Statistics of Regenerative Digital Transmission

An ideal regenerative digital repeater is defined here as a nonlinear time-varying device with response expressed as the product of a staircase o u t p u t vs. input function such as shown in Fig. 1(a) and a timesampling function such as shown in Fig. 1(b). T h e first function (a) converts a continuous range of possible input values to a discrete or quantized set of o u t p u t values. T h e second function (b) is a time-varying response function which ideally makes the repeater sensitive only during infinitesimally narrow time intervals with regular spacing T = l / / r . T a k e n together, the two functions quantize and sample the input wave at uniformly spaced instants of time. T h e input wave consists of a discrete-valued message component plus noise and distortion accumulated 1501 1502 T H E BELL SYSTEM T E C H N I C A L J O U l t N A L , N O V E M B E R 1958 in transmission f r o m the previous repeater. If the noise and distortion are sufficiently small to confine the departure from the proper discrete message value within one quantization interval at the sampling instants, the regenerated wave matches the message and errorless transmission is achieved for any number of repeater spans. To economize bandwidth, the actual pulses transmitted are not infinitesimally sharp, nor even rectangular as indicated in Fig. 1(b), but are multiples of a standard finite width pulse g(t) which m a y be considered to be generated by a linear network from the sharp samples. A typical outgoing wave from the repeater is therefore expressible b y : x(t) = Z ang(t - ?iT),