{"id":"57f5e4aa-c453-47e7-92f1-b50c75ce3d2e","arxiv_id":"2411.17759","paper_version":1,"verdict":"REJECT","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"high","formal_verification":"none","parameter_count":2,"one_line_summary":"A non-reductionist state-space PLL model is simulated over parameter grids to map synchronization and noise robustness, but key phase and VCO approximations are not justified.","lead":"This paper replaces the usual phase-error-only model of a phase-locked loop with a four-state nonlinear state-space model that keeps the oscillator and filter voltages explicit. The authors simulate parameter sweeps and noise to map where the loop locks, which is aimed at helping PLL designers choose gains without relying on phase-reduction assumptions.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Eq. (8) is not a phase estimator: it gives atan(-omega tan(theta)) instead of theta, so the synchronization metrics in (12) and the parameter maps in Figs. 10-12, 15-17 are unsupported as written.","rationale":"The paper's central claim is that the full state-space model, together with the phase estimate (8), enables direct numerical determination of synchronization regions and design-relevant parameter choices. That claim depends on psi_o being a faithful phase of the VCO output. Eq. (8) is the only estimator provided, and it is mathematically wrong for the stated oscillator equations. The consequences are concrete: the frequency-entrainment metric f and the phase-error metrics e, m, and s are all computed from psi_o, so every parameter-space diagram in Sections 3 and 4 inherits the sign and distortion error. Even if an unstated atan2/unwrapping correction were used in the simulations, the paper does not disclose it, making the results unreproducible and the claims unsupported. The manuscript itself indirectly signals trouble by attributing the spikes in Fig. 6 to 'the discontinuity of the atan function' while continuing to use (8) for all subsequent metrics. A secondary concern is that the VCO model in (1)-(2) omits the phase-integral structure of a true voltage-controlled oscillator and is inconsistent for time-varying vc(t); this compounds the problem, but the phase-extraction error alone is sufficient to invalidate the central parameter maps. No code, data, or experimental validation is provided, so there is no independent check on the reported floors. For these reasons the reader's REJECT verdict is appropriate, and no adjustment to that verdict is needed.","tokens_in":11584,"tokens_out":8218,"duration_ms":77678,"concrete_test":"Generate a pure sinusoid z1(t) = cos(omega t), z2(t) = -omega sin(omega t) with omega = 1.02, apply Eq. (8) with the unwrapping needed to make psi_o continuous, and compare the result with omega t. The unwrapped Eq. (8) should have slope approximately -1.02, giving f = |1 - omega/Omega_o| approximately 2 rather than the 10^-3 floor in Fig. 10. Then recompute Example 3 using the corrected estimator psi_o(t) = atan2(-z2/omega_inst, z1) unwrapped (equivalently, the phase of the analytic signal z1 + i z2/omega_inst) and regenerate Figs. 10-12 and 15-17. If the blue synchronization floors shift, disappear, or change shape, the published maps are artifacts of the incorrect formula; if they persist, the authors must publish the corrected phase formula and code.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The load-bearing weakness is the phase extraction formula (8). For the oscillator (1) with z1 = A cos(theta) and z2 = -A omega sin(theta), substituting into (8) gives psi_o = atan(z2/z1) = atan(-omega tan(theta)), which is not theta: it carries a sign reversal and a frequency-dependent distortion. After unwrapping, this estimate has average slope approximately -omega, not +omega. Consequently, in a locked steady state with theta = omega_i t, the average phase growth rate Omega_o in (11) would be approximately -omega_i, and the frequency-entrainment measure f in (12) would be about 2, not the 10^-3 floor reported in Fig. 10. The small values of de/dt in Fig. 6 cannot be reproduced from (8) as written unless an unstated correction - for example, atan2(-z2/omega_inst, z1) with unwrapping - is silently applied. Since the metrics f, e, m, and s in (12), the thresholds in (10), and all parameter-space diagrams in Figs. 10-12 and their noisy counterparts in Figs. 15-17 are built from psi_o, this error propagates into the paper's central evidence. This is a correctness flaw in the stated model, not a disagreement with prior consensus, and it is not resolved by any self-contained limitation statement in the manuscript.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a four-state 'non reductionist' state-space model of a phase-locked loop, comprising a harmonic-oscillator VCO, a second-order loop filter, and a multiplicative phase detector. It uses this model to estimate the VCO output phase via Eq. (8) and defines synchronization metrics f, e, m, s in Eq. (12) to map capture/lock-in regions in parameter space, with and without noise. The claimed contribution is that these maps provide design-relevant synchronization boundaries without the usual phase-reduction assumptions.","tokens_in":11914,"tokens_out":10099,"duration_ms":94191,"significance":"If the model and phase estimates were sound, the paper would offer a useful numerical design tool for third-order PLLs, including nonideal filtering and noise. The manuscript provides explicit state equations, a clear simulation protocol, and parameter sweeps without fitting parameters to a target result; those are genuine strengths. The central quantitative claims, however, rest on a phase estimator that is mathematically inconsistent with the oscillator state equations, and on a VCO representation that is not the standard FM relation when the control voltage varies. No experimental or circuit-level validation is provided, so the practical claims rest entirely on the simulation model, which as written is not reliable.","major_comments":[{"comment":"The phase estimator psi_o(t)=tan^{-1}(z2/z1) is not the output phase of the oscillator (1). If z1=A cos(phi) and z2=-A omega_inst sin(phi), then psi_o=tan^{-1}(-omega_inst tan(phi)), not phi. When unwrapped, this estimate has a mean slope of approximately -omega_inst, so in a locked state with phi approximately omega_i t the average Omega_o in Eq. (11) would be about -omega_i and the frequency-entrainment measure f in Eq. (12) would be of order 2, rather than the 1e-3 floor reported in Figure 10. Consequently, the small e-dot values in Figure 6, the thresholds in Eq. (10), and all parameter-space maps in Figures 10-12 and 15-17 are not reproduced by the formulas as stated. A corrected four-quadrant estimator such as atan2(-z2/omega_inst, z1) with unwrapping would have to be specified and the simulations regenerated.","section":"Section 2, Eq. (8); Section 3, Eqs. (10)-(12)"},{"comment":"The text states that the oscillator (1) has solution z1=A cos(omega_inst t) and that this represents an FM VCO. This is only valid if omega_inst is constant. When vc(t) varies, the solution of (1) is not A cos(omega_inst t); writing z1=A cos(phi) and differentiating shows that (1) requires phi_ddot=0 and phi_dot=omega_inst, which is incompatible with a time-varying control voltage. The correct FM phase relation is phi(t)=integral_0^t [omega0+Kv vc(tau)] d tau. The model as written is therefore not the standard FM VCO used in PLL analysis, and the physical interpretation in the surrounding text is not supported.","section":"Section 2, Eqs. (1)-(2)"},{"comment":"Equation (9) introduces an integral term Ki times the integral of vc in omega_inst, but no corresponding state is present in the proposed state-space model (6). Example 2 states that the VCO is simulated with (2) 'and with an integral term', and Example 3 uses Ki=0.22 to produce Figures 10-12. The simulations therefore are not of the model (6) that the paper derives and claims to analyze; they are of a different dynamical system whose equations are not given. This is a consistency gap between the mathematical model and the numerical evidence.","section":"Section 3, Eq. (9) and model (6)"}],"minor_comments":[{"comment":"The parameter grid is described only as yielding 520 combinations; the number of points in each dimension and the step sizes should be stated for reproducibility.","section":"Section 3, Example 3"},{"comment":"The implementation of noise in the central frequency omega0 and in the input u(t) within the state equations (6) is not specified (e.g., continuous-time white noise, sample-and-hold noise, or random initial perturbations). A precise stochastic definition is needed to reproduce the Monte Carlo results.","section":"Section 4"},{"comment":"The expression z1=A cos[(omega0+Kv vc(t))t] is dangerous notation: for time-varying vc(t) the argument should be the integral of (omega0+Kv vc(tau)) d tau, not the product (omega0+Kv vc(t))t.","section":"Section 2, after Eq. (2)"},{"comment":"The derivative e_dot in Eq. (12) is computed from a phase estimate with branch cuts; the unwrapping procedure used to obtain Figures 10-12 should be stated explicitly, because the reported floors depend on it.","section":"Figure 6 and Eq. (12)"},{"comment":"The claim that the state-space model 'takes into account the nonlinearity of the sine function' is not precisely tied to Eq. (5), where the phase detector is a product of two sinusoidal signals; the relationship between this product and the usual sine phase-detector characteristic should be clarified.","section":"Section 1 and Section 5"}],"recommendation":"reject","confidential_remarks":"I concur with the reader's assessment that the phase-estimation error in Eq. (8) is the central technical obstacle. The paper does not exhibit circular parameter fitting, but the self-referential use of the same phase estimate both to define synchronization and to judge it is problematic. The VCO modeling issue and the inconsistency between Eq. (6) and Eq. (9) further support rejection. A corrected version would require substantial reworking of the model and re-running of all parameter sweeps; I would treat any such resubmission as a new paper."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Good idea, wrong execution. The paper proposes a full-state model of a third-order PLL to map synchronization regions directly from simulation, without the usual phase-reduction assumptions. That is a useful direction: nonideal filtering, high-frequency components, and noise are all handled naturally in the time domain. The parameter-space maps in Figs. 10-12 and 15-17 would be valuable if they were reliable.\n\nBut they are not, because the phase estimate in Eq. (8) is algebraically wrong. With (1), z1 = A cos θ and z2 = -Aω sin θ, so ψ_o = atan(z2/z1) = atan(-ω tan θ), which carries a sign reversal and frequency distortion. In a locked state θ = ω_i t, ψ_o has average slope roughly -ω_i, so the entrainment measure f = |1 - ω_i/Ω_o| in (12) would be near 2, not the 1e-3 floor in Fig. 10. The tiny de/dt in Fig. 6 and the blue regions in Figs. 10-12 cannot be produced from (8) as written. Some unstated correction, like atan2(-z2/ω, z1) with unwrapping, would be needed. Since the whole performance analysis uses ψ_o, the maps and noise results are unsupported.\n\nThere is a second modeling issue. The oscillator (1)-(2), with ω_inst(t) = ω0 + Kv vc(t), is not a standard VCO for time-varying vc; z1 = A cos[(ω0 + Kv vc)t] is not a solution of (1) in general. The paper treats ω_inst as both a constant in the solution and a time-varying parameter in the equation. A proper VCO should have the phase (or frequency integral) as a state. Related, Example 2 introduces an integral term (9) that is not in the state-space model (6), so the simulation differs from the stated model.\n\nCredit where due: the idea of full-state numerical parameter-space analysis is the right way to escape some phase-reduction limitations, and the noise-robustness tests are a sensible extension. The paper also engages with prior literature on third-order PLLs. But the central quantitative claims do not survive contact with the stated equations. I would reject this version. The approach is salvageable with a corrected phase estimator, a proper VCO model, and released code/data; then the maps might become a real design tool. A serious editor could send this to peer review, but the authors need to fix the math and reprocess everything.","headline":"The non-reductionist idea is sensible, but the phase estimator in Eq. (8) is wrong, so the synchronization maps and noise results are unsupported as written.","tokens_in":12408,"tokens_out":4478,"would_cite":false,"duration_ms":39958,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":["02.30.Oz","02.60.Cb"],"model":"deepseek-v4-flash","headline":"A four-state model of the phase-locked loop maps synchronization and desynchronization regions directly, without phase-error reduction.","keywords":["phase-locked loop","state-space model","phase reduction","synchronization","capture range","lock-in range","noise robustness","parameter space"],"falsifier":"Recompute the parameter maps of Figures 10-12 with a phase estimate that accounts for the VCO amplitude and frequency, for example $\\psi_o = \\operatorname{atan2}(-z_2/(\\omega_0+K_v v_c), z_1)$ with unwrapping, and compare the locations and floor levels of the synchronization regions; if the blue plateaus move or change shape, the reported boundaries are artifacts of the Eq. (8) estimator rather than properties of the model.","tokens_in":11324,"feed_emoji":"📡","tokens_out":8619,"duration_ms":77816,"temperature":0.7,"pith_summary":"This paper aims to show that a phase-locked loop can be modeled, without the usual phase-error reduction, as a four-state dynamical system whose variables are the actual voltages and oscillator signals in the circuit. The authors' point is that this state-space model keeps effects that design-oriented phase models discard: mismatched input and output frequencies, nonideal low-pass filtering, residual high-frequency components from the multiplier phase detector, and noise injected at the VCO or the input. On that basis they argue that capture and lock-in ranges, and the quality of synchronization, can be read directly from numerically computed parameter maps. If this is right, a designer can choose loop gains and filter parameters from time-domain simulation rather than from analytical phase-plane or bifurcation analysis.","feed_headline":"State-space PLL model pinpoints lock-in ranges without phase reduction","feed_subtitle":"Full-state simulation shows where a PLL locks and how noise shrinks that region, no phase-reduction shortcut needed.","key_machinery":"The carrying object is the four-state state-space model in Eq. (6): $\\dot{x}_1=x_2$, $\\dot{x}_2=-a_0x_1-a_1x_2+K_d z_1 u(t)$, $\\dot{z}_1=z_2$, and $\\dot{z}_2=-[\\omega_0+K_v(b_0x_1+b_1x_2)]^2 z_1$. It couples a second-order loop filter, states $x_1,x_2$, with a VCO oscillator, states $z_1,z_2$, through the multiplicative phase detector. The work this does is to keep the loop's actual oscillating signals as state variables, so high-frequency components survive in the filter output and noise can be inserted at circuit-accessible points. The companion machinery is the phase-recovery formula $\\psi_o=\\tan^{-1}(z_2/z_1)$ and the metrics in Eq. (12), which turn simulation traces into synchronization maps.","core_discovery":"The central claim is that the four-state model in Eq. (6), with filter states $x_1,x_2$ and oscillator states $z_1,z_2$, represents a third-order PLL node more faithfully than phase-reduction models because the multiplier phase detector $v_d = K_d z_1 u(t)$ and the VCO frequency $\\omega_{\\rm inst}=\\omega_0+K_v(b_0x_1+b_1x_2)$ are kept as they are, with no averaging away of high-frequency terms and no assumption of equal input and output frequencies. The output phase is estimated as $\\psi_o = \\tan^{-1}(z_2/z_1)$, and synchronization performance is quantified by frequency-entrainment error $f$, phase-error magnitude $e$, mean absolute phase-error derivative $m$, and its standard deviation $s$. Parameter scans over input frequency $\\omega_i$ and loop gain $K_d=K_v$ produce maps whose flat floors mark the capture and lock-in regions. The maps show a broad synchronization band around $\\omega_i\\approx\\omega_0$ with gain between 1 and 2, and noisy runs show that noise shrinks and asymmetrizes the locking region while roughly preserving its floor level.","pith_inferences":["Implicit in the paper, the same parameter-scan strategy could be applied to networks of PLL nodes by coupling several copies of Eq. (6), replacing analytical network stability conditions with searchable maps.","The synchronization maps are only as trustworthy as the phase estimate: redoing Figures 10-12 with a corrected phase recovery such as $\\operatorname{atan2}(-z_2/\\omega_{\\rm inst}, z_1)$ with unwrapping could shift the reported boundaries, so the regions should be treated as qualitative until that check is done.","The apparent stabilizing effect visible in the noisy $s$ map is worth testing separately, since $s$ measures variability of the phase-error derivative rather than synchronization itself; varying the noise variance in a controlled sweep would show whether the effect is genuine or an artifact of the metric."],"forward_implications":["Capture and lock-in ranges become visible as floors in parameter maps of $f$, $m$, and $s$, so they can be found by numerical integration instead of analytical bifurcation or phase-plane calculations.","Because the model does not assume ideal filtering, design choices such as filter bandwidth and roll-off can be tested directly for their effect on residual high-frequency ripple in the control voltage.","Different VCO structures, such as adding an integral term to the frequency control, can be simulated without changing the modeling framework, enabling zero steady-state frequency error designs.","Noise can be injected at the VCO central frequency or at the input, and Monte Carlo runs can be used to check how much the synchronization region shrinks or becomes asymmetric.","The resulting maps agree in broad terms with earlier bifurcation-based studies of third-order PLLs while exposing transition zones between synchronized and desynchronized regimes in more detail."],"supporting_citations":[{"why":"Supplies the phase reduction approach for synchronizing oscillators that the paper deliberately replaces.","marker":"[13]"},{"why":"Provides the earlier treatment of second-harmonic terms in the phase detector that motivates keeping high-frequency components.","marker":"[14]"},{"why":"Reviews the reductionist phase-locked loop model whose simplifying assumptions the state-space model avoids.","marker":"[15]"},{"why":"Defines the synchronization thresholds and average phase-growth rate that the metrics f, e, m, and s adapt.","marker":"[16]"},{"why":"Supplies the noise-analysis basis used to interpret the Monte Carlo simulations and their compatibility with classical results.","marker":"[20]"},{"why":"Offers a study of phase-error oscillations in third-order optical PLLs used as a comparison for the new parameter maps.","marker":"[11]"},{"why":"Presents the Hopf bifurcation and chaos analysis for third-order PLLs that the new maps complement.","marker":"[12]"},{"why":"Gives a previous bifurcation-based determination of lock-in ranges whose diagrams the new figures extend.","marker":"[18]"}],"fun_headline_variants":["Full-state PLL model maps lock-in regions without shortcuts","Noise shrinks PLL lock-in region in full-state simulation","PLL lock-in map from full state-space without phase reduction","Full-state model maps PLL lock-in zones including noise"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the output phase recovered from the oscillator states through Eq. (8) is the actual VCO phase; since every synchronization metric in Eq. (12) is built from this estimate, a systematic error in that recovery would shift the capture and lock-in boundaries in Figures 10-12.","fun_headline_variants_meta":{"raw":{"variants":["Full-state PLL model maps lock-in regions without shortcuts","Noise shrinks PLL lock-in region in full-state simulation","PLL lock-in map from full state-space without phase reduction","Full-state model maps PLL lock-in zones including noise"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00101,"raw_usage":{"total_tokens":4296,"prompt_tokens":998,"completion_tokens":3298,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":614,"completion_tokens_details":{"reasoning_tokens":3228}},"tokens_in":614,"tokens_out":3298,"duration_ms":21698,"temperature":1.0,"reasoning_tokens":3228,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T12:38:04.482770+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Recompute the parameter maps of Figures 10-12 with a phase estimate that accounts for the VCO amplitude and frequency, for example $\\psi_o = \\operatorname{atan2}(-z_2/(\\omega_0+K_v v_c), z_1)$ with unwrapping, and compare the locations and floor levels of the synchronization regions; if the blue plateaus move or change shape, the reported boundaries are artifacts of the Eq. (8) estimator rather than properties of the model.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the phase reduction approach for synchronizing oscillators that the paper deliberately replaces."},{"cited_title":"50(2003), pp","cited_arxiv_id":null,"evidence_quote":"Provides the earlier treatment of second-harmonic terms in the phase detector that motivates keeping high-frequency components."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Reviews the reductionist phase-locked loop model whose simplifying assumptions the state-space model avoids."},{"cited_title":"V ., Kurths, J., and Zhou, C., Synchronization in Oscillatory Networks, Springer, Berlin (2007)","cited_arxiv_id":null,"evidence_quote":"Defines the synchronization thresholds and average phase-growth rate that the metrics f, e, m, and s adapt."},{"cited_title":"Viterbi, Phase-Locked Loop Dynamics in the Presence o f Noise by Fokker-Planck Techniques, Proceedings of the IEEE, 12 (1963), pp","cited_arxiv_id":null,"evidence_quote":"Supplies the noise-analysis basis used to interpret the Monte Carlo simulations and their compatibility with classical results."},{"cited_title":"066101-2–066101-20","cited_arxiv_id":null,"evidence_quote":"Offers a study of phase-error oscillations in third-order optical PLLs used as a comparison for the new parameter maps."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Presents the Hopf bifurcation and chaos analysis for third-order PLLs that the new maps complement."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives a previous bifurcation-based determination of lock-in ranges whose diagrams the new figures extend."}],"review_version":1}