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

Fan Liu, Yifeng Xiong, Shihang Lu, Shuangyang Li, Weijie Yuan, Christos Masouros, Shi Jin, Giuseppe Caire

TL;DR

This work analyzes the sensing performance of random data-carrying ISAC signals under Nyquist pulse shaping, introducing an Iceberg Theorem that decouples the ACF into a pulse-driven mean (iceberg) and a data-driven variance (sea level). It derives a closed-form expression for the expectation of the squared ACF, shows OFDM is globally optimal for sub-Gaussian constellations, and provides design guidelines across modulation, constellation, and pulse shaping. The authors propose an iceberg-shaped pulse design and demonstrate via numerical results that sidelobes can be substantially reduced compared with standard RRC shaping, enabling improved ranging in multi-target scenarios. The findings offer actionable strategies for optimizing joint sensing and communication performance in 6G ISAC systems, balancing data rate and ranging accuracy through waveform-level design.

Abstract

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.

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

TL;DR

This work analyzes the sensing performance of random data-carrying ISAC signals under Nyquist pulse shaping, introducing an Iceberg Theorem that decouples the ACF into a pulse-driven mean (iceberg) and a data-driven variance (sea level). It derives a closed-form expression for the expectation of the squared ACF, shows OFDM is globally optimal for sub-Gaussian constellations, and provides design guidelines across modulation, constellation, and pulse shaping. The authors propose an iceberg-shaped pulse design and demonstrate via numerical results that sidelobes can be substantially reduced compared with standard RRC shaping, enabling improved ranging in multi-target scenarios. The findings offer actionable strategies for optimizing joint sensing and communication performance in 6G ISAC systems, balancing data rate and ranging accuracy through waveform-level design.

Abstract

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.
Paper Structure (27 sections, 8 theorems, 68 equations, 7 figures, 1 table)

This paper contains 27 sections, 8 theorems, 68 equations, 7 figures, 1 table.

Table of Contents

  1. Introduction
  2. System Model
  3. Constellation Model
  4. Modulation Basis
  5. Pulse Shaping Model
  6. Sensing Signal Processing based on Matched Filtering
  7. The ACF of Pulse-Shaped Random ISAC Signals
  8. Frequency-Domain Representation of the ACF
  9. Characterization of the Average Squared ACF
  10. Coherent Integration
  11. Iceberg Theorem Inspired ISAC Transmission Design
  12. Optimal Modulation Basis
  13. Constellation Design
  14. Mainlobe Level
  15. Integration Efficiency
  16. ...and 12 more sections

Key Result

Lemma 1

A size-$N$ circulant matrix $\mathbf{C}$ can be diagonalized by the DFT matrix in the form of where $\mathbf{c}$ is the first column of $\mathbf{C}$, and $\mathbf{F}_N$ is the normalized DFT matrix of size $N$, with its $(m,n)$-th entry being defined as $\frac{1}{\sqrt{N}}e^{-\frac{j2\pi(m-1)(n-1)}{N}}$.

Figures (7)

  • Figure 1: The average squared ACF and its coherent integration version of an SC signal, with 16-QAM constellation and $\alpha = 0.35$ RRC pulse shaping, $N = 128$, $L = 10$, $M = 100$.
  • Figure 2: The average squared ACF of SC, CDMA, and OFDM signals, with 16-QAM constellation and $\alpha = 0.35$ RRC pulse shaping, $N = 128$, $L = 10$.
  • Figure 3: The average squared ACF of SC and OFDM signals under $M = 100$ coherent integration, with 16-QAM constellation and $\alpha = 0.35$ RRC pulse shaping, $N = 128$, $L = 10$.
  • Figure 4: The average squared ACF of OFDM signals under PSK, 16-QAM, 1024-QAM, and Gaussian constellations under $\alpha = 0.35$ RRC pulse shaping, $N = 128$, $L = 10$.
  • Figure 5: The average squared ACFs with 250,000 coherent integrations under OFDM signaling with 16-QAM constellation, $N = 128$, $\alpha = 0.35$, $L = 10$. (a) Iceberg Shaping approach under PSL objective function, with the delay region of interest being $k \in \left[5,15\right]$. (b) RRC pulse shaping. (c) Direct comparison of coherently integrated ACFs under the proposed iceberg shaping and RRC pulse shaping using OFDM modulation and 16-QAM constellation. (d) The squared spectra of the designed pulse and the RRC.
  • ...and 2 more figures

Theorems & Definitions (16)

  • Definition 1: Sub-Gaussian Constellation
  • Definition 2: Super-Gaussian Constellation
  • Lemma 1
  • Theorem 1: Iceberg Theorem
  • proof
  • Corollary 1
  • proof
  • Corollary 2: Sea Level Reduction via Coherent Integration
  • proof
  • Corollary 3
  • ...and 6 more