Modulation Methods/Double-Sideband Amplitude Modulation - Revision history

Hwang at 12:42, 18 January 2023

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Guenter at 17:35, 12 January 2023

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Hwang at 19:49, 2 January 2023

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Hwang at 19:42, 2 January 2023

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Hwang at 17:24, 2 January 2023

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Hwang at 17:15, 2 January 2023

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Hwang at 10:14, 17 November 2022

2022-11-17T10:14:03Z

Reed at 14:40, 24 March 2022

2022-03-24T14:40:22Z

@@ Line 124: / Line 124: @@
 *The Fourier series representation of the periodic rectangular signal&nbsp;$z_{\rm R}(t)$&nbsp; is:
 :$$z_{\rm R}(t) = \frac{4}{\pi} \cdot \cos(\omega_{\rm T}\cdot t)-\frac{4}{3\pi} \cdot \cos(3\omega_{\rm T}\cdot t) +\frac{4}{5\pi} \cdot \cos(5\omega_{\rm T}\cdot t)- \text{ ...}$$
-*The associated spectral function consists of Dirac lines at &nbsp;$±f_{\rm T}, ±3f_{\rm T}, ±5f_{\rm T}$, etc.&nbsp; Convolution with &nbsp;$Q(f)$&nbsp; leads to the spectrum <br>(the subscript stands for "ring modulator"):
+*The associated spectral function consists of Dirac delta lines at &nbsp;$±f_{\rm T}, ±3f_{\rm T}, ±5f_{\rm T}$, etc.&nbsp; Convolution with &nbsp;$Q(f)$&nbsp; leads to the spectrum <br>(the subscript stands for "ring modulator"):
 :$$S_{\rm RM}(f) = \frac{2}{\pi} \cdot Q (f \pm f_{\rm T})-\frac{2}{3\pi} \cdot Q (f \pm 3f_{\rm
   T})+\frac{2}{5\pi} \cdot Q (f \pm 5f_{\rm T}) -\text{ ...} \hspace{0.05cm}$$
@@ Line 154: / Line 154: @@
 This result,&nbsp; which would also have been arrived at via convolution,&nbsp; states:
-#The spectrum consists of four Dirac lines at frequencies &nbsp;$±(f_{\rm T} + f_{\rm N})$&nbsp; and &nbsp;$±(f_{\rm T} - f_{\rm N})$.&nbsp;
+#The spectrum consists of four Dirac delta lines at frequencies &nbsp;$±(f_{\rm T} + f_{\rm N})$&nbsp; and &nbsp;$±(f_{\rm T} - f_{\rm N})$.&nbsp;
 #In both bracket expressions, the first Dirac delta function indicates the one for positive frequencies.
 #The weights of all Dirac delta functions are equal and each is &nbsp;$A_{\rm N}/4$.&nbsp;
 #The sum of these weights &nbsp; - that is, the integral over&nbsp;  $S(f)$ - &nbsp; is equal to the signal value&nbsp; $s(t = 0) = A_{\rm N}$.
-#The Dirac lines remain for &nbsp;$ϕ_{\rm T} ≠ 0$&nbsp; and/or &nbsp;$ϕ_{\rm N} ≠ 0$&nbsp; at the same frequencies.&nbsp; However, complex rotation factors must then be added to the weights &nbsp;$A_{\rm N}/4$&nbsp;.
+#The Dirac delta lines remain for &nbsp;$ϕ_{\rm T} ≠ 0$&nbsp; and/or &nbsp;$ϕ_{\rm N} ≠ 0$&nbsp; at the same frequencies.&nbsp; However, complex rotation factors must then be added to the weights &nbsp;$A_{\rm N}/4$&nbsp;.
 {{GraueBox|TEXT=
-$\text{Example 3:}$&nbsp; The following diagram shows the spectral functions &nbsp;$S(f)$&nbsp; for different values of &nbsp;$ϕ_{\rm T}$&nbsp; and &nbsp;$ϕ_{\rm N}$,&nbsp; respectively.&nbsp; The other parameters are assumed to be &nbsp;$f_{\rm T} = 50\text{ kHz}$, &nbsp;$f_{\rm N} = 10\text{ kHz}$&nbsp; and &nbsp;$A_{\rm N} = 4\text{ V}$.&nbsp; Thus,&nbsp; the magnitudes of all Dirac lines are&nbsp;$A_{\rm N}/4 = 1\text{ V}$.
+$\text{Example 3:}$&nbsp; The following diagram shows the spectral functions &nbsp;$S(f)$&nbsp; for different values of &nbsp;$ϕ_{\rm T}$&nbsp; and &nbsp;$ϕ_{\rm N}$,&nbsp; respectively.&nbsp; The other parameters are assumed to be &nbsp;$f_{\rm T} = 50\text{ kHz}$, &nbsp;$f_{\rm N} = 10\text{ kHz}$&nbsp; and &nbsp;$A_{\rm N} = 4\text{ V}$.&nbsp; Thus,&nbsp; the magnitudes of all Dirac delta lines are&nbsp;$A_{\rm N}/4 = 1\text{ V}$.
 [[File:EN_Mod_T_2_1_S4.png|right|frame|Typical spectra for DSB-AM]]

@@ Line 8: / Line 8: @@
 == # OVERVIEW OF THE SECOND MAIN CHAPTER # ==
 <br>
-After some general explanations of modulation and demodulation,&nbsp; now a detailed description of&nbsp; "amplitude modulation"&nbsp; and the associated&nbsp; "demodulators".&nbsp;This chapter deals in detail with:
+After some general explanations of modulation and demodulation,&nbsp; now a detailed description of&nbsp; '''&raquo;amplitude modulation&laquo;'''&nbsp; and the associated&nbsp; '''&raquo;demodulators&laquo;'''.&nbsp;
-*the description and realization of&nbsp; "double-sideband amplitude modulation"&nbsp;  $\text{(DSB–AM)}$&nbsp; in frequency and time domain,
-*the characteristics of a&nbsp; "synchronous demodulator"&nbsp; and the possible applications of an&nbsp; "envelope demodulator",
+This chapter deals in detail with:
-*the similarities/differences of&nbsp; "single-sideband modulation"&nbsp; $\text{(SSB–AM)}$&nbsp; compared to DSB-AM and &nbsp; "modified AM methods".
+*the description and realization of&nbsp; &raquo;double-sideband amplitude modulation&laquo;&nbsp;  $\text{(DSB–AM)}$&nbsp; in frequency and time domain,
@@ Line 17: / Line 21: @@
 <br>
 We consider the following problem:&nbsp; a source signal&nbsp;$q(t)$,&nbsp; whose spectrum &nbsp;$Q(f)$&nbsp; is bandlimited to the range &nbsp;$\pm B_{\rm NF}$&nbsp; (subscript&nbsp; "NF"&nbsp; from German "Niederfrequenz" &nbsp; ⇒ &nbsp; low frequency),&nbsp;
 * is to be shifted to a higher frequency range where the channel frequency response&nbsp; $H_{\rm K}(f)$&nbsp; has favorable characteristics,&nbsp;
 *using a harmonic oscillation of frequency &nbsp;$f_{\rm T}$, which we will refer to as the carrier signal&nbsp; $z(t)$.
@@ Line 25: / Line 30: @@
 [[File:P_ID974__Mod_T_2_1_S1a_neu.png |right|frame|Representation of amplitude modulation in the frequency domain]]
 *The spectrum &nbsp;$Q(f)$&nbsp; drawn here is schematic.&nbsp; It states that only spectral components in the range &nbsp;$|f| ≤ B_{\rm NF}$&nbsp; are included in &nbsp;$q(t)$.&nbsp; $Q(f)$&nbsp; could also be a line spectrum.
 *Let the channel be ideal in a bandwidth range &nbsp;$B_{\rm K}$&nbsp; around frequency&nbsp; $f_{\rm M}$,&nbsp; that is, let &nbsp;$H_{\rm K}(f) = 1$&nbsp; for &nbsp;$|f - f_{\rm M}| ≤ B_{\rm K}/2.$&nbsp; Impairments by noise are ignored for now.
 *Let the carrier signal be cosine &nbsp; $($phase &nbsp;$ϕ_{\rm T} = 0)$&nbsp; and have amplitude&nbsp; $A_{\rm T} = 1$&nbsp; (without a unit).&nbsp; Let the carrier frequency &nbsp;$f_{\rm T}$&nbsp; be equal to the center frequency of the transmission band.
 *Thus,&nbsp; the spectrum of the carrier signal &nbsp;$z(t) = \cos(ω_{\rm T} · t)$&nbsp; is <br>(plotted in green in the graph):
 :$$Z(f) = {1}/{2} \cdot \delta (f + f_{\rm T})+{1}/{2} \cdot \delta (f - f_{\rm T})\hspace{0.05cm}.$$
@@ Line 41: / Line 49: @@
 The diagram displays the result.&nbsp; One can identify the following characteristics:
 [[File:EN_Mod_T_2_1_S1b.png|right|frame|Spectrum of double-sideband amplitude modulation without carrier; <br>other name:&nbsp; "double-sideband amplitude modulation with carrier suppression"]]
 *Due to the system-theoretic approach with positive and negative frequencies,&nbsp;$S(f)$&nbsp; is composed of two parts around &nbsp;$\pm f_{\rm T}$,&nbsp; each of which have the same shape as &nbsp;$Q(f)$.
 *The factor &nbsp;$1/2$&nbsp; results from the carrier amplitude &nbsp;$A_{\rm T} = 1$.&nbsp; Thus,&nbsp; $s(t = 0) = q(t = 0)$&nbsp;  and the integrals over their spectral functions&nbsp; $S(f)$&nbsp; and &nbsp;$Q(f)$&nbsp; must also be equal.
-*The channel bandwidth &nbsp;$B_{\rm K}$&nbsp;  must be at least twice the signal bandwidth &nbsp;$B_{\rm NF}$,&nbsp; which gives the name&nbsp; <br>&raquo;'''Double-Sideband Amplitude Modulation'''&laquo;&nbsp; $\text{(DSB–AM)}$.
+*The factor &nbsp;$1/2$&nbsp; results from the carrier amplitude &nbsp;$A_{\rm T} = 1$.&nbsp; Thus,&nbsp; $s(t = 0) = q(t = 0)$&nbsp;  and the integrals over their spectral functions&nbsp; $S(f)$&nbsp; and &nbsp;$Q(f)$&nbsp; must also be equal.
 *It should be noted that&nbsp;$B_{\rm NF}$ and $B_{\rm K}$&nbsp; are absolute and &nbsp;[[Signal_Representation/Fourier_Transform_Theorems#Reciprocity_Theorem_of_time_duration_and_bandwidth|$\text{non-equivalent bandwidths}$]].&nbsp; The latter are defined over rectangles of equal area and are denoted  in our tutorial by&nbsp; $Δf_q$&nbsp; and&nbsp; $Δf_{\rm K}$,&nbsp; resp.
 *The spectral function &nbsp;$S(f)$&nbsp; does not include any Dirac-lines at the carrier frequency &nbsp;$(\pm f_{\rm T})$.&nbsp; Therefore, this method is also referred to as&nbsp; "DSB-AM '''without carrier'''".
 *The frequency components
 :*above the carrier frequency  &nbsp;$f_{\rm T}$&nbsp; are called the&nbsp; "upper sideband"&nbsp; $\rm  (USB)$,
@@ Line 63: / Line 76: @@
 According to this equation, two models can be given for double-sideband amplitude modulation.&nbsp;  These are to be interpreted as follows:
 *The upper model directly describes the relationship given above,&nbsp; where the carrier &nbsp;$z(t) = \cos(ω_{\rm T}t + ϕ_{\rm T})$&nbsp; is applied without a unit.
 *The lower model is more in line with the physical condition&nbsp; "each signal also has a unit".&nbsp;  If &nbsp;$q(t)$&nbsp; and &nbsp;$z(t)$&nbsp; are voltages,&nbsp; the model still needs to provide a scaling with the modulator constant &nbsp;$K_{\rm AM}$&nbsp; with unit&nbsp; &nbsp;$\rm 1/V$,&nbsp; so that the output signal &nbsp;$s(t)$&nbsp; also represents a voltage waveform.
 *If we set &nbsp;$K_{\rm AM} = 1/A_{\rm T}$,&nbsp; both models are the same.&nbsp; In the following,&nbsp; we will always assume the lower, simpler model.
 <br clear=all>
@@ Line 82: / Line 97: @@
 [[File:P_ID978__Mod_T_2_1_S3a_neu.png|right|frame|Ring modulator to realize DSB–AM without carrier]]
 *Let the amplitude of the harmonic carrier oscillation&nbsp; $z(t)$&nbsp; be much larger than the maximum value &nbsp;$q_{\rm max}$&nbsp; of the source signal&nbsp; $q(t)$.&nbsp; Thus,&nbsp; all diodes are operated as switches.
 *When the half-wave of the carrier is positive &nbsp;$(z(t) > 0)$&nbsp; the two magenta diodes conduct while the light green ones block. Thus, without considering losses,&nbsp; it holds&nbsp; $s(t) = q(t)$.
 *For a negative half-wave,&nbsp; the light green diodes conduct and the diodes in the longitudinal branches block.&nbsp; As can be seen on the right,&nbsp; $s(t) = \ – q(t)$&nbsp; holds for this lower switch position.
 *Due to the operation of this switch,&nbsp; the harmonic oscillation&nbsp; $z(t)$&nbsp; can also be replaced by a periodic&nbsp; (rectangular)&nbsp; square wave signal with identical period duration:
 :$$z_{\rm R}(t) = \left\{ \begin{array}{c} +1 \\ -1 \\  \end{array} \right.\quad \begin{array}{*{10}c}    {\rm{for}} \\  {\rm{for}}   \\ \end{array}\begin{array}{*{20}c} {z(t) >0,}  \\ {z(t) <0.}  \\ \end{array}$$
-*The modulated signal&nbsp; $s(t)$&nbsp; is then obtained as the product of the source signal &nbsp;$q(t)$&nbsp; and this square wave '''KORREKTUR: rectangular''' signal  &nbsp;$z_{\rm R}(t)$,&nbsp; whereas in ideal DSB-AM one multiplies&nbsp;$q(t)$&nbsp; by a cosine signal.&nbsp; The carrier &nbsp;$z(t)$&nbsp; is not itself included in the signal&nbsp;$s(t)$.&nbsp; Since this is supplied via the center taps of the transformers, the induced voltages cancel out &nbsp; &rArr; &nbsp;  &raquo;'''DSB-AM without carrier'''&laquo;.
@@ Line 118: / Line 137: @@
 Please note: &nbsp; Since we are describing modulation processes,&nbsp; the phase term is used with a plus sign in the above equations.
 *Thus, &nbsp;$ϕ_{\rm N} = - 90^\circ$&nbsp; represents a sinusoidal input signal &nbsp;$q(t)$&nbsp; and&nbsp; $ϕ_{\rm T} = - 90^\circ$&nbsp; denotes a sinusoidal carrier signal&nbsp; $z(t)$.
 *Therefore,&nbsp; the equation for the modulated signal is:
 :$$s(t) = q(t) \cdot z(t) = A_{\rm N} \cdot \cos(\omega_{\rm N} t + \phi_{\rm
@@ Line 134: / Line 154: @@
 This result,&nbsp; which would also have been arrived at via convolution,&nbsp; states:
-*The spectrum consists of four Dirac lines at frequencies &nbsp;$±(f_{\rm T} + f_{\rm N})$&nbsp; and &nbsp;$±(f_{\rm T} - f_{\rm N})$.&nbsp;
+#The spectrum consists of four Dirac lines at frequencies &nbsp;$±(f_{\rm T} + f_{\rm N})$&nbsp; and &nbsp;$±(f_{\rm T} - f_{\rm N})$.&nbsp;
-*In both bracket expressions, the first Dirac delta function indicates the one for positive frequencies.
+#In both bracket expressions, the first Dirac delta function indicates the one for positive frequencies.
-*The weights of all Dirac delta functions are equal and each is &nbsp;$A_{\rm N}/4$.&nbsp;
+#The weights of all Dirac delta functions are equal and each is &nbsp;$A_{\rm N}/4$.&nbsp;
-*The sum of these weights &nbsp; - that is, the integral over&nbsp;  $S(f)$ - &nbsp; is equal to the signal value&nbsp; $s(t = 0) = A_{\rm N}$.
+#The sum of these weights &nbsp; - that is, the integral over&nbsp;  $S(f)$ - &nbsp; is equal to the signal value&nbsp; $s(t = 0) = A_{\rm N}$.
-*The Dirac lines remain for &nbsp;$ϕ_{\rm T} ≠ 0$&nbsp; and/or &nbsp;$ϕ_{\rm N} ≠ 0$&nbsp; at the same frequencies.&nbsp; However, complex rotation factors must then be added to the weights &nbsp;$A_{\rm N}/4$&nbsp;.
+#The Dirac lines remain for &nbsp;$ϕ_{\rm T} ≠ 0$&nbsp; and/or &nbsp;$ϕ_{\rm N} ≠ 0$&nbsp; at the same frequencies.&nbsp; However, complex rotation factors must then be added to the weights &nbsp;$A_{\rm N}/4$&nbsp;.

@@ Line 210: / Line 210: @@
 <br>
 [[File:Mod_T_2_1_S6_version2.png|right|frame|Spectrum of the analytical signal in two different viewpoints]]
-In the further course of this chapter,&nbsp; for the sake of simplifying the graphs,&nbsp; the spectrum &nbsp;$S_+(f)$&nbsp; of the &nbsp;[[Modulation_Methods/General_Model_of_Modulation#Beschreibung_des_physikalischen_Signals_mit_Hilfe_des_analytischen_Signals|$\text{analytical signal}$]]&nbsp;  is usually given instead of the actual, physical spectrum &nbsp;$S(f)$.
+In the further course of this chapter,&nbsp; for the sake of simplifying the graphs,&nbsp; the spectrum &nbsp;$S_+(f)$&nbsp; of the &nbsp;[[Modulation_Methods/General_Model_of_Modulation#Describing_the_physical_signal_using_the_analytic_signal|$\text{analytical signal}$]]&nbsp;  is usually given instead of the actual, physical spectrum &nbsp;$S(f)$.
 As an example,&nbsp; let us consider&nbsp; "DSB-AM with carrier"&nbsp; and the following signals:

@@ Line 137: / Line 137: @@
 *In both bracket expressions, the first Dirac delta function indicates the one for positive frequencies.
 *The weights of all Dirac delta functions are equal and each is &nbsp;$A_{\rm N}/4$.&nbsp;
-*The sum of these weights &nbsp; - that is, the integral over&nbsp;  $S(f)$ –&nbsp; is equal to the signal value&nbsp; $s(t = 0) = A_{\rm N}$.
+*The sum of these weights &nbsp; - that is, the integral over&nbsp;  $S(f)$ - &nbsp; is equal to the signal value&nbsp; $s(t = 0) = A_{\rm N}$.
 *The Dirac lines remain for &nbsp;$ϕ_{\rm T} ≠ 0$&nbsp; and/or &nbsp;$ϕ_{\rm N} ≠ 0$&nbsp; at the same frequencies.&nbsp; However, complex rotation factors must then be added to the weights &nbsp;$A_{\rm N}/4$&nbsp;.

@@ Line 86: / Line 86: @@
 *Due to the operation of this switch,&nbsp; the harmonic oscillation&nbsp; $z(t)$&nbsp; can also be replaced by a periodic&nbsp; (rectangular)&nbsp; square wave signal with identical period duration:
 :$$z_{\rm R}(t) = \left\{ \begin{array}{c} +1 \\ -1 \\  \end{array} \right.\quad \begin{array}{*{10}c}    {\rm{for}} \\  {\rm{for}}   \\ \end{array}\begin{array}{*{20}c} {z(t) >0,}  \\ {z(t) <0.}  \\ \end{array}$$
-*The modulated signal&nbsp; $s(t)$&nbsp; is then obtained as the product of the source signal &nbsp;$q(t)$&nbsp; and this square wave signal  &nbsp;$z_{\rm R}(t)$,&nbsp; whereas in ideal DSB-AM one multiplies&nbsp;$q(t)$&nbsp; by a cosine signal.&nbsp; The carrier &nbsp;$z(t)$&nbsp; is not itself included in the signal&nbsp;$s(t)$.&nbsp; Since this is supplied via the center taps of the transformers, the induced voltages cancel out &nbsp; &rArr; &nbsp;  &raquo;'''DSB-AM without carrier'''&laquo;.
+*The modulated signal&nbsp; $s(t)$&nbsp; is then obtained as the product of the source signal &nbsp;$q(t)$&nbsp; and this square wave '''KORREKTUR: rectangular''' signal  &nbsp;$z_{\rm R}(t)$,&nbsp; whereas in ideal DSB-AM one multiplies&nbsp;$q(t)$&nbsp; by a cosine signal.&nbsp; The carrier &nbsp;$z(t)$&nbsp; is not itself included in the signal&nbsp;$s(t)$.&nbsp; Since this is supplied via the center taps of the transformers, the induced voltages cancel out &nbsp; &rArr; &nbsp;  &raquo;'''DSB-AM without carrier'''&laquo;.

@@ Line 58: / Line 58: @@
 This result is still true if the restrictions made in the last section are removed:
 #real-valued spectrum &nbsp;$Q(f)$,&nbsp;
-#carrier phase &nbsp;$ϕ_{\rm T} = 0$&nbsp;
+#carrier phase &nbsp;$ϕ_{\rm T} = 0$.&nbsp;
 In general,&nbsp; this results in a complex-valued spectrum  &nbsp;$S(f)$.

@@ Line 30: / Line 30: @@
 :$$Z(f) = {1}/{2} \cdot \delta (f + f_{\rm T})+{1}/{2} \cdot \delta (f - f_{\rm T})\hspace{0.05cm}.$$
-Those familiar with the &nbsp;[[Signal_Representation/Fourier_Transform_Laws|laws of spectral transformation]]&nbsp; and in particular with the &nbsp;[[Signal_Representation/The_Convolution_Theorem_and_Operation|Convolution Theorem]]&nbsp; can immediately give a solution for the spectrum &nbsp;$S(f)$&nbsp; of the modulator output signal:
+Those familiar with the &nbsp;[[Signal_Representation/Fourier_Transform_Laws|$\text{laws of spectral transformation}$]]&nbsp; and in particular with the &nbsp;[[Signal_Representation/The_Convolution_Theorem_and_Operation|$\text{Convolution Theorem}$]]&nbsp; can immediately give a solution for the spectrum &nbsp;$S(f)$&nbsp; of the modulator output signal:
 :$$S(f)= Z(f) \star Q(f) = 1/2 \cdot \delta (f + f_{\rm T})\star Q(f)+1/2 \cdot \delta (f - f_{\rm T})\star Q(f) = 1/2 \cdot Q (f + f_{\rm T})+ 1/2 \cdot Q(f - f_{\rm T})
   \hspace{0.05cm}.$$
@@ Line 36: / Line 36: @@
 {{BlaueBox|TEXT=
 $\text{Please note:}$&nbsp; This equation takes into account
-*that the convolution of a shifted Dirac delta function  &nbsp;$δ(x - x_0)$&nbsp; with an arbitrary function&nbsp;$f(x)$&nbsp; yields the &nbsp; '''shifted function''' &nbsp;$f(x - x_0)$.}}
+*that the convolution of a shifted Dirac delta function  &nbsp;$δ(x - x_0)$&nbsp; with an arbitrary function&nbsp;$f(x)$&nbsp; yields the &nbsp; &raquo;'''shifted function'''&laquo; &nbsp;$f(x - x_0)$.}}
@@ Line 43: / Line 43: @@
 *Due to the system-theoretic approach with positive and negative frequencies,&nbsp;$S(f)$&nbsp; is composed of two parts around &nbsp;$\pm f_{\rm T}$,&nbsp; each of which have the same shape as &nbsp;$Q(f)$.
 *The factor &nbsp;$1/2$&nbsp; results from the carrier amplitude &nbsp;$A_{\rm T} = 1$.&nbsp; Thus,&nbsp; $s(t = 0) = q(t = 0)$&nbsp;  and the integrals over their spectral functions&nbsp; $S(f)$&nbsp; and &nbsp;$Q(f)$&nbsp; must also be equal.
-*The channel bandwidth &nbsp;$B_{\rm K}$&nbsp;  must be at least twice the signal bandwidth &nbsp;$B_{\rm NF}$,&nbsp; which gives the name&nbsp; <br>'''Double-Sideband Amplitude Modulation'''&nbsp; $\text{(DSB–AM)}$.
+*The channel bandwidth &nbsp;$B_{\rm K}$&nbsp;  must be at least twice the signal bandwidth &nbsp;$B_{\rm NF}$,&nbsp; which gives the name&nbsp; <br>&raquo;'''Double-Sideband Amplitude Modulation'''&laquo;&nbsp; $\text{(DSB–AM)}$.
-*It should be noted that&nbsp;$B_{\rm NF}$ and $B_{\rm K}$&nbsp; are absolute and &nbsp;[[Signal_Representation/Fourier_Transform_Laws#Reziprozit.C3.A4tsgesetz_von_Zeitdauer_und_Bandbreite|non-equivalent bandwidths]].&nbsp; The latter are defined over rectangles of equal area and are denoted  in our tutorial by&nbsp; $Δf_q$&nbsp; and&nbsp; $Δf_{\rm K}$,&nbsp; resp.
+*It should be noted that&nbsp;$B_{\rm NF}$ and $B_{\rm K}$&nbsp; are absolute and &nbsp;[[Signal_Representation/Fourier_Transform_Laws#Reziprozit.C3.A4tsgesetz_von_Zeitdauer_und_Bandbreite|$\text{non-equivalent bandwidths}$]].&nbsp; The latter are defined over rectangles of equal area and are denoted  in our tutorial by&nbsp; $Δf_q$&nbsp; and&nbsp; $Δf_{\rm K}$,&nbsp; resp.
 *The spectral function &nbsp;$S(f)$&nbsp; does not include any Dirac-lines at the carrier frequency &nbsp;$(\pm f_{\rm T})$.&nbsp; Therefore, this method is also referred to as&nbsp; "DSB-AM '''without carrier'''".
 *The frequency components
@@ Line 77: / Line 77: @@
 ==Ring modulator==
 <br>
-One possibility to realize&nbsp; "double-sideband amplitude modulation with carrier suppression"&nbsp; is offered by a so-called&nbsp; '''ring modulator''',&nbsp; also known as&nbsp; "double push-pull diode modulator".&nbsp;  Below you can see the circuit on the left and a simple functional diagram on the right.
+One possibility to realize&nbsp; "double-sideband amplitude modulation with carrier suppression"&nbsp; is offered by a so-called&nbsp; &raquo;'''ring modulator'''&laquo;,&nbsp; also known as&nbsp; "double push-pull diode modulator".&nbsp;  Below you can see the circuit on the left and a simple functional diagram on the right.
 Without claiming to be complete, the principle can be stated as follows:
@@ Line 86: / Line 86: @@
 *Due to the operation of this switch,&nbsp; the harmonic oscillation&nbsp; $z(t)$&nbsp; can also be replaced by a periodic&nbsp; (rectangular)&nbsp; square wave signal with identical period duration:
 :$$z_{\rm R}(t) = \left\{ \begin{array}{c} +1 \\ -1 \\  \end{array} \right.\quad \begin{array}{*{10}c}    {\rm{for}} \\  {\rm{for}}   \\ \end{array}\begin{array}{*{20}c} {z(t) >0,}  \\ {z(t) <0.}  \\ \end{array}$$
-*The modulated signal&nbsp; $s(t)$&nbsp; is then obtained as the product of the source signal &nbsp;$q(t)$&nbsp; and this square wave signal  &nbsp;$z_{\rm R}(t)$,&nbsp; whereas in ideal DSB-AM one multiplies&nbsp;$q(t)$&nbsp; by a cosine signal.&nbsp; The carrier &nbsp;$z(t)$&nbsp; is not itself included in the signal&nbsp;$s(t)$.&nbsp; Since this is supplied via the center taps of the transformers, the induced voltages cancel out &nbsp; &rArr; &nbsp;  '''DSB-AM without carrier'''.
+*The modulated signal&nbsp; $s(t)$&nbsp; is then obtained as the product of the source signal &nbsp;$q(t)$&nbsp; and this square wave signal  &nbsp;$z_{\rm R}(t)$,&nbsp; whereas in ideal DSB-AM one multiplies&nbsp;$q(t)$&nbsp; by a cosine signal.&nbsp; The carrier &nbsp;$z(t)$&nbsp; is not itself included in the signal&nbsp;$s(t)$.&nbsp; Since this is supplied via the center taps of the transformers, the induced voltages cancel out &nbsp; &rArr; &nbsp;  &raquo;'''DSB-AM without carrier'''&laquo;.
@@ Line 195: / Line 195: @@
 A comparison of these signal waveforms shows:
 *By adding the DC component &nbsp;$A_{\rm T}$&nbsp; the signal &nbsp;$q(t)$&nbsp; can now be seen in the envelope of &nbsp;$s(t)$.
-*Thus,&nbsp; [[Modulation_Methods/Envelope_Demodulation|envelope demodulation]]&nbsp; can be applied,&nbsp; which is easier and cheaper to implement than coherent &nbsp;[[Modulation_Methods/Synchronous_Demodulation|synchronous demodulation]].
+*Thus,&nbsp; [[Modulation_Methods/Envelope_Demodulation|$\text{envelope demodulation}$]]&nbsp; can be applied,&nbsp; which is easier and cheaper to implement than coherent &nbsp;[[Modulation_Methods/Synchronous_Demodulation|$\text{synchronous demodulation}$]].
 *However,&nbsp; a prerequisite for the application of the envelope demodulator is a modulation depth &nbsp;$m <1$.&nbsp;
 *This parameter is defined as follows:
@@ Line 210: / Line 210: @@
 <br>
 [[File:Mod_T_2_1_S6_version2.png|right|frame|Spectrum of the analytical signal in two different viewpoints]]
-In the further course of this chapter,&nbsp; for the sake of simplifying the graphs,&nbsp; the spectrum &nbsp;$S_+(f)$&nbsp; of the &nbsp;[[Modulation_Methods/General_Model_of_Modulation#Beschreibung_des_physikalischen_Signals_mit_Hilfe_des_analytischen_Signals|analytical signal]]&nbsp;  is usually given instead of the actual, physical spectrum &nbsp;$S(f)$.
+In the further course of this chapter,&nbsp; for the sake of simplifying the graphs,&nbsp; the spectrum &nbsp;$S_+(f)$&nbsp; of the &nbsp;[[Modulation_Methods/General_Model_of_Modulation#Beschreibung_des_physikalischen_Signals_mit_Hilfe_des_analytischen_Signals|$\text{analytical signal}$]]&nbsp;  is usually given instead of the actual, physical spectrum &nbsp;$S(f)$.
 As an example,&nbsp; let us consider&nbsp; "DSB-AM with carrier"&nbsp; and the following signals:
@@ Line 228: / Line 228: @@
 ==Amplitude modulation with a quadratic characteristic curve==
 <br>
-Nonlinearities are usually undesirable and troublesome in Communications Engineering.&nbsp; As explained in the chapter &nbsp;[[Linear_and_Time_Invariant_Systems/Nonlinear_Distortions|Nonlinear Distortions]] &nbsp; of the book&nbsp; "Linear and Time Invariant Systems”,&nbsp; they lead to the facts that:
+Nonlinearities are usually undesirable and troublesome in Communications Engineering.&nbsp; As explained in the chapter &nbsp;[[Linear_and_Time_Invariant_Systems/Nonlinear_Distortions|"Nonlinear Distortions"]] &nbsp; of the book&nbsp; "Linear and Time Invariant Systems”,&nbsp; they lead to the facts that:
 *the superposition principle is no longer applicable,

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 :$$S(f)  = Z(f) \star Q(f)\hspace{0.2cm}\bullet\!\!-\!\!\!-\!\!\!-\!\!\circ\, \hspace{0.2cm} s(t) = q(t) \cdot z(t) = q(t) \cdot \cos(\omega_{\rm T}\cdot t + \phi_{\rm T})\hspace{0.05cm}.$$
 [[File:Mod_T_2_1_S2a_version2.png|right|frame|Models of DSB–AM without carrier]]
-This result is still true if the restrictions made on the last page are removed:
+This result is still true if the restrictions made in the last section are removed:
 #real-valued spectrum &nbsp;$Q(f)$,&nbsp;
 #carrier phase &nbsp;$ϕ_{\rm T} = 0$&nbsp;

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 This result,&nbsp; which would also have been arrived at via convolution,&nbsp; states:
 *The spectrum consists of four Dirac lines at frequencies &nbsp;$±(f_{\rm T} + f_{\rm N})$&nbsp; and &nbsp;$±(f_{\rm T} - f_{\rm N})$.&nbsp;
-*In both bracket expressions the first Dirac delta function indicates the one for positive frequencies.
+*In both bracket expressions, the first Dirac delta function indicates the one for positive frequencies.
 *The weights of all Dirac delta functions are equal and each is &nbsp;$A_{\rm N}/4$.&nbsp;
 *The sum of these weights &nbsp; - that is, the integral over&nbsp;  $S(f)$ –&nbsp; is equal to the signal value&nbsp; $s(t = 0) = A_{\rm N}$.
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 [[File:EN_Mod_T_2_1_S4.png|right|frame|Typical spectra for DSB-AM]]
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-*The upper left diagram shows the case just discussed:&nbsp; Both the carrier and the source signal are cosine.&nbsp; Thus,&nbsp; the amplitude modulated signal &nbsp;$s(t)$&nbsp; is composed of two cosine oscillations with &nbsp;$ω_{60} = 2 π · 60\text{ kHz}$&nbsp; and &nbsp;$ω_{40} = 2 π · 40\text{ kHz}$.
+*The upper left diagram shows the case just discussed:&nbsp; Both the carrier and the source signal are cosine.&nbsp; Thus,&nbsp; the amplitude-modulated signal &nbsp;$s(t)$&nbsp; is composed of two cosine oscillations with &nbsp;$ω_{60} = 2 π · 60\text{ kHz}$&nbsp; and &nbsp;$ω_{40} = 2 π · 40\text{ kHz}$.
-*For the other three constellations,&nbsp; at least one of the signals &nbsp;$q(t)$&nbsp; or &nbsp;$z(t)$&nbsp; is sinusoidal,&nbsp; so that &nbsp;$s(0) =  0$&nbsp; always holds.&nbsp; Thus,&nbsp; for these spectra,&nbsp; the sum of the four impulse weights each add up to zero.
+*For the other three constellations,&nbsp; at least one of the signals &nbsp;$q(t)$&nbsp; or &nbsp;$z(t)$&nbsp; is sinusoidal,&nbsp; so that &nbsp;$s(0) =  0$&nbsp; always holds.&nbsp; Thus,&nbsp; for these spectra,&nbsp; the sum of the four pulse weights each add up to zero.