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arxiv: 2604.04674 · v1 · submitted 2026-04-06 · 💻 cs.IT · math.IT

Identification for Colored Gaussian Channels

Mohammad Javad Salariseddigh This is my paper

Pith reviewed 2026-05-10 19:14 UTC · model grok-4.3

classification 💻 cs.IT math.IT
keywords identification capacitycolored Gaussian channelsinter-symbol interferencesuper-exponential growthpeak power constraintMahalanobis decodercorrelated noise
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The pith

Colored Gaussian channels with polynomially bounded noise spectra support super-exponential identification codebooks even when ISI memory grows sub-linearly.

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

The paper analyzes identification over discrete-time Gaussian channels with correlated noise and inter-symbol interference under deterministic encoding and a peak power limit. It proves that when the noise covariance singular values scale between n to the power of minus mu and n to the mu over 2 for mu below one half, and ISI memory length scales as n to the kappa with kappa plus mu below one half, the largest codebook still grows like 2 to the power of R n log n. Bounds on the resulting identification capacity are obtained via a Mahalanobis-distance decoder that exploits the known noise correlations. This scaling matters because identification only requires the receiver to verify a specific message rather than recover any possible one.

Core claim

Even when the ISI memory length grows sub-linearly with n as n^kappa where kappa is in [0,1/2) and kappa plus mu is in [0,1/2), the codebook size for identification continues to exhibit super-exponential growth of order 2 to the power of (n log n) R, with R the associated rate, for channels whose noise covariance matrix has polynomially bounded singular values in the interval [n^{-mu}, n^{mu/2}]. Bounds on the identification capacity are characterized using the Mahalanobis-distance decoder induced by the colored noise statistics.

What carries the argument

The polynomially bounded singular value spectrum of the noise covariance matrix together with the Mahalanobis-distance decoder that accounts for noise correlations to achieve the super-exponential scaling under sub-linear ISI.

If this is right

  • Identification remains feasible at super-exponential rates for channels whose interference memory grows slower than the square root of block length.
  • The achievable rates can be expressed explicitly in terms of the spectrum exponent mu and the memory exponent kappa.
  • Deterministic encoders suffice to attain the super-exponential scaling without requiring randomization.
  • The Mahalanobis decoder yields concrete upper and lower bounds that tighten when mu and kappa approach zero.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The result suggests identification may tolerate moderate channel memory better than standard message transmission in non-stationary noise settings.
  • Similar scaling could be tested for average-power constraints or randomized encoding to see whether the exponent R increases.
  • The framework might extend to other linear channels whose effective noise operator has controlled singular-value growth.

Load-bearing premise

The noise covariance matrix must have singular values bounded polynomially between n to the minus mu and n to the mu over 2 for mu less than one half, together with deterministic encoding under peak power.

What would settle it

Compute the largest codebook size explicitly for a covariance matrix whose singular values violate the polynomial bound (for example by growing faster than n to the mu over 2) and check whether the growth rate drops below super-exponential for some valid kappa and mu.

read the original abstract

We study the identification capacity of discrete-time Gaussian channels impaired by correlated noise and inter-symbol interference (ISI). Our analysis is formulated for deterministic encoding functions subject to a peak power constraint and colored noise whose covariance matrix features a polynomially bounded singular value spectrum, i.e., $\sim [n^{-\mu} , n^{\mu/2}]$ where $n$ is the codeword length and $\mu \in [0,1/2)$ is the spectrum rate. A central result establishes that, even when the ISI memory length grows sub-linearly with $n,$ i.e., $\sim n^{\kappa}$ where $\kappa \in [0,1/2)$ and $\kappa + \mu \in [0,1/2),$ the codebook size continues to exhibit super-exponential growth in $n$, i.e., $\sim 2^{(n \log n)R},$ with $R$ representing the associated coding rate. Moreover, by employing the well-known Mahalanobis-distance decoder induced by colored Gaussian noise statistics, we characterize bounds on the identification capacity, with the resulting bounds parameterized by $\kappa$ and $\mu.$

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 0 minor

Summary. The manuscript studies the identification capacity of discrete-time Gaussian channels with colored noise (covariance matrix Σ having singular values polynomially bounded in [n^{-μ}, n^{μ/2}]) and inter-symbol interference (ISI matrix H of bandwidth ~n^κ). Under deterministic encoding and a peak power constraint, it claims that when κ ∈ [0,1/2) and κ + μ < 1/2 the codebook size still achieves super-exponential growth ~2^{(n log n)R}. Bounds on the identification capacity are derived using a Mahalanobis-distance decoder induced by the colored Gaussian noise statistics, with the resulting expressions parameterized by κ and μ.

Significance. If the central claims hold with rigorous error analysis, the result would establish robustness of super-exponential identification rates to sublinearly growing ISI and polynomially correlated noise, extending prior Gaussian identification results. The explicit parameterization of the bounds by the exponents κ and μ is a strength, as is the use of standard Mahalanobis decoding induced by the noise covariance.

major comments (1)
  1. Abstract: the decoder is described only as the 'Mahalanobis-distance decoder induced by colored Gaussian noise statistics.' For the model y = Hx + z the correct ML metric is (y − Hx_m)^T Σ^{-1}(y − Hx_m). If the proof employs a metric depending only on Σ (omitting the cross term −2y^T Σ^{-1} H x_m), the decision regions do not center on the actual means when κ > 0; the polynomial bounds on Σ alone then do not guarantee the required separation for 2^{n log n R} codewords. This is load-bearing for the claimed growth rate under the stated conditions on κ and μ.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address the major comment below.

read point-by-point responses

  1. Referee: Abstract: the decoder is described only as the 'Mahalanobis-distance decoder induced by colored Gaussian noise statistics.' For the model y = Hx + z the correct ML metric is (y − Hx_m)^T Σ^{-1}(y − Hx_m). If the proof employs a metric depending only on Σ (omitting the cross term −2y^T Σ^{-1} H x_m), the decision regions do not center on the actual means when κ > 0; the polynomial bounds on Σ alone then do not guarantee the required separation for 2^{n log n R} codewords. This is load-bearing for the claimed growth rate under the stated conditions on κ and μ.

    Authors: We thank the referee for identifying this ambiguity. The decoder used throughout the manuscript is the maximum-likelihood decoder for the model y = Hx + z, which minimizes the full quadratic form (y − H x_m)^T Σ^{-1} (y − H x_m). This metric includes both the cross term −2 y^T Σ^{-1} H x_m and the quadratic term in x_m. The abstract description is abbreviated and refers to the use of the noise covariance Σ in the Mahalanobis distance; it does not imply omission of the signal-dependent terms. The error analysis in the proofs accounts for the ISI matrix H under the stated conditions on κ and μ (specifically κ + μ < 1/2), which ensure that the effective separation between codeword means remains sufficient for the super-exponential growth rate. We will revise the abstract and the decoder definition section to state the complete ML metric explicitly and add a brief remark on how the bounds incorporate H. revision: yes

Circularity Check

0 steps flagged

No circularity; derivation uses standard identification bounds and noise-induced decoder

full rationale

The paper derives super-exponential identification rates under sublinear ISI (n^κ) and polynomially bounded noise spectrum using deterministic encoding, peak power, and the Mahalanobis decoder induced by Σ. The conditions κ + μ < 1/2 control error probabilities via standard random coding arguments for identification capacity; no equation or step reduces a claimed prediction to a fitted parameter, self-defined quantity, or load-bearing self-citation by construction. The result is self-contained against external information-theoretic benchmarks and does not rename known patterns or smuggle ansatzes via prior work.

Axiom & Free-Parameter Ledger

3 free parameters · 1 axioms · 0 invented entities

The result rests on standard asymptotic information theory assumptions for identification capacity and the specific polynomial spectrum bound on the noise covariance; no new entities are postulated.

free parameters (3)
  • R
    Associated coding rate in the super-exponential growth expression; appears as a parameter in the capacity bounds.
  • κ
    Exponent controlling sub-linear ISI memory growth; restricts the regime where the result holds.
  • μ
    Spectrum rate parameter for the noise covariance singular values; defines the allowed noise correlation strength.
axioms (1)
  • domain assumption Standard properties of Gaussian channels and identification capacity under peak power constraints hold for the colored noise model.
    Invoked implicitly when applying Mahalanobis decoder and claiming super-exponential growth.

pith-pipeline@v0.9.0 · 5493 in / 1313 out tokens · 47963 ms · 2026-05-10T19:14:46.476453+00:00 · methodology

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Reference graph

Works this paper leans on

34 extracted references · 34 canonical work pages · 1 internal anchor

  1. [1]

    Identification is Easier Than Decoding,

    J. J ´aJ´a, “Identification is Easier Than Decoding,” inAnnual Symposium on Foundations of Computer Science, 1985, pp. 43–50

    work page 1985

  2. [2]

    Identification via Channels,

    R. Ahlswede and G. Dueck, “Identification via Channels,”IEEE Transaction Information Theory, vol. 35, no. 1, pp. 15–29, 1989

    work page 1989

  3. [3]

    General Theory of Information Transfer: Updated,

    R. Ahlswede, “General Theory of Information Transfer: Updated,”Discrete Applied Mathematics, vol. 156, no. 9, pp. 1348–1388, 2008

    work page 2008

  4. [4]

    A Mathematical Theory of Communication,

    C. E. Shannon, “A Mathematical Theory of Communication,”Bell System Technical Journal, vol. 27, no. 3, pp. 379–423, 1948

    work page 1948

  5. [5]

    Identification Without Randomization,

    R. Ahlswede and N. Cai, “Identification Without Randomization,”IEEE Transaction Information Theory, vol. 45, no. 7, pp. 2636–2642, 1999

    work page 1999

  6. [6]

    Deterministic Identification For Molecular Communications,

    M. J. Salariseddigh, “Deterministic Identification For Molecular Communications,” Ph.D. dissertation, Technical University of Munich, 2023. [Online]. Available: https://mediatum.ub.tum.de/?id=1743195

    work page 2023

  7. [7]

    Deterministic Identification Over Channels With Power Constraints,

    M. J. Salariseddigh, U. Pereg, H. Boche, and C. Deppe, “Deterministic Identification Over Channels With Power Constraints,” IEEE Transaction Information Theory, vol. 68, no. 1, pp. 1–24, 2022

    work page 2022

  8. [8]

    Deterministic Identification Codes for Fading Channels,

    I. V orobyev, C. Deppe, and H. Boche, “Deterministic Identification Codes for Fading Channels,”IEEE Transactions on Communications, pp. 1–1, 2025

    work page 2025

  9. [9]

    Deterministic Identification Over Channels Without CSI,

    Y . Li, X. Wang, H. Zhang, J. Wang, W. Tong, G. Yan, and Z. Ma, “Deterministic Identification Over Channels Without CSI,” in IEEE Information Theory Workshop, 2022, pp. 332–337

    work page 2022

  10. [10]

    Deterministic Identification For Molecular Communications Over The Poisson Channel,

    M. J. Salariseddigh, V . Jamali, U. Pereg, H. Boche, C. Deppe, and R. Schober, “Deterministic Identification For Molecular Communications Over The Poisson Channel,”IEEE Transactions on Molecular, Biological, and Multi-Scale Communications, vol. 9, no. 4, pp. 408–424, 2023

    work page 2023

  11. [11]

    Deterministic K-Identification For MC Poisson Channel With Inter-Symbol Interference,

    ——, “Deterministic K-Identification For MC Poisson Channel With Inter-Symbol Interference,”IEEE Open Journal of the Communications Society, pp. 1–1, 2024

    work page 2024

  12. [12]

    Identification over Affine Poisson Channels: Application to Molecular Mixtures Communication Systems,

    M. J. Salariseddigh, H. K ¨oppl, H. Boche, and V . Jamali, “Identification over Affine Poisson Channels: Application to Molecular Mixtures Communication Systems,” in2025 IEEE Information Theory Workshop, 2025, pp. 1–6

    work page 2025

  13. [13]

    Deterministic Identification For MC Binomial Channel,

    M. J. Salariseddigh, V . Jamali, H. Boche, C. Deppe, and R. Schober, “Deterministic Identification For MC Binomial Channel,” in IEEE International Symposium on Information Theory, 2023, pp. 448–453

    work page 2023

  14. [14]

    Deterministic K-Identification for Future Communication Networks: The Binary Symmetric Channel Results,

    M. J. Salariseddigh, O. Dabbabi, C. Deppe, and H. Boche, “Deterministic K-Identification for Future Communication Networks: The Binary Symmetric Channel Results,”Future Internet, vol. 16, no. 3, 2024. [Online]. Available: https://www.mdpi.com/1999-5903/16/3/78

    work page 2024

  15. [15]

    Codes for Identification via Channels: Tutorial for Communications Generalists,

    C. von Lengerke, J. A. Cabrera, M. Reisslein, and F. H. Fitzek, “Codes for Identification via Channels: Tutorial for Communications Generalists,”IEEE Communications Surveys & Tutorials, 2025

    work page 2025

  16. [16]

    Identification Codes via Prime Numbers,

    E. Zinoghli and M. J. Salariseddigh, “Identification Codes via Prime Numbers,”arXiv preprint arXiv:2408.12455, 2024. [Online]. Available: http://arxiv.org/abs/2408.12455

    work page arXiv 2024

  17. [17]

    Ahlswede, I

    A. Ahlswede, I. Alth ¨ofer, C. Deppe, and U. Tamm (Eds.),Identification and Other Probabilistic Models, Rudolf Ahlswede’s Lectures on Information Theory 6, 1st ed., ser. Foundations in Signal Processing, Communications and Networking. Springer Verlag, 2021, vol. 16

    work page 2021

  18. [18]

    J. G. Proakis and M. Salehi,Digital Communications. McGraw-hill New York, 2001, vol. 4

    work page 2001

  19. [19]

    Goldsmith,Wireless Communications

    A. Goldsmith,Wireless Communications. Cambridge university press, 2005

    work page 2005

  20. [20]

    R. G. Gallager,Information Theory and Reliable Communication. New York, NY , USA: John Wiley & Sons, Inc., 1968. 19

    work page 1968

  21. [21]

    Capacity and Information Rates of Discrete-Time Channels with Memory,

    W. Hirt, “Capacity and Information Rates of Discrete-Time Channels with Memory,” Ph.D. dissertation, ETH Zurich, 1988. [Online]. Available: https://www.research-collection.ethz.ch/server/api/core/bitstreams/7d140bc3-6d6b-4b97-9fc7-e1ced8f34c71/ content

    work page 1988

  22. [22]

    Capacity of the Discrete-Time Gaussian Channel with Intersymbol Interference,

    W. Hirt and J. L. Massey, “Capacity of the Discrete-Time Gaussian Channel with Intersymbol Interference,”IEEE Transactions on Information Theory, vol. 34, no. 3, pp. 38–38, 2002

    work page 2002

  23. [23]

    The Capacity Region of Broadcast Channels with Intersymbol Interference and Colored Gaussian Noise,

    A. J. Goldsmith and M. Effros, “The Capacity Region of Broadcast Channels with Intersymbol Interference and Colored Gaussian Noise,”IEEE Transactions on Information Theory, vol. 47, no. 1, pp. 219–240, 2002

    work page 2002

  24. [24]

    Gaussian Multiaccess Channels with ISI: Capacity Region and Multiuser Water-Filling,

    R. S. Cheng and S. Verd ´u, “Gaussian Multiaccess Channels with ISI: Capacity Region and Multiuser Water-Filling,”IEEE Transactions on Information Theory, vol. 39, no. 3, pp. 773–785, 1993

    work page 1993

  25. [25]

    On a Class of Time-Varying Gaussian ISI Channels,

    K. Moshksar, “On a Class of Time-Varying Gaussian ISI Channels,”IEEE Transactions on Information Theory, vol. 70, no. 2, pp. 1147–1166, 2024

    work page 2024

  26. [26]

    Identification for ISI Gaussian Channels

    M. J. Salariseddigh, “Identification for ISI Gaussian Channels,” 2026. [Online]. Available: https://arxiv.org/abs/2603.14246

    work page internal anchor Pith review Pith/arXiv arXiv 2026

  27. [27]

    On Concepts of Performance Parameters For Channels,

    R. Ahlswede, “On Concepts of Performance Parameters For Channels,” inGeneral Theory of Information Transfer and Combinatorics. Berlin, Heidelberg, Germany: Springer, 2006, pp. 639–663

    work page 2006

  28. [28]

    J. H. Conway and N. J. A. Sloane,Sphere Packings, Lattices and Groups. New York, NY , USA: Springer, 2013

    work page 2013

  29. [29]

    Feller,An Introduction to Probability Theory and Its Applications

    W. Feller,An Introduction to Probability Theory and Its Applications. John Wiley & Sons, 1966

    work page 1966

  30. [30]

    On the Generalized Distance in Statistics,

    P. C. Mahalanobis, “On the Generalized Distance in Statistics,”Sankhy ¯a: The Indian Journal of Statistics, Series A (2008-), vol. 80, pp. S1–S7, 2018

    work page 2008

  31. [31]

    Papoulis and S

    A. Papoulis and S. U. Pillai,Probability, Random Variables, and Stochastic Processes. Boston, MA, McGraw-Hill, 2002

    work page 2002

  32. [32]

    Identification over Affine Poisson Channels: Applications to Molecular Mixtures Communication Systems,

    M. J. Salariseddigh, H. K ¨oppl, H. Boche, and V . Jamali, “Identification over Affine Poisson Channels: Applications to Molecular Mixtures Communication Systems,”arXiv preprint arXiv:2410.11569, 2024. [Online]. Available: http: //arxiv.org/abs/2410.11569.pdf

    work page arXiv 2024

  33. [33]

    K. M. Hoffman and R. Kunze,Linear Algebra, 2nd ed. Englewood Cliffs, NJ: Prentice-Hall, 1971

    work page 1971

  34. [34]

    Deterministic Identification Over Fading Channels,

    M. J. Salariseddigh, U. Pereg, H. Boche, and C. Deppe, “Deterministic Identification Over Fading Channels,” inIEEE Information Theory Workshop, 2021, pp. 1–5

    work page 2021