Aufgaben:Exercise 1.6: Root Nyquist System - Revision history

Guenter at 15:51, 13 March 2023

2023-03-13T15:51:06Z

Guenter at 15:49, 13 March 2023

2023-03-13T15:49:58Z

Hwang at 13:28, 10 May 2022

2022-05-10T13:28:32Z

Guenter at 08:52, 4 May 2022

2022-05-04T08:52:08Z

Guenter at 08:45, 4 May 2022

2022-05-04T08:45:24Z

Guenter at 13:58, 19 March 2022

2022-03-19T13:58:00Z

Guenter: Guenter moved page Aufgaben:Aufgabe 1.6: Wurzel-Nyquist-System to Aufgaben:Exercise 1.6: Root Nyquist System

2022-03-19T13:54:08Z

Guenter moved page Aufgabe 1.6: Wurzel-Nyquist-System to Exercise 1.6: Root Nyquist System

Hwang at 12:37, 18 March 2022

2022-03-18T12:37:49Z

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Javier: Text replacement - "Category:Aufgaben zu Digitalsignalübertragung" to "Category:Digital Signal Transmission: Exercises"

2021-03-23T12:51:46Z

Text replacement - "Category:Aufgaben zu Digitalsignalübertragung" to "Category:Digital Signal Transmission: Exercises"

Javier: Text replacement - "[[Digitalsignalübertragung" to "[[Digital_Signal_Transmission"

2020-07-09T12:45:18Z

Text replacement - "[[Digitalsignalübertragung" to "[[Digital_Signal_Transmission"

@@ Line 67: / Line 67: @@
 ===Solution===
 {{ML-Kopf}}
-'''(1)'''&nbsp; With the functions&nbsp; $G_{s}(f)$&nbsp; and&nbsp; $H_{\rm E}(f)$,&nbsp; the spectrum of the basic transmission pulse for&nbsp; $|f| \leq f_{2}$:
+'''(1)'''&nbsp; With the functions&nbsp; $G_{s}(f)$&nbsp; and&nbsp; $H_{\rm E}(f)$,&nbsp; the spectrum of the basic detection pulse for&nbsp; $|f| \leq f_{2}$:
 :$$G_d(f)  =   G_s(f) \cdot H_{\rm E}(f) = A \cdot \cos^2 \left( \frac {\pi \cdot f}{2 \cdot f_2} \right).$$
 *According to the general definition of the cosine rolloff spectrum,&nbsp; the corner frequencies are&nbsp; $f_{1} = 0$&nbsp; and&nbsp; $f_{2} = 1\ \rm MHz$.

@@ Line 67: / Line 67: @@
 ===Solution===
 {{ML-Kopf}}
-'''(1)'''&nbsp; With the functions&nbsp; $G_{s}(f)$&nbsp; and&nbsp; $H_{\rm E}(f)$,&nbsp; the spectrum of the basic detection pulse '''KORREKTUR: basic transmitter pulse''' for&nbsp; $|f| \leq f_{2}$:
+'''(1)'''&nbsp; With the functions&nbsp; $G_{s}(f)$&nbsp; and&nbsp; $H_{\rm E}(f)$,&nbsp; the spectrum of the basic transmission pulse for&nbsp; $|f| \leq f_{2}$:
 :$$G_d(f)  =   G_s(f) \cdot H_{\rm E}(f) = A \cdot \cos^2 \left( \frac {\pi \cdot f}{2 \cdot f_2} \right).$$
 *According to the general definition of the cosine rolloff spectrum,&nbsp; the corner frequencies are&nbsp; $f_{1} = 0$&nbsp; and&nbsp; $f_{2} = 1\ \rm MHz$.

@@ Line 67: / Line 67: @@
 ===Solution===
 {{ML-Kopf}}
-'''(1)'''&nbsp; With the functions&nbsp; $G_{s}(f)$&nbsp; and&nbsp; $H_{\rm E}(f)$,&nbsp; the spectrum of the basic detection pulse for&nbsp; $|f| \leq f_{2}$:
+'''(1)'''&nbsp; With the functions&nbsp; $G_{s}(f)$&nbsp; and&nbsp; $H_{\rm E}(f)$,&nbsp; the spectrum of the basic detection pulse '''KORREKTUR: basic transmitter pulse''' for&nbsp; $|f| \leq f_{2}$:
 :$$G_d(f)  =   G_s(f) \cdot H_{\rm E}(f) = A \cdot \cos^2 \left( \frac {\pi \cdot f}{2 \cdot f_2} \right).$$
 *According to the general definition of the cosine rolloff spectrum,&nbsp; the corner frequencies are&nbsp; $f_{1} = 0$&nbsp; and&nbsp; $f_{2} = 1\ \rm MHz$.

@@ Line 67: / Line 67: @@
 ===Solution===
 {{ML-Kopf}}
-'''(1)'''&nbsp; With the functions $G_{s}(f)$ and $H_{\rm E}(f)$, the spectrum of the basic transmitter pulse for $|f| \leq f_{2}$:
+'''(1)'''&nbsp; With the functions&nbsp; $G_{s}(f)$&nbsp; and&nbsp; $H_{\rm E}(f)$,&nbsp; the spectrum of the basic detection pulse for&nbsp; $|f| \leq f_{2}$:
 :$$G_d(f)  =   G_s(f) \cdot H_{\rm E}(f) = A \cdot \cos^2 \left( \frac {\pi \cdot f}{2 \cdot f_2} \right).$$
-*According to the general definition of the cosine rolloff spectrum, the corner frequencies $f_{1} = 0$ and $f_{2} = 1\ \rm MHz$ are obtained.
+*According to the general definition of the cosine rolloff spectrum,&nbsp; the corner frequencies are&nbsp; $f_{1} = 0$&nbsp; and&nbsp; $f_{2} = 1\ \rm MHz$.
-* From this follows for the Nyquist frequency (symmetry point with respect to the rolloff):
+* From this follows for the Nyquist frequency&nbsp; (symmetry point with respect to the rolloff):
 :$$f_{\rm Nyq} =  \frac{f_1 +f_2 }
 {2 } \hspace{0.1cm}\underline { = 0.5\,{\rm MHz}}\hspace{0.05cm}.$$
@@ Line 79: / Line 79: @@
-'''(2)'''&nbsp; The relationship between Nyquist frequency and symbol duration $T$ is $f_{\rm Nyq} = 1/(2T)$.
+'''(2)'''&nbsp; The relationship between Nyquist frequency and symbol duration&nbsp; $T$&nbsp; is&nbsp; $f_{\rm Nyq} = 1/(2T)$.
-*From this it follows for the bit rate $R = 1/T = 2 \cdot f_{\rm Nyq}\ \underline{= 1 \ \rm Mbit/s}$.
+*From this it follows for the bit rate&nbsp; $R = 1/T = 2 \cdot f_{\rm Nyq}\ \underline{= 1 \ \rm Mbit/s}$.
 *Note the different units for frequency and bit rate.
-'''(3)'''&nbsp; The <u>first and third solutions</u> are correct:
+'''(3)'''&nbsp; The&nbsp; <u>first and third solutions</u>&nbsp; are correct:
 *This is an optimal binary system under the constraint of power limitation.
-*The crest factor is not important under power limitation. With the conditions given here, $C_{\rm S} > 1$ would apply.
+*The crest factor is not important under power limitation.&nbsp; With the conditions given here,&nbsp; $C_{\rm S} > 1$ would apply.
@@ Line 93: / Line 93: @@
 '''(4)'''&nbsp; The bit error probability of an optimal system can be calculated as follows:
 :$$p_{\rm B} =   {\rm Q} \left( \sqrt{{2 \cdot E_{\rm B}}/{N_0}}\right)\hspace{0.05cm}.$$
-*In the given example, we obtain for the average energy per bit:
+*In the given example,&nbsp; we obtain for the average energy per bit:
 :$$E_{\rm B}  = \
   \int_{-\infty}^{+\infty}|G_s(f)|^2 \,{\rm d} f  =
   A^2 \cdot \int_{-1/T}^{+1/T} H_{\rm Nyq}(f) \,{\rm d} f
    = \ \frac {A^2}{T} = \frac {(10^{-6}\,{\rm V/Hz})^2}{10^{-6}\,{\rm s}} = 10^{-6}\,{\rm V^2s}\hspace{0.05cm}.$$
-*With $N_{0} = 8 \cdot 10^{–8} \ \rm V^{2}/Hz$, this further gives:
+*With&nbsp; $N_{0} = 8 \cdot 10^{–8} \ \rm V^{2}/Hz$,&nbsp; this further gives:
 :$$p_{\rm B} =   {\rm Q} \left( \sqrt{\frac{2 \cdot 10^{-6}\,{\rm V^2s}}{8 \cdot 10^{-8}\,{\rm
   V^2/Hz}}}\right)=

@@ Line 4: / Line 4: @@
-[[File:P_ID1292__Dig_A_1_6.png|right|frame|Transmitter & receiver: Cosine spectrum]]
+[[File:P_ID1292__Dig_A_1_6.png|right|frame|Cosine spectrum (transmitter & receiver)]]
 The diagram on the right shows
 *the spectrum &nbsp;$G_{s}(f)$&nbsp; of the basic transmission pulse,
@@ Line 23: / Line 23: @@
 In the whole exercise &nbsp;$A = 10^{–6} \ \rm V/Hz$&nbsp; and &nbsp;$f_{2} = 1 \ \rm MHz$ are valid.
-*Assuming that the bit rate &nbsp;$R = 1/T$&nbsp; is chosen correctly, the basic transmitter pulse &nbsp;$g_{d}(t) = g_{s}(t) ∗ h_{\rm E}(t)$&nbsp; satisfies the first Nyquist criterion.
+*Assuming that the bit rate &nbsp;$R = 1/T$&nbsp; is chosen correctly,&nbsp; the basic transmitter pulse &nbsp;$g_{d}(t) = g_{s}(t) ∗ h_{\rm E}(t)$&nbsp; satisfies the first Nyquist criterion.
 *For the associated spectral function &nbsp;$G_{d}(f)$,&nbsp; the rolloff thereby occurs cosinusoidally similar to a cosine rolloff spectrum.
 *The rolloff factor &nbsp;$r$&nbsp; is to be determined in this exercise.
@@ Line 30: / Line 30: @@
 *The crest factor is the quotient of the maximum and the rms value of the transmitted signal and thus a measure of the intersymbol interfering at the transmitting end:
 :$$C_{\rm S} =  \frac{s_0}{\sqrt{E_{\rm B}/T}} = \frac{{\rm Max}[s(t)]}{\sqrt{{\rm E}[s^2(t)]}}=  {s_0}/{s_{\rm eff}}.$$
@@ Line 43: / Line 42: @@
 <quiz display=simple>
-{Calculate the Nyquist spectrum &nbsp;$G_{d}(f)$. What is the Nyquist frequency and the rolloff factor?
+{Calculate the Nyquist spectrum &nbsp;$G_{d}(f)$.&nbsp; What is the Nyquist frequency and the rolloff factor?
 |type="{}"}
 $f_{\rm Nyq} \ = \ ${ 0.5 3% } $\ \rm MHz$
@@ Line 52: / Line 51: @@
 $R \ = \ $ { 1 3% } $\ \rm Mbit/s$
-{Why is it an optimal system under the constraint "power limitation"?
+{Why is it an optimal system under the constraint&nbsp; "power limitation"?
 |type="[]"}
 +The overall system satisfies the Nyquist condition.

← Older revision		Revision as of 13:58, 19 March 2022
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−	[[Category:Digital Signal Transmission: Exercises\|^1.4 ~~Optimierung der Basisbandsysteme~~^]]	+	[[Category:Digital Signal Transmission: Exercises\|^1.4 Optimization of Baseband Systems^]]

← Older revision	Revision as of 13:54, 19 March 2022
(No difference)

← Older revision		Revision as of 12:51, 23 March 2021
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−	[[Category:~~Aufgaben zu Digitalsignalübertragung~~\|^1.4 Optimierung der Basisbandsysteme^]]	+	[[Category:Digital Signal Transmission: Exercises\|^1.4 Optimierung der Basisbandsysteme^]]

@@ Line 33: / Line 33: @@
 ''Hinweise:''
-*Die Aufgabe gehört zum  Kapitel  &nbsp;[[Digitalsignalübertragung/Optimierung_der_Basisbandübertragungssysteme|Optimierung der Basisbandübertragungssysteme]].
+*Die Aufgabe gehört zum  Kapitel  &nbsp;[[Digital_Signal_Transmission/Optimierung_der_Basisbandübertragungssysteme|Optimierung der Basisbandübertragungssysteme]].
 *Zahlenwerte der Q–Funktion liefert zum Beispiel das interaktive Applet &nbsp;[[Applets:Komplementäre_Gaußsche_Fehlerfunktionen|Komplementäre Gaußsche Fehlerfunktionen]].