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a/semester3/numcs/parts/01_interpolation/01_trigonometric/00_fourier.tex b/semester3/numcs/parts/01_interpolation/01_trigonometric/00_fourier.tex index ff397d5..f87e40b 100644 --- a/semester3/numcs/parts/01_interpolation/01_trigonometric/00_fourier.tex +++ b/semester3/numcs/parts/01_interpolation/01_trigonometric/00_fourier.tex @@ -126,3 +126,27 @@ wobei $l = 0, 1 \ldots, N - 1$ und $N$ die Anzahl der Intervalle ist: \begin{align*} \hat{f}_N(k) := \frac{1}{N} \sum_{l = 0}^{N - 1} f(t_l) e^{-2\pi ikt_l} \approx \hat{f}(k) \end{align*} + +% TODO: Consider if we should use the below + +% \begin{tikzpicture} +% \begin{axis}[ +% legend pos=outer north east, +% title=Function plot of $f(x)$ (parts coloured), +% axis lines = box, +% xlabel = $x$, +% ylabel = $y$, +% variable = t, +% trig format plots = rad, +% ] +% \addplot [ +% domain=1:4, +% samples=70, +% color=blue, +% ] +% {log2(x)}; +% \addlegendentry{$ y=x^2 - x - 0.5$} +% \end{axis} +% \node (0) at (0, 0) {}; +% \end{tikzpicture} + diff --git a/semester3/numcs/parts/01_interpolation/01_trigonometric/02_fft.tex b/semester3/numcs/parts/01_interpolation/01_trigonometric/02_fft.tex index 983d471..e0976a0 100644 --- a/semester3/numcs/parts/01_interpolation/01_trigonometric/02_fft.tex +++ b/semester3/numcs/parts/01_interpolation/01_trigonometric/02_fft.tex @@ -1,2 +1,38 @@ \newsection \subsection{Schnelle Fourier Transformation} +Da es viele Anwendungen für die Fourier-Transformation gibt, ist ein Algorithmus mit guter Laufzeit sehr wichtig. +Während eine naive version des DFT-Algorithmus eine Laufzeit von $\tco{N^2}$ hat, +so hat der Fast Fourier Transform Algorithmus nur eine Laufzeit von $\tco{N \log(N)}$, +was bei $N = 1024$ bereits eine Laufzeitsverbesserung von $100\times$ mit sich bringt ($\tco{10\,000}$ vs $\tco{1\,000\,000}$ Operationen)! +Die untenstehende Abbildung \ref{fig:trigo-interp-fft-runtimes} findet sich, zusammen mit dem Code, +mit der sie produziert wurde im Skript auf Seite 86-88 + +\begin{figure}[h!] + \begin{center} + \includegraphics[width=0.7\textwidth]{assets/01_interpolation/01_trigonometric/fft-runtimes.png} + \end{center} + \caption{Vergleich der Laufzeit von verschiedenen Fourier-Transformations-Algorithmen} + \label{fig:trigo-interp-fft-runtimes} +\end{figure} + +Der hier besprochene Cooley-Tukey-Algorithmus wurde ursprünglich von Gauss 1805 entdeckt, dann vergessen und schliesslich 1965 von Cooley und Tukey wiederentdeckt. +Der Algorithmus verwendet einen ``Divide and Conquer'' Approach, also ist logischerweise die Idee, +dass man die Berechnung einer DFT der Länge $n$ auf die Berechnung vieler DFTs kleinerer Längen zurückführen kann. + +Für den Algorithmus müssen folgende vier Optionen betrachtet werden: +\begin{enumerate}[label=\Roman*] + \item Vektoren der Länge $N = 2m \Longrightarrow$ Laufzeit ideal + \item Vektoren der Länge $N = 2^L \Longrightarrow$ Laufzeit ideal + \item Vektoren der Länge $N = pq$ mit $p, q \in \Z \Longrightarrow$ Etwas langsamer + \item Vektoren der Länge $N$, mit $N$ prim $\Longrightarrow$ ca. $\tco{N^2}$, besonders für $N$ gross +\end{enumerate} +Wir formen die Fourier-Transformation um für den ersten Fall ($N = 2m$): +\begin{align*} + c_k & = \sum_{j = 0}^{N - 1} y_j e^{- \frac{2\pi i}{N} jk} \\ + & = \sum_{j = 0}^{m - 1} y_{2j} e^{-\frac{2 \pi i}{N}2jk} + \sum_{j = 0}^{m - 1} y_{2j + 1} e^{-\frac{2\pi i}{N}(2j + 1)k} \\ + & = \sum_{j = 0}^{m - 1} \left( y_{2j} e^{-\frac{2 \pi i}{N \div 2}jk} \right) + + e^{- \frac{2\pi}{N} k} \left( \sum_{j = 0}^{m - 1} y_{2j + 1} e^{-\frac{2\pi i}{N \div 2}jk} \right) +\end{align*} +Der zweite Fall ist einfach eine rekursive Weiterführung des ersten Falls, +bei welchem dann das $m$ kontinuierlich weiter dividiert wird bis zum Trivialfall mit einer $1 \times 1$-Matrix. +