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

\begin{document}
\maketitle

\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}
