bearing-gen-formula/main.tex

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\title{Synthetic Bearing-Fault Signal Generation\\ for Target-Normal-Only Synthetic-to-Real Diagnosis}
\author{}
\date{}

\begin{document}
\maketitle

\begin{abstract}
Reliable bearing fault diagnosis is difficult when real fault data cannot be collected without risking machine damage. We study synthetic-to-real diagnosis under a strict inductive setting in which training uses labelled synthetic faults and healthy data from the target machine, while no target-fault samples, labelled or unlabelled, are available before evaluation. This avoids the transductive use of fault examples from the test domain adopted in many domain-adaptation studies and preserves a clear separation between adaptation and testing. We propose a healthy-baseline residual framework that subtracts the mean healthy spectrum of each domain to suppress shared background structure and emphasize transferable fault signatures. The residual is combined with the log-magnitude spectrum and processed by a cosine-prototype classifier with conditional domain adaptation. Across multiple bearing benchmarks, the framework recovered every real fault class that remained unseen during training and, on most benchmarks, outperformed established adaptation and physics-guided baselines while remaining competitive on the rest. Ablation experiments attributed the main improvement to the residual representation and its interaction with the angular classifier rather than to the adaptation procedure alone. These findings show that healthy-only target data can support cross-domain bearing diagnosis when the framework isolates physically meaningful fault structure without relying on target-fault observations.
\end{abstract}


\section{Setting}
We train a fault-type classifier using only (i) \emph{synthetic} fault signals and
(ii) \emph{real healthy} recordings from the target machine; no real fault labels are
used (target-normal-only). Synthetic faults are produced by injecting
physically-parameterized impulse trains into a real healthy carrier, following the
expert-knowledge model of Wang, Taal and Fink
(\texttt{arXiv:2107.01849}, Eqs.~5--6).

\section{Generation model}
Let $n(t)$ be a measured healthy window from the target machine, normalized to unit
standard deviation,
\begin{equation}
  n \leftarrow n / \big(\sigma(n)+\varepsilon_0\big).
\end{equation}
The synthetic signal of a given fault class is
\begin{equation}
  \boxed{\;\varepsilon(t) \;=\; \sum_{i} A_i\, s\!\big(t - t_i\big) \;+\; \beta\, n(t)\;}
  \label{eq:gen}
\end{equation}

\paragraph{Impulse timing.}
Impacts recur at the fault period $T$ (in samples) with random slip:
\begin{equation}
  t_i = i\,T + \delta_i,\qquad
  T = \frac{f_s}{f_c},\qquad
  f_c = o_{\text{fault}}\cdot f_r,\qquad
  \delta_i \sim \mathcal{N}\!\big(0,(0.01\,T)^2\big),
\end{equation}
where $f_s$ is the sampling rate, $f_r$ the shaft rate (Hz) and
$o_{\text{fault}}$ the characteristic fault order (BPFO, BPFI, $2\times$BSF, \dots),
obtained from the bearing geometry.

\paragraph{Single impact.}
Each impact is a short Hann window of duty $5\%$ of $T$, passed through a wide-band
band-pass filter (a proxy for the structural resonance), and scaled by the
\textbf{impact gain} $G$:
\begin{equation}
  s(\tau) = G\cdot \mathrm{BP}_{[f_1,f_2]}\!\Big( w_{\text{Hann}}(\tau)\Big),
  \qquad
  \operatorname{len}(w) = \max\big(10,\ \lceil 0.05\,T\rceil\big).
  \label{eq:impact}
\end{equation}

\paragraph{Amplitude modulation (load-zone passage).}
\begin{equation}
  A_i = \gamma_i \cdot \sum_{k=0}^{K}\alpha_k
        \cos\!\Big(\tfrac{2\pi k\, t_i}{Q}\Big),
  \qquad \gamma_i \sim \mathcal{N}(1,0.1^2),
  \label{eq:mod}
\end{equation}
with $K=4$ and $\alpha=(1,\,0.76,\,0.38,\,0.11,\,0.05)$. The modulation period $Q$
depends on the fault type:
\begin{center}
\begin{tabular}{lll}
\toprule
Fault & Modulation & $Q$ \\
\midrule
Outer race (stationary) & none, $A_i=\gamma_i$ & $Q\to\infty$ \\
Inner race & shaft rate & $Q = f_s/f_r$ \\
Rolling element (ball) & cage rate & $Q = f_s/(o_{\text{FTF}}\,f_r)$ \\
\bottomrule
\end{tabular}
\end{center}

\paragraph{Background level.}
\begin{equation}
  \beta \sim \mathcal{U}(0.25,\,1).
\end{equation}

\paragraph{Normal class.}
The synthetic normal is the healthy carrier itself, $\varepsilon(t)=\beta\,n(t)$, so the
synthetic-normal distribution matches the real-normal distribution by construction.

\section{The impact gain $G$}
$G$ in \eqref{eq:impact} sets the impact-to-carrier amplitude ratio, i.e.\ the
\emph{signal-to-noise ratio of the synthetic fault}. Because the carrier $n$ has unit
variance, $G$ is, in principle, machine-independent.

\paragraph{Honest use (this work).}
$G$ is treated as a \textbf{fixed hyperparameter}, swept over $G\in\{15,40,100\}$.
It is \emph{not} calibrated against any real fault recording. Calibrating $G$ to match a
measured fault peak,
\begin{equation}
  G^\star = \arg\min_{G}\Big|\, \operatorname{peak}_{o_{\text{fault}}}
  \big(\text{synthetic}(G)\big)
  - \operatorname{peak}_{o_{\text{fault}}}\big(\text{\emph{real fault}}\big)\Big|,
  \label{eq:leak}
\end{equation}
would inject target-fault information into a target-normal-only protocol and is therefore
disallowed.

\paragraph{Domain-randomization alternative.}
Equivalently, $G$ need not be chosen at all: applying data-agnostic augmentations to the
synthetic during training --- random gain $g\sim\mathcal{U}(0.5,1.5)$ and additive noise
at a random SNR $\sim\mathcal{U}(5,30)\,\text{dB}$, among others --- randomizes the fault
SNR over a range, removing $G$ as a tunable and keeping the pipeline leak-free.

\section{Representation}
Each window (real or synthetic) is mapped to an \emph{envelope--order spectrum}:
\begin{equation}
  x \;\xrightarrow{\ \mathrm{BP}_{[500,4000]\,\text{Hz}}\ }\;
  \big|\mathcal{H}\{\cdot\}\big| \;\xrightarrow{\text{standardize}}\;
  \big|\mathcal{F}\{\cdot\}\big| \;\xrightarrow{\ f\mapsto f/f_r\ }\;
  \text{resample to } [0,30]\ \text{orders},\ 1000\ \text{bins},
\end{equation}
where $\mathcal{H}$ is the Hilbert-envelope operator and the order axis normalizes out the
shaft speed.

\end{document}