Spectral recovery by analytic continuation in crossing-based spectrum analysis.
By the use of analytic continuation, the correct spectrum of an undersampled analog input signal f(a) (t) of a true bandwidth B is recovered from an aliased Fourier spectrum that is computed directly from a data set consisting of sinusoid-crossing locations {t(i)}, where the signal f(a) (t) intersects with a reference sinusoid r(t) with a frequency of W <B/2 and an amplitude of A. If A ≥ ‖f(a) (t)‖ within the sampling period T, then a crossing exists within each time interval Δ = 1/2W, and a total of 2WT = 2M sinusoid crossings are detected, where M is a positive integer. The cut-off frequency for sampling is W = ±M/T. In a crossing detector, a trade-off exists between the size of Δ and the accuracy with which a crossing can be located within it because the detector has infinite response time. Low-accuracy detection of the crossing positions degrades the detection limit of the detector and results in a computed Fourier spectrum that contains spurious wideband frequencies. We show however that, if f(a)(t) has a known compact support within T, then sampling at a frequency of W < B/2 may still be possible because the correct f(a)(t) spectrum can be recovered from the aliased spectrum by means of analytic continuation. The technique is demonstrated for an interferogram test signal in both the absence and presence of additive Gaussian noise.
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