[Paper Review] The Unambiguous Distance in a Phase-based Ranging System with Hopping Frequencies

Yue Zhang, Wangdong Qi+1 more|arXiv (Cornell University)|Mar 8, 2014

Advanced Wireless Communication Techniques3 references3 citations

TL;DR

This paper proposes characterizing the unambiguous distance (UD) in a phase-based ranging system with hopping frequencies (PRSHF) by the probability that it achieves its maximum value, which depends only on the number of frequencies used. Using analytic number theory, the authors derive that this probability approaches 1/ζ(M) as bandwidth grows, with over 99.9% probability of maximum UD when M ≥ 10, regardless of bandwidth distribution or segment count.

ABSTRACT

It is a challenge to specify unambiguous distance (UD) in a phase-based ranging system with hopping frequencies (PRSHF). In this letter, we propose to characterize the UD in a PRSHF by the probability that it takes on its maximum value. We obtain a very simple and elegant expression of the probability with growth estimation techniques from analytic number theory. It is revealed that the UD in a PRSHF usually takes on the maximum value with as few as 10 frequencies in measurement, almost independent of the specific distribution of available bandwidth.

Motivation & Objective

• To address the challenge of specifying unambiguous distance (UD) in phase-based ranging systems with randomly selected hopping frequencies (PRSHF), where traditional fixed-frequency methods fail due to frequency randomness and bandwidth discontinuity.
• To characterize UD not by its full distribution but by the probability it attains its maximum value, which corresponds to the integers representing carrier frequencies being relatively prime.
• To develop a scalable, bandwidth-agnostic metric for UD in PRSHF that enables performance evaluation independent of specific spectral allocation or segment structure.
• To establish a theoretical foundation using analytic number theory to estimate the probability of relatively prime integers selected from segmented, discontinuous frequency sets.

Proposed method

• Model the carrier frequencies as integer multiples of a minimum frequency interval f_min, mapping them to a set of positive integers N with L disjoint segments.
• Define the UD as c/(k f_min), where k is the GCD of the frequency indices; maximum UD occurs when k = 1, i.e., when the indices are relatively prime.
• Use the Möbius function μ(j) and inclusion-exclusion principle to express the count of M-tuples of integers from N that are relatively prime as Z = Σ μ(j) x_j^M, where x_j is the number of integers in N divisible by j.
• Apply growth estimates from analytic number theory to approximate x_j ≈ N/j, leading to Z ≈ Σ μ(j)(N/j)^M, with error bounds O(N^{M-1}/j^{M-1}) and O(N^2).
• Derive the probability P = Z / N^M ≈ 1/ζ(M) + O(N^{-1}), showing that the probability depends only on M when N is large.
• Validate the approximation via simulations across different numbers of frequency segments (L = 1, 7, 12) and M from 3 to 13, confirming close agreement with theoretical predictions.

Experimental results

Research questions

• RQ1 What is the probability that the unambiguous distance in a PRSHF achieves its maximum value, given M randomly selected hopping frequencies?
• RQ2 How does this probability depend on the number of frequencies M, the total number of available frequencies N, and the number of bandwidth segments L?
• RQ3 Can the probability of achieving maximum UD be approximated independently of the specific distribution of available bandwidth or the starting frequency point?
• RQ4 To what extent does the Riemann zeta function ζ(M) govern the asymptotic behavior of the UD probability in PRSHF?

Key findings

• The probability that the unambiguous distance in a PRSHF achieves its maximum value is approximately 1/ζ(M), where ζ(M) is the Riemann zeta function, and this approximation becomes highly accurate as the number of available frequencies N increases.
• When M ≥ 10, the probability exceeds 0.999, meaning the UD is almost certainly at its maximum value, regardless of the number of frequency segments L or the specific distribution of available bandwidth.
• The probability depends only on M and not on L or the spectral distribution, making it a robust, scalable metric for PRSHF design and performance evaluation.
• The theoretical approximation P ≈ 1/ζ(M) is validated by simulations across multiple configurations (L = 1, 7, 12), showing excellent agreement with empirical results.
• The error term O(N^{-1}) in the probability estimate becomes negligible for large N, confirming the practical irrelevance of bandwidth distribution and segment count in high-bandwidth systems.