From aafeba34e4cd4cc56816a03ada99e9dab490620c Mon Sep 17 00:00:00 2001
From: Overleaf Sync <overleaf-sync@uncoded.ru>
Date: Wed, 15 Jul 2026 17:20:23 +0000
Subject: [PATCH] Overleaf snapshot 2026-07-15T17:20:23Z

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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}
+\documentclass{article}
+\usepackage{graphicx} % Required for inserting images
 
-\title{Synthetic Bearing-Fault Signal Generation\\ for Target-Normal-Only Synthetic-to-Real Diagnosis}
-\author{}
-\date{}
+\title{gen-formula}
+\author{comradeyar }
+\date{July 2026}
 
 \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.
+\section{Introduction}
 
 \end{document}