Aufgaben:Exercise 1.4: "Pointer diagram" and "Locality Curve" - Revision history

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 *This exercise belongs to the chapter &nbsp; [[Modulation_Methods/General_Model_of_Modulation|General Model of Modulation]].
 *Particular reference is made to the page&nbsp;  [[Modulation_Methods/General_Model_of_Modulation#Describing_the_physical_signal_using_the_equivalent_low-pass_signal|Describing the physical signal using the equivalent low-pass signal]].
-*You will find further information on these topics in these chapters of the book „Signal Representation”:
+*You will find further information on these topics in these chapters of the book "Signal Representation":
 ::(1) &nbsp; [[ Signal_Representation/Harmonic_Oscillation|Harmonic Oscillation]],
 ::(2)&nbsp; [[Signal_Representation/Analytical_Signal_and_Its_Spectral_Function|Analytical Signal and its Spectral Function]],&nbsp;

@@ Line 16: / Line 16: @@
-Reference will also be made to the &nbsp;[[Modulation_Methods/Allgemeines_Modell_der_Modulation#Beschreibung_von_s.28t.29_mit_Hilfe_des_analytischen_Signals|equivalent lowpass signal]] &nbsp;$s_{\rm TP}(t)$, with the following relationship to the analytical signal:
+Reference will also be made to the &nbsp;[[Modulation_Methods/General_Model_of_Modulation#Describing_the_physical_signal_using_the_equivalent_low-pass_signal|equivalent lowpass signal]] &nbsp;$s_{\rm TP}(t)$, with the following relationship to the analytical signal:
 :$$s_{\rm TP}(t) = s_+(t) \cdot {\rm e}^{-{\rm j}\hspace{0.03cm} \cdot \hspace{0.03cm}\omega_{\rm T}\cdot \hspace{0.05cm}t} \hspace{0.05cm}.$$
@@ Line 23: / Line 23: @@
 Hints:
-*This exercise belongs to the chapter &nbsp; [[Modulation_Methods/Allgemeines_Modell_der_Modulation|General Model of Modulation]].
+*This exercise belongs to the chapter &nbsp; [[Modulation_Methods/General_Model_of_Modulation|General Model of Modulation]].
 *Particular reference is made to the page&nbsp;  [[Modulation_Methods/General_Model_of_Modulation#Describing_the_physical_signal_using_the_equivalent_low-pass_signal|Describing the physical signal using the equivalent low-pass signal]].
 *You will find further information on these topics in these chapters of the book „Signal Representation”:

@@ Line 24: / Line 24: @@
 Hints:
 *This exercise belongs to the chapter &nbsp; [[Modulation_Methods/Allgemeines_Modell_der_Modulation|General Model of Modulation]].
-*Particular reference is made to the page&nbsp;  [[Modulation_Methods/General_Model_of_Modulation#Describing_the_physical_signal_using_the_equivalent_low-pass_signal|Describing the physical signal using the equivalent lowpass signal]].
+*Particular reference is made to the page&nbsp;  [[Modulation_Methods/General_Model_of_Modulation#Describing_the_physical_signal_using_the_equivalent_low-pass_signal|Describing the physical signal using the equivalent low-pass signal]].
 *You will find further information on these topics in these chapters of the book „Signal Representation”:
 ::(1) &nbsp; [[ Signal_Representation/Harmonic_Oscillation|Harmonic Oscillation]],

@@ Line 24: / Line 24: @@
 Hints:
 *This exercise belongs to the chapter &nbsp; [[Modulation_Methods/Allgemeines_Modell_der_Modulation|General Model of Modulation]].
-*Particular reference is made to the page&nbsp;  [[Modulation_Methods/Allgemeines_Modell_der_Modulation#Beschreibung_des_physikalischen_Signals_mit_Hilfe_des_.C3.A4quivalenten_TP-Signals|Describing the physical signal using the equivalent lowpass signal]].
+*Particular reference is made to the page&nbsp;  [[Modulation_Methods/General_Model_of_Modulation#Describing_the_physical_signal_using_the_equivalent_low-pass_signal|Describing the physical signal using the equivalent lowpass signal]].
 *You will find further information on these topics in these chapters of the book „Signal Representation”:
 ::(1) &nbsp; [[ Signal_Representation/Harmonic_Oscillation|Harmonic Oscillation]],

@@ Line 31: / Line 31: @@
 *In our tutorial $\rm LNTwww$, the plot of the analytical signal&nbsp;$s_+(t)$&nbsp; n the complex plane is sometimes referred to as the "pointer diagram", while the "locus curve" gives the time course of the equivalent lowpass signal &nbsp;$s_{\rm TP}(t)$&nbsp;. We refer you to the corresponding interactive Applets
-::(1) &nbsp;[[Applets:Physical_Signal_%26_Analytic_Signal|Physical & Analytic Signal]],
+::(1) &nbsp;[[Applets:Physical_Signal_%26_Analytic_Signal|Physical and analytic signal]],
-::(2) &nbsp;[[Applets:Physical_Signal_%26_Equivalent_Lowpass_Signal|Physical Signal & Equivalent Low-pass Signal]].
+::(2) &nbsp;[[Applets:Physical_Signal_%26_Equivalent_Lowpass_Signal|Physical signal and equivalent low-pass signal]].

@@ Line 22: / Line 22: @@
 Hints:
 *This exercise belongs to the chapter &nbsp; [[Modulation_Methods/Allgemeines_Modell_der_Modulation|General Model of Modulation]].
 *Particular reference is made to the page&nbsp;  [[Modulation_Methods/Allgemeines_Modell_der_Modulation#Beschreibung_des_physikalischen_Signals_mit_Hilfe_des_.C3.A4quivalenten_TP-Signals|Describing the physical signal using the equivalent lowpass signal]].
@@ Line 76: / Line 72: @@
 :$$s_{\rm TP}(t) = s_+(t) \cdot {\rm e}^{-{\rm j}\hspace{0.03cm} \cdot \hspace{0.03cm}\omega_{\rm T}\cdot \hspace{0.05cm}t} = A_{\rm T}+ {A_0} \cdot \cos (\omega_{\rm 0}\cdot t) \hspace{0.05cm}.$$
 Therefore the <u>last answer</u> is correct.
-*In the chapter "Envelope Demodulation" of the present book we will see that we are dealing with ''double sideband amplitude modulation of a cosine signal with a cosine carrier''.
+*In the chapter&nbsp;  "Envelope Demodulation"&nbsp; of the present book we will see that we are dealing with&nbsp; "DSB-AM of a cosine signal with a cosine carrier".
-'''(2)'''&nbsp; The period of the analytical signal &nbsp; $s_+(t)$&nbsp; is&nbsp; $T_0 = 50$&nbsp; μs.
+'''(2)'''&nbsp; The period of the analytical signal&nbsp; $s_+(t)$&nbsp; is&nbsp; $T_0 = 50$&nbsp; μs.
 *The physical signal&nbsp; $s(t)$&nbsp; has the same period duration.
-*Assuming that &nbsp; $f_{\rm T}$&nbsp;is an integer multiple of &nbsp; $f_0$&nbsp;(which always first needs confirming, but is true for this example), this results in
+*Assuming that&nbsp; $f_{\rm T}$&nbsp;is an integer multiple of&nbsp; $f_0$&nbsp; (which always first needs confirming, but is true for this example),&nbsp; this results in
 :$$f_0 = 1/T_0 \hspace{0.15cm}\underline{ = 20 \ \rm kHz}.$$
-'''(3)'''&nbsp; At the given times (multiples of $5$&nbsp; μs), the following applies to the complex rotating pointer of the carrier:
+'''(3)'''&nbsp; At the given times&nbsp; (multiples of $5$&nbsp; μs),&nbsp; the following applies to the complex rotating pointer of the carrier:
 :$${\rm e}^{\hspace{0.05cm}{\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm} 2 \pi \cdot \hspace{0.05cm} {100\,{\rm kHz}}\cdot \hspace{0.05cm}(k \hspace{0.05cm}\cdot \hspace{0.05cm} 5\,{\rm \mu s})} = {\rm e}^{\hspace{0.05cm}{\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm}k \hspace{0.03cm} \cdot \hspace{0.05cm} \pi } = \left\{ \begin{array}{c} +1 \\ -1 \\ \end{array} \right. \begin{array}{*{10}c} {\rm{if}} \\ {\rm{if}} \\ \end{array}\begin{array}{*{20}c} k \hspace{0.1cm}{\rm even} , \\ k \hspace{0.1cm}{\rm odd} . \\ \end{array}$$
-*Therefore, it follows from the equation calculated in Question&nbsp; '''(1)''' :
+*Therefore,&nbsp; it follows from the equation calculated in Question&nbsp; '''(1)''':
 :$$k = 0  \Rightarrow  \hspace{0.2cm} s_{\rm +}(t = 0) = A_{\rm T}+ {A_0} \cdot \cos (\omega_{\rm 0}\cdot 0) = A_{\rm T}+ {A_0} \hspace{0.05cm},$$
 :$$k = 5  \Rightarrow \hspace{0.2cm} s_{\rm +}(t = 25\;{\rm &micro; s}) = - \left[ A_{\rm T}+ {A_0} \cdot \cos (\omega_{\rm 0}\cdot {T_0}/{2}) \right] = -A_{\rm T}+ {A_0} \hspace{0.05cm}.$$
@@ Line 95: / Line 91: @@
 :$$ s_{\rm +}(t = 0)  =  A_{\rm T}+ {A_0}=1.5 \hspace{0.05cm}, $$
 :$$ s_{\rm +}(t = 25\;{\rm \mu s})  =  -A_{\rm T}+ {A_0} = -0.5 \hspace{0.05cm}.$$
-*This gives &nbsp; $A_{\rm T} \hspace{0.15cm}\underline{ = 1}$&nbsp; and&nbsp; $A_0 \hspace{0.15cm}\underline{ = 0.5}$.
+*This gives&nbsp; $A_{\rm T} \hspace{0.15cm}\underline{ = 1}$&nbsp; and&nbsp; $A_0 \hspace{0.15cm}\underline{ = 0.5}$.
-'''(4)'''&nbsp; At time&nbsp; $t = 15$ μs&nbsp; $(k = 3$, odd$)$&nbsp; it holds that:
+'''(4)'''&nbsp; At time&nbsp; $t = 15$ μs&nbsp; $(k = 3$,&nbsp; odd$)$&nbsp; it holds that:
 :$$ s_{\rm +}(t = 15\;{\rm &micro; s})  =  - \left[ 1+ 0.5 \cdot \cos (2 \pi \cdot 20\,{\rm kHz} \cdot 0.015\,{\rm ms}) \right] \hspace{0.05cm}  =  -1- 0.5 \cdot \cos (108^{\circ})\hspace{0.15cm}\underline {= -0.845} \hspace{0.05cm}.$$
-*In contrast, for time &nbsp; $t = 20$&nbsp; μs&nbsp; $(k = 4$, even$)$:
+*In contrast, for time &nbsp; $t = 20$&nbsp; μs&nbsp; $(k = 4$,&nbsp; even$)$:
 :$$ s_{\rm +}(t = 20\;{\rm &micro; s}) = 1 + 0.5 \cdot \cos (144^{\circ})\hspace{0.15cm}\underline {= 0.595} \hspace{0.05cm}.$$
 At all these considered time points, the physical signal $s(t) = {\rm Re}[s_+(t)]$ is exactly as large.

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 {{quiz-Header|Buchseite=Modulation_Methods/General_Model_of_Modulation}}
-[[File:P_ID966__Mod_A_1_4_neu.png|right|frame|Vorgegebenes analytisches Signal in der komplexen Ebene]]
+[[File:P_ID966__Mod_A_1_4_neu.png|right|frame|A given analytical signal in the complex plane]]
 The accompanying graph shows the analytical signal &nbsp;$s_+(t)$&nbsp; in the complex plane.
 *The numbers shown in the rectangles indicate the time points in microseconds.