Zero-Overhead Unambiguous Velocity Estimation in Multiband ISAC Systems Under Random Traffic

1. Problem Statement

Integrated Sensing and Communication (ISAC) systems aim to reuse communication signals for environmental sensing (e.g., target velocity estimation via Doppler shift). However, traditional Doppler estimation faces two critical challenges in ISAC environments:

• Irregular Sampling: Unlike dedicated radar systems that use fixed Pulse Repetition Frequencies (PRF), ISAC systems rely on opportunistic communication traffic. Packet arrival times are random and non-deterministic, violating the uniform sampling requirements of standard Discrete Fourier Transform (DFT) or subspace methods (like MUSIC).
• Doppler Ambiguity: Irregular and sparse sampling intervals lead to a low effective sampling rate, causing aliasing. This results in Doppler ambiguity, where the true velocity of a target cannot be distinguished from aliased velocities, significantly degrading estimation accuracy.
• Overhead Trade-off: Traditional solutions to resolve ambiguity involve either increasing the sampling rate (requiring dense traffic) or injecting dedicated sensing packets. Both approaches increase system overhead and reduce communication efficiency, which contradicts the "zero-overhead" goal of ISAC.

2. Methodology

The authors propose a novel framework that leverages frequency diversity across multiple carrier bands to resolve velocity ambiguity without imposing constraints on packet timing or requiring dedicated sensing resources.

Core Concept

Instead of relying on temporal regularity, the method exploits the fact that Doppler shifts are proportional to the carrier frequency ($f_q$). By observing the same target across multiple frequency bands ($Q$ bands) with random packet arrival times, the system can reconstruct the true velocity even if individual band measurements are ambiguous.

Technical Formulation

1. System Model:

   • The system operates in $Q$ distinct frequency bands with carrier frequencies $f_q$.
   • It utilizes Channel Impulse Response (CIR) estimates derived from the preamble of standard communication packets.
   • For a target with radial velocity $v$, the Doppler frequency in band $q$ is $f^D_{k,q} = 2v f_q / c$.

2. Phase Difference Extraction:

   • The algorithm computes instantaneous phase differences ($\phi_{n,q}$) between all possible pairs of received packets within each band.
   • The total phase evolution ($\theta_{n,q}$) over the Time Difference of Arrival (TDoA, $\Delta T_{n,q}$) is unbounded, but the receiver only observes the wrapped phase: $\phi_{n,q} = \theta_{n,q} \mod 2\pi$.
   • This creates an integer ambiguity problem: $\theta_{n,q} = \phi_{n,q} + 2\pi R_{n,q}$, where $R_{n,q}$ is an unknown integer representing the number of $2\pi$ rotations.

3. Mixed-Integer Quadratic Programming (MIQP):

   • The problem is formulated as a system of linear equations where the unknowns are the target velocity $v$ and the integer ambiguity vectors $R$.
   • Key Constraint: The method requires at least one unambiguous phase measurement (typically from the lowest frequency band with the shortest TDoA) to anchor the solution.
   • The system solves the following optimization problem:
     $$\arg \min_{v, r} \| y - vA\mathbf{1} + 2\pi r \|_2^2$$
     Where $y$ is the vector of observed phase differences, $A$ contains frequency and time coefficients, and $r$ is the vector of integer ambiguities.
   • This is solved using efficient integer least-squares algorithms (e.g., LAMBDA or branch-and-bound).

3. Key Contributions

1. Zero-Overhead Solution: The method achieves unambiguous velocity estimation using only existing communication traffic. It requires no dedicated sensing packets and imposes no restrictions on the timing or regularity of packet arrivals.
2. Multiband MIQP Formulation: The paper formulates the velocity estimation problem as a Mixed-Integer Quadratic Program (MIQP) that jointly utilizes phase differences from all possible pairwise packet combinations across multiple frequency bands.
3. Robustness to Random Traffic: By decoupling velocity estimation from strict temporal assumptions, the framework remains effective under sporadic and irregular traffic patterns typical of real-world networks.
4. Frequency Diversity Exploitation: The approach demonstrates that combining low-frequency bands (high unambiguous range) with high-frequency bands (high resolution) allows for accurate reconstruction of velocity even when individual bands suffer from severe aliasing.

4. Numerical Results

The authors evaluated the algorithm using realistic Wi-Fi traffic traces and simulated two-band systems (e.g., 2.4 GHz and 5–60 GHz).

• Accuracy: The proposed method achieves a relative estimation error between $10^{-3}$ and $10^{-4}$ in typical conditions.
• Impact of Frequency Separation: Estimation accuracy improves significantly as the frequency separation between bands increases. Higher frequency diversity provides more distinctive phase evolution, aiding in ambiguity resolution.
• Impact of Time Intervals: Performance improves with larger maximum inter-packet time intervals ($T_{max}$), consistent with the Doppler Cramér-Rao Bound (CRB), as longer observation times provide better resolution.
• Packet Density: Increasing the number of packets ($N$) quadratically increases the number of phase difference measurements, further reducing noise sensitivity and error.
• Comparison with Benchmarks:
  • vs. Single-Band: The single-band approach fails at high frequencies (e.g., 60 GHz) due to high ambiguity and lack of unambiguous pairs.
  • vs. Iterative Maximum Likelihood (IML): The IML benchmark often locks onto incorrect integer ambiguities when frequency separation is large, whereas the proposed MIQP approach successfully resolves these ambiguities.
• Multi-Component Targets: The algorithm remains robust even when targets have multiple scattering components (e.g., a human with moving limbs), provided one component is dominant.

5. Significance

This work addresses a fundamental bottleneck in the deployment of ISAC systems: the conflict between the need for high-precision sensing and the opportunistic nature of communication traffic.

• Practicality: It enables ISAC systems to function as robust sensing tools in dynamic, real-world networks without sacrificing communication throughput or requiring hardware modifications.
• Scalability: The approach is applicable across diverse technologies, from sub-6 GHz to millimeter-wave (mmWave) bands, making it suitable for future 6G networks.
• Paradigm Shift: It moves away from the traditional "dedicated sensing" paradigm toward a "sensing-by-communication" paradigm, proving that irregular, opportunistic data can yield precise, unambiguous physical measurements through advanced signal processing and optimization.