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| -rw-r--r-- | komfour_zf.tex | 205 |
1 files changed, 193 insertions, 12 deletions
diff --git a/komfour_zf.tex b/komfour_zf.tex index 7d3786b..933698a 100644 --- a/komfour_zf.tex +++ b/komfour_zf.tex @@ -3,7 +3,7 @@ % !TeX root = komfour_zf.tex %% TODO: publish to CTAN -\documentclass[twocolumn, margin=normal]{tex/hsrzf} +\documentclass[twocolumn, margin=small]{tex/hsrzf} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Packages @@ -57,6 +57,9 @@ \newcommand\Rset{\mathbb{R}} \newcommand\Cset{\mathbb{C}} +%% Missing operators +\DeclareMathOperator\sgn{sgn} + %% Complex operators \DeclareMathOperator\cjs{cjs} \newcommand\cjsl[1]{\cos #1 + j\sin #1} @@ -141,7 +144,7 @@ \end{lemma} \begin{definition}[Real and imaginary part and conjugation] - Let \(z = a + jb\). The \emph{real} part of \(z\) is \(\Re(z) = a\), similarly the \emph{imaginary} part is \(\Im(z) = b\). We can thus define the \emph{complex conjugate} \(\conj{z}\). + Let \(z = a + jb\). The \emph{real} part of \(z\) is \(\Re(z) = a\), similarly the \emph{imaginary} part is \(\Im(z) = b\). We can thus define the \emph{complex conjugate} \(\conj{z}\) of \(z\) to be \[ z = \Re(z) + j\Im(z) \quad @@ -154,14 +157,14 @@ \end{definition} \begin{lemma}[Properties of absolute value] - Let \(z,w\in\Cset\). We have \(\len{z} \in\Rset^+_0\), \(z\conj{z} = \len{z}^2\) and as a consequence \(\len{zw} = \len{z}\cdot\len{w}\) and \(\len{\conj{z}} = \len{z}\). In addition we have the inequalities + Let \(z,w\in\Cset\). We have \(z\conj{z} = \len{z}^2\) and as a consequence \(\len{zw} = \len{z}\cdot\len{w}\) and \(\len{\conj{z}} = \len{z}\). In addition we have the inequalities \begin{align*} -\len{z} \leq &\Re(z) \leq \len{z} & \len{z} &\leq \len{\Re(z)} + \len{\Im(z)} \\ -\len{z} \leq &\Im(z) \leq \len{z} & \len{z + w} &\leq \len{z} + \len{w} \end{align*} - The last one is the \emph{triangle inequality}. + The last one is the \emph{triangle inequality}. Notice that \(\len{z} \in\Rset^+_0\). \end{lemma} \begin{definition}[Reciprocal and quotients] @@ -169,7 +172,7 @@ It is now possible to define \(z/w = zw^{-1}\) with \(z,w \in\Cset\) and \(w \neq 0\). \end{definition} -\begin{lemma}[More properties of conjugation] +\begin{lemma}[Properties of conjugation] Let \(z,w \in\Cset\). \(\conj{z} = z\) iff \(z \in \Rset\) and \(\conj{z} = \conj{w}\) iff \(z = w\). Furthermore: @@ -183,14 +186,14 @@ \end{align*} \end{lemma} -\begin{definition}[Polar notation] - An alternative representation of a complex number \(z = a + jb\) is its \emph{polar form} \(z = r \angle \phi\), where \(r = \len{z}\) and \(\phi = \arg{z} = r(\cos\phi + j\sin\phi)\). +\begin{definition}[Argument and polar notation] + An alternative representation of a complex number \(z = a + jb\) is its \emph{polar form} \(z = r \angle \phi\), where \(r = \len{z}\) and \(\phi = \arg{z}\). \begin{align*} a &= r\cos\phi & b &= r \sin\phi & r &= \sqrt{z\conj{z}} \end{align*} - For \(a = 0\) we define \(\phi = \lim_{a\to 0} \arctan(b/a)\) and otherwise + For \(a = 0\) we define \(\phi = \lim_{a\to 0} \arctan(b/a) = \pm\pi/2\) and otherwise \begin{align*} \phi = \arg(z) &= \begin{cases} @@ -293,7 +296,7 @@ \end{theorem} \begin{lemma} - From the previous theorem follows that a polynomial of \emph{odd} degree, has always at least one real solution because \(r \in\Rset \iff r = \conj{r}\). + From the previous theorem follows that a polynomial with real coefficients of \emph{odd} degree, has \emph{always} at least one real solution because \(r \in\Rset \iff r = \conj{r}\). \end{lemma} \begin{theorem} @@ -318,7 +321,8 @@ \] \end{theorem} -\begin{lemma} Let \(a,b \in\Cset\) and \(k\in\Zset\) +\begin{lemma}[Rules for exponents] + Let \(a,b \in\Cset\) and \(k\in\Zset\), we can show that \[ e^a e^b = e^{a+b} \quad e^a / e^b = e^{a-b} \quad @@ -327,8 +331,40 @@ \end{lemma} \begin{definition}[Trigonometric functions] + When \(z\) is a complex number we define + \begin{align*} + \cos z &= \frac{e^{jz} + e^{-jz}}{2} & + \sin z &= \frac{e^{jz} - e^{-jz}}{2j} + \end{align*} + like the (real) hyperbolic trigonometric functions + \begin{align*} + \cosh z &= \left( e^z + e^{-z} \right)/2 & + \sinh z &= \left( e^z - e^{-z} \right)/2 + \end{align*} + Notice that the sinus function is point symmetric to \(\pi/2\), because \(\sin(\pi/2 - z) = \sin(\pi/2 + z)\). \end{definition} +\begin{lemma}[Some trigonometric identities] Let \(x,a,b \in\Rset\) and \(\alpha,\beta \in\Cset\) + \begin{align*} + \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) \\ + 2\sin(\alpha)\sin(\beta) &= \cos(\alpha - \beta) - \cos(\alpha + \beta) \\ + 2\sin(\alpha)\cos(\beta) &= \sin(\alpha - \beta) + \sin(\alpha + \beta) + \end{align*} +\end{lemma} + +\begin{lemma}[Superposition of sinuses] + Let \(s(t) = A\sin(\omega t + \varphi)\) be a sinusoidal wave. + We can rewrite \(s\) in complex form with + \[ + s(t) = \Im\left(Ae^{j(\omega t + \varphi)}\right) = \Im + Ae^{j\varphi}\cdot e^{j\omega t} + \] + If we now wish to sum \(N\) sinusoids with the same frequency \(\omega\), we can set + % TODO: Satz 18 +\end{lemma} + \begin{definition}[Logarithm] Because \(w = e^z\) defined from \(\Cset \to \Cset\) is not a bijection (\(e^{z + 2\pi j} = e^z\)), unless we restrict the imaginary part of the domain to \((\pi, \pi]\), we get only an equivalence relationship because \[ @@ -351,15 +387,160 @@ &= \exp\big(\ln\len{z}\cdot k/m)\exp((\arg z + 2\pi n)jk/m\big) \\ &= \len{z}^{k/m}\exp\big((\arg z + 2\pi n)jk/m\big)= \sqrt[m]{z^k} \end{align*} - like in the reals, except that we have \(m\) values instead of 1 or 2. If we let \(w \in\Cset\) the expression \(z^w\) cannot be equal to an unique value because + like in the reals, except that we have \(m\) values because of the \(m\)-th root. If we let \(w \in\Cset\) the expression \(z^w\) cannot be equal to an unique value because \begin{align*} z^w = e^{w \ln z} &= \exp\big( w (\ln\len{z} + j \arg{z} + 2\pi nj)\big) \\ &= e^{w(\ln\len{z} + j\arg z)} e^{w2\pi nj} \end{align*} - instead it is said to be \emph{multivalued}. + instead it is said to be \emph{multivalued}. This means that there are no general exponentiation rules. \end{lemma} \section{Fourier Series} +\begin{definition}[Real trigonometric polynomial] + Let \(\omega = 2\pi/T \in\Rset\) and \(A_n, B_n\) be sequences in \(\Rset\). + We define a \emph{real trigonometric polynomial} of degree \(N\) to be + \[ + \tau_N(t) = \frac{A_0}{2} + \sum_{n=1}^N A_n \cos(n\omega t) + B_n \sin(n\omega t) + \] +\end{definition} + +\begin{lemma}[Orthogonality of the basis functions] + Let \(m,n \in\Nset_0\) + \begin{align*} + \int\limits_0^T \cos(m\omega t)\cos(n\omega t) + &= \begin{cases} + T & m = n = 0 \\ + T/2 & m = n > 0 \\ + 0 & m \neq n + \end{cases} \\ + \int\limits_0^T \sin(m\omega t)\sin(n\omega t) + &= \begin{cases} + T/2 & m = n \wedge n \neq 0 \\ + 0 & m \neq n \\ + 0 & m = 0 \vee n = 0 + \end{cases} \\ + \int\limits_0^T \cos(m\omega t)\sin(n\omega t) &= 0 + \end{align*} +\end{lemma} + +\begin{definition} + We denote with \(\Omega\) the space of real valued, \(T\)-periodic, piecewise continuous functions, that have only a finite number of discontinuities, in which both the right and left limit exist, within the interval \([0,T)\). +\end{definition} + +\begin{theorem}[Fourier coefficients] + For any \(f\in\Omega\) we can now define the \emph{Fourier coefficients} + \begin{align*} + a_n &= \frac{2}{T}\int\limits_0^T f(t)\cos(n\omega t)\di{t} & a_0 &= \frac{2}{T}\int\limits_0^T f(t)\di{t} \\ + b_n &= \frac{2}{T}\int\limits_0^T f(t)\sin(n\omega t)\di{t} & b_0 &= 0 + \end{align*} + Worth noting are the special cases when \(n=0\). +\end{theorem} + +\begin{definition}[Fourier Polynomial] + We can now use the Fourier coefficients as sequences for a trigonometric polynomial to obtain a \emph{Fourier Polynomial} + \[ + S_N(t) = \frac{a_0}{2} + \sum_{n=1}^N a_n\cos(n\omega t) + b_n\sin(n\omega t) + \] +\end{definition} + +\begin{lemma} + A trigonometric polynomial has the smallest distance (by the \(L^2\) metric) from a function \(f\in\Omega\), iff \(A_n = a_n\) and \(B_n = b_n\), in other words iff it is a Fourier Polynomial. +\end{lemma} + +\begin{definition}[Fourier Series] + We can finally define the \emph{Fourier Series} to be the infinite Fourier Polynomial, by letting \(N\to\infty\) + \[ + S(t) = \frac{a_0}{2} + \sum_{n=1}^N a_n\cos(n\omega t) + b_n\sin(n\omega t) + \] +\end{definition} + +\begin{theorem}[Fourier coefficients of even and odd functions] + Recall that a function is said to be \emph{even} if \(f(-x) = f(x)\) or \emph{odd} if \(f(-x) = -f(x)\). We can show that if a function is + \begin{itemize} + \item odd, then \(b_n = 0\) for all \(n\), and + \[ + a_n = \frac{4}{T}\int\limits_0^{T/2} f(t)\cos(n\omega t)\di{t} + \] + \item even, then \(a_n = 0\) for all \(n\), and + \[ + b_n = \frac{4}{T}\int\limits_0^{T/2} f(t)\sin(n\omega t)\di{t} + \] + \end{itemize} +\end{theorem} + +\begin{lemma}[Linearity of Fourier coefficients] + Recall that linearity means \(L(\mu x + \lambda y) = \(\mu L(x) + \lambda L(y)\). We then let \(f,g \in\Omega\) be functions with Fourier series and \(h = \mu f + \lambda g\) where \(\mu,\lambda\in\Rset\) are constants. + By denoting with \(a_n^{(f)}\) the Fourier coefficient \(a_n\) of the function \(f\), and similarly with \(b_n^{(f)}\), it is easily shown that + \begin{align*} + a_n^{(h)} &= \mu a_n^{(f)} + \lambda a_n^{(g)} & + b_n^{(h)} &= \mu b_n^{(f)} + \lambda b_n^{(g)} + \end{align*} +\end{lemma} + +\begin{lemma}[Fourier coefficients after time dilation] + Let \(f\in\Omega\) be a function with a Fourier Series and \(g(t) = f(rt)\) with \(0 \neq r \in\Rset\). It follows that + \(a_n^{(g)} = a_n^{(f)}\) and \(b_n^{(g)} = \sgn(r) \cdot b_n^{(f)}\). +\end{lemma} + +\begin{lemma}[Fourier coefficients after time translation] + Let \(f\in\Omega\) be a function with a Fourier Series and \(g(t) = f(t + \tau)\) with \(\tau\in\Rset\). It follows that + \begin{align*} + a_n^{(g)} &= \cos(n\omega \tau)\cdot a_n^{(f)} + \sin(n\omega \tau)\cdot b_n^{(f)} & n &\geq 0\\ + b_n^{(g)} &= -\sin(n\omega \tau)\cdot a_n^{(f)} + \cos(n\omega \tau)\cdot b_n^{(f)} & n &> 0 + \end{align*} +\end{lemma} + +\begin{theorem}[Fourier theorem] + For any \(f\in\Omega\) the Fourier series of \(f\) converges in \(L^2\) metric to \(f\). + \[ + \lim_{N\to\infty} \left\lVert + \frac{a_0}{2} + \sum_{n=0}^N a_n \cos(n\omega t) + b_n \sin(n\omega t) - f(t) + \right\rVert = 0 + \] +\end{theorem} + +\begin{theorem}[Plancherel Parselval theorem] + Let \(f\in\Omega\) with a Fourier Series with coefficients \(a_n\) and \(b_n\). + \[ + \frac{a_0}{2} + \sum_{n=1}^\infty \left(a_n^2 + b_n^2\right) + \leqq \frac{2}{T} \int\limits_0^T \lvert f(t)\rvert^2 \di{t} = \left\lVert f \right\rVert^2 + \] +\end{theorem} + +\begin{theorem} Both sequences \(a_n, b_n\) for the Fourier coefficients of a function \(f\in\Omega\) converge to zero. + \begin{align*} + \lim_{n\to\infty} a_n + &= \lim_{n\to\infty} \frac{2}{T} + \int\limits_0^T f(t) \cos(n\omega t) \di{t} = 0 \\ + \lim_{n\to\infty} b_n + &= \lim_{n\to\infty} \frac{2}{T} + \int\limits_0^T f(t) \sin(n\omega t) \di{t} = 0 + \end{align*} +\end{theorem} + +\begin{theorem}[Rate of convergence of Fourier coefficients] + If \(f\) is a \(T\)-periodic, \((m-2)\) times differentiable, continuous function. And if its \((m-1)\)-th derivative is pieceweise monotonous and \(\in \Omega\), then there exists a constant \(c \in\Rset\) such that + \[ + \len{a_n} \leq \frac{c}{n^m} \qquad \len{b_n} \leq \frac{c}{n^m} \qquad m,n\in\Nset + \] +\end{theorem} + +\begin{theorem}[Integration and differentiation of the Fourier series] + It is possible to show from the previous theorem (and others before) that when \(m\geq 2\) the Fouriers converges \emph{uniformly}. This means that it is possible to integrate or differentiate the series term by term. + \[ + f'(t) = \sum_{n=1}^\infty b_n n\omega\cos(n\omega t) - a_n n\omega\sin(n\omega t) + \] + and + \begin{align*} + \int\limits_0^t f(\tau) \di{\tau} &= + \left(\sum_{n=1}^\infty \frac{b_n}{n\omega} \right) + + \frac{a_0}{2} t \\ + &+ \left(\sum_{n=1}^\infty + \frac{a_n}{n\omega}\sin(n\omega t) + - \frac{b_n}{n\omega}\cos(n\omega t) + \right) + \end{align*} +\end{theorem} \section{Fourier Transform} |