Francis R. Verdun

Effects of Noise, Time-Domain Damping, Zero-Filling and the FFT Algorithm on the “Exact” Interpolation of Fast Fourier Transform Spectra

A frequency-domain Lorentzian spectrum can be derived from the Fourier transform of a time-domain exponentially damped sinusoid of infinite duration. Remarkably, it has been shown that even when such a noiseless time-domain signal is truncated to zero amplitude after a finite observation period, one can determine the correct frequency of its corresponding magnitude-mode spectral peak maximum by fitting as few as three spectral data points to a magnitude-mode Lorentzian spectrum. In this paper, we show how the accuracy of such a procedure depends upon the ratio of time-domain acquisition period to exponential damping time constant, number of time-domain data points, computer word length, and number of time-domain zero-fillings. In particular, we show that extended zero-filling (e.g., a “zoom” transform) actually reduces the accuracy with which the spectral peak position can be determined. We also examine the effects of frequency-domain random noise and roundoff errors in the fast Fourier transformation (FFT) of time-domain data of limited discrete data word length (e.g., 20 bit/word at single and double precision). Our main conclusions are: (1) even in the presence of noise, a three-point fit of a magnitude-mode spectrum to a magnitude-mode Lorentzian line shape can offer an accurate estimate of peak position in Fourier transform spectroscopy; (2) the results can be more accurate (by a factor of up to 10) when the FFT processor operates with floating-point (preferably double-precision) rather than fixed-point arithmetic; and (3) FFT roundoff errors can be made negligible by use of sufficiently large (> 16 K) data sets.

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

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.