Aufgaben:Exercise 4.3: Different Frequencies - Revision history

Guenter at 14:56, 13 July 2022

2022-07-13T14:56:32Z

Guenter at 14:48, 13 July 2022

2022-07-13T14:48:01Z

Guenter at 14:47, 13 July 2022

2022-07-13T14:47:13Z

Hwang at 20:43, 14 June 2022

2022-06-14T20:43:21Z

Show changes

Guenter at 11:55, 8 June 2022

2022-06-08T11:55:58Z

Guenter: Guenter moved page Aufgaben:Aufgabe 4.3: Unterschiedliche Frequenzen to Aufgaben:Exercise 4.3: Different Frequencies

2022-06-08T11:51:06Z

Guenter moved page Aufgabe 4.3: Unterschiedliche Frequenzen to Exercise 4.3: Different Frequencies

Javier: Text replacement - "Category:Aufgaben zu Digitalsignalübertragung" to "Category:Digital Signal Transmission: Exercises"

2021-03-23T12:50:09Z

Text replacement - "Category:Aufgaben zu Digitalsignalübertragung" to "Category:Digital Signal Transmission: Exercises"

Guenter at 08:51, 11 March 2019

2019-03-11T08:51:34Z

Guenter at 08:48, 11 March 2019

2019-03-11T08:48:47Z

Guenter at 08:36, 11 March 2019

2019-03-11T08:36:59Z

@@ Line 53: / Line 53: @@
 ===Solution===
 {{ML-Kopf}}
-'''(1)'''&nbsp; Correct is the <u>solution 2</u>:
+'''(1)'''&nbsp; Correct is the&nbsp; <u>solution 2</u>:
-* This takes into account the different frequencies and the limitation to the range $0 &#8804; t < T$.
+* This takes into account the different frequencies and the limitation to the range&nbsp; $0 &#8804; t < T$.
 *The signals $s_i(t)$ according to suggestion 3, on the other hand, do not differ with respect to frequency, but have different phase positions.
-'''(2)'''&nbsp; The energy-limited signals $s_i(t) = A \cdot \cos {(2\pi \cdot i \cdot t/T)}$ are all orthogonal to each other, that is, the inner product of two signals $s_i(t)$ and $s_k(t)$ with $i &ne; k$ is always $0$:
+'''(2)'''&nbsp; The energy-limited signals &nbsp; $s_i(t) = A \cdot \cos {(2\pi \cdot i \cdot t/T)}$ &nbsp; are orthogonal to each other &nbsp; &rArr; &nbsp; the inner product of two signals&nbsp; $s_i(t)$,&nbsp; $s_k(t)$&nbsp; with&nbsp; $i &ne; k$&nbsp; is always&nbsp; $0$:
 :$$< \hspace{-0.1cm}s_i(t), \hspace{0.1cm} s_k(t)\hspace{-0.1cm} > \hspace{0.1cm} \hspace{-0.1cm} \ = \ \hspace{-0.1cm} A^2 \cdot \int_{0}^{T}\cos(2\pi \cdot i \cdot t/T) \cdot \cos(2\pi \cdot k \cdot t/T)\,{\rm d} t $$
 :$$ \Rightarrow \hspace{0.3cm} < \hspace{-0.1cm}s_i(t), \hspace{0.1cm} s_k(t)\hspace{-0.1cm} > \hspace{0.1cm} \hspace{-0.1cm}  {A^2}/{2} \cdot \int_{0}^{T}\cos(2\pi (i-k) t/T) \,{\rm d} t +
@@ Line 65: / Line 66: @@
    \hspace{0.05cm}.$$
-*With $i &#8712; \{0, \ \text{...} \ , 4\}$ and $k &#8712; \{0, \ \text{...}\ , 4\}$ as well as $i &ne; j$, both $i \, &ndash; k$ is integer not equal to $0$, as is the sum $i + k$.
+*With&nbsp; $i &#8712; \{0, \ \text{...} \ , 4\}$&nbsp; and&nbsp; $k &#8712; \{0, \ \text{...}\ , 4\}$&nbsp; as well as&nbsp; $i &ne; j$,&nbsp; both&nbsp; $i \, - k$&nbsp; is integer&nbsp; $\ne0$,&nbsp; as is the&nbsp; sum $i + k$.
-*Thus, both integrals yield the result zero:
+*Thus,&nbsp; both integrals yield the result zero:
 :$$< \hspace{-0.1cm}s_i(t), \hspace{0.1cm} s_k(t)\hspace{-0.1cm} > \hspace{0.1cm} \hspace{-0.1cm}= 0
   \hspace{0.3cm}\Rightarrow \hspace{0.3cm}  \hspace{0.05cm}\hspace{0.15cm}\underline {N = M = 5}
@@ Line 72: / Line 73: @@
-'''(3)'''&nbsp; The energy of the signal $s_0(t)$, which is constant within $T$, is equal to
+'''(3)'''&nbsp; The energy of the signal&nbsp; $s_0(t)$,&nbsp; which is constant within&nbsp; $T$,&nbsp; is equal to
 :$$E_0 = ||s_0(t)||^2 = A^2 \cdot T
   \hspace{0.3cm}\Rightarrow \hspace{0.3cm} ||s_0(t)|| = A \cdot \sqrt{T}  \hspace{0.3cm}
@@ Line 81: / Line 82: @@
 \\  {\rm otherwise}\hspace{0.05cm}. \\ \end{array}$$
-Therefore, <u>solution 2</u> is correct.
+Therefore,&nbsp; <u>solution 2</u>&nbsp; is correct.
-'''(4)'''&nbsp; The <u>last solution</u> is correct because of
+'''(4)'''&nbsp; The&nbsp; <u>last solution</u>&nbsp; is correct because of
 :$$E_1 = ||s_1(t)||^2 = \frac{A^2 \cdot T}{2}
   \hspace{0.3cm}\Rightarrow \hspace{0.3cm} ||s_1(t)|| = A \cdot \sqrt{{T}/{2}} \hspace{0.3cm}

@@ Line 29: / Line 29: @@
 <quiz display=simple>
 {Describe the signal set&nbsp; $\{s_i(t)\}$&nbsp; with&nbsp;  $0 &#8804; i &#8804; 4$&nbsp; as compactly as possible. <br>Which description form is correct?
-|type="[]"}
+|type="()"}
 - $s_i(t) = A \cdot \cos {(2\pi \cdot i \cdot t/T)}$.
 + $s_i(t) = A \cdot \cos {(2\pi \cdot i \cdot t/T)}$&nbsp; for &nbsp;$0 &#8804; t < T$, &nbsp;otherwise $0$.

@@ Line 3: / Line 3: @@
 [[File:P_ID1999__Dig_A_4_3.png|right|frame|Given signal set &nbsp;$\{s_i(t)\}$]]
-In the diagram&nbsp; $M = 5$&nbsp; different signals&nbsp; $s_i(t)$&nbsp; are shown. Contrary to the nomenclature in the theory section, the indexing variable&nbsp; $i$&nbsp; can have the values&nbsp; $0, \ \text{...} \ , M-1$.&nbsp;
+In the diagram&nbsp; $M = 5$&nbsp; different signals&nbsp; $s_i(t)$&nbsp; are shown.&nbsp; Contrary to the nomenclature in the theory section,&nbsp; the indexing variable&nbsp; $i$&nbsp; can have the values&nbsp; $0, \ \text{...} \ , M-1$.&nbsp;
 To be noted:
-* All signals are time-limited to&nbsp; $0$&nbsp; to&nbsp; $T$; thus the energy of all signals is also finite.
+* All signals are time-limited to&nbsp; $0$&nbsp; ... &nbsp; $T$;&nbsp; thus the energy of all signals is finite.
-In this exercise we are looking for the&nbsp; $N$&nbsp; basis functions, which are numbered here with &nbsp;$j = 0, \ \text{...} \ , N-1$.&nbsp;
+* The signals&nbsp; $s_i(t)$&nbsp; with &nbsp;$i &ne; 0$&nbsp; are cosine oscillations with frequency&nbsp; $i \cdot f_0$.
-''Note:''
+Note:&nbsp; The exercise belongs to the chapter&nbsp;  [[Digital_Signal_Transmission/Signals,_Basis_Functions_and_Vector_Spaces|"Signals, Basis Functions and Vector Spaces"]].
@@ Line 38: / Line 39: @@
 {What is the basis function&nbsp; $\varphi_0(t)$ that is equal in form to&nbsp; $s_0(t)$?&nbsp;
-|type="[]"}
+|type="()"}
 - $\varphi_0(t) = s_0(t)$,
-+ $\varphi_0(t) = \sqrt{1/T}$ for $0 &#8804; t < T$, &nbsp;outside &nbsp;$0$.
++ $\varphi_0(t) = \sqrt{1/T}$&nbsp; for&nbsp; $0 &#8804; t < T$, &nbsp;outside &nbsp;$0$.
-- $\varphi_0(t) = \sqrt{2/T}$ for $0 &#8804; t < T$, &nbsp;outside &nbsp;$0$.
+- $\varphi_0(t) = \sqrt{2/T}$&nbsp; for&nbsp; $0 &#8804; t < T$, &nbsp;outside &nbsp;$0$.
-{What is the basis function&nbsp; $\varphi_1(t)$ that is equal in form to&nbsp;  $s_1(t)$?&nbsp;
+{What is the basis function&nbsp; $\varphi_1(t)$&nbsp; that is equal in form to&nbsp;  $s_1(t)$?&nbsp;
-|type="[]"}
+|type="()"}
 - $\varphi_1(t) = s_1(t)$,
 - $\varphi_1(t) = \sqrt{1/T} \cdot \cos {(2\pi t/T)}$ for $0 &#8804; t < T$, &nbsp;outside $0$.

← Older revision		Revision as of 11:55, 8 June 2022
Line 95:		Line 95:


−	[[Category:Digital Signal Transmission: Exercises\|^4.1 ~~Basisfunktionen~~ & ~~Vektorräume~~^]]	+	[[Category:Digital Signal Transmission: Exercises\|^4.1 Basis Functions & Vector Spaces^]]

← Older revision		Revision as of 12:50, 23 March 2021
Line 95:		Line 95:


−	[[Category:~~Aufgaben zu Digitalsignalübertragung~~\|^4.1 Basisfunktionen & Vektorräume^]]	+	[[Category:Digital Signal Transmission: Exercises\|^4.1 Basisfunktionen & Vektorräume^]]

@@ Line 55: / Line 55: @@
 * Dieser berücksichtigt die unterschiedlichen Frequenzen und die Begrenzung auf den Bereich $0 &#8804; t < T$.
 *Die Signale $s_i(t)$ gemäß Vorschlag 3 unterscheiden sich dagegen nicht bezüglich der Frequenz, sondern weisen unterschiedliche Phasenlagen auf.
@@ Line 63: / Line 64: @@
    \hspace{0.05cm}.$$
-Mit $i &#8712; \{0, \ \text{...} \ , 4\}$ und $k &#8712; \{0, \ \text{...}\ , 4\}$ sowie $i &ne; j$ ist sowohl $i \, &ndash; k$ ganzzahlig ungleich $0$, ebenso die Summe $i + k$. Dadurch liefern beide Integrale das Ergebnis $0$:
+*Mit $i &#8712; \{0, \ \text{...} \ , 4\}$ und $k &#8712; \{0, \ \text{...}\ , 4\}$ sowie $i &ne; j$ ist sowohl $i \, &ndash; k$ ganzzahlig ungleich $0$, ebenso die Summe $i + k$.
 :$$< \hspace{-0.1cm}s_i(t), \hspace{0.1cm} s_k(t)\hspace{-0.1cm} > \hspace{0.1cm} \hspace{-0.1cm}= 0
   \hspace{0.3cm}\Rightarrow \hspace{0.3cm}  \hspace{0.05cm}\hspace{0.15cm}\underline {N = M = 5}
    \hspace{0.05cm}.$$
 '''(3)'''&nbsp; Die Energie des innerhalb $T$ konstanten Signals $s_0(t)$ ist gleich

@@ Line 30: / Line 30: @@
 |type="[]"}
 - $s_i(t) = A \cdot \cos {(2\pi \cdot i \cdot t/T)}$.
-+ $s_i(t) = A \cdot \cos {(2\pi \cdot i \cdot t/T)}$ für $0 &#8804; t < T$, sonst $0$.
++ $s_i(t) = A \cdot \cos {(2\pi \cdot i \cdot t/T)}$&nbsp; für &nbsp;$0 &#8804; t < T$, &nbsp;sonst $0$.
-- $s_i(t) = A \cdot \cos {(2\pi t/T \, &ndash; \, i \cdot \pi/2)}$ für $0 &#8804; t < T$, sonst $0$.
+- $s_i(t) = A \cdot \cos {(2\pi t/T \, &ndash; \, i \cdot \pi/2)}$&nbsp; für &nbsp;$0 &#8804; t < T$, &nbsp;sonst $0$.
 {Geben Sie die Anzahl&nbsp; $N$&nbsp; der erforderlichen Basisfunktionen an.
@@ Line 37: / Line 37: @@
 $N \ = \ $ { 5 3% }
-{Wie lautet die Basisfunktion&nbsp; $\varphi_0(t)$, die formgleich&nbsp; $s_0(t)$&nbsp; ist?
+{Wie lautet die Basisfunktion&nbsp; $\varphi_0(t)$, die formgleich mit&nbsp; $s_0(t)$&nbsp; ist?
 |type="[]"}
 - $\varphi_0(t) = s_0(t)$,
-+ $\varphi_0(t) = \sqrt{1/T}$ für $0 &#8804; t < T$, außerhalb $0$.
++ $\varphi_0(t) = \sqrt{1/T}$ für $0 &#8804; t < T$, &nbsp;außerhalb &nbsp;$0$.
-- $\varphi_0(t) = \sqrt{2/T}$ für $0 &#8804; t < T$, außerhalb $0$.
+- $\varphi_0(t) = \sqrt{2/T}$ für $0 &#8804; t < T$, &nbsp;außerhalb &nbsp;$0$.
-{Wie lautet die Basisfunktion $\varphi_1(t)$, die formgleich $s_1(t)$ ist?
+{Wie lautet die Basisfunktion&nbsp; $\varphi_1(t)$, die formgleich mit&nbsp;  $s_1(t)$&nbsp; ist?
 |type="[]"}
 - $\varphi_1(t) = s_1(t)$,
-- $\varphi_1(t) = \sqrt{1/T} \cdot \cos {(2\pi t/T)}$ für $0 &#8804; t < T$, außerhalb $0$.
+- $\varphi_1(t) = \sqrt{1/T} \cdot \cos {(2\pi t/T)}$ für $0 &#8804; t < T$, &nbsp;außerhalb $0$.
-+ $\varphi_1(t) =\sqrt{2/T} \cdot \cos {(2\pi t/T)}$ für $0 &#8804; t < T$, außerhalb $0$.
++ $\varphi_1(t) =\sqrt{2/T} \cdot \cos {(2\pi t/T)}$ für $0 &#8804; t < T$, &nbsp;außerhalb $0$.
 </quiz>

@@ Line 2: / Line 2: @@
 {{quiz-Header|Buchseite=Digitalsignalübertragung/Signale, Basisfunktionen und Vektorräume}}
-[[File:P_ID1999__Dig_A_4_3.png|right|frame|Vorgegebene Signalmenge $\{s_i(t)\}$]]
+[[File:P_ID1999__Dig_A_4_3.png|right|frame|Vorgegebene Signalmenge &nbsp;$\{s_i(t)\}$]]
-In der Grafik sind $M = 5$ verschiedene Signale $s_i(t)$ dargestellt. Entgegen der Nomenklatur im Theorieteil sind für die Laufvariable $i$ die Werte $0, \ \text{...} \ , M&ndash;1$ möglich. Anzumerken ist:
+In der Grafik sind&nbsp; $M = 5$&nbsp; verschiedene Signale&nbsp; $s_i(t)$&nbsp; dargestellt. Entgegen der Nomenklatur im Theorieteil sind für die Laufvariable&nbsp; $i$&nbsp; die Werte&nbsp; $0, \ \text{...} \ , M-1$&nbsp; möglich.
-''Hinweise:''
 *Die Aufgabe gehört zum  Kapitel  [[Digitalsignal%C3%BCbertragung/Signale,_Basisfunktionen_und_Vektorr%C3%A4ume| Signale, Basisfunktionen und Vektorräume]].
@@ Line 21: / Line 27: @@
 ===Fragebogen===
 <quiz display=simple>
-{Beschreiben Sie die Signalmenge $\{s_i(t)\}$ mit  $0 &#8804; i &#8804; 4$ möglichst kompakt. <br>Welche Beschreibungsform ist richtig?
+{Beschreiben Sie die Signalmenge&nbsp; $\{s_i(t)\}$&nbsp; mit&nbsp;  $0 &#8804; i &#8804; 4$&nbsp; möglichst kompakt. <br>Welche Beschreibungsform ist richtig?
 |type="[]"}
 - $s_i(t) = A \cdot \cos {(2\pi \cdot i \cdot t/T)}$.
@@ Line 27: / Line 33: @@
 - $s_i(t) = A \cdot \cos {(2\pi t/T \, &ndash; \, i \cdot \pi/2)}$ für $0 &#8804; t < T$, sonst $0$.
-{Geben Sie die Anzahl $N$ der erforderlichen Basisfunktionen an.
+{Geben Sie die Anzahl&nbsp; $N$&nbsp; der erforderlichen Basisfunktionen an.
 |type="{}"}
 $N \ = \ $ { 5 3% }
-{Wie lautet die Basisfunktion $\varphi_0(t)$, die formgleich $s_0(t)$ ist?
+{Wie lautet die Basisfunktion&nbsp; $\varphi_0(t)$, die formgleich&nbsp; $s_0(t)$&nbsp; ist?
 |type="[]"}
 - $\varphi_0(t) = s_0(t)$,