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In a multi-carrier phase ranging (MPR) system, the distance that radio signaltravels is estimated through phase shift of multiple carrier frequencies. Dueto phase ambiguity, a unique estimation can only be obtained within theunambiguous distance (UD), which depends on the carrier frequencies used forranging. Without external information, the maximum measurable distance of anMPR system is defined by its UD. The MPR system employing random frequencyhopping (FH) waveform has a strong anti-jamming capability and sees promisingpotentials in many fields. However, it is challenging to depict its measurabledistance. Different from current MPR system employing deterministicfrequencies, the carrier frequency under the FH waveform hops randomly withinthe occupied bandwidth. Consequently, the UD is a random variable. In thispaper, we try to find a deterministic value to depict the UD of an MPR systemunder the random spaced frequencies (RSF) configuration, serving as a metric ofits measurable distance in engineering applications. It is safe to adopt thelower bound of the random UD as the metric, but the measurable distance may bedrastically underestimated. Alternatively, we propose to adopt the upper boundof the random UD as the metric, because when the RSF set contains more than adozen of carriers, i) we prove the probability that the random UD obtains itsupper bound is very close to 1 if phase noise is not introduced; ii)simulations show that the upper bound can also be obtained reliably in thepresence of phase noise. As a comparison, the practical measurable range of theMPR system under the traditional linearly spaced frequencies (LSF)configuration is only a fraction of its theoretical UD because the UD is moresensitive to phase error.

On the Unambiguous Distance of Multi-Carrier Phase Ranging with Random Hopped Frequencie