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REVIEW 2 major objections 4 minor 19 references

Gauss-Ramanujan Functions: Constructions, Properties, and Applications in Communications and Signal Processing

T0 review · 2 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read The paper claims that a normalized difference of two Gaussian pulses yields a pulse family with near-orthogonality, tunable spectral nulls, constant-envelope GRSK modulation, and a mother wavelet tighter in time-frequency than Hermite.

desk verdict A careful paper whose headline GRSK/GRM claims are undone by the integral in Eq (77); the rest is mostly repackaged Gaussian-pulse mathematics. read the letter →

arxiv 2505.21691 v2 pith:HJ6UVIW2 submitted 2025-05-27 eess.SP

Sainath Bitragunta This is my paper · ORCID
classification eess.SP
keywords Gauss-RamanujanfunctionsRamanujansequencesGaussianpulsecontinuous-phasemodulationshiftkeyingwaveletsOTFSspectralnulls
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 paper sets out to show that one analytic pulse—the normalized difference of a Gaussian and its delayed copy, with weights taken from Ramanujan sequences—can serve as a building block for both communications waveforms and wavelet analysis. It claims that the first-order Gauss-Ramanujan function has a closed-form spectrum with tunable nulls, near-orthogonality between delayed copies, a Hilbert transform pair whose inner product is zero, a constant-envelope modulation called GRSK, and a valid mother wavelet. The paper also derives exact and approximate delay-averaged overlap formulas, linking the near-orthogonality condition to Gaussian overlap. If these claims are right, GRSK would give next-generation systems such as OTFS a delay-adaptive, constant-envelope waveform, and the wavelet would give signal analysts a tunable alternative to Hermite wavelets with tighter joint time-frequency containment.

What carries the argument

The load-bearing object is the first-order Gauss-Ramanujan function $GRI(t;T_0)=\frac{1}{\sqrt{2}}(e^{-\pi t^2}-e^{-\pi(t-T_0)^2})$, a weighted difference of a Gaussian pulse and its delayed copy. Its Fourier transform factors as $G(f)(1-e^{-j2\pi fT_0})/\sqrt{2}$, so every application follows from two identities: the inner product of a Gaussian and its delayed copy is $\frac{1}{\sqrt{2}}e^{-\pi T_0^2/2}$, giving a closed-form near-orthogonality condition, and the spectrum is $2e^{-\pi f^2}|\sin(\pi fT_0)|$, giving nulls at multiples of $1/T_0$. The same difference-of-Gaussians form, after normalization, makes the zero-mean mother wavelet; the Hilbert transform is expressed through Dawson functions, and the GRSK frequency pulse is the truncated finite version of the same difference.

What would settle it

Evaluate the definite integral in Eq. (77) at $t=T$ for representative choices of $T_0$ and $T$ (for example $T_0=1$, $T=5$). If the result is near zero rather than $1/2$, the phase trajectory in GRSK does not accumulate the promised modulation index, and the scheme as described cannot support constant-envelope continuous-phase modulation; a direct symbol-error-rate comparison with GMSK would then separate the pulse-shaping benefit from the invalid phase normalization.

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Extended reading notes

Core claim

The central discovery is the first-order Gauss-Ramanujan function $GRI(t;T_0)=\frac{1}{\sqrt{2}}(e^{-\pi t^2}-e^{-\pi(t-T_0)^2})$, a single closed-form pulse built by subtracting a delayed Gaussian from an undelayed one. Its Fourier transform is $\frac{1}{\sqrt{2}}e^{-\pi f^2}(1-e^{-j2\pi fT_0})$, so the magnitude spectrum is $2e^{-\pi f^2}|\sin(\pi fT_0)|$, giving deterministic spectral nulls at integer multiples of $1/T_0$. The paper further derives a closed-form Hilbert transform in terms of Dawson functions, reports that the function and its Hilbert transform have zero inner product, and normalizes the same difference-of-Gaussians form into a mother wavelet with time-frequency product $0.760$ versus $0.866$ for the first-order Hermite wavelet. From the same pulse it constructs the continuous-wave Gauss-Ramanujan modulation and the GRSK scheme, where the truncated Gauss-Ramanujan pulse serves as the frequency pulse in a continuous-phase modulation format.

Load-bearing premise

The load-bearing premise is that the truncated Gauss-Ramanujan frequency pulse integrates, after normalization, to a phase waveform $q_{GR}(t)=1/2$ over one symbol interval as claimed in Eq. (78); but since the pulse is a difference of two equal-area Gaussians, its total area is zero and the integral is near zero rather than $1/2$ unless the truncation or normalization changes that cancellation.

Editorial extensions

If this is right

  • If GRSK is valid, orthogonal time frequency space and other delay-Doppler systems gain a constant-envelope waveform whose pulse shape has a tunable delay parameter $T_0$ for spectral-null placement.
  • The closed-form overlap condition $T_0=\sqrt{-4\sigma^2\ln(\sqrt{2}\epsilon)}$ gives a direct way to choose symbol spacing for a target inter-symbol interference level, including values robust to delay jitter.
  • The first-order Gauss-Ramanujan wavelet can be used in time-frequency analysis, denoising, and feature extraction with a smaller uncertainty product (0.760) than the first-order Hermite wavelet (0.866).
  • GRM and GRSK both maintain constant envelope and continuous phase, so they are compatible with nonlinear power amplifiers and can be integrated into hybrid next-generation systems.

Reading between the lines

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

  • An unstated extension: because nulls occur at multiples of $1/T_0$, the delay parameter could be adapted in real time to notch occupied subbands, turning the pulse family into a dynamic spectrum-access tool; the paper does not simulate this adaptation.
  • The delay-jitter overlap formulas depend only on Gaussian overlap, so they transfer to any transmitted reference using Gaussian pulses, not just GRSK.
  • A direct way to test the wavelet advantage is to run denoising or pulse-detection benchmarks with the Gauss-Ramanujan wavelet against the Hermite wavelet; the paper gives autocorrelation functions but stops short of application-level validation.
  • The paper only analyzes the first-order function; higher-order GauRam functions using longer Ramanujan sequences could be tested for forming orthogonal pulse sets in multicarrier systems.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 4 minor

Summary. The paper constructs 'Gauss-Ramanujan' functions by combining a Gaussian pulse and its delayed versions with weights drawn from Ramanujan sequences, then analyzes their overlap, spectral, and Hilbert-transform properties. It applies these functions to three claimed contributions: a continuous-wave Gauss-Ramanujan modulation (GRM), a Gauss-Ramanujan shift keying (GRSK) constant-envelope CPM scheme, and a Gauss-Ramanujan mother wavelet with time-frequency containment compared against the Hermite wavelet. The paper also derives exact and approximate expressions for deterministic and stochastic GP-DGP overlap and reports near-orthogonality conditions.

Significance. The closed-form derivations for GP-DGP overlap (Results 1 and 2), spectral magnitude (Eq. 36), Hilbert transform (Eq. 51), and wavelet normalization (Eqs. 91-92) are largely correct and are genuine strengths of the manuscript. However, the GRSK scheme, one of the two headline communications applications, is invalid as specified because the phase-pulse integral is computed incorrectly and the claimed normalization qGR(T)=1/2 is false. The GRM modulation index is also wrong, and the proposed GRM envelope is not constant. These errors undermine the central claims of new modulation schemes for next-generation systems; the wavelet contribution may be salvageable, but the paper in its present form does not establish its main applications.

major comments (2)
  1. [§5.2.1, Eqs. (74)-(78)] The GRSK phase pulse is not normalized and Eq. (77) is incorrect. For the second Gaussian term, ∫0^t e^{-π(τ-T0)^2}dτ = (1/2)[erf(√π(t-T0)) + erf(√πT0)], not (1/2)erf(√π(t-T0)). Consequently qGR(T) = κ/(2√2)[erf(√πT) − erf(√π(T−T0)) − erf(√πT0)], which tends to −κ erf(√πT0)/(2√2) as T→∞ and is not 1/2. Thus Eq. (78) is contradicted by Eq. (77); the phase accumulation per symbol in the CPM signal (79) is unsupported, and no choice of the unspecified constant κ can make qGR(t)=1/2 on [0,T]. The GRSK scheme as defined is therefore not a valid constant-envelope continuous-phase modulation.
  2. [§5.1.1, Eqs. (59)-(63)] The modulation index derivation is incomplete. From φ(t)=−arctan(e^{π(2T0t−T0^2)}), the supremum of |φ(t)| is π/2 (approached as t→∞), not π/4; the evaluation at t=T0/2 gives only one point. Hence m_GR≈π/4 is wrong, and the GRM characterization as a π/4 modulation is incorrect. In addition, the GRM envelope in Eq. (53) is not constant because I^2(t)+Q^2(t)=1/2(e^{-2πt^2}+e^{-2π(t-T0)^2}) is time-varying, contradicting the section's constant-envelope motivation.
minor comments (4)
  1. [§4.1, remark iv] The reported group delay τg=πT0 is incorrect. For the branch ψ(f)=π/2−πfT0, τg=−dψ/dω=T0/2, and the phase changes by π at the zeros of |sin(πfT0)|, so the phase response is only piecewise affine, not globally affine.
  2. [Eq. (24)] The exponent should contain απδ^2 rather than βπδ^2 to be consistent with Eq. (13).
  3. [§5.2.3, Fig. 8] The title of §5.2.3 reads 'GMSK vs. GPSK pulses' but should be 'GRSK'; the caption of Fig. 8 describes 'pulse shapes' although the plot shows normalized PSD.
  4. [Eqs. (74) and (82)] The normalization constant κ in Eq. (74) is never specified, and Eq. (82) defines a different untruncated GRSK pulse for the PSD comparison; the relation between the CPM pulse and the PSD curve should be clarified.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found: all core results are explicit derivations from stated definitions; the GRSK Eq. (77)/(78) phase-normalization error is a correctness defect, not a circular reduction.

full rationale

The derivation chain is self-contained and none of the headline results reduces to its own inputs. GauRam functions are explicit constructions (Eqs. 27-32): a Gaussian, its delayed copy, and normalized Ramanujan-sequence weights. Every claimed property is derived by standard calculus from those definitions or numerically evaluated: the spectral nulls at f = m/T0 follow from Eq. (36); the near-orthogonality delay T0(sigma,epsilon) in Result 1 is an inversion of the exact inner product Eq. (5); the delay-averaged overlap in Result 2 is derived from the uniform-delay model, with constants alpha, beta, gamma taken from the externally published Q-function approximation of ref. [11] and validated against the exact expression (Fig. 2, 2.6% error at delta=0.09), not fitted to the target curve; the Hilbert transform Eq. (49) follows linearly from the standard Dawson-function result Eq. (38); the wavelet constant B in Eq. (92) is solved from the unit-energy condition; Table 1's uncertainty products are computed against the external Hermite benchmark. There are no self-citations (refs. [1]-[19] are all external), no imported uniqueness theorem, and no fitted parameter relabeled as a prediction. Because the construction is transparently a scaled difference of two Gaussians, novelty inflation is an attribution concern, not a hidden reduction. Per the review rule, I flag one omitted-support item: in Sec. 5.2.1, Eq. (77) drops the erf(sqrt(pi)*T0) lower-limit term when integrating the delayed Gaussian over [0,t], and Eq. (78) asserts the normalized phase waveform equals 1/2 without deriving it from Eq. (77); direct evaluation gives qGR(T) approximately -kappa*erf(sqrt(pi)*T0)/(2*sqrt(2)) approximately -0.354*kappa for T0=2.45, so the GRSK phase-accumulation claim is unsupported, and in Sec. 5.1.1 the modulation index mGR approximately pi/4 undercounts the true supremum of |phi(t)| = pi/2. These are internal mathematical inconsistencies (correctness risks), not circularity steps, so the circularity score is 0.

Assumptions & free parameters 4 free parameters · 6 assumptions · 0 invented entities

The central derivations rest on standard Gaussian and error-function identities and the Ramanujan orthogonality property cited from [10]. The stochastic overlap approximation relies on the external Q-function approximation of [11]. The GRSK application implicitly assumes a CPM normalization condition that the zero-area difference-of-Gaussians pulse cannot satisfy; this is a false ad hoc axiom. No new physical entities are introduced.

free parameters (4)
  • T0 = 1.8, 2.45 used in figures
    Delay spacing between Gaussian components; controls overlap, spectral null spacing, and wavelet scale. It is a user-selected design parameter, not derived.
  • alpha, beta, gamma = alpha=0.3842, beta=0.7640, gamma=0.6964
    Coefficients in the Q-function approximation used in Eq (13); the paper states they were chosen to minimize sum of squared errors. They come from an external fit, not derived here.
  • eta (time-bandwidth product) = varies in Fig. 8
    Pulse width parameter in the GRSK and GMSK PSD comparison; selected per plot, not estimated.
  • kappa = unspecified
    Normalization constant in the GRSK frequency pulse Eq (74) is never assigned; because the pulse has zero area, no normalization makes qGR(infinity) equal to 1/2.
assumptions (6)
  • standard math Ramanujan sum orthogonality property Eq (2) from [10]
    Used to motivate the RSE weights; not proved in the paper.
  • standard math Standard Gaussian integrals and Q-function identities
    Used throughout for overlap, energy, and stochastic averaging.
  • domain assumption Exponential Q-function approximation from [11] is valid
    Underpins the closed-form mean-overlap approximation Eq (13); it is approximate, not exact.
  • ad hoc to paper CPM frequency pulse must have total area 1/2 and q(t) must be the integral of the normalized pulse
    The GRSK section assumes g_GR yields qGR = 1/2 over [0,T] in Eq (78), but g_GR integrates to zero, so the axiom is false.
  • standard math Hilbert transform of the Gaussian equals (2/sqrt(pi)) D+(sqrt(pi) t) from [14]
    Used in Results 3 and 4 to obtain the Hilbert transform of the first-order GauRam function.
  • domain assumption Uniform delay jitter model tau0 ~ U[T0-delta, T0+delta]
    Stochastic overlap analysis assumes this distribution; it is physically plausible but not validated experimentally.

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Cite this review

Pith. "Pith review of Gauss-Ramanujan Functions: Constructions, Properties, and Applications in Communications and Signal Processing." pith.science (2026). https://pith.science/paper/HJ6UVIW2

@misc{pith2026250521691,
  author       = {Pith},
  title        = {Pith review of: Gauss-Ramanujan Functions: Constructions, Properties, and Applications in Communications and Signal Processing},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HJ6UVIW2}},
  note         = {Machine review of arXiv:2505.21691}
}
read the original abstract

In this article, I construct a new set of functions based on Ramanujan sequences (RSEs), Gaussian pulse (GP), and its delayed Gaussian pulse (DGP). The motivation for this construction is based on the special properties of RSEs, GP, and DGP. First, I present a procedure for constructing Gauss-Ramanujan (GauRam) functions using selected RSEs. I develop an insightful analysis for deterministic and stochastic overlap between GP and DGP. Specifically, I present exact and closed form approximation expressions for delay-averaged GP and DGP overlap and then evaluate them numerically. Later, I derive and analyze the mathematical (spectral) properties of selected GauRam functions. I extend the analysis by analyzing the Hilbert transform of the first-order GauRam function and validating orthogonality and its usefulness in analytic signal representations. Furthermore, I present insightful applications of these functions in communications and signal processing. Specifically, I present the continuous-wave Gauss-Ramanujan modulation (GRM) scheme, Gauss-Ramanujan Shift Keying (GRSK) scheme, and Gauss-Ramanujan wavelets and their analysis and comparisons with benchmarking. The desirable properties of these novel modulation schemes and wavelets enable their use in next-generation hybrid and energy-efficient communication systems and signal processing.

Figures

Figures reproduced from arXiv: 2505.21691 by the authors.

Figure 1
Figure 1. An illustration of 6σ separation between GP and DGP. • The result on T0 (8) is significant in applications such as multi-carrier communications where near￾orthogonality between time-shifted pulses is desired. 2.2.1 Stochastic Modeling of Delay in DGP Imperfections in physical realization could cause an offset δ in T0. Let τ0 represent a uniformly distributed random variable. Suppose τ0 ∼ U[T0 − δ, T0 + δ], 0 ≤ δ << … view at source ↗
Figure 2
Figure 2. Mean overlap as a function of δ: Exact versus closed form approximation. Remarks on the accuracy of approximation [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Illustration of GRI (t; T0) (T0 = 1.8). The first-order GauRam function GRII (t; T0) is plotted in 4. -5 -4 -3 -2 -1 0 1 2 3 4 5 t -0.5 0 0.5 1 GRII(t; T 0 ) [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figures from the paper (6 more)
Figure 4
Figure 4. Figure 4: An illustration of GRII (t; T0) (T0 = 1.8). Remarks: • Using the above approach, one can construct higher-order (third-order, fourth-order, and above) Gau￾Ram functions. For instance, the third-order GauRam function is given by GRIII (t; T0) = 1 √ 2 g(t) − 1 √ 2 g(t − …
Figure 5
Figure 5. Figure 5: Magnitude spectrum of GRI (t; T0) (T0 = 1.8). Remarks: i) I see that |GRI (f; T0)| has zero crossing at f = m T0 , m ∈ Z. ii) The phase response ψ(f) is given by ψ(f) = −πfT0 + π 2 . (37) 9 [PITH_FULL_IMAGE:figures/full_fig_p009_5.png]
Figure 6
Figure 6. Figure 6: GRdI (t; T0) and its Hilbert transform (T0 = 2.45.) [PITH_FULL_IMAGE:figures/full_fig_p012_6.png]
Figure 7
Figure 7. Figure 7: Modulated waveform GRMI (t; fc, T0) (T0 = 1.8 ns, fc = 1 GHz). Remarks: i) Using the standard CTFT pairs, I find that the frequency-domain representation of GRMI (t; fc, T0) is given by GRMI (f; fc, T0) = 1 2 √ 2  e −π(f−fc) 2 + e −π(f+fc) 2  − 1 2 √ 2 e −j(2πfT0− π …
Figure 8
Figure 8. Figure 8: GMSK vs. GRSK for different bandwidth-time product [PITH_FULL_IMAGE:figures/full_fig_p017_8.png]
Figure 9
Figure 9. Figure 9: Comparison of Hermite and Gauss-Ramanujan wavele [PITH_FULL_IMAGE:figures/full_fig_p019_9.png]

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Works this paper leans on

19 extracted references · 18 canonical work pages

  1. [1]

    Haykin, Digital communication systems , John Wiley & Sons, 2014

    S. Haykin, Digital communication systems , John Wiley & Sons, 2014

    work page 2014

  2. [2]

    A. V. Oppenheim and R. W. Schafer, Discrete-Time Signal Processing , 2nd ed. Prentice Hall, 1999

    work page 1999

  3. [3]

    Atlas and P

    L. Atlas and P. Duhamel, ”Recent developments in the core of digit al signal processing,” IEEE Signal Proc. Mag., vol. 16, pp. 16–31, Jan. 1999

    work page 1999

  4. [4]

    Orthogonal time frequency space modulation,

    R. Hadani et al. , “Orthogonal time frequency space modulation,” Proc. IEEE WCNC , 2017, pp. 1–6

    work page 2017

  5. [5]

    Interferen ce cancellation and iterative detection for orthogonal time frequency space modulation,

    P. Raviteja, K. T. Phan, Y. Hong, and E. Viterbo, “Interferen ce cancellation and iterative detection for orthogonal time frequency space modulation,” IEEE Trans. Wireless Commun. , vol. 17, no. 10, pp. 6501–6515, Oct. 2018

    work page 2018

  6. [6]

    G. D. Surabhi, R. M. Augustine and A. Chockalingam, ”Peak-to-a verage power ratio of OTFS modu- lation,” IEEE Communications Letters , vol. 23, no. 6, pp. 999–1002, June 2019

    work page 2019

  7. [7]

    Aldababsa, S

    M. Aldababsa, S. ¨Ozyurt, G. K. Kurt and O. Kucur, ”A Survey on orthogonal time fr equency space modulation,” IEEE Open Journal of the Communications Society , vol. 5, pp. 4483–4518, 2024

    work page 2024

  8. [8]

    Mallat, a Wavelet Tour of Signal Processing

    S. Mallat, a Wavelet Tour of Signal Processing . Academic Press, 1999

    work page 1999

Show all 19 references
  1. [9]

    Structured compressed sensing : from theory to applications,

    M. F. Duarte and Y. C. Eldar, “Structured compressed sensing : from theory to applications,” IEEE Trans. Signal Process., vol. 59, no. 9, pp. 4053–4085, Sep. 2011

    2011

  2. [10]

    P. P. Vaidyanathan and S. Tenneti, ”Properties of Ramanujan filter banks,” in Proc. EUSIPCO, Dec. 2015

    2015

  3. [11]

    L´ opez-Ben ´ ıtez and F

    M. L´ opez-Ben ´ ıtez and F. Casadevall, ”Versatile, accurate,and analytically tractable approximation for the Gaussian Q-function,” IEEE Trans. Commun. , vol. 59, pp. 917–922, Apr. 2011

    2011

  4. [12]

    R. N. Bracewell, The Fourier Transform and Its Applications . 3rd ed. McGraw-Hill, 1999. 21

    1999

  5. [13]

    Boashash, Time-frequency signal analysis and processing|a comprehensive reference

    B. Boashash, Time-frequency signal analysis and processing|a comprehensive reference. Elsevier Science, 2003

    2003

  6. [14]

    F. W. King, *Hilbert Transforms*, Encyclopedia of Mathematics and its Applications. Cambridge, U.K.: Cambridge University Press, 2009

    2009

  7. [15]

    Abramowitz and I

    M. Abramowitz and I. Stegun, Handbook of mathematical functions with formulas, graphs, and mathe- matical tables. Dover, 9 ed., 1972

    1972

  8. [16]

    J. G. Proakis, Digital Communications . McGraw-Hill, 2008

    2008

  9. [17]

    A. J. Goldsmith, Wireless Communications. Cambridge University Press, 2005

    2005

  10. [18]

    G. Matz, D. Schafhuber, K. Grochenig, M. Hartmann and F. Hla watsch, ”Analysis, optimization, and implementation of low-interference wireless multicarrier systems,” IEEE Trans. Wireless Commun. , vol. 6, pp. 1921–1931, May 2007

    1921

  11. [19]

    C. S. Burrus et al. , Introduction to Wavelets and Wavelet Transforms: A Primer . Prentice Hall, 1998. 22

    1998

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