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CP-OFDM Achieves the Lowest Average Ranging Sidelobe Under QAM/PSK Constellations

Fan Liu, Ying Zhang, Yifeng Xiong, Shuangyang Li, Weijie Yuan, Feifei Gao, Shi Jin, Giuseppe Caire

arXiv:2407.06691v3cs.ITeess.SPmath.IT

TL;DR

The paper asks which communication-centric ISAC waveform achieves the best ranging performance under random signaling. The paper develops a generic framework to analyze sensing performance of communication-centric ISAC waveforms built on orthonormal signaling bases and random data. With CP, OFDM is uniquely globally optimal for QAM/PSK under P-ACF EISL and each sidelobe; without CP, it is locally EISL-optimal under A-ACF.

  • Problem

    The paper asks which communication-centric ISAC waveform achieves the best ranging performance under random signaling.

    Source: §I · p. 2

  • Method

    The paper develops a generic framework to analyze sensing performance of communication-centric ISAC waveforms built on orthonormal signaling bases and random data.

    Source: §I · p. 2

  • Results

    With CP, OFDM is uniquely globally optimal for QAM/PSK under P-ACF EISL and each sidelobe; without CP, it is locally EISL-optimal under A-ACF.

    Source: §I · p. 2

  • Takeaways & Limitations

    Numerical results find no other orthogonal waveform with lower QAM/PSK ranging sidelobes for either A-ACF or P-ACF.

    Source: §VI · p. 12

    Maximum sidelobe level analysis is left for future work because its nonsmoothness makes analytical treatment difficult.

    Source: 2) Periodic ACF (P-ACF) · p. 5

Abstract

from arXiv · show

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.

I. INTRODUCTION

The introduction asks which communication-centric waveform best supports ranging when communication symbols are random. The paper develops an analysis framework and establishes OFDM’s optimality for QAM/PSK under specified CP and no-CP conditions.

I. INTRODUCTION

  • Communication-centric ISAC reuses communication signaling for sensing, but random data symbols may degrade sensing performance.The introduction frames minimizing this loss as the waveform-design question.Source: §I · pp. 1–2
  • The paper derives closed-form P-ACF and A-ACF sidelobe expressions for random signals using sub-Gaussian and super-Gaussian constellations.It also defines EISL as a ranging-performance metric.Source: §I · p. 2
  • With a CP, OFDM is the only globally optimal waveform for QAM/PSK under both EISL and each individual P-ACF sidelobe.The paper also proves CP-SC achieves the lowest Doppler sidelobe level.Source: §I · p. 2
  • Without a CP, OFDM is locally EISL-optimal for the A-ACF under QAM/PSK; global optimality for sub-Gaussian constellations is conjectured.The conjecture is not presented as a proven result.Source: §I · p. 2

A. ISAC Signal Model

  • The signal model modulates random symbols over an orthonormal basis; CP-based periodic processing and non-CP linear processing are both analyzed.The paper notes that long-range targets beyond CP coverage may call for a sufficiently small duty cycle and a zero-padding waveform.Source: A. ISAC Signal Model · pp. 3–4

B. Communication System Model

The paper relates communication-channel performance to the choice of signaling basis, then evaluates ranging through periodic and aperiodic autocorrelation. For QAM/PSK, OFDM minimizes EISL in both cases and uniquely minimizes each P-ACF sidelobe.

B. Communication System Model

  • OFDM achieves optimal mutual information in LTI multipath channels because the channel matrix is diagonalized by its signaling basis.In AWGN, any orthonormal basis preserves communication rate.Source: B. Communication System Model · p. 4

C. Sensing Performance Metric

  • The ACF is the ranging indicator, with aperiodic or periodic self-convolution corresponding to signals without or with a CP, respectively.The paper treats these as linear and periodic matched-filter processing cases.Source: C. Sensing Performance Metric · p. 4
  • EISL averages the sidelobe energy across nonzero lags for the random signal, providing an objective for selecting the signaling basis.The paper leaves maximum sidelobe level for future work because its nonsmoothness complicates analysis.Source: 2) Periodic ACF (P-ACF) · p. 5

1) Aperiodic ACF (A-ACF):

  • For QAM/PSK, OFDM minimizes EISL for the A-ACF, the paper’s no-CP ranging case.The A-ACF result is stated as minimization, not global uniqueness.Source: 2) Periodic ACF (P-ACF) · p. 5

2) Periodic ACF (P-ACF):

  • For QAM/PSK, OFDM is the unique global minimizer of P-ACF EISL and of every individual nonzero-lag sidelobe.This covers both the aggregate sidelobe objective and each lag separately.Source: 2) Periodic ACF (P-ACF) · p. 5

III. THE P-ACF CASE

The P-ACF analysis derives average sidelobe measures and establishes OFDM’s optimality for sub-Gaussian constellations, with additional results for other constellation regimes and CP-SC.

III. THE P-ACF CASE

  • The P-ACF analysis uses the Wiener–Khinchin theorem to obtain a frequency-domain representation, and treats this case as simpler than the A-ACF counterpart.Source: §III · p. 5

A. Main Results

  • The P-ACF results include closed-form expressions for average squared correlation and EISL, with the EISL expressed using a matrix ℓ4 norm.Source: A. Main Results · pp. 5–6
  • The average squared P-ACF sidelobe can be reduced by increasing N or decreasing kurtosis, with constellation shaping offered as a way to lower kurtosis.Source: A. Main Results · p. 5

B. Optimal Signaling Schemes

  • For sub-Gaussian constellations, OFDM is the only waveform achieving the lowest P-ACF EISL and the lowest average sidelobe at every delay index.The individual-lag result follows from the same OFDM basis that achieves the EISL minimum.Source: B. Optimal Signaling Schemes · p. 6
  • For PSK, OFDM’s P-ACF is a unit impulse with zero individual and integrated sidelobes, while PSK has nonzero A-ACF sidelobes.The P-ACF result follows from Fourier duality and holds for any PSK realization; the A-ACF does not share this property.Source: B. Optimal Signaling Schemes · pp. 6–7
  • For BPSK, OFDM need not be the unique zero-sidelobe basis; the paper constructs alternative unitary bases satisfying the constant-amplitude-spectrum condition.The authors leave further possible constructions for future investigation.Source: B. Optimal Signaling Schemes · p. 7
  • CP-SC is the only waveform achieving the lowest P-ACF Doppler sidelobe for sub-Gaussian constellations, by Fourier duality with OFDM’s delay result.Source: B. Optimal Signaling Schemes · p. 7
  • For super-Gaussian constellations, CP-SC achieves the lowest P-ACF EISL, while for Gaussian constellations the EISL and individual sidelobe levels are invariant to the basis.Source: B. Optimal Signaling Schemes · p. 7

IV. THE A-ACF CASE

This section investigates random ISAC signals’ ranging performance through the sidelobes of the A-ACF.

IV. THE A-ACF CASE

  • The section evaluates ranging performance for random ISAC signals by investigating A-ACF sidelobe levels.Source: §IV · p. 7

A. Main Results

The A-ACF results characterize average sidelobes and EISL, then establish OFDM's local optimality for sub-Gaussian constellations while identifying a global result for super-Gaussian constellations. Global optimality for the sub-Gaussian, CP-free case remains a conjecture.

A. Main Results

  • The average squared A-ACF is characterized by viewing it as the P-ACF of zero-padded sequences.This substitution yields the stated expression after algebraic manipulation.Source: A. Main Results · p. 7
  • The A-ACF average mainlobe matches the P-ACF average mainlobe, and its EISL has a characterization using a trimmed 2N-point DFT matrix.The matrix contains the first N columns of the size-2N DFT matrix.Source: A. Main Results · pp. 7–8

B. Optimal Signaling Schemes

  • For sub-Gaussian constellations, OFDM is a local minimum of the A-ACF EISL.The proof establishes a local maximum of the associated ℓ4 objective at the identity, using zero gradient and geodesic concavity nearby.Source: B. Optimal Signaling Schemes · p. 8
  • The authors conjecture that OFDM globally minimizes A-ACF EISL among CP-free signals under sub-Gaussian constellations, but global optimality is not theoretically guaranteed.Numerical results suggest no other CP-free waveform achieves a lower sidelobe level.Source: B. Optimal Signaling Schemes · p. 8
  • For super-Gaussian constellations, the SC waveform achieves the lowest A-ACF EISL globally.The proof minimizes the relevant ℓ4 norm over unitary matrices; the identity matrix attains the minimum.Source: B. Optimal Signaling Schemes · p. 8

V. NUMERICAL RESULTS

Numerical comparisons examine SC, OFDM, and Walsh-CDMA across QAM, PSK, and a designed SG-64-APSK constellation, averaging results over 1,000 random realizations. The results compare sidelobe levels, VISML, and practical two-target ranging performance.

A. Average Sidelobe Level Analysis

B. Variance of the Integrated Sidelobe and Mainlobe Level

C. Ranging Performance Analysis

VI. CONCLUSIONS

  • The conclusion identifies practical power allocation, pulse shaping, and nonlinear modulation as directions for further waveform-design work.The paper concludes that OFDM is globally optimal with CP and locally optimal without CP for the studied ranging sidelobes.Source: §VI · p. 12

APPENDIX A PROOF OF PROPOSITION 1

The appendix derives expressions used in the paper's propositions and corollary through moment calculations, Parseval's theorem, and A-ACF identities.

APPENDIX A PROOF OF PROPOSITION 1

  • The proof enumerates fourth-order symbol-moment cases, decomposes S, and substitutes the resulting expressions to obtain the stated compact form.The cases distinguish coincident indices, paired indices, and other index combinations.Source: Appendix A · p. 12

APPENDIX B PROOF OF PROPOSITION 2

  • The proof identifies an entry of b_k as the DFT of a row of V and applies Parseval's theorem.Source: Appendix B · p. 13

APPENDIX C PROOF OF COROLLARY 4

  • The corollary proof expresses the A-ACF through component identities, uses a commutation matrix to reuse Proposition 1, and derives the average squared A-ACF.It then substitutes an ISL expression to obtain the stated result.Source: Appendix C · p. 13

APPENDIX D PROOF OF COROLLARY 5

The proof relates an A-ACF integrated sidelobe level to an equivalent expression for a deterministic sequence, then uses that relation to obtain equation (41).

APPENDIX D PROOF OF COROLLARY 5

  • The proof begins from the result in (40) and identifies the integrated sidelobe level of the A-ACF of u_n.Source: Appendix D · p. 13
  • For a deterministic sequence u_n, the proof rewrites its ISL in an equivalent form.Source: Appendix D · p. 13
  • Substituting that expression into the earlier result yields equation (41), completing the proof.Source: Appendix D · p. 14

APPENDIX E PROOF OF THEOREM 3

The proof reformulates its objective and shows that V = I is a local maximum by checking the gradient and geodesic curvature at that point.

APPENDIX E PROOF OF THEOREM 3

  • The proof first recasts the objective function.Source: Appendix E · p. 14
  • Because the objective is defined over the unitary group, the proof reduces local maximality at V = I to a zero gradient and geodesic concavity there.Source: Appendix E · p. 14
  • The required conditions are stated for every geodesic through I, with its generator in the tangent space at I.Source: Appendix E · p. 14

1) Computing the First- and Second-Order Derivatives:

The proof sets up first- and second-order derivatives by exploiting the structure of the DFT matrices and a geodesic expansion around the identity. It also uses the invariance of the ℓ4 norm under row permutations.

1) Computing the First- and Second-Order Derivatives:

  • The ℓ4 norm is preserved under permutations, allowing the rows of ˜F2N to be rearranged.The proof separates odd and even rows using a permutation matrix.Source: 1) Computing the First- and… · p. 14
  • The DFT structure yields a circular matrix Cα from the diagonal matrix Dα.Dα has entries defined using α, and Cα is formed as 1/2 F_N Dα F_N^H.Source: 1) Computing the First- and… · p. 14
  • A geodesic through the identity has a skew-Hermitian tangent Γ = jH, with H Hermitian, and is expanded as a power series around I.This expansion is used to analyze the derivatives.Source: 1) Computing the First- and… · p. 14

2) First-Order Derivative is Zero:

The proof establishes the first-order derivative result through matrix rearrangements and a lemma, using properties of Hermitian matrices and treating even N without loss of generality. It also expresses terms using circular convolution.

2) First-Order Derivative is Zero:

  • The proof uses the fact that diagonal entries of a Hermitian matrix are real when handling terms in the first-order derivative argument.The text applies similar arguments to another term in the derivation.Source: 2) First-Order Derivative is Zero · pp. 14–15
  • The derivation rewrites terms using a permutation matrix and two-dimensional circular convolution, which is defined entrywise.It also defines one-dimensional circular convolution for length-N vectors.Source: 2) First-Order Derivative is Zero · p. 15
  • The proof assumes N is even without loss of generality and states that a similar argument holds when N is odd.It then proceeds to consider the remaining part of the derivative expression.Source: 2) First-Order Derivative is Zero · pp. 14–15

3) Second-Order Derivative is Non-Positive:

The proof represents derivative terms with circular convolution and establishes non-positivity of the second-order derivative for any Hermitian H. This makes V = I a local maximum of f(V).

3) Second-Order Derivative is Non-Positive:

  • The proof simplifies (88a) using two-dimensional circular-convolution representations introduced earlier.It also constructs a matrix G and applies the same technique to another expression.Source: 3) Second-Order Derivative is… · p. 16
  • The final positive-semidefinite step relies on showing that a symmetric matrix is diagonally dominant.The proof notes that each entry of Q is at least −1.Source: 3) Second-Order Derivative is… · p. 16
  • The diagonal-dominance argument makes the quadratic form in (121) non-negative for any H, so the second-order derivative is non-positive and V = I is a local maximum of f(V).The proof establishes the matrix condition using symmetry and non-negative entries of Q + 11T.Source: 3) Second-Order Derivative is… · p. 16

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