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z6Z9{^LMC`|I#=MIkA<58YajTu+VKDH1d+2DxDy`(^$QOd8xI!}HMNwAG}8Y7!z70V diff --git a/semester3/numcs/parts/01_interpolation/01_trigonometric/01_dft/00_intro.tex b/semester3/numcs/parts/01_interpolation/01_trigonometric/01_dft/00_intro.tex index 510568c..ebbd331 100644 --- a/semester3/numcs/parts/01_interpolation/01_trigonometric/01_dft/00_intro.tex +++ b/semester3/numcs/parts/01_interpolation/01_trigonometric/01_dft/00_intro.tex @@ -1,3 +1,4 @@ +\newsection \subsection{Diskrete Fourier Transformation} % NOTE: I'll do these 2 subchapters. Based on the lecture, we can leave out quite a lot here. @@ -8,3 +9,37 @@ % 2.3.4 relativ wichtig (Einführung Faltungen), tendenziell viel % 2.4 FFT wichtig, aber kurz + +\subsubsection{Motivation} + +Nutzen wir die Trapezregel um approximativ die Fourierkoeffizienten $\hat{f}_N(k)$ auf äquidistanten Punkten $l_t=\frac{l}{N}$ $(0 \leq l \leq N-1)$ zu bestimmen, erhalten wir tatsächlich ein Polynom $p_{N-1}$ welches die Interpolationsbedingung erfüllt: +\begin{align*} + p_{N-1}(t) = \sum_{k=-\frac{N}{2}}^{\frac{N}{2}-1} \hat{f}_N(k)e^{2\pi ikt} +\end{align*} + +Der Beweis hierfür ist im Skript auf p. $71$. Die $N$-te Einheitswurzel wird hier definiert: + +\fancydef{$N$-te Einheitswurzel} $\omega_N := \exp(\frac{-2\pi i}{N})$ + +\fancyremark{Eigenschaften von $\omega_N$} +\begin{multicols}{3} +\begin{align*} + \forall j,k \in \mathbb{Z}:\quad & \omega_N^{k+jN}=\omega_N^k \\ + \forall k \in \mathbb{Z}, t \in \mathbb{R}:\quad & \omega_N^{t+kN}=\omega_N^t +\end{align*} + +\newcolumn +\begin{align*} + \omega_N^N=1 \\ + \omega_N^{N/2}=-1 +\end{align*} + +\newcolumn +\begin{align*} + \sum_{k=0}^{N-1} \omega_N^{kj} = \begin{cases} + N, & j \equiv_N 0 \\ + 0, & \text{ sonst} + \end{cases} +\end{align*} +\newcolumn +\end{multicols} \ No newline at end of file diff --git a/semester3/numcs/parts/01_interpolation/01_trigonometric/01_dft/01_construction.tex b/semester3/numcs/parts/01_interpolation/01_trigonometric/01_dft/01_construction.tex index e69de29..1ac157b 100644 --- a/semester3/numcs/parts/01_interpolation/01_trigonometric/01_dft/01_construction.tex +++ b/semester3/numcs/parts/01_interpolation/01_trigonometric/01_dft/01_construction.tex @@ -0,0 +1,26 @@ +\subsubsection{Konstruktion} + +\fancydef{Trigonometrische Basis} $\{v_0, v_{N-1}\}$ ist eine Basis von $\mathbb{C}^N$, wobei $v_k = \begin{bmatrix} + \omega_N^{0\cdot k} \\ \omega_N^{1\cdot k}1 \\ \vdots \\ \omega_N^{(N-1)\cdot k} +\end{bmatrix} \in \mathbb{C}^N$ + +Die symmetrische, nicht hermitesche Matrix $V = [v_0,\ \ldots\ , v_{N-1}]$ ist dann eine orthogonale Basis für $\mathbb{C}^N$: $V^HV = N\cdot I_N$. + +Ebenfalls ist $V$ die Basiswechsel Matrix Trigonometrische Basis ($z$) $\mapsto$ Standardbasis ($y$). Algebraisch: +\begin{align*} + y = Vz \implies z = V^{-1}y = \frac{1}{N}V^Hy = \frac{1}{N}\underbrace{F_N}_{:= V^H} y +\end{align*} + +\fancydef{Fourier-Matrix} $F_N := V^H = \begin{bmatrix} + \omega_N^0 & \omega_N^0 & \cdots & \omega_N^0 \\ + \omega_N^0 & \omega_N^1 & \cdots & \omega_N^{N-1} \\ + \omega_N^0 & \omega_N^2 & \cdots & \omega_N^{2(N-1)} \\ + \vdots & \vdots & & \vdots \\ + \omega_N^0 & \omega_N^{N-1} &\cdots & \omega_N^{(N-1)^2} +\end{bmatrix} += \begin{bmatrix} + \omega_N^{jk} +\end{bmatrix}^{N-1}_{j,k = 0} \in \mathbb{C}^N +$ + +\fancydef{Diskrete Fourier Transformation} $\mathcal{F}_N: \mathbb{C} \to \mathbb{C}$ s.d. $\mathcal{F}_N(y) = F_N y$ \ No newline at end of file