Level Crossing Rate Bounds for Products of Independent Nonstationary Fading Processes

Source abstract

We develop a unified framework for the instantaneous total level crossing rate (LCR) of products of independent positive nonstationary fading processes. Universal upper and lower bounds are first derived from an exact decomposition of the product derivative and signed crossing imbalance, requiring only factor-wise crossing statistics. We then introduce the drifting Gaussian-derivative (DGD) class, in which each conditional derivative is Gaussian with possibly nonzero drift. For products of DGD factors, an exact Gaussian-mixture representation yields bounds that jointly capture aggregate drift and derivative fluctuations. The lower bound reduces under stationarity to a known $\ell_2$ bound, whereas, to the best of the author's knowledge, the upper bound is new even in the stationary case. Furthermore, the bounds are asymptotically exact in drift-dominated regimes. The framework is specialized to nonstationary Double-Rayleigh fading and to quasi-identically distributed DGD factors, with closed forms for products of $α$-$μ$ envelopes. Numerical results show that the DGD bounds closely bracket the exact LCR.