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

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{{Header
 
{{Header
|Untermenü=Signalverzerrungen und Entzerrung
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|Untermenü=Signal Distortion and Equalization
|Vorherige Seite=Nichtlineare Verzerrungen
+
|Vorherige Seite=Nonlinear_Distortions
|Nächste Seite=Folgerungen aus dem Zuordnungssatz
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|Nächste Seite=Conclusions_from_the_Allocation_Theorem
 
}}
 
}}
==Zusammenstellung wichtiger Beschreibungsgrößen==
+
==Compilation of important descriptive quantities==
 
<br>
 
<br>
[[File:P_ID899__LZI_T_2_3_S1_neu.png |frame|Lineare Systembeschreibung|class=fit]]
+
[[File:P_ID899__LZI_T_2_3_S1_neu.png |frame|Linear system description|class=fit]]
Nun werden nichtlineare Verzerrungen ausgeschlossen, so dass das System durch den Frequenzgang&nbsp; $H(f)$&nbsp; vollständig beschrieben wird.  
+
Now nonlinear distortions are excluded so that the system is fully described by the frequency response&nbsp; $H(f)$&nbsp;.  
  
Der im Allgemeinen komplexe Frequenzgang kann auch wie folgt dargestellt werden:  
+
The generally complex frequency response can also be formulated as follows:  
 
:$$H(f) = |H(f)| \cdot {\rm e}^{-{\rm j} \hspace{0.05cm} \cdot
 
:$$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} 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)}.$$
  
Daraus ergeben sich folgende Beschreibungsgrößen:  
+
This yields the following descriptive quantities:  
*Der Betrag&nbsp; $|H(f)|$&nbsp; wird als&nbsp; '''Amplitudengang'''&nbsp; und in logarithmierter Form als ''Dämpfungsverlauf''&nbsp; bezeichnet:  
+
*The magnitude&nbsp; $|H(f)|$&nbsp; is referred to as&nbsp; '''amplitude response'''&nbsp; and in logarithmic form as&nbsp; "attenuation curve":  
 
:$$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) }.$$
+
\hspace{0.1cm}decibel \hspace{0.1cm}(dB) }.$$
*Der&nbsp; '''Phasengang'''&nbsp; $b(f)$&nbsp; gibt den negativen, von&nbsp; $f$&nbsp; abhängigen Winkel von&nbsp; $H(f)$&nbsp; in der komplexen Ebene an, bezogen auf die reelle Achse:  
+
*The&nbsp; '''phase response'''&nbsp; $b(f)$&nbsp; specifies the negative angle of&nbsp; $H(f)$&nbsp; dependent on&nbsp; $f$&nbsp; 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==
+
==Requirements for distortion-free systems==
 
<br>
 
<br>
Nach den Ausführungen im Kapitel &nbsp;[[Linear_and_Time_Invariant_Systems/Klassifizierung_der_Verzerrungen|Klassifizierung der Verzerrungen]]&nbsp; liegt genau dann ein verzerrungsfreies System vor, wenn alle Frequenzanteile gleichmäßig gedämpft und verzögert werden:  
+
According to the explanations in the chapter &nbsp;[[Linear_and_Time_Invariant_Systems/Classification_of_the_Distortions|Classification of the Distortions]] &nbsp; there is a distortion-free system at hand if and only if all frequency components are uniformly attenuated and delayed:  
 
:$$y(t) = \alpha \cdot x(t - \tau).$$
 
:$$y(t) = \alpha \cdot x(t - \tau).$$
Nach den Gesetzmäßigkeiten der Systemtheorie muss deshalb für den Frequenzgang
+
According to the systems theory laws,&nbsp; the following must thus hold for the frequency response:
:$$H(f) = \alpha \cdot {\rm e}^{-{\rm j}\hspace{0.04cm}2 \pi f \tau}$$
+
:$$H(f) = \alpha \cdot {\rm e}^{-{\rm j}\hspace{0.04cm}2 \pi f \tau},$$
gelten, oder ausgedrückt mit den Funktionen &nbsp;$a(f)$&nbsp; und &nbsp;$b(f)$:
+
or expressed with the functions &nbsp;$a(f)$&nbsp; and &nbsp;$b(f)$:
*Der Dämpfungsverlauf muss für alle im Eingangssignal enthaltenen Frequenzen konstant sein:  
+
*The attenuation curve must be constant for all frequencies contained in the input signal:  
 
:$$a(f) = - \ln |H(f)| = - \ln \ \alpha = {\rm const.}$$
 
:$$a(f) = - \ln |H(f)| = - \ln \ \alpha = {\rm const.}$$
*Der Phasenverlauf muss im interessierenden Bereich entweder Null sein (System ohne Laufzeit) oder linear mit der Frequenz ansteigen&nbsp; $(τ$&nbsp; gibt dabei die Laufzeit des Systems an):
+
*The phase response must either be zero in the region of interest&nbsp; (system with no transit time)&nbsp; or increase linearly with frequency&nbsp; $(τ$&nbsp; indicates the transit time):
 
:$$b(f) = 2 \pi f \tau = {\rm const.} \cdot f.$$
 
:$$b(f) = 2 \pi f \tau = {\rm const.} \cdot f.$$
  
 
{{BlaueBox|TEXT=   
 
{{BlaueBox|TEXT=   
$\text{Definitionen:}$&nbsp;
+
$\text{Definitions:}$&nbsp;
Bei einem verzerrungsfreien System müssen beide Forderungen gleichzeitig erfüllt sein. Bei Verletzung auch nur einer dieser beiden Bedingungen kommt es zu linearen Verzerrungen, die entsprechend ihrer Ursache unterschieden werden.  
+
For a distortion-free system, both requirements must be satisfied simultaneously.  
*Es kommt zu&nbsp; '''Dämpfungsverzerrungen''', wenn im interessierenden Frequenzbereich der Dämpfungsverlauf nicht konstant ist:
+
 
 +
Violation of even one of these two conditions results in&nbsp; '''linear distortions'''&nbsp; which are distinguished according to their cause:
 +
*It comes to&nbsp; '''attenuation distortions'''&nbsp; if in the frequency range of interest the attenuation curve is not constant:
 
:$$a(f) \ne {\rm const.}$$
 
:$$a(f) \ne {\rm const.}$$
*Dagegen liegen&nbsp; '''Phasenverzerrungen'''&nbsp; dann vor, wenn die Phasenfunktion nicht linear bezüglich $f$&nbsp; ist:  
+
*In contrast to this, there are&nbsp; '''phase distortions'''&nbsp; if the phase function is not linear with respect to $f$&nbsp;:  
 
:$$b(f) \ne {\rm const.} \cdot f.$$}}
 
:$$b(f) \ne {\rm const.} \cdot f.$$}}
  
  
Anzumerken ist, dass bei allen realisierbaren Systemen – insbesondere den im&nbsp; [[Linear_and_Time_Invariant_Systems/Folgerungen_aus_dem_Zuordnungssatz#D.C3.A4mpfung_und_Phase_von_Minimum.E2.80.93Phasen.E2.80.93Systemen|Kapitel 3]]&nbsp; beschriebenen „minimalphasigen Systemen” meist beide Verzerrungsformen gleichzeitig auftreten.  
+
It should be noted that in all&nbsp; "realizable systems" – in particular, in the&nbsp; "minimum-phase systems"&nbsp; described in&nbsp; [[Linear_and_Time_Invariant_Systems/Conclusions_from_the_Allocation_Theorem#Dämpfung_und_Phase_von_Minimum–Phasen–Systemen|Chapter 3]]&nbsp; – both forms of distortions usually occur simultaneously.  
  
 
{{BlaueBox|TEXT=   
 
{{BlaueBox|TEXT=   
$\text{Definition:}$&nbsp; Im Zeitbereich lautet die Bedingung für ein&nbsp; '''verzerrungsfreies System''':
+
$\text{Definition:}$&nbsp; In the time domain, the condition for a&nbsp; '''distortion-free system''' is:
 
:$$h(t) = \alpha \cdot \delta(t - \tau),\hspace{0.4cm}\alpha \ne 0.$$
 
:$$h(t) = \alpha \cdot \delta(t - \tau),\hspace{0.4cm}\alpha \ne 0.$$
  
Ist zudem&nbsp; $α = 1$&nbsp; und&nbsp; $τ = 0$, so liegt ein&nbsp; '''ideales Übertragungssystem'''&nbsp; vor. Dagegen gibt es immer dann lineare Verzerrungen, wenn
+
Moreover,&nbsp; if&nbsp; $α = 1$&nbsp; and&nbsp; $τ = 0$&nbsp; hold,&nbsp; then there is an&nbsp; '''ideal transmission system'''.&nbsp; In contrast, there are linear distortions whenever
*$h(t)$&nbsp; eine zeitkontinuierliche Funktion ist, oder
+
*The impulse response&nbsp; $h(t)$&nbsp; is a continuous-time function, or
*$h(t)$&nbsp; sich aus mehr als einer Diracfunktion zusammensetzt.}}
+
*the time-dicrete impulse response&nbsp;$h(t)$&nbsp; is composed of more than one Dirac functions.}}
  
  
 
{{GraueBox|TEXT=   
 
{{GraueBox|TEXT=   
$\text{Beispiel 1:}$&nbsp;
+
$\text{Example 1:}$&nbsp;
Die folgende Skizze zeigt den Dämpfungsverlauf &nbsp;$a(f)$&nbsp; und den Phasenverlauf &nbsp;$b(f)$&nbsp; eines verzerrungsfreien Systems.  
+
The following sketch shows the attenuation curve&nbsp; $a(f)$&nbsp; and the phase response&nbsp; $b(f)$&nbsp; of a distortion-free system.  
[[File:P_ID900__LZI_T_2_3_S2_neu.png|right|frame| Voraussetzung für einen nichtverzerrenden Kanal|right|class=fit]]
+
[[File:P_ID900__LZI_T_2_3_S2_neu.png|right|frame|Requirements for a non-distorting channel: <br> &nbsp; $f_{\rm U}$:&nbsp;  lower&nbsp; $($German:&nbsp; "untere" &nbsp; &rArr; &nbsp; "U"$)$&nbsp; range limit, <br> &nbsp; $f_{\rm O}$:&nbsp;  upper&nbsp; $($German:&nbsp; "obere" &nbsp; &rArr; &nbsp; "O"$)$&nbsp; range limit, <br> &nbsp; $f_{\rm T}$:&nbsp;  carrier&nbsp; $($German:&nbsp; "Träger" &nbsp; &rArr; &nbsp; "T"$)$&nbsp; frequency|right|class=fit]]
 +
 
 +
*In a range from&nbsp; $f_{\rm U}$&nbsp; to&nbsp; $f_{\rm O}$&nbsp; around the carrier frequency&nbsp; $f_{\rm T}$,&nbsp; in which the signal&nbsp;$x(t)$&nbsp; has components, &nbsp;$a(f)$&nbsp; is constant.
 +
*From the given constant attenuation value&nbsp; $6 \ \rm dB$&nbsp; it follows for the amplitude response: &nbsp;$\vert H(f)\vert = 0.5$.
 +
*The output spectrum&nbsp; $Y(f)$&nbsp; is thus half as large in magnitude as the spectral components&nbsp;$X(f)$&nbsp; of the input signal.
 +
*The phase response&nbsp; $b(f)$&nbsp; 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 &nbsp;$τ$,&nbsp; where &nbsp;$τ$&nbsp; is fixed by the slope of&nbsp; $b(f)$&nbsp;.
 +
*With&nbsp; $b(f) = 0$&nbsp; a transit time-free system would arise as a result &nbsp; &rArr; &nbsp; $τ = 0$.
 +
 
 +
 
 +
Furthermore,&nbsp; the following generally valid properties can be identified from the graph:
 +
*The attenuation curve&nbsp; $a(f) = a(\hspace{-0.01cm}-\hspace{-0.08cm} f)$&nbsp; is an even function in&nbsp; $f$.
 +
*The phase curve&nbsp; $b(f) = \hspace{0.1cm}–\hspace{-0.01cm} b(\hspace{-0.01cm}-\hspace{-0.01cm}f)$&nbsp; is an odd function in&nbsp;$f$.
  
*In einem Bereich von&nbsp; $f_{\rm U}$&nbsp; bis&nbsp; $f_{\rm O}$&nbsp; um die Trägerfrequenz&nbsp; $f_{\rm T}$, in dem das Signal &nbsp;$x(t)$&nbsp; Anteile besitzt, ist &nbsp;$a(f)$&nbsp; konstant.
 
*Aus dem angegebenen konstanten Dämpfungswert&nbsp; $6 \ \rm dB$&nbsp; folgt für den Amplitudengang: &nbsp;$\vert H(f)\vert = 0.5$.
 
* Das Ausgangsspektrum &nbsp;$Y(f)$&nbsp; ist somit betragsmäßig halb so groß wie die Spektralanteile &nbsp;$X(f)$&nbsp; des Eingangssignals.
 
*Der Phasenverlauf &nbsp;$b(f)$&nbsp; steigt zwischen $f_{\rm U}$ und $f_{\rm O}$ linear mit der Frequenz an.
 
*Dies hat zur Folge, dass alle Frequenzanteile um die gleiche Phasenlaufzeit &nbsp;$τ$&nbsp; verzögert werden, wobei &nbsp;$τ$&nbsp; durch die Steigung von&nbsp; $b(f)$&nbsp; festliegt.
 
*Mit &nbsp;$b(f) = 0$&nbsp; würde sich ein laufzeitfreies System ergeben  &nbsp; &rArr; &nbsp; $τ = 0$.
 
<br clear=all>
 
Weiter erkennt man aus der Grafik folgende allgemeingültige Eigenschaften:
 
*Der Dämpfungsverlauf &nbsp;$a(f) = a(\hspace{-0.01cm}-\hspace{-0.08cm} f)$&nbsp; ist eine gerade Funktion in&nbsp; $f$.
 
*Der Phasenverlauf &nbsp;$b(f) = \hspace{-0.01cm}–\hspace{-0.01cm} b(\hspace{-0.01cm}-\hspace{-0.01cm}f)$&nbsp; ist eine ungerade Funktion in &nbsp;$f$.
 
  
 +
Outside the frequency band occupied by&nbsp;$x(t)$&nbsp; the&nbsp; "constant attenuation"&nbsp; and&nbsp; "linear phase"&nbsp; conditions do not need to be satisfied.
  
Außerhalb des durch &nbsp;$x(t)$&nbsp; belegten Frequenzbandes müssen die Bedingungen „konstante Dämpfung” und „lineare Phase” nicht eingehalten werden. Man erkennt aus dem gestrichelt eingezeichneten Verlauf von &nbsp;$a(f)$, dass hier sogar eine sehr viel höhere Dämpfung zweckmäßig ist, da dadurch die stets vorhandenen – in diesem Abschnitt aber nicht betrachteten Rauschanteile außerhalb der Nutzbandbreite besser unterdrückt werden. }}
+
It can be seen from the dashed curve of&nbsp;$a(f)$ 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==
+
==Attenuation distortions==
 
<br>
 
<br>
Wir betrachten im Folgenden als Eingangssignal die Summe zweier harmonischer Schwingungen:
+
In the following,&nbsp; 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).$$
  
Ist das Ausgangssignal in der 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),$$
darstellbar und gilt gleichzeitig &nbsp;$α_1 ≠ α_2$, so liegen&nbsp; '''ausschließlich Dämpfungsverzerrungen'''&nbsp; vor, da die Phasenwerte &nbsp;$\varphi_1$&nbsp; und &nbsp;$\varphi_2$&nbsp; durch das System nicht verändert werden.
+
:and at the same time&nbsp; $α_1 ≠ α_2$&nbsp; is valid,&nbsp; then&nbsp; '''exclusively attenuation distortions'''&nbsp; are existent since the phase values&nbsp; $\varphi_1$&nbsp; and&nbsp; $\varphi_2$&nbsp; are not changed by the system.
  
Die Dämpfungskonstanten &nbsp;$α_1$&nbsp; und &nbsp;$α_2$&nbsp; können aus dem Amplitudengang &nbsp;$|H(f)|$&nbsp; ermittelt werden:
+
*The attenuation constants&nbsp; $α_1$&nbsp; and&nbsp; $α_2$&nbsp; can be determined from the amplitude response&nbsp; $|H(f)|$&nbsp;:
 
:$$\alpha_1 = |H(f_1)|,\hspace{0.4cm}\alpha_2 = |H(f_2)|.$$
 
:$$\alpha_1 = |H(f_1)|,\hspace{0.4cm}\alpha_2 = |H(f_2)|.$$
  
Ist der Dämpfungsverlauf &nbsp;$a(f)$&nbsp; in Neper gegeben, so gilt gleichermaßen &nbsp;$(1 \ \rm dB$ entspricht $0.1151 \ \rm  Np)$:
+
*If the attenuation curve&nbsp; $a(f)$&nbsp; is given in Neper,&nbsp; then likewise the following holds &nbsp;$(1 \ \rm dB$ corresponds to $0.1151 \ \rm  Np)$:
 
:$$ \alpha_1 =  {\rm e}^{-{\rm a}(f_1)},\hspace{0.4cm}\alpha_2  = {\rm e}^{-{\rm a}(f_2)}.$$
 
:$$ \alpha_1 =  {\rm e}^{-{\rm a}(f_1)},\hspace{0.4cm}\alpha_2  = {\rm e}^{-{\rm a}(f_2)}.$$
  
''Hinweis:'' &nbsp; Bei manchen Zeichenfonts sind "$a$" und "$α$" (alpha) schwer zu unterscheiden.
+
Please note: &nbsp; In some character fonts,&nbsp; "$a$"&nbsp; and&nbsp; "$α$"&nbsp; (alpha)&nbsp; are difficult to distinguish.
  
[[File:P_ID901__LZI_T_2_3_S3_neu.png |frame| Auswirkungen von Dämpfungsverzerrungen | right|class=fit]]
+
[[File:P_ID901__LZI_T_2_3_S3_neu.png |frame| Effects of attenuation distortions | right|class=fit]]
 
{{GraueBox|TEXT=   
 
{{GraueBox|TEXT=   
$\text{Beispiel 2:}$&nbsp;
+
$\text{Example 2:}$&nbsp;
Die Grafik zeigt das mit &nbsp;$T_0 = 1\ \rm  ms$&nbsp; periodische Eingangssignal (blauer Verlauf)
+
The graph shows the input signal&nbsp; (blue curve)&nbsp; which is periodic with&nbsp; $T_0 = 1\ \rm  ms$&nbsp;
 
:$$x(t) =  {1\, \rm V} \cdot \cos(2 \pi \cdot {1\, \rm kHz}\cdot  t) + {1\, \rm V} \cdot \sin(2 \pi \cdot {2\, \rm kHz}\cdot  t)$$
 
:$$x(t) =  {1\, \rm V} \cdot \cos(2 \pi \cdot {1\, \rm kHz}\cdot  t) + {1\, \rm V} \cdot \sin(2 \pi \cdot {2\, \rm kHz}\cdot  t)$$
und das mit &nbsp;$α_1 = 0.2$, &nbsp;$α_2 = 0.5$&nbsp; dämpfungsverzerrte Signal &nbsp;$y(t)$.  
+
and the signal&nbsp; $y(t)$&nbsp; (red curve)&nbsp; which is attenuation-distorted with &nbsp;$α_1 = 0.2$, &nbsp;$α_2 = 0.5$.  
  
Man erkennt signifikante Auswirkungen dieser Dämpfungsverzerrungen:  
+
Significant effects of these attenuation distortions can be perceived:  
*$y(t)$&nbsp; hat nur noch wenig Ähnlichkeit mit &nbsp;$x(t)$.  
+
*$y(t)$&nbsp; bears little resemblance to&nbsp; $x(t)$.  
*Dagegen würde sich mit &nbsp;$α_1 = α_2 = α$&nbsp; das verzerrungsfreie Signal &nbsp;$y(t) = α · x(t)$&nbsp; ergeben, aus dem man durch Verstärkung um &nbsp;$1/α$&nbsp; das ursprüngliche Signal &nbsp;$x(t)$&nbsp; wieder rekonstruieren könnte.}}
+
*In contrast, considering &nbsp;$α_1 = α_2 = α$&nbsp; the distortion-free signal&nbsp; $y(t) = α · x(t)$&nbsp; would be obtained from which the original signal&nbsp; $x(t)$&nbsp; could be reconstructed by amplifying it by&nbsp;$1/α$.
  
  
Wir weisen hier ausdrücklich auf das interaktive Applet&nbsp; [[Applets:Lineare_Verzerrungen_periodischer_Signale|Lineare Verzerrungen periodischer Signale]] hin.
+
We refer here explicitly to the interactive applet&nbsp; [[Applets:Linear_Distortions_of_Periodic_Signals|Linear distortions of periodic signals]]. }}
  
  
  
==Phasenlaufzeit==
+
==Phase delay time==
 
<br>
 
<br>
[[File: EN_LZI_T_2_3_S4.png|frame|Zur Definition der Phasenlaufzeit|class=fit]]
+
We consider a system with&nbsp; $|H(f)| = 1$&nbsp; such that the following holds for the frequency response:
Wir betrachten ein System mit &nbsp;$|H(f)| = 1$, so dass für den Frequenzgang gilt:
 
 
:$$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; $b(f)$.&nbsp; 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&nbsp; $2π$&nbsp; with respect to&nbsp; $b(f)$&nbsp; in the abscissa.  
  
  
 
+
If the harmonic oscillation at the input is
Liegt am Eingang die harmonische Schwingung
+
:$$x(t) =  C \cdot \cos(2 \pi  f_0  t - \varphi)\hspace{0.15cm}
:$$x(t) =  C \cdot \cos(2 \pi  f_0  t - \varphi)
 
 
\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:
+
[[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:
 
:$$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
Line 134: Line 139:
 
\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:
 
:$$y(t) =  C \cdot \cos(2 \pi  f_0 (  t - \tau_{\rm P}(f_0)) - \varphi).$$
 
:$$y(t) =  C \cdot \cos(2 \pi  f_0 (  t - \tau_{\rm P}(f_0)) - \varphi).$$
 +
<br clear=all>
  
 
{{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.&nbsp; 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;$b(ω)$&nbsp; representation,&nbsp; 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,&nbsp; an oscillation with a different frequency will also result in a different phase delay time.&nbsp; 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; $b(f) = 2π · τ_{\rm P} · f$&nbsp; is true,&nbsp; then all frequencies have the same phase delay time&nbsp; $τ_{\rm P}$.&nbsp; 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:Linear_Distortions_of_Periodic_Signals|Linear distortions of periodic signals]].
  
==Unterschied zwischen Phasen- und Gruppenlaufzeit==
+
==Difference between phase and group delay time==
 
<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  $\tau_{\rm G}$|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&nbsp; '''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.  
+
&rArr; &nbsp; This quantity corresponds to the slope of the straight line drawn in green in the 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&nbsp; '''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 =
Line 186: Line 192:
 
  \approx {80\, \rm &micro; s}.$$}}
 
  \approx {80\, \rm &micro; s}.$$}}
  
==Phasenverzerrungen==
+
==Phase distortions==
 
<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&nbsp; (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 with period&nbsp; $T_0$ &nbsp; &rArr; &nbsp; blue curve
 
:$$x(t) =  {1\, \rm V} \cdot \cos(2 \pi \cdot {1\, \rm kHz}\cdot  t) + {1\, \rm V} \cdot \sin(2 \pi \cdot {2\, \rm kHz}\cdot  t)$$
 
:$$x(t) =  {1\, \rm V} \cdot \cos(2 \pi \cdot {1\, \rm kHz}\cdot  t) + {1\, \rm V} \cdot \sin(2 \pi \cdot {2\, \rm kHz}\cdot  t)$$
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.  
+
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
 
:$$y(t) = x(t - τ).$$}}
 
  
 +
*With &nbsp; $τ_1 = τ_2 = τ$ &nbsp; the following distortion-free signal would arise as a result:
 +
:$$y(t) = x(t - τ).$$
  
Wir weisen nochmals  auf das interaktive Applet&nbsp; [[Applets:Lineare_Verzerrungen_periodischer_Signale|Lineare Verzerrungen periodischer Signale]] &nbsp;hin.
+
We refer once more to the interactive applet&nbsp; [[Applets:Linear_Distortions_of_Periodic_Signals|Linear distortions of periodic signals]]. &nbsp;}}
  
  
 
   
 
   
==Entzerrungsverfahren==
+
==Equalization methods==
 
<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|Equalization 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,&nbsp; will only be briefly touched upon here.&nbsp; 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, because of&nbsp; "distortion"&nbsp; &rArr; &nbsp; German:&nbsp; "Verzerrung" &nbsp; &rArr; &nbsp; "V",  
*während &nbsp;$S_{\rm E}$&nbsp; der Entzerrung dient.  
+
*while&nbsp;$S_{\rm E}$&nbsp; serves for equalization &nbsp; &rArr; &nbsp; "E".  
  
  
Zu dieser Konstellation ist anzumerken:  
+
Regarding this constellation,&nbsp; the following should be noted:  
*Ist die Verzerrung nichtlinear, so muss auch die Entzerrung nichtlinear erfolgen.  
+
*If the distortion is nonlinear,&nbsp; the equalization 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,&nbsp; nonlinear equalization methods are used, e.g.&nbsp; "Decision Feedback Equalisation"&nbsp; in digital systems.&nbsp; The advantage over linear equalization 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; $H_{\rm V}(f)$,&nbsp; then the distortions can be completely eliminated with the&nbsp; '''inverse frequency response'''&nbsp; $H_{\rm E}(f) = 1/H_{\rm V}(f)$&nbsp; and&nbsp; $z(t) = x(t)$&nbsp; holds.&nbsp; However,&nbsp; 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,&nbsp; otherwise infinity points would be required for&nbsp; $H_{\rm E}(f)$&nbsp;.  
*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.  
+
*For&nbsp; '''analog systems''',&nbsp; complete equalization 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 ''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&nbsp; '''digital systems''',&nbsp; the criterion for complete equalization is less strict.&nbsp; It must then only be ensured that the signals&nbsp; $x(t)$&nbsp; and&nbsp; $z(t)$&nbsp; coincide at the detection times.&nbsp; In this context,&nbsp; one deals with&nbsp; [[Digital_Signal_Transmission/Eigenschaften_von_Nyquistsystemen|Nyquist systems]].
*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]].
 
  
==Aufgaben zum Kapitel==
+
==Exercises for the chapter==
 
<br>
 
<br>
[[Aufgaben:2.5_Verzerrung_und_Entzerrung|Aufgabe 2.5: Verzerrung und Entzerrung]]
+
[[Aufgaben:Exercise_2.5:_Distortion_and_Equalization|Exercise 2.5: Distortion and Equalization]]
  
[[Aufgaben:Aufgabe_2.5Z:_Nyquistentzerrung|Aufgabe 2.5Z: Nyquistentzerrung]]
+
[[Aufgaben:Exercise_2.5Z:_Nyquist_Equalization|Exercise 2.5Z: Nyquist Equalization]]
  
[[Aufgaben:2.6_Zweiwegekanal|Aufgabe 2.6: Zweiwegekanal]]
+
[[Aufgaben:Exercise_2.6:_Two-Way_Channel|Exercise 2.6: Two-Way Channel]]
  
[[Aufgaben:2.6Z_Synchrondemodulator|Aufgabe 2.6Z: Synchrondemodulator]]
+
[[Aufgaben:Exercise_2.6Z:_Synchronous_Demodulator|Exercise 2.6Z: Synchronous Demodulator]]
  
[[Aufgaben:2.7_Nochmals_Zweiwegekanal|Aufgabe 2.7: Nochmals Zweiwegekanal]]
+
[[Aufgaben:Exercise_2.7:_Two-Way_Channel_once_more|Exercise 2.7: Two-Way Channel once more]]
  
 
{{Display}}
 
{{Display}}

Latest revision as of 15:41, 5 October 2021

Compilation of important descriptive quantities


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:
$$b(f) = - {\rm arc} \hspace{0.1cm}H(f) \hspace{0.2cm}{\rm in \hspace{0.1cm}radian \hspace{0.1cm}(rad)}.$$

Requirements for distortion-free systems


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 attenuated and delayed:

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

According to the systems theory laws,  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 the 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):
$$b(f) = 2 \pi f \tau = {\rm const.} \cdot f.$$

$\text{Definitions:}$  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  "realizable systems" – in particular, in the  "minimum-phase systems"  described in  Chapter 3  – both forms of distortions 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

  • 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.


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

Requirements for a non-distorting channel:
  $f_{\rm U}$:  lower  $($German:  "untere"   ⇒   "U"$)$  range limit,
  $f_{\rm O}$:  upper  $($German:  "obere"   ⇒   "O"$)$  range limit,
  $f_{\rm T}$:  carrier  $($German:  "Träger"   ⇒   "T"$)$  frequency
  • 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  $b(f) = \hspace{0.1cm}–\hspace{-0.01cm} b(\hspace{-0.01cm}-\hspace{-0.01cm}f)$  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.

Attenuation distortions


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)  which is periodic with  $T_0 = 1\ \rm ms$ 

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

and the signal  $y(t)$  (red curve)  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.


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

$$x(t) = C \cdot \cos(2 \pi f_0 t - \varphi)\hspace{0.15cm} \circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\, X(f ) = {C}/{2}\cdot {\rm e}^{{\rm j}\hspace{0.05cm} \cdot \hspace{0.05cm} \varphi} \cdot \delta(f + f_0) \hspace{0.01cm} + \hspace{0.01cm}{C}/{2}\cdot {\rm e}^{-{\rm j}\hspace{0.05cm} \cdot \hspace{0.05cm} \varphi} \cdot \delta(f - f_0),$$
On the definition of the phase delay time $\tau_{\rm P}$

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.

Difference between phase and group delay time


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 $\tau_{\rm G}$

$\text{Example 3:}$  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 graph.

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

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

Phase distortions


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 with period  $T_0$   ⇒   blue curve

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

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 once more to the interactive applet  Linear distortions of periodic signals.  


Equalization methods


Equalization 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, because of  "distortion"  ⇒   German:  "Verzerrung"   ⇒   "V",
  • while $S_{\rm E}$  serves for equalization   ⇒   "E".


Regarding this constellation,  the following should be noted:

  • 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"  in digital systems.  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.  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  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  digital systems,  the criterion for complete equalization 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


Exercise 2.5: Distortion and Equalization

Exercise 2.5Z: Nyquist Equalization

Exercise 2.6: Two-Way Channel

Exercise 2.6Z: Synchronous Demodulator

Exercise 2.7: Two-Way Channel once more