I. Introduction

The free induction decay (FID) transient signals collected in nuclear magnetic resonance (NMR) are subject to a number of stochastic disturbances. Among these, the one most often discussed is the stationary 'receiver noise' which originates both in the sample and in the receiver system and determines the achievable signal-to-noise ratio. The Author has often carried out a statistical analysis of this noise either as a part of an instrument check-up or as a self-standing study, born out of the awareness that if it were not Gaussian, one could devise signal estimators superior to standard arithmetic averaging of repeated scans. Apart from situations linked to specific instrument defects, and apart from low-probability 'spikes' due to external disturbances, the receiver noise has always turned out to be normal, ruling out any conceptual improvement of the classical data accumulation.

There are, however, disturbances of other types which always limit the performance of an instrument.
In principle, these fall into two categories:

   1) Random fluctuations of the main magnetic field and/or of the phase-detector reference frequency.
These two effects can't be separated from each other because what one normally detects are just the relative beats between the RF reference and the field-dependent Larmor frequency of the nuclei. Since it is easy to carry out very precise bench tests of the phase noise of any RF source, this parameter can be kept in check much more easily than the magnetic field noise for which precise testing methods other than NMR do not exist. Consequently, the field noise is usually much more important than the RF phase noise (especially when using digital RF synthesis techniques).

   2) Random fluctuations of magnetic field inhomogeneity across the sample.

For simplicity, the first type of fluctuations shall be called field-noise, while the second type shall be referred to as field-inhomogeneity noise or, in the presence of imposed gradients, field-gradient noise.

One must keep in mind that, like the receiver noise, the field noise is always present. Hardware engineers and manufacturers of magnets and NMR instruments of course strive hard to produce magnetic fields as stable as possible and, wherever feasible, screen them from external disturbances. Over decades, such efforts have led to remarkable improvements in field stability which, however, have been more than matched by equally remarkable increases in the requirements imposed by the evolving NMR and MRI methodologies. Moreover, there are situations where field noise is inherently rather bad. The two situations which come to mind in his context are fast-field-cycling NMR relaxometry (FFC NMR) with its highly dynamic magnets and all kinds of ex-situ NMR techniques, including well-logging and the NMR MOUSE where, almost by definition, the field within the area of interest can be neither screened nor stabilized.

The question is therefore not whether the field is unstable but rather what are the quantitative characteristics of its fluctuations, how they affect the acquired data, and whether there are ways of suppressing such artifacts by suitable acquisition methods and/or post-acquisition corrections.

Terminology and assumptions

- At any moment, there is a distribution of the magnetic field induction B(r,t) across the constituent voxels of the sample. The average B(t) = <B(r,t)>_v over the sample volume is the main magnetic field while the deviations H(r,t) = B(r,t)-B(t) constitute the magnetic field inhomogeneity.

- Both B(t) and H(r,t) are time-dependent. The time-averages B = <B(t)>_t and H(r) = <H(r,t)>_t are, respectively, the static main magnetic field and the static field inhomogeneity. The deviations b(t) = B(t)-B and h(r,t) = H(r,t) - H(r) are the respective main field noise and the inhomogeneity noise.

- We always intend the Z-axis to be aligned with the direction of the static main field B.

- The fluctuations b(t) and h(r,t) are random, time-dependent vector fields whose magnitudes are much smaller than B. Consequently, effects due to their x- and y-components can be neglected compared with those due to their z-components, denoted as b(t) = b_z(t) and h(r,t) = h_z(r,t).

- The static field inhomogeneity H(r) is related to magnet geometry and its imperfections, with additional corrections achieved in some applications by means of an active field-shim system. In all NMR applications, |H(r)| is supposed to be many orders of magnitude (typically 4 to 9) smaller than the main magnetic field B. By design, any small instability in B(t) originating within the magnet system is expected to to have a negligible effect on field inhomogeneity with its contribution to h(r,t) being orders of magnitude smaller than its contribution to b(t). External contributions to B(t), such as stray magnetic fields from distant sources, are expected to be quite homogeneous across the sample volume and thus unlikely to contribute appreciably to h(r,t).

Throughout this study, we assume that h(r,t) can be neglected with respect to b(t). Even if this were not true, however, the two effects would be probably poorly correlated so that it would be possible to study their respective effects independently of each other. This study concentrates on the effects stemming from b(t).

We shall find it convenient to distinguish between two types of field noise: those which are the same for every sample voxel (type A) and those which, despite a temporal coherence over the whole sample volume, have a non-uniform amplitude distribution (type B).

Typical sources of magnetic-field instabilities

a) Intrinsic (type A)

- Random noise in the current sources of air-coil magnets and electromagnets. For example, air-coil systems with top dynamic requirements such as those of fast field cycling relaxometers (FFC-NMR) are at present characterized by field noise of the order of 1 - 10 µT r.m.s. with correlation times of the order of 0.1 -1 ms. In a classical HR-NMR electromagnet the same parameters are likely to be of the order of 0.1 -1 µT and 1 - 10 ms.

- Residual brum and/or ripple in the current sources of air-coil magnets and electromagnets.

- Electronic noise in field stabilizers. In systems with active field-control systems such as flux stabilizers and/or NMR lock systems, the perturbations listed above are to a large extent suppressed. Such control loops, however, are not always applicable and, in any case, they have a noise of their own. They are often operated just below the limit of oscillations which makes them prone to fluctuations clustered around particular frequencies (quasi-periodic instabilities).

b) Environmental (type A)

- Stray alternating fields from mains power wiring, both external and internal to the instrument. Bear in mind that a wire carrying 1A of AC current located 2m from an magnetically unshielded sample generates within the latter a field modulation of 0.1 µT, corresponding to a proton Larmor-frequency modulation of 4.25 Hz!

- Stray alternating fields from magnetic devices such as AC power transformers (including those located within the instrument's own power supplies!).

c) Motion induced (type B)

- Sample rotation in HR-NMR spectroscopy is a classic example of a periodic motion across an inhomogeneous field which subjects individual sample voxels to periodic field modulation.

- Sample motions induced by environmental vibrations (cooling pumps, acoustic waves, floor tremble, etc.) or by turbulence in the gas flows through auxiliary devices which involve the sample-assembly (spinner, temperature controller, decoupler coils cooling, etc.). Again, motion-induced field instabilities depend upon field inhomogeneity and affect different sample voxels in a different way.

Just as it is impossible to completely suppress the receiver noise, no NMR instrument can be completely immune from field noise and from at least some of the other instabilities listed above. The resulting artifacts include:

In high-resolution NMR spectroscopy (HRNMR):

- Rotational sidebands.
- Sidebands at integer multiples of the mains frequency.
- Broadening of spectral peaks during repeated-scans averaging.
- Reduced efficiency of noise suppression by repeated-scans averaging.
- t1-noise in 2D spectra.

In magnetic resonance imaging (MRI):

- Image fringes related to the mains frequency.
- Reduced efficiency of image improvement by repeated-scans averaging.

In low-resolution NMR (LRNMR) and FFC relaxometry (FFC-NMR):

- Irreproducibility of individual FID's.
- Deformation of FID shapes after repeated-scans averaging.

While all such artifacts are well known from practice, their statistical properties have rarely been analyzed in detail and there is only a limited empirical knowledge of their propagation during the data averaging process.

One point should be stressed before proceeding with the analysis. Since this Note deals only with the main field noise and not with field-inhomogeneity noise, the obvious way of avoiding its effects consists in using one of the signal-detection methods insensitive to RF phase and offset (diode detection, power detection, envelope detection). While this such approaches are not acceptable in HR-NMR and MRI where offset-sensitive signals are a must, they may be viable in LR-NMR and in FFC relaxometry.

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V. Conclusions

We have shown how the phase noise due to magnetic field instabilities (or, alternatively but less likely, the phase noise of the reference frequency) propagate into NMR data such as FID's and their Fourier transforms. We have also delimited several typical types of magnetic field noise (random, periodic, and mixed) and derived a number of novel specific formulae covering the individual cases.

However, there is much which still needs to be done. So far discussed only the artifacts due to the statistical bias of signal phase projections which can be appreciated in averaged FID's. We should also try and establish some of the statistical characteristics of single-scan FID's and spectra or, more generally, those which pertain to the averages obtained after a limited number of scans.

This means asking questions like: Having acquired N scans,
a) what is the expected error in the FID signal intensity at time t,
b) what are the probability distribution functions for spectral peak heights,
c) how much do spectral line shapes fluctuate between individual scans,
d) what are the probability distribution functions for artifacts such as modulation sidebands.

According to Eq.[3], question (a) boils down to the probability distribution of exp[jφ_f(t)] which is relatively easy to handle.
The other questions, however, may be a bit more difficult to tackle.

Another unfinished chapter regards the practical consequences of the theory presented in this paper and the ways it can be used to remove or suppress field-noise artifacts. There are several avenues how this task can be tackled using alternative accumulation strategies (modulus & phase accumulation) or temporarily storing all individual scans and applying evaluation algorithms other than plain averaging at the end of the acquisition period. The latter approach has now become feasible thanks to continuing rapid advances of electronics, particularly in terms of memory capacity and mass-data evaluation speeds.

Clearly, there is much more work to be done. The promise is a significant increase in spectral resolution and a corresponding increase in spectral sensitivity (ratio of peak height to noise amplitude), removal of rotational sidebands, and correction of a number of other artifacts linked to random and periodic magnetic-field variations.