View source for Applets:Physical Signal & Analytic Signal

{{LntAppletLink|physAnSignal_en}}  &nbsp; &nbsp; &nbsp; &nbsp; [https://www.lntwww.de/Applets:Physikalisches_Signal_%26_Analytisches_Signal '''English Applet with German WIKI description''']
==Applet Description==
<br>
This applet shows the relationship between the physical band-pass signal $x(t)$ and the associated analytic signal $x_+(t)$. It is assumed that the band-pass signal $x(t)$ has a frequency-discrete spectrum $X(f)$:
:$$x(t) = x_{\rm U}(t) + x_{\rm T}(t) + x_{\rm O}(t) = A_{\rm U}\cdot \cos\left(2\pi f_{\rm U}\cdot t- \varphi_{\rm U}\right)+A_{\rm T}\cdot \cos\left(2\pi f_{\rm T}\cdot t- \varphi_{\rm T}\right)+A_{\rm O}\cdot \cos\left(2\pi f_{\rm O}\cdot t- \varphi_{\rm O}\right). $$
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 $x_{\rm N}(t) = A_{\rm N}\cdot \cos\left(2\pi f_{\rm N}\cdot t- \varphi_{\rm N}\right)$ &nbsp; &rArr; &nbsp; in German: &nbsp;  '''N'''achrichtensignal 
*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.
 

The nomenclature is also adapted to this case:
* $x_{\rm O}(t)$ denotes the "upper sideband" &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 $\varphi_{\rm O} = \varphi_{\rm T} + \varphi_{\rm N}$.
*Similarly, for the "lower sideband" &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 associated analytic signal is:

:$$x_+(t) = x_{\rm U+}(t) + x_{\rm T+}(t) + x_{\rm O+}(t) = A_{\rm U}\cdot {\rm e}^{\hspace{0.05cm}{\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}(2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm U}\hspace{0.05cm}\cdot \hspace{0.05cm}t- \varphi_{\rm U})}
\hspace{0.1cm}+ \hspace{0.1cm}A_{\rm T}\cdot {\rm e}^{\hspace{0.05cm}{\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}(2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm T}\hspace{0.05cm}\cdot \hspace{0.05cm}t- \varphi_{\rm T})}
\hspace{0.1cm}+\hspace{0.1cm} A_{\rm O}\cdot {\rm e}^{\hspace{0.05cm}{\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}(2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm O}\hspace{0.05cm}\cdot \hspace{0.05cm}t- \varphi_{\rm O})}. $$

[[File:Zeigerdiagramm_2a_version2.png|right|frame|Analytic signal at the time $t=0$]]
The program displays $x_+(t)$ as the vectorial sum of three rotating pointers (all with counterclockwise) as a violet dot (see figure for start time $t=0$):

*The (red) pointer of the carrier $x_{\rm T+}(t)$ with length $A_{\rm T}$ and zero phase position $\varphi_{\rm T} = 0$ rotates at constant angular velocity $2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm T}$ (one revolution in time $1/f_{\rm T})$.

*The (blue) pointer of the upper sideband $x_{\rm O+}(t)$ with length $A_{\rm O}$ and zero phase position $\varphi_{\rm O}$ rotates at the angular velocity $2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm O}$, which is slightly faster than $x_{\rm T+}(t)$.

*The (green) pointer of the lower sideband $x_{\rm U+}(t)$ with length $A_{\rm U}$ and zero phase position $\varphi_{\rm U}$ rotates at the angular velocity $2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm U}$, which is slightly slower than $x_{\rm T+}(t)$.


The time trace of $x_+(t)$ is also referred to below as ''Pointer Diagram''. The relationship between the physical band-pass signal $x(t)$ and the associated analytic signal $x_+(t)$ is:

:$$x(t) = {\rm Re}\big [x_+(t)\big ].$$

''Note:'' &nbsp; In the figure $\varphi_{\rm O} = +30^\circ$. This leads to the angle with respect to the coordinate system at $t=0$: &nbsp; $\phi_{\rm O}=-\varphi_{\rm O}=-30^\circ$. Similarly, the zero phase angle $\varphi_{\rm U}=-30^\circ$ of the lower sideband leads to the phase angle to be considered in the complex plane: &nbsp; $\phi_{\rm U}=+30^\circ$.



==Theoretical Background==
<br>
===Description of Band-pass Signals===
[[File:Zeigerdiagramm_1a.png|right|frame|Band-pass spectrum $X(f)$ |class=fit]]
We consider '''band-pass 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 band-pass 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 &nbsp; &rArr; &nbsp; $x(-t)=x(t)$, $X(f)$ is real and even.


Besides the physical signal $x(t)\ \circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\, \ X(f)$, one can also use the following descriptions of band-pass signals:
*the analytic signal $x_+(t)\ \circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\, \ X_+(f)$, see next section,
*the equivalent low-pass signal &nbsp; (in German: &nbsp; äquivalentes '''T'''ief '''P'''ass&ndash;Signal) $x_{\rm TP}(t)\ \circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\, \ X_{\rm TP}(f)$, <br>see Applet [[Applets:Physical_Signal_%26_Equivalent_Low-pass_Signal|"Physical Signal & Equivalent Low-pass signal"]].
<br><br>

===Analytic Signal &ndash; Frequency Domain===

The '''analytic signal''' $x_+(t)$ belonging to the physical signal $x(t)$ is the time function whose spectrum fulfills the following property:
[[File:EN_Sig_T_4_2_S1a.png|right|frame|Construction of the spectral function $X_+(f)$|class=fit]]

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

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 band-pass 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.
<br clear=all>

===Analytic Signal &ndash; Time Domain===
At this point, it is necessary to briefly discuss another spectral transformation.

{{BlaueBox|TEXT=
$\text{Definition:}$&nbsp;
For the '''Hilbert transform''' $ {\rm H}\left\{x(t)\right\}$ of a time function $x(t)$ we have:

:$$y(t) = {\rm H}\left\{x(t)\right\} = \frac{1}{ {\rm \pi} } \cdot
\hspace{0.03cm}\int_{-\infty}^{+\infty}\frac{x(\tau)}{ {t -
\tau} }\hspace{0.15cm} {\rm d}\tau.$$

This particular integral is not solvable in a simple, conventional way, but must be evaluated using the [https://en.wikipedia.org/wiki/Cauchy_principal_value "Cauchy principal value theorem"].

Accordingly, in the frequency domain:
:$$Y(f) =  {\rm -j \cdot sign}(f) \cdot X(f) \hspace{0.05cm} .$$}}


The above result can be summarized with this definition as follows:
* The analytic signal $x_+(t)$ is obtained from the physical band-pass signal $x(t)$ by adding an imaginary part to $x(t)$ according to the Hilbert transform:

:$$x_+(t) = x(t)+{\rm j} \cdot {\rm H}\left\{x(t)\right\} .$$

*$\text{H}\{x(t)\}$ disappears only for the case $x(t) = \rm const.$ &nbsp; &rArr; &nbsp; the same signal. For all other signal forms, the analytic signal $x_+(t)$ is complex.


* From the analytic signal $x_+(t)$, the physical band-pass signal can be easily determined by the following operation:
:$$x(t) = {\rm Re}\big[x_+(t)\big] .$$

{{GraueBox|TEXT=
$\text{Example 1:}$&nbsp; The principle of the Hilbert transformation should be further clarified by the following graphic:
*After the left representation $\rm(A)$ one gets from the physical signal $x(t)$ to the analytic signal $x_+(t)$, by adding an imaginary part ${\rm j} \cdot y(t)$.
*Here $y(t) = {\rm H}\left\{x(t)\right\}$ is a real time function that can be indicated in the spectral domain by multiplying the spectrum $X(f)$ with ${\rm - j} \cdot \sign(f)$.


[[File:P_ID2729__Sig_T_4_2_S2b_neu.png|center|frame|To clarify the Hilbert transform]]

The right representation $\rm(B)$ is equivalent to $\rm(A)$. Now $x_+(t) = x(t) + z(t)$ stand with the purely imaginary function $z(t)$. A comparison of the two figures shows that in fact $z(t) = {\rm j} \cdot y(t)$.}}
<br><br>

===Representation of the Harmonic Oscillation as an Analytic Signal===

The spectral function $X(f)$ of a harmonic oscillation $x(t) = A\cdot\text{cos}(2\pi f_{\rm T}\cdot t - \varphi)$ is known to consist of two Dirac delta functions at the frequencies
* $+f_{\rm T}$ with the complex weight $A/2 \cdot \text{e}^{-\text{j}\hspace{0.05cm}\varphi}$,
* $-f_{\rm T}$ with the complex weight $A/2 \cdot \text{e}^{+\text{j}\hspace{0.05cm}\varphi}$.


Thus, the spectrum of the analytic signal (that is, without the Dirac delta function at the frequency $f =-f_{\rm T}$, but doubling at $f =+f_{\rm T}$):

:$$X_+(f) = A \cdot {\rm e}^{-{\rm j} \hspace{0.05cm}\varphi}\cdot\delta (f - f_{\rm
T}) .$$

The associated time function is obtained by applying the Displacement Law:

:$$x_+(t) = x_{\rm U+}(t) + x_{\rm T+}(t) + x_{\rm O+}(t) = A_{\rm U}\cdot {\rm e}^{\hspace{0.05cm}{\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}(2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm U}\hspace{0.05cm}\cdot \hspace{0.05cm}t- \varphi_{\rm U})}
\hspace{0.1cm}+ \hspace{0.1cm}A_{\rm T}\cdot {\rm e}^{\hspace{0.05cm}{\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}(2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm T}\hspace{0.05cm}\cdot \hspace{0.05cm}t- \varphi_{\rm T})}
\hspace{0.1cm}+\hspace{0.1cm} A_{\rm O}\cdot {\rm e}^{\hspace{0.05cm}{\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}(2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm O}\hspace{0.05cm}\cdot \hspace{0.05cm}t- \varphi_{\rm O})}. $$

This equation describes a pointer rotating at constant angular velocity $\omega_{\rm T} = 2\pi f_{\rm T}$.

{{GraueBox|TEXT=
$\text{Example 2:}$&nbsp; Here the coordinate system is rotated by $90^\circ$ (real part up, imaginary part to the left) contrary to the usual representation.

[[File:P_ID712__Sig_T_4_2_S3.png|center|frame|Pointer diagram of a harmonic oscillation]]

Based on this graphic, the following statements are possible:
* At time $t = 0$, the pointer of length $A$ (signal amplitude) lies with the angle $-\varphi$ in the complex plane. In the example shown, $\varphi=45^\circ$.
* For times $t>0$, the constant angular velocity vector $\omega_{\rm T}$ rotates in a mathematically positive direction, that is, counterclockwise.
* The tip of the pointer is thus always on a circle with radius $A$ and needs exactly the time $T_0$, i.e. the period of the harmonic oscillation $x(t)$ for one revolution.
* The projection of the analytic signal $x_+(t)$ on the real axis, marked by red dots, gives the instantaneous values of $x(t)$.}}
<br><br>

===Analytic Signal Representation of a Sum of Three Harmonic Oscillations===

In our applet, we always assume a set of three rotating pointers. The physical signal is:
:$$x(t) = x_{\rm U}(t) + x_{\rm T}(t) + x_{\rm O}(t) = A_{\rm U}\cdot \cos\left(2\pi f_{\rm U}\cdot t- \varphi_{\rm U}\right)+A_{\rm T}\cdot \cos\left(2\pi f_{\rm T}\cdot t- \varphi_{\rm T}\right)+A_{\rm O}\cdot \cos\left(2\pi f_{\rm O}\cdot t- \varphi_{\rm O}\right). $$
* 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 ''Double-sideband Amplitude Modulation'' method. "T" stands for "carrier", "U" for "lower sideband" and "O" for "upper Sideband". 
*Accordingly, $f_{\rm U} < f_{\rm T}$ and $f_{\rm O} > f_{\rm T}$. There are no restrictions for the amplitudes and phases.


The associated analytic signal is:
:$$x_+(t) = x_{\rm U+}(t) + x_{\rm T+}(t) + x_{\rm O+}(t) = A_{\rm U}\cdot {\rm e}^{\hspace{0.05cm}{\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}(2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm U}\hspace{0.05cm}\cdot \hspace{0.05cm}t- \varphi_{\rm U})}
\hspace{0.1cm}+ \hspace{0.1cm}A_{\rm T}\cdot {\rm e}^{\hspace{0.05cm}{\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}(2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm T}\hspace{0.05cm}\cdot \hspace{0.05cm}t- \varphi_{\rm T})}
\hspace{0.1cm}+\hspace{0.1cm} A_{\rm O}\cdot {\rm e}^{\hspace{0.05cm}{\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}(2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm O}\hspace{0.05cm}\cdot \hspace{0.05cm}t- \varphi_{\rm O})}. $$

{{GraueBox|TEXT=
$\text{Example 3:}$&nbsp;
Shown the constellation arises i.e. in the [https://en.wikipedia.org/wiki/Sideband "Double-sideband Amplitude Modulation"] (with carrier) 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.


There are some limitations to the program parameters in this approach:
* For the frequencies, it always applies $f_{\rm O} = f_{\rm T} + f_{\rm N}$ and $f_{\rm U} = f_{\rm T} - f_{\rm N}$.

*Without distortions the amplitude of the sidebands are $A_{\rm O}= A_{\rm U}= A_{\rm N}/2$.
*The respective phase relationships can be seen in the following graphic.

[[File:Zeigerdiagramm_5.png|center|frame|Spectrum $X_+(f)$ of the analytic signal for different phase constellations |class=fit]]}}

==Exercises==
[[File:Zeigerdiagramm_aufgabe_2.png|right]]
*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 further explanation of the applet.
<br clear=all>
In the following, $\rm Green$ denotes the lower sideband &nbsp; &rArr; &nbsp; $\big (A_{\rm U}, f_{\rm U}, \varphi_{\rm U}\big )$, &nbsp;
$\rm Red$ the carrier &nbsp; &rArr; &nbsp; $\big (A_{\rm T}, f_{\rm T}, \varphi_{\rm T}\big )$ and
$\rm Blue$ the upper sideband &nbsp; &rArr; &nbsp; $\big (A_{\rm O}, f_{\rm O}, \varphi_{\rm O}\big )$.
{{BlaueBox|TEXT=
'''(1)''' &nbsp; Consider and interpret the analytic signal  $x_+(t)$ for $\text{Red:} \hspace{0.15cm} A_{\rm T} = 1.5\ \text{V}, \ f_{\rm T} = 50 \ \text{kHz}, \ \varphi_{\rm T} = 0^\circ$, $A_{\rm U} = A_{\rm O} = 0$.

:Which signal values $x_+(t)$ result for $t = 0$, $t = 5 \ \rm &micro; s$ and $t = 20 \ \rm &micro; s$? What are the corresponding signal values for $x(t)$? }}

::&nbsp;For a cosine signal, let $x_+(t= 0) = A_{\rm T} = 1.5\ \text{V}$. Then $x_+(t)$ rotates in a mathematically positive direction (one revolution per period  $T_0 = 1/f_{\rm T}$):

::&nbsp;$x_+(t= 20 \ {\rm &micro; s}) = x_+(t= 0) =  1.5\ \text{V}\hspace{0.3cm}\Rightarrow\hspace{0.3cm}x(t= 20 \ {\rm &micro; s})  =  1.5\ \text{V,}$ 
::&nbsp;$x_+(t= 5 \ {\rm &micro; s})  =  {\rm j} \cdot 1.5\ \text{V}\hspace{0.3cm}\Rightarrow\hspace{0.3cm}x(t= 5 \ {\rm &micro; s}) = {\rm Re}[x_+(t= 5 \ {\rm &micro; s})] =  0$.

{{BlaueBox|TEXT=
'''(2)''' &nbsp; How do the ratios change for $\text{Red:} \hspace{0.15cm} A_{\rm T} = 1.0\ \text{V}, \ f_{\rm T} = 100 \ \text{kHz}, \ \varphi_{\rm T} = 90^\circ$?}}

::The signal $x(t)$ is now a sine signal with a smaller amplitude. The analytic signal now starts because of $\varphi_{\rm T} = 90^\circ$ &nbsp; &rArr; &nbsp; $\phi_{\rm T} = -90^\circ$ at $x_+(t= 0) = -{\rm j} \cdot A_{\rm T}$. <br>After that, $x_+(t)$ rotates again in a mathematically positive direction, but twice as fast because of $T_0 = 10 \ \rm &micro; s$ as in $\rm (1)$.

{{BlaueBox|TEXT=
'''(3)''' &nbsp; Now &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;   $\text{Blue:} \hspace{0.15cm} A_{\rm O} = 0.4\ \text{V}, \ f_{\rm O} = 120 \ \text{kHz}, \ \varphi_{\rm O} = 0^\circ$.

:Consider and interpret the physical signal $x(t)$ and the analytic signal $x_+(t)$.}}

::The Signal $x(t)$ results in the [https://en.wikipedia.org/wiki/Sideband Double-sideband Amplitude Modulation] (DSB&ndash;AM) of the message signal $A_{\rm N}\cdot \cos\left(2\pi f_{\rm N}\cdot t\right)$ with $A_{\rm N} = 0.8\ \text{V}$, $f_{\rm N} = 20\ \text{kHz}$. The carrier $x_{\rm T}(t)$ with $f_{\rm T} = 100\ \text{kHz}$ is also cosinusoidal. The degree of modulation is $m = A_{\rm N}/A_{\rm T} = 0.8$ and the period $T_{\rm 0} = 50\ \text{&micro;s}$.

::In the phase diagram, the (red) carrier rotates faster than the (green) lower sideband and slower than the (blue) upper sideband. The analytic signal $x_+(t)$ results as the geometric sum of the three rotating pointers. It seems that the blue pointer is leading the carrier and the green pointer is following the carrier.

{{BlaueBox|TEXT=
'''(4)''' &nbsp; The settings of task '''(3)''' still apply. Which signal values are obtained at $t=0$, $t=2.5 \ \rm &micro; s$, $t= 5 \ \rm &micro; s$ and $t=10 \ \rm &micro; s$? }}

::At time $t=0$, all pointers are in the direction of the real axis, so that $x(t=0) = {\rm Re}\big [x+(t= 0)\big] =  A_{\rm U} + A_{\rm T} + A_{\rm O}  =  1.8\ \text{V}$.

::Until the time $t=2.5 \ \rm &micro; s$, the red carrier has rotated by $90^\circ$, the blue one by $108^\circ$ and the green one by $72^\circ$. We have $x(t=2.5 \ \rm &micro; s) = {\rm Re}\big [x_+(t= 2.5 \ \rm &micro; s)\big] = 0$, because now the pointer group points in the direction of the imaginary axis. The other sought signal values are $x(t=5 \ \rm &micro; s) = {\rm Re}\big [x_+(t= 5 \ \rm &micro; s)\big] = -1.647\ \text{V}$ and $x(t=10 \ \rm &micro; s) = {\rm Re}\big [x_+(t= 10 \ \rm &micro; s)\big] = 1.247\ \text{V}$.
::For $x_+(t)$ a spiral shape results, alternating with a smaller radius and then with a larger radius.

{{BlaueBox|TEXT=
'''(5)''' &nbsp; How should the phase parameters $\varphi_{\rm T}$, $\varphi_{\rm U}$ and $\varphi_{\rm O}$ be set if both the carrier $x_{\rm T}(t)$ and the message signal $x_{\rm N}(t)$ are sinusoidal?}}

::The parameter selection $\varphi_{\rm T} = \varphi_{\rm U} = \varphi_{\rm O}=90^\circ$ describes the signals $x_{\rm T}(t) = A_{\rm T}\cdot \sin\left(2\pi f_{\rm T}\cdot t\right)$ and $x_{\rm N}(t) = A_{\rm N}\cdot \cos\left(2\pi f_{\rm N}\cdot t\right)$. If, in addition, the message $x_{\rm N}(t)$ is sinusoidal, then $\varphi_{\rm O}=\varphi_{\rm T} - 90^\circ = 0$ and $\varphi_{\rm U}=\varphi_{\rm T} + 90^\circ = 180^\circ$ must be set.

{{BlaueBox|TEXT=
'''(6)''' &nbsp; The settings of task '''(3)''' apply except $A_{\rm T} = 0.6\ \text{V}$. Which modulation method is described here?

: What are the consequences of this? What changes with $A_{\rm T} = 0$? }}

::It is a [https://en.wikipedia.org/wiki/Sideband Double-sideband Amplitude Modulation] (DSB&ndash;AM with carrier) with the modulation degree $m=0.8/0.6 = 1.333$. For $m > 1$, however,  [https://www.radio-electronics.com/info/rf-technology-design/am-reception/synchronous-demodulator-demodulation-detector.php Synchronous Demodulation] is required. [https://en.wikipedia.org/wiki/Envelope_detector Envelope Detection] no longer works. One reason for this is that now the zero crossings of $x(t)$ are no longer equidistant from $5\ \rm &micro; s$ &nbsp; &rArr; &nbsp; additional phase modulation.

::With $A_{\rm T} = 0$ &nbsp; &rArr; &nbsp; $m \to \infty$ results in a [https://en.wikipedia.org/wiki/Double-sideband_suppressed-carrier_transmission ''DSB&ndash;AM without carrier''].  For this, one also needs coherent demodulation.

{{BlaueBox|TEXT=
'''(7)''' &nbsp; &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;  $\text{Green:} \hspace{0.15cm} A_{\rm U} = 0$, &nbsp;   $\text{Blue:} \hspace{0.15cm} A_{\rm O} = 0.8\ \text{V}, \ f_{\rm O} = 120 \ \text{kHz}, \ \varphi_{\rm O} = 90^\circ$.

:Which constellation is described here? Which figure is given for the equivalent low-pass signal $x_{\rm TP}(t)$? &nbsp; &rArr; &nbsp; "locus"? <br>What changes with $A_{\rm U} = 0.8\ \text{V}$ and $A_{\rm O} = 0$?}}

::In both cases, it is a [https://en.wikipedia.org/wiki/Single-sideband_modulation Single-sideband Amplitude Modulation] (SSB&ndash;AM) with the modulation degree $\mu = 0.8$ (in SSB we denote the degree of modulation with $\mu$ instead of $m$). The carrier signal is cosinusoidal and the message signal is sinusoidal. The equivalent low-pass signal $x_{\rm TP}(t)$ has a circular course in the complex plane.

:: $A_{\rm O} = 0.8\ \text{V}$, $A_{\rm U} = 0$ is an OSB modulation. The green pointer is missing and the blue pointer rotates faster compared to the red carrier.

:: $A_{\rm U} = 0.8\ \text{V}$, $A_{\rm O} = 0$ is a USB modulation. The blue pointer is missing and the green pointer rotates slower compared to the red carrier.

{{BlaueBox|TEXT=
'''(8)''' &nbsp; Now&nbsp; $\text{Red:} \hspace{0.05cm} A_{\rm T} = 1\ \text{V}, \ f_{\rm T} = 100 \ \text{kHz}, \ \varphi_{\rm T} = 0^\circ$, &nbsp;  $\text{Green:} \hspace{0.05cm} A_{\rm U} = 0.4\ \text{V}, \ f_{\rm U} = 80 \ \text{kHz}, \ \varphi_{\rm U} = -90^\circ$,  &nbsp;   $\text{Blue:} \hspace{0.05cm} A_{\rm O} = 0.2\ \text{V}, \ f_{\rm O} = 120 \ \text{kHz}, \ \varphi_{\rm O} = +90^\circ$.

:Which constellation could be described here? Which shape results for the equivalent low-pass signal $x_{\rm TP}(t)$?}}

::It could be a DSB&ndash;AM of a sinusoidal signal with cosinusoidal carrier and modulation degree $m=0.8$, in which the upper sideband is attenuated by a factor of 2. The equivalent low-pass signal $x_{\rm TP}(t)$ has an elliptical trace in the complex plane.

==Applet Manual==
<br>
[[File:Zeigerdiagramm_abzug.png|right|Screenshot]]

* The red parameters $(A_{\rm T}, \ f_{\rm T}, \ \varphi_{\rm T})$  and the red pointer mark the ''Carrier'' <br>(German: &nbsp; '''T'''räger).
* The green parameters $(A_{\rm U}, \ f_{\rm U} < f_{\rm T}, \ \varphi_{\rm U})$ mark the ''Lower sideband'' <br>(German: &nbsp;'''U'''nteres Seitenband).
* The blue parameters $(A_{\rm O}, \ f_{\rm O} > f_{\rm T}, \ \varphi_{\rm O})$  mark the ''Upper sideband'' <br>(German: &nbsp;'''O'''beres Seitenband).
*All pointers rotate in a mathematically positive direction (counterclockwise).


Meaning of the letters in the adjacent graphic:

&nbsp; &nbsp; '''(A)''' &nbsp; &nbsp; Plot of the analytic signal $x_{\rm +}(t)$

&nbsp; &nbsp; '''(B)''' &nbsp; &nbsp; Plot of the physical signal $x(t)$

&nbsp; &nbsp; '''(C)''' &nbsp; &nbsp; Parameter input via slider: amplitudes, frequencies, phase values

&nbsp; &nbsp; '''(D)''' &nbsp; &nbsp; Control elements: &nbsp; Start &ndash; Step &ndash; Pause/Continue &ndash; Reset

&nbsp; &nbsp; '''(E)''' &nbsp; &nbsp; Speed of animation: &nbsp; "Speed" &nbsp; &rArr; &nbsp; Values: 1, 2, 3

&nbsp; &nbsp; '''(F)''' &nbsp; &nbsp; "Trace" &nbsp; &rArr; &nbsp;  On or Off, trace of complex signal values $x_{\rm +}(t)$

&nbsp; &nbsp; '''(G)''' &nbsp; &nbsp; Numerical output of the time $t$ and the signal values &nbsp;${\rm Re}[x_{\rm +}(t)] = x(t)$&nbsp; and &nbsp;${\rm Im}[x_{\rm +}(t)]$

&nbsp; &nbsp; '''(H)''' &nbsp; &nbsp; Variations for the graphical representation

$\hspace{1.5cm}$Zoom&ndash;Functions "$+$" (Enlarge), "$-$" (Decrease) and $\rm o$ (Reset to default)

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

&nbsp; &nbsp; '''(I)''' &nbsp; &nbsp; Experiment section:&nbsp; Task selection and task

&nbsp; &nbsp; '''(J)''' &nbsp; &nbsp; Experiment section:&nbsp; solution

In all applets top right:&nbsp; &nbsp; Changeable graphical interface design &nbsp; &rArr; &nbsp; '''Theme''':
* Dark: &nbsp; black background&nbsp; (recommended by the authors).
* Bright: &nbsp; white background&nbsp; (recommended for beamers and printouts)
* Deuteranopia: &nbsp; for users with pronounced green&ndash;visual impairment
* Protanopia: &nbsp; for users with pronounced red&ndash;visual impairment
<br clear=all>

==About the Authors==
This interactive calculation was designed and realized at the&nbsp;  [https://www.ei.tum.de/en/lnt/home//startseite Institute for Communications Engineering]&nbsp; of the&nbsp;  [https://www.tum.de/ Technical University of Munich] .
*The original version was created in 2005 by&nbsp; [[Biographies_and_Bibliographies/An_LNTwww_beteiligte_Studierende#Ji_Li_.28Bachelorarbeit_EI_2003.2C_Diplomarbeit_EI_2005.29|Ji Li]]&nbsp; as part of her Diploma thesis using  "FlashMX&ndash;Actionscript"&nbsp; (Supervisor:&nbsp; [[Biographies_and_Bibliographies/An_LNTwww_beteiligte_Mitarbeiter_und_Dozenten#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28am_LNT_seit_1974.29|Günter Söder]]).
*In 2018 this Applet was redesigned and updated to "HTML5" by&nbsp; [[Biographies_and_Bibliographies/An_LNTwww_beteiligte_Studierende#Xiaohan_Liu_.28Bachelorarbeit_2018.29|Xiaohan Liu]]&nbsp; as part of her Bachelor's thesis (Supervisor: [[Biographies_and_Bibliographies/Beteiligte_der_Professur_Leitungsgebundene_%C3%9Cbertragungstechnik#Tasn.C3.A1d_Kernetzky.2C_M.Sc._.28bei_L.C3.9CT_seit_2014.29|Tasnád Kernetzky]]).

==Once again:&nbsp; Open Applet in new Tab==

{{LntAppletLink|physAnSignal_en}}  &nbsp; &nbsp; &nbsp; &nbsp; [https://www.lntwww.de/Applets:Physikalisches_Signal_%26_Analytisches_Signal '''English Applet with German WIKI description''']