Uncovering the Iceberg in the Sea: Fundamentals of Pulse Shaping and Modulation Design for Random ISAC Signals

Integrated Sensing and Communications (ISAC) is expected to play a pivotal role in future 6G networks. To maximize time-frequency resource utilization, 6G ISAC systems must exploit data payload signals, that are inherently random, for both communication and sensing tasks. This paper provides a comprehensive analysis of the sensing performance of such communication-centric ISAC signals, with a focus on modulation and pulse shaping design to reshape the statistical properties of their auto-correlation functions (ACFs), thereby improving the target ranging performance. We derive a closed-form expression for the expectation of the squared ACF of random ISAC signals, considering arbitrary modulation bases and constellation mappings within the Nyquist pulse shaping framework. The structure is metaphorically described as an ``iceberg hidden in the sea", where the ``iceberg'' represents the squared mean of the ACF of random ISAC signals, that is determined by the pulse shaping filter, and the ``sea level'' characterizes the corresponding variance, caused by the randomness of the data payload. Our analysis shows that, for QAM/PSK constellations with Nyquist pulse shaping, Orthogonal Frequency Division Multiplexing (OFDM) achieves the lowest ranging sidelobe level across all lags. Building on these insights, we propose a novel Nyquist pulse shaping design to enhance the sensing performance of random ISAC signals. Numerical results validate our theoretical findings, showing that the proposed pulse shaping significantly reduces ranging sidelobes compared to conventional root-raised cosine (RRC) pulse shaping, thereby improving the ranging performance.

Pulse Shaping for Random ISAC Signals: The Ambiguity Function Between Symbols Matters

Integrated sensing and communications (ISAC) has emerged as a pivotal enabling technology for next-generation wireless networks. Despite the distinct signal design requirements of sensing and communication (S&C) systems, shifting the symbol-wise pulse shaping (SWiPS) framework from communication-only systems to ISAC poses significant challenges in signal design and processing This paper addresses these challenges by examining the ambiguity function (AF) of the SWiPS ISAC signal and introducing a novel pulse shaping design for single-carrier ISAC transmission. We formulate optimization problems to minimize the average integrated sidelobe level (ISL) of the AF, as well as the weighted ISL (WISL) while satisfying inter-symbol interference (ISI), out-of-band emission (OOBE), and power constraints. Our contributions include establishing the relationship between the AFs of both the random data symbols and signaling pulses, analyzing the statistical characteristics of the AF, and developing algorithmic frameworks for pulse shaping optimization using successive convex approximation (SCA) and alternating direction method of multipliers (ADMM) approaches. Numerical results are provided to validate our theoretical analysis, which demonstrate significant performance improvements in the proposed SWiPS design compared to the root-raised cosine (RRC) pulse shaping for conventional communication systems.

OFDM Achieves the Lowest Ranging Sidelobe Under Random ISAC Signaling

This paper aims to answer a fundamental question in the area of Integrated Sensing and Communications (ISAC): What is the optimal communication-centric ISAC waveform for ranging? Towards that end, we first established a generic framework to analyze the sensing performance of communication-centric ISAC waveforms built upon orthonormal signaling bases and random data symbols. Then, we evaluated their ranging performance by adopting both the periodic and aperiodic auto-correlation functions (P-ACF and A-ACF), and defined the expectation of the integrated sidelobe level (EISL) as a sensing performance metric. On top of that, we proved that among all communication waveforms with cyclic prefix (CP), the orthogonal frequency division multiplexing (OFDM) modulation is the only globally optimal waveform that achieves the lowest ranging sidelobe for quadrature amplitude modulation (QAM) and phase shift keying (PSK) constellations, in terms of both the EISL and the sidelobe level at each individual lag of the P-ACF. As a step forward, we proved that among all communication waveforms without CP, OFDM is a locally optimal waveform for QAM/PSK in the sense that it achieves a local minimum of the EISL of the A-ACF. Finally, we demonstrated by numerical results that under QAM/PSK constellations, there is no other orthogonal communication-centric waveform that achieves a lower ranging sidelobe level than that of the OFDM, in terms of both P-ACF and A-ACF cases.

Improving the Ranging Performance of Random ISAC Signals Through Pulse Shaping Design

In this paper, we propose a novel pulse shaping design for single-carrier integrated sensing and communication (ISAC) transmission. Due to the communication information embedded in the ISAC signal, the resulting auto-correlation function (ACF) is determined by both the information-conveying random symbol sequence and the signaling pulse, where the former leads to random fluctuations in the sidelobes of the ACF, impairing the range estimation performance. To overcome this challenge, we first analyze the statistical characteristics of the random ACF under the symbol-wise pulse shaping (SWPS) regime. As a step further, we formulate an optimization problem to design ISAC pulse shaping filters, which minimizes the average integrated sidelobe level ratio (ISLR) while meeting the Nyquist criterion, subject to power and bandwidth constraints. We then show that the problem can be recast as a convex quadratic program by expressing it in the frequency domain, which can be readily solved through standard tools. Numerical results demonstrate that the proposed pulse shaping design achieves substantial ranging sidelobe reduction compared to the celebrated root-raised cosine (RRC) pulse shaping, given that the communication throughput is unchanged.

Fundamental Limits of Communication-Assisted Sensing in ISAC Systems

Integrated sensing and communication (ISAC) system stands out as a pivotal usage scenario of 6G. To explore the coordination gains offered by the ISAC technique, this paper introduces a novel communication-assisted sensing (CAS) system. The CAS system can endow users with beyond-line-of-sight sensing capability, wherein the base station with favorable visibility senses device-free targets, simultaneously transmitting the acquired sensory information to users. Within the CAS framework, we characterize the fundamental limits to reveal the achievable distortion between the state of the targets of interest and their reconstruction at the users' end. Finally, within the confines of this theoretical framework, we employ a typical application as an illustrative example to demonstrate the minimization of distortion through dual-functional waveform design, showcasing the potential of CAS in enhancing sensing capabilities.

Performance Bounds for Passive Sensing in Asynchronous ISAC Systems -- Appendices

This document contains the appendices for our paper titled ``Performance Bounds for Passive Sensing in Asynchronous ISAC Systems." The appendices include rigorous derivations of key formulas, detailed proofs of the theorems and propositions introduced in the paper, and details of the algorithm tested in the numerical simulation for validation. These appendices aim to support and elaborate on the findings and methodologies presented in the main text. All external references to equations, theorems, and so forth, are directed towards the corresponding elements within the main paper.

Secure ISAC MIMO Systems: Exploiting Interference With Bayesian Cramér-Rao Bound Optimization

In this paper, we present a signaling design for secure integrated sensing and communication (ISAC) systems comprising a dual-functional multi-input multi-output (MIMO) base station (BS) that simultaneously communicates with multiple users while detecting targets present in their vicinity, which are regarded as potential eavesdroppers. In particular, assuming that the distribution of each parameter to be estimated is known \textit{a priori}, we focus on optimizing the targets' sensing performance. To this end, we derive and minimize the Bayesian Cram\'er-Rao bound (BCRB), while ensuring certain communication quality of service (QoS) by exploiting constructive interference (CI). The latter scheme enforces that the received signals at the eavesdropping targets fall into the destructive region of the signal constellation, to deteriorate their decoding probability, thus enhancing the ISAC's system physical-layer security (PLS) capability. To tackle the nonconvexity of the formulated problem, a tailored successive convex approximation method is proposed for its efficient solution. Our extensive numerical results verify the effectiveness of the proposed secure ISAC design showing that the proposed algorithm outperforms block-level precoding techniques.

Reshaping the ISAC Tradeoff Under OFDM Signaling: A Probabilistic Constellation Shaping Approach

Integrated sensing and communications is regarded as a key enabling technology in the sixth generation networks, where a unified waveform, such as orthogonal frequency division multiplexing (OFDM) signal, is adopted to facilitate both sensing and communications (S&C). However, the random communication data embedded in the OFDM signal results in severe variability in the sidelobes of its ambiguity function (AF), which leads to missed detection of weak targets and false detection of ghost targets, thereby impairing the sensing performance. Therefore, balancing between preserving communication capability (i.e., the randomness) while improving sensing performance remains a challenging task. To cope with this issue, we characterize the random AF of OFDM communication signals, and demonstrate that the AF variance is determined by the fourth-moment of the constellation amplitudes. Subsequently, we propose an optimal probabilistic constellation shaping (PCS) approach by maximizing the achievable information rate (AIR) under the fourth-moment, power and probability constraints, where the optimal input distribution may be numerically specified through a modified Blahut-Arimoto algorithm. To reduce the computational overheads, we further propose a heuristic PCS approach by actively controlling the value of the fourth-moment, without involving the communication metric in the optimization model, despite that the AIR is passively scaled with the variation of the input distribution. Numerical results show that both approaches strike a scalable performance tradeoff between S&C, where the superiority of the PCS-enabled constellations over conventional uniform constellations is also verified. Notably, the heuristic approach achieves very close performance to the optimal counterpart, at a much lower computational complexity.

Communication-Assisted Sensing in 6G Networks

The exploration of coordination gain achieved through the synergy of sensing and communication (S&C) functions plays a vital role in improving the performance of integrated sensing and communication systems. This paper focuses on the optimal waveform design for communication-assisted sensing (CAS) systems within the context of 6G perceptive networks. In the CAS process, the base station actively senses the targets through device-free wireless sensing and simultaneously transmits the pertinent information to end-users. In our research, we establish a CAS framework grounded in the principles of rate-distortion theory and the source-channel separation theorem (SCT) in lossy data transmission. This framework provides a comprehensive understanding of the interplay between distortion, coding rate, and channel capacity. The purpose of waveform design is to minimize the sensing distortion at the user end while adhering to the SCT and power budget constraints. In the context of target response matrix estimation, we propose two distinct waveform strategies: the separated S&C and dual-functional waveform schemes. In the former strategy, we develop a simple one-dimensional search algorithm, shedding light on a notable power allocation tradeoff between the S&C waveform. In the latter scheme, we conceive a heuristic mutual information optimization algorithm for the general case, alongside a modified gradient projection algorithm tailored for the scenarios with independent sensing sub-channels. Additionally, we identify the presence of both subspace tradeoff and water-filling tradeoff. Finally, we validate the effectiveness of the proposed algorithms through numerical simulations.

Random ISAC Signals Deserve Dedicated Precoding

Radar systems typically employ well-designed deterministic signals for target sensing, while integrated sensing and communications (ISAC) systems have to adopt random signals to convey useful information. This paper analyzes the sensing and ISAC performance relying on random signaling in a multiantenna system. Towards this end, we define a new sensing performance metric, namely, ergodic linear minimum mean square error (ELMMSE), which characterizes the estimation error averaged over random ISAC signals. Then, we investigate a data-dependent precoding (DDP) scheme to minimize the ELMMSE in sensing-only scenarios, which attains the optimized performance at the cost of high implementation overhead. To reduce the cost, we present an alternative data-independent precoding (DIP) scheme by stochastic gradient projection (SGP). Moreover, we shed light on the optimal structures of both sensing-only DDP and DIP precoders. As a further step, we extend the proposed DDP and DIP approaches to ISAC scenarios, which are solved via a tailored penalty-based alternating optimization algorithm. Our numerical results demonstrate that the proposed DDP and DIP methods achieve substantial performance gains over conventional ISAC signaling schemes that treat the signal sample covariance matrix as deterministic, which proves that random ISAC signals deserve dedicated precoding designs.