B.S.T.J. Briefs: Approximation of the Error Probability in a RegenerativeRepeater with Quantized Feedback

Recently, Zador 1 gave a clever functional iteration procedure for determining the error probability in a binary regenerative repeater with quantized feedback. Unfortunately, quantitative results for the long pulse sequences of interest are difficult to come by due to the prohibitive amount of computer time required to carry out the iterations. We have found a simple approximate procedure that breaks the computational bottleneck in all cases of practical interest. The crux of our approach is the approximation of the functional iteration by a difference equation. For clarity, we use only a few terms of a Taylor series in establishing the difference equation approximation. More terms can be used to obtain a better approximation if needed. I I . R E C A P O F ZAOOR'S W O R K In Ref. 1, Zador shows that the kth iterate of the transformation Uf(x) -- denoted by Ukf(x) = TJk~l[Uj{x)] -- when evaluated at x = 0 yields the average bit error probability p(k), for the last bit in a random sequence of k + 1 bits processed by a regenerative repeater with quantized feedback. The transformation Uf(x) is given by [Zador's (14)] Uf(x) where pi(.t) = p[N(-go - a:)] p2(x) = 1 - pi(x) -- pz(x) p3(x) = g[l - N(g0 - .r)] f(x) = Pi(x) + p3(x) (2) = Pl(x)f(rx