View source for Modulation Methods/Further AM Variants

 
{{Header
|Untermenü=Amplitude Modulation and Demodulation
|Vorherige Seite=Single-Sideband Modulation
|Nächste Seite=Phase Modulation (PM)
}}
==Vestigial sideband amplitude modulation==
<br>
When transmitting signals using single-sideband modulation&nbsp; $\rm (SSB–AM)$&nbsp; the following problems occur: 
*To suppress the unwanted sideband,&nbsp; a filter with a very high edge slope must be used.
*Such steep-edged filters exhibit strong group delay distortions,&nbsp; especially at the limit of the passband. 


[[File:EN_Mod_T_2_5_S1.png|right|frame|Spectrum&nbsp; (of the analytical signal)&nbsp; for vestigial sideband amplitude modulation]]
The problem can be greatly mitigated if instead of single-sideband AM one uses &nbsp; "vestigial sideband amplitude modulation"&nbsp; $\rm (VSB–AM)$,&nbsp; as shown in the adjacent graph. 

The present description is based on the textbook &nbsp; [Mäu88]<ref>Mäusl, R.:&nbsp; Analoge Modulationsverfahren.&nbsp;  Heidelberg: Dr. Hüthig, 1988.</ref>.&nbsp;  According to it,&nbsp; the VSB-AM can be characterized as follows: 
*A certain frequency range of the actually suppressed sideband – in the considered example of the LSB – is additionally used with a relatively flat decreasing transfer function.
*On the receiver side,&nbsp; a  selection curve linearly increasing in frequency with a so-called&nbsp; "Nyquist edge"&nbsp; is used in the transition range from the suppressed sideband to the transmitted sideband.
*The demodulation performs a convolution of the sidebands around the carrier,&nbsp; so that as a result the message content of a band with the same amplitude for all frequencies is obtained.


{{GraueBox|TEXT=
$\text{Example 1:}$&nbsp; The vestigial sideband method is used for&nbsp; (analog)&nbsp; color television,&nbsp; whose frequency spectrum according to the CCIR standard is shown in the graphic.&nbsp; The frequencies given refer to the &nbsp;[https://en.wikipedia.org/wiki/PAL PAL–B/G Television format]&nbsp; used in Germany. 

[[File:EN_Mod_T_2_5_S1b.png|right|frame|Illustrating the Nyquist edge for PAL–B/G Television]]
In this schematic representation,&nbsp; one recognises: 
*The radiated spectrum&nbsp; (only positive frequencies are drawn)&nbsp; ranges from &nbsp;$f_{\rm T} - 1.25 \ \rm MHz$&nbsp; to &nbsp;$f_{\rm T} + 5.75 \ \rm MHz$.&nbsp; The lower vestigial sideband including Nyquist edge is &nbsp;$\approx 1.25 \ \rm MHz$&nbsp; wide. 
*The green dashed line shows the receiver passband.&nbsp; The image carrier&nbsp; '''(B)'''&nbsp; &ndash; abbreviation&nbsp; "B"&nbsp; from German "Bild" &nbsp; &rArr; &nbsp;  "image") &ndash; &nbsp; at carrier frequency  &nbsp;$f_{\rm T}$&nbsp; is centered on the Nyquist edge. 
*The luminance signal &nbsp; '''(L)'''&nbsp; goes up to about &nbsp;$5 \ \rm MHz$.&nbsp;  It contains the information for the image brightness and for the color&nbsp; "green".
*The chrominance signal &nbsp; '''(C)'''&nbsp; is embedded in the upper part.  Two orthogonal carriers are [[Modulation_Methods/Further_AM_Variants#Quadrature_Amplitude_Modulation_.28QAM.29|QAM-modulated]]&nbsp; at &nbsp;$4.43 \ \rm MHz$&nbsp; for&nbsp; "red"&nbsp; and&nbsp; "blue";&nbsp; the carrier is suppressed.
*The audio carrier &nbsp; '''(T)'''&nbsp; &ndash; abbreviation&nbsp; "T"&nbsp; from German "Ton" &nbsp; &rArr; &nbsp; "sound/audio") &ndash; &nbsp; is at &nbsp;$f_{\rm T} + 5.5 \ \rm MHz$&nbsp; and is &nbsp;$12  \ \rm  dB$&nbsp; lower than the image carrier.  
*If there is a two-channel sound&nbsp; (stereo)&nbsp; transmission &nbsp; &rArr; &nbsp; a second &nbsp; '''(T)'''&nbsp; carrier at &nbsp;$5.75 \ \rm MHz$.}}


==Quadrature Amplitude Modulation (QAM)==
<br>
By exploiting the orthogonality of cosine and sine functions,&nbsp; a channel can be used twice for simultaneous transmission of two source signals &nbsp;$q_1(t)$&nbsp; and &nbsp;$q_2(t)$&nbsp; without mutual interference.&nbsp; This method is called&nbsp; "quadrature amplitude modulation"&nbsp; $\rm (QAM)$.

[[File:P_ID1053__Mod_T_2_5_S2_neu.png |right|frame|Model of quadrature amplitude modulation]]

The QAM system has the following characteristics: 
*The transmitted signal is composed of two mutually orthogonal components: 
:$$s(t) = q_1(t)  \cdot \cos (\omega_{\rm T}\cdot t) - q_2(t)  \cdot \sin (\omega_{\rm T}\cdot t)\hspace{0.05cm}.$$
*For frequency and phase synchronous demodulation,&nbsp; the signal in the upper branch before the low-pass &nbsp;$H_{\rm E1}(f)$&nbsp;  is:
:$$b_1(t) = q_1(t)  \cdot 2 \cdot \cos^2 (\omega_{\rm T}\cdot t)  - q_2(t)  \cdot 2 \cdot
 \cos (\omega_{\rm T}\cdot t)\cdot \sin (\omega_{\rm T}\cdot t)  $$
:$$\Rightarrow \hspace{0.3cm} b_1(t) =  q_1(t)\cdot \big[ 1 + \cos (2 \omega_{\rm T}\cdot t) \big] - q_2(t)\cdot \sin (2 \omega_{\rm T}\cdot t) \hspace{0.05cm}.$$
*Thus,&nbsp; by limiting to frequencies &nbsp;$|f| < f_{\rm T}$,&nbsp; we obtain in the upper and lower branches,&nbsp; resp.:
:$$v_1(t) = q_1(t),$$
:$$v_2(t) = q_2(t)\hspace{0.05cm}.$$
*When there is a phase offset &nbsp;$Δϕ_{\rm T}$&nbsp; between the transmitted and received carrier signals,&nbsp; in addition to attenuation of the intended participant,<br>crosstalk from the second participant occurs,&nbsp; resulting in nonlinear distortion:
:$$v_1(t) = \alpha_{11} \cdot q_1(t)+ \alpha_{12} \cdot q_2(t) \hspace{0.05cm}, $$
:$$v_2(t) = \alpha_{21} \cdot q_1(t)+ \alpha_{22} \cdot q_2(t)$$
:$$\Rightarrow\hspace{0.3cm}\alpha_{11} =  \alpha_{22} = \cos(\Delta \phi_{\rm T}) \hspace{0.05cm},  \hspace{0.3cm} \alpha_{12} =  -\alpha_{21} = \sin(\Delta \phi_{\rm T}) \hspace{0.05cm}.$$

==Incoherent (non-coherent) Demodulation==
<br>
{{BlaueBox|TEXT=
$\text{Definition:}$&nbsp; Demodulators can be classified in the following way: 
*A demodulator is said to be&nbsp; '''coherent''',&nbsp; if in addition to the required frequency synchronization,&nbsp; it requires accurate information about the phase of the transmit-side carrier signal &nbsp;$z(t)$&nbsp; to reconstruct the source signal.

*If this phase information is not required,&nbsp; the demodulator is said to be an&nbsp;  '''incoherent demodulator'''&nbsp; or&nbsp;  "non-coherent demodulator".}} 


An example of an incoherent demodulator is the&nbsp; [[Modulation_Methods/Envelope_Demodulation|envelope demodulator]].

{{GraueBox|TEXT=
$\text{Example 2:}$&nbsp; A second example is shown in the following block diagram.&nbsp; In contrast to quadrature amplitude modulation,&nbsp;  here the orthogonality between cosine and sine functions is not used for the simultaneous transmission of a second source signal,&nbsp; but rather to simplify the receiver-side device.  

[[File:P_ID1054__Mod_T_2_5_S3_neu.png |right|frame|Incoherent demodulation with&nbsp; $\rm DSB-AM$]]

It should be further noted regarding this arrangement:
*The receiver-side carrier signals can have an arbitrary and also time-dependent phase offset &nbsp;$Δϕ_{\rm T}$&nbsp; with respect to the carrier signals at the transmitter, as long as the phase difference between the two branches remains exactly &nbsp;$90^\circ$&nbsp;.
*For the signals in the upper and lower branches – after the multiplier and low-pass filtering, respectively – it holds:
:$$b_1(t) = \cos(\Delta \phi_{\rm T}) \cdot q(t), $$
:$$b_2(t) = -\sin(\Delta \phi_{\rm T}) \cdot q(t).$$
*This ensures that the sink signal &nbsp;$v(t)$&nbsp; matches the source signal &nbsp;$q(t)$&nbsp; in magnitude,&nbsp; regardless of the phase offset &nbsp;$Δϕ_{\rm T}$:
:$$v(t) = \sqrt{ b_1^2(t) + b_2^2(t)} = \sqrt{ q^2(t) } = \vert q(t) \vert \hspace{0.05cm}.$$
*A prerequisite for operability&nbsp; &ndash;  that is,&nbsp; for the result &nbsp;$v(t) = q(t)$&nbsp; &ndash;&nbsp; is that at all times &nbsp;$q(t) ≥ 0$.&nbsp;  In an analog transmission system,&nbsp; this fact could be enforced using the&nbsp; "DSB-AM with carrier"&nbsp; modulation method,&nbsp; for example.
*This form of non-coherent demodulation&nbsp; (or modifications of it)&nbsp; is mainly applied in some&nbsp; '''digital modulation methods''',&nbsp; which are discussed in detail in the fourth chapter of&nbsp; [[Modulation_Methods|this book]].}} 

==Exercises for the chapter==
<br>
[[Aufgaben:Exercise_2.12:_Non-coherent_Demodulation|Exercise 2.12: Non-coherent Demodulation]]

[[Aufgaben:Exercise_2.13:_Quadrature_Amplitude_Modulation|Exercise 2.13: Quadrature Amplitude Modulation (QAM)]]


==References==
<references/>

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