A simple and efficient approximation to the modified Bessel functions and its applications to Rician fading

In recent days, relay assisted cellular networks are gaining more importance in research and development because of the recent adoption of new communication standards with relaying and cooperation communication. This has introduced a multichannel diversity along with the multiuser diversity and the channel aware dynamic resource allocation models. The issue of the optimal location of relays has risen especially when dedicated relays are used as the standard proposes instead of the cooperative model of the users. In this paper, we study the optimal location of a single relay. Furthermore, we study the effect of changing the number of users on the optimal location of the relay. The effect of adding multiple relays to the system is examined. The optimal locations are examined when the relay channels are the only channels to be used by the system and when the direct channel (DT) is also available. The problem formulation assumes, Rayleigh block faded channels, half duplex regenerative (repetition coding) decode-and-forward (DF) relaying strategy, long-term average total transmitted power constraint and orthogonal multiplexing of the users messages within the channel blocks. New simple and accurate approximations to the modified Bessel functions of the first kind, zeroth order I0 (z) and first order I1 (z) are presented. The new proposed approximations are given as a simple finite sum of scaled exponential functions. Comparisons are made between the exact functions, classic approximations, and the new approximation in terms of simplicity and accuracy. The new approximation proves to be sufficiently accurate to bridge the gap between the classic large and small argument approximations and has potential applications in allowing one to analytically evaluate integrals containing Modified Bessel Functions, yielding simple closed-form solutions. A generalized closed-form expression for the average bit error rate over Nakagami-n (Rice) fading, and Rayleigh fading as a special case, are derived as sample applications, and the results are compared with Monte Carlo Simulation, where a very good matching is achieved.