Aufgaben:Exercise 1.2Z: Linear Distorting System - Revision history

Maintenance script: Fix interlanguage link: resolve redirect chain

2026-03-16T15:58:09Z

Fix interlanguage link: resolve redirect chain

← Older revision		Revision as of 17:58, 16 March 2026
Line 115:		Line 115:

	[[Category:Modulation Methods: Exercises\|^1.2 Quality Criteria^]]		[[Category:Modulation Methods: Exercises\|^1.2 Quality Criteria^]]
	[[de:Aufgaben:1.~~2Z_Linear_verzerrendes_System~~]]		[[de:Aufgaben:Aufgabe 1.2Z: Linear verzerrendes System]]

Maintenance script: Add German interlanguage link

2026-03-16T12:30:13Z

Add German interlanguage link

← Older revision		Revision as of 14:30, 16 March 2026
Line 115:		Line 115:

	[[Category:Modulation Methods: Exercises\|^1.2 Quality Criteria^]]		[[Category:Modulation Methods: Exercises\|^1.2 Quality Criteria^]]
			[[de:Aufgaben:1.2Z_Linear_verzerrendes_System]]

Reed at 16:50, 23 March 2022

2022-03-23T16:50:30Z

← Older revision		Revision as of 18:50, 23 March 2022
Line 76:		Line 76:
	===Solution===		===Solution===
	{{ML-Kopf}}		{{ML-Kopf}}
	'''(1)'''   In general,  convolution of the periodic rectangular signal   $q(t)$  with the similarly rectangular impulse response  $h(t)$  ~~only~~ yields a triangular signal   $v(t)$,  only if the rectangles involved have equal width.  It follows:		'''(1)'''   In general,  convolution of the periodic rectangular signal   $q(t)$  with the similarly rectangular impulse response  $h(t)$  yields a triangular signal   $v(t)$,  only if the rectangles involved have equal width.  It follows:
	:$$\Delta t_1 = T_0 /2 \hspace{0.3cm}\Rightarrow \hspace{0.3cm} \Delta t_1 / T_0\hspace{0.15cm}\underline {= 0.5} \hspace{0.05cm}.$$		:$$\Delta t_1 = T_0 /2 \hspace{0.3cm}\Rightarrow \hspace{0.3cm} \Delta t_1 / T_0\hspace{0.15cm}\underline {= 0.5} \hspace{0.05cm}.$$

Reed at 16:48, 23 March 2022

2022-03-23T16:48:33Z

← Older revision		Revision as of 18:48, 23 March 2022
Line 9:		Line 9:
	:$$ h(t) = \left\{ \begin{array}{c} 1/\Delta t \\ 1/(2\Delta t) \\ 0 \\ \end{array} \right. \begin{array}{{4}c} {\rm{for}} \\ {\rm{for}} \\ {\rm{for}} \\ \end{array}\begin{array}{{20}c} {\left\| \hspace{0.005cm} t\hspace{0.05cm} \right\| < \Delta t/2,} \\ {\left\| \hspace{0.005cm}t\hspace{0.05cm} \right\| = \Delta t/2,} \\ {\left\|\hspace{0.005cm} t \hspace{0.05cm} \right\| > \Delta t/2.} \\ \end{array}$$		:$$ h(t) = \left\{ \begin{array}{c} 1/\Delta t \\ 1/(2\Delta t) \\ 0 \\ \end{array} \right. \begin{array}{{4}c} {\rm{for}} \\ {\rm{for}} \\ {\rm{for}} \\ \end{array}\begin{array}{{20}c} {\left\| \hspace{0.005cm} t\hspace{0.05cm} \right\| < \Delta t/2,} \\ {\left\| \hspace{0.005cm}t\hspace{0.05cm} \right\| = \Delta t/2,} \\ {\left\|\hspace{0.005cm} t \hspace{0.05cm} \right\| > \Delta t/2.} \\ \end{array}$$

	This is a rectangular-in-time low-pass filter,  as discussed in the chapter  [[Linear_and_Time_Invariant_Systems/Some_Low-Pass_Functions_in_Systems_Theory#Slit_low-pass_filter_.E2.80.93_Rectangular-in-time\|Some Low-Pass Functions in Systems Theory]]  in the book ~~„Linear~~ and Time Invariant ~~Systems”~~.		This is a rectangular-in-time low-pass filter,  as discussed in the chapter  [[Linear_and_Time_Invariant_Systems/Some_Low-Pass_Functions_in_Systems_Theory#Slit_low-pass_filter_.E2.80.93_Rectangular-in-time\|Some Low-Pass Functions in Systems Theory]]  in the book "Linear and Time Invariant Systems".

	The periodic square wave signal  $q(t)$  of period  $T_0$  is applied at the system input.  Thus, the duration of each rectangle and each gap is  $T_0/2$.  The height of the rectangles is  $2\ \rm V$.		The periodic square wave signal  $q(t)$  of period  $T_0$  is applied at the system input.  Thus, the duration of each rectangle and each gap is  $T_0/2$.  The height of the rectangles is  $2\ \rm V$.

Reed at 10:10, 24 November 2021

2021-11-24T10:10:47Z

← Older revision		Revision as of 12:10, 24 November 2021
Line 29:		Line 29:

	Hints:		Hints:
	*This exercise belongs to the chapter  [[Modulation_Methods/~~Qualitätskriterien~~\|Quality Criteria]].  Particular reference is made to the page  [[Modulation_Methods/Quality_Criteria#Signal.E2.80.93to.E2.80.93noise_.28power.29_ratio\|Signal–to–noise (power) ratio]] and to the chapter  [[Linear_and_Time_Invariant_Systems/Linear_Distortions\|Linear Distortions]]  in the book  "Linear and Time Invarian Systems".		*This exercise belongs to the chapter  [[Modulation_Methods/Quality_Criteria\|Quality Criteria]].  Particular reference is made to the page  [[Modulation_Methods/Quality_Criteria#Signal.E2.80.93to.E2.80.93noise_.28power.29_ratio\|Signal–to–noise (power) ratio]] and to the chapter  [[Linear_and_Time_Invariant_Systems/Linear_Distortions\|Linear Distortions]]  in the book  "Linear and Time Invarian Systems".
	*The powers  $P_q$  and  $P_ε$  are the root mean square values of the signals  $q(t)$  and  $ε(t)$  and can be determined for periodic signals with period duration  $T_0$  as follows:		*The powers  $P_q$  and  $P_ε$  are the root mean square values of the signals  $q(t)$  and  $ε(t)$  and can be determined for periodic signals with period duration  $T_0$  as follows:
	:$$P_{q} = \overline{q(t)^2} = \frac{1}{T_{\rm 0}} \cdot \int_{0}^{ T_{\rm 0}} {q(t)^2 }\hspace{0.1cm}{\rm d}t \hspace{0.05cm}, \hspace{0.5cm} P_{\varepsilon} = \overline{\varepsilon(t)^2} = \frac{1}{T_{\rm 0}} \cdot \int_{0}^{ T_{\rm 0}} {\varepsilon(t)^2 }\hspace{0.1cm}{\rm d}t \hspace{0.05cm}.$$		:$$P_{q} = \overline{q(t)^2} = \frac{1}{T_{\rm 0}} \cdot \int_{0}^{ T_{\rm 0}} {q(t)^2 }\hspace{0.1cm}{\rm d}t \hspace{0.05cm}, \hspace{0.5cm} P_{\varepsilon} = \overline{\varepsilon(t)^2} = \frac{1}{T_{\rm 0}} \cdot \int_{0}^{ T_{\rm 0}} {\varepsilon(t)^2 }\hspace{0.1cm}{\rm d}t \hspace{0.05cm}.$$

Guenter at 11:12, 16 November 2021

2021-11-16T11:12:14Z

← Older revision		Revision as of 13:12, 16 November 2021
Line 76:		Line 76:
	===Solution===		===Solution===
	{{ML-Kopf}}		{{ML-Kopf}}
	'''(1)'''   In general, ~~Convolution~~ of the periodic rectangular signal   $q(t)$  with the similarly rectangular impulse response  $h(t)$  only yields a triangular signal   $v(t)$, only if the rectangles involved have equal width. It follows:		'''(1)'''   In general,  convolution of the periodic rectangular signal   $q(t)$  with the similarly rectangular impulse response  $h(t)$  only yields a triangular signal   $v(t)$,  only if the rectangles involved have equal width.  It follows:
	:$$\Delta t_1 = T_0 /2 \hspace{0.3cm}\Rightarrow \hspace{0.3cm} \Delta t_1 / T_0\hspace{0.15cm}\underline {= 0.5} \hspace{0.05cm}.$$		:$$\Delta t_1 = T_0 /2 \hspace{0.3cm}\Rightarrow \hspace{0.3cm} \Delta t_1 / T_0\hspace{0.15cm}\underline {= 0.5} \hspace{0.05cm}.$$

			[[File:P_ID958__Mod_Z_1_2_b.png\|right\|frame\|Error signals for the two receive filters of different widths]]

	[[File:P_ID958__Mod_Z_1_2_b.png\|right\|frame\|~~Fehlersignale bei den beiden betrachteten Empfangsfiltern unterschiedlicher Breite~~]]
	'''(2)'''   The error signal   $ε_1(t)$  is shown in the adjacent graph.  It can be seen that  $ε_1(t)$ can assume all values in the range   $±1 \ \rm V$ :		'''(2)'''   The error signal   $ε_1(t)$  is shown in the adjacent graph.  It can be seen that  $ε_1(t)$ can assume all values in the range   $±1 \ \rm V$ :
	:$${\varepsilon}_\text{ 1, max} \hspace{0.15cm}\underline {= {1}\;{\rm V}} \hspace{0.05cm}.$$		:$${\varepsilon}_\text{ 1, max} \hspace{0.15cm}\underline {= {1}\;{\rm V}} \hspace{0.05cm}.$$

			'''(3)'''   It is sufficient to average over the time range from  $t = 0$  to  $t =T_0/4$,  since all other subintervals contribute identically:

	'''(3)'''   It is sufficient to average over the time range from  $t = 0$  to  $t =T_0/4$, since all other subintervals contribute identically:
	:$$P_{\varepsilon{\rm 1}} = \frac{1}{T_{\rm 0}/4} \hspace{-0.05cm}\cdot \hspace{-0.05cm}\int_{0}^{ T_{\rm 0}/4} {\varepsilon_1(t)^2 }\hspace{0.1cm}{\rm d}t = \frac{1 \,{\rm V}^2}{T_{\rm 0}/4} \hspace{-0.05cm}\cdot \hspace{-0.05cm} \int_{0}^{ T_{\rm 0}/4} {\left( 1 - \frac{t}{T_{\rm 0}/4}\right)^2 }\hspace{0.1cm}{\rm d}t \hspace{0.05cm}.$$		:$$P_{\varepsilon{\rm 1}} = \frac{1}{T_{\rm 0}/4} \hspace{-0.05cm}\cdot \hspace{-0.05cm}\int_{0}^{ T_{\rm 0}/4} {\varepsilon_1(t)^2 }\hspace{0.1cm}{\rm d}t = \frac{1 \,{\rm V}^2}{T_{\rm 0}/4} \hspace{-0.05cm}\cdot \hspace{-0.05cm} \int_{0}^{ T_{\rm 0}/4} {\left( 1 - \frac{t}{T_{\rm 0}/4}\right)^2 }\hspace{0.1cm}{\rm d}t \hspace{0.05cm}.$$
	*Substituting   $x = 4 · t/T_0$ , this can also be written as:		*Substituting   $x = 4 · t/T_0$ , this can also be written as:
	:$$P_{\varepsilon{\rm 1}} = 1 \,{\rm V}^2 \hspace{-0.05cm}\cdot \hspace{-0.05cm} \int_{0}^{ 1} \hspace{-0.2cm}{\left( 1 - 2x + x^2\right)}\hspace{0.1cm}{\rm d}x \hspace{0.05cm}= 1 \,{\rm V}^2 \hspace{-0.05cm}\cdot \hspace{-0.05cm} \left( 1 - 1 + \frac{1}{3}\right)\hspace{0.15cm}\underline {= 0.333} \,{\rm V}^2\hspace{0.05cm}.$$		:$$P_{\varepsilon{\rm 1}} = 1 \,{\rm V}^2 \hspace{-0.05cm}\cdot \hspace{-0.05cm} \int_{0}^{ 1} \hspace{-0.2cm}{\left( 1 - 2x + x^2\right)}\hspace{0.1cm}{\rm d}x \hspace{0.05cm}= 1 \,{\rm V}^2 \hspace{-0.05cm}\cdot \hspace{-0.05cm} \left( 1 - 1 + \frac{1}{3}\right)\hspace{0.15cm}\underline {= 0.333} \,{\rm V}^2\hspace{0.05cm}.$$



	'''(4)'''   Averaging over one period of the squared source signal yields:		'''(4)'''   Averaging over one period of the squared source signal yields:
Line 99:		Line 93:
	:$$\rho_{v{\rm 1}} = \frac{P_{q}}{P_{\varepsilon {\rm 1}}} = \frac{2 \,{\rm V}^2}{0.333 \,{\rm V}^2}\hspace{0.15cm}\underline {= 6} \hspace{0.05cm}.$$		:$$\rho_{v{\rm 1}} = \frac{P_{q}}{P_{\varepsilon {\rm 1}}} = \frac{2 \,{\rm V}^2}{0.333 \,{\rm V}^2}\hspace{0.15cm}\underline {= 6} \hspace{0.05cm}.$$

			'''(5)'''   According to the sketch presented above,  a rectangle of duration   $0.5 \cdot T_0$  now becomes a trapezoid of absolute duration  $0.75 · T_0$.
			*Thus,  according to the laws of convolution,  it is obvious that the equivalent pulse duration must be  $Δt_2/T_0\hspace{0.15cm}\underline { = 0.25}$ .


	~~'''(5)'''   According to the sketch presented above, a rectangle of duration   $0.5 \cdot T_0$  now becomes a trapezoid of absolute duration  $0.75 · T_0$.~~		'''(6)'''   The lower plot in the above graph shows that   $ε_2(t)$  is composed of four triangles within a period of   $T_0$  just like  $ε_1(t)$,  though they are only half as wide.
	*Thus, according to the laws of convolution, it is obvious that the equivalent pulse duration must be  $Δt_2/T_0\hspace{0.15cm}\underline { = 0.25}$ .		*Thus,  in half the time,   $ε_2(t) = 0$.
			*Because of  $ε_\text{2, max} = ε_\text{1, max} = 1 \ \rm V$,  one obtains:


	'''(6)'''   The lower plot in the above graph shows that   $ε_2(t)$  is composed of four triangles within a period of   $T_0$  just like  $ε_1(t)$ , though they are only half as wide.
	*Thus, in half the time,   $ε_2(t) = 0$.

	*Because of  $ε_\text{2, max} = ε_\text{1, max} = 1 \ \rm V$ , one obtains:
	:$$P_{\varepsilon{\rm 2}} ={P_{\varepsilon{\rm 1}}}/{2} \hspace{0.15cm}\underline {= 0.167} \,{\rm V}^2 \hspace{0.3cm}\Rightarrow \hspace{0.3cm} \rho_{v{\rm 2}} = {P_{q}}/{P_{\varepsilon {\rm 2}}}\hspace{0.15cm}\underline {= 12} \hspace{0.05cm}.$$		:$$P_{\varepsilon{\rm 2}} ={P_{\varepsilon{\rm 1}}}/{2} \hspace{0.15cm}\underline {= 0.167} \,{\rm V}^2 \hspace{0.3cm}\Rightarrow \hspace{0.3cm} \rho_{v{\rm 2}} = {P_{q}}/{P_{\varepsilon {\rm 2}}}\hspace{0.15cm}\underline {= 12} \hspace{0.05cm}.$$


			'''(7)'''   For  $Δt = T_0/2$  the distortion power  $P_{ε1} = 1/3 \ \rm V^{ 2 }$ was calculated in subtask  '''(3)'''.
	'''(7)'''   For   $Δt = T_0/2$  the distortion power  $P_{ε1} = 1/3 \ \rm V^{ 2 }$ was calculated in subtask   '''(3)'''~~ ~~ .		*In subtask  '''(6)'''  it was shown,  that for   $Δt = T_0/4$  the distortion power  $P_{ε2}$  is only half.
	*In subtask   '''(6)'''  it was shown, that for   $Δt = T_0/4$  the distortion power  $P_{ε2}$  is only half.		*It was clearly illustrated that a linear relationship holds.  For  $Δt ≤ T_0/2$  we get the following empirical equations:

	*It was clearly illustrated that a linear relationship holds.  For   $Δt ≤ T_0/2$  we get the following empirical equations:
	:$$P_{\varepsilon} = \frac{2 \,{\rm V}^2}{3} \cdot \frac{\Delta t}{T_0} \hspace{0.3cm}\Rightarrow \hspace{0.3cm} \rho_{v} = \frac{P_{q}}{P_{\varepsilon }}= \frac{3}{\Delta t/T_0} \hspace{0.05cm}.$$		:$$P_{\varepsilon} = \frac{2 \,{\rm V}^2}{3} \cdot \frac{\Delta t}{T_0} \hspace{0.3cm}\Rightarrow \hspace{0.3cm} \rho_{v} = \frac{P_{q}}{P_{\varepsilon }}= \frac{3}{\Delta t/T_0} \hspace{0.05cm}.$$
	*Thus, the special case  $Δt = T_0/20$  results in:		*Thus,  the special case  $Δt = T_0/20$  results in:
	:$$P_{\varepsilon{\rm 3}} = \frac{2 \,{\rm V}^2}{60} \hspace{0.3cm}\Rightarrow \hspace{0.3cm} \rho_{v{\rm 3}} = \frac{P_{q}}{P_{\varepsilon {\rm 3}}}\hspace{0.15cm}\underline {= 60} \hspace{0.05cm}.$$		:$$P_{\varepsilon{\rm 3}} = \frac{2 \,{\rm V}^2}{60} \hspace{0.3cm}\Rightarrow \hspace{0.3cm} \rho_{v{\rm 3}} = \frac{P_{q}}{P_{\varepsilon {\rm 3}}}\hspace{0.15cm}\underline {= 60} \hspace{0.05cm}.$$

Guenter at 10:57, 16 November 2021

2021-11-16T10:57:40Z

← Older revision		Revision as of 12:57, 16 November 2021
Line 29:		Line 29:

	Hints:		Hints:
	*This exercise belongs to the chapter  [[Modulation_Methods/Qualitätskriterien\|Quality Criteria]].  Particular reference is made to the page  [[Modulation_Methods/Quality_Criteria#Signal.E2.80.93to.E2.80.93noise_.28power.29_ratio\|Signal–to–noise (power) ratio]] and to the chapter  [[Linear_and_Time_Invariant_Systems/Linear_Distortions\|Linear Distortions]] in the book  "Linear and Time Invarian Systems".		*This exercise belongs to the chapter  [[Modulation_Methods/Qualitätskriterien\|Quality Criteria]].  Particular reference is made to the page  [[Modulation_Methods/Quality_Criteria#Signal.E2.80.93to.E2.80.93noise_.28power.29_ratio\|Signal–to–noise (power) ratio]] and to the chapter  [[Linear_and_Time_Invariant_Systems/Linear_Distortions\|Linear Distortions]]  in the book  "Linear and Time Invarian Systems".
	*The powers  $P_q$  and  $P_ε$  are the root mean square values of the signals  $q(t)$  and  $ε(t)$  and can be determined for periodic signals with period duration  $T_0$  as follows:		*The powers  $P_q$  and  $P_ε$  are the root mean square values of the signals  $q(t)$  and  $ε(t)$  and can be determined for periodic signals with period duration  $T_0$  as follows:
	:$$P_{q} = \overline{q(t)^2} = \frac{1}{T_{\rm 0}} \cdot \int_{0}^{ T_{\rm 0}} {q(t)^2 }\hspace{0.1cm}{\rm d}t \hspace{0.05cm}, \hspace{0.5cm} P_{\varepsilon} = \overline{\varepsilon(t)^2} = \frac{1}{T_{\rm 0}} \cdot \int_{0}^{ T_{\rm 0}} {\varepsilon(t)^2 }\hspace{0.1cm}{\rm d}t \hspace{0.05cm}.$$		:$$P_{q} = \overline{q(t)^2} = \frac{1}{T_{\rm 0}} \cdot \int_{0}^{ T_{\rm 0}} {q(t)^2 }\hspace{0.1cm}{\rm d}t \hspace{0.05cm}, \hspace{0.5cm} P_{\varepsilon} = \overline{\varepsilon(t)^2} = \frac{1}{T_{\rm 0}} \cdot \int_{0}^{ T_{\rm 0}} {\varepsilon(t)^2 }\hspace{0.1cm}{\rm d}t \hspace{0.05cm}.$$

Guenter at 10:57, 16 November 2021

2021-11-16T10:57:09Z

← Older revision		Revision as of 12:57, 16 November 2021
Line 29:		Line 29:

	Hints:		Hints:
	*This exercise belongs to the chapter  [[Modulation_Methods/Qualitätskriterien\|Quality Criteria]].  Particular reference is made to the page  [[Modulation_Methods/Quality_Criteria#Signal.E2.80.93to.E2.80.93noise_.28power.29_ratio\|Signal–to–noise (power) ratio]] and to the chapter  [[Linear_and_Time_Invariant_Systems/Linear_Distortions\|Linear Distortions]] ~~im Buch~~ "Linear and Time Invarian Systems".		*This exercise belongs to the chapter  [[Modulation_Methods/Qualitätskriterien\|Quality Criteria]].  Particular reference is made to the page  [[Modulation_Methods/Quality_Criteria#Signal.E2.80.93to.E2.80.93noise_.28power.29_ratio\|Signal–to–noise (power) ratio]] and to the chapter  [[Linear_and_Time_Invariant_Systems/Linear_Distortions\|Linear Distortions]] in the book  "Linear and Time Invarian Systems".
	*The powers  $P_q$  and  $P_ε$  are the root mean square values of the signals  $q(t)$  and  $ε(t)$  and can be determined for periodic signals with period duration  $T_0$  as follows:		*The powers  $P_q$  and  $P_ε$  are the root mean square values of the signals  $q(t)$  and  $ε(t)$  and can be determined for periodic signals with period duration  $T_0$  as follows:
	:$$P_{q} = \overline{q(t)^2} = \frac{1}{T_{\rm 0}} \cdot \int_{0}^{ T_{\rm 0}} {q(t)^2 }\hspace{0.1cm}{\rm d}t \hspace{0.05cm}, \hspace{0.5cm} P_{\varepsilon} = \overline{\varepsilon(t)^2} = \frac{1}{T_{\rm 0}} \cdot \int_{0}^{ T_{\rm 0}} {\varepsilon(t)^2 }\hspace{0.1cm}{\rm d}t \hspace{0.05cm}.$$		:$$P_{q} = \overline{q(t)^2} = \frac{1}{T_{\rm 0}} \cdot \int_{0}^{ T_{\rm 0}} {q(t)^2 }\hspace{0.1cm}{\rm d}t \hspace{0.05cm}, \hspace{0.5cm} P_{\varepsilon} = \overline{\varepsilon(t)^2} = \frac{1}{T_{\rm 0}} \cdot \int_{0}^{ T_{\rm 0}} {\varepsilon(t)^2 }\hspace{0.1cm}{\rm d}t \hspace{0.05cm}.$$

Guenter at 10:56, 16 November 2021

2021-11-16T10:56:01Z

← Older revision		Revision as of 12:56, 16 November 2021
Line 28:		Line 28:


			Hints:
			*This exercise belongs to the chapter  [[Modulation_Methods/Qualitätskriterien\|Quality Criteria]].  Particular reference is made to the page  [[Modulation_Methods/Quality_Criteria#Signal.E2.80.93to.E2.80.93noise_.28power.29_ratio\|Signal–to–noise (power) ratio]] and to the chapter  [[Linear_and_Time_Invariant_Systems/Linear_Distortions\|Linear Distortions]] im Buch "Linear and Time Invarian Systems".


	''Hints:''
	*This exercise belongs to the chapter  [[Modulation_Methods/Qualitätskriterien\|Quality Criteria]].  Particular reference is made to the page  [[Modulation_Methods/~~Qualitätskriterien~~#Signal.E2.80.~~93zu.E2.80.93St.C3.B6r~~.E2.80.~~93Leistungsverh~~.C3.~~A4ltnis~~\|Signal–to–noise (power) ratio]] and to the chapter  [[Linear_and_Time_Invariant_Systems/~~Lineare_Verzerrungen~~\|Linear Distortions]] im Buch "Linear and Time Invarian Systems".
	*The powers  $P_q$  and  $P_ε$  are the root mean square values of the signals  $q(t)$  and  $ε(t)$  and can be determined for periodic signals with period duration  $T_0$  as follows:		*The powers  $P_q$  and  $P_ε$  are the root mean square values of the signals  $q(t)$  and  $ε(t)$  and can be determined for periodic signals with period duration  $T_0$  as follows:
	:$$P_{q} = \overline{q(t)^2} = \frac{1}{T_{\rm 0}} \cdot \int_{0}^{ T_{\rm 0}} {q(t)^2 }\hspace{0.1cm}{\rm d}t \hspace{0.05cm}, \hspace{0.5cm} P_{\varepsilon} = \overline{\varepsilon(t)^2} = \frac{1}{T_{\rm 0}} \cdot \int_{0}^{ T_{\rm 0}} {\varepsilon(t)^2 }\hspace{0.1cm}{\rm d}t \hspace{0.05cm}.$$		:$$P_{q} = \overline{q(t)^2} = \frac{1}{T_{\rm 0}} \cdot \int_{0}^{ T_{\rm 0}} {q(t)^2 }\hspace{0.1cm}{\rm d}t \hspace{0.05cm}, \hspace{0.5cm} P_{\varepsilon} = \overline{\varepsilon(t)^2} = \frac{1}{T_{\rm 0}} \cdot \int_{0}^{ T_{\rm 0}} {\varepsilon(t)^2 }\hspace{0.1cm}{\rm d}t \hspace{0.05cm}.$$

Guenter at 10:52, 16 November 2021

2021-11-16T10:52:57Z

← Older revision		Revision as of 12:52, 16 November 2021
Line 9:		Line 9:
	:$$ h(t) = \left\{ \begin{array}{c} 1/\Delta t \\ 1/(2\Delta t) \\ 0 \\ \end{array} \right. \begin{array}{{4}c} {\rm{for}} \\ {\rm{for}} \\ {\rm{for}} \\ \end{array}\begin{array}{{20}c} {\left\| \hspace{0.005cm} t\hspace{0.05cm} \right\| < \Delta t/2,} \\ {\left\| \hspace{0.005cm}t\hspace{0.05cm} \right\| = \Delta t/2,} \\ {\left\|\hspace{0.005cm} t \hspace{0.05cm} \right\| > \Delta t/2.} \\ \end{array}$$		:$$ h(t) = \left\{ \begin{array}{c} 1/\Delta t \\ 1/(2\Delta t) \\ 0 \\ \end{array} \right. \begin{array}{{4}c} {\rm{for}} \\ {\rm{for}} \\ {\rm{for}} \\ \end{array}\begin{array}{{20}c} {\left\| \hspace{0.005cm} t\hspace{0.05cm} \right\| < \Delta t/2,} \\ {\left\| \hspace{0.005cm}t\hspace{0.05cm} \right\| = \Delta t/2,} \\ {\left\|\hspace{0.005cm} t \hspace{0.05cm} \right\| > \Delta t/2.} \\ \end{array}$$

	This is a ~~slit~~ low-pass filter, as discussed in the chapter  [[Linear_and_Time_Invariant_Systems/Some_Low-Pass_Functions_in_Systems_Theory#Slit_low-pass_filter_.E2.80.93_Rectangular-in-time\|Some Low-Pass Functions in Systems Theory]]  in the book „Linear and Time Invariant Systems”.		This is a rectangular-in-time low-pass filter,  as discussed in the chapter  [[Linear_and_Time_Invariant_Systems/Some_Low-Pass_Functions_in_Systems_Theory#Slit_low-pass_filter_.E2.80.93_Rectangular-in-time\|Some Low-Pass Functions in Systems Theory]]  in the book „Linear and Time Invariant Systems”.

	The periodic square wave signal  $q(t)$  of period  $T_0$  is applied at the system input.  Thus, the duration of each rectangle and each gap is  $T_0/2$.  The height of the rectangles is  $2\ \rm V$.		The periodic square wave signal  $q(t)$  of period  $T_0$  is applied at the system input.  Thus, the duration of each rectangle and each gap is  $T_0/2$.  The height of the rectangles is  $2\ \rm V$.

	The signal  $v(t)$  at the system output is called the sink signal.  This is represented in the graph for two different parameter values ~~with~~ equivalent pulse duration (red waveforms):		The signal  $v(t)$  at the system output is called the sink signal.  This is represented in the graph for two different parameter values for the equivalent pulse duration (red waveforms):
	* The signal  $v_1(t)$  results when the equivalent pulse duration of  $h(t)$  is exactly  $Δt_1$ .		* The signal  $v_1(t)$  results when the equivalent pulse duration of  $h(t)$  is exactly  $Δt_1$ .
	* Accordingly, the signal  $v_2(t)$  is obtained with the equivalent pulse duration of  $Δt_2$.		* Accordingly, the signal  $v_2(t)$  is obtained with the equivalent pulse duration of  $Δt_2$.


	The change from the square wave signal  $q(t)$  to the triangular or trapezoidal sink signal  $v(t)$  is due to linear distortions and is captured by the error signal  $ε(t) = v(t) - q(t)$~~ .~~		The change from the square wave signal  $q(t)$  to the triangular or trapezoidal sink signal  $v(t)$  is due to linear distortions and is captured by the error signal
			:$$ε(t) = v(t) - q(t).$$

	Using the signal powers  $P_q$  and  $P_ε$  of  $q(t)$  and  $ε(t)$ , respectively, the sink SNR can be calculated:		Using the signal powers  $P_q$  and  $P_ε$  of  $q(t)$  and  $ε(t)$,  respectively, the sink SNR can be calculated:

	:$$\rho_{v} =P_{q}/{P_{\varepsilon }} \hspace{0.05cm}.$$		:$$\rho_{v} =P_{q}/{P_{\varepsilon }} \hspace{0.05cm}.$$
Line 35:		Line 36:
	*The powers  $P_q$  and  $P_ε$  are the root mean square values of the signals  $q(t)$  and  $ε(t)$  and can be determined for periodic signals with period duration  $T_0$  as follows:		*The powers  $P_q$  and  $P_ε$  are the root mean square values of the signals  $q(t)$  and  $ε(t)$  and can be determined for periodic signals with period duration  $T_0$  as follows:
	:$$P_{q} = \overline{q(t)^2} = \frac{1}{T_{\rm 0}} \cdot \int_{0}^{ T_{\rm 0}} {q(t)^2 }\hspace{0.1cm}{\rm d}t \hspace{0.05cm}, \hspace{0.5cm} P_{\varepsilon} = \overline{\varepsilon(t)^2} = \frac{1}{T_{\rm 0}} \cdot \int_{0}^{ T_{\rm 0}} {\varepsilon(t)^2 }\hspace{0.1cm}{\rm d}t \hspace{0.05cm}.$$		:$$P_{q} = \overline{q(t)^2} = \frac{1}{T_{\rm 0}} \cdot \int_{0}^{ T_{\rm 0}} {q(t)^2 }\hspace{0.1cm}{\rm d}t \hspace{0.05cm}, \hspace{0.5cm} P_{\varepsilon} = \overline{\varepsilon(t)^2} = \frac{1}{T_{\rm 0}} \cdot \int_{0}^{ T_{\rm 0}} {\varepsilon(t)^2 }\hspace{0.1cm}{\rm d}t \hspace{0.05cm}.$$
	*Specifying the powers in  $\rm V^2$ , means that the signals refer to a resistance of  $R = 1\ \rm \Omega$ .		*Specifying the powers in  $\rm V^2$,  means that the signals refer to a resistance of  $R = 1\ \rm \Omega$ .
	*~~'''~~si~~'''~~		*For abbreviation we define the following functions:
			# $\text{sinc–function}$  (predominantly used in Anglo-American literature)     ⇒     ${\rm sinc}( x ) = {\sin (\pi x) }/(\pi x ),$
			# $\text{si–function}$  or $\text{splitting function}$  (predominantly used in German literature)     ⇒     ${\rm si}\left( x \right) = \sin \left( x \right)/x = {\rm sinc}(x/\pi ).$


Line 43:		Line 46:

	<quiz display=simple>		<quiz display=simple>
	{How long is the equivalent pulse duration  $Δt_1$  within the signal  $v_1(t)$, relative to the period   $T_0$?		{How long is the equivalent pulse duration  $Δt_1$  within the signal  $v_1(t)$,  relative to the period   $T_0$?
	\|type="{}"}		\|type="{}"}
	$Δt_1/T_0 \ = \ $ { 0.5 3% }		$Δt_1/T_0 \ = \ $ { 0.5 3% }
Line 52:		Line 55:
	$ε_\text{1, max} \ = \ $ { 1 3% } $\ \rm V$		$ε_\text{1, max} \ = \ $ { 1 3% } $\ \rm V$

	{What is the "power"  $P_{ε1}$  of the error signal, i.e., the mean square deviation between  $v_1(t)$  and  $q(t)$?		{What is the "power"  $P_{ε1}$  of the error signal,  i.e.,  the mean square deviation between  $v_1(t)$  and  $q(t)$?
	\|type="{}"}		\|type="{}"}
	$P_{ε1} \ = \ $ { 0.333 3% } $\ \rm V^2$		$P_{ε1} \ = \ $ { 0.333 3% } $\ \rm V^2$
Line 61:		Line 64:
	$ρ_{v1} \ = \ $ { 6 3% }		$ρ_{v1} \ = \ $ { 6 3% }

	{How long is the equivalent pulse duration  $Δt_2$  within the signal  $v_2(t)$, relative to the period  $T_0$?		{How long is the equivalent pulse duration  $Δt_2$  within the signal  $v_2(t)$,  relative to the period  $T_0$?
	\|type="{}"}		\|type="{}"}
	$Δt_2/T_0 \ = \ $ { 0.25 3% }		$Δt_2/T_0 \ = \ $ { 0.25 3% }

	{Determine the error signal  $ε_2(t) = v_2(t) - q(t)$, the distortion power  $P_{ε2}$  and the sink SNR  $ρ_{v2}$.		{Determine the error signal  $ε_2(t) = v_2(t) - q(t)$,  the distortion power  $P_{ε2}$  and the sink SNR  $ρ_{v2}$.
	\|type="{}"}		\|type="{}"}
	$P_{ε2} \ = \ $ { 0.167 3% } $\ \rm V^2$		$P_{ε2} \ = \ $ { 0.167 3% } $\ \rm V^2$