Linear and Time Invariant Systems/Linear Distortions - Revision history

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@@ Line 250: / Line 250: @@
 #If the distortion is nonlinear,&nbsp; the equalization must also be nonlinear.
 #But even with linear distortion,&nbsp; nonlinear equalization methods are used, e.g.&nbsp; &raquo;Decision Feedback Equalisation&laquo;&nbsp;  $\rm (DFE)$&nbsp; in digital systems.&nbsp; <br>The advantage over linear equalization is that there is no increase in noise power.
-#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; &raquo;'''inverse frequency response'''&laquo;&nbsp; $H_{\rm E}(f) = 1/H_{\rm V}(f)$&nbsp; and&nbsp; $z(t) = x(t)$&nbsp; holds.&nbsp;
+#If&nbsp; $S_{\rm V}$&nbsp; is a linear system with frequency response&nbsp; $H_{\rm V}(f)$,&nbsp; <br>then the distortions can be completely eliminated with the&nbsp; &raquo;'''inverse frequency response'''&laquo;&nbsp; $H_{\rm E}(f) = 1/H_{\rm V}(f)$&nbsp; and&nbsp; $z(t) = x(t)$&nbsp; holds.&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; as otherwise&nbsp; $H_{\rm E}(f)$&nbsp; would result in infinity points.
 #For&nbsp; &raquo;analog systems&laquo;,&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.

@@ Line 130: / Line 130: @@
 :$$H(f) =  {\rm e}^{-{\rm j} \hspace{0.05cm} \cdot \hspace{0.05cm}
 b(f)}.$$
 *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;  $b(\hspace{-0.01cm}-\hspace{-0.08cm}f) = \hspace{0.08cm}-b(f)$.
 *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.
@@ Line 163: / Line 164: @@
 The following should be noted regarding this definition:
 *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.
 *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.
 *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.
-We refer here again to the interactive applet&nbsp; [[Applets:Linear_Distortions_of_Periodic_Signals|"Linear distortions of periodic signals"]].
+We refer here again to the interactive applet&nbsp; [[Applets:Linear_Distortions_of_Periodic_Signals|&raquo;Linear distortions of periodic signals&laquo;]].
 ==Difference between phase and group delay time==
@@ Line 178: / Line 181: @@
 The&nbsp; &raquo;'''group delay time'''&laquo;&nbsp; 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. }}
-[[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=
+[[File: EN_LZI_T_2_3_S5.png| right|frame|On the definition of the group delay time  $\tau_{\rm G}$|class=fit]]
 $\text{Example 3:}$&nbsp;
 The graph shows the examplary phase function:
 :$$b(ω) = \arctan (ω/ω_0).$$
-*This increases monotonically from zero&nbsp; $($at &nbsp; $ω = 0)$&nbsp; to &nbsp;$π/2$&nbsp; $($for &nbsp; $ω → ∞)$.
+#This increases monotonically from zero&nbsp; $($at &nbsp; $ω = 0)$&nbsp; to &nbsp;$π/2$&nbsp; $($for &nbsp; $ω → ∞)$.
-*The function value at&nbsp; $ω = ω_0$&nbsp; is&nbsp; $π/4$.
+#The function value at&nbsp; $ω = ω_0$&nbsp; is&nbsp; $π/4$.
-If we set&nbsp; $ω_0 = 2π · 1 \ \rm kHz$, we obtain for the&nbsp; &raquo;'''phase delay time:'''&laquo;
+If we set&nbsp; $ω_0 = 2π · 1 \ \rm kHz$,&nbsp; we obtain for the&nbsp; &raquo;'''phase delay time:'''&laquo;
 :$$\tau_{\rm P}(\omega_0) = \frac{b(\omega_0)}{\omega_0}= \frac{\pi / 4}{2 \pi \cdot{1\, \rm kHz} } =  {125\, \rm &micro; s}.$$
 &rArr; &nbsp; 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&nbsp; &raquo;'''group delay time:'''&laquo;
+In contrast,&nbsp; the lesser slope of the tangent line shown in red denotes the&nbsp; &raquo;'''group delay time:'''&laquo;
 :$$\tau_{\rm G}(\omega_0) = \left[ \frac{ {\rm d}b(\omega)}{ {\rm d}\omega}\right ]_{\omega = \omega_0} =
   \left[ \frac{1}{\omega_0} \cdot \frac{1}{1 + \left(\omega / \omega_0\right]^2} \right ]_{\omega =
@@ Line 204: / Line 208: @@
 ==Phase distortions==
 <br>
-To illustrate this point, we again consider the sum of two harmonic oscillations as input signal:
+To illustrate this point,&nbsp; we again consider the sum of two harmonic oscillations as input signal:
 :$$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).$$
@@ Line 211: / Line 215: @@
 :$$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)$$
-:and at the same time&nbsp; $τ_1 ≠ τ_2$ is valid, then &nbsp; &raquo;'''exclusively phase distortions'''&laquo;&nbsp; are existent.
+:and at the same time&nbsp; $τ_1 ≠ τ_2$ is valid,&nbsp; then &nbsp; &raquo;'''exclusively phase distortions'''&laquo;&nbsp; are existent.
-*The two phase delay times&nbsp; $τ_1 ≠ τ_2$&nbsp; can be determined from the phase response&nbsp; (in radian):
+*The two phase delay times&nbsp; $τ_1 ≠ τ_2$&nbsp; can be determined from the phase response&nbsp; $($in radian$)$:
 :$$\tau_1 =  \frac{b(f_1)}{2\pi f_1} , \hspace{0.4cm}\tau_2 =  \frac{b(f_2)}{2\pi
   f_2}.$$
-[[File: P_ID905__LZI_T_2_3_S6_neu.png|right|frame|Effects of phase distortions|class=fit]]
+{{GraueBox|TEXT=
-{{GraueBox|TEXT=
+[[File: P_ID905__LZI_T_2_3_S6_neu.png|right|frame|Effects of phase distortions|class=fit]]
 $\text{Example 4:}$&nbsp;
 The graph shows the periodic signal with period&nbsp; $T_0$ &nbsp; &rArr; &nbsp; blue curve
@@ Line 228: / Line 232: @@
 :$$y(t) = x(t - τ).$$
-We refer once more to the interactive applet&nbsp; [[Applets:Linear_Distortions_of_Periodic_Signals|"Linear distortions of periodic signals"]]. &nbsp;}}
+We refer once more to the interactive applet&nbsp; [[Applets:Linear_Distortions_of_Periodic_Signals|&raquo;Linear distortions of periodic signals&laquo;]]. &nbsp;}}
@@ Line 235: / Line 239: @@
 <br>
 [[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,&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"]].
+This method,&nbsp; 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|&raquo;Modulation Methods&laquo;]]&nbsp; and&nbsp; [[Digital_Signal_Transmission|&raquo;Digital Signal Transmission&laquo;]].
-−
-For this brief description, we assume the outlined constellation:
+For this brief description,&nbsp; we assume the outlined constellation:
-*$S_{\rm V}$&nbsp; denotes a distorting system, because of&nbsp; "distortion"&nbsp; &rArr; &nbsp; German:&nbsp; "Verzerrung" &nbsp; &rArr; &nbsp; "V",
+*$S_{\rm V}$&nbsp; denotes a distorting system, because of&nbsp; &raquo;distortion&laquo;&nbsp; &rArr; &nbsp; German:&nbsp; "Verzerrung" &nbsp; &rArr; &nbsp; "V".
-*while&nbsp;$S_{\rm E}$&nbsp; serves for equalization &nbsp; &rArr; &nbsp; "E".
 Regarding this constellation,&nbsp; the following should be noted:
-*If the distortion is nonlinear,&nbsp; the equalization must also be nonlinear.
+#If the distortion is nonlinear,&nbsp; the equalization must also be nonlinear.
-*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.
+#But even with linear distortion,&nbsp; nonlinear equalization methods are used, e.g.&nbsp; &raquo;Decision Feedback Equalisation&laquo;&nbsp;  $\rm (DFE)$&nbsp; in digital systems.&nbsp; <br>The advantage over linear equalization is that there is no increase in noise power.
-*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; &raquo;'''inverse frequency response'''&laquo;&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;.
+#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; &raquo;'''inverse frequency response'''&laquo;&nbsp; $H_{\rm E}(f) = 1/H_{\rm V}(f)$&nbsp; and&nbsp; $z(t) = x(t)$&nbsp; holds.&nbsp;
-*For&nbsp; &raquo;'''analog systems'''&laquo;,&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.
+#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; as otherwise&nbsp; $H_{\rm E}(f)$&nbsp; would result in infinity points.
-*For&nbsp; &raquo;'''digital systems'''&laquo;,&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|$\text{Nyquist systems}$]].
+#For&nbsp; &raquo;analog systems&laquo;,&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.
 ==Exercises for the chapter==

@@ Line 21: / Line 21: @@
 \hspace{0.1cm}(Np) } = - 20 \cdot \lg \vert H(f)\vert \hspace{0.2cm}{\rm in
 \hspace{0.1cm}decibel \hspace{0.1cm}(dB) }.$$
-*The&nbsp; &raquo;'''phase response'''&laquo;&nbsp; $b(f)$&nbsp; specifies the negative&nbsp; $(f$&ndash;dependent$)$&nbsp;  angle of&nbsp; $H(f)$&nbsp; in the complex plane and with respect to the real axis:
+*The&nbsp; &raquo;'''phase response'''&laquo;&nbsp; $b(f)$&nbsp; specifies the negative&nbsp; $(f$&ndash;dependent$)$&nbsp;  angle of&nbsp; $H(f)$&nbsp; in the complex plane with respect to the real axis:
 :$$b(f) = - {\rm arc} \hspace{0.1cm}H(f) \hspace{0.2cm}{\rm in
 \hspace{0.1cm}radian \hspace{0.1cm}(rad)}.$$}}
@@ Line 29: / Line 29: @@
 ==Requirements for distortion-free systems==
 <br>
-According to the explanations in the chapter &nbsp;[[Linear_and_Time_Invariant_Systems/Classification_of_the_Distortions|"Classification of the Distortions"]] &nbsp; there is a distortion-free system at hand if and only if all frequency components are uniformly attenuated and delayed:
+According to the explanations in the chapter &nbsp;[[Linear_and_Time_Invariant_Systems/Classification_of_the_Distortions|&raquo;Classification of the Distortions&raquo;]] &nbsp; there is a&nbsp; &raquo;distortion-free system&laquo;&nbsp; at hand if and only if all frequency components are uniformly attenuated and delayed:
 :$$y(t) = \alpha \cdot x(t - \tau).$$
 According to the systems theory laws,&nbsp; the following must thus hold for the frequency response:
@@ Line 36: / Line 36: @@
 *The attenuation curve must be constant for all frequencies contained in the input signal:
 :$$a(f) = - \ln |H(f)| = - \ln \ \alpha = {\rm const.}$$
-*The phase response must either be zero in the region of interest&nbsp; (system with no transit time)&nbsp; or increase linearly with frequency&nbsp; $(τ$&nbsp; indicates the transit time):
+*The phase response must either be zero in the region of interest&nbsp; $($system with no transit time$)$&nbsp; or increase linearly with frequency&nbsp; $(τ$&nbsp; indicates the transit time$)$:
 :$$b(f) = 2 \pi f \tau = {\rm const.} \cdot f.$$
 {{BlaueBox|TEXT=
 $\text{Definitions:}$&nbsp;
-For a distortion-free system, both requirements must be satisfied simultaneously.
+For a&nbsp; &raquo;'''distortion-free system'''&laquo;,&nbsp; both requirements must be satisfied simultaneously.
 Violation of even one of these two conditions results in&nbsp; &raquo;'''linear distortions'''&laquo;&nbsp; which are distinguished according to their cause:
 *It comes to&nbsp; &raquo;'''attenuation distortions'''&laquo;&nbsp; if in the frequency range of interest the attenuation curve is not constant:
 :$$a(f) \ne {\rm const.}$$
-*In contrast to this, there are&nbsp; &raquo;'''phase distortions'''&laquo;&nbsp; if the phase function is not linear with respect to $f$&nbsp;:
+*In contrast to this,&nbsp; there are&nbsp; &raquo;'''phase distortions'''&laquo;&nbsp; if the phase function is not linear with respect to $f$&nbsp;:
 :$$b(f) \ne {\rm const.} \cdot f.$$}}
-It should be noted that in all&nbsp; "realizable systems" – in particular, in the&nbsp; "minimum-phase systems"&nbsp; described in&nbsp; [[Linear_and_Time_Invariant_Systems/Conclusions_from_the_Allocation_Theorem#Attenuation_and_phase_of_minimum-phase_systems|"Chapter 3"]]&nbsp; – both forms of distortions usually occur simultaneously.
+It should be noted that in all&nbsp; &raquo;realizable systems&laquo; – in particular, in the&nbsp; &raquo;minimum-phase systems&laquo;&nbsp; described in&nbsp; [[Linear_and_Time_Invariant_Systems/Conclusions_from_the_Allocation_Theorem#Attenuation_and_phase_of_minimum-phase_systems|&raquo;Chapter 3&laquo;]]&nbsp; – both forms of distortions usually occur simultaneously.
 {{BlaueBox|TEXT=
-$\text{Definition:}$&nbsp; In the time domain, the condition for a&nbsp; &raquo;'''distortion-free system'''&laquo; is:
+$\text{Definition:}$&nbsp; In the time domain,&nbsp; the condition for a&nbsp; &raquo;'''distortion-free system'''&laquo; is:
 :$$h(t) = \alpha \cdot \delta(t - \tau),\hspace{0.4cm}\alpha \ne 0.$$
-Moreover,&nbsp; if&nbsp; $α = 1$&nbsp; and&nbsp; $τ = 0$&nbsp; hold,&nbsp; then there is an&nbsp; &raquo;'''ideal transmission system'''&laquo;.&nbsp; In contrast, there are linear distortions whenever
+Moreover,&nbsp; if&nbsp; $α = 1$&nbsp; and&nbsp; $τ = 0$&nbsp; hold,&nbsp; then there is an&nbsp; &raquo;'''ideal transmission system'''&laquo;.&nbsp; In contrast,&nbsp; there are linear distortions whenever
-*the impulse response&nbsp; $h(t)$&nbsp; is a continuous-time function, or
+*the impulse response&nbsp; $h(t)$&nbsp; is a continuous-time function,&nbsp; or
 *the time-dicrete impulse response&nbsp;$h(t)$&nbsp; is composed of more than one Dirac delta functions.}}
@@ Line 66: / Line 67: @@
 [[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$.
-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.
+Outside the frequency band occupied by&nbsp;$x(t)$&nbsp; the&nbsp; &raquo;constant attenuation&laquo;&nbsp; and&nbsp; &raquo;linear phase&laquo;&nbsp; conditions do not need to be satisfied.
 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 better suppressed. }}
@@ Line 97: / Line 104: @@
 :$$\alpha_1 = |H(f_1)|,\hspace{0.4cm}\alpha_2 = |H(f_2)|.$$
-*If the attenuation curve&nbsp; $a(f)$&nbsp; is given in Neper,&nbsp; then likewise the following holds &nbsp;$(1 \ \rm dB$ corresponds to $0.1151 \ \rm  Np)$:
+*If the attenuation curve&nbsp; $a(f)$&nbsp; is given in Neper,&nbsp; then likewise the following holds &nbsp;$(1 \ \rm dB$&nbsp; corresponds to $0.1151 \ \rm  Np)$:
 :$$ \alpha_1 =  {\rm e}^{-{\rm a}(f_1)},\hspace{0.4cm}\alpha_2  = {\rm e}^{-{\rm a}(f_2)}.$$
-Please note: &nbsp; In some character fonts,&nbsp; "$a$"&nbsp; and&nbsp; "$α$"&nbsp; (alpha)&nbsp; are difficult to distinguish.
+Please note: &nbsp; In some character fonts,&nbsp; "${\rm a}$"&nbsp; and&nbsp; "$α$"&nbsp; $($alpha$)$&nbsp; are difficult to distinguish.
-[[File:P_ID901__LZI_T_2_3_S3_neu.png |frame| Effects of attenuation distortions | right|class=fit]]
+{{GraueBox|TEXT=
-{{GraueBox|TEXT=
+[[File:P_ID901__LZI_T_2_3_S3_neu.png |frame| Effects of attenuation distortions | right|class=fit]]
 $\text{Example 2:}$&nbsp;
-The graph shows the input signal&nbsp; (blue curve)&nbsp; which is periodic with&nbsp; $T_0 = 1\ \rm  ms$&nbsp;
+The graph shows the input signal&nbsp; $($blue curve$)$&nbsp; which is periodic with&nbsp; $T_0 = 1\ \rm  ms$,&nbsp;
-:$$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),$$
-and the signal&nbsp; $y(t)$&nbsp; (red curve)&nbsp; which is attenuation-distorted with &nbsp;$α_1 = 0.2$, &nbsp;$α_2 = 0.5$.
+and the red curve signal&nbsp; $y(t)$&nbsp;  which is attenuation-distorted with &nbsp;$α_1 = 0.2$, &nbsp;$α_2 = 0.5$.
-Significant effects of these attenuation distortions can be perceived:
+&rArr; &nbsp; Significant effects of these attenuation distortions can be perceived:
-*$y(t)$&nbsp; bears little resemblance to&nbsp; $x(t)$.
+#$y(t)$&nbsp; bears little resemblance to&nbsp; $x(t)$.
-*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;$1/α$.
+#In contrast,&nbsp; 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;$1/α$.
-We refer here explicitly to the interactive applet&nbsp; [[Applets:Linear_Distortions_of_Periodic_Signals|"Linear distortions of periodic signals"]]. }}
+We refer here explicitly to the interactive applet&nbsp; [[Applets:Linear_Distortions_of_Periodic_Signals|&raquo;Linear distortions of periodic signals&laquo;]]. }}

@@ Line 7: / Line 7: @@
 ==Compilation of important descriptive quantities==
 <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&nbsp; $H(f)$&nbsp;.
-The generally complex frequency response can also be formulated as follows:
+{{BlaueBox|TEXT=
-:$$H(f) = |H(f)| \cdot {\rm e}^{-{\rm j} \hspace{0.05cm} \cdot
+[[File:P_ID899__LZI_T_2_3_S1_neu.png |frame|Linear system description|class=fit]]
-\hspace{0.05cm} b(f)} = {\rm e}^{-a(f)}\cdot {\rm e}^{-{\rm j}
+$\text{System model:}$&nbsp;
 \hspace{0.05cm} \cdot \hspace{0.05cm} b(f)}.$$
 This yields the following descriptive quantities:
-*The magnitude&nbsp; $|H(f)|$&nbsp; is referred to as&nbsp; &raquo;'''amplitude response'''&laquo;&nbsp; and in logarithmic form as&nbsp; "attenuation curve":
+*The magnitude&nbsp; $\vert H(f)\vert $&nbsp; is referred to as&nbsp; &raquo;'''amplitude response'''&laquo;&nbsp; and in logarithmic form as&nbsp; &raquo;'''attenuation curve'''&laquo;:
-:$$a(f) = - \ln |H(f)|\hspace{0.2cm}{\rm in \hspace{0.1cm}Neper
+:$$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) }.$$
-*The&nbsp; &raquo;'''phase response'''&laquo;&nbsp; $b(f)$&nbsp; specifies the negative angle of&nbsp; $H(f)$&nbsp; dependent on&nbsp; $f$&nbsp; in the complex plane and with respect to the real axis:
+*The&nbsp; &raquo;'''phase response'''&laquo;&nbsp; $b(f)$&nbsp; specifies the negative&nbsp; $(f$&ndash;dependent$)$&nbsp;  angle of&nbsp; $H(f)$&nbsp; in the complex plane and with respect to the real axis:
 :$$b(f) = - {\rm arc} \hspace{0.1cm}H(f) \hspace{0.2cm}{\rm in
-\hspace{0.1cm}radian \hspace{0.1cm}(rad)}.$$
+\hspace{0.1cm}radian \hspace{0.1cm}(rad)}.$$}}
 ==Requirements for distortion-free systems==

@@ Line 225: / Line 225: @@
 <br>
 [[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,&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"]].
+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"]].
 For this brief description, we assume the outlined constellation:

@@ Line 78: / Line 78: @@
 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.
-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. }}
+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 better suppressed. }}
 ==Attenuation distortions==

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

← Older revision		Revision as of 14:37, 30 December 2022
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−	It should be noted that in all  "realizable systems" – in particular, in the  "minimum-phase systems"  described in  [[Linear_and_Time_Invariant_Systems/Conclusions_from_the_Allocation_Theorem#~~Dämpfung_und_Phase_von_Minimum–Phasen–Systemen~~\|"Chapter 3"]]  – both forms of distortions usually occur simultaneously.	+	It should be noted that in all  "realizable systems" – in particular, in the  "minimum-phase systems"  described in  [[Linear_and_Time_Invariant_Systems/Conclusions_from_the_Allocation_Theorem#Attenuation_and_phase_of_minimum-phase_systems\|"Chapter 3"]]  – both forms of distortions usually occur simultaneously.

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