On the Effect of Shadow Fading on Wireless Geolocation in Mixed LoS/NLoS Environments

Abstract

<para xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> This paper considers the wireless non-line-of-sight (NLoS) geolocation in mixed LoS/NLoS environments by using the information of time-of-arrival. We derive the CramÉr–Rao bound (CRB) for a deterministic shadowing, the asymptotic CRB (ACRB) based on the statistical average of a random shadowing, a generalization of the modified CRB (MCRB) called a simplified Bayesian CRB (SBCRB), and the Bayesian CRB (BCRB) when the <emphasis emphasistype="italic">a priori</emphasis> knowledge of the shadowing probability density function is available. In the deterministic case, numerical examples show that for the effective bandwidth in the order of kHz, the CRB almost does not change with the additional length of the NLoS path except for a small interval of the length, in which the CRB changes dramatically. For the effective bandwidth in the order of MHz, the CRB decreases monotonously with the additional length of the NLoS path and finally converges to a constant as the additional length of the NLoS path approaches the infinity. In the random shadowing scenario, the shadowing exponent is modeled by <formula formulatype="inline"><tex Notation="TeX">$\varsigma =u\sigma $</tex></formula>, where <formula formulatype="inline"><tex Notation="TeX">$u$</tex> </formula> is a Gaussian random variable with zero mean and unit variance and <formula formulatype="inline"><tex Notation="TeX">$\sigma $</tex></formula> is another Gaussian random variable with mean <formula formulatype="inline"> <tex Notation="TeX">$\mu _{\sigma }$</tex></formula> and standard deviation <formula formulatype="inline"><tex Notation="TeX">$\sigma _{\sigma }$</tex></formula>. When <formula formulatype="inline"><tex Notation="TeX">$\mu _{\sigma }$</tex> </formula> is large, the ACRB considerably increases with <formula formulatype="inline"> <tex Notation="TeX">$\sigma _{\sigma }$</tex></formula>, whereas the SBCRB gradually decreases with <formula formulatype="inline"><tex Notation="TeX">$\sigma _{\sigma }$</tex></formula>. In addition, the SBCRB can well approximate the BCRB. </para>