Convergence of Power Control Algorithms for Wireless Networks with and without Fading

This paper presents a geometric analysis of the convergence condition for the Foschini-Miljanic power control algorithm with and without fading. In the fading case, a condition on the Lyapunov exponent is provided as the convergence condition. The Möbius transform is exploited for the first time to analyze the convergence conditions for both non-fading and fading cases. For the two-transmitter case, it reveals the connection between the eigenvalue condition of the non-fading case and the Lyapunov exponent condition of the fading case. An analysis of the power levels in linear networks is provided and applied to cognitive radio networks. The peak power constraint of wireless networks is also analyzed theoretically, and applications to random networks are explored in detail. Numerical studies confirm the accuracy of our results.