Derivation of CRB and Refined SINR Expressions for OTFS-RSMA LEO ISAC Systems

Bruno Felipe Costa; Taufik Abr\~ao

REVIEW 3 major objections 4 minor 9 references

Derivation of CRB and Refined SINR Expressions for OTFS-RSMA LEO ISAC Systems

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

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read This paper derives closed-form CRB formulas for delay and Doppler, and refined SINR expressions, for OTFS-RSMA LEO satellite integrated sensing and communication.

desk verdict A clean but narrowly-scoped supplement whose celebrated 'exact' CRBs silently assume the target reflectivity is known, which makes the delay/Doppler bounds optimistic for monostatic sensing. read the letter →

arxiv 2506.02771 v1 pith:2KGIO4NB submitted 2025-06-03 eess.SP

Bruno Felipe Costa , Taufik Abrão This is my paper · ORCID

classification eess.SP

keywords OTFSRSMALEOsatelliteISACCramér-Raobounddelay-DopplerestimationimperfectCSISICFisherinformationmatrix

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

This supplement establishes closed-form Cramér-Rao bounds for delay and Doppler estimation in an OTFS-RSMA downlink used for LEO satellite integrated sensing and communication, under a model where the echo gain varies with delay. It also derives refined signal-to-interference-plus-noise ratio expressions for common and private streams when channel state information and successive interference cancellation are imperfect. If the derivations are correct, system designers can evaluate the sensing accuracy floor and communication reliability of the framework directly from the transmitted OTFS delay-Doppler symbols and the channel-estimation error statistics, without Monte Carlo simulation. The central mathematical object is the $2\times2$ Fisher information matrix whose inversion yields the exact CRB formulas, combined with LMMSE filter outputs that feed the SINR denominators.

What carries the argument

The load-bearing object is the $2\times2$ Fisher information matrix assembled from derivative vectors of the delay-Doppler mean signal $\mu[l,k]$ with respect to the sensing parameters. The paper defines $d_\nu$ as the Doppler derivative, splits the delay derivative as $d_\tau=d_{\rm gain}+d_{{\rm phase},\tau}$ with $d_{\rm gain}=-2\mu/\tau_T$, and condenses all DD-grid sums into six scalars: $S_n$, $S_i$, $P_\mu$, $C_{\mu\tau}$, $C_{\mu\nu}$, and $C_{\tau\nu}$. The CRB matrix is $(I^{\rm exact})^{-1}$, whose diagonal yields the delay and Doppler bounds; for the communication leg, the machinery is the LMMSE filter designed from the estimated channel $\hat H_k=H_k+E_k$, turning ICSI into an effective noise term and ISIC into a residual-interference term.

What would settle it

Compute $I_{\tau\tau}$ from a numerically or experimentally measured delay-gain curve $\alpha(\tau_T)$ for a representative LEO monostatic echo and compare it with Eq. (20) evaluated under the assumed derivative; a disagreement beyond numerical or measurement tolerance would show that the variable-gain CRB formulas are not exact for that scenario.

Watch

Extended reading notes

Core claim

The paper claims that, for the monostatic sensing leg of the OTFS-RSMA downlink with a delay-dependent echo gain $\alpha(\tau_T)$, the exact Cramér-Rao bounds on delay and Doppler estimation variance are the diagonal entries of the inverse of the $2\times2$ Fisher information matrix, written explicitly as Eqs. (23) and (24). The derivation splits the delay derivative into a gain part $d_{\rm gain}=-2\mu/\tau_T$ and a phase part $d_{{\rm phase},\tau}$, then assembles the FIM from the standard complex-Gaussian formula $[I]_{ij}=(2/\sigma^2)\,{\rm Re}\{(d_i)^H d_j\}$. For communication, the paper claims that the common-stream and private-stream SINRs are given by Eqs. (26) and (28), where the estimated channel $\hat H_k$ is used for desired-signal power and filter design, imperfect CSI contributes an effective noise term $\sigma_e^2\|w\|^2 P_{\rm tot}$, and imperfect SIC contributes the residual term $\Theta_k|w_{p,k}\hat H_k P_c|^2$.

Load-bearing premise

The derivation rests on the assumed gain-slope identity $\frac{d\alpha(\tau_T)}{d\tau_T}=-\frac{2\alpha(\tau_T)}{\tau_T}$; if the physical echo gain changes with delay differently, Eqs. (23) and (24) are not the exact CRB for that physical channel.

Editorial extensions

If this is right

Equations (23) and (24) give computable lower bounds on delay and Doppler estimation variance for any OTFS delay-Doppler symbol set $X[n,i]$, expressed through the six aggregate sums $S_n$, $S_i$, $P_\mu$, $C_{\mu\tau}$, $C_{\mu\nu}$, and $C_{\tau\nu}$.
The refined SINR expressions (26) and (28) make common- and private-stream rates depend explicitly on the channel-estimation error variance $\sigma_e^2$ and the SIC imperfection $\Theta_k$, so performance predictions for the OTFS-RSMA downlink can be corrected for both impairments.
Because the FIM off-diagonal element $I_{\tau\nu}$ couples delay and Doppler, the formulas quantify how joint estimation degrades each individual bound relative to a decoupled measurement.
The CRB formulas are exact with respect to the stated variable-gain model, so they provide a benchmark against which any practical delay/Doppler estimator for this OTFS-RSMA setup can be compared.

Reading between the lines

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

An implication the authors leave implicit is that their gain-slope identity corresponds to a power-law dependence $\alpha(\tau_T)\propto\tau_T^{-2}$; replacing $-2$ by a general exponent would generalize Eqs. (23) and (24) in the same derivation, a testable extension for other path-loss models.
The CRB is derived for a deterministic and known $\alpha$; if the echo gain were random or uncertain, the Fisher information would need an extra prior term, and the stated bounds would be optimistic.
A pragmatic validation would be a Monte Carlo comparison of the empirical mean-squared error of a maximum-likelihood delay/Doppler estimator on the OTFS-RSMA echo model against Eqs. (23) and (24); systematic excess would signal misspecification of the gain model rather than estimator suboptimality.

Share X Bluesky LinkedIn Reddit HN

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

Let me give you the short version: this is a supplementary derivation document for the authors' accepted SPAWC paper, not a self-contained research contribution. The CRB part is a textbook application of FIM calculus to a specific signal model, and the SINR part is a repackaging of known approximations. What the paper does well is lay out the steps clearly—definitions, derivatives, FIM elements, and final expressions are easy to follow. The algebra in (19)–(24) is internally consistent under the stated model.

The soft spots are substantive. First, the paper calls (23) and (24) the exact CRBs for delay and Doppler, but the 2x2 FIM is built by differentiating only with respect to tau_T and nu_T, holding the complex reflectivity beta_T fixed. In a monostatic sensing problem beta_T is normally an unknown nuisance. Unless the authors explicitly assume beta_T is known to the receiver—a physically odd assumption they never state—the marginal CRBs should come from the inverse of a 4x4 FIM over [tau, nu, beta_R, beta_I]. The off-diagonal blocks don't vanish, so the 2x2 inverse is optimistic. The M=N=1 limiting case in the stress-test note makes this concrete: two real observations, four real unknowns, singular FIM, infinite CRB; (23) and (24) return finite values. That's a load-bearing flaw, not a stylistic quibble.

Second, the gain derivative d_alpha/d_tau = -2 alpha/tau_T is asserted in Eq. (5) without physical derivation or citation. It appears in the delay derivatives and therefore in I_tau_tau and I_tau_nu. If the true gain slope is different, the 'exact' bounds are not exact for the modeled system. Third, there is no numerical validation anywhere—no simulation, no Monte Carlo check, nothing to confirm these closed forms against finite-sample performance. Given the paper claims 'exact,' that omission is hard to ignore.

The SINR part is weaker but proportionate. Equations (26) and (28) are effectively model definitions: the noise-enhancement term sigma_e^2 ||w||^2 P_tot is borrowed from the listed references, not derived. The ISIC term Theta_k scales residual interference but the text does not show how Theta_k relates to SIC implementation. That is fine as an engineering approximation, but the section is titled 'Derivation,' which oversells it.

Who gets value from this? A reader actively working on OTFS-RSMA LEO ISAC who wants to reproduce the main paper's formulas and adapt them. For that purpose the CRB steps are useful as a template, though the unknown-reflectivity issue will carry over to any adaptation. The citation pattern is honest—it cites the prior work it builds on, and the self-citation to [1] is appropriate for a supplement.

If I were refereeing this as a standalone, I would ask for the unknown-beta treatment before accepting. The fix is conceptually straightforward: either state the known-reflectivity assumption explicitly and justify it, or derive the 4x4 FIM and give the marginal bounds. The gain derivative also needs a physical argument or an explicit 'for example, alpha ∝ 1/tau' assumption. No simulations are strictly required for a derivation supplement, but one simple check would catch sign errors.

Recommendation: engage with it, but conditionally. The paper deserves a serious referee because the CRB issue is fixable and the derivations are clear enough to verify. My own verdict would be 'reject in current form, consider after revision.'

Referee Report

3 major / 4 minor

Summary. This supplement to an accepted SPAWC paper derives two sets of analytical results for an OTFS-RSMA LEO ISAC system: (i) Cramér-Rao bounds (CRB) for delay and Doppler estimation under a variable echo-gain model, obtained by constructing a 2x2 Fisher information matrix (FIM) from the Gaussian observation model and inverting it (Eqs. (6)-(24)); and (ii) refined SINR expressions for common and private streams under imperfect CSI and imperfect SIC, given by (26) and (28), respectively. The CRB derivation is algebraically explicit, while the SINR section assembles expressions from an estimated-channel desired-signal term, residual-interference terms, and an effective noise term proportional to the channel-estimation error variance.

Significance. If the claimed results were fully valid, the closed-form CRBs in (23)-(24) would provide useful lower bounds for OTFS-based ISAC sensing with delay-dependent gain, and the SINR expressions would extend RSMA performance analysis to imperfect CSI and SIC. The manuscript is transparent in exposing its modeling choices, and the chain from the mean model (1) to the FIM elements (19)-(21) is traceable as algebra. However, the central claim of 'exact' CRBs is undermined by an unstated known-reflectivity assumption, the gain-derivative model is asserted without physical justification, and the SINR expressions are effectively definitions rather than derived results. No numerical or simulation validation is provided, which limits the practical trustworthiness of the expressions as predictions.

major comments (3)

[Section III.D, Eqs. (23)-(24)] The paper calls (23) and (24) exact CRBs for tau_T and nu_T, but they are the diagonal elements of the inverse of a 2x2 FIM that treats the complex target reflectivity beta_T as a known constant. In the monostatic sensing model of Eq. (1), beta_T is a parameter of the mean, and if it is unknown the delay/Doppler CRBs must be obtained from the inverse of the full 4x4 FIM for [tau_T, nu_T, beta_R, beta_I]. The off-diagonal blocks of that full FIM do not vanish, so the diagonal entries of the 2x2 inverse are not the marginal CRBs and are optimistic. A concrete limiting case makes this internal inconsistency explicit: for M=N=1 and X[0,0]=1, the observation is a single complex sample with four real unknowns, the full FIM is singular, and the delay/Doppler CRB is unbounded, whereas (23) and (24) return finite values. The manuscript nowhere states that beta_T is assumed known, so the 'exact' label is not justified for the stated model.
[Section III.B.2, Eq. (5)] The gain derivative d(alpha)/d(tau_T) = -2 alpha(tau_T)/tau_T is asserted without physical derivation or reference to an underlying propagation or scattering model. This derivative is load-bearing: it enters I_tau_tau and I_tau_nu through d_gain (defined after Eq. (5)), C_mu_tau in Eq. (16), and the final CRBs in (23) and (24). If the true gain slope differs from the assumed functional form, the expressions are not exact for the physical system. The authors should either derive this relation from a concrete model (e.g., free-space path loss with delay-dependent range and RCS) or explicitly present it as an assumed model rather than as a consequence of the variable-gain framework.
[Section IV.B, Eqs. (26) and (28)] The 'refined SINR expressions' are not derived from the LMMSE filtering statistics; they are assembled by writing the desired-signal power as |w H_hat P|^2 and adding an ad hoc effective noise term sigma_e^2 ||w||^2 P_tot (Eqs. (25) and (27)) and a residual common-interference term Theta_k |w H_hat P_c|^2. The paper cites prior work for these approximations, but does not show how the filtered estimation error depends on the channel-error matrix E_k, nor why the ICSI contribution collapses to a single additive noise-enhancement term with coefficient one. As written, (26) and (28) are model definitions rather than derived expressions. To claim a derivation, the manuscript should either carry out the expectation over E_k in the SINR definition or clearly state that these are proposed approximations requiring calibration.

minor comments (4)

[Abstract] The abstract contains a duplicated period: 'main discussion..' should be 'main discussion.'.
[Section II] The description of the LMMSE filter vectors w_c,k and w_p,k is ambiguous: '1xNdd row vectors (or Ndd x 1 column vectors, adjusting subsequent math)' should be resolved to a single consistent convention, since the norms and inner products in Section IV depend on the orientation.
[Eq. (5)] The ellipses '(...)' in the phase term make the derivative with respect to tau_T difficult to follow; writing the phase explicitly would improve readability and verifiability.
[Eqs. (13)-(18)] The definitions of S_n, S_i, C_tau_nu, C_mu_tau, C_mu_nu, and P_mu use index ranges that are not fully consistent with the DD-domain indexing in Eq. (2); for example, the sums over l and k appear interchanged relative to the SFFT definition, which should be checked for consistency.

Circularity Check

0 steps flagged · score 2.0 of 10

No material circularity: the CRB is a parameter-free application of standard FIM inversion and the SINR expressions are explicit model definitions; the only self-citation is introductory and not load-bearing.

full rationale

The CRB derivation (Sections III.A-III.D) starts from the stated echo model (1) and (3), differentiates with respect to tau_T and nu_T, forms the Gaussian FIM via the standard formula (6), and algebraically inverts it in (10)-(12). No estimated or fitted quantity is fed back into the bound, and the resulting expressions (23)-(24) are not equivalent to any assumed output by construction. The derivative d alpha/d tau_T = -2 alpha(tau_T)/tau_T in Eq. (5) is an explicit premise, not a derived prediction; it may be physically unsupported, but that is a modeling or robustness concern, not circularity. The refined SINR expressions (26) and (28) are obtained by inserting assumed models for imperfect CSI and imperfect SIC into the definition of SINR; they are model definitions rather than empirical predictions, so there is no fitted-input-renamed-as-prediction pattern. The self-citation [1] is used only to place the supplement in the context of the main OTFS-RSMA framework and is not invoked to justify the FIM algebra or the SINR approximations. The potential omission of the unknown reflectivity beta_T as a nuisance parameter in the FIM is a correctness issue about whether (23)-(24) are the true marginal CRBs, not a circularity issue. Overall, the derivations are self-contained given their stated assumptions.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

The derivation relies on standard CRB and SINR machinery. The main introduced quantities are modeling assumptions such as the gain derivative, Theta_k, and the unit ICSI coefficient, which are not fitted to data. No new physical entities are postulated.

free parameters (3)

Gain derivative exponent = -2
Section III.B.2 sets d(alpha)/d(tau_T) = -2 alpha(tau_T)/tau_T without physical derivation; the exact CRB expressions depend on this choice.
SIC imperfection factor Theta_k = Theta_k in [0,1]
Section IV.A.1 scales residual common-stream interference by Theta_k; value is system-specific and not estimated here.
ICSI noise enhancement coefficient = 1
Section IV.A.2 assumes the effective noise variance is exactly sigma_e^2 ||w||^2 P_tot with unit coefficient, without derivation.

assumptions (4)

standard math The SFFT transforms TF-domain signal into DD-domain signal as in Eq (2).
Used to obtain the mean received signal mu[l,k] in Eq (3); relies on standard OTFS transform definitions from [4].
standard math For complex Gaussian observations with parameter-independent covariance, the FIM is I_ij = (2/sigma^2) Re{(d_mu/d_theta_i)^H d_mu/d_theta_j}.
Used in Eq (6); standard result from estimation theory [2].
ad hoc to paper The echo gain alpha(tau_T) satisfies d(alpha)/d(tau_T) = -2 alpha(tau_T)/tau_T.
Introduced in Section III.B.2 without derivation; determines I_tau_tau and I_tau_nu.
domain assumption LMMSE filters are designed using the estimated channel H_hat_k, and total ICSI distortion equals sigma_e^2 ||w||^2 P_tot.
Stated in Section IV.A.2 with citations [7]-[9]; not derived in this document, and the SINR expressions depend on this approximation.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Derivation of CRB and Refined SINR Expressions for OTFS-RSMA LEO ISAC Systems." pith.science (2026). https://pith.science/paper/2KGIO4NB

@misc{pith2026250602771,
  author       = {Pith},
  title        = {Pith review of: Derivation of CRB and Refined SINR Expressions for OTFS-RSMA LEO ISAC Systems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2KGIO4NB}},
  note         = {Machine review of arXiv:2506.02771}
}

read the original abstract

This document provides detailed step-by-step derivations for the Cram\'er-Rao Bounds (CRB) for sensing parameters and the refined Signal-to-Interference-plus-Noise Ratio (SINR) expressions under imperfect Channel State Information (CSI) and imperfect Successive Interference Cancellation (SIC) for the Orthogonal Time Frequency Space (OTFS) Rate-Splitting Multiple Access (RSMA) framework presented in our main work "An Integrated OTFS-RSMA Framework for LEO Satellite ISAC: Modeling, Metrics, and Potential". These derivations support the analytical expressions and models in the broad main discussion.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

9 extracted references · 8 canonical work pages

[1]

Refined metrics, sensing limits, and resource allocation in OTFS-RSMA LEO ISAC,

B. F. Costa and T. Abrão, “Refined metrics, sensing limits, and resource allocation in OTFS-RSMA LEO ISAC,” inProc. IEEE International Workshop on Signal Processing Advances in Wireless Communications (SPAWC), 2025, accepted for publication

work page 2025
[2]

S. M. Kay,Fundamentals of Statistical Signal Processing: Estimation Theory. Upper Saddle River, NJ: Prentice-Hall, 1993

1993
[3]

Cramer–rao lower bound for soop-r-based root-zone soil moisture remote sensing,

D. R. Boyd, A. C. Gurbuz, M. Kurum, J. L. Garrison, B. R. Nold, J. R. Piepmeier, M. Vega, and R. Bindlish, “Cramer–rao lower bound for soop-r-based root-zone soil moisture remote sensing,”IEEE Journal of Selected Topics in Applied Earth Observations and Remote Sensing, vol. 13, pp. 6101–6114, 2020

work page 2020
[4]

Orthogonal time frequency space modulation,

R. e. a. Hadani, “Orthogonal time frequency space modulation,” inProc. IEEE Wireless Commun. Netw. Conf. (WCNC), 2017, pp. 1–6

work page 2017
[5]

Optimal ber minimum precoder design for otfs-based isac systems,

J. Wu, W. Yuan, Z. Wei, J. Yan, and D. W. K. Ng, “Optimal ber minimum precoder design for otfs-based isac systems,” inICASSP 2024 - 2024 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), 2024, pp. 12 966–12 970

work page 2024
[6]

Uplink rate splitting multiple access with imperfect channel state information and interference cancellation,

F. Karim, N. H. Mahmood, A. S. de Sena, D. Kumar, B. Clerckx, and M. Latva-aho, “Uplink rate splitting multiple access with imperfect channel state information and interference cancellation,”IEEE Wireless Communications Letters, pp. 1–1, 2025

work page 2025
[7]

Robust design of rate-splitting multiple access with imperfect csi for cell-free mimo systems,

D. Yu, S.-H. Park, O. Simeone, and S. Shamai Shitz, “Robust design of rate-splitting multiple access with imperfect csi for cell-free mimo systems,” in 2022 IEEE International Conference on Communications Workshops (ICC Workshops), 2022, pp. 604–609

work page 2022
[8]

Low-complexity zf/mmse mimo-otfs receivers for high-speed vehicular communication,

P. Singh, A. Gupta, H. B. Mishra, and R. Budhiraja, “Low-complexity zf/mmse mimo-otfs receivers for high-speed vehicular communication,”IEEE Open Journal of the Communications Society, vol. 3, pp. 209–227, 2022

work page 2022

Show all 9 references

[9]

Rate-splitting multiple access for downlink multiuser MIMO: Precoder optimization and PHY-layer design,

A. Mishra, Y . Mao, O. Dizdar, and B. Clerckx, “Rate-splitting multiple access for downlink multiuser MIMO: Precoder optimization and PHY-layer design,”IEEE Trans. Commun., vol. 70, no. 2, pp. 874–890, 2022

2022

Pith tools

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

[1] [1]

Refined metrics, sensing limits, and resource allocation in OTFS-RSMA LEO ISAC,

B. F. Costa and T. Abrão, “Refined metrics, sensing limits, and resource allocation in OTFS-RSMA LEO ISAC,” inProc. IEEE International Workshop on Signal Processing Advances in Wireless Communications (SPAWC), 2025, accepted for publication

work page 2025

[2] [2]

S. M. Kay,Fundamentals of Statistical Signal Processing: Estimation Theory. Upper Saddle River, NJ: Prentice-Hall, 1993

1993

[3] [3]

Cramer–rao lower bound for soop-r-based root-zone soil moisture remote sensing,

D. R. Boyd, A. C. Gurbuz, M. Kurum, J. L. Garrison, B. R. Nold, J. R. Piepmeier, M. Vega, and R. Bindlish, “Cramer–rao lower bound for soop-r-based root-zone soil moisture remote sensing,”IEEE Journal of Selected Topics in Applied Earth Observations and Remote Sensing, vol. 13, pp. 6101–6114, 2020

work page 2020

[4] [4]

Orthogonal time frequency space modulation,

R. e. a. Hadani, “Orthogonal time frequency space modulation,” inProc. IEEE Wireless Commun. Netw. Conf. (WCNC), 2017, pp. 1–6

work page 2017

[5] [5]

Optimal ber minimum precoder design for otfs-based isac systems,

J. Wu, W. Yuan, Z. Wei, J. Yan, and D. W. K. Ng, “Optimal ber minimum precoder design for otfs-based isac systems,” inICASSP 2024 - 2024 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), 2024, pp. 12 966–12 970

work page 2024

[6] [6]

Uplink rate splitting multiple access with imperfect channel state information and interference cancellation,

F. Karim, N. H. Mahmood, A. S. de Sena, D. Kumar, B. Clerckx, and M. Latva-aho, “Uplink rate splitting multiple access with imperfect channel state information and interference cancellation,”IEEE Wireless Communications Letters, pp. 1–1, 2025

work page 2025

[7] [7]

Robust design of rate-splitting multiple access with imperfect csi for cell-free mimo systems,

D. Yu, S.-H. Park, O. Simeone, and S. Shamai Shitz, “Robust design of rate-splitting multiple access with imperfect csi for cell-free mimo systems,” in 2022 IEEE International Conference on Communications Workshops (ICC Workshops), 2022, pp. 604–609

work page 2022

[8] [8]

Low-complexity zf/mmse mimo-otfs receivers for high-speed vehicular communication,

P. Singh, A. Gupta, H. B. Mishra, and R. Budhiraja, “Low-complexity zf/mmse mimo-otfs receivers for high-speed vehicular communication,”IEEE Open Journal of the Communications Society, vol. 3, pp. 209–227, 2022

work page 2022