Series solutions of companding problems

Abstract

A formal power series solution (i) x(t) = Σ1^∞ m^k xk(t) is given for the companding problem (ii) Bf{x(t)} = my(t), B{x(t)} = x(t), where B is the bandlimiting operator defined by Bg = (Bg)(t) = ∫ g(s)[sin λ(t − s)]/[π(t − s)]ds and f(t) has a Taylor series with f(0) = 0, f′(0) ≠ 0. Expressions for the xk are given in terms of the coefficients of f, and operations on y, and in a different form in terms of the coefficients of the inverse function φ, φ{(x)} = x. A series development is given for a bandlimited z(t), Bz = z, such that the solution of (ii) is given by x = Bφ(z). Also a series development is given for the “approximate identity”, x ≐ Bφ{Bf(x)}, where x = x(t), Bx = x, which is shown to be a good approximation to x for fairly linear f(x), not necessarily having a Taylor series expansion. As an example of one application of the results, a few terms are given for correction of the “inband” distortion arising in envelope detection of “full-carrier” single-sideband signals. The results should prove useful in correcting small distortions in other transmission systems. Finally, it is shown that the formal series solution (i) actually converges for sufficiently small |m|. This involves proving that the companding problem (ii) has a unique solution for arbitrary complex-valued y(t) and complex m of sufficiently small magnitude, the solution x(t; m) being, for each t, an analytic function of the complex variable m in a neighborhood of the origin. It is a curious fact, as shown by an interesting example, that the series (i) may converge for values of m for which it is not a solution of (ii).