Synthetic generation model with impact gain G

Wang-carrier impulse injection; G as fixed hyperparameter (not fault-calibrated).

Co-Authored-By: Claude Opus 4.8 <noreply@anthropic.com>

main.tex

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+\documentclass[11pt]{article}
+\usepackage[T1]{fontenc}
+\usepackage[utf8]{inputenc}
+\usepackage[english]{babel}
+\usepackage{amsmath,amssymb}
+\usepackage{geometry}
+\geometry{a4paper,margin=25mm}
+\usepackage{booktabs}
+\usepackage{siunitx}
+
+\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}