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Statistical Analysis of 3D Ambiguity Functions for Random MIMO-OFDM Signals

Pinjun Zeng, Yuyang Lu, Jialan Tang, Xiaoyang Chen, Yifeng Xiong, Fan Liu, Shi Jin

arXiv:2610.12162v1eess.SP

TL;DR

Random payload symbols make matched-filter sidelobes data-dependent, while existing statistical analyses largely omit the angular effects of MIMO arrays and precoding. The paper models the 3D-AF as a quadratic form, derives its expected statistics and sidelobe metrics, and extends the analysis to coherent multi-symbol FST processing.

  • Problem

    Random payloads make the matched-filter response realization-dependent, while prior statistical analyses largely cover delay-Doppler rather than joint angle and MIMO precoding.

    Source: §I · p. 1

  • Method

    The paper expresses the point-target 3D-AF as a quadratic form in payload symbols and derives expected squared AF and sidelobe metrics, extending the framework to coherent multi-symbol FST processing.

    Source: §I · p. 2, §V · pp. 12–13

  • Results

    Sampled EISL rises from QPSK through 16-QAM and 256-QAM to proper Gaussian symbols; peak-normalized ESL decreases with N while preserving that constellation ordering.

    Source: §V · p. 12

  • Takeaways & Limitations

    Equal physical angular offsets need not produce identical AF cuts at different target directions, because the array response depends on sin θ.

    Source: C. 3D-AF and Performance Metrics · p. 5

    The analysis assumes i.i.d. proper symbols and point targets; multi-symbol FST additionally requires small within-block phase, and modeled delays must not exceed the CP.

    Source: 4) Scope and Limitations · p. 12

Abstract

from arXiv · show

Random communication symbols allow integrated sensing and communications (ISAC) systems to exploit payload signals for sensing, but the resulting matched-filter response varies with the transmitted data. Existing statistical analyses of sensing with random orthogonal frequency division multiplexing (OFDM) waveforms have largely focused on delay and Doppler, while the spatial dimension introduced by multi-input multi-output (MIMO) arrays and precoding has received much less attention. This paper develops a three-dimensional ambiguity function (3D-AF) for random MIMO-OFDM waveforms over delay, Doppler, and angle domains. For point-target sensing, we express the 3D-AF as a quadratic form of the random communication symbols, with a deterministic matrix that captures the delay and Doppler shifts, the transmit and receive array responses, and the precoders on different subcarriers. Based on this representation, we derive a closed-form expression for the expected squared 3D-AF and show explicitly how the constellation fourth-order moment (or kurtosis) affects the ambiguity sidelobes. The same characterization yields expected integrated sidelobe level (EISL) and expected sidelobe level (ESL) metrics over mixed delay-Doppler-angle search domains. The framework is further extended to coherent multi-symbol processing with slow-time Doppler, leading to the corresponding fast-slow time (FST) approximation. These results provide a statistical characterization of the sensing ambiguity induced by random MIMO-OFDM communication signals in delay, Doppler, and angle.

I. INTRODUCTION

Random payload symbols make sensing sidelobes realization-dependent, while MIMO arrays and precoding add angular effects not captured by delay-Doppler-only statistical analyses. The paper formulates a 3D ambiguity function and analyzes its statistics for single- and multi-symbol processing.

I. INTRODUCTION

  • Random payloads make sidelobes vary across realizations, and prior statistical analyses do not jointly characterize angle and MIMO precoding.Existing random-waveform analyses focus on delay-Doppler behavior; the spatial dimension motivates a joint statistical characterization.Source: §I · pp. 1–2
  • The 3D-AF is a quadratic form in payload symbols, with a deterministic matrix that captures delay-Doppler shifts, array responses, and precoding.The formulation accounts for spatial processing at both transmitter and receiver, including subcarrier-dependent precoders.Source: §I · p. 2
  • A closed-form expected squared 3D-AF exposes the constellation fourth moment's effect on sidelobes and yields EISL and ESL over the joint search region.Source: §I · p. 2
  • The coherent multi-symbol analysis uses an FST model; with small intra-block Doppler phase and shared precoders across OFDM symbols, its AF matrix has Kronecker structure.Source: §I · p. 2

II. SYSTEM MODEL AND PERFORMANCE METRICS

  • The system model defines a single-symbol 3D-AF over integer cyclic-delay and Doppler grids and a continuous angular interval, using far-field ULAs with half-wavelength spacing.The cyclic prefix is assumed no shorter than the maximum excess delay in the channel model.Source: §II · p. 2

A. Random MIMO-OFDM Signal and Array Model

B. Point-Target Sensing Model

The point-target model compares a candidate template with target delay, Doppler, and direction coordinates, producing a search that is discrete in delay and Doppler and continuous in angle.

B. Point-Target Sensing Model

  • Each far-field point target is specified by a complex scattering coefficient and cyclic delay, intra-symbol Doppler, and direction; the receiver tests a candidate template across those coordinates.Source: B. Point-Target Sensing Model · p. 3
  • Under the monostatic far-field assumption, transmit and receive steering vectors define a rank-one spatial response, with the scattering coefficient kept separate.Source: B. Point-Target Sensing Model · p. 4
  • The delay and Doppler indices are cyclic; the model uses equivalent frequency- and time-domain delay operators and a frequency-shift matrix.Source: B. Point-Target Sensing Model · p. 4
  • The echo model applies cyclic delay to CP-free samples after CP insertion and removal; modeled delays must fit within the CP, and fractional delay and beam squint are outside scope.Source: B. Point-Target Sensing Model · p. 4

C. 3D-AF and Performance Metrics

The point-target matched-filter response is expressed through a cross 3D ambiguity function and a quadratic form in the random symbols. The section defines sidelobe metrics over a mixed delay-Doppler-angle search domain.

C. 3D-AF and Performance Metrics

  • The cross 3D-AF relates a template at (k, q, θ) to a unit-scattering reference target at (0, 0, ϑ) and underlies the matched-filter expansion.The target and test angles have receive-array angular correlation ρr(θ, ϑ).Source: C. 3D-AF and Performance Metrics · pp. 4–5
  • Cyclic delay and Doppler shifts translate the response magnitude profile, but angular cuts need not match for equal physical offsets because ULA response depends on sin θ.The angular dependence includes receive-array correlation and transmit-side responses.Source: C. 3D-AF and Performance Metrics · p. 5
  • Combining the delay-Doppler kernel and deterministic 3D-AF kernel matrix gives a quadratic-form representation of the random-symbol-dependent response.The kernel matrix is defined for a delay-Doppler cell and test and target angles.Source: C. 3D-AF and Performance Metrics · p. 5
  • The section defines ISL, EISL, and ESL over sidelobe regions in the mixed discrete-continuous domain, with ESL requiring finite, nonzero sidelobe measure.Comparisons require the same delay-Doppler domain, angular mainlobe boundary, angular weights, and normalization.Source: C. 3D-AF and Performance Metrics · p. 5

D. Multi-Symbol and FST Models

The paper extends its 3D-AF model to coherent processing across OFDM blocks, separating slow-time and fast-time Doppler. The FST approximation retains slow-time rotation while neglecting a within-block phase under a stated condition.

D. Multi-Symbol and FST Models

  • Coherent processing across M OFDM blocks follows the FST route, separating Doppler into slow-time phase across blocks and fast-time phase within each block.The construction makes cyclic-prefix operations explicit before separating the phases.Source: D. Multi-Symbol and FST Models · pp. 5–6
  • The FST condition requires η ≪ 1 for every modeled target; η measures Doppler-phase change over N sample intervals within one block.Slow-time rotation across blocks is retained while the residual within-block phase is neglected.Source: D. Multi-Symbol and FST Models · p. 6
  • In the FST model, q denotes a slow-time Doppler bin in ZM, distinct from the single-symbol intra-symbol Doppler index in ZN.The serialized-grid slow-time bin is the residue of the serialized Doppler index modulo M.Source: D. Multi-Symbol and FST Models · p. 6
  • The coherent FST matched-filter output is represented using the FST cross 3D-AF, and the section defines corresponding ISL, EISL, and ESL metrics.The ESL is defined when the sidelobe region has finite, nonzero measure.Source: D. Multi-Symbol and FST Models · pp. 6–7

III. STATISTICAL CHARACTERIZATION OF THE 3D-AF

The statistical analysis proceeds from the single-symbol kernel structure to second-order statistics, then extends to coherent multi-symbol processing and the FST approximation.

III. STATISTICAL CHARACTERIZATION OF THE 3D-AF

  • The analysis first examines the single-symbol kernel’s deterministic block structure, derives its second-order statistics, and then extends to coherent multi-symbol processing and the FST approximation.The section introduction identifies this as the sequence of the statistical characterization.Source: §III · p. 7

A. Single-Symbol Quadratic-Form Structure

The deterministic kernel matrix separates the random symbols from delay-Doppler and spatial processing. Its subcarrier blocks encode mismatch-dependent coupling weights.

A. Single-Symbol Quadratic-Form Structure

  • The deterministic matrix separates symbol randomness from sensing and transmission parameters: Ψk,q carries delay-Doppler dependence, while ρr and GT carry array and precoding dependence.Its block structure follows from the block-diagonal precoder matrix.Source: A. Single-Symbol Quadratic-Form… · p. 7
  • On the integer cyclic grid, Doppler mismatch selects a subcarrier pair and delay sets its phase; receive-array correlation and two precoders weight their angular responses.Each sH contribution to the matched-filter 3D-AF has a deterministic coupling weight M(k,q) for its subcarrier pair.Source: A. Single-Symbol Quadratic-Form… · p. 7

B. Expected Squared 3D-AF and Sidelobe Metrics

The section derives expected squared 3D ambiguity responses and sidelobe metrics from a quadratic-form representation, identifying how constellation kurtosis and precoding shape the result. It extends this statistical characterization to coherent fast-slow-time processing under an explicit small intra-block Doppler-phase condition.

B. Expected Squared 3D-AF and Sidelobe Metrics

  • Theorem 1 gives the expected squared cross 3D-AF from the quadratic-form representation under the system model and symbol assumptions.The result is built from the mean quadratic form and its covariance.Source: B. Expected Squared 3D-AF and Sidelobe… · p. 7
  • The expected-power decomposition separates the coherent mean term, total kernel energy, and a diagonal correction controlled by constellation kurtosis.The kurtosis correction vanishes for proper Gaussian symbols and is negative for proper QAM and constant-modulus PSK, for which κ < 2.Source: B. Expected Squared 3D-AF and Sidelobe… · p. 7
  • Precoding can reshape all three decomposition terms, whereas constellation shaping changes only the kurtosis-dependent diagonal contribution.The resulting distinction is made explicit over the three-dimensional search domain.Source: B. Expected Squared 3D-AF and Sidelobe… · p. 7
  • Corollary 1 evaluates 3D EISL by integrating the expected squared response over the sidelobe domain, with ESL obtained by normalization by the domain measure.The metric uses the theorem's system and symbol conditions.Source: B. Expected Squared 3D-AF and Sidelobe… · p. 8

C. FST Statistical Characterization

  • The FST analysis separates the random symbol vector from deterministic sensing operators to obtain a quadratic-form representation.Proposition 2 specifies the deterministic matrix, and Theorem 2 applies the quadratic-form moment identity to derive expected squared FST 3D-AF.Source: C. FST Statistical Characterization · p. 8
  • The FST characterization also yields deterministic EISL and ESL, while identical precoders across OFDM symbols give the AF matrix a Kronecker structure.The EISL is obtained by integrating the expected FST response over the FST sidelobe region.Source: C. FST Statistical Characterization · p. 8

IV. NUMERICAL RESULTS AND DISCUSSION

The numerical results compare constellation-dependent ambiguity sidelobes across delay, Doppler, and angle, and validate the theoretical expectations against Monte Carlo simulations. They show that aggregate EISL ordering follows constellation fourth moments, while the observed cut-level differences depend on the coordinate and configuration.

IV. NUMERICAL RESULTS AND DISCUSSION

  • The numerical study uses 32 subcarriers, 8-element transmit and receive ULAs, unit-power 16-QAM, 32 FST blocks, and 5,000 payload trials by default.The sampled angular interval is −60° to 60° in 2° increments, and the boresight angular mainlobe is approximately −14.48° to 14.48°.Source: §IV · p. 8

A. Joint Delay-Doppler-Angle Structure

  • The delay-Doppler slice has a central peak above an expected sidelobe floor, while the delay-angle and Doppler-angle slices combine grid mismatch with array and precoder effects.The angular cut at matched delay and Doppler isolates the array and precoder response; conditional slices do not establish the largest response over the full search region.Source: A. Joint Delay-Doppler-Angle Structure · pp. 8–9, Figure 2 · p. 9

B. Constellation Effects in Single-Symbol and FST Processing

  • At fixed N, theoretical EISL increases from QPSK through 16-QAM and 256-QAM to proper Gaussian symbols, and FST retains the same ordering.In the single-symbol comparison, numerical curves follow the overall theoretical trends with some deviations; FST numerical curves closely track predictions.Source: B. Constellation Effects in… · pp. 9–10, Figure 3 · p. 10
  • Both panels show increasing aggregate EISL with N, but their absolute levels are not directly comparable because their Doppler grids and processing intervals differ.A sensing-gain interpretation would additionally require common energy and grid normalization.Source: B. Constellation Effects in… · p. 10, Figure 3 · p. 10
  • Peak-normalized ESL divides average sidelobe power by expected matched-cell power, and theoretical curves closely align with Monte Carlo markers in both processing panels.Within each panel, QPSK is generally lowest, proper Gaussian highest, and QAM lies between them; ESL values are compared only within their respective panels.Source: B. Constellation Effects in… · p. 10, Figure 4 · p. 10
  • In single-symbol delay cuts, normalized sidelobes rise from 16-QAM to 1024-QAM to proper Gaussian symbols, while Doppler and angle cuts nearly overlap at the plotted scale.Theory and Monte Carlo agree across the three displayed cuts; the angular shape is primarily set by the array-precoder response under the stated configuration and normalization.Source: B. Constellation Effects in… · pp. 10–11, Figure 5 · p. 11

C. FST Approximation Accuracy

The FST approximation is evaluated by comparing serialized and FST ambiguity responses using the same payload. The measured pointwise discrepancy increases overall with η for the tested Doppler indices, consistent with FST discarding within-block Doppler phase.

C. FST Approximation Accuracy

  • The test varies target Doppler on an integer grid, with q0 setting the discarded within-block phase and Δq denoting slow-time Doppler-bin mismatch.The matched cell is (Δk, Δq) = (0, 0); the selected CP length is NCP = 6.Source: C. FST Approximation Accuracy · p. 11
  • The pointwise error numerator is the mean-square difference between serialized and FST AFs evaluated with the same payload, not the difference between expected powers.The comparison uses a single zero-delay boresight target, unit-power 16-QAM, and 100 Monte Carlo trials.Source: C. FST Approximation Accuracy · p. 11
  • For target Doppler indices q0 = 1, 4, and 16, η is approximately 0.165, 0.661, and 2.646, and relative AF discrepancy increases overall with η.The trend is consistent with the growing within-block Doppler phase discarded by FST.Source: C. FST Approximation Accuracy · p. 11

D. Design Implications and Modeling Scope

The design discussion clarifies what EISL and ESL capture, how precoding and constellation statistics enter EISL, and which model assumptions bound the analysis. The conclusions summarize single-symbol and FST findings and identify extensions beyond the current scope.

1) Interpretation of EISL and Sidelobe Statistics:

  • EISL integrates expected sidelobe power over the prescribed 3D sidelobe region, while ESL divides by its measure; neither bounds an individual payload’s largest sidelobe.A peak guarantee would require separate analysis of peak statistics or exceedance probabilities.Source: 1) Interpretation of EISL and Sidelobe… · pp. 11–12
  • Boresight EISL need not characterize targets elsewhere because the angular response is not translation invariant; sector-wide design can aggregate over desired target angles.The aggregation may use an average or worst-case value.Source: 1) Interpretation of EISL and Sidelobe… · p. 12

2) Role of Precoding and Constellation Shaping:

  • EISL can serve as a precoder-design objective: precoders alter the full kernel, while constellation statistics affect it through the diagonal kernel-energy correction.The illustrative comparison fixes total transmit power; the resulting objective is generally nonconvex, and solving the joint design problem is outside the paper’s scope.Source: 2) Role of Precoding and Constellation… · p. 12

3) Frequency-Dependent Extended Targets:

  • For deterministic frequency-dependent target responses, target and receiver-template responses weight the kernel; template mismatch changes it, and random target responses require an additional expectation.The extension is conditional on those deterministic responses.Source: 3) Frequency-Dependent Extended Targets · p. 12

4) Scope and Limitations:

  • The model assumes narrowband far-field steering, aligned cyclic delays within the CP, and i.i.d. proper symbols with a prescribed fourth moment; FST additionally requires small within-block phase.Wideband beam squint, delays beyond the CP, non-i.i.d. symbol statistics, clutter, array imperfections, and angle-dependent scattering are outside the stated model.Source: 4) Scope and Limitations · p. 12

V. CONCLUSIONS

  • Sampled EISL increases from QPSK through 16-QAM and 256-QAM to proper Gaussian symbols, while peak-normalized ESL decreases with N and preserves that constellation ordering.The AF cuts show the constellation effect most clearly along delay for the adopted precoder and normalization; Doppler and angle cuts nearly overlap.Source: §V · p. 12
  • With identical precoders across OFDM blocks, the FST AF matrix separates slow-time Doppler from the delay-angle kernel; its approximation error increases overall with η.The approximation neglects within-block phase variation under the small-phase condition η ≪1; FST EISL and normalized ESL follow the predicted constellation ordering.Source: §V · pp. 12–13

APPENDIX A QUADRATIC-FORM MOMENTS AND SIDELOBE METRICS

The appendices derive the quadratic-form moments underlying the sidelobe metrics, then construct the coherent multi-symbol FST model and its approximation. They also extend the single-symbol kernel to deterministic frequency-dependent target and receiver responses.

APPENDIX A QUADRATIC-FORM MOMENTS AND SIDELOBE METRICS

  • Independence and zero mean restrict fourth-order moment contributions to paired indices, while properness eliminates pairings of two unconjugated or two conjugated symbols.The surviving conjugate pairings are i = j, k = l and i = k, j = l.Source: Appendix A · p. 13
  • When all four indices coincide, the two pairings contribute 2 rather than the actual fourth moment κ, so the derivation adds the correction κ−2.This correction enters the quadratic-form moment expression.Source: Appendix A · p. 13
  • Setting P = Q = M and adding |E[sᴴMs]|² yields the second-moment expression used for the 3D-AF.Substituting M = M_k,q(θ, ϑ) gives the stated theorem.Source: Appendix A · p. 13
  • The nonnegative sidelobe integrand permits expectation and integration to be interchanged; integrating the second moment gives EISL, and dividing by sidelobe-region measure gives ESL.The derivation identifies this division as the ESL in Corollary 1.Source: Appendix A · p. 13

APPENDIX B MULTI-SYMBOL AND FST DERIVATIONS

  • The FST derivation builds a serialized coherent processing interval from CP-free blocks and forms the cross-AF as a quadratic form in the stacked symbols.For identical precoders across blocks, the single-block kernel is independent of block index; the fourth-order identity then yields multi-symbol sidelobe statistics.Source: Appendix B · pp. 13–14
  • Under the small-η condition, the FST Doppler approximation retains slow-time phase rotation while discarding within-block phase variation.The derivation bounds phase deviation across the CP-extended block and absorbs the reference-sample phase into the target coefficient.Source: Appendix B · p. 14
  • For aligned blocks with integer target delay no greater than the CP, CP removal gives a cyclic-delay identity used with the Doppler approximation to form the received CPI model.The stacked transmit and receive blocks feed the coherent matched-filter output for each test cell.Source: Appendix B · p. 14

APPENDIX C FREQUENCY-DEPENDENT EXTENDED-TARGET

The appendix extends the single-symbol grid model to deterministic frequency-dependent target responses and a receiver template. It applies the statistical results conditionally on those responses.

APPENDIX C FREQUENCY-DEPENDENT EXTENDED-TARGET

  • A deterministic subcarrier response describes the extended target, while the receiver may use a different deterministic frequency response as its template.The circular filtering operator represents the receiver’s frequency response in the time domain.Source: Appendix C · p. 14
  • The conditional extended-target AF is formed for a target response at the delay-Doppler origin and a test cell (k, q, θ).The cyclic-delay and target-filtering operators commute, while the Doppler operator retains its position in the matched-filter product.Source: Appendix C · p. 14
  • Conditional on deterministic target and receiver responses, the statistical results apply after replacing the kernel; random target responses require an outer expectation.The target response and receiver template explicitly represent frequency selectivity and template mismatch.Source: Appendix C · p. 14
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