Aufgaben:Exercise 4.7: Spectra of ASK and BPSK - Revision history

Hwang at 12:31, 18 January 2023

2023-01-18T12:31:03Z

Guenter at 12:31, 12 April 2022

2022-04-12T12:31:27Z

Guenter at 10:19, 12 April 2022

2022-04-12T10:19:37Z

Guenter at 10:09, 12 April 2022

2022-04-12T10:09:00Z

Hwang at 15:40, 21 March 2022

2022-03-21T15:40:23Z

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Guenter at 14:01, 16 March 2022

2022-03-16T14:01:34Z

Guenter: Guenter moved page Aufgaben:Aufgabe 4.7: Spektren von ASK und BPSK to Aufgaben:Exercise 4.7: Spectra of ASK and BPSK

2022-03-16T13:55:36Z

Guenter moved page Aufgabe 4.7: Spektren von ASK und BPSK to Exercise 4.7: Spectra of ASK and BPSK

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"

Javier: Text replacement - "[[Digitalsignalübertragung" to "[[Digital_Signal_Transmission"

2020-07-09T12:58:55Z

Text replacement - "[[Digitalsignalübertragung" to "[[Digital_Signal_Transmission"

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

2020-07-09T12:43:32Z

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

@@ Line 58: / Line 58: @@
 |type="[]"}
 + The continuous part of &nbsp;$ {\it Φ}_q(f)$&nbsp; is equal in form to &nbsp;$|G_q(f)|^2$.
-- ${\it Φ}_q(f)$&nbsp; contains a single Dirac line at ASK&nbsp; for BPSK $($at $f = 0)$.
+- ${\it Φ}_q(f)$&nbsp; contains a single Dirac delta line at ASK&nbsp; for BPSK $($at $f = 0)$.
-- ${\it Φ}_q(f)$ contains a single Dirac line at BPSK&nbsp; for BPSK $($at $f = 0)$.
+- ${\it Φ}_q(f)$ contains a single Dirac delta line at BPSK&nbsp; for BPSK $($at $f = 0)$.
 </quiz>
@@ Line 76: / Line 76: @@
 '''(2)'''&nbsp; The spectrum&nbsp; $Z(f)$&nbsp; of a cosine signal&nbsp; $z(t)$&nbsp; consists of two Dirac delta functions at&nbsp; $\pm f_{\rm T}$,&nbsp; each with weight&nbsp; $1/2$.
-*The power-spectral density&nbsp; ${\it Φ}_z(f)$&nbsp; also consists of the two Dirac delta  functions,&nbsp; but now with respective weights&nbsp; $1/4$.
+*The power-spectral density&nbsp; ${\it Φ}_z(f)$&nbsp; also consists of the two Dirac delta functions,&nbsp; but now with respective weights&nbsp; $1/4$.
 *The convolution&nbsp; ${\it Φ}_q(f) ∗ {\it Φ}_z(f)$&nbsp; gives the power-spectral density&nbsp; ${\it Φ}_s(f)$&nbsp; of the transmitted signal.&nbsp; It follows that:
 :$$C =  {A}/{4} \hspace{0.15cm}\underline { = 0.25 \cdot 10^{-6} \,{\rm
@@ Line 92: / Line 92: @@
 '''(3)'''&nbsp; For BPSK, the source signal&nbsp; $q(t)$&nbsp; is to be bipolar.
-*Therefore,&nbsp; the Dirac line is missing in the power-spectral density&nbsp;  ⇒  &nbsp; $\underline{B = 0}$.
+*Therefore,&nbsp; the Dirac delta line is missing in the power-spectral density&nbsp;  ⇒  &nbsp; $\underline{B = 0}$.
 *   The continuous PSD component is four times larger than with ASK:
 :$$A =  {s_0^2 \cdot T }\hspace{0.15cm}\underline { = 4 \cdot 10^{-6} \,{\rm V^{2}/Hz}}.$$
@@ Line 105: / Line 105: @@
 '''(5)'''&nbsp; Only the&nbsp; <u>first statement</u>&nbsp; is correct:
-* For BPSK,&nbsp;  ${\it Φ}_q(f)$&nbsp; does not contain a single Dirac line even if&nbsp; $g_q(t)$&nbsp; deviates from the rectangular form&nbsp; (assuming equally probable symbols).
+* For BPSK,&nbsp;  ${\it Φ}_q(f)$&nbsp; does not contain a single Dirac delta line even if&nbsp; $g_q(t)$&nbsp; deviates from the rectangular form&nbsp; (assuming equally probable symbols).
-*In contrast, the unipolar ASK source signal contains infinitely many Dirac lines at all multiples of&nbsp; $1/T$.
+*In contrast, the unipolar ASK source signal contains infinitely many Dirac delta lines at all multiples of&nbsp; $1/T$.
 *For more information on this topic, see the page&nbsp;  "ACF and PSD for unipolar binary signals" in the book "Digital Signal Transmission".

@@ Line 92: / Line 92: @@
 '''(3)'''&nbsp; For BPSK, the source signal&nbsp; $q(t)$&nbsp; is to be bipolar.
-*Therefore, the Dirac line is missing in the power-spectral density&nbsp;  ⇒  &nbsp; $\underline{B = 0}$.
+*Therefore,&nbsp; the Dirac line is missing in the power-spectral density&nbsp;  ⇒  &nbsp; $\underline{B = 0}$.
-*   The continuous PSD component is four times larger than in the ASK:
+*   The continuous PSD component is four times larger than with ASK:
 :$$A =  {s_0^2 \cdot T }\hspace{0.15cm}\underline { = 4 \cdot 10^{-6} \,{\rm V^{2}/Hz}}.$$
-'''(4)'''&nbsp; For the PSD parameters of the BPSK transmitted signal, the following applies analogously to the ASK:
+'''(4)'''&nbsp; For the PSD parameters of the BPSK transmitted signal,&nbsp; the following applies analogously to the ASK:
 :$$C = \frac {A}{4}\hspace{0.15cm}\underline { = 10^{-6} \,{\rm V^{2}/Hz}},\hspace{0.2cm}D =
 \frac {B}{4} \hspace{0.15cm}\underline {= 0}.$$
@@ Line 104: / Line 104: @@
-'''(5)'''&nbsp; Only the <u>first statement</u> is correct:
+'''(5)'''&nbsp; Only the&nbsp; <u>first statement</u>&nbsp; is correct:
-* For BPSK&nbsp; (bipolar source signal)&nbsp; &nbsp; ${\it Φ}_q(f)$&nbsp; does not contain a single Dirac line even if&nbsp; $g_q(t)$&nbsp; deviates from the rectangular form&nbsp; (assuming equally probable symbols).
+* For BPSK,&nbsp;  ${\it Φ}_q(f)$&nbsp; does not contain a single Dirac line even if&nbsp; $g_q(t)$&nbsp; deviates from the rectangular form&nbsp; (assuming equally probable symbols).
 *In contrast, the unipolar ASK source signal contains infinitely many Dirac lines at all multiples of&nbsp; $1/T$.
+*For more information on this topic, see the page&nbsp;  "ACF and PSD for unipolar binary signals" in the book "Digital Signal Transmission".
 {{ML-Fuß}}

@@ Line 65: / Line 65: @@
 ===Solution===
 {{ML-Kopf}}
-'''(1)'''&nbsp; The DC component of the unipolar redundancy-free source signal is&nbsp; $m_q = s_0/2$.&nbsp; Thus, the Dirac weight is&nbsp; $B = m_q^2 = s_0^2/4\hspace{0.15cm}\underline{ = 1 \ \rm V^2}$.
+'''(1)'''&nbsp; The DC component of the unipolar redundancy-free source signal is&nbsp; $m_q = s_0/2$.&nbsp; Thus,&nbsp; the Dirac weight is&nbsp; $B = m_q^2 = s_0^2/4\hspace{0.15cm}\underline{ = 1 \ \rm V^2}$.
-*Without this DC component, the stochastic square wave signal&nbsp; $q(t) - m_q ∈ \{+s_0/2, -s_0/2\}$ would be obtained.
+*Without this DC component,&nbsp; the stochastic  signal&nbsp; $q(t) - m_q ∈ \{+s_0/2, -s_0/2\}$ would be obtained.
-*This DC-free signal has the continuous PSD component&nbsp; $(s_0/2)^2 · T · {\rm si}^2(πfT)$.
+*This DC-free signal has the continuous PSD component&nbsp; $(s_0/2)^2 · T · {\rm sinc}^2(fT)$.
-*From this, the value we are looking for at frequency&nbsp; $f = 0$&nbsp; can be determined:
+*From this,&nbsp; the value we are looking for at frequency&nbsp; $f = 0$&nbsp; can be determined:
 :$$A = \frac {s_0^2 \cdot T }{4} = \frac {(2\,{\rm V})^2 \cdot
 ^{-6} \,{\rm s}}{4}\hspace{0.15cm}\underline {= 10^{-6} \,{\rm V^{2}/Hz}}.$$
@@ Line 75: / Line 75: @@
-'''(2)'''&nbsp; The spectrum&nbsp; $Z(f)$&nbsp; of a cosine signal&nbsp; $z(t)$&nbsp; consists of two Dirac functions at&nbsp; $\pm f_{\rm T}$, each with weight&nbsp; $1/2$.
+'''(2)'''&nbsp; The spectrum&nbsp; $Z(f)$&nbsp; of a cosine signal&nbsp; $z(t)$&nbsp; consists of two Dirac delta functions at&nbsp; $\pm f_{\rm T}$,&nbsp; each with weight&nbsp; $1/2$.
-*The power-spectral density&nbsp; ${\it Φ}_z(f)$&nbsp; also consists of the two Dirac functions, but now with respective weights&nbsp; $1/4$.
+*The power-spectral density&nbsp; ${\it Φ}_z(f)$&nbsp; also consists of the two Dirac delta  functions,&nbsp; but now with respective weights&nbsp; $1/4$.
 *The convolution&nbsp; ${\it Φ}_q(f) ∗ {\it Φ}_z(f)$&nbsp; gives the power-spectral density&nbsp; ${\it Φ}_s(f)$&nbsp; of the transmitted signal.&nbsp; It follows that:
 :$$C =  {A}/{4} \hspace{0.15cm}\underline { = 0.25 \cdot 10^{-6} \,{\rm
 V^{2}/Hz}},\hspace{0.2cm}D = {B}/{4}\hspace{0.15cm}\underline { = 0.25 \,{\rm V^{2}}}.$$
-''Note:'' &nbsp; The power per bit is obtained as the integral over&nbsp; ${\it Φ}_s(f)$:
+Note: &nbsp; The power per bit is obtained as the integral over&nbsp; ${\it Φ}_s(f)$:
 :$$P_{\rm S}  = \int_{ - \infty }^\infty \hspace{-0.3cm}  {{\it \Phi}_s(f)}\hspace{0.1cm} {\rm d}f
-  = 2 \cdot \int_{ 0 }^\infty \hspace{-0.3cm}  {\left [ C \cdot {\rm si}^2(\pi f T) + D \cdot \delta (f - f_{\rm T}]\right ]}\hspace{0.1cm} {\rm
+  = 2 \cdot \int_{ 0 }^\infty \hspace{-0.3cm}  {\left [ C \cdot {\rm sinc}^2(f T) + D \cdot \delta (f - f_{\rm T}]\right ]}\hspace{0.1cm} {\rm
   d}f= 2 \cdot \left [ \frac{C}{T}  + D \right ] =
 \cdot \left [ \frac{0.25 \cdot 10^{-6} \,{\rm

@@ Line 4: / Line 4: @@
 [[File:P_ID1701__Mod_A_4_6.png|right|frame|Power-spectral densities of&nbsp; $q(t)$&nbsp; and&nbsp; $s(t)$&nbsp;  &ndash; valid for ASK and BPSK]]
-The transmitted signals of ASK&nbsp; (''Amplitude Shift Keying'')&nbsp; and BPSK&nbsp; (''Binary Phase Shift Keying'')&nbsp; can both be expressed in the form
+The transmitted signals of&nbsp; $\rm ASK$&nbsp; ("Amplitude Shift Keying")&nbsp; and&nbsp; $\rm BPSK$&nbsp; ("Binary Phase Shift Keying")&nbsp; can both be expressed in the form
-:$$s(t) = q(t) · z(t)$$
+:$$s(t) = q(t) · z(t),$$
-where &nbsp;$z(t)$&nbsp; represents a harmonic oscillation with frequency &nbsp;$f_{\rm T}$&nbsp; and amplitude &nbsp;$1$.&nbsp; The carrier phase &nbsp;$ϕ_{\rm T}$&nbsp; is not important for the power-spectral densities considered here.
+where the carrier &nbsp;$z(t)$&nbsp; represents a harmonic oscillation with frequency &nbsp;$f_{\rm T}$&nbsp; and amplitude &nbsp;$1$.&nbsp; The carrier phase &nbsp;$ϕ_{\rm T}$&nbsp; is not important for the power-spectral densities considered here.
-*In each case, the source is redundancy-free, which means that the two possible symbols $±1$ are equally probable and the symbols are statistically independent of each other.
+*In each case,&nbsp; the source is redundancy-free,&nbsp; which means that the two possible symbols are equally probable and the symbols are statistically independent of each other.
-*For ASK, unipolar amplitude coefficients - that is: &nbsp;$a_ν ∈ \{0, 1\}$&nbsp; – of the source signal are
+*For ASK,&nbsp; the unipolar amplitude coefficients &ndash; that is: &nbsp;$a_ν ∈ \{0, 1\}$&nbsp; &ndash; of the source signal are
 :$$ q(t) = \sum_{\nu = - \infty}^{+\infty}a_\nu \cdot g_q (t - \nu \cdot T)$$
 :while in the case of BPSK &nbsp;$a_ν ∈ \{-1, +1\}$&nbsp; has to be considered.
-In the diagram, the power-spectral densities &nbsp;${\it Φ}_q(f)$&nbsp; and &nbsp;${\it Φ}_s(f)$&nbsp; of source signal and transmitted signal are given, respectively, for an NRZ rectangular pulse &nbsp;$g_q(t)$&nbsp; with amplitude &nbsp;$s_0 = 2 \ \rm V$&nbsp; and duration &nbsp;$T = 1 \ \rm &micro; s$.&nbsp; Thus the spectral function is:
+In the diagram,&nbsp; the power-spectral densities &nbsp;${\it Φ}_q(f)$&nbsp; and &nbsp;${\it Φ}_s(f)$&nbsp; of the source signal and the transmitted signal are given,&nbsp; respectively,&nbsp; for an NRZ rectangular pulse &nbsp;$g_q(t)$&nbsp; with amplitude &nbsp;$s_0 = 2 \ \rm V$&nbsp; and duration &nbsp;$T = 1 \ \rm &micro; s$.&nbsp; Thus the spectral function is:
 :$$G_q(f) = s_0 \cdot T \cdot {\rm si}(\pi f T)\hspace{0.05cm}.$$
 The constants &nbsp;$A$, &nbsp;$B$, &nbsp;$C$&nbsp; and &nbsp;$D$&nbsp; for the&nbsp; $\rm ASK$&nbsp; and&nbsp; $\rm BPSK$ modulation methods are to be determined.
@@ Line 23: / Line 23: @@
 Notes:
 *The exercise belongs to the chapter&nbsp; [[Modulation_Methods/Linear_Digital_Modulation|Linear Digital Modulation]].
-*However, reference is also made to the chapter&nbsp;  [[Digital_Signal_Transmission/Grundlagen_der_codierten_Übertragung|Basics of Coded Transmission]]&nbsp;  in the book "Digital Signal Transmission".
+*Reference is also made to the chapter&nbsp;  [[Digital_Signal_Transmission/Grundlagen_der_codierten_Übertragung|Basics of Coded Transmission]]&nbsp;  in the book&nbsp; "Digital Signal Transmission".
 *The powers are to be specified in  &nbsp;$\rm V^2$;&nbsp; they thus refer to the reference resistance &nbsp;$R = 1 \ \rm \Omega$.
@@ Line 37: / Line 34: @@
 <quiz display=simple>
-{Which values result for ASK for the parameters &nbsp;$A = {\it Φ}_q(f = 0)$&nbsp; and &nbsp;$B$&nbsp; $($Dirac weight at &nbsp;$f = 0)$?
+{Which values result with ASK for the parameters &nbsp;$A = {\it Φ}_q(f = 0)$&nbsp; and &nbsp;$B$&nbsp; $($Dirac weight at &nbsp;$f = 0)$?
 |type="{}"}
 $A \ = \ $ { 1 3% } $\ \cdot 10^{-6} \  \rm V^2/Hz$
@@ Line 48: / Line 45: @@
 $D \ = \ $ { 0.25 3% } $\ \rm V^2$
-{What are the values of BPSK for parameters &nbsp;$A$&nbsp; and &nbsp;$B$?
+{What are the values for parameters &nbsp;$A$&nbsp; and &nbsp;$B$&nbsp; for BPSK?
 |type="{}"}
 $A \ = \ $ { 4 3% }  $\ \cdot 10^{-6} \  \rm V^2/Hz$
 $B \ = \ $ { 0. } $\ \rm V^2$
-{What are the values of parameters &nbsp;$C$&nbsp; and &nbsp;$D$ for BPSK?
+{What are the values of parameters &nbsp;$C$&nbsp; and &nbsp;$D$&nbsp; for BPSK?
 |type="{}"}
 $C \ = \ $ { 1 3% } $\ \cdot 10^{-6} \  \rm V^2/Hz$
 $D \ = \ $ { 0. }  $\ \rm V^2$
-{Which statements are always true, i.e., even if &nbsp;$g_q(t)$&nbsp; is not an NRZ rectangular pulse?
+{Which statements are always true  for BPSK even if &nbsp;$g_q(t)$&nbsp; is not an NRZ rectangular pulse?
 |type="[]"}
 + The continuous part of &nbsp;$ {\it Φ}_q(f)$&nbsp; is equal in form to &nbsp;$|G_q(f)|^2$.
-- ${\it Φ}_q(f)$&nbsp; contains a single Dirac line at ASK $($at $f = 0)$.
+- ${\it Φ}_q(f)$&nbsp; contains a single Dirac line at ASK&nbsp; for BPSK $($at $f = 0)$.
-- ${\it Φ}_q(f)$ contains a single Dirac line at BPSK $($at $f = 0)$.
+- ${\it Φ}_q(f)$ contains a single Dirac line at BPSK&nbsp; for BPSK $($at $f = 0)$.
 </quiz>

← Older revision		Revision as of 14:01, 16 March 2022
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−	[[Category:Modulation Methods: Exercises\|^4.2 ~~Lineare digitale~~ Modulation^]]	+	[[Category:Modulation Methods: Exercises\|^4.2 Linear Digital Modulation^]]

← Older revision		Revision as of 12:46, 23 March 2021
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−	[[Category:~~Aufgaben zu Modulationsverfahren~~\|^4.2 Lineare digitale Modulation^]]	+	[[Category:Modulation Methods: Exercises\|^4.2 Lineare digitale Modulation^]]

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 ''Hinweise:''
 *Die Aufgabe gehört zum  Kapitel&nbsp; [[Modulation_Methods/Lineare_digitale_Modulation|Lineare digitale Modulation]].
-*Bezug genommen wird aber auch auf das Kapitel&nbsp;  [[Digitalsignalübertragung/Grundlagen_der_codierten_Übertragung|Grundlagen der codierten Übertragung]]&nbsp;  im Buch „Digitalsignalübertragung”.
+*Bezug genommen wird aber auch auf das Kapitel&nbsp;  [[Digital_Signal_Transmission/Grundlagen_der_codierten_Übertragung|Grundlagen der codierten Übertragung]]&nbsp;  im Buch „Digitalsignalübertragung”.
 *Die Leistungen sind in  &nbsp;$\rm V^2$&nbsp; anzugeben;&nbsp; sie beziehen sich somit auf den Bezugswiderstand &nbsp;$R = 1 \ \rm \Omega$.