Aufgaben:Exercise 5.8Z: Cyclic Prefix and Guard Interval - Revision history

Guenter at 14:52, 31 January 2022

2022-01-31T14:52:36Z

Guenter at 17:33, 24 January 2022

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Guenter at 17:23, 24 January 2022

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Guenter at 17:15, 24 January 2022

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Hwang at 11:11, 29 December 2021

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Guenter at 15:59, 14 December 2021

2021-12-14T15:59:46Z

Guenter: Guenter moved page Aufgaben:Aufgabe 5.8Z: Zyklisches Präfix und Guard–Intervall to Aufgaben:Exercise 5.8Z: Cyclic Prefix and Guard Interval

2021-12-14T15:57:23Z

Guenter moved page Aufgabe 5.8Z: Zyklisches Präfix und Guard–Intervall to Exercise 5.8Z: Cyclic Prefix and Guard Interval

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2021-05-28T14:42:58Z

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

2021-03-23T12:46:23Z

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

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 }}
-[[File:P_ID1664__Z_5_8.png|right|frame|OFDM scheme with cyclic prefix  '''KORREKTUR''': guard interval, output symbol]]
+[[File:EN_Z_5_8.png|right|frame|OFDM scheme with cyclic prefix]]
 In this exercise,&nbsp; we assume an&nbsp; $\rm OFDM$ system with &nbsp;$N = 8$&nbsp; carriers and cyclic prefix.&nbsp; Let the subcarrier spacing be &nbsp;$f_0 = 4 \ \rm kHz$ &nbsp; &rArr; &nbsp; basic symbol duration:&nbsp; $T=1/f_0$.&nbsp; The diagram shows the principle of the cyclic prefix.
 *Transmission is via a two-way channel,&nbsp; with both paths delayed.&nbsp; The channel impulse response is thus with&nbsp; $τ_1 = \ \rm 50 &micro;s$&nbsp; and &nbsp;$τ_2 = 125\ \rm  &micro;s$:

@@ Line 4: / Line 4: @@
 [[File:P_ID1664__Z_5_8.png|right|frame|OFDM scheme with cyclic prefix  '''KORREKTUR''': guard interval, output symbol]]
-In this exercise,&nbsp; we assume an&nbsp; $\rm OFDM$ system with &nbsp;$N = 8$&nbsp; carriers and cyclic prefix.&nbsp; Let the subcarrier spacing be &nbsp;$f_0 = 4 \ \rm kHz$.&nbsp; The diagram shows the principle of the cyclic prefix.
+In this exercise,&nbsp; we assume an&nbsp; $\rm OFDM$ system with &nbsp;$N = 8$&nbsp; carriers and cyclic prefix.&nbsp; Let the subcarrier spacing be &nbsp;$f_0 = 4 \ \rm kHz$ &nbsp; &rArr; &nbsp; basic symbol duration:&nbsp; $T=1/f_0$.&nbsp; The diagram shows the principle of the cyclic prefix.
 *Transmission is via a two-way channel,&nbsp; with both paths delayed.&nbsp; The channel impulse response is thus with&nbsp; $τ_1 = \ \rm 50 &micro;s$&nbsp; and &nbsp;$τ_2 = 125\ \rm  &micro;s$:
 :$$ h(t) = h_1 \cdot \delta (t- \tau_1) + h_2 \cdot \delta (t- \tau_2).$$
@@ Line 69: / Line 69: @@
 $\beta\ = \ $ { 0.667 3% }
-{What is the associated SNR loss &nbsp;$10 · \lg \ Δ_ρ$&nbsp; (in dB) assuming the matched filter approach?
+{What is the associated SNR loss &nbsp;$10 · \lg \ Δρ$&nbsp; (in dB) assuming the matched filter approach?
 |type="{}"}
 $10 · \lg \ Δρ \ = \ $ { 1.76 3% } $\ \rm dB$

@@ Line 76: / Line 76: @@
 ===Solution===
 {{ML-Kopf}}
-'''(1)'''&nbsp;   The core symbol duration is equal to the reciprocal of the carrier spacing: &nbsp;
+'''(1)'''&nbsp;   The basic symbol duration is equal to the reciprocal of the carrier spacing: &nbsp;
 :$$ T = {1}/{f_0} \hspace{0.15cm}\underline {= 250\,\,{\rm &micro; s}}.$$
-'''(2)'''&nbsp;   To avoid interference, the duration&nbsp; $T_{\rm G}$&nbsp;  of the guard interval should be at least as long as the maximum channel delay&nbsp; $($here: $τ_2 = 125\ \rm  &micro; s)$:
+'''(2)'''&nbsp;   To avoid interference,&nbsp; the duration&nbsp; $T_{\rm G}$&nbsp;  of the guard interval should be at least as long as the maximum channel delay&nbsp; $($here: $τ_2 = 125\ \rm  &micro; s)$:
 :$$ T_{\rm G} \hspace{0.15cm}\underline {= 125\,\,{\rm &micro; s}}.$$
-'''(3)'''&nbsp;   Thus, for the frame duration:
+'''(3)'''&nbsp;   Thus,&nbsp; for the frame duration:
 :$$ T_{\rm{R}} = T + T_{\rm G}\hspace{0.15cm}\underline {= 375\,\,{\rm &micro; s}}.$$
-'''(4)'''&nbsp;   <u>Solution 2</u> is correct:
+'''(4)'''&nbsp;   <u>Solution 2</u>&nbsp; is correct:
 *Only intersymbol interference&nbsp; $\rm (ISI)$&nbsp; can be avoided by a guard gap of suitable length.
 *The gap duration&nbsp; $T_{\rm G}$&nbsp; must be chosen so large that the current symbol is not affected by the predecessor symbol.
-*In the present example, it must be &nbsp; $T_{\rm G}≥ 125\ \rm  &micro; s$.&nbsp;
+*In the present example,&nbsp; it must be &nbsp; $T_{\rm G}≥ 125\ \rm  &micro; s$.&nbsp;
-'''(5)'''&nbsp;    <u>Both solutions</u> are applicable:
+'''(5)'''&nbsp;    <u>Both solutions</u>&nbsp; are applicable:
 *A cyclic prefix of suitable length also suppresses intercarrier interference&nbsp; $\rm (ICI)$.&nbsp;
-*This ensures that a complete and undistorted oscillation occurs for all carriers within the core symbol duration&nbsp; $T$,&nbsp; even if other carriers are active.
+*This ensures that a complete and undistorted oscillation occurs for all carriers within the basic symbol duration&nbsp; $T$,&nbsp; even if other carriers are active.
-'''(6)'''&nbsp;   The number of samples within the core symbol is equal to the number of carriers &nbsp; &rArr; &nbsp; $\underline{N=8}$.
+'''(6)'''&nbsp;   The number of samples within the basic symbol is equal to the number of carriers &nbsp; &rArr; &nbsp; $\underline{N=8}$.
-*Because of&nbsp; $T_{\rm G}= T/2$&nbsp;, &nbsp; $N_{\rm G}\hspace{0.15cm}\underline {= 4}$&nbsp; and thus&nbsp; $N_{\rm R} = N + N_{\rm G}\hspace{0.15cm}\underline {= 12}$ holds.
+*Because of&nbsp; $T_{\rm G}= T/2$&nbsp;, &nbsp; $N_{\rm G}\hspace{0.15cm}\underline {= 4}$&nbsp; and thus&nbsp; $N_{\rm R} = N + N_{\rm G}\hspace{0.15cm}\underline {= 12}$&nbsp; holds.
-'''(7)'''&nbsp;   Assigning the coefficient "-1" to the first carrier&nbsp; $($frequency $f_0)$&nbsp; results in the samples
+'''(7)'''&nbsp;   Assigning the coefficient&nbsp; "-1"&nbsp; to the first carrier&nbsp; $($frequency $f_0)$&nbsp; results in the samples
 :$$d_0 = -1, \hspace{0.3cm}d_1 = -0.707 - {\rm j} \cdot 0.707, \hspace{0.3cm}d_2 =  -{\rm j} ,\hspace{0.3cm} d_3 = +0.707 -{\rm j} \cdot 0.707, $$
 :$$d_4 = +1, \hspace{0.3cm}d_5 = +0.707 + {\rm j} \cdot 0.707, \hspace{0.3cm}d_6 =  +{\rm j} ,\hspace{0.3cm} d_7 = -0.707 +{\rm j} \cdot 0.707. $$
@@ Line 115: / Line 115: @@
-'''(8)'''&nbsp;   According to the given equation, the bandwidth efficiency is equal to
+'''(8)'''&nbsp;   According to the given equation,&nbsp; the bandwidth efficiency is equal to
 :$$\beta = \frac{1}{1 + {T_{\rm{G}}}/{T}} = \frac{1}{1 + ({125\,\,{\rm \mu s}})/({250\,\,{\rm \mu s}})} \hspace{0.15cm}\underline {= 0.667}.$$

@@ Line 4: / Line 4: @@
 [[File:P_ID1664__Z_5_8.png|right|frame|OFDM scheme with cyclic prefix  '''KORREKTUR''': guard interval, output symbol]]
-In this exercise, we assume an&nbsp; $\rm OFDM$ system with &nbsp;$N = 8$&nbsp; carriers and cyclic prefix.&nbsp; Let the subcarrier spacing be &nbsp;$f_0 = 4 \ \rm kHz$.&nbsp; The diagram shows the principle of the cyclic prefix.
+In this exercise,&nbsp; we assume an&nbsp; $\rm OFDM$ system with &nbsp;$N = 8$&nbsp; carriers and cyclic prefix.&nbsp; Let the subcarrier spacing be &nbsp;$f_0 = 4 \ \rm kHz$.&nbsp; The diagram shows the principle of the cyclic prefix.
-*Transmission is via a two-way channel, with both paths delayed.&nbsp; The channel impulse response is thus with&nbsp; $τ_1 = \ \rm 50 &micro;s$&nbsp; and &nbsp;$τ_2 = 125\ \rm  &micro;s$:
+*Transmission is via a two-way channel,&nbsp; with both paths delayed.&nbsp; The channel impulse response is thus with&nbsp; $τ_1 = \ \rm 50 &micro;s$&nbsp; and &nbsp;$τ_2 = 125\ \rm  &micro;s$:
 :$$ h(t) = h_1 \cdot \delta (t- \tau_1) + h_2 \cdot \delta (t- \tau_2).$$
-*However, the use of such a cyclic prefix decreases the bandwidth efficiency&nbsp; $($ratio of symbol rate to bandwidth$)$&nbsp; by a factor of
+*However,&nbsp; the use of such a cyclic prefix decreases the bandwidth efficiency&nbsp; $($ratio of symbol rate to bandwidth$)$&nbsp; by a factor of
 :$$ \beta = \frac{1}{{1 + T_{\rm{G}} /T}} $$
-:and also leads to a reduction of the signal-to-noise ratio by this value <i>&beta;</i> as well.
+:and leads also to a reduction of the signal-to-noise ratio by this value&nbsp; $\beta$&nbsp; as well.
-*However, a prerequisite for the validity of the SNR loss given here is that the impulse responses &nbsp;$g_{\rm S}(t)$&nbsp; and &nbsp;$g_{\rm E}(t)$&nbsp; of the transmit and receive filters are matched to the symbol duration &nbsp;$T$&nbsp; &nbsp; $($matched&ndash;filter approach$)$.
+*However,&nbsp; a prerequisite for the validity of the SNR loss given here is that the impulse responses &nbsp;$g_{\rm S}(t)$&nbsp; and &nbsp;$g_{\rm E}(t)$&nbsp; of the transmit filter&nbsp; (German:&nbsp; "Sendefilter" &nbsp; &rArr;  &nbsp; subscript:&nbsp; "S")&nbsp; and the receive filter&nbsp; (German:&nbsp; "Empfangsfilter" &nbsp; &rArr;  &nbsp; subscript:&nbsp; "E")&nbsp; are matched to the symbol duration &nbsp;$T$&nbsp;  $($matched&ndash;filter approach$)$.
@@ Line 16: / Line 16: @@
 Notes:
+*The exercise belongs to the chapter&nbsp; [[Modulation_Methods/Realisierung_von_OFDM-Systemen|Implementation of OFDM Systems]].
 *Reference is made in particular to the pages&nbsp;  [[Modulation_Methods/Implementation_of_OFDM_Systems#Cyclic_Prefix|Cyclic Prefix]]&nbsp; and &nbsp;[[Modulation_Methods/Implementation_of_OFDM_Systems#OFDM_system_with_cyclic_prefix|OFDM system with cyclic prefix]].
@@ Line 29: / Line 26: @@
 <quiz display=simple>
-{Specify the core symbol duration &nbsp;$T$&nbsp; an.
+{Specify the basic symbol duration &nbsp;$T$.
 |type="{}"}
 $T \ = \ $ { 250 3% } $\ \rm &micro; s$
@@ Line 41: / Line 38: @@
 $T_{\rm R}\ = \ $ { 375 3% } $\ \rm &micro; s$
-{Which statements are correct?&nbsp; By using a guard gap, i.e. setting the OFDM signal to zero in the guard interval, it is possible to
+{Which statements are correct?&nbsp; By using a guard gap,&nbsp; i.e. setting the OFDM signal to zero in the guard interval,&nbsp; it is possible to
 |type="[]"}
 - suppress intercarrier interference&nbsp; $\rm (ICI)$,&nbsp;
 + suppress intersymbol interference&nbsp; $\rm (ISI)$.&nbsp;
-{Which statements are correct?&nbsp; By using a cyclic prefix, i.e. a cyclic extension of the OFDM signal in the guard interval, it is possible to
+{Which statements are correct?&nbsp; By using a cyclic prefix,&nbsp; i.e. a cyclic extension of the OFDM signal in the guard interval,&nbsp; it is possible to
 |type="[]"}
 + suppress intercarrier interference&nbsp; $\rm (ICI)$,&nbsp;
 + suppress intersymbol interference&nbsp; $\rm (ISI)$.&nbsp;
-{State the respective number of samples for the core symbol &nbsp;$(N)$,&nbsp; $\rm (ICI)$&nbsp; the guard interval &nbsp;$(N_{\rm G})$&nbsp; and the entire frame &nbsp;$(N_{\rm R})$.
+{State the respective number of samples for the basic symbol &nbsp;$(N)$,&nbsp;  the guard interval &nbsp;$(N_{\rm G})$&nbsp; and the entire frame &nbsp;$(N_{\rm R})$.
 |type="{}"}
 $N \hspace{0.35cm} = \ $ { 8 }
@@ Line 57: / Line 54: @@
 $N_{\rm R} \ = \ $ { 12 }
-{Specify the guard interval samples, assuming that only the first carrier with carrier coefficient &nbsp;$-1$&nbsp; is used.
+{Specify the guard interval samples, assuming that only the first carrier  is used with carrier coefficient &nbsp;$-1$.
 |type="{}"}
 $\text{Re}\big[d_{-1}\big] \ = \ $ { -714--0.700 }
@@ Line 72: / Line 69: @@
 $\beta\ = \ $ { 0.667 3% }
-{What is the associated SNR loss &nbsp;$10 · \lg \ Δ_ρ$&nbsp; (in dB) assuming the matched-filter approach?
+{What is the associated SNR loss &nbsp;$10 · \lg \ Δ_ρ$&nbsp; (in dB) assuming the matched filter approach?
 |type="{}"}
-$10 · \lg \ Δ_ρ \ = \ $ { 1.76 3% } $\ \rm dB$
+$10 · \lg \ Δρ \ = \ $ { 1.76 3% } $\ \rm dB$
 </quiz>

← Older revision		Revision as of 15:59, 14 December 2021
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−	[[Category:Modulation Methods: Exercises\|^5.6 ~~Realisierung von~~ OFDM~~-Systemen~~^]]	+	[[Category:Modulation Methods: Exercises\|^5.6 Realization of OFDM Systems^]]

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-'''(7)'''&nbsp;   Die Belegung des ersten Trägers&nbsp; $($Frequenz $f_0)$&nbsp; mit dem Koeffizienten &bdquo;–1" führt zu den Abtastwerten
+'''(7)'''&nbsp;   Die Belegung des ersten Trägers&nbsp; $($Frequenz $f_0)$&nbsp; mit dem Koeffizienten "–1" führt zu den Abtastwerten
 :$$d_0 = -1, \hspace{0.3cm}d_1 = -0.707 - {\rm j} \cdot 0.707, \hspace{0.3cm}d_2 =  -{\rm j} ,\hspace{0.3cm} d_3 = +0.707 -{\rm j} \cdot 0.707, $$
 :$$d_4 = +1, \hspace{0.3cm}d_5 = +0.707 + {\rm j} \cdot 0.707, \hspace{0.3cm}d_6 =  +{\rm j} ,\hspace{0.3cm} d_7 = -0.707 +{\rm j} \cdot 0.707. $$

← Older revision		Revision as of 12:46, 23 March 2021
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−	[[Category:~~Aufgaben zu Modulationsverfahren~~\|^5.6 Realisierung von OFDM-Systemen^]]	+	[[Category:Modulation Methods: Exercises\|^5.6 Realisierung von OFDM-Systemen^]]