Three-Axis Independent Phase Adjustment

• Three-axis independent phase adjustment is a method that realigns phase offsets on orthogonal signals, ensuring each axis meets a predefined zero-phase target.
• It employs DFT-based phase extraction and computed time shifts per axis to minimize distortion while enhancing signal integrity.
• Empirical evaluations demonstrate improved fault diagnosis accuracy and enhanced electro-optic modulator stability, affirming its practical effectiveness.

Three-axis independent phase adjustment refers to the explicit, axis-wise manipulation and alignment of phase information in multidimensional signals, ensuring that each of three orthogonal axes achieves a pre-designated phase target—usually zero—at a selected spectral component or system operating point. The technique is critical in domains such as fault diagnosis of rotating machinery using three-axis accelerometers and the stabilization of dual-parallel electro-optic (I/Q) modulators, where uncontrolled or random phase offsets can severely degrade subsequent signal processing or classification.

1. Mathematical Formulation and Core Principles

In rotating machinery condition monitoring, the three-axis accelerometer captures synchronized vibration signals along $X$, $Y$, and $Z$ directions: $s(t) = [x(t), y(t), z(t)]^\top$, sampled at rate $f_s$. After windowing, the discrete Fourier transform (DFT) is applied independently to each axis for each window, yielding $X^{(i)}[k]$, $Y^{(i)}[k]$, $Z^{(i)}[k]$. At the dominant (known or estimated) frequency $f_d$, the DFT phase $\phi_x^{(i)} = \arg X^{(i)}[k_d]$ (similarly for $y$ and $z$) encodes the axis-specific phase offset.

Independent phase alignment is realized by time-shifting each axis’s signal by $\Delta t_a^{(i)} = -\phi_a^{(i)} / (2\pi f_d)$, for $a \in \{x, y, z\}$. The shifted segment $a'^{(i)}(n) = a^{(i)}( n + \Delta t_a^{(i)} f_s )$ ensures, post-adjustment, that the phase at $k_d$ is zero for all axes. Practical shift implementation utilizes linear or higher-order interpolation in the time domain or phase multiplication in the frequency domain, guaranteeing minimal distortion and efficient computation (Nagahama et al., 17 Dec 2025).

Analogously, in dual-parallel (I/Q) electro-optic modulators, each arm’s phase bias (sub-MZI biases $\Phi_A$, $\Phi_B$, outer MZI $\Phi_P$) is stabilized independently. By modulating each internal bias with an auxiliary RF tone (here, 2 MHz, $\Omega_{LF}$), demodulating the output photocurrent yields error signals $V_{err,A}$, $V_{err,B}$, $V_{err,P}$ orthogonal to each other and linear in bias error, enabling three independent closed feedback loops to achieve precise bias stabilization (Wald et al., 2022).

2. Algorithmic Workflow and Implementation

The three-axis independent adjustment pipeline comprises the following stages:

1. Preprocessing: Removal of initial transients (e.g., sensor warm-up) and extraction of analysis segments.
2. Windowing: Segment the multichannel data into frames of $L$ samples (e.g., $L=512$).
3. DFT Computation: Compute the FFT per axis per window.
4. Dominant Frequency Estimation: Select or estimate $f_d$ and determine $k_d = \mathrm{round}( f_d \cdot L / f_s )$.
5. Phase Extraction and Time-Shift Calculation: For each axis, extract the phase at $k_d$ and compute the corresponding shift.
6. Shift Application: Apply axis-wise time shifts via chosen implementation.
7. Assembly: Construct the adjusted, zero-phase-aligned window.

Pseudocode for this algorithm and parameter definitions are provided in (Nagahama et al., 17 Dec 2025).

For analog I/Q modulator stabilization, RF signal processing delivers three error signals via mixers and low-pass filters, each routed to a slow integrator driving the respective DC bias port. The architecture ensures minimal cross-coupling and phase margin $\sim 60^\circ$ for stable lock (Wald et al., 2022).

3. Computational Demands and Calibration

For vibration analysis, the computational complexity per window is $O(L \log L)$ per axis, dominated by the FFT and, if implemented in the frequency domain, the inverse FFT following phase correction. At typical settings (300 windows, $L=512$), total operations per file remain feasible for real-time or batch processing on modern hardware.

Key calibration parameters include:

• Dominant Frequency ($f_d$): Must match machine speed; can be estimated via spectral analysis.
• Window Size ($L$) & Hop: Affect temporal and spectral resolution.
• Interpolation Order: Higher-order methods may better preserve signal integrity for non-integer shift values.
• Adaptive Extensions: For variable-speed machinery, $f_d$ and, hence, $\Delta t_a$ must be recomputed per segment. The paper discusses the prospect of multi-frequency alignment as a future extension (Nagahama et al., 17 Dec 2025).

In analog stabilization, loop bandwidth (set by integrator time constant, $f_c \approx 100$ Hz) and communication parameters (mixing frequencies/phases, loop filter gains) are critical. Hardware component choices—including photodiode, mixers, and op-amps—directly impact residual noise, bandwidth, and long-term stability (Wald et al., 2022).

4. Empirical Performance and Comparative Analysis

In fault classification experiments employing a two-stage framework (self-supervised time-series prediction followed by SVM classification), three-axis independent adjustment yields architecture-agnostic gains across six deep learning models. The baseline (no adjustment) accuracy for the Transformer model is $90.8\% \pm 3.2\%$; independent adjustment improves this to $93.5\% \pm 0.5\%$ (+2.7 percentage points) (Nagahama et al., 17 Dec 2025).

Performance summary (Transformer architecture):

Preprocessing Method	Accuracy (%) ± std	Gain
No Adjustment	90.8 ± 3.2	—
Three-axis Independent	93.5 ± 0.5	+2.7 pp
X-axis Reference	96.2 ± 1.1	+5.4 pp
Y-axis Reference	95.2 ± 1.3	+4.4 pp
Z-axis Reference	96.1 ± 0.5	+5.3 pp

Cross-architecture results echo this trend, with independent adjustment conferring $2$–$7$ percentage points improvement over non-phase-aligned baselines.

In electro-optic modulator applications, three-axis stabilization allows suppression of the carrier and unwanted sideband to below $-27\,\mathrm{dB}$ relative to the desired sideband, with residual phase error noise $<0.1$ mrad RMS and multi-hour stability under ambient lab conditions (Wald et al., 2022).

5. Practical Considerations, Limitations, and Integration

Three-axis independent phase adjustment robustly removes random phase jitter arising from arbitrary window or sampling offsets but does so at the expense of discarding inter-axis phase differences. Such cross-axis phase relationships can encode essential spatial or dynamic properties—vital, for example, in detecting faults manifesting as relative phase changes across axes (e.g., asymmetric rotor mass). Empirical results show that preserving inter-axis phase (via single-axis reference alignment) recovers an additional $\sim$2 percentage points in classification accuracy; thus, full exploitation of tri-axial sensors favors reference-based methods when spatial phase cues are informative (Nagahama et al., 17 Dec 2025).

Integration as a preprocessing step is straightforward: the procedure fits naturally after segmentation and can be batched or pipelined on parallel architectures (multi-core CPU, GPU). Calibration must ensure accurate tracking of $f_d$, especially in variable-speed contexts.

In analog phase control, sequence and tuning of loop closure, demodulation phases, and DC bias offsets are essential to achieve and maintain independent locking. Hardware limitations (thermal drift, device imperfections) set ultimate stability bounds. System design (loop bandwidth, filter cut-offs) must balance noise rejection with immunity to mechanical/thermal resonances.

6. Future Directions and Extensions

A principal trajectory for extension is multi-frequency phase alignment, where axes are aligned to multiple harmonics or spectral components, potentially with adaptive windowing or joint-trainable alignment layers in end-to-end deep networks. In analog stabilization, digital or hybrid feedback loops may offer enhanced adaptivity and integration with control and monitoring systems. The systematic study of trade-offs between independent and reference-based alignment, and adaptive strategies in environments with non-stationary phase structure, remains an open research topic (Nagahama et al., 17 Dec 2025).