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Ambiguity Function Analysis of OFDM Signals With Pilots and Data Payloads

Jialin Wu, Fan Liu, Ying Zhang, Yifeng Xiong, Jie Yang, Kawon Han, Shi Jin

arXiv:2609.16691v1eess.SP

TL;DR

Practical OFDM frames combine deterministic pilots and random data, but their joint ambiguity behavior has not been systematically characterized. The paper derives expected squared DP-AF and FST-AF expressions and analyzes pilot-pattern effects. FST-AF sidelobes are uniform, while DP-AF pilot design cannot minimize all sidelobes in a two-dimensional delay-Doppler region.

  • Problem

    Existing OFDM ambiguity analyses assume independent, identically distributed data and do not systematically characterize pilot-embedded frames, whose distinct pilot and data statistics create cross-correlation terms.

    Source: §I · p. 2

  • Method

    The paper develops an analysis framework for pilot-embedded OFDM covering DP-AF and FST-AF, deriving closed-form expected squared expressions and examining pilot patterns.

    Source: §I · p. 2

  • Results

    FST-AF sidelobes are uniform, while no DP-AF pilot design can minimize all sidelobes in a two-dimensional delay-Doppler region.

    Source: §I · p. 2

  • Takeaways & Limitations

    Randomly placed pilots can suppress high sidelobes and improve target-estimation performance compared with the examined regular patterns.

    Source: §I · p. 3, §V · p. 10

Abstract

from arXiv · show

Practical orthogonal frequency division multiplexing (OFDM) communication frames contain both deterministic pilots and random data payloads, motivating the joint ambiguity function (AF) analysis of the two components when the entire frame is reused for integrated sensing and communication (ISAC). This paper characterizes two discrete AF formulations for different Doppler regimes, namely the discrete periodic AF (DP-AF) and fast-slow-time AF (FST-AF), and derives closed-form expressions for their expected squared values. For the FST-AF, the expected sidelobe level (ESL) is uniform over the delay-Doppler plane and depends only on the pilot count, constellation kurtosis and total number of time-frequency resources, but not on the pilot symbols or pattern. For the DP-AF, we establish attainable lower and upper ESL bounds and show that no pilot design can minimize all sidelobes simultaneously. We further prove that attaining the lower bound at non-zero Doppler requires a periodic pilot pattern, while equally spaced chirp pilots, including Zadoff-Chu (ZC) sequences, maximize the numbers of sidelobes attaining the lower and upper bounds simultaneously. Two representative ZC pilot patterns widely encountered in communication frames are then examined: contiguous placement produces delay-Doppler ridges described by squared Dirichlet kernels, whereas equally spaced placement generates periodic peak-and-notch structures. Both regular patterns exhibit pronounced high sidelobes, suggesting that communication-oriented pilot patterns should be re-designed for delay-Doppler estimation in the context of ISAC. Numerical results validate the analysis and show that irregular pilot placement can suppress high sidelobes and improve target estimation performance.

I. INTRODUCTION

Practical OFDM frames combine deterministic pilots and random data, but their joint ambiguity behavior had not been systematically characterized. This work analyzes pilot-embedded OFDM using DP-AF and FST-AF formulations.

1) 1D OFDM Signaling:

  • The paper motivates using data-bearing resources for sensing because conventional reference signals occupy about 10% of resources, leaving most unused for sensing.The pilot symbols are normalized to the same energy as data symbols for the analysis.Source: §I · p. 1, 1) 1D OFDM Signaling · p. 3

I. INTRODUCTION

  • Earlier OFDM ambiguity analyses did not systematically characterize practical frames that jointly process deterministic pilots and random data.Distinct pilot and data statistics create cross-correlation terms that existing frameworks do not handle.Source: §I · p. 2
  • The analysis derives expected squared DP-AF and FST-AF expressions, with FST-AF sidelobes uniform and DP-AF sidelobes affected by pilot design.For DP-AF, no pilot design minimizes all sidelobes in a two-dimensional delay-Doppler region; equally spaced chirp pilots maximize the number attaining the bounds.Source: §I · p. 2

A. OFDM Waveform with Embedded Pilots

  • The waveform is described both as serial time-domain samples and as a time-frequency lattice with subcarriers and symbol periods as its axes.The two descriptions provide the signal models used for the ambiguity analysis.Source: A. OFDM Waveform with Embedded Pilots · p. 3

1) 1D OFDM Signaling:

  • A 1D OFDM block has N entries, with L fixed pilot symbols and N − L mutually independent data symbols drawn from a common constellation.The data symbols are assumed zero-mean, unit-power, and to have zero pseudo-variance; most common QAM and PSK formats satisfy the stated conditions, with BPSK and 8-QAM noted as exceptions.Source: 1) 1D OFDM Signaling · p. 3

2) 2D OFDM Signaling:

The 2D signal model places pilot and data symbols across fast-time and slow-time slots. The section then introduces discrete ambiguity metrics and their delay and Doppler indexing.

2) 2D OFDM Signaling:

  • The 2D waveform spans N fast-time slots and M slow-time slots, with L unit-amplitude pilots and MN − L independently drawn data entries.The data entries use the same constellation constraints as the 1D model.Source: 2) 2D OFDM Signaling · p. 3

B. Evaluation Metrics for OFDM Sensing

  • The paper adopts discrete ambiguity functions established for 1D and 2D signaling as metrics for evaluating OFDM sensing performance.The expected local sidelobe power and total expected sidelobe energy are the ESL and EISL, respectively.Source: B. Evaluation Metrics for OFDM Sensing · p. 4

1) DP-AF associated with the 1D Signal:

  • For the 1D DP-AF, CP insertion and removal make the effective delay shift periodic, while k and q index delay and Doppler.A Doppler shift indexed by q is represented by a frequency-domain diagonal matrix.Source: 1) DP-AF associated with the 1D Signal · p. 4

2) FST-AF associated with the 2D Signal:

The FST-AF formulation treats Doppler as approximately constant within each fast-time block and varying across slow-time pulses.

2) FST-AF associated with the 2D Signal:

  • Under sufficiently small Doppler, target-induced phase rotation is approximately uniform over each N-sample fast-time block, enabling the waveform to be arranged into M columns.A cyclic prefix is prepended to each slow-time segment to cover the largest expected delay.Source: 2) FST-AF associated with the 2D Signal · p. 4
  • After matched filtering, the signal-dependent term is called the FST-AF.Its compact representation is given for the 2D OFDM waveform.Source: 2) FST-AF associated with the 2D Signal · p. 4

III. STATISTICAL AF ANALYSIS OF PILOT-EMBEDDED OFDM SIGNALS

The section defines expected sidelobe and integrated sidelobe measures, then derives the expected squared DP-AF and its dependence on pilots and constellation statistics. It also shows that the total sidelobe sum at non-zero Doppler is invariant to pilot design.

III. STATISTICAL AF ANALYSIS OF PILOT-EMBEDDED OFDM SIGNALS

  • The ESL measures expected local sidelobe power, while the EISL sums expected energy over all sidelobes.Both measures are used to examine dependence on pilot count, pilot symbols, pilot patterns, and kurtosis.Source: §III · p. 4

A. Analytical Expression for the Expected Squared DP-AF

  • The expected squared DP-AF has a zero-Doppler expression governed by signal length, pilot count, and constellation kurtosis, while non-zero-Doppler values depend on the pilot pattern and symbols.At zero Doppler the ESL is (κ − 1)(N − L); it is zero when L = N or κ = 1, and the non-zero-Doppler ESL is independent of κ.Source: A. Analytical Expression for the… · pp. 4–5
  • The DP-AF expression separates pilot-symbol-and-pattern dependence from a term determined solely by the pilot pattern.The pattern-only term is the cyclic autocorrelation of the pilot-pattern mask, with cyclic shift indexed by Doppler.Source: A. Analytical Expression for the… · p. 5

B. Sum of the Expected Sidelobe Levels for the DP-AF

  • For any non-zero Doppler index, the sum of expected DP-AF sidelobe levels over delay is constant under a given pilot count and constellation kurtosis.Consequently, adjusting pilot design alone cannot optimize that sum.Source: B. Sum of the Expected Sidelobe Levels… · p. 5

C. Impact of Pilot Patterns on the Expected Squared DP-AF

For nonzero Doppler, the expected squared DP-AF has attainable upper and lower sidelobe bounds, but no pilot design can minimize all sidelobes across a two-dimensional delay-Doppler region. Periodic patterns are required to attain either bound, and equally spaced chirp pilots maximize the numbers of sidelobes attaining both bounds simultaneously.

C. Impact of Pilot Patterns on the Expected Squared DP-AF

  • Cyclically shifting the pilot pattern leaves the expected squared DP-AF unchanged.This permits flexibility in pilot placement without changing sidelobe behavior.Source: C. Impact of Pilot Patterns on the… · p. 5
  • For nonzero Doppler, the attainable expected squared DP-AF sidelobe bounds are N − L below and N + L^2 − L above.At zero Doppler, the ESL is (κ − 1)(N − L); for L < N it is above the lower bound for super-Gaussian constellations and attains the global minimum for sub-Gaussian constellations.Source: C. Impact of Pilot Patterns on the… · p. 5
  • A nonzero-Doppler sidelobe can attain either bound only if the pilot pattern is periodic; nonempty Dh requires gcd(N, L) > 1 and period Tw = N/λN,L.The pattern must be invariant under cyclic shifts by the relevant Doppler index.Source: C. Impact of Pilot Patterns on the… · pp. 5–6
  • The number of eligible nonzero Doppler indices is at most L − 1, with equality only for equally spaced pilots of period N/L when L divides N.Equally spaced chirp pilots make the upper- and lower-bound attainment counts simultaneously maximal.Source: C. Impact of Pilot Patterns on the… · p. 6
  • With equally spaced chirp pilots, each qualifying Doppler index has N/L upper-bound sidelobes and N − N/L lower-bound sidelobes.The chirp sequence reduces to Zadoff–Chu when A = 1 and ν = 0.Source: C. Impact of Pilot Patterns on the… · p. 6
  • No pilot design can minimize every DP-AF sidelobe in a two-dimensional delay-Doppler region, although equally spaced chirp pilots can minimize sidelobes on certain one-dimensional subsets.Such subset minimization includes certain adjacent sidelobes at nonzero Doppler q = m(N/L).Source: C. Impact of Pilot Patterns on the… · p. 6
  • When L < N, no 2 × 2 delay-Doppler region can attain the minimum sidelobes.Adjacent eligible Doppler indices would require L = N.Source: C. Impact of Pilot Patterns on the… · p. 6

D. Statistical Characterization of the FST-AF

The expected squared FST-AF has identical sidelobe levels throughout the delay-Doppler plane. These levels depend on pilot count, constellation kurtosis, and the number of time-frequency resources, but not on pilot symbols or pattern.

D. Statistical Characterization of the FST-AF

  • The expected squared FST-AF equals M^2N^2δk,0δq,0 + (κ − 1)(MN − L), giving uniform sidelobe levels across delay and Doppler.Its ESL is (κ − 1)(MN − L), which is zero when L = MN or κ = 1.Source: D. Statistical Characterization of the… · p. 6
  • FST-AF expected sidelobes are independent of pilot pattern and symbols, and depend on pilot count, constellation kurtosis, and total time-frequency resources.The small-Doppler assumption in 2D OFDM models may fail to capture pilot effects on realistic sensing performance.Source: D. Statistical Characterization of the… · p. 6

IV. CASE STUDY ON USING ZC SEQUENCES AS PILOTS

The case study examines odd-length ZC pilots under equally spaced and contiguous placements. For equally spaced pilots, the expected squared DP-AF forms a delay comb on selected Doppler bins, with peaks and notches governed by a congruence condition.

IV. CASE STUDY ON USING ZC SEQUENCES AS PILOTS

  • Odd-length ZC sequences are assigned in ascending order to the selected pilot subcarriers, with the sequence root usually chosen coprime to its length.Source: §IV · p. 7

A. Equally Spaced Placement as the Pilot Pattern

B. Contiguous Placement as the Pilot Pattern

Contiguous ZC pilot placement produces delay-Doppler ridges described by Dirichlet kernels, with flat data-floor regions where shifted pilots do not overlap. The ridge pattern changes when the pilot length exceeds half the bandwidth, and both regular placements exhibit prominent sidelobes.

B. Contiguous Placement as the Pilot Pattern

V. SIMULATION RESULTS

Simulations validate the ambiguity-function analysis and show that pilot placement affects sidelobes and sensing accuracy. Irregular placement suppresses high sidelobes, while the conclusion identifies communication metrics for future joint optimization.

V. SIMULATION RESULTS

  • Zero-Doppler DP-AF sidelobes decrease as L/N increases; doubling N at the same L/N reduces normalized sidelobe level by approximately 3 dB.256-QAM has nearly 1 dB higher sidelobes than 16-QAM.Source: Figure 7 · p. 9, §V · pp. 9–10
  • For contiguous pilots, zero-delay cuts have dominant local Doppler peaks; increasing L from 15 to 45 lowers the minimum sidelobe by 2.86 dB but raises the maximum by 2.24 dB.A flat data floor appears for L = 15 and 25, but vanishes for L = 45 > N/2.Source: §V · p. 9, Figure 6 · p. 9
  • Randomly placed pilots reduce high DP-AF sidelobes relative to equally spaced and contiguous patterns, producing a flatter ambiguity profile.The conclusion also reports improved target-estimation performance with irregular placement.Source: Figure 8 · p. 9, §V · p. 10, §VI · p. 11
  • Equally spaced pilots produce periodic zero-delay Doppler dips at q = m(N/L), reaching the lower bound and separated by a constant data floor.The reported cases have L = 15 and L = 25, both dividing N.Source: §V · p. 9, Figure 6 · p. 9
  • Random pilot placement outperforms equally spaced and contiguous ZC patterns in range and velocity RMSE, while using data with pilots improves RMSE over pilots alone.The comparison evaluates weak-target estimation.Source: Figure 9 · p. 10, §V · p. 10, §VI · p. 11
  • FST-AF sidelobes are uniform; doubling MN reduces normalized sidelobe level by approximately 3 dB, and 256-QAM produces higher sidelobes than 16-QAM.The constellation difference is attributed to larger kurtosis.Source: Figure 11 · p. 10, Figure 10 · p. 10, §V · p. 10

VI. CONCLUSION

  • The analysis concludes that no pilot design minimizes every DP-AF sidelobe simultaneously, and regular ZC patterns exhibit pronounced high sidelobes.Equally spaced placement yields a comb-like profile, while contiguous placement produces delay-Doppler ridges.Source: §VI · p. 11
  • Future joint ISAC optimization will incorporate achievable rate and channel-estimation accuracy alongside pilot and data resource design.Source: §VI · p. 11

APPENDIX A PROOF OF THEOREM 1

The appendix begins the DP-AF expectation proof by expressing the squared ambiguity function through the data-plus-pilot signal. It then relates pilot-set shift overlaps to periodicity.

APPENDIX A PROOF OF THEOREM 1

  • The squared DP-AF is written as the squared magnitude of the quadratic form (d + p)^H B(d + p).Source: Appendix A · p. 11
  • The data vector contains a random communication symbol where w_n = 0 and zero where w_n = 1.Source: Appendix A · p. 11
  • The expectation is decomposed, and separate expressions for expected squared DP-AF are derived for q ≠ 0 and q = 0, matching Theorem 1.Source: Appendix A · p. 11

APPENDIX B PROOF OF PROPOSITION 2

  • The overlap h(q) counts pilots shared by the pilot set and its cyclic shift; h(q) = L means that shift leaves the set unchanged.Source: Appendix B · p. 11
  • When nonzero shifts preserve the pilot set, the pattern is periodic; conversely, a common divisor of N and L permits construction of such a periodic pattern.Source: Appendix B · p. 11

APPENDIX C PROOF OF PROPOSITION 3

The proof characterizes how pilot patterns and pilot symbols attain sidelobe bounds. It identifies equally spaced patterns as a bound-attaining case and derives pilot-symbol expressions for odd and even L.

APPENDIX C PROOF OF PROPOSITION 3

  • When the number of lower-bound sidelobes is N − N/L, the remaining N/L sidelobes attain the upper bound, and neither count can be increased in that case.Source: Appendix C · p. 12
  • If Card(Dh) = L − 1 and L divides N, the pilot pattern is equally spaced with period N/L.Source: Appendix C · p. 12
  • The proof states that unit-modulus chirp parameters maximize the numbers of sidelobes attaining both bounds, then gives pilot-symbol expressions separately for odd and even L.Source: Appendix C · p. 12

APPENDIX D PROOF OF THEOREM 2

The appendix verifies the FST-AF expectation against Theorem 2 and derives DP-AF expressions by considering pilot-count regimes and delay cases.

APPENDIX D PROOF OF THEOREM 2

  • The FST-AF expectation calculation matches the expression in Theorem 2.Source: Appendix D · p. 12

APPENDIX E PROOF OF PROPOSITION 4

  • For L ≤ N/2, the proof derives the expression for q > N − L and obtains q < L through central symmetry.Source: Appendix E · pp. 12–13
  • For L > N/2, the proof treats N − L < q < L separately and combines the cases to obtain the expression; boundary cases approach the corresponding limit.Source: Appendix E · p. 13
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