Linear and Time Invariant Systems/Linear Distortions - Revision history

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	$\text{System model:}$		$\text{System model:}$
	The  »'''generally complex frequency response'''« can also be formulated as follows:		The  »'''generally complex frequency response'''« can also be formulated as follows:
	:$$H(f) = \vert H(f) \vert \cdot {\rm e}^{- {\rm j} \hspace{0.05cm} \cdot		:$$H(f) = \vert H(f) \vert \cdot {\rm e}^{- {\rm j} \hspace{0.05cm} \cdot\hspace{0.05cm} b(f)} = {\rm e}^{- a(f)}\cdot {\rm e}^{- {\rm j}\hspace{0.05cm} \cdot \hspace{0.05cm} b(f)}.$$
	\hspace{0.05cm} b(f)} = {\rm e}^{- a(f)}\cdot {\rm e}^{- {\rm j}
	\hspace{0.05cm} \cdot \hspace{0.05cm} b(f)}.$$
	This yields the following descriptive quantities:		This yields the following descriptive quantities:
	*The magnitude  $\vert H(f)\vert $  is referred to as  »'''amplitude response'''«  and in logarithmic form as  »'''attenuation curve'''«:		*The magnitude  $\vert H(f)\vert $  is referred to as  »'''amplitude response'''«  and in logarithmic form as  »'''attenuation curve'''«:
	:$$a(f) = - \ln \vert H(f)\vert \hspace{0.2cm}{\rm in \hspace{0.1cm}Neper		:$$a(f) = - \ln \vert H(f)\vert \hspace{0.2cm}{\rm in \hspace{0.1cm}Neper\hspace{0.1cm}(Np) } = - 20 \cdot \lg \vert H(f)\vert \hspace{0.2cm}{\rm in\hspace{0.1cm}decibel \hspace{0.1cm}(dB) }.$$
	\hspace{0.1cm}(Np) } = - 20 \cdot \lg \vert H(f)\vert \hspace{0.2cm}{\rm in
	\hspace{0.1cm}decibel \hspace{0.1cm}(dB) }.$$
	*The  »'''phase response'''«  $b(f)$  specifies the negative  $(f$–dependent$)$  angle of  $H(f)$  in the complex plane with respect to the real axis:		*The  »'''phase response'''«  $b(f)$  specifies the negative  $(f$–dependent$)$  angle of  $H(f)$  in the complex plane with respect to the real axis:
	:$$b(f) = - {\rm arc} \hspace{0.1cm}H(f) \hspace{0.2cm}{\rm in		:$$b(f) = - {\rm arc} \hspace{0.1cm}H(f) \hspace{0.2cm}{\rm in\hspace{0.1cm}radian \hspace{0.1cm}(rad)}.$$}}
	\hspace{0.1cm}radian \hspace{0.1cm}(rad)}.$$}}


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	<br>		<br>
	In the following,  we consider the sum of two harmonic oscillations as input signal:		In the following,  we consider the sum of two harmonic oscillations as input signal:
	:$$x(t) = A_1 \cdot \cos(2 \pi f_1 \cdot t - \varphi_1) + A_2 \cdot \cos(2 \pi f_2 \cdot t -		:$$x(t) = A_1 \cdot \cos(2 \pi f_1 \cdot t - \varphi_1) + A_2 \cdot \cos(2 \pi f_2 \cdot t -\varphi_2).$$
	\varphi_2).$$

	*If the output signal can be represented in the form		*If the output signal can be represented in the form
	:$$y(t) = \alpha_1 \cdot A_1 \cdot \cos(2 \pi f_1 \cdot t - \varphi_1) + \alpha_2 \cdot A_2 \cdot \cos(2 \pi f_2 \cdot t -		:$$y(t) = \alpha_1 \cdot A_1 \cdot \cos(2 \pi f_1 \cdot t - \varphi_1) + \alpha_2 \cdot A_2 \cdot \cos(2 \pi f_2 \cdot t -\varphi_2),$$
	\varphi_2),$$
	:and at the same time  $α_1 ≠ α_2$  is valid,  then  »'''exclusively attenuation distortions'''«  are existent since the phase values  $\varphi_1$  and  $\varphi_2$  are not changed by the system.		:and at the same time  $α_1 ≠ α_2$  is valid,  then  »'''exclusively attenuation distortions'''«  are existent since the phase values  $\varphi_1$  and  $\varphi_2$  are not changed by the system.

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	<br>		<br>
	We consider a system with  $\|H(f)\| = 1$  such that the following holds for the frequency response:		We consider a system with  $\|H(f)\| = 1$  such that the following holds for the frequency response:
	:$$H(f) = {\rm e}^{-{\rm j} \hspace{0.05cm} \cdot \hspace{0.05cm}		:$$H(f) = {\rm e}^{-{\rm j} \hspace{0.05cm} \cdot \hspace{0.05cm}b(f)}.$$
	b(f)}.$$
	*The left graph shows an exemplary phase response  $b(f)$.  Such a phase response is always an odd function with respect to the frequency $f$:   $b(\hspace{-0.01cm}-\hspace{-0.08cm}f) = \hspace{0.08cm}-b(f)$.		*The left graph shows an exemplary phase response  $b(f)$.  Such a phase response is always an odd function with respect to the frequency $f$:   $b(\hspace{-0.01cm}-\hspace{-0.08cm}f) = \hspace{0.08cm}-b(f)$.

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	If the harmonic oscillation at the input is		If the harmonic oscillation at the input is
	:$$x(t) = C \cdot \cos(2 \pi f_0 t - \varphi)\hspace{0.15cm}		:$$x(t) = C \cdot \cos(2 \pi f_0 t - \varphi)\hspace{0.15cm}\circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\, X(f ) = {C}/{2}\cdot{\rm e}^{{\rm j}\hspace{0.05cm} \cdot \hspace{0.05cm} \varphi} \cdot \delta(f + f_0)\hspace{0.01cm} + \hspace{0.01cm}{C}/{2}\cdot {\rm e}^{-{\rm j}\hspace{0.05cm} \cdot \hspace{0.05cm} \varphi} \cdot \delta(f - f_0),$$
	\circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\, X(f ) = {C}/{2}\cdot
	{\rm e}^{{\rm j}\hspace{0.05cm} \cdot \hspace{0.05cm} \varphi} \cdot \delta(f + f_0)
	\hspace{0.01cm} + \hspace{0.01cm}{C}/{2}\cdot {\rm e}^{-{\rm
	j}\hspace{0.05cm} \cdot \hspace{0.05cm} \varphi} \cdot \delta(f - f_0),$$
	[[File: EN_LZI_T_2_3_S4.png\|frame\|On the definition of the phase delay time $\tau_{\rm P}$\|class=fit]]		[[File: EN_LZI_T_2_3_S4.png\|frame\|On the definition of the phase delay time $\tau_{\rm P}$\|class=fit]]

	then the following arises as a result for the spectral function at the output:		then the following arises as a result for the spectral function at the output:
	:$$Y(f ) = {C}/{2}\cdot		:$$Y(f ) = {C}/{2}\cdot{\rm e}^{{\rm j}\hspace{0.05cm} \cdot \hspace{0.05cm} \varphi} \cdot {\rm e}^{\hspace{0.05cm}{\rm j}\hspace{0.05cm} \cdot \hspace{0.05cm} b(f_0)} \cdot \delta(f + f_0) \hspace{0.05cm} +\hspace{0.05cm}{C}/{2}\cdot {\rm e}^{-{\rm j}\hspace{0.05cm} \cdot \hspace{0.05cm}\varphi}\cdot {\rm e}^{-{\rm j}\hspace{0.05cm} \cdot \hspace{0.05cm} b(f_0)} \cdot\delta(f - f_0).$$
	{\rm e}^{{\rm j}\hspace{0.05cm} \cdot \hspace{0.05cm} \varphi} \cdot {\rm e}^{\hspace{0.05cm}{\rm
	j}\hspace{0.05cm} \cdot \hspace{0.05cm} b(f_0)} \cdot \delta(f + f_0) \hspace{0.05cm} +
	\hspace{0.05cm}{C}/{2}\cdot {\rm e}^{-{\rm j}\hspace{0.05cm} \cdot \hspace{0.05cm}
	\varphi}\cdot {\rm e}^{-{\rm j}\hspace{0.05cm} \cdot \hspace{0.05cm} b(f_0)} \cdot
	\delta(f - f_0).$$
	Thus, the output signal is:		Thus, the output signal is:
	:$$y(t) = C \cdot \cos(2 \pi f_0 t - b(f_0) - \varphi).$$		:$$y(t) = C \cdot \cos(2 \pi f_0 t - b(f_0) - \varphi).$$
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	$\text{Definition:}$		$\text{Definition:}$
	The  »'''phase delay time'''«  indicates the delay experienced by a harmonic oscillation of frequency  $f_0$  through the system.  For phase-distorting systems, the phase delay time is frequency-dependent:		The  »'''phase delay time'''«  indicates the delay experienced by a harmonic oscillation of frequency  $f_0$  through the system.  For phase-distorting systems, the phase delay time is frequency-dependent:
	:$$\tau_{\rm P}(f_0) = \frac{b(f_0)}{2\pi f_0} \hspace{0.4cm}{\rm or} \hspace{0.4cm}		:$$\tau_{\rm P}(f_0) = \frac{b(f_0)}{2\pi f_0} \hspace{0.4cm}{\rm or} \hspace{0.4cm}\tau_{\rm P}(\omega_0) = \frac{b(\omega_0)}{\omega_0}.$$}}
	\tau_{\rm P}(\omega_0) = \frac{b(\omega_0)}{\omega_0}.$$}}


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	In contrast,  the lesser slope of the tangent line shown in red denotes the  »'''group delay time:'''«		In contrast,  the lesser slope of the tangent line shown in red denotes the  »'''group delay time:'''«
	:$$\tau_{\rm G}(\omega_0) = \left[ \frac{ {\rm d}b(\omega)}{ {\rm d}\omega}\right ]_{\omega = \omega_0} =		:$$\tau_{\rm G}(\omega_0) = \left[ \frac{ {\rm d}b(\omega)}{ {\rm d}\omega}\right ]_{\omega = \omega_0} =\left[ \frac{1}{\omega_0} \cdot \frac{1}{1 + \left(\omega / \omega_0\right]^2} \right ]_{\omega =\omega_0}= \frac{1}{2\omega_0}= \frac{1}{4 \pi \cdot{1\, \rm kHz} }\approx {80\, \rm µ s}.$$}}
	\left[ \frac{1}{\omega_0} \cdot \frac{1}{1 + \left(\omega / \omega_0\right]^2} \right ]_{\omega =
	\omega_0}
	= \frac{1}{2\omega_0}= \frac{1}{4 \pi \cdot{1\, \rm kHz} }
	\approx {80\, \rm µ s}.$$}}

	==Phase distortions==		==Phase distortions==
	<br>		<br>
	To illustrate this point,  we again consider the sum of two harmonic oscillations as input signal:		To illustrate this point,  we again consider the sum of two harmonic oscillations as input signal:
	:$$x(t) = A_1 \cdot \cos(2 \pi f_1 \cdot t - \varphi_1) + A_2 \cdot \cos(2 \pi f_2 \cdot t -		:$$x(t) = A_1 \cdot \cos(2 \pi f_1 \cdot t - \varphi_1) + A_2 \cdot \cos(2 \pi f_2 \cdot t -\varphi_2).$$
	\varphi_2).$$

	*If the output signal for this input signal can be represented in the form		*If the output signal for this input signal can be represented in the form
	:$$y(t) = A_1 \cdot \cos(2 \pi f_1 \cdot (t - \tau_1) - \varphi_1) + A_2 \cdot \cos(2 \pi f_2 \cdot (t - \tau_2) -		:$$y(t) = A_1 \cdot \cos(2 \pi f_1 \cdot (t - \tau_1) - \varphi_1) + A_2 \cdot \cos(2 \pi f_2 \cdot (t - \tau_2) -\varphi_2)$$
	\varphi_2)$$
	:and at the same time  $τ_1 ≠ τ_2$ is valid,  then   »'''exclusively phase distortions'''«  are existent.		:and at the same time  $τ_1 ≠ τ_2$ is valid,  then   »'''exclusively phase distortions'''«  are existent.

	*The two phase delay times  $τ_1 ≠ τ_2$  can be determined from the phase response  $($in radian$)$:		*The two phase delay times  $τ_1 ≠ τ_2$  can be determined from the phase response  $($in radian$)$:
	:$$\tau_1 = \frac{b(f_1)}{2\pi f_1} , \hspace{0.4cm}\tau_2 = \frac{b(f_2)}{2\pi		:$$\tau_1 = \frac{b(f_1)}{2\pi f_1} , \hspace{0.4cm}\tau_2 = \frac{b(f_2)}{2\pif_2}.$$
	~~f_2~~}.$$

	{{GraueBox\|TEXT=		{{GraueBox\|TEXT=

Maintenance script: Add German interlanguage link

2026-03-16T12:27:54Z

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	{{Display}}		{{Display}}
			[[de:Lineare_zeitinvariante_Systeme/Lineare_Verzerrungen]]

Guenter at 17:47, 14 November 2023

2023-11-14T17:47:02Z

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	#If the distortion is nonlinear,  the equalization must also be nonlinear.		#If the distortion is nonlinear,  the equalization must also be nonlinear.
	#But even with linear distortion,  nonlinear equalization methods are used, e.g.  »Decision Feedback Equalisation«  $\rm (DFE)$  in digital systems.  <br>The advantage over linear equalization is that there is no increase in noise power.		#But even with linear distortion,  nonlinear equalization methods are used, e.g.  »Decision Feedback Equalisation«  $\rm (DFE)$  in digital systems.  <br>The advantage over linear equalization is that there is no increase in noise power.
	#If  $S_{\rm V}$  is a linear system with frequency response  $H_{\rm V}(f)$,  then the distortions can be completely eliminated with the  »'''inverse frequency response'''«  $H_{\rm E}(f) = 1/H_{\rm V}(f)$  and  $z(t) = x(t)$  holds.		#If  $S_{\rm V}$  is a linear system with frequency response  $H_{\rm V}(f)$,  <br>then the distortions can be completely eliminated with the  »'''inverse frequency response'''«  $H_{\rm E}(f) = 1/H_{\rm V}(f)$  and  $z(t) = x(t)$  holds.
	#A prerequisite for this is that the frequency response  $H_{\rm V}(f)$  has no zeros  in the spectral range of interest,  as otherwise  $H_{\rm E}(f)$  would result in infinity points.		#A prerequisite for this is that the frequency response  $H_{\rm V}(f)$  has no zeros  in the spectral range of interest,  as otherwise  $H_{\rm E}(f)$  would result in infinity points.
	#For  »analog systems«,  complete equalization means that  $z(t)$  differs from  $x(t)$  only by the unavoidable noise components and possibly by a transit time.		#For  »analog systems«,  complete equalization means that  $z(t)$  differs from  $x(t)$  only by the unavoidable noise components and possibly by a transit time.

Guenter at 17:45, 14 November 2023

2023-11-14T17:45:20Z

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Guenter at 17:05, 14 November 2023

2023-11-14T17:05:41Z

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Guenter at 16:28, 13 November 2023

2023-11-13T16:28:09Z

← Older revision		Revision as of 18:28, 13 November 2023
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	==Compilation of important descriptive quantities==		==Compilation of important descriptive quantities==
	<br>		<br>
	~~[[File:P_ID899__LZI_T_2_3_S1_neu.png \|frame\|Linear system description\|class=fit]]~~		Now nonlinear distortions are excluded so that the system is fully described by the frequency response  $H(f)$ .
	Now nonlinear distortions are excluded so that the system is fully described by the frequency response  $H(f)$ .

	The generally complex frequency response can also be formulated as follows:		{{BlaueBox\|TEXT=
	:$$H(f) = \|H(f)\| \cdot {\rm e}^{-{\rm j} \hspace{0.05cm} \cdot		[[File:P_ID899__LZI_T_2_3_S1_neu.png \|frame\|Linear system description\|class=fit]]
	\hspace{0.05cm} b(f)} = {\rm e}^{-a(f)}\cdot {\rm e}^{-{\rm j}		$\text{System model:}$
			The  »'''generally complex frequency response'''« can also be formulated as follows:
			:$$H(f) = \vert H(f) \vert \cdot {\rm e}^{- {\rm j} \hspace{0.05cm} \cdot
			\hspace{0.05cm} b(f)} = {\rm e}^{- a(f)}\cdot {\rm e}^{- {\rm j}
	\hspace{0.05cm} \cdot \hspace{0.05cm} b(f)}.$$		\hspace{0.05cm} \cdot \hspace{0.05cm} b(f)}.$$

	This yields the following descriptive quantities:		This yields the following descriptive quantities:
	*The magnitude  $\|H(f)\|$  is referred to as  »'''amplitude response'''«  and in logarithmic form as  "attenuation curve":		*The magnitude  $\vert H(f)\vert $  is referred to as  »'''amplitude response'''«  and in logarithmic form as  »'''attenuation curve'''«:
	:$$a(f) = - \ln \|H(f)\|\hspace{0.2cm}{\rm in \hspace{0.1cm}Neper		:$$a(f) = - \ln \vert H(f)\vert \hspace{0.2cm}{\rm in \hspace{0.1cm}Neper
	\hspace{0.1cm}(Np) } = - 20 \cdot \lg \|H(f)\|\hspace{0.2cm}{\rm in		\hspace{0.1cm}(Np) } = - 20 \cdot \lg \vert H(f)\vert \hspace{0.2cm}{\rm in
	\hspace{0.1cm}decibel \hspace{0.1cm}(dB) }.$$		\hspace{0.1cm}decibel \hspace{0.1cm}(dB) }.$$
	*The  »'''phase response'''«  $b(f)$  specifies the negative ~~angle of~~  $H(f)$  ~~dependent on~~  $f$  in the complex plane and with respect to the real axis:		*The  »'''phase response'''«  $b(f)$  specifies the negative  $(f$–dependent$)$  angle of  $H(f)$  in the complex plane and with respect to the real axis:
	:$$b(f) = - {\rm arc} \hspace{0.1cm}H(f) \hspace{0.2cm}{\rm in		:$$b(f) = - {\rm arc} \hspace{0.1cm}H(f) \hspace{0.2cm}{\rm in
	\hspace{0.1cm}radian \hspace{0.1cm}(rad)}.$$		\hspace{0.1cm}radian \hspace{0.1cm}(rad)}.$$}}



	==Requirements for distortion-free systems==		==Requirements for distortion-free systems==

Hwang at 14:53, 30 December 2022

2022-12-30T14:53:05Z

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	<br>		<br>
	[[File:P_ID906__LZI_T_2_3_S7_neu.png\|right\|frame\|Equalization of signals\|class=fit]]		[[File:P_ID906__LZI_T_2_3_S7_neu.png\|right\|frame\|Equalization of signals\|class=fit]]
	This method, which is very important for ~~Communications Engineering~~,  will only be briefly touched upon here.  More detailed information can be found in the books  [[Modulation_Methods\|"Modulation Methods"]]  and  [[Digital_Signal_Transmission\|"Digital Signal Transmission"]].		This method, which is very important for communications engineering,  will only be briefly touched upon here.  More detailed information can be found in the books  [[Modulation_Methods\|"Modulation Methods"]]  and  [[Digital_Signal_Transmission\|"Digital Signal Transmission"]].

	For this brief description, we assume the outlined constellation:		For this brief description, we assume the outlined constellation:

Hwang at 14:44, 30 December 2022

2022-12-30T14:44:31Z

← Older revision		Revision as of 16:44, 30 December 2022
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	Outside the frequency band occupied by $x(t)$  the  "constant attenuation"  and  "linear phase"  conditions do not need to be satisfied.		Outside the frequency band occupied by $x(t)$  the  "constant attenuation"  and  "linear phase"  conditions do not need to be satisfied.

	It can be seen from the dashed curve of $a(f)$ that even a much higher attenuation is purposeful here because as a consequence the always-existing noise components outside the useful bandwidth – which are not considered in this section – are suppressed ~~better~~. }}		It can be seen from the dashed curve of $a(f)$ that even a much higher attenuation is purposeful here because as a consequence the always-existing noise components outside the useful bandwidth – which are not considered in this section – are better suppressed. }}

	==Attenuation distortions==		==Attenuation distortions==

Hwang at 14:40, 30 December 2022

2022-12-30T14:40:50Z

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	Moreover,  if  $α = 1$  and  $τ = 0$  hold,  then there is an  »'''ideal transmission system'''«.  In contrast, there are linear distortions whenever		Moreover,  if  $α = 1$  and  $τ = 0$  hold,  then there is an  »'''ideal transmission system'''«.  In contrast, there are linear distortions whenever
	*the impulse response  $h(t)$  is a continuous-time function, or		*the impulse response  $h(t)$  is a continuous-time function, or
	*the time-dicrete impulse response $h(t)$  is composed of more than one Dirac functions.}}		*the time-dicrete impulse response $h(t)$  is composed of more than one Dirac delta functions.}}

Hwang at 14:38, 30 December 2022

2022-12-30T14:38:39Z

← Older revision		Revision as of 16:38, 30 December 2022
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	Moreover,  if  $α = 1$  and  $τ = 0$  hold,  then there is an  »'''ideal transmission system'''«.  In contrast, there are linear distortions whenever		Moreover,  if  $α = 1$  and  $τ = 0$  hold,  then there is an  »'''ideal transmission system'''«.  In contrast, there are linear distortions whenever
	*~~The~~ impulse response  $h(t)$  is a continuous-time function, or		*the impulse response  $h(t)$  is a continuous-time function, or
	*the time-dicrete impulse response $h(t)$  is composed of more than one Dirac functions.}}		*the time-dicrete impulse response $h(t)$  is composed of more than one Dirac functions.}}