to the much more difficult

case of multiple exponential decays by splitting the data up into more blocks and solving

the increased number of simultaneous equations by matrix methods. A preliminary

study of two decays suggested, however, that the errors in the time constants are

large and that other algebraic methods (7) may be better for the multiple-decay case.

Although other methods may be better for their analysis, testing for the presence of

multiple decays can be done by the present method simply by using a range of NA/T

values from about 2.5 to 7. With a single exponential decay, the values of z obtained

from Eq. [4] are independent of NA/z, but if there is more than one decay present, the

value of z wil l steadily increase with increasing NA/z. Such a systematic trend is easily

detectable even if it is not much bigger than the noise level, so the method is quite a

sensitive indicator of the presence of multiple decays.

ACKNOWLEDGMENTS

The author wishes to thank M r. G. J. McC al lum for wi l lingly doing the computer programming, andDr. D. C. Robinson for calculating the syste ma tic error expression.

REFERENCES

1. W . S. MOORE AND T . YALCIN , J. Magn. Resonance l&50 (1973).

2. M . R . SMITH AND H. A . BUCKMASTER, J. Magn. Resonance 17,29 (1975).

3. G. G. MCDONALD AND J. S. LEIGH, J. Magn. Resonance 9,358 (1973).

4. D. C. ROBINSON, UKAEA Report AERE-R5911 (1968).

5. D. C. ROBINSON, personal communicat ion.

6. J . BUTTERWORTH, D. E. MACLAUG HLIN, AND B. C. M o s s , Rev. Sci. Instrum. 44, 1029 (1967).

7 . 0. CAPRANI, E. SVEINSDOTTIR, AND N. LASSEN, J. Theor. Biof. 25,299 (1975).

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Algebraic Analysis of Noisy Exponential Decays