Accurate I/O Jitter Estimation by Fractional-Exponent Q-Scale Polynomial

William Wu; Jong-Ru Guo

doi:10.20944/preprints202609.1118.v1

Submitted:

12 September 2026

Posted:

15 September 2026

You are already at the latest version

Abstract

Accurate estimation of total jitter (TJ) at low bit error rates (BERs) is essential for high-speed serial interfaces, where direct measurement is often impractical. The dual-Dirac model is widely used due to its simplicity and linear TJ-Q relationship, but its assumption that deterministic jitter (DJ) consists of two Dirac impulses does not capture practical jitter distributions. Simulations show that this linear TJ-Q prediction can introduce significant errors. Prior work using higher-order polynomial fitting improves accuracy but depends strongly on input data range selection. This work investigates the impact of DJ distributions on TJ-Q behavior through a convolution-based numerical jitter simulation framework. A fractional-exponent Q-scale polynomial model is proposed to better align with physical nature of the jitter distribution, improve low-BER jitter prediction accuracy while maintaining similar computational complexity. Results show that, with measurements down to 10⁻⁶ BER, the proposed method predicts jitter at 10⁻¹² BER with over 90% error reduction (10× accuracy improvement) over the linear method and 60–80%+ improvement over fourth-order polynomial fitting. Measurements down to 10⁻⁹ BER further enable prediction at 10⁻²⁰ and 10⁻²⁷ BER levels with excellent accuracy, extending extrapolation by up to 18 orders of magnitude.

Keywords:

bit error rate

;

deterministic jitter

;

dual-Dirac model

;

high-speed serial interfaces

;

jitter modeling

;

Q-scale

;

signal integrity

Subject:

Engineering - Electrical and Electronic Engineering

I. Introduction

With the continuous increase of data rates in modern high-speed communication systems, timing uncertainty has become one of the critical limitations in achieving reliable signal transmission. Accurate estimation of total jitter (TJ) at low bit error rate (BER) levels is therefore essential for system design, validation, and compliance testing.

However, directly measuring jitter at extremely low BER levels is impractical due to the enormous amount of data required. For example, a single error occurrence for a BER of

10 ⁻ ² ⁰

requires observation of approximately

10 ² ⁰

transmitted bits. For a reasonably accurate BER measurement, it requires many error occurrences. Consequently, statistical jitter estimation techniques are required to predict system performance beyond experimentally measurable regions.

The dual-Dirac model is one of the most widely used approaches for jitter analysis [1], [2]. By modeling deterministic jitter as two discrete impulses and random jitter as Gaussian noise, the dual-Dirac model derives a total jitter (TJ), deterministic jitter (DJ), and random jitter (RJ) relationship:

T J = D J + 2 Q σ

(1)

where σ represents the random jitter standard deviation and Q corresponds to the target BER through the inverse complementary error function by

Q = \sqrt{2} {e r f c}^{- 1} (2 \cdot B E R)

(2)

Although the dual-Dirac model has been successfully applied in many high-speed I/O applications, its accuracy depends on the assumption that deterministic jitter can be approximated by two discrete locations. In practical systems, deterministic jitter may result from various physical mechanisms, including periodic modulation, bounded interference, and rare system disturbances. These mechanisms may generate deterministic jitter distributions with characteristics significantly different from the dual-Dirac assumption.

Several approaches have been proposed to improve jitter estimation accuracy. Li et al. introduced a distribution-tail fitting method that extracts jitter characteristics by analyzing the statistical behavior of measured jitter distributions [3], [4]. These works demonstrated the importance of distribution shape and tail behavior in jitter decomposition. Subsequent studies extended this approach to more complex systems [5], [6], [7].

Hong and Cheng further improved low-BER jitter estimation by introducing polynomial fitting of the TJ-Q relationship [8]. Instead of relying on the linear dual-Dirac assumption, their method uses up to a fourth-order polynomial model to approximate the relationship between Q-factor and total jitter.

Jitter decomposition and fitting have become integral parts of modern measurement systems [9,10,11,12,13]. However, directly fitting TJ as a function of Q does not explicitly consider the physical behavior of different jitter distributions. As shown in this work, certain deterministic jitter distributions produce sublinear TJ-Q relationships, causing conventional polynomial fitting to become sensitive to fitting-range selection and the extrapolation region.

In this paper, a distribution-based jitter simulation framework is developed to investigate the limitations of existing TJ-Q prediction methods. A fractional-exponent Q-scale polynomial (FEQP) model is proposed to better represent the nonlinear relationship between jitter distribution characteristics and BER extrapolation.

The remainder of this paper is organized as follows. Section II introduces the theoretical background of jitter analysis, including jitter convolution and existing estimation methods. Section III presents the numerical jitter simulation framework used to evaluate arbitrary jitter distributions. Section IV discusses the limitations of the conventional linear TJ-Q prediction method under non-dual-Dirac deterministic jitter distributions. Section V introduces the proposed FEQP Model. Section VI presents simulation results and validation across multiple deterministic jitter distributions. Section VII evaluates the prediction stability and sensitivity of different methods to fitting data range selection. Section VIII discusses practical applications of the proposed method, and Section IX concludes this paper.

II. Background

A. Combining Jitter Distributions Using Convolution

Total jitter is generated by the combination of multiple independent jitter sources.

For deterministic jitter DJ(t) and random jitter RJ(t), the total jitter probability density function can be obtained through convolution:

p_{T J} (t) = \int_{- \infty}^{+ \infty} p_{D J} (τ) p_{R J} (t - τ) d τ

(3)

The convolution operation is required because total timing error is the sum of independent jitter contributions:

For TJ = DJ + RJ, for a given total timing error value t, if the deterministic jitter component is τ, the corresponding random jitter component must be

t - τ

. Because all possible values of τ can contribute to the total jitter value t, the contributions must be integrated over the entire range of τ, resulting in the convolution relationship above.

B. Dual-Dirac Jitter Model

Li et al. proposed analyzing jitter distribution tails to extract RJ/DJ decomposition for TJ estimation. Their method recognizes that the statistical behavior of the distribution tail contains important information for predicting BER performance. However, the method primarily focuses on extracting jitter components from measured distributions rather than modeling the relationship between arbitrary deterministic jitter distributions and BER extrapolation.

The dual-Dirac model was introduced as a tool for quickly estimating total jitter defined at a BER. The model approximates deterministic jitter distribution as two impulse functions:

p_{D J} (t) = \frac{1}{2} δ (t + μ_{L}) + \frac{1}{2} δ (t - μ_{R})

(4)

Random jitter is modeled as a Gaussian distribution:

p_{R J} (t) = e^{{- t}^{2} / (2 σ^{2})} / (σ \sqrt{2 π})

(5)

After convolution, the resulting total jitter distribution consists of two Gaussian distributions shifted by

μ_{L}

and

μ_{R}

. At a given BER level, the distance between the left and right distribution boundaries gives total jitter:

T J = μ_{L} + μ_{R} + 2 Q σ

(6)

which means DJ is

μ_{L} + μ_{R}

. This relationship predicts a linear TJ-Q relationship [1], [2], [14].

Since its introduction, the dual-Dirac method has gained wide industry adoption. It has become the standard method for high-speed serial I/O such as PCIe, UCIe, and Ethernet SERDES [2], [15], [16], [17].

C. Previous Improvement Methods

Hong and Cheng proposed replacing the linear TJ-Q relationship with up to a fourth-order polynomial fitting model:

T J (Q) = a_{0} + a_{1} Q + a_{2} Q^{2} + a_{3} Q^{3} + a_{4} Q^{4}

(7)

This approach improves accuracy when measured jitter behavior deviates from the dual-Dirac assumption. The results show that, for a system affected by both RJ and DJ, fourth-order fitting can accurately estimate TJ at the 10⁻¹² BER level. However, polynomial fitting directly in Q may not match the physical scaling behavior of arbitrary jitter distributions, especially when extrapolating significantly beyond the available measurement range.

III. Numerical Jitter Simulation Method

To evaluate the validity of the dual-Dirac approximation, a Python-based simulator was developed.

The simulator accepts arbitrary deterministic and random jitter probability distributions and calculates the resulting total jitter distribution.

The simulation procedure is:

1. Generate a Gaussian random jitter PDF.

2. Generate deterministic jitter PDF.

3. Perform numerical convolution.

4. Compute the cumulative distribution function (CDF).

5. Determine left and right timing boundaries at a target BER.

6. Calculate total jitter from the left and right timing boundaries using (8).

T J = t_{r i g h t} - t_{l e f t}

(8)

The corresponding Q-factor is calculated from the target BER.

This approach allows evaluation of jitter behavior at extremely low BER levels that are difficult or impossible to measure experimentally.

IV. Limitations of Linear TJ-Q Prediction

The simulator was first applied to non-dual-Dirac deterministic jitter distributions.

Figure 1 shows a uniform deterministic jitter distribution combined with Gaussian random jitter. Upon detailed examination, the resulting TJ-Q curve exhibits concave-down behavior.

This deviation occurs because the deterministic jitter distribution no longer consists of two discrete impulses. Instead, the shape of the distribution influences the probability density and, consequently, the behavior of the distribution tails. Concave-down behavior is likely to occur in light-tailed deterministic jitter distributions. In theory, all bounded jitter distributions fall into this category when measured at low enough BER. On the other hand, heavy-tailed distributions, such as the exponential distribution associated with Poisson Process, produce faster-than-linear growth at high Q values. These observations indicate that the deterministic jitter distribution has a direct impact on low-BER extrapolation accuracy.

V. Proposed Fractional-Exponent Q-Scale Polynomial Model

While the fourth-order polynomial fit proposed by Hong and Cheng improves the estimation accuracy, it has some limitations [8]. Because the model directly fits Q, its behavior remains dependent on the selected fitting range and polynomial order. Furthermore, high-order polynomial functions may exhibit unstable extrapolation outside the measured region.

First, a polynomial’s lowest-order behavior is linear, making it unsuitable for distributions where TJ grows sublinearly with Q.

Second, higher-order polynomial models may become unstable outside the fitting region. In particular, fitting a concave-down TJ distribution can result in a negative leading coefficient, which can produce unrealistic divergence at large Q values.

To better match the observed behavior of jitter distributions, this work proposes fitting TJ as a polynomial function of fractional-exponent Q:

T J (Q) = a_{0} + a_{1} Q^{1 / 2} + a_{2} Q + a_{3} Q^{3 / 2}

(9)

This model maintains the computational complexity of a third-order polynomial fit by performing a cubic polynomial expansion in the transformed variable

\sqrt{Q}

, while providing additional flexibility to represent sublinear, linear, and superlinear TJ-Q behavior. The proposed formulation better aligns with the statistical characteristics of different jitter distribution tails, including light-tailed, Gaussian-like, and heavy-tailed distributions. Specifically, the term allows the model to capture sublinear growth commonly observed in light-tailed distributions, while the

Q^{3 / 2}

term provides additional flexibility to represent faster-than-linear growth associated with heavy-tailed distributions.

VI. Simulation Validation

The proposed model was evaluated using several deterministic jitter distributions:

• Dual-Dirac distribution

• Uniform distribution: a commonly used DJ model

• Sinusoidal jitter: the distribution generated by sinusoidal jitter, as shown in ref. [2]. This is another commonly used DJ model, as practical systems often have one or more dominant frequencies in power-supply noise.

{D J}_{s i n u s o i d a l} (t) = \{\begin{matrix} 1 / (π \sqrt{\frac{m}{2} - \frac{{2 t}^{2}}{m}}), f o r |t| < \frac{m}{2} \\ 0, o t h e r w i s e \end{matrix}

(10)

• Laplace distribution: The Laplace distribution was included to represent heavy-tail behavior caused by rare abnormal events in practical systems.

For each distribution:

• Total jitter distributions were generated using convolution.

• TJ values were calculated at BER levels

10 ⁻ ² ⁷

and

10 ⁻ ² ⁰

.

• Linear dual-Dirac fitting, fourth-order polynomial fitting, and the proposed FEQP model were compared.

Figure 2, like Figure 1, shows a uniform deterministic jitter distribution combined with Gaussian random jitter. As can be seen, the resulting fit from the proposed FEQP demonstrates a much better fit of the concave-down behavior than the linear fit in Figure 1.

All jitter values in the above tables are normalized to the standard deviation of Gaussian random jitter (σ), allowing the results to be directly scaled to practical systems. For example, when σ=1 ps, all jitter values are represented in picoseconds; similarly, when σ=1% of the unit interval (UI) in a high-speed I/O system, the results can be interpreted as percentages of UI. The parameters of the uniform and sinusoidal deterministic jitter distributions were selected such that deterministic jitter contributes approximately 50% of total jitter at a BER of

10 ⁻ ⁶

, which is consistent with practical high-speed I/O standards such as PCIe [15].

Table 1 and Table 2 compare the proposed FEQP method with the linear method. Table 3 and Table 4 compare the proposed method with the fourth-order polynomial (Poly4) fitting method. The proposed method achieves significantly improved prediction accuracy across different jitter distributions, reducing prediction error by approximately 90% in most cases. In other words, it provides a ~10× improvement over existing approaches.

VII. Prediction Stability and Data-Range Sensitivity

An additional evaluation was performed to investigate the sensitivity of different prediction methods to the selected fitting range. Since practical jitter measurements are often limited to relatively high BER levels, a robust extrapolation method should maintain accuracy when predicting significantly lower BER levels from limited data.

The proposed method was evaluated using different fitting ranges and compared with existing approaches. Using data from

10 ⁻ ²

to

10 ⁻ ⁶

BER, the proposed method was used to predict jitter at

10 ⁻ ¹ ²

BER, as shown in Table 5. The proposed method achieves over 90% error reduction compared with the linear method and 60%–80%+ improvement over the Poly4 method.

Additional tests were performed by including higher-BER points (

10 ⁻ ¹

) in the fitting range to simulate non-ideal measurement conditions. The proposed method exhibited lower variation compared with linear and fourth-order polynomial approaches, demonstrating improved robustness against fitting range selection. The same trend was observed for predictions at

10 ⁻ ² ⁰

and

10 ⁻ ² ⁷

BER levels presented in the previous section.

VIII. Practical Applications

The proposed method is applicable to high-speed I/O systems requiring extremely low BER operation.

The evaluated BER levels correspond to realistic system requirements. A BER of

10 ⁻ ² ⁰

represents extremely long-duration high-speed operation, while

10 ⁻ ² ⁷

corresponds to emerging high-reliability interconnect requirements, such as UCIe [18].

As mentioned earlier, the analysis is normalized to a Gaussian standard deviation. For example, in PCIe 5, the data rate is 32 Gb/s, with unit interval t=31.25 ps. For a standard deviation of 1% UI, that means all numbers presented are scaled by 0.3125 picoseconds [15].

Because deterministic jitter distributions are typically unknown in real systems, accurate extrapolation models are essential for estimating low-BER performance from limited measurements. The results show that, with measurements available down to a BER of

10 ⁻ ⁶

, the proposed method can accurately predict jitter at a BER level of

10 ⁻ ¹ ²

and achieve better accuracy than existing methods. Furthermore, measurements down to

10 ⁻ ⁹

enable prediction at BER levels of

10 ⁻ ² ⁰

and

10 ⁻ ² ⁷

, representing an extrapolation range of up to 18 orders of magnitude.

IX. Conclusion

This paper investigated the limitations of existing low-BER jitter estimation methods under arbitrary deterministic jitter distributions. Through convolution-based simulation, it was demonstrated that the conventional linear TJ-Q relationship may significantly deviate when deterministic jitter does not follow the dual-Dirac assumption.

Building on previous distribution-based decomposition and polynomial fitting approaches, this work proposes a fractional-exponent Q-scale polynomial model. The proposed method better aligns with the physical nature of jitter distributions and provides improved accuracy across multiple deterministic jitter distributions while maintaining low computational complexity.

The proposed approach provides a practical framework for more accurate low-BER jitter estimation in future high-speed I/O systems.

References

Stephens, R. Jitter Analysis: The Dual-Dirac Model, RJ/DJ, and Q-Scale. In Agilent Technologies, White Paper; 2004. [Google Scholar]
Derickson, D.; Muller, M. Digital Communications Test and Measurement: High-Speed Physical Layer Characterization; Prentice Hall, 2007. [Google Scholar]
Li, M. P.; Wilstrup, J.; Mason, R. J.; Bronson, J. E. A New Method for Jitter Decomposition Through Its Distribution Tail Fitting. Proc. IEEE Int. Test Conf., Atlantic City, NJ, USA, 1999. [Google Scholar] [CrossRef]
Erb, S.; Pribyl, W. Comparison of jitter decomposition methods for BER analysis of high-speed serial links. Proc. 13th IEEE Int. Symp. Design Diagn. Electron. Circuits Syst. (DDECS), Vienna, Austria, 2010. [Google Scholar] [CrossRef]
Oh, D.; Ren, J.; Chang, S. Hybrid statistical link simulation technique. IEEE Trans. Compon. Packag. Manuf. Technol. 2011, vol. 1(no. 5), 772–783. [Google Scholar] [CrossRef]
Dou, Y.; Jiao, D.; Yan, J.; Zhu, J. A cluster-based method for analysis of jitter effect on signaling performance of nonlinear high-speed links. IEEE Trans. Microw. Theory Techn. 2024, vol. 72(no. 9), 5106–5119. [Google Scholar] [CrossRef]
Fang, S.; Pan, W.; Pan, D.; Xu, Z.; Gao, Z.; Zhao, T. Jitter analysis based on kernel density estimation and dual-Dirac. Proc. 2024 IEEE Int. Conf. Signal, Inf. Data Process. (ICSIDP), Zhuhai, China, 2024. [Google Scholar] [CrossRef]
Hong, D.; Cheng, K.-T. An Accurate Jitter Estimation Technique for Efficient High Speed I/O Testing. Proc. IEEE 16th Asian Test Symp. (ATS), Beijing, China, 2007. [Google Scholar] [CrossRef]
Chun, J. H.; Lee, J. W.; Abraham, J. A. A novel characterization technique for high speed I/O mixed signal circuit components using random jitter injection. Proc. 15th Asia South Pacific Design Autom. Conf. (ASP-DAC), Taipei, Taiwan, 2010. [Google Scholar] [CrossRef]
Keysight Technologies, “Jitter fundamentals: Sources, types, and characteristics,” Appl. Note 5992-1637EN, 2021. Available online: https://www.keysight.com/us/en/assets/7018-05301/application-notes/5992-1637.pdf.
Agilent Technologies. “Using clock jitter analysis to reduce BER in serial data applications,” Appl. Note 5989-5718EN, Dec. 2006. Available online: https://www.keysight.com/us/en/assets/7018-01450/application-notes/5989-5718.pdf.
Wavecrest Corporation, “Understanding jitter,” Wavecrest Corp., Eden Prairie, MN, USA; White Paper, 13 Apr 2001.
Foster, G. Tech. Note SR-TN045; Dual-Dirac, scope histograms and BERTScan measurements – A primer. SyntheSys Research, Inc., Sep 2005.
Stephens, R. What the dual-Dirac model is and what it is not. In Tektronix, Inc., Jitter 360° Knowledge Series; White Paper, Oct 2006; Volume pt. 2. [Google Scholar]
PCI-SIG, PCI Express base specification revision 7.0. Jun 2025. Available online: https://members.pcisig.com/wg/PCI-SIG/document/22464.
UCIe Consortium. Universal Chiplet Interconnect Express (UCIe) Specification Revision 3.0; Beaverton: OR, USA, Aug 2025; Available online: https://members.uciexpress.org/wg/Members/document/1358.
Wu, Z.; Guo, J.R. Analysis of UCIe 48/64-GT/s Electrical Links. IEEE Open J. Solid-State Circuits Soc. 2025, vol. 5, 401–409. [Google Scholar] [CrossRef]
Sharma, D. D.; Pasdast, G.; Qian, Z.; Aygun, K. Universal Chiplet Interconnect Express (UCIe): An Open Industry Standard for Innovations with Chiplets at Package Level. IEEE Trans. Compon. Packag. Manuf. Technol. 2022, vol. 12(issue 9), 1423–1431. [Google Scholar] [CrossRef]

Short Biography of Authors

WILLIAM WU is a high school senior at Valley Christian High School, San Jose, CA, USA. His research interests include electrical engineering, high-speed interfaces, nanotechnology, and applied mathematics. His current research focuses on timing-jitter modeling and low bit-error-rate (BER) prediction for high-speed electrical I/O systems.

He has earned 10 national and international mathematics honors, including qualification for the American Invitational Mathematics Examination (AIME) through the AMC 12, and is a 2027 National Merit Scholarship Program Semifinalist. At Penn’s 2026 Engineering Summer Academy (ESAP), he co-proposed an original microfluidic robot for gastric cancer screening, designed a key detection method, substantiated the project’s feasibility, and received the Outstanding Self-Drive Award. He also holds Silver division status in the USA Computing Olympiad (USACO), has earned semifinalist and finalist honors at national debate tournaments, and is active in STEM mentorship.

JONG-RU GUO earned his master’s degree in electrical engineering from National Tsing-Hua University, Hsinchu, Taiwan, in 1996. He received his Ph.D. in electrical engineering from Rensselaer Polytechnic Institute, Albany, New York, USA, in 2005.

He is currently with ARM, Austin, Texas, USA, as a High-Speed Interface Architect since 2025. He works on developing I/O architecture and circuit development for ML/AI computing applications. He was with Intel Corporation, Hillsboro, Oregon, USA, since 2012. He worked on developing industry standards such as PCIe and UCIe, focusing on electrical specification, I/O architecture, and circuit developments. He also delivered the prototyped CXL disaggregated memory system supporting the latest high-performance computing applications. Prior to joining Intel, he was an Advisory Engineer with High-Speed Serial link (HSS) design group in IBM at E. Fishkill, NY, starting in 2005. During this time, he developed the key circuits in the high-speed serial I/Os operating at data rates ranging from 6.4 Gbps to 28/32 Gbps in both CMOS and SOI-CMOS technologies.

He holds more than 40 patents and has published papers in IEEE conferences such as CICC, JSSC, Photonics Journal, Proceedings of the IEEE, and Journal on Exploratory Solid-State Computational Devices and Circuits. Dr. Guo was the co-recipient of 2011 IEEE Custom Integrated Circuits Conference (CICC) Best Regular Paper Award. He also received multiple Intel and Altera awards.

Figure 1. Tj-Q linear model for uniform deterministic-jitter distribution.

Figure 2. TJ-Q FEQP model for a uniform deterministic-jitter distribution.

Table 1. Linear and FEQP Comparison.

Distribution	Parameters	Exact TJ	Linear TJ	FEQP TJ	Linear error (%)	FEQP error (%)	Error reduction (%)
Dual-Dirac	μ (-6, 4)	31.6999	31.6999	31.6998	0.00	0.00	—
Uniform	leftBound = -6, rightBound = 4	30.9582	31.9216	31.0678	3.11	0.35	88.62
Sinusoidal	Amplitude = 5	31.2688	31.8365	31.3363	1.82	0.22	88.11
Laplace	scale = 0.3, center = 1	40.6352	28.6589	40.77	29.47	0.33	98.87

Table 2. Linear and FEQP Comparison.

Distribution	Parameters	Exact TJ	Linear TJ	FEQP TJ	Linear error (%)	FEQP error (%)	Error reduction (%)
Dual-Dirac	μ (-6, 4)	28.5251	28.5251	28.525	0.00	0.00	—
Uniform	leftBound = -6, rightBound = 4	27.6902	28.3247	27.7345	2.29	0.16	93.02
Sinusoidal	Amplitude = 5	28.0397	28.4136	28.0671	1.33	0.10	92.67
Laplace	scale = 0.3, center = 1	30.9644	24.3027	31.0319	21.51	0.22	98.99

Table 3. Poly4 and FEQP Comparison.

Distribution	Parameters	Exact TJ	Poly4 TJ	FEQP TJ	Poly4 error (%)	FEQP error (%)	Error reduction (%)
Dual-Dirac	μ (-6, 4)	31.6999	31.6999	31.6998	0.00	0.00	—
Uniform	leftBound = -6, rightBound = 4	30.9582	28.4879	31.0678	7.98	0.35	95.56
Sinusoidal	Amplitude = 5	31.2688	30.379	31.3363	2.85	0.22	92.41
Laplace	scale = 0.3, center = 1	40.6352	38.7918	40.77	4.54	0.33	92.69

Table 4. BER = 10⁻²⁰: Poly4 and FEQP Comparison.

Distribution	Parameters	Exact TJ	Poly4 TJ	FEQP TJ	Poly4 error (%)	FEQP error (%)	Error reduction (%)
Dual-Dirac	μ (-6, 4)	28.5251	28.525	28.525	0.00	0.00	—
Uniform	leftBound = -6, rightBound = 4	27.6902	27.2961	27.7345	1.42	0.16	88.76
Sinusoidal	Amplitude = 5	28.0397	27.8008	28.0671	0.85	0.10	88.53
Laplace	scale = 0.3, center = 1	30.9644	30.6213	31.0319	1.11	0.22	80.33

Table 5. Comparison of Linear, Poly4, and FEQP Methods.

(a) Total-jitter estimates
Distribution	Parameters	Exact TJ		Linear TJ	Poly4 TJ		FEQP TJ
Dual-Dirac	μ (-6, 4)	24.0693		24.0693	24.0693		24.0692
Uniform	leftBound = -6, rightBound = 4	23.0475		23.5083	22.9213		23.0634
Sinusoidal	Amplitude = 5	23.4748		23.7465	23.396		23.4853
Laplace	scale = 0.3, center = 1	19.9119		17.0748	20.4604		19.709
(b) Prediction errors and error reductions
Distribution	Linear error (%)	Poly4 error (%)	FEQP error (%)			Linear reduction (%)	Poly4 reduction (%)
Dual-Dirac	0.00	0.00	0.00			—	—
Uniform	2.00	0.55	0.07			96.55	87.40
Sinusoidal	1.16	0.34	0.04			96.14	86.68
Laplace	14.25	2.75	1.02			92.85	63.01

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content.

© 2026 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

Copyright: This open access article is published under a Creative Commons CC BY 4.0 license, which permit the free download, distribution, and reuse, provided that the author and preprint are cited in any reuse.