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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
The paper addresses how random communication payloads affect sensing performance in communication-centric ISAC, especially through modulation and pulse shaping. It derives the expected squared ACF under general Nyquist-pulse-shaped signaling, finds OFDM optimal for QAM/PSK ranging sidelobes, and proposes iceberg shaping to improve ranging.
Problem
Random communication payloads make ISAC ACFs statistical, motivating analysis of how modulation and pulse shaping affect ranging sidelobes and target detection.
Method
The paper derives a closed-form expectation for the squared ACF of random ISAC signals with arbitrary modulation bases and constellation mappings under Nyquist pulse shaping.
Results
For QAM/PSK constellations, OFDM achieves the lowest ranging sidelobe level at every lag, while iceberg shaping improves performance by 80% over RRC at SNR = 35 dB.
Takeaways & Limitations
After coherent integration suppresses the variance component, pulse design becomes central, motivating Nyquist-pulse optimization for lower ranging sidelobes.
Abstract
from arXiv · showhide
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.
I. INTRODUCTION
The paper studies communication-centric ISAC, using random data payloads for sensing while preserving communication-oriented signal structures. It analyzes how modulation, constellation, and pulse shaping determine random-signal ACF statistics and ranging sidelobes, then proposes iceberg shaping.
- Communication-centric ISAC reuses transmitted data payloads for sensing, avoiding complex waveform redesign while improving resource efficiency and compatibility with existing standards.
- Communication signals combine constellation symbols, an orthonormal modulation basis, and a pulse-shaping filter, each of which can affect sensing performance.
- Prior work showed that OFDM is globally optimal for QAM and PSK under cyclic prefixes, but its framework did not fully account for pulse-shaping effects.
- The paper derives the expected squared ACF for arbitrary modulation bases and constellation mappings under Nyquist pulse shaping, separating pulse-determined squared mean from data-induced variance.
- Coherent integration by M reduces the variance component by a factor of M, making the pulse-dependent ACF geometry increasingly important for ranging.
- For QAM/PSK constellations, OFDM achieves the lowest ranging sidelobe level at every lag across Nyquist pulse-shaping filters.
- The proposed iceberg shaping technique designs Nyquist pulses to minimize ACF sidelobes within a specified delay region after coherent integration suppresses the variance component.
B. Modulation Basis
The modulation-basis model represents communication symbols over an orthonormal basis, then applies Nyquist pulse shaping to produce a band-limited continuous-time signal. The formulation also supports sampled and circular-convolution implementations for practical ranging analysis.
- The symbol vector is modulated over an orthonormal basis represented by a unitary matrix U, producing discrete time-domain samples x.
- This generic unitary model encompasses SC, OFDM, CDMA, OTFS, and AFDM signaling schemes, often with a cyclic prefix to address multipath-induced inter-symbol interference.
- Pulse shaping restricts signal bandwidth and eliminates inter-symbol interference among time-domain samples using a band-limited Nyquist prototype pulse.
- With a cyclic prefix, the pulse-shaped signal can be represented as a circular convolution of the upsampled discrete signal and sampled pulse.
- The analysis uses oversampling with integer ratio L and samples the pulse-shaped waveform at rate f_s = 1/T_s.
- The sampled sequence approximates the continuous waveform for ranging analysis, particularly when the oversampling ratio is high.
D. Sensing Signal Processing based on Matched Filtering
The section formulates matched-filter sensing for multi-target ISAC and evaluates ranging through the statistical properties of the random signal's periodic ACF. Under arbitrary orthogonal modulation and Nyquist pulse shaping, the analysis targets high matched-filter peaks at target delays and low sidelobes elsewhere.
- Signal and channel model: The received sampled echo for Q targets is modeled using target reflection coefficients, delays, and periodic time-shift matrices induced by the cyclic prefix.The target delays are assumed sufficiently aligned to the sampling grid for the considered formulation.
- Matched-filter sensing: Matched filtering produces a linear combination of time-shifted ACFs plus noise, so target detection requires high peaks at target delays and small sidelobes elsewhere.The squared matched-filter output is used because the ACF is complex-valued.
- Statistical performance metric: The sensing metric is the expected squared ACF, E(|R_k|^2), because random communication symbols make the ACF a random function.This statistical metric evaluates average mainlobe and sidelobe behavior rather than a single symbol realization.
- General analytical framework: The analysis derives this metric for arbitrary orthogonal waveforms, Nyquist pulses, and i.i.d. symbols from any proper constellation.The resulting iceberg interpretation separates pulse-induced ACF structure from communication-data randomness.
B. Characterization of the Average Squared ACF
The average squared ACF decomposes into an iceberg, given by the squared mean, and a sea level, given by the variance. The iceberg equals the squared ACF of the pulse-shaping filter, while the sea level captures randomness-dependent variation.
- Iceberg decomposition: Theorem 1 decomposes the average squared ACF into |E(R_k)|^2 and var(R_k), interpreted as the iceberg and sea level, respectively.This closed-form structure separates coherent pulse-dependent behavior from random fluctuations.
- Pulse-dependent component: The iceberg is the squared IDFT of a zero-padded sequence and is equivalent to the squared ACF of the pulse-shaping filter.The equivalence follows from the frequency-domain representation of the Nyquist pulse.
- Mainlobe behavior: Under Nyquist pulse shaping, the average mainlobe level depends only on the constellation kurtosis.This corollary isolates constellation statistics as the determinant of the average mainlobe level.
- Illustrative example: For a 16-QAM SC signal with α = 0.35 RRC shaping, the average squared ACF closely matches the pulse ACF within delay region [−2, 2].This region is described as the tip of the iceberg in the illustrative example.
C. Coherent Integration
Coherent integration averages ACFs from multiple independent symbol realizations while assuming stationary targets across transmission slots. It leaves the expectation unchanged and reduces the variance component, exposing more of the pulse-determined iceberg.
- Integration setup: Coherent integration averages matched-filter outputs from M i.i.d. symbol sequences under the assumption that targets remain stationary across transmission slots.The method combines multiple random signal realizations before evaluating sensing performance.
- Sea-level reduction: Coherent integration reduces the sea-level variance component of the average squared ACF by a factor of M.The expectation remains unchanged because the integrated ACF is a sample mean.
- Mainlobe behavior: The coherent-integration analysis separately characterizes the resulting mainlobe level after integration by M times.The mainlobe is treated as a distinct quantity from the reduced sea-level variance.
- Illustrative result: With M = 100 coherent integrations, the illustrative SC signal exhibits a 20 dB reduction in sea level and reveals more of the iceberg within delay region [−4, 4].The example uses 16-QAM, α = 0.35 RRC shaping, N = 128, and L = 10.
IV. ICEBERG THEOREM INSPIRED ISAC TRANSMISSION DESIGN
The analysis separates random-signal ranging sidelobes into modulation-dependent and pulse-dependent components, identifying the optimal modulation basis by constellation kurtosis. For sub-Gaussian constellations such as QAM and PSK, OFDM uniquely minimizes sidelobes at every lag.
- The average squared ACF depends on the modulation basis, constellation, and pulse shaping filter.
- The optimal modulation basis is determined by the sign of the constellation’s excess kurtosis.
- For sub-Gaussian constellations, OFDM is the only modulation basis achieving the lowest ranging sidelobe level at every lag.
- For super-Gaussian constellations, SC modulation achieves the lowest ranging sidelobe level for every lag.
- With Gaussian constellations, the sidelobe becomes independent of the modulation basis because the Gaussian distribution is unitary invariant.
B. Constellation Design
Constellation kurtosis controls both the ACF’s mainlobe and its random-data variance, or “sea level.” Lower-kurtosis constellations improve sensing-related sidelobe behavior but can reduce communication rate, creating a sensing–communication tradeoff.
- The constellation affects the average squared ACF exclusively through its kurtosis.
- 2) Integration Efficiency: PSK has the lowest possible sea level because its kurtosis is µ4 = 1.
- 1) Mainlobe Level: Larger kurtosis increases the mainlobe level, although its contribution becomes negligible when N is sufficiently large.
- 2) Integration Efficiency: For OFDM, the sea level is proportional to (µ4 −1)/M, giving 16-QAM an integration efficiency of 3.125 versus 1 for Gaussian signaling.
- 2) Integration Efficiency: Lower-kurtosis constellations may not support high communication rates; PSK generally has lower rates than same-order QAM.
2) Sea Waves:
The pulse’s ACF forms the “iceberg,” while roll-off-dependent oscillations form “sea waves” in the sidelobe region. Iceberg shaping therefore minimizes sidelobes over a selected delay region through a convex pulse-design problem.
- 2) Sea Waves: Periodic sidelobe ripples arise from a cosine term, with amplitude proportional to Σ_n g_n(1−g_n).
- 2) Sea Waves: Larger roll-off factors generally produce higher sidelobe ripples, whereas a sinc pulse yields a completely flat sea level.
- 3) Iceberg Shaping: Coherent integration reduces only the sea level, so sufficiently integrated sensing performance depends mainly on the pulse-ACF geometry.
- 3) Iceberg Shaping: The design minimizes either integrated or peak sidelobe level within a specified delay region using convex objectives in g_n.
- 3) Iceberg Shaping: The pulse-design constraints enforce a fixed non-roll-off spectrum, monotonic roll-off behavior, and constant pulse energy.
- 3) Iceberg Shaping: The resulting iceberg-shaping problem is a convex quadratic program solvable with standard numerical tools.
V. NUMERICAL RESULTS
Numerical results validate the theoretical modulation and pulse-shaping analysis. OFDM lowers sidelobes among tested modulation schemes, while iceberg shaping improves sidelobe suppression and multi-target range estimation over RRC baselines.
- Modulation comparison: With 16-QAM and RRC shaping, OFDM has the lowest sidelobe at every lag and achieves a 5 dB reduction versus SC and CDMA.
- Pulse shaping: The iceberg-shaped pulse produces a uniform −80 dB sidelobe level over k ∈[5, 15] under the PSL objective.
- Pulse shaping: Reaching −80 dB requires M = 250,000 coherent integrations, reducing the sea level by 54 dB relative to no integration.
- Pulse shaping: The designed pulse and RRC both satisfy the folded spectrum criterion, but the designed pulse uses a stepped roll-off spectrum.
- Ranging performance: The proposed technique improves range estimation by more than 58% versus SC at high SNRs and by 80% versus RRC at SNR = 35 dB.
- Ranging performance: For OFDM with 16-QAM and M = 1000, iceberg shaping achieves a 50% improvement over RRC and enables weak-target peak identification.
APPENDIX A PROOF OF THEOREM 1
The proof expands the squared ACF and uses a constellation-dependent lemma to simplify its computation. It then separates the mean and variance of Rk before concluding.
- The proof starts by expressing the squared ACF and substituting previously derived relations.
- The matrix S is decomposed into structured components, including repeated vectors and all-zero blocks.One component is explicitly written using c and zero matrices of size N^2 × N.
- Lemma 2 represents ˜s as vec(ssH) and provides a structured form for constellations satisfying Assumptions 1 and 2.The lemma uses the constellation kurtosis µ4 and all-zero vectors of length N.
- The resulting derivation identifies the mean and variance of Rk, completing the proof.
APPENDIX B PROOF OF THEOREM 2
The proof minimizes the sidelobe sea level using convex relaxation and majorization. For the relevant optimum, the signaling basis reduces to OFDM up to subcarrier permutations and phase rotations.
- For µ4 < 2, the proof formulates minimization of the sidelobe sea level over unistochastic matrices.
- The unistochastic optimization is relaxed to bistochastic matrices, with tightness established for the resulting optimum.
- Majorization and Schur-convexity show that equality requires the relevant bistochastic matrix to be a permutation matrix.
- The optimal signaling basis consists of a Fourier-based basis with arbitrary unit-modulus phases and subcarrier permutations.
- OFDM is the only modulation basis achieving the lowest ranging sidelobe, with standard OFDM obtained for identity permutation and unit phases.
APPENDIX C PROOF OF THEOREM 3
For µ4 > 2, the proof identifies a unitary constant-modulus optimum. Choosing the IDFT matrix yields the optimal signaling basis for super-Gaussian constellations and produces SC modulation.
- For µ4 > 2, the proof reformulates sidelobe-sea-level minimization over unistochastic matrices.
- A uniform bistochastic matrix is invariant under multiplication by any bistochastic matrix, enabling the optimization reformulation.
- Majorization and Schur-convexity indicate that the optimum should be unitary and have constant modulus.
- The resulting optimal signaling basis for super-Gaussian constellations is obtained by choosing the IDFT matrix and leads to SC modulation.