Digital filters for low-field NMR

We review the theory and operation of digital filters in modern nuclear magnetic resonance (NMR) spectrometers with fully digital receivers. Custom digital filters tailored for particular experimental requirements offer substantial improvements in signal-to-noise ratio (SNR), sensitivity, pulse sequence timing, and rejection of heteronuclear contamination. Pass-band filters are designed and applied in the frequency domain. In high-field imaging and spectroscopy, the impact of the filter is straight forward to visualize. However, low-field NMR data acquired on bench-top magnets are typically analyzed in the time-domain where the influence of a frequency-domain filter is not obvious and largely overlooked by end-users. We provide practical guidance on the design and implementation of digital filters for bench-top NMR applications, with examples of data acquired at 2.4 and 12.9 MHz. We discuss the compromise between speed (filter settling time) and noise rejection, and consider the special case of ¹⁹F signal contamination in ¹H measurements. We suggest filter designs for narrow-line liquid samples, broad-line samples, and imaging.