Spectral estimation of NMR relaxation

In this paper, spectral estimation of NMR relaxation is constructed as an e xtension of Fourier Transform (FT) theory as it is practiced in NMR or MRI, where multidimensional FT theory is used. no NMR strives to separate overl apping resonances, so the treatment given here deals primarily with monoexp onential decay. In the domain of real error, it is shown how optimal estima tion based on prior knowledge can be derived. Assuming small Gaussian error , the estimation variance and bias are derived. Minimum bias and minimum va riance are shown to be contradictory experimental design objectives. The an alytical continuation of spectral estimation is constructed in an optimal m anner. An important property of spectral estimation is that it is phase inv ariant. Hence, hypercomplex data storage is unnecessary. It is shown that, under reasonable assumptions, spectral estimation is unbiased in the contex t of complex error and its variance is reduced because the modulus of the w hole signal is used. Because of phase invariance, the labor of phasing and any error due to imperfect phase can be avoided. A comparison of spectral e stimation with nonlinear least squares (NLS) estimation is made analyticall y and with numerical examples. Compared to conventional sampling for NLS es timation, spectral estimation would typically provide estimation values of comparable precision in one-quarter to one-tenth of the spectrometer time w hen SIN is high. When SIN is low, the time saved can be used for signal ave raging at the sampled points to give better precision. NLS typically provid es one estimate at a time, whereas spectral estimation is inherently parall el. The frequency dimensions of conventional nD FT NMR may be denoted D-1, D-2, etc. As an extension of no FT NMR, one can view spectral estimation of NMR relaxation as an extension into the zeroth dimension. In no NMR, the i nformation content of a spectrum can be extracted as a set of n-tuples (ome ga(1),...omega(n)), corresponding to the peak maxima, Spectral estimation o f NMR relaxation allows this information content to be extended to a set of (n + 1)-tuples (lambda, omega(1),...omega(n)), where lambda is the relaxat ion rate. (C) 2000 Academic Press.