Aufgaben:Exercise 5.2Z: About PN Modulation - Revision history

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Guenter at 15:25, 15 November 2021

2021-11-15T15:25:45Z

Guenter: Guenter moved page Aufgaben:Aufgabe 5.2Z: Zur PN–Modulation to Aufgaben:Exercise 5.2Z: About PN Modulation

2021-11-15T15:24:55Z

Guenter moved page Aufgabe 5.2Z: Zur PN–Modulation to Exercise 5.2Z: About PN Modulation

@@ Line 69: / Line 69: @@
 '''(2)'''&nbsp; Again the&nbsp; <u>last solution</u>&nbsp; is correct:
-* In the noise&ndash; and interference-free case &nbsp; ⇒ &nbsp; $n(t) = 0$,&nbsp; the twofold multiplication by&nbsp; $c(t) ∈ \{+1, –1\}$&nbsp; can be omitted,
+* In the noise-free and interference-free case &nbsp; ⇒ &nbsp; $n(t) = 0$,&nbsp; the twofold multiplication by&nbsp; $c(t) ∈ \{+1, –1\}$&nbsp; can be omitted,
 *so that the upper model is identical to the lower model.

@@ Line 3: / Line 3: @@
 }}
-[[File:P_ID1871__Mod_Z_5_2.png|right|frame|Models of PN modulation (top) and BPSK (bottom)]]
+[[File:EN_Bei_A_4_5.png|right|frame|Models of PN modulation (top) and BPSK (bottom)]]
 The upper diagram shows the equivalent circuit of&nbsp; $\rm PN$&nbsp; modulation&nbsp; $($Direct-Sequence Spread Spectrum, abbreviated&nbsp; $\rm DS–SS)$&nbsp; in the equivalent low-pass range,&nbsp; based on AWGN noise &nbsp;$n(t)$.&nbsp;

@@ Line 6: / Line 6: @@
 The upper diagram shows the equivalent circuit of&nbsp; $\rm PN$&nbsp; modulation&nbsp; $($Direct-Sequence Spread Spectrum, abbreviated&nbsp; $\rm DS–SS)$&nbsp; in the equivalent low-pass range,&nbsp; based on AWGN noise &nbsp;$n(t)$.&nbsp;
-Shown below is the low-pass model of binary phase shift keying&nbsp; $\rm (BPSK)$.&nbsp; The low-pass transmitted signal &nbsp;$s(t)$&nbsp; is set equal to the rectangular source signal &nbsp;$q(t) ∈ \{+1, –1\}$&nbsp; with rectangular duration &nbsp;$T$&nbsp; for reasons of uniformity.
+Shown below is the low-pass model of binary phase shift keying&nbsp; $\rm (BPSK)$.&nbsp;
-The function of the integrator can be described as follows:
+*The function of the integrator can be described as follows:
 :$$d (\nu T) = \frac{1}{T} \cdot \hspace{-0.1cm} \int_{(\nu -1 )T }^{\nu T} \hspace{-0.3cm} b (t )\hspace{0.1cm} {\rm d}t \hspace{0.05cm}.$$
-The two models differ by multiplication with the &nbsp;$±1$ spreading signal &nbsp;$c(t)$&nbsp; at transmitter and receiver,&nbsp; where only the spreading factor &nbsp;$J$&nbsp; is known from &nbsp;$c(t)$.&nbsp;
+*The two models differ by multiplication with the &nbsp;$±1$ spreading signal &nbsp;$c(t)$&nbsp; at transmitter and receiver,&nbsp; where only the spreading factor &nbsp;$J$&nbsp; is known from &nbsp;$c(t)$.&nbsp;
 It has to be investigated whether the lower BPSK model can also be used for PN modulation and whether the BPSK error probability
@@ Line 23: / Line 25: @@
 Notes:
 *This exercise belongs to the chapter&nbsp; [[Modulation_Methods/Direct-Sequence_Spread_Spectrum_Modulation|Direct-Sequence Spread Spectrum Modulation]].
 *For the solution of this exercise,&nbsp; the specification of the specific spreading sequence&nbsp; $($M-sequence or Walsh function$)$&nbsp; is not important.

@@ Line 80: / Line 80: @@
 '''(4)'''&nbsp; The&nbsp; <u>last solution</u>&nbsp; is correct:
 *Multiplying the AWGN noise by the high-frequency&nbsp; $±1$ signal&nbsp; $c(t)$,&nbsp; the product is also Gaussian and white.
-*Because of &nbsp;${\rm E}\big[c^2(t)\big] = 1$,&nbsp; the noise variance is not changed either.
+*Because of &nbsp;${\rm E}\big[c^2(t)\big] = 1$,&nbsp; the noise variance is not changed either.&nbsp; Thus:
-*Thus,&nbsp; the equation&nbsp; $p_{\rm B} = {\rm Q} \left( \hspace{-0.05cm} \sqrt {{2 E_{\rm B}}/{N_{\rm 0}} } \hspace{0.05cm} \right )$&nbsp; valid for BPSK is also applicable for PN modulation,&nbsp; independent of the spreading factor&nbsp; $J$&nbsp; and the specific spreading sequence.
+*The equation&nbsp; $p_{\rm B} = {\rm Q} \left( \hspace{-0.05cm} \sqrt {{2 E_{\rm B}}/{N_{\rm 0}} } \hspace{0.05cm} \right )$&nbsp; valid for BPSK is also applicable for PN modulation,&nbsp; independent of spreading factor&nbsp; $J$&nbsp; and specific spreading sequence.
-*Ergo: &nbsp; &nbsp; For AWGN noise,&nbsp; band spreading neither increases nor decreases the error probability.
+*Ergo:&nbsp; For AWGN noise,&nbsp; band spreading neither increases nor decreases the error probability.
 {{ML-Fuß}}

@@ Line 56: / Line 56: @@
 ===Solution===
 {{ML-Kopf}}
-'''(1)'''&nbsp; The <u>last solution</u> is correct:
+'''(1)'''&nbsp; The&nbsp; <u>last solution</u>&nbsp; is correct:
 *We are dealing here with an optimal receiver.
-*Without noise, the signal&nbsp; $b(t)$&nbsp; within each bit is constantly equal to&nbsp; $+1$&nbsp; or&nbsp; $-1$.
+*Without noise,&nbsp; the signal&nbsp; $b(t)$&nbsp; within each bit is constantly equal to&nbsp; $+1$&nbsp; or&nbsp; $-1$.
 *From the given equation for the integrator
 :$$d (\nu T) = \frac{1}{T} \cdot \hspace{-0.1cm} \int_{(\nu -1 )T }^{\nu T} \hspace{-0.3cm} b (t )\hspace{0.1cm} {\rm d}t $$
@@ Line 65: / Line 65: @@
-'''(2)'''&nbsp; Again the <u>last solution</u> is correct:
+'''(2)'''&nbsp; Again the&nbsp; <u>last solution</u>&nbsp; is correct:
 * In the noise&ndash; and interference-free case &nbsp; ⇒ &nbsp; $n(t) = 0$,&nbsp; the twofold multiplication by&nbsp; $c(t) ∈ \{+1, –1\}$&nbsp; can be omitted,
 *so that the upper model is identical to the lower model.
@@ Line 71: / Line 71: @@
-'''(3)'''&nbsp; <u>Solution 1</u> is correct:
+'''(3)'''&nbsp; <u>Solution 1</u>&nbsp; is correct:
-*Since both models are identical in the noise-free case, only the noise signal has to be adjusted: &nbsp; $n'(t) = n(t) · c(t)$.
+*Since both models are identical in the noise-free case,&nbsp; only the noise signal has to be adjusted: &nbsp; $n'(t) = n(t) · c(t)$.
-*In contrast, the other two solutions are not applicable:
+*In contrast,&nbsp; the other two solutions are not applicable:
 *The integration must still be done over&nbsp; $T = J · T_c$&nbsp; and the PN modulation does not reduce the AWGN noise.
-'''(4)'''&nbsp; The <u>last solution</u> is correct:
+'''(4)'''&nbsp; The&nbsp; <u>last solution</u>&nbsp; is correct:
-*Multiplying the AWGN noise by the high-frequency&nbsp; $±1$ signal&nbsp; $c(t)$, the product is also Gaussian and white.
+*Multiplying the AWGN noise by the high-frequency&nbsp; $±1$ signal&nbsp; $c(t)$,&nbsp; the product is also Gaussian and white.
 *Because of &nbsp;${\rm E}\big[c^2(t)\big] = 1$,&nbsp; the noise variance is not changed either.
-*Thus, the equation&nbsp; $p_{\rm B} = {\rm Q} \left( \hspace{-0.05cm} \sqrt {{2 E_{\rm B}}/{N_{\rm 0}} } \hspace{0.05cm} \right )$&nbsp; valid for BPSK is also applicable for PN modulation, independent of the spreading factor&nbsp; $J$&nbsp; and the specific spreading sequence.
+*Thus,&nbsp; the equation&nbsp; $p_{\rm B} = {\rm Q} \left( \hspace{-0.05cm} \sqrt {{2 E_{\rm B}}/{N_{\rm 0}} } \hspace{0.05cm} \right )$&nbsp; valid for BPSK is also applicable for PN modulation,&nbsp; independent of the spreading factor&nbsp; $J$&nbsp; and the specific spreading sequence.
-*Ergo: &nbsp; &nbsp; For AWGN noise, band spreading neither increases nor decreases the error probability.
+*Ergo: &nbsp; &nbsp; For AWGN noise,&nbsp; band spreading neither increases nor decreases the error probability.
 {{ML-Fuß}}

@@ Line 4: / Line 4: @@
 [[File:P_ID1871__Mod_Z_5_2.png|right|frame|Models of PN modulation (top) and BPSK (bottom)]]
-The diagram shows the equivalent circuit of&nbsp; $\rm PN$ modulation&nbsp; $($Direct Sequence Spread Spectrum, abbreviated&nbsp; $\rm DS–SS)$&nbsp; in the equivalent low-pass range, based on AWGN noise &nbsp;$n(t)$.&nbsp; Shown below is the low-pass model of binary phase modulation&nbsp; $\rm (BPSK)$.
+The upper diagram shows the equivalent circuit of&nbsp; $\rm PN$&nbsp; modulation&nbsp; $($Direct-Sequence Spread Spectrum, abbreviated&nbsp; $\rm DS–SS)$&nbsp; in the equivalent low-pass range,&nbsp; based on AWGN noise &nbsp;$n(t)$.&nbsp;
-The low-pass transmitted signal &nbsp;$s(t)$&nbsp; is set equal to the rectangular source signal &nbsp;$q(t) ∈ \{+1, –1\}$&nbsp; with rectangular duration &nbsp;$T$&nbsp; for reasons of uniformity.
+Shown below is the low-pass model of binary phase shift keying&nbsp; $\rm (BPSK)$.&nbsp; The low-pass transmitted signal &nbsp;$s(t)$&nbsp; is set equal to the rectangular source signal &nbsp;$q(t) ∈ \{+1, –1\}$&nbsp; with rectangular duration &nbsp;$T$&nbsp; for reasons of uniformity.
 The function of the integrator can be described as follows:
 :$$d (\nu T) = \frac{1}{T} \cdot \hspace{-0.1cm} \int_{(\nu -1 )T }^{\nu T} \hspace{-0.3cm} b (t )\hspace{0.1cm} {\rm d}t \hspace{0.05cm}.$$
-The two models differ by multiplication with the &nbsp;$±1$ spread signal &nbsp;$c(t)$&nbsp; at transmitter and receiver, where only the degree of spread &nbsp;$J$&nbsp; is known from &nbsp;$c(t)$.&nbsp;
+The two models differ by multiplication with the &nbsp;$±1$ spreading signal &nbsp;$c(t)$&nbsp; at transmitter and receiver,&nbsp; where only the spreading factor &nbsp;$J$&nbsp; is known from &nbsp;$c(t)$.&nbsp;
 It has to be investigated whether the lower BPSK model can also be used for 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 given equation should be modified.
+is also valid for PN modulation,&nbsp; or how the given equation should be modified.
@@ Line 21: / Line 21: @@
 Notes:
 *This exercise belongs to the chapter&nbsp; [[Modulation_Methods/Direct-Sequence_Spread_Spectrum_Modulation|Direct-Sequence Spread Spectrum Modulation]].
-*For the solution of this exercise, the specification of the specific spreading sequence&nbsp; $($M-sequence or Walsh function$)$&nbsp; is not important.
+*For the solution of this exercise,&nbsp; the specification of the specific spreading sequence&nbsp; $($M-sequence or Walsh function$)$&nbsp; is not important.
@@ Line 32: / Line 29: @@
 <quiz display=simple>
-{Which detection signal values are possible with BPSK (in the noise-free case)?
+{Which detection signal values are possible with BPSK&nbsp; (in the noise-free case)?
 |type="[]"}
 - $d(νT)$&nbsp; can be Gaussian distributed.
@@ Line 38: / Line 35: @@
 + Only the values &nbsp;$d(νT) = +1$&nbsp; and &nbsp;$d(νT) = -1$&nbsp; are possible.
-{Which values are possible in PN modulation (in the noise-free) case?
+{Which values are possible in PN modulation&nbsp; (in the noise-free)&nbsp; case?
 |type="[]"}
 - $d(νT)$&nbsp; can be Gaussian distributed.
@@ Line 50: / Line 47: @@
 - The noise power &nbsp;$σ_n^2$&nbsp; must be reduced by a factor of &nbsp;$J$.&nbsp;
-{What is the bit error probability &nbsp;$p_{\rm B}$&nbsp; for &nbsp;$10 \lg \  (E_{\rm B}/N_0) = 6\ \rm  dB$&nbsp; for PN modulation?&nbsp; ''Note:'' &nbsp; For BPSK, the following applies in this case: &nbsp; $p_{\rm B} ≈ 2.3 · 10^{–3}$.
+{What is the bit error probability &nbsp;$p_{\rm B}$&nbsp; for &nbsp;$10 \lg \  (E_{\rm B}/N_0) = 6\ \rm  dB$&nbsp; for PN modulation?&nbsp; <br>Note: &nbsp; For BPSK, the following applies in this case: &nbsp; $p_{\rm B} ≈ 2.3 · 10^{–3}$.
 |type="()"}
 - The larger &nbsp;$J$&nbsp; is chosen, the smaller &nbsp;$p_{\rm B}$ is.

← Older revision		Revision as of 15:25, 15 November 2021
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−	[[Category:Modulation Methods: Exercises\|^5.2 ~~PN–Modulation~~^]]	+	[[Category:Modulation Methods: Exercises\|^5.2 PN Modulation^]]