Importance of Coherence and Phase in Vibration Testing

The Critical Importance of Measurement Synchronization and Signal Integrity

Spectral Dynamics, Inc. PANTHER Vibration Control System

Introduction
Modern vibration control systems represent sophisticated measurement and feedback mechanisms that must accurately characterize dynamic response across multiple measurement points while maintaining precise temporal relationships between signals. Whether conducting swept sine resonance searches, controlling complex sine-on-random environments, or replicating field-measured time histories, the quality of vibration testing fundamentally depends on two often-overlooked parameters: coherence and phase matching between measurement channels.

Coherence quantifies the degree of linear correlation between two signals at each frequency, providing critical insight into measurement quality, system linearity, and control loop performance. Phase matching ensures that multiple measurement channels maintain accurate temporal relationships, enabling proper interpretation of structural response and preventing measurement artifacts that can corrupt control algorithms or lead to incorrect test conclusions.

This technical paper examines why coherence and phase matching matter in single-axis vibration control, explains the technical mechanisms underlying these parameters, and demonstrates how PANTHER's superior specifications—including better than ±1 degree phase accuracy up to 100 kHz—enable more accurate testing across diverse applications.

Understanding Coherence in Vibration Measurements

The Coherence Function

Coherence is a frequency-domain function that measures the linear correlation between two signals, taking values from 0 to 1 at each frequency. A coherence value of 1 indicates perfect linear correlation—the signals maintain a constant phase relationship and amplitude ratio at that frequency. A coherence value of 0 indicates no correlation—the phase relationship varies randomly and no linear relationship exists between the signals.

Mathematically, coherence is computed from the power spectral densities and cross-spectral density of the two signals. The coherence function removes phase information through magnitude-squared operations, focusing purely on the consistency of the relationship rather than the specific phase angle. This makes coherence particularly valuable as a diagnostic tool—it reveals whether signals are related without requiring knowledge of what that relationship should be.

Physical Interpretation

In vibration testing, coherence between drive and control signals provides fundamental insight into control system performance and test quality:

•    High Coherence (>0.95): Indicates excellent linear relationship, low noise, and reliable control. The control signal responds predictably to drive input with minimal contamination.
•    Moderate Coherence (0.8-0.95): Suggests acceptable correlation but with some noise interference, nonlinearity, or time-varying characteristics. Control quality remains good but not optimal.
•    Low Coherence (<0.8): Indicates serious measurement problems such as excessive noise, significant system nonlinearity, loose connections, or control channel interference. Control accuracy is compromised.

The PANTHER system uses coherence thresholds during compensation and system identification processes. When setting up a test, the controller measures coherence between excitation and response to ensure the compensation function accurately characterizes system dynamics. Spectral lines with coherence below the threshold (typically 0.90-0.95) are excluded from compensation calculations, preventing unreliable measurements from corrupting the control function.

Common Causes of Poor Coherence

Low coherence between vibration control system channels can arise from several sources:

•    Excessive Noise: Electrical noise, electromagnetic interference, or ground loops corrupt the measurement signal, reducing correlation with the clean drive signal.
•    System Nonlinearity: Friction, backlash, loose joints, or material nonlinearity causes the response to deviate from simple linear scaling of the input.
•    Mechanical Issues: Loose accelerometer mounting, cracked fixtures, or failing shaker components introduce unrelated vibration that reduces coherence.
•    Measurement Range Problems: Signals too small relative to noise floor or overloading at high amplitudes both degrade coherence.
•    Time-Varying Systems: Systems with dynamics that change during measurement (thermal effects, progressive damage, settling) exhibit reduced coherence due to non-stationary behavior.

Phase Matching and Temporal Synchronization

What is Phase Matching?

Phase matching refers to the accurate preservation of temporal relationships between multiple measurement channels within a vibration control system. In a perfectly phase-matched system, when a physical event occurs simultaneously at two transducer locations, the digitized signals from those transducers maintain precisely synchronized timing relationships through the entire measurement chain—from sensor output through signal conditioning, analog-to-digital conversion, and digital processing.

Phase accuracy is typically specified in degrees at a reference frequency. PANTHER specifications state phase matching of better than ±1 degree up to 100 kHz between channels within a single unit, and between multiple units when expanded to 32 channels. This represents exceptional temporal synchronization—at 100 kHz, one degree of phase corresponds to approximately 27.8 nanoseconds of time difference.

Sources of Phase Error

Multiple mechanisms can introduce phase errors between measurement channels:

•    ADC Sampling Skew: If channels are not sampled truly simultaneously, time delays between samples appear as phase errors. Sequential sampling architectures suffer from this unless carefully compensated.
•    Analog Filter Variations: Anti-aliasing filters and signal conditioning circuits introduce frequency-dependent phase shifts. Channel-to-channel variations in filter characteristics cause phase mismatch.
•    Cable Length Differences: Different cable lengths between transducers and the controller introduce propagation delays, particularly significant at high frequencies.
•    Digital Processing Delays: Digital filters and decimation can introduce phase shifts that vary between channels if not carefully implemented.
•    Clock Synchronization: When expanding to multiple hardware units, imperfect synchronization of master/slave clocks creates systematic phase errors.

PANTHER addresses these challenges through simultaneous sampling on all channels, carefully matched analog filtering, and precision clock distribution in multi-box configurations. The system employs advanced calibration and compensation techniques to achieve the specified ±1 degree phase accuracy throughout the measurement chain.

Critical Applications: Swept Sine Testing

Tracking Filters and Phase Measurement

Swept sine vibration testing represents one of the most demanding applications for phase accuracy and coherence. During a swept sine test, the controller outputs a single sinusoidal tone that continuously changes frequency, typically sweeping linearly or logarithmically from a low frequency (5-20 Hz) to a high frequency (2000-10000 Hz) over a specified time duration.

To measure the response at the instantaneous sweep frequency, vibration controllers employ tracking filters—digital bandpass filters whose center frequency continuously follows the sweep frequency. The tracking filter rejects energy at all frequencies except a narrow band around the current tone frequency, effectively isolating the response to the controlled excitation from background noise and harmonic distortion.

Critically, tracking filters can measure both amplitude and phase of the response signal relative to the drive. This phase measurement capability distinguishes tracking filter methods from simpler RMS or peak detection approaches, which only measure magnitude. Phase information proves essential for several key swept sine applications:

•    Resonance Tracking: Resonances exhibit characteristic phase behavior, with phase transitioning through 90 degrees at the resonant frequency. Accurate phase measurement enables the controller to precisely lock onto and track a shifting resonance frequency during fatigue testing.
•    Transfer Function Measurement: Transfer functions require both magnitude and phase data. Phase errors directly corrupt the transfer function, leading to incorrect dynamic characterization.
•    Modal Analysis: Mode shape determination requires accurate phase relationships between measurement points to distinguish in-phase and out-of-phase motion.

Poor phase matching between drive and control channels introduces systematic errors in phase measurements. A 5-degree phase error might seem small, but at a high-Q resonance where phase changes rapidly with frequency, this translates to significant frequency uncertainty and control inaccuracy. PANTHER's ±1 degree specification ensures phase measurements remain reliable even at high frequencies and sharp resonances.

Coherence in Sine Testing

During swept sine testing, coherence between drive and control provides real-time quality assurance. High coherence (>0.95) confirms that the control signal responds linearly to the drive excitation with minimal noise contamination. This validation is particularly important when sweeping through structural resonances, where response amplitudes increase dramatically and system linearity may be stressed.

Coherence degradation during a sine sweep often indicates mechanical problems. A sudden drop in coherence at specific frequencies might reveal loose mounting hardware, fixture resonances, or incipient component failure.

The tracking filter bandwidth affects coherence measurements. Narrower filters provide better noise rejection and higher coherence but require more stable frequency control. Wider filters are more tolerant of sweep rate variations but include more out-of-band noise. PANTHER's adaptive tracking filters optimize this trade-off, maintaining high coherence while accommodating the rapid frequency changes characteristic of logarithmic sweeps.
 
Sine-on-Random Control

Simultaneous Narrowband and Broadband Excitation

Sine-on-random (SoR) vibration testing combines a sinusoidal tone (possibly sweeping) with broadband random vibration, simulating environments where deterministic periodic excitation coexists with random background vibration. Common applications include rocket motor vibration (periodic combustion pressure oscillations superimposed on turbulent flow noise), aircraft structures during flight (engine tones on aerodynamic buffeting), and rotating machinery mounted on platforms.

Controlling sine-on-random environments presents unique challenges for coherence and phase management. The controller must simultaneously:

•    Extract the narrowband sine component from the composite signal for amplitude and phase control
•    Measure the broadband random component and maintain its spectral shape
•    Prevent the random component from interfering with sine measurements
•    Avoid introducing spectral holes in the random spectrum at the sine frequency

Signal Separation and Phase Preservation

PANTHER employs proprietary multiple filters optimized for SoR generation and control. These specialized filters achieve clean separation between sine and random components while preserving phase relationships critical for accurate control. The sine component uses narrow tracking filters similar to swept sine testing, isolating the tone for amplitude and phase measurement. The random component uses complementary filtering that maintains spectral integrity across the full frequency range.

Phase matching between measurement channels proves critical for SoR control. The controller must accurately measure the phase of the sine tone relative to the drive reference while simultaneously monitoring random vibration levels. Phase errors corrupt sine phase control, causing the tone to drift from its intended phase relationship (leading to distortion of the tone). This becomes particularly problematic when the sine frequency sweeps through the random bandwidth, requiring seamless transition between control modes.

Coherence monitoring in SoR testing serves multiple purposes. The coherence between drive and control at the sine frequency indicates tone control quality—high coherence confirms the sine component maintains stable phase and amplitude. Coherence computed across the random spectrum validates random control fidelity. Deviations from expected coherence patterns can reveal nonlinear behavior, measurement problems, or control algorithm issues.

Avoiding Spectral Artifacts

One of the most challenging aspects of SoR control involves preventing spectral artifacts at the sine frequency. Naive implementations that simply add sine and random components or use aggressive notch filters to separate them create undesirable side effects—spectral holes where random energy is completely suppressed, or spectral peaks where components add constructively. Furthermore, sweeping tonal components often suffer from spectral leakage effects when using discrete FFT methods.

PANTHER's advanced filtering maintains natural random spectral density even at frequencies occupied by sine tones. This is achieved through sophisticated phase and amplitude management that allows sine and random components to coexist without destructive interference. The result is a controlled spectrum that faithfully reproduces the specified environment without artificial discontinuities.

Poor phase matching between channels would make this careful balance impossible. If different channels measure different phase relationships for the same physical signal, the controller cannot distinguish between intentional phase patterns and measurement artifacts. The ±1 degree phase accuracy specification ensures that all channels report consistent phase information, enabling the advanced SoR algorithms to function correctly.

Field Data Replication and Time History Control

Reproducing Real-World Environments

Field data replication (FDR), also known as time history replication or time waveform replication, represents the ultimate goal of vibration testing—exactly reproducing the acceleration-time history measured in an actual operating environment. Rather than approximating field conditions with sine sweeps or random spectra, FDR directly replays recorded time-domain data on a laboratory shaker, preserving every transient event, peak acceleration, and temporal relationship present in the original environment.

The motivation for FDR stems from recognition that conventional test methods based on spectral averaging may not adequately represent real environments. Random vibration testing, by definition, uses an averaged power spectral density that discounts infrequently occurring events. A brief but severe shock embedded in otherwise moderate random vibration might be completely missed by PSD-based specifications. FDR captures such events exactly as they occurred, ensuring the laboratory test includes all damage mechanisms present in service.

Coherence Requirements for Accurate Replication

Field data replication places extreme demands on coherence between drive and control. Unlike random or sine testing where the controller generates the excitation signal and can tolerate some measurement imperfections, FDR requires the controller to precisely match a pre-defined time history. Any measurement error directly corrupts control accuracy leading to distorted time domain response signals.

During FDR system identification and compensation, the controller determines the transfer function between drive output and control response by applying a broadband identification signal and measuring the resulting response. This process fundamentally relies on coherence:

•    High Coherence (>0.95): Indicates the measured response accurately represents the system's linear transfer function. Compensation based on high-coherence data will reliably equalize the system.
•    Low Coherence (<0.90): Suggests noise, nonlinearity, or time-varying behavior that prevents accurate characterization. Compensation based on low-coherence data will be unreliable or even unstable.
•    Coherence Blanking: PANTHER uses coherence thresholds to exclude unreliable spectral data from compensation calculations.

Only frequencies meeting the coherence threshold contribute to the transfer function, ensuring robust control.
The PANTHER documentation explicitly addresses this in the compensation parameters section, noting that coherence serves as the primary tool for evaluating compensation quality. When coherence falls below the specified threshold (typically 0.90-0.95), those spectral lines are excluded from real-time compensation updates to prevent degradation of the control function.

Phase Accuracy in Time Domain Control

Phase matching proves absolutely critical for time history replication. Unlike frequency-domain control methods that primarily care about spectral magnitude, time-domain replication must preserve the exact temporal structure of the waveform—every peak, valley, and transition must occur at precisely the correct instant.

Consider a recorded time history containing multiple transient impacts. If phase errors shift these impacts by even a few milliseconds relative to each other, the reproduced waveform no longer accurately represents the original environment. Impacts that occurred sequentially might overlap, or events that should coincide might be separated. The structural response will differ from what occurred in service.

At high frequencies, phase matching becomes increasingly stringent. A ±1 degree phase specification at 10 kHz corresponds to ±2.78 microseconds of timing accuracy. At 100 kHz, this tightens to ±278 nanoseconds. PANTHER's specification of better than ±1 degree up to 100 kHz ensures that even high-frequency content in field recordings—shock impacts, high-speed machinery vibration, acoustic coupling—reproduces with temporal fidelity.

This exceptional phase accuracy throughout the frequency range distinguishes professional-grade vibration controllers from basic systems. Systems with poor phase matching may appear to reproduce the general character of a waveform but fail to preserve critical timing relationships that determine structural response and fatigue damage accumulation.

Adaptive Control and Multiple Filters

PANTHER employs adaptive control algorithms rather than simple iterative correction for field data replication. This adaptive approach continuously monitors the relationship between drive and response, updating compensation in real-time to maintain accurate replication despite changing conditions such as thermal effects, progressive damage, or fixture settling.

The multiple filters implemented in PANTHER's FDR control (described as "multiple filters for adaptive control of time domain replication waveforms" in the system specifications) work in concert to handle the diverse spectral content present in real-world recordings. Low-frequency content requires different filtering strategies than high-frequency transients. The filters maintain phase linearity across the frequency range, preventing phase distortion that would corrupt waveform shape.

Coherence monitoring during FDR testing provides real-time quality assurance. If coherence begins degrading during a long-duration replication test, this signals developing problems—perhaps fixture loosening, thermal effects, or progressive damage. The operator can intervene before complete loss of control, and post-test analysis of coherence trends helps identify when test validity may have been compromised.
 
PANTHER System Capabilities

Hardware Specifications for Phase Matching

PANTHER achieves its exceptional phase matching specifications through careful hardware design at multiple levels:

•    Simultaneous Sampling: All input channels sample simultaneously using a shared clock, eliminating skew between channels inherent in multiplexed or sequential sampling architectures.
•    Matched Signal Paths: Analog signal conditioning, anti-aliasing filters, and ADC circuitry are carefully matched across channels to within ±0.25 dB amplitude and ±1.0 degree phase up to 100 kHz.
•    24-Bit ADCs: High-resolution 24-bit analog-to-digital converters with >110 dB dynamic range ensure clean measurements across the full amplitude range without noise-induced phase jitter.
•    262,144 Samples/Second: The maximum sample rate provides Nyquist frequency of 131 kHz, enabling accurate phase measurement well into the ultrasonic range.
•    Multi-Box Synchronization: When expanding to 32 channels across multiple hardware units, precision clock distribution maintains ±1 degree phase accuracy between units throughout the frequency range.

The specifications explicitly state "Phase Synch < 1 degree" and "Expandable to 32 channels fully phase synched accurate to < 1 degree." This guarantee of phase accuracy across expanded channel counts distinguishes PANTHER from systems that achieve good phase matching within a single hardware box but degrade significantly when multiple units are combined.

Digital Signal Processing and Filtering

Beyond hardware, PANTHER's DSP implementation maintains phase accuracy through advanced filtering techniques:

•    Linear Phase Filters: Shock and shock synthesis applications employ specific linear phase filters that preserve waveform shape without introducing frequency-dependent phase distortion.
•    Optimal Adaptive Tracking: Swept sine testing uses optimal adaptive tracking filters that maintain phase accuracy while following rapid frequency changes during logarithmic sweeps.
•    Proprietary Multiple Filters: Sine-on-random control employs proprietary filtering that cleanly separates components while maintaining phase relationships.
•    Adaptive Control Algorithms: Time domain replication uses adaptive control rather than simple iteration, enabling real-time compensation updates while preserving phase accuracy.

These digital processing capabilities leverage the hardware's phase-matched foundation, ensuring that sophisticated control algorithms enhance rather than degrade measurement accuracy.

Coherence-Based Quality Assurance

PANTHER integrates coherence measurement throughout its control applications, providing continuous quality monitoring:

•    System Identification: During compensation runs, coherence validates that the measured transfer function accurately represents system dynamics. Spectral lines below the coherence threshold are excluded from compensation.
•    Real-Time Monitoring: During test execution, coherence provides immediate feedback on control quality and measurement integrity.
•    Diagnostic Tool: Coherence plots help identify mechanical problems, electrical interference, or system nonlinearities that compromise test validity.
•    Post-Test Analysis: Recorded coherence data enables verification that test conditions remained stable and measurements remained reliable throughout the duration.

The coherence threshold parameters in PANTHER (Coh Threshold % and Coh Blanking level) provide user control over how strictly coherence requirements are enforced. Typical settings aim for 90-95% of spectral lines meeting the 0.95 coherence threshold, balancing rigorous quality standards against practical test system limitations.

Practical Implications and Best Practices

Verifying System Performance

Test engineers should regularly verify that their vibration control systems maintain specified coherence and phase accuracy:

•    Phase Verification: Apply identical signals to multiple input channels and verify that measured phase differences remain within specification across the frequency range.
•    Coherence Testing: Run system identification sweeps and examine coherence plots to ensure measurements meet >0.95 coherence across the test bandwidth.
•    Transfer Function Quality: Repeated transfer function measurements should yield consistent results—significant variations indicate coherence or phase problems.
•    Fixture Validation: Sine sweeps with multiple accelerometers should show expected phase relationships based on fixture geometry and modal behavior.

Troubleshooting Poor Coherence

When coherence falls below acceptable levels, systematic troubleshooting identifies and corrects the problem:

•    Check Cable Integrity: Damaged cables, loose connections, or improper shielding often cause intermittent coherence problems.
•    Verify Accelerometer Mounting: Loose accelerometers or poor mounting surfaces introduce measurement artifacts that reduce coherence.
•    Examine Fixture Condition: Cracks, loose bolts, or worn components in fixtures introduce nonlinearities that degrade coherence.
•    Assess Measurement Range: Signals too small relative to noise floor or approaching overload both compromise coherence. Adjust accelerometer sensitivity or gain settings appropriately.
•    Investigate Ground Loops: Multiple ground paths can introduce noise that reduces coherence, particularly at low frequencies.

Optimizing Test Setup

Proper test setup maximizes coherence and preserves phase accuracy:

•    Match Cable Lengths: Use equal-length cables between transducers and the controller to minimize phase differences, particularly important at high frequencies.
•    Quality Cables and Connectors: Use low-noise cables with proper shielding and high-quality connectors to minimize electrical interference.
•    Proper Grounding: Establish single-point grounds and avoid ground loops that introduce noise.
•    Appropriate Transducers: Select accelerometer sensitivity and frequency range appropriate for the test, ensuring adequate signal-to-noise ratio without overload risk.
•    Calibration: Regular calibration of transducers and verification of system gains ensures measurements remain accurate and phase-matched.

Conclusion

Coherence and phase matching represent fundamental requirements for accurate single-axis vibration control, yet these parameters often receive insufficient attention compared to more obvious specifications like dynamic range or channel count. This oversight can lead to subtle but significant testing errors that compromise product qualification, corrupt modal analysis data, or fail to reproduce critical aspects of service environments.

Swept sine testing requires phase accuracy to measure transfer functions, track resonances, and characterize modal behavior. Sine-on-random control demands both coherence and phase matching to cleanly separate signal components while maintaining spectral integrity. Field data replication places the most stringent requirements of all—every aspect of the recorded waveform must be faithfully reproduced, requiring exceptional coherence during system identification and sub-microsecond phase accuracy for high-frequency content.

PANTHER's specifications—better than ±1 degree phase accuracy up to 100 kHz, simultaneous sampling on all channels, >110 dB dynamic range, and coherence-based quality assurance—directly address these demanding requirements. The system employs matched signal paths, advanced digital filtering, and adaptive control algorithms that preserve measurement fidelity throughout the test process.

These capabilities enable confident testing across diverse applications. Aerospace engineers can accurately characterize flight hardware response knowing that phase measurements will correctly identify modal behavior. Automotive test facilities can replicate proving ground data with certainty that high-frequency impacts and peak accelerations occur at precisely the right instants. Research laboratories can extract meaningful transfer functions from swept sine testing without phase errors corrupting the results.

The investment in superior coherence and phase matching pays dividends through more reliable test results, fewer repeated tests due to questionable data, earlier detection of mechanical problems, and ultimately greater confidence that laboratory testing accurately represents service environments. When product reliability and test validity matter, the technical rigor embodied in PANTHER's coherence and phase specifications proves essential rather than merely desirable.

References

1. Spectral Dynamics, Inc. "PANTHER Complete MISO Software Suite." Technical specifications and product literature.

2. Spectral Dynamics, Inc. "Understanding Phase and Coherence in Vibration Control Testing." Technical Library. http://spectraldynamics.com/support/technical-library/

3. Data Physics Corporation. "Evaluating Controller Performance – Part 1 – Swept Sine." (June 13, 2024).

4. Vibration Research. "Coherence Graph - Advanced Graph Function." VRU Technical Documentation. (November 10, 2023).

5. Crystal Instruments. "Swept Sine Test Measurement Strategy." Technical White Paper.

6. Zheng, R.H., et al. "Control strategy for multi-axial swept sine on random mixed vibration testing." Journal of Sound and Vibration, Volume 520, 2022. ScienceDirect.

7. Vibration Research. "Field Data Replication (FDR)." Technical Documentation. (July 1, 2025).

8. Unholtz-Dickie Corporation. "Vibration Time History Testing." Application Note.