Prüfung: &=
nbsp; &nbs=
p; Informationstechnik
Termin:  =
; &n=
bsp; Mon=
tag,
02.02.2004
11:00 – 13:00
Prüfer: &=
nbsp; &nbs=
p; Prof.
J. Walter
Hilfsmittel: &=
nbsp; &nbs=
p; beliebig
/ Kein Internetzugang
=
Name: &=
nbsp; &nbs=
p; _________________________
Vorname: &=
nbsp; ______________=
___________
Bemerkung: &=
nbsp; _________________________<=
/b>
<=
span
style=3D'mso-bidi-font-family:Arial'>bitte keine rote Farbe verwenden<=
/o:p>
<=
span
style=3D'mso-bidi-font-family:Arial'>
<=
span
style=3D'mso-bidi-font-family:Arial'>(nicht ausfüllen) !
Aufgabe
|
mögl.
Punkte
|
erreichte
Punkte
|
1
|
10
|
|
2
|
11
|
|
3
|
10
|
|
4
|
10
|
|
5
|
9
|
|
|
|
|
|
|
|
Gesamt
|
50
|
|
|
|
|
|
Note
|
|
Bearbeiten
Sie die Aufgaben nur, falls Sie keine gesundheitlichen Beschwerden haben. <=
o:p>
Viel
Erfolg
Bemerkung:
Sie können die Vorder- und Rückseite benutzten. Es werden =
nur
die auf den Prüfungsblättern vorhandenen oder fest mit den
Prüfungsblättern verbundenen Ergebnisse gewertet.
1. Gauß'sches Prinzip der kleinsten
Fehlerquadrate (10 Punkte)
Die Funktion: soll im Bereich optimal dur=
ch
eine Gerade angenä=
hert
werden.
a) Bestimmen Sie die Gleichung der Geraden<=
/o:p>
b) Skizzieren Sie das Ergebnis=
<=
/span>
<=
/span>
<=
/span>
Lösung:
<=
o:p>
Lösung mit Maple:
> 0=3Ddiff(int(((cos(x)+sin(=
x))-(a*x+b))^2,
x=3D0.9..4),a);
> 0=3Ddiff(int(
((cos(x)+sin(x))-(a*x+b))^2, x=3D0.9..4 ),b);<=
/span>
> b=3Dsolve( 0 =3D
6.747127478+42.18066666*a+15.19*b, b );<=
/span>
> a=3Dsolve( 0 =3D
.5297516309+15.19*a+6.2*b, a );
> b=3Dsolve( b =3D
-.4441821908-2.776870748*(-.3487502508e-1-.4081632653*b), b );
> a=3Dsolve(a =3D
-.3487502508e-1-.4081632653*(2.603414197),a);<=
/span>
> plot([sin(x)+cos(x),
-1.097*x+2.603], x=3D0.9..4, color=3D[red,blue], style=3D[point,line]);
>
2. DFT&nbs=
p;
(11 Punkte)
=
Ein C=
osinus
(Amplitudenwerte +1, -1) mit der Frequenz 50 Hz wird mit der
Blockgröße N=3D8 abgetastet. Die Messzeit ist 20ms. Infolge eines
Softwarefehlers wurde ein Punkt falsch – nämlich doppelt –
gespeichert.
=
a)
1P Trage=
n Sie
die Abtastwerte aus der nachfolgenden Tabelle in die Zeichnung ein. =
o:p>
b)
1P Skizz=
ieren
Sie den „korrekten“ Cosinus.
c)
7P Berec=
hnen
Sie aus den Abtastwerten die skalierte DFT für m=3D0, m=3D1, m=3D2, m=
=3D3, m=3D4
d)
2P Zeich=
nen
Sie das Amplitudenspektrum für den „korrekten“ und
„fälschlicherweise“ abgetasteten Cosinus.
<=
o:p>
<=
span
style=3D'mso-spacerun:yes'>
Bild 1: Cosinus „falsch“ und
„richtig“
Der rote Punkt ist fehlerhaft.
n=3D=
|
f(n)=
|
t/ms
|
0=
|
1
|
0
|
1=
|
+0.707=
|
2,5
|
2=
|
+0.707=
|
5
|
3=
|
-0.707=
|
7,5
|
4=
|
-1
|
10
|
5=
|
-0.707=
|
12,5=
p>
|
6=
|
0
|
15
|
7=
|
+0.707=
|
17,5=
p>
|
&n=
bsp;
&n=
bsp;
Lö=
sung c :
m=3D0
|
0,088<=
/p>
|
0
|
m=3D1<=
/p>
|
1,016<=
/p>
|
1
|
m=3D2<=
/p>
|
0,177<=
/p>
|
0
|
m=3D3<=
/p>
|
0,177<=
/p>
|
0
|
m=3D4<=
/p>
|
0,177<=
/p>
|
0
|
Die Werte wurden mit der Formel fü=
;r die
skalierte DFT berechnet :
Der Mittelwert wurde m=3D0 wurde extra berec=
hnet.
<=
o:p>
Lö=
sung d
&n=
bsp;
&n=
bsp;
Bemerkung:
Das Amplitudendichtespektrum wurde aus der
skalierten DFT gewonnen. Selbstverständlich muss bei einer Messung die
Amplitude unabhängig von der gewählten Anzahl der Abtastpunkte se=
in.
3.
DGL - Übertragungsfunktion - Systemantwort (10 Punkte)
Erstellen
Sie für die nachfolgende Schaltung die Übertragungsfunktion.
Schaltung mit R
und C
a) Erstellen
Sie die Übertragungsfunktion G1=
(s)
b) Erstellen
Sie die Übertragungsfunktion G(s) für die normierten Werte R=3D1,=
C=3D1
c) Erstellen
Sie die Differentialgleichung für den Zeitbereich ( System ist am Anfa=
ng
in Ruhe)
d) Bestimmen
Sie die Impulsantwort für die normierten Werte R=3D1, C=3D1,
e) Skizzieren
Sie die Impulsantwort für t=3D0 bis 6 =
L&o=
uml;sung
Aufgabe 3a
L&o=
uml;sung b
L&o=
uml;sung c
L&o=
uml;sung
Aufgabe d
Mit
Maple:
>
with(inttrans):
> invlaplace((s/2+(1/(2*R*C)=
))/(s+(3/(4*R*C))),
s, t);
>
plot(1/2*Dirac(t)+1/8*exp(-3/4*t), t=3D0..5=
);=
<=
o:p>
<=
o:p>
Führt man den=
Plot
nur in Maple aus wird dieser falsch, da der Dirac-Stoß nicht
eingezeichnet wird.
4 FIR-Filter<=
span
style=3D'mso-spacerun:yes'> (10 Punkte)
E=
in
Bandpass mit den Grenzfrequenzen fgoben =3D 120Hz und fgunten=3D80Hz ist als
FIR-Filter für N=3D5 zu entwerfen. Die Abtastfrequenz beträgt fa=
=3D1,2
kHz.
<=
o:p>
a) Berechnen
Sie die Filtergleichung für das FIR-Filter
b) Berechnen
und skizzieren Sie die Impuls-Antwort des FIR-Filters.
Lösung:<=
/b>
Formel für TP
Bandpass=3DTiefpass_Oben-Tiefpass_Unten<=
/o:p>
akBP=3DakTPo-akTPu=
( 1 Punkt )
<=
span
style=3D'mso-bidi-font-family:Arial'>-0,055
|
<=
span
style=3D'mso-bidi-font-family:Arial'>a5
|
<=
span
style=3D'mso-bidi-font-family:Arial'>a-5
|
<=
span
style=3D'mso-bidi-font-family:Arial'>-0,032
|
<=
span
style=3D'mso-bidi-font-family:Arial'>a4=
span>
|
<=
span
style=3D'mso-bidi-font-family:Arial'>a-4
|
<=
span
style=3D'mso-bidi-font-family:Arial'>0
|
<=
span
style=3D'mso-bidi-font-family:Arial'>a3
|
<=
span
style=3D'mso-bidi-font-family:Arial'>a-3
|
<=
span
style=3D'mso-bidi-font-family:Arial'>0,033
|
<=
span
style=3D'mso-bidi-font-family:Arial'>a2
|
<=
span
style=3D'mso-bidi-font-family:Arial'>a-2
|
<=
span
style=3D'mso-bidi-font-family:Arial'>0,058
|
<=
span
style=3D'mso-bidi-font-family:Arial'>a1
|
<=
span
style=3D'mso-bidi-font-family:Arial'>a-1
|
<=
span
style=3D'mso-bidi-font-family:Arial'>0,067
|
<=
span
style=3D'mso-bidi-font-family:Arial'>
|
<=
span
style=3D'mso-bidi-font-family:Arial'>a_0
|
<=
o:p>
( 5=
Punkte )
Berechnung mit Excel:
<=
span
style=3D'mso-bidi-font-family:Arial'>n
|
<=
span
style=3D'mso-bidi-font-family:Arial'>xn – Eingang=
|
<=
span
style=3D'mso-bidi-font-family:Arial'>yn-Ausgang
|
<=
span
style=3D'mso-bidi-font-family:Arial'>-10
|
<=
span
style=3D'mso-bidi-font-family:Arial'>0
|
<=
span
style=3D'mso-bidi-font-family:Arial'>0
|
<=
span
style=3D'mso-bidi-font-family:Arial'>-9
|
<=
span
style=3D'mso-bidi-font-family:Arial'>0
|
<=
span
style=3D'mso-bidi-font-family:Arial'>0
|
<=
span
style=3D'mso-bidi-font-family:Arial'>-8
|
<=
span
style=3D'mso-bidi-font-family:Arial'>0
|
<=
span
style=3D'mso-bidi-font-family:Arial'>0
|
<=
span
style=3D'mso-bidi-font-family:Arial'>-7
|
<=
span
style=3D'mso-bidi-font-family:Arial'>0
|
<=
span
style=3D'mso-bidi-font-family:Arial'>0
|
<=
span
style=3D'mso-bidi-font-family:Arial'>-6
|
<=
span
style=3D'mso-bidi-font-family:Arial'>0
|
<=
span
style=3D'mso-bidi-font-family:Arial'>0
|
<=
span
style=3D'mso-bidi-font-family:Arial'>-5
|
<=
span
style=3D'mso-bidi-font-family:Arial'>0
|
<=
span
style=3D'mso-bidi-font-family:Arial'>-0,055
|
<=
span
style=3D'mso-bidi-font-family:Arial'>-4
|
<=
span
style=3D'mso-bidi-font-family:Arial'>0
|
<=
span
style=3D'mso-bidi-font-family:Arial'>-0,032
|
<=
span
style=3D'mso-bidi-font-family:Arial'>-3
|
<=
span
style=3D'mso-bidi-font-family:Arial'>0
|
<=
span
style=3D'mso-bidi-font-family:Arial'>0
|
<=
span
style=3D'mso-bidi-font-family:Arial'>-2
|
<=
span
style=3D'mso-bidi-font-family:Arial'>0
|
<=
span
style=3D'mso-bidi-font-family:Arial'>0,033
|
<=
span
style=3D'mso-bidi-font-family:Arial'>-1
|
<=
span
style=3D'mso-bidi-font-family:Arial'>0
|
<=
span
style=3D'mso-bidi-font-family:Arial'>0,058
|
<=
span
style=3D'mso-bidi-font-family:Arial'>0
|
<=
span
style=3D'mso-bidi-font-family:Arial'>1
|
<=
span
style=3D'mso-bidi-font-family:Arial'>0,067
|
<=
span
style=3D'mso-bidi-font-family:Arial'>1
|
<=
span
style=3D'mso-bidi-font-family:Arial'>0
|
<=
span
style=3D'mso-bidi-font-family:Arial'>0,058
|
<=
span
style=3D'mso-bidi-font-family:Arial'>2
|
<=
span
style=3D'mso-bidi-font-family:Arial'>0
|
<=
span
style=3D'mso-bidi-font-family:Arial'>0,033
|
<=
span
style=3D'mso-bidi-font-family:Arial'>3
|
<=
span
style=3D'mso-bidi-font-family:Arial'>0
|
<=
span
style=3D'mso-bidi-font-family:Arial'>0
|
<=
span
style=3D'mso-bidi-font-family:Arial'>4
|
<=
span
style=3D'mso-bidi-font-family:Arial'>0
|
<=
span
style=3D'mso-bidi-font-family:Arial'>-0,032
|
<=
span
style=3D'mso-bidi-font-family:Arial'>5
|
<=
span
style=3D'mso-bidi-font-family:Arial'>0
|
<=
span
style=3D'mso-bidi-font-family:Arial'>-0,055
|
<=
span
style=3D'mso-bidi-font-family:Arial'>6
|
<=
span
style=3D'mso-bidi-font-family:Arial'>0
|
<=
span
style=3D'mso-bidi-font-family:Arial'>0
|
<=
span
style=3D'mso-bidi-font-family:Arial'>7
|
<=
span
style=3D'mso-bidi-font-family:Arial'>0
|
<=
span
style=3D'mso-bidi-font-family:Arial'>0
|
<=
span
style=3D'mso-bidi-font-family:Arial'>8
|
<=
span
style=3D'mso-bidi-font-family:Arial'>0
|
<=
span
style=3D'mso-bidi-font-family:Arial'>0
|
<=
span
style=3D'mso-bidi-font-family:Arial'>9
|
<=
span
style=3D'mso-bidi-font-family:Arial'>0
|
<=
span
style=3D'mso-bidi-font-family:Arial'>0
|
<=
span
style=3D'mso-bidi-font-family:Arial'>10
|
<=
span
style=3D'mso-bidi-font-family:Arial'>0
|
<=
span
style=3D'mso-bidi-font-family:Arial'>0
|
5 Skizzieren Sie die Kurv=
en nach
den vorgegebenen Werten:
Cosinus mit Rauschen mit Amplitude 0.2
Die Faltung zweier Rechtecke: 6 Punkte
Die Amplitude des entstehenden Dreiecks bere=
chnet
sich über die Blockgröße N=3D256.
Die Amplitude ist die Hälfte der
Blockgröße=3D128. Die Breite des Impulses ist 20ms.
diskrete Fa=
ltung
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