Series Solutions of Companding Problems

Series Solutions of Companding Problems By B. F. LOGAN, Jr.* (Manuscript received March 10, 1982) A formal power series solution (i) x(t) = mkXk(t) is given for the companding problem (ii) Bfx(t)j = my(t), Bx(t) = x(t), where B is the bandlimiting operator defined by Bg = (Bg){t) = /!!=* g(s)[sin (t - s)]/[ir(t s)]ds and f(t) has a Taylor series with /(0) = 0, /'(0) ^ 0. Expressions for the xk are given in terms of the coefficients of /, and operations on y, and in a different form in terms of the coefficients of the inverse function > $j/(x)j = t, x. A series development is given for a bandlimited z(t), Bz = z, such that the solution of (ii) is given by x = B(z). Also a series development is given for the "approximate identity", x -- Bct>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.