latex: Match with Overleaf current work

latex/chapters/id/03_methodology/steps/preprocessing/index.tex

@@ -0,0 +1,66 @@
+\subsection{Signal Normalization}
+Each raw acceleration time series 
+\(\mathbf{a}_{k}(n)\), \(n = 0,1,\dots,N-1\) with \(N=262144\) samples (collected at \(f_s=1024\) Hz over 256 s) :contentReference[oaicite:0]{index=0} is first standardized to zero mean and unit variance:
+\[
+\tilde a_{k}(n)
+=\frac{a_{k}(n)-\mu_{k}}{\sigma_{k}},
+\quad
+\mu_{k}=\frac{1}{N}\sum_{n=0}^{N-1}a_{k}(n),
+\quad
+\sigma_{k}=\sqrt{\frac{1}{N}\sum_{n=0}^{N-1}\bigl(a_{k}(n)-\mu_{k}\bigr)^{2}}.
+\]
+
+\subsection{Framing and Windowing}
+The normalized signal \(\tilde a_{k}(n)\) is chopped into overlapping frames of length \(W\) samples with hop size \(H\). The \(p\)-th frame is
+\[
+x_{k,p}[m]
+=\tilde a_{k}(pH + m)\,w[m],
+\quad
+m=0,1,\dots,W-1,
+\]
+where \(w[m]\) is a chosen window function (e.g., Hamming).
+
+\subsection{Short-Time Fourier Transform (STFT)}
+For each frame \(x_{k,p}[m]\), compute its STFT:
+\[
+S_{k}(f,p)
+=\sum_{m=0}^{W-1}x_{k,p}[m]\;e^{-j2\pi\,f\,m/W},
+\]
+where \(f=0,1,\dots,W-1\) indexes frequency bins :contentReference[oaicite:1]{index=1}.
+
+\subsection{Spectrogram and Log-Magnitude}
+Form the magnitude spectrogram
+\[
+M_{k}(f,p)
+=\bigl|S_{k}(f,p)\bigr|,
+\]
+and apply log scaling for numerical stability:
+\[
+L_{k}(f,p)
+=\log\bigl(1 + M_{k}(f,p)^{2}\bigr).
+\]
+This yields a time–frequency representation
+\(\mathbf{L}_{k}\in\mathbb{R}^{F\times P}\), with \(F\) frequency bins and \(P\) frames.
+
+\subsection{Feature Matrix Assembly}
+For each column \(j\in\{1,\dots,5\}\), select only the two endpoint sensors:
+\[
+\mathbf{L}_{\text{bot},j} = \mathbf{L}_{(j)},\quad
+\mathbf{L}_{\text{top},j} = \mathbf{L}_{(25+j)},
+\]
+and stack them:
+\[
+\mathbf{F}_{j}
+=
+\begin{bmatrix}
+\mathbf{L}_{\text{bot},j} \\[6pt]
+\mathbf{L}_{\text{top},j}
+\end{bmatrix}
+\;\in\mathbb{R}^{2F\times P}.
+\]
+Finally, flatten into a feature vector:
+\[
+\mathbf{f}_{j}
+=\operatorname{vec}\bigl(\mathbf{F}_{j}\bigr)
+\;\in\mathbb{R}^{2FP}.
+\]