The extended Fourier transform for 2D spectral estimation.
We present a linear algebraic method, named the eXtended Fourier Transform (XFT), for spectral estimation from truncated time signals. The method is a hybrid of the discrete Fourier transform (DFT) and the regularized resolvent transform (RRT) (J. Chen et al., J. Magn. Reson. 147, 129-137 (2000)). Namely, it estimates the remainder of a finite DFT by RRT. The RRT estimation corresponds to solution of an ill-conditioned problem, which requires regularization. The regularization depends on a parameter, q, that essentially controls the resolution. By varying q from 0 to infinity one can "tune" the spectrum between a high-resolution spectral estimate and the finite DFT. The optimal value of q is chosen according to how well the data fits the form of a sum of complex sinusoids and, in particular, the signal-to-noise ratio. Both 1D and 2D XFT are presented with applications to experimental NMR signals.
Armstrong
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Chemistry Department, University of California-Irvine, Irvine, CA 92697-2025, USA. garmstro@uci.edu
Mandelshtam
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