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Abstract:

DRAFT-- do not cite or circulate We present a method for recovering (to within a constant factor) periodic, octave band-limited signals given the times of the zero-crossings. Recovery involves taking the singular-value decomposition of a size N \Theta 2M matrix, where N is the number of zero-crossings within one period and M is product of the octave bandwidth and the period length. We also discuss approximate approaches which can be used to reconstruct aperiodic or very-long-period signals. Our algorithm achieves an inversion of Logan's theorem in the case where such is possible. 1 Logan's Theorem Sampling theorems provide conditions under which continuous signals may be represented by countable sets of real numbers. In the usual setting, we agree on a regular grid of time points and provide samples of the signal amplitude at those times. The Nyquist-Shannon theorems tell us that a lowpass signal may be reconstructed exactly, so long as the times of samples are spaced at least as densely as half the period of the highest frequency. However, we could also agree upon a regular grid of amplitude levels and provide the times at which the signal crossed those levels. This has been dubbed implicit sampling by Bar-David [Bond and Cahn, 1958, Bar-David, 1974]; although there is no clear theory relating the frequency content of the signal and the number of levels required. Logan's theorem [Logan, Jr., 1977] addresses a special case of this general problem of signal reconstruction from level crossings. It states that if a signal is band-limited to a single octave then the times of the zero crossings are sufficient to reconstruct the signal--to within a constant factor of course. 1

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