Aufgaben:Exercise 4.13Z: AMI Code - Revision history

Guenter at 16:33, 25 March 2022

2022-03-25T16:33:59Z

Guenter at 16:16, 25 March 2022

2022-03-25T16:16:47Z

Guenter at 16:07, 25 March 2022

2022-03-25T16:07:21Z

Noah at 20:03, 7 March 2022

2022-03-07T20:03:16Z

Noah at 19:49, 7 March 2022

2022-03-07T19:49:44Z

Guenter at 09:43, 28 February 2022

2022-02-28T09:43:32Z

Guenter: Guenter moved page Aufgaben:Aufgabe 4.13Z: AMI-Code to Aufgaben:Exercise 4.13Z: AMI Code

2022-02-28T09:35:06Z

Guenter moved page Aufgabe 4.13Z: AMI-Code to Exercise 4.13Z: AMI Code

Javier: Text replacement - "Category:Aufgaben zu Stochastische Signaltheorie" to "Category:Theory of Stochastic Signals: Exercises"

2021-03-23T12:43:42Z

Text replacement - "Category:Aufgaben zu Stochastische Signaltheorie" to "Category:Theory of Stochastic Signals: Exercises"

Javier: Text replacement - "[[Stochastische_Signaltheorie/" to "[[Theory_of_Stochastic_Signals/"

2020-07-09T09:53:15Z

Text replacement - "[[Stochastische_Signaltheorie/" to "[[Theory_of_Stochastic_Signals/"

Guenter at 14:42, 2 December 2019

2019-12-02T14:42:00Z

@@ Line 88: / Line 88: @@
 ===Solution===
 {{ML-Kopf}}
-'''(1)'''&nbsp; The discrete ACF value for&nbsp; $k = 0$&nbsp; gives the root mean square (here equal to the variance) of the source symbols.
+'''(1)'''&nbsp; The discrete ACF value for&nbsp; $k = 0$&nbsp; gives the variance of the source symbols.
-*Since&nbsp; $q_\nu$&nbsp; can only take the values&nbsp; $-1$&nbsp; and&nbsp; $+1$&nbsp; ,&nbsp; $\varphi_q(k=0)\hspace{0.15cm}\underline{= 1}$.
+*Since&nbsp; $q_\nu$&nbsp; can only take the values&nbsp; $-1$&nbsp; and&nbsp; $+1$ &nbsp;  &rArr; &nbsp; $\varphi_q(k=0)\hspace{0.15cm}\underline{= 1}$.
@@ Line 96: / Line 96: @@
 *The discrete-time ACF and its Fourier transform are:
 :$${\rm A} \{ \varphi_q ( \tau ) \} =  \varphi_q ( k = 0) \cdot T \cdot \delta (\tau) \hspace{0.3cm} \circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\, \hspace{0.3cm} {\rm P} \{{\it \Phi_q}( f) \} =  \varphi_q ( k = 0) \cdot T = T.$$
-*It is considered that&nbsp; $\varphi_q(k=0)= \sigma_q^2= 1$&nbsp; This means: &nbsp;
+*It is considered that&nbsp; $\varphi_q(k=0)= \sigma_q^2= 1$.&nbsp; This means: &nbsp; The periodic continuation of&nbsp; ${\rm P} \{ {\it \Phi}_q(f) \}$&nbsp; thus gives the same value for all frequencies.
 *In contrast, the continuous-time ACF can be represented as follows: &nbsp;
 :$$ \varphi_q ( \tau ) = {\rm A} \{ \varphi_q ( \tau ) \} \star ( {\rm \delta} ( \tau) / T ).$$
-*The associated Power spectral density spectrum (Fourier transform of the ACF) is then the product of the Fourier transforms of the two convolution terms: &nbsp;
+*The associated power-spectral density&nbsp; (Fourier transform of the ACF)&nbsp; is then the product of the Fourier transforms of the two convolution terms: &nbsp;
-:$$ {\it \Phi_q} ( f) = {\rm P} \{ {\it \Phi_q}( f) \} \cdot {\rm si}^2 (\pi f T ) = T \cdot {\rm si}^2 (\pi f T ) .$$
+:$$ {\it \Phi_q} ( f) = {\rm P} \{ {\it \Phi_q}( f) \} \cdot {\rm sinc}^2 (f T ) = T \cdot {\rm sinc}^2 (f T ) .$$
-*Based on the chosen ACF interpolation (with straight line intercepts) from their samples, a&nbsp; $\rm si^2$-shaped PSD is obtained.
+*Based on the chosen ACF interpolation&nbsp; (with straight line intercepts)&nbsp; from their samples, a&nbsp; $\rm sinc^2$-shaped PSD is obtained.
-*A rectangular spectrum according to the proposed solution&nbsp; '''(2)'''&nbsp; would only occur with&nbsp; $\rm si$-shaped interpolation.
+*A rectangular spectrum according to the second proposed solution would only occur with&nbsp; $\rm sinc$-shaped interpolation.
-'''(3)'''&nbsp; The coded sequence is: &nbsp; $\langle +1, \ 0, -1, +1, \ 0, -1, +1, \ 0, \ 0, \ 0 \rangle$.&nbsp; Thus the 6th symbol is&nbsp; $c_6\hspace{0.15cm}\underline{= -1}$.
+'''(3)'''&nbsp; The coded sequence is: &nbsp; $\langle +1, \ 0, -1, +1, \ 0, -1, +1, \ 0, \ 0, \ 0 \rangle$.&nbsp; Thus the 6th code symbol is&nbsp; $c_6\hspace{0.15cm}\underline{= -1}$.
-'''(4)'''&nbsp; The probabilities of occurrence of the values&nbsp; $-1$&nbsp;, $\ 0$&nbsp; and $+1$&nbsp; are&nbsp; $0.25, 0.5, 0.25$.&nbsp; It follows:
+'''(4)'''&nbsp; The occurrence probabilities of the values&nbsp; $-1$, &nbsp; $\ 0$ &nbsp; and&nbsp; $+1$&nbsp; are&nbsp; $0.25, &nbsp; 0.5, &nbsp; 0.25$.&nbsp; It follows:
 :$$\varphi_c ( k = 0) = 0.25 \cdot (-1)^2 + 0.5 \cdot 0^2 +0.25 \cdot (+1)^2\hspace{0.15cm}\underline{ = 0.5}. $$
-'''(5)'''&nbsp; For the ACF value at&nbsp; $k = 1$&nbsp; consider the product&nbsp; $c_{\nu} \cdot c_{\nu+1}$.&nbsp; The combinations shown on the right are obtained.
+'''(5)'''&nbsp; For the ACF value at&nbsp; $k = 1$&nbsp; consider the product&nbsp; $c_{\nu} \cdot c_{\nu+1}$.&nbsp; The combinations shown in the table are obtained.
 *Only products&nbsp; $c_{\nu} \cdot c_{\nu+1} \ne 0$&nbsp; with&nbsp; ${\rm Pr}\big[c_{\nu} \cdot c_{\nu+1}\big] \ne 0$:
 :$$\varphi_c ( k = 1) = {\rm Pr} \big [( c_{\nu} = +1) \cap ( c_{\nu + 1} = -1) \big ] \cdot (+1) \cdot (-1) + {\rm Pr} \big [ ( c_{\nu} = -1) \cap ( c_{\nu + 1} = +1) \big ] \cdot (-1) \cdot (+1).$$
 [[File:P_ID428__Sto_Z_4_13_e.png|right|frame|For ACF calculation of AMI code]]
-*In the table, these terms are marked in red. Further:
+*In the table,&nbsp; these terms are marked in red.&nbsp; Further:
 :$$ {\rm Pr} \big [ ( c_{\nu} = +1) \cap ( c_{\nu + 1} = -1) \big ] = $$
 :$$ = {\rm Pr} ( c_{\nu} = +1) \cdot {\rm Pr} \left ( c_{\nu + 1} = -1 | c_{\nu } = +1) \right ) = \frac{1}{4} \cdot \frac{1}{2}= \frac{1}{8} . $$
-:Here it is assumed that&nbsp; $+1$&nbsp; occurs with probability&nbsp; $0.25$&nbsp; and is followed by&nbsp; $-1$&nbsp; only in half of the cases.
+:It is assumed that&nbsp; $+1$&nbsp; occurs with probability&nbsp; $0.25$&nbsp; and is followed by&nbsp; $-1$&nbsp; only in half of the cases.
 *The same result is obtained for the second contribution. Thus applies:
-$$varphi_c ( k = 1) = \frac {1}{8} \cdot (+1)\cdot (-1) + \frac {1}{8} \cdot (-1)\cdot (+1) \hspace{0.15cm}\underline{= -0.25}.$$
+:$$\varphi_c ( k = 1) = \frac {1}{8} \cdot (+1)\cdot (-1) + \frac {1}{8} \cdot (-1)\cdot (+1) \hspace{0.15cm}\underline{= -0.25}.$$
 :$$\varphi_c ( k = -1) = \varphi_c ( k = 1) \hspace{0.15cm}\underline{= -0.25}.$$
 *To calculate&nbsp; $\varphi_c ( k = 2)$&nbsp; it is necessary to average over&nbsp; $3^3 = 27$&nbsp; combinations. The result is zero.
@@ Line 133: / Line 131: @@
 :$$P \{{\it \Phi_c}( f) \} =  T\cdot   \varphi_c ( k = 0) +2T \cdot \varphi_c ( k = 1) \cdot {\rm cos} ( 2 \pi f T ).$$
-*With the result of the last subtask, it follows:
+*With the result of the last subtask,&nbsp; it follows:
 :$$P \{{\it \Phi}_c( f) \} =  \frac {T}{2} (1 - {\rm cos} ( 2 \pi f T ) )= T \cdot {\rm sin}^2 ( \pi f T ).$$
-*As shown in item&nbsp; '''(2)''', then, for the PSD &ndash; that is, the Fourier transform of&nbsp; $\varphi_c(\tau)$:
+*As shown in item&nbsp; '''(2)''':&nbsp; For the PSD &ndash; that is, the Fourier transform of&nbsp; $\varphi_c(\tau)$:
-:$${\it \Phi_c}( f) = T \cdot {\rm sin}^2 ( \pi f T ) \cdot {\rm si}^2 ( \pi f T ) = T \cdot \frac {{\rm sin}^4 ( \pi f T )}{( \pi f T )^2 } .$$
+:$${\it \Phi_c}( f) = T \cdot {\rm sin}^2 ( \pi f T ) \cdot {\rm sinc}^2 ( f T ) = T \cdot \frac {{\rm sin}^4 ( \pi f T )}{( \pi f T )^2 } .$$
 :$$\Rightarrow \hspace{0.3cm} {\it \Phi_c}( f = 0) \hspace{0.15cm}\underline{= 0}, \hspace{0.8cm}
 {\it \Phi_c}( f = {\rm500 \hspace{0.1cm}kHz}) = T \cdot \frac {{\rm sin}^4 ( \pi /2 )}{( \pi /2 )^2 } = \frac {4 T}{\pi^2} \rm \hspace{0.15cm}\underline{= 0.405 \cdot 10^{-6} \ {1}/{Hz}}.$$

@@ Line 80: / Line 80: @@
 ${\it \Phi}_c(f = 0) \ = \ $ { 0. } $\ \cdot 10^{-6} \ \rm 1/Hz$
 ${\it \Phi}_c(f = 500 \hspace{0.08cm} \rm kHz)\ = \ $ { 0.405 3% } $\ \cdot 10^{-6} \ \rm 1/Hz$
-$c_6 \ = \ $ { -1.01--0.99 }

@@ Line 3: / Line 3: @@
 }}
-[[File:P_ID427__Sto_Z_4_13.png|right|frame|ACF at AMI coding]]
+[[File:P_ID427__Sto_Z_4_13.png|right|frame|Auto-correlation functions at the input and output of AMI coding]]
-For spectral adaptation (shaping) of a digital signal to the characteristics of the channel, one uses so-called&nbsp; <i>pseudo-ternary codes</i>.&nbsp; In these codes, the binary source symbol sequence&nbsp; $\langle q_\nu \rangle$&nbsp; is converted to a sequence&nbsp; $\langle c_\nu \rangle$&nbsp; of ternary symbols according to a fixed rule:
+For spectral adaptation&nbsp; (shaping)&nbsp; of a digital signal to the characteristics of the channel,&nbsp; one uses so-called&nbsp; "pseudo-ternary codes".&nbsp; With these codes,&nbsp; the binary source symbol sequence&nbsp; $\langle q_\nu \rangle$&nbsp; is converted to a sequence&nbsp; $\langle c_\nu \rangle$&nbsp; of ternary symbols according to a fixed rule:
 :$$q_{\nu} \in \{ -1,\hspace{0.1cm} +1 \} \hspace{0.3cm}\Rightarrow \hspace{0.3cm} c_{\nu} \in \{ -1, \hspace{0.1cm}0, \hspace{0.1cm}+1 \} .$$
-The best known representative of this code class is the AMI code (from <i>Alternate Mark Inversion</i>).&nbsp; Here.
+The best known representative of this code class is the AMI code&nbsp; (from&nbsp; "Alternate Mark Inversion").&nbsp; Here
 *the binary value&nbsp; $q_\nu = -1$&nbsp; is always mapped to&nbsp; $c_\nu = 0$&nbsp;,
-*while&nbsp; $q_\nu = +1$&nbsp; is alternately represented by the ternary values&nbsp; $c_\nu = +1$&nbsp; and&nbsp; $c_\nu = -1$&nbsp;.
+*while&nbsp; $q_\nu = +1$&nbsp; is alternately represented by the ternary values&nbsp; $c_\nu = +1$&nbsp; and&nbsp; $c_\nu = -1$.
-By convention, the ternary symbol&nbsp; $c_\nu = +1$&nbsp; shall be selected at the first occurrence of&nbsp; $q_\nu = +1$&nbsp;.
+By convention,&nbsp; the ternary symbol&nbsp; $c_\nu = +1$&nbsp; shall be selected at the first occurrence of&nbsp; $q_\nu = +1$&nbsp;.
 It is further assumed that
@@ Line 19: / Line 19: @@
-Thus, all discrete ACF values are zero except&nbsp; $\varphi_q(k=0)$:
+Thus,&nbsp; all discrete ACF values are zero except&nbsp; $\varphi_q(k=0)$:
-:$$\varphi_q ( k \cdot T) = 0 \hspace{0.5cm} {\rm f alls} \hspace{0.5cm} k \not= 0.$$
+:$$\varphi_q ( k \cdot T) = 0 \hspace{0.5cm} {\rm if} \hspace{0.5cm} k \not= 0.$$
-Here&nbsp; $T$&nbsp; denotes the distance between sources&ndash; or code symbols.&nbsp; Use the value&nbsp; $T = 1 \hspace{0.05cm} \rm &micro; s$.
+Here&nbsp; $T$&nbsp; denotes the time distance between the source symbols.  Use&nbsp; $T = 1 \hspace{0.05cm} \rm &micro; s$. The code symbols have the same spacing.
-The figure shows the given auto-correlation functions.&nbsp; Please note:
+The graphic shows the given auto-correlation functions.&nbsp; Please note:
-* In red are respectively the discrete-time representations&nbsp; ${\rm A} \{ \varphi_q(\tau) \}$&nbsp; and&nbsp; ${\rm A} \{ \varphi_c(\tau) \}$&nbsp; of the auto-correlation functions, each with the reference value&nbsp; $T$.
+* In red are respectively the discrete-time representations&nbsp; ${\rm A} \{ \varphi_q(\tau) \}$&nbsp; and&nbsp; ${\rm A} \{ \varphi_c(\tau) \}$&nbsp; of the auto-correlation functions,&nbsp; each with the reference value&nbsp; $T$.
-* The functions shown in blue indicate the continuous-time progressions&nbsp; $\varphi_q(\tau)$&nbsp; and&nbsp; $\varphi_c(\tau)$&nbsp; of the ACF, assuming square-wave pulses.
+* The functions shown in blue indicate the continuous-time progressions&nbsp; $\varphi_q(\tau)$&nbsp; and&nbsp; $\varphi_c(\tau)$&nbsp; of the ACF,&nbsp; assuming square-wave pulses.
@@ Line 38: / Line 38: @@
 *This exercise belongs to the chapter&nbsp; [[Theory_of_Stochastic_Signals/Power-Spectral_Density|Power-Spectral Density]].
 *Reference is also made to the chapter&nbsp; [[Theory_of_Stochastic_Signals/Auto-Correlation_Function|Auto-Correlation Function]]&nbsp; as well as to the page&nbsp; [[Theory_of_Stochastic_Signals/Power-Spectral_Density#Numerical_PSD_determination|Numerical PSD determination]].
+*Use the following Fourier correspondence, where&nbsp; ${\rm \Delta} (t)$&nbsp; denotes  a triangular pulse symmetric about&nbsp; $t = 0$&nbsp; with&nbsp; ${\rm \Delta} (t= 0) = 1$&nbsp; and&nbsp; ${\rm \Delta} (t) = 0$&nbsp; for&nbsp; $|t| \ge T$:
 :$${\rm \Delta} (t) \hspace{0.3cm} \circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\, \hspace{0.3cm} T \cdot {\rm si}^2 ( \pi f T).$$
@@ Line 47: / Line 46: @@
 <quiz display=simple>
-{What is the discrete ACF&ndash;value of the source symbols for&nbsp; $k = 0$?
+{What is the discrete ACF value of the source symbols for&nbsp; $k = 0$?
 |type="{}"}
 $\varphi_q(k=0) \ = \ $ { 1 3% }
-{Which statements are valid for the PSD&ndash;functions&nbsp; ${\it \Phi}_q(f)$&nbsp; and&nbsp; ${\rm P} \{ {\it \Phi}_q(f) \}$?
+{Which statements are valid for the PSD functions&nbsp; ${\it \Phi}_q(f)$&nbsp; and&nbsp; ${\rm P} \{ {\it \Phi}_q(f) \}$?
 |type="[]"}
 + ${\rm P} \{ {\it \Phi}_q(f) \}$&nbsp; is a constant for all frequencies.
 - ${\it \Phi}_q(f)$&nbsp; is constant for&nbsp; $|f \cdot T| < 0.5$&nbsp; and outside zero.
-+ ${\it \Phi}_q(f)$&nbsp; proceeds&nbsp; $\rm si^2$-shaped.
++ ${\it \Phi}_q(f)$&nbsp; is&nbsp; $\rm sinc^2$-shaped.
@@ Line 65: / Line 64: @@
-{What is the discrete ACF&ndash;value of the code symbols for&nbsp; $k = 0$.
+{What is the discrete ACF value of the code symbols for&nbsp; $k = 0$.
 |type="{}"}
 $\varphi_c(k=0) \ = \ $ { 0.5 3% }
@@ Line 77: / Line 76: @@
-{What power spectral density&nbsp; ${\it \Phi}_c(f)$&nbsp; results for frequency&nbsp;$f=0$ or for&nbsp;$f = 500 \hspace{0.08cm}  \rm kHz$. &nbsp; <i>Note:</i> &nbsp; For&nbsp; $|k| \ge 2$&nbsp; all ACF&ndash;values&nbsp; $\varphi_c(k) are \equiv 0$.
+{What power-spectral density&nbsp; ${\it \Phi}_c(f)$&nbsp; results for frequencies&nbsp;$f=0$&nbsp; and&nbsp;$f = 500 \hspace{0.08cm}  \rm kHz$. &nbsp; <br>Note: &nbsp; For&nbsp; $|k| \ge 2$ &nbsp; &rArr; &nbsp; all ACF&ndash;values&nbsp; $\varphi_c(k)  = 0$.
 |type="{}"}
 ${\it \Phi}_c(f = 0) \ = \ $ { 0. } $\ \cdot 10^{-6} \ \rm 1/Hz$

@@ Line 89: / Line 89: @@
 ===Solution===
 {{ML-Kopf}}
-'''(1)'''&nbsp; Der diskrete AKF-Wert f&uuml;r&nbsp; $k = 0$&nbsp; gibt den quadratischen Mittelwert (hier gleich der Varianz) der Quellensymbole an.
+'''(1)'''&nbsp; The discrete ACF value for&nbsp; $k = 0$&nbsp; gives the root mean square (here equal to the variance) of the source symbols.
-*Da&nbsp; $q_\nu$&nbsp; nur die Werte&nbsp; $-1$&nbsp; und&nbsp; $+1$&nbsp; annehmen kann, ist&nbsp; $\varphi_q(k=0)\hspace{0.15cm}\underline{= 1}$.
+*Since&nbsp; $q_\nu$&nbsp; can only take the values&nbsp; $-1$&nbsp; and&nbsp; $+1$&nbsp; ,&nbsp; $\varphi_q(k=0)\hspace{0.15cm}\underline{= 1}$.
-'''(2)'''&nbsp; Richtig sind <u>die Lösungsvorschläge 1 und 3</u>:
+'''(2)'''&nbsp; Correct are <u>the proposed solutions 1 and 3</u>:
-*Die zeitdiskrete AKF und deren Fouriertransformierte lauten:
+*The discrete-time ACF and its Fourier transform are:
 :$${\rm A} \{ \varphi_q ( \tau ) \} =  \varphi_q ( k = 0) \cdot T \cdot \delta (\tau) \hspace{0.3cm} \circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\, \hspace{0.3cm} {\rm P} \{{\it \Phi_q}( f) \} =  \varphi_q ( k = 0) \cdot T = T.$$
-*Es ist ber&uuml;cksichtigt, dass&nbsp; $\varphi_q(k=0)= \sigma_q^2= 1$&nbsp; ist.&nbsp; Das bedeutet:  &nbsp;
+*It is considered that&nbsp; $\varphi_q(k=0)= \sigma_q^2= 1$&nbsp; This means: &nbsp;
-:Die periodische Fortsetzung von&nbsp; ${\rm P} \{ {\it \Phi}_q(f) \}$&nbsp; ergibt somit f&uuml;r alle Frequenzen den gleichen Wert.
+:The periodic continuation of&nbsp; ${\rm P} \{ {\it \Phi}_q(f) \}$&nbsp; thus gives the same value for all frequencies.
-*Dagegen kann die zeitkontinuierliche AKF wie folgt dargestellt werden: &nbsp;
+*In contrast, the continuous-time ACF can be represented as follows: &nbsp;
-:$$ \varphi_q ( \tau ) = {\rm A} \{ \varphi_q ( \tau ) \} \star ( {\rm \Delta} ( \tau) / T ).$$
+:$$ \varphi_q ( \tau ) = {\rm A} \{ \varphi_q ( \tau ) \} \star ( {\rm \delta} ( \tau) / T ).$$
-*Das dazugeh&ouml;rige Leistungsdichtespektrum (Fouriertransformierte der AKF) ist dann das Produkt der Fouriertransformierten der beiden Faltungsterme: &nbsp;
+*The associated Power spectral density spectrum (Fourier transform of the ACF) is then the product of the Fourier transforms of the two convolution terms: &nbsp;
-:$$ {\it \Phi_q} ( f) =  {\rm P} \{ {\it \Phi_q}( f) \} \cdot {\rm si}^2 (\pi f T ) = T \cdot {\rm si}^2 (\pi f T ) .$$
+:$$ {\it \Phi_q} ( f) = {\rm P} \{ {\it \Phi_q}( f) \} \cdot {\rm si}^2 (\pi f T ) = T \cdot {\rm si}^2 (\pi f T ) .$$
-*Aufgrund der gew&auml;hlten AKF-Interpolation (mit Geradenabschnitten) aus ihren Abtastwerten ergibt sich ein&nbsp; $\rm si^2$-f&ouml;rmiges LDS.
+*Based on the chosen ACF interpolation (with straight line intercepts) from their samples, a&nbsp; $\rm si^2$-shaped PSD is obtained.
-*Ein rechteckförmiges Spektrum gemäß L&ouml;sungsvorschlag&nbsp; '''(2)'''&nbsp; w&uuml;rde sich nur bei&nbsp; $\rm si$-f&ouml;rmiger Interpolation einstellen.
+*A rectangular spectrum according to the proposed solution&nbsp; '''(2)'''&nbsp; would only occur with&nbsp; $\rm si$-shaped interpolation.
-'''(3)'''&nbsp; Die codierte Folge lautet: &nbsp; $\langle +1, \ 0, -1, +1, \ 0, -1, +1, \ 0, \ 0, \ 0 \rangle$.&nbsp; Das 6. Symbol ist somit&nbsp; $c_6\hspace{0.15cm}\underline{= -1}$.
+'''(3)'''&nbsp; The coded sequence is: &nbsp; $\langle +1, \ 0, -1, +1, \ 0, -1, +1, \ 0, \ 0, \ 0 \rangle$.&nbsp; Thus the 6th symbol is&nbsp; $c_6\hspace{0.15cm}\underline{= -1}$.
-'''(4)'''&nbsp; Die Auftrittswahrscheinlichkeiten der Werte&nbsp; $-1$&nbsp;, $\ 0$&nbsp; und $+1$&nbsp; sind&nbsp; $0.25, 0.5, 0.25$.&nbsp; Daraus folgt:
+'''(4)'''&nbsp; The probabilities of occurrence of the values&nbsp; $-1$&nbsp;, $\ 0$&nbsp; and $+1$&nbsp; are&nbsp; $0.25, 0.5, 0.25$.&nbsp; It follows:
 :$$\varphi_c ( k = 0) = 0.25 \cdot (-1)^2 + 0.5 \cdot 0^2 +0.25 \cdot (+1)^2\hspace{0.15cm}\underline{ = 0.5}. $$
-'''(5)'''&nbsp; F&uuml;r den AKF-Wert bei&nbsp; $k = 1$&nbsp; betrachtet man das Produkt&nbsp; $c_{\nu} \cdot c_{\nu+1}$.&nbsp; Es ergeben sich die rechts gezeigten Kombinationen.
+'''(5)'''&nbsp; For the ACF value at&nbsp; $k = 1$&nbsp; consider the product&nbsp; $c_{\nu} \cdot c_{\nu+1}$.&nbsp; The combinations shown on the right are obtained.
-*Einen Beitrag liefern nur Produkte&nbsp; $c_{\nu} \cdot c_{\nu+1} \ne 0$&nbsp; mit&nbsp; ${\rm Pr}\big[c_{\nu} \cdot c_{\nu+1}\big] \ne 0$:
+*Only products&nbsp; $c_{\nu} \cdot c_{\nu+1} \ne 0$&nbsp; with&nbsp; ${\rm Pr}\big[c_{\nu} \cdot c_{\nu+1}\big] \ne 0$:
 :$$\varphi_c ( k = 1) = {\rm Pr} \big [( c_{\nu} = +1) \cap ( c_{\nu + 1} = -1) \big ] \cdot (+1) \cdot (-1) + {\rm Pr} \big [ ( c_{\nu} = -1) \cap ( c_{\nu + 1} = +1) \big ] \cdot (-1) \cdot (+1).$$
-[[File:P_ID428__Sto_Z_4_13_e.png|right|frame|Zur AKF-Berechnung des AMI-Codes]]
+[[File:P_ID428__Sto_Z_4_13_e.png|right|frame|For ACF calculation of AMI code]]
-*In der Tabelle sind diese Terme rot gekennzeichnet. Weiter gilt:
+*In the table, these terms are marked in red. Further:
 :$$ {\rm Pr} \big [ ( c_{\nu} = +1) \cap ( c_{\nu + 1} = -1) \big ] = $$
-:$$  = {\rm Pr}  ( c_{\nu} = +1)  \cdot {\rm Pr} \left (  c_{\nu + 1} = -1 | c_{\nu } = +1) \right )  = \frac{1}{4} \cdot \frac{1}{2}= \frac{1}{8} . $$
+:$$ = {\rm Pr} ( c_{\nu} = +1) \cdot {\rm Pr} \left ( c_{\nu + 1} = -1 | c_{\nu } = +1) \right ) = \frac{1}{4} \cdot \frac{1}{2}= \frac{1}{8} . $$
-:Hierbei ist vorausgesetzt, dass&nbsp; $+1$&nbsp; mit der Wahrscheinlichkeit&nbsp; $0.25$&nbsp; auftritt und danach&nbsp; $-1$&nbsp; nur in der H&auml;lfte der F&auml;lle folgt.
+:Here it is assumed that&nbsp; $+1$&nbsp; occurs with probability&nbsp; $0.25$&nbsp; and is followed by&nbsp; $-1$&nbsp; only in half of the cases.
-*Das gleiche Ergebnis erh&auml;lt man f&uuml;r den zweiten Beitrag. Damit gilt:
+*The same result is obtained for the second contribution. Thus applies:
-:$$\varphi_c ( k = 1) = \frac {1}{8} \cdot (+1)\cdot (-1) + \frac {1}{8} \cdot (-1)\cdot (+1) \hspace{0.15cm}\underline{= -0.25}.$$
+$$varphi_c ( k = 1) = \frac {1}{8} \cdot (+1)\cdot (-1) + \frac {1}{8} \cdot (-1)\cdot (+1) \hspace{0.15cm}\underline{= -0.25}.$$
 :$$\varphi_c ( k = -1) = \varphi_c ( k = 1) \hspace{0.15cm}\underline{= -0.25}.$$
-*Zur Berechnung von&nbsp; $\varphi_c ( k = 2)$&nbsp;  muss &uuml;ber&nbsp; $3^3 = 27$&nbsp; Kombinationen gemittelt werden. Das Ergebnis ist Null.
+*To calculate&nbsp; $\varphi_c ( k = 2)$&nbsp; it is necessary to average over&nbsp; $3^3 = 27$&nbsp; combinations. The result is zero.
-'''(6)'''&nbsp; Die Fouriertransformierte der zeitdiskreten AKF&nbsp; ${\rm A} \{ \varphi_c(\tau) \}$&nbsp; lautet:
+'''(6)'''&nbsp; The Fourier transform of the discrete-time ACF&nbsp; ${\rm A} \{ \varphi_c(\tau) \}$&nbsp; is:
 :$$P \{{\it \Phi_c}( f) \} =  T\cdot   \varphi_c ( k = 0) +2T \cdot \varphi_c ( k = 1) \cdot {\rm cos} ( 2 \pi f T ).$$
-*Mit dem Ergebnis der letzten Teilaufgabe folgt daraus:
+*With the result of the last subtask, it follows:
 :$$P \{{\it \Phi}_c( f) \} =  \frac {T}{2} (1 - {\rm cos} ( 2 \pi f T ) )= T \cdot {\rm sin}^2 ( \pi f T ).$$
-*Wie unter Punkt&nbsp; '''(2)''' gezeigt, gilt dann f&uuml;r das LDS &ndash; also die Fouriertransformierte von&nbsp; $\varphi_c(\tau)$:
+*As shown in item&nbsp; '''(2)''', then, for the PSD &ndash; that is, the Fourier transform of&nbsp; $\varphi_c(\tau)$:
-:$${\it \Phi_c}( f) = T \cdot {\rm sin}^2 ( \pi f T ) \cdot  {\rm si}^2 ( \pi f T ) = T \cdot \frac {{\rm sin}^4 ( \pi f T )}{( \pi f T )^2 } .$$
+:$${\it \Phi_c}( f) = T \cdot {\rm sin}^2 ( \pi f T ) \cdot {\rm si}^2 ( \pi f T ) = T \cdot \frac {{\rm sin}^4 ( \pi f T )}{( \pi f T )^2 } .$$
-:$$\Rightarrow  \hspace{0.3cm} {\it \Phi_c}( f = 0) \hspace{0.15cm}\underline{= 0}, \hspace{0.8cm}
+:$$\Rightarrow \hspace{0.3cm} {\it \Phi_c}( f = 0) \hspace{0.15cm}\underline{= 0}, \hspace{0.8cm}
-{\it \Phi_c}( f = {\rm500 \hspace{0.1cm}kHz}) = T \cdot \frac {{\rm sin}^4 ( \pi /2 )}{( \pi /2 )^2 } = \frac {4 T}{\pi^2} \rm \hspace{0.15cm}\underline{= 0.405 \cdot 10^{-6} \  {1}/{Hz}}.$$
+{\it \Phi_c}( f = {\rm500 \hspace{0.1cm}kHz}) = T \cdot \frac {{\rm sin}^4 ( \pi /2 )}{( \pi /2 )^2 } = \frac {4 T}{\pi^2} \rm \hspace{0.15cm}\underline{= 0.405 \cdot 10^{-6} \ {1}/{Hz}}.$$
 {{ML-Fuß}}

@@ Line 1: / Line 1: @@
-{{quiz-Header|Buchseite=Stochastische Signaltheorie/Leistungsdichtespektrum (LDS)
+{{quiz-Header|Buchseite=Theory_of_Stochastic_Signals/Power-Spectral_Density
 }}
-[[File:P_ID427__Sto_Z_4_13.png|right|frame|AKF bei AMI-Codierung]]
+[[File:P_ID427__Sto_Z_4_13.png|right|frame|ACF at AMI coding]]
-Zur Spektralanpassung (Formung) eines Digitalsignals an die Eigenschaften des Kanals verwendet man so genannte&nbsp; <i>Pseudotern&auml;rcodes</i>.&nbsp; Bei diesen Codes wird die bin&auml;re Quellensymbolfolge&nbsp; $\langle q_\nu  \rangle$&nbsp; nach einer festen Vorschrift in eine Folge&nbsp; $\langle c_\nu  \rangle$&nbsp;  von Tern&auml;rsymbolen umgesetzt:
+For spectral adaptation (shaping) of a digital signal to the characteristics of the channel, one uses so-called&nbsp; <i>pseudo-ternary codes</i>.&nbsp; In these codes, the binary source symbol sequence&nbsp; $\langle q_\nu \rangle$&nbsp; is converted to a sequence&nbsp; $\langle c_\nu \rangle$&nbsp; of ternary symbols according to a fixed rule:
 :$$q_{\nu} \in \{ -1,\hspace{0.1cm} +1 \} \hspace{0.3cm}\Rightarrow \hspace{0.3cm} c_{\nu} \in \{ -1, \hspace{0.1cm}0, \hspace{0.1cm}+1 \} .$$
-Der bekannteste Vertreter dieser Codeklasse  ist der AMI-Code (von <i>Alternate Mark Inversion</i>).&nbsp; Hier wird
+The best known representative of this code class is the AMI code (from <i>Alternate Mark Inversion</i>).&nbsp; Here.
-*der Bin&auml;rwert&nbsp; $q_\nu  = -1$&nbsp; stets auf&nbsp; $c_\nu  = 0$&nbsp; abgebildet,
+*the binary value&nbsp; $q_\nu = -1$&nbsp; is always mapped to&nbsp; $c_\nu = 0$&nbsp;,
-*w&auml;hrend&nbsp; $q_\nu  = +1$&nbsp; abwechselnd (alternierend) durch die Tern&auml;rwerte&nbsp; $c_\nu  = +1$&nbsp; und&nbsp; $c_\nu  = -1$&nbsp; dargestellt wird.
+*while&nbsp; $q_\nu = +1$&nbsp; is alternately represented by the ternary values&nbsp; $c_\nu = +1$&nbsp; and&nbsp; $c_\nu = -1$&nbsp;.
-Vereinbarungsgemäß soll beim ersten Auftreten von&nbsp; $q_\nu  = +1$&nbsp; das Tern&auml;rsymbol&nbsp; $c_\nu  = +1$&nbsp; ausgew&auml;hlt werden.
+By convention, the ternary symbol&nbsp; $c_\nu = +1$&nbsp; shall be selected at the first occurrence of&nbsp; $q_\nu = +1$&nbsp;.
-Weiter wird vorausgesetzt, dass
+It is further assumed that
-*die zwei m&ouml;glichen Quellensymbole jeweils gleichwahrscheinlich sind und
+*the two possible source symbols are each equally probable and
-*die Quellensymbolfolge&nbsp; $\langle q_\nu  \rangle&nbsp;$ keine inneren statistischen Bindungen aufweist.
+*the source symbol sequence&nbsp; $\langle q_\nu \rangle&nbsp;$ has no internal statistical bindings.
-Somit sind alle diskreten AKF-Werte gleich Null mit Ausnahme von&nbsp; $\varphi_q(k=0)$:
+Thus, all discrete ACF values are zero except&nbsp; $\varphi_q(k=0)$:
 :$$\varphi_q ( k \cdot T) = 0 \hspace{0.5cm} {\rm f alls} \hspace{0.5cm} k \not= 0.$$
-Hierbei  bezeichnet&nbsp; $T$&nbsp; den Abstand der Quellen&ndash; bzw. Codesymbole.&nbsp; Verwenden Sie den Wert&nbsp; $T = 1 \hspace{0.05cm} \rm &micro; s$.
+Here&nbsp; $T$&nbsp; denotes the distance between sources&ndash; or code symbols.&nbsp; Use the value&nbsp; $T = 1 \hspace{0.05cm} \rm &micro; s$.
-Das Bild zeigt die gegebenen Autokorrelationsfunktionen.&nbsp; Bitte beachten Sie:
+The figure shows the given auto-correlation functions.&nbsp; Please note:
-* Rot eingezeichnet sind jeweils die zeitdiskreten Darstellungen&nbsp; ${\rm A} \{ \varphi_q(\tau) \}$&nbsp; und&nbsp; ${\rm A} \{ \varphi_c(\tau) \}$&nbsp; der Autokorrelationsfunktionen, jeweils mit dem Bezugswert&nbsp; $T$.
+* In red are respectively the discrete-time representations&nbsp; ${\rm A} \{ \varphi_q(\tau) \}$&nbsp; and&nbsp; ${\rm A} \{ \varphi_c(\tau) \}$&nbsp; of the auto-correlation functions, each with the reference value&nbsp; $T$.
-* Die blau dargestellten Funktionen zeigen die zeitkontinuierlichen Verläufe&nbsp; $\varphi_q(\tau)$&nbsp; und&nbsp; $\varphi_c(\tau)$&nbsp; der AKF, wobei Rechteckimpulse vorausgesetzt sind.
+* The functions shown in blue indicate the continuous-time progressions&nbsp; $\varphi_q(\tau)$&nbsp; and&nbsp; $\varphi_c(\tau)$&nbsp; of the ACF, assuming square-wave pulses.
@@ Line 35: / Line 35: @@
-''Hinweise:''
+Hints:
-*Die Aufgabe gehört zum  Kapitel&nbsp; [[Theory_of_Stochastic_Signals/Leistungsdichtespektrum_(LDS)|Leistungsdichtespektrum]].
+*This exercise belongs to the chapter&nbsp; [[Theory_of_Stochastic_Signals/Power-Spectral_Density|Power-Spectral Density]].
-*Bezug genommen wird auch auf das  Kapitel&nbsp; [[Theory_of_Stochastic_Signals/Autokorrelationsfunktion_(AKF)|Autokorrelationsfunktion]]&nbsp; sowie auf die Seite&nbsp; [[Theory_of_Stochastic_Signals/Leistungsdichtespektrum_(LDS)#Numerische_LDS-Ermittlung|Numerische_LDS-Ermittlung]].
+*Reference is also made to the chapter&nbsp; [[Theory_of_Stochastic_Signals/Auto-Correlation_Function|Auto-Correlation Function]]&nbsp; as well as to the page&nbsp; [[Theory_of_Stochastic_Signals/Power-Spectral_Density#Numerical_PSD_determination|Numerical PSD determination]].
 *Benutzen Sie die folgende Fourierkorrespondenz, wobei&nbsp; ${\rm \Delta} (t)$&nbsp; einen um&nbsp; $t = 0$&nbsp; symmetrischen Dreieckimpuls mit&nbsp; ${\rm \Delta} (t= 0) = 1$&nbsp; und&nbsp; ${\rm \Delta} (t) = 0$&nbsp; für&nbsp; $|t| \ge T$&nbsp;  bezeichnet:
@@ Line 44: / Line 44: @@
-===Fragebogen===
+===Questions===
 <quiz display=simple>
-{Wie gro&szlig; ist der diskrete AKF&ndash;Wert der Quellensymbole f&uuml;r&nbsp; $k = 0$?
+{What is the discrete ACF&ndash;value of the source symbols for&nbsp; $k = 0$?
 |type="{}"}
 $\varphi_q(k=0) \ = \ $ { 1 3% }
-{Welche Aussagen gelten für die LDS&ndash;Funktionen&nbsp; ${\it \Phi}_q(f)$&nbsp;  und&nbsp; ${\rm P} \{ {\it \Phi}_q(f) \}$?
+{Which statements are valid for the PSD&ndash;functions&nbsp; ${\it \Phi}_q(f)$&nbsp; and&nbsp; ${\rm P} \{ {\it \Phi}_q(f) \}$?
 |type="[]"}
-+ ${\rm P} \{ {\it \Phi}_q(f) \}$&nbsp; ist f&uuml;r alle Frequenzen eine Konstante.
++ ${\rm P} \{ {\it \Phi}_q(f) \}$&nbsp; is a constant for all frequencies.
-- ${\it \Phi}_q(f)$&nbsp; ist f&uuml;r&nbsp; $|f \cdot T| < 0.5$&nbsp; konstant und au&szlig;erhalb Null.
+- ${\it \Phi}_q(f)$&nbsp; is constant for&nbsp; $|f \cdot T| < 0.5$&nbsp; and outside zero.
-+ ${\it \Phi}_q(f)$&nbsp; verl&auml;uft&nbsp; $\rm si^2$-f&ouml;rmig.
++ ${\it \Phi}_q(f)$&nbsp; proceeds&nbsp; $\rm si^2$-shaped.
-{Die Quellensymbolfolge sei&nbsp; $\langle q_\nu  \rangle = \langle +1, -1, +1, +1, -1, +1, +1, -1, -1, -1  \rangle$.
+{The source symbol sequence is&nbsp; $\langle q_\nu  \rangle = \langle +1, -1, +1, +1, -1, +1, +1, -1, -1, -1  \rangle$.
-<br>Wie lauten die Codesymbole&nbsp; $c_\nu$&nbsp;? Geben Sie das Codesymbol&nbsp; $c_6$&nbsp; ein.
+<br>What are the code symbols&nbsp; $c_\nu$&nbsp;? Enter the code symbol&nbsp; $c_6$&nbsp;.
 |type="{}"}
-$c_6 \ = \ $ { -1.01--0.99  }
+$c_6 \ = \ $ { -1.01--0.99 }
-{Wie gro&szlig; ist der diskrete AKF&ndash;Wert der Codesymbole f&uuml;r&nbsp; $k = 0$.
+{What is the discrete ACF&ndash;value of the code symbols for&nbsp; $k = 0$.
 |type="{}"}
 $\varphi_c(k=0) \ = \ $ { 0.5 3% }
-{Berechnen Sie die AKF-Werte&nbsp; $\varphi_c(k=+1)$&nbsp; und&nbsp; $\varphi_c(k=-1)$.
+{Calculate the ACF values&nbsp; $\varphi_c(k=+1)$&nbsp; and&nbsp; $\varphi_c(k=-1)$.
 |type="{}"}
 $\varphi_c(k=+1) \ = \ $ { -0.26--0.24 }
@@ Line 77: / Line 77: @@
-{Welche spektrale Leistungsdichte&nbsp; ${\it \Phi}_c(f)$&nbsp; ergibt sich für die Frequenz&nbsp;$f=0$  bzw. für&nbsp;$f = 500  \hspace{0.08cm} \rm kHz$. &nbsp; <i>Hinweis:</i> &nbsp; F&uuml;r&nbsp; $|k| \ge 2$&nbsp; sind alle AKF&ndash;Werte&nbsp; $\varphi_c(k) \equiv 0$.
+{What power spectral density&nbsp; ${\it \Phi}_c(f)$&nbsp; results for frequency&nbsp;$f=0$ or for&nbsp;$f = 500 \hspace{0.08cm}  \rm kHz$. &nbsp; <i>Note:</i> &nbsp; For&nbsp; $|k| \ge 2$&nbsp; all ACF&ndash;values&nbsp; $\varphi_c(k) are \equiv 0$.
 |type="{}"}
 ${\it \Phi}_c(f = 0) \ = \ $ { 0. } $\ \cdot 10^{-6} \ \rm 1/Hz$
-${\it \Phi}_c(f = 500  \hspace{0.08cm} \rm kHz)\ = \ $ { 0.405 3% } $\ \cdot 10^{-6} \ \rm 1/Hz$
+${\it \Phi}_c(f = 500 \hspace{0.08cm} \rm kHz)\ = \ $ { 0.405 3% } $\ \cdot 10^{-6} \ \rm 1/Hz$
@@ Line 86: / Line 87: @@
 </quiz>
-===Musterlösung===
+===Solution===
 {{ML-Kopf}}
 '''(1)'''&nbsp; Der diskrete AKF-Wert f&uuml;r&nbsp; $k = 0$&nbsp; gibt den quadratischen Mittelwert (hier gleich der Varianz) der Quellensymbole an.

← Older revision		Revision as of 09:43, 28 February 2022
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−	[[Category:Theory of Stochastic Signals: Exercises\|^4.5 ~~Leistungsdichtespektrum (LDS)~~^]]	+	[[Category:Theory of Stochastic Signals: Exercises\|^4.5 Power-Spectral Density^]]

← Older revision		Revision as of 12:43, 23 March 2021
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−	[[Category:~~Aufgaben zu Stochastische Signaltheorie~~\|^4.5 Leistungsdichtespektrum (LDS)^]]	+	[[Category:Theory of Stochastic Signals: Exercises\|^4.5 Leistungsdichtespektrum (LDS)^]]