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Several methods have been proposed to determine a high-resolution NNR spectrum from limited data. This paper attempts to improve and extend the NMR spectrum estimation method using a complex Kalman filter, which was pro- posed previously by Nishlyama and Mita. In the proposed NMR spectrum estimation using the Kalman filter, extra data were required in addition to that to be used in the spectrum estimation, in the approximate determination of the spin-spin relaxation time. Hence, the previously reported NMR spectrum estimation is not based strictly on the same condition as that of the discrete Fourier transform. This paper improves this point, and the extra data are obtained by extrapolation by the AR model. This makes it possible to compare the method with the discrete Fourier transform based on the same number of data. It is shown that when the spectrum is limited to a certain frequency band, the proposed method gives an NMR spectrum with twice higher frequency resolution by reducing the effect of the spin-spin relaxation time and the effect of discrete approximation. In the proposed method, the spin-spin relaxation time of the nucleus, corresponding to the frequency resolution that is twice higher than usual, is determined by a linear interpolation of the spin-spin (relaxation time which is obtained approximately by discrete Fourier transform. [ABSTRACT FROM AUTHOR]

As a method to estimate the frequency spectrum, the discrete Fourier transform is employed widely, since a high-speed algorithm is available. However, the discrete Fourier transform has the following problems. The presence of noise is not considered, and the resolution of the frequency spectrum is determined uniquely by the sample duration. Other problems are that the sampling inter- val must be kept constant and the decay of the signal is not considered. This paper proposes a high-resolution frequency spectrum estimation where the complex Kalman filter is applied to the observed signal represented by the complex stochastic system, effectively utilizing the a priori knowledge of the signal source. The method is applied to the measured data and to the simulation, and it is demonstrated that a high-resolution frequency spectrum is obtained by suppressing the effect of the discrete approximation of the frequency component, and, when the frequency spectrum is concentrated, the resolution can be improved approximately by a factor of two, without iterating the measurement. It is shown also that the sampling interval can be set arbitrarily in the proposed method, and the noise and the decay can be considered. A theoretical discussion is made on the relation between the special case of the proposed method and the discrete Fourier transform. [ABSTRACT FROM AUTHOR]

The Fourier transform NMR (nuclear magnetic resonance) spectroscopy is used widely at present as a noninvasive method in the structural analysis of macromolecules in organic chemistry and in the metabolic study of tissue and cells in biochemistry. We believe, however, that the Fourier transform is not adequate in the determination of the density distribution of the nucleus. The NMR signal contains essentially the relaxation (a kind of damping) phenomena, but the Fourier transform does not take this fact into consideration. Consequently, the obtained NNR spectrum is degraded both in intensity and resolution, providing apparent nucleus density distribution which is different from the true value. This paper proposes an estimation of the NNR spectrum, which positively takes the relaxation phenomenon into consideration. The method is derived by extending the Kalman filter to the complex stochastic system and applying the result to the NMR signal represented by the complex stochastic system. The proposed method is applied to the NMR signal of proton (114) obtained from mayonnaise in 2[T] static magnetic field. The result is compared with the NMR spectrum obtained by the Fourier transform, and it is shown that the resolution and the intensity are improved drastically. [ABSTRACT FROM AUTHOR]