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-rw-r--r--komfour_zf.tex45
1 files changed, 41 insertions, 4 deletions
diff --git a/komfour_zf.tex b/komfour_zf.tex
index 933698a..485eb1e 100644
--- a/komfour_zf.tex
+++ b/komfour_zf.tex
@@ -346,6 +346,7 @@
\begin{lemma}[Some trigonometric identities] Let \(x,a,b \in\Rset\) and \(\alpha,\beta \in\Cset\)
\begin{align*}
+ \sin(x + \pi/2) &= \cos(x) \qquad \cos(x - \pi/2) = \sin(x) \\
\sinh(jx) &= j\sin(x) \qquad \cosh(jx) = \cos(x) \\
\sin(a + jb) &= \sin(a)\cosh(b) + j\cos(a)\sinh(b) \\
\cos(a + jb) &= \cos(a)\cosh(b) + j\sin(a)\sinh(b) \\
@@ -361,8 +362,12 @@
s(t) = \Im\left(Ae^{j(\omega t + \varphi)}\right) = \Im
Ae^{j\varphi}\cdot e^{j\omega t}
\]
- If we now wish to sum \(N\) sinusoids with the same frequency \(\omega\), we can set
- % TODO: Satz 18
+ If we now wish to sum \(N\) sinusoids with the same frequency \(\omega\), the resulting sinusoid \(A\sin(\omega t + \varphi)\) has
+ \[
+ A = \left\lvert \sum_{n=1}^N A_n e^{j\varphi_n} \right\lvert
+ \quad
+ \varphi = \arg\sum_{n=1}^N A_n e^{j\varphi_n}
+ \]
\end{lemma}
\begin{definition}[Logarithm]
@@ -450,7 +455,7 @@
\begin{definition}[Fourier Series]
We can finally define the \emph{Fourier Series} to be the infinite Fourier Polynomial, by letting \(N\to\infty\)
\[
- S(t) = \frac{a_0}{2} + \sum_{n=1}^N a_n\cos(n\omega t) + b_n\sin(n\omega t)
+ S(t) = \frac{a_0}{2} + \sum_{n=1}^\infty a_n\cos(n\omega t) + b_n\sin(n\omega t)
\]
\end{definition}
@@ -542,7 +547,39 @@
\end{align*}
\end{theorem}
-\section{Fourier Transform}
+\begin{theorem}[Dirichlet pointwise convergence]
+ Let \(f\in\Omega\), then it is known that its Fourier series converges to
+ \[
+ \lim_{\epsilon\to 0}\frac{f(t-\epsilon) + f(t+\epsilon)}{2}
+ \]
+ for every \(t\), where the left and right derivative \(f'(t-\epsilon)\), \(f'(t+\epsilon)\), with \(\epsilon\to 0\), exist.
+
+ In the special case where \(f\) is continuous at \(t\), and the derivatives exist, there the Fourier series converges exactly to \(f(t)\), i.e. the value of the function at \(t\).
+\end{theorem}
+
+\begin{definition}[Complex representation of the Fourier coefficients]
+ By letting \(n\in\Zset\) and
+ \[
+ c_n = \conj{c_{-n}} = \frac{a_n - jb_n}{2} = \frac{1}{T}\int\limits_0^T f(t) e^{-jn\omega t} \di{t}
+ \]
+ using a notational trick for negative indices. We can compactly write a Fourier series or polynomial as
+ \[
+ S(t) = \sum_{n=-\infty}^\infty c_n e^{jn\omega t}
+ \]
+\end{definition}
+
+\begin{theorem}[Complex Fourier coefficients of even and odd functions]
+ By the definition of \(c_n\) and the previous similar theorem for the real coefficients, it is clear that when a function \(f\in\Omega\) is \emph{even}, then \(\Im(c_k) = 0\), whereas when \(f\) is \emph{odd} \(\Re(c_k) = 0\) (\(k\in\Zset\)).
+\end{theorem}
+
+\begin{theorem}[Complex Fourier coeffients after time translation]
+ Similarly to the previous theorem, we can now compactly write that if \(f\in\Omega\) has a Fourier series with coefficients \(c_k^{(f)}\), and \(g(t) = f(t + \tau)\), then
+ \[
+ c_k^{(g)} = e^{jk\omega \tau} c_k^{(f)} \qquad k \in \Zset
+ \]
+\end{theorem}
+
+%% TODO: fourier transform
\section{License}
\doclicenseThis