Applets:Physical Signal & Equivalent Lowpass Signal: Unterschied zwischen den Versionen

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==Applet Description==
 
==Applet Description==
 
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This applet shows the relationship between the physical bandpass&ndash;signal $x(t)$ and the associated equivalent low pass&ndash;signal $x_{\rm TP}(t)$. The starting point is always a bandpass&ndash;signal $x(t)$ with a frequency-discrete spectrum $X(f)$:
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This applet shows the relationship between the physical bandpass signal $x(t)$ and the associated equivalent lowpass signal $x_{\rm TP}(t)$. It is assumed that the bandpass signal $x(t)$ has a frequency-discrete spectrum $X(f)$:
:$$x(t) = x_{\rm T}(t) + x_{\rm O}(t) + x_{\rm U}(t) = A_{\rm T}\cdot \cos\left(2\pi f_{\rm T}\cdot t- \varphi_{\rm T}\right)+A_{\rm O}\cdot \cos\left(2\pi f_{\rm O}\cdot t- \varphi_{\rm O}\right)+ A_{\rm U}\cdot \cos\left(2\pi f_{\rm U}\cdot t- \varphi_{\rm U}\right). $$
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:$$x(t) = x_{\rm T}(t) + x_{\rm O}(t)+ x_{\rm U}(t) = A_{\rm T}\cdot \cos\left(2\pi f_{\rm T}\cdot t- \varphi_{\rm T}\right)+A_{\rm O}\cdot \cos\left(2\pi f_{\rm O}\cdot t- \varphi_{\rm O}\right) + A_{\rm U}\cdot \cos\left(2\pi f_{\rm U}\cdot t- \varphi_{\rm U}\right). $$
The physical signal $x(t)$ is thus composed of three [[Signaldarstellung/Harmonische_Schwingung|harmonic oscillations]], a constellation which, for example, in the [[Modulationsverfahren/Zweiseitenband-Amplitudenmodulation#AM-Signale_und_-Spektren_bei_harmonischem_Eingangssignal|double-sideband amplitude modulation]] of the message signal $x_{\rm N}(t) = A_{\rm N}\cdot \cos\left(2\pi f_{\rm N}\cdot t- \varphi_{\rm N}\right)$ with the carrier signal $x_{\rm T}(t) = A_{\rm T}\cdot \cos\left(2\pi f_{\rm T}\cdot t - \varphi_{\rm T}\right)$ returns. The nomenclature is also adapted to this case:
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The physical signal $x(t)$ is thus composed of three harmonic oscillations, a constellation that can be found, for example, in the ''Double-sideband Amplitude Modulation''
* $x_{\rm O}(t)$ denotes the &bdquo;upper sideband&rdquo; with the amplitude $A_{\rm O}= A_{\rm N}/2$, the frequency $f_{\rm O} = f_{\rm T} + f_{\rm N}$ and the phase $\varphi_{\rm O} = \varphi_{\rm T} + \varphi_{\rm N}$.
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*of the message signal $x_{\rm N}(t) = A_{\rm N}\cdot \cos\left(2\pi f_{\rm N}\cdot t- \varphi_{\rm N}\right)$ &nbsp; &rArr; &nbsp; in German: &nbsp;  '''N'''achrichtensignal
*Similarly, for the &bdquo;lower Sideband&rdquo; $x_{\rm U}(t)$ with $f_{\rm U} = f_{\rm T} + f_{\rm N}$, $A_{\rm U}= A_{\rm O}$ and $\varphi_{\rm U} = -\varphi_{\rm O}$.
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*with the carrier signal $x_{\rm T}(t) = A_{\rm T}\cdot \cos\left(2\pi f_{\rm T}\cdot t - \varphi_{\rm T}\right)$ &nbsp; &rArr; &nbsp; in German: &nbsp; '''T'''rägersignal.
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The nomenclature is also adapted to this case:
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* $x_{\rm O}(t)$ denotes the &bdquo;upper sideband&rdquo; &nbsp; (in German: &nbsp; '''O'''beres Seitenband) with the amplitude $A_{\rm O}= A_{\rm N}/2$, the frequency $f_{\rm O} = f_{\rm T} + f_{\rm N}$ and the phase $\varphi_{\rm O} = \varphi_{\rm T} + \varphi_{\rm N}$.
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*Similarly, for the &bdquo;lower sideband&rdquo; &nbsp; (in German: &nbsp; '''U'''nteres Seitenband) $x_{\rm U}(t)$ with $f_{\rm U} = f_{\rm T} - f_{\rm N}$, $A_{\rm U}= A_{\rm O}$ and $\varphi_{\rm U} = -\varphi_{\rm O}$.
  
The corresponding equivalent low-pass&ndash;signal is $f_{\rm O}\hspace{0.01cm}' = f_{\rm O}- f_{\rm T} > 0$, &nbsp; $f_{\rm U}\hspace{0.01cm}' = f_{\rm U}- f_{\rm T} < 0$ &nbsp;and &nbsp;$f_{\rm T}\hspace{0.01cm}' =  0$:
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The associated equivalent lowpass&ndash;signal is $f_{\rm O}\hspace{0.01cm}' = f_{\rm O}- f_{\rm T} > 0$, &nbsp; $f_{\rm U}\hspace{0.01cm}' = f_{\rm U}- f_{\rm T} < 0$ &nbsp;and &nbsp;$f_{\rm T}\hspace{0.01cm}' =  0$:
  
 
:$$x_{\rm TP}(t) = x_\text{TP, T}(t) + x_\text{TP, O}(t) + x_\text{TP, U}(t) = A_{\rm T}\cdot {\rm e}^{-{\rm j} \varphi_{\rm T} } \hspace{0.1cm}+ \hspace{0.1cm} A_{\rm O}\cdot {\rm e}^{-{\rm j} \varphi_{\rm O} } \cdot {\rm e}^{ {\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm O}\hspace{0.01cm}'\hspace{0.05cm}\cdot \hspace{0.05cm}t}\hspace{0.1cm}+ \hspace{0.1cm}
 
:$$x_{\rm TP}(t) = x_\text{TP, T}(t) + x_\text{TP, O}(t) + x_\text{TP, U}(t) = A_{\rm T}\cdot {\rm e}^{-{\rm j} \varphi_{\rm T} } \hspace{0.1cm}+ \hspace{0.1cm} A_{\rm O}\cdot {\rm e}^{-{\rm j} \varphi_{\rm O} } \cdot {\rm e}^{ {\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm O}\hspace{0.01cm}'\hspace{0.05cm}\cdot \hspace{0.05cm}t}\hspace{0.1cm}+ \hspace{0.1cm}
 
A_{\rm U}\cdot {\rm e}^{-{\rm j} \varphi_{\rm U} } \cdot {\rm e}^{ {\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm U}\hspace{0.01cm}'\hspace{0.05cm}\cdot \hspace{0.05cm}t} . $$
 
A_{\rm U}\cdot {\rm e}^{-{\rm j} \varphi_{\rm U} } \cdot {\rm e}^{ {\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm U}\hspace{0.01cm}'\hspace{0.05cm}\cdot \hspace{0.05cm}t} . $$
  
[[Datei:Ortskurve_1.png|right|frame|Equivalent lowpass&ndash;signal currently $t=0$ for cosinusoidal carrier &nbsp; &rArr; &nbsp; $\varphi_{\rm T} = 0$]]
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[[Datei:Ortskurve_1.png|right|frame|Equivalent low-pass signal currently $t=0$ for cosinusoidal carrier &nbsp; &rArr; &nbsp; $\varphi_{\rm T} = 0$]]
The program shows $x_{\rm TP}(t)$ as the vectorial sum of three rotation pointers as a violet point (see example graph for start time $t=0$ and cosinusoidal support):
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The program shows $x_{\rm TP}(t)$ as the vectorial sum of three rotation pointers as a violet dot (see figure for start time $t=0$ and cosinusoidal carrier):
  
*The (red) pointer of the carrier $x_\text{TP, T}(t)$ with the length $A_{\rm T}$ and the zero phase position $\varphi_{\rm T}=0$ lies in the complex plane firmly. So it applies to all times $t$: &nbsp; $x_{\rm TP}(t)= A_{\rm T}\cdot {\rm e}^{-{\rm j} \varphi_{\rm T} }$.
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*The (red) pointer of the carrier $x_\text{TP, T}(t)$ with the length $A_{\rm T}$ and the zero phase position $\varphi_{\rm T}=0$ is fixed in the complex plane. So it applies to all times $t$: &nbsp; $x_{\rm TP}(t)= A_{\rm T}\cdot {\rm e}^{-{\rm j} \varphi_{\rm T} }$.
  
 
*The (blue) pointer of the upper sideband $x_\text{TP, O}(t)$ with the length $A_{\rm O}$ and the zero phase position $\varphi_{\rm O}$ rotates at the angular velocity $2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm O}\hspace{0.01cm}'$ in mathematically positive direction (one revolution in time $1/f_{\rm O}\hspace{0.01cm}')$.
 
*The (blue) pointer of the upper sideband $x_\text{TP, O}(t)$ with the length $A_{\rm O}$ and the zero phase position $\varphi_{\rm O}$ rotates at the angular velocity $2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm O}\hspace{0.01cm}'$ in mathematically positive direction (one revolution in time $1/f_{\rm O}\hspace{0.01cm}')$.
Zeile 24: Zeile 29:
 
*The (green) pointer of the lower sideband $x_{\rm U+}(t)$ with the length $A_{\rm U}$ and the zero phase position $\varphi_{\rm U}$ rotates at the angular velocity $2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm U}\hspace{0.01cm}'$, because of $f_{\rm U}\hspace{0.01cm}'<0$ counterclockwise.
 
*The (green) pointer of the lower sideband $x_{\rm U+}(t)$ with the length $A_{\rm U}$ and the zero phase position $\varphi_{\rm U}$ rotates at the angular velocity $2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm U}\hspace{0.01cm}'$, because of $f_{\rm U}\hspace{0.01cm}'<0$ counterclockwise.
  
*With $f_{\rm U}\hspace{0.01cm}' = -f_{\rm O}\hspace{0.01cm}'$ the blue and green pointers will spin at the same speed but in different directions. Also, if $A_{\rm O} = A_{\rm U}$ and $\varphi_{\rm U} = -\varphi_{\rm O}$, then $x_{\rm TP}(t)$ on a line with a slope of $\varphi_{\rm T}$.
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*With $f_{\rm U}\hspace{0.01cm}' = -f_{\rm O}\hspace{0.01cm}'$ the blue and green pointers will spin at the same speed but in different directions. Also, if $A_{\rm O} = A_{\rm U}$ and $\varphi_{\rm U} = -\varphi_{\rm O}$, then $x_{\rm TP}(t)$ moves on a straight line with a incline of $\varphi_{\rm T}$.
  
  
''Hinweis:'' &nbsp; The graphic applies to $\varphi_{\rm O} = +30^\circ$. From this follows for the start time $t=0$ the angle of the blue pointer (OSB)  with respect to the coordinate system: &nbsp; $\phi_{\rm O} = -\varphi_{\rm O} = -30^\circ$. Likewise, the null phantom $\varphi_{\rm U} = -30^\circ$ of the lower sideband (USB, grüner Zeiger) follows for the phase angle to be considered in the complex plane: &nbsp; $\phi_{\rm U} = +30^\circ$.
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''Note:'' &nbsp; In the figure $\varphi_{\rm O} = +30^\circ$. From this follows for the start time $t=0$ the angle of the upper sideband (OSB, blue pointer)  with respect to the coordinate system: &nbsp; $\phi_{\rm O} = -\varphi_{\rm O} = -30^\circ$. Likewise, the zero phase position $\varphi_{\rm U} = -30^\circ$ of the lower sideband (USB, green pointer) follows for the phase angle to be considered in the complex plane: &nbsp; $\phi_{\rm U} = +30^\circ$.
  
  
The temporal course of $x_{\rm TP}(t)$ is also referred to below as '''locus'''. The relationship between $x_{\rm TP}(t)$ and the physical bandpass &ndash;signal $x(t)$ is given in the section [[???]] and the associated analytic signal is $x_+(t)$ :
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The temporal process of $x_{\rm TP}(t)$ is also referred to below as &bdquo;locus&rdquo;. The relationship between $x_{\rm TP}(t)$ and the physical bandpass signal $x(t)$ is given in the section and the associated analytic signal is $x_+(t)$ :
  
 
:$$x_{\rm TP}(t) = x_{\rm +}(t)\cdot {\rm e}^{-{\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm}2 \pi \cdot f_{\rm T}\cdot \hspace{0.05cm}t},$$
 
:$$x_{\rm TP}(t) = x_{\rm +}(t)\cdot {\rm e}^{-{\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm}2 \pi \cdot f_{\rm T}\cdot \hspace{0.05cm}t},$$
Zeile 43: Zeile 48:
 
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===Description of Bandpass Signals===
 
===Description of Bandpass Signals===
[[Datei:Zeigerdiagramm_1a.png|right|frame|bandpass&ndash;spectrum $X(f)$ |class=fit]]
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[[Datei:Zeigerdiagramm_1a.png|right|frame|bandpass spectrum $X(f)$ |class=fit]]
 
We consider '''bandpass signals''' $x(t)$ with the property that their spectra $X(f)$ are not in the range around the frequency $f=0$, but around a carrier frequency $f_{\rm T}$. In most cases it can also be assumed that the bandwidth is $B \ll f_{\rm T}$.
 
We consider '''bandpass signals''' $x(t)$ with the property that their spectra $X(f)$ are not in the range around the frequency $f=0$, but around a carrier frequency $f_{\rm T}$. In most cases it can also be assumed that the bandwidth is $B \ll f_{\rm T}$.
  
Zeile 50: Zeile 55:
  
 
Beside the physical signal $x(t)\ \circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\, \ X(f)$, one can also use the following descriptions of bandpass signals:
 
Beside the physical signal $x(t)\ \circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\, \ X(f)$, one can also use the following descriptions of bandpass signals:
*the analytic signal $x_+(t)\ \circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\, \ X_+(f)$, see applet [[Applets:Physikalisches_Signal_%26_Analytisches_Signal|Physikalisches Signal & Analytisches Signal]],
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*the analytic signal $x_+(t)\ \circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\, \ X_+(f)$, see applet [[Applets:Physical_Signal_%26_Analytic_Signal|Physical Signal & Analytic Signal]],
*the equivalent low-pass&ndash;signal $x_{\rm TP}(t)\ \circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\, \ X_{\rm TP}(f)$, see next page
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*the equivalent lowpass signal $x_{\rm TP}(t)\ \circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\, \ X_{\rm TP}(f)$, see next page
 
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===Spectral Functions of the Analytic and the Equivalent Low-Pass Signal===
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===Spectral Functions of the Analytic and the Equivalent Lowpass Signal===
  
The '''analytische Signal''' $x_+(t)$ belonging to the physical signal $x(t)$ is the time function whose spectrum fulfills the following property:
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The '''analytic signal''' $x_+(t)$ belonging to the physical signal $x(t)$ is the time function whose spectrum fulfills the following property:
[[Datei:Ortskurve_2.png|right|frame|spectral functions $X_+(f)$ and $X_{\rm TP}(f)$ |class=fit]]
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[[Datei:Ortskurve_2.png|right|frame|spectral functions $X(f)$, $X_+(f)$ and $X_{\rm TP}(f)$ |class=fit]]
 
:$$X_+(f)=\big[1+{\rm sign}(f)\big] \cdot X(f) = \left\{ {2 \cdot
 
:$$X_+(f)=\big[1+{\rm sign}(f)\big] \cdot X(f) = \left\{ {2 \cdot
 
X(f) \; \hspace{0.2cm}\rm for\hspace{0.2cm} {\it f} > 0, \atop {\,\,\,\, \rm 0 \; \hspace{0.9cm}\rm for\hspace{0.2cm} {\it f} < 0.} }\right.$$
 
X(f) \; \hspace{0.2cm}\rm for\hspace{0.2cm} {\it f} > 0, \atop {\,\,\,\, \rm 0 \; \hspace{0.9cm}\rm for\hspace{0.2cm} {\it f} < 0.} }\right.$$
  
The ''Signumfunktion'' is for positive values of $f$ equal to $+1$ and for negative $f$ values equal to $-1$.
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The ''Signum function'' is for positive values of $f$ equal to $+1$ and for negative $f$ values equal to $-1$.
 
*The (double-sided) limit returns $\sign(0) = 0$.
 
*The (double-sided) limit returns $\sign(0) = 0$.
 
*The index „+” should make it clear that $X_+(f)$ only has parts at positive frequencies.
 
*The index „+” should make it clear that $X_+(f)$ only has parts at positive frequencies.
  
  
From the graph you can see the calculation rule for $X_+(f)$: <br> The actual BP spectrum $X(f)$ becomes
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From the graph you can see the calculation rule for $X_+(f)$: &nbsp; The actual bandpass spectrum $X(f)$ becomes
 
*doubled at the positive frequencies, and
 
*doubled at the positive frequencies, and
 
*set to zero at the negative frequencies.
 
*set to zero at the negative frequencies.
Zeile 74: Zeile 79:
  
  
The spectrum $X_{\rm TP}(f)$ of the equivalent low-pass signal is obtained by shifting $X_+(f)$ to the left by the carrier frequency $f_{\rm T}$:
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The spectrum $X_{\rm TP}(f)$ of the equivalent lowpass signal is obtained by shifting $X_+(f)$ to the left by the carrier frequency $f_{\rm T}$:
 
:$$X_{\rm TP}(f)= X_+(f+f_{\rm T}).$$
 
:$$X_{\rm TP}(f)= X_+(f+f_{\rm T}).$$
  
Zeile 80: Zeile 85:
 
:$$x_{\rm TP}(t) = x_{\rm +}(t)\cdot {\rm e}^{-{\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm}2 \pi \cdot f_{\rm T}\cdot \hspace{0.05cm}t}.$$   
 
:$$x_{\rm TP}(t) = x_{\rm +}(t)\cdot {\rm e}^{-{\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm}2 \pi \cdot f_{\rm T}\cdot \hspace{0.05cm}t}.$$   
  
It can be seen that $x_{\rm TP}(t)$ s generally complex. But if $X_+(f)$ is symmetric about the carrier frequency $f_{\rm T}$, $X_{\rm TP}(f)$ is symmetric about the frequency $f=0$ and the result is accordingly real time function $x_{\rm TP}(t)$.
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It can be seen that $x_{\rm TP}(t)$ is generally complex. But if $X_+(f)$ is symmetric about the carrier frequency $f_{\rm T}$, $X_{\rm TP}(f)$ is symmetric about the frequency $f=0$ and there is accordingly a real time function $x_{\rm TP}(t)$.
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<br><br>
  
===$x_{\rm TP}(t)$&ndash;Representation of a Sum of Three Harmonic Oscillations===
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===$x_{\rm TP}(t)$ Representation of a Sum of Three Harmonic Oscillations===
  
 
In our applet, we always assume a set of three rotating pointers. The physical signal is:
 
In our applet, we always assume a set of three rotating pointers. The physical signal is:
:$$x(t) = x_{\rm U}(t) + x_{\rm T}(t) + x_{\rm O}(t) = A_{\rm U}\cdot \cos\left(2\pi f_{\rm U}\cdot t- \varphi_{\rm U}\right)+A_{\rm T}\cdot \cos\left(2\pi f_{\rm T}\cdot t- \varphi_{\rm T}\right)+A_{\rm O}\cdot \cos\left(2\pi f_{\rm O}\cdot t- \varphi_{\rm O}\right). $$
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:$$x(t) = x_{\rm T}(t) + x_{\rm O}(t) + x_{\rm U}(t) = A_{\rm T}\cdot \cos\left(2\pi f_{\rm T}\cdot t- \varphi_{\rm T}\right)+A_{\rm O}\cdot \cos\left(2\pi f_{\rm O}\cdot t- \varphi_{\rm O}\right) + A_{\rm U}\cdot \cos\left(2\pi f_{\rm U}\cdot t- \varphi_{\rm U}\right). $$
 
* Each of the three harmonic oscillations $x_{\rm T}(t)$, $x_{\rm U}(t)$ and $x_{\rm O}(t)$ is represented by an amplitude $(A)$, a frequency $(f)$ and a phase value $(\varphi)$.
 
* Each of the three harmonic oscillations $x_{\rm T}(t)$, $x_{\rm U}(t)$ and $x_{\rm O}(t)$ is represented by an amplitude $(A)$, a frequency $(f)$ and a phase value $(\varphi)$.
*The indices are based on the modulation method [[Modulationsverfahren/Zweiseitenband-Amplitudenmodulation|double sideband &ndash;amplitude modulation]]. &bdquo;T&rdquo; stands for &bdquo;carrier&rdquo;, &bdquo;U&rdquo; for &bdquo;lowei sideband&rdquo; and &bdquo;O&rdquo; for &bdquo;upper Sideband&rdquo;. Similarly, $f_{\rm U} < f_{\rm T}$ and $f_{\rm O} > f_{\rm T}$. There are no restrictions for the amplitudes and phases.
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*The indices are based on the modulation method [[Modulationsverfahren/Zweiseitenband-Amplitudenmodulation|double sideband amplitude modulation]]. &bdquo;T&rdquo; stands for &bdquo;carrier&rdquo;, &bdquo;U&rdquo; for &bdquo;lower sideband&rdquo; and &bdquo;O&rdquo; for &bdquo;upper Sideband&rdquo;. Similarly, $f_{\rm U} < f_{\rm T}$ and $f_{\rm O} > f_{\rm T}$. There are no restrictions for the amplitudes and phases.
  
  
Zeile 97: Zeile 103:
 
{{GraueBox|TEXT=
 
{{GraueBox|TEXT=
 
$\text{Example 1:}$&nbsp;
 
$\text{Example 1:}$&nbsp;
The constellation given here results, for example, in the [[Modulationsverfahren/Zweiseitenband-Amplitudenmodulation#AM-Signale_und_-Spektren_bei_harmonischem_Eingangssignal|double sideband amplitude modulation]] of the message signal $x_{\rm N}(t) = A_{\rm N}\cdot \cos\left(2\pi f_{\rm N}\cdot t- \varphi_{\rm N}\right)$ with the carrier signal $x_{\rm T}(t) = A_{\rm T}\cdot \cos\left(2\pi f_{\rm T}\cdot t - \varphi_{\rm T}\right)$. This is discussed more frequently in the experimental procedure.
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The constellation given here results, for example, in the [[Modulationsverfahren/Zweiseitenband-Amplitudenmodulation#AM-Signale_und_-Spektren_bei_harmonischem_Eingangssignal|double sideband amplitude modulation]] of the message signal $x_{\rm N}(t) = A_{\rm N}\cdot \cos\left(2\pi f_{\rm N}\cdot t- \varphi_{\rm N}\right)$ with the carrier signal $x_{\rm T}(t) = A_{\rm T}\cdot \cos\left(2\pi f_{\rm T}\cdot t - \varphi_{\rm T}\right)$. This is discussed frequently in the exercises.
  
[[Datei:Ortskurve_4.png|center|frame|Spectrum $X_{\rm TP}(f)$ of the equivalent low&ndash;pass signal for different phase constellations |class=fit]]
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[[Datei:Ortskurve_5.png|center|frame|Spectrum $X_{\rm TP}(f)$ of the equivalent low&ndash;pass signal for different phase constellations |class=fit]]
  
 
There are some limitations to the program parameters in this approach:
 
There are some limitations to the program parameters in this approach:
 
* The frequencies are always $f\hspace{0.05cm}'_{\rm O} =  f_{\rm N}$ and $f\hspace{0.05cm}'_{\rm U} =  -f_{\rm N}$.
 
* The frequencies are always $f\hspace{0.05cm}'_{\rm O} =  f_{\rm N}$ and $f\hspace{0.05cm}'_{\rm U} =  -f_{\rm N}$.
 
*Without distortion, the amplitude of the sidebands is $A_{\rm O}= A_{\rm U}= A_{\rm N}/2$.
 
*Without distortion, the amplitude of the sidebands is $A_{\rm O}= A_{\rm U}= A_{\rm N}/2$.
*The respective phase relationships can be seen from the graph.
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*The respective phase relationships can be seen in the following graphic.
  
 
}}
 
}}
 +
<br><br>
  
===Representation of the Equivalent Low-Pass Signal by Magnitude and Phase===
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===Representation of the Equivalent Lowpass Signal by Magnitude and Phase===
  
The generally complex valued equivalent low-pass signal  
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The generally complex equivalent lowpass signal  
 
:$$x_{\rm TP}(t) = a(t) \cdot {\rm e}^{ {\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm} \phi(t) }$$
 
:$$x_{\rm TP}(t) = a(t) \cdot {\rm e}^{ {\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm} \phi(t) }$$
 
can be split into a magnitude function $a(t)$ and a phase function $\phi(t)$ according to the equation given here, where:
 
can be split into a magnitude function $a(t)$ and a phase function $\phi(t)$ according to the equation given here, where:
Zeile 116: Zeile 123:
 
:$$\phi(t) = \text{arc }x_{\rm TP}(t) = \arctan \frac{{\rm Im}\big [x_{\rm TP}(t)\big ]}{{\rm Re}\big [x_{\rm TP}(t)\big ]}.$$
 
:$$\phi(t) = \text{arc }x_{\rm TP}(t) = \arctan \frac{{\rm Im}\big [x_{\rm TP}(t)\big ]}{{\rm Re}\big [x_{\rm TP}(t)\big ]}.$$
  
The reason that a bandpass&ndash;signal $x(t)$ is usually described by the equivalent low-pass signal $x_{\rm TP}(t)$ is that the functions $a(t)$ and $\phi(t)$ are interpretable in both representations:
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The reason for this is that a bandpass signal $x(t)$ is usually described by the equivalent lowpass signal $x_{\rm TP}(t)$ that the functions $a(t)$ and $\phi(t)$ are interpretable in both representations:
*The amount $a(t)$ of the equivalent low-pass signal $x_{\rm TP}(t)$ indicates the (time-dependent) envelope of $x(t)$.
+
*The amount $a(t)$ of the equivalent lowpass signal $x_{\rm TP}(t)$ indicates the (time-dependent) envelope of $x(t)$.
 
*The phase $\phi(t)$ of $x_{\rm TP}(t)$ denotes the location of the zero crossings of $x(t)$, where:
 
*The phase $\phi(t)$ of $x_{\rm TP}(t)$ denotes the location of the zero crossings of $x(t)$, where:
 
:&ndash; &nbsp; For $\phi(t)>0$ the zero crossing is earlier than its nominal position &nbsp; &rArr; &nbsp; the signal is leading here.
 
:&ndash; &nbsp; For $\phi(t)>0$ the zero crossing is earlier than its nominal position &nbsp; &rArr; &nbsp; the signal is leading here.
 
:&ndash; &nbsp;When $\phi(t)<0$, the zero crossing is later than its target position &nbsp; &rArr; &nbsp; the signal is trailing here.
 
:&ndash; &nbsp;When $\phi(t)<0$, the zero crossing is later than its target position &nbsp; &rArr; &nbsp; the signal is trailing here.
 +
  
 
{{GraueBox|TEXT=
 
{{GraueBox|TEXT=
Zeile 126: Zeile 134:
 
The graph is intended to illustrate this relationship, assuming $A_{\rm U} > A_{\rm O}$ &nbsp; &rArr; &nbsp;  the green pointer (for the lower sideband) is longer than the blue pointer (upper sideband). This is a snapshot at time $t_0$:
 
The graph is intended to illustrate this relationship, assuming $A_{\rm U} > A_{\rm O}$ &nbsp; &rArr; &nbsp;  the green pointer (for the lower sideband) is longer than the blue pointer (upper sideband). This is a snapshot at time $t_0$:
  
[[Datei:Ortskurve_3_neu.png|center|frame|bandpass&ndash;Spectrum $X(f)$ |class=fit]]
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[[Datei:Ortskurve_3_neu.png|center|frame|bandpass spectrum $X(f)$ |class=fit]]
  
 
*For these system parameters, the top of the pointer cluster $x_{\rm TP}(t)$ &ndash; that is, the geometric sum of red, blue and green pointers &ndash; on an ellipse.  
 
*For these system parameters, the top of the pointer cluster $x_{\rm TP}(t)$ &ndash; that is, the geometric sum of red, blue and green pointers &ndash; on an ellipse.  
Zeile 151: Zeile 159:
 
'''(1)''' &nbsp; Let &nbsp; $\text{Red:} \hspace{0.15cm} A_{\rm T} = 1\ \text{V},  f_{\rm T} = 100 \ \text{kHz},  \varphi_{\rm T} = 0^\circ$, &nbsp;  $\text{Green:} \hspace{0.15cm} A_{\rm U} = 0.4 \text{V}, \ f_{\rm U} = 80 \ \text{kHz},  \varphi_{\rm U} = -90^\circ$,  &nbsp;  $\text{Blue:} \hspace{0.15cm} A_{\rm O} = 0.4\ \text{V},  f_{\rm O} = 120 \ \text{kHz},  \varphi_{\rm O} = 90^\circ$.
 
'''(1)''' &nbsp; Let &nbsp; $\text{Red:} \hspace{0.15cm} A_{\rm T} = 1\ \text{V},  f_{\rm T} = 100 \ \text{kHz},  \varphi_{\rm T} = 0^\circ$, &nbsp;  $\text{Green:} \hspace{0.15cm} A_{\rm U} = 0.4 \text{V}, \ f_{\rm U} = 80 \ \text{kHz},  \varphi_{\rm U} = -90^\circ$,  &nbsp;  $\text{Blue:} \hspace{0.15cm} A_{\rm O} = 0.4\ \text{V},  f_{\rm O} = 120 \ \text{kHz},  \varphi_{\rm O} = 90^\circ$.
  
:Consider and interpret the equivalent low-pass signal $x_{\rm TP}(t)$ and the physical signal $x(t)$. Which period $T_0$ does one recognize?}}
+
:Consider and interpret the equivalent lowpass signal $x_{\rm TP}(t)$ and the physical signal $x(t)$. Which period $T_0$ recognizable?}}
  
::&nbsp;The equivalent low-pass signal $x_{\rm TP}(t)$ takes from $x_{\rm TP}(t=0)=1\ \text{V}$ on the real axis values between $0.2\ \text{V}$ and $1.8\ \text{V}$  &nbsp; &rArr; &nbsp; phase $\phi(t) \equiv 0$.<br>&nbsp;The amount $|x_{\rm TP}(t)|$ indicates the envelope $a(t)$ of the physical signal $x(t)$. It holds $A_{\rm N} = 0.8\ \text{V}$ and $f_{\rm N} = 20\ \text{kHz}$: &nbsp; $a(t) = A_{\rm T}+ A_{\rm N} \cdot \sin(2\pi\cdot f_{\rm N} \cdot t)$.<br>&nbsp;Both $x_{\rm TP}(t)$ and $x(t)$ are periodic with the period $T_0 = 1/f_{\rm N} = 50\ \rm &micro; s$.
+
::&nbsp;The equivalent lowpass signal $x_{\rm TP}(t)$ takes from $x_{\rm TP}(t=0)=1\ \text{V}$ on the real axis values between $0.2\ \text{V}$ and $1.8\ \text{V}$  &nbsp; &rArr; &nbsp; phase $\phi(t) \equiv 0$.<br>&nbsp;The amount $|x_{\rm TP}(t)|$ indicates the envelope $a(t)$ of the physical signal $x(t)$. It holds $A_{\rm N} = 0.8\ \text{V}$ and $f_{\rm N} = 20\ \text{kHz}$: &nbsp; $a(t) = A_{\rm T}+ A_{\rm N} \cdot \sin(2\pi\cdot f_{\rm N} \cdot t)$.<br>&nbsp;Both $x_{\rm TP}(t)$ and $x(t)$ are periodically with the period $T_0 = 1/f_{\rm N} = 50\ \rm &micro; s$.
  
  
Zeile 159: Zeile 167:
 
'''(2)''' &nbsp; How do the ratios change to '''(1)''' with $f_{\rm U} = 99 \ \text{kHz}$ and $f_{\rm O} = 101 \ \text{kHz}$&nbsp;? How could $x(t)$ have arisen?}}
 
'''(2)''' &nbsp; How do the ratios change to '''(1)''' with $f_{\rm U} = 99 \ \text{kHz}$ and $f_{\rm O} = 101 \ \text{kHz}$&nbsp;? How could $x(t)$ have arisen?}}
  
::&nbsp;For the envelope $a(t)$ of the signal $x(t)$ we still have $a(t) = A_{\rm T}+ A_{\rm N} \cdot \sin(2\pi\cdot f_{\rm N} \cdot t)$, but now $f_{\rm N} = 1\ \text{kHz}$. Even though it can not be recognized:<br>&nbsp;$x_{\rm TP}(t)$ and $x(t)$ are still periodic: &nbsp; $T_0 = 1\ \rm ms$. Example: Double sideband - Amplitude modulation '''(ZSB&ndash;AM)''' of a sine signal with cosine&ndash;carrier.  
+
::&nbsp;For the envelope $a(t)$ of the signal $x(t)$ we still have $a(t) = A_{\rm T}+ A_{\rm N} \cdot \sin(2\pi\cdot f_{\rm N} \cdot t)$, but now $f_{\rm N} = 1\ \text{kHz}$. Even though it can not be recognized:<br>&nbsp;$x_{\rm TP}(t)$ and $x(t)$ are still periodically: &nbsp; $T_0 = 1\ \rm ms$. Example: Double sideband Amplitude modulation '''(DSB&ndash;AM)''' of a sine signal with cosine carrier.  
  
  
 
{{BlaueBox|TEXT=
 
{{BlaueBox|TEXT=
'''(3)''' &nbsp; Which settings have to be changed from '''(2)''' in order to arrive at the ZSB&ndash;AM of a cosine signal with sine&ndash;carrier. What changes over '''(2)'''?}}
+
'''(3)''' &nbsp; Which settings have to be changed from '''(2)''' in order to arrive at the DSB&ndash;AM of a cosine signal with sine carrier. What changes over '''(2)'''?}}
  
::The carrier phase must be changed to $\varphi_{\rm T} = 90^\circ$ &nbsp; &rArr; &nbsp; sine&ndash;carrier. Similarly, $\varphi_{\rm O} =\varphi_{\rm U} =\varphi_{\rm T} = 90^\circ$ must be set &nbsp; &rArr; &nbsp; cosine-shaped message<br>&nbsp;The locus now lies on the imaginary axis&nbsp; &rArr; &nbsp; $\phi(t) \equiv -90^\circ$. At the beginning we have $x_{\rm TP}(t=0)= - {\rm j} \cdot 1.8 \ \text{V}$.
+
::The carrier phase must be changed to $\varphi_{\rm T} = 90^\circ$ &nbsp; &rArr; &nbsp; sine carrier. Similarly, $\varphi_{\rm O} =\varphi_{\rm U} =\varphi_{\rm T} = 90^\circ$ must be set &nbsp; &rArr; &nbsp; cosinusoidal message<br>&nbsp;The locus now lies on the imaginary axis&nbsp; &rArr; &nbsp; $\phi(t) \equiv -90^\circ$. At the beginning $x_{\rm TP}(t=0)= - {\rm j} \cdot 1.8 \ \text{V}$.
  
  
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'''(4)''' &nbsp; Now let &nbsp; $\text{Red:} \hspace{0.15cm} A_{\rm T} = 1\ \text{V}, \ f_{\rm T} = 100 \ \text{kHz}, \ \varphi_{\rm T} = 0^\circ$, &nbsp;  $\text{Green:} \hspace{0.15cm} A_{\rm U} = 0.4 \text{V}, \ f_{\rm U} = 80 \ \text{kHz}, \  \varphi_{\rm U} = 0^\circ$,  &nbsp;  $\text{Blue:} \hspace{0.15cm} A_{\rm O} = 0.4\ \text{V}, \ f_{\rm O} = 120 \ \text{kHz}, \ \varphi_{\rm O} = 0^\circ$.  
 
'''(4)''' &nbsp; Now let &nbsp; $\text{Red:} \hspace{0.15cm} A_{\rm T} = 1\ \text{V}, \ f_{\rm T} = 100 \ \text{kHz}, \ \varphi_{\rm T} = 0^\circ$, &nbsp;  $\text{Green:} \hspace{0.15cm} A_{\rm U} = 0.4 \text{V}, \ f_{\rm U} = 80 \ \text{kHz}, \  \varphi_{\rm U} = 0^\circ$,  &nbsp;  $\text{Blue:} \hspace{0.15cm} A_{\rm O} = 0.4\ \text{V}, \ f_{\rm O} = 120 \ \text{kHz}, \ \varphi_{\rm O} = 0^\circ$.  
  
:What are the characteristics of this system &bdquo;ZSB&ndash;AM, where the message signal and carrier are respectively cosinusoidal&rdquo;? What is the modulation depth $m$?}}
+
:What are the characteristics of this system &bdquo;DSB&ndash;AM, where the message signal and carrier are respectively cosinusoidal&rdquo;? What is the degree of modulation $m$?}}
  
::&nbsp;The equivalent low-pass signal $x_{\rm TP}(t)$ takes from $x_{\rm TP}(t=0)=1.8\ \text{V}$ on the real axis values between $0.2\ \text{V}$ and $1.8\ \text{V}$ &nbsp; &rArr; &nbsp; phase $\phi(t) \equiv 0$.<br>&nbsp;Except for the start state $x_{\rm TP}(t=0)$ same behavior as with the setting '''(1)'''. The modulation depth is $m = 0.8$.  
+
::&nbsp;The equivalent lowpass signal $x_{\rm TP}(t)$ takes from $x_{\rm TP}(t=0)=1.8\ \text{V}$ on the real axis values between $0.2\ \text{V}$ and $1.8\ \text{V}$ &nbsp; &rArr; &nbsp; phase $\phi(t) \equiv 0$.<br>&nbsp;Except for the start state $x_{\rm TP}(t=0)$ same behavior as at the setting '''(1)'''. The degree of modulation is $m = 0.8$.  
  
  
 
{{BlaueBox|TEXT=
 
{{BlaueBox|TEXT=
'''(5)''' &nbsp; The parameters are still valid according to '''(4)''' with the exception of $A_{\rm T}= 0.6 \text{V}$. How big is the degree of modulation $m$? What are the consequences?}}
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'''(5)''' &nbsp; The parameters are still valid according to '''(4)''' with the exception of $A_{\rm T}= 0.6 \text{V}$. What is the degree of modulation $m$? What are the consequences?}}
  
::&nbsp;There is now a ZSB&ndash;AM with modulation degree $m = 1.333$. For $m > 1$, the simpler  [[Modulationsverfahren/Hüllkurvendemodulation|envelope demodulation]] is not applicable, since the phase function $\phi(t) \in \{ 0, \ \pm 180^\circ\}$ is no longer constant and the envelope $a(t)$ no longer matches the message signal. Rather, the more complex  [[Modulationsverfahren/Synchrondemodulation|synchronous demodulation]] must be used. Envelope detection would produce nonlinear distortions.
+
::&nbsp;There is now a DSB&ndash;AM with modulation degree $m = 1.333$. For $m > 1$, the simpler  ''Envelope Demodulation'' is not applicable, since the phase function $\phi(t) \in \{ 0, \ \pm 180^\circ\}$ is no more constant and the envelope $a(t)$ no more matches the message signal. Rather, the complex  ''Synchronous Demodulation'' must be used. Envelope detection would produce nonlinear distortions.
  
  
Zeile 185: Zeile 193:
 
'''(6)''' &nbsp; The parameters are still valid according to '''(4)''' or '''(5)''' with the exception from $A_{\rm T}= 0$ on &nbsp; &rArr; &nbsp; $m \to \infty$. Which modulation method is described in this way?}}
 
'''(6)''' &nbsp; The parameters are still valid according to '''(4)''' or '''(5)''' with the exception from $A_{\rm T}= 0$ on &nbsp; &rArr; &nbsp; $m \to \infty$. Which modulation method is described in this way?}}
  
::It is a '''ZSB&ndash;AM without carrier''' and requires synchronous demodulation. The equivalent low-pass signal $x_{\rm TP}(t)$ is on the real axis, but not only in the right half-plane. Thus, the phase function $\phi(t) \in \{ 0, \ \pm 180^\circ\}$, also applies here, which means that envelope demodulation is not applicable.
+
::It is a '''DSB&ndash;AM without carrier''' and a synchronous demodulation is required. The equivalent lowpass signal $x_{\rm TP}(t)$ is on the real axis, but not only in the right half-plane. Thus, the phase function $\phi(t) \in \{ 0, \ \pm 180^\circ\}$, also applies here, which means that ''Envelope Demodulation'' is not applicable.
  
  
Zeile 193: Zeile 201:
 
:Which constellation is described here? Which characteristics of this procedure can be recognized from the graphic?}}
 
:Which constellation is described here? Which characteristics of this procedure can be recognized from the graphic?}}
  
::It is a [[Modulationsverfahren/Einseitenbandmodulation|single-sideband modulation]] '''(ESB&ndash;AM)''', more specifically an '''OSB&ndash;AM''': the red carrier is fixed, the green pointer missing and the blue pointer (OSB) turns counterclockwise. The degree of modulation is $\mu = 0.8$ (in the case of ESB we denote the degree of modulation with $\mu$ instead of $m$). The carrier signal is cosinusoidal and the message signal sinusoidal.<br>The locus is a circle. $x_{\rm TP}(t)$ moves in a mathematically positive direction. Because of $\phi(t) \ne \text{const.}$, The envelope demodulation is not applicable here: &nbsp;This can be seen by the fact that the envelope $a(t)$ is not cosinusoidal.  Rather, the lower half-wave is sharper than the upper &nbsp; &rArr; &nbsp; strong linear distortions.
+
::It is a [[Modulationsverfahren/Einseitenbandmodulation|single-sideband modulation]] '''(SSB&ndash;AM)''', more specifically an '''OSB&ndash;AM''': the red carrier is fixed, the green pointer missing and the blue pointer (OSB) turns counterclockwise. The degree of modulation is $\mu = 0.8$ (in the case of SSB we denote the degree of modulation with $\mu$ instead of $m$). The carrier signal is cosinusoidal and the message signal sinusoidal.<br>The locus is a circle. $x_{\rm TP}(t)$ moves in a mathematically positive direction. Because of $\phi(t) \ne \text{const.}$ the envelope demodulation is not applicable here: &nbsp;This can be seen by the fact that the envelope $a(t)$ is not cosinusoidal.  Rather, the lower half-wave is sharper than the upper one &nbsp; &rArr; &nbsp; strong linear distortions.
  
  
Zeile 199: Zeile 207:
 
'''(8)''' &nbsp; The parameters are still valid according to '''(7)''' with the exception of $A_{\rm O}= 0$ and $A_{\rm U}= 0.8 \text{ V}$. What differences arise opposite '''(7)'''?}}
 
'''(8)''' &nbsp; The parameters are still valid according to '''(7)''' with the exception of $A_{\rm O}= 0$ and $A_{\rm U}= 0.8 \text{ V}$. What differences arise opposite '''(7)'''?}}
  
::Now it is a '''USB&ndash;AM''': The red carrier is fixed, the blue pointer is missing and the green pointer (USB) rotates clockwise. All other statements of '''(7)''' apply here as well.
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::Now it is a '''LSB&ndash;AM''': The red carrier is fixed, the blue pointer is missing and the green pointer (LSB) rotates clockwise. All other statements of '''(7)''' apply here as well.
  
  
Zeile 205: Zeile 213:
 
'''(9)''' &nbsp; The parameters according to '''(7)''' are still valid with the exception of $A_{\rm O} = 0.2 \text{ V} \ne A_{\rm U} = 0.4 \text{ V} $. What are the differences from '''(7)'''?}}
 
'''(9)''' &nbsp; The parameters according to '''(7)''' are still valid with the exception of $A_{\rm O} = 0.2 \text{ V} \ne A_{\rm U} = 0.4 \text{ V} $. What are the differences from '''(7)'''?}}
  
::The locus $x_{\rm TP}(t)$ is not a horizontal straight line, but an ellipse with the real part between $0.4 \text{ V}$ and $1.6 \text{ V}$ and the imaginary part in the range $\pm 0.2  \text{ V}$. Because of $\phi(t) \ne \text{const.}$ , Envelope demodulation would lead to non-linear distortions here too.<br> The constellation simulated here describes the situation of  '''(4)''', namely a ZSB&ndash;AM with modulation degree $m = 0.8$, where the upper sideband is reduced to $50\%$ due to channel attenuation.
+
::The locus $x_{\rm TP}(t)$ is not a horizontal straight line, but an ellipse with the real part between $0.4 \text{ V}$ and $1.6 \text{ V}$ and the imaginary part in the range $\pm 0.2  \text{ V}$. Because of $\phi(t) \ne \text{const.}$ , Envelope demodulation would lead to non-linear distortions here too.<br> The constellation simulated here describes the situation of  '''(4)''', namely a DSB&ndash;AM with modulation degree $m = 0.8$, where the upper sideband is reduced to $50\%$ due to channel attenuation.
  
 
==Applet Manual==
 
==Applet Manual==
 
[[Datei:Ortskurve_abzug3.png|right]]
 
[[Datei:Ortskurve_abzug3.png|right]]
* The red parameters $(A_{\rm T}, \ f_{\rm T}, \ \varphi_{\rm T})$  and the red pointer mark the ''Carrier''. (German: '''T'''räger).
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* The red parameters $(A_{\rm T}, \ f_{\rm T}, \ \varphi_{\rm T})$  and the red pointer mark the ''Carrier'' <br>(German: '''T'''räger).
* The green parameters $(A_{\rm U}, \ f_{\rm U} < f_{\rm T}, \ \varphi_{\rm U})$  mark the ''Lower sideband''. (German: '''U'''ntere Seitenband).
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* The green parameters $(A_{\rm U}, \ f_{\rm U} < f_{\rm T}, \ \varphi_{\rm U})$  mark the ''Lower sideband'' <br>(German: '''U'''ntere Seitenband).
* The blue parameters $(A_{\rm O}, \ f_{\rm O} > f_{\rm T}, \ \varphi_{\rm O})$  mark the '' Upper sideband''. (German: '''O'''bere Seitenband).
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* The blue parameters $(A_{\rm O}, \ f_{\rm O} > f_{\rm T}, \ \varphi_{\rm O})$  mark the '' Upper sideband'' <br>(German: '''O'''bere Seitenband).
 
* The red pointer does not turn.
 
* The red pointer does not turn.
 
* The green pointer rotates in a mathematically negative direction (clockwise).
 
* The green pointer rotates in a mathematically negative direction (clockwise).
 
* The blue pointer turns counterclockwise.
 
* The blue pointer turns counterclockwise.
 +
  
 
<u>Meaning of the letters in the adjacent graphic:</u>
 
<u>Meaning of the letters in the adjacent graphic:</u>
  
&nbsp; &nbsp; '''(A)''' &nbsp; &nbsp; Plot of the equivalent low-pass Signal $x_{\rm TP}(t)$
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&nbsp; &nbsp; '''(A)''' &nbsp; &nbsp; Plot of the equivalent lowpass signal $x_{\rm TP}(t)$
  
 
&nbsp; &nbsp; '''(B)''' &nbsp; &nbsp; Plot of the physical signal $x(t)$
 
&nbsp; &nbsp; '''(B)''' &nbsp; &nbsp; Plot of the physical signal $x(t)$
Zeile 228: Zeile 237:
 
&nbsp; &nbsp; '''(E)''' &nbsp; &nbsp; Speed of animation: &nbsp; &bdquo;Speed&rdquo; &nbsp; &rArr; &nbsp; Values: 1, 2 oder 3
 
&nbsp; &nbsp; '''(E)''' &nbsp; &nbsp; Speed of animation: &nbsp; &bdquo;Speed&rdquo; &nbsp; &rArr; &nbsp; Values: 1, 2 oder 3
  
&nbsp; &nbsp; '''(F)''' &nbsp; &nbsp; &bdquo;Trace&rdquo; &nbsp; &rArr; &nbsp;  On or Off, trace of equivalent low-pass Signal &nbsp; $x_{\rm TP}(t)$
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&nbsp; &nbsp; '''(F)''' &nbsp; &nbsp; &bdquo;Trace&rdquo; &nbsp; &rArr; &nbsp;  On or Off, trace of equivalent lowpass signal &nbsp; $x_{\rm TP}(t)$
  
 
&nbsp; &nbsp; '''(G)''' &nbsp; &nbsp; Numerical output: &nbsp; time $t$, the signal values &nbsp;${\rm Re}[x_{\rm TP}(t)]$ &nbsp;and&nbsp; ${\rm Im}[x_{\rm TP}(t)]$,
 
&nbsp; &nbsp; '''(G)''' &nbsp; &nbsp; Numerical output: &nbsp; time $t$, the signal values &nbsp;${\rm Re}[x_{\rm TP}(t)]$ &nbsp;and&nbsp; ${\rm Im}[x_{\rm TP}(t)]$,
  
$\text{}\hspace{4.2cm}$ &nbsp; Hüllkurve $a(t) = |x_{\rm TP}(t)|$ &nbsp;und&nbsp; Phase $\phi(t) = {\rm arc} \ x_{\rm TP}(t)$
+
$\text{}\hspace{4.2cm}$ &nbsp; envelope $a(t) = |x_{\rm TP}(t)|$ &nbsp;and&nbsp; phase $\phi(t) = {\rm arc} \ x_{\rm TP}(t)$
  
 
&nbsp; &nbsp; '''(H)''' &nbsp; &nbsp; Variations for the graphical representation
 
&nbsp; &nbsp; '''(H)''' &nbsp; &nbsp; Variations for the graphical representation
  
$\hspace{1.5cm}$Zoom&ndash;Functions &bdquo;$+$&rdquo; (Enlarge), &bdquo;$-$&rdquo; (Decrease) und $\rm o$ (Reset to default)
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$\hspace{1.5cm}$Zoom&ndash;Functions &bdquo;$+$&rdquo; (Enlarge), &bdquo;$-$&rdquo; (Decrease) and $\rm o$ (Reset to default)
  
 
$\hspace{1.5cm}$Move with &bdquo;$\leftarrow$&rdquo; (Section to the left, ordinate to the right),  &bdquo;$\uparrow$&rdquo; &bdquo;$\downarrow$&rdquo; &bdquo;$\rightarrow$&rdquo;
 
$\hspace{1.5cm}$Move with &bdquo;$\leftarrow$&rdquo; (Section to the left, ordinate to the right),  &bdquo;$\uparrow$&rdquo; &bdquo;$\downarrow$&rdquo; &bdquo;$\rightarrow$&rdquo;
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==Once again: Open Applet in new Tab==
 
==Once again: Open Applet in new Tab==
  
{{LntAppletLinkEn|physAnSignal}}
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{{LntAppletLinkEn|physAnLPSignal_en}}
  
 
[[Category:Applets|^Verzerrungen^]]
 
[[Category:Applets|^Verzerrungen^]]

Version vom 9. Juni 2020, 21:04 Uhr

Open Applet in new Tab

Applet Description


This applet shows the relationship between the physical bandpass signal $x(t)$ and the associated equivalent lowpass signal $x_{\rm TP}(t)$. It is assumed that the bandpass signal $x(t)$ has a frequency-discrete spectrum $X(f)$:

$$x(t) = x_{\rm T}(t) + x_{\rm O}(t)+ x_{\rm U}(t) = A_{\rm T}\cdot \cos\left(2\pi f_{\rm T}\cdot t- \varphi_{\rm T}\right)+A_{\rm O}\cdot \cos\left(2\pi f_{\rm O}\cdot t- \varphi_{\rm O}\right) + A_{\rm U}\cdot \cos\left(2\pi f_{\rm U}\cdot t- \varphi_{\rm U}\right). $$

The physical signal $x(t)$ is thus composed of three harmonic oscillations, a constellation that can be found, for example, in the Double-sideband Amplitude Modulation

  • of the message signal $x_{\rm N}(t) = A_{\rm N}\cdot \cos\left(2\pi f_{\rm N}\cdot t- \varphi_{\rm N}\right)$   ⇒   in German:   Nachrichtensignal
  • with the carrier signal $x_{\rm T}(t) = A_{\rm T}\cdot \cos\left(2\pi f_{\rm T}\cdot t - \varphi_{\rm T}\right)$   ⇒   in German:   Trägersignal.


The nomenclature is also adapted to this case:

  • $x_{\rm O}(t)$ denotes the „upper sideband”   (in German:   Oberes Seitenband) with the amplitude $A_{\rm O}= A_{\rm N}/2$, the frequency $f_{\rm O} = f_{\rm T} + f_{\rm N}$ and the phase $\varphi_{\rm O} = \varphi_{\rm T} + \varphi_{\rm N}$.
  • Similarly, for the „lower sideband”   (in German:   Unteres Seitenband) $x_{\rm U}(t)$ with $f_{\rm U} = f_{\rm T} - f_{\rm N}$, $A_{\rm U}= A_{\rm O}$ and $\varphi_{\rm U} = -\varphi_{\rm O}$.


The associated equivalent lowpass–signal is $f_{\rm O}\hspace{0.01cm}' = f_{\rm O}- f_{\rm T} > 0$,   $f_{\rm U}\hspace{0.01cm}' = f_{\rm U}- f_{\rm T} < 0$  and  $f_{\rm T}\hspace{0.01cm}' = 0$:

$$x_{\rm TP}(t) = x_\text{TP, T}(t) + x_\text{TP, O}(t) + x_\text{TP, U}(t) = A_{\rm T}\cdot {\rm e}^{-{\rm j} \varphi_{\rm T} } \hspace{0.1cm}+ \hspace{0.1cm} A_{\rm O}\cdot {\rm e}^{-{\rm j} \varphi_{\rm O} } \cdot {\rm e}^{ {\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm O}\hspace{0.01cm}'\hspace{0.05cm}\cdot \hspace{0.05cm}t}\hspace{0.1cm}+ \hspace{0.1cm} A_{\rm U}\cdot {\rm e}^{-{\rm j} \varphi_{\rm U} } \cdot {\rm e}^{ {\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm U}\hspace{0.01cm}'\hspace{0.05cm}\cdot \hspace{0.05cm}t} . $$
Equivalent low-pass signal currently $t=0$ for cosinusoidal carrier   ⇒   $\varphi_{\rm T} = 0$

The program shows $x_{\rm TP}(t)$ as the vectorial sum of three rotation pointers as a violet dot (see figure for start time $t=0$ and cosinusoidal carrier):

  • The (red) pointer of the carrier $x_\text{TP, T}(t)$ with the length $A_{\rm T}$ and the zero phase position $\varphi_{\rm T}=0$ is fixed in the complex plane. So it applies to all times $t$:   $x_{\rm TP}(t)= A_{\rm T}\cdot {\rm e}^{-{\rm j} \varphi_{\rm T} }$.
  • The (blue) pointer of the upper sideband $x_\text{TP, O}(t)$ with the length $A_{\rm O}$ and the zero phase position $\varphi_{\rm O}$ rotates at the angular velocity $2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm O}\hspace{0.01cm}'$ in mathematically positive direction (one revolution in time $1/f_{\rm O}\hspace{0.01cm}')$.
  • The (green) pointer of the lower sideband $x_{\rm U+}(t)$ with the length $A_{\rm U}$ and the zero phase position $\varphi_{\rm U}$ rotates at the angular velocity $2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm U}\hspace{0.01cm}'$, because of $f_{\rm U}\hspace{0.01cm}'<0$ counterclockwise.
  • With $f_{\rm U}\hspace{0.01cm}' = -f_{\rm O}\hspace{0.01cm}'$ the blue and green pointers will spin at the same speed but in different directions. Also, if $A_{\rm O} = A_{\rm U}$ and $\varphi_{\rm U} = -\varphi_{\rm O}$, then $x_{\rm TP}(t)$ moves on a straight line with a incline of $\varphi_{\rm T}$.


Note:   In the figure $\varphi_{\rm O} = +30^\circ$. From this follows for the start time $t=0$ the angle of the upper sideband (OSB, blue pointer) with respect to the coordinate system:   $\phi_{\rm O} = -\varphi_{\rm O} = -30^\circ$. Likewise, the zero phase position $\varphi_{\rm U} = -30^\circ$ of the lower sideband (USB, green pointer) follows for the phase angle to be considered in the complex plane:   $\phi_{\rm U} = +30^\circ$.


The temporal process of $x_{\rm TP}(t)$ is also referred to below as „locus”. The relationship between $x_{\rm TP}(t)$ and the physical bandpass signal $x(t)$ is given in the section and the associated analytic signal is $x_+(t)$ :

$$x_{\rm TP}(t) = x_{\rm +}(t)\cdot {\rm e}^{-{\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm}2 \pi \cdot f_{\rm T}\cdot \hspace{0.05cm}t},$$
$$x_{\rm +}(t) = x_{\rm TP}(t)\cdot {\rm e}^{+{\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm}2 \pi \cdot f_{\rm T}\cdot \hspace{0.05cm}t}.$$



German description

Theoretical Background


Description of Bandpass Signals

bandpass spectrum $X(f)$

We consider bandpass signals $x(t)$ with the property that their spectra $X(f)$ are not in the range around the frequency $f=0$, but around a carrier frequency $f_{\rm T}$. In most cases it can also be assumed that the bandwidth is $B \ll f_{\rm T}$.

The figure shows such a bandpass spectrum $X(f)$. Assuming that the associated $x(t)$ is a physical signal and thus real, the spectral function $X(f)$ has a symmetry with respect to the frequency $f = 0$, if $x(t)$ is an even function   ⇒   $x(-t)=x(t)$, $X(f)$ is real and even.


Beside the physical signal $x(t)\ \circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\, \ X(f)$, one can also use the following descriptions of bandpass signals:

  • the analytic signal $x_+(t)\ \circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\, \ X_+(f)$, see applet Physical Signal & Analytic Signal,
  • the equivalent lowpass signal $x_{\rm TP}(t)\ \circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\, \ X_{\rm TP}(f)$, see next page



Spectral Functions of the Analytic and the Equivalent Lowpass Signal

The analytic signal $x_+(t)$ belonging to the physical signal $x(t)$ is the time function whose spectrum fulfills the following property:

spectral functions $X(f)$, $X_+(f)$ and $X_{\rm TP}(f)$
$$X_+(f)=\big[1+{\rm sign}(f)\big] \cdot X(f) = \left\{ {2 \cdot X(f) \; \hspace{0.2cm}\rm for\hspace{0.2cm} {\it f} > 0, \atop {\,\,\,\, \rm 0 \; \hspace{0.9cm}\rm for\hspace{0.2cm} {\it f} < 0.} }\right.$$

The Signum function is for positive values of $f$ equal to $+1$ and for negative $f$ values equal to $-1$.

  • The (double-sided) limit returns $\sign(0) = 0$.
  • The index „+” should make it clear that $X_+(f)$ only has parts at positive frequencies.


From the graph you can see the calculation rule for $X_+(f)$:   The actual bandpass spectrum $X(f)$ becomes

  • doubled at the positive frequencies, and
  • set to zero at the negative frequencies.


Due to the asymmetry of $X_+(f)$ with respect to the frequency $f = 0$, it can already be said that the time function $x_+(t)$ except for a trivial special case $x_+(t)= 0 \ \ \circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\,\ \ X_+(f)= 0$ is always complex.


The spectrum $X_{\rm TP}(f)$ of the equivalent lowpass signal is obtained by shifting $X_+(f)$ to the left by the carrier frequency $f_{\rm T}$:

$$X_{\rm TP}(f)= X_+(f+f_{\rm T}).$$

In the time domain this operation corresponds to the multiplication of $x_{\rm +}(t)$ with the complex exponential function with negative exponent:

$$x_{\rm TP}(t) = x_{\rm +}(t)\cdot {\rm e}^{-{\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm}2 \pi \cdot f_{\rm T}\cdot \hspace{0.05cm}t}.$$

It can be seen that $x_{\rm TP}(t)$ is generally complex. But if $X_+(f)$ is symmetric about the carrier frequency $f_{\rm T}$, $X_{\rm TP}(f)$ is symmetric about the frequency $f=0$ and there is accordingly a real time function $x_{\rm TP}(t)$.

$x_{\rm TP}(t)$ Representation of a Sum of Three Harmonic Oscillations

In our applet, we always assume a set of three rotating pointers. The physical signal is:

$$x(t) = x_{\rm T}(t) + x_{\rm O}(t) + x_{\rm U}(t) = A_{\rm T}\cdot \cos\left(2\pi f_{\rm T}\cdot t- \varphi_{\rm T}\right)+A_{\rm O}\cdot \cos\left(2\pi f_{\rm O}\cdot t- \varphi_{\rm O}\right) + A_{\rm U}\cdot \cos\left(2\pi f_{\rm U}\cdot t- \varphi_{\rm U}\right). $$
  • Each of the three harmonic oscillations $x_{\rm T}(t)$, $x_{\rm U}(t)$ and $x_{\rm O}(t)$ is represented by an amplitude $(A)$, a frequency $(f)$ and a phase value $(\varphi)$.
  • The indices are based on the modulation method double sideband amplitude modulation. „T” stands for „carrier”, „U” for „lower sideband” and „O” for „upper Sideband”. Similarly, $f_{\rm U} < f_{\rm T}$ and $f_{\rm O} > f_{\rm T}$. There are no restrictions for the amplitudes and phases.


The associated equivalent low-pass signal is with $f_{\rm O}\hspace{0.01cm}' = f_{\rm O}- f_{\rm T} > 0$,   $f_{\rm U}\hspace{0.01cm}' = f_{\rm U}- f_{\rm T} < 0$  and  $f_{\rm T}\hspace{0.01cm}' = 0$:

$$x_{\rm TP}(t) = x_\text{TP, T}(t) + x_\text{TP, O}(t) + x_\text{TP, U}(t) = A_{\rm T}\cdot {\rm e}^{-{\rm j} \varphi_{\rm T} } \hspace{0.1cm}+ \hspace{0.1cm} A_{\rm O}\cdot {\rm e}^{-{\rm j} \varphi_{\rm O} } \cdot {\rm e}^{ {\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm O}\hspace{0.01cm}'\hspace{0.05cm}\cdot \hspace{0.05cm}t}\hspace{0.1cm}+ \hspace{0.1cm} A_{\rm U}\cdot {\rm e}^{-{\rm j} \varphi_{\rm U} } \cdot {\rm e}^{ {\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm U}\hspace{0.01cm}'\hspace{0.05cm}\cdot \hspace{0.05cm}t} . $$

$\text{Example 1:}$  The constellation given here results, for example, in the double sideband amplitude modulation of the message signal $x_{\rm N}(t) = A_{\rm N}\cdot \cos\left(2\pi f_{\rm N}\cdot t- \varphi_{\rm N}\right)$ with the carrier signal $x_{\rm T}(t) = A_{\rm T}\cdot \cos\left(2\pi f_{\rm T}\cdot t - \varphi_{\rm T}\right)$. This is discussed frequently in the exercises.

Spectrum $X_{\rm TP}(f)$ of the equivalent low–pass signal for different phase constellations

There are some limitations to the program parameters in this approach:

  • The frequencies are always $f\hspace{0.05cm}'_{\rm O} = f_{\rm N}$ and $f\hspace{0.05cm}'_{\rm U} = -f_{\rm N}$.
  • Without distortion, the amplitude of the sidebands is $A_{\rm O}= A_{\rm U}= A_{\rm N}/2$.
  • The respective phase relationships can be seen in the following graphic.



Representation of the Equivalent Lowpass Signal by Magnitude and Phase

The generally complex equivalent lowpass signal

$$x_{\rm TP}(t) = a(t) \cdot {\rm e}^{ {\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm} \phi(t) }$$

can be split into a magnitude function $a(t)$ and a phase function $\phi(t)$ according to the equation given here, where:

$$a(t) = \vert x_{\rm TP}(t)\vert = \sqrt{ {\rm Re}^2\big [x_{\rm TP}(t)\big ] + {\rm Im}^2\big [x_{\rm TP}(t)\big ] }\hspace{0.05cm},$$
$$\phi(t) = \text{arc }x_{\rm TP}(t) = \arctan \frac{{\rm Im}\big [x_{\rm TP}(t)\big ]}{{\rm Re}\big [x_{\rm TP}(t)\big ]}.$$

The reason for this is that a bandpass signal $x(t)$ is usually described by the equivalent lowpass signal $x_{\rm TP}(t)$ that the functions $a(t)$ and $\phi(t)$ are interpretable in both representations:

  • The amount $a(t)$ of the equivalent lowpass signal $x_{\rm TP}(t)$ indicates the (time-dependent) envelope of $x(t)$.
  • The phase $\phi(t)$ of $x_{\rm TP}(t)$ denotes the location of the zero crossings of $x(t)$, where:
–   For $\phi(t)>0$ the zero crossing is earlier than its nominal position   ⇒   the signal is leading here.
–  When $\phi(t)<0$, the zero crossing is later than its target position   ⇒   the signal is trailing here.


$\text{Example 2:}$  The graph is intended to illustrate this relationship, assuming $A_{\rm U} > A_{\rm O}$   ⇒   the green pointer (for the lower sideband) is longer than the blue pointer (upper sideband). This is a snapshot at time $t_0$:

bandpass spectrum $X(f)$
  • For these system parameters, the top of the pointer cluster $x_{\rm TP}(t)$ – that is, the geometric sum of red, blue and green pointers – on an ellipse.
  • The amount $a(t_0) = \vert x_{\rm TP}(t_0) \vert$ is drawn in black in the left-hand diagram and the phase value $\phi(t_0) = \text{arc }x_{\rm TP}(t_0) > 0$ is indicated in brown color.
  • In the graph on the right, the amount $a(t_0) = \vert x_{\rm TP}(t_0) \vert$ of the equivalent low-pass signal indicates the envelope of the physical signal $x(t)$.
  • At $\phi(t) \equiv 0$, all zero crossings of $x(t)$ would occur at equidistant intervals. Because of $\phi(t_0) > 0$, the signal is leading at the time $t_0$, that is: the zero crossings come earlier than the grid dictates.

Exercises

Exercises verzerrungen.png
  • First select the task number.
  • A task description is displayed.
  • Parameter values are adjusted.
  • Solution after pressing „Hide solition”.


The number „0” will reset the program and output a text with the further explanation of the applet.


In the following, $\rm Green$ denotes the lower sideband   ⇒   $\big (A_{\rm U}, f_{\rm U}, \varphi_{\rm U}\big )$,   $\rm Red$ the carrier   ⇒   $\big (A_{\rm T}, f_{\rm T}, \varphi_{\rm T}\big )$ and $\rm Blue$ the upper sideband   ⇒   $\big (A_{\rm O}, f_{\rm O}, \varphi_{\rm O}\big )$.

(1)   Let   $\text{Red:} \hspace{0.15cm} A_{\rm T} = 1\ \text{V}, f_{\rm T} = 100 \ \text{kHz}, \varphi_{\rm T} = 0^\circ$,   $\text{Green:} \hspace{0.15cm} A_{\rm U} = 0.4 \text{V}, \ f_{\rm U} = 80 \ \text{kHz}, \varphi_{\rm U} = -90^\circ$,   $\text{Blue:} \hspace{0.15cm} A_{\rm O} = 0.4\ \text{V}, f_{\rm O} = 120 \ \text{kHz}, \varphi_{\rm O} = 90^\circ$.

Consider and interpret the equivalent lowpass signal $x_{\rm TP}(t)$ and the physical signal $x(t)$. Which period $T_0$ recognizable?
 The equivalent lowpass signal $x_{\rm TP}(t)$ takes from $x_{\rm TP}(t=0)=1\ \text{V}$ on the real axis values between $0.2\ \text{V}$ and $1.8\ \text{V}$   ⇒   phase $\phi(t) \equiv 0$.
 The amount $|x_{\rm TP}(t)|$ indicates the envelope $a(t)$ of the physical signal $x(t)$. It holds $A_{\rm N} = 0.8\ \text{V}$ and $f_{\rm N} = 20\ \text{kHz}$:   $a(t) = A_{\rm T}+ A_{\rm N} \cdot \sin(2\pi\cdot f_{\rm N} \cdot t)$.
 Both $x_{\rm TP}(t)$ and $x(t)$ are periodically with the period $T_0 = 1/f_{\rm N} = 50\ \rm µ s$.


(2)   How do the ratios change to (1) with $f_{\rm U} = 99 \ \text{kHz}$ and $f_{\rm O} = 101 \ \text{kHz}$ ? How could $x(t)$ have arisen?

 For the envelope $a(t)$ of the signal $x(t)$ we still have $a(t) = A_{\rm T}+ A_{\rm N} \cdot \sin(2\pi\cdot f_{\rm N} \cdot t)$, but now $f_{\rm N} = 1\ \text{kHz}$. Even though it can not be recognized:
 $x_{\rm TP}(t)$ and $x(t)$ are still periodically:   $T_0 = 1\ \rm ms$. Example: Double sideband Amplitude modulation (DSB–AM) of a sine signal with cosine carrier.


(3)   Which settings have to be changed from (2) in order to arrive at the DSB–AM of a cosine signal with sine carrier. What changes over (2)?

The carrier phase must be changed to $\varphi_{\rm T} = 90^\circ$   ⇒   sine carrier. Similarly, $\varphi_{\rm O} =\varphi_{\rm U} =\varphi_{\rm T} = 90^\circ$ must be set   ⇒   cosinusoidal message
 The locus now lies on the imaginary axis  ⇒   $\phi(t) \equiv -90^\circ$. At the beginning $x_{\rm TP}(t=0)= - {\rm j} \cdot 1.8 \ \text{V}$.


(4)   Now let   $\text{Red:} \hspace{0.15cm} A_{\rm T} = 1\ \text{V}, \ f_{\rm T} = 100 \ \text{kHz}, \ \varphi_{\rm T} = 0^\circ$,   $\text{Green:} \hspace{0.15cm} A_{\rm U} = 0.4 \text{V}, \ f_{\rm U} = 80 \ \text{kHz}, \ \varphi_{\rm U} = 0^\circ$,   $\text{Blue:} \hspace{0.15cm} A_{\rm O} = 0.4\ \text{V}, \ f_{\rm O} = 120 \ \text{kHz}, \ \varphi_{\rm O} = 0^\circ$.

What are the characteristics of this system „DSB–AM, where the message signal and carrier are respectively cosinusoidal”? What is the degree of modulation $m$?
 The equivalent lowpass signal $x_{\rm TP}(t)$ takes from $x_{\rm TP}(t=0)=1.8\ \text{V}$ on the real axis values between $0.2\ \text{V}$ and $1.8\ \text{V}$   ⇒   phase $\phi(t) \equiv 0$.
 Except for the start state $x_{\rm TP}(t=0)$ same behavior as at the setting (1). The degree of modulation is $m = 0.8$.


(5)   The parameters are still valid according to (4) with the exception of $A_{\rm T}= 0.6 \text{V}$. What is the degree of modulation $m$? What are the consequences?

 There is now a DSB–AM with modulation degree $m = 1.333$. For $m > 1$, the simpler Envelope Demodulation is not applicable, since the phase function $\phi(t) \in \{ 0, \ \pm 180^\circ\}$ is no more constant and the envelope $a(t)$ no more matches the message signal. Rather, the complex Synchronous Demodulation must be used. Envelope detection would produce nonlinear distortions.


(6)   The parameters are still valid according to (4) or (5) with the exception from $A_{\rm T}= 0$ on   ⇒   $m \to \infty$. Which modulation method is described in this way?

It is a DSB–AM without carrier and a synchronous demodulation is required. The equivalent lowpass signal $x_{\rm TP}(t)$ is on the real axis, but not only in the right half-plane. Thus, the phase function $\phi(t) \in \{ 0, \ \pm 180^\circ\}$, also applies here, which means that Envelope Demodulation is not applicable.


(7)   Now let   $\text{Red:} \hspace{0.15cm} A_{\rm T} = 1\ \text{V}, f_{\rm T} = 100 \ \text{kHz}, \varphi_{\rm T} = 0^\circ$,   $\text{Green:} \hspace{0.15cm} A_{\rm U} = 0, \ f_{\rm U} = 80 \ \text{kHz}, \varphi_{\rm U} = -90^\circ$,   $\text{Blue:} \hspace{0.15cm} A_{\rm O} = 0.8\ \text{V}, f_{\rm O} = 120 \ \text{kHz}, \varphi_{\rm O} = 90^\circ$.

Which constellation is described here? Which characteristics of this procedure can be recognized from the graphic?
It is a single-sideband modulation (SSB–AM), more specifically an OSB–AM: the red carrier is fixed, the green pointer missing and the blue pointer (OSB) turns counterclockwise. The degree of modulation is $\mu = 0.8$ (in the case of SSB we denote the degree of modulation with $\mu$ instead of $m$). The carrier signal is cosinusoidal and the message signal sinusoidal.
The locus is a circle. $x_{\rm TP}(t)$ moves in a mathematically positive direction. Because of $\phi(t) \ne \text{const.}$ the envelope demodulation is not applicable here:  This can be seen by the fact that the envelope $a(t)$ is not cosinusoidal. Rather, the lower half-wave is sharper than the upper one   ⇒   strong linear distortions.


(8)   The parameters are still valid according to (7) with the exception of $A_{\rm O}= 0$ and $A_{\rm U}= 0.8 \text{ V}$. What differences arise opposite (7)?

Now it is a LSB–AM: The red carrier is fixed, the blue pointer is missing and the green pointer (LSB) rotates clockwise. All other statements of (7) apply here as well.


(9)   The parameters according to (7) are still valid with the exception of $A_{\rm O} = 0.2 \text{ V} \ne A_{\rm U} = 0.4 \text{ V} $. What are the differences from (7)?

The locus $x_{\rm TP}(t)$ is not a horizontal straight line, but an ellipse with the real part between $0.4 \text{ V}$ and $1.6 \text{ V}$ and the imaginary part in the range $\pm 0.2 \text{ V}$. Because of $\phi(t) \ne \text{const.}$ , Envelope demodulation would lead to non-linear distortions here too.
The constellation simulated here describes the situation of (4), namely a DSB–AM with modulation degree $m = 0.8$, where the upper sideband is reduced to $50\%$ due to channel attenuation.

Applet Manual

Ortskurve abzug3.png
  • The red parameters $(A_{\rm T}, \ f_{\rm T}, \ \varphi_{\rm T})$ and the red pointer mark the Carrier
    (German: Träger).
  • The green parameters $(A_{\rm U}, \ f_{\rm U} < f_{\rm T}, \ \varphi_{\rm U})$ mark the Lower sideband
    (German: Untere Seitenband).
  • The blue parameters $(A_{\rm O}, \ f_{\rm O} > f_{\rm T}, \ \varphi_{\rm O})$ mark the Upper sideband
    (German: Obere Seitenband).
  • The red pointer does not turn.
  • The green pointer rotates in a mathematically negative direction (clockwise).
  • The blue pointer turns counterclockwise.


Meaning of the letters in the adjacent graphic:

    (A)     Plot of the equivalent lowpass signal $x_{\rm TP}(t)$

    (B)     Plot of the physical signal $x(t)$

    (C)     Parameter input via slider:   amplitudes, frequencies, phase values

    (D)     Control elements:   Start – Step – Pause/Continue – Reset

    (E)     Speed of animation:   „Speed”   ⇒   Values: 1, 2 oder 3

    (F)     „Trace”   ⇒   On or Off, trace of equivalent lowpass signal   $x_{\rm TP}(t)$

    (G)     Numerical output:   time $t$, the signal values  ${\rm Re}[x_{\rm TP}(t)]$  and  ${\rm Im}[x_{\rm TP}(t)]$,

$\text{}\hspace{4.2cm}$   envelope $a(t) = |x_{\rm TP}(t)|$  and  phase $\phi(t) = {\rm arc} \ x_{\rm TP}(t)$

    (H)     Variations for the graphical representation

$\hspace{1.5cm}$Zoom–Functions „$+$” (Enlarge), „$-$” (Decrease) and $\rm o$ (Reset to default)

$\hspace{1.5cm}$Move with „$\leftarrow$” (Section to the left, ordinate to the right), „$\uparrow$” „$\downarrow$” „$\rightarrow$”

    (I)     Experiment section:   Task selection and task

    (J)     Experiment section:  solution

About the Authors

This interactive calculation was designed and realized at the Lehrstuhl für Nachrichtentechnik of the Technical University of Munich .

  • The original version was created in 2005 by Ji Li as part of her Diploma thesis using „FlashMX–Actionscript” (Supervisor: Günter Söder).
  • In 2018 this Applet was redesigned and updated to „HTML5” by Xiaohan Liu as part of her Bachelor's thesis (Supervisor: Tasnád Kernetzky).

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