Aufgaben:Exercise 4.1Z: Other Basis Functions - Revision history

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	[[Category:Digital Signal Transmission: Exercises\|^4.1 Basis Functions & Vector Spaces^]]		[[Category:Digital Signal Transmission: Exercises\|^4.1 Basis Functions & Vector Spaces^]]
	[[de:~~Zusatzaufgaben~~:4.~~1_Andere_Basisfunktionen~~]]		[[de:Aufgaben:Aufgabe 4.1Z: Andere Basisfunktionen]]

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	For  $M = 4$  energy-limited signals  $s_i(t)$  with  $i = 1, \ \text{...} \ , 4$,  the  $N$  required orthonormal basis functions  $\varphi_{\it j}(t)$  are to be found,  which must satisfy the following condition:		For  $M = 4$  energy-limited signals  $s_i(t)$  with  $i = 1, \ \text{...} \ , 4$,  the  $N$  required orthonormal basis functions  $\varphi_{\it j}(t)$  are to be found,  which must satisfy the following condition:
	:$$< \hspace{-0.1cm} \varphi_j(t), \hspace{0.1cm}\varphi_k(t) \hspace{-0.1cm} > \hspace{0.1cm} \hspace{-0.1cm} \ = \ \hspace{-0.1cm} \int_{-\infty}^{+\infty}\varphi_j(t) \cdot \varphi_k(t)\, {\rm d} t = {\rm \delta}_{jk} =		:$$< \hspace{-0.1cm} \varphi_j(t), \hspace{0.1cm}\varphi_k(t) \hspace{-0.1cm} > \hspace{0.1cm} \hspace{-0.1cm} \ = \ \hspace{-0.1cm} \int_{-\infty}^{+\infty}\varphi_j(t) \cdot \varphi_k(t)\, {\rm d} t = {\rm \delta}_{jk} =\left\{ \begin{array}{c} 1 \\0 \end{array} \right.\quad\begin{array}{*{1}c} j = k\\ j \ne k \\ \end{array}\hspace{0.05cm}.$$
	\left\{ \begin{array}{c} 1 \\
	0 \end{array} \right.\quad
	\begin{array}{*{1}c} j = k
	\\ j \ne k \\ \end{array}
	\hspace{0.05cm}.$$

	With  $M$  transmitted signals  $s_i(t)$,  already fewer basis functions  $\varphi_{\it j}(t)$  can suffice,  namely  $N$.  Thus,  in general,  $N ≤ M$.		With  $M$  transmitted signals  $s_i(t)$,  already fewer basis functions  $\varphi_{\it j}(t)$  can suffice,  namely  $N$.  Thus,  in general,  $N ≤ M$.

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	*Furthermore holds:		*Furthermore holds:
	:$$s_{4}(t) = s_{41}\cdot \varphi_1(t) + s_{42}\cdot \varphi_2(t) + s_{43}\cdot \varphi_3(t)$$		:$$s_{4}(t) = s_{41}\cdot \varphi_1(t) + s_{42}\cdot \varphi_2(t) + s_{43}\cdot \varphi_3(t)$$
	:$$\Rightarrow \hspace{0.3cm}s_{41} = K \hspace{0.1cm}\hspace{0.15cm}\underline {= 10^{-3}\sqrt{\rm Ws}}\hspace{0.05cm}, \hspace{0.2cm}s_{42} = -K \hspace{0.1cm}\hspace{0.15cm}\underline {= -10^{-3}\sqrt{\rm Ws}}\hspace{0.05cm},		:$$\Rightarrow \hspace{0.3cm}s_{41} = K \hspace{0.1cm}\hspace{0.15cm}\underline {= 10^{-3}\sqrt{\rm Ws}}\hspace{0.05cm}, \hspace{0.2cm}s_{42} = -K \hspace{0.1cm}\hspace{0.15cm}\underline {= -10^{-3}\sqrt{\rm Ws}}\hspace{0.05cm},\hspace{0.2cm}s_{43} \hspace{0.1cm}\hspace{0.15cm}\underline { = 0}\hspace{0.05cm}. $$
	\hspace{0.2cm}s_{43} \hspace{0.1cm}\hspace{0.15cm}\underline { = 0}\hspace{0.05cm}. $$
	{{ML-Fuß}}		{{ML-Fuß}}

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	[[Category:Digital Signal Transmission: Exercises\|^4.1 Basis Functions & Vector Spaces^]]		[[Category:Digital Signal Transmission: Exercises\|^4.1 Basis Functions & Vector Spaces^]]
			[[de:Zusatzaufgaben:4.1_Andere_Basisfunktionen]]

Hwang at 08:36, 12 August 2022

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	*However,  there are probably  $($because of  $N = 3)$  only  $3! = 6$  possible sets of basis functions.		*However,  there are probably  $($because of  $N = 3)$  only  $3! = 6$  possible sets of basis functions.

	*As can be seen from the  [[Aufgaben:~~Eercise_4~~.1:_About_the_Gram-~~Schmidt_Method~~\|"solution"]]  to  "Exercise 4.1",  the same basis functions will result with the order  $s_1(t),\ s_2(t),\ s_4(t),\ s_3(t)$  as with  $s_1(t),\ s_2(t),\ s_3(t),\ s_4(t)$.  However,  this is only a conjecture of the authors;  we have not checked it.		*As can be seen from the  [[Aufgaben:Exercise_4.1:_About_the_Gram-Schmidt_Process\|"solution"]]  to  "Exercise 4.1",  the same basis functions will result with the order  $s_1(t),\ s_2(t),\ s_4(t),\ s_3(t)$  as with  $s_1(t),\ s_2(t),\ s_3(t),\ s_4(t)$.  However,  this is only a conjecture of the authors;  we have not checked it.

	* Statement 3 cannot be true simply because of the different units of  $s_i(t)$  and  $\varphi_{\it j}(t)$.  Like  $A$,  the signals have the unit  $\sqrt{\rm W}$,  the basis functions the unit $\sqrt{\rm 1/s}$.		* Statement 3 cannot be true simply because of the different units of  $s_i(t)$  and  $\varphi_{\it j}(t)$.  Like  $A$,  the signals have the unit  $\sqrt{\rm W}$,  the basis functions the unit $\sqrt{\rm 1/s}$.

Hwang at 08:34, 12 August 2022

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	With  $M$  transmitted signals  $s_i(t)$,  already fewer basis functions  $\varphi_{\it j}(t)$  can suffice,  namely  $N$.  Thus,  in general,  $N ≤ M$.		With  $M$  transmitted signals  $s_i(t)$,  already fewer basis functions  $\varphi_{\it j}(t)$  can suffice,  namely  $N$.  Thus,  in general,  $N ≤ M$.

	These are exactly the same energy-limited signals  $s_i(t)$  as in  [[Aufgaben:~~Aufgabe_4~~.1:~~_Zum_Gram~~-~~Schmidt-Verfahren~~\|"Exercise 4.1"]]:		These are exactly the same energy-limited signals  $s_i(t)$  as in  [[Aufgaben:Exercise_4.1:_About_the_Gram-Schmidt_Process\|"Exercise 4.1"]]:
	*The difference is the different order of the signals  $s_i(t)$.		*The difference is the different order of the signals  $s_i(t)$.

Guenter: Guenter moved page Aufgaben:Exercise 4.1Z: Other Basic Functions to Aufgaben:Exercise 4.1Z: Other Basis Functions

2022-07-13T12:37:46Z

Guenter moved page Aufgaben:Exercise 4.1Z: Other Basic Functions to Aufgaben:Exercise 4.1Z: Other Basis Functions

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(No difference)

Guenter at 12:36, 13 July 2022

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	===Solution===		===Solution===
	{{ML-Kopf}}		{{ML-Kopf}}
	'''(1)'''  The only difference to Exercise 4.1 is the different numbering of the signals $s_i(t)$.		'''(1)'''  The only difference to Exercise 4.1 is the different numbering of the signals  $s_i(t)$.
	*Thus it is obvious that $\underline {N = 3}$ must hold here as well.		*Thus it is obvious that  $\underline {N = 3}$  must hold here as well.


	'''(2)'''  The 2–norm gives the root of the signal energy and is comparable to the rms value for power-limited signals.		'''(2)'''  The  "2–norm"  gives the root of the signal energy and is comparable to the  "rms value"  for power-limited signals.
	*The first three signals all have the 2–norm		*The first three signals all have the same 2–norm:
	:$$\|\|s_1(t)\|\| = \|\|s_2(t)\|\| = \|\|s_3(t)\|\| = \sqrt{A^2 \cdot T}\hspace{0.1cm}\hspace{0.15cm}\underline { = 10^{-3}\sqrt{\rm Ws}} \hspace{0.05cm}.$$		:$$\|\|s_1(t)\|\| = \|\|s_2(t)\|\| = \|\|s_3(t)\|\| = \sqrt{A^2 \cdot T}\hspace{0.1cm}\hspace{0.15cm}\underline { = 10^{-3}\sqrt{\rm Ws}} \hspace{0.05cm}.$$

	*The norm of the last signal is larger by a factor of $\sqrt{2}$:		*The norm of the last signal is larger by a factor of  $\sqrt{2}$:
	:$$\|\|s_4(t)\|\| \hspace{0.1cm}\hspace{0.15cm}\underline { = 1.414 \cdot 10^{-3}\sqrt{\rm Ws}} \hspace{0.05cm}.$$		:$$\|\|s_4(t)\|\| \hspace{0.1cm}\hspace{0.15cm}\underline { = 1.414 \cdot 10^{-3}\sqrt{\rm Ws}} \hspace{0.05cm}.$$

	~~''' Basisfunktionssatz'''~~
	'''(3)'''  The <u>first and last statements are true</u> in contrast to statements 2 and 3:		'''(3)'''  The  <u>first and last statements are true</u>  in contrast to statements 2 and 3:
	* It would be completely illogical if the basis functions found should no longer hold when the signals $s_i(t)$ are sorted differently.		* It would be completely illogical if the basis functions found should no longer hold when the signals  $s_i(t)$  are sorted differently.
	* The Gram–Schmidt process yields only one possible set $\{\varphi_{\it j}(t)\}$ of basis functions. A different sorting (possibly) yields a different ~~one~~.
	*The number of permutations of  $M = 4$  signals is  $4! = 24$. In any case, there cannot be more basis function sets   ⇒   solution 2 is wrong.		* The Gram–Schmidt process yields only one possible set  $\{\varphi_{\it j}(t)\}$  of basis functions.  A different sorting  (possibly)  yields a different basis function.
	*However, there are probably (because of $N = 3$) only $3! = 6$ possible sets of basis functions. As can be seen from the ~~sample~~ [[Aufgaben:Eercise_4.1:_About_the_Gram-Schmidt_Method\|"solution"]] to Exercise 4.1, the same basis functions will result with the order $s_1(t), s_2(t), s_4(t), s_3(t)$ as with $s_1(t), s_2(t), s_3(t), s_4(t)$. However, this is only a conjecture of the authors; we have not checked it.
	* Statement 3 cannot be true simply because of the different units of $s_i(t)$ and $\varphi_{\it j}(t)$. Like $A$, the signals have the unit $\sqrt{\rm W}$, the basis functions the unit $\sqrt{\rm 1/s}$.		*The number of permutations of   $M = 4$   signals is   $4! = 24$.  In any case,  there cannot be more basis function sets   ⇒   solution 2 is wrong.
	* Thus, the last solution is correct, where for $K$ holds:
			*However,  there are probably  $($because of  $N = 3)$  only  $3! = 6$  possible sets of basis functions.

			*As can be seen from the  [[Aufgaben:Eercise_4.1:_About_the_Gram-Schmidt_Method\|"solution"]]  to  "Exercise 4.1",  the same basis functions will result with the order  $s_1(t),\ s_2(t),\ s_4(t),\ s_3(t)$  as with  $s_1(t),\ s_2(t),\ s_3(t),\ s_4(t)$.  However,  this is only a conjecture of the authors;  we have not checked it.

			* Statement 3 cannot be true simply because of the different units of  $s_i(t)$  and  $\varphi_{\it j}(t)$.  Like  $A$,  the signals have the unit  $\sqrt{\rm W}$,  the basis functions the unit $\sqrt{\rm 1/s}$.

			* Thus,  the last solution is correct,  where for  $K$  holds:
	:$$K = \|\|s_1(t)\|\| = \|\|s_2(t)\|\| = \|\|s_3(t)\|\| = 10^{-3}\sqrt{\rm Ws} \hspace{0.05cm}.$$		:$$K = \|\|s_1(t)\|\| = \|\|s_2(t)\|\| = \|\|s_3(t)\|\| = 10^{-3}\sqrt{\rm Ws} \hspace{0.05cm}.$$
	~~''' Ende Basisfunktionssatz'''~~

	'''(4)'''  From the comparison of the diagrams in the specification section we can see:		'''(4)'''  From the comparison of the diagrams in the specification section we can see:

Guenter at 12:22, 13 July 2022

2022-07-13T12:22:01Z

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	*The exercise belongs to the chapter  [[Digital_Signal_Transmission/Signals,_Basis_Functions_and_Vector_Spaces\|"Signals, Basis Functions and Vector Spaces"]].		*The exercise belongs to the chapter  [[Digital_Signal_Transmission/Signals,_Basis_Functions_and_Vector_Spaces\|"Signals, Basis Functions and Vector Spaces"]].

	*For numerical calculations,  use:   $A = 1 \sqrt{\rm W} , \hspace{0.2cm} T = 1\,{\rm µ s} \hspace{0.05cm}. $		*For numerical calculations,  use  $A = 1 \sqrt{\rm W} , \hspace{0.2cm} T = 1\,{\rm µ s} \hspace{0.05cm}. $

Guenter at 12:20, 13 July 2022

2022-07-13T12:20:39Z

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	[[File:P_ID1996__Dig_Z_4_1.png\|right\|frame\|Energy-limited signals]]		[[File:P_ID1996__Dig_Z_4_1.png\|right\|frame\|Energy-limited signals]]
	This exercise pursues exactly the same goal as  [[Aufgaben:~~Aufgabe_4~~.1:~~_Zum_Gram~~-~~Schmidt-Verfahren~~\|"Exercise 4.1"]]:		This exercise pursues exactly the same goal as  [[Aufgaben:Eercise_4.1:_About_the_Gram-Schmidt_Process\|"Exercise 4.1"]]:

	For  $M = 4$  energy-limited signals  $s_i(t)$  with  $i = 1, \ \text{...} \ , 4$,  the  $N$  required orthonormal basis functions  $\varphi_{\it j}(t)$  are to be found,  which must satisfy the following condition:		For  $M = 4$  energy-limited signals  $s_i(t)$  with  $i = 1, \ \text{...} \ , 4$,  the  $N$  required orthonormal basis functions  $\varphi_{\it j}(t)$  are to be found,  which must satisfy the following condition: