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

Hwang at 08:36, 12 August 2022

2022-08-12T08:36:19Z

Hwang at 08:34, 12 August 2022

2022-08-12T08:34:18Z

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 Exercise 4.1Z: Other Basic Functions to Exercise 4.1Z: Other Basis Functions

Guenter at 12:36, 13 July 2022

2022-07-13T12:36:18Z

Guenter at 12:22, 13 July 2022

2022-07-13T12:22:01Z

Guenter at 12:20, 13 July 2022

2022-07-13T12:20:39Z

Guenter at 12:19, 13 July 2022

2022-07-13T12:19:29Z

Hwang at 06:39, 20 June 2022

2022-06-20T06:39:34Z

Guenter at 12:33, 18 June 2022

2022-06-18T12:33:54Z

Hwang at 12:01, 14 June 2022

2022-06-14T12:01:19Z

@@ Line 79: / Line 79: @@
 *However,&nbsp; there are probably&nbsp; $($because of&nbsp; $N = 3)$&nbsp; only&nbsp; $3! = 6$&nbsp; possible sets of basis functions.&nbsp;
-*As can be seen from the&nbsp; [[Aufgaben:Eercise_4.1:_About_the_Gram-Schmidt_Method|"solution"]]&nbsp; to&nbsp; "Exercise 4.1",&nbsp; the same basis functions will result with the order&nbsp; $s_1(t),\ s_2(t),\ s_4(t),\ s_3(t)$&nbsp; as with&nbsp; $s_1(t),\ s_2(t),\ s_3(t),\ s_4(t)$.&nbsp; However,&nbsp; this is only a conjecture of the authors;&nbsp; we have not checked it.
+*As can be seen from the&nbsp; [[Aufgaben:Exercise_4.1:_About_the_Gram-Schmidt_Process|"solution"]]&nbsp; to&nbsp; "Exercise 4.1",&nbsp; the same basis functions will result with the order&nbsp; $s_1(t),\ s_2(t),\ s_4(t),\ s_3(t)$&nbsp; as with&nbsp; $s_1(t),\ s_2(t),\ s_3(t),\ s_4(t)$.&nbsp; However,&nbsp; this is only a conjecture of the authors;&nbsp; we have not checked it.
 * Statement 3 cannot be true simply because of the different units of&nbsp; $s_i(t)$&nbsp; and&nbsp; $\varphi_{\it j}(t)$.&nbsp; Like&nbsp; $A$,&nbsp; the signals have the unit&nbsp; $\sqrt{\rm W}$,&nbsp; the basis functions the unit $\sqrt{\rm 1/s}$.

@@ Line 15: / Line 15: @@
 With&nbsp; $M$&nbsp; transmitted signals&nbsp; $s_i(t)$,&nbsp; already fewer basis functions&nbsp; $\varphi_{\it j}(t)$&nbsp; can suffice,&nbsp; namely&nbsp; $N$.&nbsp; Thus,&nbsp; in general,&nbsp; $N &#8804; M$.
-These are exactly the same energy-limited signals&nbsp; $s_i(t)$&nbsp; as in&nbsp; [[Aufgaben:Aufgabe_4.1:_Zum_Gram-Schmidt-Verfahren|"Exercise 4.1"]]:
+These are exactly the same energy-limited signals&nbsp; $s_i(t)$&nbsp; as in&nbsp; [[Aufgaben:Exercise_4.1:_About_the_Gram-Schmidt_Process|"Exercise 4.1"]]:
 *The difference is the different order of the signals&nbsp; $s_i(t)$.

@@ Line 58: / Line 58: @@
 ===Solution===
 {{ML-Kopf}}
-'''(1)'''&nbsp; The only difference to Exercise 4.1 is the different numbering of the signals $s_i(t)$.
+'''(1)'''&nbsp; The only difference to Exercise 4.1 is the different numbering of the signals&nbsp; $s_i(t)$.
-*Thus it is obvious that $\underline {N = 3}$ must hold here as well.
+*Thus it is obvious that&nbsp; $\underline {N = 3}$&nbsp; must hold here as well.
-'''(2)'''&nbsp; The 2&ndash;norm gives the root of the signal energy and is comparable to the rms value for power-limited signals.
+'''(2)'''&nbsp; The&nbsp; "2&ndash;norm"&nbsp; gives the root of the signal energy and is comparable to the&nbsp; "rms value"&nbsp; for power-limited signals.
-*The first three signals all have the 2&ndash;norm
+*The first three signals all have the same 2&ndash;norm:
 :$$||s_1(t)|| = ||s_2(t)|| = ||s_3(t)|| = \sqrt{A^2 \cdot T}\hspace{0.1cm}\hspace{0.15cm}\underline {  = 10^{-3}\sqrt{\rm Ws}} \hspace{0.05cm}.$$
-*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&nbsp; $\sqrt{2}$:
 :$$||s_4(t)|| \hspace{0.1cm}\hspace{0.15cm}\underline { = 1.414 \cdot 10^{-3}\sqrt{\rm Ws}} \hspace{0.05cm}.$$
-''' Basisfunktionssatz'''
-'''(3)'''&nbsp; The <u>first and last statements are true</u> in contrast to statements 2 and 3:
+'''(3)'''&nbsp; The&nbsp; <u>first and last statements are true</u>&nbsp; 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&nbsp; $s_i(t)$&nbsp; are sorted differently.
-* The Gram&ndash;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 &nbsp;$M = 4$&nbsp; signals is &nbsp;$4! = 24$. In any case, there cannot be more basis function sets &nbsp; &rArr; &nbsp; solution 2 is wrong.
+* The Gram&ndash;Schmidt process yields only one possible set&nbsp; $\{\varphi_{\it j}(t)\}$&nbsp; of basis functions.&nbsp; A different sorting&nbsp; (possibly)&nbsp; 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 &nbsp; $M = 4$ &nbsp; signals is &nbsp; $4! = 24$.&nbsp; In any case,&nbsp; there cannot be more basis function sets &nbsp; &rArr; &nbsp; solution 2 is wrong.
 * Thus, the last solution is correct, where for $K$ holds:
 :$$K = ||s_1(t)|| = ||s_2(t)|| = ||s_3(t)|| = 10^{-3}\sqrt{\rm Ws} \hspace{0.05cm}.$$
-''' Ende Basisfunktionssatz'''
 '''(4)'''&nbsp; From the comparison of the diagrams in the specification section we can see:

← Older revision		Revision as of 12:22, 13 July 2022
Line 26:		Line 26:
	*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}. $

@@ Line 3: / Line 3: @@
 [[File:P_ID1996__Dig_Z_4_1.png|right|frame|Energy-limited signals]]
-This exercise pursues exactly the same goal as&nbsp; [[Aufgaben:Aufgabe_4.1:_Zum_Gram-Schmidt-Verfahren|"Exercise 4.1"]]:
+This exercise pursues exactly the same goal as&nbsp; [[Aufgaben:Eercise_4.1:_About_the_Gram-Schmidt_Process|"Exercise 4.1"]]:
 For&nbsp; $M = 4$&nbsp; energy-limited signals&nbsp; $s_i(t)$&nbsp; with&nbsp; $i = 1, \ \text{...} \ , 4$,&nbsp; the&nbsp; $N$&nbsp; required orthonormal basis functions&nbsp; $\varphi_{\it j}(t)$&nbsp; are to be found,&nbsp; which must satisfy the following condition:

← Older revision	Revision as of 12:37, 13 July 2022
(No difference)

@@ Line 2: / Line 2: @@
 {{quiz-Header|Buchseite=Digital_Signal_Transmission/Signals,_Basis_Functions_and_Vector_Spaces}}
-[[File:P_ID1996__Dig_Z_4_1.png|right|frame|Some energy-limited signals]]
+[[File:P_ID1996__Dig_Z_4_1.png|right|frame|Energy-limited signals]]
 This exercise pursues exactly the same goal as&nbsp; [[Aufgaben:Aufgabe_4.1:_Zum_Gram-Schmidt-Verfahren|"Exercise 4.1"]]:
-For&nbsp; $M = 4$&nbsp; energy-limited signals&nbsp; $s_i(t)$&nbsp; with&nbsp; $i = 1, \ \text{...} \ , 4$,&nbsp; the&nbsp; $N$&nbsp; required orthonormal basis functions&nbsp; $\varphi_{\it j}(t)$&nbsp; are to be found, which must satisfy the following condition:
+For&nbsp; $M = 4$&nbsp; energy-limited signals&nbsp; $s_i(t)$&nbsp; with&nbsp; $i = 1, \ \text{...} \ , 4$,&nbsp; the&nbsp; $N$&nbsp; required orthonormal basis functions&nbsp; $\varphi_{\it j}(t)$&nbsp; are to be found,&nbsp; which must satisfy the following condition:
 :$$< \hspace{-0.1cm} \varphi_j(t), \hspace{0.1cm}\varphi_k(t) \hspace{-0.1cm} > \hspace{0.1cm} \hspace{-0.1cm} \ = \ \hspace{-0.1cm} \int_{-\infty}^{+\infty}\varphi_j(t) \cdot \varphi_k(t)\, {\rm d} t = {\rm \delta}_{jk} =
 \left\{ \begin{array}{c} 1 \\
@@ Line 13: / Line 13: @@
   \hspace{0.05cm}.$$
-With&nbsp; $M$&nbsp; transmitted signals&nbsp; $s_i(t)$,&nbsp; already fewer basis functions&nbsp; $\varphi_{\it j}(t)$&nbsp; can suffice, namely&nbsp; $N$. Thus, in general,&nbsp; $N &#8804; M$.
+With&nbsp; $M$&nbsp; transmitted signals&nbsp; $s_i(t)$,&nbsp; already fewer basis functions&nbsp; $\varphi_{\it j}(t)$&nbsp; can suffice,&nbsp; namely&nbsp; $N$.&nbsp; Thus,&nbsp; in general,&nbsp; $N &#8804; M$.
 These are exactly the same energy-limited signals&nbsp; $s_i(t)$&nbsp; as in&nbsp; [[Aufgaben:Aufgabe_4.1:_Zum_Gram-Schmidt-Verfahren|"Exercise 4.1"]]:
 *The difference is the different order of the signals&nbsp; $s_i(t)$.
 *In this exercise, these are sorted in such a way that the basis functions can be found without using the more cumbersome&nbsp; [[Digital_Signal_Transmission/Signals,_Basis_Functions_and_Vector_Spaces#The_Gram-Schmidt_process|"Gram-Schmidt process"]].&nbsp;
+*In this exercise,&nbsp; these are sorted in such a way that the basis functions can be found without using the more cumbersome&nbsp; [[Digital_Signal_Transmission/Signals,_Basis_Functions_and_Vector_Spaces#The_Gram-Schmidt_process|"Gram-Schmidt process"]].&nbsp;
-''Notes:''
+Notes:
 *The exercise belongs to the chapter&nbsp;  [[Digital_Signal_Transmission/Signals,_Basis_Functions_and_Vector_Spaces|"Signals, Basis Functions and Vector Spaces"]].
-*For numerical calculations, use: &nbsp; $A = 1 \sqrt{\rm W} ,  \hspace{0.2cm} T = 1\,{\rm &micro; s}  \hspace{0.05cm}.  $
+*For numerical calculations,&nbsp; use: &nbsp; $A = 1 \sqrt{\rm W} ,  \hspace{0.2cm} T = 1\,{\rm &micro; s}  \hspace{0.05cm}.  $
 ===Questions===
 <quiz display=simple>
-{In Exercise 4.1, the Gram-Schmidt process resulted in&nbsp; $N = 3$&nbsp; basis functions. How many basis functions are needed here?
+{In Exercise 4.1,&nbsp; the Gram-Schmidt process resulted in&nbsp; $N = 3$&nbsp; basis functions.&nbsp; How many basis functions are needed here?
 |type="{}"}
 $N \ = \ $  { 3 3% }
-{Give the 2&ndash;norm of all signals:
+{Give the 2&ndash;norm of all these signals:
 |type="{}"}
 $||s_1(t)|| \ = \ $ { 1 3% } $\ \cdot \ 10^{\rm &ndash;3} \ \rm \sqrt{Ws}$
@@ Line 45: / Line 44: @@
 {Which statements are true for the basis functions&nbsp; $\varphi_1(t)$,&nbsp; $\varphi_2(t)$&nbsp; and&nbsp; $\varphi_3(t)$?
 |type="[]"}
-+ The basis functions computed in Exericse 4.1 are also appropriate here.
++ The basis functions computed in&nbsp; "Exericse 4.1"&nbsp; are also appropriate here.
-- There are infinitely many possibilities for&nbsp; $\{\varphi_1(t),&nbsp; \varphi_2(t),&nbsp; \varphi_3(t)\}$.
+- There are infinitely many possibilities for&nbsp; $\{\varphi_1(t),\ \varphi_2(t),\ \varphi_3(t)\}$.
-- A possible theorem is&nbsp; $\{\varphi_{\it j}(t)\} = \{s_{\it j}(t)\}$, with&nbsp; $j = 1, 2, 3$.
+- A possible set is&nbsp; $\{\varphi_{\it j}(t)\} = \{s_{\it j}(t)\}$,&nbsp; with&nbsp; $j = 1,\ 2,\ 3$.
-+ A possible theorem is&nbsp; $\{\varphi_{\it j}(t)\} = \{s_{\it j}(t)/K\}$, with&nbsp; $j = 1, 2, 3$.
++ A possible set is&nbsp; $\{\varphi_{\it j}(t)\} = \{s_{\it j}(t)/K\}$,&nbsp; with&nbsp; $j = 1,\ 2,\ 3$.
-{What are the coefficients of the signal&nbsp; $s_4(t)$, with respect to the basis functions&nbsp; $\{\varphi_{\it j}(t)\} = \{s_{\it j}(t)/K\}$, with&nbsp; $j = 1, 2, 3$?
+{What are the coefficients of the signal&nbsp; $s_4(t)$&nbsp; with respect to the basis functions&nbsp; $\{\varphi_{\it j}(t)\} = \{s_{\it j}(t)/K\}$, with&nbsp; $j = 1,\ 2,\ 3$?
 |type="{}"}
 $s_{\rm 41} \ = \ $ { 1 3% } $\ \cdot \ 10^{\rm &ndash;3} \ \rm \sqrt{Ws}$