Aufgaben:Exercise 3.7: PN Modulation - Revision history

Javier: Text replacement - "Category:Exercises for Mobile Communications" to "Category:Mobile Communications: Exercises"

2021-03-23T12:38:15Z

Text replacement - "Category:Exercises for Mobile Communications" to "Category:Mobile Communications: Exercises"

Guenter at 17:17, 24 February 2021

2021-02-24T17:17:02Z

Guenter at 13:11, 25 January 2021

2021-01-25T13:11:19Z

Guenter at 12:57, 25 January 2021

2021-01-25T12:57:08Z

Guenter at 12:55, 25 January 2021

2021-01-25T12:55:58Z

Javier: Text replacement - "===Sample solution===" to "===Solution==="

2020-12-16T13:16:30Z

Text replacement - "===Sample solution===" to "===Solution==="

Javier: Text replacement - "Mobile Kommunikation/Die Charakteristika von UMTS" to "Mobile_Communications/Characteristics_of_UMTS"

2020-12-15T14:04:22Z

Text replacement - "Mobile Kommunikation/Die Charakteristika von UMTS" to "Mobile_Communications/Characteristics_of_UMTS"

Javier: Text replacement - "transmit signal" to "transmitted signal"

2020-08-25T09:37:44Z

Text replacement - "transmit signal" to "transmitted signal"

Javier: Text replacement - "[[Modulationsverfahren" to "[[Modulation_Methods"

2020-07-09T12:41:29Z

Text replacement - "[[Modulationsverfahren" to "[[Modulation_Methods"

Javier: Text replacement - "Mobile_Kommunikation/" to "Mobile_Communications/"

2020-07-09T09:13:39Z

Text replacement - "Mobile_Kommunikation/" to "Mobile_Communications/"

← Older revision		Revision as of 12:38, 23 March 2021
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−	[[Category:~~Exercises for~~ Mobile Communications\|^3.4 Characteristics of UMTS^]]	+	[[Category:Mobile Communications: Exercises\|^3.4 Characteristics of UMTS^]]

@@ Line 4: / Line 4: @@
 [[File:P_ID2259__Mod_Z_5_2.png|right|frame|Equivalent models for "PN modulation" and "BPSK"]]
-The figure shows the equivalent block diagram of "PN modulation" or "Direct Sequence Spread Spectrum", abbreviated&nbsp; $\rm DS-SS$&nbsp; in the equivalent low-pass range, where&nbsp; $n(t)$&nbsp; stands for AWGN noise.
+The figure shows the equivalent block diagram of&nbsp; "PN modulation"&nbsp; or&nbsp; "Direct Sequence Spread Spectrum", abbreviated&nbsp; $\rm DS-SS$&nbsp; in the equivalent low-pass range, where&nbsp; $n(t)$&nbsp; stands for AWGN noise.
 Below, the low-pass model of binary phase modulation&nbsp; $\rm (BPSK)$&nbsp; is sketched.
 *The low-pass transmission signal&nbsp; $s(t)$&nbsp; is equal to the rectangular source signal&nbsp; $q(t) ∈ \{+1, -1\}$&nbsp; with rectangular duration&nbsp; $T$&nbsp; only for reasons of uniformity.
@@ Line 11: / Line 11: @@
 *The two models differ in the multiplication by the&nbsp; $±1$&nbsp; spreading signal&nbsp; $c(t)$&nbsp; at the transmitter and the receiver.&nbsp; Of the signal&nbsp; $c(t)$&nbsp; only the spread degree $J$&nbsp; is known.
-*The specification of the specific spreading sequence&nbsp; (M sequence or Walsh function)&nbsp; is not important for the solution of this task.
+*The specification of the specific spreading sequence&nbsp; $($"M sequence"&nbsp; or&nbsp; "Walsh function"$)$&nbsp; is not important for the solution of this task.
-It has to be examined whether the lower BPSK model can also be applied with PN modulation and whether the BPSK error probability
+It has to be examined whether the lower BPSK model can also be applied with&nbsp; "PN modulation"&nbsp; and whether the BPSK error probability
 :$$p_{\rm B} = {\rm Q} \left( \hspace{-0.05cm} \sqrt { {2 \cdot E_{\rm B}}/{N_{\rm 0}} } \hspace{0.05cm} \right )$$
-is also valid for PN modulation, or how the given equation should be modified.
+is also valid for&nbsp; "PN modulation", or how the given equation should be modified.
@@ Line 40: / Line 40: @@
 + Only the values&nbsp; $d(\nu T) = +1$&nbsp; and&nbsp; $d(\nu T) = -1$&nbsp; are possible.
-{Which values are possible with PN modulation in a noise-free case??
+{Which values are possible with PN modulation in a noise-free case?
 |type="[]"}
 - $d(\nu T)$&nbsp; is Gaussian  distributed.
@@ Line 54: / Line 54: @@
 {The following applies &nbsp; $10 \cdot {\rm lg}\ (E_{\rm B}/N_0) = 6 \ \rm dB$. &nbsp;What error probability&nbsp; $p_{\rm B}$&nbsp; results with PN modulation? <br>''Hint'': &nbsp; For BPSK, the result is&nbsp; $p_{\rm B} \approx 2.3 \cdot 10^{-3}$.
 |type="[]"}
-- The larger&nbsp; $J$&nbsp; is selected, the smaller  $p_{\rm B}$ is.
+- The larger&nbsp; $J$&nbsp; is selected, the smaller&nbsp;  $p_{\rm B}$&nbsp; is.
-- The larger&nbsp; $J$&nbsp; is selected, the larger  $p_{\rm B}$.
+- The larger&nbsp; $J$&nbsp; is selected, the larger&nbsp;  $p_{\rm B}$.
 + It results  always the value&nbsp; $p_{\rm B} = 2.3 \cdot 10^{-3}$ results, independently from&nbsp; $J$.
@@ Line 71: / Line 71: @@
-'''(2)'''&nbsp; Correct is the <u>solution 3</u>.&nbsp; In the noise-free case &nbsp; &rArr; &nbsp; $n(t) = 0$, the twofold multiplication by&nbsp; $c(t) ∈ \{+1, -1\}$ &nbsp; &rArr; &nbsp; $c(t)^{2} = 1$&nbsp; can be omitted, so that the upper model is identical to the lower model.
+'''(2)'''&nbsp; Correct is the <u>solution 3</u>.&nbsp;
@@ Line 80: / Line 82: @@
 '''(4)'''&nbsp; Correct is the <u>solution 3</u>:
-*The equation valid for BPSK and AWGN channel
+*The equation valid for BPSK and AWGN channel is also applicable to PN modulation, independent of the spreading factor&nbsp; $J$&nbsp; and the specific spreading sequence:
-:$$p_{\rm B} = {\rm Q} \left( \hspace{-0.05cm} \sqrt { \frac{2 \cdot E_{\rm B}}{N_{\rm 0}} } \hspace{0.05cm} \right )$$
+:$$p_{\rm B} = {\rm Q} \left( \hspace{-0.05cm} \sqrt { {2 \cdot E_{\rm B}}/{N_{\rm 0}} } \hspace{0.05cm} \right )$$
 *For AWGN noise, the error probability is neither increased nor decreased by band spreading.

@@ Line 64: / Line 64: @@
 '''(1)'''&nbsp; Correct is the <u>solution 3</u>:
-*This is an optimal receiver.
+*This is an optimal receiver.&nbsp;
-*Without noise, the signal $b(t)$ within each bit is constantly equal to $+1$ or $-1$.
+*Without noise, the signal&nbsp; $b(t)$&nbsp;  is constantly equal to&nbsp; $+1$&nbsp; or&nbsp; $-1$&nbsp; within each bit.
 *From the equation given for the integrator
 :$$d (\nu T) = \frac{1}{T} \cdot \hspace{-0.03cm} \int_{(\nu -1 )T }^{\nu T} \hspace{-0.3cm} b (t )\hspace{0.1cm} {\rm d}t$$
-:follows that $d(\nu T)$ can only take the values $±1$.
+:follows that&nbsp; $d(\nu T)$&nbsp; can only take the values&nbsp; $±1$.
-'''(2)'''&nbsp; Again, the  <u>solution 3</u> is correct:
+'''(2)'''&nbsp; Correct is the <u>solution 3</u>.&nbsp; In the noise-free case &nbsp; &rArr; &nbsp; $n(t) = 0$, the twofold multiplication by&nbsp; $c(t) ∈ \{+1, -1\}$ &nbsp; &rArr; &nbsp; $c(t)^{2} = 1$&nbsp; can be omitted, so that the upper model is identical to the lower model.
 '''(3)'''&nbsp; The <u>solution 1</u> is applicable:
-*Since both models are identical in the noiseless case, only the noise signal has to be adjusted: $n'(t) = n(t) \cdot c(t)$.
+*Since both models are identical in the noiseless case, only the noise signal has to be adjusted:&nbsp; $n'(t) = n(t) \cdot c(t)$.
-*The solutions 2 and 3, on the other hand, are not applicable: Integration must still be done via $T = J \cdot T_{c}$ (not via $J \cdot T$) and PN modulation does not reduce the AWGN noise.
+*The solutions 2 and 3  are not applicable:&nbsp; Integration must still be done via&nbsp; $T = J \cdot T_{c}$&nbsp; $($not via&nbsp; $J \cdot T)$&nbsp; and PN modulation does not reduce the AWGN noise.
-'''(4)'''&nbsp; The <u>solution 3</u> is correct:
+'''(4)'''&nbsp; Correct is the <u>solution 3</u>:
 *The equation valid for BPSK and AWGN channel
 :$$p_{\rm B} = {\rm Q} \left( \hspace{-0.05cm} \sqrt { \frac{2 \cdot E_{\rm B}}{N_{\rm 0}} } \hspace{0.05cm} \right )$$
-:is therefore also applicable to PN modulation, independent of the spreading factor $J$ and the specific spreading sequence.
+:is therefore also applicable to PN modulation, independent of the spreading factor&nbsp; $J$&nbsp; and the specific spreading sequence.
 *For AWGN noise, the error probability is neither increased nor decreased by band spreading.

@@ Line 28: / Line 28: @@
 *This task refers to the chapter&nbsp; [[Mobile_Communications/Characteristics_of_UMTS|Characteristics of UMTS]].
 *The CDMA method used for UMTS is also known as "PN modulation".
-*The nomenclature used in this task is partly based on the page&nbsp; [[Modulation_Methods/Direct-Sequence_Spread_Spectrum_Modulation|PN modulation]]&nbsp; of the book "Modulation Methods".
+*The nomenclature used in this task is partly based on the page&nbsp; [[Modulation_Methods/Direct-Sequence_Spread_Spectrum_Modulation|Direct-sequence spread spectrum modulation]]&nbsp; of the book "Modulation Methods".

@@ Line 3: / Line 3: @@
 }}
-[[File:P_ID2259__Mod_Z_5_2.png|right|frame|Equivalent circuit diagrams of &bdquo;PN modulation&rdquo; and &bdquo;BPSK&rdquo;]]
+[[File:P_ID2259__Mod_Z_5_2.png|right|frame|Equivalent models for "PN modulation" and "BPSK"]]
-The upper figure shows the equivalent circuit diagram of PN modulation (&nbsp; ''Direct Sequence Spread Spectrum'', abbreviated DS-SS) in the equivalent low-pass range, where&bdquo; $n(t)$&nbsp; stands for AWGN noise.
+The figure shows the equivalent block diagram of "PN modulation" or "Direct Sequence Spread Spectrum", abbreviated&nbsp; $\rm DS-SS$&nbsp; in the equivalent low-pass range, where&nbsp; $n(t)$&nbsp; stands for AWGN noise.
-Below, the low-pass model of binary phase modulation (BPSK) is sketched
+Below, the low-pass model of binary phase modulation&nbsp; $\rm (BPSK)$&nbsp; is sketched.
-*The low-pass transmitted signal&nbsp; $s(t)$&nbsp; is equal to the rectangular source signal&nbsp; $q(t) ∈ \{+1, -1\}$&nbsp; with rectangular duration&nbsp; $T$&nbsp; only for reasons of uniformity.
+*The low-pass transmission signal&nbsp; $s(t)$&nbsp; is equal to the rectangular source signal&nbsp; $q(t) ∈ \{+1, -1\}$&nbsp; with rectangular duration&nbsp; $T$&nbsp; only for reasons of uniformity.
 *The function of the integrator can be written as follows:
 :$$d (\nu T) = \frac{1}{T} \cdot \int_{(\nu -1 )T }^{\nu T} \hspace{-0.03cm} b (t )\hspace{0.1cm} {\rm d}t \hspace{0.05cm}.$$
-*The two models differ in the multiplication by the&nbsp; $±1$-spreading signal&nbsp; $c(t)$&nbsp; at the transmitter and receiver, whereas of this signal&nbsp; $c(t)$&nbsp; only the spread degree &nbsp; $J$&nbsp; is known.
+*The two models differ in the multiplication by the&nbsp; $±1$&nbsp; spreading signal&nbsp; $c(t)$&nbsp; at the transmitter and the receiver.&nbsp; Of the signal&nbsp; $c(t)$&nbsp; only the spread degree $J$&nbsp; is known.
-*The specification of the specific spreading sequence (M sequence or Walsh function) is not important for the solution of this task.
+*The specification of the specific spreading sequence&nbsp; (M sequence or Walsh function)&nbsp; is not important for the solution of this task.
 It has to be examined whether the lower BPSK model can also be applied with PN modulation and whether the BPSK error probability
 :$$p_{\rm B} = {\rm Q} \left( \hspace{-0.05cm} \sqrt { {2 \cdot E_{\rm B}}/{N_{\rm 0}} } \hspace{0.05cm} \right )$$
-is also valid for PN modulation, or how the specified equation should be modified.
+is also valid for PN modulation, or how the given equation should be modified.
@@ Line 26: / Line 26: @@
 ''Notes:''
-*This exercise belongs to the chapter&nbsp;  [[Mobile_Communications/Die_Charakteristika_von_UMTS|Die Charakteristika von UMTS]].
+*This task refers to the chapter&nbsp; [[Mobile_Communications/Characteristics_of_UMTS|Characteristics of UMTS]].
 *The CDMA method used for UMTS is also known as "PN modulation".
-*The nomenclature used in this task is also partly based on the&nbsp; [[Modulation_Methods/PN–Modulation|PN–Modulation]]&nbsp; im Buch „Modulationsverfahren”.
+*The nomenclature used in this task is partly based on the page&nbsp; [[Modulation_Methods/Direct-Sequence_Spread_Spectrum_Modulation|PN modulation]]&nbsp; of the book "Modulation Methods".
@@ Line 35: / Line 34: @@
 <quiz display=simple>
-{Which detection values are possible with BPSK (without noise)?
+{Which values for the detection  signal&nbsp; $d(t)$&nbsp; are possible with BPSK (without noise)?
 |type="[]"}
-- $d(\nu T)$&nbsp; is gauss distributed.
+- $d(\nu T)$&nbsp; is Gaussian distributed.
-- $d(\nu T)$&nbsp; can take the values &nbsp; $+1$,&nbsp; $0$&nbsp; und&nbsp; $-1$&nbsp;.
+- $d(\nu T)$&nbsp; can take the values&nbsp; $+1$,&nbsp; $0$&nbsp; and&nbsp; $-1$&nbsp;.
-+ Only the values&nbsp; $d(\nu T) = +1$&nbsp; und&nbsp; $d(\nu T) = -1$&nbsp; are possible.
++ Only the values&nbsp; $d(\nu T) = +1$&nbsp; and&nbsp; $d(\nu T) = -1$&nbsp; are possible.
 {Which values are possible with PN modulation in a noise-free case??
 |type="[]"}
-- $d(\nu T)$&nbsp; is gauss distributed.
+- $d(\nu T)$&nbsp; is Gaussian  distributed.
 - $d(\nu T)$&nbsp; can take the values&nbsp; $+1$,&nbsp; $0$&nbsp; und&nbsp; $-1$&nbsp;.
 + Only the values&nbsp; $d(\nu T) = +1$&nbsp; und&nbsp; $d(\nu T) = -1$&nbsp; are possible.
-{What modification must be made to the BPSK model so that it can also be used for PN modulation?
+{What modification must be done to the BPSK model so that it can also be used for PN modulation?
 |type="[]"}
-+ The noise &nbsp; $n(t)$&nbsp; must be replaced by&nbsp; $n\hspace{0.05cm}'(t) = n(t) \cdot c(t)$&nbsp;.
++ The noise&nbsp; $n(t)$&nbsp; must be replaced by&nbsp; $n\hspace{0.05cm}'(t) = n(t) \cdot c(t)$.
-- The integration must now be done via&nbsp; $J \cdot T$&nbsp;.
+- The integration must now be done via&nbsp; $J \cdot T$.
 - The noise power must be reduced by the factor&nbsp; $J$&nbsp;.
-{The following applies &nbsp; $10 \cdot {\rm lg}\ (E_{\rm B}/N_0) = 6 \ \rm dB$. &nbsp;what error probability nbsp; $p_{\rm B}$&nbsp; results with PN modulation? <br>''Hint'': &nbsp; For BPSK, the result is&nbsp; $p_{\rm B} \approx 2.3 \cdot 10^{-3}$.
+{The following applies &nbsp; $10 \cdot {\rm lg}\ (E_{\rm B}/N_0) = 6 \ \rm dB$. &nbsp;What error probability&nbsp; $p_{\rm B}$&nbsp; results with PN modulation? <br>''Hint'': &nbsp; For BPSK, the result is&nbsp; $p_{\rm B} \approx 2.3 \cdot 10^{-3}$.
 |type="[]"}
-- The larger&nbsp; $J$&nbsp; is selected, the smaller&nbsp; $p_{\rm B}$ is.
+- The larger&nbsp; $J$&nbsp; is selected, the smaller  $p_{\rm B}$ is.
-- The larger&nbsp; $J$&nbsp; is selected, the larger&nbsp; $p_{\rm B}$.
+- The larger&nbsp; $J$&nbsp; is selected, the larger  $p_{\rm B}$.
-+ It results independently from&nbsp; $J$&nbsp;, always the value&nbsp; $p_{\rm B} = 2.3 \cdot 10^{-3}$ results.
++ It results  always the value&nbsp; $p_{\rm B} = 2.3 \cdot 10^{-3}$ results, independently from&nbsp; $J$.
 </quiz>

@@ Line 61: / Line 61: @@
 </quiz>
-===Sample solution===
+===Solution===
 {{ML-Kopf}}

← Older revision		Revision as of 14:04, 15 December 2020
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−	{{quiz-Header\|Buchseite=~~Mobile Kommunikation~~/~~Die Charakteristika von UMTS~~	+	{{quiz-Header\|Buchseite=Mobile_Communications/Characteristics_of_UMTS
	}}		}}

← Older revision		Revision as of 12:41, 9 July 2020
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	*The CDMA method used for UMTS is also known as "PN modulation".		*The CDMA method used for UMTS is also known as "PN modulation".
−	*The nomenclature used in this task is also partly based on the  [[~~Modulationsverfahren~~/PN–Modulation\|PN–Modulation]]  im Buch „Modulationsverfahren”.	+	*The nomenclature used in this task is also partly based on the  [[Modulation_Methods/PN–Modulation\|PN–Modulation]]  im Buch „Modulationsverfahren”.