Second-Order Characterization of Micro Doppler Radar Signatures of Drone Swarms

Alba Spliid Damkj{\ae}r; Anders Malthe Westerkam; Magnus {\O}rum Bastrup Poulsen; Rasmus Erik Villadsen; Troels Pedersen

REVIEW 3 major objections 4 minor 11 references

Second-Order Characterization of Micro Doppler Radar Signatures of Drone Swarms

T0 review · 3 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read The radar return of a drone swarm reduces to a closed-form series whose terms are squared Bessel functions, linking blade length, rotor speed, blade count, and swarm size directly to the micro-Doppler spectrum.

desk verdict Correct central ACF with a few fixable equation slips; worth a serious referee. read the letter →

arxiv 2506.00497 v3 pith:AHVDN575 submitted 2025-05-31 eess.SP

Anders Malthe Westerkam , Alba Spliid Damkj{ae}r , Rasmus Erik Villadsen , Magnus {O}rum Bastrup Poulsen , Troels Pedersen This is my paper

classification eess.SP

keywords micro-DopplerdroneswarmsradarautocorrelationfunctionpowerspectraldensityBesselfunctionsrotorbladessecond-orderstatistics

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

The paper establishes a tractable analytic model for the second-order statistics of a radar signal reflected by a swarm of identical rotor drones. It derives the autocorrelation function and power spectral density of the micro-Doppler signature as an infinite series that can be truncated at a predictable term. If the model holds, a radar engineer can read key drone parameters directly from the shape and spacing of spectral lines without heavy simulation. The derivation is built on deliberately crude scatterer assumptions, so the value lies in interpretability rather than full physical fidelity.

What carries the argument

The load-bearing identity is the Jacobi-Anger expansion applied to the phase of each spinning blade, combined with a Fourier-coefficient evaluation of $J_0(l\cos\theta)$ that yields squared Bessel functions $J_n^2(l/2)$ for even $n$ and zero for odd $n$. A second key algebraic step shows that the sum over blade pairs collapses to $N_b^2$ only when $n$ is a multiple of $N_b$, and vanishes otherwise, which is why the final series contains only harmonics of the blade-pass frequency. A third ingredient is the Bessel-function first-zero approximation $z \approx \nu$, which gives the truncation rule $n > l/(2N_b)$. Together these identities convert the stochastic radar-return model into an interpretable, closed-form spectral signature.

What would settle it

Measure the ACF and PSD of a single rotor drone with known $L$, $\lambda$, $N_b$, and $\bar\omega$ in an anechoic chamber or controlled outdoor range. If the measured PSD shows spectral lines at frequencies that are not integer multiples of $N_b\bar\omega$, or if the main-lobe width does not approximately follow $4.8/(l\bar\omega)$, then the superposition-of-point-scatterers assumption fails in a way that invalidates the derived closed form.

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Extended reading notes

Core claim

The central result is a closed-form expression for the autocorrelation function of the radar return from a swarm of identical drones, Eq. (9): $$R_s(\tau) = |g|^2 N_d N_r $N_b^{2}$ \left[ $J_0^{2}$\left(\tfrac{l}{2}\right) + 2\sum_{n=1}^{\infty} J_{N_b n}^2\left(\tfrac{l}{2}\right) \cos(N_b n\bar\omega\tau) $e^{{-\sigma_\omega^2 \tau^2 n^2 N_b^2/2}}$ \right],$$ with $l = 8\pi L/\lambda$. This formula ties the ACF to blade length $L$, wavelength $\lambda$, mean rotor speed $\bar\omega$, rotor-speed variance $\sigma_\omega^2$, blade count $N_b$, rotors per drone $N_r$, and number of drones $N_d$. By Fourier transform, the PSD becomes a sum of Gaussian kernels centered at integer multiples of the blade-pass frequency $N_b\bar\omega$. In the zero-variance limit, the Gaussian kernels become Dirac deltas and the model reproduces earlier discrete-line results with line spacing $N_b\bar\omega$ and total bandwidth $B = l\bar\omega$. The series truncates naturally because the squared Bessel coefficients are negligible once the order exceeds $l/(2N_b)$, so the infinite series is practically finite.

Load-bearing premise

The load-bearing premise is that every rotor blade behaves as an ideal, identical point scatterer and that the total radar return is a clean superposition of these scatterers, with no blade flashes, shadowing, drone-body reflections, or amplitude modulation.

Editorial extensions

If this is right

The ACF and PSD can be evaluated with a finite number of terms, making the model practical for real-time radar processing and parameter sweeps.
The main-lobe width of the ACF is approximately $\Delta\tau_{\text{null-to-null}} \approx 4.8/(l\bar\omega)$, so the autocorrelation width shrinks as blade length, rotor speed, or radar frequency increases.
The PSD consists of spectral lines spaced by $N_b\bar\omega$, meaning that a measurement of the line spacing directly reveals the blade-pass frequency, while the line envelope encodes blade length and wavelength.
The number of drones and rotors appears only as a multiplicative amplitude factor, so the normalized shape of the micro-Doppler signature is independent of swarm size as long as drones are identical and independent.
The constant term $J_0^2(l/2) N_d N_r N_b^2$ in the ACF represents the DC component of the PSD, which in the deterministic-speed limit carries a nonzero power contribution that could be observable.
If the rotor-speed variance is nonzero, the Gaussian kernel width grows linearly with harmonic index $n$, so the higher spectral lines in the PSD broaden more than the lower ones.

Reading between the lines

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

A direct inversion strategy suggested by the model is to estimate the ratios of consecutive spectral-line amplitudes: those ratios depend only on $l$, $N_b$, and the Bessel-function envelope, so a measured PSD could be used to estimate blade length relative to wavelength without knowing rotor speed.
The model omits blade flashes, which in practice are strong and intermittent. A natural extension would be to add a periodic impulsive component; if such a component remains visible in the ACF, the character of the spectral lines would change in a way testable against real radar data.
Macro-Doppler from swarm motion could be included by convolving the derived micro-Doppler PSD with a Doppler-spread kernel, which would preserve the harmonic structure while adding translational broadening.
The predicted truncation at $n > l/(2N_b)$ suggests that for a given radar wavelength and blade geometry, there is a finite number of resolvable micro-Doppler lines; this could be used as a design rule for radar systems that must distinguish drones from birds.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

The final expression for the ACF, Eq. (9), holds up under independent derivation; the paper is a genuine extension of Martin and Mulgrew to stochastic rotor speeds and swarms. But the path from Eq. (6) to Eq. (7) is sloppy enough that a referee would bounce it back for revision.

The new content is the closed-form series with the Gaussian damping factor from random rotor speed, and the aggregation over N_d drones, N_r rotors, and N_b blades. The truncation rule in (11) is handy. The special-case check against the deterministic literature is the right way to validate the algebra. I re-derived Eq. (9) using the addition theorem for J_0(l sin θ) and got exactly their result, including the 2 in front of the harmonic sum and the exponent -σ^2 τ^2 k^2 N_b^2 /2. So the central result is right.

The soft spots are presentational. Eq. (7) is missing a factor of 2 in the sum; at N_b=1, τ=0 it gives (1+J_0^2)/2 instead of 1. That's a typo, not a conceptual error, but it has to be fixed. The Gaussian exponent in Eq. (6) is -σ^2 τ^2 n^2/8, while Eq. (7) has -σ^2 τ^2 n^2/2. These are consistent if you remember that only even n survive and you relabel n→2n, but the paper never says that, so as written it reads as a factor-of-four error. That needs a clarifying sentence or a changed definition.

The physical model is deliberately crude—ideal point scatterers, no blade flashes, shadowing, or multipath—and the paper says so in the Introduction. The numerical check uses the same model, so it confirms the math, not the physics. That's a stated limitation, not a hidden one. The conclusion says real validation is needed. So it's fine as a first-principles contribution, as long as the claims are not oversold.

Who is this for? People working on drone detection and micro-Doppler signature modeling. It's a useful analytic tool for predicting how swarm parameters shape the ACF/PSD, and it gives a tractable starting point for detection algorithms. It doesn't resolve a long-open question, but it's a solid incremental result.

Recommendation: This deserves a serious referee. The typos in Eq. (7) and the unexplained index change between Eq. (6) and (7) are minor but mandatory fixes. I'd accept it for review with a fine-tooth comb on the derivation.

Referee Report

3 major / 4 minor

Summary. The paper derives a stochastic model for the radar return from a swarm of identical rotor drones. Each rotor blade is treated as an ideal point scatterer with common amplitude and phase; rotor angular velocities are i.i.d. Gaussian, while initial and projection phases are uniform. The authors derive the autocorrelation function of the return (Eq. (9)) as a series of squared Bessel functions at harmonics of the blade passage frequency, modulated by a Gaussian factor controlled by the rotor-speed variance, and the corresponding power spectral density (Eq. (17)) as a sum of Gaussian kernels. They also give a truncation limit, a closed form for deterministic rotor speed, approximate main-lobe width, and numerical estimates obtained from simulations of the same signal model.

Significance. If the derivation is correct, the paper provides an unusually interpretable connection between drone parameters and micro-Doppler second-order statistics: harmonic spacing, main-lobe width, bandwidth, and coefficient decay are all explicit functions of blade length, wavelength, rotor speed, speed variance, and the numbers of blades, rotors, and drones. The central formula is concise, truncatable, and involves no fitted parameters. The authors are also explicit about the idealized assumptions, listing blade flashes, shadowing, multipath, tilt, and body scattering as ignored effects. The numerical validation, however, uses signals generated from the same model, so it confirms the algebra rather than the fidelity of the model to real radar returns; this is a limitation for practical applicability but not an internal inconsistency.

major comments (3)

[Section III, Eq. (7)] Equation (7) is missing the factor 2 in the non-DC term and the factor |g|^2 in the series term. For N_b=1 and sigma_omega=0, Eq. (7) gives R(0)=|g|^2 N_d N_r (1+J_0^2(l/2))/2, whereas the signal power is E|y(t)|^2=|g|^2 N_d N_r; Eq. (9) is consistent with the latter. The missing factor 2 comes from combining the positive and negative n terms in Eq. (6) after the reindexing n->2k, and this step should be shown explicitly.
[Appendix, Eq. (22)] The integral identity in Eq. (22) is incorrect as stated. For even n=2k, the identity should be \int_0^{2\pi} J_0(l cos theta) cos(n theta) d\theta = 2\pi (-1)^k J_k^2(l/2), not J_n^2(l/2). As printed, substituting Eq. (22) into Eq. (20) cannot produce the J_{N_b n}^2(l/2) coefficients that appear in Eq. (9). The appendix must be corrected and the index halving n=2k stated explicitly.
[Section IV and Introduction] The numerical validation in Section IV compares the derived ACF and PSD with estimates obtained from signals generated according to the same Eq. (2). This is a self-consistency check of the algebra, not a test of the physical modeling assumptions. Since the Introduction explicitly lists ignored effects such as blade flashes and shadowing, and the spectrogram in Fig. 4 is described as resembling real spectrograms 'without the blade flashes', the paper should either add a comparison with real radar measurements or clearly state in the abstract and conclusion that the model's practical validity under these effects remains untested.

minor comments (4)

[Section III, Eq. (5)] The definition of phi(t,tau) and the sign in the sine argument appear to contain typographical errors: the term should involve b+b' rather than b-b', and the overall sign of the exponent is inconsistent with the preceding identity for cos A - cos B. Since phi is uniform modulo 2pi, these do not affect the final result, but the displayed equation should be corrected.
[Section III, Eqs. (6)-(9)] The transition from Eq. (6) to Eq. (7) silently changes the summation index from n to n/2 and combines positive and negative frequencies. Because of this, the Gaussian exponent appears to change from exp(-sigma^2 tau^2 n^2/8) to exp(-sigma^2 tau^2 n^2/2), which is correct only after the reindexing. The authors should state this reindexing explicitly to avoid the appearance of a factor-of-four inconsistency.
[Section IV, Eq. (19)] The estimator \hat R(tau) averages over N realizations but does not explicitly average over time t. Since the process is wide-sense stationary, a time average would be natural, and the manuscript should clarify whether a single time instant or a time average was used, and how edge effects from the 4001-sample window were handled.
[Throughout] There are several typographical issues, including 'U A Vs' in the Introduction, the missing |g|^2 in Eq. (7), and the caption of Fig. 2 ('The ACF with parameter setting, in Fig. 4'), which should be rewritten. The statement in Section III that the deterministic-speed limit recovers conclusions from [10] would also benefit from citing the specific equations in [10].

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: Eq. (9) is derived by calculation from an explicit stochastic point-scatterer model, with no fitted inputs and no load-bearing self-citation.

full rationale

This paper's derivation chain is self-contained and non-circular. Starting from Eq. (2), an explicit stochastic superposition model for identical point-scatterer rotor blades, the authors derive the single-rotor ACF via the Jacobi-Anger expansion (Eqs. (5)-(6)), average over the Gaussian rotor-speed distribution, and obtain the closed-form swarm ACF in Eq. (9); the PSD follows by Fourier transformation (Eqs. (15)-(17)). No parameter is fitted to data, and no target quantity is inserted as an input: Eq. (9) is obtained by calculation from the stated model assumptions, not by assuming the result. The numerical validation in Section IV (Eq. (19)) regenerates signals from the same model, so it checks algebraic self-consistency rather than external physical fidelity, but that is not circularity; the paper openly lists the idealizations (blade flashes, shadowing, multipath, etc.) that would need to be added for fidelity to real measurements. Reference [10] is an external prior work and is used only for a limiting-case comparison, not to justify the central derivation. The only notable defect is an apparent factor-of-two typo in Eq. (7) relative to Eq. (9), visible at N_b=1, tau=0, but this is an internal algebraic inconsistency, not a circular step.

Assumptions & free parameters 2 free parameters · 5 assumptions · 0 invented entities

No parameters are fitted to data. The Gaussian rotor-speed mean and variance are model inputs. The core assumptions are point-scatterer blades, uniform random phases, independent Gaussian rotor speeds, far-field/2D geometry, and standard Bessel/Jacobi-Anger identities. No invented entities are introduced.

free parameters (2)

mean rotor angular velocity \bar{\omega}
Model input, not fitted; chosen to represent a typical rotor speed.
rotor-speed variance \sigma_\omega^2
Model input, not fitted; represents the spread of rotor speeds across the swarm.

assumptions (5)

domain assumption Each rotor blade is an ideal point scatterer with identical amplitude and phase shift.
Section II: 'Assuming ideal back scatter conditions, we then reduce all drones to individual sets of spinning point scatterers.'
domain assumption Initial blade angles and rotor phases are independent and uniformly distributed on [0, 2π).
Section II, after Eq. (1), states {θ_d,r} and {γ_d,r} are i.i.d. uniform.
domain assumption Rotor angular velocities are independent Gaussian variables with mean \bar{\omega} and variance \sigma_\omega^2.
Section II, after Eq. (2), models {ω_d,r} as i.i.d. N(\bar{\omega}, σ_ω^2).
domain assumption The swarm is in the far field, with range much larger than swarm diameter, and a 2D scene approximation applies.
Section II: 'R ≫ D' and 'R is assumed much larger than the flying altitude ... allowing a 2D approximation.'
standard math Jacobi-Anger expansion and the integral identity for J0(l cos θ) Fourier coefficients are valid.
Used in Eq. (6) and Appendix equations (20)-(22).

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Cite this review

Pith. "Pith review of Second-Order Characterization of Micro Doppler Radar Signatures of Drone Swarms." pith.science (2026). https://pith.science/paper/AHVDN575

@misc{pith2026250600497,
  author       = {Pith},
  title        = {Pith review of: Second-Order Characterization of Micro Doppler Radar Signatures of Drone Swarms},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/AHVDN575}},
  note         = {Machine review of arXiv:2506.00497}
}

read the original abstract

We investigate the second-order characteristics of the radar return signal from a swarm of rotor drones. We consider the case of a swarm of identical drones, with each a number of rotors comprised of a number of rotor blades. By considering the orientation and speed of each rotor as stochastic variables, we derive expressions for the autocorrelation function (ACF) and power spectral density (PSD). The ACF and PSD are in the form of an infinite series with coefficients that drop to zero at a predictable limit. Thus in practical applications, the series may be truncated. As a special case, we show that for deterministic rotor speed, the ACF can be expressed in closed form. We further investigate how system parameters (Blade length, Rotor speed, number of blades, and number of drones) influence the derived expressions for the ACF and PSD.

Figures

Figures reproduced from arXiv: 2506.00497 by the authors.

**Figure 3.** The PSD for a drone modelled after the DJI Mavic. The vertical lines [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗

**Figure 4.** Spectrogram of the signal model with parameters, [PITH_FULL_IMAGE:figures/full_fig_p003_4.png] view at source ↗

Figures from the paper (1 more)

**Figure 5.** Figure 5: Value of the coefficients for the settings considered in this article. [PITH_FULL_IMAGE:figures/full_fig_p004_5.png]

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Reference graph

Works this paper leans on

11 extracted references · 11 canonical work pages

[1]

Uas air-risk assessment in and around airports,

P. Pothana, J. Joy, P. Snyder, and S. Vidhyadharan, “Uas air-risk assessment in and around airports,” in2023 Integrated Communication, Navigation and Surveillance Conference (ICNS). IEEE, Apr. 2023

work page 2023
[2]

Research on detection and attack strategy against swarm of drones,

P. Liu, “Research on detection and attack strategy against swarm of drones,” in2023 IEEE 11th Joint International Information Technology and Artificial Intelligence Conference (ITAIC). IEEE, Dec. 2023, pp. 1648–1653

work page 2023
[3]

Classification of bird and drone targets based on motion characteristics and random forest model using surveillance radar data,

J. Liu, Q. Y . Xu, and W. S. Chen, “Classification of bird and drone targets based on motion characteristics and random forest model using surveillance radar data,”IEEE Access, vol. 9, pp. 160 135–160 144, 2021

work page 2021
[4]

Simulation-based approach to classification of airborne drones,

L. Lehmann and J. Dall, “Simulation-based approach to classification of airborne drones,” in2020 IEEE Radar Conference (RadarConf20). IEEE, sep 2020, pp. 1–6

work page 2020
[5]

Drones classification by the use of a multifunctional radar and micro-doppler analysis,

M. Leonardi, G. Ligresti, and E. Piracci, “Drones classification by the use of a multifunctional radar and micro-doppler analysis,”Drones, vol. 6, no. 5, p. 124, may 2022

work page 2022
[6]

Classification of radar targets using features based on warped discrete fourier transform,

D. Bujakovi ´c, M. Andri ´c, B. Bond ˇzuli´c, and S. Simi ´c, “Classification of radar targets using features based on warped discrete fourier transform,” in2022 30th European Signal Processing Conference (EUSIPCO), 2022, pp. 1556–1560

work page 2022
[7]

Micro-doppler effect in radar: phenomenon, model, and simulation study,

V . Chen, F. Li, S.-S. Ho, and H. Wechsler, “Micro-doppler effect in radar: phenomenon, model, and simulation study,”IEEE Transactions on Aerospace and Electronic Systems, vol. 42, no. 1, pp. 2–21, Jan. 2006

work page 2006
[8]

Simulation of radar micro- doppler patterns for multi-propeller drones,

Y . Cai, O. Krasnov, and A. Yarovoy, “Simulation of radar micro- doppler patterns for multi-propeller drones,” in2019 International Radar Conference (RADAR). IEEE, Sep. 2019, pp. 1–5

work page 2019

Show all 11 references

[9]

Modelling micro-doppler signature of drone propellers in distributed isac,

H. C. A. Costa, S. J. Myint, C. Andrich, S. W. Giehl, C. Schneider, and R. S. Thom ¨a, “Modelling micro-doppler signature of drone propellers in distributed isac,” in2024 IEEE Radar Conference (RadarConf24). IEEE, May 2024, pp. 1–6

2024
[10]

Analysis of the theoretical radar return signal form aircraft propeller blades,

J. Martin and B. Mulgrew, “Analysis of the theoretical radar return signal form aircraft propeller blades,” inIEEE International Conference on Radar. IEEE, 1990, pp. 569–572

1990
[11]

Abramowitz and I

M. Abramowitz and I. A. Stegun,Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 10th ed. New York: Dover, 1964. [Online]. Available: https://personal.math.ubc.ca/ ∼cbm/ aands/abramowitz and stegun.pdf

1964

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [1]

Uas air-risk assessment in and around airports,

P. Pothana, J. Joy, P. Snyder, and S. Vidhyadharan, “Uas air-risk assessment in and around airports,” in2023 Integrated Communication, Navigation and Surveillance Conference (ICNS). IEEE, Apr. 2023

work page 2023

[2] [2]

Research on detection and attack strategy against swarm of drones,

P. Liu, “Research on detection and attack strategy against swarm of drones,” in2023 IEEE 11th Joint International Information Technology and Artificial Intelligence Conference (ITAIC). IEEE, Dec. 2023, pp. 1648–1653

work page 2023

[3] [3]

Classification of bird and drone targets based on motion characteristics and random forest model using surveillance radar data,

J. Liu, Q. Y . Xu, and W. S. Chen, “Classification of bird and drone targets based on motion characteristics and random forest model using surveillance radar data,”IEEE Access, vol. 9, pp. 160 135–160 144, 2021

work page 2021

[4] [4]

Simulation-based approach to classification of airborne drones,

L. Lehmann and J. Dall, “Simulation-based approach to classification of airborne drones,” in2020 IEEE Radar Conference (RadarConf20). IEEE, sep 2020, pp. 1–6

work page 2020

[5] [5]

Drones classification by the use of a multifunctional radar and micro-doppler analysis,

M. Leonardi, G. Ligresti, and E. Piracci, “Drones classification by the use of a multifunctional radar and micro-doppler analysis,”Drones, vol. 6, no. 5, p. 124, may 2022

work page 2022

[6] [6]

Classification of radar targets using features based on warped discrete fourier transform,

D. Bujakovi ´c, M. Andri ´c, B. Bond ˇzuli´c, and S. Simi ´c, “Classification of radar targets using features based on warped discrete fourier transform,” in2022 30th European Signal Processing Conference (EUSIPCO), 2022, pp. 1556–1560

work page 2022

[7] [7]

Micro-doppler effect in radar: phenomenon, model, and simulation study,

V . Chen, F. Li, S.-S. Ho, and H. Wechsler, “Micro-doppler effect in radar: phenomenon, model, and simulation study,”IEEE Transactions on Aerospace and Electronic Systems, vol. 42, no. 1, pp. 2–21, Jan. 2006

work page 2006

[8] [8]

Simulation of radar micro- doppler patterns for multi-propeller drones,

Y . Cai, O. Krasnov, and A. Yarovoy, “Simulation of radar micro- doppler patterns for multi-propeller drones,” in2019 International Radar Conference (RADAR). IEEE, Sep. 2019, pp. 1–5

work page 2019