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| -rw-r--r-- | komfour_zf.tex | 45 |
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 |