comparison of models for parameter estimation in broad-line nmr spectra

JOURNAL OF MAGNETIC RESONANCE 93, l-1 1 ( 199 1)

Comparison of Models for Parameter Estimation in Broad-line NMR Spectra

SEHOKANG,MOHANNAMBOODIRI,AND DANIELFIAT*

Department of Physiology and Biophysics and Department of Bioengineering, College of Medicine, University qf Illinois at Chicago, P.O. Box 6998. Chicago, Illinois 60680

Received July 6, 1989; revised November 6, 1990

The effects of finite acquisition time, improperly chosen sampling time, and spectrometer dead time on broad-line NMR spectra are examined. A discrete Fourier-transform model equation for a nonlinear least-squares analysis that overcomes these limitations is derived. A nonlinear regression method is used to perform the nonlinear least-squares fit of the simulated and experimental data. The accuracy of the proposed model is compared with those of the conventional Lorentzian and the time-domain models. It is shown that the proposed discrete Fourier-transform model yields higher accuracy than the conventional Lorentzian line. The accuracy of the estimation of the proposed discrete Fourier-transform model is similar to that of time-domain model: however. its uncertainty level is lower. c’ I YY I Academic Press. Inc.

Recently “0 NMR studies have been increasingly utilized for analyzing amino acids, peptides, and related model compounds in solution (I) and in the solid state (2). Topics of research have included solute-solvent interactions, hydration, and sol- vation (3)) pH effects (4)) dynamics of molecular motion, and binding of paramagnetic ions (5). “0 lines of solution are broader than ‘H and 13C lines by orders of magnitude due to the nuclear quadrupolar relaxation of “0 nuclei. In solutions containing paramagnetic ions, additional broadening of the lines occurs. The hyperfine coupling con- stant is orders of magnitude larger than those of ‘H and 13C and thus the “0 linewidths are much larger. The combined effect of broad lines and the low natural abundance of “0 (0.037%) results in a poor signal-to-noise ratio (S/N). Because of the high noise level, parameter values cannot be estimated visually. To overcome these diffi- culties, we usually study “O-enriched materials, pay close attention to optimizing experimental parameters, and carry out extensive signal averaging. Even with such procedures, visual parameter estimation remains problematic. Therefore, least-squares analysis has been applied to these noisy, distorted, and poorly resolved spectra, yielding estimates of the four parameters of interest, i.e., amplitude, peak position, spin-spin relaxation time, and phase angle. One of the pivotal factors in determining the validity of the results of least-squares estimation is the model equation; significant errors occur if it does not fit the data adequately (6). To test the accuracy of nonlinear model

* To whom correspondence should be addressed.

0022-236419 I $3.00 Copyngbt 0 1991 by Academic Press, Inc. All rights of reproduction m any form reserved.

2 KANG, NAMBOODIRI, AND FIAT

equations, we applied them to simulated noisy spectra with known parameters and compared the results, as well as to experimental data.

The least-squares fit of Lorentzian lines in the frequency domain is widely used for lines which are the combination of the real and imaginary parts of the Fourier transform of a free induction decay. It was reported that inherent limitations of the discrete Fourier transform affect analysis of NMR spectra. These limitations include poor resolution and spectral response leakage due to the finite acquisition times, spectral distortion as a result of data lost during spectrometer dead time, and foldover phenomena induced by discrete sampling of the time-domain signal ( 7, 8). These limitations, to some extent, may cause errors in parameter values. Recently, high-resolution parameter-estimation techniques in the time domain which do not use the discrete Fourier transform have been utilized. The methods include ( 1) maximum entropy, (2) maximum likelihood, and (3) linear prediction and singular-value decomposition (LPSVD) . Those methods enhance the resolution of poorly resolved high-resolution spectra. However, the disadvantage of the methods is a large estimation bias at low S/N. In this paper, we have studied the effects of the inherent limitations of the discrete Fourier transform of FID and proposed a model equation for the nonlinear regression analysis of NMR spectra. For comparison purposes, we focused on studies of a broad line with a single peak. We compared the proposed model with conventional Lorentzian and time-domain models.

METHODS

The continuous FID signalf( t) observed in a pulsed FT NMR experiment is written f(t) = Ae-t/TzeJ boI+b), ill

where A, T2, w. , and 4 are amplitude, transverse relaxation time, peak-position frequency, and phase angle, respectively. The detected spectrum in a pulsed Fourier- transform experiment is the real part of Fourier transform of Eq. [l] and is given by

R(w) = A{ T,cos(c,b) + T:(w - wo)sin(4))

1 + T;(u - wo)* > I21

where T2 is 1 /(r X linewidth). Equation [ 21 is the most widely used nonlinear frequency-domain model equation.

FT NMR involves data acquisition of discrete FID values and discrete Fourier transform (DFT) of the FID. The truncation of data acquisition results in spectral distortion and the finite sampling time brings about foldovers. The spectrometer dead time results in phase-angle distortion.

First, we consider truncation effects. Equation [ 3 ] gives the Fourier transform of a truncated FID for 4 = 0,

R,(w) = A[l/T, - l/T2Ucos{(w - wo)T} + U(w - wo)sin{(w - wo)T}]

(l/T212 + (w - wo)* > [31

where U = epTIT2. The truncated FID with acquisition time T is equivalent to the multiplication of the nontruncated FID with a rectangular window whose range is

PARAMETER ESTIMATION IN BROAD-LINE SPECTRA 3

between 0 and T. Accordingly, Eq. [ 3 ] is equivalent to the convolution of the continuous nontruncated Fourier spectrum with the Fourier transform of the rectangular window function. The side lobes of the transformed window function, when convolved with the Fourier spectrum of the FID, induce changes in the lineshape. This effect is due to the truncation of the acquisition time in the time domain (9). Figure 1 depicts the truncation effect given by Eq. [ 31: for T -C 3T2, the peak intensity is attenuated, the line is broadened, and the wings of the line have an oscillatory character. The truncated lines coincide with the theoretical Lorentzian line for T > (5-6) T2.

The sampling rate is a second significant parameter determining the degree of lineshape distortion in the frequency domain. Following the Nyquist theorem, spectral foldovers (aliasing) occur for sampling rates that are smaller than twice the difference between the line and the carrier frequencies. However, the Nyquist frequency for a Lorentzian line is infinite since its amplitude does not decrease to zero unless the frequency is infinite and thus the theorem does not provide the criterion for choosing

I I

2500 5000 7500 1c

FREQUENCY (Hz) 00

FIG, 1, The effect of the truncation of acquisition time (T) on the Lorentzian lineshape. Simulated linewidth and peak position are 2000 and 5000 Hz, respectively ( 1, T = KZ ; 2, T = ZT,; 3, T = 3T,; 4, T = ST,: 5. T = 6T,).


a minimum sampling frequency ( 10). Therefore, for a finite sampling time, the DFT introduces a certain number of foldovers which alter the lineshape; the best one can do is to choose a sampling time which minimizes severe aliasing. An exact equation for the DFT which takes the above phenomena into consideration can be used as a nonlinear least-squares model. In practice, the time-domain signal f( t) in Eq. [l] is sampled asf( kt,) where k = 0, 1, 2, . . . , N - 1 and t, is the sampling time and N is the number of sampling points. The discrete Fourier transform off(&) is given by the equation ( 7)

N-l

F[n/Nt,] = t, C f(kts)e-j2ank’N. k=O

[41

To see the aliasing effect as a function of sampling time, we assume infinite acquisition time. Equation [4] then reduces to

F[wl = At,{1 - RP-jRQ}(cos(4) +jW4)1 (1 - RP)2 f (RQ)2 ’

whereR=exp(-t,/T,),P=cos{(o- >t} w. S ,andQ=sin{(o-wo)t,}.Thenonlinear model equation consists of the real part of Eq. [ 5 ] :

R,(w) = At& 1 - RpbsW + RQ sin(d} ( 1 - RP)* + (RQ)2 . [61

The effect of the sampling time t, (Eq. [ 61) on the lineshape is given in Fig. [ 21. The principal effects of aliasing on broad-line spectra are the increase of ( 1) baseline, (2) peak intensity, and (3) linewidth. As seen in the figure, the discrete time spectrum approaches the Lorentzian as the sampling rate increases.

The third effect studied here is distortion in the phase of the signal. Phase correction is required to compensate for the transfer function of the detection system. The phase angle results from the hardware filter network, the finite duration of the pulse, and the spectrometer dead time necessary to avoid pulse breakthrough ( 1 I). In Eq. [ 11, phase angle 4 can be divided into #, which represents an inherent phase shift due to the hardware filter network, and &, which varies with the dead time. The latter can be shown to have the form

$d = UOtdr [71

where o. and td are peak-position frequency and spectrometer dead time, respectively (12). Given this expression for @)d, we can easily determine 4’. Figure 3 shows the spectral distortion of a simulated NMR spectrum with a 2000 Hz linewidth as a function of the spectrometer dead time. This distortion results in changes in the baseline, reduction of intensity, and broadening of the linewidth.

From the three major effects, the truncation effect can be ignored if we set the acquisition time longer than 5 T2. Equations [ 61 and [ 7 1, which mathematically de- scribe the aliasing phenomena and spectrometer dead time effect, respectively, can be combined into the equation


0 I 0 2500 5000 7500

FREQUENCY (Hz)

FIG. 2. The effect of the aliasing due to improperly chosen sampling time (t,) on the Lorentzian lineshape. Simulated linewidth and peak position are 2000 and 5000 Hz, respectively ( 1. t, = 0 ps: 2, t, = 5 ps: 3, t, = 15 fis: 4. I, = 30 /.a).

R,(w) = Ae- ‘d’T2tS{ ( 1 - RP)cos(@ + w&) + RQ sin($’ + w&)}

( 1 - RP)2 + (RQ)2 [81

In this study, model Eqs. [ 21, [ 81, and [ 91 are compared. Equation [ 91 consists of the real part of Eq. [I] and includes the spectrometer dead time term:

f(t) = A&t+b)/r2 cos{wo(t + ld) + c#l’}. [91

To find the best-fit parameters, one typically minimizes an objective function that consists of the sum of the squared differences (errors) between the observed and pre- dicted values. The objective function is minimized by the Gauss-Newton method ( 13). The accuracy of estimation in the simulation study was represented as a function of the bias (the absolute error). The bias measures the deviation of the estimated value of a parameter from the expected value. The standard error is a measure of uncertainty level.

KANG, NAMBOODIRI, AND FIAT

-1 I I I 0 2500 5000 7500 1c

FREQUENCY (Hz) 00

FIG. 3. The effect of phase distortion due the spectrometer dead time ( td) on the Lorentzian lineshape. Simulated linewidth and peak position are 2000 and 5000 Hz, respectively ( I, td = 0 ps; 2, td = 10 ps; 3, td = 30ps;4,t, = 5Ops).

Simulation of the data was carried out in on the main-frame IBM computer at the University of Illinois at Chicago, using a time-domain FID model equation. The noise was generated by a random-number-generating subroutine (mean, 0.0 1979; standard error, 0.0 1057; range, 0- 1) . Signal-to-noise ratios of 5, 10, and 20 in the time domain were used as well as a noise-free signal. To get the proper signal-to-noise ratios, the same noise sequence was multiplied by the appropriate scale factor to ensure the consistency of the noise source. The linewidth of the simulated spectra was selected as 3000 Hz. Peak intensity was fixed at 10,000 and the inherent phase angle at 0.75 rad with spectrometer dead times of 20 and 100 ps. The given dwell times are 10 and 50 ps. Simulated data were Fourier transformed for use with two frequency-domain models, and zero filling was done to make a total of 8 192 points, exceeding 6T2 acquisition time. Curve fitting was performed by the nonlinear regression analysis (NLIN) method in the Statistical Software Analysis (SAS) package (14).


TABLE 1

Results of Parameter Estimation for Simulated NMR Spectra

Parameter SIN

Amplitude (10,000)~

Peak position (5000 Hz)

Linewidth (3000 Hz)

Phase angle (0.75 rad)

2”o 10

5

Fl 10

5

G 10

5

2”o 10

5

Type of model

Lorentzian (Eq. [2]) Proposed DFI (Eq. [S]) Time domain (Eq. [9])

9689 -t 0.5 10,000 f 0.0 10,000 f 0.0 9687 i 16.0 9,997 + 16.8 9,985 i 56.4 9686 t 31.6 9,994 f 33.6 9,972 f 112.2 9685 +- 62.9 9,989 +- 66.9 9,948 f 222.4

5000 +- 0.0 5,000 k 0.0 5,000 f 0.0 5002 f 0.5 5,001 f 0.5 5,000 f 1.8 5003 + 1.1 5,001 + 1.1 4,999 2k 3.6 5004 + 2.3 5,003 f 2.1 4,998 + 7.1

3003 :c 2.3 3,000 f 0.0 3,000 -+ 0.0 2980 zk 7.9 2,999 f 6.8 2,999 f 23.2 2905 :t 17.7 2,980 f 13.4 2,982 + 45.2 2840 :t 32.3 2,950 I!Z 26.8 2,948 f 90.7

1.378 i 0.000 0.750 f 0.000 0.750 f 0.000 1.374 f 0.003 0.75 I f 0.002 0.753 + 0.005 I .379 -+ 0.004 0.753 + 0.003 0.756 f 0.011 1.380 + 0.007 0.756 + 0.006 0.762 + 0.021

u ( ) in the parameter column stands for the expected (known) value. Note. The given amplitude, peak position, linewidth, and phase angle are 10,000, 5000 Hz, 3000 Hz, and

0.75 rad, respectively. To derive the pure spectrometer dead time effect, a short sampling time (10 PCS) is chosen, and 20 PCS of spectrometer dead time is given. Estimated parameter values and their standard errors are shown in the table (estimated value -t standard error).

The experimental spectra were obtained with an NTC-200 spectrometer using a 12 mm sample tube. The aqueous MnS04 solution of “0 (natural abundance, 0.037%) was used to obtain the broad-line spectra with about 2000 Hz of linewidth. To get the spectra with different S/N, time-averaging techniques were used with different numbers of scans. Spectrometer frequency was 27.128 MHz, and 90” pulse duration was 50 PUS. Several sampling times are chosen; 10, 30, and 50 ps, to see the effects of foldovers on the parameter estimation. Spectrometer dead time was given as 100 ps.

RESULTS AND DISCUSSION

The truncation effects become practically negligible when the acquisition time is longer than ( 5-6) T2. However, the improper finite sampling time and the spectrometer dead time are major contributors to the spectral distortion that affects the amplitude, peak position, linewidth and phase angle. In this study, we derived expressions that allowed us to overcome the distortion.

KANG, NAMBOODIRI, AND PIAT

TABLE 2

Results of Parameter Estimation for Simulated NMR Spectra

Type of model

Parameter SIN

Amplitude ( 10.000)

Peak position (5000 Hz)

Linewidth (3000 Hz)

Phase angle (0.75 rad)

2”o 10 5

zl 10 5

2”o 10 5

2”o 10 5

Lorentzian (Eq. [2]) Proposed DFT (Eq. [8]) Time domain (Eq. [9])

9059 + 17.0 10,000 f 0.0 10,000 It 0.0 8968 + 44.9 9,873 k 53.8 9,988 f 147.6 8882 f 83.5 9,754 f 105.2 9,958 _+ 294.1 8716 f 161.2 9,537 f 201.9 9,920 f 583.4

5014 It 0.7 5,000 f 0.0 5,000 f 0.0 5015 f 1.7 5,001 f 1.5 4,998 f 4.6 5016 + 3.2 5,002 f 3.0 4,996 f 9.1 5018 f 6.1 5,003 * 5.7 4,992 f 17.9

2890 + 7.8 3,000 f 0.0 3,000 f 0.0 2814 + 20.2 2,998 z!z 18.3 2,999 f 55.2 2750 +- 40.3 2,982 k 36.6 2,982 ?I 100.5 2640 + 80.2 2,960 f 73.2 2,958 f 200.4

3.823 f 0.002 0.750 f 0.000 0.750 f 0.000 3.819 f 0.005 0.744 + 0.006 0.752 f 0.002 3.814 f 0.010 0.738 f 0.011 0.754 f 0.003 3.803 f 0.019 0.728 f 0.022 0.757 + 0.059

a ( ) in the parameter column stands for the expected (known) value. Note. The given amplitude, peak position, linewidth, and phase angle are 10,000, 5000 Hz, 3000 Hz, and

0.75 rad, respectively. To derive the aliasing effect, a relatively long sampling time (50 p.s) is chosen, and 100 ps of spectrometer dead time is given. Estimated parameter values and their standard errors are shown in the table (estimated value f standard error).

Analysis of simulated data. To determine the spectrometer dead time effects, a short finite sampling time ( 10 ps) was chosen. The spectrometer dead time was 20 p.s. Table 1 shows the comparison of three model equations for the simulated spectra. When the finite sampling time is short enough to avoid aliasing, the estimates of peak position and linewidth are accurate for the three models, as can be seen in Table 1. However, the conventional Lorentzian line model yields a biased estimate of the amplitude and phase angle, whereas the modified DFT and time-domain models result in unbiased estimation.

To determine the aliasing effects, a relatively long sampling time (50 PS) was selected. Table 2 shows the comparison of the three models. The Lorentzian line model resulted in biased values in the amplitude, peak position, linewidth, and phase angle, whereas the DFT and time-domain models resulted in lower estimation errors.

The absolute errors of all four parameters increase as the S/N decreases (see Tables 1 and 2). The standard errors were found to be inversely proportional to the S/N. Thus, standard errors computed by this nonlinear regression analysis can be used to


TABLE 3

Results of Parameter Estimation for Experimental NMR Spectra

Type of model Sampling time -

Parameter (b4 Lorentzian (Eq. [2]) Modified DFT (Eq. [S]) Time domain (Eq. [9])

Amplitude”

Peak position (Hz)

Linewidth (Hz)

10 30 50

10 30 50

10 30 50

2.126 f 0.026 14.133 f 0.361 14.157 + 0.496 2.220 f 0.029 14.732 f 0.383 14.723 k 0.520 2.212 f 0.024 15.073 + 0.35 1 15.144 f 0.537

--6025 + 2.1 -6024 5 3.9 -5993 i 7.0 --6023 + 2.4 -6024 + 2.4 -5994 f 6.6 m-6019 f 2.1 -6018 k 2.1 -6020 + 8.3

2401 i- 40.3 2420 k 27.6 2548 f 139.1 2350 t 45.2 2424 ? 46.2 2598 f 135.3 2215 III 29.8 2418 f 41.9 2560 +- 172.2

Phase angle (ml) 10 -3.835 k 0.037 ~ 1.747 f 0.038 -1.899 + 0.013

30 -3.116 + 0.033 - 1.47 I -+ 0.028 -1.619 f 0.015 50 -3.219 f 0.036 -1.231 f 0.024 -1.344 f 0.014

a Amplitude is computed as (estimated value/number of scans). Nofe. The aqueous MnSo, solution with natural abundance of “0 was used to obtain the spectra. Several

sampling times are given (IO, 30, and 50 ps), and a spectrometer dead time of 100 Fsec is chosen. Estimated parameter values and their standard errors are shown in the table (estimated value f standard error).

determine the relative S/N. We have found that the DFT model results in an accuracy similar to that of the time-domain model. However, the DFT model yields parameter values with an uncertainty smaller than that of the time-domain model.

Analysis of experimental data. Experimental “0 NMR data of aqueous MnS04 solution (a single peak) were collected and analyzed as a function of the sampling time and the number of scans.

Table 3 indicates the result of a nonlinear least-squares fit for NMR spectra of aqueous MnS04 solution as a function of the sampling and spectrometer dead times. As in the simulation study, the proposed DFT model and time-domain model produce undistorted amplitude and phase-angle parameter estimates, whereas the conventional Lorentzian line model yields inaccurate estimates due to the spectrometer dead time effect. The Lorentzian line model and DFT model estimate the peak position with the same accuracy, while the time-domain model yields low values for the peak position. It is obvious that only the DFT model provides a consistent parameter estimation of the linewidth irrespective of improper sampling rates which result in foldovers.

Table 4 displays the results of the parameter estimation as a function of the number of scans. The Lorentzian and the DFT models produce similar peak-position and linewidth parameter values. We have found (Table 3) that the uncertainty level, as measured by the standard error, decreases as the number of scans increases. This finding is in agreement with the simulation study. However, the results of the time-


TABLE 4

Results of Parameter Estimation for Experimental NMR Spectra

Parameter Number of scans Lorentzian (Eq. [2])

Type of model

Proposed DFT (Eq. [S]) Time domain (Eq. [9])

Amplitude” 2.126 + 0.049 14.143 f 0.625 14.212 + 0.858 10,000 2.221 2 0.034 14.764 + 0.442 14.684 f 0.607 20,000 2.126 + 0.026 14.134 + 0.361 14.157 + 0.496 30,000

Peak position (Hz) 10,000 -6030 f 3.8 -6030 f 3.8 -6001 f 9.7

20,000 -6019 k 2.8 -6019 f 2.8 -5998 + 8.4 30,000 -6025 + 2.1 -6025 + 3.9 -5993 + 7.0

Linewidth 0-W 10,000 2315 f 72.2 2427 -c 71.5 2542 -r- 193.9

20,000 2422 f 52.7 2429 + 52.6 2569 k 166.0 30,000 2406 f 39.8 2419 f 21.2 2548 + 139.1

Phase angle (rad.) 10,oOiJ -3.733 f 0.050 - 1.647 f 0.045 - 1.839 f 0.024

20,000 -3.712 f 0.042 - 1.828 f 0.032 -1.955 * 0.017 30,000 -3.835 f 0.037 - 1.747 f 0.038 -1.899 f 0.013

’ Amplitude is computed as (estimated value/number of scans). Note. The aqueous MnSo, solution with natural abundance “0 was used to obtain the spectra. Different

numbers of scans are chosen (10,000, 20,000, and 30,000). Sampling time and spectrometer dead time are given as 20 and 100 ps, respectively. Estimated parameter values and their standard errors are shown in the table (estimated value + standard error).

domain model (Table 4) are quite different from those of the Lorentzian and DFT models. The Lorentzian line model yields distorted amplitude and phase-angle values due to the spectrometer dead time.

The estimated values of four parameters in the simulated DFT model study are similar to those in the time-domain model except that the latter has an uncertainty level higher than that of the former. The peak position and linewidth of the time- domain model estimates are found to be different from those of the DFT model estimates because of abnormal values of data sequences due to pulse breakthrough. Therefore, the proposed DFT model estimates are the most consistent and nondistorted.

SUMMARY AND CONCLUSIONS

Three major phenomena that occur in the discrete Fourier-transform analysis for broad NMR lines have been reviewed and their mathematical expressions used to derive proper model equations in the frequency and time domains. The proposed DFT model equation was compared with the conventional Lorentzian and the time- domain models. We have found that, by choosing an acquisition time longer than 5 to 6 T2, the spectral distortion is minimized. The foldovers due to improper sampling time and the phase distortion due to spectrometer dead time can be minimized by use of the proposed DFT equation.


The proposed DFI’ and time-domain model equations produced more accurate and unbiased estimation of the amplitude, peak position, linewidth, and phase angle than the conventional Lorentzian line model. The proposed DFT model results in an accuracy similar to that of the time-domain model; however, the uncertainty level of estimation of the former is smaller than that of latter. We conclude that the proposed DFT model equation yields better estimates than the Lorentzian and time-domain model equations.

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comparison of models for parameter estimation in broad-line nmr spectra

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