spectral_analysis.tex

\input{prefix}

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\begin{document}


% links between Fourier transforms
\begin{tikzpicture}[auto, node distance=4cm]
    \node (cont) {$x(t)$};
    \node [right of=cont] (F) {$X(j \omega)$};
    \draw [o-*] (cont)--(F) node[midway, above]{$\mathcal{F}$};
    
    \node [below of=cont, yshift=2.2cm] (samp) {$x[k]$};
    \node [right of=samp] (DTFT) {$X(e^{j \Omega})$};
    \draw [o-*] (samp)--(DTFT) node[midway, above]{$\mathcal{F}_*$};
    \draw [->] (cont)--(samp) node[midway, left]{\shortstack[r]{sampling\\$x[k] = x(k T)$}};
    
    \node [below of=samp, yshift=2.2cm] (wind) {$x_N[k]$};
    \node [right of=wind] (DTFTN) {$X_N(e^{j \Omega})$};
    \draw [o-*] (wind)--(DTFTN) node[midway, above]{$\mathcal{F}_*$};
    \draw [->] (samp)--(wind) node[midway, left]{\shortstack[r]{windowing\\$x_N[k] = x[k] \cdot w[k]$}};;
    
    \node [below of=wind, yshift=2.2cm] (disc) {$x_N[k]$};
    \node [right of=disc] (DFT) {$X_N[\mu]$};
    \draw [o-*] (disc)--(DFT) node[midway, above]{$\text{DFT}_N$};
    \draw [->] (DTFTN)--(DFT) node[midway, right]{\shortstack[l]{sampling\\$X_N[\mu] = X_N(e^{j \frac{2 \pi}{N} \mu})$}};
    
\end{tikzpicture}


\end{document}