An implicit sampling theorem for bounded bandlimited functions

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The following sampling theorem is proved: Let f(t) be a bounded band-limited function, possibly a sample of a nonstationary stochastic process, such that |f(t)| < B. Denote by wo the appropriately defined bandwidth of f(t). Let tk denote the set of instants for which f(t) = C cos 2πwt, with C > B and w > w0. Then f(0) and tk determine uniquely f(t). Namely f(t) is represented, up to a multiplicative constant, by its sine-wave-crossings, i.e., by the set of its argument values at which it crosses a given sinusoid the amplitude and the frequency of which exceed, respectively, the bound on f(t) and the limit on its band.

The reconstruction of f(t) from f(0) and tk is a noncausal operation. A practical feedback scheme that interpolates a causal estimate of f(t) from the set of its past sine-wave-crossings and from f(0) is introduced. The input to the circuit is a binary waveform: its phase changes occur at tk and its amplitude is linear in f(0).

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The work reported in Section II has been completed while the author was with the Department of Electrical Engineering, City College of New York, and has been supported by NASA Grants NGR 33-013-048 and NGR 33-013-063.