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### Beating the Nyquist Limit by Means of Interleaved Alternated Delay Sampling: Extension of Lower Mass Limit in Direct-Mode Fourier Transform Ion Cyclotron Resonance Mass Spectrometry

*Volume 42, Number 2 (Feb. 1988) Page 199-203*

Verdun, Francis R.; Ricca, Tom L.; Marshall, Alan G.

According to the Nyquist theorem, the highest signal frequency which can be represented without foldover (aliasing) in a Fourier transform frequency-domain discrete spectrum is one-half of the time-domain sampling frequency. For example, since ion cyclotron resonance (ICR) frequency is inversely related to ionic mass-to-charge ratio, *m/z*, the highest ICR frequency (corresponding to the lowest correctly represented *m/z*) in direct-mode Fourier transform ICR mass spectrometry is restricted to one-half of the maximum sampling frequency, or about *m/z* ≥ 18 at 3.058 tesla (T) for a maximum sampling frequency of about 5.2 MHz. In this paper, we show that interleaved addition of two digitized time-domain transient signals, one of which is delayed by one-half of one sampling period (i.e., half of one cycle of the time-domain sampling frequency) with respect to the other, generates a time-domain discrete waveform which is indistinguishable from a single waveform produced by sampling at twice the original sampling rate. Thus, provided that the two transients have (or have been normalized to) the same magnitude, one can double the Nyquist-limited frequency range. If the sampling period is divided into three or more equal parts, with interleaved addition of three or more correspondingly delayed transients, the same method can further increase the upper frequency limit. The method is applied to the experimental doubling or quadrupling of FT/ICR direct-mode frequency range, as for example in the extension of the lower mass limit to below *m/z* = 12 at 3.058 T with a sampling rate of only 4.0 MHz.