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arxiv: 2604.08177 · v1 · submitted 2026-04-09 · 📡 eess.SP

Estimating PLL Phase Noise Parameters from Measurements for System-Level Modeling

Carl Collmann , Ahmad Nimr , Gerhard Fettweis This is my paper

Pith reviewed 2026-05-10 17:52 UTC · model grok-4.3

classification 📡 eess.SP
keywords phase noisePLLparameter estimationleast squaresOrnstein-UhlenbeckMIMOfrequency synthesizer
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The pith

A least-squares method estimates PLL phase noise parameters directly from spectrum measurements.

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

This paper develops a practical way to turn raw phase noise measurements into the specific numbers needed for computer models of phase-locked loops. In MIMO wireless systems, phase noise from these loops can limit overall performance, so good models are needed to test compensation methods. The approach uses a least squares fit to match a standard mathematical description of the noise process to the measured spectrum. By applying it to two commercial PLL chips, the authors obtain usable values for oscillator constants and loop bandwidth. These values then support both time-step and frequency-domain simulations of the noise.

Core claim

The paper presents a least squares method for estimating PLL parameters such as oscillator constants or PLL bandwidth from a measured phase noise spectrum. The phase noise process at a PLL output is represented by an Ornstein-Uhlenbeck process whose spectrum is captured by a multi-pole/zero model. The method is applied on the MAX2870 and MAX2871 PLL chips and parameter estimates such as oscillator constants and PLL bandwidths are provided. The resulting parameter set enables both time- and frequency-domain numerical simulations.

What carries the argument

Least-squares optimization that fits the coefficients of a multi-pole/zero spectral model, derived from an Ornstein-Uhlenbeck phase noise process, to measured data.

If this is right

  • The estimated parameters enable numerical simulations of phase noise effects in MIMO mobile communication systems.
  • Concrete oscillator constants and PLL bandwidth values are obtained for the MAX2870 and MAX2871 chips.
  • The parameter sets support both time-domain and frequency-domain modeling for testing phase noise compensation techniques.

Where Pith is reading between the lines

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

  • The same fitting procedure could be applied to other PLL designs or frequency synthesizers to generate chip-specific models without relying on generic datasheet values.
  • Using these measured parameters in end-to-end link simulations would allow direct comparison of modeled versus observed bit-error rates or throughput under phase noise.
  • The method might be adapted to extract parameters from time-domain jitter measurements rather than frequency-domain spectra.

Load-bearing premise

The phase noise process at the PLL output follows an Ornstein-Uhlenbeck process whose spectrum is accurately captured by a multi-pole/zero model.

What would settle it

Simulated phase noise spectra generated from the fitted parameters deviate substantially from fresh independent measurements on the same MAX2870 or MAX2871 chips.

Figures

Figures reproduced from arXiv: 2604.08177 by Ahmad Nimr, Carl Collmann, Gerhard Fettweis.

Figure 1
Figure 1. Figure 1: Setup for PSD measurement of MAX2870/71 PLL synthesizers [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Measured phase noise spectrum for USRP 2944R with UBX [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Measured phase noise spectrum of USRP 2944R with parameter [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗
read the original abstract

In current MIMO mobile communication systems, phase noise can significantly impair performance. To allow for compensation of these impairments, accurate phase noise modeling is necessary. Numerical modeling of the phase noise process at a phase-locked loop (PLL) output is established in the literature and commonly represented by an Ornstein-Uhlenbeck (OU) process. The corresponding spectrum can be represented by a multi-pole/zero model. This work presents a least squares (LS) method for estimating the PLL parameters such as oscillator constants or PLL bandwidth from a measured phase noise spectrum. The method is applied on the MAX2870 and MAX2871 PLL chips and parameter estimates such as oscillator constants and PLL bandwidths are provided. The resulting parameter set enables both time- and frequency-domain numerical simulations.

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

2 major / 3 minor

Summary. The paper presents a least-squares (LS) fitting method to extract PLL phase noise model parameters (oscillator constants and loop bandwidth) from measured power spectral density data. The underlying model is the closed-form spectrum of an Ornstein-Uhlenbeck process expressed via a multi-pole/zero transfer function. The method is applied to laboratory measurements from the MAX2870 and MAX2871 PLL chips, with the resulting parameter values intended to support both time-domain and frequency-domain numerical simulations for MIMO communication systems.

Significance. If the fitting procedure is accurate and the extracted parameters are shown to be reliable, the work supplies a practical, reproducible route for obtaining device-specific phase-noise parameters needed for system-level simulations. This is useful in MIMO contexts where phase noise is a performance limiter, and the direct use of an established OU model plus standard LS fitting is a strength for accessibility and reproducibility.

major comments (2)
  1. [Results] Results section: The manuscript reports the LS-derived parameter estimates for the MAX2870 and MAX2871 but supplies no quantitative fit-quality metrics (residual norm, R^{2}, or cross-validation error), no parameter uncertainty intervals, and no comparison against alternative estimators. This absence makes it impossible to assess whether the extracted values are sufficiently accurate for the claimed simulation use.
  2. [Method] Method section: The multi-pole/zero spectrum model is adopted without discussion of how the number of poles and zeros is selected or whether the LS problem is well-conditioned; if the model order is chosen post hoc to improve the fit, the resulting parameters may not be unique or physically interpretable.
minor comments (3)
  1. [Abstract] Abstract: The abstract states that parameter estimates are provided but does not include the numerical values or their orders of magnitude; a short table or explicit listing would improve immediate utility.
  2. Figures: Captions and legends for measured-versus-fitted spectra should explicitly label axes, units, and which trace is data versus model to aid reader interpretation.
  3. Notation: Ensure consistent symbols for the phase-noise PSD and the fitted parameters across equations, text, and tables.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive feedback on our manuscript. We address each major comment below and have revised the manuscript to incorporate additional quantitative metrics and expanded methodological discussion as suggested.

read point-by-point responses

  1. Referee: [Results] Results section: The manuscript reports the LS-derived parameter estimates for the MAX2870 and MAX2871 but supplies no quantitative fit-quality metrics (residual norm, R^{2}, or cross-validation error), no parameter uncertainty intervals, and no comparison against alternative estimators. This absence makes it impossible to assess whether the extracted values are sufficiently accurate for the claimed simulation use.

    Authors: We agree that explicit quantitative fit-quality metrics strengthen the results. In the revised manuscript we now report the residual norm and R^{2} for the least-squares fits to both the MAX2870 and MAX2871 measured spectra. Parameter uncertainty intervals are added via the diagonal of the covariance matrix obtained from the normal equations of the linear least-squares problem. Cross-validation and direct comparison against alternative estimators (e.g., maximum-likelihood) are not included, as the paper focuses on the practical accessibility of the standard LS procedure for the log-linearized spectrum model; we believe the added metrics suffice to allow readers to judge suitability for system-level simulations. revision: yes

  2. Referee: [Method] Method section: The multi-pole/zero spectrum model is adopted without discussion of how the number of poles and zeros is selected or whether the LS problem is well-conditioned; if the model order is chosen post hoc to improve the fit, the resulting parameters may not be unique or physically interpretable.

    Authors: The pole/zero count is not chosen post hoc; it is fixed by the closed-form power spectral density of the Ornstein-Uhlenbeck process that models the PLL phase noise (two poles and one zero for the standard second-order loop). We have expanded the Method section with the explicit derivation from the OU stochastic differential equation and a brief analysis showing that the resulting design matrix is well-conditioned over the measured frequency range because the basis functions are linearly independent and parameters are constrained to positive physical values. This guarantees uniqueness and interpretability. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper presents a standard least-squares fitting procedure that estimates PLL parameters (oscillator constants, bandwidth) by matching a known closed-form Ornstein-Uhlenbeck multi-pole/zero spectrum model to independent laboratory phase-noise measurements from commercial chips. No derivation step reduces algebraically to its own inputs, no fitted quantity is relabeled as a prediction, and no load-bearing premise rests on self-citation chains. The method is a direct, externally falsifiable application of an established model to new data, yielding parameter sets for subsequent simulation use.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the domain assumption that PLL phase noise follows an Ornstein-Uhlenbeck process whose spectrum is exactly described by a multi-pole/zero model; no free parameters or invented entities are introduced beyond the quantities being estimated from data.

axioms (1)
  • domain assumption Phase noise at the PLL output follows an Ornstein-Uhlenbeck process whose spectrum is captured by a multi-pole/zero model
    Explicitly stated in the abstract as the established representation used for numerical modeling.

pith-pipeline@v0.9.0 · 5425 in / 1102 out tokens · 29139 ms · 2026-05-10T17:52:58.189097+00:00 · methodology

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Reference graph

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