Signals designed for recovery after clipping — III: Generalizations

Let S(b, c) be the class of all real-valued bounded functions s(t) of the form <tex>$$s(t) = g(t) + \cos ct, \eqno{\hbox{(i)}}$$</tex> where g is bandlimited to [−b, b] and 0 < b < c < ∞ and such that <tex>$$(-1)^{k} s(k \pi/c) > 0, k = 0, \pm 1, \pm 2, \ldots, \eqno{\hbox{(ii)}}$$</tex> a condition that is always satisfied if |g(t)| < 1. In earlier papers we showed that such functions could be reconstructed from a knowledge of their zeros in the interval (t − T, t + T) to within an accuracy 0(e^−λT), where λ = c − b. This paper generalizes these results to functions of the form (i) satisfying the condition that s(i) have only real zeros, a condition which is weaker than (ii). The bounds on the accuracy of the reconstruction obtained are weaker. This paper also shows that every interval of length greater than 2π/λ, where λ = c − b > 0, must contain at least one zero of s(i), and that s(t) satisfies <tex>$$\vert s(t)\vert \le 2^{P-1}, -\infty < t < \infty,$$</tex> where p = 2c/λ.