From a9f6febfc42a678884566fb702c8177dfcc8dbfb Mon Sep 17 00:00:00 2001 From: "matthias@quintern.xyz" Date: Sun, 4 Aug 2024 10:10:17 +0200 Subject: [PATCH] finish vl11 --- img/vl11-block-addierer.png | Bin 0 -> 84628 bytes img/vl11-block-multiplizierer.png | Bin 0 -> 47348 bytes img/vl11-block-verzögerung.png | Bin 0 -> 86562 bytes img/vl11-bsp-nichtrekursiv-H.png | Bin 0 -> 29757 bytes img/vl11-bsp-nichtrekursiv.png | Bin 0 -> 165139 bytes img/vl11-bsp-rekursiv-kaskadiert.png | Bin 0 -> 141737 bytes img/vl11-bsp-rekursiv.png | Bin 0 -> 143550 bytes img/vl11-z-beispiel.png | Bin 0 -> 124130 bytes main.tex | 662 ++++++++++++++++----------- 9 files changed, 390 insertions(+), 272 deletions(-) create mode 100644 img/vl11-block-addierer.png create mode 100644 img/vl11-block-multiplizierer.png create mode 100644 img/vl11-block-verzögerung.png create mode 100644 img/vl11-bsp-nichtrekursiv-H.png create mode 100644 img/vl11-bsp-nichtrekursiv.png create mode 100644 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z5iETgq$%uT^IFJ3G6t<=bXlOqEKD#9cly6CZgTQBA&9Db`fvCCe4#aICtfDyrgIyN zd1l=lj@Z!(Q?B<_;%1)&5OAIjrl)5L$tOuBsEPH!qM(heEK1Xm@Go)PviJmaboG1K zjN(U#V9|q6edm%fp?Z}NMNowl{$oq}46I``tEX2($FNiVkOtf34;@&~zzM6|Fqn%U@I&FOLl_-2*)mG*}i-dS$f;iu@a4 z^9POS!nas_#cJQB6pLSf-1sIZOe|*?{eE#4og0h))5;L+#b{?#a{yT^i34Yi|aJIS1IYMNOu>PTq- z-mzKoFSSr~+eA?6cHi#_jgC$!ilpXI1yk)wwcGuF6uv%?rQNCgp=RTGJF(vv@BA!< nR=%|RBHhSIN&0?$_V|bjyFR}dW*F6n1J82>b@>`uqrm?IOP59( literal 0 HcmV?d00001 diff --git a/main.tex b/main.tex index fa16291..b43daae 100644 --- a/main.tex +++ b/main.tex @@ -527,11 +527,16 @@ Frequenzbereich}\label{beschreibung-linearer-systeme-im-frequenzbereich} \end{equation} Zeitsignalunktion nach Amplitudendichtespektrum: \begin{equation} - F(\omega) = \frac{1}{2\pi}\int_{-\infty}^{\infty} g(t)\e^{-j\omega t} d\omega + F(\omega) = \frac{1}{2\pi}\int_{-\infty}^{\infty} f(t)\e^{-j\omega t} d t \end{equation} - Schreibweisen: - Fourier-Trafo: $\mathcal{F}\{f(t)\}=F(\omega)$ - - Rück-Trafo: $\mathcal{F}^{-1}\{F(\omega)\}=f(t)$ - - $f(t) \text{ist korrespondierende}\,\, o- F(\omega)$ + + \def\rmapsto{\reflectbox{\multimap}} + Schreibweisen: + \begin{itemize} + \item Fourier-Trafo: $\mathcal{F}\{f(t)\}=F(\omega)$ + \item Rück-Trafo: $\mathcal{F}^{-1}\{F(\omega)\}=f(t)$ + \item $f(t) \rmapsto F(\omega)$ (\textit{$\rmapsto$ bedeutet "ist korrespondierende"}) + \end{itemize} \begin{enumerate} \def\labelenumi{\alph{enumi}.} @@ -552,9 +557,9 @@ Frequenzbereich}\label{beschreibung-linearer-systeme-im-frequenzbereich} Sonderfall: undendlich lang andauernde Schwingung, d.h. $T\to\infty$ Grenzübergang: \begin{align} - \lim_{T\to\infty} F(\omega)= \lim_{T\to\infty}\Big[ 2\pi\frac{sin[(\omega_0-\omega)T]}{\pi(\omega_0-\omega)}\Big] = 2\pi\delta(\omega_0 - \omega) = 2\pi\delta(\omega-\omega_0) \\ + \lim_{T\to\infty} F(\omega)= \lim_{T\to\infty}\Big[ 2\pi\frac{sin[(\omega_0-\omega)T]}{\pi(\omega_0-\omega)}\Big] &= 2\pi\delta(\omega_0 - \omega) = 2\pi\delta(\omega-\omega_0) &\\ \intertext{d.h. es ist:} - \mathcal{F}\{\e^{\pm j \omega_0 t}\} = 2\pi \delta(\omega \mp \omega_0) + \mathcal{F}\{\e^{\pm j \omega_0 t}\} &= 2\pi \delta(\omega \mp \omega_0) \end{align} Damit Analyse für verschiedene andere Funktionen möglich: \begin{enumerate} @@ -766,7 +771,7 @@ Frequenzbereich}\label{beschreibung-linearer-systeme-im-frequenzbereich} \item kausale Zeitreihe: $x \equiv 0$ für $n < 0$ \item - akausale Zeitreihe: $x \ge 0$ für $n \ge 0$ + akausale Zeitreihe: $x \equiv 0$ für $n < 0$ \item zweiseitige Zeitreihe: Ist immer über die Kombination von kausaler und akausaler Zeitreihe möglich \item @@ -1029,301 +1034,414 @@ Frequenzbereich}\label{beschreibung-linearer-systeme-im-frequenzbereich} \end{itemize} \Rightarrow System von nicht-rekursiven Operatoren einfacher, keine Stabilitätsbetrachtung erfolrderlich - \subsubsection{Die z-Transformation} - Zunächst Betrachtung der Fourer-Transformation - \begin{equation} - F(\omega) = \int_{-\infty}^\infty f(t) \e^{-j\omega t} \d t - \end{equation} - Frage: Was ändert sich hier, wenn $f(t)$ ein zeitdiskretes (abgetastetes) Signal ist, das nur zu den Zeitpunkten $t=nT$ definiert ist? - \\DTFT (\neq DFT!) \textit{discrete-time Fourier transform}: - \begin{equation} - \boxed{ - X(\e^{j\omega}) = \sum_{n=-\infty}^{\infty} x(n) \e^{-jn\omega} - } - \end{equation} - wobei $w$ die digitale Frequenz $-\pi \le \omega \le +\pi$ darstellt, mit $\omega \in \R$. - \\ - IDTFT (inverse): - \begin{equation} - \boxed{ - x(n) = \frac{1}{2\pi} \int_{-\pi}^{+\pi}X(\e^{j\omega}) \e^{-jn\omega} \d \omega - } - \end{equation} + \subsection{Die z-Transformation} + Zunächst Betrachtung der Fourer-Transformation + \begin{equation} + F(\omega) = \int_{-\infty}^\infty f(t) \e^{-j\omega t} \d t + \end{equation} + Frage: Was ändert sich hier, wenn $f(t)$ ein zeitdiskretes (abgetastetes) Signal ist, das nur zu den Zeitpunkten $t=nT$ definiert ist? + \\DTFT (\neq DFT!) \textit{discrete-time Fourier transform}: + \begin{equation} + \boxed{ + X(\e^{j\omega}) = \sum_{n=-\infty}^{\infty} x(n) \e^{-jn\omega} + } + \end{equation} + wobei $w$ die digitale Frequenz $-\pi \le \omega \le +\pi$ darstellt, mit $\omega \in \R$. + \\ + IDTFT (inverse): + \begin{equation} + \boxed{ + x(n) = \frac{1}{2\pi} \int_{-\pi}^{+\pi}X(\e^{j\omega}) \e^{-jn\omega} \d \omega + } + \end{equation} - \begin{itemize} - \item - Integration wird Summation, da nur Beiträge zu den Abtastzeitpunkten (dazwischen nicht definiert) - \item - Statt kontinuierlicher Frequenz $\omega$ nun digitale Frequenz, da Spektrum periodisch, d-h- für $\omega \lessgtr \pi$ Wiederholung - \\Um die Periodizität anzudeuten wird im Argument von $X$ $\omega$ durch $\e^{j\omega}$ ersetzt, $\e^{j\omega}$ ist ja mit $2\pi$ periodisch - \end{itemize} + \begin{itemize} + \item + Integration wird Summation, da nur Beiträge zu den Abtastzeitpunkten (dazwischen nicht definiert) + \item + Statt kontinuierlicher Frequenz $\omega$ nun digitale Frequenz, da Spektrum periodisch, d-h- für $\omega \lessgtr \pi$ Wiederholung + \\Um die Periodizität anzudeuten wird im Argument von $X$ $\omega$ durch $\e^{j\omega}$ ersetzt, $\e^{j\omega}$ ist ja mit $2\pi$ periodisch + \end{itemize} - \textbf{Problem}: Für viele nichtperiodische Funktionen konvergiert das Integral $\int_{-\infty}^{\infty} f(t) e^{-j\omega t} \d t$ nicht - \\ Deshalb (im kausalen Fall) Dämpfung der Zeitfunktion mit $\e^{-\sigma t}$. Dann ist in Abhängigkeit von $f(t)$ für bestimmte Werte von $\sigma \ge 0$ Konvergenz herstellbar. - D.h. neues Integral: - \begin{equation} - F_\sigma(\omega) = \int_0^\infty f(t) \e^-{j\omega t} \d t - \end{equation} - Nun Einführung der Komplexen Variablen $s=\sigma + j \omega$. - $F_\sigma(\omega)$ stellt eine ganze Schar von Spektraldichten dar, von denen einige nicht existieren können. - Betrachtung in der $s$-Ebene: - \begin{figure}[H] - \centering - \includegraphics[width=0.8\textwidth]{img/vl09-s-ebene.png} - \caption{$s$-Ebene} - \end{figure} + \textbf{Problem}: Für viele nichtperiodische Funktionen konvergiert das Integral $\int_{-\infty}^{\infty} f(t) e^{-j\omega t} \d t$ nicht + \\ Deshalb (im kausalen Fall) Dämpfung der Zeitfunktion mit $\e^{-\sigma t}$. Dann ist in Abhängigkeit von $f(t)$ für bestimmte Werte von $\sigma \ge 0$ Konvergenz herstellbar. + D.h. neues Integral: + \begin{equation} + F_\sigma(\omega) = \int_0^\infty f(t) \e^-{j\omega t} \d t + \end{equation} + Nun Einführung der Komplexen Variablen $s=\sigma + j \omega$. + $F_\sigma(\omega)$ stellt eine ganze Schar von Spektraldichten dar, von denen einige nicht existieren können. + Betrachtung in der $s$-Ebene: + \begin{figure}[H] + \centering + \includegraphics[width=0.8\textwidth]{img/vl09-s-ebene.png} + \caption{$s$-Ebene} + \end{figure} - Schreibt man - \begin{equation} - F(s) = \sigma_0^\infty f(t) \e^{(-\sigma + j\omega)t} \d t - \end{equation} - dann stellt $F(s)$ bei festem $\sigma$ die Spketraldichte der mit $\e^{-\sigma t}$ gedämpften Zeitfunktion $f(t)$ dar. Man bezeichnet $F(s)$ als einseitige \textbf{Laplace-Transformation} der Zeitfunktion $f(t)$. - \\ - Die L-Transformation verfügt im wesenetlichen über die gleichen Eigenschaften wie die F-Transformation, deckt aber die meisten in der Praxis vorkommenden Fälle nichtperiodischer Funktionen ab. + Schreibt man + \begin{equation} + F(s) = \sigma_0^\infty f(t) \e^{(-\sigma + j\omega)t} \d t + \end{equation} + dann stellt $F(s)$ bei festem $\sigma$ die Spketraldichte der mit $\e^{-\sigma t}$ gedämpften Zeitfunktion $f(t)$ dar. Man bezeichnet $F(s)$ als einseitige \textbf{Laplace-Transformation} der Zeitfunktion $f(t)$. + \\ + Die L-Transformation verfügt im wesenetlichen über die gleichen Eigenschaften wie die F-Transformation, deckt aber die meisten in der Praxis vorkommenden Fälle nichtperiodischer Funktionen ab. - Schreibt man die L-Transformierte - \begin{equation} - F(s) = \int_{0/-\infty}^{+\infty} f(t) \e^{-s t} - \end{equation} - (wobei die die Integrationsgrenze $0$ die einseitige und $-\infty$ die zweiseitige bezeichnet) für zeitdiskrete Signale auf, so hat man (im zweiseitigen Fall): - \begin{equation} - F(s) = \sum_{n=-\infty}^{\infty} f(n) \e^{-s n T} - \end{equation} - und mit der Abkürzung $z=\e^{sT}$ wird daraus die \textbf{z-Transformation}: - \def\Z{\mathcal{Z}} - \begin{equation} - \boxed{ - X(z) = \sum_{n=-\infty}^{\infty} x(n) z^{-n} = \Z\left\{x(n)\right\} - } - \end{equation} + Schreibt man die L-Transformierte + \begin{equation} + F(s) = \int_{0/-\infty}^{+\infty} f(t) \e^{-s t} + \end{equation} + (wobei die die Integrationsgrenze $0$ die einseitige und $-\infty$ die zweiseitige bezeichnet) für zeitdiskrete Signale auf, so hat man (im zweiseitigen Fall): + \begin{equation} + F(s) = \sum_{n=-\infty}^{\infty} f(n) \e^{-s n T} + \end{equation} + und mit der Abkürzung $z=\e^{sT}$ wird daraus die \textbf{z-Transformation}: + \def\Z{\mathcal{Z}} + \begin{equation} + \boxed{ + X(z) = \sum_{n=-\infty}^{\infty} x(n) z^{-n} = \Z\left\{x(n)\right\} + } + \end{equation} \lecture{10} - \begin{figure}[H] - \centering - \includegraphics[width=0.8\textwidth]{img/vl10-z.png} - \end{figure} - Die z-Transformation bildet eine diskrete Zahlenfolge in den z-Bereich ab. - Die Vorteile der Laplace-Transformation im kontinuierlichen Fall bleiben erhalten. - Insbesondere: Ersetzung der Berechnung von Faltungssummen durch die Multiplikation von Spektren im z-Bereich. + \begin{figure}[H] + \centering + \includegraphics[width=0.8\textwidth]{img/vl10-z.png} + \end{figure} + Die z-Transformation bildet eine diskrete Zahlenfolge in den z-Bereich ab. + Die Vorteile der Laplace-Transformation im kontinuierlichen Fall bleiben erhalten. + Insbesondere: Ersetzung der Berechnung von Faltungssummen durch die Multiplikation von Spektren im z-Bereich. - \paragraph{Rücktransformation} - \begin{equation} - x(n) = \Z^{-1}\left\{X(z)\right\} = \frac{1}{2\pi j} \oint_c X(z) z^{n-1} \d z - \end{equation} - Komplexes Integral: Integration über geschlossenen Weg in der Komplexen Ebene (\rightarrow Funktionentheorie, Residuensatz). + \subsubsection{Rücktransformation} + \begin{equation} + x(n) = \Z^{-1}\left\{X(z)\right\} = \frac{1}{2\pi j} \oint_c X(z) z^{n-1} \d z + \end{equation} + Komplexes Integral: Integration über geschlossenen Weg in der Komplexen Ebene (\rightarrow Funktionentheorie, Residuensatz). - z-Transformation ist die Entwicklung von $x(n)$ in eine \textbf{Laurent-Reihe} zum $z=0$ (keine Potenzreihe, nur negative Exponenten für $z$). - \begin{equation} - \Z\left\{x(n)\right\} = x(0) + \frac{x(1)}{z} + \frac{x(2)}{z^2} + \dots - \end{equation} - \paragraph{Beispiel}: z-Transformation der Zeitreihen - \begin{align} - &\left\begin{array}{l} - h(n) = \{1,2,2,0,0,0,\dots\}\\ - x(n) = \{1,1,0,1,0,0,\dots\} - \end{array}\right\} \text{siehe obiges zur Faltung} \\ - &H(z) = 1 + 2z^{-1} + 2z^{-2} \\ - &X(z) = 1+ z^{-1} + z^{-3} - \end{align} + z-Transformation ist die Entwicklung von $x(n)$ in eine \textbf{Laurent-Reihe} zum $z=0$ (keine Potenzreihe, nur negative Exponenten für $z$). + \begin{equation} + \Z\left\{x(n)\right\} = x(0) + \frac{x(1)}{z} + \frac{x(2)}{z^2} + \dots + \end{equation} + \paragraph{Beispiel}: z-Transformation der Zeitreihen + \begin{align} + &\left\begin{array}{l} + h(n) = \{1,2,2,0,0,0,\dots\}\\ + x(n) = \{1,1,0,1,0,0,\dots\} + \end{array}\right\} \text{siehe obiges zur Faltung} \\ + &H(z) = 1 + 2z^{-1} + 2z^{-2} \\ + &X(z) = 1+ z^{-1} + z^{-3} + \end{align} - Jetzt Multiplikation der Spektren: Polynommultiplikation - \begin{equation} - H(z)\cdot X(z) = 1 + 3z^{-1} + 4z^{-2} + 3z^{-3} + 2z^{-4} + 2z^{-5} = Y(z) - \end{equation} - Das entspricht Offensichtlich: der z-Transformation der Zeitreihe $y(n) = \{1,3,4,3,2,2,0,0,\dots\}$ \Rightarrow siehe Bsp. zur Faltung + Jetzt Multiplikation der Spektren: Polynommultiplikation + \begin{equation} + H(z)\cdot X(z) = 1 + 3z^{-1} + 4z^{-2} + 3z^{-3} + 2z^{-4} + 2z^{-5} = Y(z) + \end{equation} + Das entspricht Offensichtlich: der z-Transformation der Zeitreihe $y(n) = \{1,3,4,3,2,2,0,0,\dots\}$ \Rightarrow siehe Bsp. zur Faltung - Beobachtung: - \begin{enumerate}[a)] - \item Der Wert vor dem Ausdruck $z^{-n}$ ist offensichtlich der Wert der Reihe zum Zeitpunkt $n$ - \item Die Spektren $H(z)$ und $X(z)$ sind $<\infty$ für alle Werte von $\neq0$ \Rightarrow Konvergenz der z-Transformation - \end{enumerate} + Beobachtung: + \begin{enumerate}[a)] + \item Der Wert vor dem Ausdruck $z^{-n}$ ist offensichtlich der Wert der Reihe zum Zeitpunkt $n$ + \item Die Spektren $H(z)$ und $X(z)$ sind $<\infty$ für alle Werte von $\neq0$ \Rightarrow Konvergenz der z-Transformation + \end{enumerate} - \subsubsection{Konvergenz der z-Transformation} - Die z-Transformation ist nur sinnvoll, wenn sie beschränkt ist. - Daher ist die Konvergenzuntersuchung der Reihe notwendig. + \subsubsection{Konvergenz der z-Transformation} + Die z-Transformation ist nur sinnvoll, wenn sie beschränkt ist. + Daher ist die Konvergenzuntersuchung der Reihe notwendig. - Definition eines Konvergenzgebietes: - \begin{equation} - R = \left\{z: \left\lvert \sum_{n=-\infty}^{\infty} x(n) z^{-n}\right\rvert < \infty\right\} - \end{equation} - Konvergenzregion $R$ umfasst alle Werte von $z$, für die der Betragswert der Summe endlich ist. - Um die Analyse zu strukturieren, kann man die Reihe in eine kausale Reihe und eine akausale Reihe aufteilen: - \begin{equation} - X(z) = \sum_{n=0}^{\infty} x(n) z^{-n} + \sum_{n=-\infty}^{-1} x(n)z^{-n} = X_\text{c}(z) + X_\text{a}(z) - \end{equation} - Es gilt: - \begin{itemize} - \item $\abs*{X_\text{c}(z)} < \infty$ für $\abs*{z} > r_\text{c}$, d.h für alle $z$ außerhalb eines Konvergenzradius $r_\text{c}$ - \item $\abs*{X_\text{a}(z)} < \infty$ für $\abs*{z} < r_\text{<}$, d.h für alle $z$ innerhalb eines Konvergenzradius $r_\text{a}$ - \end{itemize} - Wobei $r_\text{c},\,r_\text{a}\in \R^+$. + Definition eines Konvergenzgebietes: + \begin{equation} + R = \left\{z: \left\lvert \sum_{n=-\infty}^{\infty} x(n) z^{-n}\right\rvert < \infty\right\} + \end{equation} + Konvergenzregion $R$ umfasst alle Werte von $z$, für die der Betragswert der Summe endlich ist. + Um die Analyse zu strukturieren, kann man die Reihe in eine kausale Reihe und eine akausale Reihe aufteilen: + \begin{equation} + X(z) = \sum_{n=0}^{\infty} x(n) z^{-n} + \sum_{n=-\infty}^{-1} x(n)z^{-n} = X_\text{c}(z) + X_\text{a}(z) + \end{equation} + Es gilt: + \begin{itemize} + \item $\abs*{X_\text{c}(z)} < \infty$ für $\abs*{z} > r_\text{c}$, d.h für alle $z$ außerhalb eines Konvergenzradius $r_\text{c}$ + \item $\abs*{X_\text{a}(z)} < \infty$ für $\abs*{z} < r_\text{<}$, d.h für alle $z$ innerhalb eines Konvergenzradius $r_\text{a}$ + \end{itemize} + Wobei $r_\text{c},\,r_\text{a}\in \R^+$. - In der z-Ebene: - \begin{figure}[H] - \centering - \includegraphics[width=0.48\textwidth]{img/vl10-z-konvergenz-kausal.png} - \includegraphics[width=0.48\textwidth]{img/vl10-z-konvergenz-akausal.png}\\ - \includegraphics[width=0.96\textwidth]{img/vl10-z-konvergenz-beide.png}\\ - \end{figure} + In der z-Ebene: + \begin{figure}[H] + \centering + \includegraphics[width=0.48\textwidth]{img/vl10-z-konvergenz-kausal.png} + \includegraphics[width=0.48\textwidth]{img/vl10-z-konvergenz-akausal.png}\\ + \includegraphics[width=0.96\textwidth]{img/vl10-z-konvergenz-beide.png}\\ + \end{figure} - \paragraph{Beispiel für Konvergenzregion} - \begin{enumerate}[a)] - \item $x(n) = a^n$ wo $n\gt0$ sei (kausaler Fall) \\ - \begin{align} - X(z) &= \sum_{n=0}^{\infty} a^nz^{-n} = \sum_{n=0}^{\infty} (az^{-1})^n = \\ - \shortintertext{geometrische Reihe für $\abs*{\frac{a}{z}} <1$ \Rightarrow $\abs*{a} < \abs*{z}$} - &= \frac{1}{1-\frac{a}{z}} - \end{align} - \begin{center} - \includegraphics[width=0.5\textwidth]{img/vl10-beispiel-kausal.png} - \end{center} + \paragraph{Beispiel für Konvergenzregion} + \begin{enumerate}[a)] + \item $x(n) = a^n$ wo $n\gt0$ sei (kausaler Fall) \\ + \begin{align} + X(z) &= \sum_{n=0}^{\infty} a^nz^{-n} = \sum_{n=0}^{\infty} (az^{-1})^n = \\ + \shortintertext{geometrische Reihe für $\abs*{\frac{a}{z}} <1$ \Rightarrow $\abs*{a} < \abs*{z}$} + &= \frac{1}{1-\frac{a}{z}} + \end{align} + \begin{center} + \includegraphics[width=0.5\textwidth]{img/vl10-beispiel-kausal.png} + \end{center} - \item $x(n) = b^n$ für $n<0$ - \begin{align} - X(z) &= \sum_{n=-\infty}^{-1} b^nz^{-n} = -1 + \sum_{n=0}^{\infty} \left( \frac{z}{b}\right)^n = - \shortintertext{geometrische Reihe für $\abs*{\frac{z}{b}} <1$ \Rightarrow $\abs*{z} < \abs*{b}$} - &= -1 + \frac{1}{1-\frac{z}{b}} - \end{align} - \begin{center} - \includegraphics[width=0.5\textwidth]{img/vl10-beispiel-akausal.png} - \end{center} + \item $x(n) = b^n$ für $n<0$ + \begin{align} + X(z) &= \sum_{n=-\infty}^{-1} b^nz^{-n} = -1 + \sum_{n=0}^{\infty} \left( \frac{z}{b}\right)^n = + \shortintertext{geometrische Reihe für $\abs*{\frac{z}{b}} <1$ \Rightarrow $\abs*{z} < \abs*{b}$} + &= -1 + \frac{1}{1-\frac{z}{b}} + \end{align} + \begin{center} + \includegraphics[width=0.5\textwidth]{img/vl10-beispiel-akausal.png} + \end{center} - \item $x(n) = \cos(n\omega) ) \frac{1}{2}(\e^{jn\omega}+\e^{-jn\omega})$ - \begin{align} - X(z) &= \frac{1}{2} \sum_{n=0}^{\infty} (\e^{jn\omega}+\e^{-jn\omega})z^{-n} - = \frac{1}{2}\left[\sum_{n=0}^{\infty} \left(\frac{\e^{j\omega}}{z}\right)^n + \sum_{n=0}^{\infty} \left(\frac{\e^{-j\omega}}{z}\right)^{-n}\right] =\\ - \shortintertext{$\abs*{z}>1$} - &= \frac{1}{2}\left[\frac{1}{1-\frac{\e^{j\omega}}{z}} + \frac{1}{1-\frac{\e^{-j\omega}}{z}}\right] = \frac{z(z-\cos\omega)}{z^2-2z\cos\omega + 1} - \end{align} - \begin{center} - \includegraphics[width=0.5\textwidth]{img/vl10-beispiel-cos.png} - \end{center} - Sonderfall $\omega=0$ \Rightarrow $x(n)=1$ für $n\gt1,\,x(n)=u(n)$ - \begin{equation} - X(Z) = \frac{z(z-1)}{z^2-2z+1}=\frac{z}{z-1} - \end{equation} - \end{enumerate} + \item $x(n) = \cos(n\omega) ) \frac{1}{2}(\e^{jn\omega}+\e^{-jn\omega})$ + \begin{align} + X(z) &= \frac{1}{2} \sum_{n=0}^{\infty} (\e^{jn\omega}+\e^{-jn\omega})z^{-n} + = \frac{1}{2}\left[\sum_{n=0}^{\infty} \left(\frac{\e^{j\omega}}{z}\right)^n + \sum_{n=0}^{\infty} \left(\frac{\e^{-j\omega}}{z}\right)^{-n}\right] =\\ + \shortintertext{$\abs*{z}>1$} + &= \frac{1}{2}\left[\frac{1}{1-\frac{\e^{j\omega}}{z}} + \frac{1}{1-\frac{\e^{-j\omega}}{z}}\right] = \frac{z(z-\cos\omega)}{z^2-2z\cos\omega + 1} + \end{align} + \begin{center} + \includegraphics[width=0.5\textwidth]{img/vl10-beispiel-cos.png} + \end{center} + Sonderfall $\omega=0$ \Rightarrow $x(n)=1$ für $n\gt1,\,x(n)=u(n)$ + \begin{equation} + X(Z) = \frac{z(z-1)}{z^2-2z+1}=\frac{z}{z-1} + \end{equation} + \end{enumerate} - \noindent\Rightarrow Lage von Polen und Nullstellen in der z-Ebene charakterisisert die Eigenschaften von Systemen - hinsichtlich Stabilität! + \noindent\Rightarrow Lage von Polen und Nullstellen in der z-Ebene charakterisisert die Eigenschaften von Systemen + hinsichtlich Stabilität! - \subsubsection{Stabilitätskriterium im Frequenzbereich} - Bei einem stabilen System war im Zeitbereich ein Kriterium: - \begin{equation} - \sum_{n=-\infty}^{\infty} \abs*{h(n)} < \infty - \end{equation} - Für die Übertragungsfunktion eines Kausalen Systems gilt: - \begin{align} - \abs*{H(z)} &= \abs*{\sum_{n=0}^{\infty} h(n)z^{-n}} \,\explUnder[\le]{Dreicksungleichung}\,\sum_{n=0}^{\infty} \abs*{h(n)}\cdot\abs*{z^{-n}} \\ - \shortintertext{also ist für $\abs*{z} \gt 1$} - \abs*{H(z)} &\le \abs*{\sum_{n=0}^{\infty} h(n)} - \end{align} - \Rightarrow bei stabilem System ist $H(z)$ sicher beschränkt für $\abs{z}\gt1$. Ein Kausales System, das stabil ist, hat keine Pole auf oder außerhalb des Einheitskreises. + \subsubsection{Stabilitätskriterium im Frequenzbereich} + Bei einem stabilen System war im Zeitbereich ein Kriterium: + \begin{equation} + \sum_{n=-\infty}^{\infty} \abs*{h(n)} < \infty + \end{equation} + Für die Übertragungsfunktion eines Kausalen Systems gilt: + \begin{align} + \abs*{H(z)} &= \abs*{\sum_{n=0}^{\infty} h(n)z^{-n}} \,\explUnder[\le]{Dreicksungleichung}\,\sum_{n=0}^{\infty} \abs*{h(n)}\cdot\abs*{z^{-n}} \\ + \shortintertext{also ist für $\abs*{z} \gt 1$} + \abs*{H(z)} &\le \abs*{\sum_{n=0}^{\infty} h(n)} + \end{align} + \Rightarrow bei stabilem System ist $H(z)$ sicher beschränkt für $\abs{z}\gt1$. Ein Kausales System, das stabil ist, hat keine Pole auf oder außerhalb des Einheitskreises. - Für unsere Beispiele gilt entsprechend: - \begin{enumerate}[a)] - \item stabil für $a<1$, offensichtlich klar auch im Zeitbereich - \item stabil für $b>1$ - \item Pole für $z^2-2z \cos\omega+1=0$ \Rightarrow $z_{1/2} = \cos\omega\pm j\sin\omega$ - \begin{center} - \includegraphics[width=0.4\textwidth]{img/vl10-beispiel-pole.png} - \end{center} - \end{enumerate} - Wenn man bei c) die Transformierte als Systemantwort interpretiert, dass ist dies die Reaktion des Systems auf einen Einheitsimpuls. Reaktion ist eine $\cos$-Schwingung undendlicher Dauer (ungedämpft).\\ - Für eine gedämpften $\cos$: - \begin{equation} - \Z \left\{a^n\cos n\omega\right\} = \frac{z(z-a\cos\omega)}{z^-2az\cos\omega + a^2} - \end{equation} - \Rightarrow Pol bei $z_{1/2} = a\cos\omega \pm j a\sin\omega$ \Rightarrow stabil für $a<1$ + Für unsere Beispiele gilt entsprechend: + \begin{enumerate}[a)] + \item stabil für $a<1$, offensichtlich klar auch im Zeitbereich + \item stabil für $b>1$ + \item Pole für $z^2-2z \cos\omega+1=0$ \Rightarrow $z_{1/2} = \cos\omega\pm j\sin\omega$ + \begin{center} + \includegraphics[width=0.4\textwidth]{img/vl10-beispiel-pole.png} + \end{center} + \end{enumerate} + Wenn man bei c) die Transformierte als Systemantwort interpretiert, dass ist dies die Reaktion des Systems auf einen Einheitsimpuls. Reaktion ist eine $\cos$-Schwingung undendlicher Dauer (ungedämpft).\\ + Für eine gedämpften $\cos$: + \begin{equation} + \Z \left\{a^n\cos n\omega\right\} = \frac{z(z-a\cos\omega)}{z^-2az\cos\omega + a^2} + \end{equation} + \Rightarrow Pol bei $z_{1/2} = a\cos\omega \pm j a\sin\omega$ \Rightarrow stabil für $a<1$ - \subsubsection{Einige Korrespondenzen der z-Transformation} - \newcolumntype{C}{>{$\displaystyle}c<{$}} - \begin{center} - \begingroup - \renewcommand{\arraystretch}{3} - \begin{tabular}{C|C} - f(n) & F(z) \\\hline\hline - \delta(n) & 1 \\\hline - u(n) & \frac{z}{z-1} \\\hline - u(n)\,a^n & \frac{z}{z-a} \\\hline - u(n)\,\cos n\omega & \frac{z(z-\cos\omega)}{z^2-2z-\cos\omega+1} \\\hline - u(n)\,\sin n\omega & \frac{z\sin\omega}{z^2-2z\cos\omega + 1} \\\hline - u(n)\,n & \frac{z}{(z-1)^2} \\\hline - u(n)\,\cosh n\beta & \frac{z(z-\cosh\beta)}{z^2-2z\cosh\beta+1} \\\hline - u(n)\, \frac{1}{n!} & \e^\frac{1}{z} - \end{tabular} - \endgroup - \end{center} + \subsubsection{Einige Korrespondenzen der z-Transformation} + \newcolumntype{C}{>{$\displaystyle}c<{$}} + \begin{center} + \begingroup + \renewcommand{\arraystretch}{3} + \begin{tabular}{C|C} + f(n) & F(z) \\\hline\hline + \delta(n) & 1 \\\hline + u(n) & \frac{z}{z-1} \\\hline + u(n)\,a^n & \frac{z}{z-a} \\\hline + u(n)\,\cos n\omega & \frac{z(z-\cos\omega)}{z^2-2z-\cos\omega+1} \\\hline + u(n)\,\sin n\omega & \frac{z\sin\omega}{z^2-2z\cos\omega + 1} \\\hline + u(n)\,n & \frac{z}{(z-1)^2} \\\hline + u(n)\,\cosh n\beta & \frac{z(z-\cosh\beta)}{z^2-2z\cosh\beta+1} \\\hline + u(n)\, \frac{1}{n!} & \e^\frac{1}{z} + \end{tabular} + \endgroup + \end{center} \lecture{11} - \subsubsection{Eigenschaften der einseitigen z-Tranformation} - \begin{enumerate}[a)] - \item \textbf{Linearität}: Linearkombination von k Zeitreihen - \begin{equation} - \Zvon{\sum_{k=-1}^{i} c_k x_k(n)} = \sum_{k=1}^{i}c_k \Zvon{x_k(n)} - \end{equation} - \item \textbf{Verschiebung im Zeitbereich} - Sei y(n) die um $m$ Zeitpunkte nach rechts (später) verschobene kausale Zeitreihe $x(n)$, also $y(n) = x(n-m)$. - Dann ist - \begin{align} - Y(z) &= \sum_{n=0}^{\infty}[x(n-m)]z^{-n} \explUnder[=]{subst $k=n-m$} \sum_{k=-m}^{\infty}x(k) z^{-m+k} \\ - &= z^{-m} \explUnder[\sum_{k=0}^{\infty}]{da Kausalität vorausgesetzt kein Beitrag für $k=-m\dots-1$} x(k) z^{-k} = z^{-m} X(z) - \end{align} - Bei Linksverschiebung (früher) $y(n) = x(n+m)$: - \begin{align} - Y(z) &= \sum_{n=0}^{\infty} x(n+m) z^{-n} \explUnder{k=n+m} \sum_{k=m}^{\infty}x(k) z^{+(m-k)} \\ - &= z^{+m} \Big[\underbrace{\sum_{k=0}^{\infty} x(k)z^{-k}}_{X(z)} - \sum_{k=0}^{m-1} x(k) z^{-k} \Big] \\ - \Rightarrow Y(z) &= z^{+m} \left(X(z)-\sum_{n=0}^{m-1}x(n)z^{-n}\right) - \end{align} - \includegraphics[width=0.8\textwidth]{img/vl11-z-verschoben.png} + \subsubsection{Eigenschaften der einseitigen z-Tranformation} + \begin{enumerate}[a)] + \item \textbf{Linearität}: Linearkombination von k Zeitreihen + \begin{equation} + \Zvon{\sum_{k=-1}^{i} c_k x_k(n)} = \sum_{k=1}^{i}c_k \Zvon{x_k(n)} + \end{equation} + \item \textbf{Verschiebung im Zeitbereich} + Sei y(n) die um $m$ Zeitpunkte nach rechts (später) verschobene kausale Zeitreihe $x(n)$, also $y(n) = x(n-m)$. + Dann ist + \begin{align} + Y(z) &= \sum_{n=0}^{\infty}[x(n-m)]z^{-n} \explUnder[=]{subst $k=n-m$} \sum_{k=-m}^{\infty}x(k) z^{-m+k} \\ + &= z^{-m} \explUnder[\sum_{k=0}^{\infty}]{da Kausalität vorausgesetzt kein Beitrag für $k=-m\dots-1$} x(k) z^{-k} = z^{-m} X(z) + \end{align} + Bei Linksverschiebung (früher) $y(n) = x(n+m)$: + \begin{align} + Y(z) &= \sum_{n=0}^{\infty} x(n+m) z^{-n} \explUnder{k=n+m} \sum_{k=m}^{\infty}x(k) z^{+(m-k)} \\ + &= z^{+m} \Big[\underbrace{\sum_{k=0}^{\infty} x(k)z^{-k}}_{X(z)} - \sum_{k=0}^{m-1} x(k) z^{-k} \Big] \\ + \Rightarrow Y(z) &= z^{+m} \left(X(z)-\sum_{n=0}^{m-1}x(n)z^{-n}\right) + \end{align} + \includegraphics[width=0.8\textwidth]{img/vl11-z-verschoben.png} - \item Ähnlichkeit: - \begin{align} - \Zvon{a^n x(n)} &= \sum_{n=0}^{\infty}a^n x(n) z^{-n} = \sum_{n=0}^{\infty}x(n)z(z\,a^{-1})^{-n} \\ - &= X(z\, a^{-1}) - \end{align} - \item \textbf{Faltungssatz}: - \begin{equation} - \Zvon{f(n) * g(n)} = F(z) \cdot G(z) - \end{equation} - Bei linearen Systemen ist - \begin{align} - y(n) &= h(n) * x(n) \\ - Y(z) &= H(z) \cdot X(z) - \end{align} - wobei $H(z)$ die Überlappungsfunktion des Systems ist - \item Differentation der Bildfunktion: - \begin{align} - -z \odv{F(z)}{z} = \Zvon{n\cdot f(n)} \\ - \shortintertext{allgemein} - \left(-z \odv{}{z} \right)^k F(z) = \Zvon{n^k f(n)} - \end{align} + \item Ähnlichkeit: + \begin{align} + \Zvon{a^n x(n)} &= \sum_{n=0}^{\infty}a^n x(n) z^{-n} = \sum_{n=0}^{\infty}x(n)z(z\,a^{-1})^{-n} \\ + &= X(z\, a^{-1}) + \end{align} + \item \textbf{Faltungssatz}: + \begin{equation} + \Zvon{f(n) * g(n)} = F(z) \cdot G(z) + \end{equation} + Bei linearen Systemen ist + \begin{align} + y(n) &= h(n) * x(n) \\ + Y(z) &= H(z) \cdot X(z) + \end{align} + wobei $H(z)$ die Überlappungsfunktion des Systems ist + \item Differentation der Bildfunktion: + \begin{align} + -z \odv{F(z)}{z} = \Zvon{n\cdot f(n)} \\ + \shortintertext{allgemein} + \left(-z \odv{}{z} \right)^k F(z) = \Zvon{n^k f(n)} + \end{align} + \end{enumerate} + + \subsubsection{Die z-Transformation bei linearen rekursiven Systemen} + Die Klasse der kausalen linearen rekusriven Systeme ist beschreibbar durch: + \begin{equation} + \sum_{k=0}^{p}a_k y(n-k) = \sum_{k=0}^{q} b_k x(n-k) + \end{equation} + Nunmehr z-Transformation auf beiden Seiten: + \begin{align} + \Zvon{\sum_{k=0}^{p}a_k y(n-k)} &= \Zvon{\sum_{k=0}^{q} b_k x(n-k)} \\ + \shortintertext{mit Linearitätssatz} + \sum_{k=0}^{p}a_k \Zvon{y(n-k)} &= \sum_{k=0}^{q} b_k \Zvon{x(n-k)} \\ + \shortintertext{mit Vertauschungssatz} + \sum_{k=0}^{p}Y(z) z^{-k} &= \sum_{k=0}^{q} b_k X(z) z^{-k} + \end{align} + Daraus: + \begin{align} + H(z) &= \frac{Y(z)}{X(z)} = \frac{\sum_{k=0}^{q}b_k z^{-k}}{\sum_{k=0}^{p} a_k z^{-k}} \\ + \Aboxed{ + H(z) &= \frac{b_0 + b_1z^{-1} b_2z^{-2} + \dots + b_qz^{-q}}{a_0 + a_1z^{-1} a_2z^{-2} + \dots + a_pz^{-p}} + } + \end{align} + \Rightarrow Ein lineares, rekursives System hat immer eine in $z$ rationale Übertragungsfunktion. + Mit dem Hauptsatz der linearen Algebra lässt dies umschreiben zu + + \begin{align} + H(z) &= c \frac{\prod_{k=1}^{q} \left(1-z_kz^{-1}\right)}{\prod_{k=1}^{p} \left(1-p_kz^{-1}\right)} \\ + &= c \frac{\left(1-z_1 z^{-1}\right) \left(1-z_2 z^{-1}\right) \dots \left(1-z_q z^{-1}\right)} + {\left(1-p_1z^{-1}\right) \left(1-p_2z^{-1}\right) \dots \left(1-p_pz^{-1}\right)} + \end{align} + + Für + \begin{itemize} + \item $z=z_k$: $H(z) = 0$ \Rightarrow Übertragungsfunktion hat eine Nullstelle, Systemausgang $=0$ + \item $z\to p_k$: $H(z) \to \infty$ \Rightarrow Übertragungsfunktion hat einen Pol, Systemausgang ist unendlich hoch + \end{itemize} + + \paragraph{Beispiel} + \begin{align} + H_1(z) &= \frac{b_0 + b_1 z^{-1} + b_2 z^{-2}}{a_0 + a_1 z^{-1} + a_2 z^{-2}} \\ + &= \frac{b_0 z^2 + b_1 z + b_2}{a_0 z^2 + a_1 z + a_2} \\ + &= \frac{(z-z_1)(z-z_2)}{(z-p_1) + (z-p_2)} = \frac{(1-z_1 z^{-1})(1-z_2z^{-1})}{(1-p_1z^{-1})(1-p_2z^{-1})} + \shortintertext{wo:} + z_{1,2} &= \frac{-b_1\pm \sqrt{b_1^2-4b_0b_2}}{2b_0} \\ + p_{1,2} &= \frac{-a_1\pm \sqrt{a_1^2-4a_0a_2}}{2a_0} + \end{align} + + \begin{enumerate} + \item $b_1 = b_2 = 0$\\\Rightarrow $H(z)$ hat zwei Pole bei $z=p_1$ und $z=p_2$ \Rightarrow ``Allpol-System'' + \item $b_2 = 0$\\\Rightarrow $H(z)$ hat zwei Pole (wie bei Fall 1), eine Nullstelle bei $z=0$ und eine Nullstelle bei $z=\frac{-b_1}{b_0}$ + \item keine Einschränkungen, d.h. Pole wie Fall 1, Nullstellen für $z=z_1$ und $z=z_2$ \end{enumerate} - \subsubsection{Die z-Transformation bei linearen rekursiven Systemen} - Die Klasse der kausalen linearen rekusriven Systeme ist beschreibbar durch: - \begin{equation} - \sum_{k=0}^{p}a_k y(n-k) = \sum_{k=0}^{q} b_k x(n-k) - \end{equation} - Nunmehr z-Transformation auf beiden Seiten: + \subsubsection{Realisierungen} + \begin{center} + \includegraphics[width=0.8\textwidth]{img/vl11-z-beispiel.png} + \end{center} + Für das allgemeine, rekursive, lineare System ist die Beschreibungsgleichung \begin{align} - \Zvon{\sum_{k=0}^{p}a_k y(n-k)} &= \Zvon{\sum_{k=0}^{q} b_k x(n-k)} \\ - \shortintertext{mit Linearitätssatz} - \sum_{k=0}^{p}a_k \Zvon{y(n-k)} &= \sum_{k=0}^{q} b_k \Zvon{x(n-k)} \\ - \shortintertext{mit Vertauschungssatz} - \sum_{k=0}^{p}Y(z) z^{-k} &= \sum_{k=0}^{q} b_k X(z) z^{-k} - \end{align} - Daraus: - \begin{align} - H(z) &= \frac{Y(z)}{X(z)} = \frac{\sum_{k=0}^{q}b_k z^{k})}{\sum_{k=0}^{p} a_k z^{-k}} \\ - \Aboxed{ - H(z) &= \frac{b_0 + b_1z^{-1} b_2z^{-2} + \dots + b_qz^{-q}}{a_0 + a_1z^{-1} a_2z^{-2} + \dots + a_pz^{-p}} - } + \sum_{k=0}^{p} a_k y(n-k) = \sum_{k=0}^{q} b_k x(n-k) + \shortintertext{bzw:} + Y(z) \sum_{k=0}^{p} a_k z^{-k} = X(z) \sum_{k=0}^{q} b_k z^{-k} \end{align} + Aus der Gleichung ist sofort ersichtlich, dass 3 Grundoperationen ausreichen: + \begin{enumerate} + \item Multiplikation mit einer Konstanten + \item Summation (Subtraktion) + \item Verzögerung um \textbf{einen} Abtasttakt + \end{enumerate} + Realisierung eines gewünschten Umformalgorithmus durch ein Rechenprogramm oder direkt durch Hardwarebausteine Multiplizierer, Summierer und Schieberegister, die in bestimmter Weise verknüpft werden.\\ + Prinzipiell können unterschiedliche Strukturen den gleichen Algorithmus realisieren. + Bei endlicher Rechengenauigkeit bleiben jedoch Unterschiede bestehen. - + \paragraph{Blockdiagramme} geben die Struktur einer Realisierung an. 3 Bauelemente decken alle Anwendungen ab: + \begin{enumerate} + \item Multiplizierer mit einer Konstanten $a$ \\ + \includegraphics[width=0.5\textwidth]{img/vl11-block-Multiplizierer.png} + \item Addierer (Subtrahierer) \\ + \includegraphics[width=0.5\textwidth]{img/vl11-block-addierer.png} + \item Verzögerung, Zeitverschiebung um einen Abtasttakt\\ + \includegraphics[width=0.5\textwidth]{img/vl11-block-verzögerung.png} + \end{enumerate} + + \paragraph{Beispiele} + \begin{itemize} + \item Nichtrekursives System \\ + \includegraphics[width=0.8\textwidth]{img/vl11-bsp-nichtrekursiv.png} \\ + \begin{align} + y(n) &= x(n) + x(n-1) - a\,x(n-2) \\ + Y(z) &= X(z) (1 + z^{-1} - a\,z^{-2}) \\ + H(z) &= \frac{Y(z)}{X(z)} = 1 + z^{-1} - a\,z^{-2} \\ + \mathcal{Z}^{-1}\{H(z)\} &= \delta(n) + \delta(n-1) - a\,\delta(n-2) + \end{align} + \includegraphics[width=0.5\textwidth]{img/vl11-bsp-nichtrekursiv-H.png} + \begin{itemize} + \item System mit endlicher Impulsantwort, FIR + \item System hat zwei Verzögerungselemente, entspricht Grad des Polynoms + \end{itemize} + + \item Rekursives System 1. Ordnung \\ + \includegraphics[width=0.8\textwidth]{img/vl11-bsp-rekursiv.png} \\ + \begin{align} + y(n) &= x(n) + a\, y(n-1) \\ + \end{align} + Impulsantwort: \\ + \newcolumntype{C}{>{$}c<{$}} % math-mode version of "c" column type + \begin{tabular}{C|C|C} + n & x(n) & y(n) \\ + 0 & 1 & 1 \\ + 1 & 0 & a \\ + 2 & 0 & a^2 \\ + 3 & 0 & a^3 \\ + 4 & 0 & a^4 \\ + 5 & 0 & a^5 \\ + \end{tabular}\\ + + \Rightarrow System mit Impulsantwort von unendlich langer Dauer: + \begin{align} + Y(z) &= X(z) + a\,z^{-1} Y(z) \\ + H(z) &= \frac{1}{1-a\,z^{-1}} = \frac{z}{z-a} \\ + \mathcal{Z}^{-1}\{H(z)\} &= a^n u(n) + \end{align} + Beachte: der Addierer darf nur Werte erhalten, die bereits berechnet wurden. + Deshalb, in jedem Zyklus mindestens ein Verzögerer erforderlich! + + Kaskadierung dieses Systems ergibt rekursives System 2. Ordnung:\\ + \includegraphics[width=0.8\textwidth]{img/vl11-bsp-rekursiv-kaskadiert.png} \\ + Zwei Gleichungen + \begin{align} + w(n) &= x(n) + a\,w(n-1) \\ + y(n) &= w(n) + a\,y(n-1) \\ + \Rightarrow y(n) &= x(n) + 2a\,y(n-1) - a^2 y(n-2) \\ + Y(z) &= X(z) + 2a\,z^{-1} Y(z) - a^2z^{-2} Y(z) \\ + \Rightarrow H(z) &= \frac{1}{1 - 2a\,z^{-1} + a^2z^{-2}} = \frac{1}{(1-a\,z^{-1})^2} + \end{align} + \Rightarrow Kaskadierung bedeutet Multiplikation der Übertragungsfunktion + \end{itemize} \end{document}