^{1}, Arindam Ghosh

^{2}, Fengli Zhang

^{1}, Thomas Szyperski

^{2}and Rafael Brüschweiler

^{1,a)}

### Abstract

Due to the limited sensitivity of many nuclear magnetic resonance(NMR) applications, careful consideration must be given to the effect of NMRdata processing on spectralnoise. This work presents analytical relationships as well as simulated and experimental results characterizing the propagation of noise by unsymmetric covariance NMR processing, which concatenates two NMRspectra along a common dimension, resulting in a new spectrum showing spin correlations as cross peaks that are not directly measured in either of the two input spectra. It is shown how the unsymmetric covariance spectrum possesses an inhomogeneous noise distribution across the spectrum with the least amount of noise in regions whose rows and columns do not contain any cross or diagonal peaks and with the largest amount of noise on top of signal peaks. Therefore, methods of noise estimation commonly used in Fourier transform spectroscopy underestimate the amount of uncertainty in unsymmetric covariance spectra. Different data processing procedures, including the -matrix formalism, thresholding, and maxima ratio scaling, are described to assess noise contributions and to reduce noise inhomogeneity. In particular, determination of a score, which measures the difference in standard deviations of a statistic from its mean, for each spectral point yields a matrix, which indicates whether a given peak intensity above a threshold arises from the covariance of signals in the input spectra or whether it is likely to be caused by noise. Application to an unsymmetric covariance spectrum, obtained by concatenating two 2D heteronuclear, single quantum coherence (HSQC) and heteronuclear, multiple bond correlation (HMBC) spectra of a metabolite mixture along their common proton dimension, reveals that for sufficiently sensitive input spectra the reduction in sensitivity due to covariance processing is modest.

This work was supported by the National Science Foundation (Grant No. MCB 0416899 to T.S.) and the National Institutes of Health (Grant No. GM 066041 to R.B.). The NMR experiments were conducted at the National High Magnetic Field Laboratory (NHMFL) supported by cooperative agreement DMR 0654118 between the NSF and the State of Florida.

INTRODUCTION

THEORY

Thresholding

Maxima ratio scaling

MATERIALS AND METHODS

Simulations

Experimental

RESULTS AND DISCUSSION

Theory

Simulations

Experiment

CONCLUSION

### Key Topics

- Nuclear magnetic resonance
- 19.0
- Fourier transform spectroscopy
- 7.0
- Molecular spectra
- 5.0
- Multiple resonance spectra
- 5.0
- Random noise
- 5.0

## Figures

Signal-to-noise (S∕N) ratio of a peak arising from the covariance of a pair of peaks, computed using Eq. (9) as a function of and the S∕N ratio of the input peaks. (a) S∕N for the covariance between peaks each having the indicated (5, 10, and 20) S∕N values. (b) S∕N for the covariance between a peak with and a peak with the indicated S∕N. Note that the lower the signal to noise of the weaker peak, the lower the signal to noise of the covariance peak. However, in the limit where the weaker peak is much weaker than the stronger peak, so long as is small, the signal to noise of the covariance peak approaches that of the weaker peak. The sensitivity of an unsymmetric covariance spectrum, for small values of , is not that much lower than that of the less sensitive of the two spectra subject to covariance with signal-to-noise values decreasing at most by a factor from that of the least sensitive of the two input spectra.

Signal-to-noise (S∕N) ratio of a peak arising from the covariance of a pair of peaks, computed using Eq. (9) as a function of and the S∕N ratio of the input peaks. (a) S∕N for the covariance between peaks each having the indicated (5, 10, and 20) S∕N values. (b) S∕N for the covariance between a peak with and a peak with the indicated S∕N. Note that the lower the signal to noise of the weaker peak, the lower the signal to noise of the covariance peak. However, in the limit where the weaker peak is much weaker than the stronger peak, so long as is small, the signal to noise of the covariance peak approaches that of the weaker peak. The sensitivity of an unsymmetric covariance spectrum, for small values of , is not that much lower than that of the less sensitive of the two spectra subject to covariance with signal-to-noise values decreasing at most by a factor from that of the least sensitive of the two input spectra.

Noise propagation through unsymmetric covariance. (a) Simulated (noisy) input spectrum with . (b) Covariance spectrum , where has the same signal peak as and the same noise level. (c) The variance calculated using Eq. (10) at each point of the covariance spectrum. (d) matrix calculated according to Eq. (11). (e) same as (d) after setting all elements to zero with a S∕N ratio less than (4.85), the score belonging to the critical value for which Dunn–Sidak correction yields a spectrum-wide . (f) The covariance spectrum produced by thresholding by setting all elements of and less than to zero prior to covariance. In (a), (b), (d), (e), and (f) the cross peak is truncated to highlight noise features: the actual peak heights are 44, 2112, 33, 33, and 2101, respectively.

Noise propagation through unsymmetric covariance. (a) Simulated (noisy) input spectrum with . (b) Covariance spectrum , where has the same signal peak as and the same noise level. (c) The variance calculated using Eq. (10) at each point of the covariance spectrum. (d) matrix calculated according to Eq. (11). (e) same as (d) after setting all elements to zero with a S∕N ratio less than (4.85), the score belonging to the critical value for which Dunn–Sidak correction yields a spectrum-wide . (f) The covariance spectrum produced by thresholding by setting all elements of and less than to zero prior to covariance. In (a), (b), (d), (e), and (f) the cross peak is truncated to highlight noise features: the actual peak heights are 44, 2112, 33, 33, and 2101, respectively.

Covariance of simulated spectra (as described in text) subjected to maxima ratio scaling (mrs). The peak is truncated and has an actual height of 2059.

Covariance of simulated spectra (as described in text) subjected to maxima ratio scaling (mrs). The peak is truncated and has an actual height of 2059.

Noise propagation through unsymmetric covariance. (a) Simulated input spectrum A as in Fig. 2 with . (b) Covariance spectrum , where has the same signal peak as and the same noise level. (c) matrix calculated according to Eq. (11). (d) same as (c) after setting all elements to zero with a S∕N ratio less than (4.85), cf. Fig. 2. (e) The covariance spectrum produced by thresholding by setting all elements of and less than to zero prior to covariance. (f) Covariance of simulated spectra subjected to maxima ratio scaling (mrs). In each panel, the cross peak is truncated to highlight noise features: the actual peak heights are 48, 2387, 5, 5, 2170, and 2246, respectively.

Noise propagation through unsymmetric covariance. (a) Simulated input spectrum A as in Fig. 2 with . (b) Covariance spectrum , where has the same signal peak as and the same noise level. (c) matrix calculated according to Eq. (11). (d) same as (c) after setting all elements to zero with a S∕N ratio less than (4.85), cf. Fig. 2. (e) The covariance spectrum produced by thresholding by setting all elements of and less than to zero prior to covariance. (f) Covariance of simulated spectra subjected to maxima ratio scaling (mrs). In each panel, the cross peak is truncated to highlight noise features: the actual peak heights are 48, 2387, 5, 5, 2170, and 2246, respectively.

Selected spectral region taken from an experimental unsymmetric HSQC-HMBC covariance spectrum of metabolite mixture using different processing schemes. (a) Covariance spectrum computed according to Eq. (1). (b) The variance calculated, by Eq. (10) at each point in the covariance spectrum. (c) matrix calculated according to Eq. (11). (d) as (c), after setting all elements to zero having a S∕N ratio less than (5.85), the score belonging to the critical value for which Dunn–Sidak correction yields a spectrum-wide . (e) Spectrum computed using thresholding at applied to the input HMBC and HSQC spectra. (f) Spectrum computed using maxima ratio scaling according to Eqs. (13) and (14). In (a), (c), (d), (e), and (f), the cross peak (corresponding to peak 2 in text and tables) has been clipped: the maximum amplitude of this peak is 263 [(a) and (e)], 294 [(c) and (d)], and 144 (f).

Selected spectral region taken from an experimental unsymmetric HSQC-HMBC covariance spectrum of metabolite mixture using different processing schemes. (a) Covariance spectrum computed according to Eq. (1). (b) The variance calculated, by Eq. (10) at each point in the covariance spectrum. (c) matrix calculated according to Eq. (11). (d) as (c), after setting all elements to zero having a S∕N ratio less than (5.85), the score belonging to the critical value for which Dunn–Sidak correction yields a spectrum-wide . (e) Spectrum computed using thresholding at applied to the input HMBC and HSQC spectra. (f) Spectrum computed using maxima ratio scaling according to Eqs. (13) and (14). In (a), (c), (d), (e), and (f), the cross peak (corresponding to peak 2 in text and tables) has been clipped: the maximum amplitude of this peak is 263 [(a) and (e)], 294 [(c) and (d)], and 144 (f).

## Tables

Variance of noise intensities in a simple unsymmetric covariance spectrum: simulation vs theory.

Variance of noise intensities in a simple unsymmetric covariance spectrum: simulation vs theory.

Variances of column∕row noise and peak intensity for two covariance peaks.

Variances of column∕row noise and peak intensity for two covariance peaks.

Expected signal-to-noise ratios for two covariance peaks.

Expected signal-to-noise ratios for two covariance peaks.

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