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Position localization of transmitters can be carried out by an adversary owning a network of pervasive receivers, which can pinpoint the victim mobile nodes' locations with high temporal and spatial accuracy, such that pseudonym changing and higher-layer obfuscation are insufficient to protect their location privacy. Our contribution is to consider covert beam patterns, generated using multiple antennas to do adaptive beamforming, and so reducing radio signature. We architect such a privacy-enhancing system, give an informal security analysis, and develop an evaluation framework to analyze its location privacy. We performed simulations using wireless LAN parameters, and found that signal-to-noise-ratios for successful direction-finding are more stringent than those required for mere communications. We composed an end-to-end integrated radio and mobility simulation, and compared location privacy performance of omnidirectional versus adaptive beamforming antennas. Our proposal is shown to perform better. In addition, our evaluation framework is flexible and extensible.