Modulation Methods/Frequency Modulation (FM) - Revision history

Guenter at 13:35, 13 January 2023

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@@ Line 217: / Line 217: @@
 ==Realization of a FM modulator==
 <br>
-A frequency modulation is obtained when the oscillation frequency of an oscillator is changed to the rhythm of the modulating '''KORREKTUR: modulated?''' signal.  RC elements or oscillating circuits usually serve as frequency-determining elements. The left graph shows the form of a possible circuit realization; the exact circuit description can be found in [Mäu 88]<ref name= "Mäu 88">Mäusl, R.:&nbsp; Analoge Modulationsverfahren.&nbsp;  Heidelberg: Dr. Hüthig, 1988.</ref>.&nbsp;  The idealized frequency-voltage characteristic is shown on the right.
+A frequency modulation&nbsp; $s(t)$&nbsp; is obtained when the oscillation frequency of an oscillator is changed to the rhythm of the modulating  signal&nbsp; $u(t)$.&nbsp;  RC elements or oscillating circuits usually serve as frequency-determining elements. The left graph shows the form of a possible circuit realization; the exact circuit description can be found in [Mäu 88]<ref name= "Mäu 88">Mäusl, R.:&nbsp; Analoge Modulationsverfahren.&nbsp;  Heidelberg: Dr. Hüthig, 1988.</ref>.&nbsp;  The idealized frequency-voltage characteristic is shown on the right.
 [[File:EN_Mod_T_3_2_S6.png |right|frame| Realization of a FM modulator and its characteristic curve]]
@@ Line 236: / Line 236: @@
 In short,&nbsp; this circuit,&nbsp; which operates as a &nbsp; $($"Phase–Locked–Loop",&nbsp; $\rm PLL)$,&nbsp; can be described as follows:
-*The phase detector determines the phase differences&nbsp; (distances of zero intercepts '''KORREKTUR: crossings''')&nbsp; between  the received signal &nbsp;$r(t)$&nbsp; and the comparison signal provided by the VCO.
+*The phase detector determines the phase differences&nbsp; (distances of zero crossings)&nbsp; between  the received signal &nbsp;$r(t)$&nbsp; and the comparison signal provided by the VCO.
 *After low-pass filtering and amplification,&nbsp; the output signal &nbsp;$v(t)$&nbsp; is approximately equal to the source signal&nbsp; $q(t)$&nbsp; if it has been FM-modulated at the transmit end.
 *The output signal &nbsp;$v(t)$&nbsp; is simultaneously applied to the input  of the&nbsp; "Voltage Controlled Oscillator",&nbsp; abbreviated as&nbsp; "$\rm VCO$".

@@ Line 217: / Line 217: @@
 ==Realization of a FM modulator==
 <br>
-A frequency modulation is obtained when the oscillation frequency of an oscillator is changed to the rhythm of the modulating signal.  RC elements or oscillating circuits usually serve as frequency-determining elements. The left graph shows the form of a possible circuit realization; the exact circuit description can be found in [Mäu 88]<ref name= "Mäu 88">Mäusl, R.:&nbsp; Analoge Modulationsverfahren.&nbsp;  Heidelberg: Dr. Hüthig, 1988.</ref>.&nbsp;  The idealized frequency-voltage characteristic is shown on the right.
+A frequency modulation is obtained when the oscillation frequency of an oscillator is changed to the rhythm of the modulating '''KORREKTUR: modulated?''' signal.  RC elements or oscillating circuits usually serve as frequency-determining elements. The left graph shows the form of a possible circuit realization; the exact circuit description can be found in [Mäu 88]<ref name= "Mäu 88">Mäusl, R.:&nbsp; Analoge Modulationsverfahren.&nbsp;  Heidelberg: Dr. Hüthig, 1988.</ref>.&nbsp;  The idealized frequency-voltage characteristic is shown on the right.
 [[File:EN_Mod_T_3_2_S6.png |right|frame| Realization of a FM modulator and its characteristic curve]]

@@ Line 236: / Line 236: @@
 In short,&nbsp; this circuit,&nbsp; which operates as a &nbsp; $($"Phase–Locked–Loop",&nbsp; $\rm PLL)$,&nbsp; can be described as follows:
-*The phase detector determines the phase differences&nbsp; (distances of zero intercepts)&nbsp; between  the received signal &nbsp;$r(t)$&nbsp; and the comparison signal provided by the VCO.
+*The phase detector determines the phase differences&nbsp; (distances of zero intercepts '''KORREKTUR: crossings''')&nbsp; between  the received signal &nbsp;$r(t)$&nbsp; and the comparison signal provided by the VCO.
 *After low-pass filtering and amplification,&nbsp; the output signal &nbsp;$v(t)$&nbsp; is approximately equal to the source signal&nbsp; $q(t)$&nbsp; if it has been FM-modulated at the transmit end.
 *The output signal &nbsp;$v(t)$&nbsp; is simultaneously applied to the input  of the&nbsp; "Voltage Controlled Oscillator",&nbsp; abbreviated as&nbsp; "$\rm VCO$".

@@ Line 16: / Line 16: @@
 {{BlaueBox|TEXT=
 $\text{Definitions:}$&nbsp;
-*The&nbsp; '''instantaneous angular frequency'''&nbsp; is the derivative of the angular function with respect to time:
+*The&nbsp; &raquo;'''instantaneous angular frequency'''&laquo;&nbsp; is the derivative of the angular function with respect to time:
 :$$\omega_{\rm A}(t) = \frac{ {\rm d}\psi(t)}{ {\rm d}t}\hspace{0.05cm}.$$
-*Accordingly, the '''instantaneous frequency''' is captured by:
+*Accordingly, the &raquo;'''instantaneous frequency'''&laquo; is captured by:
 :$$f_{\rm A}(t) = \frac{\omega_{\rm A}(t)}{2\pi} = \frac{1}{2\pi} \cdot \frac{ {\rm d}\hspace{0.03cm}\psi(t)}{ {\rm d}t}\hspace{0.05cm}.$$
-*The&nbsp; '''frequency deviation'''&nbsp; is the maximum deviation &nbsp;$Δf_{\rm A}$&nbsp; of the time-dependent instantaneous frequency &nbsp;$f_{\rm A}(t)$&nbsp; from the constant carrier frequency &nbsp;$f_{\rm T}$. }}
+*The&nbsp; &raquo;'''frequency deviation'''&laquo;&nbsp; is the maximum deviation &nbsp;$Δf_{\rm A}$&nbsp; of the time-dependent instantaneous frequency &nbsp;$f_{\rm A}(t)$&nbsp; from the constant carrier frequency &nbsp;$f_{\rm T}$. }}
@@ Line 62: / Line 62: @@
 $\text{Therefore:}$&nbsp;
-The&nbsp; '''instantaneous frequency &nbsp;$f_{\rm A}(t)$&nbsp; is thus not a physically measurable frequency'''&nbsp; in the conventional sense,&nbsp; but only an&nbsp; '''abstract mathematical quantity''',&nbsp; namely the derivative of the angular function &nbsp;$ψ(t)$.}}
+The&nbsp; '''instantaneous frequency &nbsp;$f_{\rm A}(t)$&nbsp; is thus not a physically measurable frequency'''&nbsp; in the conventional sense,&nbsp; but only an&nbsp; &raquo;'''abstract mathematical quantity'''&laquo;,&nbsp; namely the derivative of the angular function &nbsp;$ψ(t)$.}}
 ==Signal characteristics with frequency modulation==
 <br>
-As in the chapter &nbsp;[[Modulation_Methods/Phase_Modulation_(PM)|Phase Modulation]]&nbsp; we will assume that the carrier signal &nbsp;$z(t)$&nbsp; has a cosine waveform and the source signal &nbsp;$q(t)$&nbsp; is peak-limited.
+As in the chapter &nbsp;[[Modulation_Methods/Phase_Modulation_(PM)|"Phase Modulation"]]&nbsp; we will assume that the carrier signal &nbsp;$z(t)$&nbsp; has a cosine waveform and the source signal &nbsp;$q(t)$&nbsp; is peak-limited.
 {{BlaueBox|TEXT=
-$\text{Definition:}$&nbsp; If the instantaneous angular frequency&nbsp;$ω_{\rm A}(t)$&nbsp; of a communication system is linearly dependent on the instantaneous value of the source signal  &nbsp;$q(t)$,&nbsp; we call this&nbsp; '''frequency modulation'''&nbsp; $\rm (FM)$:
+$\text{Definition:}$&nbsp; If the instantaneous angular frequency&nbsp;$ω_{\rm A}(t)$&nbsp; of a communication system is linearly dependent on the instantaneous value of the source signal  &nbsp;$q(t)$,&nbsp; we call this&nbsp; &raquo;'''frequency modulation'''&laquo;&nbsp; $\rm (FM)$:
 :$$\omega_{\rm A}(t) = 2 \pi \cdot f_{\rm A}(t) = \omega_{\rm T} + K_{\rm FM} \cdot q(t)
 \hspace{0.05cm}.$$
@@ Line 97: / Line 97: @@
 *In frequency modulation,&nbsp; the conversion
 :$$s(t) = a(t) \cdot \cos (\omega (t) \cdot t + \phi(t)) \hspace{0.3cm}\Rightarrow \hspace{0.3cm} s(t) = A_{\rm T} \cdot \cos (\omega (t) \cdot t + \phi_{\rm T})$$
-:is only allowed sometimes,&nbsp; such as in the non-linear digital modulation method &nbsp;[[Modulation_Methods/Nonlinear_Digital_Modulation#FSK_.E2.80.93_Frequency_Shift_Keying|"Frequency Shift Keying"]]&nbsp; $\rm (FSK)$&nbsp; with rectangular basic pulse.}}
+:is only allowed sometimes,&nbsp; such as in the non-linear digital modulation method &nbsp;[[Modulation_Methods/Nonlinear_Digital_Modulation#FSK_.E2.80.93_Frequency_Shift_Keying|$\text{Frequency Shift Keying}$]]&nbsp; $\rm (FSK)$&nbsp; with rectangular basic pulse.}}
 ==Frequency modulation of a cosine signal==
@@ Line 107: / Line 107: @@
 :$$\psi(t) = \omega_{\rm T} \cdot t + \frac {K_{\rm FM} \cdot A_{\rm N}}{\omega_{\rm N}} \cdot \sin (\omega_{\rm N} \cdot t) \hspace{0.05cm}.$$
-A comparison with the propositions in the chapter &nbsp;[[Modulation_Methods/Phase_Modulation_(PM)|Phase Modulation]]&nbsp;  makes clear:
+A comparison with the propositions in the chapter &nbsp;[[Modulation_Methods/Phase_Modulation_(PM)|"Phase Modulation"]]&nbsp;  makes clear:
 *The frequency modulation of a cosine signal yields a qualitatively identical transmitted signal &nbsp;$s(t)$&nbsp; as the phase modulation of a sinusoidal source signal&nbsp;$q(t)$.&nbsp; However,&nbsp; this requires that the modulator constants are matched according to the ratio  &nbsp;$K_{\rm FM}/K_{\rm PM} = ω_{\rm N}$&nbsp;.
 *The transmitted signal &nbsp;$s(t)$&nbsp; can thus be described uniformly for the two configurations&nbsp; "PM – sine signal"&nbsp; as well as&nbsp; "FM – cosine signal":
@@ Line 114: / Line 114: @@
 :$$\eta_{\rm PM} = {K_{\rm PM} \cdot A_{\rm N}},$$
 :$$\eta_{\rm FM} = \frac {K_{\rm FM} \cdot A_{\rm N}}{\omega_{\rm N}} \hspace{0.05cm}.$$
-*If the source signal is not a harmonic oscillation,&nbsp; but is instead composed of several frequencies,&nbsp; the time signals also differ qualitatively for phase and frequency modulation.&nbsp; This can be seen in the &nbsp;[[Modulation_Methods/General_Model_of_Modulation#Modulated_signals_with_a_digital_source_signal|earlier comparison of PSK and FSK]].
+*If the source signal is not a harmonic oscillation,&nbsp; but is instead composed of several frequencies,&nbsp; the time signals also differ qualitatively for phase and frequency modulation.&nbsp; This can be seen in the &nbsp;[[Modulation_Methods/General_Model_of_Modulation#Modulated_signals_with_a_digital_source_signal|$\text{earlier comparison of PSK and FSK}$]].
@@ Line 153: / Line 153: @@
 This equation can be explained as follows:
 *In the section &nbsp;[[Modulation_Methods/Phase_Modulation_(PM)#Equivalent_low-pass_signal_in_phase_modulation|"Equivalent low-pass signal in PM"]]&nbsp; this equation was derived for a phase-modulated sine signal,&nbsp; where &nbsp;$η = A_{\rm N} · K_{\rm PM}$&nbsp; denotes the modulation index and &nbsp;${\rm J}_n(η)$&nbsp; denotes the &nbsp; $n$&ndash;th order Bessel functions of the first kind.&nbsp; $K_{\rm PM}$&nbsp; is the modulator constant.
-*When the message phase &nbsp;$ϕ_{\rm N}$&nbsp; changes,&nbsp; only the phase function &nbsp;${\rm arc} \ S_+(f)$ changes,&nbsp; but not the magnitude spectrum &nbsp;$|S_+(f)|$.&nbsp; This important result is confirmed in [[Aufgaben:Exercise_3.3Z:_Characteristics_Determination|Exercise 3.3Z]].
+*When the message phase &nbsp;$ϕ_{\rm N}$&nbsp; changes,&nbsp; only the phase function &nbsp;${\rm arc} \ S_+(f)$ changes,&nbsp; but not the magnitude spectrum &nbsp;$|S_+(f)|$.&nbsp; This important result is confirmed in [[Aufgaben:Exercise_3.3Z:_Characteristics_Determination|"Exercise 3.3Z"]].
 *In the&nbsp;[[Modulation_Methods/Frequency_Modulation_(FM)#Frequency_modulation_of_a_cosine_signal|"Frequency modulation of a cosine signal"]]&nbsp; section,&nbsp; it was shown that a frequency-modulated signal can be represented in the same way as a phase-modulated signal if the modulation index &nbsp;$η = K_{\rm FM} · A_{\rm N}/ω_{\rm N}$&nbsp; is used.&nbsp; Consequently,&nbsp; the magnitude spectra for&nbsp; $\rm PM$&nbsp; and&nbsp; $\rm FM$&nbsp; can also be represented in the same way.
@@ Line 196: / Line 196: @@
 #As a result,&nbsp; new frequencies&nbsp; ("harmonics")&nbsp; are created,&nbsp; which were not in the original source signal &nbsp;$q(t)$.
 #The smaller the available bandwidth&nbsp;$B_{\rm HF}$&nbsp; and the larger the chosen modulation index &nbsp;$η$,&nbsp; the larger becomes the distortion factor &nbsp; $K$,&nbsp; which captures nonlinear distortions.&nbsp;
-#As a rule of thumb for the required high-frequency bandwidth for a desired distortion factor &nbsp;$K$,&nbsp; one can use the so-called &nbsp; '''Carson Rule''':
+#As a rule of thumb for the required high-frequency bandwidth for a desired distortion factor &nbsp;$K$,&nbsp; one can use the so-called &nbsp; &raquo;'''Carson Rule'''&laquo;:
 ::$$K < 10\%\text{:}\hspace{0.6cm} B_{\rm HF}  \ge  2 \cdot f_{\rm N} \cdot (\eta +1)\hspace{0.05cm},$$
 ::$$K < 1\%\text{:}\hspace{0.72cm} B_{\rm HF} \ge 2 \cdot f_{\rm N} \cdot(\eta +2)\hspace{0.05cm}.$$

@@ Line 93: / Line 93: @@
 {{BlaueBox|TEXT=
 $\text{It can also be seen from the above equation:}$&nbsp;
-* The equation encountered earlier on the page &nbsp;[[Modulation_Methods/General_Model_of_Modulation#A_very_simple_.28though_unfortunately_not_always_correct.29_modulator_equation|"A very simple (though unfortunately not always correct) modulator equation"]] &nbsp; in the first chapter of this book,&nbsp; will only hold for frequency modulation in very special cases.
+* The equation encountered earlier in the section &nbsp;[[Modulation_Methods/General_Model_of_Modulation#A_very_simple_.28though_unfortunately_not_always_correct.29_modulator_equation|"A very simple (though unfortunately not always correct) modulator equation"]] &nbsp; in the first chapter of this book,&nbsp; will only hold for frequency modulation in very special cases.
 *In frequency modulation,&nbsp; the conversion
@@ Line 154: / Line 154: @@
 *In the section &nbsp;[[Modulation_Methods/Phase_Modulation_(PM)#Equivalent_low-pass_signal_in_phase_modulation|"Equivalent low-pass signal in PM"]]&nbsp; this equation was derived for a phase-modulated sine signal,&nbsp; where &nbsp;$η = A_{\rm N} · K_{\rm PM}$&nbsp; denotes the modulation index and &nbsp;${\rm J}_n(η)$&nbsp; denotes the &nbsp; $n$&ndash;th order Bessel functions of the first kind.&nbsp; $K_{\rm PM}$&nbsp; is the modulator constant.
 *When the message phase &nbsp;$ϕ_{\rm N}$&nbsp; changes,&nbsp; only the phase function &nbsp;${\rm arc} \ S_+(f)$ changes,&nbsp; but not the magnitude spectrum &nbsp;$|S_+(f)|$.&nbsp; This important result is confirmed in [[Aufgaben:Exercise_3.3Z:_Characteristics_Determination|Exercise 3.3Z]].
-*On the&nbsp;[[Modulation_Methods/Frequency_Modulation_(FM)#Frequency_modulation_of_a_cosine_signal|"Frequency modulation of a cosine signal"]]&nbsp; page,&nbsp; it was shown that a frequency-modulated signal can be represented in the same way as a phase-modulated signal if the modulation index &nbsp;$η = K_{\rm FM} · A_{\rm N}/ω_{\rm N}$&nbsp; is used.&nbsp; Consequently,&nbsp; the magnitude spectra for&nbsp; $\rm PM$&nbsp; and&nbsp; $\rm FM$&nbsp; can also be represented in the same way.
+*In the&nbsp;[[Modulation_Methods/Frequency_Modulation_(FM)#Frequency_modulation_of_a_cosine_signal|"Frequency modulation of a cosine signal"]]&nbsp; section,&nbsp; it was shown that a frequency-modulated signal can be represented in the same way as a phase-modulated signal if the modulation index &nbsp;$η = K_{\rm FM} · A_{\rm N}/ω_{\rm N}$&nbsp; is used.&nbsp; Consequently,&nbsp; the magnitude spectra for&nbsp; $\rm PM$&nbsp; and&nbsp; $\rm FM$&nbsp; can also be represented in the same way.

@@ Line 78: / Line 78: @@
 :$$\psi(t) = \omega_{\rm T} \cdot t + K_{\rm FM} \cdot \int q(t)\hspace{0.1cm}{\rm d}t \hspace{0.05cm}\hspace{0.3cm}\Rightarrow \hspace{0.3cm}s(t) = A_{\rm T} \cdot \cos \big[\psi(t)\big] \hspace{0.05cm}.$$
-[[File:EN_Mod_T_3_2_S2_v1.png |right|frame|Relationship between phase and frequency modulation<b>KORREKTUR Kanal ⇒ channel </b>]]
+[[File:EN_Mod_T_3_2_S2_v1.png |right|frame|Relationship between phase and frequency modulation]]
 From these equations,&nbsp; we can immediately see:

@@ Line 2: / Line 2: @@
 {{Header
 |Untermenü=Angle Modulation and Demodulation
-|Vorherige Seite=Phasenmodulation (PM)
+|Vorherige Seite=Phase Modulation (PM)
-|Nächste Seite=Rauscheinfluss bei Winkelmodulation
+|Nächste Seite=Influence of Noise on Systems with Angle Modulation
 }}
 ==Instantaneous frequency==