Sensitivity enhancement of ESR spectra by integration

JOURNAL

OF MAGNETIC

RESONANCE

14,129-140

(1974)

Sensitivity Enhancementof ESR Spectra by Integration D. W. POSENER Division of Applied Physics, Commonwealth Scientific and Industrial Research Organization, Chippendale, Australia 2008

ReceivedSeptemberlo,1973 It is shown that integration of ESR derivative spectraproducesabsorption profiles with enhanced signal-to-noise ratios. Under appropriate conditions, the effective distortion is negligible. The processis useful for reducing the effects of noise when qualitative examination of the spectrum is required and, in contrast to conventional enhancementmethods, has the advantage that no lossof precision is introduced when objectivemeasurementsare required.

1. INTRODUCTION

of derivative spectra, such as those commonly observed in ESR, is shown in this paper to have properties which can be of considerable benefit in the analysis of appropriate spectra and which are advantageous when compared with methods previously reported. The analytical integration procedures discussed below should not be confused with the accumulation methods, often called “integration”, Integration

associated with the experimental superposition of multiply recorded spectra in a time-

averaging system (I, 2). In what follows, “sensitivity” is regarded as synonymous with the ratio R of signal height to noise rms value, or signal-to-noise ratio (SNR). Many ways have been described in the literature for increasing the sensitivity of spectroscopic and similar observations, with a number of approaches that are applicable to magnetic resonance being reviewed by Ernst (3). All of these may be classified as (a) prerecording techniques

(such as instrumental electronic filtering, time-averaging, etc.) or (b) postrecording methods (mainly digital filtering, moving-average smoothing, etc.) or a combination of both. The work described here is in the class (b) (see also (4-9)), and supposes that the most favorable experimental conditions have been used, so producing the best spectrum practicable for the available equipment and for the particular problem being investi-

gated. The aim of an experimenter influences the choice of means to handle a specific

problem. If he is searching for a weak unknown spectrum, or comparing weak spectra under different conditions, then discrimination against noise may be one of the more overriding considerations. This is the “signal detection problem” of communications theory (10) and seeks to maximize the SNR (so allowing easier subjective examination and spectrum recognition) but usually at the cost (3, 7, 8) of distorting the spectrum. When the experimenter wants to measure line heights, positions, etc., using some objective method such as least-squares fitting, the quantitative requirements of desired measurement precision may constrain the acceptable distortion (II) and consequently (3) limit the sensitivity enhancement obtainable by conventional methods to a value Copyright 0 1974 by Academic Press, Inc. All rights of reproduction in any form reserved. Printed in Great Britain

129

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D. W.

POSENER

well below the otherwise achievable maximum. But in this case the aim of the experimenter is such that the SNR is not an adequate criterion for the “goodness” of his spectrum. It follows that procedures which are sometimes described as “optimum” are not necessarily so in many casesof practical importance, and a compromise between enhancement and distortion is often made (3). The present work shows that integration of derivative spectra enhances the SNR without introducing distortion that may affect measurement precision significantly, an advantage over conventional enhancement methods. It is especially useful when high measurement precision is sought (11) because of the practical simplicity of the integration process and because the applicable conditions result in enhancement similar to the best otherwise obtainable (with the “optimum” or “matched” filter). The theory given below is in the context of such a case and, in particular, assumes that the “most favorable experimental conditions” used include the constraint that instrumental lineshape distortion is sufficiently small (II). Treatment of the theory is first given for continuous spectra, being most straightforward, although long-term spectrometer and integrator stability may present some difficulties with analog integration (12). Objective least-squares fitting to a spectrum (implied by the requirement for high measurement precision) needs numerical quantities, so the numerical integration of digitized spectra-essentially the use of a postrecording digital filter-is then discussed and indicates how operating conditions should be chosen to achieve the best results in the context of the kind of problem primarily considered. For simplicity of discussion, a single slow-speed scan is assumed, but is not a limitation on the method. Then the requirement that there is small instrumental lineshape distortion is in practice basically related to sweep time (and thus the slowest sweep rate) available, having regard to spectrometer stability or other factors. The linewidths of the profiles in the spectrum together with the spectrometer output filter characteristics (known approximately) then determine what is the smallest filter bandwidth, fc, that can be used without introducing errors greater than the measurement precision desired. This determination is easily made for a simple RC filter (13, 14). Thus, in the subsequent discussion, fc is just an experimental constant chosen by practical quantitative considerations to reduce output noise as much as possible without significantly affecting measurement precision. Outside the scope of this paper, but relevant to comparison with conventional methodology, is the qualitative advantage which derivative spectra have for certain problems, in that overlapping profiles are sometimes more readily examined subjectively than are absorption profiles (such as result from integration). However, as will be shown, when objective measurement is required, both modes of presentation give the same precision, and the “resolution enhancement” commonly associated with a derivative spectrum is purely subjective. Which approach is more suited to any specific problem is thus a matter of critcal evaluation by the experimenter, and the present work may assist his judgment. 2. SPECTRUM

INTEGRATION

c

Conventional postrecording enhancement methods (3) attempt to reduce noise effects while causing minimal alterations to the desired profiles in the data. Integration changes the appearance of the profiles markedly; however, the information contained

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ENHANCEMENT

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131

in the data can be preserved in the integral and may be just as readily available from it. For example, typical ESR spectrometers produce the derivative of an absorption spectrum as a function of time-swept magnetic field, and integration of such a derivative will regenerate the absorption spectrum; from the latter, profile parameters can be obtained by objective measurement, such as least-squares fitting (II) if the lineshape is known sufficiently well, as easily as from the original data. For simplicity of initial discussion, assume that (a) the observed spectrum consists only of derivative profiles perturbed by white noise, and (b) the iineshape is known. Denote an absorption profile by A(xi,z), where the xi are parameters such as A0 (peak amplitude), z,, (line position), and W (linewidth between half-amplitude points); z is the spectrum variable (e.g., magnetic field in ESR). Let the corresponding derivative profile be D(xt,z) = &4/az. To a good approximation, common lineshapes are described by profiles such as Gaussian and Lorentzian with --to < z < +a~, and integration is defined by A(Xj, z) = j-zD(x*, Z)dZ.

-m

PI

Let DO be the peak-to-peak amplitude of the derivative profile. Then, as may be verified for common lineshapes, Ad W = DoIP, PI where p is a lineshape-dependent constant; ,u = ,~o = (32 In 2/e)“’ z 2.857 for a Gaussian, ,u= pu, = (27/4)“” w 2.598 for a Lorentzian (and p = 71for a sinusoid if one cycle is regarded as a “profile”). For a spectrum swept at a rate S = dzldt,

131

and with independent variable 1, Eq. [2] may be replaced by SA,I W = Do/p.

141 A description, through Fourier transformation, of a time-swept profile by representation in frequencies5 or angular frequencies w = 27rf, is appropriate when considering SNR enhancement and smoothing. Then a convenient parameter is the “profile characteristic frequency” (II) fp= l/T,= S/219: PI where the “characteristic period” Tp is the time required to sweep through two linewidths. The common definition of SNR as the ratio of signal height to noise rms value will be used here for ease of treatment, and leads to conclusions closely approximating those obtained from a different approach (see Appendix). The derivative SNR is then & = Dohi,

M

R, = &In,,

[71

while that of an absorption profile is in which nd and n, are rms values of noise. Integration of a signal is a process where the ratio (output height)/(input

height)

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D. W. POSENER

may be defined as the system amplitude gain. For a time-swept derivative profile integrated with respect to time, this gain is, from Eqs. [4] and [5], AdDo = WI@ = 1/Wa,

PI

which is the same as the amplitude gain l/o, for a sinusoid of frequency fo = w4fp.

[91

The spectrum noise will have an effective upper frequency bound& due to spectrometer limitation by, say, a low-pass filter, and an effective lower bound,

DOI

fi = W’I~

due to the finite spectrum scan time Tl (2,3), plus a dc component, so the rms value nd of non-dc noise in the data is given by

nd”= I‘;df = J(fc - fi),

[Ill

fl

where J is the noise power spectral density (and constant, because of the white-noise assumption). In the integrated spectrum, the noise power spectral density will be J/CL?, and the rms value n, of the output noise will be given by n’, = IfcWoi-)&=

JCL -fd/~l~c,

t14

fl

so for the white noise the gain from integration will be nolnd = 1/0:/~ ~0:~~.

[I31

The SNR enhancement is, from Eqs. [8] and [13], E = &I&

= (&lD,Mnalncd

= We/f@“‘.

[I41

This increases as the sweep time l/fi is decreased (if other factors are kept constant). Because a sweep covering at least about two linewidths is desirable to define a profile, a reasonable (though somewhat arbitrary) minimum sweep is such thatf, =fO; whether the spectrum is actually swept through (an/p) W, or in a more extensive spectrum only this amount of profile is examined, is immaterial in practice. For such a “minimum sweep”, the effective enhancement is E,,, = (fe/fo)“2 = (nl,@“(T,fJ””

= (d4”2~;‘2,

VW

where 1, is the “information density” (II); 1, >lOO is obtainable experimentally (14). The factor (x/CL>“~ is lineshape dependent, and approximately unity for common profiles. The property of an integrator of enhancing the SNR by a time-bandwidth product T, fc is well-known (15) (see also (26)), and the enhancement given by a “matched filter” (I, 17) is also this amount because (II) I, ‘12 = (total profile energy/noise spectral density)1’2; however, the use of this property has not previously been reported in spectroscopy. Equation [15] says that the effective enhancement is, roughly, the square root of the

INTEGRATION

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133

ratio of the time to sweep through two linewidths to the spectrometer filter time constant. Integration enhances profile SNRs by attenuating high-frequency noise (with componentsf>fJ and accentuating low-frequency noise (f
SPECTRUM

INTEGRATION

Suppose that the continuous spectrum data appearing at the spectrometer output is sampled at a constant rate f, = l/At, 1161 where At is the period between samples, so giving a sequence of M+ 1 derivative amplitude measurements vO,vl, . . ., vM, at intervals AZ = SAt.

P71 Let the sampled data be numerically integrated with respect to elapsed time by using the trapezoidal rule. The values V,(k = 1, 2, . . ., M) of the integral, at successive times t = kdt, are then F’k= Vk-, + 3 @k-l+ v&At = Vo + i: J&,+, + v,)At, #?I=1

[I81

where VOis an arbitrary integration constant and is normally zero in practice. Alternatively, some continuous analog recording, such as a chart recorder plot, could be digitally sampled at intervals AZ. In any case, integration to successive points may be carried out off-line using the sampling intervals AZ = z,, - zKel

[191

instead of the At. The properties of the numerical integration process can be found by examining the behavior of a sinusoidal component of the sampled data, such as the typical Fourier component Y,W = qr sin b t 1, PO1 where o, = 27$ = 2mlMAt,

andr=l,2,.

WI

. ., M/2. Use of Eq. [18] gives the integral ofy,, to time t = kdt, as Y, (kdt) = C, - Q,cos(w, kdt),

P4

where Qr = 34, At cot Wr At), v31 and C, is an integration constant. Since Eq. [22] describes the kth point of a cosine

134

D. W. POSENJZR

function with angular frequency CO,,numerical integration introduces a phase lag +r, exactly, as for analytical integration, and because this phase shift is independent of r, there is no phase distortion of the numerical integral. The amplitude gain of the numerical integration process is [241 and approaches the analytical integration limit 1/CO,when CO:4 12fs. Fourier analysis of common derivative profiles shows that the amplitude of a component of frequency fi, relative to the amplitude atfO (which is near the frequency of the largest component), is of the form (f/‘J exp (-fr2/‘$ (Gaussian) and (jJfO) exp (-fr,&) (Lorentzian), so components of frequencies much above& do not contribute significantly to the profile. Thus if the amplitude distortion of an integrated profile is to be kept small, the sampling rate must be high compared tofo : or

fZ B (7r2/3).%

If this condition applies, amplitude distortion is small and numerical integration behaves very much like analytical integration. Integration of a derivative profile then gives an absorption profile whose peak amplitude is underestimated by &4,,, where ~A,/& = (7r2/3WW2. In this, Eqs. [16], [17], [7], and [5] give L/f0 = @71/P)w/w

= CWPK,

1261 ~271

where K= W/AZ

PI

and is the sampling density. The maximum useful sampling rate is (II) the Nyquist rate

114 = 2fc,

WI

2fcKl= cwPL)G

[301

6A,,/A,w ,u2/12K2,

[311

for which Thus Eq. [26] may be written

with a minimum relative error, for Nyquist sampling, (dA,,/A,),w

p2/121;.

1321

It may be noted that the sweep parameter (23) a = S/4fc w

is related to the information

1331

density through 0. = l/21,,

[341

= (P~/~)~~/(K/Z,)~,

1351

and then U,lAo

so that low filter distortion of a profile is accompanied by low amplitude distortion through numerical integration if sampling is at or near the Nyquist rate. In the above it is assumed that digitization and other numerical errors are negligible.

INTEGRATION

ENHANCEMENT

135

OF ESR SPECTRA

It will be appreciated that practical application of the integration process, as defined in Eqs. [18] and [19], is an extremely simple matter. 4. EFFECT

OF

INTEGRATION

ON

MEASUREMENT

PRECISION

In principle, measurement of the profile parameters xi can be carried out objectively by least-squares estimation, and then with a precision proportional to the SNR (II). However, as will be shown here, enhancement of the SNR by integration does not lead to improvement in measurement precision because the proportionality is not the same for both data and integrated data. Least-squares fitting of derivative profiles to the data involves equally weighted “errors”, since the assumption of white noise implies that the errors are uncorrelated, so the weight matrix (20) is the unit matrix and the precision is (II)

Let the kth value of the sampled input noise be Z,, which is approximated by the corresponding data residual (difference between data value and least-squares-fitted derivative profile value). If the output noise value is X,, integration of noise is described by X, = X,-, +Z, At. 1371 Equation [37] is that of a “random walk”, for which the covariance is (21) f+= min (i,j)02,

[381

where hf+1

c2= 1 z;/@f+ k-l

1 -1;>,

[391

and there are L parameters to be estimated. From Eq. [38], it follows that the variancecovariance matrix (20) has the form 1 1 1 I....... 1 2 2 2.. , . . . . 1401

with inverse

Least-squares fitting with correlated errors requires (20) using the weight matrix 02U-l when forming the normal equations. Equation [41] then leads to the conclusion that fitting to integrated data is equivalent to fitting differences and so, in the limit of sufficiently small sampling intervals, to fitting derivatives. Thus the answers

136

D. W. POSENER

(parameters) and error estimates (standard deviations) should be the same as for the original data, and the available measurement precision is unchanged by integration. This equivalence was confirmed by numerical analysis of synthetic Gaussian derivative profiles to which synthetic white noise had been added, using (for a given SNR) 100 different noise samples. Parameters and precisions obtained by fitting derivative profiles to the data were compared with those obtained by fitting absorption profiles to the integrated data (a) assuming equally weighted errors (unit weight matrix) and (b) using the weight matrix a2U-‘. The precisions from (b) agreed with those for the derivative fittings. Although the individual precisions from (a) were greater (by a factor E) than for the data, the statistics of 100 estimates gave precisions agreeing with the derivative data (and with (b)), so showing that those calculated by (a) were incorrect. It is shown in (II) that the available precision of objective least-squares estimation of parameters is generally invariant to linear transformations. Integration is a linear transformation, and thus is a special case for which the invariance of measurement precision can be demonstrated directly. If P’(xi) is the precision of measurement from the integrated spectrum, then (cf. Eq. 1361) P’(Xi) = C; K”‘R, = P(Xi), ~421 where c; = q/E,

[431

and E is given by Eq. [14]. 5. DISCUSSION

Integration smooths spectra by reducing the high-frequency noise present in the data. If the data noise is nonwhite, and if the noise spectral density is defined, then by the methods used in Eqs. [l l] and [12] the noise processing gain can be calculated and the resulting enhancement determined. In practice, most nonwhite-noise spectral densities are proportional tofep, (p > 0), and then the resulting SNR enhancement will always be less than that given by Eq. [14] for white noise. SNR enhancement is useful in problems of signal detection (IO); reduction of the rms noise over a range of a few linewidths makes a weak profile stand out and thus be more easily recognized subjectively. Subjective measurement of line parameters is also easier, though not necessarily more precise, when the noise is reduced. The visual improvement obtained is illustrated in Fig. 1, showing, to larger scale, part of Fig. 2 of (14); in this, E,,, x 12. Conventional enhancement methods distort the profile in that the shape after smoothing is no longer easily describable by some explicit function. If integration is used to smoothderivative spectra, the resulting absorption profiles are usually expressible by functions that are as simple as those describing the derivatives; in this sense integration does not distort a spectrum, although it can also be regarded as a deconvolutor of the instrumental function (II). When weak profiles overlap, this absence of distortion permits better subjective estimation of profile parameters than can be achieved when significant line distortion is present.

137

INTEGRATION ENHANCEMENT OF ESR SPECTRA

When the lineshape is known, least-squares fitting to the data determines the xi and thus the total absorption due to a spectrum. For unknown lineshapes, double integration also gives the total absorption, but its accuracy may be limited by effects due to background spectra.

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t

t

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(a). ) 78

75

B (mT)--

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lb)

75

78

B (mT)-

1. ESR line of NaCl : MI? (14) from nearest-neighbor dipoles // <1 IO>, B //
138

D. W. POSENER

APPENDIX

A common noise measure is its rms value, the mean square being associated with “power”. Thus it is consistent to define SNR as a power ratio, y, such that Y=

(mean square signal power)1’2 rms noise

HI

Integration of a sinusoid exp(-iot) (where i = 1/(-l)) produces a power gain 2/c?. Integration of a derivative profile gives a power gain 2 Wz/~2S2, where /3= /?d = (8 ln2)‘12 w 2.355 for a Gaussian, fi=pL = 2 for a Lorentzian. Equating this to 2/w; defines (cf. Eq. [9]) fo = Wdf,WI Integration is equivalent to using a system with impulse response function (21) h(r) =

1 forO< It/
[A31

for which the frequency response function is T112

H(f) =

S exp (-iot)dt

= sin (nflf)/($),

[A41

so if the input noise power spectral density is J, the output spectral density is JIH(f)12=

Jsin’Wl!Y(~f~‘. For input noise bandlimited to an upper frequencyf,, the rms value is given by nj

=

I“Jdf

= Jfc.

[A51

0

Integration leads to output noise with rms value given by

If fc B fi (as is common in practice), the last integral is well-approximated dx = 7112,and n: % J/2fi.

by {in2 x/x2 0

[A71

Then the integration power gain for the noise is nalnd z 1/(2fif,)“‘.

WI

When analyzing real spectra, it is usually convenient to ignore, or to subtract off, any bias. If the background were due to white noise alone, this would be equivalent to taking the mean noise to be zero (as expected for T,-+ 03). In effect, contributions from

INTEGRATION

ENHANCEMENT

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frequencies belowf, are not readily distinguishable from dc (2,3). Adequate definition of a baseline (e.g., by some fitting process) thus removes much of the noise contributions in this frequency region, and the mean square value of the apparent noise is then better described by the integrals of Eqs. [A51 and [A61 with lower limits replaced byf,. Then

ni=JCfi-.fd=Jfe

(forA 9 fi).

L491

The lower frequencies account for a large fraction of the integrated white noise

i’JIH(l)l’df=(J/~~~)jsinlxi*idx. b 0 and from tables (22,23) s

II sin2 x/x2 dx = Si(2n) x 1.4182 % 2r/‘,

0

where Si(u) is the sine integral, so m sin2xIx2dxw J z

n/2 - 21t2z 1/27c,

and n:xJ/2n'fi,

[A101

so that

n,/n, w (2/o, ~0,)~‘~

[All1

and the enhancement is nearly the same as given by Eq. [14] within the small factor relating the definitions Eqs. [9] and [A21 off,. ACKNOWLEDGMENTS

The author is grateful to C. H. Burton for critical discussionson the subjectof noise. REFERENCES 1. 2. 3.

M. R. R.

P. KLEIN AND G. W. BARTON, Rev. Sci. Instrum. 34, 754 (1963). R. ERNST, Rev. Sci. Instrum. 36,1659 (1965). R. ERNST, in “Advancesin Magnetic Resonance” (J. S. Waugh, Ed.), Vol. 2, AcademicPress,

New York, 1966, 4. A. SAV~T~KY AND M. J. E. GOLAY, Anal. Chem. 36,1627 (1964). 5. K. YAMASHITA AND S. MINAMI, Jap. J. Appl. Phys. 8,1505 (1969); lo,1097 (1971). 6. J. P. PORCHET AND H. H. G~~NTHARD, J. Phys. E3,261(1970). 7. R. R. ERNST AND H. BENZ, IEEE Trans. Audio Electroacoust. AU-l& 380 (1970). 8. M. CAPRIM, S. C&N-SFETCU, AND A. M. MANOF, IEEE Trans. Audio Electroacorrst. AU-18,389 (1970).

9. C. E. BRYSON, Rev. Sci. Instrum. 42,1504 (1971). IO. D. MIDDLETON, “Introduction to StatisticalCommunication Theory”, McGraw-Hill, New York, 1960. II. D. W. POSENER, J. Magn. Resonance 14, 121 (1974). 12. M. L. RANDOLPH, Rev. Sci. Instrum. 31,949 (1960). 13. D. W. POSENER, J. Appl. Phys. 43,3117 (1972). 14. D. W. POSENER, J. Magn. Resonrmce 13, 102 (1974). 15. K. S. MILLER AND R. I. BERNSEEIN, IRE Trans. Information Theory IT-3,237 (1957). 16. T. H. GLISSON, C. I. BLACK, AND A. P. SAGE, IEEE Trans. Audio Electroacoust. AU-B, 271(1970).

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17. G. L. TURIN, IRE Trans. Information Theory IT-6,311 (1960). Z8. A. BAIJDER AND R. J. MYERS, J. Mol. Spectrosc. 27, 110 (1968). 19. R. B. D. FRASER AND E. SUZUKI, in “Spectral Analysis” (J. A.

20. 21. 22,

23.

Blackburn, Ed.), Marcel Dekker, New York, 1970. W. C. HAMILTON, “Statistics in Physical Science”, Ronald Press, New York, 1964. G. M. JENKINS AND D. G. WATTS, “Spectral Analysis and its Applications”, Holden-Day, San Francisco, 1969. I. S. GRADSHTEYN ANI) I. M. RYZIK, “Tables of Integrals, Series, and Products”, 4th ed., Academic Press, New York, 1965. M. ABRAMOWITZ AND I. A. STEGUN, “Handbook of Mathematical Functions”, U.S. G.P.O., Washington, 1964.