{"id":"606834bb-cfff-46ef-9800-298f55cf9d32","arxiv_id":"2505.21691","paper_version":2,"verdict":"REJECT","confidence":"HIGH","novelty_score":3.0,"correctness_risk":"high","formal_verification":"none","parameter_count":4,"one_line_summary":"The paper defines Gauss-Ramanujan pulses from shifted Gaussians and applies them to modulation and wavelets, but the GRSK modulation is inconsistent with its own pulse equations.","lead":"This paper defines Gauss-Ramanujan functions by weighting delayed Gaussian pulses with Ramanujan sequence coefficients, then derives overlap, spectral, and Hilbert-transform properties. It applies them to a continuous-wave modulation, a shift-keying scheme, and a wavelet, but the shift-keying scheme's frequency pulse integrates to zero and cannot work as described.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"GRSK is invalid as specified: Eq. (77) integrates to a negative value, not the asserted qGR(T)=1/2, so the CPM/constant-envelope data-modulation claim in §5.2 is unsupported.","rationale":"The reader's REJECT verdict is well supported. The central advertised applications are GRSK and GRM, and the GRSK phase-pulse normalization is not a matter of tuning: for any T>T0, qGR(T) is negative (tending to −κ erf(√πT0)/(2√2)), so Eq. (78) is false. This is an internal inconsistency, not a disagreement with consensus. It cannot be fixed by citing the correct GP-DGP overlap or wavelet normalization, because those results do not feed into Eq. (78). The GRM section has an independent modulation-index error (max |φ|=π/2, not π/4), which further weakens the applications, but my headline objection is the GRSK failure. I note that the reader's statement that qGR(T) is 'near zero' is imprecise; the truncated integral is approximately −0.35κ, not near zero, but this does not affect the conclusion. The concrete test is a direct quadrature of Eq. (77), which any reader can run in a few lines.","tokens_in":15421,"tokens_out":8546,"duration_ms":81942,"concrete_test":"Evaluate Eq. (77) at t=T for a representative configuration, e.g., T=4, T0=2.45, and compare with the asserted qGR(T)=1/2. Direct computation gives qGR(4)=κ/(2√2)[erf(7.09)−erf(2.75)−erf(4.34)]≈−0.354κ, not +0.5. Run the same quadrature in the T→∞ limit; it remains −κ erf(√πT0)/(2√2), never reaching 0.5. If the author responds by redefining κ to force qGR(T)=0.5, then verify that the renormalized gGR differs from the pulse used in Eq. (82) and Fig. 8, and recompute the PSD comparisons under the corrected normalization.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The most load-bearing defect is in §5.2. A CPM waveform Eq. (73)/(79) is valid only if the phase pulse q(t) is the integral of the frequency pulse and reaches 1/2 over one symbol. The paper defines gGR in Eq. (74) and derives qGR in Eq. (77), then asserts in Eq. (78) that the normalized phase waveform is 1/2 on [0,T]. But evaluating Eq. (77) at t=T using the identity ∫0^T e^{-π(t-T0)^2}dt = 0.5[erf(√π(T−T0)) + erf(√πT0)] gives qGR(T) = κ/(2√2)[erf(√πT) − erf(√π(T−T0)) − erf(√πT0)]. For T0=2.45 and T=4 this is approximately −0.354κ, not +0.5; in the T→∞ limit it tends to −κ erf(√πT0)/(2√2), also about −0.354κ, never reaching 0.5. Hence Eq. (78) is not the integral of Eq. (74), the phase accumulation per symbol is incorrect, and the GRSK claim of a valid constant-envelope continuous-phase modulation is unsupported. The GRM section has a separate modulation-index error (max |φ|=π/2, not π/4), but the GRSK normalization failure alone breaks a headline application.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":15682,"tokens_out":17434,"duration_ms":171709,"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":[{"comment":"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.","section":"§5.2.1, Eqs. (74)-(78)"},{"comment":"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.","section":"§5.1.1, Eqs. (59)-(63)"}],"minor_comments":[{"comment":"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.","section":"§4.1, remark iv"},{"comment":"The exponent should contain απδ^2 rather than βπδ^2 to be consistent with Eq. (13).","section":"Eq. (24)"},{"comment":"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.","section":"§5.2.3, Fig. 8"},{"comment":"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.","section":"Eqs. (74) and (82)"}],"recommendation":"reject","confidential_remarks":"To the editor: the GRSK error is not a matter of missing rigor; Eq. (77) omits an integration constant that changes the value by approximately −0.35κ at T, and Eq. (78) is plainly inconsistent with it. The GRM modulation-index error is similarly clear-cut. Because the paper's main applied claims fail on these points, I recommend rejection despite the correct ancillary derivations. The author could potentially resubmit a revised manuscript that removes or completely reworks GRSK and corrects GRM, focusing on the wavelet and overlap results."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: the first-order GauRam function is a difference of Gaussians, and every spectral, energy, and Hilbert property follows linearly from standard Gaussian facts. The higher-order Ramanujan-weighted sums are finite linear combinations that the paper never uses to derive anything beyond what those combinations inherit. So the novelty claim is overstated.\n\nWhat it does well: the GP-DGP overlap integral, the delay-averaged overlap (exact and via the López-Benítez–Casadevall Q-function approximation), the Hilbert transform through Dawson functions, and the wavelet normalization are all derived correctly and check out. The text is honest about which constants are fitted and where the Q-approximation comes from. There is no circularity.\n\nThe soft spot is load-bearing. In §5.2, g_GR is a difference of two equal-area Gaussians, so its total integral is zero. Integrating from 0 to T gives q_GR(T) = κ/(2√2)[erf(√πT) − erf(√π(T−T0)) − erf(√πT0)], which for T0=2.45 and T=4 is about −0.354κ, not the asserted 1/2 in Eq (78). The T→∞ limit gives the same negative value. So the phase accumulation in the GRSK CPM waveform is wrong, and the constant-envelope continuous-phase claim is unsupported. The modulation-index analysis in §5.1 has a similar issue: φ(t)→−π/2 as t→∞, so the maximum phase deviation is π/2, not π/4. These are not cosmetic. They are the proposed applications.\n\nStrip out GRSK and GRM, and what remains is a small collection of exact formulas for difference-of-Gaussians pulses. I would not cite the modulation claims. I would send the paper to a referee rather than desk reject, because the flaw is pinpointable and the rest of the derivations are careful. A referee can require the applications to be corrected or removed. If that happens, a shorter note on GauRam functions and their Gaussian-pulse properties might be publishable. As written, the headline applications do not work.","headline":"A careful paper whose headline GRSK/GRM claims are undone by the integral in Eq (77); the rest is mostly repackaged Gaussian-pulse mathematics.","tokens_in":16251,"tokens_out":4170,"would_cite":false,"duration_ms":39681,"reading_group":"no","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["Gauss-Ramanujan functions","Ramanujan sequences","Gaussian pulse","continuous-phase modulation","Gauss-Ramanujan shift keying","wavelets","OTFS","spectral nulls"],"falsifier":"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.","tokens_in":15126,"feed_emoji":"📡","tokens_out":10463,"duration_ms":106454,"temperature":0.7,"pith_summary":"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.","feed_headline":"Two shifted Gaussians make a new modulation and wavelet","feed_subtitle":"GRSK gains tunable spectral nulls from a delay parameter; the wavelet tightens time-frequency containment over Hermite.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the Gaussian pulse identities (unit area, self-Fourier property, Rayleigh energy) and the Q-function machinery used throughout the overlap and spectral analysis.","marker":"[1]"},{"why":"Provides the Ramanujan sums and their orthogonality property, which motivates the choice of weights in the GauRam construction.","marker":"[10]"},{"why":"Supplies the exponential approximation of the Gaussian Q-function used to convert the exact mean-overlap integral into a closed form.","marker":"[11]"},{"why":"Gives the Dawson-function representation of the Hilbert transform of a Gaussian, which is the basis for the Hilbert transform result for the first-order GauRam function.","marker":"[14]"},{"why":"Provides the standard integrals used to derive the Hilbert transform and the error-function expression for the GRSK phase waveform.","marker":"[15]"},{"why":"Supplies the general continuous-phase modulation signal model that defines the GRSK waveform and its phase accumulation.","marker":"[16]"},{"why":"Supplies the GMSK pulse as the comparison baseline for the normalized PSD analysis.","marker":"[17]"},{"why":"Supports the claim that Gaussian pulses have superior time-frequency localization, motivating the wavelet comparison.","marker":"[18]"},{"why":"Supplies the mother-wavelet admissibility conditions (zero mean and unit energy) used to validate the Gauss-Ramanujan wavelet.","marker":"[19]"}],"fun_headline_variants":["Difference of two Gaussians yields new modulations and wavelets","Gauss-Ramanujan pulse: tunable spectral nulls for wavelets","New function from shifted Gaussians improves time-frequency","A single pulse from two Gaussians powers new comms schemes","Wavelet from two Gaussians sharpens time-frequency over Hermite"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Difference of two Gaussians yields new modulations and wavelets","Gauss-Ramanujan pulse: tunable spectral nulls for wavelets","New function from shifted Gaussians improves time-frequency","A single pulse from two Gaussians powers new comms schemes","Wavelet from two Gaussians sharpens time-frequency over Hermite"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000848,"raw_usage":{"total_tokens":3727,"prompt_tokens":1019,"completion_tokens":2708,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":635,"completion_tokens_details":{"reasoning_tokens":2621}},"tokens_in":635,"tokens_out":2708,"duration_ms":21759,"temperature":1.0,"reasoning_tokens":2621,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T13:26:07.422260+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Haykin, Digital communication systems , John Wiley & Sons, 2014","cited_arxiv_id":null,"evidence_quote":"Supplies the Gaussian pulse identities (unit area, self-Fourier property, Rayleigh energy) and the Q-function machinery used throughout the overlap and spectral analysis."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the Ramanujan sums and their orthogonality property, which motivates the choice of weights in the GauRam construction."},{"cited_title":"L´ opez-Ben ´ ıtez and F","cited_arxiv_id":null,"evidence_quote":"Supplies the exponential approximation of the Gaussian Q-function used to convert the exact mean-overlap integral into a closed form."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the Dawson-function representation of the Hilbert transform of a Gaussian, which is the basis for the Hilbert transform result for the first-order GauRam function."},{"cited_title":"Abramowitz and I","cited_arxiv_id":null,"evidence_quote":"Provides the standard integrals used to derive the Hilbert transform and the error-function expression for the GRSK phase waveform."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the general continuous-phase modulation signal model that defines the GRSK waveform and its phase accumulation."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supports the claim that Gaussian pulses have superior time-frequency localization, motivating the wavelet comparison."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the mother-wavelet admissibility conditions (zero mean and unit energy) used to validate the Gauss-Ramanujan wavelet."}],"review_version":1}