Linear and Time Invariant Systems/Nonlinear Distortions - Revision history

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Hwang at 14:03, 30 December 2022

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@@ Line 8: / Line 8: @@
 ==Properties of nonlinear systems==
 <br>
-The system description by means of the frequency response&nbsp; $H(f)$&nbsp; and/or the impulse response&nbsp; $h(t)$&nbsp; is only possible for an&nbsp; [[Linear_and_Time_Invariant_Systems/System_Description_in_Frequency_Domain|$\text{LTI System}$]].&nbsp; &nbsp; However, if the system contains nonlinear components, as it is assumed for this chapter, no frequency response and no impulse response can be stated.&nbsp; The model must be designed in a more general way.
+The system description by means of the frequency response&nbsp; $H(f)$&nbsp; and/or the impulse response&nbsp; $h(t)$&nbsp; is only possible for an&nbsp; [[Linear_and_Time_Invariant_Systems/System_Description_in_Frequency_Domain|&raquo;'''LTI system'''&laquo;]].&nbsp; &nbsp; However,&nbsp; if the system contains nonlinear components,&nbsp; as it is assumed for this chapter,&nbsp; no frequency response and no impulse response can be stated.&nbsp; The model must be designed in a more general way.
-[[File:EN_LZI_T_2_2_S1.png|frame| Description of a non-linear system|class=fit]]
+[[File:EN_LZI_T_2_2_S1.png|frame| Description of a nonlinear system|class=fit]]
-Also in this nonlinear system,&nbsp; we denote the signals at the input and the output by&nbsp; $x(t)$&nbsp; and&nbsp; $y(t)$&nbsp; respectively,&nbsp; and the corresponding spectral functions by&nbsp; $X(f)$&nbsp; and&nbsp; $Y(f)$.
+Also in this nonlinear system,&nbsp; we denote the signals at the input and the output by&nbsp; $x(t)$&nbsp; resp. &nbsp; $y(t)$&nbsp;  and the corresponding spectral functions by&nbsp; $X(f)$&nbsp; and&nbsp; $Y(f)$.
 An observer will note the following here:
-*The transmission characteristics are now also&nbsp; '''dependent on the amplitude of the input signal'''.&nbsp; If&nbsp; $x(t)$&nbsp; results in the output signal&nbsp; $y(t)$,&nbsp; it can now no longer be concluded that the input signal&nbsp; $K · x(t)$&nbsp; will always result in the signal&nbsp; $K · y(t)$.
+*The transmission characteristics are now also&nbsp; '''dependent on the amplitude of the input signal'''.&nbsp; <br>If&nbsp; $x(t)$&nbsp; results in the output signal&nbsp; $y(t)$,&nbsp; it can now no longer be concluded that the input signal&nbsp; $K · x(t)$&nbsp; will always result in the signal&nbsp; $K · y(t)$.
 *This also implies that the&nbsp; '''superposition principle is no longer applicable'''.&nbsp; Consequently, the result&nbsp; $x_1(t) + x_2(t)$ &nbsp; ⇒ &nbsp; $y_1(t) + y_2(t)$&nbsp; cannot be reasoned from the two correspondences &nbsp; $x_1(t) ⇒ y_1(t)$&nbsp; and&nbsp; $x_2(t) ⇒ y_2(t)$.
-*Due to nonlinearities&nbsp; '''new frequencies occur'''.&nbsp; If&nbsp; $x(t)$&nbsp; is a harmonic oscillation with the frequency&nbsp; $f_0$, the output signal&nbsp; $y(t)$&nbsp; also contains components at multiples of&nbsp; $f_0$.&nbsp; In communications engineering, these are referred to as&nbsp; &raquo;'''harmonics'''&laquo;.
+*This also implies that the&nbsp; '''superposition principle is no longer applicable'''.&nbsp; <br>Consequently,&nbsp; the result&nbsp; $x_1(t) + x_2(t)$ &nbsp; ⇒ &nbsp; $y_1(t) + y_2(t)$&nbsp; cannot be reasoned from the two correspondences &nbsp; $x_1(t) ⇒ y_1(t)$&nbsp; and&nbsp; $x_2(t) ⇒ y_2(t)$.
-*In practice, an information signal usually contains many frequency components. The harmonics of the low-frequency signal components now fall into the range of higher-frequency useful components.&nbsp; This results in &raquo;'''nonreversible signal falsifications'''&laquo;.
-Before mentioning&nbsp; [[Linear_and_Time_Invariant_Systems/Nonlinear_Distortions#Constellations_which_result_in_nonlinear_distortions|$\text{constellations which result in nonlinear distortions}$]]&nbsp; at the end of the section,&nbsp; the problem of nonlinear distortions is captured mathematically.
+Before mentioning&nbsp; [[Linear_and_Time_Invariant_Systems/Nonlinear_Distortions#Constellations_which_result_in_nonlinear_distortions|&raquo;constellations which result in nonlinear distortions&laquo;]]&nbsp; at the end of this section,&nbsp; the problem of nonlinear distortions is captured mathematically.
-*We assume here that&nbsp; '''the system has no memory''' &nbsp; so that the output value&nbsp; $y = y(t_0)$&nbsp; depends only on the instantaneous input value&nbsp; $x = x(t_0)$,&nbsp;
-*but not on the signal curve&nbsp; $x(t)$&nbsp; for&nbsp; $t < t_0$.
+{{BlaueBox|TEXT=
 ==Description of nonlinear systems==
@@ Line 29: / Line 37: @@
 {{BlaueBox|TEXT=
 $\text{Definition:}$&nbsp;
-A system is said to be&nbsp; &raquo;'''nonlinear'''&laquo; if the following relationship exists between the signal value &nbsp;$x = x(t)$&nbsp; at the input and the output &nbsp;$y = y(t)$&nbsp;:
+A system is said to be&nbsp; &raquo;'''nonlinear'''&laquo;&nbsp; if the following relationship exists between the signal value &nbsp;$x = x(t)$&nbsp; at the input and the output value &nbsp;$y = y(t)$&nbsp;:
 $$y = g(x) \ne {\rm const.}  \cdot x.$$
 The curve shape &nbsp;$y = g(x)$&nbsp; is called the&nbsp; &raquo;'''nonlinear characteristic curve'''&laquo;&nbsp; of the system.}}
@@ Line 36: / Line 44: @@
 [[File:P_ID888__LZI_T_2_2_S2_neu.png |right|frame| Nonlinear characteristic curve|class=fit]]
-*In the diagram,&nbsp; as an example,&nbsp; the green curve is the nonlinear characteristic curve&nbsp; $y = g(x)$&nbsp; which is shaped according to the first quarter of a sine function.
+*As an example,&nbsp; in the diagram  the green curve is the nonlinear characteristic curve&nbsp; $y = g(x)$&nbsp; which is shaped according to the first quarter of a sine function.
 *In this diagram the special case of a linear system with the characteristic curve&nbsp; $y = x$&nbsp; can be seen dashed in red.
@@ Line 42: / Line 51: @@
 Since every characteristic curve can be developed into a Taylor series around the operating point the output signal can also be represented as follows:
 $$y(t) = \sum_{i=0}^{\infty}\hspace{0.1cm} c_i \cdot x^{i}(t)  = c_0 + c_1 \cdot x(t) + c_2 \cdot x^{2}(t) + c_3 \cdot x^{3}(t) + \hspace{0.05cm}\text{...}$$
-If&nbsp; $x(t)$&nbsp; has a unit,&nbsp; for exmple "Volt”,&nbsp; then the coefficients of the Taylor series also have appropriate and different units:
+If&nbsp; $x(t)$&nbsp; has a unit,&nbsp; e.g. "Volt”,&nbsp; then the coefficients of the Taylor series also have appropriate and different units:
-*$c_0$&nbsp; with&nbsp; $\rm V$,
+#$c_0$ &nbsp; &rArr; &nbsp; "Volt”,
-*$c_1$&nbsp; without units,
+#$c_1$&nbsp; without unit,
-*$c_2$&nbsp; with&nbsp; $\rm 1/V$, etc.
+#$c_2$ &nbsp; &rArr; &nbsp; "1/Volt”, etc.

@@ Line 155: / Line 155: @@
 Nonlinear distortions of the sink signal&nbsp; $v(t)$&nbsp; with respect to the source signal&nbsp; $q(t)$&nbsp; occur when
-*nonlinear distortions already occur on the channel – i.e. with respect to the transmission signal&nbsp; $s(t)$&nbsp; and the received signal&nbsp; $r(t)$,
+*nonlinear distortions already occur on the channel – i.e. with respect to the transmitted signal&nbsp; $s(t)$&nbsp; and the received signal&nbsp; $r(t)$,
 *an envelope demodulator is used for&nbsp; [[Modulation_Methods/Double-Sideband_Amplitude_Modulation#Double-Sideband_Amplitude_Modulation_with_carrier|$\text{double-sideband amplitude modulation}$]]&nbsp; (DSB–AM)&nbsp; with modulation factor&nbsp; $m > 1$,
 *for DSB–AM and envelope demodulation there is a linearly distorting channel,&nbsp; even with a modulation factor&nbsp; $m < 1$,

@@ Line 58: / Line 58: @@
-[[File:P_ID889__LZI_T_2_2_S2b_neu.png|center|frame|Effects of a nonlinear characteristic curve|class=fit]]
+[[File:P_ID889__LZI_T_2_2_S2b_neu.png|right|frame|Effects of a nonlinear characteristic curve|class=fit]]
-The outer diagrams show – each in blue – cosine-shaped input signals&nbsp; $x(t)$&nbsp; with different amplitudes&nbsp; $A$&nbsp; and the corresponding distorted output signals&nbsp; $y(t)$ in red.&nbsp; It can be seen that the nonlinear distortions increase with increasing amplitude,&nbsp; which are quantified by the distortion factor&nbsp; $K$&nbsp; defined in the next section.
+The outer diagrams show – each in blue – cosine-shaped input signals&nbsp; $x(t)$&nbsp; with different amplitudes&nbsp; $A$&nbsp; and the corresponding distorted output signals&nbsp; $y(t)$ in red.&nbsp;
 *The diagram on the upper right-hand corner for&nbsp; $A = 1.5$&nbsp; clearly shows that&nbsp; $y(t)$&nbsp; is no longer cosine-shaped;&nbsp; the half-waves run rounder than the ones of the cosine function.
@@ Line 69: / Line 71: @@
 ==The distortion factor==
 <br>
-To quantitatively capture the nonlinear distortions we assume that the input signal&nbsp; $x(t)$&nbsp; is cosine-shaped with the amplitude&nbsp; $A_x$.&nbsp; The output signal contains harmonics due to the nonlinear distortions and the following is generally true: &nbsp; &nbsp; &nbsp; $y(t) =  A_0 + A_1 \cdot \cos(\omega_0 t) + A_2 \cdot \cos(2\omega_0 t) +
+[[File:EN_LZI_T_2_2_S3.png|right|frame|Definition of the distortion factor|class=fit]]
-  A_3 \cdot \cos(3\omega_0 t) + \hspace{0.05cm}\text{...}$
+To quantitatively capture the nonlinear distortions we assume that the input signal&nbsp; $x(t)$&nbsp; is cosine-shaped with the amplitude&nbsp; $A_x$.&nbsp;
+*The output signal contains harmonics due to the nonlinear distortions and the following is generally true: &nbsp; &nbsp; &nbsp;
-[[File:EN_LZI_T_2_2_S3.png|center|frame|Definition of the distortion factor|class=fit]]
+:$$y(t)\hspace{-0.03cm} =\hspace{-0.03cm}  A_0 \hspace{-0.03cm}+\hspace{-0.03cm} A_1 \cdot \cos(\omega_0 t) \hspace{-0.03cm}+\hspace{-0.03cm} A_2 \cdot \cos(2\omega_0 t) \hspace{-0.03cm}+\hspace{-0.03cm}
+  A_3 \cdot \cos(3\omega_0 t) \hspace{-0.03cm}+\hspace{-0.03cm} \hspace{0.05cm}\text{...}$$
 {{BlaueBox|TEXT=
 $\text{Definition:}$&nbsp;

@@ Line 156: / Line 156: @@
 *for DSB–AM and envelope demodulation there is a linearly distorting channel,&nbsp; even with a modulation factor&nbsp; $m < 1$,
 *the combination of&nbsp; [[Modulation_Methods/Single-Sideband_Modulation|$\text{single-sideband modulation}$]]&nbsp; and&nbsp; [[Modulation_Methods/Hüllkurvendemodulation|$\text{envelope demodulation}$]]&nbsp; is used (regardless of the sideband–to–carrier ratio),
-*an&nbsp; [[Modulation_Methods/Phasenmodulation_(PM)#Gemeinsamkeiten_zwischen_Phasen.E2.80.93_und_Frequenzmodulation|$\text{angle modulation}$]]&nbsp; (generic term for frequency and phase modulation)&nbsp; is applied and the available bandwidth is finite.
+*an&nbsp; [[Modulation_Methods/Phase_Modulation_(PM)#Similarities_between_phase_and_frequency_modulation|$\text{angle modulation}$]]&nbsp; (generic term for frequency and phase modulation)&nbsp; is applied and the available bandwidth is finite.
 ==Exercises for the chapter==

@@ Line 153: / Line 153: @@
 Nonlinear distortions of the sink signal&nbsp; $v(t)$&nbsp; with respect to the source signal&nbsp; $q(t)$&nbsp; occur when
 *nonlinear distortions already occur on the channel – i.e. with respect to the transmission signal&nbsp; $s(t)$&nbsp; and the received signal&nbsp; $r(t)$,
-*an envelope demodulator is used for&nbsp; [[Modulation_Methods/Zweiseitenband-Amplitudenmodulation#ZSB-Amplitudenmodulation_mit_Tr.C3.A4ger|$\text{double-sideband amplitude modulation}$]]&nbsp; (DSB–AM)&nbsp; with modulation factor&nbsp; $m > 1$,
+*an envelope demodulator is used for&nbsp; [[Modulation_Methods/Double-Sideband_Amplitude_Modulation#Double-Sideband_Amplitude_Modulation_with_carrier|$\text{double-sideband amplitude modulation}$]]&nbsp; (DSB–AM)&nbsp; with modulation factor&nbsp; $m > 1$,
 *for DSB–AM and envelope demodulation there is a linearly distorting channel,&nbsp; even with a modulation factor&nbsp; $m < 1$,
 *the combination of&nbsp; [[Modulation_Methods/Single-Sideband_Modulation|$\text{single-sideband modulation}$]]&nbsp; and&nbsp; [[Modulation_Methods/Hüllkurvendemodulation|$\text{envelope demodulation}$]]&nbsp; is used (regardless of the sideband–to–carrier ratio),

@@ Line 142: / Line 142: @@
 These result solely from frequency conversion products&nbsp; ("intermodulation components")&nbsp; of different spectral components, i.e. from nonlinear distortions.
-By varying the centre frequency&nbsp; $f_{\rm M}$&nbsp; and integrating over all these small interfering components the distortion power can thus be determined. More details on this method can be found, for example, in&nbsp; [Kam04]<ref>Kammeyer, K.D.:&nbsp; Nachrichtenübertragung. Stuttgart: B.G. Teubner, 4. Auflage, 2004.</ref>.
+By varying the center frequency&nbsp; $f_{\rm M}$&nbsp; and integrating over all these small interfering components the distortion power can thus be determined. More details on this method can be found, for example, in&nbsp; [Kam04]<ref>Kammeyer, K.D.:&nbsp; Nachrichtenübertragung. Stuttgart: B.G. Teubner, 4. Auflage, 2004.</ref>.
 <br clear=all>

← Older revision		Revision as of 14:11, 30 December 2022
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	*In the so-called clirr measurement the signal  $x(t)$  to be transmitted is modelled by white noise with the noise power density  ${\it \Phi}_x(f)$.		*In the so-called clirr measurement the signal  $x(t)$  to be transmitted is modelled by white noise with the noise power density  ${\it \Phi}_x(f)$.
−	*In addition, a narrow band-stop filter  $\rm (BS)$  with ~~centre~~ frequency  $f_{\rm M}$  and  (very small)  bandwidth  $B_{\rm BS}$  is introduced into the system.	+	*In addition, a narrow band-stop filter  $\rm (BS)$  with center frequency  $f_{\rm M}$  and  (very small)  bandwidth  $B_{\rm BS}$  is introduced into the system.

@@ Line 91: / Line 91: @@
-A comparison with the section&nbsp; [[Linear_and_Time_Invariant_Systems/Klassifizierung_der_Verzerrungen#Ber.C3.BCcksichtigung_von_D.C3.A4mpfung_und_Laufzeit|"Consideration of attenuation and runtime"]]&nbsp;  reveals that for the special case of a cosine-shaped input signal the signal–to–distortion–power ratio defined is equal to the reciprocal of the distortion factor squared:
+A comparison with the section&nbsp; [[Linear_and_Time_Invariant_Systems/Classification_of_the_Distortions#Elimination_of_damping_factor_and_transit_time|"Consideration of attenuation and runtime"]]&nbsp;  reveals that for the special case of a cosine-shaped input signal the signal–to–distortion–power ratio defined is equal to the reciprocal of the distortion factor squared:
 :$$\rho_{\rm V} = \frac{ \alpha^2 \cdot P_{x}}{P_{\rm V}} = \left(\frac{  A_{1}}{A_x} \right)^2 \cdot
   \frac{ {1}/{2} \cdot A_{x}^2}{{1}/{2} \cdot (A_{2}^2 + A_{3}^2 + A_{4}^2 + \hspace{0.05cm}...) } = \frac{1}{K^2}\hspace{0.05cm}.$$

@@ Line 52: / Line 52: @@
 {{GraueBox|TEXT=
 $\text{Example 1:}$&nbsp;
-The properties of nonlinear systems listed in the&nbsp; [[Linear_and_Time_Invariant_Systems/Nonlinear_Distortions#Properties_of_nonlinear_systems|"first section of this page"]]&nbsp; are illustrated here using the characteristic curve &nbsp;$y = g(x) = \sin(x)$&nbsp; shown in the centre of the diagram.
+The properties of nonlinear systems listed in the&nbsp; [[Linear_and_Time_Invariant_Systems/Nonlinear_Distortions#Properties_of_nonlinear_systems|"first section of this page"]]&nbsp; are illustrated here using the characteristic curve &nbsp;$y = g(x) = \sin(x)$&nbsp; shown in the center of the diagram.
 *Here, the direct (DC) signal &nbsp;$x(t) = 0.5$&nbsp; results in the constant output signal &nbsp;$y(t) = 0.479$&nbsp;.
 * For &nbsp;$x(t) = 1$&nbsp; the input signal results in the output signal &nbsp;$y(t) = 0.841 ≠ 2 · 0.479$.