A Framework for the Lower Bound on the BER of DCSK Systems Over Multi-Path Nakagami-m Fading Channels

In this brief, a generalized closed-form approach for the evaluation of the lower bound on the BER of differential chaos-shift-keying (DCSK) systems over multi-path Nakagami-m fading channels is presented. A closed-form lower bound is derived for all configurations of the Nakagami-m channel model which extends to all fading profiles covered by the Nakagami model. In contrast to previous works where very specific models were targeted and only numerical analyses of the DCSK systems were presented, we provide an approach that analytically unifies the performance analysis of DCSK systems over more generalized fading models. A similar analytical development is generalized for Quadrature Chaos Shift Keying (QCSK) and Multi-carrier Differential Chaos Shift Keying (MC-DCSK) systems. A lower bound is applied on the Q-function to extract an exponential expression of the BER. Afterwards, a bound on the exponential function is used to derive a simple polynomial expression which is easily integrable using basic integration techniques for all multi-path Nakagami-m fading channels. The final expression of the newly derived lower bound is forwarded in terms of the Gaussian hypergeometric functions. Comparative simulations conducted for the different systems under consideration show a tight gap between the BER of the simulated systems and their respective lower bounds.