Optimized derivative fast Fourier transform with high resolution and low noise from encoded time signals: Ovarian NMR spectroscopy

The unfiltered derivative fast Fourier transform (dFFT) of degrees higher than two fails flagrantly for encoded time signals. These data are always dominated by noise at larger times of encodings. Such a breakdown is due to processing the unweighted product of the time signal and the time power function. The latter is generated by the frequency derivative operator applied to the fast Fourier transform (FFT). As a result, the unfiltered dFFT cannot separate the overlapped resonances and it dramatically decreases signal-to-noise ratio (SNR) relative to the FFT. This problem is solved by a derivative-adapted optimization with the properly attenuated filters. The ensuing optimized dFFT achieves the long sought simultaneous enhancement of both resolution and SNR. It uncovers the genuine resonances hidden within overlapping peaks to enable quantitative interpretations. It does not impose any model on the input time signals nor on the output lineshape in the spectra. It is computationally expedient as it uses the Cooley-Tukey fast algorithm. The present applications deal with time signals encoded by in vitro NMR spectroscopy from human malignant ovarian cyst fluid. A remarkably successful performance of the optimized dFFT is demonstrated for reconstructed spectra of potentially added value in clinical decision-making.