Aufgaben:Exercise 1.1: Basic Transmission Pulses - Revision history

Hwang at 13:13, 9 May 2022

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Hwang at 13:12, 9 May 2022

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Guenter at 15:37, 27 April 2022

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Guenter at 15:18, 27 April 2022

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Guenter: Guenter moved page Aufgaben:Exercise 1.1: Basic Transmitter Pulses to Aufgaben:Exercise 1.1: Basic Transmission Pulses over redirect

2022-04-27T11:44:25Z

Guenter moved page Exercise 1.1: Basic Transmitter Pulses to Exercise 1.1: Basic Transmission Pulses over redirect

Guenter: Guenter moved page Aufgaben:Exercise 1.1: Basic Transmission Pulses to Aufgaben:Exercise 1.1: Basic Transmitter Pulses

2022-03-05T16:54:21Z

Guenter moved page Exercise 1.1: Basic Transmission Pulses to Exercise 1.1: Basic Transmitter Pulses

Hwang at 16:14, 10 February 2022

2022-02-10T16:14:24Z

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Guenter at 15:16, 2 February 2022

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Guenter: Guenter moved page Aufgaben:Aufgabe 1.1: Sendegrundimpulse to Aufgaben:Exercise 1.1: Basic Transmission Pulses

2022-02-02T15:12:45Z

Guenter moved page Aufgabe 1.1: Sendegrundimpulse to Exercise 1.1: Basic Transmission Pulses

@@ Line 87: / Line 87: @@
 :$$g_s(t)  =   \left\{ \begin{array}{c} s_0 \cdot \cos^2(\pi \cdot \frac{t}{T})  \\
 \\  \end{array} \right.\quad
-\begin{array}{*{1}c} {\rm{f\ddot{u}r}}
+\begin{array}{*{1}c} {\rm{for}}
 \\    \\ \end{array}\begin{array}{*{20}c}
 -T/2 \le t \le +T/2 \hspace{0.05cm}, \\

@@ Line 51: / Line 51: @@
 |type="{}"}
 $\text{For the signal}\ \ s_{\rm R}(t) \text{:} \ \ \Delta t_{\rm S}/T \ = \ $ { 1 3% }
-$\text{for the signal}\ \ s_{\rm C}(t) \text{:} \ \ \Delta t_{\rm S}/T \ = \ $ { 0.5 3% }
+$\text{For the signal}\ \ s_{\rm C}(t) \text{:} \ \ \Delta t_{\rm S}/T \ = \ $ { 0.5 3% }
 {What is the energy of the rectangular basic transmission pulse &nbsp;$g_s(t)$?

@@ Line 74: / Line 74: @@
 ===Solution===
 {{ML-Kopf}}
-'''(1)'''&nbsp;  <u>Solution 2</u> is correct:
+'''(1)'''&nbsp;  <u>Solution 2</u>&nbsp; is correct:
-*In both cases, the transmitted signal can be represented in the following form:
+*In both cases,&nbsp; the transmitted signal can be represented in the following form:
 :$$s(t) = \sum_{(\nu)} a_\nu \cdot g_s ( t - \nu \cdot T)$$
-*For the signal $s_{\rm R}(t)$, the amplitude coefficients $a_ν$ are either $0$ or $1$. Thus, a unipolar signal is present.
+*For the signal $s_{\rm R}(t)$,&nbsp; the amplitude coefficients&nbsp; $a_ν$&nbsp; are either&nbsp; $0$&nbsp; or&nbsp; $1$.&nbsp; Thus,&nbsp; a unipolar signal is present.
-*In contrast, for the bipolar signal $s_{\rm R}(t)$, $a_ν ∈ \{–1, +1\}$ holds.
+*In contrast,&nbsp; for the bipolar signal $s_{\rm R}(t)$ &nbsp; &rArr;  &nbsp;  $a_ν ∈ \{–1, +1\}$ holds.
-'''(2)'''&nbsp; The signal $s_{\rm R}(t)$ is NRZ rectangular.
+'''(2)'''&nbsp; The signal&nbsp; $s_{\rm R}(t)$&nbsp; is NRZ&nbsp; ("non-return-to-zero")&nbsp; rectangular.
-*Accordingly, both the absolute pulse duration $T_{\rm S}$ and the equivalent pulse duration $\Delta t_{\rm S}$ are equal to the symbol duration $T$:
+*Accordingly,&nbsp; both the absolute pulse duration&nbsp; $T_{\rm S}$&nbsp; and the equivalent pulse duration&nbsp; $\Delta t_{\rm S}$&nbsp; are equal to the symbol duration&nbsp; $T$:
 :$$T_{\rm S} / T = 1\hspace{0.05cm},\hspace{0.3cm}\Delta t_{\rm S} / T \hspace{0.1cm}\underline{ = 1 }\hspace{0.05cm}.$$
-*The basic transmission pulse for the signal $s_{\rm C}(t)$ is:
+*The basic transmission pulse for the signal&nbsp; $s_{\rm C}(t)$&nbsp; is:
 :$$g_s(t)  =   \left\{ \begin{array}{c} s_0 \cdot \cos^2(\pi \cdot \frac{t}{T})  \\
 \\  \end{array} \right.\quad
@@ Line 90: / Line 90: @@
 \\    \\ \end{array}\begin{array}{*{20}c}
 -T/2 \le t \le +T/2 \hspace{0.05cm}, \\
-{\rm sonst} \hspace{0.05cm}.  \\
+{\rm else} \hspace{0.05cm}.  \\
 \end{array}$$
-*From the diagram on the information section, we can see that the following values apply to the $\cos^2$ pulse:
+*From the diagram on the information section,&nbsp; we can see that the following values apply to the cosine&ndash;square pulse:
 :$$T_{\rm S} / T = 1\hspace{0.05cm},\hspace{0.3cm}\Delta t_{\rm S} / T \hspace{0.1cm}\underline{ = 0.5} \hspace{0.05cm}.$$
@@ Line 102: / Line 102: @@
-'''(4)'''&nbsp; For a bipolar square wave signal, the following would apply:
+'''(4)'''&nbsp; For a bipolar rectangular signal,&nbsp; the following would apply:
 :$$s_{\rm R}^2(t)= s_0^2 = {\rm const.} \hspace{0.3cm}\Rightarrow \hspace{0.3cm} P_s =  s_0^2 \cdot
   \lim_{T_{\rm M} \to \infty} \frac{1}{T_{\rm M}} \cdot \int ^{T_{\rm M}/2} _{-T_{\rm M}/2} \,{\rm
   d}t = s_0^2 \hspace{0.05cm}.$$
-*However, since the signal $s_{\rm R}(t)$ is unipolar here, in half the time $s_{\rm R}(t)= 0$. Thus, we get:
+*However,&nbsp; since the signal&nbsp; $s_{\rm R}(t)$ is&nbsp; unipolar here,&nbsp; in half the time $s_{\rm R}(t)= 0$.&nbsp; Thus,&nbsp; we get:
 :$$P_{\rm S} = {1}/{2} \cdot s_0^2 \hspace{0.1cm}\underline{= 0.25 \,{\rm
   W}}  \hspace{0.05cm}.$$
-'''(5)'''&nbsp; For the energy of the $\cos^2$ pulse holds:
+'''(5)'''&nbsp; For the energy of the cosine&ndash;square pulse holds:
 :$$E_g =   \int^{+\infty} _{-\infty} g_s^2(t)\,{\rm
   d}t = 2 \cdot s_0^2 \cdot \int^{T/2} _{0} \cos^4(\pi \cdot {t}/{T})\,{\rm
   d}t \hspace{0.05cm}.$$
-*Here, the formula derived in point '''(3)''' and the symmetry of $g_s(t)$ about time $t = 0$ are considered.
+*Here,&nbsp; the formula derived in subtask&nbsp; '''(3)'''&nbsp; and the symmetry of&nbsp; $g_s(t)$&nbsp; about time&nbsp; $t = 0$&nbsp; are considered.
-*The integral is given in the task description, where $a = π/T$ is to be set:
+*The integral is given in the task description,&nbsp; where $a = π/T$&nbsp; is to be set:
 :$$E_g =    2  \cdot s_0^2 \cdot \left [ \frac{3}{8} \cdot t + \frac{T}{4\pi} \cdot \sin(2 \pi \frac{t}{T})+ \frac{T}{32\pi} \cdot
   \sin(4 \pi \frac{t}{T})\right ]_{0}^{T/2}\hspace{0.05cm}.$$
-*The lower bound $t = 0$ always yields the result $0$. With respect to the upper bound, only the first term yields a result different from $0$. Thus:
+*The lower bound&nbsp; $t = 0$&nbsp; always yields the result&nbsp; $0$. &nbsp; With respect to the upper bound,&nbsp; only the first term yields a result different from&nbsp; $0$.&nbsp; Thus:
 :$$E_g =    2  \cdot s_0^2 \cdot  \frac{3}{8} \cdot \frac{T}{2} = \frac{3}{8} \cdot 5 \cdot 10^{-7}\, {\rm
   Ws} \hspace{0.1cm}\underline{ = 0.1875 \cdot 10^{-6}\, {\rm

@@ Line 21: / Line 21: @@
 Always assume in your calculations that the two possible amplitude coefficients are equally likely and that the distance between adjacent symbols is &nbsp;$T = 1 \ \rm  &micro; s$.&nbsp; This corresponds to a bit rate of &nbsp;$R = 1 \ \rm Mbit/s$.
-*The (positive) maximum value of the transmitted signal is the same in both cases
+*The&nbsp; (positive)&nbsp; maximum value of the transmitted signal is the same in both cases:
 :$$s_0 =  \sqrt{0.5\, {\rm W}}  \hspace{0.05cm}.$$
-*Assuming that the transmitter is terminated with a &nbsp;$50\ \rm  Ω$&nbsp; resistor, this corresponds to the following voltage value:
+*Assuming that the transmitter is terminated with a &nbsp;$50\ \rm  Ω$&nbsp; resistor,&nbsp; this corresponds to the following voltage value:
 :$$s_0^2 =  0.5\, {\rm W} \cdot 50\, {\rm \Omega} = 25\, {\rm V}^2 \hspace{0.15cm} \Rightarrow \hspace{0.15cm} s_0 =5\, {\rm V} \hspace{0.05cm}.$$
@@ Line 32: / Line 32: @@
 Notes:
 *The exercise belongs to the chapter&nbsp; [[Digital_Signal_Transmission/System_Components_of_a_Baseband_Transmission_System|"System Components of a Baseband Transmission System"]].
 *Reference is made in particular to the section&nbsp; [[Digital_Signal_Transmission/System_Components_of_a_Baseband_Transmission_System#Characteristics_of_the_digital_transmitter|"Characteristics of the digital transmitter"]].
 *Given is the following indefinite integral:
 :$$\int \cos^4(a  x)\,{\rm d}x = \frac{3}{8} \cdot x + \frac{1}{4a} \cdot \sin(2 a  x)+ \frac{1}{32a} \cdot \sin(4 a

@@ Line 5: / Line 5: @@
 [[File:P_ID1256__Dig_A_1_1.png|right|frame|Considered basic transmission pulses]]
-In this exercise, we examine the two transmitted signals &nbsp;$s_{\rm R}(t)$&nbsp; and &nbsp;$s_{\rm C}(t)$&nbsp; with square-wave and &nbsp;$\cos^2$ basic transmission pulse, respectively, shown in the diagram. In particular, the following characteristics are to be calculated for the respective pulses &nbsp;$g_s(t)$:&nbsp;
+In this exercise,&nbsp; we examine the two transmitted signals &nbsp;$s_{\rm R}(t)$&nbsp; and &nbsp;$s_{\rm C}(t)$&nbsp; with rectangular resp. cosine&ndash;square basic transmission pulse,&nbsp; shown in the diagram.
 *the equivalent pulse duration of &nbsp;$g_s(t)$:
 :$$\Delta t_{\rm S} =   \frac {\int ^{+\infty} _{-\infty} \hspace{0.15cm} g_s(t)\,{\rm
   d}t}{{\rm Max} \hspace{0.05cm}[g_s(t)]} \hspace{0.05cm},$$
-*the energy of the basic transmission pulse &nbsp;$g_s(t)$:
+*the energy of &nbsp;$g_s(t)$:
 :$$E_g =   \int^{+\infty} _{-\infty} g_s^2(t)\,{\rm
   d}t \hspace{0.05cm},$$
@@ Line 22: / Line 24: @@
 :$$s_0 =  \sqrt{0.5\, {\rm W}}  \hspace{0.05cm}.$$
 *Assuming that the transmitter is terminated with a &nbsp;$50\ \rm  Ω$&nbsp; resistor, this corresponds to the following voltage value:
-:$$s_0 =  \sqrt{0.5\, {\rm W}}  \hspace{0.05cm}.$$
+:$$s_0^2 =  0.5\, {\rm W} \cdot 50\, {\rm \Omega} = 25\, {\rm V}^2 \hspace{0.15cm} \Rightarrow \hspace{0.15cm} s_0 =5\, {\rm V} \hspace{0.05cm}.$$
@@ Line 29: / Line 30: @@
-''Notes:''
+Notes:
-*The exercise belongs to the chapter&nbsp; [[Digital_Signal_Transmission/System_Components_of_a_Baseband_Transmission_System|System Components of a Baseband Transmission System]].
+*The exercise belongs to the chapter&nbsp; [[Digital_Signal_Transmission/System_Components_of_a_Baseband_Transmission_System|"System Components of a Baseband Transmission System"]].
-*Reference is made in particular to the section&nbsp; [[Digital_Signal_Transmission/System_Components_of_a_Baseband_Transmission_System#Characteristics_of_the_digital_transmitter|Characteristics of the digital transmitter]].
+*Reference is made in particular to the section&nbsp; [[Digital_Signal_Transmission/System_Components_of_a_Baseband_Transmission_System#Characteristics_of_the_digital_transmitter|"Characteristics of the digital transmitter"]].
 *Given is the following indefinite integral:
@@ Line 48: / Line 49: @@
-{What is the equivalent pulse duration &nbsp;$\Delta t_{\rm S}$, normalized to the symbol duration &nbsp;$T$?
+{What is the equivalent pulse duration &nbsp;$\Delta t_{\rm S}$,&nbsp; normalized to the symbol duration &nbsp;$T$?
 |type="{}"}
-$\text{for the signal}\ \ s_{\rm R}(t) \text{:} \ \ \Delta t_{\rm S}/T \ = \ $ { 1 3% }
+$\text{For the signal}\ \ s_{\rm R}(t) \text{:} \ \ \Delta t_{\rm S}/T \ = \ $ { 1 3% }
 $\text{for the signal}\ \ s_{\rm C}(t) \text{:} \ \ \Delta t_{\rm S}/T \ = \ $ { 0.5 3% }
-{What is the energy of the rectangular basic transmission pulse??
+{What is the energy of the rectangular basic transmission pulse &nbsp;$g_s(t)$?
 |type="{}"}
 $E_g \ = \ $ { 0.5 } $\ \cdot 10^{-6}\ \rm Ws$
@@ Line 61: / Line 62: @@
 $P_{\rm S} \ = \ $ { 0.25 3% } $\ \rm W$
-{What is the energy of the &nbsp;$\cos^2$ basic transmission pulse?
+{What is the energy of the cosine&ndash;square basic transmission pulse &nbsp;$g_s(t)$?
 |type="{}"}
 $E_g \ = \ $ { 0.1875 3% } $\ \cdot 10^{-6}\ \rm Ws$

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