Mean level signal crossing rate for an arbitrary stochastic process: reply to comment

摘要

In a comment by Santos Filho et al. [1] some comments and concerns regarding a recent article [2] have been raised. In this article we addressed the issue of finding an engineering approach to the average number of level crossings for a signal with an arbitrary probability density function (PDF) for its signal level. This article was based on the seminal work by Rice [3] in the 1940s. The authors of the above comment have concurrently published an article on the second-order statistics of Nakagami fading simulators [4]. The details for arriving at the results for the average number of level crossings and thus the average time for a surge and a fade can be found in the articles but will here be recapped in short. Our approach [2] was based on a nonlinear, memoryless transformation of the PDF under investigation into a zero-mean, unit variance Gaussian PDF from which we know the number of level crossings [3]. The transformation being bijective, and knowing that the correlation between the signal and its first derivative is zero, the signal and its first derivative were considered independent. We showed that one could in many cases obtain sufficient agreement between the analytical results and the results arrived at by simulations. As stated in the article, for some PDFs--especially Nakagami distributions with low order (exponential and Rayleigh distributions)--we observed a discrepancy. Here, the authors of this comment have in a good way addressed our concern in [2], where we "laid open future discussion" of this discrepancy. And we agree on their explanation for this discrepancy for these distributions, and thank them for their effort.