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

## TL;DR

This paper investigates the unambiguous distance in multi-carrier phase ranging systems with random frequency hopping, proposing a reliable upper bound metric that accurately reflects the measurable distance despite phase noise.

## Contribution

It introduces a method to determine a deterministic upper bound for the unambiguous distance in random frequency hopping MPR systems, improving measurement reliability.

## Key findings

- The probability of the UD reaching its upper bound approaches 1 with more than a dozen carriers.
- The upper bound metric remains reliable even with phase noise.
- Traditional linearly spaced frequency systems have a smaller practical range due to phase error sensitivity.

## Abstract

In a multi-carrier phase ranging (MPR) system, the distance that radio signal travels is estimated through phase shift of multiple carrier frequencies. Due to phase ambiguity, a unique estimation can only be obtained within the unambiguous distance (UD), which depends on the carrier frequencies used for ranging. Without external information, the maximum measurable distance of an MPR system is defined by its UD. The MPR system employing random frequency hopping (FH) waveform has a strong anti-jamming capability and sees promising potentials in many fields. However, it is challenging to depict its measurable distance. Different from current MPR system employing deterministic frequencies, the carrier frequency under the FH waveform hops randomly within the occupied bandwidth. Consequently, the UD is a random variable. In this paper, we try to find a deterministic value to depict the UD of an MPR system under the random spaced frequencies (RSF) configuration, serving as a metric of its measurable distance in engineering applications. It is safe to adopt the lower bound of the random UD as the metric, but the measurable distance may be drastically underestimated. Alternatively, we propose to adopt the upper bound of the random UD as the metric, because when the RSF set contains more than a dozen of carriers, i) we prove the probability that the random UD obtains its upper 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 the presence of phase noise. As a comparison, the practical measurable range of the MPR system under the traditional linearly spaced frequencies (LSF) configuration is only a fraction of its theoretical UD because the UD is more sensitive to phase error.

---
Source: https://tomesphere.com/paper/1702.05616