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The LPM local positioning system is based on a bi-static frequency-modulated continuous-wave radar where the frequency ramps are generated by two different units of circuitry: one of higher quality in the base station (BS) and one optimized for low power consumption in the transponder (TP). This difference in circuitry causes a small chirp rate difference (CRD) between the ramp incline of the BS and that of the TP, which has previously been considered negligible. We show that it is imperative to take account of this CRD in the intermediate frequency signal. We derived a method based on a nonlinear least squares (NLS) estimation principle to estimate its value. The NLS method can further be used for fault detection and to mitigate the effect of multipath signals, which we validate with both simulated and measured data.