Aufgaben:Exercise 4.12: Root-Nyquist Systems - Revision history

Guenter at 12:07, 19 April 2022

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Guenter at 11:58, 19 April 2022

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Guenter at 11:35, 19 April 2022

2022-04-19T11:35:25Z

Guenter: Guenter moved page Aufgaben:Exercise 4.12: Root Nyquist Systems to Aufgaben:Exercise 4.12: Root-Nyquist Systems

2022-04-19T09:24:25Z

Guenter moved page Exercise 4.12: Root Nyquist Systems to Exercise 4.12: Root-Nyquist Systems

Guenter at 09:24, 19 April 2022

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Reed: /* Solution */

2022-03-19T17:55:15Z

Solution

Reed at 17:54, 19 March 2022

2022-03-19T17:54:45Z

← Older revision		Revision as of 12:07, 19 April 2022
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	*In this case,  the basic detection pulse  $g_d(t)$  satisfies the  [[Digital_Signal_Transmission/Properties_of_Nyquist_Systems#First_Nyquist_criterion_in_the_time_domain\|first Nyquist criterion]],  since  $G_d(f)$  is point-symmetric about the so-called  "Nyquist frequency"  $f_{\rm Nyq} = 1/T$ .		*In this case,  the basic detection pulse  $g_d(t)$  satisfies the  [[Digital_Signal_Transmission/Properties_of_Nyquist_Systems#First_Nyquist_criterion_in_the_time_domain\|first Nyquist criterion]],  since  $G_d(f)$  is point-symmetric about the so-called  "Nyquist frequency"  $f_{\rm Nyq} = 1/T$ .
−	*$G_d(f)$  is a  [[Linear_and_Time_Invariant_Systems/Some_Low-Pass_Functions_in_Systems_Theory#Raised-cosine_low-pass_filter\|cosine ~~rolloff~~ spectrum]],  where the rolloff factor  $r$  can take values from $0$  to  $1$  (including these limits).	+	*$G_d(f)$  is a  [[Linear_and_Time_Invariant_Systems/Some_Low-Pass_Functions_in_Systems_Theory#Raised-cosine_low-pass_filter\|raised-cosine spectrum]],  where the rolloff factor  $r$  can take values from $0$  to  $1$  (including these limits).

← Older revision		Revision as of 12:06, 19 April 2022
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	*In this case,  the basic detection pulse  $g_d(t)$  satisfies the  [[Digital_Signal_Transmission/Properties_of_Nyquist_Systems#First_Nyquist_criterion_in_the_time_domain\|first Nyquist criterion]],  since  $G_d(f)$  is point-symmetric about the so-called  "Nyquist frequency"  $f_{\rm Nyq} = 1/T$ .		*In this case,  the basic detection pulse  $g_d(t)$  satisfies the  [[Digital_Signal_Transmission/Properties_of_Nyquist_Systems#First_Nyquist_criterion_in_the_time_domain\|first Nyquist criterion]],  since  $G_d(f)$  is point-symmetric about the so-called  "Nyquist frequency"  $f_{\rm Nyq} = 1/T$ .
−	*$G_d(f)$  is a  [[Linear_and_Time_Invariant_Systems/Some_Low-Pass_Functions_in_Systems_Theory#~~Cosine.E2.80.93square_low~~-pass_filter\|cosine rolloff spectrum]],  where the rolloff factor  $r$  can take values from $0$  to  $1$  (including these limits).	+	*$G_d(f)$  is a  [[Linear_and_Time_Invariant_Systems/Some_Low-Pass_Functions_in_Systems_Theory#Raised-cosine_low-pass_filter\|cosine rolloff spectrum]],  where the rolloff factor  $r$  can take values from $0$  to  $1$  (including these limits).

@@ Line 83: / Line 83: @@
 ===Solution===
 {{ML-Kopf}}
-'''(1)'''&nbsp; If we substitute &nbsp; $r = 0$&nbsp; into the given equation, the first terms in the numerator and denominator disappear and we get:
+'''(1)'''&nbsp; If we substitute&nbsp; $r = 0$&nbsp; into the given equation,&nbsp; the first terms in the numerator and denominator disappear and we get:
-: $$g_s(t) = g_0 \cdot \frac{\sin \left (\pi \cdot t/T \right )}{\pi \cdot t/T} = g_0 \cdot {\rm si} \left (\pi \cdot {t}/{T} \right )\hspace{0.05cm}.$$
+: $$g_s(t) = g_0 \cdot \frac{\sin \left (\pi \cdot t/T \right )}{\pi \cdot t/T} = g_0 \cdot {\rm sinc} \left ( {t}/{T} \right )\hspace{0.05cm}.$$
-*At time&nbsp; $t = 0$&nbsp;, the &nbsp; $\rm si$–impulse is equal to&nbsp; $g_0$: &nbsp;
+*At time&nbsp; $t = 0$,&nbsp;  ${\rm sinc} \left ( {t}/{T} \right ) =g_0$: &nbsp;
 :$$ g_s(t) \hspace{0.15cm}\underline { = 1.0 } \cdot g_0 \hspace{0.05cm}.$$
-'''(2)'''&nbsp; When&nbsp; $r = 1$&nbsp;, the given equation simplies as follows:
+'''(2)'''&nbsp; When&nbsp; $r = 1$,&nbsp; the given equation simplies as follows:
 :$$g_s(t) = \frac{4 \cdot g_0}{\pi} \cdot \frac{ \cos \left (2 \pi \cdot t/T \right )}{\left [1- (4 t/T)^2 \right ] }\hspace{0.3cm}\Rightarrow \hspace{0.3cm} g_s(t = 0) = \frac{4 \cdot g_0}{\pi} \hspace{0.15cm}\underline {= 1.273 }\cdot g_0 \hspace{0.05cm}.$$
-'''(3)'''&nbsp; The <u>last answer</u> is correct:
+'''(3)'''&nbsp; The&nbsp; <u>last answer</u>&nbsp; is correct:
-*Zero intercepts are only possible for&nbsp; $r = 1$&nbsp; if the cosine function in the numerator is zero, that is, for all integer values of &nbsp; $k$:
+*Zero intercepts are only possible for&nbsp; $r = 1$&nbsp; if the cosine function in the numerator is zero,&nbsp; that is,&nbsp; for all integer values of &nbsp; $k$:
 :$$2 \pi \cdot t/T = {\pi}/{2} + k \cdot \pi \hspace{0.3cm}\Rightarrow \hspace{0.3cm} t = \pm 0.25T, \hspace{0.15cm} \pm 0.75T, \hspace{0.15cm}\pm 1.25T, \hspace{0.15cm} ...$$
-*However, only the last answer is correct, since the zero values at &nbsp; $±0.25T$&nbsp; are cancelled by the zero in the denominator.
+*However,&nbsp; only the last answer is correct,&nbsp; since the zero values at &nbsp; $±0.25T$&nbsp; are cancelled by the zero in the denominator.
 *Applying de l'Hospital's rule yields &nbsp; $g_s(t = ± 0.25T) = g_0$.
@@ Line 104: / Line 104: @@
-'''(4)'''&nbsp;  With&nbsp; $r = 0.5$&nbsp; and the shortcut&nbsp; $x = t/T$&nbsp;, one gets:
+'''(4)'''&nbsp;  With&nbsp; $r = 0.5$&nbsp; and the shortcut&nbsp; $x = t/T$,&nbsp; one gets:
 :$$g_s(x) = \frac{g_0}{\pi} \cdot \frac{2 \cdot x \cdot \cos \left (1.5\pi \cdot x \right )+ \sin \left (0.5\pi \cdot x \right )}{\left (1- 4 \cdot x^2 \right ) \cdot x}\hspace{0.05cm}.$$
-*For the calculation at time &nbsp; $t = 0$&nbsp;, de l'Hospital's rule must be applied.
+*For the calculation at time&nbsp; $t = 0$,&nbsp; de l'Hospital's rule must be applied.
 *The derivatives of the numerator and denominator give:
 :$$Z'(x)  =  2 \cdot \cos \left (1.5\pi \cdot x \right ) - 3 \pi \cdot x \cdot \sin \left (1.5\pi \cdot x \right ) + 0.5 \pi \cdot \cos \left (0.5\pi \cdot x \right ),$$
-[[File:P_ID1723__Mod_A_4_11b.png|right|frame|Fundamental transmission pulse (root Nyquist) and base detection pulse (Nyquist)]]
+[[File:P_ID1723__Mod_A_4_11b.png|right|frame|Basic transmission pulse&nbsp; (root-Nyquist)&nbsp; and basic detection pulse&nbsp; (Nyquist)]]
 :$$N'(x)  =  \left (1- 4 \cdot x^2 \right ) - 8 \cdot x^2 \hspace{0.05cm}.$$
@@ Line 118: / Line 118: @@
 :$$g_s(t=0) = \frac{g_0}{\pi} \cdot \left ( 2 +{\pi }/{2} \right ) = {g_0} \cdot \left ( 0.5 + {2}/{\pi } \right )\hspace{0.15cm}\underline {= 1.137} \cdot g_0 \hspace{0.05cm}.$$
-Here, the graph illustrates the results calculated again:
+Here,&nbsp; the graph illustrates the results calculated again:
-*The impulse&nbsp; $g_d(t)$&nbsp; is a Nyquist pulse, meaning that it has zero crossings at least at all multiples of the symbol duration $T$ (and possibly others depending on the rolloff factor).
+*$g_d(t)$&nbsp; is a Nyquist pulse,&nbsp; meaning that it has zero crossings at least at all multiples of the symbol duration&nbsp; $T$&nbsp; (and possibly others depending on the rolloff factor).
-*On the other hand, the impulse&nbsp; $g_s(t)$&nbsp; does not satisfy the Nyquist criterion.
+*On the other hand,&nbsp; the  pulse&nbsp; $g_s(t)$&nbsp; does not satisfy the Nyquist criterion.&nbsp; Moreover,&nbsp; from this plot one can once again see that for &nbsp; $r ≠ 0$&nbsp; the pulse amplitude $g_s(t = 0)$&nbsp; is always larger than $g_0$.
-'''(5)'''&nbsp; The <u>last answer</u>&nbsp; is correct $($the first answer is ruled out from the results in questions&nbsp; '''(2)'''&nbsp; and&nbsp; '''(4)'''&nbsp;$)$.&nbsp; The validity of the lower bound &nbsp; $g_0$&nbsp; and the upper bound &nbsp; $4g_0/π$&nbsp; can be proved as follows:
+'''(5)'''&nbsp; The&nbsp; <u>last answer</u>&nbsp; is correct&nbsp; $($the first answer is ruled out from the results in questions&nbsp; '''(2)'''&nbsp; and&nbsp; '''(4)'''&nbsp;$)$.&nbsp; The validity of the lower bound &nbsp; $g_0$&nbsp; and the upper bound &nbsp; $4g_0/π$&nbsp; can be proved as follows:
 * The pulse amplitude&nbsp; $g_s(t = 0)$&nbsp; is generally equal to the area under the spectral function&nbsp; $G_s(f)$.
 * The smallest area is obtained for&nbsp; $r = 0$.&nbsp; Here, &nbsp; $G_s(f) = g_0 · T$&nbsp; is in the range&nbsp; $|f| < ±1/(2T)$.&nbsp; Thus, the area is equal to&nbsp; $g_0$.
-* The largest area is obtained for&nbsp; $r = 1$. Here &nbsp; $G_s(f)$&nbsp; extends to the range &nbsp; $±1/T$&nbsp; and has a cosine shape.
+* The largest area is obtained for&nbsp; $r = 1$. Here, &nbsp; $G_s(f)$&nbsp; extends to the range &nbsp; $±1/T$&nbsp; and has a cosine shape.
 *The result&nbsp; $g_s(t = 0) = 4g_0/π$&nbsp; was already calculated in question &nbsp; '''(3)'''&nbsp;.&nbsp; Though it still holds that:
 :$$g_s(t=0)  =  2 \cdot {g_0} \cdot \int_{ 0 }^{1/T} {\cos\left(\frac{\pi }{2}\cdot f \cdot T \right)}\hspace{0.1cm} {\rm d}f = \frac{4 g_0}{\pi} \cdot \int_{ 0 }^{\pi/2} {\cos\left(x \right)}\hspace{0.1cm} {\rm d}x = {4 g_0}/{\pi} \cdot \big[\sin(\pi/2) - \sin(0) \big] = {4 g_0}/{\pi}\hspace{0.05cm}.$$
@@ Line 134: / Line 133: @@
-'''(6)'''&nbsp; The energy of the fundamental transmission pulse &nbsp; $g_s(t)$&nbsp; can be found in the time or frequency domain according to Parseval's theorem:
+'''(6)'''&nbsp; The energy of the basic transmission pulse &nbsp; $g_s(t)$&nbsp; can be found in the time or frequency domain according to Parseval's theorem:
 :$$E_{g_s} = \int_{ -\infty }^{+\infty} {[g_s(t)]^2}\hspace{0.1cm} {\rm d}t = \int_{ -\infty }^{+\infty} {|G_s(f)|^2}\hspace{0.1cm} {\rm d}f \hspace{0.05cm}.$$
-*From the equations and the graph on the exercise page, we can see that &nbsp; $|G_s(f)|^2$&nbsp; is the same shape as &nbsp; $G_d(f)$&nbsp;, except that the height is now &nbsp; $(g_0 · T)^2$&nbsp; instead of &nbsp; $g_0 · T$&nbsp;:
+*From the equations and graph on the exercise page,&nbsp; we can see that &nbsp; $|G_s(f)|^2$&nbsp; has the same shape as &nbsp; $G_d(f)$,&nbsp; but the height is now &nbsp; $(g_0 · T)^2$&nbsp; instead of &nbsp; $g_0 · T$:
 :$$E_{g_s} = \int_{ -\infty }^{+\infty} {|G_s(f)|^2}\hspace{0.1cm} {\rm d}f = \frac{g_0^2 \cdot T^2}{g_0 \cdot T} \cdot \int_{ -\infty }^{+\infty} {G_d(f)}\hspace{0.1cm} {\rm d}f \hspace{0.05cm}.$$
-*Due to the Nyquist form of&nbsp; $G_d(f)$&nbsp;, it holds independently of &nbsp; $r$ that:
+*Due to the Nyquist form of&nbsp; $G_d(f)$,&nbsp; it holds independently of &nbsp; $r$:
 :$$\int_{ -\infty }^{+\infty} {G_d(f)}\hspace{0.1cm} {\rm d}f = g_0 \hspace{0.05cm}.$$
-*Thus, the impulse energy is also independent of &nbsp; $r$, so it is also valid for &nbsp; $r = 0$&nbsp; and&nbsp; $r = 1$.&nbsp;. In <u>both cases</u>,  $E_ {g_s}\hspace{0.15cm}\underline { = 1.0} · g_0^2 · T.$
+*Thus,&nbsp; the pulse energy is also independent of&nbsp; $r$,&nbsp; so it is also valid for &nbsp; $r = 0$&nbsp; and&nbsp; $r = 1$.&nbsp; In&nbsp; <u>both cases</u>,&nbsp;  $E_ {g_s}\hspace{0.15cm}\underline { = 1.0} · g_0^2 · T.$
 {{ML-Fuß}}

@@ Line 39: / Line 39: @@
 Hints:
 *This exercise belongs to the chapter&nbsp; [[Modulation_Methods/Quadrature_Amplitude_Modulation|"Quadrature Amplitude Modulation"]].
-*Particular reference is made to the page &nbsp; [[Modulation_Methods/Quadrature_Amplitude_Modulation#Nyquist_and_root-Nyquist_QAM_systems|"Nyquist and root-Nyquist systems"]]&nbsp; in this chapter.
+*Particular reference is made to the page &nbsp; [[Modulation_Methods/Quadrature_Amplitude_Modulation#Nyquist_and_Root-Nyquist_QAM_systems|"Nyquist and Root-Nyquist systems"]]&nbsp; in this chapter.
 *Further useful informations can be found in the chapter&nbsp; [[Digital_Signal_Transmission/Properties_of_Nyquist_Systems|Properties of Nyquist Systems]]&nbsp; in the book&nbsp; "Digital Signal Transmission".
 * '''[Kam04]'''&nbsp; refers to the textbook&nbsp; "Kammeyer, K.D.:&nbsp; Nachrichtenübertragung.&nbsp; Stuttgart: B.G. Teubner, 4. Auflage, 2004".

@@ Line 39: / Line 39: @@
 Hints:
 *This exercise belongs to the chapter&nbsp; [[Modulation_Methods/Quadrature_Amplitude_Modulation|"Quadrature Amplitude Modulation"]].
-*Particular reference is made to the page &nbsp; [[Modulation_Methods/Quadrature_Amplitude_Modulation#Nyquist_and_root_Nyquist_QAM_systems|"Nyquist and root-Nyquist systems"]]&nbsp; in this chapter.
+*Particular reference is made to the page &nbsp; [[Modulation_Methods/Quadrature_Amplitude_Modulation#Nyquist_and_root-Nyquist_QAM_systems|"Nyquist and root-Nyquist systems"]]&nbsp; in this chapter.
 *Further useful informations can be found in the chapter&nbsp; [[Digital_Signal_Transmission/Properties_of_Nyquist_Systems|Properties of Nyquist Systems]]&nbsp; in the book&nbsp; "Digital Signal Transmission".
 * '''[Kam04]'''&nbsp; refers to the textbook&nbsp; "Kammeyer, K.D.:&nbsp; Nachrichtenübertragung.&nbsp; Stuttgart: B.G. Teubner, 4. Auflage, 2004".

@@ Line 4: / Line 4: @@
 [[File:P_ID1722__Mod_A_4_11.png|right|frame|Spectra of transmission pulse (above)  and detection pulse (below)]]
-In quadrature amplitude modulation systems, the ''root-Nyquist variant'' is often chosen instead of a rectangular basic transmission pulse, which gets its name from the spectral range. The reason for this is the significantly smaller bandwidth.
+In&nbsp; "quadrature amplitude modulation"&nbsp; $\rm (QAM)$&nbsp; systems,&nbsp; the&nbsp; "root-Nyquist variant"&nbsp; is often chosen&nbsp; (which gets its name from the spectral range)&nbsp; instead of a rectangular basic transmission pulse.&nbsp; The reason for this is the significantly smaller bandwidth.
-*In this case, base detection pulse &nbsp;$g_d(t)$&nbsp; satisfies the &nbsp;[[Digital_Signal_Transmission/Properties_of_Nyquist_Systems#First_Nyquist_criterion_in_the_time_domain|first Nyquist criterion]],&nbsp; since &nbsp;$G_d(f)$&nbsp; is point-symmetric about the so-called Nyquist frequency &nbsp;$f_{\rm Nyq} = 1/T$&nbsp;.
+*In this case,&nbsp; the basic detection pulse &nbsp;$g_d(t)$&nbsp; satisfies the &nbsp;[[Digital_Signal_Transmission/Properties_of_Nyquist_Systems#First_Nyquist_criterion_in_the_time_domain|first Nyquist criterion]],&nbsp; since &nbsp;$G_d(f)$&nbsp; is point-symmetric about the so-called&nbsp; "Nyquist frequency" &nbsp;$f_{\rm Nyq} = 1/T$&nbsp;.
-*The spectral function $G_d(f)$&nbsp; is a &nbsp;[[Linear_and_Time_Invariant_Systems/Some_Low-Pass_Functions_in_Systems_Theory#Cosine.E2.80.93square_low-pass_filter|cosine rolloff spectrum]], where the rolloff factor &nbsp;$r$&nbsp; can take values from&nbsp;$0$&nbsp; to &nbsp;$1$&nbsp; (including these limits).
+*$G_d(f)$&nbsp; is a &nbsp;[[Linear_and_Time_Invariant_Systems/Some_Low-Pass_Functions_in_Systems_Theory#Cosine.E2.80.93square_low-pass_filter|cosine rolloff spectrum]],&nbsp; where the rolloff factor &nbsp;$r$&nbsp; can take values from&nbsp;$0$&nbsp; to &nbsp;$1$&nbsp; (including these limits).
-Furthermore, the following holds for the Nyquist frequency response:
+Furthermore,&nbsp; the following holds for the Nyquist frequency response:
-* When &nbsp;$|f| < f_1 = f_{\rm Nyq} · (1 – r)$&nbsp;, &nbsp;$G_d(f)$&nbsp; is constant and equal to &nbsp;$g_0 · T$.
+* When &nbsp;$|f| < f_1 = f_{\rm Nyq} · (1 – r)$ &nbsp; &rArr; &nbsp; $G_d(f)$&nbsp; is constant and equal to &nbsp;$g_0 · T$.
-* At frequencies greater tha &nbsp;$f_2 = f_{\rm Nyq} · (1 + r)$&nbsp;,&nbsp;$G_d(f)$&nbsp; has no components.
+* At frequencies greater than &nbsp;$f_2 = f_{\rm Nyq} · (1 + r)$ &nbsp; &rArr; &nbsp; $G_d(f)$&nbsp; has no components.
-* In between, the slope is cosine.
+* In between,&nbsp; the slope is cosine.
 The optimization of digital communication systems requires that the receiver frequency response &nbsp;$H_{\rm E}(f)$&nbsp; should be of the same shape as the transmission spectrum&nbsp;$G_s(f)$&nbsp;.
-To obtain dimensionally correct spectral functions for this task and the graph, it is assumed that:
+To obtain dimensionally correct spectral functions for this task and the graph,&nbsp; it is assumed that
-:$$G_s(f) = \sqrt{g_0 \cdot T \cdot G_d(f)},\hspace{0.4cm} H_{\rm E}(f) = \frac{1}{g_0 \cdot T}\cdot G_s(f)\hspace{0.05cm}.$$
+:$$G_s(f) = \sqrt{g_0 \cdot T \cdot G_d(f)},$$
-The top graph shows the transmission spectrum &nbsp;$G_s(f)$&nbsp; for the rolloff factors.
+The top graph shows the transmission spectrum &nbsp;$G_s(f)$&nbsp; for the rolloff factors
 *$r = 0$ &nbsp; (green dotted rectangle),
 *$r = 0.5$ &nbsp; (blue solid curve),
@@ Line 27: / Line 28: @@
-Below, the spectrum &nbsp; $G_d(f)$&nbsp; before the decider is shown in the same colors.
+Below,&nbsp; the spectrum&nbsp; $G_d(f)$&nbsp; of the basic detection pulse before the decider is shown in the same colors.
-*The associated impulse &nbsp; $g_d(t)$&nbsp; is a [[Digital_Signal_Transmission/Properties_of_Nyquist_Systems#First_Nyquist_criterion_in_the_time_domain|Nyquist impulse]] pulse for all valid rolloff factors &nbsp; $(0 ≤ r ≤ 1)$&nbsp; as opposed to the fundamental transmission pulse &nbsp; $g_s(t)$.
+*The associated pulse &nbsp; $g_d(t)$&nbsp; is a [[Digital_Signal_Transmission/Properties_of_Nyquist_Systems#First_Nyquist_criterion_in_the_time_domain|Nyquist pulse]]&nbsp; for all valid rolloff factors &nbsp; $(0 ≤ r ≤ 1)$&nbsp; as opposed to the basic transmission pulse &nbsp; $g_s(t)$.
-*For this, the following equation is given in the literature - for example in&nbsp; '''[Kam04]''' :
+*For this,&nbsp; the following equation is given in the literature - for example in&nbsp; '''[Kam04]''' :
 :$$g_s(t) = g_0 \cdot \frac{4 r t/T \cdot \cos \left [\pi \cdot (1+r) \cdot t/T \right ]+ \sin \left [\pi \cdot (1-r) \cdot t/T \right ]}{\left [1- (4 r t/T)^2 \right ] \cdot \pi \cdot t/T}\hspace{0.05cm}.$$
@@ Line 36: / Line 37: @@
 Hints:
+*This exercise belongs to the chapter&nbsp; [[Modulation_Methods/Quadrature_Amplitude_Modulation|"Quadrature Amplitude Modulation"]].
+*Particular reference is made to the page &nbsp; [[Modulation_Methods/Quadrature_Amplitude_Modulation#Nyquist_and_root_Nyquist_QAM_systems|"Nyquist and root-Nyquist systems"]]&nbsp; in this chapter.
+*Further useful informations can be found in the chapter&nbsp; [[Digital_Signal_Transmission/Properties_of_Nyquist_Systems|Properties of Nyquist Systems]]&nbsp; in the book&nbsp; "Digital Signal Transmission".
+* '''[Kam04]'''&nbsp; refers to the textbook&nbsp; "Kammeyer, K.D.:&nbsp; Nachrichtenübertragung.&nbsp; Stuttgart: B.G. Teubner, 4. Auflage, 2004".
-''Hints:''
+*Energies are to be specified in&nbsp; $\rm V^2s$;&nbsp; they thus refer to the reference resistance &nbsp;$R = 1 \ \rm \Omega$.
@@ Line 54: / Line 50: @@
 <quiz display=simple>
-{What is the fundamental transmission pulse &nbsp;$g_s(t)$&nbsp; for the rolloff factor &nbsp;$r = 0$?&nbsp;?  What is the signal value at time &nbsp;$t = 0$?
+{What is the basic transmission pulse &nbsp;$g_s(t)$&nbsp; for the rolloff factor &nbsp;$r = 0$?&nbsp;  What is the signal value at time &nbsp;$t = 0$?
 |type="{}"}
 $g_s(t = 0) \ = \ $  { 1 3% } $\ \cdot g_0$
-{What is the fundamental transmission pulse &nbsp;$g_s(t)$&nbsp; for the rolloff factor&nbsp;$r = 1$?&nbsp; What is the signal value at time &nbsp;$t = 0$?
+{What is the basic transmission pulse &nbsp;$g_s(t)$&nbsp; for the rolloff factor&nbsp;$r = 1$?&nbsp; What is the signal value at time &nbsp;$t = 0$?
 |type="{}"}
 $g_s(t = 0) \ = \ $ { 1.273 3% }  $\ \cdot g_0$
@@ Line 68: / Line 64: @@
 + At &nbsp;$t = ±0.75 T, \ ±1.25 T,\  ±1.75 T$, ...
-{What is the fundamental transmission pulse &nbsp;$g_s(t)$&nbsp; for the rolloff factor &nbsp;$r = 0.5$?&nbsp; Welcher Signalwert ergibt sich zum Zeitpunkt &nbsp;$t = 0$?
+{What is the basic transmission pulse &nbsp;$g_s(t)$&nbsp; for the rolloff factor &nbsp;$r = 0.5$?&nbsp; What is the signal value at time &nbsp;$t = 0$?
 |type="{}"}
 $g_s(t = 0) \ = \ $ { 1.137 3% } $\ \cdot g_0$
-{Which statements are valid for the signal amplitude, independent of &nbsp;$r$&nbsp;?&nbsp; Solve using the frequency domain.
+{Which statements are valid for the pulse amplitude, independent of &nbsp;$r$&nbsp;?&nbsp; Solve using the frequency domain.
 |type="()"}
-- The transmit pulse amplitude can take any value in the range &nbsp; $0 ≤ g_s(t = 0) ≤ g_0$ &nbsp;.
+- The pulse amplitude can take any value in the range &nbsp; $0 ≤ g_s(t = 0) ≤ g_0$ &nbsp;.
-- The transmit pulse amplitude can take any value in the range &nbsp; $g_0 ≤ g_s(t = 0) ≤ 2 g_0$ &nbsp;.
+- The pulse amplitude can take any value in the range &nbsp; $g_0 ≤ g_s(t = 0) ≤ 2 g_0$ &nbsp;.
-+ The transmit pulse amplitude can take any value in the range &nbsp; $g_0 ≤ g_s(t = 0) ≤ 4 g_0/π$ &nbsp;.
++ The pulse amplitude can take any value in the range &nbsp; $g_0 ≤ g_s(t = 0) ≤ 4 g_0/π$ &nbsp;.
-{What is the energy&nbsp;$E_{g_s}$&nbsp; of the fundamental transmission pulse &nbsp;$g_s(t)$&nbsp; when &nbsp;$r = 0$&nbsp; and &nbsp;$r = 1$?
+{What is the energy&nbsp; $E_{g_s}$&nbsp; of the basic transmission pulse &nbsp;$g_s(t)$&nbsp; when &nbsp;$r = 0$&nbsp; and &nbsp;$r = 1$?
 |type="{}"}
 $r = 0\text{:} \ \ \ \  E_{g_s} \ = \ $ { 1 3% } $\ \cdot g_0^2 \cdot T$