Frequency-Domain Stationarity Correction

• Frequency-Domain Stationarity Correction is a collection of methods that adjust non-stationary time-series by leveraging spectral transforms like Fourier and fractional Fourier to model drift and transient artifacts.
• It employs advanced techniques including LTV impulse response modeling, window function corrections, and frequency variogram analysis to achieve reliable spectral estimation.
• Applications span electrochemical impedance spectroscopy, spatio-temporal Gaussian processes, and deep multivariate forecasting, enhancing model robustness and data integrity.

Frequency-domain stationarity correction encompasses a collection of methodologies for adjusting, estimating, or modeling time-series and spatio-temporal data that violate the classical assumption of stationarity, by operating primarily in the frequency (or spectral) domain. The core principle is to use spectral analysis—typically through the discrete or continuous Fourier (or fractional Fourier) transform—to model, detect, and correct the effects of time-variation, drift, or transient artifacts. These corrections are critical in both applied fields such as electrochemical impedance spectroscopy (EIS) and data-driven machine learning pipelines for multivariate time series, as well as in theoretical contexts including spatio-temporal covariance estimation and stochastic process theory.

1. Breakdown of Classical Stationarity Assumptions

Classical linear systems and time-series analysis frequently presuppose (i) time-invariance of system properties and (ii) stationarity of time-series inputs—meaning constant first and second moments and time-invariant impulse responses. Under these assumptions, convolution and spectral division (e.g., $Z(\omega) = V(\omega)/I(\omega)$ for EIS) yield physically meaningful transfer functions or impedance spectra. However, practical measurements commonly encounter processes with evolving operating points, environmental drifts, or structural changes in their second-order statistics, leading to violations of strict stationarity. For example, in battery systems under charge or discharge, the open-circuit voltage and internal dynamics are time-varying, causing artifacts such as frequency-domain "skirts" in naive spectral analyses (Hallemans et al., 2023).

In the frequency domain, three major manifestations of non-stationarity are distinguished:

• Time-varying means, drifts, or trends corrupting low-frequency spectral content;
• Time-varying covariances or impulse responses yielding nonstationary transfer functions;
• Nonstationary boundary or windowing artifacts when observing or modeling systems over finite intervals (Martini et al., 2019).

2. Mathematical Foundations for Frequency-Domain Correction

Central to frequency-domain stationarity correction is extending the spectral framework to encompass linear time-varying (LTV) models and to account for window-induced or process-induced nonstationarities.

2.1 LTV Impulse Response Models

A time-varying impulse response $z(\tau,t)$ replaces the LTI $z(\tau)$, yielding the generalized convolution

$$v(t) = v_0(t) + \int_{-\infty}^\infty z(\tau,t) i_\mathrm{exc}(t-\tau)\, d\tau,$$

with $v_0(t)$ capturing slow drift. The associated time-varying impedance spectrum is

$$Z(\omega, t) = \int_{-\infty}^{\infty} z(\tau, t) e^{-j\omega \tau} d\tau,$$

so $$v(t)=v_0(t) + \mathcal{F}^{-1}\{Z(\omega, t) I_\mathrm{exc}(\omega)\}$$ (Hallemans et al., 2023).

2.2 Window Functions and Correction Terms

When analyzing data on finite intervals, windowing introduces correction terms. For a window $w(t)$ and observed time-series $x(t)$,

$$X_w(j\omega) = \int_0^T w(t)x(t) e^{-j\omega t} dt,$$

the relationship between frequency-domain input and output involves additional transient terms $\Delta T(j\omega)$: $$H(j\omega) U_w(j\omega) - Y_w(j\omega) = \Delta T(j\omega),$$ where $\Delta T(j\omega)$ aggregates contributions from window derivative-weighted data terms, computable analytically and supporting rapid, parameter-efficient least squares estimation (Martini et al., 2019).

2.3 Intrinsic Stationarity and Frequency Variogram

In spatio-temporal analysis, intrinsic stationarity—requiring only stationary increments—can be leveraged by differencing and analyzing the frequency variogram

$$G_{\mathbf{h}}(\omega) = \mathbb{E}|X_\ell(\omega) - X_m(\omega)|^2,$$

allowing spectral-domain parametric modeling on increments whose variance is shift-invariant, even when the underlying process covariance is not (Rao et al., 2016).

2.4 Stationarity in Fractional Fourier Domains

For stochastic processes, the generalized frequency representation via the fractional Fourier transform (FRFT) allows analysis of stationarity properties in fractional domains. Necessary (but not sufficient) conditions for stationarity after FRFT include zero mean, and spectral "whitening" in the given fractional domain may be achieved via tailored spectral-domain filtering (Shafie et al., 2012).

3. Correction Methodologies and Algorithmic Procedures

A broad spectrum of algorithmic approaches address frequency-domain correction for non-stationarity, ranging from local windowing to global regression and modern deep learning modules.

Method	Key Idea	Reference
STFT-EIS	Short-time Fourier transform (FFT) over sliding windows to capture local stationarity; autocorrelation estimated using local spectra	(Hallemans et al., 2023)
DMFA	Frequency-domain bandpass filtering, followed by time reconstruction and local transfer function estimation	(Hallemans et al., 2023)
Operando EIS	Series regression using low-order basis expansion (e.g., Legendre polynomials) in time, explicit correction for drift and time-variation	(Hallemans et al., 2023)
Windowed ODE correction	Computation of window-induced correction terms, analytic or via recurrences, included in frequency-domain system identification	(Martini et al., 2019)
Frequency Variogram	Parametric modeling of stationary increments in Fourier space across spatial (or spatio-temporal) lags	(Rao et al., 2016)
Fractional PSD Filtering	FRFT, estimation and flattening of fractional PSD, and inverse FRFT for whitened outputs	(Shafie et al., 2012)
F-Corr (FSC) in D-CTNet	FFT of feature maps, computation of input and predicted autocorrelation via power spectrum, scaling predicted features to match input spectral shape	(Wang et al., 30 Nov 2025)

3.1 Multisine/Multiband Excitation and Correction

For EIS, the design of odd-random phase multisines ensures rich frequency coverage, minimizes nonlinear distortion, and supports statistical weighting and drift estimation via frequency domain regression (Hallemans et al., 2023).

3.2 Patch-wise Frequency Alignment in Deep Learning

The Frequency-Domain Stationarity Correction module (F-Corr) in D-CTNet implements FFT/IFFT-based autocorrelation comparison between input and predicted feature maps, scaling features to achieve spectral-autocorrelation alignment and empirically improving generalization under shifting distributions (Wang et al., 30 Nov 2025).

4. Practical Implementation and Case Studies

4.1 Electrochemical Impedance Spectroscopy

In a 48X 4.8 Ah Li-ion cell, classical EIS under stationarity shows FFT power at discrete harmonics. Under charging (nonstationary drift), "skirts" around excited lines indicate time variation in system dynamics and spurious spectral mixing. Application of operando EIS regression using Legendre polynomial time bases enables simultaneous modeling of dynamic impedance and drift, yielding robust recovery of physically meaningful, time-resolved equivalent circuit parameters across the charge/discharge trajectory (Hallemans et al., 2023).

4.2 Spatio-Temporal Gaussian Processes

Frequency-domain correction via the variogram approach enables estimation of covariance parameters on increments, thus sidestepping nonstationarity in the raw process and allowing computationally efficient, asymptotically valid likelihood-based inference in large spatio-temporal networks (Rao et al., 2016).

4.3 Deep Multivariate Time-Series Forecasting

In D-CTNet, training with frequency-domain stationarity correction leads to measurable improvements in predictive MSE and MAE under a wide range of real-world industrial datasets, surpassing established baselines. Ablations confirm that removing F-Corr degrades generalization under non-stationarity, evidencing the empirical relevance of spectral alignment in learning pipelines (Wang et al., 30 Nov 2025).

5. Statistical Weighting, Filtering, and Spectral Calibration

Post-correction procedures enhance robustness and inferential clarity:

• Spectral weighting suppresses frequency bands dominated by noise or drift.
• SNR-adaptive weighting schemes optimize least-squares estimation or regression.
• Inclusion of drift terms or low-order basis expansions explicitly accounts for time-varying means or trends without resorting to raw time-domain differencing.
• Spectral calibration via parametric or non-parametric estimation of the frequency variogram, or via direct inspection of window-induced artifacts, further refines corrected estimators (Hallemans et al., 2023, Rao et al., 2016, Martini et al., 2019).

6. Theoretical and Computational Properties

Distinctive features of frequency-domain stationarity correction include:

• Exploitation of the asymptotic independence and (for increments) Gaussianity of DFT coefficients at distinct frequencies.
• Avoidance of large-matrix inversions in covariance estimation via Whittle-type approximations.
• Reduction of aliasing and leakage error through analytic correction for windowed data instead of ad hoc polynomial fitting.
• Extension to fractional and multivariate spectral domains, with explicit formulae dictating necessary and sufficient conditions for stationarity of transforms or filtered outputs (Shafie et al., 2012, Martini et al., 2019).

7. Outlook and Significance

Frequency-domain stationarity correction provides a flexible, computationally efficient, and physically informed toolkit for analyzing nonstationary systems and datasets. Its theoretical motivation spans stochastic process theory, system identification, and spectral analysis, while practical implementations in fields ranging from electrochemistry to machine learning demonstrate its versatility and empirical strength. The ongoing integration with deep learning architectures (e.g., D-CTNet) and adaptation to non-Euclidean or irregularly sampled domains suggests a sustained trajectory of methodological innovation (Wang et al., 30 Nov 2025, Rao et al., 2016).