Fourier transformation and spectrum analysis of sparsely sampled signals

Alternative methods for deriving Fourier transforms and power spectral estimates from sparse signal samples are evaluated by analysis and computer simulations. The results confirm that randomization of the sample interval can effectively circumvent the well-known Nyquist sampling theorem and make possible spectrum analysis at frequencies well above the average sample rate. The paper compares spectrum analysis by direct Fourier transformation of the time-domain signal samples with spectrum analysis via the Dirac comb correlation-function method recently proposed by Gaster and Roberts (1975, 1977), and introduces random sample processing methods that are based on simultaneous-equation regression analysis. Careful consideration is given to frequency-resolution (spectral window) issues and the confidence factors associated with the measured frequency spectra.