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ISAC Super-Resolution Receiver via Lifted Atomic Norm Minimization

Iman Valiulahi, Christos Masouros, Athina P. Petropulu

TL;DR

An off-the-grid estimator for ISAC systems based on the lifted atomic norm minimization (LANM) is introduced and it is proved that LANM can simultaneously achieve localization of radar targets and decoding of communication symbols, when the number of observations is proportional to the degrees of freedom in the ISAC systems.

Abstract

This paper introduces an off-the-grid estimator for integrated sensing and communication (ISAC) systems, utilizing lifted atomic norm minimization (LANM). The key challenge in this scenario is that neither the transmit signals nor the radar-and-communication channels are known. We prove that LANM can simultaneously achieve localization of radar targets and decoding of communication symbols, when the number of observations is proportional to the degrees of freedom in the ISAC systems. Despite the inherent ill-posed nature of the problem, we employ the lifting technique to initially encode the transmit signals. Then, we leverage the atomic norm to promote the structured low-rankness for the ISAC channel. We utilize a dual technique to transform the LANM into an infinite-dimensional search over the signal domain. Subsequently, we use semidefinite relaxation (SDR) to implement the dual problem. We extend our approach to practical scenarios where received signals are contaminated by additive white Gaussian noise (AWGN) and jamming signals. Furthermore, we derive the computational complexity of the proposed estimator and demonstrate that it is equivalent to the conventional pilot-aided ANM for estimating the channel parameters. Our simulation experiments demonstrate the ability of the proposed LANM approach to estimate both communication data and target parameters with a performance comparable to traditional radar-only super-resolution techniques.

ISAC Super-Resolution Receiver via Lifted Atomic Norm Minimization

TL;DR

An off-the-grid estimator for ISAC systems based on the lifted atomic norm minimization (LANM) is introduced and it is proved that LANM can simultaneously achieve localization of radar targets and decoding of communication symbols, when the number of observations is proportional to the degrees of freedom in the ISAC systems.

Abstract

This paper introduces an off-the-grid estimator for integrated sensing and communication (ISAC) systems, utilizing lifted atomic norm minimization (LANM). The key challenge in this scenario is that neither the transmit signals nor the radar-and-communication channels are known. We prove that LANM can simultaneously achieve localization of radar targets and decoding of communication symbols, when the number of observations is proportional to the degrees of freedom in the ISAC systems. Despite the inherent ill-posed nature of the problem, we employ the lifting technique to initially encode the transmit signals. Then, we leverage the atomic norm to promote the structured low-rankness for the ISAC channel. We utilize a dual technique to transform the LANM into an infinite-dimensional search over the signal domain. Subsequently, we use semidefinite relaxation (SDR) to implement the dual problem. We extend our approach to practical scenarios where received signals are contaminated by additive white Gaussian noise (AWGN) and jamming signals. Furthermore, we derive the computational complexity of the proposed estimator and demonstrate that it is equivalent to the conventional pilot-aided ANM for estimating the channel parameters. Our simulation experiments demonstrate the ability of the proposed LANM approach to estimate both communication data and target parameters with a performance comparable to traditional radar-only super-resolution techniques.

Paper Structure

This paper contains 18 sections, 11 theorems, 84 equations, 7 figures.

Table of Contents

  1. Introduction
  2. System Model and Problem Formulation
  3. Conditions of Solvability of the LANM Problem
  4. Optimality Conditions and SDR
  5. Robustness Against AWGN Noise and Jamming Signals
  6. Complexity Analysis
  7. NUMERICAL Results
  8. Conclusion
  9. Appendices
  10. Proof of Theorem \ref{['maintheorem']}
  11. Construction of the dual polynomial
  12. Certifying (18b)
  13. Proof of Proposition \ref{['firstpro']}
  14. Proof of Lemma \ref{['lem2']}
  15. Proof of lemma \ref{['lem4']}
  16. ...and 3 more sections

Key Result

Theorem 1

Consider the linear system in (1) and its sampled version in (2) and assume that the unknown waveforms vectors can be written as $\bm{x}_{k} = \bm{D}\bm{h}_{k}$ where $\bm{D}$ satisfies Assumption assum1 while $\bm{h}_{k}$ follows Assumption assumption2. Further, let the unknown shifts satisfy the m one can recover $\bm{U}$ using problem (problemmain). The proof is given in the Appendix proofthepo

Figures (7)

  • Figure 1: System model.
  • Figure 2: The magnitude of the dual polynomial in $(\phi, \rho)$ and $(\tau, v)$ domains in Figs. \ref{['fig.bound45']} and \ref{['fig.bound5']}, respectively. Note that red circles represent the recovered radar parameters, respectively.
  • Figure 3: The NMSE and SER of the proposed estimator compared to the conventional approaches in Figs. \ref{['fig.bound451']} and \ref{['fig.bound52']}, respectively.
  • Figure 4: The average success rate of LANM with respect to the number of targets and the subspace dimension $T$ when the number of measurements varies. Figs. \ref{['fig.bound412']}, \ref{['fig.bound412']}, and \ref{['fig.bound124']} represent the results for $K=1$, $K=2$, $K=1$ with one jammer, respectively.
  • Figure 5: SER versus NMSE for different numbers of observations.
  • ...and 2 more figures

Theorems & Definitions (11)

  • Theorem 1
  • Proposition 1
  • Proposition 2
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • Lemma 5
  • Lemma 6
  • Lemma 7
  • Lemma 8
  • ...and 1 more