Rapid estimation of spectra from irregularly sampled records

Records of physical quantities often arise as continuous electrical signals. Spectral estimates may be formed either by analogue means or from digitised samples that are then processed on a computer, When the samples are provided at regularly spaced time instants, this can be achieved very quickly with the aid of the f.f.t (fast Fourier transform) algorithm. There are situations, however, where the data is known only at random time instants, and the paper is concerned with the computation of spectral estimates from such data. When the sample times are Poisson distributed, it has been shown, in previous papers, that unbiased alias-free estimates can be formed, either through the correlation function or by a direct Fourier transform of short blocks of data. Random sampling introduces additional variability in these spectral estimates, and it is consequently necessary to process a large amount of data in order to achieve stable results. Unfortunately, this is very time consuming, most of the computer effort being spent evaluating sine and cosine functions which are then multiplied by the data samples. Here, two methods that can be used to simplify this operation are discussed. It is shown that when the sine and cosine functions are replaced by their equivalent rectangular waveforms, the resulting estimates can be related to spectral estimates through the Fourier expansion for the rectangular waves. A second way of speeding up the processing of Gaussian signals can be achieved by quantising the data to a sign bit and using the ‘arc-sine’ rule to transform the autocorrelation function to that of the full signal. It is shown that when both techniques are used together, and the processing reduced to 1-bit logical operations, valid spectral estimates can indeed be formed. These ideas are tested on various simulated sets of data.