Aufgaben:Exercise 4.5: Pseudo Noise Modulation - Revision history

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	[[Category:Examples of Communication Systems: Exercises\|^4.3 Telecommunications Aspects^]]		[[Category:Examples of Communication Systems: Exercises\|^4.3 Telecommunications Aspects^]]
	[[de:Aufgaben:4.~~5_PN~~-Modulation]]		[[de:Aufgaben:Aufgabe 4.5: Pseudo Noise-Modulation]]

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	[[Category:Examples of Communication Systems: Exercises\|^4.3 Telecommunications Aspects^]]		[[Category:Examples of Communication Systems: Exercises\|^4.3 Telecommunications Aspects^]]
			[[de:Aufgaben:4.5_PN-Modulation]]

Guenter at 17:22, 4 March 2023

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	[[Category:Examples of Communication Systems: Exercises\|^4.3 Telecommunications Aspects]]		[[Category:Examples of Communication Systems: Exercises\|^4.3 Telecommunications Aspects^]]

Guenter at 16:59, 4 March 2023

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	[[Category:~~Modulation Methods~~: Exercises\|^5.~~2 PN Modulation^~~]]		[[Category:Examples of Communication Systems: Exercises\|^4.3 Telecommunications Aspects]]

Guenter at 16:56, 4 March 2023

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Guenter at 16:51, 4 March 2023

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	- The noise power  $σ_n^2$  must be reduced by a factor of  $J$.		- The noise power  $σ_n^2$  must be reduced by a factor of  $J$.

	{What is the bit error probability  $p_{\rm B}$  for  $10 \lg \ (E_{\rm B}/N_0) = 6\ \rm dB$  for PN modulation?  <br>Note:   For BPSK~~, the following~~ applies in this case:   $p_{\rm B} ≈ 2.3 · 10^{–3}$.		{What is the bit error probability  $(p_{\rm B})$  for  $10 \lg \ (E_{\rm B}/N_0) = 6\ \rm dB$  for PN modulation?   <u>Note:</u>   For BPSK applies in this case:   $p_{\rm B} ≈ 2.3 · 10^{–3}$.
	\|type="()"}		\|type="()"}
	- The larger  $J$  is chosen, the smaller  $p_{\rm B}$ is.		- The larger  $J$  is chosen, the smaller  $p_{\rm B}$ is.

Guenter at 16:47, 4 March 2023

2023-03-04T16:47:35Z

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Noah at 18:08, 2 March 2023

2023-03-02T18:08:20Z

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	[[File:EN_Bei_A_4_5.png\|right\|frame\|Equivalent circuit of PN modulation and BPSK]]		[[File:EN_Bei_A_4_5.png\|right\|frame\|Equivalent circuit of PN modulation and BPSK]]
	The graph above shows the equivalent circuit of "pseudo-noise" modulation  ( ''Direct Sequence Spread Spectrum'', abbreviated  '''DS-SS''')  in the equivalent low-pass region. $n(t)$  denotes AWGN noise.		The graph above shows the equivalent circuit of "pseudo-noise" modulation  (''Direct Sequence Spread Spectrum'', abbreviated  '''DS-SS''')  in the equivalent low-pass region. $n(t)$  denotes AWGN noise.

	Below is sketched the low pass model of  [[Modulation_Methods/Linear_Digital_Modulation#BPSK_.E2.80.93_Binary_Phase_Shift_Keying\|"Binary Phase Shift Keying"]].		Below is sketched the low pass model of  [[Modulation_Methods/Linear_Digital_Modulation#BPSK_.E2.80.93_Binary_Phase_Shift_Keying\|"Binary Phase Shift Keying"]].

Noah at 18:08, 2 March 2023

2023-03-02T18:08:14Z

← Older revision		Revision as of 20:08, 2 March 2023
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	[[File:EN_Bei_A_4_5.png\|right\|frame\|Equivalent circuit of PN modulation and BPSK]]		[[File:EN_Bei_A_4_5.png\|right\|frame\|Equivalent circuit of PN modulation and BPSK]]
	The graph above shows the equivalent circuit of "pseudo-noise" modulation  (~~ ~~''Direct Sequence Spread Spectrum'', abbreviated  '''DS-SS''')  in the equivalent low-pass region. $n(t)$  denotes AWGN noise.		The graph above shows the equivalent circuit of "pseudo-noise" modulation  ( ''Direct Sequence Spread Spectrum'', abbreviated  '''DS-SS''')  in the equivalent low-pass region. $n(t)$  denotes AWGN noise.

	Below is sketched the low pass model of  [[Modulation_Methods/Linear_Digital_Modulation#BPSK_.E2.80.93_Binary_Phase_Shift_Keying\|"Binary Phase Shift Keying"]].		Below is sketched the low pass model of  [[Modulation_Methods/Linear_Digital_Modulation#BPSK_.E2.80.93_Binary_Phase_Shift_Keying\|"Binary Phase Shift Keying"]].

Noah at 18:08, 2 March 2023

2023-03-02T18:08:04Z

← Older revision		Revision as of 20:08, 2 March 2023
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	[[File:EN_Bei_A_4_5.png\|right\|frame\|Equivalent circuit of PN modulation and BPSK]]		[[File:EN_Bei_A_4_5.png\|right\|frame\|Equivalent circuit of PN modulation and BPSK]]
	The graph above shows the equivalent circuit of "pseudo-noise" modulation  (  ''Direct Sequence Spread Spectrum'', abbreviated  '''DS-SS''')  in the equivalent low-pass region. $n(t)$  denotes AWGN noise.		The graph above shows the equivalent circuit of "pseudo-noise" modulation  ( ''Direct Sequence Spread Spectrum'', abbreviated  '''DS-SS''')  in the equivalent low-pass region. $n(t)$  denotes AWGN noise.

	Below is sketched the low pass model of  [[Modulation_Methods/Linear_Digital_Modulation#BPSK_.E2.80.93_Binary_Phase_Shift_Keying\|"Binary Phase Shift Keying"]].		Below is sketched the low pass model of  [[Modulation_Methods/Linear_Digital_Modulation#BPSK_.E2.80.93_Binary_Phase_Shift_Keying\|"Binary Phase Shift Keying"]].

← Older revision		Revision as of 18:56, 4 March 2023
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	'''(1)'''  The  <u>last solution</u>  is correct:		'''(1)'''  The  <u>last solution</u>  is correct:
	*We are dealing here with an optimal receiver.		*We are dealing here with an optimal receiver.

	*Without noise,  the signal  $b(t)$  within each bit is constantly equal to  $+1$  or  $-1$.		*Without noise,  the signal  $b(t)$  within each bit is constantly equal to  $+1$  or  $-1$.

	*From the given equation for the integrator		*From the given equation for the integrator
	:$$d (\nu T) = \frac{1}{T} \cdot \hspace{-0.1cm} \int_{(\nu -1 )T }^{\nu T} \hspace{-0.3cm} b (t )\hspace{0.1cm} {\rm d}t $$		:$$d (\nu T) = \frac{1}{T} \cdot \hspace{-0.1cm} \int_{(\nu -1 )T }^{\nu T} \hspace{-0.3cm} b (t )\hspace{0.1cm} {\rm d}t $$
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	'''(2)'''  Again the  <u>last solution</u>  is correct:		'''(2)'''  Again the  <u>last solution</u>  is correct:
	* In the noise~~–~~ and interference-free case   ⇒   $n(t) = 0$,  the twofold multiplication by  $c(t) ∈ \{+1, –1\}$  can be omitted,		* In the noise-free and interference-free case   ⇒   $n(t) = 0$,  the twofold multiplication by  $c(t) ∈ \{+1, –1\}$  can be omitted,

	*so that the upper model is identical to the lower model.		*so that the upper model is identical to the lower model.

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	'''(3)'''  <u>Solution 1</u>  is correct:		'''(3)'''  <u>Solution 1</u>  is correct:
	*Since both models are identical in the noise-free case,  only the noise signal has to be adjusted:   $n'(t) = n(t) · c(t)$.		*Since both models are identical in the noise-free case,  only the noise signal has to be adjusted:   $n'(t) = n(t) · c(t)$.

	*In contrast,  the other two solutions are not applicable:		*In contrast,  the other two solutions are not applicable:

	*The integration must still be done over  $T = J · T_c$  and the PN modulation does not reduce the AWGN noise.		*The integration must still be done over  $T = J · T_c$  and the PN modulation does not reduce the AWGN noise.

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	'''(4)'''  The  <u>last solution</u>  is correct:		'''(4)'''  The  <u>last solution</u>  is correct:
	*Multiplying the AWGN noise by the high-frequency  $±1$ signal  $c(t)$,  the product is also Gaussian and white.		*Multiplying the AWGN noise by the high-frequency  $±1$ signal  $c(t)$,  the product is also Gaussian and white.
	*Because of  ${\rm E}\big[c^2(t)\big] = 1$,  the noise variance is not changed either.  Thus:
			*Because of  ${\rm E}\big[c^2(t)\big] = 1$,  the noise variance is not changed either.  Thus:

	*The equation  $p_{\rm B} = {\rm Q} \left( \hspace{-0.05cm} \sqrt {{2 E_{\rm B}}/{N_{\rm 0}} } \hspace{0.05cm} \right )$  valid for BPSK is also applicable for PN modulation,  independent of spreading factor  $J$  and specific spreading sequence.		*The equation  $p_{\rm B} = {\rm Q} \left( \hspace{-0.05cm} \sqrt {{2 E_{\rm B}}/{N_{\rm 0}} } \hspace{0.05cm} \right )$  valid for BPSK is also applicable for PN modulation,  independent of spreading factor  $J$  and specific spreading sequence.

	*Ergo:  For AWGN noise,  band spreading neither increases nor decreases the error probability.		*Ergo:  For AWGN noise,  band spreading neither increases nor decreases the error probability.
	{{ML-Fuß}}		{{ML-Fuß}}