Quantifying Stochastic Resonance in Precision Propagation Timing Models

The Challenge of Quantifying Stochastic Resonance in Timing Models

In precision propagation timing—the backbone of systems like GPS, network time synchronization, and distributed radar—stochastic resonance (SR) presents both an opportunity and a conundrum. Unlike conventional noise mitigation, SR exploits ambient noise to enhance weak periodic signals, potentially improving timing accuracy in low-signal regimes. However, quantifying this phenomenon in propagation timing models is fraught with complexity: the metrics that work in idealized bi-stable systems often fail when applied to the multi-path, dynamic, and band-limited channels typical of real-world timing infrastructure. Practitioners face a critical gap: how do you measure SR's effect on timing precision with sufficient rigor to decide whether to embrace or reject noise in a given system?

This guide addresses that gap head-on. We assume you are familiar with the basic concept of SR—where noise pushes a system over a threshold, amplifying a sub-threshold periodic signal—and are now seeking robust, quantifiable methods to evaluate its presence and magnitude in propagation timing models. The stakes are high: misidentifying SR can lead to degraded synchronization, increased jitter, or false confidence in timing accuracy. Conversely, properly harnessed SR can improve detection limits in challenging propagation environments, such as underground or underwater timing links.

We will explore three core quantification frameworks: signal-to-noise ratio (SNR) gain analysis, residence-time distribution (RTD) characterization, and spectral coherence measures. Each offers distinct insights and trade-offs. SNR gain directly measures signal improvement but requires careful baseline estimation. RTD captures the dynamical fingerprint of SR but demands high-resolution sampling. Spectral coherence provides a frequency-domain view but may obscure transient effects. Throughout, we emphasize practical application: how to set up simulations, collect empirical data, and interpret results without overfitting to noise models.

A key caveat: SR is highly sensitive to system parameters—threshold levels, noise intensity, and signal frequency. Quantification must account for these dependencies, often requiring multi-dimensional sweeps. We'll illustrate with a composite scenario typical of a long-haul fiber timing link where environmental temperature fluctuations introduce quasi-periodic phase noise. In such a system, SR might manifest as a narrow peak in timing jitter reduction at a specific noise amplitude, but only if the signal's period aligns with the system's intrinsic relaxation time. Misidentifying this peak as a resonance when it is merely a statistical fluctuation is a real risk. Our quantification methods must include statistical significance testing and cross-validation across independent noise realizations.

The remainder of this guide provides step-by-step workflows, tool comparisons, and common pitfalls, all aimed at helping you confidently quantify SR in your own precision propagation timing models.

Core Frameworks for Quantifying Stochastic Resonance

Three principal frameworks dominate the quantitative analysis of stochastic resonance in timing systems. Each framework offers a different lens through which to view the interaction of noise, signal, and system dynamics. Understanding their strengths and limitations is crucial for selecting the appropriate metric for your specific timing model.

Signal-to-Noise Ratio Gain Analysis

SNR gain is the most intuitive metric: compare the output SNR (signal power at the driving frequency divided by noise power in a narrow band) to the input SNR. In a classic SR system, the output SNR exhibits a non-monotonic peak as noise intensity increases, indicating optimal resonance. For timing models, we compute SNR from the timing error signal or the recovered clock phase. A typical workflow involves injecting a known periodic timing offset (e.g., a sinusoidal modulation of the time stamp) and measuring the phase-locked loop's error signal variance. The SNR gain is then the ratio of the output peak amplitude to the noise floor, normalized by the input signal-to-noise ratio. This method is straightforward but sensitive to the choice of bandwidth and the stationarity of noise. In practice, we use Welch's method to estimate power spectral density, ensuring sufficient averaging over multiple segments. A composite scenario: in a simulated GPS receiver tracking a weak signal, we inject band-limited noise and observe a 3 dB improvement in tracking jitter at an optimal noise intensity of 0.15 (normalized units). Below that, the signal remains sub-threshold; above, noise overwhelms the loop. The SNR gain curve reveals a clear maximum, but statistical significance requires at least 20 independent Monte Carlo runs to confirm the peak is not due to random fluctuation.

Residence-Time Distribution Characterization

Residence-time distributions (RTDs) examine the time intervals the system spends in each of its two (or more) stable states before switching. In SR, the distribution of residence times shows a peak at half the driving period, indicating that noise-induced transitions are synchronized with the signal. For timing models, the "states" might correspond to two phase positions of a clock (e.g., early vs. late relative to a reference). By recording the times between threshold crossings of the phase error, we construct histograms of residence times. A clear peak at half the signal period (or integer multiples) is a hallmark of SR. This method is powerful because it directly reveals the dynamical synchronization, but it requires high temporal resolution—often sub-microsecond sampling for typical timing signals. In practice, we use a zero-crossing detector on the phase error and log timestamps. We then bin the intervals and compare to the theoretical distribution for a noiseless system. A composite example: in an atomic clock disciplining loop, we observe that the residence time histogram shows a prominent peak at 0.5 seconds when the noise amplitude is set to 0.2 (relative to threshold), matching the driving signal's half-period. This peak disappears at both lower and higher noise levels, confirming SR. However, care must be taken to avoid artifacts from finite sampling—we recommend at least 10,000 transitions for robust statistics.

Spectral Coherence Measures

Spectral coherence quantifies the correlation between the input and output signals in the frequency domain. For SR, coherence at the driving frequency should peak at the optimal noise level. We compute the magnitude-squared coherence using cross-spectral density estimates. This metric is less sensitive to baseline noise than SNR gain and can detect SR even when the output SNR is dominated by noise. In timing models, we use the input periodic modulation as the reference and the recovered timing signal as the output. Coherence values near 1 indicate strong phase locking. A typical implementation uses the Welch method with overlapping windows and a Hanning taper. For a composite scenario—a distributed acoustic sensing timing link—coherence at the signal frequency rises from 0.2 (no injected noise) to 0.85 at optimal noise, then falls to 0.3 at high noise. This clear peak provides robust evidence of SR. However, coherence can be inflated by common noise sources; we recommend also computing coherence at off-signal frequencies as a control. Each framework has its place: SNR gain for direct signal improvement, RTD for mechanistic insight, and coherence for robust detection. In practice, we advocate using at least two methods in concert to cross-validate findings.

Practical Workflows for Quantification

Translating these frameworks into repeatable workflows requires careful experimental design, data collection, and statistical validation. Below, we outline a step-by-step process that has proven effective in both simulation and laboratory settings for precision propagation timing models. This workflow assumes you have a timing system—such as a phase-locked loop, delay-locked loop, or two-way time transfer link—that can be perturbed by controllable noise.

Step 1: Define the System and Signal Parameters

Begin by characterizing your timing system's baseline dynamics: its natural frequency, damping factor, and threshold levels. For a phase-locked loop, the threshold might be the phase detector's linear range. For a time transfer system, it could be the discrimination threshold of the time-of-arrival estimator. Choose a periodic signal—typically a sinusoidal modulation of the reference timing offset—with amplitude below the threshold (sub-threshold). The frequency should be within the system's passband, often near its relaxation frequency. Document these parameters: signal amplitude A_s, signal frequency f_s, and threshold value V_th. In a composite scenario, a typical long-haul fiber link might have a relaxation frequency of 1 Hz due to thermal time constants; we set f_s = 0.8 Hz and A_s = 0.7 * V_th. This ensures the signal alone cannot trigger threshold crossings.

Step 2: Inject Controlled Noise and Record Output

Inject band-limited white Gaussian noise into the system, varying its intensity (standard deviation) across a range that spans from well below the threshold to several times above it. For each noise level, record the output timing error signal (or recovered phase) for a duration long enough to capture at least 100 cycles of the driving signal. A typical experiment might use 10 noise levels logarithmically spaced from 0.01 to 5.0 (normalized to threshold). For each level, run at least 10 independent trials with different noise seeds to assess variability. In simulation, this is straightforward; in hardware, use a programmable noise generator and a high-speed digitizer. Ensure the recording duration is at least 10 times the system's settling time to avoid transient effects. For example, with a 0.8 Hz signal, record for at least 200 seconds per trial.

Step 3: Compute Quantification Metrics

For each noise level and trial, compute the three metrics: SNR gain, residence-time distribution, and spectral coherence. For SNR gain, estimate the power spectral density of the output timing error using Welch's method (e.g., 8 segments with 50% overlap, Hanning window). Extract the peak power at f_s and the noise floor in a narrow band around f_s (e.g., ±0.05 Hz). Compute output SNR = peak / noise floor. Then SNR gain = output SNR / input SNR (where input SNR is the injected signal power divided by the noise power at the system input). For RTD, apply a threshold crossing detector to the output timing error (using the same threshold V_th). Log the times of upcrossings and downcrossings. Compute residence times as intervals between successive crossings of the same type (e.g., upcrossing to next upcrossing). Bin these intervals and look for a peak at 1/(2f_s) and its harmonics. For spectral coherence, compute the magnitude-squared coherence between the input signal (the known modulation) and the output timing error using the same Welch parameters. Extract the coherence value at f_s.

Step 4: Identify Optimal Noise Level and Validate

Plot each metric as a function of noise intensity. Look for a non-monotonic peak: SNR gain should rise, peak, then fall; RTD should show a clear peak at half-period only near the optimal noise; coherence should peak at the same noise level. The optimal noise intensity is where these metrics coincide. Validate by performing a permutation test: shuffle the output trials across noise levels and recompute the metric; if the observed peak is higher than 95% of shuffled peaks, it is statistically significant. Cross-validate using a second metric—e.g., if SNR gain peaks at noise 0.15, verify that RTD also shows a matching peak. In our fiber link example, the optimal noise intensity was 0.12, with SNR gain improving by 2.5 dB, RTD showing a clear half-period peak, and coherence rising from 0.3 to 0.78. These consistent results confirm SR.

Tools, Stack, and Economic Considerations

Quantifying stochastic resonance in timing models requires a combination of simulation tools, data acquisition hardware, and computational resources. The choice of tools depends on whether you are in a research, development, or deployment phase. Below, we compare three primary options: Python-based simulation stacks, MATLAB toolboxes, and dedicated hardware testbeds. Each has distinct trade-offs in cost, flexibility, and accuracy.

Python Simulation Stack (NumPy, SciPy, and Custom Libraries)

Python offers a flexible, open-source environment for SR quantification. Key libraries include NumPy for array operations, SciPy for signal processing (spectral estimation, filtering), and matplotlib for visualization. For timing-specific models, the `control` library can simulate linear and nonlinear system dynamics. A typical workflow involves defining the timing system as a differential equation (e.g., phase-locked loop model) and solving it with `scipy.integrate.solve_ivp`. Noise injection is straightforward using `numpy.random.randn`. The primary advantages are cost (free), extensibility, and community support. However, performance can be a bottleneck for Monte Carlo simulations with many trials; using Numba (just-in-time compilation) or parallel processing (joblib) can mitigate this. In a composite scenario, a research team used Python to simulate a two-way satellite time transfer link, running 100 noise levels with 50 trials each. With Numba acceleration, the full sweep completed in 4 hours on a standard workstation. The main drawback is the lack of built-in SR-specific functions—practitioners must implement metrics from scratch, which increases the risk of errors. We recommend validating your code against a known benchmark (e.g., a simple bi-stable system) before applying it to timing models.

MATLAB with Signal Processing and Simulink Toolboxes

MATLAB provides a more integrated environment, with toolboxes for signal processing, system identification, and Simulink for block-diagram modeling. The `pwelch` and `mscohere` functions directly compute power spectra and coherence, reducing implementation effort. The Optimization Toolbox can automate parameter sweeps. For hardware-in-the-loop testing, Simulink can interface with data acquisition devices. The main advantages are reliability, documentation, and built-in statistical tests. However, MATLAB is expensive (commercial licenses range from $2,000 to $5,000 per year per seat), which may be prohibitive for smaller teams or startups. In a composite scenario, a defense contractor used MATLAB to quantify SR in a GPS disciplined oscillator, leveraging Simulink's noise generator and the Signal Processing Toolbox's coherence analysis. The project required a one-time license cost of $3,500, but development time was reduced by 30% compared to Python. The trade-off is vendor lock-in and limited customizability for novel metrics.

Dedicated Hardware Testbeds (FPGA-Based Noise Injection and Data Logging)

For production systems or field deployment, software simulation may not capture hardware-specific effects (e.g., quantization noise, nonlinearities). Dedicated testbeds use FPGAs or microcontrollers to inject controllable noise and record timing signals with high precision (sub-nanosecond resolution). A typical setup includes a noise generator (e.g., using a linear feedback shift register with a digital-to-analog converter), a timing error detector (e.g., a time-to-digital converter), and a data logger. The advantages are realism and the ability to test under actual operating conditions. However, cost can be significant: a high-resolution time-to-digital converter module alone may cost $2,000–$10,000, and FPGA development boards add $500–$2,000. Development time is also longer, often requiring firmware expertise. In a composite scenario, a telecom equipment manufacturer built an FPGA-based testbed to validate SR in a Precision Time Protocol (PTP) grandmaster clock. The hardware cost was approximately $15,000, but it allowed them to test noise injection effects at nanosecond granularity, revealing SR benefits that simulation had missed due to idealized noise models. For teams without hardware expertise, we recommend starting with simulation and only moving to hardware when simulation results are promising and need empirical validation.

Growth Mechanics: Positioning, Traffic, and Persistence

Quantifying stochastic resonance in timing models is a niche but growing field, driven by the increasing demand for ultra-precise timing in 5G/6G networks, distributed sensing, and financial trading. For practitioners and organizations, building expertise in this area can unlock competitive advantages and thought leadership. However, growth—whether in career, research impact, or product adoption—requires strategic positioning, consistent content creation, and persistence through the steep learning curve.

Positioning Your Work for Maximum Impact

The key is to frame SR quantification not as a theoretical curiosity but as a practical tool for solving real timing challenges. When presenting your findings, emphasize the performance gains (e.g., improved tracking jitter, extended holdover) and the conditions under which SR is beneficial. Target audiences include system architects at telecom companies, defense contractors working on GPS-denied navigation, and researchers in time-frequency metrology. Publish detailed case studies (anonymized) that show concrete before-and-after metrics. For example, a white paper describing how SR injection improved two-way time transfer stability by 40% in a specific fiber link can attract attention from industry peers. Avoid overhyping; instead, present balanced results including the conditions where SR did not help (e.g., in high-noise environments). This honesty builds credibility. For online visibility, use specific keywords in your titles and abstracts: "stochastic resonance timing jitter reduction," "noise-enhanced synchronization," "phase-locked loop noise optimization." These terms have moderate search volume but high relevance for technical audiences.

Content Strategy for Traffic and Authority

A blog or a series of technical articles on SR quantification can establish you as a go-to resource. Write in-depth tutorials that walk through the workflow described in this guide, using open-source code examples (e.g., Python notebooks hosted on GitHub). Each article should target a specific aspect: "How to Compute Residence-Time Distributions for PLLs" or "Using Spectral Coherence to Detect SR in Time Transfer." Cross-link between articles to build a knowledge base. Engage with communities on platforms like IEEE Xplore, ResearchGate, and relevant subreddits (e.g., r/FPGA, r/DSP). Present at conferences such as the Precise Time and Time Interval (PTTI) meeting or the International Frequency Control Symposium. Over time, your accumulated content will attract organic search traffic from engineers searching for solutions. Persistence is crucial: the field is small, so even a few high-quality articles can make you a recognized voice. Track your progress with metrics like citation counts, download statistics, and direct inquiries from industry partners.

Overcoming the Learning Curve and Maintaining Momentum

The mathematics underpinning SR quantification—stochastic differential equations, spectral estimation, statistical hypothesis testing—can be intimidating. To persist, break the learning process into small milestones: first, implement a simple bi-stable system in Python and reproduce the classic SR peak. Then, extend to a timing model. Join a study group or find a mentor with experience in stochastic processes. Many universities offer online courses in stochastic resonance (e.g., through Coursera or MIT OpenCourseWare). Budget at least three months of dedicated effort before expecting publishable results. In a composite scenario, a postdoctoral researcher spent six months developing a robust SR quantification pipeline for optical lattice clocks; the initial attempts failed due to insufficient averaging, but persistence led to a breakthrough that was published in a top metrology journal. The key is to treat failures as data—each failed experiment teaches you about the system's parameter sensitivity. Finally, remember that the field is evolving; new metrics (e.g., mutual information, Fisher information) are being adapted for timing systems. Stay current by subscribing to journals like IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control.

Risks, Pitfalls, and Mitigations

Quantifying stochastic resonance is fraught with subtle traps that can lead to false positives, wasted effort, or incorrect system designs. Awareness of these pitfalls is essential for robust results. Here, we catalog the most common mistakes and provide concrete mitigations, drawn from composite experiences in timing system development.

Pitfall 1: Mistaking Statistical Fluctuations for Resonance

The most frequent error is interpreting a random peak in the SNR gain curve as evidence of SR. With limited trials, noise can produce spurious maxima. Mitigation: always run a sufficient number of independent trials (at least 20 per noise level) and use a permutation test to assess significance. For example, if you observe a 2 dB peak at noise level 0.15, shuffle the trial labels and recompute the metric 1000 times. If fewer than 5% of shuffled datasets produce a peak as high, you can reject the null hypothesis. Additionally, require that the peak appears in at least two of the three metrics (SNR gain, RTD, coherence) at the same noise level. This cross-validation dramatically reduces false positives. In a composite scenario, a team initially thought they had found SR in a fiber link, but after applying the permutation test, the p-value was 0.12—not significant. They increased the number of trials from 10 to 30, and the peak disappeared, confirming it was a fluctuation.

Pitfall 2: Parameter Mismatch and Tuning Artifacts

SR is highly sensitive to the driving frequency relative to the system's characteristic time scales. If the signal frequency is too high or too low, SR may not occur, or the optimal noise level may shift. Many practitioners fix the signal frequency arbitrarily and miss SR. Mitigation: perform a two-dimensional sweep of both noise intensity and signal frequency, covering at least a decade around the system's relaxation frequency. Plot a heatmap of SNR gain as a function of both parameters. The optimal region should appear as a distinct island. In our fiber link example, the optimal frequency was 0.8 Hz, but at 0.5 Hz the SR peak shifted to higher noise and lower gain. Without this sweep, they might have concluded SR was absent. Similarly, the threshold level must be precisely characterized; a 10% error in threshold can shift the optimal noise by a factor of two. Use calibration runs with a known signal to measure the threshold accurately.

Pitfall 3: Ignoring Non-Stationarity and Colored Noise

Real-world noise is often non-stationary or colored, deviating from the white Gaussian assumption. SR quantification methods assume stationary noise; violations can produce misleading results. Mitigation: before each experiment, test the noise for stationarity using a run test or the Augmented Dickey-Fuller test. If non-stationarity is detected, apply a high-pass filter to remove low-frequency drift, but be aware that filtering can alter the noise spectrum and affect SR. For colored noise (e.g., 1/f noise), SR can still occur but the optimal noise level may differ. Use a noise generator that allows spectral shaping, or model the noise as an autoregressive process. In a composite scenario, a team working on underwater acoustic timing encountered strong 1/f noise; they initially saw no SR peak. After modeling the noise and adjusting the threshold, they found SR at a higher noise intensity than expected. The lesson: do not assume white noise; characterize your noise environment and adapt your analysis accordingly.

Pitfall 4: Overlooking System Nonlinearities and Saturation

Many timing systems have nonlinearities (e.g., phase detector dead zones, voltage-controlled oscillator saturation) that can mimic or mask SR. For instance, saturation at high noise can limit the output, creating a false plateau in SNR gain. Mitigation: operate the system well within its linear range during baseline characterization. Use a nonlinearity test: inject a known signal and measure the output distortion (e.g., harmonic distortion). If total harmonic distortion exceeds 10%, the system's nonlinearity may interfere with SR quantification. Consider using a Volterra series model to separate nonlinear effects from SR. If saturation is unavoidable, restrict the noise range to avoid it, or model the saturation explicitly in your simulations.

Decision Checklist and Mini-FAQ

Before committing resources to a stochastic resonance quantification project, use this decision checklist to ensure you have addressed the key considerations. The checklist is followed by answers to common questions that arise during implementation.

Decision Checklist

• System Characterization: Have you measured the threshold level, relaxation frequency, and linear range of your timing system? (Yes/No—if No, complete this first.)
• Signal Selection: Is the periodic signal amplitude sub-threshold and frequency within the passband? (Recommended: A_s
• Noise Control: Can you inject band-limited white noise with adjustable intensity? (If not, consider using a hardware noise generator or a software noise source in simulation.)
• Trial Count: Will you run at least 20 independent trials per noise level? (For high-stakes decisions, target 50 trials.)
• Metric Suite: Will you compute at least two of the three metrics (SNR gain, RTD, spectral coherence)? (Cross-validation is essential.)
• Statistical Test: Will you perform a permutation test to assess significance? (Yes/No—if No, your results may be unreliable.)
• Parameter Sweep: Will you sweep both noise intensity and signal frequency? (A 2D sweep avoids missing SR due to parameter mismatch.)
• Stationarity Check: Have you tested the noise for stationarity and colored spectrum? (If non-stationary, apply high-pass filtering or model the noise.)
• Nonlinearity Check: Is the total harmonic distortion below 10% at the operating point? (If higher, consider modeling nonlinearities.)
• Documentation: Will you document all parameters, including threshold, noise bandwidth, and signal amplitude, for reproducibility? (Yes/No—reproducibility is key for credibility.)

If you answered "No" to any item, address that gap before proceeding. This checklist reduces the risk of wasted experiments and false conclusions.

Mini-FAQ

Q: Can stochastic resonance improve timing accuracy in any system? A: No. SR is most beneficial in systems with a threshold and a sub-threshold periodic signal. Systems that already operate with high SNR or are linear may not show improvement. It is also sensitive to parameter tuning; in some cases, noise degrades performance. Always quantify before assuming SR will help.

Q: How many noise levels should I test? A: At least 10–15 levels log-spaced over a range of 0.01 to 5 times the threshold. This provides enough resolution to identify the peak. For coarse screening, 8 levels may suffice, but 12–15 is recommended for publication-grade results.

Q: What if my system has multiple thresholds? A: Some timing systems (e.g., multi-level phase detectors) have more than two states. In that case, SR can manifest as synchronization between transitions across different thresholds. The RTD method generalizes to multiple states by analyzing the distribution of times between successive transitions of any type. The spectral coherence method remains applicable as long as you define the output as the state index.

Q: How do I know if the noise I inject is realistic? A: Realistic noise depends on your application. For GPS timing, atmospheric scintillation produces log-normal amplitude fluctuations; for fiber links, thermal noise is Gaussian but band-limited. Use a noise model that matches your environment, or collect actual noise data from the field and replay it in simulation. The key is to test SR under conditions representative of operation.

Q: Can I use commercial software like Simulink for SR quantification? A: Yes, but be aware that built-in noise generators may produce ideal white noise. Ensure you can control noise bandwidth and amplitude precisely. Also, verify that the spectral estimation functions use consistent parameters (e.g., windowing, overlap) across runs. Commercial tools often hide these details, which can lead to irreproducible results.

Synthesis and Next Actions

Quantifying stochastic resonance in precision propagation timing models is a nuanced endeavor that demands rigorous methodology, cross-validation, and a deep understanding of both the system and the noise. This guide has walked you through the core frameworks—SNR gain, residence-time distributions, and spectral coherence—and provided a practical workflow for implementing them. We have compared tools (Python, MATLAB, hardware testbeds) to help you choose the right stack for your budget and accuracy needs, and we have highlighted common pitfalls so you can avoid wasted effort. The decision checklist and FAQ address the most frequent questions that arise in practice.

As a next action, we recommend starting with a simulation of a simple timing system—such as a first-order phase-locked loop—using the Python stack with Numba acceleration. Implement the three metrics and run a two-dimensional parameter sweep over noise intensity and signal frequency. Verify that you can reproduce the classic SR peak (non-monotonic SNR gain, clear RTD peak, coherence peak). This exercise will build your intuition and help you debug your analysis pipeline before moving to more complex models or hardware. Document every step meticulously, including parameter values, trial counts, and statistical test results. Once you have a working pipeline, apply it to a real timing system from your domain, starting with a controlled environment (e.g., a laboratory fiber link) before field deployment.

Remember that SR is not a universal panacea; it is a tool that, when properly quantified, can provide a measurable improvement in specific conditions. The field is still evolving, with ongoing research into metrics like Fisher information and mutual information for timing applications. Stay engaged with the community through conferences and journals, and consider publishing your findings even if they are negative—knowing when SR does not work is equally valuable. By following the methods in this guide, you can contribute to a more robust understanding of noise-enhanced synchronization in precision timing.

Finally, we encourage you to share your experiences and challenges with the broader timing community. The collective knowledge built from careful quantification will accelerate the adoption of stochastic resonance in next-generation timing infrastructures.