Aufgaben:Exercise 4.6Z: Locality Curve for Phase Modulation - Revision history

Guenter at 13:02, 24 May 2021

2021-05-24T13:02:27Z

Guenter at 10:52, 12 May 2021

2021-05-12T10:52:28Z

Guenter at 10:43, 12 May 2021

2021-05-12T10:43:13Z

Guenter: Guenter moved page Aufgaben:Exercise 4.6Z: Locus Curve for Phase Modulation to Aufgaben:Exercise 4.6Z: Locality Curve for Phase Modulation

2021-05-10T12:56:34Z

Guenter moved page Exercise 4.6Z: Locus Curve for Phase Modulation to Exercise 4.6Z: Locality Curve for Phase Modulation

Javier: Text replacement - "Low Pass" to "Low-Pass"

2021-05-04T07:33:28Z

Text replacement - "Low Pass" to "Low-Pass"

Guenter at 08:54, 23 April 2021

2021-04-23T08:54:40Z

Noah at 11:09, 26 March 2021

2021-03-26T11:09:46Z

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Javier: Javier moved page Aufgaben:Exercise 4.6Z: Loot Curve for Phase Modulation to Aufgaben:Exercise 4.6Z: Locus Curve for Phase Modulation

2021-03-23T13:18:12Z

Javier moved page Exercise 4.6Z: Loot Curve for Phase Modulation to Exercise 4.6Z: Locus Curve for Phase Modulation

Javier: Text replacement - "Category:Exercises for Signal Representation" to "Category:Signal Representation: Exercises"

2021-03-23T12:39:40Z

Text replacement - "Category:Exercises for Signal Representation" to "Category:Signal Representation: Exercises"

Oezdemir: Oezdemir moved page Aufgaben:Aufgabe 4.6Z: Ortskure bei Phasenmodulation to Aufgaben:Exercise 4.6Z: Loot Curve for Phase Modulation

2020-11-26T15:15:14Z

Oezdemir moved page Aufgabe 4.6Z: Ortskure bei Phasenmodulation to Exercise 4.6Z: Loot Curve for Phase Modulation

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−	{{quiz-Header\|Buchseite=Signal_Representation/~~Equivalent_Low_Pass_Signal_and_Its_Spectral_Function~~	+	{{quiz-Header\|Buchseite=Signal_Representation/Equivalent Low-Pass Signal and its Spectral Function
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 ===Solution===
 {{ML-Kopf}}
-'''(1)'''&nbsp;  The locus curve is a circular arc with radius&nbsp; $2$. Therefore, the magnitude function is constant&nbsp;  $\underline{a(t) = 2}$.
+'''(1)'''&nbsp;  The locality curve is a circular arc with radius&nbsp; $2$.&nbsp; Therefore, the magnitude function is constant&nbsp;  $\underline{a(t) = 2}$.
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 '''(3)'''&nbsp; In general, the relation&nbsp; $s_{\rm TP}(t) = a(t) \cdot {\rm e}^{{\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}
-\phi(t)}.$ applies here. A comparison with the given function yields:
+\phi(t)}$&nbsp; applies here.&nbsp; A comparison with the given function yields:
 :$$\phi(t) = \eta \cdot q(t).$$
-*The maximum phase value&nbsp; $\phi_{\rm max} = +\pi \; \Rightarrow  \; {180^\circ}$&nbsp; is obtained for the signal amplitude&nbsp; $q_{\rm max} = 1$. From this follows directly&nbsp; ${\eta = \pi} \; \underline{\approx 3.14}$.
+*The maximum phase value&nbsp; $\phi_{\rm max} = +\pi \; \Rightarrow  \; {180^\circ}$&nbsp; is obtained for the signal amplitude&nbsp; $q_{\rm max} = 1$.&nbsp; From this follows directly&nbsp; ${\eta = \pi} \; \underline{\approx 3.1415}$.
 *This modulation index is confirmed by the values&nbsp; $\phi_{\rm min} = -\pi /2$&nbsp; and&nbsp; $q_{\rm min} = -0.5$&nbsp;.
-[[File:P_ID769__Sig_Z_4_6_d_neu.png|right|frame|Locus curve (phase diagram) for a rectangular signal]]
+[[File:P_ID769__Sig_Z_4_6_d_neu.png|right|frame|Locality curve (phase diagram) for a rectangular source signal]]
 '''(4)'''&nbsp;  <u>The second and third proposed solutions</u> are correct:
 *If&nbsp; $q(t) = \text{const.} =-0.5$, the phase function is also constant:
 :$$\phi(t) = \eta \cdot q(t) = - {\pi}/{2}\hspace{0.3cm}
 \Rightarrow \hspace{0.3cm} s_{\rm TP}(t) = - {\rm j} \cdot s_0  = - 2{\rm j}.$$
-*Thus, for the actual, physical signal:
+*Thus, for the actual physical signal:
 :$$s(t) = s_0 \cdot  {\cos} (  \omega_{\rm T}\hspace{0.05cm} t -
   {\pi}/{2}) = 2 \cdot  {\sin} (  \omega_{\rm T} \hspace{0.05cm} t ).$$
 *In contrast,&nbsp; $q(t) = +0.5$&nbsp; leads to &nbsp;$\phi (t) = \pi /2$&nbsp; and to &nbsp;$s_{\rm TP}(t) = 2{\rm j}$.
-*If&nbsp; $q(t)$&nbsp; is a rectangular signal that alternates between&nbsp; $+0.5$&nbsp; and&nbsp; $–0.5$&nbsp; , then the locus curve consists of only two points on the imaginary axis, regardless of how long the intervals with &nbsp; $+0.5$&nbsp; and&nbsp; $–0.5$ last.
+*If&nbsp; $q(t)$&nbsp; is a rectangular signal that alternates between&nbsp; $+0.5$&nbsp; and&nbsp; $–0.5$&nbsp; , then the locality curve consists of only two points on the imaginary axis, regardless of how long the intervals with &nbsp; $+0.5$&nbsp; and&nbsp; $–0.5$&nbsp; last.
 *If, on the other hand,&nbsp; $q(t) = \pm 1$, then the possible phase values&nbsp; $+\pi$&nbsp; and&nbsp; $-\pi$ result purely formally, but they are identical.
-*The „locus curve” then consists of only one point: &nbsp; $s_{\rm TP}(t) = - s_0$ &nbsp; <br>&rArr; &nbsp;  the signal&nbsp; $s(t)$&nbsp; is  „minus-cosine” for all times&nbsp; $t$&nbsp;.
+*The locality curve then consists of only one point: &nbsp; $s_{\rm TP}(t) = - s_0$ &nbsp; &rArr; &nbsp;  the signal&nbsp; $s(t)$&nbsp; is&nbsp;  "minus-cosine"&nbsp; for all times&nbsp; $t$.

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 }}
-[[File:P_ID768__Sig_Z_4_6.png|right|frame|A possible locus curve with phase modulation]]
+[[File:P_ID768__Sig_Z_4_6.png|right|frame|A possible locality curve with phase modulation]]
-We assume here a message signal&nbsp; $q(t)$&nbsp;, which is considered normalised (dimensionless).
+We assume a source signal&nbsp; $q(t)$, which is considered normalised.
 *The maximum value of this signal is&nbsp; $q_{\rm max} = 1$&nbsp; and the minimum signal value is&nbsp; $q_{\rm min} = -0.5$.
-*Otherwise nothing is known about&nbsp; $q(t)$&nbsp;.
+*Otherwise nothing is known about&nbsp; $q(t)$.
-The modulated signal with phase modulation is:
+The modulated signal with phase modulation &nbsp; &rArr; &nbsp; "transmission signal"&nbsp; is:
 :$$s(t) = s_0 \cdot  {\cos} (  \omega_{\rm T}\hspace{0.05cm} t + \eta \cdot q(t)).$$
-Here&nbsp; $\eta$&nbsp; denotes the so-called modulation index. Let the constant envelope&nbsp; $s_0$&nbsp; also be a dimensionless quantity, which is set to&nbsp; $s_0 = 2$&nbsp; in the following (see diagram).
+Here&nbsp; $\eta$&nbsp; denotes the so-called&nbsp; "modulation index".&nbsp; Let the constant envelope&nbsp; $s_0$&nbsp; also be a normalise quantity, which is set to&nbsp; $s_0 = 2$&nbsp; in the following (see diagram).
 If one replaces the cosine function with the complex exponential function, one arrives at the analytical signal
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 <quiz display=simple>
-{What is the magnitude function&nbsp; $a(t) = |s_{\rm TP}(t)|$? Which value is valid for&nbsp; $t = 0$?
+{What is the magnitude function&nbsp; $a(t) = |s_{\rm TP}(t)|$?&nbsp; Which value is valid for&nbsp; $t = 0$?
 |type="{}"}
 $a(t = 0)\ = \ $  { 2 3% }
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 |type="[]"}
 - From&nbsp; $q(t) = -0.5 = \text{const.}$&nbsp; follows&nbsp; $s(t) = s_0 \cdot \cos (\omega_T \cdot t)$.
-+ With a rectangular signal&nbsp; $q(t)$&nbsp; $($with only two possible signal values&nbsp; $\pm 0.5)$&nbsp; the locus curve degenerates to two points.
++ With a rectangular signal&nbsp;  $($with only two possible signal values&nbsp; $q(t)=\pm 0.5)$&nbsp; the locality curve degenerates to two points.
-+ With the signal values&nbsp; $\pm 1$&nbsp; $(q_{\rm min} = -0.5$&nbsp; is then no longer valid$)$ the locus curve degenerates to one point: &nbsp; $s_{\rm TP}(t) = -s_0$.
++ With the signal values&nbsp; $\pm 1$&nbsp; $(q_{\rm min} = -0.5$&nbsp; is then no longer valid$)$ the locality curve degenerates to one point: &nbsp; $s_{\rm TP}(t) = -s_0$.

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 ''Hints:''
-*This task belongs to the chapter&nbsp; [[Signal_Representation/Equivalent_Low_Pass_Signal_and_Its_Spectral_Function|Equivalent Low Pass Signal and Its Spectral Function]].
+*This task belongs to the chapter&nbsp; [[Signal_Representation/Equivalent_Low_Pass_Signal_and_Its_Spectral_Function|Equivalent Low-Pass Signal and Its Spectral Function]].
-*You can check your solution with the interactive applet&nbsp; [[Applets:Physikalisches_Signal_%26_Äquivalentes_TP-Signal|Physical Signal and Equivalent Low Pass Signal]] &nbsp; &rArr; &nbsp; locus curve,
+*You can check your solution with the interactive applet&nbsp; [[Applets:Physikalisches_Signal_%26_Äquivalentes_TP-Signal|Physical Signal and Equivalent Low-Pass Signal]] &nbsp; &rArr; &nbsp; locus curve,

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← Older revision	Revision as of 15:15, 26 November 2020
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