Modulation Methods/Objectives of Modulation and Demodulation - Revision history

Hwang at 13:09, 23 January 2023

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Guenter at 12:56, 13 January 2023

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Hwang at 15:45, 2 January 2023

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 *A common problem is that the channel is often unsuitable for direct transmission of the source signal&nbsp;$q(t)$&nbsp;because it contains&nbsp; "inconvenient frequencies".&nbsp;  For example,&nbsp; a music signal with frequencies up to about $\text{15 kHz}$&nbsp; cannot be transmitted directly by radio,&nbsp; since radio propagation is only possible from around $\text{100 kHz}$.
-*The only solution here is a signal conversion at the transmitter known as&nbsp; &raquo;'''modulation'''&laquo;.&nbsp; The output signal of the modulator will be uniformly referred to as the transmission signal &nbsp;$s(t)$&nbsp; in the following.&nbsp; This is generally at higher frequencies than the source signal &nbsp;$q(t)$.
+*The only solution here is a signal conversion at the transmitter known as&nbsp; &raquo;'''modulation'''&laquo;.&nbsp; The output signal of the modulator will be uniformly referred to as the transmitted signal &nbsp;$s(t)$&nbsp; in the following.&nbsp; This is generally at higher frequencies than the source signal &nbsp;$q(t)$.
 *The signal reset at the receiver to recover the low-frequency sink signal &nbsp;$v(t) ≈ q(t)$&nbsp; from the high-frequency received signal &nbsp;$r(t)$&nbsp; is called&nbsp; &raquo;'''demodulation'''&laquo;.&nbsp; With real channels, the desired result&nbsp;$v(t) \equiv q(t)$&nbsp; is not possible due to the noise &nbsp;$n(t)$&nbsp; that is always present.
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 The diagram illustrates the situation. &nbsp;$K$&nbsp; signals are to be transmitted simultaneously via a physical channel of appropriate bandwidth.&nbsp;  The subchannels &nbsp; (German:&nbsp; "Teilkanäle" &nbsp; &rArr;  "T")&nbsp; are marked here with &nbsp;$T_1$, ... , $T_K$&nbsp;.&nbsp;  One proceeds as follows:
 *One modulates the source signals &nbsp;$q_1(t)$, &nbsp;$q_2(t)$, ... , &nbsp;$q_k(t)$, ... , &nbsp;$q_K(t)$&nbsp; of users with different carrier frequencies&nbsp; $f_1$, &nbsp;$f_2$, ... , &nbsp;$f_K$.
-*One combines the transmission signals &nbsp;$s_1(t)$, &nbsp;$s_2(t)$, ... , &nbsp;$s_k(t)$, ... , &nbsp;$s_K(t)$&nbsp; to form a total signal 𝑠(𝑡),&nbsp; enabling multiple use of the transmission set-up.
+*One combines the transmitted signals &nbsp;$s_1(t)$, &nbsp;$s_2(t)$, ... , &nbsp;$s_k(t)$, ... , &nbsp;$s_K(t)$&nbsp; to form a total signal 𝑠(𝑡),&nbsp; enabling multiple use of the transmission set-up.
 *To demodulate the signal &nbsp;$s_k(t)$,&nbsp; one uses the special carrier frequency &nbsp;$f_k$.&nbsp; Subsequent filtering achieves &nbsp; $v_k(t) = q_k(t)$,&nbsp; but only with negligible noise.&nbsp;   This process is called&nbsp; &raquo;'''channel separation'''&laquo;.

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 # &raquo;linear" and&nbsp; &raquo;non-linear distortions&laquo;&nbsp; due to modulation/demodulation,
 # degradation in the presence of&nbsp; &raquo;stochastic interference&laquo;,&nbsp; such as&nbsp; &raquo;noise&laquo;,
-# a&nbsp; &raquo;unified model&Laquo;&nbsp; for describing of amplitude and angle modulation,
+# a&nbsp; &raquo;unified model&laquo;&nbsp; for describing of amplitude and angle modulation,
-# descriptions using the&nbsp; &raquo;analytical signal"&nbsp; and the&nbsp; "equivalent low&ndash;pass signal".
+# descriptions using the&nbsp; &raquo;analytical signal&laquo;&nbsp; and the&nbsp; &raquo;equivalent low&ndash;pass signal&laquo;.
 ==The communication system under consideration ==

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 This chapter deals in detail with:
-* the&nbsp; "objectives"&nbsp; of modulation and demodulation,
+# the&nbsp; &raquo;objectives&laquo;&nbsp; of modulation and demodulation,
-* the&nbsp; "differences and similarities"&nbsp; between analog and digital modulation techniques,
+# the&nbsp; &raquo;differences and similarities&laquo;&nbsp; between analog and digital modulation techniques,
-* the&nbsp; "signal-to-noise power ratio"&nbsp; as a very general quality criterion,
+# the&nbsp; &raquo;signal-to-noise power ratio&laquo;&nbsp; as a very general quality criterion,
-* "linear" and&nbsp; "non-linear distortions"&nbsp; due to modulation/demodulation,
+# &raquo;linear" and&nbsp; &raquo;non-linear distortions&laquo;&nbsp; due to modulation/demodulation,
-* degradation in the presence of&nbsp; "stochastic interference",&nbsp; such as&nbsp; "noise",
+# degradation in the presence of&nbsp; &raquo;stochastic interference&laquo;,&nbsp; such as&nbsp; &raquo;noise&laquo;,
-* a&nbsp; "unified model"&nbsp; for describing of amplitude and angle modulation,
+# a&nbsp; &raquo;unified model&Laquo;&nbsp; for describing of amplitude and angle modulation,
-* descriptions using the&nbsp; "analytical signal"&nbsp; and the&nbsp; "equivalent low&ndash;pass signal".
+# descriptions using the&nbsp; &raquo;analytical signal"&nbsp; and the&nbsp; "equivalent low&ndash;pass signal".
 ==The communication system under consideration ==

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 [[File:EN_Mod_T_1_1_S4b_neu.png|right|frame|Example signals for analog and digital amplitude modulation]]
-*In the analog transmission system (above), the information about the analog source signal &nbsp;$q(t)$&nbsp; is directly in the amplitude (envelope) of the modulated signal &nbsp;$s(t)$.&nbsp; This is the analog modulation method [[Modulation_Methods/Double-Sideband_Amplitude_Modulation#Double-Sideband_Amplitude_Modulation_with_carrier|$\text{double-sideband amplitude modulation&nbsp; $\text{(DSB&ndash;AM)}$&nbsp; (with carrier)}$]].
+*In the analog transmission system (above), the information about the analog source signal &nbsp;$q(t)$&nbsp; is directly in the amplitude (envelope) of the modulated signal &nbsp;$s(t)$.&nbsp; This is the analog modulation method&nbsp; [[Modulation_Methods/Double-Sideband_Amplitude_Modulation#Double-Sideband_Amplitude_Modulation_with_carrier|"double-sideband amplitude modulation with carrier"]] &nbsp; &rArr; &nbsp; $\text{(DSB-AM)}$.
 *The modulated signal &nbsp;$s(t)$&nbsp; shows that the modulator also has similar functionality in the digital transmission system as in the&nbsp; (above)&nbsp; analog transmission system.}}

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 <br>The two block diagrams illustrate the main differences between an analog and a digital transmission system.&nbsp; It becomes clear that:
-*In analog modulation, the modulating '''KORREKTUR: source?''' signal &nbsp;$q(t)$&nbsp; is always an analog signal and thus both continuous in value and in time.
+*In analog modulation,&nbsp; the modulating source signal &nbsp;$q(t)$&nbsp; is always an analog signal and thus both continuous in value and in time.
 *In contrast, in digital modulation, the modulator's input signal &nbsp;$q_{\rm D}(t)$&nbsp; is always digital, thus both discrete in value and discrete in time.

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 *The user signals&nbsp; $q_1(t)$,&nbsp; $q_2(t)$&nbsp; and&nbsp; $q_3(t)$&nbsp; are binary and are fully characterized by amplitude coefficients&nbsp; $(0$&nbsp; or&nbsp; $1)$. &nbsp; A&nbsp; [[Signal_Representation/Time_Discrete_Signal_Representation|$\text{time-discrete signal representation}$]]&nbsp; is present &nbsp; $($symbol duration &nbsp;$T = 1\ \rm &micro; s)$.
-* For the bit rates of the first two signals, &nbsp;$R_1 = R_2 = 1/T = \text{1 Mbit/s}$.&nbsp; applies in each case.&nbsp;  In contrast,&nbsp; the bit rate of &nbsp;$q_3(t)$&nbsp; is twice as large, i.e. &nbsp; $R_3 =  \text{2 Mbit/s}$.
+* For the bit rates of the first two signals, &nbsp;$R_1 = R_2 = 1/T = \text{1 Mbit/s}$&nbsp; applies in each case.&nbsp;  In contrast,&nbsp; the bit rate of &nbsp;$q_3(t)$&nbsp; is twice as large, i.e. &nbsp; $R_3 =  \text{2 Mbit/s}$.
 *The common time-division multiplexed output signal &nbsp;$q(t)$&nbsp; is shown below.&nbsp;  The reference to the input signals is color-coded.&nbsp; The total bit rate is &nbsp;$R = R_1 + R_2 + R_3 = \text{4 Mbit/s}$.
 *After transmitting &nbsp;$q(t)$&nbsp; via the physical channel,&nbsp; the partial signals &nbsp;$v_1(t)$, ... ,&nbsp; $v_3(t)$&nbsp; are separated again at the receiver.&nbsp;  This functional unit is called&nbsp; "demultiplexer".

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 *In contrast, in digital modulation, the modulator's input signal &nbsp;$q_{\rm D}(t)$&nbsp; is always digital, thus both discrete in value and discrete in time.
-*When digitally modulating an analog audio or video signal &nbsp;$q(t)$,&nbsp; it must first be&nbsp; $\rm A/D$&nbsp; converted:  &nbsp; $q(t) \ \rightarrow \ q_{\rm D}(t)$.&nbsp; This is referred to as&nbsp; [[Modulation_Methods/Pulse_Code_Modulation|$\text{pulse code modulation}$]].&nbsp; This requires the following actions on the transmission side: &nbsp;[[Modulation_Methods/Pulse_Code_Modulation#Abtastung_und_Signalrekonstruktion|$\text{sampling}$]]&nbsp; –&nbsp; [[Modulation_Methods/Pulse_Code_Modulation#Quantisierung_und_Quantisierungsrauschen|$\text{quantization}$]]&nbsp; –&nbsp; [[Modulation_Methods/Pulse_Code_Modulation#PCM.E2.80.93Codierung_und_.E2.80.93Decodierung|$\text{PCM encoding}$]].
+*When digitally modulating an analog audio or video signal &nbsp;$q(t)$,&nbsp; it must first be&nbsp; $\rm A/D$&nbsp; converted:  &nbsp; $q(t) \ \rightarrow \ q_{\rm D}(t)$.&nbsp; This is referred to as&nbsp; [[Modulation_Methods/Pulse_Code_Modulation|$\text{pulse code modulation}$]].&nbsp; This requires the following actions on the transmission side: &nbsp;[[Modulation_Methods/Pulse_Code_Modulation#Sampling_and_signal_reconstruction|$\text{sampling}$]]&nbsp; –&nbsp; [[Modulation_Methods/Pulse_Code_Modulation#Quantization_and_quantization_noise|$\text{quantization}$]]&nbsp; –&nbsp; [[Modulation_Methods/Pulse_Code_Modulation#PCM_encoding_and_decoding|$\text{PCM encoding}$]].
 *Functionally, the modulator of the digital system&nbsp; $\rm (Mod)$&nbsp; does not differ from the modulator of the analog transmission system.&nbsp; But  the two demodulators differ in principle:

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 <br>
 The primary task of modulation&nbsp; (in the sense meant here)&nbsp; is adding a higher frequency carrier signal with carrier frequency &nbsp;$f_{\rm T}$&nbsp; to shift the source signal to a different frequency position with
-*a more favourable channel frequency response&nbsp; $H_{\rm K}(f)$&nbsp; and/or
+*a more favorable channel frequency response&nbsp; $H_{\rm K}(f)$&nbsp; and/or
-*a more favourable noise power-spectral density&nbsp; ${\it Φ}_n(f)$.
+*a more favorable noise power-spectral density&nbsp; ${\it Φ}_n(f)$.
@@ Line 60: / Line 60: @@
   ]]
-*The attenuation curve &nbsp;$a_{\rm K}(f) = \ –\ln \ \vert H_{\rm K}(f) \vert $&nbsp; of the channel is plotted in green,&nbsp; which here shows favourable characteristics with constant low attenuation in a sufficiently large frequency range.
+*The attenuation curve &nbsp;$a_{\rm K}(f) = \ –\ln \ \vert H_{\rm K}(f) \vert $&nbsp; of the channel is plotted in green,&nbsp; which here shows favorable characteristics with constant low attenuation in a sufficiently large frequency range.
 *In dark yellow,&nbsp; you can see the noise power-spectral density&nbsp;${\it Φ}_n(f)$,&nbsp; which does not disappear throughout the frequency range due to thermal noise and,&nbsp; in our constructed example,&nbsp; takes on particularly large values around the (center) frequency&nbsp;$f_{\rm N}$&nbsp; due to external interferences.
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 *The transmission quality is usually very good,&nbsp; since noise is only noticeable if it is greater than a predefined threshold.
 *Because signals can be regenerated at regular intervals by so-called regenerators,&nbsp; very large distances can be bridged with sufficiently good transmission quality.
-*Data transmission&nbsp; – e.g. between server and client –&nbsp; is best done digitally,&nbsp; since every data signal is already digital.&nbsp; Analog signals must first be digitalised.
+*Data transmission&nbsp; – e.g. between server and client –&nbsp; is best done digitally,&nbsp; since every data signal is already digital.&nbsp; Analog signals must first be digitalized.
 *Uniform transmission of voice, image and data signals makes it possible to build a shared,&nbsp; high-performance network for many telecommunications services.
 *Simple and highly efficient encryption and data security mechanisms exist for digital signals,&nbsp; which is an important prerequisite for safety-critical applications.