Level Crossing Rate Inequalities for Product Processes and Applications to Fading Multichannels

This paper presents <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">sharp</i> inequalities for the level crossing rate of a stochastic process composed of the product of independent stochastic processes. The inequalities, which are simple to state, enlighten the contribution of each process individually. Additionally, we derive an exact formulation when the conditioned pointwise derivative of each process follows a Gaussian distribution. As application examples, the results are exercised in different scenarios using the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\alpha\text{-}\mu $ </tex-math></inline-formula> and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\kappa\text{-}\mu$ </tex-math></inline-formula> fading models, which encompass several other well-known models such as Semi-Gaussian, Rayleigh, Rice, Nakagami-<italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</i>, and Weibull. We provide unprecedented closed-form formulations for the level crossing rate bounds of the product of these widely-known fading processes. Notably, there are no closed forms for the level crossing rate of these products. Therefore, our bounds supply a benchmark for this metric, with one of them serving as an excellent approximation of the exact metric.