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