Upper bounds for the average error probability of a time-hopping wideband system

Ultra-wideband technology has been proposed as a viable solution for high-speed indoor short-range wireless communication systems because of its robustness to severe multi-path and multi-user conditions, low cost, and low power implementation. This paper proposes and analyzes a multi-user time-hopping system in which symbols are sent multiple times. A single-user receiver structure is proposed in which collisions with interfering users are discarded. Two formulas for the probability of error are derived. The first formula is a finite sum in which the number of terms grows with the number of symbol repetitions. The second formula is an integral that is well-suited to Chebyshev-Gauss quadrature as well as for allowing the derivation of bounds on the probability of error. Asymptotic formulas and upper bounds for the error probability are derived by letting various system parameters go to infinity. We evaluate the bounds numerically for some reasonable parameters and study their interplay with other parameters of interest.