Time-Frequency Coherence

Time-frequency coherence is a normalized, bounded measure that quantifies the strength of linear coupling between two non-stationary processes in the joint time-frequency plane. It is computed as the ratio of the cross time-frequency spectrum to the product of the auto time-frequency spectra of the individual signals. The resulting coherence function takes values between 0 (no linear relationship) and 1 (perfect linear coupling) at each time-frequency point. This is a direct extension of ordinary magnitude-squared coherence, replacing stationary spectral estimates with time-frequency distributions such as the Short-Time Fourier Transform or Continuous Wavelet Transform.

Wavelet-based coherence is a widely adopted method that uses the Continuous Wavelet Transform (CWT) to decompose signals into time-scale space before computing coherence. The Morlet wavelet is the preferred mother wavelet due to its optimal joint time-frequency localization. The estimation process involves:

• Computing the CWT for each signal individually.
• Calculating the cross-wavelet transform to identify regions of high common power.
• Smoothing the wavelet spectra in both time and scale to produce a consistent estimator.
• Normalizing the smoothed cross-wavelet spectrum by the smoothed auto-wavelet spectra. This approach naturally provides multi-resolution analysis, offering high frequency resolution at low frequencies and high time resolution at high frequencies.

A critical output of time-frequency coherence analysis is the local phase difference between the two signals. The phase angle of the complex-valued cross time-frequency spectrum reveals the instantaneous lead-lag relationship. This is typically visualized as arrows overlaid on a coherence magnitude plot:

• Right-pointing arrows indicate the signals are in-phase (positively correlated).
• Left-pointing arrows indicate anti-phase (negatively correlated).
• Down-pointing arrows indicate the first signal leads the second by 90 degrees.
• Up-pointing arrows indicate the second signal leads the first by 90 degrees. This directional information is essential for inferring causal coupling dynamics in physiological and physical systems.

Coherence estimates are inherently biased and must be tested against a null hypothesis of no linear correlation. The standard approach uses Monte Carlo simulations to generate surrogate data pairs with identical spectral properties but no coherence. Specifically:

• A large ensemble of red noise processes (AR(1) models) is simulated, matching the autocorrelation structure of the original signals.
• Coherence is computed for each surrogate pair.
• The 95th percentile of the surrogate coherence distribution defines the significance threshold at each time-frequency point.
• Only coherence values exceeding this threshold are considered statistically significant, protecting against spurious correlations caused by spectral leakage or finite sample effects.

Time-frequency coherence is a foundational tool in neuroscience for analyzing the dynamic coupling between brain regions. Key applications include:

• Event-Related Coherence: Quantifying transient synchronization between EEG or MEG channels time-locked to a stimulus or cognitive event.
• Corticomuscular Coherence: Measuring the linear coupling between brain activity (EEG/MEG) and muscle activity (EMG) to study motor control pathways.
• Cross-Frequency Coupling Analysis: Detecting interactions where the phase of a low-frequency oscillation modulates the amplitude of a high-frequency oscillation, a phenomenon linked to neural communication and memory consolidation. These analyses reveal how functional brain networks form and dissolve on millisecond timescales.

While ordinary coherence is derived from the cross-spectral density (CSD) — a purely frequency-domain measure — time-frequency coherence captures transient, non-stationary coupling. The CSD assumes signal statistics are stationary over the analysis window, averaging out time-varying dynamics. Time-frequency coherence, by contrast, preserves temporal localization. This is critical for analyzing signals with abrupt state changes, such as epileptic seizure onset in EEG, fault transients in machinery vibration, or burst-mode communication signals. The trade-off is governed by the Heisenberg-Gabor uncertainty principle, where improved time resolution necessarily degrades frequency resolution.

The foundational joint-domain transform that computes the Fourier spectrum of successive windowed signal segments. By dividing a longer signal into shorter, equal-length frames, the STFT reveals how sinusoidal frequency content evolves over time. It serves as the primary input for computing the time-frequency coherence function, where the cross-spectral density and auto-spectral densities are estimated from the complex STFT coefficients of two signals.

A measure of the shared power between two signals as a function of frequency. In the time-frequency domain, it is estimated by multiplying the STFT of one signal by the complex conjugate of the STFT of the other. The magnitude and phase of this cross-term are the core components of time-frequency coherence, indicating not just if two signals are correlated, but the phase lag between them at each time-frequency point.

A visual representation of the squared magnitude of the STFT, displaying signal energy as a two-dimensional heatmap. While a spectrogram shows the power distribution of a single signal, time-frequency coherence can be viewed as a normalized, bounded spectrogram-like map (values between 0 and 1) that reveals the degree of linear coupling between two signals, independent of their individual amplitude scales.

An alternative to Fourier-based coherence that uses the Continuous Wavelet Transform (CWT) instead of the STFT. This method provides a multi-resolution analysis of the correlation between two signals, offering better time resolution for high-frequency events and better frequency resolution for low-frequency oscillations. It is particularly effective for analyzing transient, non-stationary coupling in geophysical and biomedical signals.

A quadratic time-frequency distribution offering the highest possible joint resolution. Coherence can be defined using the WVD's cross-term formulation, but this approach is highly susceptible to cross-term interference artifacts. Advanced kernels from Cohen's Class, like the Choi-Williams distribution, are often applied to suppress these spurious oscillations before computing a clean coherence estimate for multi-component signals.

The time-varying phase angle of an analytic signal, derived via the Hilbert transform. Phase synchronization between two signals, a concept closely related to coherence, is often quantified by examining the difference in their instantaneous phases. While magnitude-squared coherence measures amplitude correlation, phase coherence specifically tracks the stability of the phase relationship over time, a key metric in neural coupling studies.