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b/semester3/numcs/parts/01_interpolation/01_trigonometric/01_dft/01_construction.tex @@ -37,8 +37,6 @@ Die diskreten Fourier-Koeffizienten $\gamma_k$ sind eine Umsortierung der trigon \end{align*} \end{multicols} -\newpage - \fancydef{Fourier-Matrix}\\ \begin{align*} F_N := V^H = [v_0, \ldots, v_{N-1}]^H = \begin{bmatrix} diff --git a/semester3/numcs/parts/01_interpolation/01_trigonometric/01_dft/02_fftshift.tex b/semester3/numcs/parts/01_interpolation/01_trigonometric/01_dft/02_fftshift.tex index 25fda00..4989c0b 100644 --- a/semester3/numcs/parts/01_interpolation/01_trigonometric/01_dft/02_fftshift.tex +++ b/semester3/numcs/parts/01_interpolation/01_trigonometric/01_dft/02_fftshift.tex @@ -1,4 +1,4 @@ -\subsection{DFT in Numpy} +\subsubsection{DFT in Numpy} Sei $y$ in der Standardbasis, und $c = \mathcal{F}_N(y)$, also $y$ in der trig. Basis. $$ @@ -10,7 +10,6 @@ Seien $z = \frac{1}{N} F_N y$ und $\zeta = \verb|fft.fftshift|(z)$. \begin{align*} f(x) \approx \underbrace{\sum_{k=-N/2}^{N/2-1} \zeta_k \cdot e^{2 \pi ikx} }_{\text{Form des trig. Polynoms}} \end{align*} - \setcounter{all}{13} \inlineremark Man kann mit dieser Approximation einfach die $L^2$-Norm und Ableitungen berechnen: \begin{multicols}{2} @@ -23,6 +22,4 @@ Seien $z = \frac{1}{N} F_N y$ und $\zeta = \verb|fft.fftshift|(z)$. \begin{align*} f'(t) \approx \sum_{k=-N/2}^{N/2-1} (2\pi ik) \zeta_k \cdot e^{2 \pi ikx} \end{align*} -\end{multicols} - -Im Skript S. 78 - 83 befinden sich einige sehr gute Anwendungsbeispiele. \ No newline at end of file +\end{multicols} \ No newline at end of file diff --git a/semester3/numcs/parts/01_interpolation/01_trigonometric/01_dft/03_linalg.tex b/semester3/numcs/parts/01_interpolation/01_trigonometric/01_dft/03_linalg.tex index a6c86bb..73b9fad 100644 --- a/semester3/numcs/parts/01_interpolation/01_trigonometric/01_dft/03_linalg.tex +++ b/semester3/numcs/parts/01_interpolation/01_trigonometric/01_dft/03_linalg.tex @@ -1,5 +1,5 @@ \newpage -\subsection{DFT \& Lineare Algebra} +\subsubsection{DFT \& Lineare Algebra} \setcounter{all}{25} \fancydef{Zirkulant} Für einen vektor $c \in \mathbb{R}^N$ hat der Zirkulant $C \in \mathbb{R}^{N \times N}$ die Form: @@ -98,7 +98,7 @@ Im Fall von $T$-periodischen Funktionen gilt: $(g * h)(x) = \frac{1}{T}\displays \begin{align*} c = a \circledast b = Ab = F_N^{-1}p(D)F_Nb \quad \quad \quad (p(D) \text{ ist Diagonalmatrix der } \lambda \text{ von } C ) \end{align*} -Man erhält so letzendlich das Faltungs-Theorem: Die $F_N$-Transformierte einer Faltung ist genau das gleiche wie die Multiplikation zweier $F_N$-Transormierten. Da die DFT in $\mathcal{O}(n\log(n))$ (Kap. 3.5) geht, gilt dies nun auch für die Faltung. +Man erhält so letzendlich das Faltungs-Theorem: Die $F_N$-Transformierte einer Faltung ist genau das gleiche wie die Multiplikation zweier $F_N$-Transormierten. Da die DFT in $\mathcal{O}(n\log(n))$ (Kap. 3.3) geht, gilt dies nun auch für die Faltung. \begin{align*} F_Nc = \text{diag}(F_N a) F_N b \end{align*} \ No newline at end of file