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Ambiguity Function Analysis of Pilot-Embedded Random OFDM Signals

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Pith's one-line read The paper derives exact formulas for the mean squared ambiguity function of OFDM signals with deterministic pilots embedded among random data, showing that pilot placement and symbols shape delay-Doppler sidelobes in one formulation while o

desk verdict A clean analytical extension of the random-OFDM AF framework to pilot-embedded frames; the DP-AF formula is new and the proofs hold up under stated assumptions, though the scope is deliberately narrow. read the letter →

arxiv 2607.17663 v1 pith:2HB7KMXT submitted 2026-07-20 eess.SP

Jialin Wu , Fan Liu , Ying Zhang , Yifeng Xiong , Jie Yang , Shi Jin This is my paper · ORCID
classification eess.SP MSC 94A1294A13
keywords OFDMIntegratedsensingandcommunicationAmbiguityfunctionDiscreteperiodicFast-slow-timePilotdesignExpectedsidelobelevelRandomwaveforms
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

In integrated sensing and communication, the same OFDM waveform carries both known pilot symbols and random data, and the paper asks how those pilots change the waveform's sensing ambiguity function—the delay-Doppler response that determines how well targets can be separated. The authors derive closed-form expressions for the mean squared discrete periodic ambiguity function (DP-AF) and the fast-slow-time ambiguity function (FST-AF) of pilot-embedded random OFDM signals. The central result is a split: under the DP-AF formulation, sidelobes away from zero Doppler depend on exactly where pilots sit and what symbols they carry, while under the FST-AF formulation every off-mainlobe sidelobe equals the same constant, determined only by the number of pilots, not their pattern or values. Giving designers a formula for this split matters because it turns pilot placement into a quantitative tool for shaping sensing sidelobes without changing the communication payload.

What carries the argument

The argument rests on splitting the transmitted symbol vector into a deterministic pilot component p and a random data component d, encoded by a binary indicator vector w, and then expanding the fourth-order moment E(s_l^* s_r s_m s_n^*) of the mixed signal. Using constellation symmetry assumptions that force odd and certain cross moments to vanish, the expansion collapses to diagonal terms controlled by the kurtosis κ plus pilot-pilot terms controlled by the filtered pilot sequence. This is what produces the DP-AF formula whose off-Doppler part depends on pilot pattern and symbols. For the FST-AF, the key simplification is the identity A_FST = √(MN) F_N^H |S|^2 F_M, which shows the ambiguit

What would settle it

Transmit a 1D OFDM signal with BPSK data symbols and L pilots, average |A_DP(k,q)|^2 over many trials for q≠0, and compare with the paper's formula: for BPSK the moment E(|s|^2 s) is nonzero, so the pilot-data cross terms the proof sets to zero will appear as a systematic discrepancy. The same test can be run analytically by inserting the measured fourth-order moments into the expansion before Eq. (37) and checking that the residual matches the simulation.

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Extended reading notes

Core claim

For a length-N OFDM symbol containing L unit-modulus pilots and N−L i.i.d. data symbols drawn from a constellation with kurtosis κ, the paper proves E[|A_DP(k,q)|^2] equals N^2 + (κ−1)(N−L) at the mainlobe (k,q)=(0,0), equals (κ−1)(N−L) along the zero-Doppler axis q=0, and for q≠0 equals a pilot-dependent term: the squared ambiguity of the pilot sub-sequence alone, |p^H F_N D_{N,q} J_{N,k} F_N^H p|^2, plus a residual N − w^H F_N D_{N,q} F_N^H w, where w marks pilot locations and p holds pilot symbols. Thus off-Doppler DP-AF sidelobes are shaped by pilot pattern and pilot symbols. For the two-dimensional fast-slow-time formulation with M OFDM symbols, the paper proves E[|A_FST(k,q)|^2] = L +

Load-bearing premise

The DP-AF derivation holds only when the random data constellation is symmetric enough that E(|s|^2 s)=E(|s|^2 s*)=0, and the FST-AF result additionally assumes Doppler is small enough that phase is constant within each OFDM block; if either condition fails, the corresponding formula is incomplete or approximate.

Editorial extensions

If this is right

  • For a fixed OFDM frame and constellation, adding pilots does not change the total expected integrated sidelobe power of the DP-AF—it redistributes sidelobes, lowering some delay-Doppler regions at the cost of raising others.
  • The zero-Doppler sidelobe level of the DP-AF is (κ−1)(N−L), independent of pilot pattern or pilot symbols, so only the number of pilots and the constellation kurtosis govern that cut.
  • In the off-Doppler region q≠0, the DP-AF sidelobes are governed by a pilot-pattern term plus a residual, so choosing pilot positions and pilot symbols is a direct design degree of freedom for shaping the delay-Doppler response.
  • Under the small-Doppler FST-AF approximation, no pilot pattern or pilot symbol choice can shape the sidelobes; only the pilot count matters, so pattern-dependent sensing improvements should be sought in the exact DP-AF domain.
  • Comb-type pilot patterns produce periodic peaks and troughs along delay and Doppler, while block-type patterns create clustered high- and low-sidelobe regions, offering qualitatively different trade-offs in practice.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The DP-AF formula supplies a ready-made cost function for pilot design: one could optimize the indicator vector w and pilot symbols p to push sidelobes into delay-Doppler regions that least interfere with known targets or clutter, something the paper demonstrates qualitatively but does not optimize.
  • Because the off-Doppler DP-AF term is an ambiguity of the pilot sequence itself, familiar complementary-sequence or ambiguity-shaping ideas could be imported to design pilot sets that null or suppress specific sidelobe ridges; the paper stops short of testing such designs.
  • The FST-AF's complete independence from pilot pattern is a consequence of the small-Doppler block-constant approximation; a natural extension is to ask where pattern dependence re-emerges as Doppler grows, by evaluating the exact DP-AF of the full MN-length signal rather than the block-wise approximation.
  • The stated DP-AF formula relies on the symmetry assumption E(|s|^2 s)=E(|s|^2 s*)=0, which fails for BPSK and asymmetric constellations; extending the derivation to those cases would require carrying the extra fourth-order cross moments instead of dropping them.
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Editorial analysis

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Desk editor's note, referee report, and a circularity audit.

Referee Report

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Summary. This manuscript studies OFDM signals composed of deterministic unit-modulus pilots and random data symbols, and derives closed-form expressions for the mean squared discrete periodic ambiguity function (DP-AF) and fast-slow-time ambiguity function (FST-AF). Proposition 1 (Eq. (21)) gives a piecewise expression for E[|A_DP(k,q)|^2]: N^2+(κ−1)(N−L) at the mainlobe, (κ−1)(N−L) on the zero-Doppler sidelobe, and a pilot-pattern/pilot-symbol-dependent term f_I(k,q) elsewhere. Proposition 2 (Eq. (26)) gives E[|A_FST(k,q)|^2] = L+(MN−L)κ + M^2N^2δ_{k,0}δ_{q,0} − MN, depending only on the number of pilots. The formulas are validated with 1000-trial Monte Carlo simulations using 16-QAM and ZC pilots, and the paper discusses implications for pilot design in communication-centric ISAC systems.

Significance. If correct, the paper is a useful and nontrivial extension of the random-waveform ambiguity analysis in [8] to practical hybrid pilot-data frames. The derivations are self-contained, and the results are parameter-free in the sense that they depend only on the known constellation kurtosis κ, the pilot count L, and the given pilot pattern/symbols; no fitting is involved. The L=0 and L=N limits are consistent, and the DP-AF/FST-AF distinction is practically relevant for ISAC pilot design. The explicit assumptions—Eq. (2) for the third-order data moments and the small-Doppler approximation before Eq. (14)—appropriately scope the claims. The paper deserves publication after minor presentation fixes.

minor comments (5)
  1. [Section II-A, Eq. (2)] The additional assumption E(|s_c|^2 s_c)=E(|s_c|^2 s_c^*)=0 is stated to hold for 'most symmetric constellations'. Please state the precise symmetry condition, or at least verify it explicitly for the constellations used (16-QAM and M-PSK with M≥4). This will prevent misapplication to asymmetric constellations that satisfy Assumption 1 but not Eq. (2).
  2. [Fig. 3 caption] The caption reads 'N=64, M=20, L=16M'. This is ambiguous: is L=320 total pilots (16 per OFDM symbol), or should it be L=16 with M=20? Please clarify the exact pilot count used in the FST-AF simulation.
  3. [Appendix B, Eq. (42)] The transition from the g/h sum to the cosine sum is terse. Adding a one-line identity, e.g., Σ_{n,l} e^{j2πk(n−l)/N}=N^2δ_{k,0} and similarly for the M-dimension, would improve readability and make the M^2N^2δ_{k,0}δ_{q,0} term transparent.
  4. [Section IV-A] The phrase 'binary valued' for the no-pilot DP-AF could be made precise: the theoretical mean squared sidelobe equals N for q≠0 and (κ−1)N for q=0, k≠0. This would help the reader map the qualitative description to Eqs. (21)–(23).
  5. [References] Reference [8] is cited as an arXiv preprint. If a peer-reviewed version is now available, please update the citation.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: DP-AF/FST-AF expectations are derived from explicit statistical assumptions; self-citations to [8] are for definitions and are not load-bearing.

full rationale

The central formulas (Proposition 1, Eqs. 21-22; Proposition 2, Eq. 26) are derived in the appendices from the signal model and stated moment assumptions, not assumed as inputs. No parameter is fitted to make the formulas match simulation: kappa is a known constellation statistic (Eq. 3), L and the pilot pattern/symbols are given inputs, and the ZC root in the numerical examples is fixed (u=1) rather than optimized against the target. The pilot-pattern dependence of the DP-AF appears as an explicit pilot-only term |p^H F_N D_{N,q} J_{N,k} F_N^H p|^2 in f_I(k,q), which is a consequence of the derivation rather than a restatement. The FST-AF result follows from |F_N X|^2 = |S|^2 and unit-modulus pilots, so only the pilot count L enters. Reliance on the authors' prior work [8] for DP-AF/FST-AF definitions, the channel model, and ESL/EISL metrics is transparent: these are the analytical framework, not the predicted quantities. The additional moment assumption E(|s_c|^2 s_c)=E(|s_c|^2 s_c^*)=0 (Eq. 2) is explicitly stated and scopes the result; it is a limitation for asymmetric or 1D constellations, not a circular step. The small-Doppler approximation behind FST-AF is also acknowledged in the model. No step reduces to its own input by construction.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The central formulas depend only on stated statistical assumptions and system dimensions (N, M, L, kappa); no constants are fitted to data and no new physical entities are introduced. The ZC root and pilot patterns used in simulations are illustrative choices, not fitted parameters.

assumptions (6)
  • domain assumption Random communication symbols have unit power, zero mean, and zero pseudo-variance (Eq. 1).
    Used throughout Appendix A to evaluate expectations of products of data symbols.
  • domain assumption Third-order moments vanish: E(|s_c|^2 s_c)=E(|s_c|^2 s_c^*)=0 (Eq. 2).
    Required for the pilot-data cross terms to drop in the DP-AF proof; holds for symmetric constellations but not universally.
  • domain assumption Pilots are unit-modulus and equal-power to data symbols.
    Stated before Eq. (5); makes |p_i|^2=1, used in the simplification to Proposition 1 and in the zero-Doppler pilot-term cancellation.
  • domain assumption Communication symbols are i.i.d. across subcarriers and slow-time slots.
    Assumed in the signal models in Eqs. (6) and (7); factorizes fourth-order expectations.
  • domain assumption Small-Doppler approximation: phase shift is invariant across each block of N fast-time samples (Eq. 14).
    Defines the FST-AF framework and limits the applicability of Proposition 2 to low Doppler.
  • standard math Volume identity for the DP-AF: sum_{k,q} |A_DP(k,q)|^2 = N ||x||_2^4 (Eq. 24).
    Taken from [8] and used to derive EISL_DP without summing the individual ESL formulas.

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Pith. "Pith review of Ambiguity Function Analysis of Pilot-Embedded Random OFDM Signals." pith.science (2026). https://pith.science/paper/2HB7KMXT

@misc{pith2026260717663,
  author       = {Pith},
  title        = {Pith review of: Ambiguity Function Analysis of Pilot-Embedded Random OFDM Signals},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2HB7KMXT}},
  note         = {Machine review of arXiv:2607.17663}
}
read the original abstract

This paper investigates the statistical ambiguity functions (AFs) of orthogonal frequency division multiplexing (OFDM) waveforms that incorporate deterministic unit-modulus pilot symbols and random data payloads for integrated sensing and communication (ISAC). We derive analytical expressions for the mean squared discrete periodic ambiguity function (DP-AF) and fast-slow-time ambiguity function (FST-AF) of such pilot-embedded OFDM signals. Our analysis demonstrates that, under a fixed signal length and constellation scheme, the mean squared DP-AF depends jointly on the pilot patterns, pilot symbols and number of pilots, while the mean squared FST-AF relies only on the number of pilots. Numerical simulations closely match the theoretical expressions. Furthermore, in numerical results, we show that different pilot patterns correspond to DP-AF with distinct characteristics, offering relevant considerations for pilot design in communication-centric ISAC systems.

Figures

Figures reproduced from arXiv: 2607.17663 by the authors.

Figure 1
Figure 1. Two different types of pilot patterns. (a) Without Pilot, Theoretical (b) Without Pilot, Simulated (c) Block-type Pilot, Theoretical (d) Block-type Pilot, Simulated (e) Comb-type Pilot, Theoretical (f) Comb-type Pilot, Simulated [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. The mean squared DP-AF of OFDM signals with/without pilots, where N = 64, L = 16. ZC sequences are adopted as pilots, with block-type and comb-type pattern. A. Effect of Pilots on the DP-AF of 1D OFDM signals For the configuration of pilot signals, we adopt Zadoff–Chu (ZC) sequence [10], which is widely used as pilot symbols in OFDM communication systems due to their favorable autocorrelation properties. The ZC sequ… view at source ↗
Figure 3
Figure 3. The mean squared FST-AF of OFDM signals with/without pilots, where N = 64, M = 20, L = 16M. According to (13), the mean squared DP-AF is given by E [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

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Pith tools

Reviewed August 1, 2026 · model on record in the stance chip above.