Many potentially interesting and useful classes of NMR experi-However, the Fourier transform is only strictly applicable to ments generate data for which conventional spectral estimation the limited subset of ''complete '' signals, i.e., t Å 0 to ϱ and quantification via the Fourier transform are unsatisfactory. . Fourier transformation of signals that are truncated, In particular, recently introduced solid-state NMR experiments at either the start or the end of the decay, leads to familiar which involve long delays before data acquisition fall into this spectral distortions, baseline roll, and ''sinc-wiggles,'' recategory, as the free induction decays are heavily ''truncated'' spectively. Besides, there is no built-in mechanism for noise and have low signal-to-noise ratios. A novel detection-estimation suppression. The linear nature of the Fourier transform imscheme is introduced in order to analyze data from such experiplies that reducing these problems or improving apparent ments and others where the sensitivity is low and/or the data resolution can only be done at the expense of spectral resolurecord is strongly damped or truncated. Based on the assumption of exponential data modeling, the number of signals present is tion and/or sensitivity. first detected using criteria derived from information theory and
The goal of NMR spectral estimation is to obtain an estithe spectral parameters are then estimated using the matrix pencil mate of the frequency response function of the underlying method. Monte Carlo simulations and experimental applications spin system from the measured free induction decay (FID). are carried out to demonstrate its superior statistical and computa-A particular FID can be characterized in terms of a model tional performances and its general applicability to delayed acquifunction with a set of free parameters. A crucial problem in sition data. Over the range of noise levels investigated, it is found NMR spectral estimation is, therefore, the detection of the that this approach is essentially near-optimal in the sense that the signal model and the estimation of the spectral parameters estimated spectral parameters have biases almost equal to zero (e.g., damping factor, frequency, amplitude, and phase). The and variances very close to their theoretical Crame ´r-Rao lower bounds. Compared to the popular method of linear prediction with difficulty of the detection-estimation problem is increased singular value decomposition, this method not only improves the by the low sensitivity inherent in NMR spectroscopy. Beestimation accuracy (by a factor of 2-4) with a lower ''breakcause of the computational complexity and noise interferdown'' signal-to-noise threshold ( É1.5 dB), but also reduces the ence, the problem is usually solved in two steps. The model computational cost by about an order of magnitude. It also holds function is first chosen and verified on physical grounds or great promise in effectively reducing truncation artifacts. It is by statistical tests. After successful signal modeling, the free concluded that this approach not only facilitates the analysis of parameters of the signals are then estimated. delayed acquisition data, but can also become a valuable tool in Detection theory refers to the selection of the physical or the routine quantification of general NMR spectra. A listing of mathematical model that best describes the measured pheprograms is also included in the Appendix. ᭧ 1997 Academic Press
nomena. The model function must be chosen with care; if the number of parameters is too large, many of them will be spurious, particularly if one must contend with noise,