Cardinal series to filter oversampled truncated magnetic resonance signals

Abstract

Digital low pass filters are routinely used to improve the signal-to-noise ratio of NMR signals, e.g. FID or echoes, when pass band widths of the available analogue filters do not correspond to the spectral width of the signals. Applying digital filters will always necessitate an oversampling of the signal to filter. The digital filters with which the commercial spectrometers are nowadays equipped and most of those known to date from literature were designed to be applied to signals in the time domain. Nevertheless, most of them are aimed at optimising the filtering of signals in the frequency domain and tend to distort them in the time domain, especially when applied to truncated signals. Herein we propose a low pass filter that preserves all the features of the signal in both domains. The method consists in fitting raw NMR data with a finite sum of truncated cardinal sine functions and requires nothing but the signal being a band-limited function. We devised sensible and, in practice, hardly restrictive rules for setting parameters of the filter and applied it to various computer-simulated and experimentally measured truncated data sets to demonstrate its success in filtering both FID and echo signals.