Signals Designed for Recovery After Clipping-II. Fourier Transform Theory of Recovery

Abstract

This paper develops a Fourier transform theory for the recovery of signals of a certain class from their zeros. The class, denoted by S(b, c), consists of real-valued signals of the form s(t) = g(t) + cos et, where g is bandlimited to [-b, b], 0 < b < e < 00, and such that (-l)ks(k-n/e) > 0, which is satisfied, for example, if Ig(t) I < 1. A very simple method of recovery is given for the case e> 3b, and a somewhat more complicated method is given for the case e > 2b.The theory also suggests a novel method of effecting amplitude modulation, which has advantages in high-power applications.

Reconstruction of multidimensional signals from zero crossings

“…In fact, we can permit a more generalized form of threshold crossings by allowing crossings of an arbitrary periodic function. This form of threshold crossings can be considered as a generalization of sine-wave crossings as used in other work such as [16]. also generalizes previous work with sine-wave crossings mentioned earlier.…”

“…Logan's result has been extended to two dimensions [2,15] by requiring a one-dimensional signal derived from the original two-dimensional signal to satisfy the constraints of Logan's theorem. In addition, one-dimensional results on reconstruction from sine-wave crossings have been extended to two-dimensional problems [16]. However, as mentioned earlier, the two-dimensional problem is fundamentally different from the one-dimensional problem since in two dimensions, the zero crossings" are actually zero crossing contours and not isolated points as in the one-dimensional case.…”

Signals Designed for Recovery After Clipping-III: Generalizations

“…2, generalizations of the basic recovery formula were developed, showing how s(t) could be recovered from bandlimited versions of h(t) or, equivalently, from bandlimited versions of (8) Here we wish to extend the validity of the previous results by removing the alternation condition (lb), simply requiring that s(t) of the form (1) In order to obtain a bound on Ih(t) I we consider (9) and show that h' (r) is a high-pass distribution with no spectrum in (-A, A). To do this we must first show that the total mass of the distribution in any interval of fixed length T is uniformly bounded.…”

“…The recovery procedures involve operations on the so-called fundamental function associated with the zeros ltkl of s, (2) where J(t) is a jump function increasing by 7[ at each zero tk, J(O) = 0. The linearly decreasing term -ct just offsets the growth of J(t) so that -7[ < h(t) < 7[, -00 < t < 00.…”

An Impossibility Result for Linear Signal Processing Under Thresholding

“…In general, is only an approximation of , and we want the reconstructed signal to be close to if is sufficiently small. We use the peak reconstruction error (2) to measure this closeness. Since all signals are uniquely determined by their samples, and the series , , uses all "important" samples of the signal, i.e., all samples that are larger or equal than , one could expect to be a good approximation for , at least if is small.…”