Modulation Methods/General Model of Modulation - Revision history

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@@ Line 11: / Line 11: @@
 [[File:P_ID955__Mod_T_1_3_S1_neu.png | frame|Gemeinsame Beschreibung von Amplituden&ndash; und Winkelmodulation | Joint description of amplitude and angle modulation]]
 The two input signals and the output signal have the following characteristics:
 *The&nbsp; &raquo;'''source signal'''&laquo; &nbsp; (German:&nbsp; "Quellensignal" &nbsp; &rArr; &nbsp; letter&nbsp; "q") &nbsp; &rArr; &nbsp; $q(t)$&nbsp; is the low-frequency message signal and has the spectrum &nbsp;$Q(f)$.&nbsp; This signal is continuous in value and time and limited to the frequency range &nbsp;$|f| ≤ B_{\rm NF}$&nbsp; <br>("NF"&nbsp; from German&nbsp; "Niederfrequenz"&nbsp; i.e.&nbsp; "low-frequency").
 *The&nbsp; &raquo;'''carrier signal'''&laquo; &nbsp;$z(t)$&nbsp; is a harmonic oscillation of the form &nbsp; (subscript&nbsp; "T"&nbsp; from German&nbsp; "Träger"&nbsp; i.e.&nbsp; "carrier"):
 :$$z(t) = A_{\rm T} \cdot \cos(2 \pi f_{\rm T} t - \varphi_{\rm T})= A_{\rm T} \cdot \cos(2 \pi f_{\rm T} t + \phi_{\rm T})\hspace{0.05cm}.$$
 *The&nbsp; &raquo;'''transmitted signal'''&laquo;&nbsp;  (German:&nbsp;  "Sendesignal" &nbsp;  ⇒ &nbsp;  letter  "s") &nbsp;  ⇒ &nbsp;$s(t)$&nbsp; is a higher frequency signal,&nbsp; whose spectrum &nbsp;$S(f)$&nbsp;is in the range around the carrier frequency&nbsp;$f_{\rm T}$.
@@ Line 20: / Line 22: @@
 {{BlaueBox|TEXT=$\text{Definition:}$&nbsp; Each &nbsp;[[Signal_Representation/Harmonic_Oscillation|$\text{harmonic oscillation}$]] &nbsp;$z(t)$&nbsp; can be described by
 *the amplitude &nbsp;$A_{\rm T}$,
 *the frequency &nbsp;$f_{\rm T}$&nbsp; and
 *the zero phase position &nbsp;${\it ϕ}_{\rm T}$ .
@@ Line 40: / Line 44: @@
 *In&nbsp; &raquo;'''amplitude modulation'''&laquo;&nbsp; $\rm (AM)$,&nbsp; the time-dependent amplitude &nbsp;$a(t)$&nbsp; changes according to the signal&nbsp; $q(t)$,&nbsp; while the other two signal parameters stay constant:
 :$$\omega(t) = \omega_{\rm T} = {\rm const.}\hspace{0.08cm}, \hspace{0.2cm}\phi(t) = \phi_{\rm T} = {\rm const.}\hspace{0.08cm}, \hspace{0.2cm} a(t) = {\rm function \hspace{0.15cm}of}\hspace{0.15cm}q(t) .$$
 *In&nbsp; &raquo;'''frequency modulation'''&laquo;&nbsp; $\rm (FM)$,&nbsp; only the instantaneous&nbsp; (circular)&nbsp; frequency&nbsp;$\omega(t)$&nbsp; is determined by the signal&nbsp; $q(t)$:
 :$$a(t) = A_{\rm T} = {\rm const.}\hspace{0.08cm}, \hspace{0.2cm}\phi(t) = \phi_{\rm T} = {\rm const.}\hspace{0.08cm}, \hspace{0.2cm} \omega(t)= {\rm function \hspace{0.15cm}of}\hspace{0.15cm}q(t) .$$
 *In&nbsp; &raquo;'''phase modulation'''&laquo;&nbsp; $\rm (PM)$,&nbsp;the phase&nbsp;$\phi(t)$&nbsp; varies according to the signal&nbsp; $q(t)$:
 :$$a(t) = A_{\rm T} = {\rm const.}\hspace{0.08cm}, \hspace{0.2cm}\omega(t) = \omega_{\rm T} = {\rm const.}\hspace{0.08cm}, \hspace{0.2cm} \phi(t) = {\rm function \hspace{0.15cm}of}\hspace{0.15cm}q(t) .$$
@@ Line 75: / Line 81: @@
 *The analytical signal&nbsp;$s_+(t)$&nbsp; is obtained from the real physical signal &nbsp;$s(t)$,&nbsp; by adding to it&nbsp; (as an imaginary part)&nbsp; its [[Signal_Representation/Analytical_Signal_and_Its_Spectral_Function#Representation_with_Hilbert_transform|$\text{Hilbert transform}$]]:
 :$$s_+(t) = s(t) + {\rm j} \cdot {\rm H}\{ s(t)\}\hspace{0.05cm}.$$
 *The analytical signal &nbsp;$s_+(t)$&nbsp;   is therefore always complex.&nbsp;  The following simple relationship holds between the two time signals:
 :$$s(t) = {\rm Re} \big[s_+(t)\big] \hspace{0.05cm}.$$
 *The spectrum &nbsp;$S_+(f)$&nbsp; of the analytic signal is obtained from the two-sided spectrum &nbsp;$S(f)$ by doubling it for positive frequencies and setting it to zero for negative frequencies:
 :$$S_+(f) =\big[ 1 + {\rm sign}(f)\big]  \cdot S(f)  = \left\{ \begin{array}{c} 2 \cdot S(f) \\ 0 \\  \end{array} \right.\quad \begin{array}{*{10}c}    {\rm{for}} \\  {\rm{for}}   \\ \end{array}\begin{array}{*{20}c} f>0 \hspace{0.05cm}, \\ f<0  \hspace{0.05cm}, \\ \end{array} \hspace{1.3cm}
@@ Line 87: / Line 95: @@
 *The axial symmetry of the spectral function &nbsp;$S(f)$&nbsp;  with respect to the frequency &nbsp;$f=0$: &nbsp;
 :$${\rm Re}\big[S( - f)\big] = {\rm Re}\big[S(f)\big].$$
 *If the spectrum of the actual band-pass signal &nbsp;$s(t)$&nbsp; has an imaginary part,&nbsp; it would be point-symmetric about &nbsp;$f=0$:
 :$${\rm Im}\big[S( - f)\big] = - {\rm Im}\big[S(f)\big].$$
@@ Line 92: / Line 101: @@
 The spectrum&nbsp;$S_+(f)$&nbsp; of the corresponding analytical signal &nbsp;$s_+(t)$&nbsp; is shown below.&nbsp; This is obtained from &nbsp;$S(f)$&nbsp; by
 *truncating the negative frequency components:  &nbsp; $S_+(f) \equiv 0$ &nbsp;for&nbsp; $f<0$,
 *doubling the positive frequency components:   &nbsp; $S_+(f ) = 2 \cdot S(f )$ &nbsp;for&nbsp; $f \ge 0$.
@@ Line 104: / Line 114: @@
 The second equation describes a rotating pointer with the following properties:
 *The pointer length denotes the signal amplitude &nbsp;$A_{\rm T}$.&nbsp; At time &nbsp;$t = 0$,&nbsp; the pointer lies in the complex plane with an angle of &nbsp;$ϕ_{\rm T}$.
 *For &nbsp;$t > 0$,&nbsp; the pointer rotates with constant angular velocity &nbsp;$ω_{\rm T}$&nbsp;  in a mathematically positive direction (counterclockwise).
 *The pointer tip always lies on a circle with radius &nbsp;$A_{\rm T}$&nbsp; and requires exactly the period &nbsp;$T_0$ for one rotation.
 <br clear=all>
 {{BlaueBox|TEXT=
 $\text{The individual modulation methods can now be represented as follows:}$
 *In&nbsp; &raquo;'''amplitude modulation'''&laquo;&nbsp; the pointer length &nbsp;$a(t) = \vert s_+(t)\vert $&nbsp; changes  according to the source signal &nbsp;$q(t)$. <br>The angular velocity &nbsp;$ω(t)$&nbsp; remains constant in this case.
 *During&nbsp; &raquo;'''frequency modulation'''&laquo;&nbsp; the angular velocity &nbsp;$ω(t)$&nbsp; of the rotating pointer changes according to &nbsp;$q(t)$. <br>The pointer length &nbsp;$a(t) = A_{\rm T}$&nbsp; stays unchanged.
 *In&nbsp; &raquo;'''phase modulation'''&laquo;,&nbsp; the phase &nbsp;$ϕ(t)$&nbsp; is time-dependent according to the source signal &nbsp;$q(t)$. <br>There are many similarities with frequency modulation,&nbsp; which also belongs to the class of angle modulation.}}
@@ Line 149: / Line 163: @@
 Upper graph &nbsp; &rArr; &nbsp; &raquo;'''Amplitude modulation'''&laquo;&nbsp; $\rm (AM)$:
 *Here,&nbsp; the source signal &nbsp;$q(t)$&nbsp; can be seen in the envelope &nbsp;$a(t)$.
 *Since the zero crossings of the carrier &nbsp;$z(t)$&nbsp; stay the same: &nbsp;$ϕ(t) = 0$.&nbsp; The equivalent low-pass signal &nbsp;$s_{\rm TP}(t) = a(t)$&nbsp; is real.
@@ Line 156: / Line 171: @@
 Bottom graph &nbsp; &rArr; &nbsp; &raquo;'''Angle modulation'''&laquo;&nbsp; $\rm (WM)$:
 *Here,&nbsp; the envelope &nbsp;$a(t)$&nbsp; is constant &nbsp; &rArr; &nbsp; the equivalent low-pass signal &nbsp;$s_{\rm TP}(t) = A_{\rm T} · e^{\rm j·ϕ(t)}$&nbsp; describes a circular arc.
 *The information about the source signal &nbsp;$q(t)$&nbsp; is found here in the location of the zero crossings &nbsp;$s(t)$.
 *More details about this modulation method can be found in the chapter &nbsp;[[Modulation_Methods/Phase_Modulation_(PM)#Similarities_between_phase_and_frequency_modulation|"Similarities between phase and frequency modulation"]].}}
@@ Line 162: / Line 179: @@
 {{BlaueBox|TEXT=
 $\text{Notes:}$
 *In the following,&nbsp; we also refer to the time-dependent course of&nbsp; $s_{\rm TP}(t)$&nbsp; in the complex plane as the&nbsp; &raquo;'''locus curve'''&laquo;.&nbsp;
-*The&nbsp; &raquo;'''pointer diagram'''&laquo;&nbsp; describes the course of the analytical signal&nbsp; $s_+(t)$&nbsp;.
 *The topic presented here is illustrated in the time domain with two interactive&nbsp; "HTML 5/JS"&nbsp; applets:
 ::(1) &nbsp;[[Applets:Physikalisches_Signal_%26_Analytisches_Signal|"Physical & Analytic Signal"]],

@@ Line 162: / Line 162: @@
 {{BlaueBox|TEXT=
 $\text{Notes:}$
-*In the following,&nbsp; we also refer to the time-dependent course of&nbsp; $s_{\rm TP}(t)$&nbsp; in the complex plane as the&nbsp; &raquo;'''locus curve'''&laquo;.,&nbsp;
+*In the following,&nbsp; we also refer to the time-dependent course of&nbsp; $s_{\rm TP}(t)$&nbsp; in the complex plane as the&nbsp; &raquo;'''locus curve'''&laquo;.&nbsp;
 *The&nbsp; &raquo;'''pointer diagram'''&laquo;&nbsp; describes the course of the analytical signal&nbsp; $s_+(t)$&nbsp;.
 *The topic presented here is illustrated in the time domain with two interactive&nbsp; "HTML 5/JS"&nbsp; applets:

@@ Line 117: / Line 117: @@
 ==Describing the physical signal using the equivalent low-pass signal==
 <br>
-Some facts concerning modulation at the transmitter and demodulation at the receiver can be explained best by means of the [[Signal_Representation/Equivalent_Low_Pass_Signal_and_Its_Spectral_Function|$\text{equivalent low-pass signal}$]]&nbsp; (German:&nbsp; "äquivalentes Tiefpass&ndash;Signal" &nbsp; &rArr; &nbsp; subscript&nbsp; "TP")&nbsp; according to the definition given in the book&nbsp; [[Signal_Representation/Equivalent_Low-Pass_Signal_and_its_Spectral_Function|"Signal Representation"]].&nbsp;
+Some facts concerning modulation at the transmitter and demodulation at the receiver can be explained best by means of the [[Signal_Representation/Equivalent_Low-Pass_Signal_and_its_Spectral_Function|$\text{equivalent low-pass signal}$]]&nbsp; (German:&nbsp; "äquivalentes Tiefpass&ndash;Signal" &nbsp; &rArr; &nbsp; subscript&nbsp; "TP")&nbsp; according to the definition given in the book&nbsp; [[Signal_Representation/Equivalent_Low-Pass_Signal_and_its_Spectral_Function|"Signal Representation"]].&nbsp;
 [[File:Mod_T_1_3_S5a_version2.png|right|frame|The equivalent lowpass signal in the frequency domain]]

@@ Line 48: / Line 48: @@
 In these basic methods,&nbsp; two of the three oscillation parameters are thus always kept constant.
 *Additionally,&nbsp; there are also variants with more than one time dependency for amplitude,&nbsp; frequency,&nbsp; and phase, resp.
-*An example of this is &nbsp;[[Modulation_Methods/Single-Sideband_Modulation|$\text{single-sideband modulation}$]],&nbsp; where both &nbsp;$a(t)$&nbsp; and &nbsp;${\it ϕ}(t)$&nbsp; are affected by the source signal &nbsp;$q(t)$.
+*An example of this is &nbsp;[[Modulation_Methods/Single-Sideband_Modulation|$\text{Single-Sideband Modulation}$]],&nbsp; where both &nbsp;$a(t)$&nbsp; and &nbsp;${\it ϕ}(t)$&nbsp; are affected by the source signal &nbsp;$q(t)$.
@@ Line 55: / Line 55: @@
 When describing &nbsp; $\rm AM$,&nbsp; $\rm FM$&nbsp; and&nbsp; $\rm PM$,&nbsp; the source signal  &nbsp;$q(t)$&nbsp; is usually assumed to be continuous in time and value.
 *However,&nbsp; the above equations can also be applied to a rectangular source signal.
-*In this case,&nbsp; $q(t)$&nbsp; is continuous in time but discrete in value.&nbsp;  Thus,&nbsp; it also describes methods for&nbsp; [[Modulation_Methods/Linear_Digital_Modulation|$\text{linear digital modulation}$]].
+*In this case,&nbsp; $q(t)$&nbsp; is continuous in time but discrete in value.&nbsp;  Thus,&nbsp; it also describes methods for&nbsp; [[Modulation_Methods/Linear_Digital_Modulation|$\text{Linear Digital Modulation}$]].

@@ Line 163: / Line 163: @@
 $\text{Notes:}$
 *In the following,&nbsp; we also refer to the time-dependent course of&nbsp; $s_{\rm TP}(t)$&nbsp; in the complex plane as the&nbsp; '''locus curve'''.,&nbsp;
-*The&nbsp; &raquo;'''pointer diagram'''&laquo;&nbsp; describes the course of the analytical signal&nbsp; $s_+(t)$&nbsp;.
+*The&nbsp; '''pointer diagram'''&nbsp; describes the course of the analytical signal&nbsp; $s_+(t)$&nbsp;.
 *The topic presented here is illustrated in the time domain with two interactive&nbsp; "HTML 5/JS"&nbsp; applets:
 ::(1) &nbsp;[[Applets:Physikalisches_Signal_%26_Analytisches_Signal|Physical & Analytic Signal ]],

@@ Line 147: / Line 147: @@
 [[File:EN_Mod_T_1_3_S5b_neu.png |right|frame| Transmitted signals for amplitude and angle modulation]]
-Upper graphic &nbsp; &rArr; &nbsp; '''Amplitude modulation'''&nbsp; $\rm (AM)$:
+Upper graph &nbsp; &rArr; &nbsp; '''Amplitude modulation'''&nbsp; $\rm (AM)$:
 *Here,&nbsp; the source signal &nbsp;$q(t)$&nbsp; can be seen in the envelope &nbsp;$a(t)$.
 *Since the zero crossings of the carrier &nbsp;$z(t)$&nbsp; stay the same: &nbsp;$ϕ(t) = 0$.&nbsp; The equivalent low-pass signal &nbsp;$s_{\rm TP}(t) = a(t)$&nbsp; is real.
@@ Line 154: / Line 154: @@
-Bottom graphic &nbsp; &rArr; &nbsp; '''Angle modulation'''&nbsp; $\rm (WM)$:
+Bottom graph &nbsp; &rArr; &nbsp; '''Angle modulation'''&nbsp; $\rm (WM)$:
 *Here,&nbsp; the envelope &nbsp;$a(t)$&nbsp; is constant &nbsp; &rArr; &nbsp; the equivalent low-pass signal &nbsp;$s_{\rm TP}(t) = A_{\rm T} · e^{\rm j·ϕ(t)}$&nbsp; describes a circular arc.
 *The information about the source signal &nbsp;$q(t)$&nbsp; is found here in the location of the zero crossings &nbsp;$s(t)$.