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elektrische Größe beeinflussen, die weiter verarbeitet werden können. - Eine Umwandlung der Energieform der Biosignale ist notwendig. Selbst bei - Sensoren für elektrische Größen ist eine Umwandlung (von Ionenleitung - zur Elektronenleitung) nötig. - - Weitere Sensorengruppen, wie Temperatur-und chemische Sensoren werden - hier nicht behandelt, da ihre Dynamik aus Sicht der BSA vernachlässigbar - gering ist. - - - \subsection{Klassifikation von Sensoren} - - Ein Sensor (lateinisch 'sensus': Gefühl) oder Fühler ist ein technisches - Bauteil, das die physikalischen oder chemischen Eigenschaften (z.B. - Wärmestrahlung, Temperatur, Feuchtigkeit, Druck, Schall, Helligkeit, - Magnetismus, Beschleunigung, Kraft, elektrisches Potential) erfassen und - in ein elektronisches oder ein anderes geeignetes Signal umwandeln kann. - + Ein Sensor ist ein technisches Bauteil, das die physikalischen oder chemischen Eigenschaften erfassen und in ein elektronisches oder ein anderes geeignetes Signal umwandeln kann. Man unterscheidet zwischen aktiven und passiven Sensoren - - \begin{itemize*} - \item Aktiver Sensor: gibt eine Spannung oder einen Strom ab, wobei er für seine Funktion elektrische Energie benötigt oder eine Energieart in die elektrische umwandelt. Er wirkt wie eine elektrische Signalquelle (z.B. Phototransistor) - \item Passiver Sensor: ändert elekttrische Größen (z.B. den Widerstand des Dehnmessstreifens in Abhängigkeit von der Dehnung) ohne Energiezufuhr von außen. - \end{itemize*} + \begin{description*} + \item[Aktiv] gibt Spannung/Strom ab, wobei er für Funktion Energie benötigt/umwandelt. wirkt wie elektrische Signalquelle + \item[Passiv] ändert elektrische Größen (z.B. Widerstand) ohne Energiezufuhr von außen + \end{description*} Auflösung von Sensoren: - - \begin{itemize*} - \item temporal: Zeitabstand zwischen Messungen (z.B. Geigerzähler, Aktionspotentiale) - \item spektral: Abstand von Spektrallinien (z.B. Wärmebildkamera, Infrarotsensor, Spektralphotometer) - \item räumlich: räumlicher Abstand (z.B. EEG, Ultraschall, CT/MRT) - \item und alle Kombinationen (z.B. spatialtemporale Auflösung in einem Frequenzband) - \end{itemize*} + \begin{description*} + \item[temporal] Zeitabstand zwischen Messungen (z.B. Aktionspotentiale) + \item[spektral] Abstand von Spektrallinien (z.B. Wärmebildkamera) + \item[räumlich] räumlicher Abstand (z.B. EEG, Ultraschall) + \item[...] Kombinationen (z.B. spatialtemporale Auflösung in Frequenzband) + \end{description*} Klassifikation nach Messgröße: - - \begin{itemize*} - \item Physikalisch: Kraft, Druck, Moment, Durchfluss - \item Elektrizität: Potential, Strom, Impedanz - \item Magnetismus: Fluss, Induktion - \item Optik/Licht: spektrale Dämpfung, Extinktion - \item Chemisch: Partialdruck von Gasen, Zucker, Hämoglobin - \item Akustik: Herzschalltöne, Atmung - \item Temperatur: Körpertemperatur - \end{itemize*} + \begin{description*} + \item[Physikalisch] Kraft, Druck, Moment, Durchfluss + \item[Elektrizität] Potential, Strom, Impedanz + \item[Magnetismus] Fluss, Induktion + \item[Optik/Licht] spektrale Dämpfung, Extinktion + \item[Chemisch] Partialdruck von Gasen, Zucker, Hämoglobin + \item[Akustik] Herzschalltöne, Atmung + \item[Temperatur] Körpertemperatur + \end{description*} \subsection{Druck, Dehnung und Kraft} - Dehnmessstreifen (DMS) - \begin{itemize*} \item Messprinzip: Dehnungsabhängiger Widerstand - \item Realisierung: Widerstandsdraht oder Halbleiter als Gitter auf Träger + \item Realisierung: Widerstandsdraht/Halbleiter Gitter auf Träger \item Messbare Größen: Kraft, Druck, Moment - \item Anwendung in der Medizintechnik: Atmungsdiagnostik, Fahrradergometer, Diagnostik des Bewegungsapparates - \item Signaleigenschaftem - \begin{itemize*} - \item passiver Sensor - thermisches und/oder Halbleiter-Rauschen - \item empfindlich gegen NF-elektrische/magnetische und HF-elektromagnetische Felder - \item temperaturabhängig vor allem Halbleiter - \item Übertragungseigenschaften abhängig von der Sensorkopplung - \item Signaldynamik abhängig von Masse und Technologie \end{itemize*} + \item Anwendung: Atmungsdiagnostik, Fahrradergometer + \end{itemize*} + Signaleigenschaftem + \begin{itemize*} + \item passiver Sensor - thermisches und/oder Halbleiter-Rauschen + \item empfindlich gegen NF-elektrische/magnetische und HF-elektromagnetische Felder + \item temperaturabhängig sind vor allem Halbleiter + \item Übertragungseigenschaften abhängig von der Sensorkopplung + \item Signaldynamik abhängig von Masse und Technologie \end{itemize*} - Grundlage ist der piezoresestive Widerstandseffekt: Durch Zug/Druck - nimmt der elektrische Widerstand zu/ab - + Grundlage ist der piezoresestive Widerstandseffekt \begin{itemize*} - \item \$R=\textbackslash ro\textbackslash frac\{l\}\{A\}=\textbackslash ro\textbackslash frac\{4l\}\{d\^{}2\textbackslash pi\}\$, \$\textbackslash ro\$ spez. Widerstand, \$l\$ Drahtlänge, \$d\$ Drahtdurchmesser - \item \$R+\textbackslash delta R=(\textbackslash ro+\textbackslash delta\textbackslash ro)\textbackslash frac\{4(l+\textbackslash delta l)\}\{(d-\textbackslash delta d)\^{}2\textbackslash pi\}\$; davon ist \$(l+\textbackslash delta l)\$ relevant für die Dehnungsmessung + \item $R=\rho\frac{l}{A}=\rho\frac{4l}{d^2\pi}$, $\rho$ Widerstand, $l$ Länge, $d$ Durchmesser + \item $R+\delta R=(\rho+\delta\rho)\frac{4(l+\delta l)}{(d-\delta d)^2\pi}$ + \item davon ist $(l+\delta l)$ relevant für die Dehnungsmessung + \item Temp. linear abhängig von spez. Widerstand + \item nicht linear abhängig von mechanischen Änderungen \end{itemize*} - - Die Widerstandsänderung ist linear abhängig von der - Temperaturabhängigkeit des spez. Widerstandes, jedoch nicht linear - abhängig von der mechanisch bedingten Änderung der Abmessungen. - Natürlich hängen die Längenänderung und die des Durchmessers zusammen. - Der konkrete Zusammenhang ist jedoch durch die Konstruktion und das - Material gegeben und kann nicht verallgemeinert werden. Da für die - Messung allein die Längenänderung von Interesse ist, wird im weiteren - nur diese betrachtet. - + % Die Widerstandsänderung ist linear abhängig von der Temperaturabhängigkeit des spez. Widerstandes, jedoch nicht linear abhängig von der mechanisch bedingten Änderung der Abmessungen. Natürlich hängen die Längenänderung und die des Durchmessers zusammen. + %Der konkrete Zusammenhang ist jedoch durch die Konstruktion und das Material gegeben und kann nicht verallgemeinert werden. Da für die Messung allein die Längenänderung von Interesse ist, wird im weiteren nur diese betrachtet. \begin{itemize*} - \item \$\textbackslash frac\{\textbackslash delta R\}\{R\}=k\textbackslash frac\{\textbackslash delta l\}\{l\}=k*\textbackslash epsilon\$, \$\textbackslash epsilon\$-relative Dehnung - \item \$\textbackslash epsilon=\textbackslash frac\{F\}\{EA\}\$, \$F=\textbackslash frac\{\textbackslash delta R\}\{R\}*\textbackslash frac\{EA\}\{k\}\$, E-Elastizitätsmodul - \item In der Praxis aus Kostengründen und wegen geringer Temperaturabhängigkeit meistens Konstant an (54\% Cu, 45\% Ni, 1\% Mn mit \$k=2,05\$) verwendet + \item $\frac{\delta R}{R}=k\frac{\delta l}{l}=k*\epsilon$, $\epsilon$-relative Dehnung + \item $\epsilon=\frac{F}{EA}$, $F=\frac{\delta R}{R}*\frac{EA}{k}$, E-Elastizitätsmodul + %\item In der Praxis aus Kostengründen und wegen geringer Temperaturabhängigkeit meistens Konstant an (54\% Cu, 45\% Ni, 1\% Mn mit $k=2,05$) verwendet \end{itemize*} Messverfahren: Widerstandsmessung mit Brückenschaltung - \begin{itemize*} - \item Wheatstone'sche Brücke: \$U\_A\textbackslash rightarrow 0\textbar\_R \textbackslash Rightarrow \textbackslash frac\{R\_x\}\{R\_V\}=\textbackslash frac\{R\_2\}\{R\_1\}\$ - \item \$R\_X=R\_V\textbackslash frac\{R\_2\}\{R\_1\}\$ - \item \includegraphics[width=.5\linewidth]{./Assets/Biosignalverarbeitung-wheatstone-brücke.png} - \item Warum wird Rx mit Wheatstone-Brücke und nicht klassisch über Stromeinspeisung und Spannungsmessung bestimmt? - \begin{itemize*} - \item Das Messsignal (Ua) ist stärker bzw. die Empfindlichkeit der Messanordnung ist höher als bei einer reinen Strom/Spannungsmessung, außerdem ist Temperaturkompensation möglich. - \item Starkes Messsignal ist sinnvoll wegen der immer vorhandenen Störungen vom Netz und Leitungen, die direkt auf die Kabel der Messanordnung wirken. \end{itemize*} + \item Wheatstone'sche Brücke: $U_A\rightarrow 0|_R \Rightarrow \frac{R_x}{R_V}=\frac{R_2}{R_1}$ + \item $R_X=R_V\frac{R_2}{R_1}$ + \end{itemize*} + %\includegraphics[width=.5\linewidth]{./Assets/Biosignalverarbeitung-wheatstone-brücke.png} + %Warum wird Rx mit Wheatstone-Brücke und nicht klassisch über Stromeinspeisung und Spannungsmessung bestimmt? + \begin{itemize*} + \item Empfindlichkeit der Messanordnung höher als bei reinen Strom/Spannungsmessung + \item Temperaturkompensation möglich + \item Starkes Messsignal sinnvoll wegen Störungen vom Netz und Leitungen%, die direkt auf die Kabel der Messanordnung wirken. \end{itemize*} Folien-DMS: \begin{itemize*} - \item Widerstandsdraht mit ca \$20\textbackslash mu m\$ Durchmesser oder Halbleiter + \item Widerstandsdraht mit ca $20\mu m$ Durchmesser oder Halbleiter \item Träger Acrylharz, Epoxydharz, Polyamid - \item \includegraphics[width=.5\linewidth]{Assets/Biosignalverarbeitung-folien-dms.png} + %\includegraphics[width=.5\linewidth]{Assets/Biosignalverarbeitung-folien-dms.png} + \item Dehnungsmessrosette: Messung in drei Richtungen + %\includegraphics[width=.5\linewidth]{Assets/Biosignalverarbeitung-Dehnungsmessrosette.png} + %Wie man an diesen Konstruktionsbeispielen gut erkennen kann, bilden die Leitungen ungewollterweise eine Antenne, die alle vorhandenen Störungen aus der Umgebung aufnimmt, vor allem Netzeinstreuung, Mobilfunk und Computernetze. \end{itemize*} - Dehnungsmessrosette: - \begin{itemize*} - \item Messung in drei Richtungen - \item \includegraphics[width=.5\linewidth]{Assets/Biosignalverarbeitung-Dehnungsmessrosette.png} - \end{itemize*} + %Aufbau von Massebezogenen und Massefreien Messungen: + %\includegraphics[width=\linewidth]{Assets/Biosignalverarbeitung-masse-messung.png} + %Messspannung von $U_{R5}$ in der massebezogenen Schaltung + %\includegraphics[width=.5\linewidth]{Assets/Biosignalverarbeitung-messspannung-massebezogen.png} + Bei massebezogener Messung - auch Single-End genannt - werden die Störungen direkt dem Messsignal überlagert, so dass später eine Trennung ohne aufwendige Signalverarbeitung kaum möglich ist. - Wie man an diesen Konstruktionsbeispielen gut erkennen kann, bilden die - Leitungen ungewollterweise eine Antenne, die alle vorhandenen Störungen - aus der Umgebung aufnimmt, vor allem Netzeinstreuung, Mobilfunk und - Computernetze. - - Aufbau von Massebezogenen und Massefreien Messungen: - \includegraphics[width=.5\linewidth]{Assets/Biosignalverarbeitung-masse-messung.png} - - Messspannung von \$U\_\{R5\}\$ in der massebezogenen Schaltung - \includegraphics[width=.5\linewidth]{Assets/Biosignalverarbeitung-messspannung-massebezogen.png} - Bei massebezogener Messung - auch Single-End genannt, da gegen Masse - - werden die Störungen direkt dem Messsignal überlagert, so dass später - eine Trennung ohne aufwendige Signalverarbeitung kaum möglich ist. - - Massebezogne eBrückenspannung (rot, blau) und Indikatorspannung \$U\_d\$ - (grün) - \includegraphics[width=.5\linewidth]{Assets/Biosignalverarbeitung-massebezogen-brückenspannung.png} - Einen großen Teil der Netzstörung bilden die elektrostatischen - (kapazitiv eingekoppelten) Felder, die Gleichtaktcharakter haben. Diese - lassen sich also durch Differenzbildung -hier mit einer Wheatstonschen - Brücke -zum Teil eliminieren. + %Massebezogne Brückenspannung (rot, blau) und Indikatorspannung $U\_d$ (grün) + %\includegraphics[width=.5\linewidth]{Assets/Biosignalverarbeitung-massebezogen-brückenspannung.png} + Einen großen Teil der Netzstörung bilden die elektrostatischen Felder, die Gleichtaktcharakter haben. Diese lassen sich also durch Differenzbildung (Wheatstonschen Brücke) zum Teil eliminieren. \subsection{Durchfluss, Volumen}\label{durchfluss-volumen} - + Massendurchfluss \begin{itemize*} - \item Massendurchfluss + \item $\dot m=\frac{dm}{dt}$ + \item $[\dot m]=\frac{kg}{h};\frac{g}{s}$ + \item industriell relevant z.B. Kraftstoff, Luftverbrauch im Motor + \end{itemize*} + Volumendurchfluss + \begin{itemize*} + \item $\dot V=\frac{dV}{dt}$ + \item $[\dot V]=\frac{m^3}{h};\frac{l}{min}$ + \item wichtige Messgröße: Blutfluss, Atmung, Gastrointestinalapparat + \end{itemize*} + %\item Der Durchfluss eines Mediums ist eine der wichtigsten Größen in der technischen und medizinischen Messtechnik. Technisch vor allem der Massendurchfluss, medizinisch der Volumendurchfluss, da medizinisch grundsätzlich die Volumina diagnostisch relevante Größe darstellen. + Bei bekannter durchflossener Fläche wird der Volumenfluss über die Geschwindigkeitsmessung ermittelt + \begin{itemize*} + \item $\dot V=\frac{dV}{dt}=\frac{A*dl}{dt}= A*v$ + \item Reale Verteilung der Geschwindigkeit ist Parabel %mit Maximum in der Mitte + \item[$\rightarrow$] Gemessene Geschwindigkeit ist die mittlere Geschwindigkeit + \end{itemize*} + %\item In der Medizin kann weder eine Geschwindigkeitsverteilung - wie in der Technik - erzwungen werden, noch kann sie vollständig erfasst werden. Daher misst man tatsächlich nur eine ,,mittlere'' Geschwindigkeit, wobei der Begriff ,,mittlere'' hier nicht ganz korrekt ist, da die tatsächliche Verteilung nach wie vor unbekannt ist. + + Druckdifferentmessung nach Gesetz von Hagen-Poiseuille + \begin{itemize*} + \item $\dot V=\frac{dV}{dt}\frac{\pi d^4}{128\mu}*\frac{\delta p}{l}$ + \item $d$: Durchmesser Kappilare, $l$: Länge der Kapillare, + \item $\delta p=p_A - p_B$: Druckdifferenz über Kapillare%, abhängigkeit von der Strömungsgeschwindigkeit + , $\mu$ - Viskosität %des Mediums + %\item \includegraphics[width=.5\linewidth]{Assets/Biosignalverarbeitung-pneumotachograph.png} + %\item Bsp: 10\% Verengung der Kapillare $\rightarrow$ 34\% Reduktion des Durchsatzes, d.h. im Blutkreislauf Anstieg des Blutdrucks um 34\% + %\item Bei externen Sensoren der Durchflussmessung kann man die Messbedingungen relativ klar vorgeben, z.B. im Pneumotachographen. Man erzwingt kapillare Strömung, der Strömungswiderstand und die Fläche sind bekannt, so dass aus der Druckdifferenz direkt auf den Durchfluss geschlossen werden kann. + \end{itemize*} + Anwendung in der Medizintechnik + \begin{itemize*} + \item Messung aller vitaler Lungenvolumina + \item Messung des Blutflusses + \end{itemize*} + Nachteile des Messprinzips von Hagen-Poiseuille + \begin{itemize*} + \item zusätzlicher Strömungswiderstand verfälscht das Ergebnis + \item bei Temperaturunterschieden Kappilaren Tröpfchenbildung + \item geringer Dynamikbereich (1:10) + \item niedrige Messgenauigkeit wegen Turbulenzen an Kapillarenden + \item direkter Kontakt mit Medium nötig + \end{itemize*} + + Ultraschall-Geschwindigkeitsmessung nach dem Laufzeitverfahren + \begin{itemize*} + \item $v=\frac{T_2-T_1}{T_1 T_2}*\frac{L}{2\ cos\ \alpha}$ + \item $v$ - mittlere Strömungsgeschwindigkeit des Mediums + \item $T_1$ - Laufzeit mit der Strömung + \item $T_2$ - Laufzeit gegen die Strömung + \item $L$: Pfadlänge, $\alpha$: Winkel Strömung-Pfad + %\item \includegraphics[width=.5\linewidth]{Assets/Biosignalverarbeitung-ultraschall-geschwindigkeit.png} + \item Vorteile \begin{itemize*} - \item \$\textbackslash dot m=\textbackslash frac\{dm\}\{dt\}\$ - \item \${[}\textbackslash dot m{]}=\textbackslash frac\{kg\}\{h\};\textbackslash frac\{g\}\{s\} - \item industriell relevante Größe, z.B. Kraftstoffe, Luftverbrauch im Motor + \item kein Kontakt mit dem Medium, insbesondere Blutbahnen + \item Installation und Messung ohne Unterbrechnung des Flusses \end{itemize*} - \item Volumendurchfluss + \item Nachteile \begin{itemize*} - \item \$\textbackslash dot V=\textbackslash frac\{dV\}\{dt\}\$ - \item \${[}\textbackslash dot V{]}=\textbackslash frac\{m\^{}3\}\{h\};\textbackslash frac\{l\}\{min\}\$ - \item wichtige Messgröße in der medizinischen Messtechnik: Blutfluss, Atmung, Gastrointestinalapparat + \item invasive Methode, da Blutgefäß freigelegt werden muss + \item Ungenauigkeit wegen der Verfomung der Blutgefäße \end{itemize*} - \item Der Durchfluss eines Mediums ist eine der wichtigsten Größen in der technischen und medizinischen Messtechnik. Technisch vor allem der Massendurchfluss, medizinisch der Volumendurchfluss, da medizinisch grundsätzlich die Volumina diagnostisch relevante Größe darstellen. - \item Bei bekannter durchflossener Fläche wird der Volumenfluss über die Geschwindigkeitsmessung ermittelt + \item Signaleigenschafte \begin{itemize*} - \item \$\textbackslash dot V=\textbackslash frac\{dV\}\{dt\}=\textbackslash frac\{A\emph{dl\}\{dt\}= A}v\$ - \item Reale Verteilung der Geschwindigkeit ist Parabel mit Maximum in der Mitte \$\textbackslash rightarrow\$ Gemessene Geschwindigkeit ist die mittlere Geschwindigkeit - \end{itemize*} - \item In der Medizin kann weder eine Geschwindigkeitsverteilung - wie in der Technik - erzwungen werden, noch kann sie vollständig erfasst werden. Daher misst man tatsächlich nur eine ,,mittlere'' Geschwindigkeit, wobei der Begriff ,,mittlere'' hier nicht ganz korrekt ist, da die tatsächliche Verteilung nach wie vor unbekannt ist. - \item Messprinzip Druckdifferentmessung nach Gesetz von Hagen-Poiseuille - \begin{itemize*} - \item \$\textbackslash dot V=\textbackslash frac\{dV\}\{dt\}\textbackslash frac\{\textbackslash pi d\^{}4\}\{128\textbackslash mu\}*\textbackslash frac\{\textbackslash delta p\}\{l\}\$ - \item \$d\$ - Durchmesser der Kappilare - \item \$\textbackslash delta p=p\_A - p\_B\$ - Druckdifferenz über der Kapillare, abhängigkeit von der Strömungsgeschwindigkeit - \item \$l\$ - Länge der Kapilalre - \item \$\textbackslash mu\$ - Viskosität des Mediums - \item \includegraphics[width=.5\linewidth]{Assets/Biosignalverarbeitung-pneumotachograph.png} - \item Bsp: 10\% Verengung der Kapillare \$\textbackslash rightarrow\$ 34\% Reduktion des Durchsatzes, d.h. im Blutkreislauf Anstieg des Blutdrucks um 34\% - \item Bei externen Sensoren der Durchflussmessung kann man die Messbedingungen relativ klar vorgeben, z.B. im Pneumotachographen. Man erzwingt kapillare Strömung, der Strömungswiderstand und die Fläche sind bekannt, so dass aus der Druckdifferenz direkt auf den Durchfluss geschlossen werden kann. - \end{itemize*} - \item Anwendung in der Medizintechnik - \begin{itemize*} - \item Messung aller vitaler Lungenvolumina - \item Messung des Blutflusses - \end{itemize*} - \item Nachteile des Messprinzips - \begin{itemize*} - \item zusätzlicher Strömungswiderstand verfälscht das Ergebnis - \item bei Temperaturunterschieden Kappilaren-Medium Tröpfchenbildung - \item geringer Dynamikbereich (1:10) - \item niedrige Messgenauigkeit wegen Turbulenzen an Kapillarenden - \item direkter Kontakt mit Medium - \end{itemize*} - \item Ultraschall-Geschwindigkeitsmessung nach dem Laufzeitverfahren - \begin{itemize*} - \item \$v=\textbackslash frac\{T\_2-T\_1\}\{T\_1 T\_2\}*\textbackslash frac\{L\}\{2\textbackslash{} cos\textbackslash{} \textbackslash alpha\}\$ - \item \$v\$ - mittlere Strömungsgeschwindigkeit des Mediums - \item \$T\_1\$ - Laufzeit des Ultraschalls mit der Strömung - \item \$T\_2\$ - Laufzeit des Ultraschalls gegen die Strömung - \item \$L\$ - Länge des Ultraschall-Pfades - \item \$\textbackslash alpha\$ - Winkel zwischen der Strömung und dem Ultraschall-Pfad - \item \includegraphics[width=.5\linewidth]{Assets/Biosignalverarbeitung-ultraschall-geschwindigkeit.png} - \item Vorteile - \begin{itemize*} - \item kein Kontakt mit dem Medium, insbesondere Blutbahnen - \item Installation und Messung ohne Unterbrechnung des Flusses - \end{itemize*} - \item Nachteile - \begin{itemize*} - \item invasive Methode, da Blutgefäß freigelegt werden muss - \item Ungenauigkeit wegen der Verfomung der Blutgefäße - \end{itemize*} - \item Signaleigenschafte - \begin{itemize*} - \item verrauscht wegen Streuung im Medium, Sensorrauschen - \item Echo statisch verteilt wegen Geschwindigkeitsprofil - \end{itemize*} - \end{itemize*} - \item Ultraschall-Geschwindigkeitsmessung nach dem Dopplerverfahren - \begin{itemize*} - \item \$f\_D=f\textbackslash frac\{c\}\{c-v\} \textbackslash Rightarrow v=c\textbackslash frac\{f-f\_D\}\{f\_D\}\$ - \item \$c\$ - Ausbreitungsgeschwindigkeit des Ultraschalls im Medium - \item \$f\$ - Originalfrequenz der Signalquelle - \item \$f\_D\$ - gemessene Frequenz (Beobachter) - \item \$v\$ - Geschwindigkeit der Signalquelle - \item \includegraphics[width=.5\linewidth]{Assets/Biosignalverarbeitung-ultraschall-doppler.png} - \item \includegraphics[width=.5\linewidth]{Assets/Biosignalverarbeitung-ultraschall-doppler-2.png} - \item Anm: Feste Blutbestandteile (Blutkörperchen) reflektieren die Schallwellen und sind somit für den Ultraschall-Empfänger bewegte Signalquellen - \end{itemize*} - \item signalanalytisch relevante Eigenschaften - \begin{itemize*} - \item Flussgeschwindigkeit ungleichmäßig verteilt - \item im technischen Bereich konstruktiv beherrschbar (Messkammer, Durchmesser, Material) - \item im medizinischen Bereich kein Einfluss auf die Gefäße, daher relativ ungenaue Messung der mittleren Geschwindgkeit + \item verrauscht wegen Streuung im Medium, Sensorrauschen + \item Echo statisch verteilt wegen Geschwindigkeitsprofil \end{itemize*} \end{itemize*} + Ultraschall-Geschwindigkeitsmessung nach dem Dopplerverfahren + \begin{itemize*} + \item $f_D=f\frac{c}{c-v} \Rightarrow v=c\frac{f-f_D}{f_D}$ + \item $c$ - Ausbreitungsgeschwindigkeit + \item $f$ - Originalfrequenz Signalquelle + \item $f_D$ - gemessene Frequenz + \item $v$ - Geschwindigkeit Signalquelle + %\item \includegraphics[width=.5\linewidth]{Assets/Biosignalverarbeitung-ultraschall-doppler.png} + %\item \includegraphics[width=.5\linewidth]{Assets/Biosignalverarbeitung-ultraschall-doppler-2.png} + \item Anm: Feste Blutbestandteile (Blutkörperchen) reflektieren die Schallwellen und sind somit für den Ultraschall-Empfänger bewegte Signalquellen + \end{itemize*} + signalanalytisch relevante Eigenschaften + \begin{itemize*} + \item Flussgeschwindigkeit ungleichmäßig verteilt + \item im technischen Bereich konstruktiv beherrschbar %(Messkammer, Durchmesser, Material) + \item im medizinischen Bereich kein Einfluss auf die Gefäße, daher relativ ungenaue Messung der mittleren Geschwindgkeit + \end{itemize*} + \subsection{Optische Sensoren}\label{optische-sensoren} - - Optische und Strahlungsquellen - - \begin{itemize*} - \item Kaltlichtquelle: in der Endoskopie, bläuliches Tageslicht wegen der Farbtreue - \item Diagnostische Laser: in der Ophthalmologie, Urologie, inneren Medizin, Dermatologie - \item Leuchtdioden: in der Photoplethysomographie (Pulsoximetrie) - \item Röntgen-, Gamma-, UV- und IR-Strahler: in der diagnostischen Bildgebung - \item Inspektionslicht: in der HNO (Halogenstrahler) - \end{itemize*} + \begin{description*} + \item[Kaltlichtquelle] Endoskopie; bläuliches Tageslicht wegen Farbtreue + \item[Diagnostische Laser] Ophthalmologie, Urologie, Dermatologie + \item[Leuchtdioden] Photoplethysomographie (Pulsoximetrie) + \item[Röntgen-/Gamma-/UV-/IR-Strahler] diagnostische Bildgebung + \item[Inspektionslicht] in der HNO (Halogenstrahler) + \end{description*} Signalanalytisch wichtige Eigenschafte - - \begin{itemize*} - \item Temperaturstrahler: sind träge, daher statisches, konstantes Licht - \item Halbleiter (Leuchtdioden), Laser und Leuchtstoffröhren sind gepulste Quellen - mit dem Auge nicht wahrnehmbar, aber analytisch unter Umständen sehr problematisch - \end{itemize*} + \begin{description*} + \item[Temperaturstrahler] sind träge, daher statisches, konstantes Licht + \item[Halbleiter] (Leuchtdioden) Laser/Leuchtstoffröhren sind Gepulst + \end{description*} Optische Sensoren - - \begin{itemize*} - \item Phototransistor: in Flachbilddetektoren der Radiologie - \item Kamerachips: in den Endoskopen - \item Szintillatoren: in Gamma-Kameras - \item Photovervielfacher (SEV) in Laser-Fluroszenzsystemen - \end{itemize*} + \begin{description*} + \item[Phototransistor] in Flachbilddetektoren der Radiologie + \item[Kamerachips] in den Endoskopen + \item[Szintillatoren] in Gamma-Kameras + \item[Photovervielfacher] (SEV) in Laser-Fluroszenzsystemen + \end{description*} Sensoreigenschaften - \begin{itemize*} \item starkes Eigenrauschen, typisch für Halbleiter, ,,Dunkelstrom'' - \item hohe Temperaturabhängigkeit, ist materialbedingt, variable Parameter - \item ungünstige Dynamikeigenschaften, Nachleuchten durch Trägheit, systemanalytisch lange Impulsantwort + \item hohe Temperaturabhängigkeit, ist materialbedingt%, variable Parameter + \item ungünstige Dynamikeigenschaften, Nachleuchten durch Trägheit + \item lange Impulsantwort + \item nicht invasiv, daher patientenfreundlich \end{itemize*} - Beispiel Optischer Sensor CMOS Kamera LOGLUX i5 + %Optische Messmethoden gewinnen in der Medizin immer mehr an Bedeutung, vor allem, weil sie nichtinvasiv sind und daher patientenfreundlich. Mit der Kombination von Spektralfotometrie und Photoplethysmografie kann die Sauerstoffsättigung bestimmt werden. Dazu ist es notwendig, Gewebe durchzustrahlen, welches mit arteriellem Blut versorgt wird. Sehr verbreiten ist die Transmissionsmessung -d.h., das Gewebe wird durchstrahlt, was den Anforderungen an eine Messanordnung entsprechend der Theorie noch am nächsten kommt. Eine Alternative wurde notwendig, da der Finger u.U. nicht versorgt wird, z.B. beim Schock: Die Reflexionsmessung, bei der das Licht über einem Flächenknochen eingestrahlt und das reflektierte erfasst wird. - \begin{itemize*} - \item wahlfreier Pixelzugriff - \item CameraLink oder FireWire Datenschnittstelle - \item Auflösung 1280x1024 Pixel, 10 bit Graustufen - \item \$\textgreater100\$ dB Kontrast-/Dynamikumfang - \item ca 36 fps bei Vollbild; höhere Bildrate bei kleinerem Bildfeld bis ca 1500 fps - \item Vorverarbeitung der Bilddaten mittels integrierter LUT (look-up-tables) möglich - \item spektraler Arbeitsbereich 400-1000nm - \end{itemize*} - - Optische Messmethoden gewinnen in der Medizin immer mehr an Bedeutung, - vor allem, weil sie nichtinvasiv sind und daher patientenfreundlich. Mit - der Kombination von Spektralfotometrie und Photoplethysmografie kann die - Sauerstoffsättigung bestimmt werden. Dazu ist es notwendig, Gewebe - durchzustrahlen, welches mit arteriellem Blut versorgt wird. Sehr - verbreiten ist die Transmissionsmessung -d.h., das Gewebe wird - durchstrahlt, was den Anforderungen an eine Messanordnung entsprechend - der Theorie noch am nächsten kommt. Eine Alternative wurde notwendig, da - der Finger u.U. nicht versorgt wird, z.B. beim Schock: Die - Reflexionsmessung, bei der das Licht über einem Flächenknochen - eingestrahlt und das reflektierte erfasst wird. - - \begin{itemize*} - \item \includegraphics[width=.5\linewidth]{Assets/Biosignalverarbeitung-pulsoxy-1.png} - \item \includegraphics[width=.5\linewidth]{Assets/Biosignalverarbeitung-pulsoxy-2.png} - \item \includegraphics[width=.5\linewidth]{Assets/Biosignalverarbeitung-pulsoxy-3.png} - \end{itemize*} - - Signal am Photodetektor + % \includegraphics[width=.5\linewidth]{Assets/Biosignalverarbeitung-pulsoxy-1.png} + %\includegraphics[width=.5\linewidth]{Assets/Biosignalverarbeitung-pulsoxy-2.png} + % \includegraphics[width=.5\linewidth]{Assets/Biosignalverarbeitung-pulsoxy-3.png} + Pulsoxymetrie: Signal am Photodetektor \begin{itemize*} \item Multiplex bzw. sequentielle Abtastung - \item Rauschen (Optoelektronik) - \item Umgebungslicht, insbesondere Leuchtstoffröhren - \item \includegraphics[width=.5\linewidth]{Assets/Biosignalverarbeitung-pulsoxy-4.png} + \item Störung durch Rauschen und Umgebungslicht%, insbesondere Leuchtstoffröhren + \item Nutzsignal im unteren Prozentbereich des Signals + %\item \includegraphics[width=.5\linewidth]{Assets/Biosignalverarbeitung-pulsoxy-4.png} \end{itemize*} Signal am Demultiplexer - \begin{itemize*} \item DC ca 95-98\% \item AC nach DC Subtraktion verstärkt - \item \includegraphics[width=.5\linewidth]{Assets/Biosignalverarbeitung-pulsoxy-5.png} + %\item \includegraphics[width=.5\linewidth]{Assets/Biosignalverarbeitung-pulsoxy-5.png} \end{itemize*} - Für die Signalverarbeitung bedeutet die Analyse des empfangenen Signals - eine komplexe Herausforderung: Die Störungen, das Rauschen und das - Umgebungslicht (vor allem im OP), sind enorm stark, so dass ihre - Trennung vom Signal schwierig ist. Hinzu kommt, dass das Nutzsignal im - unteren Prozentbereich des gesamten empfangenen Signals liegt, so dass - hier das SNR um weitere zwei Dekaden schlechter wird. + % Für die Signalverarbeitung bedeutet die Analyse des empfangenen Signals eine komplexe Herausforderung: Die Störungen, das Rauschen und das Umgebungslicht (vor allem im OP), sind enorm stark, so dass ihre Trennung vom Signal schwierig ist. Hinzu kommt, dass das Nutzsignal im unteren Prozentbereich des gesamten empfangenen Signals liegt, so dass hier das SNR um weitere zwei Dekaden schlechter wird. - An diesem Beispiel eines realen Pulsoximetriesignals kann man die realen - Eigenschaften erkennen: - - \begin{itemize*} - \item \includegraphics[width=.5\linewidth]{Assets/Biosignalverarbeitung-pulsoxy-6.png} - \begin{itemize*} - \item Der DC-Anteil, der im Grunde durch eine Tiefpassfilterung gewonnen wird, ist real ein stark schwankender gleitender Mittelwert (unterer Verlauf). - \item Der AC-Anteil (oberer Verlauf) zeigt ebenfalls starke Schwankungen. Um dem Mediziner einen einigermaßen stabilen Messwert zu bieten, sind mehrere Schritte der SV notwendig - \end{itemize*} - \item \includegraphics[width=.5\linewidth]{Assets/Biosignalverarbeitung-pulsoxy-7.png} - \begin{itemize*} - \item Pulsbreite 1ms, analoger Tiefpass 10kHz, Abtastrate 10 ksps - \item Zunächst müssen aus dem Multiplexsignal die aktuellen Signalpegel für das rote und infrarote Licht sowie für das Umgebungslicht gewonnen werden: Durch die 10fache Überabtastung stehen für Rot und Infrarot zunächst elf Messwerte zur Verfügung. Dieser Umfang an Messdaten ist für eine Pegelbestimmung mit dem Mittelwert zu gering, daher wird der Median verwendet. Nach der Medianbildung liegen die Signalpegel für weitere Berechnung vor. - \end{itemize*} - \item \includegraphics[width=.5\linewidth]{Assets/Biosignalverarbeitung-pulsoxy-8.png} - \begin{itemize*} - \item Die gewonnenen Signalpegel werden nun einer Signalanalyse unterzogen. Die Analyse bei einer Wellenlänge ist ausreichend, da die Signalform bei allen qualitativ identisch ist. Für die Bestimmung des AC-Pegels werden die Extrema detektiert. Aus der Physiologie ist bekannt, dass die Anstiegszeit der Pulswelle höchstens 30\% der Gesamtzeit beträgt, so dass eine Prüfung im Zeitfenster folgt. Weiterhin ist der Bereich der Periode bekannt, diese Prüfung folgt im nächsten Schritt. Durch Artefakte, vor allem durch Bewegung, entstehen Schwankungen der Basislinie. Nach einem empirische ermittelten Kriterium wird ein Trend von bis zu 30\% vor der Berechnung akzeptiert. - \end{itemize*} - \end{itemize*} + %Beispiel eines realen Pulsoximetriesignals + %\begin{itemize*} + %\item \includegraphics[width=.5\linewidth]{Assets/Biosignalverarbeitung-pulsoxy-6.png} + %\item AC/DC Anteil stark schwankender gleitender Mittelwert + %\item Der DC-Anteil, der im Grunde durch eine Tiefpassfilterung gewonnen wird, ist real ein stark schwankender gleitender Mittelwert (unterer Verlauf) + %\item Der AC-Anteil (oberer Verlauf) zeigt ebenfalls starke Schwankungen. Um dem Mediziner einen einigermaßen stabilen Messwert zu bieten, sind mehrere Schritte der SV notwendig + %\item \includegraphics[width=.5\linewidth]{Assets/Biosignalverarbeitung-pulsoxy-7.png} + %\item Pulsbreite 1ms, analoger Tiefpass 10kHz, Abtastrate 10 ksps + %\item Zunächst müssen aus dem Multiplexsignal die aktuellen Signalpegel für das rote und infrarote Licht sowie für das Umgebungslicht gewonnen werden: Durch die 10fache Überabtastung stehen für Rot und Infrarot zunächst elf Messwerte zur Verfügung. Dieser Umfang an Messdaten ist für eine Pegelbestimmung mit dem Mittelwert zu gering, daher wird der Median verwendet. Nach der Medianbildung liegen die Signalpegel für weitere Berechnung vor. + %\item \includegraphics[width=.5\linewidth]{Assets/Biosignalverarbeitung-pulsoxy-8.png} + %\item Die gewonnenen Signalpegel werden nun einer Signalanalyse unterzogen. Die Analyse bei einer Wellenlänge ist ausreichend, da die Signalform bei allen qualitativ identisch ist. Für die Bestimmung des AC-Pegels werden die Extrema detektiert. Aus der Physiologie ist bekannt, dass die Anstiegszeit der Pulswelle höchstens 30\% der Gesamtzeit beträgt, so dass eine Prüfung im Zeitfenster folgt. Weiterhin ist der Bereich der Periode bekannt, diese Prüfung folgt im nächsten Schritt. Durch Artefakte, vor allem durch Bewegung, entstehen Schwankungen der Basislinie. Nach einem empirische ermittelten Kriterium wird ein Trend von bis zu 30\% vor der Berechnung akzeptiert. + %\end{itemize*} \subsection{Akustische Sensoren}\label{akustische-sensoren} - - Physiologischer Schall (Herztöne, Atmungsapparat) liegt im hörbaren - Bereich, so dass hier Methoden eingesetzt werden, die aus der - allgemeinen Akustik bekannt sind. Konventionelle Mikrophontechnik mit - spezifischer Signalverarbeitung - + Physiologischer Schall liegt im hörbaren Bereich. Konventionelle Mikrophontechnik mit spezifischer Signalverarbeitung \begin{itemize*} \item Verstärkung im tieffrequenten Bereich mit linearer Phase - \item Richtcharakteristik umschaltbar bzw einstellbar, mechanisch bereits in den ältesten Stetoskopen - \item spektrale Filterung für typische Geräusche, wie Herzklappen, Pfeifen in der Lunge, etc - \item Merkmalserkennung in computerbasierter Auswertung, Mustererkennung typischer pathologisch bedingter Geräusche + \item Richtcharakteristik umschaltbar bzw einstellbar + \item spektrale Filterung für typische Geräusche, wie Herzklappen + \item Merkmalserkennung in computerbasierter Auswertung + \item Mustererkennung typischer pathologisch bedingter Geräusche \end{itemize*} + Ultraschall Methoden + %Beim Ultraschall (CW,PW,Doppler) handelt es sich um mechanische Schwingungen bis in den zweistelligen Megahertzbereich ( ca. 30MHz). Hier müssen aufwendige Methoden der SV angewandt und entwickelt werden, die primär -d.h. bis zum Übergang in den physiologischen Bereich bzw. zur Bildgebung -eher in der Nachrichtentechnik und Stochastik ihren Ursprung haben: Signaldetektion, Korrelationsrechnung, Histogramme, Signalzerlegung. Signalanalytisch wichtige Eigenschaften: + \begin{description*} + \item[CW] (Continous Wave) keine Tiefeninformation, Information über Dopplerfrequenz mit hoher Variationsbreite, stochastischer Charakter mit viel Rauschen + \item[PW] (pulsed Wave) Auflösung von der Signalverarbeitung abhängig, physikalische Grenzen erreicht + \item[Doppler-Technologie] CW/PW vereint, Summe aller Vor- und Nachteile + \end{description*} - Beim Ultraschall (CW,PW,Doppler) handelt es sich um mechanische - Schwingungen bis in den zweistelligen Megahertzbereich ( ca. 30MHz). - Hier müssen aufwendige Methoden der SV angewandt und entwickelt werden, - die primär -d.h. bis zum Übergang in den physiologischen Bereich bzw. - zur Bildgebung -eher in der Nachrichtentechnik und Stochastik ihren - Ursprung haben: Signaldetektion, Korrelationsrechnung, Histogramme, - Signalzerlegung. Signalanalytisch wichtige Eigenschaften: - - \begin{itemize*} - \item bei CW (Continous Wave) keine Tiefeninformation verfügbar, Information über Dopplerfrequenz mit hoher Variationsbreite, stochastischer Charakter mit viel Rauschen - \item bei PW (pulsed Wave) Auflösung von der Signalverarbeitung entscheidend abhängig, da physikalische Grenzen lange erreicht - \item in der Doppler-Technologie beides (CW und PW) vereint, daher Summe aller Vor- und Nachteile - \end{itemize*} - - \subsection{Sensoren für elektrische Größen}\label{sensoren-fuxfcr-elektrische-gruxf6uxdfen} - + \subsection{Sensoren für elektrische Größen}\label{sensoren-fuxfcr-elektrische-gröuxdfen} \subsubsection{Elektrochemische Grundlagen}\label{elektrochemische-grundlagen} - \begin{itemize*} - \item Dieser Sensortyp dient der Erfassung der elektrischen Aktivität des Menschen - \item Der Mensch produziert elektrische Signale, daher ist keine Umwandung der Energieform notwendig - \item Der Mensch ist elektrisch gesehen ein Volumentleiter der 2. Art - ein Elektrolyt oder ein Ionenleiter - \item Das Messsystem ist mit metallischen Leitern aufgebaut - Leiter der 1. Art, Elektronenleiter - \item daher ist die Schaffung einer Schnittstelle notwendig - die Elektrode - \item \includegraphics[width=.5\linewidth]{Assets/Biosignalverarbeitung-elektrochemische-grundlage.png} + %\item Dieser Sensortyp dient der Erfassung der elektrischen Aktivität des Menschen + \item Mensch produziert elektrische Signale, keine Umwandung der Energieform notwendig + \item Mensch ist Volumentleiter 2. Art - Elektrolyt oder Ionenleiter + \item Messsystem mit metallischen Leitern aufgebaut - Leiter 1. Art, Elektronenleiter + \item Schnittstelle notwendig $\rightarrow$ Elektrode + %\item \includegraphics[width=.5\linewidth]{Assets/Biosignalverarbeitung-elektrochemische-grundlage.png} \begin{itemize*} - \item \$mM \textbackslash Leftrightarrow mM\^{}+ + me\^{}-\$ - \item \$K\_k A\_a\textbackslash Leftrightarrow kK\^{}+ + aA\^{}-\$ - \item \$\textbackslash leftarrow\$: Reduktion; \$\textbackslash rightarrow\$: Oxidation + \item $mM \Leftrightarrow mM^+ + me^-$ + \item $K_k A_a\Leftrightarrow kK^+ + aA^-$ + \item $\leftarrow$: Reduktion; $\rightarrow$: Oxidation \item Dynamisches Gleichgewicht an den Phasengrenzen - \item An der Phasengrenze der beiden Leitertypen entwickelt sich -ähnlich wie in einem Halbleiter -eine Raumladungszone. Die freien Elektronen im Metall und die Kationen des Elektrolyts ziehen sich an und bilden an der Grenze eine Doppelschicht. Je nach der chemischen Zusammensetzung des Elektrolyts und des Metalls finden unterschiedlich starke chemische Reaktionen statt, die beim dynamischen Gleichgewicht die sog. Elektrodenspannung bilden. Funktionell handelt es sich hierbei also um ein ungewolltes Voltaisches Element. \end{itemize*} + \item an Phasengrenze der Leitertypen Raumladungszone + \item freie Elektronen im Metall und Kationen des Elektrolyts bilden Doppelschicht + \item je nach chemischer Zusammensetzung des Elektrolyts und des Metalls unterschiedlich starke chemische Reaktionen %, die beim dynamischen Gleichgewicht die sog. Elektrodenspannung bilden. Funktionell handelt es sich hierbei also um ein ungewolltes Voltaisches Element. \end{itemize*} \subsubsection{Elektroden der Diagnostik}\label{elektroden-der-diagnostik} - \begin{itemize*} - \item aus signalanalytischer Sicht Eingangsdaten - \item aus messtechnischer Sicht Systemeingang + \item aus signalanalytischer Sicht: Eingangsdaten + \item aus messtechnischer Sicht: Systemeingang \end{itemize*} + Ziele + \begin{itemize*} + \item geringe Elektrodenspannung + \item geringer Drift der Elektrodenspannung + \item geringes Eigenrauschen + \end{itemize*} + Realisierbarkeit + \begin{itemize*} + \item Spannung durch Materialwahl (AgAgCl) + \item Drift physiologisch bedingt, daher kaum beeinflussbar + \item Eigenrauschen: Materialwahl und Technologie + \end{itemize*} + Praktikabilität Elektroden + \begin{itemize*} + \item EMG: Nadelform, aus Edelmetall, schlechte Signaleigenschaften, große Impedanz, kapazitives Verhalten, hohe Elektrodenspannung + \item EKG: große Fläche, AgAgCl, niedrige Impedanz, tieffrequent, niedrige Elektrodenspannung + \end{itemize*} + %Aus signalanalytischer Sicht sind die Ziele ganz klar vorgegeben. In der Praxis muss jedoch immer ein Kompromiss zwischen diesen Zielen und den Anforderungen der Anwendung und Praktikabilität gefunden werden: Wie diese Beispiele zeigen, hängt die Konstruktion der Elektrode von ihrer Bestimmung ab und daraus ergeben sich auch die Signaleigenschaften. So z.B. muss eine subkutane EMG-Elektrode die Form eine Nadel haben und aus einem Edelmetall sein. Dies hat zur Folge, dass die EMG-Elektroden relativ schlechte Signaleigenschaften aufweist: Riesige Elektrodenimpedanz (bis einige MOhm), stark kapazitives Verhalten, sehr hohe Elektrodenspannung (bis in den Voltbereich). Im Vergleich dazu haben die EKG-Elektroden -vor allem auf Grund ihrer großen Fläche und des Materials (AgAgCl, NaCl) -sehr günstige Eigenschaften: Niedrige Elektrodenimpedanz (kOhm-Bereich), sehr tieffrequent (bis DC), niedrige Elektrodenspannung (um 100 mV). - \begin{tabular}{l|l} - Ziele & Realisierbarkeit \\\hline - geringe Elektrodenspannung & durch Materialwahl (AgAgCl) \\ - geringer Drift der Elektrodenspannung & physiologisch bedingt, daher kaum beeinflussbar \\ - geringes Eigenrauschen & Materialwahl und Technologie \\ - \end{tabular} - - Aus signalanalytischer Sicht sind die Ziele ganz klar vorgegeben. In der - Praxis muss jedoch immer ein Kompromiss zwischen diesen Zielen und den - Anforderungen der Anwendung und Praktikabilität gefunden werden: Wie - diese Beispiele zeigen, hängt die Konstruktion der Elektrode von ihrer - Bestimmung ab und daraus ergeben sich auch die Signaleigenschaften. So - z.B. muss eine subkutane EMG-Elektrode die Form eine Nadel haben und aus - einem Edelmetall sein. Dies hat zur Folge, dass die EMG-Elektroden - relativ schlechte Signaleigenschaften aufweist: Riesige - Elektrodenimpedanz (bis einige MOhm), stark kapazitives Verhalten, sehr - hohe Elektrodenspannung (bis in den Voltbereich). Im Vergleich dazu - haben die EKG-Elektroden -vor allem auf Grund ihrer großen Fläche und - des Materials (AgAgCl, NaCl) -sehr günstige Eigenschaften: Niedrige - Elektrodenimpedanz (kOhm-Bereich), sehr tieffrequent (bis DC), niedrige - Elektrodenspannung (um 100 mV). - - \includegraphics[width=.5\linewidth]{Assets/Biosignalverarbeitung-elektroden.png} + %\includegraphics[width=.5\linewidth]{Assets/Biosignalverarbeitung-elektroden.png} \subsubsection{Elektroden der Therapie}\label{elektroden-der-therapie} - \begin{itemize*} - \item aus signalanalytischer Sicht Ausgangsdaten - \item aus messtechnischer Sicht Systemausgang + \item aus signalanalytischer Sicht: Ausgangsdaten + \item aus messtechnischer Sicht: Systemausgang + \end{itemize*} + Ziele + \begin{itemize*} + \item geringe Impedanz + \item geringer Drift der Impedanz + \item Langzeitstabilität + \end{itemize*} + Realisierbarkeit + \begin{itemize*} + \item Impedanz durch Materialwahl (beschichtet Cu) + \item Drift physiologisch bedingt + \item Stabilität durch Materialwahl und Technologie \end{itemize*} - \begin{tabular}{l|l} - Ziele & Realisierbarkeit \\\hline - geringe Impedanz & durch Materialwahl (beschichtet Cu) \\ - geringer Drift der Impedanz & physiologisch bedingt \\ - Langzeitstabilität & Materialwahl und Technologie - \end{tabular} + %Ebenso wichtig wie die Eigenschaften der diagnostischen Elektroden, sind es auch die der therapeutischen Elektroden. Dies liegt darin begründet, dass die Therapie von den zuvor analysierten diagnostischen Daten abhängt -natürlich im signalanalytischen Sinne, denn medizinisch ist es immer so. Man muss sich also bei der gewählten Therapie darauf verlassen können, dass das, was man auf die Elektrode schickt, so auch am biologischen Objekt ankommt. Diese Forderung technologisch umzusetzen ist ungleich leichter als bei diagnostischen Elektroden, denn hier können relative große Flächen mit gutem Kontaktmaterial verwendet werden. - Ebensowichtig wie die Eigenschaften der diagnostischen Elektroden, sind - es auch die der therapeutischen Elektroden. Dies liegt darin begründet, - dass die Therapie von den zuvor analysierten diagnostischen Daten - abhängt -natürlich im signalanalytischen Sinne, denn medizinisch ist es - immer so. Man muss sich also bei der gewählten Therapie darauf verlassen - können, dass das, was man auf die Elektrode schickt, so auch am - biologischen Objekt ankommt. Diese Forderung technologisch umzusetzen - ist ungleich leichter als bei diagnostischen Elektroden, denn hier - können relative große Flächen mit gutem Kontaktmaterial verwendet - werden. + % \includegraphics[width=.5\linewidth]{Assets/Biosignalverarbeitung-elektrode-therapie.png} - \includegraphics[width=.5\linewidth]{Assets/Biosignalverarbeitung-elektrode-therapie.png} - - \subsection{Sensoren für magnetische Größen}\label{sensoren-fuxfcr-magnetische-gruxf6uxdfen} - - \subsubsection{Messprinzipien}\label{messprinzipien} - - Um einen Eindruck über die Signalstärke (eher Signalschwäche) der - biomagnetischen Signale zu bekommen, wird mit dem natürlichen Erdfeld - verglichen, obwohl dieses für den Biomagnetismus eigentlich gar kein - Problem darstellt. Störend sind die vom Menschen gemachten magnetischen - Felder, vor allem die vom Stromversorgungsnetz, die jedoch weit über dem - magnetischen Erdfeld liegen. + \subsection{Sensoren für magnetische Größen}\label{sensoren-fuxfcr-magnetische-gröuxdfen} + %\subsubsection{Messprinzipien}\label{messprinzipien} + %Um einen Eindruck über die Signalstärke (eher Signalschwäche) der biomagnetischen Signale zu bekommen, wird mit dem natürlichen Erdfeld verglichen, obwohl dieses für den Biomagnetismus eigentlich gar kein Problem darstellt. Störend sind die vom Menschen gemachten magnetischen Felder, vor allem die vom Stromversorgungsnetz, die jedoch weit über dem magnetischen Erdfeld liegen. + \begin{itemize*} + \item vergleich mit natürlichen Erdfeld + \item Störend technische M.Felder, z.B. Stromversorgungsnetz + \end{itemize*} \begin{enumerate*} - \def\labelenumi{\arabic{enumi}.} - \item Das stärkste Biosignal, das MKG, liegt 6 Dekaden unter dem Erdfeld (120dB), und weitere 2...3 Dekaden unter den technischen Feldern. - \item MEG -7 Dekaden, oder 140dB, + \item stärkstes Biosignal (MKG) 6 Dekaden unter Erdfeld (120dB)%, und weitere 2...3 Dekaden unter den technischen Feldern. + \item MEG -7 Dekaden oder 140dB, \item evozierte Felder -8 Dekaden oder 160dB \end{enumerate*} \begin{itemize*} - \item \$10\^{}0T\$: MR-Tomographie-Magnete - \item \$10\^{}\{-5\}T\$: Erdfeld - \item \$10\^{}\{-6\}T\$: Zivilisationsfelder (Rauschen) - \item \$10\^{}\{-9\}T\$: magn. Kontamination der Lunge - \item \$10\^{}\{-10\}T\$: Magnetkardiogramm - \item \$10\^{}\{-12\}T\$: Magnetoenzephalogramm - \item \$10\^{}\{-13\}T\$: evozierte kortikale Aktivität - \item \$10\^{}\{-15\}T\$: SQUID System Rauschen + \item $10^{0}T$: MR-Tomographie-Magnete + \item $10^{-5}T$: Erdfeld + \item $10^{-6}T$: Zivilisationsfelder (Rauschen) + %\item $10^{-9}T$: magn. Kontamination der Lunge + %\item $10^{-10}T$: Magnetkardiogramm + %\item $10^{-12}T$: Magnetoenzephalogramm + %\item $10^{-13}T$: evozierte kortikale Aktivität + \item $10^{-15}T$: SQUID System Rauschen \end{itemize*} - Biomagnetische Signale sind sehr schwach (SNR\textless{} -120dB). + Biomagnetische Signale sind sehr schwach (SNR $< -120dB$). Mehrere Maßnahmen zur SNR-Anhebung notwendig - \begin{itemize*} - \item Abschirmung des Messkreises gegen Störfelder (dickwandige Kammer aus \$\textbackslash mu\$-Metallen) + \item Abschirmung des Messkreises gegen Störfelder \item Ausnutzung der Feldeigenschaften - Gradiometer \item Spezialtechnologie der Signalverstärker - SQUID \end{itemize*} \subsubsection{Gradiometer}\label{gradiometer} - - Prinzip: - \begin{itemize*} - \item Störfelder meist ferne Quellen, Biologische Strukuren nahe Quellen + \item Störfelder ferne Quellen, Biologische Strukuren nahe Quellen \item ferne Quellen produzieren annährend homogenes Feld - \item nahe Quellen Produzieren inhomogenes Feld - \item mit Gradiometer wird die erste bzw zweite räumliche Ableitung gebildet, dadurch wird homogenes Störfeld unterdrückt - \item \includegraphics[width=.5\linewidth]{Assets/Biosignalverarbeitung-Gradiometer.png} - \item \includegraphics[width=.5\linewidth]{Assets/Biosignalverarbeitung-gradiometer-2.png} - \item homogenes Fernfeld (Störung, blau): \$u=u\_2-u\_1=0\$ - \item inhomogenes Nahfeld (Biosignalquelle, rot): \$u=u\_2-u\_1\textless\textgreater0\$ + \item nahe Quellen produzieren inhomogenes Feld + \item Gradiometer 1./2. räumliche Ableitung, dadurch homogenes Störfeld unterdrücken + %\item \includegraphics[width=.5\linewidth]{Assets/Biosignalverarbeitung-Gradiometer.png} + %\item \includegraphics[width=.5\linewidth]{Assets/Biosignalverarbeitung-gradiometer-2.png} + %\item homogenes Fernfeld (Störung, blau): $u=u_2-u_1=0$ + %\item inhomogenes Nahfeld (Biosignalquelle, rot): $u=u_2-u_1 < > 0$ \end{itemize*} - \subsubsection{SQUID}\label{squid} - - \includegraphics[width=.5\linewidth]{Assets/Biosignalverarbeitung-squid.png} - - Das supraleitende Quanteninterferenzgerät (SQUID) besteht aus zwei - Supraleitern, die durch dünne Isolierschichten getrennt sind und zwei - parallele Josephson-Kontakte bilden. Das Gerät kann als Magnetometer - konfiguriert werden, um unglaublich kleine Magnetfelder zu erkennen - - klein genug, um die Magnetfelder in lebenden Organismen zu messen. SQUID - wurden zur Messung der Magnetfelder in Mäusehirnen verwendet, um zu - testen, ob ihr Magnetismus ausreicht, um ihre Navigationsfähigkeit auf - einen inneren Kompass zurückzuführen. - \href{http://hyperphysics.phy-astr.gsu.edu/hbase/Solids/Squid.html}{Quelle} - - \section{Verstärkung und analoge Filterung}\label{verstuxe4rkung-und-analoge-filterung} - - \subsection{Eigenschaften von Biosignalen und Störungen}\label{eigenschaften-von-biosignalen-und-stuxf6rungen} - - \subsubsection{Entstehung der Biosignale, biologische Signalquellen}\label{entstehung-der-biosignale-biologische-signalquellen} - + \subsubsection{Supraleitende Quanteninterferenzgerät (SQUID)} + %\includegraphics[width=.5\linewidth]{Assets/Biosignalverarbeitung-squid.png} \begin{itemize*} - \item Analysegegenstand: Sensorisches, motorisches und zentrales Nervensystem - \item Grundbaustein: Nervenzelle, Neuron. Einzelne Neurone kaum untersuchbar, im Einzelfall mit Mikroelektroden, dennoch für die Gesamtheit wenig Bedeutung. Wichtiger sind Untersuchungen an Neuronenverbänden und -strängen, z.B. motorische Steuerung von Muskeln in den Extremitäten. Hier haben die Nerven überschaubare und anatomisch sowie elektrophysiologisch gut bekannte Struktur. - \item am Neuronausgang - Axon: Aktionspotentiale - \item am Neuroneingang - Synapsen: EPSP/IPSP (exzitatorische und inhibitorische postsynaptische Potentiale) - \item Sensorisches System ist deutlich komplexer, vor allem das akustische und das visuelle. So hat die Retina allein mehrere Millionen Sensoren (Stäbchen und Zapfen), die mit Ganglienzellen verbunden sind und bereits vor Ort relativ einfache Informationsverarbeitung durchführen. - \item Zahlenmäßig und daher in auch in seiner Komplexität ist das größte das zentrale Nervensystem (ZNS), das aus ca. 10 Milliarden Neuronen besteht, die funktionelle und anatomische Zentren bilden aber zeitlich stark variierende Eigenschaften aufweisen. - \item Signalanalytisch ist das Grundelement das Aktionspotential (AP), das vom Neuron nach Erreichen der Reizschwelle an seinen Eingängen über das Axon nach außen bzw. an andere Neurone abgegeben wird. Die Synapsen empfangen die Aktionspotentiale von anderen Neuronen und bewerten diese je nach Zustand mit EPSP oder IPSP, die von sich aus starken Veränderungen unterliegen. Im EEG sind die AP deutlich unterrepräsentiert (nur etwa 10\% des EEG), wesentlicher Anteil bilden die PSP. Dies ist unter anderem durch den Tiefpasscharakter des Schädels bedingt, das die hochfrequenten AP unterdrückt. + \item aus zwei Supraleitern, durch dünne Isolierschichten getrennt und bildet zwei parallele Josephson-Kontakte + \item erkennt unglaublich kleine Magnetfelder + \item zur Messung der Magnetfelder in Mäusehirnen verwendet %, um zu testen, ob ihr Magnetismus ausreicht, um ihre Navigationsfähigkeit auf einen inneren Kompass zurückzuführen. + \end{itemize*} + %\href{http://hyperphysics.phy-astr.gsu.edu/hbase/Solids/Squid.html}{Quelle} + + \section{Verstärkung und analoge Filterung}\label{verstärkung-und-analoge-filterung} + %\subsection{Eigenschaften von Biosignalen und Störungen}\label{eigenschaften-von-biosignalen-und-störungen} + \subsubsection{Entstehung der Biosignale, biologische Signalquellen}\label{entstehung-der-biosignale-biologische-signalquellen} + \begin{itemize*} + %\item Analysegegenstand: Sensorisches, motorisches und zentrales Nervensystem + %\item Grundbaustein: Nervenzelle, Neuron. Einzelne Neurone kaum untersuchbar, im Einzelfall mit Mikroelektroden, dennoch für die Gesamtheit wenig Bedeutung. Wichtiger sind Untersuchungen an Neuronenverbänden und -strängen, z.B. motorische Steuerung von Muskeln in den Extremitäten. Hier haben die Nerven überschaubare und anatomisch sowie elektrophysiologisch gut bekannte Struktur. + \item Neuronausgang: Axon, Aktionspotentiale + \item Neuroneingang: Synapsen, exzitatorische/inhibitorische postsynaptische Potentiale + \item Sensorisches System deutlich komplexer%, vor allem das akustische und das visuelle. So hat die Retina allein mehrere Millionen Sensoren (Stäbchen und Zapfen), die mit Ganglienzellen verbunden sind und bereits vor Ort relativ einfache Informationsverarbeitung durchführen. + %\item Zahlenmäßig und daher in auch in seiner Komplexität ist das größte das zentrale Nervensystem (ZNS), das aus ca. 10 Milliarden Neuronen besteht, die funktionelle und anatomische Zentren bilden aber zeitlich stark variierende Eigenschaften aufweisen. + \item Signalanalytisches Grundelement ist Aktionspotential (AP)%, das vom Neuron nach Erreichen der Reizschwelle an seinen Eingängen über das Axon nach außen bzw. an andere Neurone abgegeben wird. Die Synapsen empfangen die Aktionspotentiale von anderen Neuronen und bewerten diese je nach Zustand mit EPSP oder IPSP, die von sich aus starken Veränderungen unterliegen. Im EEG sind die AP deutlich unterrepräsentiert (nur etwa 10\% des EEG), wesentlicher Anteil bilden die PSP. Dies ist unter anderem durch den Tiefpasscharakter des Schädels bedingt, das die hochfrequenten AP unterdrückt. \end{itemize*} - \includegraphics[width=.5\linewidth]{Assets/Biosignalverarbeitung-ekg.png} + %\includegraphics[width=.5\linewidth]{Assets/Biosignalverarbeitung-ekg.png} - Ein medizinisch und auch signalanalytisch besonders interessantes Signal - ist das EKG: Medizinische Indikation ergibt sich allein aus der - besonderen Stellung des Herzens in der Physiologie als des Motors des - Kreislaufs. Signalanalytisch ist es deswegen interessant, da es unter - reproduzierbaren Messbedingungen (Extremitätenableitungen) - formkonstanten Signalverlauf zeigt. Das EKG wurde entsprechend seiner - elektromedizinischen Bedeutung extensiv untersucht, zahlreiche - Erkrankungen und Schäden werden anhand typischer Formveränderungen des - EKG diagnostiziert. Die Signalquelle des EKG ist das räumlich zwar recht - komplizierte, aber anatomisch qualitativ konstante Reizleitungssystem - des Herzens. Zur Ableitung des EKG werden standardmäßig 3-, 6-oder - 12-kanalige Extremitäten-und Brustwandsysteme verwendet. + %Ein medizinisch und auch signalanalytisch besonders interessantes Signal ist das EKG: Medizinische Indikation ergibt sich allein aus der besonderen Stellung des Herzens in der Physiologie als des Motors des Kreislaufs. Signalanalytisch ist es deswegen interessant, da es unter reproduzierbaren Messbedingungen (Extremitätenableitungen) formkonstanten Signalverlauf zeigt. Das EKG wurde entsprechend seiner elektromedizinischen Bedeutung extensiv untersucht, zahlreiche Erkrankungen und Schäden werden anhand typischer Formveränderungen des EKG diagnostiziert. Die Signalquelle des EKG ist das räumlich zwar recht komplizierte, aber anatomisch qualitativ konstante Reizleitungssystem des Herzens. Zur Ableitung des EKG werden standardmäßig 3-, 6-oder 12-kanalige Extremitäten-und Brustwandsysteme verwendet. - \includegraphics[width=.5\linewidth]{Assets/Biosignalverarbeitung-herz-ekg.png} + %\includegraphics[width=.5\linewidth]{Assets/Biosignalverarbeitung-herz-ekg.png} - Projektion der Reizausbreitung auf einen längs zur Herzachse liegenden - Vektor (vertikal): Zu beachten ist, dass durch die Differenzbildung an - zwei Punkten an der Körperoberfläche damit mathematisch die erste - räumliche Ableitung (oder auch der erste Gradient) gebildet wird. Das - hat zur Folge, dass die Ableitung nicht nur in Phasen der Ruhe (vor der - P-Welle), sondern auch bei maximaler Erregung ( PQ-und ST-Strecke) Null - ist. Wellen und Zacken im EKG sind Ausdruck der räumlich-zeitlichen - Veränderung im Reizleitungssystem. + %Projektion der Reizausbreitung auf einen längs zur Herzachse liegenden Vektor (vertikal): Zu beachten ist, dass durch die Differenzbildung an zwei Punkten an der Körperoberfläche damit mathematisch die erste räumliche Ableitung (oder auch der erste Gradient) gebildet wird. Das hat zur Folge, dass die Ableitung nicht nur in Phasen der Ruhe (vor der P-Welle), sondern auch bei maximaler Erregung ( PQ-und ST-Strecke) Null ist. Wellen und Zacken im EKG sind Ausdruck der räumlich-zeitlichen Veränderung im Reizleitungssystem. - \includegraphics[width=.5\linewidth]{Assets/Biosignalverarbeitung-gehirn.png} - \includegraphics[width=.5\linewidth]{Assets/Biosignalverarbeitung-gehirn-ekg.png} + %\includegraphics[width=.5\linewidth]{Assets/Biosignalverarbeitung-gehirn.png} + %\includegraphics[width=.5\linewidth]{Assets/Biosignalverarbeitung-gehirn-ekg.png} - Zur Ableitung des EEG werden wie beim EKG standardisierte - Elektrodensysteme verwendet. Allerdings ist die anatomische Zuordnung - hier ungleich schwieriger, denn die einzigen einigermaßen stabilen - anatomischen Bezugspunkte sind das Nasion und das Inion. Es ist jedoch - bekannt, dass die Lage des Gehirns in Bezug auf diese Punkte individuell - stark unterschiedlich ist und im Zentimeterbereich liegt, so dass eine - genaue Zuordnung der Elektroden zu Funktionszentren gar nicht möglich - ist. Die Dichte der Elektroden in der Praxis liegt höchstens bei 10\% - NI, d.h. im Schnit bei etwa 3cm. Eine höhere Dichte bringt keine - zusätzliche Information, da der Schädel als räumlicher Tiefpass - funktioniert und keine höhere Auflösung erlaubt. + %Zur Ableitung des EEG werden wie beim EKG standardisierte Elektrodensysteme verwendet. Allerdings ist die anatomische Zuordnung hier ungleich schwieriger, denn die einzigen einigermaßen stabilen anatomischen Bezugspunkte sind das Nasion und das Inion. Es ist jedoch bekannt, dass die Lage des Gehirns in Bezug auf diese Punkte individuell stark unterschiedlich ist und im Zentimeterbereich liegt, so dass eine genaue Zuordnung der Elektroden zu Funktionszentren gar nicht möglich ist. Die Dichte der Elektroden in der Praxis liegt höchstens bei 10\% NI, d.h. im Schnit bei etwa 3cm. Eine höhere Dichte bringt keine zusätzliche Information, da der Schädel als räumlicher Tiefpass funktioniert und keine höhere Auflösung erlaubt. - Aus Sicht der Signalanalyse ist es besonders wichtig zu wissen, unter - welchen Messbedingungen das EEG abgeleitet wurde. Im Idealfall wird - unipolar gegen verbundene Ohren oder Hals abgeleitet. Aus unipolaren - Daten lassen sich die bipolaren Ableitungen einfach berechnen, umgekehrt - geht das jedoch nicht. Auf jeden Fall ist zu klären, wie die - Verschaltung des EEG-Verstärkers und der Elektroden realisiert wurde. - Vermeintlich elegante Tricks, wie hardwaremäßige CAR sind auf jeden Fall - zu meiden, ebenso Antialiasingfilter mit nichtlinearer Phase. + %Aus Sicht der Signalanalyse ist es besonders wichtig zu wissen, unter welchen Messbedingungen das EEG abgeleitet wurde. Im Idealfall wird unipolar gegen verbundene Ohren oder Hals abgeleitet. Aus unipolaren Daten lassen sich die bipolaren Ableitungen einfach berechnen, umgekehrt geht das jedoch nicht. Auf jeden Fall ist zu klären, wie die Verschaltung des EEG-Verstärkers und der Elektroden realisiert wurde. Vermeintlich elegante Tricks, wie hardwaremäßige CAR sind auf jeden Fall zu meiden, ebenso Antialiasingfilter mit nichtlinearer Phase. - \includegraphics[width=.5\linewidth]{Assets/Biosignalverarbeitung-gehirn-eeg.png} + %\includegraphics[width=.5\linewidth]{Assets/Biosignalverarbeitung-gehirn-eeg.png} + %Das EEG wird in in fünf typische Bereiche unterteilt: delta (0..4Hz), theta (4-7Hz), alpha (8..13Hz), beta (13..30Hz), gamma ($>30Hz$). Diese Bereiche sind typisch für bestimmte physiologischen (Schlaf, Konzentration, Entspannung) und pathologischen Bilder. Für die Signalanalyse ist wichtig, dass die Bereiche nicht gleichzeitig vorhanden sind, einer ist immer dominant, was die Analyse leicht vereinfacht. + EEG in fünf Bereiche unterteilt: + \begin{itemize*} + \item 0-4 Hz: Delta + \item 4-7 Hz: Theta + \item 8-13 Hz: Alpha + \item 13-30 Hz: Beta + \item >30 Hz: Gamma + \end{itemize*} - Das EEG wird in in fünf typische Bereiche unterteilt: delta (0..4Hz), - theta (4-7Hz), alpha (8..13Hz), beta (13..30Hz), gamma - (\textgreater30Hz). Diese Bereiche sind typisch für bestimmte - physiologischen (Schlaf, Konzentration, Entspannung) und pathologischen - Bilder. Für die Signalanalyse ist wichtig, dass die Bereiche nicht - gleichzeitig vorhanden sind, einer ist immer dominant, was die Analyse - leicht vereinfacht. + \subsubsection{Biologische und technische Störquellen}\label{biologische-und-technische-störquellen} + \begin{itemize*} + %\item Das biomedizinische Messsystem ist von vielen Störquellen umgeben, die meisten sind dem Bereich der Medienversorgung, Industrie, Verkehr und Nachrichtentechnik zuzuschreiben. Für die BSA sind periodische (Versorgungsnetz, Monitore) und quasiperiodische (rotierende Maschinen, Straßenbahn) Störungen noch ein vergleichsweise geringes Problem, denn diese lassen sich gezielt mit spektralen Filtern in der analogen Messkette oder digital nach ADC unterdrücken. + \item (quasi-)periodisch: geringes Problem, spektrale Filter + %\item Wesentlich schwieriger ist die Situation, wenn transiente Störungen vorliegen, denn diese haben im Allgemeinen einen unbekannten, einmaligen und daher nicht reproduzierbaren Verlauf. Solange die transiente Störung die Signalerfassung nicht beeinträchtigt (durch Übersteuerung des Messverstärkers) und deutlich von der Signalform abweicht (z.B. Ausgleichsvorgang mi EKG), kann sie mit relativ einfachen Mitteln beseitigt werden, dennoch im Allgemeinen ist dies kaum möglich. + \item transient: unbekannter, einmaliger, nicht reproduzierbarer Verlauf + \end{itemize*} - \subsubsection{Biologische und technische Störquellen}\label{biologische-und-technische-stuxf6rquellen} + %\includegraphics[width=.5\linewidth]{Assets/Biosignalverarbeitung-netzfrequenz-bandsperre.png} + %Die häufigste -weil immer vorhanden- ist die Netzstörung. Selbst batteriebetriebene portable Messgeräte sind von dieser Störung betroffen. Da die Frequenz der Störung aber bekannt ist, kann sie -falls keine Übersteuerung vorliegt- mit einer Bandsperre reduziert werden. Allerdings sollte nicht die früher übliche ,,50 Hz -Filter'' Taste verwendet werden, denn diese Filter haben einen nichtlinearen Phasenfrequenzgang und können das Biosignal deutlich verfälschen. Bei der heutigen Technologie werden ausschließlich digitale Filter verwendet. + + %\includegraphics[width=.5\linewidth]{Assets/Biosignalverarbeitung-trendelimination.png} + %Eine sehr häufige transiente Störung im medizinischen Bereich ist die Bewegungsartefakte. Jegliche Bewegung im Messbereich erzeugt in der empfindlichen medizinischen Messtechnik Ausgleichsvorgänge. Wenn die Signalform gut bekannt ist, wie z.B. beim EKG, so lässt sich eine langsame Artefakte durch Hochpassfilterung beseitigen. \begin{tabular}{l|l} periodische & transiente \\\hline öffentliches Stromversorgungsnetz & Spannungsspitzen im Netz \\ Straßenbahn & Bewegungen im Messbereich \\ - Monitore & Schaltvorgänge \\ - Kommunikationsnetze & Lastschwankungen \\ - Rotierende Maschinen & \\ - Sender inkl. Funktelefon & \\ + Rotierende Maschinen & Schaltvorgänge \\ + Kommunikationsnetze & Lastschwankungen \end{tabular} - 1.Das biomedizinische Messsystem ist von vielen Störquellen umgeben, die - meisten sind dem Bereich der Medienversorgung, Industrie, Verkehr und - Nachrichtentechnik zuzuschreiben. Für die BSA sind periodische - (Versorgungsnetz, Monitore) und quasiperiodische (rotierende Maschinen, - Straßenbahn) Störungen noch ein vergleichsweise geringes Problem, denn - diese lassen sich gezielt mit spektralen Filtern in der analogen - Messkette oder digital nach ADC unterdrücken. 2.Wesentlich schwieriger - ist die Situation, wenn transiente Störungen vorliegen, denn diese haben - im Allgemeinen einen unbekannten, einmaligen und daher nicht - reproduzierbaren Verlauf. Solange die transiente Störung die - Signalerfassung nicht beeinträchtigt (durch Übersteuerung des - Messverstärkers) und deutlich von der Signalform abweicht (z.B. - Ausgleichsvorgang mi EKG), kann sie mit relativ einfachen Mitteln - beseitigt werden, dennoch im Allgemeinen ist dies kaum möglich. - - \includegraphics[width=.5\linewidth]{Assets/Biosignalverarbeitung-netzfrequenz-bandsperre.png} - Die häufigste -weil immer vorhanden- ist die Netzstörung. Selbst - batteriebetriebene portable Messgeräte sind von dieser Störung - betroffen. Da die Frequenz der Störung aber bekannt ist, kann sie -falls - keine Übersteuerung vorliegt- mit einer Bandsperre reduziert werden. - Allerdings sollte nicht die früher übliche ,,50 Hz -Filter'' Taste - verwendet werden, denn diese Filter haben einen nichtlinearen - Phasenfrequenzgang und können das Biosignal deutlich verfälschen. Bei - der heutigen Technologie werden ausschließlich digitale Filter - verwendet. - - \includegraphics[width=.5\linewidth]{Assets/Biosignalverarbeitung-trendelimination.png} Eine - sehr häufige transiente Störung im medizinischen Bereich ist die - Bewegungsartefakte. Jegliche Bewegung im Messbereich erzeugt in der - empfindlichen medizinischen Messtechnik Ausgleichsvorgänge. Wenn die - Signalform gut bekannt ist, wie z.B. beim EKG, so lässt sich eine - langsame Artefakte durch Hochpassfilterung beseitigen. - + Störquellen \begin{itemize*} - \item Maximal \$f\_\{0,01\}=0,5 Hz\$ Patienten-Monitor EKG (nicht oberhalb) - \item Maximal \$f\_\{0,02\}=0,05Hz\$ Diagnostischer Monitor bei EKG (nicht oberhalb) + \item periodische Netzstörung im 50Hz Bereich $\rightarrow$ digitaler Filter + \item transiente Bewegungsartefakte $\rightarrow$ langsame A. durch Hochpass + \begin{itemize*} + \item Maximal $f_{0,01}=0,5 Hz$ Patienten-Monitor EKG + \item Maximal $f_{0,02}=0,05Hz$ Diagnostischer Monitor bei EKG + \end{itemize*} + \item Biologische Störquellen lassen sich nicht abschalten/kaum unterdrücken \end{itemize*} - Ob ein Biosignal gewollt ist oder eine Störung darstellt, ist von der - Messaufgabe abhängig: - + Ob Biosignal gewollt oder Störung, ist von Messaufgabe abhängig: \begin{itemize*} \item soll das EKG gemessen werden, ist das EMG eine Störung \item soll das EEG gemessen werden, ist das EKG eine Störung \item soll das EOG gemessen werden, ist das EEG eine Störung \end{itemize*} - Prinzipielles Problem: Biologische Störquellen lassen sich nicht - abschalten und kaum unterdrücken + % Aus Sicht der BSA gestaltet sich das Problem der Störungen wesentlich schwieriger als bei technischen Störungen. Erstens, die Biosignalquellen befinden sich innerhalb des Körpers, daher können sie weder abgeschirmt noch abgeschaltet werden. Zweitens, das biologische Signalspektrum ist für alle Biosignale in etwa gleich, streckt sich von 0 bis etwa 1kHz aus und weist ein Maximum bei etwa 100Hz auf. Daher können biologische Störsignale mit spektralen Filtern allein nicht beseitigt werden. - Aus Sicht der BSA gestaltet sich das Problem der Störungen wesentlich - schwieriger als bei technischen Störungen. Erstens, die Biosignalquellen - befinden sich innerhalb des Körpers, daher können sie weder abgeschirmt - noch abgeschaltet werden. Zweitens, das biologische Signalspektrum ist - für alle Biosignale in etwa gleich, streckt sich von 0 bis etwa 1kHz aus - und weist ein Maximum bei etwa 100Hz auf. Daher können biologische - Störsignale mit spektralen Filtern allein nicht beseitigt werden. + %Ein weiteres -messmethodisches -Problem besteh darin, dass man Biosignale nicht pauschal in Nutz-und Störsignale trennen kann. Es ist vielmehr die Messaufgabe, an Hand der man diese Klassifikation vornehmen muss. - Ein weiteres -messmethodisches -Problem besteh darin, dass man - Biosignale nicht pauschal in Nutz-und Störsignale trennen kann. Es ist - vielmehr die Messaufgabe, an Hand der man diese Klassifikation vornehmen - muss. - - \paragraph{Eigenschaften technischer Störungen}\label{eigenschaften-technischer-stuxf6rungen} - - \begin{tabular}{l|l} - periodische Störungen & transiente Störungen \\\hline - NF-magnetische Felder nicht eliminierbar durch Schirmung, erzeugen Differenzspannung & kaum eliminierbar, da Signalform unbekannt und nicht reproduzierbar \\ - NF-elektrische Felder gut beherrschbar, erzeugen Gleichtaktstörungen & bestenfalls Detektion möglich, Messdaten nicht korrigierbar \\ - HF-Felder immer mehr vorhanden (Kommunikationsnetze), Abschirmung unwirtschaftlich & \\ + \paragraph{Eigenschaften technischer Störungen} + \begin{tabular}{p{4cm}|p{4cm}} + periodische Störungen & transiente Störungen \\\hline + NF-magnetische Felder nicht eliminierbar durch Schirmung, erzeugen Differenzspannung & kaum eliminierbar, Signalform unbekannt/nicht reproduzierbar \\\hline + NF-elektrische Felder gut beherrschbar, erzeugen Gleichtaktstörungen & bestenfalls Detektion möglich, Messdaten nicht korrigierbar \\\hline + HF-Felder immer mehr vorhanden (Kommunikation), Abschirmung unwirtschaftlich & \\ \end{tabular} - \begin{enumerate*} - \def\labelenumi{\arabic{enumi}.} - \item Naturgemäß erzeugen niederfrequente magnetische Felder am Verstärkereingang Differenzspannungen, die direkt mit dem Biosignal überlagert werden, so dass sie mit der üblichen Verstäkertechnik nicht reduziert werden können. Hinzu kommt, dass auch eine Abschirmung nicht viel bringt, da in diesem Frequenzbereich mehrere 10- Zentimeter dicke Eisenplatten verwendet werden müssten, was in der Praxis nicht realisierbar ist. Da die niederfrequenten elektrischen (kapazitiv eingekoppelten) Störfelder Gleichtaktsignale sind, können sie zum Teil gut durch die Differenzverstärkertechnik reduziert werden. In immer höheren Maße stören hochfrequente Felder, vor allem aus dem Mobilfunk, Datennetzen, WLAN, Bluetooth, etc. Eine Abschirmung ist im normalen Praxisbetrieb unwirtschaftlich, so dass eine Reduktion der Störung allein durch Maßnahmen der EMV zu erreichen ist. - \item Wie schon erwähnt, transiente Störungen sind im Grunde nicht beherrschbar, da sie eigentlich nicht bekannt und nicht vorhersehbar sind. Mit Methoden der BSA ist zum Teil ihre Detektion möglich, wenn z.B. der Messbereich oder das Spektrum des Biosignals nachweislich verlassen wird. Diese Detektion kann allerdings nur dazu genutzt werden, die beeinträchtigten Daten zu verwerfen, eine Korrektur ist nicht möglich. - \end{enumerate*} + %\begin{enumerate*} + % \item Naturgemäß erzeugen niederfrequente magnetische Felder am Verstärkereingang Differenzspannungen, die direkt mit dem Biosignal überlagert werden, so dass sie mit der üblichen Verstäkertechnik nicht reduziert werden können. Hinzu kommt, dass auch eine Abschirmung nicht viel bringt, da in diesem Frequenzbereich mehrere 10- Zentimeter dicke Eisenplatten verwendet werden müssten, was in der Praxis nicht realisierbar ist. Da die niederfrequenten elektrischen (kapazitiv eingekoppelten) Störfelder Gleichtaktsignale sind, können sie zum Teil gut durch die Differenzverstärkertechnik reduziert werden. In immer höheren Maße stören hochfrequente Felder, vor allem aus dem Mobilfunk, Datennetzen, WLAN, Bluetooth, etc. Eine Abschirmung ist im normalen Praxisbetrieb unwirtschaftlich, so dass eine Reduktion der Störung allein durch Maßnahmen der EMV zu erreichen ist. + % \item Wie schon erwähnt, transiente Störungen sind im Grunde nicht beherrschbar, da sie eigentlich nicht bekannt und nicht vorhersehbar sind. Mit Methoden der BSA ist zum Teil ihre Detektion möglich, wenn z.B. der Messbereich oder das Spektrum des Biosignals nachweislich verlassen wird. Diese Detektion kann allerdings nur dazu genutzt werden, die beeinträchtigten Daten zu verwerfen, eine Korrektur ist nicht möglich. + %\end{enumerate*} - \paragraph{Eigenschaften biologischer Störungen}\label{eigenschaften-biologischer-stuxf6rungen} + \paragraph{Eigenschaften biologischer Störungen}\label{eigenschaften-biologischer-störungen} \begin{itemize*} - \item Spektral alle Biosignale im selben Band (0...100Hz) + \item Spektral alle Biosignale im selben Band (0-100Hz) \item Nichtlineare Verkopplung der Biosignale verhindern Trennung mit herkömmlichen Methoden \item Kein Biosignal deterministisch und reproduzierbar - \item Transiente bzw apperiodische und instationäre Biosignale nicht qualifizierbar - \item Eine Trennung kaum möglich, bestenfalls eine Reduktion (z.B. Abschwächung des EMG im EKG) + \item Transiente/apperiodische, instationäre Biosignale nicht qualifizierbar + \item Trennung kaum möglich, bestenfalls Reduktion (z.B. Abschwächung) + \item Problem: funktionelle Verkopplung/Überlagerung im Mensch \end{itemize*} - Das größte Problem bei der Reduktion von biologischen Störsignalen ist - ihre funktionelle Verkopplung und physikalische Überlagerung im - Volumenleiter Mensch. Die funktionelle Verkopplung (z.B. Einfluss der - Atmung auf die Herzrate) ist nicht abschaltbar, ist nichtlinear und - qualitativ unbekannt bzw. mit Methoden der BSA nicht beschreibbar. - Außerdem sind die Verkopplungen in ihrer Komplexität weitgehend - unerforscht und höchstens in Ansätzen dokumentiert. + %Das größte Problem bei der Reduktion von biologischen Störsignalen ist ihre funktionelle Verkopplung und physikalische Überlagerung im Volumenleiter Mensch. Die funktionelle Verkopplung (z.B. Einfluss der Atmung auf die Herzrate) ist nicht abschaltbar, ist nichtlinear und qualitativ unbekannt bzw. mit Methoden der BSA nicht beschreibbar. Außerdem sind die Verkopplungen in ihrer Komplexität weitgehend unerforscht und höchstens in Ansätzen dokumentiert. - Man kann im Einzelfall den Einfluss eines Biosignals auf ein anderes zum - Teil reduzieren. So z.B. ist bekannt, dass das EMG ein breitbandiges und - vor allem hochfrequentes Signals ist, während das EKG seine Hauptanteile - eher im niederfrequenten Bereich besitzt. Daher kann man den Einfluss - des EMG mit einem relativ einfachen Tiefpass reduzieren, allerdings auch - auf Kosten der Beeinträchtigung des EKG. - - \subsection{Medizinische Messverstärker}\label{medizinische-messverstuxe4rker} - - \subsubsection{Dynamik, Linearität}\label{dynamik-linearituxe4t} - - Messverstärker Anforderungen: + %Man kann im Einzelfall den Einfluss eines Biosignals auf ein anderes zum Teil reduzieren. So z.B. ist bekannt, dass das EMG ein breitbandiges und vor allem hochfrequentes Signals ist, während das EKG seine Hauptanteile eher im niederfrequenten Bereich besitzt. Daher kann man den Einfluss des EMG mit einem relativ einfachen Tiefpass reduzieren, allerdings auch auf Kosten der Beeinträchtigung des EKG. + \subsection{Medizinische Messverstärker}\label{medizinische-messverstärker} + %\subsubsection{Dynamik, Linearität}\label{dynamik-linearität} + Messverstärker Anforderungen \begin{itemize*} \item Linearität im Arbeitsbereich + \begin{itemize*} + \item statische Linearität des Verstärkers + \item statische Beziehung zw Aus-/Eingangsspannung + \end{itemize*} \item Linearer Phasenfrequenzgang - \item Geringes Eigenrauschen - \item Hohe Gleichtaktunterdrückung - \item Übersteuerungsfestigkeit + \begin{itemize*} + \item Erhaltung der Signalform bei Verstärkung + \end{itemize*} + \item Geringes Eigenrauschen des Messverstärkers + \item Hohe Gleichtaktunterdrückung, nicht unter 100dB + \item Übersteuerungsfestigkeit (100v Defi, 100W HF-Leistung) \end{itemize*} - \begin{enumerate*} - \def\labelenumi{\arabic{enumi}.} - \item Mit Linearität im Arbeitsbereich ist die statische Linearität des Verstärkers gemeint, also die statische Beziehung zwischen der Ausgans-zu der Eingangsspannung \$U\_a/U\_e\$. - \item Mit linearem Phasengang ist die dynamische Linearität gemeint, also die Erhaltung der Signalform bei der Verstärkung. Beim nichtlinearen Phasengang wird die Veränderung der Signalform fälschlicherweise auch als ,,lineare Verzerrung'' bezeichnet, wohl in Anlehnung an die nichtlinearen Verzerrungen im Arbeitsbereich. - \item Das Eigenrauschen des Messverstärkers ist ein sehr wichtiger Parameter vor allem in der medizinischen Messtechnik, denn das Rauschen liegt im Bereich der zu messenden Signale im unteren Mikrovoltbereich. Ausgerechnet das 1/f-Halbleiterrauschen liegt dort, wo die Biosignale ihren wesentlichen Spektralanteil aufweisen. - \item Wie schon erwähnt, ein wesentlicher Teil der beherrschbaren technischen Störungen bilden die Gleichtaktsignale. Daher wird von den Messverstärkern eine hohe CMRR gefordert, die nicht unter 100dB liegen sollte. - \item Die Empfindlichkeit eines Verstärkers allein ist noch kein hinreichendes Kriterium. Ein medizinischer Verstärker muss übersteuerungsfest sein, damit er nicht schon beim ersten Defibrilationsimpuls oder bei der ersten OP mit HF-Gerät seine Dienste aufgibt. Und dies zu gewährleisten ist für die Elektroniker eine echte Herausforderung: Es gilt nämlich das Ziel, einen Verstärker aufzubauen, der im Mikrovoltbereich arbeitet, dennoch bei Spannungen von mehreren 100V (Defibrilation) oder HF-Leistungen (um 100W) nicht beschädigt wird und zeitnah seinen Arbeitsbereich wiederfindet. - \end{enumerate*} + %\begin{enumerate*} + % \item Mit Linearität im Arbeitsbereich ist die statische Linearität des Verstärkers gemeint, also die statische Beziehung zwischen der Ausgans-zu der Eingangsspannung $U_a/U_e$. + % \item Mit linearem Phasengang ist die dynamische Linearität gemeint, also die Erhaltung der Signalform bei der Verstärkung. Beim nichtlinearen Phasengang wird die Veränderung der Signalform fälschlicherweise auch als ,,lineare Verzerrung'' bezeichnet, wohl in Anlehnung an die nichtlinearen Verzerrungen im Arbeitsbereich. + % \item Das Eigenrauschen des Messverstärkers ist ein sehr wichtiger Parameter vor allem in der medizinischen Messtechnik, denn das Rauschen liegt im Bereich der zu messenden Signale im unteren Mikrovoltbereich. Ausgerechnet das 1/f-Halbleiterrauschen liegt dort, wo die Biosignale ihren wesentlichen Spektralanteil aufweisen. + % \item Wie schon erwähnt, ein wesentlicher Teil der beherrschbaren technischen Störungen bilden die Gleichtaktsignale. Daher wird von den Messverstärkern eine hohe CMRR gefordert, die nicht unter 100dB liegen sollte. + % \item Die Empfindlichkeit eines Verstärkers allein ist noch kein hinreichendes Kriterium. Ein medizinischer Verstärker muss übersteuerungsfest sein, damit er nicht schon beim ersten Defibrilationsimpuls oder bei der ersten OP mit HF-Gerät seine Dienste aufgibt. Und dies zu gewährleisten ist für die Elektroniker eine echte Herausforderung: Es gilt nämlich das Ziel, einen Verstärker aufzubauen, der im Mikrovoltbereich arbeitet, dennoch bei Spannungen von mehreren 100V (Defibrilation) oder HF-Leistungen (um 100W) nicht beschädigt wird und zeitnah seinen Arbeitsbereich wiederfindet. + %\end{enumerate*} + % \includegraphics[width=.5\linewidth]{Assets/Biosignalverarbeitung-Linearität-arbeitsbereich.png} - \includegraphics[width=.5\linewidth]{Assets/Biosignalverarbeitung-Linearität-arbeitsbereich.png} - - Die Pegel der Biosignale sind gut bekannt, so dass den Arbeitsbereich - des Verstärkers vorzugeben, kein Problem darstellt. So wird dieser - Bereich für das EKG etwa zwischen - 5 und +5 mV liegen. Als Reserve bis - zur Begrenzung sollte man mindestens 50\% des Arbeitsbereiches vorsehen. + Arbeitsbereich + \begin{itemize*} + \item Pegel der Biosignale gut bekannt + \item EKG zwischen -5 und +5 mV + \item Reserve bis Begrenzung ungefähr 50\% des Arbeitsbereichs + \end{itemize*} + %Die Pegel der Biosignale sind gut bekannt, so dass den Arbeitsbereich des Verstärkers vorzugeben, kein Problem darstellt. So wird dieser Bereich für das EKG etwa zwischen - 5 und +5 mV liegen. Als Reserve bis zur Begrenzung sollte man mindestens 50\% des Arbeitsbereiches vorsehen. \subsubsection{Eigenrauschen}\label{eigenrauschen} - - \includegraphics[width=.5\linewidth]{Assets/Biosignalverarbeitung-eigenrauschen.png} - - Das Halbleiterrauschen (1/f) erreicht bei etwa 10Hz den Pegel des weißen - (Widerstands-) Rauschens. Da aber in diesem Bereich die meiste Energie - der Biosignale liegt, ist es beim Schaltungsentwurf wichtiger, geeignete - Halbleiter auszusuchen als sich auf die Minimierung des - Widerstandsrauschens zu beschränken. Da die Auswahl an guten Halbleitern - sehr begrenzt ist und dadurch den Entwicklern deutliche technologische - Grenzen gesetzt sind, versuchen einige Konstrukteure und Hersteller die - Eigenschaften ihrer Technik dadurch zu beschönigen, dass sie das - Spektrum nach unten durch einen Hochpass begrenzen und erst dann die - Rauschspannung messen und angeben. Daher muss man bei den Vergleichen - verschiedener Techniken an dieser Stelle sehr vorsichtig vorgehen. - Beispielsweise ist ein Verstärker, der angeblich nur 2uV Rauschspannung - erzeugt aber erst bei 1Hz beginnt sicher nicht besser, als einer mit 3uV - Rauschspannung dafür aber bereits ab 0.1Hz verstärkt. - - \subsubsection{Frequenzgang}\label{frequenzgang} - - Linearer Phasenfrequenzgang: Keine Formverzerrung - + %\includegraphics[width=.5\linewidth]{Assets/Biosignalverarbeitung-eigenrauschen.png} \begin{itemize*} - \item Gruppenlaufzeit: \$d(f)=const.\$ - \item Phasenfrequenzgang: \$\textbackslash phi(f)=\textbackslash int + \item Halbleiterrauschen bei 10Hz weißes (Widerstands-) Rauschen + \item Schaltungsentwurf auf minimierung des Rauschens beschränken + \item Auswahl an guten Halbleitenr sehr begrenzt + \item Herstellerangaben teils beschönigt \end{itemize*} - Die wichtigste Eigenschaft der Biosignale, die von Medizinern - diagnostisch genutzt wird, ist ihre Signalform. Daher lautet eine der - grundlegenden Anforderungen an die Messtechnik und die BSA, dass die - Signalform nicht verfälscht werden darf. Das bedeutet, dass sowohl im - analogen als auch im digitalen Teil des Messsystems die Gruppenlaufzeit - konstant sein muss. Daraus lässt sich die Forderung herleiten, dass der - Phasengang linear sein muss, zumindest im Übertragungsbereich. + %Das Halbleiterrauschen (1/f) erreicht bei etwa 10Hz den Pegel des weißen (Widerstands-) Rauschens. Da aber in diesem Bereich die meiste Energie der Biosignale liegt, ist es beim Schaltungsentwurf wichtiger, geeignete Halbleiter auszusuchen als sich auf die Minimierung des Widerstandsrauschens zu beschränken. Da die Auswahl an guten Halbleitern sehr begrenzt ist und dadurch den Entwicklern deutliche technologische Grenzen gesetzt sind, versuchen einige Konstrukteure und Hersteller die Eigenschaften ihrer Technik dadurch zu beschönigen, dass sie das Spektrum nach unten durch einen Hochpass begrenzen und erst dann die Rauschspannung messen und angeben. Daher muss man bei den Vergleichen verschiedener Techniken an dieser Stelle sehr vorsichtig vorgehen. Beispielsweise ist ein Verstärker, der angeblich nur 2uV Rauschspannung erzeugt aber erst bei 1Hz beginnt sicher nicht besser, als einer mit 3uV Rauschspannung dafür aber bereits ab 0.1Hz verstärkt. - \subsection{Differenzverstärker}\label{differenzverstuxe4rker} + \subsubsection{Frequenzgang}\label{frequenzgang} + \begin{itemize*} + \item Linearer Phasenfrequenzgang: Keine Formverzerrung + \item Signalform darf nicht verfälscht werden + \item deshalb Gruppenlaufzeit konstant $d(f)=const.$ + \item $\rightarrow$ Phasenfrequenzgang: $\phi(f)=\int d(f)df=\varphi_0*f$ + \end{itemize*} - \subsubsection{Funktionsprinzip}\label{funktionsprinzip} + % Die wichtigste Eigenschaft der Biosignale, die von Medizinern diagnostisch genutzt wird, ist ihre Signalform. Daher lautet eine der grundlegenden Anforderungen an die Messtechnik und die BSA, dass die Signalform nicht verfälscht werden darf. Das bedeutet, dass sowohl im analogen als auch im digitalen Teil des Messsystems die Gruppenlaufzeit konstant sein muss. Daraus lässt sich die Forderung herleiten, dass der Phasengang linear sein muss, zumindest im Übertragungsbereich. + \subsection{Differenzverstärker}\label{differenzverstärker} Vollkommene Symmetrie (DV und Signalanbindung) \begin{itemize*} - \item \includegraphics[width=.5\linewidth]{Assets/Biosignalverarbeitung-Differenzverstärker-funktion.png} - \item Vg ist Quelle der massebezogenen Störung. Die Störspannung gelangt auf beide Eingänge über Streukapazitäten, deren Impedanzen mit R4 und R5 simuliert werden, in gleicher Phase und im Idealfall auch mit gleichem Pegel. Die Störsignale an den Eingängen U10 und U20 sind also gleich, werden daher als Gleichtaktsignale bezeichnet. - \item Vd ist die gewünschte massefreie Spannung (aus Sicht der Signalquelle zählen R4 und R5 nicht als Masseverbindung, die ,,hängt in der Luft'', floating source). Die Signalquelle Vd liegt direkt zwischen den Eingängen an, erzeugt daher eine Differenzspannung (siehe Funktionsprinzip eines Differenzverstärkers: Durch die Verkopplung der beiden Zweige T1 und T2 hat eine Zunahme der Eingangsspannung U10 Abnahme von Ud1 und Zunahme von Ud2, analog gilt das für U20. Daher liegt zwischen Ud1 und Ud2 die verstärkte Differenz von U10 und U20 an). - \item Betrachtet man Ud1 und Ud2 massebezogen, so liegen überlagerte Gleichtakt- und Differenzspannungen an (unterer Grafik). Betrachtet man die verstärkte Spannung massefrei (also als Differenz zwischen Ud1 und Ud2), so verschwindet durch die Differenzbildung die Gleichtaktstörung und die gewünschte Differenzspannung bleibt übrig. - \item Alle bisherigen Erläuterungen gelten nur im Idealfall: Sowohl der Verstärker ist ideal symmetrisch (identische Transistoren und Widerstände), als auch die Einkopplung der Gleichtaktstörung erfolgt ideal symmetrisch (über R4 und R5). + %\item \includegraphics[width=.5\linewidth]{Assets/Biosignalverarbeitung-Differenzverstärker-funktion.png} + %\item Vg ist Quelle der massebezogenen Störung. Die Störspannung gelangt auf beide Eingänge über Streukapazitäten, deren Impedanzen mit R4 und R5 simuliert werden, in gleicher Phase und im Idealfall auch mit gleichem Pegel. Die Störsignale an den Eingängen U10 und U20 sind also gleich, werden daher als Gleichtaktsignale bezeichnet. + \item Störsignale an Eingängen Phasen/Pegel-gleich $\rightarrow$ Gleichtaktsignal + %\item Vd ist die gewünschte massefreie Spannung (aus Sicht der Signalquelle zählen R4 und R5 nicht als Masseverbindung, die ,,hängt in der Luft'', floating source). Die Signalquelle Vd liegt direkt zwischen den Eingängen an, erzeugt daher eine Differenzspannung (siehe Funktionsprinzip eines Differenzverstärkers: Durch die Verkopplung der beiden Zweige T1 und T2 hat eine Zunahme der Eingangsspannung U10 Abnahme von Ud1 und Zunahme von Ud2, analog gilt das für U20. Daher liegt zwischen Ud1 und Ud2 die verstärkte Differenz von U10 und U20 an). + \item Signalquelle erzeugt Differenzspannung + %\item Betrachtet man Ud1 und Ud2 massebezogen, so liegen überlagerte Gleichtakt- und Differenzspannungen an (unterer Grafik). Betrachtet man die verstärkte Spannung massefrei (also als Differenz zwischen Ud1 und Ud2), so verschwindet durch die Differenzbildung die Gleichtaktstörung und die gewünschte Differenzspannung bleibt übrig. + %\item Alle bisherigen Erläuterungen gelten nur im Idealfall: Sowohl der Verstärker ist ideal symmetrisch (identische Transistoren und Widerstände), als auch die Einkopplung der Gleichtaktstörung erfolgt ideal symmetrisch (über R4 und R5). + \item Verstärker ist ideal symmetrisch (identische Transistoren/R) + \item Einkopplung der Gleichtaktstörung idel symmetrisch über R \end{itemize*} Symmetrie im DV, asymmetrische (realistische) Signalanbindung - \begin{itemize*} - \item \includegraphics[width=.5\linewidth]{Assets/Biosignalverarbeitung-Differenzverstärker-asymmetrisch.png} - \item In der Realität lassen sich zwar Verstärker bauen, die an das Ideal gut herankommen. - \item Die Einkopplung der Gleichtaktstörung ist jedoch immer unsymmetrisch, es ist unmöglich, im Messkreis Symmetrie herzustellen (R4 und R5 unterschiedlich). Daher wird aus der ihrem Wesen nach Gleichtaktstörung zum Teil eine Differenzstörung. Und die Differenzstörung erscheint in der Ausgangsspannung Ud1-Ud2 zwangsläufig auch. - \item Das heißt, in der Realität wird der Gleichtaktanteil der Störung zwar unterdrückt, aber der zur Differenz gewordene Anteil bleibt am Ausgang bestehen. - \end{itemize*} + %\item \includegraphics[width=.5\linewidth]{Assets/Biosignalverarbeitung-Differenzverstärker-asymmetrisch.png} + \item Einkopplung der Gleichtaktstörung immer unsymmetrisch + \item unmöglich, im Messkreis Symmetrie herzustellen + \item Gleichtaktanteil der Störung unterdrückt aber zur Differenz gewordene Anteil am Ausgang besteht + \end{itemize*} \subsubsection{Differenz- und Gleichtaktverhalten}\label{differenz--und-gleichtaktverhalten} - - \includegraphics[width=.5\linewidth]{Assets/Biosignalverarbeitung-Diff-und-Gleichtakt.png} - + % \includegraphics[width=.5\linewidth]{Assets/Biosignalverarbeitung-Diff-und-Gleichtakt.png} \begin{itemize*} - \item \$SNR\_\{in\} = \textbackslash frac\{U\_\{d\_in\}\}\{U\_\{g\_in\}\}=\textbackslash frac\{1mV\}\{10V\}=10\^{}\{-4\}\textbackslash approx -80dB\$ - \item \$V\_g\$: Gleichtaktstörung (Netz) - \item \$V\_d\$: Nutzsignal (EKG) + \item $SNR_{in} = \frac{U_{d_in}}{U_{g_in}}=\frac{1mV}{10V}=10^{-4}\approx -80dB$ + \item $V\_g$: Gleichtaktstörung (Netz) + \item $V\_d$: Nutzsignal (EKG) \item Heute werden Differenzverstärker meistens als integrierte analoge Schaltungen mit OPVs eingesetzt. Da der Ausgang massefrei ist, folgt eine zweite Stufe zur Differenzbildung (IC3), die am Ausgang eine -wie üblich -massebezogene Spannung liefert. Diese Anordnung wird als Instrumentationsverstärker bezeichnet (instrumenation amplifier) und ist auch integriert verfügbar. \item Am Eingang liegt eine realistische Situation vor: Das gewünschte Signal hat den Pegel von 1mV (EKG), die Netzstörung erreicht (auch mehr als) 10V. Daher ist der SNR am Eingang sehr niedrig, -80dB. \item \includegraphics[width=.5\linewidth]{Assets/Biosignalverarbeitung-Diff-und-Gleichtakt2.png} - \item \$CMRR=\textbackslash frac\{U\_\{d\_out\}\}\{U\_\{g\_out\}\}*\textbackslash frac\{U\_\{g\_in\}\}\{U\_\{d\_in\}\}=\textbackslash frac\{200mV\}\{20mV\} *\textbackslash frac\{10V\}\{1mV\}=10\^{}5\textbackslash approx 100dB\$ + \item $CMRR=\frac{U_{d_out}}{U_{g_out}}*\frac{U_{g_in}}{U_{d_in}}=\frac{200mV}{20mV} *\frac{10V}{1mV}=10^5 \approx 100dB$ \item Führt man mit dem Ausgangssignal des Verstärkers Spektralanalyse durch, so stellt man fest, dass die Netzstörung am Ausgang 20mV beträgt, während das gewünschte Signal 200mV erreicht, also der SNR am Ausgang ist 10 bzw. 20dB. Da der SNR am Eingang - 80dB betrug, wurde eine SNR-Verbesserung von 100dB erreicht. Diese Verbesserung ist auf die Gleichtaktunterdrückung selbst bei Asymmetrie am Eingang zurückzuführen, so dass in diesem Fall das CMRR identisch der SNR-Verbesserung ist. (Common-Mode Rejection Ratio, Gleichtaktunterdrückung, muss in der Medizintechnik laut Katalog mindestens 100dB, besser 120dB erreichen). \end{itemize*} - \subsection{Instrumentationsverstärker}\label{instrumentationsverstuxe4rker} + \subsection{Instrumentationsverstärker}\label{instrumentationsverstärker} Der Instrumentationsverstärker (IV) ist ein mehrstufiger Verstärker, von dem in der medizinischen Messtechnik ein hoher Eingangswiderstand (besser als 100MOhm) und eine hohe CMRR (besser 100dB) gefordert wird. - \subsubsection{Mehrstufiger Verstärker}\label{mehrstufiger-verstuxe4rker} + \subsubsection{Mehrstufiger Verstärker}\label{mehrstufiger-verstärker} \begin{itemize*} \item \includegraphics[width=.5\linewidth]{Assets/Biosignalverarbeitung-mehrstufiger-verstärker.png} \item Die erste Stufe ist der Eingangs-Differenzverstärker mit massefreiem Ausgang; die Ausgangsspannung ergibt sich aus der Differenz der Ausgangsspannungen von IC1 und IC2. Die zweite Stufe verstärkt zusätzlich und bezieht die verstärkte Spannung auf Masse, so dass am Ausgang massebezogene, verstärkte Eingangsdifferenz vorliegt. - \item V1: \$u\_\{ad\}=A\emph{u\_\{ed\}+B}u\_\{eg\}\$, \$u\_\{ag\}=C*u\_\{eg\}+D+u\_\{ed\}\$, - \begin{itemize*} \item \$A/B=F\$: Diskriminationsfaktor \item \$A/C=H\$: Rejektionsfaktor \end{itemize*} + \item V1: $u_{ad}=A*u_{ed}+B*u_{eg}$, $u_{ag}=C*u_{eg}+D+u_{ed}$, + \begin{itemize*} + \item $A/B=F$: Diskriminationsfaktor + \item $A/C=H$: Rejektionsfaktor + \end{itemize*} \item V2: - \begin{itemize*} \item \$u\_a=V\_d u\_\{ed\}+\textbackslash frac\{V\_d\}\{CMR\}u\_\{eg\}=V\_d(A u\_\{ed\}+\textbackslash frac\{A\}\{F\} u\_\{eg\})+\textbackslash frac\{V\_d\}\{CMR\}\textbackslash frac\{A\}\{H\} u\_\{eg\}\$ \item \$u\_a\textbar{}\emph{\{u}\{ed\}=0\} = V\_d A(\textbackslash frac\{1\}\{F\}+\textbackslash frac\{1\}\{CMR*H\}) u\_\{eg\}\$ \item die gesamt-Gleichtaktunterdrückung eines mehrstufigen Verstärkers ist abhängig im Wesentlichen von der ersten (Eingangs-) Stufe \end{itemize*} + \begin{itemize*} + \item $u_a=V_d u_{ed}+\frac{V_d}{CMR}u_{eg}=V_d(A u_{ed}+\frac{A}{F} u_{eg})+\frac{V_d}{CMR}\frac{A}{H} u_{eg}$ + \item $u_a|_{u_{ed}=0} = V_d A(\frac{1}{F}+\frac{1}{CMR*H}) u_{eg}$ + \item die gesamt-Gleichtaktunterdrückung eines mehrstufigen Verstärkers ist abhängig im Wesentlichen von der ersten (Eingangs-) Stufe + \end{itemize*} \item Berechnet man die Ausgangsspannung in Abhängigkeit von der Eingangs-Gleichtaktspannung und von den Verstärkerparametern, so zeigt sich, dass für den CMRR die erste Stufe (wie auch bei anderen Parametern, z.B. Eigenrauschen) entscheidend ist, die folgenden Stufen sind unwesentlich beteiligt. Daher wird in der ersten Stufe der höchste Entwicklungsaufwand getrieben. \end{itemize*} @@ -956,8 +748,8 @@ \begin{itemize*} \item \includegraphics[width=.5\linewidth]{Assets/Biosignalverarbeitung-hoher-eingangswiderstand.png} - \item \$R\^{}\{(1)\}\_\{ed\}=2R\_D+R\_C\textbackslash approx 2R\_D\$ - \item \$R\^{}\{(2)\}\_\{ed\}=R\_1+R\_3\textless\textless R\_D\$ + \item $R^{(1)}_{ed}=2R_D+R_C\approx 2R_D$ + \item $R^{(2)}_{ed}=R_1+R_3 << R_D$ \item Theoretisch würde zur Ableitung von Biosignalen die zweite Stufe allein reichen, denn sie selbst verstärkt die Differenzspannung an ihrem Eingang (R1 und R3). Allerdings ist der Eingangswiderstand der zweiten Stufe für Biosignale viel zu niedrig. Eine zusätzliche Stufe mit hohem Eingangswiderstand ist daher notwendig, die außerdem noch wesentlich zur Verstärkung beiträgt. \end{itemize*} @@ -970,7 +762,7 @@ \item Gute Eigenschaften sind nur mit integrierter Technologie und getrimmten Widerständen erreichbar. Daher werden bis auf einige speziellen Ausnahmen ausschließlich integrierte IV eingesetzt. \end{itemize*} - \subsection{Isolationsverstärker}\label{isolationsverstuxe4rker} + \subsection{Isolationsverstärker}\label{isolationsverstärker} Aus Sicherheitsgründen bzw. wegen zu hoher Spannungen ist es in der Medizin, aber auch z.B. in der Leistungselektronik, oft notwendig, den @@ -1027,7 +819,7 @@ \item Problem: Zuführung des Bezugspotentials notwendig \end{itemize*} - \subsubsection{Störungsresistenz}\label{stuxf6rungsresistenz} + \subsubsection{Störungsresistenz}\label{störungsresistenz} \begin{itemize*} \item Aktive Elektroden technologisch aufwendig, haben aber Vorteile bei Störungen, die direkt auf die Messanordnung wirken @@ -1046,7 +838,7 @@ pauschal, bei aktiven Elektroden ist Differenzierung notwendig \begin{itemize*} - \item Aktive Elektroden meistens mit Verstärkung \$V=1\$ + \item Aktive Elektroden meistens mit Verstärkung $V=1$ \item daher CMR rechnerisch gleich 1, theoretisch zu niedrig \item prinzipbedingt starke Unterdrückung der Stör-Gleichtaktsignale \item daher praktisch sehr gute CMRR von 100dB und mehr @@ -1068,8 +860,8 @@ \item \includegraphics[width=.5\linewidth]{Assets/Biosignalverarbeitung-Filtertheorie.png} \item Bei spektralen Filtern werden folgende Parameter verwendet, um die Filtercharakteristik zu beschreiben: \begin{itemize*} - \item Die Eckfrequenz, oder Grenzfrequenz: Frequenz \$F\_\{pass\}\$, bei der der Durchlassbereich in die Filterflanke übergeht und bei der die Übertragung um 3dB bzw. auf 70\% der Übertragung vom Durchlassbereiche abgesunken ist. - \item Die Sperrfrequenz \$F\_\{stop\}\$, bei der die geforderte Dämpfung im Sperrbereich erreicht wird. Übergangsband Fstop-Fpass, auch transition band, ist der Übergangsbereich vom Durchlass-in das Sperrband, auch Filterflanke genannt. \item Steilheit ist Maß für die Filterflanke in dB/Hz. Grundsätzlich gilt, je steiler, umso besser. Hängt hauptsächlich von der Filterordnung ab. + \item Die Eckfrequenz, oder Grenzfrequenz: Frequenz $F\_\{pass\}$, bei der der Durchlassbereich in die Filterflanke übergeht und bei der die Übertragung um 3dB bzw. auf 70\% der Übertragung vom Durchlassbereiche abgesunken ist. + \item Die Sperrfrequenz $F\_\{stop\}$, bei der die geforderte Dämpfung im Sperrbereich erreicht wird. Übergangsband Fstop-Fpass, auch transition band, ist der Übergangsbereich vom Durchlass-in das Sperrband, auch Filterflanke genannt. \item Steilheit ist Maß für die Filterflanke in dB/Hz. Grundsätzlich gilt, je steiler, umso besser. Hängt hauptsächlich von der Filterordnung ab. \item Welligkeit im Durchlassbereich Apass gibt an, im welchen Bereich die Übertragung im Durchlassbereich schwankt. Üblich ist weniger als 1dB, um 3dB ist für niedrige Ansprüche ausreichend. \item Minimale Dämpfung Astop gibt die garantierte Dämpfung an. Hängt hauptsächlich von der Filterordnung ab. \item Fs/2 ist die halbe Abtastrate oder die Nyquistfrequenz. @@ -1077,10 +869,10 @@ \item \includegraphics[width=.5\linewidth]{Assets/Biosignalverarbeitung-Filtertheorie2.png} \item Die Filtertheorie unterscheidet vier Grundtypen, siehe oben. Die Filtertheorie bietet ein Instrumentarium zum Entwurf von Filtern, vor allem aber für den nachrichtentechnischen Bereich, d.h. L-C-Kombinationen, also schwingfähige Systeme. Im Spektralbereich der Biosignale werden fast ausschließlich RC-Filter verwendet. Die Vorgehensweise beim klassischen Filterentwurf ist über die Schaltungsanalyse, also faktisch in einem Iterationsverfahren: Grundbausteine der spektralen Filter sind bekannt und mit diesen versucht man die gewünschte Charakteristik iterativ durch hinzufügen von Elementen und anschließender Analyse zu erreichen. Im analogen Bereich ist es kaum möglich, eine Filtercharakteristik vorzugeben und nach irgendeiner Methode die Schaltung als Ergebnis zu erhalten, der Entwurf ist daher sehr intuitiv und routineorientiert. Die Schaltungssynthese reduziert sich dann lediglich auf die Entnormung der Modelle auf konkrete Bauelemente. \begin{itemize*} - \item Übertragungsfunktion \$G(j\textbackslash omega)=\textbackslash frac\{U\_2(j\textbackslash omega)\}\{U\_1(j\textbackslash omega)\}=\textbar G(j\textbackslash omega)\textbar*e\^{}\{j\textbackslash omega\textbackslash phi\}\$ - \item Amplitudenfrequenzgang \$\textbar G(j\textbackslash omega)\textbar=\textbackslash sqrt\{Re\{G(j\textbackslash omega)\}\^{}2 +Im\{G(j\textbackslash omega)\}\^{}2\}\$ - \item Phasenfrequenzgang \$\textbackslash phi(\textbackslash omega)=arctan\textbackslash frac\{IM\{G(j\textbackslash omega)\}\}\{Re\{G(j\textbackslash omega)\}\}\$ - \item Grenzfrequenz \$\textbackslash omega\_g=\textbackslash frac\{1\}\{RC\}\$ + \item Übertragungsfunktion $G(j\omega)=\frac{U_2(j\omega)}{U_1(j\omega)}=| G(j\omega)|*e^{j\omega\phi}$ + \item Amplitudenfrequenzgang $|G(j\omega)|=\sqrt{Re{G(j\omega)}^2 +Im{G(j\omega)}^2}$ + \item Phasenfrequenzgang $\phi(\omega)=arctan\frac{IM{G(j\omega)}}{Re{G(j\omega)}}$ + \item Grenzfrequenz $\omega_g=\frac{1}{RC}$ \end{itemize*} \item Üblicherweise werden die Filter über ihre Übertragungsfunktion beschrieben, wobei auch äquivalente Beschreibungen möglich sind -Impulsantwort im digitalen Bereich, Pole-Nullstellen-Diagramme, seltener Zustandsgleichungen. \item Aus Sicht der BSA sind entscheidend die Beschreibungen über den Amplituden- und Phasenfrequenzgang. @@ -1111,7 +903,12 @@ \begin{itemize*} \item \includegraphics[width=.5\linewidth]{Assets/Biosignalverarbeitung-tiefpass-2.ordnung.png} \item \includegraphics[width=.5\linewidth]{Assets/Biosignalverarbeitung-hochpass-2.ordnung.png} - \item \$R\_\{0a\}=(\textbackslash epsilon-1)R\_0\$, \$u\_a=\textbackslash epsilon*u\_e\$ \$\textbackslash Rightarrow\$ Filtertyp mit \$R\_0\$ einstellbar \textbar{} \textbar{} Bessel \textbar{} Butterworth \textbar{} Tschebyscheff (1,5dB) \textbar{} \textbar{} -\/-\/-\/-\/-\/-\/-\/-\/-\/- \textbar{} -\/-\/-\/-\/-\/-\/- \textbar{} -\/-\/-\/-\/-\/-\/-\/-\/-\/-\/- \textbar{} -\/-\/-\/-\/-\/-\/-\/-\/-\/-\/-\/-\/-\/-\/-\/-\/-\/-\/-\/-\/- \textbar{} \textbar{} \$\textbackslash epsilon\$ \textbar{} \$1,267\$ \textbar{} \$1,586\$ \textbar{} \$2,482\$ \textbar{} \textbar{} \$\textbackslash gamma\$ \textbar{} \$0,618\$ \textbar{} \$1,0\$ \textbar{} \$1,663\$ \textbar{} + \item $R_{0a}=(\epsilon-1)R_0$, $u_a=\epsilon*u_e$ $\Rightarrow$ Filtertyp mit $R_0$ einstellbar + \begin{tabular}{l|l|l|l} + & Bessel & Butterworth & Tschebyscheff (1,5dB) \\\hline + $\epsilon$ & $1,267$ & $1,586$ & $2,482$ \\ + $\gamma$ & $0,618$ & $1,0$ & $1,663$ + \end{tabular} \item Durch die Verwendung von OVs, die eine definierte Gegenkopplung ermöglichen, ist es mit relativ einfachen Mitteln möglich, effektive Filter zu konstruieren, die mit passiven Bauelementen allein um Größenordnungen komplizierter wären. Die beliebteste Entwurfstechnik ist die mit Hilfe von kaskadierten Stufen 2. Ordnung, auch im digitalen Bereich. Filter 2.Ordnung sind übersichtlich strukturiert, die Bauelemente müssen nicht mühsam ausgesucht und ausgemessen werden, die Eigenschaften sind sehr gut bekannt und bequem einstellbar, so dass eine kompliziertere Filterstruktur - z.B. Antialiasing-Tiefpass 8. Ordnung - durch einfache Kaskadierung (Serienschaltung) realisierbar ist. Natürlich muss man bei der Kaskadierung beachten, dass jede Stufe bei ihrer Grenzfrequenz 3dB-Abfall verursacht, so dass man die Kette entsprechend dimensionieren muss. \item Sehr vorteilhaft ist in dieser Schaltung, dass der Filtertyp bequem durch die Veränderung (Durchstimmung) eines einzigen Widerstandes über alle drei Basischarakteristiken eingestellt werden kann, sonst ist keine Veränderung der Schaltung notwendig. \item die Basistypen sind folgende: @@ -1126,11 +923,11 @@ \item Von allen drei hat nur Bessel konstante Gruppenlaufzeit, die in der BSA zwingend notwendig ist bei Echtzeitanwendungen. Folgen nichtkonstanter Gruppenlaufzeit sind u.a. Formverzerrungen, wie später gezeigt wird. \item \includegraphics[width=.5\linewidth]{Assets/Biosignalverarbeitung-diskreter-integrator-mit-ov.png} \begin{itemize*} - \item \$\textbackslash tau=RC\$ + \item $\tau=RC$ \end{itemize*} \item \includegraphics[width=.5\linewidth]{Assets/Biosignalverarbeitung-integrierter-integrator-mit-sc.png} \begin{itemize*} - \item \$\textbackslash tau=\textbackslash frac\{1\}\{f\_S\}*\textbackslash frac\{C\_3\}\{C\_2\}\$ + \item $\tau=\frac{1}{f_S}*\frac{C_3}{C_2}$ \end{itemize*} \item Konventionell werden aktive Filter -am häufigsten der Antialiasing-Tiefpass vor dem AD-Wandler -mit Hilfe von OVs und RC-Netzwerken realisiert. \item Eine sehr elegante Alternative bieten die Filter mit geschalteten Kapazitäten: An Stelle des Widerstandes am Eingang befindet sich eine Kapazität, die im Takt von fs zwischen Eingang und dem OV umgeschaltet wird. Der mittlere Strom, der mit C3 integriert wird, hängt also von der Schaltfrequenz und dem Kapazität C2 ab. Daher ergibt sich die Zeitkonstante aus den beiden Kapazitäten, die auf dem Chip integriert sind und aus der Abtastfrequenz. Man kann also die Zeitkonstante allein durch Veränderung der Schaltfrequenz einstellen, ohne ein Bauelement der Schaltung ändern zu müssen. @@ -1226,13 +1023,12 @@ Abtastung ist mathematisch eine Multiplikation des Signals mit einer Folge von Dirac-Pulsen: - \$y(t)=x(t)*\textbackslash sum\_\{n=-\textbackslash infty\}\^{}\{\textbackslash infty\}\textbackslash sigma(t-nT\_A)\$ - (\$T\_A\$ ist die Abtastperiode) + $y(t)=x(t)*\sum_{n=-\infty}^{\infty}\sigma(t-nT_A)$ + ($T_A$ ist die Abtastperiode) Eine Multiplikation von zwei Signalen im Zeitbereich entspricht der Faltung ihrer Spektren im Frequenzbereich (und umgekehrt): - \$Y(\textbackslash omega)=X(\textbackslash omega)*\textbackslash sum\_\{n=-\textbackslash infty\}\^{}\{\textbackslash infty\} - \textbackslash sigma(\textbackslash omega-n\textbackslash frac\{1\}\{T\_A\})\$ + $Y(\omega)=X(\omega)*\sum_{n=-\infty}^{\infty} \sigma(\omega-n\frac{1}{T_A})$ \includegraphics[width=.5\linewidth]{Assets/Biosignalverarbeitung-EKG-kontinuierlich-spektrum.png} @@ -1244,7 +1040,7 @@ \includegraphics[width=.5\linewidth]{Assets/Biosignalverarbeitung-EKG-tasten-falten.png} \begin{itemize*} - \item Um Überlappung (Aliasing) der Spektren und Mehrdeutigkeiten zu vermeiden, muss gelten: \$\textbackslash frac\{1\}\{T\_A\}\textbackslash geq 2*f\_\{max\}\$ + \item Um Überlappung (Aliasing) der Spektren und Mehrdeutigkeiten zu vermeiden, muss gelten: $\frac{1}{T_A}\geq 2*f_{max}$ \item Nyquist-Frequenz entspricht der halben Grundfrequenz der Abtastrate, sie begrenzt nach oben das bei Null beginnende sog. Basisband. Das Basisband ist der Frequenzbereich, in dem man bei der Signalanalyse arbeitet. Damit das gespiegelte Spektrum nicht bis ins Basisband reicht und dadurch das Vorhandensein real nichtexistenter Signalkomponenten vortäuscht, muss gewährleistet werden, dass die halbe AR höher liegt, als die höchste Frequenz des Signals, d.h. die AR muss mindestens doppelt so hoch sein, die die höchste vorhandene Frequenz. \item Die periodische Wiederholung des Spektrums nach der Abtastung hat folgende praktische Bedeutung: Da sich das Spektrum mit jeder Harmonischen der AR wiederholt und gespiegelt wird, kann man ein bandbegrenztes Signal ins Basisband holen. So wäre bspw. denkbar, das zwischen 500 und 625 Hz liegendes EKG abzutasten und im Basisband zu verarbeiten. Dazu später ein Beispiel mit AM-modulierten EKG \end{itemize*} @@ -1254,14 +1050,14 @@ \begin{itemize*} \item \includegraphics[width=.5\linewidth]{Assets/Biosignalverarbeitung-beispiel-film.png} - \item Annahme: Ein Rad mit 8 Speichen dreht sich so, dass der Wagen sich mit 28.3 km/h bewegt, die Drehfrequenz des Rades beträgt demnach 2.5Hz. \$v=28,3\textbackslash frac\{km\}\{s\}=7,85\textbackslash frac\{m\}\{s\}\$, \$f=\textbackslash frac\{v\}\{U\}=\textbackslash frac\{7,85m/s\}\{3,14m\}=2,5Hz\$ - \item Die Periode, mit der die Speichen einen Referenzpunkt passieren (oben) beträgt 50ms bei einer Wiederholrate von 20/s \$f\_s=8*f=20Hz\textbackslash Rightarrow T\_s=50ms\$ - \item Entsprechend der Fernsehnorm werden 25 Bilder pro Sekunde aufgenommen, die Periode ist also 40ms. \$f\_A=25fps\textbackslash Rightarrow=40ms\$ - \item Entsprechend der Faltung beim Abtasten entstehen zwei Frequenzen -5 und 20, wobei nur die -5Hz im Basisband liegen. Das bedeutet praktisch, dass man das Rad langsam nach hinten drehen sieht. \$f\_\{SA\}=f\_A\textbackslash pm f\_S=-5/20 Hz\$ + \item Annahme: Ein Rad mit 8 Speichen dreht sich so, dass der Wagen sich mit 28.3 km/h bewegt, die Drehfrequenz des Rades beträgt demnach 2.5Hz. $v=28,3\frac{km}{s}=7,85\frac{m}{s}$, $f=\frac{v}{U}=\frac{7,85m/s}{3,14m}=2,5Hz$ + \item Die Periode, mit der die Speichen einen Referenzpunkt passieren (oben) beträgt 50ms bei einer Wiederholrate von 20/s $f_s=8*f=20Hz\Rightarrow T_s=50ms$ + \item Entsprechend der Fernsehnorm werden 25 Bilder pro Sekunde aufgenommen, die Periode ist also 40ms. $f_A=25fps\Rightarrow=40ms$ + \item Entsprechend der Faltung beim Abtasten entstehen zwei Frequenzen -5 und 20, wobei nur die -5Hz im Basisband liegen. Das bedeutet praktisch, dass man das Rad langsam nach hinten drehen sieht. $f_{SA}=f_A\pm f_S=-5/20 Hz$ \item reale Effekte: Räder (Auto), Stroboskopeffekt in Industriehallen mit Leuchtstoffröhren an rotierenden Maschinen \end{itemize*} - \includegraphics[width=.5\linewidth]{Assets/Biosignalverarbeitung-rekonstruierter-sinus.png} + %\includegraphics[width=.5\linewidth]{Assets/Biosignalverarbeitung-rekonstruierter-sinus.png} Oft wird nach der Abtastung und anschließenden Signalverarbeitung das Ergebnis im ursprünglichen Bereich des Signals benötigt. Dazu ist die sog. Rekonstruktion notwendig, also eine Übertragung aus dem Analyse-in @@ -1281,30 +1077,30 @@ \includegraphics[width=.5\linewidth]{Assets/Biosignalverarbeitung-abtastung-rekonstruktion.png} \begin{itemize*} - \item \$y(t)=x(t)\emph{\textbackslash sum\^{}\{\textbackslash infty\}\_\{n=-\textbackslash infty\} \textbackslash sigma(t-nT\_A) \textbackslash Leftrightarrow Y(\textbackslash omega)=X(\textbackslash omega)}\textbackslash sum\^{}\{\textbackslash infty\}\_\{n=-\textbackslash infty\}\textbackslash sigma(\textbackslash omega-n\textbackslash frac\{1\}\{T\_A\})\$ + \item $y(t)=x(t)*\sum^{\infty}_{n=-\infty} \sigma(t-nT_A) \Leftrightarrow Y(\omega)=X(\omega)*\sum^{\infty}_{n=-\infty}\sigma(\omega-n\frac{1}{T_A})$ \item Übergang aus dem kontinuierlichen Zeitbereich in eine Folge, d.h. Entkopplung von der Abtastperiode - \item \$y(n)=y(nT\_A) \textbackslash Leftrightarrow Y(k)=Y(k\textbackslash omega\_A/M)=\textbackslash sum\_\{n=1\}\^{}M y(nT\_A)\^{}\{-jkn/M\}\$ - \item \$Y(K)\textbackslash Leftrightarrow FFT(y(n))\textbackslash Rightarrow\$ normierte Frequenz \$\textbackslash omega\textbackslash in(0,2\textbackslash pi)\textbackslash vee f\textbackslash in(0,1)\$ - \item Nyquist Frequenz \$\textbackslash omega\_N=\textbackslash pi\$, \$f\_N=0,5\$ + \item $y(n)=y(nT_A) \Leftrightarrow Y(k)=Y(k\omega_A/M)=\sum_{n=1}^M y(nT_A)^{-jkn/M}$ + \item $Y(K)\Leftrightarrow FFT(y(n))\Rightarrow$ normierte Frequenz $\omega\in(0,2\pi)\vee f\in(0,1)$ + \item Nyquist Frequenz $\omega_N=\pi$, $f_N=0,5$ \item Nach der Abtastung und Digitalisierung hat das Signal die Form einer Zahlenfolge bzw. eines Vektors oder Matrix. Ist die Abtastrate unbekannt, so ist das Signal auch nicht mehr reproduzierbar. Da sich aber Analysen und digitale Filterung grundsätzlich auch ohne Kenntnis der Abtastrate durchführen lassen, wird die sog. normierte Frequenz eingeführt, die bei der Rekonstruktion durch eine reale Abtastrate ersetzt wird. \end{itemize*} \includegraphics[width=.5\linewidth]{Assets/Biosignalverarbeitung-amplitudenmodulation.png} \begin{itemize*} - \item \$EKG\_\{AM\}=EKG*sin(\textbackslash omega\_c t)\$ + \item $EKG\_\{AM\}=EKG*sin(\omega\_c t)$ \item Beispiel zum Abtasttheorem: Das EKG wird für eine Kabelübertragung mit einem Träger bei 10kHz multipliziert, was nachrichtentechnisch einer Amplitudenmodulation entspricht. Das Spektrum spiegelt sich um den Träger herum, ähnlich wie bei der Abtastung. Hier gibt es allerdings nur eine Spiegelung, da der Träger eine Harmonische ist und somit im Spektrum nur eine Nadel darstellt. Es entstehen zwei Seitenbänder, das obere und das untere. Beide sind hinsichtlich des Informationsgehaltes völlig identisch. \item Die Frage ist nun zu beantworten, wie hoch die Abtastrate für ein solches Signal sein muss. \end{itemize*} - Abtasttheorem Kotelnikov, Channon (\$T\_A=1/2f\_\{max\}\$) + Abtasttheorem Kotelnikov, Channon ($T\_A=1/2f\_\{max\}$) \begin{itemize*} \item hinreichende/notwendige Bedingung - \item hinreichend aber nicht notwendig \$AR\textbackslash geq 22ksps\$ - \item \$Y(\textbackslash omega)=X(\textbackslash omega)*\textbackslash sum\^{}\{\textbackslash infty\}\_\{n=-\textbackslash infty\} \textbackslash sigma(\textbackslash omega-n\textbackslash frac\{1\}\{T\_A\})\$ - \item notwendig und hinreichend: \$AR\textbackslash geq 2ksps\$ - \item \includegraphics[width=.5\linewidth]{Assets/Biosignalverarbeitung-abtastung-kotelnikov.png} + \item hinreichend aber nicht notwendig $AR\geq 22ksps$ + \item $Y(\omega)=X(\omega)*\sum^{\infty}_{n=-\infty} \sigma(\omega-n\frac{1}{T_A})$ + \item notwendig und hinreichend: $AR\geq 2ksps$ + %\item \includegraphics[width=.5\linewidth]{Assets/Biosignalverarbeitung-abtastung-kotelnikov.png} \item Da die höchste Frequenz im Signal 11kHz beträgt, müsste die Abtastrate mind. 22ksps betragen. Dies ist eine hinreichende, aber keine notwendige Bedingung. Denn bezieht man sich auf die Wiederholung des Spektrums um jede Harmonische der Abtastrate, wird eine AR von 2ksps ausreichen. Damit passt eine Wiederholung des Spektrums in das Basisband, in wir ja analysieren. \end{itemize*} @@ -1316,7 +1112,7 @@ \item Zeit diskret \item Pegel analog \end{itemize*} - \item \includegraphics[width=.5\linewidth]{Assets/Biosignalverarbeitung-Pulsamplitudenmodulation.png} + %\item \includegraphics[width=.5\linewidth]{Assets/Biosignalverarbeitung-Pulsamplitudenmodulation.png} \item Die Abtastung (sample \& hold) entspricht nachrichtentechnisch der PAM: die Werte treten in definierten Abständen entsprechend dem Pulsbreite (Abtastperiode) auf und haben einen kontinuierlichen Wertebereich. Allerdings spielt diese Modulationsart in der Nachrichtentechnik keine praktische Rolle, wichtig ist sie für die Theorie. \end{itemize*} @@ -1325,7 +1121,7 @@ \begin{itemize*} \item \includegraphics[width=.5\linewidth]{Assets/Biosignalverarbeitung-Mehrkanalsysteme.png} \item Oft werden in der Technik und vor allem in der Medizin mehrkanalige Messsysteme benötigt. Für die Analyse ist von entscheidender Bedeutung, dass der zeitliche Zusammenhang der Kanäle identisch, oder zumindest bekannt ist. Beim echten Simultansampling werden alle Kanalsignale zum selben Zeitpunkt abgetastet und sequentiell digitalisiert. Damit reicht im Normalfall ein ADC für alle Kanäle. - \item \includegraphics[width=.5\linewidth]{Assets/Biosignalverarbeitung-Mehrkanalsysteme2.png} + %\item \includegraphics[width=.5\linewidth]{Assets/Biosignalverarbeitung-Mehrkanalsysteme2.png} \begin{itemize*} \item alle Kanäle im selben Augenblick abgetastet, echtes Simultansampling \end{itemize*} @@ -1333,21 +1129,19 @@ \begin{itemize*} \item Um den HW-Aufwand zu minimieren, werden die Kanalsignale sequentiell abgetastet und digitalisiert, dabei kann die Einsparung an Elektronik beachtlich sein. Signalanalytisch kann sie jedoch problematisch sein: Aus der Signalsequenz wird das Simultansignal über die Laufzeitkorrektur in der FFT zurückgerechnet. Bei zeitkritischen Vorgängen ist diese Alternative zu verwerfen, da die durch die sequentielle Abtastung verlorengegangenen Signalteile durch Rückrechnung nicht mehr zu retten sind. \end{itemize*} - \item \includegraphics[width=.5\linewidth]{Assets/Biosignalverarbeitung-Mehrkanalsysteme4.png} + %\item \includegraphics[width=.5\linewidth]{Assets/Biosignalverarbeitung-Mehrkanalsysteme4.png} \begin{itemize*} - \item Versatz der Kanäle um \$T\_A/N\$ + \item Versatz der Kanäle um $T_A/N$ \item Rechnerische Korrektur der Abtastzeit (nicht-online-fähig) - \item \$X\^{}*(j\textbackslash omega)=X(j\textbackslash omega)\^{}\{j\textbackslash omega T\_A/N\}\$ + \item $X^{}*(j\omega)=X(j\omega)^{j\omega T_A/N}$ \end{itemize*} \end{itemize*} \subsection{Digitalisierung}\label{digitalisierung} - \subsubsection{Prinzipien der AD Wandlung}\label{prinzipien-der-ad-wandlung} Einrampenverfahren, Single-Slope-Conversion - \begin{itemize*} \item \includegraphics[width=.5\linewidth]{Assets/Biosignalverarbeitung-Einrampenverfahren.png} \item Ur steigt aus dem negativen Bereich an, kreuzt es die Null, so wird K1 positiv. Da K2 noch positiv ist, da Ur unterhalb von Ue liegt, öffnet das Äquivalenzgatter = das Tor \& und der Zähler beginnt zu zählen. @@ -1356,7 +1150,6 @@ \end{itemize*} Zweirampenverfahren, Dual-Slope-Conversion - \begin{itemize*} \item \includegraphics[width=.5\linewidth]{Assets/Biosignalverarbeitung-Zweirampenverfahren.png} \item In Phase 1 Aufladung durch die Eingangsspannung Ue über konstante Zeit. Damit ist der integrierte Wert proportional zur Ue. @@ -1487,11 +1280,11 @@ \begin{itemize*} \item \includegraphics[width=.5\linewidth]{Assets/Biosignalverarbeitung-iir-1.png} - \item \$y(t)=g(t)*x(t)=\textbackslash int\_\{-\textbackslash infty\}\^{}\{\textbackslash infty\} g(\textbackslash tau)x(t-\textbackslash tau) d\textbackslash tau\$ - \item \$Y(j\textbackslash omega)=G(j\textbackslash omega)* X(j\textbackslash omega)\$ + \item $y(t)=g(t)*x(t)=\int_{-\infty}^{\infty} g(\tau)x(t-\tau) d\tau$ + \item $Y(j\omega)=G(j\omega)* X(j\omega)$ \item \includegraphics[width=.5\linewidth]{Assets/Biosignalverarbeitung-iir-2.png} - \item \$g(t)=\textbackslash frac\{1\}\{\textbackslash tau\}exp(-t/\textbackslash tau)\$ - \item \$G(j\textbackslash omega)=\textbackslash frac\{1\}\{1+j\textbackslash omega\textbackslash tau\}\$ + \item $g(t)=\frac\{1\}\{\tau\}exp(-t/\tau)$ + \item $G(j\omega)=\frac\{1\}\{1+j\omega\tau\}$ \item Im analogen Zeitbereich ergibt sich der Filterausgang aus der Faltung der Impulsantwort mit dem Eingangssignal. Entsprechend der FT ist dies äquivalent der Multiplikation von Spektren im f-Bereich. \item Am Beispiel eines TP erster Ordnung soll veranschaulicht werden, wie ein digitaler IIR entworfen werden kann: Die Impulsantwort ist eine fallende e-Funktion, im f-Bereich ein T1-Glied. \end{itemize*} @@ -1501,38 +1294,38 @@ \begin{itemize*} \item \includegraphics[width=.5\linewidth]{Assets/Biosignalverarbeitung-iir-3.png} \begin{itemize*} - \item \$q\_0=5, q=exp(-T/\textbackslash tau) = 0,61\$ + \item $q\_0=5, q=exp(-T/\tau) = 0,61$ \end{itemize*} \item zeitdiskret (nach Abtastung) \begin{itemize*} - \item \$g\_a(t)=q\_0(\textbackslash delta(t)+q\textbackslash delta(t-T)+q\^{}2\textbackslash delta(t-2T)+...)\$ - \item \$FT\{\textbackslash delta(t-T)\}=e\^{}\{-j\textbackslash omega T\}\$ - \item \$G\_a(j\textbackslash omega)=q\_0(1+qe\^{}\{-j\textbackslash omega T\}+ q\^{}2e\^{}\{-2j\textbackslash omega T\}...)\$ - \item \$G\_a(j\textbackslash omega)=q\_0\textbackslash frac\{1\}\{1-qe\^{}\{-j\textbackslash omega T\}\}\$ - \item Z-Transformation \$z=e\^{}\{j\textbackslash omega T\} \textbackslash Rightarrow G(z)=q\_0\textbackslash frac\{1\}\{1-qz\^{}\{-t\}\}\$ Verzögerung um \$T\_A\$ + \item $g_a(t)=q_0(\delta(t)+q\delta(t-T)+q^2\delta(t-2T)+...)$ + \item $FT\{\delta(t-T)\}=e^{-j\omega T}$ + \item $G_a(j\omega)=q_0(1+qe^{-j\omega T}+ q^2e^{-2j\omega T}...)$ + \item $G_a(j\omega)=q_0\frac{1}{1-qe^{-j\omega T}}$ + \item Z-Transformation $z=e^{j\omega T} \Rightarrow G(z)=q_0\frac{1}{1-qz^{-t}}$ Verzögerung um $T_A$ \end{itemize*} \item In Anlehnung an die sog. Impulsantwort-Invariant-Technik wird die IR abgetastet, es liegt also zeitdiskrete Version der IR vor. Diese lässt sich als exponentielle Folge beschreiben. \item Die FT der Zeitverschiebung ist bekannt, damit gibt es ein Äquivalent der Reihe im f-Bereich. \item Über Näherungsrechnung lässt sich die geometrische Folge im f-Bereich zu einem Quotienten zusammenfassen. \item Nun wird der Übergang aus zeitanalogem in den zeitdiskreten Bereich vollzogen -über die z-Transformation. \item Aus Sicht der Realisierung des digitalen Filters ist die z-Transformation besonders anschaulich -der Exponent über z gibt die Anzahl der Einheitsverzögerungen als Vielfaches der Abtastperiode an. - \item Filter der Ordnung N: \$G(z)=\textbackslash frac\{k\_1\}\{1-q\_1z\^{}\{-1\}\}+\textbackslash frac\{k\_2\}\{1-q\_2z\^{}\{-1\}\}+...+\textbackslash frac\{k\_N\}\{1-q\_Nz\^{}\{-1\}\}\$ - \item gemeinsamer Nenner: \$G(z)=\textbackslash frac\{b\_0+g\_1z\^{}\{-1\}+...+b\_mz\^{}\{-m\}\}\{1+a\_1z\^{}\{-1\}+a\_2z\^{}\{-2\}+...+a\_nz\^{}\{-n\}\}=\textbackslash frac\{Y(z)\}\{X(z)\}\$ - \item \$Y(z)+a\_1z\^{}\{-1\} Y(z)+... =b\_0X(z)+b\_1z\^{}\{-1\}X(z)+...\$ - \item zeitkontinuierlich \$y(t)+a\_1y(t-T)+... = b\_0x(t)+b\_1x(t-T)+...\$ - \item Sequenz: \$y(n)=b\_0x(n) +b\_1x(n-1)+.... -a\_1y(n-1)-a\_2y(n-2)-...\$ + \item Filter der Ordnung N: $G(z)=\frac{k_1}{1-q_1z^{-1}}+\frac{k_2}{1-q_2z^{-1}}+...+\frac{k_N}{1-q_Nz^{-1}}$ + \item gemeinsamer Nenner: $G(z)=\frac{b_0+g_1z^{-1}+...+b_mz^{-m}}{1+a_1z^{-1}+a_2z^{-2}+...+a_nz^{-n}}=\frac{Y(z)}{X(z)}$ + \item $Y(z)+a_1z^{-1} Y(z)+... =b_0X(z)+b_1z^{-1}X(z)+...$ + \item zeitkontinuierlich $y(t)+a_1y(t-T)+... = b_0x(t)+b_1x(t-T)+...$ + \item Sequenz: $y(n)=b_0x(n) +b_1x(n-1)+.... -a_1y(n-1)-a_2y(n-2)-...$ \item Nun wird das Vorgehen auf ein beliebiges Filter verallgemeinert: Natürlich sind die IIR im allgemeinen viel komplizierter, als ein TP 1. Ordnung. Jedoch lässt sich jede IR auf eine Summe von abklingenden e-Funktionen zurückführen (siehe Regelung), so dass die Verallgemeinerung nach diesem Schema möglich ist. \item Bringt man die Formel auf den gemeinsamen Nenner, so kann man sie in Terme für den Eingang x und den Ausgang y trennen. \item Transformiert man die Formel in den zeitanalogen Bereich zurück, so lässt sich überprüfen, ob die gewünschte Übertragungsfunktion erreicht wurde. - \item Überträgt man die Formel in den zeitdiskreten Bereich, so erhält man eine Rekursionsformel für den Ausgang \$y(n)\$. + \item Überträgt man die Formel in den zeitdiskreten Bereich, so erhält man eine Rekursionsformel für den Ausgang $y(n)$. \end{itemize*} IIR-Filter rekursiv \begin{itemize*} - \item \$y(n)=\textbackslash sum\_\{i=0\}\^{}\{N\} b\_ix(n-i) - \textbackslash sum\_\{i=1\}\{N\} a\_iy(n-i)\$ + \item $y(n)=\sum_{i=0}^{N} b_ix(n-i) - \sum_{i=1}{N} a_iy(n-i)$ \item \includegraphics[width=.5\linewidth]{Assets/Biosignalverarbeitung-iir-rekursiv.png} - \item \$y(n)=\textbackslash sum\_\{i=0\}\^{}N b\_ix(n-i) - \textbackslash sum\_\{i=1\}\^{}N a\_iy(n-i) = b\_0x(n)+ \textbackslash sum\_\{i=1\}\^{}N {[}b\_ix(n-i) - a\_iy(n-i){]}\$ + \item $y(n)=\sum_{i=0}^N b_ix(n-i) - \sum_{i=1}^N a_iy(n-i) = b_0x(n)+ \sum_{i=1}^N [b_ix(n-i) - a_iy(n-i)]$ \item \includegraphics[width=.5\linewidth]{Assets/Biosignalverarbeitung-iir-rekursiv-2.png} \end{itemize*} @@ -1540,11 +1333,11 @@ \begin{enumerate*} \def\labelenumi{\arabic{enumi}.} - \item Übertragungsfunktion \$G(j\textbackslash omega)\$ des Analogfilters - \item Kontinuierliche Impulsantwort \$g(t)\$ des Analogfilters - \item Aus der kontinuierlichen Impulsantwort \$g(t)\$ abgetastete Impulsantwort \$g(nT)\$ - \item Aus der Reihe für \$g(nT)\$ die z-Übertragungsfunktion \$G(z)\$ des gesuchten IIR-Filters - \item Aus \$G(z)\$ durch Rücktransformation in den Zeitbereich die Rekursionsformel für das Ausgangssignal \$y(n)\$ + \item Übertragungsfunktion $G(j\omega)$ des Analogfilters + \item Kontinuierliche Impulsantwort $g(t)$ des Analogfilters + \item Aus der kontinuierlichen Impulsantwort $g(t)$ abgetastete Impulsantwort $g(nT)$ + \item Aus der Reihe für $g(nT)$ die z-Übertragungsfunktion $G(z)$ des gesuchten IIR-Filters + \item Aus $G(z)$ durch Rücktransformation in den Zeitbereich die Rekursionsformel für das Ausgangssignal $y(n)$ \end{enumerate*} Entwurf eines IIR-Filter - Beispiel @@ -1592,14 +1385,14 @@ \subsection{FIR-Filter}\label{fir-filter} \begin{itemize*} - \item kein rekursiver Anteil: \$G(z)=\textbackslash frac\{b\_0+b\_1z\^{}\{-1\}+...+b\_mz\^{}\{-m\}\}\{1\}=ßfrac\{Y(z)\}\{X(z)\}\$ - \item zeitdiskrete Realisierung: \$Y(z)=b\_0X(z)+b\_1z\^{}\{-1\}X(z)+...\$ - \item analoge Faltung: \$y(t)=b\_0x(t)+b\_1x(t-T)+...\$ - \item diskrete Faltung: \$y(n)=b\_0x(n)+b\_1x(n-1)+...\$ - \item Filterkoeffizienten gleich der abgetasteten Impulsantwort: \$g(t)=b\_0\textbackslash delta(t)+b\_1\textbackslash delta(t-T)+...+b\_L\textbackslash delta(t-NT)\$ + \item kein rekursiver Anteil: $G(z)=\frac{b_0+b_1z^{-1}+...+b_mz^{-m}}{1}=\frac{Y(z)}{X(z)}$ + \item zeitdiskrete Realisierung: $Y(z)=b_0X(z)+b_1z^{-1}X(z)+...$ + \item analoge Faltung: $y(t)=b_0x(t)+b_1x(t-T)+...$ + \item diskrete Faltung: $y(n)=b_0x(n)+b_1x(n-1)+...$ + \item Filterkoeffizienten gleich der abgetasteten Impulsantwort: $g(t)=b_0\delta(t)+b_1\delta(t-T)+...+b_L\delta(t-NT)$ \end{itemize*} - \$\$y(n)=\textbackslash sum\_\{i=0\}\^{}N b\_ix(n-i)\$\$ + $$y(n)=\sum_{i=0}^N b_ix(n-i)$$ \begin{itemize*} \item \includegraphics[width=.5\linewidth]{Assets/Biosignalverarbeitung-fir-nichtrekursiv.png} @@ -1609,12 +1402,12 @@ \begin{enumerate*} \def\labelenumi{\arabic{enumi}.} - \item Definition des Freqzenzgangs \$G(j\textbackslash omega)\$ eines idealen analogen Filters - \item Berechnung der Impulsantwort \$g(t)\$ des analogen Filters - \item Abtastung der Impulsantwort \$g(nT)\$ des idealen FIR-Filters - \item Definition eines Fenstertyps (Rechteck, Hanning, Hamming) und Begrenzung der Impulsantwort \$g(nT)\$ durch das Fenster + \item Definition des Freqzenzgangs $G(j\omega)$ eines idealen analogen Filters + \item Berechnung der Impulsantwort $g(t)$ des analogen Filters + \item Abtastung der Impulsantwort $g(nT)$ des idealen FIR-Filters + \item Definition eines Fenstertyps (Rechteck, Hanning, Hamming) und Begrenzung der Impulsantwort $g(nT)$ durch das Fenster \item Verschiebung der Impulsantwort so, dass der Filter kausal wird - \item Die Filterkoeffizienten \$b\_i\$ sind identisch mit den Werten der begrenzten und verschobenen Impulsantwort + \item Die Filterkoeffizienten $b\_i$ sind identisch mit den Werten der begrenzten und verschobenen Impulsantwort \end{enumerate*} Eigenschaften von FIR-Filtern @@ -1630,7 +1423,7 @@ \begin{itemize*} \item \includegraphics[width=.5\linewidth]{Assets/Biosignalverarbeitung-fir-tiefpass-2.png} \end{itemize*} - \item Digitaler Tiefpass der Länge \$L=501\$ + \item Digitaler Tiefpass der Länge $L=501$ \begin{itemize*} \item \includegraphics[width=.5\linewidth]{Assets/Biosignalverarbeitung-fir-tiefpass-3.png} \item Durch die Beschneidung der IR-Länge weicht die Filtercharakteristik in Abhängigkeit von der tatsächlichen Länge ab. Hier wird die gewünschte Grenzfrequenz mit 501 Filterkoeffizienten noch ganz gut erreicht (bis ca. -40dB). @@ -1646,7 +1439,7 @@ \begin{itemize*} \item ideale Phase identisch Null - nullphasiger Filter, nur off-line, kausales Filter um halbe Länge zeitverschoben - \item konstante Gruppenlaufzeit - lineare Phase, on-line-fähig: \$\textbackslash phi(\textbackslash omega)=-\textbackslash omega\textbackslash tau\$ + \item konstante Gruppenlaufzeit - lineare Phase, on-line-fähig: $\phi(\omega)=-\omega\tau$ \item definierter Phasenverlauf - Allpass \item In der echtzeitfähigen Signalverarbeitung mit FIR beträgt die Gruppenlaufzeit die halbe Filterlänge, und das unabhängig von der Frequenz(!). Dies wird deutlich, wenn man sich die kanonische Form anschaut. Damit ist gewährleistet, dass der Phasenfrequenzgang linear ist und es zu keinen Formverzerrungen kommt. \end{itemize*} @@ -1656,8 +1449,8 @@ \begin{itemize*} \item Dieses Beispiel eines Tiefpasses mit 63 Filterkoeffizienten zeigt ein realisierbares Filter. \item Im Zeitverlauf des EKG vor (blau) und nach (rot) der Filterung ist die durch die halbe Filterlänge verursachte Verzögerung gut erkennbar. Für Patientenmonitoring wäre eine solche Verzögerung akzeptabel, für Aufgaben der Echtzeitanalyse z.B. im Herzschrittmacher nicht mehr. - \item Gruppenlaufzeit: \$\textbackslash tau(\textbackslash omega)=L\emph{T\_A=31}T\_A\$ - \item Phasenfrequenzband: \$\textbackslash phi(\textbackslash omega)=-2\textbackslash pi*LT\_A\$ + \item Gruppenlaufzeit: $\tau(\omega)=L\emph{T\_A=31}T\_A$ + \item Phasenfrequenzband: $\phi(\omega)=-2\pi*LT\_A$ \end{itemize*} @@ -1766,23 +1559,23 @@ sound(winfilt, Fs) \begin{itemize*} \item \includegraphics[width=.5\linewidth]{Assets/Biosignalverarbeitung-adaptiv-fir.png} \begin{itemize*} - \item FIR-Filterlänge = \$2L+1\$ - \item Filterausgang: \$y(n)=w(-L)x(n-L)+...+ w(L)x(n+L)= \textbackslash overline\{x\}\^{}T \textbackslash overline\{w\}\$ - \item Modellfunktion, Sollsignal, desired response: \$d(n)\$ - \item Fehlersignal, error: \$e(n)=d(n)-y(n)\$ + \item FIR-Filterlänge = $2L+1$ + \item Filterausgang: $y(n)=w(-L)x(n-L)+...+ w(L)x(n+L)= \overline{x}^T \overline{w}$ + \item Modellfunktion, Sollsignal, desired response: $d(n)$ + \item Fehlersignal, error: $e(n)=d(n)-y(n)$ \end{itemize*} \item Das adaptive Filter ist ein Rückgekoppeltes System mit negativer Rückkopplung, so dass -ähnlich wie bei Regelkreisen -auch Stabilitätsbedingungen eingehalten werden müssen. - \item Die einfachste Variante eines adaptiven Filters (AF) ist ein FIR mit der Länge \$2L+1\$, das man mathematisch mit einem Vektor w beschreiben kann. Grundsätzlich kann der Eingangsvektor x physisch ein Spaltenvektor sein, dann entsprechend die Filterkoeffizienten der Wichtung von Signalen in parallel liegenden Kanälen, üblich in der spatialen Signalverarbeitung, kommt im nächsten Kapitel. Oder x ist ein Zeilenvektor, d.h. er wird als Analysefenster temporal über ein Signal geschoben, ist also ein Filter im üblichen Sinne der temporalen Filterung. Die physikalische Anordnung ist jedoch für die Herleitung an dieser Stelle irrelevant, im weiteren gehen wir wegen der einheitlichen Schreibweise von einem Spaltenvektor aus, wie üblich in der Signalverarbeitung. + \item Die einfachste Variante eines adaptiven Filters (AF) ist ein FIR mit der Länge $2L+1$, das man mathematisch mit einem Vektor w beschreiben kann. Grundsätzlich kann der Eingangsvektor x physisch ein Spaltenvektor sein, dann entsprechend die Filterkoeffizienten der Wichtung von Signalen in parallel liegenden Kanälen, üblich in der spatialen Signalverarbeitung, kommt im nächsten Kapitel. Oder x ist ein Zeilenvektor, d.h. er wird als Analysefenster temporal über ein Signal geschoben, ist also ein Filter im üblichen Sinne der temporalen Filterung. Die physikalische Anordnung ist jedoch für die Herleitung an dieser Stelle irrelevant, im weiteren gehen wir wegen der einheitlichen Schreibweise von einem Spaltenvektor aus, wie üblich in der Signalverarbeitung. \item Der Ausgang entspricht der Faltung des Filtervektors mit dem Eingangssignal im Punkt n, jeweils L samples nach links und rechts bzw. nach oben und unten. - \item \$d(n)\$ ist die desired response, regelungstechnisch das Sollsignal oder analytisch das Modell. - \item Aus der Differenz von \$d(n)\$ und \$y(n)\$ wird das Errorsignal gebildet, das von einem Adaptionsalgorithmus ausgewertet wird und die Filterkoeffizienten dann von dem Algorithmus so verändert, dass der Fehler gegen Null konvergiert. - \item Errorsignal (Zeitindex weggelassen, daher auch spatial gültig): \$e=d-y=d-\textbackslash overline\{x\}\^{}T \textbackslash overline\{w\}\$ - \item Quadrat des Errorsignals: \$e\^{}2=d\^{}2-2d\textbackslash overline\{x\}\^{}T\textbackslash overline\{w\}+\textbackslash overline\{w\}\^{}T\textbackslash overline\{x\}\textbackslash overline\{x\}\^{}T\textbackslash overline\{w\}\$ - \item Erwartungswert: \$F=E\{e\^{}2\}=E\{d\^{}2\}-2E\{d\textbackslash overline\{x\}\^{}T\textbackslash overline\{w\}\}+E\{\textbackslash overline\{w\}\^{}T\textbackslash overline\{x\}\textbackslash overline\{x\}\^{}T\textbackslash overline\{w\}\}\$ - \item Wiener Filter: \$E\{d\textbackslash overline\{x\}\}=\textbackslash overline\{w\}E\{\textbackslash overline\{x\}\textbackslash overline\{x\}\^{}T\}\$ + \item $d(n)$ ist die desired response, regelungstechnisch das Sollsignal oder analytisch das Modell. + \item Aus der Differenz von $d(n)$ und $y(n)$ wird das Errorsignal gebildet, das von einem Adaptionsalgorithmus ausgewertet wird und die Filterkoeffizienten dann von dem Algorithmus so verändert, dass der Fehler gegen Null konvergiert. + \item Errorsignal (Zeitindex weggelassen, daher auch spatial gültig): $e=d-y=d-\overline{x}^T \overline{w}$ + \item Quadrat des Errorsignals: $e^2=d^2-2d\overline{x}^T\overline{w}+\overline{w}^T\overline{x}\overline{x}^T\overline{w}$ + \item Erwartungswert: $F=E\{e^2\}=E\{d^2\}-2E\{d\overline{x}^T\overline{w}\}+E\{\overline{w}^T\overline{x}\overline{x}^T\overline{w}\}$ + \item Wiener Filter: $E\{d\overline{x}\}=\overline{w}E\{\overline{x}\overline{x}^T\}$ \begin{itemize*} - \item \$\textbackslash overline\{w\}=R\^{}\{-1\}*p\$, R = Autokovarianzmatrix, p = Kreuzkovarianzvektor - \item \$W=\textbackslash frac\{p\_\{xd\}\}\{p\_\{xx\}\}\$, \$p\_xd=\$ Kreuzleistungsdichte, \$p\_\{xx\}=\$ Autoleistungsdichte + \item $\overline{w}=R^{-1}*p$, R = Autokovarianzmatrix, p = Kreuzkovarianzvektor + \item $W=\frac{p_{xd}}{p_{xx}}$, $p_{xd}=$ Kreuzleistungsdichte, $p_{xx}=$ Autoleistungsdichte \end{itemize*} \item Das Errorsignal ist ein Skalar, ergibt sich aus der Differenz der desired response (auf der Position n) und dem Skalarprodukt des Eingangsvektors mit den Filtervektor (Filterkoeffizienten) \item Man geht hier vom stationären Fall der Signalstatistik aus, so dass primär nicht der Momentanwert des Errors Null sein soll, sondern seine Energie bzw. Leistung. Dazu wird der Error zunächst quadriert (zweite Potenz ist Maß für Energie bzw. Leistung). @@ -1790,22 +1583,22 @@ sound(winfilt, Fs) \item Man bildet die erste Ableitung des Fehlers F nach den Filterkoeffizienten (w für weights) und setzt diese gleich Null. Die Lösung dieser Gleichung ergibt das Wiener-oder Optimalfilter. \item Das Wienerfilter kann im Originalbereich mit Hilfe von Auto-und Kreuzkovarianzen beschrieben werden, \item oder im Spektralbereich mit Auto-und Kreuzleistungsdichte (siehe Regelungstechnik und Modellbildung) - \item Schätzung des Erwartungswertes des Fehlerquadrats: \$E\{e\^{}2\}\textbackslash approx \textbackslash frac\{1\}\{M\}\textbackslash sum\_\{i=1\}\^{}M e\_i\^{}2\$ - \item Schätzung des Erwartungswertes der Autokorrelationsmatrix: \$R=E\{x*x\^{}T\}\textbackslash approx\textbackslash begin\{pmatrix\} x(0)x(0) \&...\& x(0)x(M-1)\textbackslash{} ...\textbackslash{} x(M-1)x(0)\& ...\& x(M-1)x(M-1)\textbackslash end\{pmatrix\}\$ - \item Schätzung des Erwartungswertes des Kreuzkorrelationsvektors: \$p=E\{dx\}\textbackslash approx dx\$ + \item Schätzung des Erwartungswertes des Fehlerquadrats: $E\{e^2\}\approx \frac{1}{M}\sum_{i=1}^M e_i^2$ + \item Schätzung des Erwartungswertes der Autokorrelationsmatrix: $R=E\{x*x^T\}\approx\begin{pmatrix} x(0)x(0) &...& x(0)x(M-1)\\ ...\\ x(M-1)x(0)& ...& x(M-1)x(M-1)\end{pmatrix}$ + \item Schätzung des Erwartungswertes des Kreuzkorrelationsvektors: $p=E\{dx\}\approx dx$ \item kontinuierliche Zeit \begin{itemize*} - \item Kreuzkorrelationsfunktion: \$r\_\{xd\}(\textbackslash tau)=lim\_\{T\textbackslash rightarrow\textbackslash infty\}\textbackslash frac\{1\}\{2T\}\textbackslash int\_\{-T\}\^{}T x(\textbackslash tau)d(t+\textbackslash tau)dt\$ - \item Autokorrelationsfunktion: \$r\_\{xx\}(\textbackslash tau)=r\_\{xd\}(\textbackslash tau)\textbar\_\{d=x\}\$ - \item Kreuzleistungsdichte: \$S\_\{xd\}(f)=\textbackslash int\_\{-\textbackslash infty\}\^{}\{\textbackslash infty\} r\_\{xd\}(\textbackslash tau)e\^{}\{-i2\textbackslash pi ft\}dt\$ - \item Autoleistungsdichte: \$S\_\{xx\}(f)=S\_\{xd\}(f)\textbar\_\{d=x\}\$ + \item Kreuzkorrelationsfunktion: $r_{xd}(\tau)=lim_{T\rightarrow\infty}\frac{1}{2T}\int_{-T}^T x(\tau)d(t+\tau)dt$ + \item Autokorrelationsfunktion: $r_{xx}(\tau)=r_{xd}(\tau)|_{d=x}$ + \item Kreuzleistungsdichte: $S_{xd}(f)=\int_{-\infty}^{\infty} r_{xd}(\tau)e^{-i2\pi ft}dt$ + \item Autoleistungsdichte: $S_{xx}(f)=S_{xd}(f)|_{d=x}$ \end{itemize*} \item diskrete Zeit \begin{itemize*} - \item Kreuzkorrelationsfunktion: \$r\_\{xd\}(m)=\textbackslash frac\{1\}\{N\}\textbackslash sum\_\{n=1\}\^{}N x(n)d(n+m)\$ - \item Autokorrelationsfunktion: \$r\_\{xx\}(m)=r\_\{xd\}(m)\textbar\_\{d=x\}\$ - \item Kreuzleistungsdichte: \$S\_\{xd\}(k)=\textbackslash frac\{1\}\{M\}\textbackslash sum\_\{m=0\}\^{}\{M-1\} r\_\{xd\}(m)e\^{}\{-i2\textbackslash pi km\}\$ - \item Autoleistungsdichte: \$S\_\{xx\}(k)=S\_\{xd\}(k)\textbar\_\{d=x\}\$ + \item Kreuzkorrelationsfunktion: $r_{xd}(m)=\frac{1}{N}\sum_{n=1}^N x(n)d(n+m)$ + \item Autokorrelationsfunktion: $r_{xx}(m)=r_{xd}(m)|_{d=x}$ + \item Kreuzleistungsdichte: $S_{xd}(k)=\frac{1}{M}\sum_{m=0}^{M-1} r_{xd}(m)e^{-i2\pi km}$ + \item Autoleistungsdichte: $S_{xx}(k)=S_{xd}(k)|_{d=x}$ \end{itemize*} \end{itemize*} @@ -1827,7 +1620,7 @@ sound(winfilt, Fs) Stochastischer Prozess: Ensemble, Sequenz von Zufallsvariablen \begin{itemize*} - \item \$X=\{X(n-m),...,X(n),....,X(n+m)\}\$ + \item $X=\{X(n-m),...,X(n),....,X(n+m)\}$ \item \includegraphics[width=.5\linewidth]{Assets/Biosignalverarbeitung-adaptiver-filter-stochastik.png} \item Starke Stationarität: die Verteilungen der Zufallsvariablen sind identisch \item Die Annahme der starken Stationarität ist zwar für viele Methoden der Signalstatistik notwendig (i.i.d. = independent identically distributed). Sie kann aber pratkisch nicht erfüllt bzw. geprüft werden. @@ -1836,9 +1629,9 @@ sound(winfilt, Fs) Schwache Stationarität \begin{itemize*} - \item \$E\{x\_t\}=\textbackslash mu\$ - \item \$var(x\_t)\textless\textbackslash infty\$ - \item \$cov(x\_\{t1\}, x\_\{t2\})\$ + \item $E\{x_t\}=\mu$ + \item $var(x_t)<\infty$ + \item $cov(x_{t1}, x_{t2})$ \item Da die Annahme der Gleichheit von Verteilungen der Zufallsgrößen real nicht geprüft werden kann, wird sie auch nicht gefordert. Faktisch müssen nur die Momente erster und zweiter Ordnung zeitlich konstant sein. \item Dies wiederum ist für die signalanalytische Praxis oft zu wenig, da Momente dritter und vierter Ordnung nicht gleich sein müssen (Schiefe, Exzess). \end{itemize*} @@ -1846,12 +1639,12 @@ sound(winfilt, Fs) Praktikable Koeffizientenberechnung - LMS \begin{itemize*} - \item alternativer Weg zum Fehlerminimum - über den Gradienten: \$\textbackslash Delta\_j=\textbackslash frac\{\textbackslash delta F(\textbackslash bar\{w\})\}\{\textbackslash delta \textbackslash bar\{w\}\}\textbar\_\{w=w\_j\}\$ - \item Schätzung des Gradienten über den aktuellen Wert: \$\textbackslash hat\{\textbackslash Delta\}\emph{j=\textbackslash frac\{\textbackslash delta(e\_j\^{}2(\textbackslash bar\{w\}))\}\{\textbackslash delta \textbackslash bar\{w\}\}\textbar{}}\{w=w\_j\} =we\_j \textbackslash frac\{\textbackslash delta e\_j\}\{\textbackslash delta \textbackslash bar\{w\}\}\textbar\_\{w=w\_j\}=-2e\_j \textbackslash bar\{x\}\_j\$ - \item Rekursionsformel für Filterkoeffizienten \$\textbackslash bar\{w\}\_\{j+1\}=\textbackslash bar\{w\}\_j + 2\textbackslash mu e\_j \textbackslash bar\{x\}\_j\$ - \item \$\textbackslash mu\$: Adoptionskonstante; \$\textbackslash lambda\_\{max\}\$: größter Eigenwert der Autokovarianzmatrix; \$\textbackslash frac\{1\}\{\textbackslash lambda\_\{max\}\}\textgreater\textbackslash mu \textgreater0\$ - \item praktisch, Obergrenze gegeben durch Signalenergie: \$\textbackslash frac\{1\}\{\textbackslash sum\_\{j=0\}\^{}N x\^{}2(j)\}\textgreater\textbackslash mu \textgreater0\$ - \item Die Fehlerfunktion \$F(w)\$ ist eine (\$2L+1\$ -dimensionale) Parabel, deren Minimum der Optimallösung entspricht. Es gibt mehrere Wege, dieses Minimum zu erreichen. An dieser Stelle leiten wir die Optimallösung mit Hilfe des sehr anschaulichen LMS-Algorithmus her (LMS - Least-Mean-Square, Methode der kleinsten Quadrate). + \item alternativer Weg zum Fehlerminimum - über den Gradienten: $\Delta_j=\frac{\delta F(\bar{w})}{\delta \bar{w}}|_{w=w_j}$ + \item Schätzung des Gradienten über den aktuellen Wert: $\hat{\Delta}_j=\frac{\delta(e_j^2(\bar{w}))}{\delta \bar{w}}|_{w=w_j} =we_j \frac{\delta e_j}{\delta \bar{w}}|_{w=w_j}=-2e_j \bar{x}_j$ + \item Rekursionsformel für Filterkoeffizienten $\bar{w}_{j+1}=\bar{w}_j + 2\mu e_j \bar{x}_j$ + \item $\mu$: Adoptionskonstante; $\lambda\_\{max\}$: größter Eigenwert der Autokovarianzmatrix; $\frac{1}{\lambda_{max}}>\mu >0$ + \item praktisch, Obergrenze gegeben durch Signalenergie: $\frac{1}{\sum_{j=0}^N x^2(j)}>\mu >0$ + \item Die Fehlerfunktion $F(w)$ ist eine ($2L+1$ -dimensionale) Parabel, deren Minimum der Optimallösung entspricht. Es gibt mehrere Wege, dieses Minimum zu erreichen. An dieser Stelle leiten wir die Optimallösung mit Hilfe des sehr anschaulichen LMS-Algorithmus her (LMS - Least-Mean-Square, Methode der kleinsten Quadrate). \item Da der Weg zum Optimum über die inverse Autokovarianzmatrix und über die Leistungsdichten verbaut ist, nähern wir uns dem Minimum der Parabel mit Hilfe des Gradienten iterativ. Der Gradient ist mathematisch über die partiellen Ableitungen der Parameter definiert. Das stößt in der Praxis -vor allem bei der Online-Analyse -bald an Grenzen, da der Erwartungswert über längere Zeit ermittelt werden müsste. \item Daher schätzt man den Gradienten aus dem aktuellen Fehler, faktisch lässt man also die Mittelwertbildung weg. Das kann man unter der Annahme der Stationarität machen. Der geschätzte Gradient ergibt sich dann allein aus dem Produkt des Fehlers und des Eingangsvektors. \item Nun kann man den Gradienten dazu nutzen, mit hinreichend kleinen, durch die Adaptionskonstante bestimmten, Schritten auf das Minimum zu konvergieren. @@ -1862,7 +1655,7 @@ sound(winfilt, Fs) \begin{itemize*} \item \includegraphics[width=.5\linewidth]{Assets/Biosignalverarbeitung-adaptiver-filter-reale-aufnahme.png} - \item Ein reales IPG wurde nachträglich mit additiven simulierten Störungen (50 Hz, 75 Hz, Rauschen) stark gestört (Amplitude der Harmonischen \$A=1\$, Effektivwert des Rauschens 20). + \item Ein reales IPG wurde nachträglich mit additiven simulierten Störungen (50 Hz, 75 Hz, Rauschen) stark gestört (Amplitude der Harmonischen $A=1$, Effektivwert des Rauschens 20). \item Eine Modellfunktion liegt daher vor. \item Das gefilterte Signal erreicht relativ schnell die Qualität des Originals. \item Basiert auf der Übung 5.4 des Buches Biosignalverarbeitung/Elektrische Biosignale in der Medizintechnik @@ -1872,12 +1665,12 @@ sound(winfilt, Fs) \begin{itemize*} \item \includegraphics[width=.5\linewidth]{Assets/Biosignalverarbeitung-adaptiver-filter-aufbau.png} - \item \$x(n)\$: reales Signal - \item \$e(n)\$: Fehler - \item \$d(n)\$: ungestörtes Muster (woher?) - \item \$y(n)\$: Filterausgang - \item Funktion: Im Ausgang \$y(n)\$ erscheinen diejenigen Signalanteile von \$x(n)\$, die gut mit \$d(n)\$ korrelieren - \item Im stationären Fall und nach erfolgter Konvergenz kann man davon ausgehen, dass der Filterausgang diejenigen Anteile von \$x(n)\$ enthält, die mit \$d(n)\$ gut korrelieren. Das funktioniert natürlich nur, wenn man das Mustersignal ganz genau kennt und vorgeben kann. + \item $x(n)$: reales Signal + \item $e(n)$: Fehler + \item $d(n)$: ungestörtes Muster (woher?) + \item $y(n)$: Filterausgang + \item Funktion: Im Ausgang $y(n)$ erscheinen diejenigen Signalanteile von $x(n)$, die gut mit $d(n)$ korrelieren + \item Im stationären Fall und nach erfolgter Konvergenz kann man davon ausgehen, dass der Filterausgang diejenigen Anteile von $x(n)$ enthält, die mit $d(n)$ gut korrelieren. Das funktioniert natürlich nur, wenn man das Mustersignal ganz genau kennt und vorgeben kann. \item Woher aber sollen wir das Mustersignal nehmen, wenn es ja gestört und verrauscht am Filtereingang vorliegt? Fazit ist, ein solches Filter ist nicht realisierbar bzw. macht keinen Sinn, wenn das Muster vorliegt. \end{itemize*} @@ -1885,10 +1678,10 @@ sound(winfilt, Fs) \begin{itemize*} \item \includegraphics[width=.5\linewidth]{Assets/Biosignalverarbeitung-adaptiver-filter-noise-canceller.png} - \item \$ref\$: Rauschreferenz - \item \$err\$: Signal - \item \$prim\$: Signal + Rauschen - \item \$out\$: Rauschen + \item $ref$: Rauschreferenz + \item $err$: Signal + \item $prim$: Signal + Rauschen + \item $out$: Rauschen \item In der praktischen Analyse ist es sehr oft so, dass man zwar kein Mustersignal hat, dafür aber eine Rauschreferenz, d.h. genügend Information über das Störsignal ohne Anteile des gewünschten Signals. Die Aufgabe jetzt heißt also, wenn wir schon kein Mustersignal haben, dann können wir versuchen, die Störung zu beseitigen, wenn wir sie kennen. Im Idealfall ist die Störung eliminiert und das gewünschte Signal bleibt übrig. \item Hierzu werden die Filteranschlüsse umfunktioniert: Der Filtereingang wird zur Rauschreferenz, hier wird die vorliegende Störung eingespeist, die allerdings keine Anteile des gewünschten Signals enthalten darf. Dies ist aber bei technischen Störungen bei Biosignalverarbeitung kein wesentliches Problem. \item Der Eingang für Mustersignal wird zum Primäreingang, in den das gestörte aber noch unbekannte Wunschsignal eingespeist wird. @@ -1900,7 +1693,7 @@ sound(winfilt, Fs) \begin{itemize*} \item IPG gestört periodisch und stochastisch (50Hz, 75 Hz, Rauschen) - \item Startwert für Gewichte \$w(50)\$ und \$w(52)\$ mit \$L=101\$ bei \$(0,0)\$ + \item Startwert für Gewichte $w(50)$ und $w(52)$ mit $L=101$ bei $(0,0)$ \item Variabilität höher bei größerer Adaptionskonstante \item Variabilität niedriger bei kleinerer Adaptionskonstante \item \includegraphics[width=.5\linewidth]{Assets/Biosignalverarbeitung-adaptiver-filter-adaption-anc.png} @@ -1925,7 +1718,7 @@ sound(winfilt, Fs) \begin{itemize*} \item \includegraphics[width=.5\linewidth]{Assets/Biosignalverarbeitung-adaptiver-filter-ekg-abdominal.png} - \item \$\textbackslash mu=0,02\$, \$L=101\$ + \item $\mu=0,02$, $L=101$ \item Der ANC wurde zur Trennung des fötalen EKG (fEKG) vom maternalen mEKG. Zur Gewinnung des fEKG wird das abdominale EKG (aEKG, vom Bauch) benötigt, das mEKG wird konventionell an Extremitäten abgeleitet. Das fEKG (untere Grafik) ist nur schlecht erkennbar und vom mEKG selbst nach zahlreichen empirischen zweidimensionalen Optimierungen der Filterlänge und der Adaptionskonstante noch immer stark gestört. Die Ursache liegt darin, dass beide Signale (Störung und gestörtes Signal) stark instationär sind, die für die Konvergenz zur optimalen Lösung notwendige Bedingung der Stationarität mindestens eines Signalanteils ist hier nicht erfüllt. \end{itemize*} @@ -1936,9 +1729,9 @@ sound(winfilt, Fs) \item In bestimmten Messsituationen (Ruhe EKG vor Fahrradergometrie) liegt eine Musterfunktion (Template) vor. \item Für signalanalytische korrekte Detektion/Filterung müssen Signal (EKG) sowie Template (Muster) weißes Spektrum haben \item Zum Prewhitening wird LMS mit binärem Gradienten verwendet. Die selben Filterkoeffizienten filtern auch das Template für das MF - \item Lineare Prädikation: \$x\_p{[}k{]}=a\_1x{[}k-1{]}+a\_2x{[}k-2{]}+...+a\_nx{[}k-n{]}\$ - \item Residualfehler ist weiß: \$x\_\{err\}{[}k{]}=x{[}k{]}-x\_p{[}k{]}\$ - \item Robuster LMS mit binärem Gradienten: \$w{[}k+1{]}=w{[}k{]}+\textbackslash mu*sng(e{[}k{]}x{[}k{]})\$ + \item Lineare Prädikation: $x_p[k]=a_1x[k-1]+a_2x[k-2]+...+a_nx[k-n]$ + \item Residualfehler ist weiß: $x_{err}[k]=x[k]-x_p[k]$ + \item Robuster LMS mit binärem Gradienten: $w[k+1]=w[k]+\mu*sng(e[k]x[k])$ \end{itemize*} \includegraphics[width=.5\linewidth]{Assets/Biosignalverarbeitung-adaptiver-filter-ekg-roh-weiß.png}

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