Aufgaben:Exercise 5.7Z: Application of the IDFT - Revision history

Guenter at 08:35, 12 January 2022

2022-01-12T08:35:13Z

Guenter at 08:30, 12 January 2022

2022-01-12T08:30:34Z

Guenter at 08:28, 12 January 2022

2022-01-12T08:28:39Z

Guenter at 08:26, 12 January 2022

2022-01-12T08:26:16Z

Hwang at 09:39, 29 December 2021

2021-12-29T09:39:40Z

Guenter at 15:58, 14 December 2021

2021-12-14T15:58:56Z

Guenter: Guenter moved page Aufgaben:Aufgabe 5.7Z: Anwendung der IDFT to Aufgaben:Exercise 5.7Z: Application of the IDFT

2021-12-14T15:55:22Z

Guenter moved page Aufgabe 5.7Z: Anwendung der IDFT to Exercise 5.7Z: Application of the IDFT

Javier: Text replacement - "„" to """

2021-05-28T14:41:23Z

Text replacement - "„" to """

Javier: Text replacement - "”" to """

2021-05-28T14:39:57Z

Text replacement - "”" to """

Javier: Text replacement - "Category:Aufgaben zu Modulationsverfahren" to "Category:Modulation Methods: Exercises"

2021-03-23T12:47:39Z

Text replacement - "Category:Aufgaben zu Modulationsverfahren" to "Category:Modulation Methods: Exercises"

@@ Line 48: / Line 48: @@
 ===Solution===
 {{ML-Kopf}}
-'''(1)'''&nbsp;  Because of&nbsp; $D(μ) = 0$&nbsp; for&nbsp; $μ ≠ 0$&nbsp; all time coefficients&nbsp; $d(ν) = D(0)= 1 - {\rm  j}$.&nbsp; Thus also holds:
+'''(1)'''&nbsp;  Because of&nbsp; $D(μ) = 0$&nbsp; for&nbsp; $μ ≠ 0$&nbsp; all time coefficients&nbsp; $d(ν) = D(0)= 1 - {\rm  j}$.&nbsp; Thus:
 :$${\rm Re}[d(1)] \hspace{0.15cm}\underline {=+ 1}, \hspace{0.3cm}{\rm Im}[d(1)]  \hspace{0.15cm}\underline {= -1}.$$
-'''(2)'''&nbsp;  Here, all spectral coefficients are zero except&nbsp; $D_1 = 1 - {\rm  j}$&nbsp; and&nbsp; $D_7 = 1 + {\rm  j}$.&nbsp; It follows for all time coefficients&nbsp; $(0 ≤ ν ≤ 7)$:
+'''(2)'''&nbsp;  Here,&nbsp; all spectral coefficients are zero except&nbsp; $D_1 = 1 - {\rm  j}$&nbsp; and&nbsp; $D_7 = 1 + {\rm  j}$.&nbsp; It follows for all time coefficients&nbsp; $(0 ≤ ν ≤ 7)$:
 :$$d(\nu) = (1 - {\rm j}) \cdot {\rm{e}}^{ - {\rm{j}}\hspace{0.04cm}\cdot \hspace{0.04cm} {\rm{\pi}}/4\hspace{0.04cm}\cdot \hspace{0.04cm}\nu} +(1 + {\rm j}) \cdot {\rm{e}}^{ - {\rm{j}}\hspace{0.04cm}\cdot \hspace{0.04cm} {7\rm{\pi}}/4\hspace{0.04cm}\cdot \hspace{0.04cm}\nu}.$$
-*However, due to periodicity, also holds:
+*However,&nbsp; due to periodicity,&nbsp; also holds:
 :$$d(\nu)  =  (1 - {\rm j}) \cdot {\rm{e}}^{ - {\rm{j}}\hspace{0.04cm}\cdot \hspace{0.04cm} {\rm{\pi}}/4\hspace{0.04cm}\cdot \hspace{0.04cm}\nu} +(1 + {\rm j}) \cdot {\rm{e}}^{ +{\rm{j}}\hspace{0.04cm}\cdot \hspace{0.04cm} {\rm{\pi}}/4\hspace{0.04cm}\cdot \hspace{0.04cm}\nu}=
   \left[ {\rm{e}}^{ + {\rm{j}}\hspace{0.04cm}\cdot \hspace{0.04cm} {\rm{\pi}}/4\hspace{0.04cm}\cdot \hspace{0.04cm}\nu} + {\rm{e}}^{ - {\rm{j}}\hspace{0.04cm}\cdot \hspace{0.04cm} {\rm{\pi}}/4\hspace{0.04cm}\cdot \hspace{0.04cm}\nu}\right]+ {\rm{j}} \cdot\left[ {\rm{e}}^{ + {\rm{j}}\hspace{0.04cm}\cdot \hspace{0.04cm} {\rm{\pi}}/4\hspace{0.04cm}\cdot \hspace{0.04cm}\nu} - {\rm{e}}^{ - {\rm{j}}\hspace{0.04cm}\cdot \hspace{0.04cm} {\rm{\pi}}/4\hspace{0.04cm}\cdot \hspace{0.04cm}\nu}\right].$$
-*Using Euler's theorem, this expression can be transformed as follows:
+*Using Euler's theorem,&nbsp; this expression can be transformed as follows:
 :$$d(\nu) = 2 \cdot \cos \left( {\pi}/{4}\cdot \nu \right)+ 2 \cdot \sin \left(  {\pi}/{4}\cdot \nu \right).$$
 *This time function&nbsp; $d(ν)$&nbsp; is purely real and characterizes a harmonic oscillation with amplitude&nbsp; $ 2 \cdot \sqrt{2}$&nbsp; and phase&nbsp; $φ = 45^\circ$.

@@ Line 21: / Line 21: @@
 *The exercise belongs to the chapter&nbsp; [[Modulation_Methods/Realisierung_von_OFDM-Systemen|Implementation of OFDM Systems]].
 *Reference is also made to the chapter&nbsp;  [[Signal_Representation/Discrete_Fourier_Transform_(DFT)|Discrete Fourier Transform]]&nbsp; in the book "Signal Representation".
-*You can check your results with the interactive applet &nbsp;[[Applets:Diskrete_Fouriertransformation_und_Inverse|Discrete Fourier Transform and Inverse]].&nbsp;
+*You can check your results with the interactive applet &nbsp;[[Applets:Discrete_Fouriertransform_and_Inverse|Discrete Fourier Transform and Inverse]].&nbsp;

@@ Line 8: / Line 8: @@
 Here, &nbsp;$w$&nbsp; abbreviates the complex rotation factor:
 :$$w = {\rm e}^{-{\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm} 2 \pi /N} = \cos \left( {2 \pi}/{N}\right)-{\rm j} \cdot \sin \left( {2 \pi}/{N}\right) \hspace{0.05cm}.$$
-Correspondingly,&nbsp; for the &nbsp;[[Signal_Representation/Discrete_Fourier_Transform_(DFT)#Inverse_Diskrete_Fouriertransformation|Inverse Discrete Fourier Transform]]&nbsp; $\rm (IDFT)$&nbsp; practically as an&nbsp; "inverse function"&nbsp; of the DFT applies:
+Correspondingly,&nbsp; for the &nbsp;[[Signal_Representation/Discrete_Fourier_Transform_(DFT)#Inverse_discrete_Fourier_transform|Inverse discrete Fourier transform]]&nbsp; $\rm (IDFT)$&nbsp; practically as an&nbsp; "inverse function"&nbsp; of the DFT applies:
 :$$d(\nu) = \sum_{\mu = 0 }^{N-1} D(\mu) \cdot {w}^{-\nu \hspace{0.08cm} \cdot \hspace{0.05cm}\mu} \hspace{0.05cm}.$$
 *In this exercise,&nbsp; the time coefficients &nbsp;$d(ν)$&nbsp; are to be determined for different complex-valued example sequences &nbsp;$D(μ)$ – which are labeled &nbsp;$\rm A$, &nbsp;$\rm B$&nbsp; and &nbsp;$\rm C$&nbsp; in the table.&nbsp;

@@ Line 4: / Line 4: @@
 [[File:P_ID1670__Mod_Z_5_7.png|right|frame|Three sets &nbsp;$\rm A$, &nbsp;$\rm B$&nbsp; and &nbsp;$\rm C$&nbsp;  <br>for the spectral coefficients]]
-In the &nbsp;[[Signal_Representation/Discrete_Fourier_Transform_(DFT)|Discrete Fourier Transform]]&nbsp; $\rm (DFT)$,&nbsp; the time sample values &nbsp;$d(ν)$&nbsp; with the control variable &nbsp;$ν = 0$, ... , $N – 1$&nbsp; the discrete spectral coefficients &nbsp;$D(μ)$&nbsp; with &nbsp;$μ = 0$, ... , $N – 1$&nbsp; is calculated as follows:
+In the &nbsp;[[Signal_Representation/Discrete_Fourier_Transform_(DFT)|Discrete Fourier Transform]]&nbsp; $\rm (DFT)$,&nbsp;  the discrete spectral coefficients &nbsp;$D(μ)$&nbsp; with control variable &nbsp;$μ = 0$, ... , $N – 1$&nbsp; are calculated from the time sample values &nbsp;$d(ν)$&nbsp; with &nbsp;$ν = 0$, ... , $N – 1$&nbsp;as follows:
 :$$D(\mu) = \frac{1}{N} \cdot \sum_{\nu = 0 }^{N-1} d(\nu)\cdot {w}^{\hspace{0.05cm}\nu \hspace{0.08cm} \cdot \hspace{0.05cm}\mu} \hspace{0.05cm}.$$
-Here, &nbsp;$w$&nbsp; abbreviates the complex rotation factor, which is defined as follows:
+Here, &nbsp;$w$&nbsp; abbreviates the complex rotation factor:
 :$$w = {\rm e}^{-{\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm} 2 \pi /N} = \cos \left( {2 \pi}/{N}\right)-{\rm j} \cdot \sin \left( {2 \pi}/{N}\right) \hspace{0.05cm}.$$
-Correspondingly, for the &nbsp;[[Signal_Representation/Discrete_Fourier_Transform_(DFT)#Inverse_Diskrete_Fouriertransformation|Inverse Discrete Fourier Transform]]&nbsp; $\rm (IDFT)$&nbsp; practically as an "inverse function" of the DFT the following applies:
+Correspondingly,&nbsp; for the &nbsp;[[Signal_Representation/Discrete_Fourier_Transform_(DFT)#Inverse_Diskrete_Fouriertransformation|Inverse Discrete Fourier Transform]]&nbsp; $\rm (IDFT)$&nbsp; practically as an&nbsp; "inverse function"&nbsp; of the DFT applies:
 :$$d(\nu) = \sum_{\mu = 0 }^{N-1} D(\mu) \cdot {w}^{-\nu \hspace{0.08cm} \cdot \hspace{0.05cm}\mu} \hspace{0.05cm}.$$
-In this exercise, the time coefficients &nbsp;$d(ν)$&nbsp; are to be determined for different complex-valued example sequences &nbsp;$D(μ)$ – which are labeled &nbsp;$\rm A$, &nbsp;$\rm B$&nbsp; and &nbsp;$\rm C$&nbsp; in the table.&nbsp; Thus, &nbsp;$N = 8$ is always valid.
+*In this exercise,&nbsp; the time coefficients &nbsp;$d(ν)$&nbsp; are to be determined for different complex-valued example sequences &nbsp;$D(μ)$ – which are labeled &nbsp;$\rm A$, &nbsp;$\rm B$&nbsp; and &nbsp;$\rm C$&nbsp; in the table.&nbsp;
@@ Line 17: / Line 18: @@
 Notes:
 *The exercise belongs to the chapter&nbsp; [[Modulation_Methods/Realisierung_von_OFDM-Systemen|Implementation of OFDM Systems]].
 *Reference is also made to the chapter&nbsp;  [[Signal_Representation/Discrete_Fourier_Transform_(DFT)|Discrete Fourier Transform]]&nbsp; in the book "Signal Representation".

@@ Line 1: / Line 1: @@
-{{quiz-Header|Buchseite=Modulationsverfahren/Realisierung von OFDM-Systemen
+{{quiz-Header|Buchseite=Modulation_Methods/Implementation_of_OFDM_Systems
 }}
-[[File:P_ID1670__Mod_Z_5_7.png|right|frame|Drei Sätze &nbsp;$\rm A$, &nbsp;$\rm B$&nbsp; und &nbsp;$\rm C$&nbsp;  <br>für die Spektralkoeffizienten]]
+[[File:P_ID1670__Mod_Z_5_7.png|right|frame|Three sets &nbsp;$\rm A$, &nbsp;$\rm B$&nbsp; and &nbsp;$\rm C$&nbsp;  <br>for the spectral coefficients]]
-Bei der &nbsp;[[Signal_Representation/Discrete_Fourier_Transform_(DFT)|Diskreten Fouriertransformation]]&nbsp; $\rm (DFT)$&nbsp; werden aus den Zeitabtastwerten &nbsp;$d(ν)$&nbsp; mit der Laufvariablen &nbsp;$ν = 0$, ... , $N – 1$&nbsp; die diskreten Spektralkoeffizienten &nbsp;$D(μ)$&nbsp; mit &nbsp;$μ = 0$, ... , $N – 1$&nbsp; wie folgt berechnet:
+In the &nbsp;[[Signal_Representation/Discrete_Fourier_Transform_(DFT)|Discrete Fourier Transform]]&nbsp; $\rm (DFT)$,&nbsp; the time sample values &nbsp;$d(ν)$&nbsp; with the control variable &nbsp;$ν = 0$, ... , $N – 1$&nbsp; the discrete spectral coefficients &nbsp;$D(μ)$&nbsp; with &nbsp;$μ = 0$, ... , $N – 1$&nbsp; is calculated as follows:
 :$$D(\mu) = \frac{1}{N} \cdot \sum_{\nu = 0 }^{N-1} d(\nu)\cdot {w}^{\hspace{0.05cm}\nu \hspace{0.08cm} \cdot \hspace{0.05cm}\mu} \hspace{0.05cm}.$$
-Hierbei ist mit &nbsp;$w$&nbsp; der komplexe Drehfaktor abgekürzt, der wie folgt definiert ist:
+Here, &nbsp;$w$&nbsp; abbreviates the complex rotation factor, which is defined as follows:
 :$$w = {\rm e}^{-{\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm} 2 \pi /N} = \cos \left( {2 \pi}/{N}\right)-{\rm j} \cdot \sin \left( {2 \pi}/{N}\right) \hspace{0.05cm}.$$
-Entsprechend gilt für die &nbsp;[[Signal_Representation/Discrete_Fourier_Transform_(DFT)#Inverse_Diskrete_Fouriertransformation|Inverse Diskrete Fouriertransformation]]&nbsp; $\rm (IDFT)$&nbsp; quasi als "Umkehrfunktion" der DFT:
+Correspondingly, for the &nbsp;[[Signal_Representation/Discrete_Fourier_Transform_(DFT)#Inverse_Diskrete_Fouriertransformation|Inverse Discrete Fourier Transform]]&nbsp; $\rm (IDFT)$&nbsp; practically as an "inverse function" of the DFT the following applies:
 :$$d(\nu) = \sum_{\mu = 0 }^{N-1} D(\mu) \cdot {w}^{-\nu \hspace{0.08cm} \cdot \hspace{0.05cm}\mu} \hspace{0.05cm}.$$
-In dieser Aufgabe sollen für verschiedene komplexwertige Beispielfolgen &nbsp;$D(μ)$ – die in der Tabelle mit &nbsp;$\rm A$, &nbsp;$\rm B$&nbsp; und &nbsp;$\rm C$&nbsp; bezeichnet sind – die Zeitkoeffizienten &nbsp;$d(ν)$&nbsp; ermittelt werden.&nbsp; Es gilt somit stets &nbsp;$N = 8$.
+In this exercise, the time coefficients &nbsp;$d(ν)$&nbsp; are to be determined for different complex-valued example sequences &nbsp;$D(μ)$ – which are labeled &nbsp;$\rm A$, &nbsp;$\rm B$&nbsp; and &nbsp;$\rm C$&nbsp; in the table.&nbsp; Thus, &nbsp;$N = 8$ is always valid.
@@ Line 20: / Line 20: @@
-''Hinweise:''
+''Notes:''
-*Die Aufgabe gehört zum  Kapitel&nbsp; [[Modulation_Methods/Realisierung_von_OFDM-Systemen|Realisierung von OFDM-Systemen]].
+*The exercise belongs to the chapter&nbsp; [[Modulation_Methods/Realisierung_von_OFDM-Systemen|Implementation of OFDM Systems]].
-*Bezug genommen wird auch  auf das Kapitel&nbsp;  [[Signal_Representation/Discrete_Fourier_Transform_(DFT)|Diskrete Fouriertransformation]]&nbsp; im Buch „Signaldarstellung”.
+*Reference is also made to the chapter&nbsp;  [[Signal_Representation/Discrete_Fourier_Transform_(DFT)|Discrete Fourier Transform]]&nbsp; in the book "Signal Representation".
-*Sie können Ihre Ergebnisse mit dem interaktiven Applet &nbsp;[[Applets:Diskrete_Fouriertransformation_und_Inverse|Diskrete Fouriertransformation und Inverse]]&nbsp; kontrollieren.
+*You can check your results with the interactive applet &nbsp;[[Applets:Diskrete_Fouriertransformation_und_Inverse|Discrete Fourier Transform and Inverse]].&nbsp;
-===Fragebogen===
+===Questions===
 <quiz display=simple>
-{Wie lauten die Zeitkoeffizienten &nbsp;$d(ν)$&nbsp; für die Spektralkoeffizienten &nbsp;$D(μ)$&nbsp; gemäß &nbsp;$\rm A$? <br> Geben Sie den ersten Koeffizienten &nbsp;$d(1)$&nbsp; mit Real&ndash; und Imaginärteil ein.
+{What are the time coefficients &nbsp;$d(ν)$&nbsp; for the spectral coefficients &nbsp;$D(μ)$&nbsp; according to &nbsp;$\rm A$? <br> Enter the first coefficient &nbsp;$d(1)$&nbsp; with real and imaginary parts.
 |type="{}"}
 ${\rm Re}\big[d(1)\big] \ = \ $ { 1 1% }
 ${\rm Im}\big[d(1)\big] \ = \ $ { -1.03--0.97 }
-{Wie lauten die Zeitkoeffizienten &nbsp;$d(ν)$&nbsp; für die Spektralkoeffizienten &nbsp;$D(μ)$&nbsp; gemäß &nbsp;$\rm B$? <br> Geben Sie den ersten Koeffizienten &nbsp;$d(1)$&nbsp; mit Real&ndash; und Imaginärteil ein.
+{What are the time coefficients &nbsp;$d(ν)$&nbsp; for the spectral coefficients &nbsp;$D(μ)$&nbsp; according to &nbsp;$\rm B$? <br> Enter the first coefficient &nbsp;$d(1)$&nbsp; with real and imaginary parts.
 |type="{}"}
 ${\rm Re}\big[d(1)\big] \ = \ $ { 2.828 1% }
 ${\rm Im}\big[d(1)\big] \ = \ $ { 0. }
-{Wie lauten die Zeitkoeffizienten &nbsp;$d(ν)$&nbsp; für die Spektralkoeffizienten &nbsp;$D(μ)$&nbsp; gemäß &nbsp;$\rm C$? <br> Geben Sie den ersten Koeffizienten &nbsp;$d(1)$&nbsp; mit Real&ndash; und Imaginärteil ein.
+{What are the time coefficients &nbsp;$d(ν)$&nbsp; for the spectral coefficients &nbsp;$D(μ)$&nbsp; according to &nbsp;$\rm C$? <br> Enter the first coefficient &nbsp;$d(1)$&nbsp; with real and imaginary parts.
 |type="{}"}
 ${\rm Re}\big[d(1)\big] \ = \ $ { -6.9--6.7 }
@@ Line 48: / Line 48: @@
 </quiz>
-===Musterlösung===
+===Solution===
 {{ML-Kopf}}
-'''(1)'''&nbsp;  Wegen&nbsp; $D(μ) = 0$&nbsp; für&nbsp; $μ ≠ 0$&nbsp; sind alle Zeitkoeffizienten&nbsp; $d(ν) = D(0)= 1 - {\rm  j}$.&nbsp; Damit gilt auch:
+'''(1)'''&nbsp;  Because of&nbsp; $D(μ) = 0$&nbsp; for&nbsp; $μ ≠ 0$&nbsp; all time coefficients&nbsp; $d(ν) = D(0)= 1 - {\rm  j}$.&nbsp; Thus also holds:
 :$${\rm Re}[d(1)] \hspace{0.15cm}\underline {=+ 1}, \hspace{0.3cm}{\rm Im}[d(1)]  \hspace{0.15cm}\underline {= -1}.$$
-'''(2)'''&nbsp;  Hier sind alle Spektralkoeffizienten Null mit Ausnahme von&nbsp; $D_1 = 1 - {\rm  j}$&nbsp; und&nbsp; $D_7 = 1 + {\rm  j}$.&nbsp; Daraus folgt für alle Zeitkoeffizienten&nbsp; $(0 ≤ ν ≤ 7)$:
+'''(2)'''&nbsp;  Here, all spectral coefficients are zero except&nbsp; $D_1 = 1 - {\rm  j}$&nbsp; and&nbsp; $D_7 = 1 + {\rm  j}$.&nbsp; It follows for all time coefficients&nbsp; $(0 ≤ ν ≤ 7)$:
 :$$d(\nu) = (1 - {\rm j}) \cdot {\rm{e}}^{ - {\rm{j}}\hspace{0.04cm}\cdot \hspace{0.04cm} {\rm{\pi}}/4\hspace{0.04cm}\cdot \hspace{0.04cm}\nu} +(1 + {\rm j}) \cdot {\rm{e}}^{ - {\rm{j}}\hspace{0.04cm}\cdot \hspace{0.04cm} {7\rm{\pi}}/4\hspace{0.04cm}\cdot \hspace{0.04cm}\nu}.$$
-*Aufgrund der Periodizität gilt aber auch:
+*However, due to periodicity, also holds:
 :$$d(\nu)  =  (1 - {\rm j}) \cdot {\rm{e}}^{ - {\rm{j}}\hspace{0.04cm}\cdot \hspace{0.04cm} {\rm{\pi}}/4\hspace{0.04cm}\cdot \hspace{0.04cm}\nu} +(1 + {\rm j}) \cdot {\rm{e}}^{ +{\rm{j}}\hspace{0.04cm}\cdot \hspace{0.04cm} {\rm{\pi}}/4\hspace{0.04cm}\cdot \hspace{0.04cm}\nu}=
   \left[ {\rm{e}}^{ + {\rm{j}}\hspace{0.04cm}\cdot \hspace{0.04cm} {\rm{\pi}}/4\hspace{0.04cm}\cdot \hspace{0.04cm}\nu} + {\rm{e}}^{ - {\rm{j}}\hspace{0.04cm}\cdot \hspace{0.04cm} {\rm{\pi}}/4\hspace{0.04cm}\cdot \hspace{0.04cm}\nu}\right]+ {\rm{j}} \cdot\left[ {\rm{e}}^{ + {\rm{j}}\hspace{0.04cm}\cdot \hspace{0.04cm} {\rm{\pi}}/4\hspace{0.04cm}\cdot \hspace{0.04cm}\nu} - {\rm{e}}^{ - {\rm{j}}\hspace{0.04cm}\cdot \hspace{0.04cm} {\rm{\pi}}/4\hspace{0.04cm}\cdot \hspace{0.04cm}\nu}\right].$$
-*Mit dem Satz von Euler lässt sich dieser Ausdruck wie folgt umformen:
+*Using Euler's theorem, this expression can be transformed as follows:
 :$$d(\nu) = 2 \cdot \cos \left( {\pi}/{4}\cdot \nu \right)+ 2 \cdot \sin \left(  {\pi}/{4}\cdot \nu \right).$$
-*Diese Zeitfunktion&nbsp; $d(ν)$&nbsp; ist rein reell und kennzeichnet eine harmonische Schwingung mit der Amplitude&nbsp; $ 2 \cdot \sqrt{2}$&nbsp; und der Phase&nbsp; $φ = 45^\circ$.
+*This time function&nbsp; $d(ν)$&nbsp; is purely real and characterizes a harmonic oscillation with amplitude&nbsp; $ 2 \cdot \sqrt{2}$&nbsp; and phase&nbsp; $φ = 45^\circ$.
-*Der Zeitkoeffizient mit Index&nbsp; $ν = 1$&nbsp; gibt das Maximum an:
+*The time coefficient with index&nbsp; $ν = 1$&nbsp; indicates the maximum:
 :$$ {\rm Re}[d(1)] = 2 \cdot \frac {\sqrt{2}}{2}+ 2 \cdot \frac {\sqrt{2}}{2} = 2 \cdot {\sqrt{2}} \hspace{0.15cm}\underline {\approx 2.828}, \hspace{0.5cm}{\rm Im}[d(1)] \hspace{0.15cm}\underline {= 0}.$$
-'''(3)'''&nbsp;  Entsprechend der allgemeinen Gleichung gilt:
+'''(3)'''&nbsp;  According to the general equation:
 :$$d(1) =  \sum\limits_{\mu = 0}^{7} D(\mu)\cdot {\rm{e}}^{ - {\rm{j}}\hspace{0.04cm}\cdot \hspace{0.04cm} {\rm{\pi}}/4\hspace{0.04cm}\cdot \hspace{0.04cm}\mu} = \left[ D(1) + D(7) \right]\cdot \cos \left( {\pi}/{4} \right) + \left[ D(3) + D(5) \right]\cdot \cos \left( {3\pi}/{4} \right)+
    {\rm j} \cdot \left[ D(2) - D(6) \right]\cdot \sin \left( {\pi}/{2} \right) + D(4) \cdot {\rm{e}}^{ - {\rm{j}}\hspace{0.04cm}\cdot \hspace{0.04cm} {\rm{\pi}}}.$$
-*Die ersten drei Terme liefern rein reelle Ergebnisse:
+*The first three terms give pure real results:
 :$${\rm Re}[d(1)]  =  (1+1) \cdot \frac{1}{\sqrt{2}}-(3+3) \cdot \frac{1}{\sqrt{2}}+ {\rm j} \cdot4{\rm j} \cdot 1 = -\frac{4}{\sqrt{2}}-4\hspace{0.15cm}\underline { \approx -6.829}.$$
-*Für den Imaginärteil ergibt sich:
+*For the imaginary part, we obtain:
 :$${\rm Im}[d(1)] = {\rm Im}\left[4 \cdot{\rm j} \cdot (-1) \right] \hspace{0.15cm}\underline {= -4}.$$

← Older revision		Revision as of 15:58, 14 December 2021
Line 79:		Line 79:


−	[[Category:Modulation Methods: Exercises\|^5.6 ~~Realisierung von~~ OFDM~~-Systemen~~^]]	+	[[Category:Modulation Methods: Exercises\|^5.6 Realization of OFDM Systems^]]

← Older revision		Revision as of 12:47, 23 March 2021
Line 79:		Line 79:


−	[[Category:~~Aufgaben zu Modulationsverfahren~~\|^5.6 Realisierung von OFDM-Systemen^]]	+	[[Category:Modulation Methods: Exercises\|^5.6 Realisierung von OFDM-Systemen^]]