Difference between revisions of "Modulation Methods/Double-Sideband Amplitude Modulation"

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{{Header
 
{{Header
 
|Untermenü=Amplitude Modulation and Demodulation
 
|Untermenü=Amplitude Modulation and Demodulation
|Vorherige Seite=Allgemeines Modell der Modulation
+
|Vorherige Seite=General Model of Modulation
|Nächste Seite=Synchrondemodulation
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|Nächste Seite=Synchronous Demodulation
 
}}
 
}}
  
== # OVERVIEW OF SECOND MAIN CHAPTER # ==
+
== # OVERVIEW OF THE SECOND MAIN CHAPTER # ==
 
<br>
 
<br>
$\Rightarrow \hspace{0.5cm}\text{We are just beginning the English translation of this chapter.}$
+
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:
 
+
*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",
After some general explanations of modulation and demodulation, a detailed description of &nbsp; ''amplitude modulation''&nbsp; and their associated&nbsp; ''demodulators''.&nbsp;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 double-sideband amplitude modulation  (DSB–AM) in the frequency and time domains,
 
*the characteristics of a synchronous demodulator and the possible applications of an envelope demodulator,
 
*the similarities/differences of single-sideband modulation compared to DSB-AM and modified AM methods.
 
  
  
 
==Description in the frequency domain==
 
==Description in the frequency domain==
 
<br>
 
<br>
We consider the following problem: a message signal&nbsp;$q(t)$, whose spectrum &nbsp;$Q(f)$&nbsp; is bandlimited to the range &nbsp;$\pm B_{\rm NF}$&nbsp; (subscript NF from German "Niederfrequenz" ⇒ low frequency), is to be shifted to a higher frequency range where the channel frequency response&nbsp;$H_{\rm K}(f)$ has favorable characteristics, using a harmonic oscillation of frequency &nbsp;$f_{\rm T}$, which we will refer to as the carrier signal&nbsp;$z(t)$&nbsp;.  
+
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; $z(t)$.  
 +
 
  
[[File:P_ID974__Mod_T_2_1_S1a_neu.png |center|frame|Representation of amplitude modulation in the frequency domain]]
+
The diagram illustrates the task, with the following simplifying assumptions:
  
The diagram illustrates the task, with the following simplifying assumptions:
+
[[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.  
+
*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; Noise interference is ignored for now.  
+
*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;$ϕ_T = 0)$&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.  
+
*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, the spectrum of the carrier signal &nbsp;$z(t) = \cos(ω_{\rm T} · t)$&nbsp; (plotted in green in the graph) is:  
+
*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):  
 
:$$Z(f) = {1}/{2} \cdot \delta (f + f_{\rm T})+{1}/{2} \cdot \delta (f - f_{\rm T})\hspace{0.05cm}.$$
 
:$$Z(f) = {1}/{2} \cdot \delta (f + f_{\rm T})+{1}/{2} \cdot \delta (f - f_{\rm T})\hspace{0.05cm}.$$
  
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{{BlaueBox|TEXT=
 
{{BlaueBox|TEXT=
$\text{Please note:}$&nbsp; This equation takes into account that the convolution of a shifted Dirac function  &nbsp;$δ(x x_0)$&nbsp; with an arbitrary function&nbsp;$f(x)$&nbsp; yields the &nbsp; '''shifted function''' &nbsp;$f(x - x_0)$&nbsp;.}}  
+
$\text{Please note:}$&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;$f(x - x_0)$.}}  
  
[[File:EN_Mod_T_2_1_S1b.png|center|frame|Spectrum of DSB-AM without a carrier]]
 
  
The diagram displays the result. One can identify the following characteristics:  
+
The diagram displays the result.&nbsp; One can identify the following characteristics:  
*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;$Q(f)$&nbsp;.  
+
[[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"]]
*The factor &nbsp;$1/2$&nbsp; results from the carrier amplitude &nbsp;$A_{\rm T} = 1$.&nbsp; Thus, &nbsp;$s(t = 0) = q(t = 0)$, and the integrals over their spectral functions &nbsp;$S(f)$&nbsp; and &nbsp;$Q(f)$&nbsp; must also be equal.  
+
*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 channel bandwidth &nbsp;$B_{\rm K}$&nbsp;  must be at least twice the signal bandwidth &nbsp;$B_{\rm NF}$, which gives the name "double-sideband amplitude modulation"(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_Laws#Reziprozit.C3.A4tsgesetz_von_Zeitdauer_und_Bandbreite|non-equivalent bandwidths]].&nbsp; The latter are defined over rectangles of equal area and are denoted by&nbsp;$Δf_q$&nbsp; and &nbsp;$Δf_{\rm K}$&nbsp; in our tutorial, respectively.  
+
*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 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 "'''DSB-AM without carrier'''".  
+
*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.  
*The frequency components above the carrier frequency  &nbsp;$f_{\rm T}$&nbsp; are called the ''upper sideband''&nbsp; (USB), and those below &nbsp;$f_{\rm T}$&nbsp; are the ''lower sideband''&nbsp; (LSB).  
+
*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; $\rm  (USB)$,  
 +
:*and those below &nbsp;$f_{\rm T}$&nbsp; are the&nbsp; "lower sideband"&nbsp; $\rm  (LSB)$.  
  
 
==Description in the time domain==
 
==Description in the time domain==
 
<br>
 
<br>
Adapting the notation and nomenclature to this problem, the convolution theorem reads:
+
Adapting the notation and nomenclature to this problem,&nbsp; the convolution theorem reads:
 +
 
 
:$$S(f)  = Z(f) \star Q(f)\hspace{0.2cm}\bullet\!\!-\!\!\!-\!\!\!-\!\!\circ\, \hspace{0.2cm} s(t) = q(t) \cdot z(t) = q(t) \cdot \cos(\omega_{\rm T}\cdot t + \phi_{\rm T})\hspace{0.05cm}.$$
 
:$$S(f)  = Z(f) \star Q(f)\hspace{0.2cm}\bullet\!\!-\!\!\!-\!\!\!-\!\!\circ\, \hspace{0.2cm} s(t) = q(t) \cdot z(t) = q(t) \cdot \cos(\omega_{\rm T}\cdot t + \phi_{\rm T})\hspace{0.05cm}.$$
This result is still true if the restrictions made on the last page $($real-valued spectrum &nbsp;$Q(f)$, carrier phase &nbsp;$ϕ_{\rm T} = 0)$&nbsp; are removed.  In general, this results in a complex-valued spectral function &nbsp;$S(f)$.
+
[[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:
 +
#real-valued spectrum &nbsp;$Q(f)$,&nbsp;
 +
#carrier phase &nbsp;$ϕ_{\rm T} = 0$&nbsp;  
 +
 
 +
In general,&nbsp; this results in a complex-valued spectrum &nbsp;$S(f)$.
  
[[File:Mod_T_2_1_S2a_version2.png|right|frame|Models of DSB–AM without a carrier]]
+
According to this equation, two models can be given for double-sideband amplitude modulation.&nbsp; These are to be interpreted as follows:  
According to this equation, two models can be given for double-sideband amplitude modulation.  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 first model directly describes the relationship given above, where here 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.  
*The second model is more in line with the physical conditions, after each signal also has a unit.  If &nbsp;$q(t)$&nbsp; and &nbsp;$z(t)$&nbsp; are voltages respectively, the model still needs to provide a scaling with the modulator constant &nbsp;$K_{\rm AM}$&nbsp; (unitso 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.
*If we set &nbsp;$K_{\rm AM} = 1/A_{\rm T}$, both models are the same.&nbsp; In the following, we will always assume the simpler model.
 
 
<br clear=all>
 
<br clear=all>
[[File:P_ID977__Mod_T_2_1_S2b_neu.png|right|frame|Signal waveforms in DSB–AM without a carrier]]
+
[[File:P_ID977__Mod_T_2_1_S2b_neu.png|right|frame|Signal waveforms in DSB–AM without carrier]]
 
{{GraueBox|TEXT=
 
{{GraueBox|TEXT=
$\text{Example 1:}$&nbsp; The graph shows in red the transmitted signals &nbsp;$s(t)$&nbsp; for DSB–AM with two different carrier frequencies.&nbsp;  
+
$\text{Example 1:}$&nbsp; The graph shows in red the transmitted signals &nbsp;$s(t)$&nbsp; for DSB–AM with two different carrier frequencies.&nbsp;
*The source signal &nbsp;$q(t)$&nbsp; with bandwidth &nbsp;$B_{\rm NF} = 4\text{ kHz}$, which is the same in both cases, is drawn in solid blue and the signal &nbsp;$-q(t)$&nbsp is dashed.
+
*The carrier signal &nbsp;$z(t)$&nbsp; has a cosine shape in both cases.  
+
The source signal &nbsp;$q(t)$&nbsp; with bandwidth &nbsp;$B_{\rm NF} = 4\text{ kHz}$,&nbsp; which is the same in both cases,&nbsp; is drawn in solid blue and the signal &nbsp;$-q(t)$&nbsp; is dashed.
*In the upper image, the carrier frequency is &nbsp;$f_{\rm T} = 20\text{ kHz}$&nbsp; and the lower image &nbsp;$f_{\rm T} = 100\text{ kHz}$.}}  
+
 
 +
The carrier signal &nbsp;$z(t)$&nbsp; has a cosine shape in both cases.&nbsp; In the upper sketch,&nbsp; the carrier frequency&nbsp; (German:&nbsp; "Trägersignal" &nbsp; &rArr; &nbsp; subscript "T")&nbsp; is &nbsp;$f_{\rm T} = 20\text{ kHz}$&nbsp; and in the lower sketch &nbsp;$f_{\rm T} = 100\text{ kHz}$.}}  
  
 
==Ring modulator==
 
==Ring modulator==
 
<br>
 
<br>
One possibility to realize "double sideband amplitude modulation with carrier suppression" is offered by a so-called ''ring modulator'', also known as ''double push-pull diode modulator''.  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; '''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.
  
[[File:P_ID978__Mod_T_2_1_S3a_neu.png|center|frame|Ring modulator to realize DSB–AM without a carrier]]
 
 
Without claiming to be complete, the principle can be stated as follows:
 
Without claiming to be complete, the principle can be stated as follows:
*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 message signal&nbsp;$q(t)$.&nbsp; Thus, all diodes are operated as switches.
+
[[File:P_ID978__Mod_T_2_1_S3a_neu.png|right|frame|Ring modulator to realize DSB–AM without carrier]]
*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;$s(t) = q(t)$ holds.  
+
*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.
*For a negative half-wave, the light green diodes conduct and the diodes in the longitudinal branches block. As can be seen from the right image,;$s(t) = \ – q(t)$ holds for this lower switch position.  
+
*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; $s(t) = q(t)$.  
*Due to the operation of this switch, the harmonic oscillation &nbsp;$z(t)$&nbsp; can also be replaced by a periodic square wave signal with identical period duration:  
+
*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:  
 
:$$z_{\rm R}(t) = \left\{ \begin{array}{c} +1 \\ -1 \\  \end{array} \right.\quad \begin{array}{*{10}c}    {\rm{for}} \\  {\rm{for}}  \\ \end{array}\begin{array}{*{20}c} {z(t) >0,}  \\ {z(t) <0.}  \\ \end{array}$$
 
:$$z_{\rm R}(t) = \left\{ \begin{array}{c} +1 \\ -1 \\  \end{array} \right.\quad \begin{array}{*{10}c}    {\rm{for}} \\  {\rm{for}}  \\ \end{array}\begin{array}{*{20}c} {z(t) >0,}  \\ {z(t) <0.}  \\ \end{array}$$
*The modulated signal&nbsp;$s(t)$&nbsp; is then obtained as the product of the message signal &nbsp;$q(t)$&nbsp; and this square wave signal  &nbsp;$z_{\rm R}(t)$, whereas in ideal DSB-AM one multiplies by a cosine signal.  
+
*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 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 ("DSB-AM without a carrier").  
 
  
  
 
{{GraueBox|TEXT=
 
{{GraueBox|TEXT=
$\text{Example 2:}$&nbsp; Now we will explain the mode of operation of a ring modulator using typical signal characteristics.&nbsp; Let the carrier frequency be &nbsp;$f_{\rm T} = 10\text{ kHz}$.
+
$\text{Example 2:}$&nbsp; Now we will explain the operation mode of a ring modulator using exemplary  signal characteristics.&nbsp; Let the carrier frequency be &nbsp;$f_{\rm T} = 10\text{ kHz}$.
  
 
[[File:EN_Mod_T_2_1_S3a.png|right|frame|Signals to illustrate a ring modulator]]  
 
[[File:EN_Mod_T_2_1_S3a.png|right|frame|Signals to illustrate a ring modulator]]  
 
+
<br><br><br><br>
*The top graph shows the signals &nbsp;$q(t)$&nbsp; and &nbsp;$-q(t)$&nbsp; as magenta and light green waveforms respectively.
+
*The top graph shows the signals &nbsp;$q(t)$&nbsp; and &nbsp;$-q(t)$&nbsp; as magenta and light green waveforms respectively.&nbsp; The bipolar square wave&nbsp; (rectangular)&nbsp; signal&nbsp;$z_{\rm R}(t)$&nbsp; is shown in blue dashes,&nbsp; and takes the values &nbsp;$±1$&nbsp;.
 
 
 
*The bipolar square wave signal&nbsp;$z_{\rm R}(t)$&nbsp; is shown in blue dashes, and takes the values &nbsp;$±1$&nbsp;.
 
 
 
 
   
 
   
 
*The middle chart shows the modulated signal from the ring modulator:  
 
*The middle chart shows the modulated signal from the ring modulator:  
 
:$$s_{\rm RM}(t) = q(t) · z_{\rm R}(t).$$
 
:$$s_{\rm RM}(t) = q(t) · z_{\rm R}(t).$$
 
 
  
 
*For comparison, the conventional DSB-AM signal is shown in the bottom graph:
 
*For comparison, the conventional DSB-AM signal is shown in the bottom graph:
 
:$$s(t) = q(t) · \cos(ω_{\rm T} · t).$$
 
:$$s(t) = q(t) · \cos(ω_{\rm T} · t).$$
<br clear=all>  
+
<br clear=all>
 
One can see significant differences, but these can be compensated for in a simple way:  
 
One can see significant differences, but these can be compensated for in a simple way:  
*The Fourier series representation of the periodic square wave signal&nbsp;$z_{\rm R}(t)$&nbsp; is:
+
*The Fourier series representation of the periodic rectangular signal&nbsp;$z_{\rm R}(t)$&nbsp; is:
 
:$$z_{\rm R}(t) = \frac{4}{\pi} \cdot \cos(\omega_{\rm T}\cdot t)-\frac{4}{3\pi} \cdot \cos(3\omega_{\rm T}\cdot t) +\frac{4}{5\pi} \cdot \cos(5\omega_{\rm T}\cdot t)- \text{ ...}$$
 
:$$z_{\rm R}(t) = \frac{4}{\pi} \cdot \cos(\omega_{\rm T}\cdot t)-\frac{4}{3\pi} \cdot \cos(3\omega_{\rm T}\cdot t) +\frac{4}{5\pi} \cdot \cos(5\omega_{\rm T}\cdot t)- \text{ ...}$$
*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 spectral function (the subscript stands for "ring modulator"):
+
*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"):
 
:$$S_{\rm RM}(f) = \frac{2}{\pi} \cdot Q (f \pm f_{\rm T})-\frac{2}{3\pi} \cdot Q (f \pm 3f_{\rm
 
:$$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}$$
 
  T})+\frac{2}{5\pi} \cdot Q (f \pm 5f_{\rm T}) -\text{ ...} \hspace{0.05cm}$$
*From this, it can be seen that by appropriately band-limiting $($for example, to &nbsp;$±2f_{\rm T})$ and attenuating with  &nbsp;$π/4 ≈ 0.785$&nbsp; the familiar DSB-AM spectrum can be obtained:
+
*From this, it can be seen that by appropriately band-limiting $($e.g. to &nbsp;$±2f_{\rm T})$ and attenuating with  &nbsp;$π/4 ≈ 0.785$&nbsp; the familiar DSB-AM spectrum can be obtained:
 
:$$S(f) = {1}/{2} \cdot Q (f \pm f_{\rm T})\hspace{0.05cm}.$$
 
:$$S(f) = {1}/{2} \cdot Q (f \pm f_{\rm T})\hspace{0.05cm}.$$
Here it must be taken into account that in the above reasoning, &nbsp;$B_{\rm NF} \ll f_{\rm T}$&nbsp; can always be assumed.}}
+
Here it must be taken into account that in the above reasoning, &nbsp;$B_{\rm NF} \ll f_{\rm T}$&nbsp; must hold.}}
  
 
==AM signals and spectra with a harmonic input signal==
 
==AM signals and spectra with a harmonic input signal==
 
<br>
 
<br>
Now we will consider a special case, which is important for testing purposes, where not only the carrier signal &nbsp;$z(t)$  is a harmonic oscillation, but also the message signal to be modulated &nbsp;$q(t)$&nbsp;:  
+
Now we consider a special case which is important for testing purposes,&nbsp; where not only the carrier&nbsp; $z(t)$&nbsp; is a harmonic oscillation,&nbsp; but also the signal&nbsp; $q(t)$&nbsp; to be modulated:  
 
:$$\begin{align*}q(t) & = A_{\rm N} \cdot \cos(\omega_{\rm N} \cdot t + \phi_{\rm N})\hspace{0.05cm}, \\ \\ z(t) & = \hspace{0.15cm}1 \hspace{0.13cm} \cdot \hspace{0.1cm}\cos(\omega_{\rm T} \cdot t + \phi_{\rm T})\hspace{0.05cm}.\end{align*}$$
 
:$$\begin{align*}q(t) & = A_{\rm N} \cdot \cos(\omega_{\rm N} \cdot t + \phi_{\rm N})\hspace{0.05cm}, \\ \\ z(t) & = \hspace{0.15cm}1 \hspace{0.13cm} \cdot \hspace{0.1cm}\cos(\omega_{\rm T} \cdot t + \phi_{\rm T})\hspace{0.05cm}.\end{align*}$$
  
Please note: &nbsp; since we are describing modulation processes, the phase term is used with a plus sign in the above equations.  
+
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 $z(t)$.  
+
*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, the equation for the modulated signal is:
+
*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
 
:$$s(t) = q(t) \cdot z(t) = A_{\rm N} \cdot \cos(\omega_{\rm N} t + \phi_{\rm
 
  N})\cdot  \cos(\omega_{\rm T} t + \phi_{\rm T})\hspace{0.05cm}.$$
 
  N})\cdot  \cos(\omega_{\rm T} t + \phi_{\rm T})\hspace{0.05cm}.$$
 
  
 
This equation can be transformed using the trigonometric addition theorem:  
 
This equation can be transformed using the trigonometric addition theorem:  
 
:$$s(t) = A_{\rm N}/{2} \cdot \cos \big [(\omega_{\rm T} +\omega_{\rm N})\cdot t + \phi_{\rm T}+ \phi_{\rm N} \big ] + A_{\rm N}/{2} \cdot \cos \big [(\omega_{\rm T} -\omega_{\rm N})\cdot t + \phi_{\rm T}- \phi_{\rm N} \big ]\hspace{0.05cm}.$$
 
:$$s(t) = A_{\rm N}/{2} \cdot \cos \big [(\omega_{\rm T} +\omega_{\rm N})\cdot t + \phi_{\rm T}+ \phi_{\rm N} \big ] + A_{\rm N}/{2} \cdot \cos \big [(\omega_{\rm T} -\omega_{\rm N})\cdot t + \phi_{\rm T}- \phi_{\rm N} \big ]\hspace{0.05cm}.$$
  
*For cosinusoidal signals &nbsp;$(ϕ_{\rm T} = ϕ_{\rm N} = 0)$&nbsp;, this equation simplifies to
+
*For cosinusoidal signals&nbsp; $(ϕ_{\rm T} = ϕ_{\rm N} = 0)$,&nbsp; this equation simplifies to
 
:$$s(t) = {A_{\rm N}}/{2} \cdot \cos\big[(\omega_{\rm T}+\omega_{\rm N})\cdot t\big]  + {A_{\rm N}}/{2} \cdot \cos\big[(\omega_{\rm T} -\omega_{\rm N})\cdot t \big]\hspace{0.05cm}.$$  
 
:$$s(t) = {A_{\rm N}}/{2} \cdot \cos\big[(\omega_{\rm T}+\omega_{\rm N})\cdot t\big]  + {A_{\rm N}}/{2} \cdot \cos\big[(\omega_{\rm T} -\omega_{\rm N})\cdot t \big]\hspace{0.05cm}.$$  
  
Line 133: Line 133:
 
  N})+\delta ( f + f_{\rm T} + f_{\rm N})\big)] +  {A_{\rm N}}/{4} \cdot \big[ \delta ( f - f_{\rm T}+ f_{\rm N})+\delta ( f+ f_{\rm T} - f_{\rm N} ) \big]\hspace{0.05cm}.$$
 
  N})+\delta ( f + f_{\rm T} + f_{\rm N})\big)] +  {A_{\rm N}}/{4} \cdot \big[ \delta ( f - f_{\rm T}+ f_{\rm N})+\delta ( f+ f_{\rm T} - f_{\rm N} ) \big]\hspace{0.05cm}.$$
  
 
+
This result,&nbsp; which would also have been arrived at via convolution,&nbsp; states:
This result, which would also have been arrived at via convolution, 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;$±(f_{\rm T} - f_{\rm N})$, where in both bracket expressions the first Dirac 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 functions are equal and each is &nbsp;$A_{\rm N}/4$.&nbsp;  The sum of these weights - that is, the integral over  $S(f)$ – is equal to the signal value &nbsp;$s(t = 0) = A_{\rm N}$.  
+
*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 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=
 
{{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;$ϕ_{\rm N}$, 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, 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 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]]
 
[[File:EN_Mod_T_2_1_S4.png|right|frame|Typical spectra for DSB-AM]]
 
<br>
 
<br>
*The upper left image shows the case just discussed: both the carrier and the message signal are cosine. Thus, the amplitude modulated signal &nbsp;$s(t)$&nbsp; s composed of two cosine oscillations with &nbsp;$ω_{60} = 2 π · 60\text{ kHz}$&nbsp; and &nbsp;$ω_{40} = 2 π · 40\text{ kHz}$&nbsp;.
+
*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, at least one of the signals &nbsp;$q(t)$&nbsp; or &nbsp;$z(t)$&nbsp; sinusförmig, is sinusoidal, so that &nbsp;$s(0) =  0$&nbsp; always holds.&nbsp; Thus, for these spectra, 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.
  
 
   
 
   
*The bottom right image depicts &nbsp;$s(t) = A_{\rm N} · \sin(ω_{\rm N} t) · \sin(ω_{\rm T}t)$.&nbsp; Multiplying two odd functions yields the even function &nbsp;$s(t)$&nbsp; and thus a real spectrum &nbsp;$S(f)$.&nbsp; In contrast, the other two constellations each result in imaginary spectral functions. }}
+
*The bottom right diagram depicts &nbsp;$s(t) = A_{\rm N} · \sin(ω_{\rm N} t) · \sin(ω_{\rm T}t)$.&nbsp; Multiplying two odd functions yields the even function &nbsp;$s(t)$&nbsp; and thus a real spectrum &nbsp;$S(f)$.&nbsp; In contrast,&nbsp; the other two constellations each result in imaginary spectral functions. }}
  
 
==Double-Sideband Amplitude Modulation with carrier==
 
==Double-Sideband Amplitude Modulation with carrier==
 
<br>
 
<br>
The following diagram shows how to get from "DSB-AM without a carrier" to the better known variant "DSB-AM with carrier".  This has the advantage that the demodulator can be realized much more easily and cheaply by a simple manipulation at the transmitter.
+
The following diagram shows how to get from&nbsp; "DSB-AM without carrier"&nbsp; to the better known variant "DSB-AM with carrier".&nbsp; This has the advantage that the demodulator can be realized much more easily and cheaply by a simple manipulation at the transmitter.
  
[[File: P_ID981__Mod_T_2_1_S5a_neu.png|center|frame|MModels of DSB–AM with carrier]]
+
[[File: P_ID981__Mod_T_2_1_S5a_neu.png|right|frame|Models of DSB–AM with carrier]]
 
 
Die Grafik ist wie folgt zu interpretieren:
 
*Die obere Darstellung zeigt das physikalische Modell der „ZSB–AM mit Träger”, wobei Veränderungen gegenüber der „ZSB–AM ohne Träger” rot hervorgehoben sind.
 
*Dem Signal &nbsp;$s(t)$&nbsp; ist additiv das Trägersignal &nbsp;$z(t) = A_{\rm T} · \cos(ω_{\rm T} · t)$&nbsp; hinzugefügt, das im Spektrum zwei zusätzliche Diracfunktionen bei &nbsp;$±f_{\rm T}$&nbsp; bewirkt,&nbsp; jeweils mit dem Impulsgewicht &nbsp;$A_{\rm T}/2$.
 
*Durch Addition des Gleichsignals &nbsp;$A_{\rm T}$&nbsp; zum Quellensignal und anschließende Multiplikation mit dem dimensionslosen Träger &nbsp;$z(t)$&nbsp; entsprechend der unteren Skizze ergibt sich das gleiche Signal &nbsp;$s(t)$&nbsp; und das gleiche Spektrum &nbsp;$S(f)$&nbsp; wie oben.
 
*Die zweite Darstellung ist also mit dem oberen Modell äquivalent.&nbsp; Die Trägerphase  ist in beiden Fällen nur aus Gründen einer vereinfachten Darstellung &nbsp;$ϕ_{\rm T} = 0$&nbsp; gesetzt.
 
  
 +
The diagram is to be interpreted as follows: 
 +
*The top plot shows the physical model of the&nbsp; "DSB-AM with carrier",&nbsp; with changes from the&nbsp; "DSB-AM without carrier"&nbsp; highlighted in red.
 +
*The carrier signal &nbsp;$z(t) = A_{\rm T} · \cos(ω_{\rm T} · t)$&nbsp;  is added to the signal &nbsp;$s(t)$,&nbsp;  which causes two additional Dirac delta functions in the spectrum &nbsp;$±f_{\rm T}$,&nbsp; each with impulse weight &nbsp;$A_{\rm T}/2$.
 +
*Adding the DC signal &nbsp;$A_{\rm T}$&nbsp; to the source signal and then multiplying by the dimensionless carrier &nbsp;$z(t)$&nbsp; as shown in the lower sketch results in the same signal &nbsp;$s(t)$&nbsp; and spectrum &nbsp;$S(f)$&nbsp; as above.
 +
*Thus, the second representation is equivalent to the upper model.&nbsp; In both cases the carrier phase is only set to &nbsp;$ϕ_{\rm T} = 0$&nbsp; for the sake of simplified presentation. 
 +
<br clear=all>
  
 
{{GraueBox|TEXT=
 
{{GraueBox|TEXT=
$\text{Beispiel 4:}$&nbsp; Die&nbsp; ''Zweiseitenband-Amplitudenmodulation mit Träger''&nbsp; findet auch heutzutage noch ihre Hauptanwendung in der Rundfunkübertragung auf
+
$\text{Example 4:}$&nbsp; The&nbsp; "double-sideband amplitude modulation with carrier"&nbsp; still finds its main application in radio transmissions for
*Langwelle&nbsp; &nbsp; $($Frequenzbereich&nbsp; $\text{30 kHz}$ ... $\text{300 kHz})$,  
+
*long wave &nbsp; &nbsp; $($frequency range&nbsp; $\text{30 kHz}$ ... $\text{300 kHz})$,  
*Mittelwelle&nbsp; $($Frequenzbereich&nbsp; $\text{300 kHz}$ ... $\text{3 MHz})$,  
+
*medium wave&nbsp; $($frequency range&nbsp; $\text{300 kHz}$ ... $\text{3 MHz})$,  
*Kurzwelle&nbsp; &nbsp; $($Frequenzbereich&nbsp; $\text{3 MHz}$ ... $\text{30 MHz})$.  
+
*short wave&nbsp; &nbsp; $($frequency range&nbsp; $\text{3 MHz}$ ... $\text{30 MHz})$.  
  
  
Diese Frequenzen werden jedoch mehr und mehr für digitale Anwendungen freigegeben, zum Beispiel für&nbsp; ''Digital Video Broadcast''&nbsp; (DVB).  
+
However,&nbsp; these frequencies are increasingly being made available for digital applications, e.g.&nbsp; "Digital Video Broadcast"&nbsp; $\rm (DVB)$.  
  
Eine Anwendung von&nbsp; ''Zweiseitenband-Amplitudenmodulation ohne Träger''&nbsp; gibt es beispielsweise beim UKW-Stereo-Rundfunk:  
+
An application of &nbsp; "double-sideband amplitude modulation without carrier"&nbsp; exists for example in FM stereo broadcasting:  
*Hier wird das Differenzsignal zwischen den beiden Stereokanälen bei&nbsp; $\text{39 kHz}$&nbsp; trägerlos amplitudenmoduliert.  
+
*Here,&nbsp; the differential signal between the two stereo channels is amplitude modulated (without carrier) at&nbsp; $\text{39 kHz}$.
*Dann werden das Summensignal der beiden Kanäle&nbsp; $($jeweils im Bereich&nbsp; $\text{30 Hz}$ ... $\text{15 kHz})$, ein Hilfsträger bei&nbsp; $\text{19 kHz}$&nbsp; sowie das Differenzsignal zusammengefasst und frequenzmoduliert.}}  
+
*Then the sum signal of the two channels&nbsp; $($each in the range&nbsp; $\text{30 Hz}$ ... $\text{15 kHz})$&nbsp; is combined with the differential signal and an auxiliary carrier at&nbsp; $\text{19 kHz}$.&nbsp; Then, this combined signal is frequency modulated.}}  
  
  
 
{{GraueBox|TEXT=
 
{{GraueBox|TEXT=
$\text{Beispiel 5:}$&nbsp; Die folgenden Signalverläufe sollen das Prinzip der „ZSB–AM mit Träger” weiter verdeutlichen.
+
$\text{Example 5:}$&nbsp; The following signal waveforms are intended to further clarify the principle of the modulation method&nbsp; "DSB-AM with carrier".
 +
 
 +
[[File:EN_Mod_T_2_1_S5b_v2.png|right|frame|Signal waveforms for&nbsp; "DSB–AM with carrier"&nbsp; and&nbsp; "DSB-AM with carrier suppression"]]
 +
 +
*In the upper diagram you see a section of the source signal &nbsp;$q(t)$&nbsp; with frequencies in the range &nbsp;$\vert f \vert \le 4\text{ kHz}$.
 +
 +
*$s(t)$&nbsp; is obtained by adding the DC component &nbsp;$A_{\rm T}$&nbsp; to &nbsp;$q(t)$&nbsp; and multiplying this summed signal by the carrier  &nbsp;$z(t)$&nbsp; of frequency  &nbsp;$f_{\rm T} = 100\text{ kHz}$ &nbsp; &rArr; &nbsp; see middle diagram.
 +
 
 +
*In the lower diagram you see  for comparison the transmitted signal of the "DSB-AM without carrier" &nbsp; &rArr; &nbsp; "DSB-AM with carrier suppression".  
  
[[File:EN_Mod_T_2_1_S5b_v2.png|right|frame|Signalverläufe bei der ZSB–AM mit Träger]]
 
<br>
 
*Oben sehen Sie einen Ausschnitt des auf Frequenzen &nbsp;$\vert f \vert \le 4\text{ kHz}$&nbsp; begrenzten Quellensignal &nbsp;$q(t)$.
 
  
+
A comparison of these signal waveforms shows:
*$s(t)$&nbsp; ergibt sich, wenn man zu &nbsp;$q(t)$&nbsp; den Gleichanteil &nbsp;$A_{\rm T}$&nbsp; addiert und  die Signalsumme mit dem Trägersignal &nbsp;$z(t)$&nbsp; der Frequenz &nbsp;$f_{\rm T} = 100\text{ kHz}$&nbsp; multipliziert.
+
*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]].
 +
*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:
 +
:$$m = \frac{q_{\rm max} }{A_{\rm T} } \hspace{0.3cm}{\rm with}\hspace{0.3cm} q_{\rm max} = \max_{t} \hspace{0.05cm} \vert q(t) \vert\hspace{0.05cm}.$$}}
 +
 
  
+
{{BlaueBox|TEXT=
*Unten ist zum Vergleich das Sendesignal der „ZSB–AM ohne Träger” dargestellt.
+
$\text{Conclusions:}$&nbsp;
<br clear=all>
+
*The advantage of a simpler demodulator is traded off with a much higher transmit power,&nbsp; since the power of the carrier cannot be used for demodulation.&nbsp;  
Ein Vergleich dieser Signalverläufe zeigt:
+
*Furthermore,&nbsp; care must be taken to ensure that the source signal does not contain a DC component,&nbsp; since this would be masked by the carrier.&nbsp;
*Durch die Zusetzung des Gleichanteils &nbsp;$A_{\rm T}$&nbsp; wurde erreicht, dass nun das Nachrichtensignal &nbsp;$q(t)$&nbsp; in der Hüllkurve von &nbsp;$s(t)$&nbsp; zu erkennen ist.
+
*For speech and music signals,&nbsp; however,&nbsp; this is not a major restriction. }}
*Dadurch kann die &nbsp;[[Modulation_Methods/Hüllkurvendemodulation|Hüllkurvendemodulation]]&nbsp; angewandt werden, die einfacher und billiger zu realisieren ist als die kohärente &nbsp;[[Modulation_Methods/Synchrondemodulation|Synchrondemodulation]].
 
*Voraussetzung für die Anwendung des Hüllkurvendemodulators ist aber ein Modulationsgrad &nbsp;$m <1$.&nbsp; Dieser ist wie folgt definiert:
 
:$$m = \frac{q_{\rm max} }{A_{\rm T} } \hspace{0.3cm}{\rm mit}\hspace{0.3cm} q_{\rm max} = \max_{t} \hspace{0.05cm} \vert q(t) \vert\hspace{0.05cm}.$$
 
*Der Vorteil eines einfacheren Demodulators muss aber durch eine deutlich höhere Sendeleistung erkauft werden, da der Leistungsbeitrag des Trägers nicht zur Demodulation genutzt werden kann.
 
*Weiter ist zu darauf zu achten, dass das Quellensignal keinen Gleichanteil beinhaltet, da dieser durch den Träger überdeckt würde.&nbsp; Bei Sprach– und Musiksignalen ist dies allerdings keine große Einschränkung. }}
 
  
==Beschreibung der ZSB-AM durch das analytische Signal==
+
==Describing DSB-AM with carrier using the analytical signal==
 
<br>
 
<br>
[[File:Mod_T_2_1_S6_version2.png|right|frame|Spektrum des analytischen Signals in zwei verschiedenen Darstellungsformen]]
+
[[File:Mod_T_2_1_S6_version2.png|right|frame|Spectrum of the analytical signal in two different viewpoints]]
Im weiteren Verlauf wird zur Vereinfachung von Grafiken meist das Spektrum &nbsp;$S_+(f)$&nbsp; des &nbsp;[[Modulation_Methods/Allgemeines_Modell_der_Modulation#Beschreibung_des_physikalischen_Signals_mit_Hilfe_des_analytischen_Signals|analytischen Signals]]&nbsp; anstelle des tatsächlichen, physikalischen Spektrums &nbsp;$S(f)$&nbsp; angegeben.&nbsp;
+
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;$S(f)$.  
  
Beispielhaft betrachten wir hier eine&nbsp; „ZSB–AM mit Träger”&nbsp; und folgende Signale:
+
As an example,&nbsp; let us consider&nbsp; "DSB-AM with carrier"&nbsp; and the following signals:
  
 
:$$\begin{align*}s(t) & = \left(q(t) + A_{\rm T}\right) \cdot \cos(\omega_{\rm T}\cdot  t + \phi_{\rm T})\hspace{0.05cm}, \\ \\ q(t) & = A_{\rm N} \cdot \cos(\omega_{\rm N}\cdot  t + \phi_{\rm N})\hspace{0.05cm}.\end{align*}$$
 
:$$\begin{align*}s(t) & = \left(q(t) + A_{\rm T}\right) \cdot \cos(\omega_{\rm T}\cdot  t + \phi_{\rm T})\hspace{0.05cm}, \\ \\ q(t) & = A_{\rm N} \cdot \cos(\omega_{\rm N}\cdot  t + \phi_{\rm N})\hspace{0.05cm}.\end{align*}$$
  
Dann lautet das dazugehörige analytische Signal:  
+
Then the corresponding analytical signal is:
 
:$$s_+(t)  = A_{\rm T} \cdot {\rm e}^{{\rm j}\hspace{0.03cm} \cdot \hspace{0.01cm}(\omega_{\rm T}\cdot \hspace{0.02cm}t \hspace{0.05cm}+ \hspace{0.05cm}\phi_{\rm T})}+  \frac{A_{\rm N}}{2} \cdot {\rm e}^{{\rm j}\hspace{0.03cm} \cdot \hspace{0.01cm}((\omega_{\rm T} \hspace{0.05cm}+ \hspace{0.05cm} \omega_{\rm N} )\hspace{0.02cm}\cdot \hspace{0.02cm}t \hspace{0.05cm}+ \hspace{0.05cm} \phi_{\rm T}+ \phi_{\rm N})} + \frac{A_{\rm N}}{2} \cdot {\rm e}^{{\rm j}\hspace{0.03cm} \cdot \hspace{0.01cm}((\omega_{\rm T} \hspace{0.05cm}- \hspace{0.05cm} \omega_{\rm N} )\hspace{0.02cm}\cdot \hspace{0.02cm}t \hspace{0.05cm}+ \hspace{0.05cm} \phi_{\rm T}- \phi_{\rm N})} \hspace{0.05cm}.$$
 
:$$s_+(t)  = A_{\rm T} \cdot {\rm e}^{{\rm j}\hspace{0.03cm} \cdot \hspace{0.01cm}(\omega_{\rm T}\cdot \hspace{0.02cm}t \hspace{0.05cm}+ \hspace{0.05cm}\phi_{\rm T})}+  \frac{A_{\rm N}}{2} \cdot {\rm e}^{{\rm j}\hspace{0.03cm} \cdot \hspace{0.01cm}((\omega_{\rm T} \hspace{0.05cm}+ \hspace{0.05cm} \omega_{\rm N} )\hspace{0.02cm}\cdot \hspace{0.02cm}t \hspace{0.05cm}+ \hspace{0.05cm} \phi_{\rm T}+ \phi_{\rm N})} + \frac{A_{\rm N}}{2} \cdot {\rm e}^{{\rm j}\hspace{0.03cm} \cdot \hspace{0.01cm}((\omega_{\rm T} \hspace{0.05cm}- \hspace{0.05cm} \omega_{\rm N} )\hspace{0.02cm}\cdot \hspace{0.02cm}t \hspace{0.05cm}+ \hspace{0.05cm} \phi_{\rm T}- \phi_{\rm N})} \hspace{0.05cm}.$$
  
Die zugehörige Spektralfunktion &nbsp;$S_+(f)$&nbsp; besteht aus drei Diraclinien mit jeweils komplexen Gewichten entsprechend der Grafik:  
+
The corresponding spectral function &nbsp;$S_+(f)$&nbsp; consists of three Dirac delta lines,&nbsp; each with complex weights corresponding to the graph:  
  
*Die linke Skizze zeigt den Betrag &nbsp;$|S_+(f)|$, wobei &nbsp;$A_{\rm T}$&nbsp; das Gewicht des Trägers angibt und &nbsp;$A_{\rm N}/2$&nbsp; die Gewichte von&nbsp; '''OSB'''&nbsp; (oberes Seitenband)&nbsp; und&nbsp; '''USB'''&nbsp; (unteres Seitenband).  
+
*The left sketch shows&nbsp; $|S_+(f)|$.&nbsp; $A_{\rm T}$&nbsp; indicates the weight of the carrier and &nbsp;$A_{\rm N}/2$&nbsp; indicates the weights of&nbsp; $\rm USB$&nbsp; (upper sideband) and&nbsp; $\rm LSB$&nbsp; (lower sideband).
*In Klammern stehen die auf &nbsp;$A_{\rm T}$&nbsp; normierten Werte.&nbsp; Da hier &nbsp;$q_{\rm max} = A_{\rm N}$&nbsp; gilt, erhält man mit dem Modulationsgrad &nbsp;$m = A_{\rm N}/A_{\rm T}$&nbsp; als normierte Gewichte von oberem und unterem Seitenband jeweils&nbsp; $m/2$.  
+
*The values normalized to &nbsp;$A_{\rm T}$&nbsp; are given in parentheses.&nbsp; Since &nbsp;$q_{\rm max} = A_{\rm N}$&nbsp; holds here,&nbsp; the modulation depth &nbsp;$m = A_{\rm N}/A_{\rm T}$&nbsp; gives &nbsp;$m/2$ as the normalized weights of both the upper and lower sideband.  
*Die rechte Skizze gibt einen Blick in Richtung der Frequenzachse und zeigt die Phasenwinkel von Träger &nbsp;$(ϕ_{\rm T})$, USB &nbsp;$(ϕ_{\rm T} ϕ_{\rm N})$&nbsp; und OSB &nbsp;$(ϕ_{\rm T} + ϕ_{\rm N})$.  
+
*The right sketch depicts the direction of the frequency axis and shows the phase angles of carrier &nbsp;$(ϕ_{\rm T})$,&nbsp; of&nbsp; $\rm LSB$&nbsp; $(ϕ_{\rm T} - ϕ_{\rm N})$&nbsp; and&nbsp; of&nbsp; $\rm USB$ &nbsp;$(ϕ_{\rm T} + ϕ_{\rm N})$.  
  
  
==Amplitudenmodulation durch quadratische Kennlinie==
+
==Amplitude modulation with a quadratic characteristic curve==
 
<br>
 
<br>
Nichtlinearitäten sind in der Nachrichtentechnik meist unerwünscht und störend.&nbsp; Wie im Kapitel &nbsp;[[Linear_and_Time_Invariant_Systems/Nichtlineare_Verzerrungen|Nichtlineare Verzerrungen]]&nbsp; des Buches „Lineare zeitinvariante Systeme” dargelegt, führen sie dazu, dass
+
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:
*das Superpositionsprinzip nicht mehr anwendbar ist,
 
*das Übertragungsverhalten von der Größe des Eingangssignals abhängt,&nbsp; und
 
*die Verzerrungen von nichtlinearer Art und damit irreversibel sind.
 
  
 +
*the superposition principle is no longer applicable,
 +
*the transmission behavior depends on the magnitude of the input signal,&nbsp; and
 +
*the distortions are of nonlinear nature and thus irreversible.
  
Eine Nichtlinearität der allgemeinen Form
+
 
 +
A nonlinearity of the general form
 
:$$y(t) = c_0 + c_1 \cdot x(t) + c_2 \cdot x^2(t)+ c_3 \cdot x^3(t) + \text{...}$$
 
:$$y(t) = c_0 + c_1 \cdot x(t) + c_2 \cdot x^2(t)+ c_3 \cdot x^3(t) + \text{...}$$
kann aber auch zur Realisierung einer ZSB–AM genutzt werden.&nbsp; Unter der Voraussetzung, dass
+
can also be used to implement&nbsp; $\text{DSB}\hspace{0.05cm}&ndash;\hspace{-0.05cm}\text{AM}$.&nbsp; Provided that
*nur die Koeffizienten &nbsp;$c_1$&nbsp; und &nbsp;$c_2$&nbsp; vorhanden sind, und
 
*das Eingangssignal &nbsp;$x(t) = q(t) + z(t)$&nbsp; angelegt wird,
 
  
 
+
*only the coefficients &nbsp;$c_1$&nbsp; and &nbsp;$c_2$&nbsp; are present,&nbsp; and
erhält man für das Ausgangssignal der Nichtlinearität:  
+
*the input signal &nbsp;$x(t) = q(t) + z(t)$&nbsp; is applied,
 +
<br>
 +
we obtain the nonlinearity's output signal:
 
:$$y(t) = c_1 \cdot q(t) + c_1 \cdot z(t) + c_2 \cdot q^2(t)+ 2 \cdot c_2 \cdot q(t)\cdot z(t)+ c_2 \cdot z^2(t)\hspace{0.05cm}.$$
 
:$$y(t) = c_1 \cdot q(t) + c_1 \cdot z(t) + c_2 \cdot q^2(t)+ 2 \cdot c_2 \cdot q(t)\cdot z(t)+ c_2 \cdot z^2(t)\hspace{0.05cm}.$$
Der erste, der dritte und der letzte Anteil liegen spektral gesehen bei &nbsp;$| f | ≤ 2 · B_{\rm NF}$&nbsp; bzw. &nbsp;$| f | = 2 · f_{\rm T}$.  
+
The first, third, and last components are in terms of spectra at &nbsp;$|\hspace{0.05cm} f \hspace{0.05cm}| ≤ 2 · B_{\rm NF}$&nbsp; and &nbsp;$|\hspace{0.05cm} f\hspace{0.05cm} | = 2 · f_{\rm T}$, respectively.  
  
Entfernt man diese Signalanteile durch einen Bandpass und berücksichtigt &nbsp;$z(t) = A_{\rm T} · \cos(ω_{\rm T} · t)$, so erhält man die für&nbsp; „ZSB–AM mit Träger”&nbsp; typische Gleichung&nbsp; <br>(nur noch der zweite und der vierte Term):  
+
Removing these signal components by a band-pass and considering &nbsp;$z(t) = A_{\rm T} · \cos(ω_{\rm T} · t)$,&nbsp; we obtain the equation typical for&nbsp; "DSB-AM with carrier" <br>(only the second and fourth terms remain):  
 
:$$s(t) = c_1 \cdot A_{\rm T} \cdot \cos(\omega_{\rm T} \cdot t ) + 2 \cdot c_2 \cdot A_{\rm T} \cdot q(t)\cdot \cos(\omega_{\rm T} \cdot t )\hspace{0.05cm}.$$
 
:$$s(t) = c_1 \cdot A_{\rm T} \cdot \cos(\omega_{\rm T} \cdot t ) + 2 \cdot c_2 \cdot A_{\rm T} \cdot q(t)\cdot \cos(\omega_{\rm T} \cdot t )\hspace{0.05cm}.$$
Der Modulationsgrad ist bei dieser Realisierungsform durch die Koeffiziente &nbsp;$c_1$&nbsp; und &nbsp;$c_2$&nbsp; veränderbar:  
+
The modulation depth in this realization form is variable by the coefficients &nbsp;$c_1$&nbsp; and &nbsp;$c_2$&nbsp;:  
 
:$$m = \frac{2 \cdot c_2 \cdot q_{\rm max}}{c_1} \hspace{0.05cm}.$$
 
:$$m = \frac{2 \cdot c_2 \cdot q_{\rm max}}{c_1} \hspace{0.05cm}.$$
Eine Diode und der Feldeffekttransistor besitzen mit guter Näherung eine solche quadratische Kennlinie und können zur Realisierung einer ZSB–AM genutzt werden.&nbsp; Kubische Anteile &nbsp;$(c_3 ≠ 0)$&nbsp; und Nichtlinearitäten höherer Ordnung führen allerdings zu (großen) nichtlinearen Verzerrungen.  
+
*A diode and a field-effect transistor both approximate such a quadratic characteristic curve and can thus be used to realize DSB-AM.
 +
*However,&nbsp; cubic components &nbsp;$(c_3 ≠ 0)$&nbsp; and higher order nonlinearities lead to&nbsp; (large)&nbsp; nonlinear distortions.
  
  
==Aufgaben zum Kapitel==
+
==Exercises for the chapter==
 
<br>
 
<br>
[[Aufgaben:Aufgabe_2.1:_ZSB-AM_mit_Cosinus%3F_Oder_mit_Sinus%3F|Aufgabe 2.1: ZSB-AM mit Cosinus? Oder mit Sinus?]]
+
[[Aufgaben:Exercise_2.1:_DSB-AM_with_Cosine%3F_Or_with_Sine%3F|Exercise 2.1: DSB-AM with Cosine? Or with Sine?]]
  
[[Aufgaben:Aufgabe_2.1Z:_ZSB-AM_ohne/mit_Träger|Aufgabe 2.1Z: ZSB-AM ohne/mit Träger]]
+
[[Aufgaben:Exercise_2.1Z:_DSB-AM_without/with_Carrier|Exercise 2.1Z: DSB-AM without/with Carrier]]
  
[[Aufgaben:Aufgabe_2.2:_Modulationsgrad|Aufgabe 2.2: Modulationsgrad]]
+
[[Aufgaben:Exercise_2.2:_Modulation_Depth|Exercise 2.2: Modulation Depth]]
  
[[Aufgaben:Aufgabe_2.2Z:_Leistungsbetrachtung|Aufgabe 2.2Z: Leistungsbetrachtung]]
+
[[Aufgaben:Exercise_2.2Z:_Power_Consideration|Exercise 2.2Z: Power Consideration]]
  
[[Aufgaben:Aufgabe_2.3:_ZSB–AM–Realisierung|Aufgabe 2.3: ZSB–AM–Realisierung]]
+
[[Aufgaben:Exercise_2.3:_DSB-AM_Realization|Exercise 2.3: DSB–AM Realization]]
  
[[Aufgaben:Aufgabe_2.3Z:_ZSB_durch_Nichtlinearität|Aufgabe 2.3Z: ZSB durch Nichtlinearität]]
+
[[Aufgaben:Exercise_2.3Z:_DSB-AM_due_to_Nonlinearity|Exercise 2.3Z: DSB-AM due to Nonlinearity]]
  
 
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Revision as of 15:40, 24 March 2022

# OVERVIEW OF THE SECOND MAIN CHAPTER #


After some general explanations of modulation and demodulation,  now a detailed description of  "amplitude modulation"  and the associated  "demodulators". This chapter deals in detail with:

  • the description and realization of  "double-sideband amplitude modulation"  $\text{(DSB–AM)}$  in frequency and time domain,
  • the characteristics of a  "synchronous demodulator"  and the possible applications of an  "envelope demodulator",
  • the similarities/differences of  "single-sideband modulation"  $\text{(SSB–AM)}$  compared to DSB-AM and   "modified AM methods".


Description in the frequency domain


We consider the following problem:  a source signal $q(t)$,  whose spectrum  $Q(f)$  is bandlimited to the range  $\pm B_{\rm NF}$  (subscript  "NF"  from German "Niederfrequenz"   ⇒   low frequency), 

  • is to be shifted to a higher frequency range where the channel frequency response  $H_{\rm K}(f)$  has favorable characteristics, 
  • using a harmonic oscillation of frequency  $f_{\rm T}$, which we will refer to as the carrier signal  $z(t)$.


The diagram illustrates the task, with the following simplifying assumptions:

Representation of amplitude modulation in the frequency domain
  • The spectrum  $Q(f)$  drawn here is schematic.  It states that only spectral components in the range  $|f| ≤ B_{\rm NF}$  are included in  $q(t)$.  $Q(f)$  could also be a line spectrum.
  • Let the channel be ideal in a bandwidth range  $B_{\rm K}$  around frequency  $f_{\rm M}$,  that is, let  $H_{\rm K}(f) = 1$  for  $|f - f_{\rm M}| ≤ B_{\rm K}/2.$  Impairments by noise are ignored for now.
  • Let the carrier signal be cosine   $($phase  $ϕ_{\rm T} = 0)$  and have amplitude  $A_{\rm T} = 1$  (without a unit).  Let the carrier frequency  $f_{\rm T}$  be equal to the center frequency of the transmission band.
  • Thus,  the spectrum of the carrier signal  $z(t) = \cos(ω_{\rm T} · t)$  is
    (plotted in green in the graph):
$$Z(f) = {1}/{2} \cdot \delta (f + f_{\rm T})+{1}/{2} \cdot \delta (f - f_{\rm T})\hspace{0.05cm}.$$

Those familiar with the  laws of spectral transformation  and in particular with the  Convolution Theorem  can immediately give a solution for the spectrum  $S(f)$  of the modulator output signal:

$$S(f)= Z(f) \star Q(f) = 1/2 \cdot \delta (f + f_{\rm T})\star Q(f)+1/2 \cdot \delta (f - f_{\rm T})\star Q(f) = 1/2 \cdot Q (f + f_{\rm T})+ 1/2 \cdot Q(f - f_{\rm T}) \hspace{0.05cm}.$$

$\text{Please note:}$  This equation takes into account

  • that the convolution of a shifted Dirac delta function  $δ(x - x_0)$  with an arbitrary function $f(x)$  yields the   shifted function  $f(x - x_0)$.


The diagram displays the result.  One can identify the following characteristics:

Spectrum of double-sideband amplitude modulation without carrier;
other name:  "double-sideband amplitude modulation with carrier suppression"
  • Due to the system-theoretic approach with positive and negative frequencies, $S(f)$  is composed of two parts around  $\pm f_{\rm T}$,  each of which have the same shape as  $Q(f)$.
  • The factor  $1/2$  results from the carrier amplitude  $A_{\rm T} = 1$.  Thus,  $s(t = 0) = q(t = 0)$  and the integrals over their spectral functions  $S(f)$  and  $Q(f)$  must also be equal.
  • The channel bandwidth  $B_{\rm K}$  must be at least twice the signal bandwidth  $B_{\rm NF}$,  which gives the name 
    Double-Sideband Amplitude Modulation  $\text{(DSB–AM)}$.
  • It should be noted that $B_{\rm NF}$ and $B_{\rm K}$  are absolute and  non-equivalent bandwidths.  The latter are defined over rectangles of equal area and are denoted in our tutorial by  $Δf_q$  and  $Δf_{\rm K}$,  resp.
  • The spectral function  $S(f)$  does not include any Dirac-lines at the carrier frequency  $(\pm f_{\rm T})$.  Therefore, this method is also referred to as  "DSB-AM without carrier".
  • The frequency components
  • above the carrier frequency  $f_{\rm T}$  are called the  "upper sideband"  $\rm (USB)$,
  • and those below  $f_{\rm T}$  are the  "lower sideband"  $\rm (LSB)$.

Description in the time domain


Adapting the notation and nomenclature to this problem,  the convolution theorem reads:

$$S(f) = Z(f) \star Q(f)\hspace{0.2cm}\bullet\!\!-\!\!\!-\!\!\!-\!\!\circ\, \hspace{0.2cm} s(t) = q(t) \cdot z(t) = q(t) \cdot \cos(\omega_{\rm T}\cdot t + \phi_{\rm T})\hspace{0.05cm}.$$
Models of DSB–AM without carrier

This result is still true if the restrictions made on the last page are removed:

  1. real-valued spectrum  $Q(f)$, 
  2. carrier phase  $ϕ_{\rm T} = 0$ 

In general,  this results in a complex-valued spectrum  $S(f)$.

According to this equation, two models can be given for double-sideband amplitude modulation.  These are to be interpreted as follows:

  • The upper model directly describes the relationship given above,  where the carrier  $z(t) = \cos(ω_{\rm T}t + ϕ_{\rm T})$  is applied without a unit.
  • The lower model is more in line with the physical condition  "each signal also has a unit".  If  $q(t)$  and  $z(t)$  are voltages,  the model still needs to provide a scaling with the modulator constant  $K_{\rm AM}$  with unit   $\rm 1/V$,  so that the output signal  $s(t)$  also represents a voltage waveform.
  • If we set  $K_{\rm AM} = 1/A_{\rm T}$,  both models are the same.  In the following,  we will always assume the lower, simpler model.


Signal waveforms in DSB–AM without carrier

$\text{Example 1:}$  The graph shows in red the transmitted signals  $s(t)$  for DSB–AM with two different carrier frequencies. 

The source signal  $q(t)$  with bandwidth  $B_{\rm NF} = 4\text{ kHz}$,  which is the same in both cases,  is drawn in solid blue and the signal  $-q(t)$  is dashed.

The carrier signal  $z(t)$  has a cosine shape in both cases.  In the upper sketch,  the carrier frequency  (German:  "Trägersignal"   ⇒   subscript "T")  is  $f_{\rm T} = 20\text{ kHz}$  and in the lower sketch  $f_{\rm T} = 100\text{ kHz}$.

Ring modulator


One possibility to realize  "double-sideband amplitude modulation with carrier suppression"  is offered by a so-called  ring modulator,  also known as  "double push-pull diode modulator".  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:

Ring modulator to realize DSB–AM without carrier
  • Let the amplitude of the harmonic carrier oscillation  $z(t)$  be much larger than the maximum value  $q_{\rm max}$  of the source signal  $q(t)$.  Thus,  all diodes are operated as switches.
  • When the half-wave of the carrier is positive  $(z(t) > 0)$  the two magenta diodes conduct while the light green ones block. Thus, without considering losses,  it holds  $s(t) = q(t)$.
  • For a negative half-wave,  the light green diodes conduct and the diodes in the longitudinal branches block.  As can be seen on the right,  $s(t) = \ – q(t)$  holds for this lower switch position.
  • Due to the operation of this switch,  the harmonic oscillation  $z(t)$  can also be replaced by a periodic  (rectangular)  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  $s(t)$  is then obtained as the product of the source signal  $q(t)$  and this square wave signal  $z_{\rm R}(t)$,  whereas in ideal DSB-AM one multiplies $q(t)$  by a cosine signal.  The carrier  $z(t)$  is not itself included in the signal $s(t)$.  Since this is supplied via the center taps of the transformers, the induced voltages cancel out   ⇒   DSB-AM without carrier.


$\text{Example 2:}$  Now we will explain the operation mode of a ring modulator using exemplary signal characteristics.  Let the carrier frequency be  $f_{\rm T} = 10\text{ kHz}$.

Signals to illustrate a ring modulator





  • The top graph shows the signals  $q(t)$  and  $-q(t)$  as magenta and light green waveforms respectively.  The bipolar square wave  (rectangular)  signal $z_{\rm R}(t)$  is shown in blue dashes,  and takes the values  $±1$ .
  • The middle chart shows the modulated signal from the ring modulator:
$$s_{\rm RM}(t) = q(t) · z_{\rm R}(t).$$
  • For comparison, the conventional DSB-AM signal is shown in the bottom graph:
$$s(t) = q(t) · \cos(ω_{\rm T} · t).$$


One can see significant differences, but these can be compensated for in a simple way:

  • The Fourier series representation of the periodic rectangular signal $z_{\rm R}(t)$  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  $±f_{\rm T}, ±3f_{\rm T}, ±5f_{\rm T}$, etc.  Convolution with  $Q(f)$  leads to the spectrum
    (the subscript stands for "ring modulator"):
$$S_{\rm RM}(f) = \frac{2}{\pi} \cdot Q (f \pm f_{\rm T})-\frac{2}{3\pi} \cdot Q (f \pm 3f_{\rm T})+\frac{2}{5\pi} \cdot Q (f \pm 5f_{\rm T}) -\text{ ...} \hspace{0.05cm}$$
  • From this, it can be seen that by appropriately band-limiting $($e.g. to  $±2f_{\rm T})$ and attenuating with  $π/4 ≈ 0.785$  the familiar DSB-AM spectrum can be obtained:
$$S(f) = {1}/{2} \cdot Q (f \pm f_{\rm T})\hspace{0.05cm}.$$

Here it must be taken into account that in the above reasoning,  $B_{\rm NF} \ll f_{\rm T}$  must hold.

AM signals and spectra with a harmonic input signal


Now we consider a special case which is important for testing purposes,  where not only the carrier  $z(t)$  is a harmonic oscillation,  but also the signal  $q(t)$  to be modulated:

$$\begin{align*}q(t) & = A_{\rm N} \cdot \cos(\omega_{\rm N} \cdot t + \phi_{\rm N})\hspace{0.05cm}, \\ \\ z(t) & = \hspace{0.15cm}1 \hspace{0.13cm} \cdot \hspace{0.1cm}\cos(\omega_{\rm T} \cdot t + \phi_{\rm T})\hspace{0.05cm}.\end{align*}$$

Please note:   Since we are describing modulation processes,  the phase term is used with a plus sign in the above equations.

  • Thus,  $ϕ_{\rm N} = - 90^\circ$  represents a sinusoidal input signal  $q(t)$  and  $ϕ_{\rm T} = - 90^\circ$  denotes a sinusoidal carrier signal  $z(t)$.
  • Therefore,  the equation for the modulated signal is:
$$s(t) = q(t) \cdot z(t) = A_{\rm N} \cdot \cos(\omega_{\rm N} t + \phi_{\rm N})\cdot \cos(\omega_{\rm T} t + \phi_{\rm T})\hspace{0.05cm}.$$

This equation can be transformed using the trigonometric addition theorem:

$$s(t) = A_{\rm N}/{2} \cdot \cos \big [(\omega_{\rm T} +\omega_{\rm N})\cdot t + \phi_{\rm T}+ \phi_{\rm N} \big ] + A_{\rm N}/{2} \cdot \cos \big [(\omega_{\rm T} -\omega_{\rm N})\cdot t + \phi_{\rm T}- \phi_{\rm N} \big ]\hspace{0.05cm}.$$
  • For cosinusoidal signals  $(ϕ_{\rm T} = ϕ_{\rm N} = 0)$,  this equation simplifies to
$$s(t) = {A_{\rm N}}/{2} \cdot \cos\big[(\omega_{\rm T}+\omega_{\rm N})\cdot t\big] + {A_{\rm N}}/{2} \cdot \cos\big[(\omega_{\rm T} -\omega_{\rm N})\cdot t \big]\hspace{0.05cm}.$$
  • Using a Fourier transform we arrive at the spectral function:
$$S(f) = {A_{\rm N}}/{4} \cdot \big[\delta ( f - f_{\rm T} - f_{\rm N})+\delta ( f + f_{\rm T} + f_{\rm N})\big)] + {A_{\rm N}}/{4} \cdot \big[ \delta ( f - f_{\rm T}+ f_{\rm N})+\delta ( f+ f_{\rm T} - f_{\rm N} ) \big]\hspace{0.05cm}.$$

This result,  which would also have been arrived at via convolution,  states:

  • The spectrum consists of four Dirac lines at frequencies  $±(f_{\rm T} + f_{\rm N})$  and  $±(f_{\rm T} - f_{\rm N})$. 
  • 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  $A_{\rm N}/4$. 
  • The sum of these weights   - that is, the integral over  $S(f)$ –  is equal to the signal value  $s(t = 0) = A_{\rm N}$.
  • The Dirac lines remain for  $ϕ_{\rm T} ≠ 0$  and/or  $ϕ_{\rm N} ≠ 0$  at the same frequencies.  However, complex rotation factors must then be added to the weights  $A_{\rm N}/4$ .


$\text{Example 3:}$  The following diagram shows the spectral functions  $S(f)$  for different values of  $ϕ_{\rm T}$  and  $ϕ_{\rm N}$,  respectively.  The other parameters are assumed to be  $f_{\rm T} = 50\text{ kHz}$,  $f_{\rm N} = 10\text{ kHz}$  and  $A_{\rm N} = 4\text{ V}$.  Thus,  the magnitudes of all Dirac lines are $A_{\rm N}/4 = 1\text{ V}$.

Typical spectra for DSB-AM


  • The upper left diagram shows the case just discussed:  Both the carrier and the source signal are cosine.  Thus,  the amplitude-modulated signal  $s(t)$  is composed of two cosine oscillations with  $ω_{60} = 2 π · 60\text{ kHz}$  and  $ω_{40} = 2 π · 40\text{ kHz}$.


  • For the other three constellations,  at least one of the signals  $q(t)$  or  $z(t)$  is sinusoidal,  so that  $s(0) = 0$  always holds.  Thus,  for these spectra,  the sum of the four pulse weights each add up to zero.


  • The bottom right diagram depicts  $s(t) = A_{\rm N} · \sin(ω_{\rm N} t) · \sin(ω_{\rm T}t)$.  Multiplying two odd functions yields the even function  $s(t)$  and thus a real spectrum  $S(f)$.  In contrast,  the other two constellations each result in imaginary spectral functions.

Double-Sideband Amplitude Modulation with carrier


The following diagram shows how to get from  "DSB-AM without carrier"  to the better known variant "DSB-AM with carrier".  This has the advantage that the demodulator can be realized much more easily and cheaply by a simple manipulation at the transmitter.

Models of DSB–AM with carrier

The diagram is to be interpreted as follows:

  • The top plot shows the physical model of the  "DSB-AM with carrier",  with changes from the  "DSB-AM without carrier"  highlighted in red.
  • The carrier signal  $z(t) = A_{\rm T} · \cos(ω_{\rm T} · t)$  is added to the signal  $s(t)$,  which causes two additional Dirac delta functions in the spectrum  $±f_{\rm T}$,  each with impulse weight  $A_{\rm T}/2$.
  • Adding the DC signal  $A_{\rm T}$  to the source signal and then multiplying by the dimensionless carrier  $z(t)$  as shown in the lower sketch results in the same signal  $s(t)$  and spectrum  $S(f)$  as above.
  • Thus, the second representation is equivalent to the upper model.  In both cases the carrier phase is only set to  $ϕ_{\rm T} = 0$  for the sake of simplified presentation.


$\text{Example 4:}$  The  "double-sideband amplitude modulation with carrier"  still finds its main application in radio transmissions for

  • long wave     $($frequency range  $\text{30 kHz}$ ... $\text{300 kHz})$,
  • medium wave  $($frequency range  $\text{300 kHz}$ ... $\text{3 MHz})$,
  • short wave    $($frequency range  $\text{3 MHz}$ ... $\text{30 MHz})$.


However,  these frequencies are increasingly being made available for digital applications, e.g.  "Digital Video Broadcast"  $\rm (DVB)$.

An application of   "double-sideband amplitude modulation without carrier"  exists for example in FM stereo broadcasting:

  • Here,  the differential signal between the two stereo channels is amplitude modulated (without carrier) at  $\text{39 kHz}$.
  • Then the sum signal of the two channels  $($each in the range  $\text{30 Hz}$ ... $\text{15 kHz})$  is combined with the differential signal and an auxiliary carrier at  $\text{19 kHz}$.  Then, this combined signal is frequency modulated.


$\text{Example 5:}$  The following signal waveforms are intended to further clarify the principle of the modulation method  "DSB-AM with carrier".

Signal waveforms for  "DSB–AM with carrier"  and  "DSB-AM with carrier suppression"
  • In the upper diagram you see a section of the source signal  $q(t)$  with frequencies in the range  $\vert f \vert \le 4\text{ kHz}$.
  • $s(t)$  is obtained by adding the DC component  $A_{\rm T}$  to  $q(t)$  and multiplying this summed signal by the carrier  $z(t)$  of frequency  $f_{\rm T} = 100\text{ kHz}$   ⇒   see middle diagram.
  • In the lower diagram you see for comparison the transmitted signal of the "DSB-AM without carrier"   ⇒   "DSB-AM with carrier suppression".


A comparison of these signal waveforms shows:

  • By adding the DC component  $A_{\rm T}$  the signal  $q(t)$  can now be seen in the envelope of  $s(t)$.
  • Thus,  envelope demodulation  can be applied,  which is easier and cheaper to implement than coherent  synchronous demodulation.
  • However,  a prerequisite for the application of the envelope demodulator is a modulation depth  $m <1$. 
  • This parameter is defined as follows:
$$m = \frac{q_{\rm max} }{A_{\rm T} } \hspace{0.3cm}{\rm with}\hspace{0.3cm} q_{\rm max} = \max_{t} \hspace{0.05cm} \vert q(t) \vert\hspace{0.05cm}.$$


$\text{Conclusions:}$ 

  • The advantage of a simpler demodulator is traded off with a much higher transmit power,  since the power of the carrier cannot be used for demodulation. 
  • Furthermore,  care must be taken to ensure that the source signal does not contain a DC component,  since this would be masked by the carrier. 
  • For speech and music signals,  however,  this is not a major restriction.

Describing DSB-AM with carrier using the analytical signal


Spectrum of the analytical signal in two different viewpoints

In the further course of this chapter,  for the sake of simplifying the graphs,  the spectrum  $S_+(f)$  of the  analytical signal  is usually given instead of the actual, physical spectrum  $S(f)$.

As an example,  let us consider  "DSB-AM with carrier"  and the following signals:

$$\begin{align*}s(t) & = \left(q(t) + A_{\rm T}\right) \cdot \cos(\omega_{\rm T}\cdot t + \phi_{\rm T})\hspace{0.05cm}, \\ \\ q(t) & = A_{\rm N} \cdot \cos(\omega_{\rm N}\cdot t + \phi_{\rm N})\hspace{0.05cm}.\end{align*}$$

Then the corresponding analytical signal is:

$$s_+(t) = A_{\rm T} \cdot {\rm e}^{{\rm j}\hspace{0.03cm} \cdot \hspace{0.01cm}(\omega_{\rm T}\cdot \hspace{0.02cm}t \hspace{0.05cm}+ \hspace{0.05cm}\phi_{\rm T})}+ \frac{A_{\rm N}}{2} \cdot {\rm e}^{{\rm j}\hspace{0.03cm} \cdot \hspace{0.01cm}((\omega_{\rm T} \hspace{0.05cm}+ \hspace{0.05cm} \omega_{\rm N} )\hspace{0.02cm}\cdot \hspace{0.02cm}t \hspace{0.05cm}+ \hspace{0.05cm} \phi_{\rm T}+ \phi_{\rm N})} + \frac{A_{\rm N}}{2} \cdot {\rm e}^{{\rm j}\hspace{0.03cm} \cdot \hspace{0.01cm}((\omega_{\rm T} \hspace{0.05cm}- \hspace{0.05cm} \omega_{\rm N} )\hspace{0.02cm}\cdot \hspace{0.02cm}t \hspace{0.05cm}+ \hspace{0.05cm} \phi_{\rm T}- \phi_{\rm N})} \hspace{0.05cm}.$$

The corresponding spectral function  $S_+(f)$  consists of three Dirac delta lines,  each with complex weights corresponding to the graph:

  • The left sketch shows  $|S_+(f)|$.  $A_{\rm T}$  indicates the weight of the carrier and  $A_{\rm N}/2$  indicates the weights of  $\rm USB$  (upper sideband) and  $\rm LSB$  (lower sideband).
  • The values normalized to  $A_{\rm T}$  are given in parentheses.  Since  $q_{\rm max} = A_{\rm N}$  holds here,  the modulation depth  $m = A_{\rm N}/A_{\rm T}$  gives  $m/2$ as the normalized weights of both the upper and lower sideband.
  • The right sketch depicts the direction of the frequency axis and shows the phase angles of carrier  $(ϕ_{\rm T})$,  of  $\rm LSB$  $(ϕ_{\rm T} - ϕ_{\rm N})$  and  of  $\rm USB$  $(ϕ_{\rm T} + ϕ_{\rm N})$.


Amplitude modulation with a quadratic characteristic curve


Nonlinearities are usually undesirable and troublesome in Communications Engineering.  As explained in the chapter  Nonlinear Distortions   of the book  "Linear and Time Invariant Systems”,  they lead to the facts that:

  • the superposition principle is no longer applicable,
  • the transmission behavior depends on the magnitude of the input signal,  and
  • the distortions are of nonlinear nature and thus irreversible.


A nonlinearity of the general form

$$y(t) = c_0 + c_1 \cdot x(t) + c_2 \cdot x^2(t)+ c_3 \cdot x^3(t) + \text{...}$$

can also be used to implement  $\text{DSB}\hspace{0.05cm}–\hspace{-0.05cm}\text{AM}$.  Provided that

  • only the coefficients  $c_1$  and  $c_2$  are present,  and
  • the input signal  $x(t) = q(t) + z(t)$  is applied,


we obtain the nonlinearity's output signal:

$$y(t) = c_1 \cdot q(t) + c_1 \cdot z(t) + c_2 \cdot q^2(t)+ 2 \cdot c_2 \cdot q(t)\cdot z(t)+ c_2 \cdot z^2(t)\hspace{0.05cm}.$$

The first, third, and last components are – in terms of spectra – at  $|\hspace{0.05cm} f \hspace{0.05cm}| ≤ 2 · B_{\rm NF}$  and  $|\hspace{0.05cm} f\hspace{0.05cm} | = 2 · f_{\rm T}$, respectively.

Removing these signal components by a band-pass and considering  $z(t) = A_{\rm T} · \cos(ω_{\rm T} · t)$,  we obtain the equation typical for  "DSB-AM with carrier"
(only the second and fourth terms remain):

$$s(t) = c_1 \cdot A_{\rm T} \cdot \cos(\omega_{\rm T} \cdot t ) + 2 \cdot c_2 \cdot A_{\rm T} \cdot q(t)\cdot \cos(\omega_{\rm T} \cdot t )\hspace{0.05cm}.$$

The modulation depth in this realization form is variable by the coefficients  $c_1$  and  $c_2$ :

$$m = \frac{2 \cdot c_2 \cdot q_{\rm max}}{c_1} \hspace{0.05cm}.$$
  • A diode and a field-effect transistor both approximate such a quadratic characteristic curve and can thus be used to realize DSB-AM.
  • However,  cubic components  $(c_3 ≠ 0)$  and higher order nonlinearities lead to  (large)  nonlinear distortions.


Exercises for the chapter


Exercise 2.1: DSB-AM with Cosine? Or with Sine?

Exercise 2.1Z: DSB-AM without/with Carrier

Exercise 2.2: Modulation Depth

Exercise 2.2Z: Power Consideration

Exercise 2.3: DSB–AM Realization

Exercise 2.3Z: DSB-AM due to Nonlinearity