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Localization in unknown environments using low-cost sensors on embedded hardware is challenging. Yet, it is a requirement for consumer robots if systematic navigation is desired. In this paper, we present a localization approach that learns the spatial variation of an observed continuous signal over the environment. We model the signal as a piecewise linear function and estimate its parameters using a simultaneous localization and mapping (SLAM) approach. By applying the concepts of the exactly sparse extended information filter (ESEIF) , a constant-time, linear-space algorithm is obtained under certain approximations. We apply our framework to a sensor measuring bearing to active beacons, where measurements are distorted because of occlusion and signal reflections. Experimental results from running GraphSLAM, extended Kalman filter SLAM, and ESEIF-SLAM on manually collected sensor measurements, as well as on data recorded on a vacuum-cleaner robot, validate our model. The ESEIF-SLAM solution is evaluated on an ARM 7 embedded board with 64-kB RAM connected to a Roomba 510 vacuum cleaner. The presented methods are also used in Evolution Robotics ' Mint Cleaner product for autonomous floor cleaning.