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Prof. J. WALTER   Info-numerische.mht  Stand: Dezember 2003
Seite 1
Numer= ische Verarbeitung digitaler Signale
•Glätten
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•drei gemessene Werte
• yn-= 1, yn, yn+= 1
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•       linearer Mittelwert
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àAus einem Satz von N Messwerten erhält man N-= 2 geglättete Werte       
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Glei= tender Mittelwert  - moving average – die neue Kurve wird durch Mittelung von drei Punkten aus der alten Kurve gewonnen.
Hat = ähnliche Wirkung wie ein Tiefpaß.
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Prof. J. WALTER   Info-numerische.mht  S= tand: Dezember 2003
Seite 2 =
Zweimaliges Glätten
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info2<= /span>
Prof. J. WALTER   Info-numerische.mht  S= tand: Dezember 2003
Seite 3 =
Beispiel
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5,87
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5,97
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5,63
-
5,63
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4,94
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1
         aus unsym. Ausgleichsf.
yn
n
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Prof. J. WALTER   Info-numerische.mht  S= tand: Dezember 2003
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Ausgleichsgerade
Bsp.:=
durch folgende 5 Messpaare x<= span style=3D'font-size:75%'>n,yn wird ein Ausgleichspolynom 1. Ordnung gelegt.
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Ausgleichspolynom f(x)=3DP(x)        = ; 

Polynom 1. Ordnun= g
f(x)= =3Da+bx
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Prof. J. WALTER   Info-numerische.mht  S= tand: Dezember 2003
Seite 5 =
Gauß‘sches Prinzip der kleinsten Quadrate
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b
b
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xn
xn+1
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Streng genommen: senkrechter Abstand muss minimiert werden
Praxis: Abzisse wird fehlerfrei angenommen
minimiert wird Ordinaten-Differenz   f(= x)-yn<= /span>
In d= er Praxis ist diese Annahme berechtigt, da der „Jitter“ im Vergleich zum Amplitudenfehler sehr gering ist. In der x-Achse kein Fehler. Der Fehler = ist nur in der Amplitude.
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info2<= /span>
Prof. J. WALTER   Info-numerische.mht  S= tand: Dezember 2003
Seite 6 =
Herleitung Gauß‘sches Fehlerquadrat
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info2<= /span>
Prof. J. WALTER   Info-numerische.mht  S= tand: Dezember 2003
Seite 7 =
Gauß‘sches Fehlerquadrat
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Prof. J. WALTER   Info-numerische.mht  S= tand: Dezember 2003
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Zum Beispiel
Ausglei= chsgerade durch 5 Punkte
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Prof. J. WALTER   Info-numerische.mht  S= tand: Dezember 2003
Seite 9 =
Skizze: Ausgleichspolynom
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info2<= /span>
Prof. J. WALTER   Info-numerische.mht  S= tand: Dezember 2003
Seite 10 =
Ausgl= eichspolynom 3.Ordnung durch 5 Punkte
xn-2
xn-1
xn
xn+1
xn+2
(xn,   )
geglätteter Wert / wahrscheinlichere Wert
n = ;
Stützstellen  k =3D -2, -1, 0, 1, 2
(xn+= k, yn+k)
In d= iesem Fall sind gerade 5 Messwerte vorhanden. Das Verfahren lässt sich aber wie= der moving average für ein Signal mit vielen Messwerten anwenden.=
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info2<= /span>
Prof. J. WALTER   Info-numerische.mht  S= tand: Dezember 2003
Seite 11 =
Polynom 3. Ordnung
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4 Gleichungen
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aus (1) und (3) folgt:
Die ersten beiden und die letzten beiden Punkte wurden nicht geglättet.
Ausgleichspolynom
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info2<= /span>
Prof. J. WALTER   Info-numerische.mht  S= tand: Dezember 2003
Seite 14 =
Unsymmetrische Ausgleichsformel
Dies= e dient zur Berechnung der Randwerte.
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file:///C:/2ACB1965/06-info-numerische-Dateien/slide0126.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Numerische Verarbeitung digitaler Signale
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info2<= /span>
Prof. J. WALTER   Info-numerische.mht  S= tand: Dezember 2003
Seite 15 =
Differenzenquotient
•Messweite yn   Variable xn
•Äquidistanter Abstan xn+= 1-xn=3Dn&= #13;
•Aus Taylorreihe folgt:
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Differenzenquotient aus den zu (xn,yn<= /span>)benachbarten Stellen
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Prof. J. WALTER   Info-numerische.mht  S= tand: Dezember 2003
Seite 16 =
Taylor‘sche Satz
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Prof. J. WALTER   Info-numerische.mht  S= tand: Dezember 2003
Seite 17 =
Quadraturformeln - Integrieren
•Der Name Quadraturformel stammt vom Delischen Problem der Quadratur des Kreises •Quadraturformeln, ermöglichen die numerische Berechnung von Flächeninhalten.
<= span style=3D'mso-special-format:bullet;color:blue;position:absolute;left:-4.11= %; top:.3em;font-size:75%'>•Visualisierung zur numerischen Integration
•
Man = versucht den Flächeninhalt eines Quadrates und den Flächeninhalt eines Kreis= es gleich groß zu machen. Dies ist aufgrund der Transzendenz der Zahl = Pi nicht möglich. Die Visualisierung der numerischen Integrationsverfah= ren ist sehr gut auf den Internetseiten www.mathematik.ch
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Prof. J. WALTER   Info-numerische.mht  S= tand: Dezember 2003
Seite 18 =
Integrieren
x1
x2
x3
x4
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xN
x = ;
y = ;
Gesamt= fläche
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Signale
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info2<= /span>
Prof. J. WALTER   Info-numerische.mht  S= tand: Dezember 2003
Seite 19 =
Trapezregel
x1
x2
x3
x4
x5
xN
y1
y2
A2
A3
h = ;
Formel für Trapezfläche
Gesamtfläche:
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Prof. J. WALTER   Info-numerische.mht  S= tand: Dezember 2003
Seite 20 =
Keplersche Fassregel
<= span style=3D'font-size:75%'>•Durch 3 Stützstellen wird Polynom 2.Ordnung gelegt
<= span style=3D'font-size:75%'>Integrationsintervall von 2h
x1
x2
x3
y1
h = ;
0 = ;
2h = ;
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Prof. J. WALTER   Info-numerische.mht  S= tand: Dezember 2003
Seite 21 =
Herleitung Kepler‘sche Fassre= gel
Ü= ;bung
Sie = haben 3 Messwerte: y1,y2,y3. Durch diese drei Messwerte legen Sie ein Polynom 2. Ordnung. Setzen Sie für x die Werte 0,h,2h ein, erhalten Sie drei Gleichungen für a,b,und c. Die Fläche unter der Kurve lässt sich über das Intergral von 0 bis 2h über y(x) bestimmen. K ist= die Lösung der Integration. In diese Gleichung setzen Sie die drei Gleichungen für a,b,c ein. Sie erhalten als Ergebnis die Kepler‘sche Fassregel.

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Content-Location: file:///C:/2ACB1965/06-info-numerische-Dateien/slide0079_image110.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhOwAMAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAMAAwA2 AAYAgAAAAAAAAAIdhBFxy+0PX5q02ouz3rz7hHziSB1ThKaqo6zuaxQAOx== ------=_NextPart_01C57E34.A03B30C0 Content-Location: file:///C:/2ACB1965/06-info-numerische-Dateien/slide0079_image111.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhAwB0AHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAQAC AHIAgAAAAAAAAAINjI+py+0Po5y02ottAQA7 ------=_NextPart_01C57E34.A03B30C0 Content-Location: file:///C:/2ACB1965/06-info-numerische-Dateien/slide0080.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Numerische Verarbeitung digitaler Signale
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Prof. J. WALTER   Info-numerische.mht  S= tand: Dezember 2003
Seite 22 =
Keplersche Fassregel
2 Streifen
Bemerk= ung:
Bei ge= rader Zahl von Messwerten: Simpson-Regel
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Prof. J. WALTER   Info-numerische.mht  S= tand: Dezember 2003
Seite 23 =
Simpson-Regel
Anst= elle der linearen wählt man hier quadratische Interpolation mit Parabelstücken; weil für das Aufstellen einer Parabelgleichung = drei Punkte nötig sind, muss man eine ungerade Zahl von Stützstellen wählen. Die Simpson-Regel basiert auf der Kepler‘schen Fassreg= el
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Prof. J. WALTER   Info-numerische.mht  S= tand: Dezember 2003
Seite 24 =
3/8-Regel
•Polynom 3.Ordnung = à 3 Streifen
•y(x)=3Da+bx+cx&sup= 2;+dx³
x1
x2
x3
y1
h = ;
0 = ;
2h = ;
x4
3h = ;
A3
drei Streifen
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Prof. J. WALTER   Info-numerische.mht  S= tand: Dezember 2003
Seite 25 =
Newton-Cotes-Formeln
= 4. Ordnung
Poly= gone höherer Ordnung neigen zu Schwingungen.
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Prof. J. WALTER   Info-numerische.mht  S= tand: Dezember 2003
Seite 26 =
Beurteilung der Integrationsverfahr= en
•Welche Verf= ahren wird man in der Praxis einsetzen?
•= Abwägung: Speicherplatz – Prozessorleistung
•= Früher waren Speicherplatz und schnelle A/D-Wandler teuer ->Prozessor + SW •Heute: Integrationsintervall klein -&g= t; Rechteckregel
<= span class=3DBB style=3D'position:absolute;left:-4.86%;top:.3em'>•= Übung: www.mathematik.ch/anwendungenmath/numint  
<= span class=3DBB style=3D'position:absolute;left:-4.86%;top:.3em'>•= Anzahl der Intervalle vergrößern.
Die = MAC-Einheit (Multiplizier- und Akkumuliereinheit) in Signalprozessoren kann in einem Rechenschritt / Operationszyklus gleichzeitig multiplizieren und addieren. Hardware ist angepasst an häufige Rechenoperationen.
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Numerische Integration
-gewöhnliche Differentialgleichungen
Lösbar mit Kepler Fassregel
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Seite 28 =
Inhomogene DGL
DGL
x(t)
y(t)
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Prof. J. WALTER   Info-numerische.mht  S= tand: Dezember 2003
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Herleitung: DGL aus Frequenzbereich=
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Prof. J. WALTER   Info-numerische.mht  S= tand: Dezember 2003
Seite 30 =
Herleitung über Fourierbereich=
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Prof. J. WALTER   Info-numerische.mht  S= tand: Dezember 2003
Seite 31 =
Polygonzugverfahren nach Euler
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info2<= /span>
Prof. J. WALTER   Info-numerische.mht  S= tand: Dezember 2003
Seite 32 =
Übung: Vergleich „Analyt= isch – Numerisch“
<= span class=3DBB style=3D'position:absolute;left:-4.35%;top:.3em'>•= Stellen Sie die beiden Kurven in Excel dar.
•= Eingangsfunktion x[n]=3D1 für 0≤t
 x[n]=3D0 f&uum= l;r 0>t
<= span class=3DBB style=3D'position:absolute;left:-4.86%;top:.3em'>•= RC=3D1; h=3D0,1
<= span class=3DBB style=3D'position:absolute;left:-4.86%;top:.3em'>•= Bereich 0…7
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info2<= /span>
Prof. J. WALTER   Info-numerische.mht  S= tand: Dezember 2003
Seite 33 =
Explizites Polygonzugverfahren - Eu= ler
y0
y(t1)
y1
t0
t1
y(t) = ;
h = ;
h* = ;
dt = ;
Steigung an der linken Grenze
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Prof. J. WALTER   Info-numerische.mht  S= tand: Dezember 2003
Seite 34 =
Implizites Polygonzugverfahren
y0
y(t1)
y1
t0
t1
y(t) = ;
h = ;
h* = ;
dt = ;
Steigung an der rechten Grenze
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info2<= /span>
Prof. J. WALTER   Info-numerische.mht  S= tand: Dezember 2003
Seite 35 =
Trapezverfahren nach Heun
y0
h = ;
t0
t1
t = ;
Integration einer DGL nach Trapezverfahren<= /div>
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Prof. J. WALTER   Info-numerische.mht  S= tand: Dezember 2003
Seite 36 =
Zwei Tiefpässe hintereinander<= /div>
C
Ue
C
U2
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R = ;
R = ;
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Seite 37 =
Z-Transformation
•Kontinuierliche         = ;            &n= bsp; Fourier-Transformation
•Zeitsignale
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•Diskrete Z-Transformation
•Zeitsignale DFT ist Spezialfall der
• Z-Transformation
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Seite 38 =
z-Tranformierte
•Z-Transformation ordnet Zeitsignal x(n Ta<= /span>)=3Dxn
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•Z-Transformierte
 = ;
•Fourier-Transformierte
 = ;
•Diskrete Fourier-Transf.
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•Im allgemeinen gilt:
Funktion        = ;   à Zuordnung à= ; komplexe Variable z
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info2<= /span>
Prof. J. WALTER   Info-numerische.mht  S= tand: Dezember 2003
Seite 39 =
Z-Tranformierte als Abbildung
Beschr&= auml;nkung auf einseitige Z-Transformation
xn<= /span>=3D0 für n<0
insta= bil
stabil=
Z-Ebe= ne
+1
s-Eben= e
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info2<= /span>
Prof. J. WALTER   Info-numerische.mht  S= tand: Dezember 2003
Seite 40 =
Eigenschaften der Z-Transformation<= /div>
= 226;ist ein Polynom von z
•Der Faktor z-n sep= ariert die Funktionswerte voneinander.
•Der Faktor z-n beinhaltet eine Verzögerung u= m n Ta<= /span> von = t=3D0 aus.
•Einheitskreis in der Z-Ebene
•
Fall= s für z die Kreisfunktionen gesetzt werden, ist die z-Tranformierte gleich der DF= T.
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=
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info2<= /span>
Prof. J. WALTER   Info-numerische.mht  S= tand: Dezember 2003
Seite 41 =
Anwendung auf nichtrekursive Filter=
ak
x(n Ta)=3Dx= k<= /span>
y(n Ta)=3Dy= k<= /span>
Blocks= chaltbild eines nicht-rekursiven Filters
Ausgangssignal hängt nur von Werten des Eingangssignals ab. Keine Rückkopplung
à immer stabil
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=
=
info2<= /span>
Prof. J. WALTER   Info-numerische.mht  S= tand: Dezember 2003
Seite 42 =
Zusammenhang: Y(z), X(z), A(z)
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Prof. J. WALTER   Info-numerische.mht  S= tand: Dezember 2003
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Prof. J. WALTER   Info-numerische.mht  S= tand: Dezember 2003
Seite 44 =
Kausaler Filter
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Prof. J. WALTER   Info-numerische.mht  S= tand: Dezember 2003
Seite 45 =
Akausaler Filter
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info2<= /span>
Prof. J. WALTER   Info-numerische.mht  S= tand: Dezember 2003
Seite 46 =
Beispiel für FiR-Filter
1
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2
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y(n*Ta)
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ak
x(n*Ta)
y(n*Ta)
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Prof. J. WALTER   Info-numerische.mht  S= tand: Dezember 2003
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Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhCwAKAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQAJ AAkAgAAAAAAAAAINBIJol8kNo5zUvbbeKgA7 ------=_NextPart_01C57E34.A03B30C0 Content-Location: file:///C:/2ACB1965/06-info-numerische-Dateien/slide0099_image336.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhCgAKAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAQAK AAkAgAAAAAAAAAIRhINokB3nFIvUTXotbGs/UwAAOw== ------=_NextPart_01C57E34.A03B30C0 Content-Location: file:///C:/2ACB1965/06-info-numerische-Dateien/slide0100.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Numerische Verarbeitung digitaler Signale
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Prof. J. WALTER   Info-numerische.mht  S= tand: Dezember 2003
Seite 49 =
Filterkoeffizienten für Kreisfunktionen
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Prof. J. WALTER   Info-numerische.mht  S= tand: Dezember 2003
Seite 50 =
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TP Übertragungsfunktion
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Idealer Tiefpass angenähert durch Koeffizienten ak
gerade Funktion
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Prof. J. WALTER   Info-numerische.mht  S= tand: Dezember 2003
Seite 53 =
Integralsinus
Kleine Übung:
Stellen Sie die Funktion si(x) im Bereich von
-20<x<20 &#= 13;
dar.
Lösungsmöglichkeiten: Maple, Excel, Simu= link, Mathlab,
HPVEE, Taschenrec= hner, plot Taschenrechner
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Prof. J. WALTER   Info-numerische.mht  S= tand: Dezember 2003
Seite 54 =
Beispiel: Berechne Tiefpass
Filter-
gleichung
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Verarbeitung digitaler Signale
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Prof. J. WALTER   Info-numerische.mht  S= tand: Dezember 2003
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Kleine Übung
•Eingangsfun= ktion für Filter Impuls der Breite 5 – Amplitude =3D 1&= #13;•= Berechnen Sie die Ausgangswerte für den TP mit Excel
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Prof. J. WALTER   Info-numerische.mht  S= tand: Dezember 2003
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Filtergleichung
Gleichspannungsverstärkung
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Prof. J. WALTER   Info-numerische.mht  S= tand: Dezember 2003
Seite 57 =
„Kleine Übung“
<= span class=3DBB style=3D'position:absolute;left:-4.43%;top:.3em'>•= Plotten Sie die Filtergleichung für x=3D-3…3
<= span style=3D'visibility:hidden'>•
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Prof. J. WALTER   Info-numerische.mht  S= tand: Dezember 2003
Seite 58 =
Übung:
•Berechnen S= ie die Filterkoeffizienten für einen Hochpass mit:
<= span class=3DBB style=3D'position:absolute;left:-4.86%;top:.3em'>•= fg=3D20Hz
<= span class=3DBB style=3D'position:absolute;left:-4.86%;top:.3em'>•= N=3D3
<= span class=3DBB style=3D'position:absolute;left:-4.86%;top:.3em'>•= fa=3D100Hz
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Prof. J. WALTER   Info-numerische.mht  S= tand: Dezember 2003
Seite 59 =
Tiefpass à Hochpass
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=
=
info2<= /span>
Prof. J. WALTER   Info-numerische.mht  S= tand: Dezember 2003
Seite 60 =
Koeffizienten für Hochpass
-0,06 a3,TP
0,09
a2<= /span>,TP
0,3
a1<= /span>,TP
0,4
a0<= /span>,TP
ak<= /span>,TP
0,06
-a3<= /span>,TP
-0,09
-a2<= /span>,TP
-0,3
- a1,TP
0,6
1-a0<= /span>,TP
ak<= /span>,HP
0
0
0
1
ak<= /span>,AP
+/-3
+/-2
+/-1
0
k
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=
=
info2<= /span>
Prof. J. WALTER   Info-numerische.mht  S= tand: Dezember 2003
Seite 61 =
Tiefpass à Bandpass
TPO<= /span>
TPU<= /span>
BP
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Rekursive Filter
Ordnung des Filters größere Zahl von M oder N
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Prof. J. WALTER   Info-numerische.mht  S= tand: Dezember 2003
Seite 65 =
Blockbild rekursiver Filter
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a0
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+ = ;
yn-1
yn
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Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhPgAKAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAMAAwA6 AAUAgAAAAAAAAAIahBOmy+0PI0Kx2uum3rz7D3IHhZVmlZzqWQAAOw== ------=_NextPart_01C57E34.A03B30C0 Content-Location: file:///C:/2ACB1965/06-info-numerische-Dateien/slide0115.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Numerische Verarbeitung digitaler Signale
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Prof. J. WALTER   Info-numerische.mht  S= tand: Dezember 2003
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Beispiel
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Abtasttheorem
•Bei allen numerischen Filterberechnungen FIR muss das Abtasttheorem eingehalten werden
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Herleitung Berechnung Koeffizienten = FIR
•Optimierung nach „Klein= stes Fehlerquadrat“
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Filter allgemein
• Filter lassen sich nach verschiedenen Kriterien entwerfen. Die Kriterien sind:
1.Welligkeit im Durchlassbereich (Frequenzbereich)
2.Welligkeit im Sperrbereich (Frequenzbereich)
3.Steilheit beim Übergang vom Durchlass- zum Sperrbereich <= /span>(Frequenzbereich)
4.Eignung zur Impulsübertragungsfunktion (Zeitbereich) =
5.
5.Vorsicht: Filterordnung im analogen Bereich nicht mit <= span style=3D'position:absolute;top:62.25%;left:18.35%;width:82.02%;height:4.25= %'>Filterordnung (Taps) im digitalen Bereich gleichse= tzen
wg
|A|
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FIR-Filter - DSP
•Digital Signal Processor
&= #8211;RISC-Prozessor
&#= 8211;Jeder Befehl wird in einem Taktzyklus ausgeführt
•Filtergleichung: Addition und Multiplikation
&= #8211;Koeffizienten multipliziert mit Messwerten
&= #8211;MAC Multiplizier- und Accumuliereinheit
•modifizierte Harvard-Architektur
–im Programmspeicher sind auch Daten – Koeffizienten
–
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Suche nach dem idealen DSP….<= /div>
•Der ideale D= SP kann in einem Taktzyklus: Koeffizienten von Speicher 1 und Messwerte von Speicher 2 einlesen, multiplizieren und Addieren
<= span class=3DBB style=3D'position:absolute;left:-4.86%;top:.3em'>•= BSP: ADSP 21xx
<= span class=3DBB style=3D'position:absolute;left:-4.86%;top:.3em'>•= BSP: TMS 320xx
<= span class=3DBB style=3D'position:absolute;left:-4.86%;top:.3em'>•= BSP: Motorola 56xxx, 96xxx
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Korrelationsmesstechnik
Messgröße x
Messgröße y
Zusammenhang? =3DKorrelation
X Ernteertrag
– Niederschlagsmenge y
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Mittelwert
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Varianz
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Kovarianz
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Korrelationskoeffizient
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Berechnungen in EXCEL
•Analyse Plug-in aktivieren
•Kleine Übung
•Sie haben zwei Messreihen: xn, yn
–xn: 1, 2, 3, 4, 5, 6
–yn: 6,5,4,3,2,1
–<= /span>Berechnen Sie den Korrelationskoeffizienten
–<= /span>Berechnen Sie den Korrelationskoeffizeinten für
–yn: 1,2,3,4,5,6
–<= /span>Berechnen Sie den Korrelationskoeffizeinten für
–yn: 1,2,3,3,2,1
–<= /span>Stellen Sie die Messreihen als Kurven dar.<= /span>
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Ergebnisse
•Korrelationkoeffzient:
&= #8211;+1 vollständige Abhängigkeit
&= #8211;  0 statistisch unabhängig = ;
–-1 vollständige Abhängigkeit aber gegenläufig
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Korrelationsfunktionen
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Prof. J. WALTER   Info-numerische.mht  S= tand: Dezember 2003
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Berechnen Sie die Kreuzkorrelationsfunktion
<= span class=3DBB style=3D'position:absolute;left:-4.86%;top:.3em'>•= 1,2,4,6,3
<= span class=3DBB style=3D'position:absolute;left:-4.86%;top:.3em'>•= 2,3,2,1,3
<= span class=3DBB style=3D'position:absolute;left:-4.86%;top:.3em'>•= Faltung – Kreuzkorrelationsfunktion
<= span style=3D'visibility:hidden'>•
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Prof. J. WALTER   Info-numerische.mht  S= tand: Dezember 2003
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Autokorrelationsfunktion
•Dien Funktion wird mit sich selbst übereinandergeschoben= und multipliziert und aufsummiert
&= #8211;symmetrisch
&#= 8211;Maximum ist bei Verschiebung ττ=3D0
•Beispiel:
–Geschwindigkeitsmessung mit zwei Sensoren
&= #8211;Autokorrelationsfunktion
•
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Kreuzkorrelationsfunktion
<= span class=3DBB style=3D'position:absolute;left:-4.86%;top:.3em'>•= eigentlich nur periodische Funktionen
<= span class=3DBB style=3D'position:absolute;left:-4.33%;top:.3em'>•= nicht periodische Funktionen T->unendlich
•= Funktion 1 wird gegen Funktion 2 um τ verschoben und jeweils miteinander multipliziert und aufsummiert
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Kreuzkorrelation – Faltung -H= PVEE
•Kreuzkorrelation xcorrelate
&= #8211;Device – Function – Signalprocessing –
•Convolve 
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KKF und AKF
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Stetige Korrelationsfunktionen
Mittelw= ert
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Kovarianz
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Kreuzkorrelationsfunktion
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Andere Schreibweise
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diskrete Korrelation – diskre= te Faltung
Beachten Sie bei der konkreten Aufgabe die = Randbedingungen und die Reduktion auf signifikante Punkte.
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Gleitender Mittelwert  - moving average – die neue= Kurve wird durch Mittelung von drei Punkten aus der alten Kurve gewonnen.
Hat ähnliche Wirkung wie ein Tiefpaß.
In der Praxis ist diese Annahme berechtigt, da der „Jitter“ im Vergleich zum Amplitudenfehler = sehr gering ist. In der x-Achse kein Fehler. Der Fehler ist nur in der Amplitud= e.
In diesem Fall sind gerade 5 Mes= swerte vorhanden. Das Verfahren lässt sich aber wie der moving average f&uum= l;r ein Signal mit vielen Messwerten anwenden.
Diese dient zur Berechnung der Randwerte.
Man versucht den Flächeninh= alt eines Quadrates und den Flächeninhalt eines Kreises gleich groß= zu machen. Dies ist aufgrund der Transzendenz der Zahl Pi nicht möglich.= Die Visualisierung der numerischen Integrationsverfahren ist sehr gut auf den = Internetseiten www.mathematik.ch
Sie haben 3 Messwerte: y1,y2,y3.= Durch diese drei Messwerte legen Sie ein Polynom 2. Ordnung. Setzen Sie für= x die Werte 0,h,2h ein, erhalten Sie drei Gleichungen für a,b,und c. Die Fläche unter der Kurve lässt sich über das Intergral von 0 = bis 2h über y(x) bestimmen. K ist die Lösung der Integration. In die= se Gleichung setzen Sie die drei Gleichungen für a,b,c ein. Sie erhalten= als Ergebnis die Kepler‘sche Fassregel.
Anstelle der linearen wählt= man hier quadratische Interpolation mit Parabelstücken; weil für das Aufstellen einer Parabelgleichung drei Punkte nötig sind, muss man ei= ne ungerade Zahl von Stützstellen wählen. Die Simpson-Regel basiert= auf der Kepler‘schen Fassregel
Polygone höherer Ordnung ne= igen zu Schwingungen.
Die MAC-Einheit (Multiplizier- u= nd Akkumuliereinheit) in Signalprozessoren kann in einem Rechenschritt / Operationszyklus gleichzeitig multiplizieren und addieren. Hardware ist angepasst an häufige Rechenoperationen.
Falls für z die Kreisfunkti= onen gesetzt werden, ist die z-Tranformierte gleich der DFT.
Angenähert nach dem Kriteri= um „Kleinstes Fehlerquadrat“
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} function GetTags(base,tag) { if(g_supportsPPTHTML) return base.all.tags(tag); else return base.getElementsByTagName(tag); } function UpdNtsPane(){ if(frames["PPTNts"]) PPTNts.location.replace( MHTMLP= refix+GetHrefObj( gCurSld ).mNtsHref ) } function UpdNavPane( sldIndex ){ if(gNavLoaded) PPTNav.UpdNav() } function UpdOtNavPane(){ if(gOtlNavLoaded) PPTOtlNav.UpdOtlNav() } function UpdOtlPane(){ if(gOtlLoaded) PPTOtl.UpdOtl() } function SetHasNts( fVal ) { if( gHasNts !=3D fVal ) { gHasNts=3DfVal UpdNavPane() } } function ToggleOtlText() { gOtlTxtExp=3D!gOtlTxtExp UpdOtlPane() } function ClearMedia() { // Clear any sounds playing before launching another browser window. Other= wise, // in fullscreen mode, you'll continue to hear the sound in the frames mod= e. if (PPTSld.pptSound) PPTSld.pptSound.loop =3D 0; } function FullScreen() {=20 if ( PPTSld.g_animUseRuntime ) PPTSld.document.body.pause(); ClearMedia(); var href =3D ( document.location.protocol =3D=3D 'mhtml:') ? FULLSCR_HREF = : FULLSCR_HREF+"#"+GetHrefObj(gCurSld).mSldHref; if(PPTNav.event.ctrlKey) { var w =3D (window.screen.availWidth * 1.0) / 2.0 var h =3D w * (PPTSld.g_origH * 1.0) / PPTSld.g_origW win =3D window.open( MHTMLPrefix+href,null,"toolbar=3D0,resizable=3D1,top= =3D0,left=3D0," + "width=3D"+ w + ",height=3D" + h ); if( win.document.body && PPTSld.g_animUseRuntime ) win.document.body.PPTSldFrameset=3Dwindow; } else { win =3D window.open( MHTMLPrefix+href,null,"fullscreen=3Dyes" ); if( win.document.body && PPTSld.g_animUseRuntime ) win.document.body.PPTSldFrameset=3Dwindow; } } function ToggleVNarration() { rObj=3DPPTSld.document.all("NSPlay") if( rObj && !PPTSld.g_animUseRuntime ) { if( (rObj.playState =3D=3D 1)||(rObj.playState =3D=3D 0) ) rObj.Play() else if( rObj.playState =3D=3D 2 ) rObj.Pause() else return; } else if( PPTSld.g_animUseRuntime ) { narObj =3D PPTSld.document.all("narrationID") if( narObj ) narObj.togglePause() } } function GetCurSldNum() { =20 obj=3DGetHrefObj(gCurSld) if( obj.mOrigVis =3D=3D 1 ) return obj.mSldIdx else =20 return gCurSld } function GetNumSlds() { =20 if( GetHrefObj(gCurSld).mOrigVis =3D=3D 1 ) return GetSldList().mNumVisSlds; else return GetSldList().mList.length } function GetSldNum( href ) { for(ii=3D0; ii 1 ) PopSldList(); else if( !IsFramesMode() ) { if( gLoopCont ) GoToFirst() else EndShow() } } function GoToPrevSld() { ii=3DgCurSld-1 if( ii > 0 ) { obj=3DGetHrefObj(ii) while ( obj && ( obj.mVis =3D=3D 0 ) && ( ii>0 ) ) obj=3DGetHrefObj(--ii) if( ii =3D=3D 0 ) ii=3D1 GoToSldNum(ii) } } function GoToFirst(){ GoToSld( GetHrefObj(1).mSldHref ) } function GoToLast() { ii=3DGetSldList().mList.length if( ii !=3D gCurSld ) GoToSld( GetHrefObj(ii).mSldHref ) } function GoToSldNum( num ) { if( PPTSld.event ) PPTSld.event.cancelBubble=3Dtrue obj =3D GetHrefObj( num ) obj.mVis=3D1 gPrevSld=3DgCurSld gCurSld =3D num; PPTSld.location.replace(MHTMLPrefix+obj.mSldHref) if( IsFramesMode() ) { UpdNavPane(); UpdOtlPane(); UpdNtsPane() } } function GoToSld( href ) { if( PPTSld.event ) PPTSld.event.cancelBubble=3Dtrue GetHrefObj( GetSldNum(href) ).mVis=3D1 PPTSld.location.replace(MHTMLPrefix+href) } function SldUpdated( id ) { if( id =3D=3D GetHrefObj(gCurSld).mSldHref ) return gPrevSld=3DgCurSld gCurSld=3DGetSldNum(id) if( IsFramesMode() ) { UpdNavPane(); UpdOtlPane(); UpdNtsPane() } } function PrevSldViewed(){ GoToSld( GetHrefObj(gPrevSld).mSldHref ) } function HasPrevSld() { return ( gIsEndShow || ( gCurSld !=3D 1 && GetHrefO= bj( gCurSld-1 ).mVis =3D=3D 1 )||( GetCurSldNum() > 1 ) ) } function HasNextSld() { return (GetCurSldNum() !=3D GetNumSlds()) } function CloseWindow() { if( HideMenu() ) return; =09 var event =3D PPTSld.event; if( !IsFramesMode() && event && (event.keyCode=3D=3D27 || event.keyCode=3D= =3D32 || event.type=3D=3D"click" ) ) window.close( self ); CatchNumKeys( self, event ); } function Unload() { gIsEndShow=3D0; } function SetupEndShow() { gIsEndShow=3D1; PPTSld.document.body.scroll=3D"no"; PPTSld.document.onkeypress=3DCloseWindow; PPTSld.document.onclick=3DCloseWindow; PPTSld.document.oncontextmenu=3D_CM; } function EndShow() { if( IsFramesMode() ) return if( PPTSld.event ) PPTSld.event.cancelBubble=3Dtrue doc=3DPPTSld.document var dir =3D doc.body.dir if( dir !=3D "rtl" ) dir =3D "ltr"; doc.open() doc.writeln('


' + ENDSHOW_MESG + '

') doc.close() } function SetSldVisited(){ GetSldList().mList[gCurSld-1].mVisited=3Dtrue } function IsSldVisited(){ return GetSldList().mList[gCurSld-1].mVisited } function hrefList( sldHref, visible, advDelay, advClk ) { this.mSldHref=3D this.mNtsHref =3D sldHref this.mOrigVis=3D this.mVis =3D visible this.mVisited=3D false this.mAdvDelay=3D advDelay this.mAdvOnClk=3D advClk } function SldList(arr,curSld,fEnd) { this.mCurSld =3D curSld; this.mList =3D new Array(); var idx =3D 1; for(ii=3D0;ii 0) { PushSldList(sldList,fEnd); gCurSld =3D 1; } else if( PPTSld.event ) PPTSld.event.cancelBubble=3Dtrue } function PushSldList(arr,fEnd) { var ii =3D gSldStack.length; gSldStack[ii] =3D new SldList(arr,gCurSld,fEnd); GoToSld( gSldStack[ii].mList[0].mSldHref ); } function PopSldList() { if (gSldStack[gSldStack.length-1].fEndShow) EndShow() else { gCurSld =3D gSldStack[gSldStack.length-1].mCurSld; gSldStack[gSldStack.length-1] =3D null; gSldStack.length--; var sldList =3D gSldStack[gSldStack.length-1]; GoToSld( sldList.mList[gCurSld - 1].mSldHref ); } } var custShowList=3Dnew Array(); /********************************************* Navigation button implementation There are 2 types of buttons: ImgBtn, TxtBtn implemented as function objects. They share a similiar interface so the event handlers can call SetActive, for example, on a button=20 object without needing to know exactly=20 what type of button it is. **********************************************/ //---------------------------------- function ImgBtn( oId,bId,w,action ) //---------------------------------- { var t=3Dthis t.Perform =3D _IBP t.SetActive =3D _IBSetA t.SetInactive=3D _IBSetI t.SetPressed =3D _IBSetP t.SetDisabled=3D _IBSetD t.Enabled =3D _IBSetE t.ChangeIcon =3D null t.UserAction =3D action t.ChgState =3D _IBUI t.mObjId =3D oId t.mBorderId=3D bId t.mWidth =3D w t.mIsOn =3D t.mCurState =3D 0 } function _IBSetA() { if( this.mIsOn ) { obj=3Dthis.ChgState( gHiliteClr,gShadowClr,2 ) obj.style.posTop=3D0 } } function _IBSetI() { if( this.mIsOn ) { obj=3Dthis.ChgState( gFaceClr,gFaceClr,1 ) obj.style.posTop=3D0=20 } } function _IBSetP() { if( this.mIsOn ) { obj=3Dthis.ChgState( gShadowClr,gHiliteClr,2 ) obj.style.posLeft+=3D1; obj.style.posTop+=3D1 } } function _IBSetD() { =20 obj=3Dthis.ChgState( gFaceClr,gFaceClr,0 ) obj.style.posTop=3D0=20 } function _IBSetE( state ) { var t=3Dthis GetObj( t.mBorderId ).style.visibility=3D"visible" if( state !=3D t.mIsOn ) { t.mIsOn=3Dstate if( state ) t.SetInactive() else t.SetDisabled() } } function _IBP() { var t=3Dthis if( t.mIsOn ) { if( t.UserAction !=3D null ) t.UserAction() if( t.ChangeIcon ) { obj=3DGetObj(t.mObjId) if( t.ChangeIcon() ) obj.style.posLeft=3Dobj.style.posLeft+(t.mCurState-4)*t.mWidth else obj.style.posLeft=3Dobj.style.posLeft+(t.mCurState-0)*t.mWidth } t.SetActive() } =20 } function _IBUI( clr1,clr2,nextState ) { var t=3Dthis SetBorder( GetObj( t.mBorderId ),clr1,clr2 ) obj=3DGetObj( t.mObjId ) obj.style.posLeft=3Dobj.style.posLeft+(t.mCurState-nextState)*t.mWidth-obj= .style.posTop t.mCurState=3DnextState return obj } //----------------------------------------- function TxtBtn( oId,oeId,action,chkState ) //----------------------------------------- { var t=3Dthis t.Perform =3D _TBP t.SetActive =3D _TBSetA t.SetInactive=3D _TBSetI t.SetPressed =3D _TBSetP t.SetDisabled=3D _TBSetD t.SetEnabled =3D _TBSetE t.GetState =3D chkState t.UserAction =3D action t.ChgState =3D _TBUI t.mObjId =3D oId t.m_elementsId=3D oeId t.mIsOn =3D 1 } function _TBSetA() { var t=3Dthis if( t.mIsOn && !t.GetState() ) t.ChgState( gHiliteClr,gShadowClr,0,0 ) } function _TBSetI() { var t=3Dthis if( t.mIsOn && !t.GetState() ) t.ChgState( gFaceClr,gFaceClr,0,0 ) } function _TBSetP() { if( this.mIsOn ) this.ChgState( gShadowClr,gHiliteClr,1,1 ) } function _TBSetD() { =20 this.ChgState( gFaceClr,gFaceClr,0,0 ) this.mIsOn =3D 0 } function _TBSetE() { var t=3Dthis if( !t.GetState() ) t.ChgState( gFaceClr,gFaceClr,0,0 ) else t.ChgState( gShadowClr,gHiliteClr,1,1 ) t.mIsOn =3D 1 } function _TBP() { var t=3Dthis if( t.mIsOn ) {=20 if( t.UserAction !=3D null ) t.UserAction() if( !t.GetState ) return if( t.GetState() ) t.SetPressed() else t.SetActive() } =20 } function _TBUI( clr1,clr2,lOffset,tOffset ) { SetBorder( GetObj( this.mObjId ),clr1,clr2 ) Offset( GetObj( this.m_elementsId ),lOffset,tOffset ) } function Offset( obj, top, left ){ obj.style.top=3Dtop; obj.style.left=3Dle= ft } function SetBorder( obj, upperLeft, lowerRight ) { s=3Dobj.style; s.borderStyle =3D "solid" s.borderWidth =3D 1=20 s.borderLeftColor =3D s.borderTopColor =3D upperLeft s.borderBottomColor=3D s.borderRightColor =3D lowerRight } function GetBtnObj(){ return gBtnArr[window.event.srcElement.id] } function BtnOnOver(){ b=3DGetBtnObj(); if( b !=3D null ) b.SetActive() } function BtnOnDown(){ b=3DGetBtnObj(); if( b !=3D null ) b.SetPressed() } function BtnOnOut(){ b=3DGetBtnObj(); if( b !=3D null ) b.SetInactive() } function BtnOnUp() { b=3DGetBtnObj() if( b !=3D null ) b.Perform() else Upd() } function GetNtsState(){ return parent.gNtsOpen } function GetOtlState(){ return parent.gOtlOpen } function GetOtlTxtState(){ return parent.gOtlTxtExp } function NtsBtnSetFlag( fVal ) { s=3Ddocument.all.item( this.m_flagId ).style s.display=3D"none" if( fVal ) s.display=3D"" else s.display=3D"none" } function _BSetA_Border(){ b =3D gBtnArr[this.mObjId]; if( b !=3D null ) b.S= etActive() } function _BSetI_Border(){ b =3D gBtnArr[this.mObjId]; if( b !=3D null ) b.S= etInactive() } function _BSetP_Border(){ b =3D gBtnArr[this.mObjId]; if( b !=3D null ) b.S= etPressed() } function _BSetA_BorderImg() {=20 b =3D gBtnArr[this.mBorderId]=20 if( b !=3D null && this.mIsOn && !b.GetState() ) { obj=3Dthis.ChgState( gHiliteClr,gShadowClr,2 ) obj.style.posTop=3D0 } } function _BSetI_BorderImg() {=20 b =3D gBtnArr[this.mBorderId] if( b !=3D null && this.mIsOn && !b.GetState() ) { obj=3Dthis.ChgState( gFaceClr,gFaceClr,1 ) obj.style.posTop=3D0 } } var gHiliteClr=3D"THREEDHIGHLIGHT",gShadowClr=3D"THREEDSHADOW",gFaceClr=3D"= THREEDFACE" var gBtnArr =3D new Array() gBtnArr["nb_otl"] =3D new TxtBtn( "nb_otl","nb_otlElem",parent.ToggleOtlPan= e,GetOtlState ) gBtnArr["nb_otlElem"] =3D new TxtBtn( "nb_otl","nb_otlElem",parent.ToggleOt= lPane,GetOtlState ) gBtnArr["nb_nts"] =3D new ImgBtn( "nb_nts","nb_ntsBorder",10,parent.ToggleN= tsPane ) gBtnArr["nb_nts"].SetActive =3D _BSetA_BorderImg; gBtnArr["nb_nts"].SetInactive =3D _BSetI_BorderImg; gBtnArr["nb_ntsBorder"] =3D new TxtBtn( "nb_ntsBorder","nb_ntsElem",parent.= ToggleNtsPane,GetNtsState ) gBtnArr["nb_ntsElem"] =3D new TxtBtn( "nb_ntsBorder","nb_ntsElem",parent.To= ggleNtsPane,GetNtsState ) gBtnArr["nb_prevBorder"] =3D gBtnArr["nb_prev"]=3D new ImgBtn( "nb_prev","n= b_prevBorder",30,parent.GoToPrevSld ) gBtnArr["nb_nextBorder"] =3D gBtnArr["nb_next"]=3D new ImgBtn( "nb_next","n= b_nextBorder",30,parent.GoToNextSld ) gBtnArr["nb_sldshw"]=3D new ImgBtn( "nb_sldshw","nb_sldshwBorder",18,parent= .FullScreen ) gBtnArr["nb_sldshwBorder"] =3D new TxtBtn( "nb_sldshw","nb_sldshwBorder",pa= rent.FullScreen,null ) gBtnArr["nb_sldshwBorder"].SetActive =3D _BSetA_Border; gBtnArr["nb_sldshwBorder"].SetInactive =3D _BSetI_Border; gBtnArr["nb_sldshwText"] =3D new TxtBtn( "nb_sldshw","nb_sldshwText",parent= .FullScreen,null ) gBtnArr["nb_sldshwText"].SetActive =3D _BSetA_Border; gBtnArr["nb_sldshwText"].SetInactive =3D _BSetI_Border; gBtnArr["nb_voice"] =3D gBtnArr["nb_voiceBorder"] =3D new ImgBtn( "nb_voice= ","nb_voiceBorder",18,parent.ToggleVNarration ) gBtnArr["nb_otlTxtBorder"] =3D gBtnArr["nb_otlTxt"]=3D new ImgBtn( "nb_otlT= xt","nb_otlTxtBorder",23,parent.ToggleOtlText ) gBtnArr["nb_ntsBorder"].m_flagId=3D "nb_nts" gBtnArr["nb_ntsBorder"].SetFlag =3D NtsBtnSetFlag gBtnArr["nb_otlTxt"].ChangeIcon=3D GetOtlTxtState /********************************************* Context menu implementation _CM() is the function that's hooked up to the oncontextmenu event. Once we're asked to show the menu, we first build it by creating DIVs on-the-fly. Then we position it=20 within the screen area so it doesn't get clipped. Creating the DIVs using createElement() means we don't have to write out any extra HTML into the slide HTML files. **********************************************/ var sNext=3D"Weiter",sPrev=3D"Zurück",sEnd=3D"Präsentation beende= n",sFont=3D"Arial",sArrow=3D"Pfeil",sFreeform=3D"Freihandform",sRect=3D"Rec= hteck",sOval=3D"Oval" function ShowMenu() { BuildMenu(); var doc=3DPPTSld.document.body,x=3DPPTSld.event.clientX+doc.scrollLeft,y= =3DPPTSld.event.clientY+doc.scrollTop m =3D PPTSld.document.all.item("ctxtmenu") m.style.pixelLeft=3Dx if( (x+m.scrollWidth > doc.clientWidth)&&(x-m.scrollWidth > 0) ) m.style.pixelLeft=3Dx-m.scrollWidth m.style.pixelTop=3Dy if( (y+m.scrollHeight > doc.clientHeight)&&(y-m.scrollHeight > 0) ) m.style.pixelTop=3Dy-m.scrollHeight m.style.display=3D"" } function _CM() { if( !parent.IsFullScrMode() ) return; if(!PPTSld.event.ctrlKey) { ShowMenu() return false } else HideMenu() } function BuildMenu() { if( PPTSld.document.all.item("ctxtmenu") ) return var mObj=3DCreateItem( PPTSld.document.body ) mObj.id=3D"ctxtmenu" mObj.style.visibility=3D"hidden" var s=3DmObj.style s.position=3D"absolute" s.cursor=3D"default" s.width=3D"120px" SetCMBorder(mObj,"menu","black") var iObj=3DCreateItem( mObj ) SetCMBorder( iObj, "threedhighlight","threedshadow" ) iObj.style.padding=3D2 CreateMenuItem( iObj,sNext,M_GoNextSld,M_True ) CreateMenuItem( iObj,sPrev,M_GoPrevSld,M_HasPrevSld ) =09 CreateSeparator( iObj ) CreateMenuItem( iObj,sEnd,M_End,M_True ) mObj.style.visibility=3D"visible" } function Cancel() { window.event.cancelBubble=3Dtrue; window.event.returnVa= lue=3Dfalse } function Highlight() { ChangeClr("activecaption","threedhighlight") } function Deselect() { ChangeClr("threedface","menutext") } function Perform() { e=3DPPTSld.event.srcElement if( e.type=3D=3D"menuitem" && e.IsActive() ) e.Action() else PPTSld.event.cancelBubble=3Dtrue } function ChangeClr( bg,clr ) { e=3DPPTSld.event.srcElement if( e.type=3D=3D"menuitem" && e.IsActive() ) { e.style.backgroundColor=3Dbg e.style.color=3Dclr } } function M_HasPrevSld() { return( parent.HasPrevSld() ) } function M_GoNextSld() { if( gIsEndShow ) M_End(); else GoToNextSld() } function M_GoPrevSld() { if( gIsEndShow ) { gIsEndShow=3D0; history.back();= PPTSld.event.cancelBubble=3Dtrue; } else GoToPrevSld() } function M_True() { return true } function M_End() { window.close( self ) } function CreateMenuItem( node,text,action,eval ) { var e=3DCreateItem( node ) e.type=3D"menuitem" e.Action=3Daction e.IsActive=3Deval e.innerHTML=3Dtext if( !e.IsActive() ) e.style.color=3D"threedshadow" e.onclick=3DPerform e.onmouseover=3DHighlight e.onmouseout=3DDeselect s=3De.style; 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charset="us-ascii" Numerische Verarbeitung digitaler Signale ------=_NextPart_01C57E34.A03B30C0 Content-Location: file:///C:/2ACB1965/06-info-numerische-Dateien/outline.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii"
 No= tizen
Bildschir= mpräsentation
Gliederung
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