New alternative series representations for the generalized Nuttall Q-function with applications

摘要

In this article, we present six recently derived distinct series representations of the generalized Nuttall Q-function with their corresponding Marcum Q-function. We investigate different application areas where these series expressions might be of interest. Using these representations, we address some of the outstanding problems that have until now resisted simple solutions. First, we show that the new series representation can be used in the performance evaluation of the periodogram using the Neyman-Pearson criterion. We show that, for the particular case of periodogram with zero padded fast Fourier transform, our series are the only existing analytical formulas capable of producing the desired numerical results. This is because of the need to compute and/or average the generalized Marcum Q-function of real orders which is not supported by commercial mathematical software such as MAPLE and MATLAB. Second, we present different expressions that can be used to evaluate the average area under the receiver operating characteristics (ROC) curve over generalized fading channels. In this case, many existing result are limited to specific fading channel and/or integer time-bandwidth product whereas, our expressions place no such restrictions. Lastly, we present the only existing closed-form solution for the evaluation of the partial area under the ROC curve which is of practical interest but has been shown to be difficult to evaluate with closed-form formulas. Numerical results have been presented for representative case and the obtained result agrees with analytical expectations. To the best of our knowledge, all the problems solved using these series representations are still open problems in the literature.