Difference between revisions of "Linear and Time Invariant Systems/Linear Distortions"

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:$$a(f) = - \ln |H(f)|\hspace{0.2cm}{\rm in \hspace{0.1cm}Neper
 
:$$a(f) = - \ln |H(f)|\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 |H(f)|\hspace{0.2cm}{\rm in
\hspace{0.1cm}Dezibel \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 angle of  $H(f)$  dependent on  $f$  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)}.$$
  
 
==Voraussetzungen für verzerrungsfreie Systeme==
 
==Voraussetzungen für verzerrungsfreie Systeme==
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*The attenuation curve must be constant for all frequencies contained in the input signal:  
 
*The attenuation curve must be constant for all frequencies contained in the input signal:  
 
:$$a(f) = - \ln |H(f)| = - \ln \ \alpha = {\rm const.}$$
 
:$$a(f) = - \ln |H(f)| = - \ln \ \alpha = {\rm const.}$$
*The phase response must either be zero in the region of interest (system with no transit time) or increase linearly with frequency  $(τ$  indicats the transit time of the system):
+
*The phase response must either be zero in the region of interest (system with no transit time) or increase linearly with frequency  $(τ$  indicates the transit time of the system):
 
:$$b(f) = 2 \pi f \tau = {\rm const.} \cdot f.$$
 
:$$b(f) = 2 \pi f \tau = {\rm const.} \cdot f.$$
  
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:$$h(t) = \alpha \cdot \delta(t - \tau),\hspace{0.4cm}\alpha \ne 0.$$
 
:$$h(t) = \alpha \cdot \delta(t - \tau),\hspace{0.4cm}\alpha \ne 0.$$
  
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  
 
*$h(t)$  is a continuous-time function or  
 
*$h(t)$  is a continuous-time function or  
 
*$h(t)$  is composed of more than one Dirac function.}}
 
*$h(t)$  is composed of more than one Dirac function.}}
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*From the given constant attenuation value  $6 \ \rm dB$  it follows for the amplitude response:  $\vert H(f)\vert = 0.5$.
 
*From the given constant attenuation value  $6 \ \rm dB$  it follows for the amplitude response:  $\vert H(f)\vert = 0.5$.
 
*The output spectrum $Y(f)$  is thus half as large in magnitude as the spectral components $X(f)$  of the input signal.
 
*The output spectrum $Y(f)$  is thus half as large in magnitude as the spectral components $X(f)$  of the input signal.
*The phase response $b(f)$  increases linearly with frequency between $f_{\rm U}$ und $f_{\rm O}$.
+
*The phase response $b(f)$  increases linearly with frequency between $f_{\rm U}$ and $f_{\rm O}$.
 
*This results in all frequency components being delayed by the same phase delay time $τ$  where  $τ$  is fixed by the slope of  $b(f)$ .
 
*This results in all frequency components being delayed by the same phase delay time $τ$  where  $τ$  is fixed by the slope of  $b(f)$ .
 
*With $b(f) = 0$  a transit time-free system would arise as a result   ⇒   $τ = 0$.  
 
*With $b(f) = 0$  a transit time-free system would arise as a result   ⇒   $τ = 0$.  
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Significant effects of these attenuation distortions can be perceived:  
 
Significant effects of these attenuation distortions can be perceived:  
 
*$y(t)$  bears little resemblance to $x(t)$.  
 
*$y(t)$  bears little resemblance to $x(t)$.  
*In contrast, considering  $α_1 = α_2 = α$  the distortion-free signal $y(t) = α · x(t)$  would be obtained from which the original signal $x(t)$  could be reconstructed by amplifying by $1/α$ .}}
+
*In contrast, considering  $α_1 = α_2 = α$  the distortion-free signal $y(t) = α · x(t)$  would be obtained from which the original signal $x(t)$  could be reconstructed by amplifying it by $1/α$ .}}
  
  
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==Phasenlaufzeit==
 
==Phasenlaufzeit==
 
<br>
 
<br>
[[File: EN_LZI_T_2_3_S4.png|frame|Zur Definition der Phasenlaufzeit|class=fit]]
+
[[File: EN_LZI_T_2_3_S4.png|frame|On the definition of the phase delay time|class=fit]]
Wir betrachten ein System mit &nbsp;$|H(f)| = 1$, so dass für den Frequenzgang gilt:
+
We consider a system with&nbsp;$|H(f)| = 1$ 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)}.$$
*Die linke Grafik zeigt einen beispielhaften Phasenverlauf &nbsp;$b(f)$. Ein solcher Phasenverlauf ist stets eine ungerade Funktion bezüglich der Frequenz &nbsp;$f$: &nbsp;  $b(\hspace{-0.01cm}-\hspace{-0.08cm}f) = \hspace{0.08cm}-b(f)$.  
+
*The left graph shows an exemplary phase response&nbsp;$b(f)$. Such a phase response is always an odd function with respect to the frequency&nbsp;$f$: &nbsp;  $b(\hspace{-0.01cm}-\hspace{-0.08cm}f) = \hspace{0.08cm}-b(f)$.  
*Rechts ist die Funktion &nbsp;$b(ω)$&nbsp; skizziert, die gegenüber &nbsp;$b(f)$&nbsp; in der Abszisse um den Faktor $2π$ gestreckt ist.  
+
*On the right, the function&nbsp;$b(ω)$&nbsp; is sketched which is dilated by a factor of $2π$ with respect to&nbsp;$b(f)$&nbsp; in the abscissa.  
  
  
  
Liegt am Eingang die harmonische Schwingung
+
If the harmonic oscillation at the input is
 
:$$x(t) =  C \cdot \cos(2 \pi  f_0  t - \varphi)
 
:$$x(t) =  C \cdot \cos(2 \pi  f_0  t - \varphi)
 
\circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\, X(f )  = {C}/{2}\cdot
 
\circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\, X(f )  = {C}/{2}\cdot
 
{\rm e}^{{\rm j}\hspace{0.05cm} \cdot \hspace{0.05cm} \varphi} \cdot \delta(f + f_0)
 
{\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
 
\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)$$
+
j}\hspace{0.05cm} \cdot \hspace{0.05cm} \varphi} \cdot \delta(f - f_0),$$
an, so ergibt sich für die Spektralfunktion am Ausgang:
+
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
 
{\rm e}^{{\rm j}\hspace{0.05cm} \cdot \hspace{0.05cm} \varphi} \cdot {\rm e}^{\hspace{0.05cm}{\rm
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\varphi}\cdot {\rm e}^{-{\rm j}\hspace{0.05cm} \cdot \hspace{0.05cm} b(f_0)} \cdot
 
\varphi}\cdot {\rm e}^{-{\rm j}\hspace{0.05cm} \cdot \hspace{0.05cm} b(f_0)} \cdot
 
\delta(f - f_0).$$
 
\delta(f - f_0).$$
Somit lautet das Ausgangssignal:
+
Thus, the output signal is:
 
:$$y(t) =  C \cdot \cos(2 \pi  f_0  t - b(f_0) - \varphi).$$
 
:$$y(t) =  C \cdot \cos(2 \pi  f_0  t - b(f_0) - \varphi).$$
Dieses Signal kann auch in folgender Form dargestellt werden:
+
This signal can also be represented in the following form:
 
:$$y(t) =  C \cdot \cos(2 \pi  f_0 (  t - \tau_{\rm P}(f_0)) - \varphi).$$
 
:$$y(t) =  C \cdot \cos(2 \pi  f_0 (  t - \tau_{\rm P}(f_0)) - \varphi).$$
  
 
{{BlaueBox|TEXT=   
 
{{BlaueBox|TEXT=   
 
$\text{Definition:}$&nbsp;
 
$\text{Definition:}$&nbsp;
Die&nbsp; '''Phasenlaufzeit'''&nbsp; gibt die Verzögerung an, die eine harmonische Schwingung mit der Frequenz &nbsp;$f_0$&nbsp; durch das System erfährt. Bei phasenverzerrenden Systemen ist die Phasenlaufzeit frequenzabhängig:
+
The&nbsp; '''phase delay time'''&nbsp; indicates the delay experienced by a harmonic oscillation of frequency&nbsp;$f_0$&nbsp; through the system. For phase-distorting systems, the phase delay time is frequency-dependent:
:$$\tau_{\rm P}(f_0) =  \frac{b(f_0)}{2\pi f_0} \hspace{0.4cm}{\rm bzw.} \hspace{0.4cm}
+
:$$\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}.$$}}
  
  
Zu dieser Definition ist anzumerken:  
+
The following should be noted regarding this definition:  
*In der &nbsp;$b(ω)$–Darstellung kann die Phasenlaufzeit &nbsp;$τ_{\rm P}$&nbsp; als die Steigung der in der obigen Grafik grün eingezeichneten Geraden auch grafisch ermittelt werden.  
+
*In the&nbsp;$b(ω)$–representation, the phase delay time&nbsp;$τ_{\rm P}$&nbsp; can also be determined graphically as the slope of the straight line drawn in green in the above graph.  
*Im Allgemeinen wird eine Schwingung mit anderer Frequenz auch eine andere Phasenlaufzeit zur Folge haben. Dies ist der physikalische Hintergrund für Phasenverzerrungen.  
+
*In general, an oscillation with a different frequency will also result in a different phase delay time. This is the physical background for phase distortions.  
*Gilt bei einem System &nbsp;$b(ω) = τ_{\rm P} · ω$ &nbsp; bzw. &nbsp; $b(f) = 2π · τ_{\rm P} · f$, so haben alle Frequenzen die gleiche Phasenlaufzeit &nbsp;$τ_{\rm P}$. Ein solches System führt nicht zu Phasenverzerrungen.
+
*If for a system&nbsp;$b(ω) = τ_{\rm P} · ω$ &nbsp; or &nbsp; $b(f) = 2π · τ_{\rm P} · f$ is true, then all frequencies have the same phase delay time&nbsp;$τ_{\rm P}$. Such a system does not cause phase distortions.
  
  
Wir verweisen hier nochmals auf das interaktive Applet &nbsp;[[Applets:Lineare_Verzerrungen_periodischer_Signale|Lineare Verzerrungen periodischer Signale]].
+
We refer here again to the interactive applet&nbsp;[[Applets:Lineare_Verzerrungen_periodischer_Signale|Linear distortions of periodic signals]].
  
 
==Unterschied zwischen Phasen- und Gruppenlaufzeit==
 
==Unterschied zwischen Phasen- und Gruppenlaufzeit==
 
<br>
 
<br>
Eine weitere wichtige Systembeschreibungsgröße ist die  Gruppenlaufzeit, die nicht mit der Phasenlaufzeit verwechselt werden darf.  
+
Another important system description quantity is the group delay time which must not be confused with the phase delay time.  
  
 
{{BlaueBox|TEXT=   
 
{{BlaueBox|TEXT=   
 
$\text{Definition:}$&nbsp;
 
$\text{Definition:}$&nbsp;
Die&nbsp; '''Gruppenlaufzeit'''&nbsp; ist wie folgt definiert:
+
The&nbsp; '''group delay time'''&nbsp; is defined as follows:
 
:$$\tau_{\rm G}(\omega_0) =  \left[ \frac{ {\rm d}b(\omega)}{ {\rm d}\omega}\right ]_{\omega = \omega_0}.$$
 
:$$\tau_{\rm G}(\omega_0) =  \left[ \frac{ {\rm d}b(\omega)}{ {\rm d}\omega}\right ]_{\omega = \omega_0}.$$
*Diese Größe wird vorwiegend zur Beschreibung von Schmalbandsystemen herangezogen.  
+
*This quantity is mainly used to describe narrow-band systems.  
*Sie gibt die Verzögerung an, welche die Hüllkurve eines Bandpass-Systems erfährt. }}
+
*It indicates the delay experienced by the envelope of a band-pass system. }}
  
  
[[File: EN_LZI_T_2_3_S5.png| right|frame|Zur Definition der Gruppenlaufzeit|class=fit]]
+
[[File: EN_LZI_T_2_3_S5.png| right|frame|On the definition of the group delay time|class=fit]]
 
{{GraueBox|TEXT=   
 
{{GraueBox|TEXT=   
$\text{Beispiel 3:}$&nbsp;
+
$\text{Example 3:}$&nbsp;
Die Grafik zeigt die beispielhafte Phasenfunktion:  
+
The graph shows the examplary phase function:  
 
:$$b(ω) = \arctan (ω/ω_0).$$
 
:$$b(ω) = \arctan (ω/ω_0).$$
*Diese steigt monoton von Null&nbsp; $($bei &nbsp;$ω = 0)$&nbsp; bis &nbsp;$π/2$&nbsp; $($für &nbsp;$ω → ∞)$.  
+
*This increases monotonically from zero&nbsp; $($at &nbsp;$ω = 0)$&nbsp; to &nbsp;$π/2$&nbsp; $($for &nbsp;$ω → ∞)$.  
*Der Funktionswert bei &nbsp;$ω = ω_0$&nbsp; beträgt &nbsp;$π/4$.
+
*The function value at&nbsp;$ω = ω_0$&nbsp; is&nbsp;$π/4$.
  
  
Setzen wir &nbsp;$ω_0 = 2π · 1 \ \rm kHz$, so erhalten wir für die ''Phasenlaufzeit:''
+
If we set&nbsp;$ω_0 = 2π · 1 \ \rm kHz$, we obtain for the ''phase delay time:''
 
:$$\tau_{\rm P}(\omega_0) = \frac{b(\omega_0)}{\omega_0}= \frac{\pi / 4}{2 \pi \cdot{1\, \rm kHz} } =  {125\, \rm &micro; s}.$$
 
:$$\tau_{\rm P}(\omega_0) = \frac{b(\omega_0)}{\omega_0}= \frac{\pi / 4}{2 \pi \cdot{1\, \rm kHz} } =  {125\, \rm &micro; s}.$$
Diese Größe entspricht der Steigung der grün eingezeichneten Geraden in obiger Grafik.  
+
This quantity corresponds to the slope of the straight line drawn in green in the above graph.  
  
Dagegen kennzeichnet die geringere Steigung der rot dargestellten Tangente die ''Gruppenlaufzeit:''
+
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 =
 
  \left[ \frac{1}{\omega_0} \cdot \frac{1}{1 + \left(\omega / \omega_0\right]^2} \right ]_{\omega =
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==Phasenverzerrungen==
 
==Phasenverzerrungen==
 
<br>
 
<br>
Zur Verdeutlichung dieses Sachverhaltes betrachten wir als Eingangssignal wieder die Summe zweier harmonischer Schwingungen:
+
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).$$
  
Ist bei diesem Eingangssignal das Ausgangssignal in der 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)$$
darstellbar und gilt gleichzeitig &nbsp;$τ_1 ≠ τ_2$, so liegen&nbsp; '''ausschließlich Phasenverzerrungen'''&nbsp; vor.  
+
and at the same time&nbsp;$τ_1 ≠ τ_2$ is valid, then&nbsp; '''exclusively phase distortions'''&nbsp; are existent.  
  
Die beiden Phasenlaufzeiten &nbsp;$τ_1 ≠ τ_2$&nbsp; können aus dem Phasenverlauf (in Radian) ermittelt werden:
+
The two phase delay times&nbsp;$τ_1 ≠ τ_2$&nbsp; 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\pi
 
  f_2}.$$
 
  f_2}.$$
  
[[File: P_ID905__LZI_T_2_3_S6_neu.png|right|frame|Auswirkungen von Phasenverzerrungen|class=fit]]
+
[[File: P_ID905__LZI_T_2_3_S6_neu.png|right|frame|Effects of phase distortions|class=fit]]
 
{{GraueBox|TEXT=   
 
{{GraueBox|TEXT=   
$\text{Beispiel 4:}$&nbsp;
+
$\text{Example 4:}$&nbsp;
Die Grafik zeigt als blauen Kurvenverlauf das mit der Periodendauer &nbsp;$T_0$&nbsp; periodische Signal
+
The graph shows the periodic signal
 
:$$x(t) =  {1\, \rm V} \cdot \cos(2 \pi \cdot {1\, \rm kHz}\cdot  t) + {1\, \rm V} \cdot \sin(2 \pi \cdot {2\, \rm kHz}\cdot  t)$$
 
:$$x(t) =  {1\, \rm V} \cdot \cos(2 \pi \cdot {1\, \rm kHz}\cdot  t) + {1\, \rm V} \cdot \sin(2 \pi \cdot {2\, \rm kHz}\cdot  t)$$
sowie das mit den Laufzeiten &nbsp;$τ_1 = 0.7 \ \rm ms$&nbsp; und &nbsp;$τ_2 = 0.3 \ \rm ms$&nbsp; phasenverzerrte Signal &nbsp;$y(t)$&nbsp; &rArr; &nbsp; rote Kurve.  
+
with period&nbsp;$T_0$&nbsp; as a blue curve as well as the signal&nbsp;$y(t)$&nbsp; which is phase-distorted with the transit times&nbsp;$τ_1 = 0.7 \ \rm ms$&nbsp; and &nbsp;$τ_2 = 0.3 \ \rm ms$&nbsp; &rArr; &nbsp; red curve.  
*Man erkennt deutlich die Auswirkungen der Phasenverzerrungen.  
+
*The effects of the phase distortions can be clearly seen.  
  
*Mit &nbsp;$τ_1 = τ_2 = τ$&nbsp; ergäbe sich das verzerrungsfreie Signal
+
*With&nbsp;$τ_1 = τ_2 = τ$&nbsp; the following distortion-free signal would arise as a result:
 
:$$y(t) = x(t - τ).$$}}
 
:$$y(t) = x(t - τ).$$}}
  
  
Wir weisen nochmals  auf das interaktive Applet&nbsp; [[Applets:Lineare_Verzerrungen_periodischer_Signale|Lineare Verzerrungen periodischer Signale]] &nbsp;hin.
+
We refer again to the interactive applet&nbsp; [[Applets:Lineare_Verzerrungen_periodischer_Signale|Linear distortions of periodic signals]]. &nbsp;
  
  
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==Entzerrungsverfahren==
 
==Entzerrungsverfahren==
 
<br>
 
<br>
[[File:P_ID906__LZI_T_2_3_S7_neu.png|right|frame| Entzerrung von Signalen|class=fit]]
+
[[File:P_ID906__LZI_T_2_3_S7_neu.png|right|frame|Equalisation of signals|class=fit]]
Dieses für die Nachrichtentechnik sehr wichtige Verfahren soll hier nur kurz angerissen werden. Nähere Informationen hierzu finden Sie in den Büchern&nbsp; [[Modulation_Methods|Modulationsverfahren]] &nbsp;und&nbsp; [[Digital_Signal_Transmission|Digitalsignalübertragung]].  
+
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&nbsp; [[Modulation_Methods|Modulation Methods]] &nbsp;and&nbsp; [[Digital_Signal_Transmission|Digital Signal Transmission]].  
  
Wir gehen für diese Kurzbeschreibung von der skizzierten Konstellation aus:
+
For this brief description, we assume the outlined constellation:
*$S_{\rm V}$&nbsp; bezeichnet ein verzerrendes System,  
+
*$S_{\rm V}$&nbsp; denotes a distorting system,  
*während &nbsp;$S_{\rm E}$&nbsp; der Entzerrung dient.  
+
*while&nbsp;$S_{\rm E}$&nbsp; serves for equalisation.  
  
  
Zu dieser Konstellation ist anzumerken:  
+
Regarding this constellation, the following should be noted:  
*Ist die Verzerrung nichtlinear, so muss auch die Entzerrung nichtlinear erfolgen.  
+
*If the distortion is nonlinear, the equalisation must also be nonlinear.  
*Aber auch bei linearen Verzerrungen werden nichtlineare Entzerrungsverfahren eingesetzt, zum Beispiel ''Decision Feedback Equalization'' bei Digitalsystemen. Der Vorteil gegenüber linearer Entzerrung ist, dass es nicht zu einer Erhöhung der Rauschleistung kommt.  
+
*But even with linear distortion, nonlinear equalisation methods are used, for example ''Decision Feedback Equalisation'' in digital systems. The advantage over linear equalisation is that there is no increase in noise power.  
*Ist &nbsp;$S_{\rm V}$&nbsp; ein lineares System mit Frequenzgang &nbsp;$H_{\rm V}(f)$, so können mit dem ''inversen Frequenzgang'' &nbsp;$H_{\rm E}(f) = 1/H_{\rm V}(f)$&nbsp; die Verzerrungen vollständig eliminiert werden, und es gilt &nbsp;$z(t) = x(t)$.  
+
*If&nbsp;$S_{\rm V}$&nbsp; is a linear system with frequency response&nbsp;$H_{\rm V}(f)$, then the distortions can be completely eliminated with the ''inverse frequency response'' &nbsp;$H_{\rm E}(f) = 1/H_{\rm V}(f)$&nbsp; and &nbsp;$z(t) = x(t)$ holds.  
*Voraussetzung hierfür ist allerdings, dass der Frequenzgang &nbsp;$H_{\rm V}(f)$&nbsp; im interessierenden Spektralbereich ''keine Nullstellen''&nbsp; besitzt, da sonst bei &nbsp;$H_{\rm E}(f)$&nbsp; Unendlichkeitsstellen erforderlich wären.  
+
*However, a prerequisite for this is that the frequency response&nbsp;$H_{\rm V}(f)$&nbsp; has ''no zeros''&nbsp; in the spectral range of interest, otherwise infinity points would be required for&nbsp;$H_{\rm E}(f)$&nbsp;.  
*Bei ''Analogsystemen''&nbsp; bedeutet eine vollständige Entzerrung, dass sich &nbsp;$z(t)$&nbsp; von &nbsp;$x(t)$&nbsp; nur durch die unvermeidbaren Rauschanteile unterscheidet, und eventuell durch eine Laufzeit.  
+
*For ''analogue systems'',&nbsp; complete equalisation means that&nbsp;$z(t)$&nbsp; differs from&nbsp;$x(t)$&nbsp; only by the unavoidable noise components and possibly by a transit time.  
*Bei ''Digitalsystemen''&nbsp; ist das Kriterium für eine vollständige Entzerrung weniger streng. Es muss dann nur sichergestellt werden, dass die Signale &nbsp;$x(t)$&nbsp; und &nbsp;$z(t)$&nbsp; zu den Detektionszeitpunkten übereinstimmen. Man spricht in diesem Zusammenhang von&nbsp; [[Digital_Signal_Transmission/Eigenschaften_von_Nyquistsystemen|Nyquistsystemen]].
+
*For ''digital systems'',&nbsp; the criterion for complete equalisation is less strict. It must then only be ensured that the signals&nbsp;$x(t)$&nbsp; and&nbsp;$z(t)$&nbsp; coincide at the detection times. In this context, one deals with&nbsp; [[Digital_Signal_Transmission/Eigenschaften_von_Nyquistsystemen|Nyquist Systems]].
  
==Aufgaben zum Kapitel==
+
==Exercises for the Chapter==
 
<br>
 
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[[Aufgaben:2.5_Verzerrung_und_Entzerrung|Aufgabe 2.5: Verzerrung und Entzerrung]]
 
[[Aufgaben:2.5_Verzerrung_und_Entzerrung|Aufgabe 2.5: Verzerrung und Entzerrung]]

Revision as of 12:34, 16 September 2021

Zusammenstellung wichtiger Beschreibungsgrößen


Linear system description

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:

$$H(f) = |H(f)| \cdot {\rm e}^{-{\rm j} \hspace{0.05cm} \cdot \hspace{0.05cm} b(f)} = {\rm e}^{-a(f)}\cdot {\rm e}^{-{\rm j} \hspace{0.05cm} \cdot \hspace{0.05cm} b(f)}.$$

This yields the following descriptive quantities:

  • The magnitude  $|H(f)|$  is referred to as  amplitude response  and in logarithmic form as attenuation curve :
$$a(f) = - \ln |H(f)|\hspace{0.2cm}{\rm in \hspace{0.1cm}Neper \hspace{0.1cm}(Np) } = - 20 \cdot \lg |H(f)|\hspace{0.2cm}{\rm in \hspace{0.1cm}decibel \hspace{0.1cm}(dB) }.$$
  • 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:
$$b(f) = - {\rm arc} \hspace{0.1cm}H(f) \hspace{0.2cm}{\rm in \hspace{0.1cm}radian \hspace{0.1cm}(rad)}.$$

Voraussetzungen für verzerrungsfreie Systeme


According to the explanations in the chapter Classification of the Distortions  there is a distortion-free system at hand if and only if all frequency components are uniformly damped and delayed:

$$y(t) = \alpha \cdot x(t - \tau).$$

According to the laws of system theory, the following must thus hold for the frequency response

$$H(f) = \alpha \cdot {\rm e}^{-{\rm j}\hspace{0.04cm}2 \pi f \tau}$$

or expressed with functions  $a(f)$  and  $b(f)$:

  • The attenuation curve must be constant for all frequencies contained in the input signal:
$$a(f) = - \ln |H(f)| = - \ln \ \alpha = {\rm const.}$$
  • The phase response must either be zero in the region of interest (system with no transit time) or increase linearly with frequency  $(τ$  indicates the transit time of the system):
$$b(f) = 2 \pi f \tau = {\rm const.} \cdot f.$$

$\text{Definitionen:}$  For a distortion-free system, both requirements must be satisfied simultaneously. Violation of even one of these two conditions results in linear distortions which are distinguished according to their cause.

  • It comes to  attenuation distortions if in the frequency range of interest the attenuation curve is not constant:
$$a(f) \ne {\rm const.}$$
  • In contrast to this, there are  phase distortions  if the phase function is not linear with respect to $f$ :
$$b(f) \ne {\rm const.} \cdot f.$$


It should be noted that in all realisable systems – in particular, in the "minimum-phase systems" described in  Chapter 3  – both forms of distortion usually occur simultaneously.

$\text{Definition:}$  In the time domain, the condition for a  distortion-free system is:

$$h(t) = \alpha \cdot \delta(t - \tau),\hspace{0.4cm}\alpha \ne 0.$$

Moreover, if  $α = 1$  and  $τ = 0$ hold, then there is an  ideal transmission system . In contrast, there are linear distortions whenever

  • $h(t)$  is a continuous-time function or
  • $h(t)$  is composed of more than one Dirac function.


$\text{Example 1:}$  The following sketch shows the attenuation curve $a(f)$  and the phase response $b(f)$  of a distortion-free system.

Requirement for a non-distorting channel
  • In a range from  $f_{\rm U}$  to  $f_{\rm O}$  around the carrier frequency  $f_{\rm T}$, in which the signal $x(t)$  has components,  $a(f)$  is constant.
  • From the given constant attenuation value  $6 \ \rm dB$  it follows for the amplitude response:  $\vert H(f)\vert = 0.5$.
  • The output spectrum $Y(f)$  is thus half as large in magnitude as the spectral components $X(f)$  of the input signal.
  • The phase response $b(f)$  increases linearly with frequency between $f_{\rm U}$ and $f_{\rm O}$.
  • This results in all frequency components being delayed by the same phase delay time $τ$  where  $τ$  is fixed by the slope of  $b(f)$ .
  • With $b(f) = 0$  a transit time-free system would arise as a result   ⇒   $τ = 0$.


Furthermore, the following generally valid properties can be identified from the graph:

  • The attenuation curve $a(f) = a(\hspace{-0.01cm}-\hspace{-0.08cm} f)$  is an even function in  $f$.
  • The phase curve $b(f) = \hspace{-0.01cm}–\hspace{-0.01cm} b(\hspace{-0.01cm}-\hspace{-0.01cm}f)$  is an odd function in $f$.


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.

Dämpfungsverzerrungen


In the following, we consider the sum of two harmonic oscillations as input signal:

$$x(t) = A_1 \cdot \cos(2 \pi f_1 \cdot t - \varphi_1) + A_2 \cdot \cos(2 \pi f_2 \cdot t - \varphi_2).$$

If the output signal can be represented in the form

$$y(t) = \alpha_1 \cdot A_1 \cdot \cos(2 \pi f_1 \cdot t - \varphi_1) + \alpha_2 \cdot A_2 \cdot \cos(2 \pi f_2 \cdot t - \varphi_2).$$

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.

The attenuation constants $α_1$  and  $α_2$  can be determined from the amplitude response $|H(f)|$ :

$$\alpha_1 = |H(f_1)|,\hspace{0.4cm}\alpha_2 = |H(f_2)|.$$

If the attenuation curve $a(f)$  is given in Neper, then likewise the following holds  $(1 \ \rm dB$ corresponds to $0.1151 \ \rm Np)$:

$$ \alpha_1 = {\rm e}^{-{\rm a}(f_1)},\hspace{0.4cm}\alpha_2 = {\rm e}^{-{\rm a}(f_2)}.$$

Please note:   In some character fonts, "$a$" and "$α$" (alpha) are difficult to distinguish.

Effects of attenuation distortions

$\text{Example 2:}$  The graph shows the input signal (blue curve)

$$x(t) = {1\, \rm V} \cdot \cos(2 \pi \cdot {1\, \rm kHz}\cdot t) + {1\, \rm V} \cdot \sin(2 \pi \cdot {2\, \rm kHz}\cdot t)$$

which is periodic with $T_0 = 1\ \rm ms$  and the signal $y(t)$ which is attenuation-distorted with  $α_1 = 0.2$,  $α_2 = 0.5$ .

Significant effects of these attenuation distortions can be perceived:

  • $y(t)$  bears little resemblance to $x(t)$.
  • In contrast, considering  $α_1 = α_2 = α$  the distortion-free signal $y(t) = α · x(t)$  would be obtained from which the original signal $x(t)$  could be reconstructed by amplifying it by $1/α$ .


We refer here explicitly to the interactive applet  Linear distortions of periodic signals.


Phasenlaufzeit


On the definition of the phase delay time

We consider a system with $|H(f)| = 1$ such that the following holds for the frequency response:

$$H(f) = {\rm e}^{-{\rm j} \hspace{0.05cm} \cdot \hspace{0.05cm} b(f)}.$$
  • The left graph shows an exemplary phase response $b(f)$. Such a phase response is always an odd function with respect to the frequency $f$:   $b(\hspace{-0.01cm}-\hspace{-0.08cm}f) = \hspace{0.08cm}-b(f)$.
  • On the right, the function $b(ω)$  is sketched which is dilated by a factor of $2π$ with respect to $b(f)$  in the abscissa.


If the harmonic oscillation at the input is

$$x(t) = C \cdot \cos(2 \pi f_0 t - \varphi) \circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\, X(f ) = {C}/{2}\cdot {\rm e}^{{\rm j}\hspace{0.05cm} \cdot \hspace{0.05cm} \varphi} \cdot \delta(f + f_0) \hspace{0.01cm} + \hspace{0.01cm}{C}/{2}\cdot {\rm e}^{-{\rm j}\hspace{0.05cm} \cdot \hspace{0.05cm} \varphi} \cdot \delta(f - f_0),$$

then the following arises as a result for the spectral function at the output:

$$Y(f ) = {C}/{2}\cdot {\rm e}^{{\rm j}\hspace{0.05cm} \cdot \hspace{0.05cm} \varphi} \cdot {\rm e}^{\hspace{0.05cm}{\rm j}\hspace{0.05cm} \cdot \hspace{0.05cm} b(f_0)} \cdot \delta(f + f_0) \hspace{0.05cm} + \hspace{0.05cm}{C}/{2}\cdot {\rm e}^{-{\rm j}\hspace{0.05cm} \cdot \hspace{0.05cm} \varphi}\cdot {\rm e}^{-{\rm j}\hspace{0.05cm} \cdot \hspace{0.05cm} b(f_0)} \cdot \delta(f - f_0).$$

Thus, the output signal is:

$$y(t) = C \cdot \cos(2 \pi f_0 t - b(f_0) - \varphi).$$

This signal can also be represented in the following form:

$$y(t) = C \cdot \cos(2 \pi f_0 ( t - \tau_{\rm P}(f_0)) - \varphi).$$

$\text{Definition:}$  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:

$$\tau_{\rm P}(f_0) = \frac{b(f_0)}{2\pi f_0} \hspace{0.4cm}{\rm or} \hspace{0.4cm} \tau_{\rm P}(\omega_0) = \frac{b(\omega_0)}{\omega_0}.$$


The following should be noted regarding this definition:

  • In the $b(ω)$–representation, the phase delay time $τ_{\rm P}$  can also be determined graphically as the slope of the straight line drawn in green in the above graph.
  • In general, an oscillation with a different frequency will also result in a different phase delay time. This is the physical background for phase distortions.
  • If for a system $b(ω) = τ_{\rm P} · ω$   or   $b(f) = 2π · τ_{\rm P} · f$ is true, then all frequencies have the same phase delay time $τ_{\rm P}$. Such a system does not cause phase distortions.


We refer here again to the interactive applet Linear distortions of periodic signals.

Unterschied zwischen Phasen- und Gruppenlaufzeit


Another important system description quantity is the group delay time which must not be confused with the phase delay time.

$\text{Definition:}$  The  group delay time  is defined as follows:

$$\tau_{\rm G}(\omega_0) = \left[ \frac{ {\rm d}b(\omega)}{ {\rm d}\omega}\right ]_{\omega = \omega_0}.$$
  • This quantity is mainly used to describe narrow-band systems.
  • It indicates the delay experienced by the envelope of a band-pass system.


On the definition of the group delay time

$\text{Example 3:}$  The graph shows the examplary phase function:

$$b(ω) = \arctan (ω/ω_0).$$
  • This increases monotonically from zero  $($at  $ω = 0)$  to  $π/2$  $($for  $ω → ∞)$.
  • The function value at $ω = ω_0$  is $π/4$.


If we set $ω_0 = 2π · 1 \ \rm kHz$, we obtain for the phase delay time:

$$\tau_{\rm P}(\omega_0) = \frac{b(\omega_0)}{\omega_0}= \frac{\pi / 4}{2 \pi \cdot{1\, \rm kHz} } = {125\, \rm µ s}.$$

This quantity corresponds to the slope of the straight line drawn in green in the above graph.

In contrast, the lesser slope of the tangent line shown in red denotes the group delay time:

$$\tau_{\rm G}(\omega_0) = \left[ \frac{ {\rm d}b(\omega)}{ {\rm d}\omega}\right ]_{\omega = \omega_0} = \left[ \frac{1}{\omega_0} \cdot \frac{1}{1 + \left(\omega / \omega_0\right]^2} \right ]_{\omega = \omega_0} = \frac{1}{2\omega_0}= \frac{1}{4 \pi \cdot{1\, \rm kHz} } \approx {80\, \rm µ s}.$$

Phasenverzerrungen


To illustrate this point, we again consider the sum of two harmonic oscillations as input signal:

$$x(t) = A_1 \cdot \cos(2 \pi f_1 \cdot t - \varphi_1) + A_2 \cdot \cos(2 \pi f_2 \cdot t - \varphi_2).$$

If the output signal for this input signal can be represented in the form

$$y(t) = A_1 \cdot \cos(2 \pi f_1 \cdot (t - \tau_1) - \varphi_1) + A_2 \cdot \cos(2 \pi f_2 \cdot (t - \tau_2) - \varphi_2)$$

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):

$$\tau_1 = \frac{b(f_1)}{2\pi f_1} , \hspace{0.4cm}\tau_2 = \frac{b(f_2)}{2\pi f_2}.$$
Effects of phase distortions

$\text{Example 4:}$  The graph shows the periodic signal

$$x(t) = {1\, \rm V} \cdot \cos(2 \pi \cdot {1\, \rm kHz}\cdot t) + {1\, \rm V} \cdot \sin(2 \pi \cdot {2\, \rm kHz}\cdot t)$$

with period $T_0$  as a blue curve as well as the signal $y(t)$  which is phase-distorted with the transit times $τ_1 = 0.7 \ \rm ms$  and  $τ_2 = 0.3 \ \rm ms$  ⇒   red curve.

  • The effects of the phase distortions can be clearly seen.
  • With $τ_1 = τ_2 = τ$  the following distortion-free signal would arise as a result:
$$y(t) = x(t - τ).$$


We refer again to the interactive applet  Linear distortions of periodic signals.  


Entzerrungsverfahren


Equalisation of signals

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  and  Digital Signal Transmission.

For this brief description, we assume the outlined constellation:

  • $S_{\rm V}$  denotes a distorting system,
  • while $S_{\rm E}$  serves for equalisation.


Regarding this constellation, the following should be noted:

  • If the distortion is nonlinear, the equalisation must also be nonlinear.
  • But even with linear distortion, nonlinear equalisation methods are used, for example Decision Feedback Equalisation in digital systems. The advantage over linear equalisation 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.
  • However, a prerequisite for this is that the frequency response $H_{\rm V}(f)$  has no zeros  in the spectral range of interest, otherwise infinity points would be required for $H_{\rm E}(f)$ .
  • For analogue systems,  complete equalisation means that $z(t)$  differs from $x(t)$  only by the unavoidable noise components and possibly by a transit time.
  • For digital systems,  the criterion for complete equalisation is less strict. It must then only be ensured that the signals $x(t)$  and $z(t)$  coincide at the detection times. In this context, one deals with  Nyquist Systems.

Exercises for the Chapter


Aufgabe 2.5: Verzerrung und Entzerrung

Aufgabe 2.5Z: Nyquistentzerrung

Aufgabe 2.6: Zweiwegekanal

Aufgabe 2.6Z: Synchrondemodulator

Aufgabe 2.7: Nochmals Zweiwegekanal