  This paper proposes the first distributed algorithm that solves the
weight-balancing problem using only finite rate and simplex communications
among nodes, compliant with the directed nature of the graph edges. It is
proved that the algorithm converges to a weight-balanced solution at sublinear
rate. The analysis builds upon a new metric inspired by positional system
representations, which characterizes the dynamics of information exchange over
the network, and on a novel step-size rule. Building on this result, a novel
distributed algorithm is proposed that solves the average consensus problem
over digraphs, using, at each timeslot, finite rate simplex communications
between adjacent nodes -- some bits for the weight-balancing problem and others
for the average consensus. Convergence of the proposed quantized consensus
algorithm to the average of the node's unquantized initial values is
established, both almost surely and in the moment generating function of the
error; and a sublinear convergence rate is proved for sufficiently large
step-sizes. Numerical results validate our theoretical findings.
