SPECTRAL RECOVERY BY ANALYTIC CONTINUATION IN CROSSING-BASED SPECTRUMANALYSIS

By the use of analytic continuation, the correct spectrum of an unders ampled analog input signal f(alpha)(t) of a true bandwidth B is recove red from an aliased Fourier spectrum that is computed directly from a data set consisting of sinusoid-crossing locations {t(i)}, where the s ignal f(alpha)(t) intersects with a reference sinusoid r(t) with a fre quency of W < B/2 and an amplitude of A. If A greater than or equal to \f(alpha)(t)\ within the sampling period T, then a crossing exists wi thin each time interval Delta = 1/2W, and a total of 2WT = 2M sinusoid crossings are detected, where Mis a positive integer. The cut-off fre quency for sampling is W = +/-M/T. In a crossing detector, a trade-off exists between the size of Delta and the accuracy with which a crossi ng can be located within it because the detector has a finite response time. Low-accuracy detection of the crossing positions degrades the d etection limit of the detector and results in a computed Fourier spect rum that contains spurious wideband frequencies. We show however that, if f(alpha)(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(al pha)(t) spectrum can be recovered from the aliased spectrum by means o f analytic continuation. The technique is demonstrated for an interfer ogram test signal in both the absence and presence of additive Gaussia n noise. (C) 1996 Optical Society of America