Linear and Time Invariant Systems/Classification of the Distortions - Revision history

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	{{BlaueBox\|TEXT=		{{BlaueBox\|TEXT=
	$\text{Definition:}$  Making use of the power  $P_x$  $($based on  $R = 1 \ Ω)$  of the input signal  $x(t)$  the   »'''signal–to–distortion (power) ratio'''«  can be given as:		$\text{Definition:}$  Making use of the power  $P_x$  $($based on  $R = 1 \ Ω)$  of the input signal  $x(t)$  the   »'''signal–to–distortion (power) ratio'''«  can be given as:
	:$$\rho_{\rm V} = \frac{ P_{x} }{P_{\rm V} } \hspace{0.3cm} \Rightarrow \hspace{0.3cm} 10 \cdot \lg \hspace{0.1cm}\rho_{\rm V} =		:$$\rho_{\rm V} = \frac{ P_{x} }{P_{\rm V} } \hspace{0.3cm} \Rightarrow \hspace{0.3cm} 10 \cdot \lg \hspace{0.1cm}\rho_{\rm V} =10 \cdot \lg \hspace{0.1cm}\frac{ P_{x} }{P_{\rm V} }\hspace{0.3cm} \left( {\rm in \hspace{0.15cm} dB} \right).$$
	10 \cdot \lg \hspace{0.1cm}\frac{ P_{x} }{P_{\rm V} }\hspace{0.3cm} \left( {\rm in \hspace{0.15cm} dB} \right).$$

	For the signals shown in the diagram above   ⇒   $P_x = 4 \ {\rm V}^2$,  $P_{\rm V} = 0.04 \ {\rm V}^2$;		For the signals shown in the diagram above   ⇒   $P_x = 4 \ {\rm V}^2$,  $P_{\rm V} = 0.04 \ {\rm V}^2$;
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	*The distortion energy is obtained by varying the unknown quantities  $α$  and  $τ$  and thus finding the minimum of the  »mean squared error«:		*The distortion energy is obtained by varying the unknown quantities  $α$  and  $τ$  and thus finding the minimum of the  »mean squared error«:
	:$$E_{\rm V} = \min_{\alpha, \ \tau} \int_{ - \infty }^{ + \infty}		:$$E_{\rm V} = \min_{\alpha, \ \tau} \int_{ - \infty }^{ + \infty}{\big[y(t) - \left(\alpha \cdot x(t - \tau) \right) \big]^2}\hspace{0.1cm}{\rm d}t.$$
	{\big[y(t) - \left(\alpha \cdot x(t - \tau) \right) \big]^2}\hspace{0.1cm}{\rm d}t.$$
	*The energy of the attenuated and delayed signal  $α · x(t - τ)$  is  $E_{\rm V} =α^2 · E_x$ independent of the transit time  $τ$.  Thus for the signal-to-distortion  $($energy or power$)$  ratio the following is applicable:		*The energy of the attenuated and delayed signal  $α · x(t - τ)$  is  $E_{\rm V} =α^2 · E_x$ independent of the transit time  $τ$.  Thus for the signal-to-distortion  $($energy or power$)$  ratio the following is applicable:
	:$$\rho_{\rm V} = \frac{ \alpha^2 \cdot E_{x}}{E_{\rm V}}\hspace{0.3cm}{\rm or}\hspace{0.3cm}\rho_{\rm V}= \frac{ \alpha^2 \cdot P_{x}}{P_{\rm V}} .$$		:$$\rho_{\rm V} = \frac{ \alpha^2 \cdot E_{x}}{E_{\rm V}}\hspace{0.3cm}{\rm or}\hspace{0.3cm}\rho_{\rm V}= \frac{ \alpha^2 \cdot P_{x}}{P_{\rm V}} .$$

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	==References==		==References==
	<references/>		<references/>
			[[de:Lineare_zeitinvariante_Systeme/Klassifizierung_der_Verzerrungen]]

Guenter at 15:30, 9 November 2023

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	We refer here to the  $($German language$)$  learning video  [[Lineare_und_nichtlineare_Verzerrungen_(Lernvideo)\|»~~Lineare_und_nichtlineare_Verzerrungen~~]]   ⇒   »Linear and nonlinear distortions«.		We refer here to the  $($German language$)$  learning video  [[Lineare_und_nichtlineare_Verzerrungen_(Lernvideo)\|»Lineare und nichtlineare Verzerrungen]]   ⇒   »Linear and nonlinear distortions«.

	==Exercises for the chapter==		==Exercises for the chapter==

Guenter at 15:30, 9 November 2023

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	We refer here to the  $($German language$)$  learning video  [[Lineare_und_nichtlineare_Verzerrungen_(Lernvideo)\|»Linear and nonlinear distortions«]].		We refer here to the  $($German language$)$  learning video  [[Lineare_und_nichtlineare_Verzerrungen_(Lernvideo)\|»Lineare_und_nichtlineare_Verzerrungen]]   ⇒   »Linear and nonlinear distortions«.

	==Exercises for the chapter==		==Exercises for the chapter==

Guenter at 15:27, 9 November 2023

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	==Elimination of attenuation factor and transit time==		==Elimination of attenuation factor and transit time==
	<br>		<br>
	The equations given in the last section do not result in applicable statements if the system is additionally affected by an attenuation factor  $α$  and/or a transit time  $τ$.  The diagram shows the attenuated, delayed and distorted signal		The equations given in the last section do not result in applicable statements if the system is additionally affected by an attenuation factor  $α$  and/or a transit time  $τ$.  The diagram shows the attenuated,  delayed and distorted signal

	[[File:P_ID876__LZI_T_2_1_S4_neu.png\|right\|frame\|Elimination of attenuation factor $α$		[[File:P_ID876__LZI_T_2_1_S4_neu.png\|right\|frame\|Elimination of attenuation factor $α$
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	:$$y(t) = \alpha \cdot x(t - \tau) + \varepsilon_1(t).$$		:$$y(t) = \alpha \cdot x(t - \tau) + \varepsilon_1(t).$$
	~~The term~~  $~~ε_1~~(t)$  ~~summarises all distortions.  It can be seen from the green area that the error signal~~  $~~ε_1~~(t)$  ~~is relatively small~~.		*Here,  instead of the  »distortion power«  the  »distortion energy«  must be considered because  $x(t)$  and  $y(t)$  are energy-limited signals.

	In contrast to this, if the attenuation factor  $α$  and the transit time  $τ$  are unknown, the following should be noted:		*The term  $ε_1(t)$  summarizes all distortions.  It can be seen from the green area that the difference signal   ⇒   »error signal« $ε_1(t)$  is relatively small.
	*~~The error~~ signal  $ε_2(t) = y(t) - x(t)$  determined in this way is relatively large despite small distortions  $ε_1(t)$.
	*Here, instead of the distortion power the distortion energy must be considered because  $x(t)$  and  $y(t)$  are energy-limited signals.
	*The distortion energy is obtained by varying the unknown quantities  $α$  and  $τ$  and thus finding the minimum of the mean squared error:		In contrast to this,  if the attenuation factor  $α$  and the transit time  $τ$  are unknown,  the following should be noted:
			*In the second example the difference signal  $ε_2(t) = y(t) - x(t)$  determined in this way is relatively large despite small distortions  $ε_1(t)$.

			*The distortion energy is obtained by varying the unknown quantities  $α$  and  $τ$  and thus finding the minimum of the  »mean squared error«:
	:$$E_{\rm V} = \min_{\alpha, \ \tau} \int_{ - \infty }^{ + \infty}		:$$E_{\rm V} = \min_{\alpha, \ \tau} \int_{ - \infty }^{ + \infty}
	{\big[y(t) - \left(\alpha \cdot x(t - \tau) \right) \big]^2}\hspace{0.1cm}{\rm d}t.$$		{\big[y(t) - \left(\alpha \cdot x(t - \tau) \right) \big]^2}\hspace{0.1cm}{\rm d}t.$$
	*The energy of the attenuated and delayed signal  $α · x(t - τ)$  is  $E_{\rm V} =α^2 · E_x$ independent of the transit time  $τ$.  Thus for the signal-to-distortion (power) ratio the following is applicable:		*The energy of the attenuated and delayed signal  $α · x(t - τ)$  is  $E_{\rm V} =α^2 · E_x$ independent of the transit time  $τ$.  Thus for the signal-to-distortion  $($energy or power$)$  ratio the following is applicable:
	:$$\rho_{\rm V} = \frac{ \alpha^2 \cdot E_{x}}{E_{\rm V}}\hspace{0.3cm}{\rm or}\hspace{0.3cm}\rho_{\rm D}= \frac{ \alpha^2 \cdot P_{x}}{P_{\rm V}} .$$		:$$\rho_{\rm V} = \frac{ \alpha^2 \cdot E_{x}}{E_{\rm V}}\hspace{0.3cm}{\rm or}\hspace{0.3cm}\rho_{\rm V}= \frac{ \alpha^2 \cdot P_{x}}{P_{\rm V}} .$$
	*The first of these two equations applies to time-limited and thus energy-limited signals, the second one to time-unlimited and thus power-limited signals according to the section  [[Signal_Representation/Signal_classification#Energy.E2.80.93limited_and_power.E2.80.93limited_signals\|"Energy-limited and power-limited signals"]]  in the book "Signal Representation".		*The first of these two equations applies to time-limited and thus energy-limited signals,  the second one to time-unlimited and thus power-limited signals according to the section  [[Signal_Representation/Signal_classification#Energy.E2.80.93limited_and_power.E2.80.93limited_signals\|»Energy-limited and power-limited signals«]]  in the book  »Signal Representation«.

	==Linear and ~~non-linear~~ distortions==		==Linear and nonlinear distortions==
	<br>		<br>
	A distinction is made between linear and nonlinear distortions:		A distinction is made between  »linear distortions«  and  »nonlinear distortions«:

	If the system is linear and time-invariant  $(\rm LTI)$, then it is fully ~~characterised~~ by its  [[Linear_and_Time_Invariant_Systems/System_Description_in_Frequency_Domain#Frequency_response_.E2.80.93_Transfer_function\|$\text{frequency response}$]]  $H(f)$ and the following can be stated:		If the system is linear and time-invariant  $(\rm LTI)$,  then it is fully characterized by its  [[Linear_and_Time_Invariant_Systems/System_Description_in_Frequency_Domain#Frequency_response_.E2.80.93_Transfer_function\|$\text{frequency response}$]]  $H(f)$  and the following can be stated:
	*According to the  $H(f)$  definition the following holds for the output spectrum:   $Y(f)=X(f) · H(f)$.   As a consequence according to the calculation rules of multiplication,  $Y(f)$  cannot contain any frequency components that are not already contained in  $X(f)$~~ ~~.		#According to the  $H(f)$  definition the following holds for the output spectrum:   $Y(f)=X(f) · H(f)$.
	*The inverse implies:   The output signal  $y(t)$  can include any frequency  $f_0$  already contained in the input  $x(t)$ . The prerequisite is therefore that  $X(f_0) ≠ 0$~~  holds~~.		#As a consequence according to the calculation rules of multiplication,  $Y(f)$  cannot contain any frequency components that are not already contained in  $X(f)$.
	*For an LTI system the absolute bandwidth  $B_y$  of the output signal is never greater than the bandwidth  $B_x$  of the input signal:   $B_y \le B_x .$		#The inverse implies:   The output signal  $y(t)$  can include any frequency  $f_0$  already contained in the input  $x(t)$ . The prerequisite is therefore that  $X(f_0) ≠ 0$.
			#For an LTI system the absolute bandwidth  $B_y$  of the output signal is never greater than the bandwidth  $B_x$  of the input signal:   $B_y \le B_x .$


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	[[File:EN_LZI_T_2_1_S5_neu.png\|frame\| Linear and nonlinear distortions\|class=fit]]		[[File:EN_LZI_T_2_1_S5_neu.png\|frame\| Linear and nonlinear distortions\|class=fit]]
	In the upper diagram  $B_y = B_x$  holds. There are  »'''linear distortions'''«  because in this frequency band  $X(f)$  and  $Y(f)$  differ.		$\rm (A)$   In the upper diagram  $B_y = B_x$  holds. There are  »'''linear distortions'''«  because in this frequency band  $X(f)$  and  $Y(f)$  differ.

	A band limitation  $(B_y < B_x)$  is a special form of linear ~~distortion~~, which will be discussed in the  [[Linear_and_Time_Invariant_Systems/Lineare_Verzerrungen\|"chapter after next"]].		:A band limitation  $(B_y < B_x)$  is a special form of linear distortions,  which will be discussed in the  [[Linear_and_Time_Invariant_Systems/Lineare_Verzerrungen\|»chapter after next«]].


			$\rm (B)$   The lower diagram shows an example of  »'''non-linear distortions'''«  $(B_y > B_x)$.  For such a system no frequency response  $H(f)$  can be given.

			:Descriptive quantities applicable for nonlinear systems will be explained in the next chapter  [[Linear_and_Time_Invariant_Systems/Nonlinear_Distortions\|»Non-linear Distortions«]] .

	~~The lower diagram shows an example of  »'''non-linear distortions'''«  $(B_y > B_x)$.  For such a system no frequency response  $H(f)$  can be given.~~

	~~Descriptive quantities applicable for non-linear systems will be explained in the next chapter  [[Linear_and_Time_Invariant_Systems/Nonlinear_Distortions\|"Non-linear Distortions"]] .~~		In most real transmission channels both linear and nonlinear distortions occur.  However,  for a whole range of problems the precise separation of the two types of distortions is essential.  In  [Kam04]<ref>Kammeyer, K.D.:  Nachrichtenübertragung.  Stuttgart: B.G. Teubner, 4. Auflage, 2004.</ref>  a corresponding substitute model is shown. }}
	~~<br clear=all>~~
	In most real transmission channels both linear and ~~non-linear~~ distortions occur.  However, for a whole range of problems the precise separation of the two types of distortions is essential.  In  [Kam04]<ref>Kammeyer, K.D.:  Nachrichtenübertragung.  Stuttgart: B.G. Teubner, 4. Auflage, 2004.</ref>  a corresponding substitute model is shown. }}


	We refer here to the  (German language)  learning video  [[Lineare_und_nichtlineare_Verzerrungen_(Lernvideo)\|"Linear and ~~non-Linear~~ distortions"]].		We refer here to the  $($German language$)$  learning video  [[Lineare_und_nichtlineare_Verzerrungen_(Lernvideo)\|»Linear and nonlinear distortions«]].

	==Exercises for the chapter==		==Exercises for the chapter==

Guenter at 17:36, 8 November 2023

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	We reference the interactive applet  [[Applets:Linear_Distortions_of_Periodic_Signals\|"Linear Distortions of Periodic Signals"]].		We reference the interactive applet  [[Applets:Linear_Distortions_of_Periodic_Signals\|»Linear Distortions of Periodic Signals«]].

	==Elimination of attenuation factor and transit time==		==Elimination of attenuation factor and transit time==

Guenter at 17:33, 8 November 2023

2023-11-08T17:33:29Z

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Guenter at 15:29, 12 January 2023

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	{{BlaueBox\|TEXT=		{{BlaueBox\|TEXT=
	$\text{Definition:}$		$\text{Definition:}$
	In general,  ~~$\text{~~distortion}$  is understood to be undesirable deterministic changes in a message signal caused by a transmission system. }}		In general,  '''»distortion«'''  is understood to be undesirable deterministic changes in a message signal caused by a transmission system. }}


	In addition to stochastic interferences  (noise, crosstalk, etc.), such deterministic distortions are a critical limitation on the transmission quality and rate for many communication systems.		In addition to stochastic interferences  $($noise,  crosstalk, etc.$)$,  such deterministic distortions are a critical limitation on the transmission quality and rate for many communication systems.

	This chapter presents these distortions in a summarizing way, in particular:		This chapter presents these distortions in a summarizing way,  in particular:

	*The quantitative ~~detection~~ of such signal falsifications via the distortion power,		#The quantitative description of such signal falsifications via the  »distortion power«,
	*the distinguishing features between nonlinear and linear distortions,		#the distinguishing features between  »nonlinear and linear distortions«,
	*the meaning and computation of the distortion factor in nonlinear systems,  and		#the meaning and computation of the  »distortion factor in nonlinear systems«,  and
	*the effects of linear attenuation and phase distortions.		#the effects of  »linear attenuation and phase distortions«.

Guenter at 09:56, 5 January 2023

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Hwang at 13:36, 30 December 2022

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	The lower diagram shows an example of  »'''non-linear distortions'''«  $(B_y > B_x)$.  For such a system no frequency response  $H(f)$  can be given.		The lower diagram shows an example of  »'''non-linear distortions'''«  $(B_y > B_x)$.  For such a system no frequency response  $H(f)$  can be given.

	Descriptive quantities applicable for non-linear systems will be explained in the next chapter  [[Linear_and_Time_Invariant_Systems/~~Nichtlineare_Verzerrungen~~\|"Non-linear Distortions"]] .		Descriptive quantities applicable for non-linear systems will be explained in the next chapter  [[Linear_and_Time_Invariant_Systems/Nonlinear_Distortions\|"Non-linear Distortions"]] .
	<br clear=all>		<br clear=all>
	In most real transmission channels both linear and non-linear distortions occur.  However, for a whole range of problems the precise separation of the two types of distortions is essential.  In  [Kam04]<ref>Kammeyer, K.D.:  Nachrichtenübertragung.  Stuttgart: B.G. Teubner, 4. Auflage, 2004.</ref>  a corresponding substitute model is shown. }}		In most real transmission channels both linear and non-linear distortions occur.  However, for a whole range of problems the precise separation of the two types of distortions is essential.  In  [Kam04]<ref>Kammeyer, K.D.:  Nachrichtenübertragung.  Stuttgart: B.G. Teubner, 4. Auflage, 2004.</ref>  a corresponding substitute model is shown. }}

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	*the distinguishing features between nonlinear and linear distortions,		*the distinguishing features between nonlinear and linear distortions,
	*the meaning and computation of the distortion factor in nonlinear systems,  and		*the meaning and computation of the distortion factor in nonlinear systems,  and
	*the effects of linear ~~damping '''KORREKTUR:~~ attenuation~~'''~~ and phase distortions.		*the effects of linear attenuation and phase distortions.


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	*It should be noted that such an ideal system does not exist in reality even if statistical disturbances and noise processes (that always exist but are not considered in this book) are disregarded.		*It should be noted that such an ideal system does not exist in reality even if statistical disturbances and noise processes (that always exist but are not considered in this book) are disregarded.
	*Every transmission medium exhibits losses  (~~damping '''KORREKTUR:~~ attenuation~~'''~~)  and transit times.  Even if these physical phenomena are very small, they are never zero.  Therefore it is necessary to introduce a somewhat less strict quality characteristic.		*Every transmission medium exhibits losses  (attenuation)  and transit times.  Even if these physical phenomena are very small, they are never zero.  Therefore it is necessary to introduce a somewhat less strict quality characteristic.


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	A  »'''distortion-free system'''«  exists if the following condition is fulfilled:		A  »'''distortion-free system'''«  exists if the following condition is fulfilled:
	:$$y(t) = \alpha \cdot x(t - \tau).$$		:$$y(t) = \alpha \cdot x(t - \tau).$$
	*Here,  $α$  describes the ~~damping '''KORREKTUR:~~ attenuation~~'''~~ factor and  $τ$  the transit time.		*Here,  $α$  describes the attenuation factor and  $τ$  the transit time.
	*If this condition is not met, the system is said to be  »''' distortive'''«.}}		*If this condition is not met, the system is said to be  »''' distortive'''«.}}

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	Note:		Note:
	*The ~~damping '''KORREKTUR:~~ attenuation~~'''~~ factor  $α$  can be completely reversed by a receiver-side gain of  $1/α = 1.25$, but it must be taken into account that this also increases any noise.		*The attenuation factor  $α$  can be completely reversed by a receiver-side gain of  $1/α = 1.25$, but it must be taken into account that this also increases any noise.
	*However, the transit time  $τ$  cannot be compensated due to  [[Signal_Representation/Signal_classification#Causal_and_non-causal_signals\|$\text{causality reasons}$]].  It now depends on the application whether such a transit time is subjectively perceived as disturbing or not. }}		*However, the transit time  $τ$  cannot be compensated due to  [[Signal_Representation/Signal_classification#Causal_and_non-causal_signals\|$\text{causality reasons}$]].  It now depends on the application whether such a transit time is subjectively perceived as disturbing or not. }}

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	We now consider a distortive system on the basis of the input and output signal.		We now consider a distortive system on the basis of the input and output signal.
	*In doing so, we assume that apart from the signal distortions there is no additional ~~damping '''KORREKTUR:~~ attenuation~~'''~~ factor   $α$  which is constant for all frequencies and no additional transit time  $τ$  that is constant for all frequencies.		*In doing so, we assume that apart from the signal distortions there is no additional attenuation factor   $α$  which is constant for all frequencies and no additional transit time  $τ$  that is constant for all frequencies.
	*These conditions are fulfilled for the signal sections sketched on the right.		*These conditions are fulfilled for the signal sections sketched on the right.

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	We reference the interactive applet  [[Applets:Linear_Distortions_of_Periodic_Signals\|"Linear Distortions of Periodic Signals"]].		We reference the interactive applet  [[Applets:Linear_Distortions_of_Periodic_Signals\|"Linear Distortions of Periodic Signals"]].

	==Elimination of ~~damping~~ factor and transit time==		==Elimination of attenuation factor and transit time==
	~~'''KORREKTUR: attenuation'''~~
	<br>		<br>
	The equations given in the last section do not result in applicable statements if the system is additionally affected by an attenuation factor  $α$  and/or a transit time  $τ$.  The diagram shows the attenuated, delayed and distorted signal		The equations given in the last section do not result in applicable statements if the system is additionally affected by an attenuation factor  $α$  and/or a transit time  $τ$.  The diagram shows the attenuated, delayed and distorted signal

	[[File:P_ID876__LZI_T_2_1_S4_neu.png\|right\|frame\|Elimination of ~~damping '''KORREKTUR:~~ attenuation~~'''~~ factor $α$		[[File:P_ID876__LZI_T_2_1_S4_neu.png\|right\|frame\|Elimination of attenuation factor $α$
	<br>and transit time  $τ$\|class=fit]]		<br>and transit time  $τ$\|class=fit]]

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	:$$E_{\rm V} = \min_{\alpha, \ \tau} \int_{ - \infty }^{ + \infty}		:$$E_{\rm V} = \min_{\alpha, \ \tau} \int_{ - \infty }^{ + \infty}
	{\big[y(t) - \left(\alpha \cdot x(t - \tau) \right) \big]^2}\hspace{0.1cm}{\rm d}t.$$		{\big[y(t) - \left(\alpha \cdot x(t - \tau) \right) \big]^2}\hspace{0.1cm}{\rm d}t.$$
	*The energy of the ~~damped '''KORREKTUR:~~ attenuated~~'''~~ and delayed signal  $α · x(t - τ)$  is  $E_{\rm V} =α^2 · E_x$ independent of the transit time  $τ$.  Thus for the signal-to-distortion (power) ratio the following is applicable:		*The energy of the attenuated and delayed signal  $α · x(t - τ)$  is  $E_{\rm V} =α^2 · E_x$ independent of the transit time  $τ$.  Thus for the signal-to-distortion (power) ratio the following is applicable:
	:$$\rho_{\rm V} = \frac{ \alpha^2 \cdot E_{x}}{E_{\rm V}}\hspace{0.3cm}{\rm or}\hspace{0.3cm}\rho_{\rm D}= \frac{ \alpha^2 \cdot P_{x}}{P_{\rm V}} .$$		:$$\rho_{\rm V} = \frac{ \alpha^2 \cdot E_{x}}{E_{\rm V}}\hspace{0.3cm}{\rm or}\hspace{0.3cm}\rho_{\rm D}= \frac{ \alpha^2 \cdot P_{x}}{P_{\rm V}} .$$
	*The first of these two equations applies to time-limited and thus energy-limited signals, the second one to time-unlimited and thus power-limited signals according to the section  [[Signal_Representation/Signal_classification#Energy.E2.80.93limited_and_power.E2.80.93limited_signals\|"Energy-limited and power-limited signals"]]  in the book "Signal Representation".		*The first of these two equations applies to time-limited and thus energy-limited signals, the second one to time-unlimited and thus power-limited signals according to the section  [[Signal_Representation/Signal_classification#Energy.E2.80.93limited_and_power.E2.80.93limited_signals\|"Energy-limited and power-limited signals"]]  in the book "Signal Representation".