[[File:P_ID319__Sig_Z_2_1.png|right|frame|Rectangular signal, triangular signal, sum signal]]
In der nebenstehenden Grafik sind die beiden periodischen Signale ${x(t)}$ und ${y(t)}$ dargestellt, aus denen das Summensignal ${s(t)}$ – im unteren Bild skizziert – sowie das Differenzsignal ${d(t)}$ gebildet werden.
The adjacent diagram shows the periodic signals ${x(t)}$ and ${y(t)}$, from which the sum ${s(t)}$ – sketched in the lower diagram – and the difference ${d(t)}$ are formed.
Weiterhin betrachten wir in dieser Aufgabe das Signal ${w(t)}$, das sich aus der Summe der beiden periodischen Signalen ${u(t)}$ und $v(t)$ ergibt. Die Grundfrequenzen der Signale seien
Furthermore, in this task we consider the signal ${w(t)}$, which results from the sum of two periodic signals ${u(t)}$ and $v(t)$ . The base frequencies of these signals are
:* $f_u = 998 \,\text{Hz},$
:* $f_u = 998 \,\text{Hz},$
:* $f_u = 1002 \,\text{Hz}.$
:* $f_v = 1002 \,\text{Hz}.$
Mehr ist von diesen Signalen ${u(t)}$ und $v(t)$ nicht bekannt.
That is all we know about the signals ${u(t)}$ and $v(t)$.
''Hinweise:''
*Die Aufgabe gehört zum Kapitel [[Signaldarstellung/Allgemeine_Beschreibung|Allgemeine Beschreibung periodischer Signale]].
*Sollte die Eingabe des Zahlenwertes „0” erforderlich sein, so geben Sie bitte „0.” ein.
===Fragebogen===
''Hints:''
*The exercise belongs to the chapter [[Signal_Representation/General_Description|General description of periodic signals]].
*With the interactive applet [[Applets:Period_Duration_of_Periodic_Signals|Period Duration of Periodic Signals]] the resulting period duration of two harmonic oscillations can be determined.
===Questions===
<quiz display=simple>
<quiz display=simple>
{Wie groß ist Periodendauer $T_x$ und Grundfrequenz $f_x$ des Signals ${x(t)}$?
{What is the period duration $T_x$ and the basic frequency $f_x$ of the signal ${x(t)}$?
|type="{}"}
|type="{}"}
$f_x$ = { 1 3% } $\text{kHz}$
$f_x\ = \ $ { 1 3% } $\text{kHz}$
{Wie groß ist Periodendauer $T_y$ und Grundfrequenz $f_y$ des Signals ${y(t)}$?
{What is the period duration $T_y$ and the basic frequency $f_y$ of the signal ${y(t)}$?
|type="{}"}
|type="{}"}
$f_y$ = { 0.4 3% } $\text{kHz}$
$f_y\ = \ $ { 0.4 3% } $\text{kHz}$
{Bestimmen Sie die Grundfrequenz $f_s$ sowie die Periodendauer $T_s$ des Summensignals ${s(t)}$ und überprüfen Sie das Ergebnis anhand der Skizze.
{Determine the basic frequency $f_s$ and the period duration $T_s$ of the sum signal ${s(t)}$. <br>Verify your results with the help of the sketched signal.
|type="{}"}
|type="{}"}
$T_s$ = { 5 3% } $\text{ms}$
$T_s\ = \ $ { 5 3% } $\text{ms}$
{Welche Periodendauer $T_d$ weist das Differenzsignal ${d(t)}$ auf?
{What is the period duration $T_d$ of the difference signal ${d(t)}$ ?
|type="{}"}
|type="{}"}
$T_d$ = { 5 3% } $\text{ms}$
$T_d\ = \ $ { 5 3% } $\text{ms}$
{Welche Periodendauer $T_w$ besitzt das Signal ${w(t)} = {u(t)} + $v(t)$?
{What is the period duration $T_w$ of the signal ${w(t)} = {u(t)} + v(t)$?
|type="{}"}
|type="{}"}
$T_w$ = { 500 3% } $\text{ms}$
$T_w\ = \ $ { 500 3% } $\text{ms}$
</quiz>
</quiz>
===Musterlösung===
===Solution===
{{ML-Kopf}}
{{ML-Kopf}}
'''1.''' Es gilt $T_x = 1 \text{ms}$ und $f_x \underline{= 1 \text{kHz}}$.
'''(1)''' The following applies to the rectangular signal: $T_x = 1 \,\text{ms}$ ⇒
'''(3)''' The basic frequency $f_s$ of the sum signal $s(t)$ is the greatest common divisor $f_x = 1 \,\text{kHz}$ and $f_y = 0.4 \,\text{kHz}$.
*From this follows $f_s = 200 \,\text{Hz}$ and the period duration $T_s\hspace{0.15cm}\underline{ = 5 \,\text{ms}}$.
* This is also evident from the graphic on the information page.
'''2.''' Es gilt $T_y = 2.5 \text{ms}$ und $f_y \underline{= 0.4 \text{kHz}}$.
'''3.''' Die Grundfrequenz $f_s$ ist der größte gemeinsame Teiler von $f_x = 1 \text{kHz}$ und $f_y = 0.4 \text{kHz}$. Daraus folgt $f_s = 200 \text{Hz}$ und die Periodendauer $T_s = 5 \text{ms}$, wie auch aus der grafischen Darstellung des Signals $\text{s(t)}$ hervorgeht.
'''(4)''' The period duration $T_d$ does not change compared to the period duration $T_s$, if the signal ${y(t)}$ is not added but subtracted:
'''4.''' Die Periodendauer $T_d$ ändert sich gegenüber der Periodendauer $T_s$ nicht, wenn das Signal $\text{y(t)}$ nicht addiert, sondern subtrahiert wird: $T_d = T_s = 5 \text{ms}$.
'''(5)''' The greatest common divisor of $f_u = 998 \,\text{Hz}$ and $f_{v} = 1002 \,\text{Hz}$ is $f_w = 2 \,\text{Hz}$. The inverse of this gives the period duration $T_w \hspace{0.15cm}\underline{= 500 \,\text{ms}}$.
[[File:P_ID320__Sig_Z_2_1_d_neu.png|center|]]
'''5.''' Der größte gemeinsame Teiler von $f_u = 0.998 \text{kHz}$ und $f_{\upsilon} = 1.002 \text{kHz}$ ist $f_w = 2 \text{Hz}$. Der Kehrwert hiervon ergibt die Periodendauer $T_w = 500 \text{ms}$.
{{ML-Fuß}}
{{ML-Fuß}}
[[Category:Aufgaben zu Signaldarstellung|^2. Periodische Signale^]]
[[Category:Signal Representation: Exercises|^2.1 Description of Periodic Signals^]]
The adjacent diagram shows the periodic signals ${x(t)}$ and ${y(t)}$, from which the sum ${s(t)}$ – sketched in the lower diagram – and the difference ${d(t)}$ are formed.
Furthermore, in this task we consider the signal ${w(t)}$, which results from the sum of two periodic signals ${u(t)}$ and $v(t)$ . The base frequencies of these signals are
$f_u = 998 \,\text{Hz},$
$f_v = 1002 \,\text{Hz}.$
That is all we know about the signals ${u(t)}$ and $v(t)$.
(5) The greatest common divisor of $f_u = 998 \,\text{Hz}$ and $f_{v} = 1002 \,\text{Hz}$ is $f_w = 2 \,\text{Hz}$. The inverse of this gives the period duration $T_w \hspace{0.15cm}\underline{= 500 \,\text{ms}}$.
