Modulation Methods/Envelope Demodulation - Revision history

Guenter at 09:44, 13 January 2023

2023-01-13T09:44:15Z

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Guenter at 09:35, 13 January 2023

2023-01-13T09:35:35Z

Hwang at 21:29, 2 January 2023

2023-01-02T21:29:16Z

Hwang at 21:28, 2 January 2023

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Hwang at 21:17, 2 January 2023

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Hwang at 21:15, 2 January 2023

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Hwang at 21:09, 2 January 2023

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Hwang at 20:59, 2 January 2023

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Hwang at 10:02, 17 November 2022

2022-11-17T10:02:48Z

Reed at 15:23, 31 March 2022

2022-03-31T15:23:26Z

@@ Line 71: / Line 71: @@
-[[File:EN_Mod_T_2_3_S2b.png|right|frame|Frequency domain representation of an envelope demodulator '''Korrektur''': ungefähr, auch bei Deutsch.]]
+[[File:EN_Mod_T_2_3_S2b.png|right|frame|Frequency domain representation of an envelope demodulator]]
 The graph on the right is intended to illustrate the operation of the envelope demodulator in the frequency domain. The spectrum &nbsp;$W(f)$&nbsp; of the signal &nbsp;$w(t)$&nbsp; at the RC parallel circuit differs from the spectrum  &nbsp;$Q(f)$&nbsp; of the source signal as follows:
 *Due to the carrier signal &nbsp;$z(t)$&nbsp; added at the transmitter,&nbsp; the spectral function &nbsp;$W(f)$&nbsp; includes a Dirac delta line at &nbsp;$f = 0$&nbsp; with weight &nbsp;$A_{\rm T}$&nbsp; (carrier amplitude).

@@ Line 252: / Line 252: @@
 We now assume the following:
 *A source signal &nbsp;$q(t)$&nbsp; – composed of two cosine components at the frequencies &nbsp;$f_1$&nbsp; and &nbsp;$f_2$&nbsp; with amplitudes &nbsp;$A_1$&nbsp; and &nbsp;$A_2$.
-*DSB–AM with carrier&nbsp; ⇒  &nbsp;  the transmit signal &nbsp;$s(t)$&nbsp; is composed of a total of five cosine oscillations at the frequencies &nbsp;$f_{\rm T}, &nbsp;f_{\rm T} ± f_1$&nbsp; and &nbsp;$f_{\rm T} ± f_2$.
+*DSB–AM with carrier&nbsp; ⇒  &nbsp;  the transmitted signal &nbsp;$s(t)$&nbsp; is composed of a total of five cosine oscillations at the frequencies &nbsp;$f_{\rm T}, &nbsp;f_{\rm T} ± f_1$&nbsp; and &nbsp;$f_{\rm T} ± f_2$.
 [[File:Mod_T_2_3_S6_version2.png|right|frame|Spectra in the equivalent low-pass region]]
 *A channel with attenuation distortions,&nbsp; symmetrical about the carrier frequency:

@@ Line 228: / Line 228: @@
 #The angle between the complex-valued &nbsp;$r_{\rm TP}(t)$&nbsp; and the coordinate origin is now no longer continuous &nbsp;$ϕ(t) \equiv 0$,&nbsp; but instead fluctuates between &nbsp;$±ϕ_{\rm max}.$
 #The maximal phase is equal to the angle of the tangent to the ellipse.&nbsp; In the physical signal, &nbsp;$ϕ(t) ≠ 0$&nbsp; leads to shifts  of the zero crossings of &nbsp;$r(t)$&nbsp; with respect to its nominal positions &ndash; as given by the carrier signal &nbsp;$z(t)$.
-#The magnitude &nbsp;$a(t) = \vert r_{\rm TP}(t) \vert$ – i.e.,&nbsp; the envelope of &nbsp;$r(t)$ – is thus no longer cosine-shaped,&nbsp; and the signal after the envelope demodulator contains harmonicsin addition to the frequency &nbsp;$f_{\rm N}$: <br> &nbsp; &nbsp; $v(t) = A_{\rm 1} \cdot \cos(\omega_{\rm N}\cdot t ) +A_{\rm 2} \cdot \cos(2\omega_{\rm N}\cdot t
+#The magnitude &nbsp;$a(t) = \vert r_{\rm TP}(t) \vert$ – i.e.,&nbsp; the envelope of &nbsp;$r(t)$ – is thus no longer cosine-shaped,&nbsp; and the signal after the envelope demodulator contains harmonics in addition to the frequency &nbsp;$f_{\rm N}$: <br> &nbsp; &nbsp; $v(t) = A_{\rm 1} \cdot \cos(\omega_{\rm N}\cdot t ) +A_{\rm 2} \cdot \cos(2\omega_{\rm N}\cdot t
   )+A_{\rm 3} \cdot \cos(3\omega_{\rm N} \cdot t )+ \text{...}$<br>
 #These lead to nonlinear distortions,&nbsp; which are captured by the distortion factor &nbsp;$K$&nbsp;. <br>When &nbsp;$A_{\rm N} = 4 \ \rm V$, &nbsp;$A_{\rm T} = 5 \ \rm  V$, &nbsp;$α_{\rm U} = 1$&nbsp; and &nbsp;$α_{\rm T} = 0.8$,&nbsp; the following numerical values are obtained as a function of the upper sideband&nbsp;$α_{\rm O}$:

@@ Line 83: / Line 83: @@
 <br>
-The graph shows the DSB-AM signals for &nbsp;$m = 0.5$&nbsp; and &nbsp;$m = 2$.&nbsp; From this picture one can recognise the following differences:
+The graph shows the DSB-AM signals for &nbsp;$m = 0.5$&nbsp; and &nbsp;$m = 2$.&nbsp; From this picture one can recognize the following differences:
 [[File:Mod_T_2_3_S3_version2.png|right|frame|Envelope demodulation  when&nbsp; $m  < 1$&nbsp; (above) and&nbsp;  $m  > 1$&nbsp; (below)]]
@@ Line 190: / Line 190: @@
 The left half of the image shows the transmitter and at the same time gives the conditions at the receiver for an ideal channel.
-#Due to the modulation depth &nbsp;$m ≤ 1$,&nbsp; the source signal &nbsp;$q(t)$ can be recognised in the envelope &nbsp;$a(t)$.
+#Due to the modulation depth &nbsp;$m ≤ 1$,&nbsp; the source signal &nbsp;$q(t)$ can be recognized in the envelope &nbsp;$a(t)$.
 #Consequently,&nbsp; envelope demodulation is applicable without distortion under certain conditions,&nbsp; as shown in &nbsp;$\text{Example 4}$.
@@ Line 206: / Line 206: @@
 r(t) \hspace{-0.05cm}=\hspace{-0.05cm}  s(t) \hspace{0.02cm}.$$
-From these representations one can recognise:
+From these representations one can recognize:
 #The red pointer of length &nbsp;$A_{\rm T}$&nbsp; depicting the carrier is fixed.
 #The upper sideband&nbsp; (in blue)&nbsp; rotates in mathematically positive direction,&nbsp; the lower sideband&nbsp; (in green)&nbsp; in the opposite direction.

@@ Line 75: / Line 75: @@
 *Due to the carrier signal &nbsp;$z(t)$&nbsp; added at the transmitter,&nbsp; the spectral function &nbsp;$W(f)$&nbsp; includes a Dirac delta line at &nbsp;$f = 0$&nbsp; with weight &nbsp;$A_{\rm T}$&nbsp; (carrier amplitude).
 * $W(f)$&nbsp; also exhibits spectral components in the region around the carrier frequency  &nbsp;$f_{\rm T}$,&nbsp; which can be explained by the jagged time course of &nbsp;$w(t)$&nbsp; (see the first graph in this section).
-*&nbsp;$W(f)$&nbsp; also differs only slightly from &nbsp;$Q(f)$&nbsp; in the Low frequency domain.&nbsp;  Here,&nbsp; the error gets smaller as the carrier frequency increases compared to the bandwidth&nbsp; $B_{\rm NF}$.
+*&nbsp;$W(f)$&nbsp; also differs only slightly from &nbsp;$Q(f)$&nbsp; in the low frequency domain.&nbsp;  Here,&nbsp; the error gets smaller as the carrier frequency increases compared to the bandwidth&nbsp; $B_{\rm NF}$.
 <br clear=all>
 The first two signal distortions are eliminated by the high-pass and low-pass filters,&nbsp; which together produce a band-pass. However,&nbsp; there also remains a slight deviation between the sink signal &nbsp;$v(t)$&nbsp; and the source signal &nbsp;$q(t)$&nbsp; in the interesting range &nbsp;$0 < f < B_{\rm NF}$,&nbsp; as shown by comparing the output spectrum &nbsp;$V(f)$&nbsp; plotted in red and the input spectrum &nbsp;$Q(f)$&nbsp;  plotted in blue.

@@ Line 48: / Line 48: @@
 The first circuit section – consisting of a diode and a parallel connection of a resistor &nbsp;$R$&nbsp; and a capacitance&nbsp;$C$&nbsp; – performs the following tasks:
-*If the&nbsp; (light)&nbsp; grey signal&nbsp; $r(t)$&nbsp;is larger than the voltage &nbsp;$w(t)$&nbsp; at &nbsp;$R$&nbsp; and &nbsp;$C$,&nbsp; the diode conducts, &nbsp;$w(t) = r(t)$&nbsp; holds,&nbsp; and the capacitance&nbsp; $C$&nbsp;  is charged.  In these regions,&nbsp; $w(t)$&nbsp; is drawn in green.
+*If the&nbsp; (light)&nbsp; gray signal&nbsp; $r(t)$&nbsp;is larger than the voltage &nbsp;$w(t)$&nbsp; at &nbsp;$R$&nbsp; and &nbsp;$C$,&nbsp; the diode conducts, &nbsp;$w(t) = r(t)$&nbsp; holds,&nbsp; and the capacitance&nbsp; $C$&nbsp;  is charged.  In these regions,&nbsp; $w(t)$&nbsp; is drawn in green.
 *If &nbsp;$r(t) < w(t)$,&nbsp; as is the case at the times marked in purple,&nbsp; the diode blocks and the capacitance discharges through resistor &nbsp;$R$.&nbsp;  The signal &nbsp;$w(t)$&nbsp; decays exponentially with time constant &nbsp;$T = R · C$.
 *From the times marked with circles, &nbsp;$r(t) > w(t)$&nbsp; holds again and the capacitance is recharged.&nbsp;  It can be seen from the sketch that &nbsp;$w(t)$&nbsp;  matches&nbsp; "approximately"&nbsp; the envelope &nbsp;$a(t)$.

@@ Line 16: / Line 16: @@
 *Thus,&nbsp; when &nbsp;$H_{\rm K}(f) = 1$&nbsp; and &nbsp;$n(t) \equiv 0$&nbsp; we get for the received signal:
 :$$r(t) = s(t) = a(t)  \cdot \cos (\omega_{\rm T}\cdot t )\hspace{0.05cm}$$
-*In this equation, &nbsp;$a(t)$&nbsp; describes the&nbsp; '''envelope'''&nbsp; of the received signal &nbsp;$r(t)$.&nbsp; The phase function is &nbsp;$\mathbf{ϕ}(t) = 0$.
+*In this equation, &nbsp;$a(t)$&nbsp; describes the&nbsp; &raquo;'''envelope'''&laquo;&nbsp; of the received signal &nbsp;$r(t)$.&nbsp; The phase function is &nbsp;$\mathbf{ϕ}(t) = 0$.
 {{BlaueBox|TEXT=
-$\text{Definition:}$ &nbsp; An &nbsp; '''envelope demodulator'''&nbsp; detects the envelope &nbsp;$a(t)$&nbsp; of its input signal &nbsp;$r(t)$&nbsp; and outputs it as the sink signal after eliminating the DC component &nbsp;$A_{\rm T}$:
+$\text{Definition:}$ &nbsp; An &nbsp; &raquo;'''envelope demodulator'''&laquo;&nbsp; detects the envelope &nbsp;$a(t)$&nbsp; of its input signal &nbsp;$r(t)$&nbsp; and outputs it as the sink signal after eliminating the DC component &nbsp;$A_{\rm T}$:
 :$$v(t) = a(t) - A_{\rm T}\hspace{0.05cm}.$$
 The removal of the DC component &nbsp;$A_{\rm T}$&nbsp; can be realized,&nbsp; for example,&nbsp; by a high-pass filter that allows all frequencies to pass unimpeded except for&nbsp; $f = 0$&nbsp;.}}
@@ Line 91: / Line 91: @@
 * In contrast,&nbsp; with &nbsp;$m > 1$&nbsp; the following relationship holds:
 ::$$a(t) = |q(t) + A_{\rm T} |\hspace{0.05cm}.$$
-#Here,&nbsp; envelope demodulation always leads to &nbsp;[[Linear_and_Time_Invariant_Systems/Nonlinear_Distortions|nonlinear distortions]].
+#Here,&nbsp; envelope demodulation always leads to &nbsp;[[Linear_and_Time_Invariant_Systems/Nonlinear_Distortions|$\text{nonlinear distortions}$]].
 #The sink signal &nbsp;$v(t)$&nbsp; now includes new frequencies,&nbsp; which were not present in&nbsp;$q(t)$.
 #For the DC component&nbsp;(expected value)&nbsp; of the envelope,  in this case it holds that: <br> &nbsp; &nbsp;${\rm E}[a(t)] \ne   A_{\rm T}\hspace{0.05cm}.$
@@ Line 98: / Line 98: @@
 ==Description using the equivalent low-pass signal==
 <br>
-Especially when the source signal &nbsp;$q(t)$&nbsp; can be represented as a sum of several harmonic oscillations,&nbsp; a signal description with the equivalent low-pass signal &nbsp;${r_{\rm TP}(t)}$&nbsp; is extremely advantageous.&nbsp;  This is described in detail in the book &nbsp;[[Signal_Representation/Equivalent_Low-Pass_Signal_and_its_Spectral_Function|Signal Representation]].
+Especially when the source signal &nbsp;$q(t)$&nbsp; can be represented as a sum of several harmonic oscillations,&nbsp; a signal description with the equivalent low-pass signal &nbsp;${r_{\rm TP}(t)}$&nbsp; is extremely advantageous.&nbsp;  This is described in detail in the book &nbsp;[[Signal_Representation/Equivalent_Low-Pass_Signal_and_its_Spectral_Function|"Signal Representation"]].
 {{BlaueBox|TEXT=
-$\text{Please note:}$&nbsp; If noise/interference is disregarded,&nbsp; the&nbsp; '''received signal'''&nbsp; can be written as:
+$\text{Please note:}$&nbsp; If noise/interference is disregarded,&nbsp; the&nbsp; &raquo;'''received signal'''&laquo;&nbsp; can be written as:
 :$$r(t) = a(t)  \cdot \cos (\omega_{\rm T}\cdot t + \phi(t))\hspace{0.05cm}.$$}}
@@ Line 112: / Line 112: @@
 {{BlaueBox|TEXT=
-$\text{In general}$:&nbsp; The associated&nbsp; '''equivalent low-pass signal'''&nbsp; (German:&nbsp; "äquivalentes Tiefpass&ndash;Signal" &nbsp; &rArr; &nbsp; subscript&nbsp; "TP")&nbsp;is complex and given as:
+$\text{In general}$:&nbsp; The associated&nbsp; &raquo;'''equivalent low-pass signal'''&laquo;&nbsp; (German:&nbsp; "äquivalentes Tiefpass&ndash;Signal" &nbsp; &rArr; &nbsp; subscript&nbsp; "TP")&nbsp;is complex and given as:
 :$$r_{\rm TP}(t) = a(t)  \cdot {\rm e}^{\hspace{0.03cm}{\rm j} \hspace{0.05cm} \cdot \hspace{0.05cm} \phi(t)}\hspace{0.05cm}.$$
 The time functions &nbsp;$a(t)$&nbsp; and &nbsp;$ϕ(t)$&nbsp; contained in the equations are identical for both representations:
-* The function&nbsp; $a(t)$&nbsp; describes the&nbsp; '''envelope'''&nbsp; (time-dependent amplitude)&nbsp; of the physical signal &nbsp;$r(t)$&nbsp; and the magnitude &nbsp;$\vert r_{\rm TP}(t) \vert $&nbsp; of the equivalent low-pass signal, respectively.&nbsp; This is detected during envelope demodulation.
+* The function&nbsp; $a(t)$&nbsp; describes the&nbsp; &raquo;'''envelope'''&laquo;&nbsp; (time-dependent amplitude)&nbsp; of the physical signal &nbsp;$r(t)$&nbsp; and the magnitude &nbsp;$\vert r_{\rm TP}(t) \vert $&nbsp; of the equivalent low-pass signal, respectively.&nbsp; This is detected during envelope demodulation.
-* The function&nbsp; $ϕ(t)$&nbsp; is the&nbsp; '''time-dependent phase'''.&nbsp; This function contains all information about the position of the zero crossings of &nbsp;$r(t)$&nbsp; and indicates whether an additional phase modulation is effective.}}
+* The function&nbsp; $ϕ(t)$&nbsp; is the&nbsp; &raquo;'''time-dependent phase'''&laquo;.&nbsp; This function contains all information about the position of the zero crossings of &nbsp;$r(t)$&nbsp; and indicates whether an additional phase modulation is effective.}}
 In the case of double sideband amplitude modulation&nbsp; $\text{(DSB-AM)}$,&nbsp; the following holds for an ideal channel:
-*The&nbsp; '''locality curve'''&nbsp; or&nbsp;  '''locus curve'''&nbsp; – by this we mean the time-dependent representation of the signal&nbsp; $r_{\rm TP}(t)$&nbsp; in the complex plane – is a horizontal straight line on the real axis.
+*The&nbsp; &raquo;'''locality curve'''&laquo;&nbsp; or&nbsp; &raquo;'''locus curve'''&laquo;&nbsp; – by this we mean the time-dependent representation of the signal&nbsp; $r_{\rm TP}(t)$&nbsp; in the complex plane – is a horizontal straight line on the real axis.
 *It further follows that the phase function can take only two values: &nbsp;$0$&nbsp; and &nbsp;$π$ &nbsp; $(180^\circ)$&nbsp;.
 :*When &nbsp;$m ≤ 1$,&nbsp; then &nbsp;$ϕ(t) ≡ 0$&nbsp; and the envelope demodulation can be applied without distortion.
@@ Line 139: / Line 139: @@
-The physical signals &nbsp;$q(t)$, &nbsp;$r(t)$ and &nbsp;$v(t)$&nbsp; corresponding to this example can be found in the graph in the&nbsp; [[Modulation_Methods/Envelope_Demodulation#Why_does_envelope_demodulation_fail_for_.7F.27.22.60UNIQ-MathJax104-QINU.60.22.27.7F.3F|previous section]].}}
+The physical signals &nbsp;$q(t)$, &nbsp;$r(t)$ and &nbsp;$v(t)$&nbsp; corresponding to this example can be found in the graph in the&nbsp; [[Modulation_Methods/Envelope_Demodulation#Why_does_envelope_demodulation_fail_for_.7F.27.22.60UNIQ-MathJax104-QINU.60.22.27.7F.3F|"previous section"]].}}
 ==Special case of a cosine-shaped message signal==
@@ Line 169: / Line 169: @@
 :$$\Rightarrow \hspace{0.3cm}A_{\rm 1} = 4.48\,{\rm V},\hspace{0.3cm}A_{\rm 2} = 0.46\,{\rm V},\hspace{0.3cm}A_{\rm 3} = -0.37\,{\rm V},\hspace{0.3cm}
   A_{\rm 4} = 0.26\,{\rm V},\hspace{0.1cm}\text{...}$$
-*Thus,&nbsp; according to the details in the book&nbsp; [[Linear_and_Time_Invariant_Systems/Nonlinear_Distortions#The_distortion_factor|Linear and Time Invariant Systems]],&nbsp; we obtain for the individual distortion factors as well as the total distortion factor:
+*Thus,&nbsp; according to the details in the book&nbsp; [[Linear_and_Time_Invariant_Systems/Nonlinear_Distortions#The_distortion_factor|"Linear and Time Invariant Systems"]],&nbsp; we obtain for the individual distortion factors as well as the total distortion factor:
 :$$K_2 ={|A_{\rm 2}|}/{A_{\rm 1}} = 0.102\hspace{0.3cm}K_3 ={|A_{\rm 3}|}/{A_{\rm 1}} = 0.082,\hspace{0.3cm}K_4 = {|A_{\rm 4}|}/{A_{\rm 1}} = 0.058,\hspace{0.1cm}\text{...}$$
 :$$\Rightarrow \hspace{0.3cm}K = \sqrt{K_2^2 + K_3^2 + K_4^2 +\text{...} } \approx 15 \%.$$
-*In the chapter &nbsp;[[Modulation_Methods/Quality_Criteria#Investigations_with_regard_to_signal_distortions|Quality Criteria]]&nbsp; it was shown,&nbsp; that this also fixes the SNR at &nbsp;$ρ_v = 1/K^2 ≈ 44$&nbsp;.&nbsp;
+*In the chapter &nbsp;[[Modulation_Methods/Quality_Criteria#Investigations_with_regard_to_signal_distortions|"Quality Criteria"]]&nbsp; it was shown,&nbsp; that this also fixes the SNR at &nbsp;$ρ_v = 1/K^2 ≈ 44$&nbsp;.&nbsp;
-*The SNR&nbsp; $ρ_v$&nbsp; $($but not &nbsp;$K)$&nbsp; can also be used as a quality criterion if &nbsp;$q(t)$&nbsp; contains more than one frequency  &nbsp; ⇒  &nbsp; derivation in the book&nbsp;[[Linear_and_Time_Invariant_Systems/Nonlinear_Distortions#The_distortion_factor|Linear and Time Invariant Systems]].
+*The SNR&nbsp; $ρ_v$&nbsp; $($but not &nbsp;$K)$&nbsp; can also be used as a quality criterion if &nbsp;$q(t)$&nbsp; contains more than one frequency &nbsp; ⇒  &nbsp; derivation in the book&nbsp;[[Linear_and_Time_Invariant_Systems/Nonlinear_Distortions#The_distortion_factor|"Linear and Time Invariant Systems"]].
 ==Considering channel distortions==
@@ Line 289: / Line 289: @@
 The result of the envelope demodulator&nbsp; $\rm (ED)$&nbsp; is marked with solid lines,&nbsp; while the dotted lines represent the synchronous demodulator&nbsp; $\rm (SD)$.
-As already derived in the section&nbsp;[[Modulation_Methods/Quality_Criteria#Investigations_with_regard_to_signal_distortions|Investigations with regard to signal distortions]],&nbsp; the results for the synchronous demodulator in this double-logarithmic representation are
+As already derived in the section&nbsp;[[Modulation_Methods/Quality_Criteria#Investigations_with_regard_to_signal_distortions|"Investigations with regard to signal distortions"]],&nbsp; the results for the synchronous demodulator in this double-logarithmic representation are
 *the angle bisector&nbsp; (for &nbsp;$m → ∞$,&nbsp; "DSB–AM without carrier"),&nbsp; and
 *a shifted line parallel to it for&nbsp; "DSB–AM with carrier", with a vertical distance of &nbsp;$10 · \lg \, (1 + 2/m^2$).
@@ Line 307: / Line 307: @@
 The most important reason for the use of the envelope demodulator is&nbsp; (better said:&nbsp; was)&nbsp; that it avoids the often laborious need for frequency and phase synchronization,&nbsp; so that it can be realized cheaply.&nbsp;  The envelope demodulator is thus an example of an&nbsp; "incoherent demodulator".
-On the other hand,&nbsp; several reasons can be given in favour of the&nbsp; [[Modulation_Methods/Synchronous_Demodulation|synchronous demodulator]]&nbsp; and against the envelope demodulator:
+On the other hand,&nbsp; several reasons can be given in favour of the&nbsp; [[Modulation_Methods/Synchronous_Demodulation|$\text{synchronous demodulator}$]]&nbsp; and against the envelope demodulator:
 *With envelope demodulation,&nbsp; an over-modulation &nbsp;$(m > 1)$&nbsp; must be avoided under all circumstances.&nbsp; This can be achieved for example by limiting the amplitude of the source signal,&nbsp; though this also results in nonlinear distortions.
 *All else being equal,&nbsp; the modulation depth &nbsp;$m < 1$&nbsp; can only be achieved by increasing the transmitted power by at least a factor of&nbsp; $3$.&nbsp; This is also problematic because of justified discussions on the topic of&nbsp; "electronic smog"&nbsp; in large parts of our society.

← Older revision		Revision as of 10:02, 17 November 2022
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−	The physical signals  $q(t)$,  $r(t)$ and  $v(t)$  corresponding to this example can be found in the graph on the  [[Modulation_Methods/Envelope_Demodulation#Why_does_envelope_demodulation_fail_for_.7F.27.22.60UNIQ-MathJax104-QINU.60.22.27.7F.3F\|previous ~~page~~]].}}	+	The physical signals  $q(t)$,  $r(t)$ and  $v(t)$  corresponding to this example can be found in the graph in the  [[Modulation_Methods/Envelope_Demodulation#Why_does_envelope_demodulation_fail_for_.7F.27.22.60UNIQ-MathJax104-QINU.60.22.27.7F.3F\|previous section]].}}

	==Special case of a cosine-shaped message signal==		==Special case of a cosine-shaped message signal==

@@ Line 310: / Line 310: @@
 *With envelope demodulation,&nbsp; an over-modulation &nbsp;$(m > 1)$&nbsp; must be avoided under all circumstances.&nbsp; This can be achieved for example by limiting the amplitude of the source signal,&nbsp; though this also results in nonlinear distortions.
 *All else being equal,&nbsp; the modulation depth &nbsp;$m < 1$&nbsp; can only be achieved by increasing the transmitted power by at least a factor of&nbsp; $3$.&nbsp; This is also problematic because of justified discussions on the topic of&nbsp; "electronic smog"&nbsp; in large parts of our society.
-*Linear channel distortions can lead in the case of an envelope demodulator to irreversible nonlinear channel distortions,&nbsp; whereas the linear distortions occurring during synchronous demodulation can possibly be compensated for by special measures taken at the receiver.
+*Linear channel distortions  in the case of an envelope demodulator can lead to irreversible nonlinear channel distortions,&nbsp; whereas the linear distortions occurring during synchronous demodulation can possibly be compensated for by special measures taken at the receiver.
 ==Exercises for the chapter==
 <br>
-[[Aufgaben:Aufgabe_2.7:_Ist_der_Modulationsgrad_zu_groß%3F|Exercise 2.7: Is the Modulation Depth Too High?]]
+[[Aufgaben:Exercise_2.7:_Is_the_Modulation_Depth_Too_High%3F|Exercise 2.7: Is the Modulation Depth Too High?]]
-[[Aufgaben:Aufgabe_2.7Z:_ZSB-AM_und_Hüllkurvendemodulator|Exercise 2.7Z: DSB-AM and Envelope Demodulator]]
+[[Aufgaben:Exercise_2.7Z:_DSB-AM_and_Envelope_Demodulator|Exercise 2.7Z: DSB-AM and Envelope Demodulator]]
-[[Aufgaben:Aufgabe_2.8:_Unsymmetrischer_Kanal|Exercise 2.8: Asymmetrical Channel]]
+[[Aufgaben:Exercise_2.8:_Asymmetrical_Channel|Exercise 2.8: Asymmetrical Channel]]
-[[Aufgaben:Aufgabe_2.9:_Symmetrische_Verzerrungen|Exercise 2.9: Symmetrical Distortions]]
+[[Aufgaben:Exercise_2.9:_Symmetrical_Distortions|Exercise 2.9: Symmetrical Distortions]]