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On the Optimality of OFDM for Aperiodic Ranging

Yifeng Xiong, Fan Liu, Ya-Feng Liu, Shi Jin, Jianhua Zhang

arXiv:2610.02843v1eess.SPcs.ITmath.IT

TL;DR

The paper asks how to choose unitary modulation for minimum expected aperiodic sidelobes at individual lags and across lag windows. It derives exact lag-wise minima and attaining modulations, then establishes when OFDM is optimal. OFDM is optimal for prefix partial EISL, while simultaneous individual-lag optimality over a prefix requires every lag to divide N.

  • Problem

    The paper asks which unitary modulations minimize ESL at a prescribed lag and whether one modulation can attain those minima across lag intervals.

    Source: A. Background and Research Questions · p. 3, B. Related Work and Remaining Questions · p. 4

  • Method

    For independent proper sub-Gaussian symbols, the paper derives exact single-lag minima and constructs attaining unitary modulations using coordinate chains.

    Source: §VI · p. 25

  • Results

    OFDM attains the single-lag ESL minimum exactly when the lag divides N, while it globally minimizes partial EISL over every positive-lag prefix.

    Source: D. Main Result and Interpretation · p. 11, §VI · p. 25

  • Takeaways & Limitations

    Simultaneous individual-lag optimality over a prefix is possible exactly when every lag in it divides N; for N ≥3, no modulation is optimal at all positive lags.

    Source: §VI · p. 25

Abstract

from arXiv · show

We study the expected aperiodic autocorrelation sidelobes of random communication symbols transmitted through a unitary modulation matrix. We consider proper sub-Gaussian constellations, namely those with fourth absolute moment $\mu_4<2$ and zero pseudo-variance, including the commonly used uniform square quadrature amplitude modulation (QAM) and uniform phase-shift keying (PSK) constellations. For every prescribed lag, we determine the exact minimum expected sidelobe level (ESL) and a unitary modulation attaining it by applying a normalized inverse discrete Fourier transform (IDFT) separately on each chain of sample indices spaced by that lag. The full orthogonal frequency division modulation (OFDM) modulation attains this minimum if and only if the lag divides the block length $N$. Combining this criterion with the complete characterization of the minimizers at lag one, we show that a unitary modulation minimizes the ESL simultaneously over the first $r$ positive lags, called the prefix $\{1,\ldots,r\}$ with $1\le r<N$, if and only if every lag in that prefix divides $N$. In particular, simultaneous optimality over all positive lags is impossible for $N\ge 3$. We also prove that OFDM globally minimizes the partial expected integrated sidelobe level (EISL), defined as the sum of the ESLs over a selected subset of positive lags, for every such prefix. This also resolves the full EISL conjecture when $r=N-1$.

I. INTRODUCTION

The introduction frames unitary modulation design for reducing expected aperiodic ranging sidelobes, posing questions about individual-lag and prefix-level optimality under a partial EISL objective. It relates these questions to prior periodic OFDM results and an open aperiodic conjecture, and uses a fourth-moment identity to cast optimization for the stated sub-Gaussian symbol class as joint lag control.

A. Background and Research Questions

  • Strong-target sidelobes can obscure weaker targets at nearby delays, motivating modulation choices that reduce expected aperiodic ranging sidelobes beyond the cyclic-prefix interval.For such delays, the receiver uses linear matched filtering on a finite transmitted block; the ESL is the expected squared magnitude of a sidelobe at a prescribed lag.Source: A. Background and Research Questions · p. 2
  • The partial EISL is the equal-weight sum of ESLs over selected positive lags; the paper focuses on prefixes {1,...,r}, where 1 ≤ r < N.The prefix length sets the relative-delay range of interest, and r = N − 1 gives the full EISL.Source: A. Background and Research Questions · p. 3

B. Related Work and Remaining Questions

  • Prior work found OFDM optimal at every nonzero periodic lag and for periodic EISL, but established only local optimality and conjectured global optimality for full aperiodic EISL [7, Theorem 3 and Conjecture 1].The introduction presents that aperiodic conjecture as motivation for examining lagwise and prefix-level optimality.Source: B. Related Work and Remaining Questions · p. 4
  • The paper asks which unitary modulations minimize ESL at each aperiodic lag, when OFDM attains those minima simultaneously over a prefix, and which modulations minimize prefix EISL.It also asks whether OFDM minimizes EISL for every positive-lag prefix and which matrices attain each minimum.Source: B. Related Work and Remaining Questions · p. 4
  • Each ESL equals a modulation-independent term plus deterministic column-correlation energy multiplied by μ4−2.This fourth-moment identity provides the starting point for comparing modulations.Source: B. Related Work and Remaining Questions · p. 4
  • Under independent zero-mean symbols with vanishing unconjugated second moments, μ4 < 2 defines the sub-Gaussian class, including uniform square QAM and uniform PSK with at least three points [7, Definition 1].Source: B. Related Work and Remaining Questions · pp. 4–5
  • Since μ4−2 is negative, ESL minimization becomes column-correlation-energy maximization; unlike periodic shifts, aperiodic boundary truncation varies by lag and requires joint control across selected lags and orthonormal columns.Periodic shifts share the Fourier basis as an eigenbasis, whereas finite-block truncation introduces lag-dependent contributions.Source: B. Related Work and Remaining Questions · p. 5

C. Contributions of This Paper

For proper sub-Gaussian symbols, the paper characterizes optimal unitary modulations for individual lags and lag prefixes, and proves OFDM is globally optimal for prefix partial EISL. Simultaneous individual-lag optimality over a prefix holds exactly when every lag in it divides N.

Assumptions

  • The results assume independent complex symbols with zero mean, vanishing unconjugated second moment, unit second absolute moment, and common fourth absolute moment µ4 < 2.The stated class includes normalized uniform square QAM and uniform PSK constellations with at least three points.Source: C. Contributions of This Paper · p. 5

1) Single lag optimality of OFDM

  • For each lag, OFDM globally minimizes ESL among unitary modulations if and only if that lag divides N.For any single lag, an optimal modulation is constructed by applying a normalized IDFT separately to each chain of sample indices spaced by that lag.Source: C. Contributions of This Paper · p. 5

2) Global optimality of OFDM for prefix partial EISL

  • OFDM globally minimizes partial EISL for every positive-lag prefix, resolving the full EISL conjecture when r = N −1.The complete family of prefix minimizers is obtained from the normalized IDFT using arbitrary subcarrier phases, a subcarrier permutation, and a common linear phase; this family is independent of prefix length.Source: C. Contributions of This Paper · p. 5

3) Simultaneous optimality over a prefix

  • A unitary modulation minimizes ESL simultaneously over {1, . . . , r} exactly when every lag in that prefix divides N; for N ≥3, all positive lags cannot be jointly optimal.This follows from combining the individual-lag criterion with the characterization of the minimizers at lag one.Source: C. Contributions of This Paper · p. 5

Notation

The section establishes notation and the random-symbol model, then defines aperiodic sidelobe objectives for unitary modulation. It frames both single-lag optimization and joint optimization over selected lag windows.

Notation

  • Bold lowercase and uppercase letters denote vectors and matrices, respectively; T and H denote transpose and Hermitian transpose, and E denotes expectation.The notation also defines vector and matrix norms, trace, rank, the identity matrix, the Kronecker delta, and complex conjugation.Source: Notation · p. 6

A. Random Symbols and Unitary Modulation

  • Independent communication symbols are mapped to a transmitted sample block by a deterministic unitary matrix selected using block length, delay window, and known symbol statistics, but fixed before realization; it preserves each realized block’s energy.Source: A. Random Symbols and Unitary Modulation · pp. 6–7
  • The normalized IDFT is the OFDM modulation, and the stated sub-Gaussian class includes normalized uniform square QAM and uniform M-ary PSK constellations.The paper calls constellations satisfying its moment conditions sub-Gaussian because their fourth absolute moment is below 2, the value for a unit-power proper complex Gaussian symbol.Source: A. Random Symbols and Unitary Modulation · pp. 6–7

B. Aperiodic Correlation and Sidelobe Criteria

  • The expected sidelobe level (ESL) is defined at a positive lag; partial EISL sums ESL over a selected nonempty set of positive lags, excluding zero lag.A prefix {1, . . . , r} corresponds to relative delays from Ts to rTs when sample spacing is Ts; r = N −1 gives the full aperiodic EISL.Source: B. Aperiodic Correlation and Sidelobe… · p. 8
  • The design questions are whether one modulation minimizes ESL at every lag in an interval and how to minimize total ESL when tradeoffs among selected lags are allowed.OFDM attains individual ESL minima exactly at lags dividing N, so full-range simultaneous ESL optimality fails for N ≥3; the paper establishes prefix partial-EISL optimality by comparing lag contributions jointly.Source: B. Aperiodic Correlation and Sidelobe… · pp. 8–9

C. Fourth Moment Reduction

The fourth-moment reduction separates symbol statistics from modulation-dependent correlation terms, enabling exact ESL and partial-EISL optimality results. Those results characterize when individual lag minima can be attained together and when OFDM is optimal.

C. Fourth Moment Reduction

  • The fourth-moment reduction separates symbol-statistics contributions from modulation-dependent correlation terms, reducing expected sidelobe optimization to deterministic correlation inequalities over unitary matrices.The same expansion remains valid for μ4<∞ when the other symbol assumptions are retained.Source: C. Fourth Moment Reduction · pp. 9–10
  • When μ4=2 the expected criteria are independent of the modulation, whereas for μ4>2 the two maximizations become minimizations.Source: C. Fourth Moment Reduction · p. 10

D. Main Result and Interpretation

III. CORRELATION ENERGY GAPS AND COEFFICIENT CONDITIONS

The paper compares a unitary modulation's correlation energies with the normalized IDFT benchmark, reducing sidelobe bounds to nonnegative correlation energy gaps. It decomposes those gaps into phase and magnitude contributions, with boundary power imbalance helping control the latter.

A. IDFT Benchmark and Correlation Energy Gap

  • For a nonempty lag set K, the correlation energy gap is defined; since 2 − µ4 > 0 by (1), each gap’s sign directly determines the candidate modulation’s comparison outcome.Source: A. IDFT Benchmark and Correlation… · p. 12

B. Gap Decomposition and Boundary Power Imbalances

  • The gap decomposition separates a nonnegative phase-cancellation loss from a magnitude-dependent contribution expressed through lag-separated entry differences.The proof sums over columns and lags because individual summands can be negative.Source: B. Gap Decomposition and Boundary… · p. 13
  • The boundary imbalance measures excess power in the first and last k entries relative to their equal-magnitude reference, counting overlap twice.When the normalized power is one, the two boundary sums total 2k.Source: B. Gap Decomposition and Boundary… · pp. 14–15

C. A Sufficient Coefficient Condition

A doubly substochastic coefficient condition supplies a sufficient bound on the boundary imbalance. The coefficient equations are solvable exactly under a gcd and rank criterion, which covers dividing single lags and prefixes.

C. A Sufficient Coefficient Condition

  • A sufficient coefficient condition bounds the boundary imbalance using a nonnegative doubly substochastic matrix whose row and column sums are at most ρ.For normalized powers, the coefficient expansion's left side equals the boundary imbalance.Source: C. A Sufficient Coefficient Condition · p. 16

D. Solvability of the Coefficient Equations

  • The balance equations have real solutions for every lag in K exactly when gcd(K) divides N and rank(A_K) = N − gcd(K).The criterion applies to dividing single lags and all prefixes considered in the paper.Source: D. Solvability of the Coefficient… · p. 17

IV. OPTIMALITY FOR SINGLE LAGS AND PREFIXES

The paper determines exact single-lag optima and constructs bounds for every lag prefix. It shows when OFDM attains the single-lag optimum and develops a coefficient construction for prefix bounds.

IV. OPTIMALITY FOR SINGLE LAGS AND PREFIXES

  • The single-lag analysis provides an explicit minimizing modulation and compares its optimum with the full IDFT benchmark.For lag k, the construction applies a normalized IDFT separately to chains of sample indices spaced by k.Source: §IV · p. 18, A. Single Lag Optimality · p. 18

A. Single Lag Optimality

  • OFDM attains the single-lag optimum if and only if k divides N.Writing N = qk + h, the proposition compares the constructed optimum with the full IDFT benchmark.Source: A. Single Lag Optimality · pp. 18–19

B. A Uniform Coefficient Bound for Prefixes

  • For every prefix, the construction seeks coefficients satisfying the allowed-difference expansion and coefficient budgets.The method expresses boundary imbalance using only allowed differences, then selects paths to meet the budgets.Source: B. A Uniform Coefficient Bound for… · p. 19

1) The Complete Graph and the Full Prefix:

  • For the full prefix r = N − 1, every edge is allowed, and the coefficient expansion and budgets already hold.The construction therefore needs no replacement of forbidden edges in this case.Source: 1) The Complete Graph and the Full… · p. 19

2) Replacing Forbidden Edges:

  • For shorter prefixes, paired forbidden-edge contributions are replaced by allowed-edge differences along increasing paths.Every resulting edge has length at most r, removing all forbidden differences.Source: 2) Replacing Forbidden Edges · pp. 20–21

3) Updating Coefficients on Allowed Edges:

The prefix proof updates coefficients to preserve the expansion after forbidden edges are replaced, then arranges paths to control edge reuse. These bounds establish nonnegative correlation-energy gaps for every unitary modulation.

3) Updating Coefficients on Allowed Edges:

  • Replacing forbidden edges can reuse allowed edges and exceed coefficient budgets, so updated coefficients preserve the expansion identity.The updated coefficient construction accounts for the original contribution on each allowed edge.Source: 3) Updating Coefficients on Allowed… · p. 21

4) Constructing Path Arrangements:

  • A bounded-congestion path arrangement yields coefficients meeting both the expansion and budget conditions, proving prefix gaps are nonnegative for every unitary U.Each path uses increasing indices and steps of length at most r; the resulting bound controls edge reuse.Source: 4) Constructing Path Arrangements · p. 22

C. All Prefix Minimizers

The paper characterizes when individual-lag and prefix ESL minima are jointly attainable, proves OFDM minimizes prefix EISL, and identifies the full family of prefix minimizers.

C. All Prefix Minimizers

  • Equality in the prefix bound forces the unitary matrix to have form (21), which includes the normalized IDFT attaining that bound.The proposition characterizes every unitary matrix attaining the prefix minimum.Source: C. All Prefix Minimizers · p. 23

A. When Simultaneous Optimality of All ESLs Is Achievable in a Prefix

B. Optimal Modulation for a Prescribed Lag

  • At lag k, applying an IDFT separately to chains of samples spaced by k attains the minimum; the full IDFT also attains it exactly when k divides N.When k does not divide N, the chain lengths differ and the construction strictly reduces ESL relative to the full IDFT.Source: B. Optimal Modulation for a Prescribed… · p. 24

C. The Cost of Improving an Individual Lag

  • Improving ESL at one lag requires a total increase over the preceding lags at least as large as that improvement.For the lag-five optimum, the reduction is (2−µ4)/12, requiring at least that total increase over lags one through four.Source: C. The Cost of Improving an Individual… · p. 25

D. Design Freedom within the Prefix Minimizing Family

  • The prefix-minimizing family allows subcarrier phases, subcarrier permutation, and a common linear phase to vary while preserving a common ESL profile and every prefix EISL minimum.These parameters permit other design requirements to be explored within the family.Source: D. Design Freedom within the Prefix… · p. 25

VI. CONCLUSION

  • Under the stated symbol assumptions, every lag has an exact ESL minimum attained by a coordinate-chain modulation; simultaneous prefix minimization requires every prefix lag to divide N.OFDM globally minimizes prefix EISL, and all prefix minimizers form the same family regardless of prefix length.Source: §VI · p. 25

APPENDIX A

The appendices supply moment calculations and proofs for the lag and prefix results, then establish a graph-based divisibility criterion.

APPENDIX A

  • Independence, zero means, and properness reduce the fourth-moment calculation to exhaustive index cases, yielding the correlation identity used in the main results.In the repeated-index case, the Kronecker terms sum to µ4.Source: Appendix A · pp. 26–27

APPENDIX B

  • The lag minimum and divisibility criterion follow by applying the fourth-moment identity to the chain construction and comparing it with the full IDFT.The appendix also derives the prefix bound and its necessary matrix form, then proves attainment for matrices of that form.Source: Appendix B · p. 27

APPENDIX C

  • The proof connects the lag-one equality characterization to the prefix criterion and rules out simultaneous optimality through lag N−1 when N ≥ 3.The argument uses preservation of correlation energy for matrices of form (21).Source: Appendix C · p. 27

APPENDIX D

  • The appendix derives its stated inequality by expanding the squares and rearranging the resulting expression.Source: Appendix D · p. 27

APPENDIX E

  • The appendix combines the assumed expansion with weighted Cauchy–Schwarz and row and column bounds to complete its estimate.Source: Appendix E · p. 28

APPENDIX F

  • For the graph condition, equality forces d connected components of size N/d and hence d | N; when d | N, residue classes provide the converse construction.Source: Appendix F · pp. 28–29

APPENDIX G

The appendix bounds lag-one correlation energy by decomposing the samples into chains, then constructs a unitary chainwise IDFT that attains the bound.

APPENDIX G

  • The chainwise estimate bounds the correlation contribution using adjacent sample pairs within each chain and the normalization of rows and columns.The proof applies Cauchy–Schwarz across chains and sums the resulting bounds.Source: Appendix G · pp. 30–31
  • A block matrix using an IDFT on each chain is unitary and attains the derived bound.Each column is supported on one chain; a chain of length one contributes zero correlation.Source: Appendix G · p. 31
  • The comparison with the full IDFT is zero exactly when h = 0, giving the divisibility criterion.Source: Appendix G · p. 31

APPENDIX H

The appendices establish a path-counting bound and derive the structure of unitary matrices attaining the prefix bound.

APPENDIX I

  • Equality in the prefix bound forces entries of equal magnitude and a constant phase increment within each column, with distinct increments forming the N Fourier phase choices.Thus the equality matrix has the form (21).Source: Appendix I · pp. 32–34
  • The proof decomposes the bound into nonnegative terms; equality forces each term to vanish, supporting the prefix-minimizer characterization.Source: Appendix I · p. 33
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