Difference between revisions of "Modulation Methods/Synchronous Demodulation"

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{{Header
 
{{Header
 
|Untermenü=Amplitude Modulation and Demodulation
 
|Untermenü=Amplitude Modulation and Demodulation
|Vorherige Seite=Zweiseitenband-Amplitudenmodulation
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|Vorherige Seite=Double-Sideband Amplitude Modulation
|Nächste Seite=Hüllkurvendemodulation
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|Nächste Seite=Envelope Demodulation
 
}}
 
}}
 
==Block diagram and time domain representation==
 
==Block diagram and time domain representation==
 
<br>
 
<br>
Modulation at the transmitter only makes sense, if it is possible to reverse this signal conversion at the receiver end, ideally without loss of information. In any form of amplitude modulation - be it double sideband (DSB) or single sideband (SSB), with or without a carrier signal - the so-called synchronous demodulator fulfils this task.
+
Modulation at the transmitter only makes sense,&nbsp;  if it is possible to reverse this signal conversion at the receiver end,&nbsp; ideally without loss of information.&nbsp; In any form of amplitude modulation&nbsp; - be it double sideband (DSB) or single sideband (SSB),&nbsp; with or without a carrier signal -&nbsp; the so-called&nbsp; &raquo;'''synchronous demodulator'''&laquo;&nbsp; fulfils this task.
  
[[File:EN_Mod_T_2_2_S1.png|center|frame| DSB–Amplitudenmodulation and synchronous demodulation]]
+
[[File:EN_Mod_T_2_2_S1.png|rirgt|frame| DSB amplitude modulation and synchronous demodulation]]
  
The following should be noted with regard to the above block diagram:
+
The following should be noted with regard to the block diagram on the right:
*For modulation, we will focus on "DSB–AM without a carrier"&nbsp; $($modulation depth&nbsp; $m → ∞)$&nbsp;.&nbsp; However, synchronous demodulation is also applicable in "DSB–AM with carrier".  
+
*For modulation,&nbsp; we will focus on&nbsp; "DSB–AM without a carrier"&nbsp; $($modulation depth&nbsp; $m → ∞)$.&nbsp; However,&nbsp; synchronous demodulation is also applicable in&nbsp; "DSB–AM with carrier".
*Let the channel be ideal and the distortions negligible,  that the received signal &nbsp;$r(t)$&nbsp; is identical to the transmitted signal &nbsp;$s(t)$&nbsp;:   
+
 +
*Let the channel be ideal and the distortions negligible,&nbsp; that the received signal&nbsp; $r(t)$&nbsp; is identical to the transmitted signal &nbsp;$s(t)$:   
 
:$$r(t) = s(t) = q(t) \cdot \cos(\omega_{\rm T} \cdot t + \phi_{\rm T})\hspace{0.05cm}.$$
 
:$$r(t) = s(t) = q(t) \cdot \cos(\omega_{\rm T} \cdot t + \phi_{\rm T})\hspace{0.05cm}.$$
*At the receiver, &nbsp;$r(t)$&nbsp; is first multiplied by the receiver-side carrier &nbsp;$z_{\rm E}(t)$&nbsp;, which is identical to the transmitter-side carrier &nbsp;$z(t)$&nbsp; but larger by a factor of &nbsp; $2$&nbsp;:
+
 
 +
*At the receiver, &nbsp;$r(t)$&nbsp; is first multiplied by the receiver-side carrier &nbsp;$z_{\rm E}(t)$,&nbsp; which is identical shaped to the transmitter-side carrier &nbsp;$z(t)$&nbsp; but larger by a factor of&nbsp; $2$:
 
:$$z_{\rm E}(t) = 2 \cdot \cos(\omega_{\rm T} \cdot t + \phi_{\rm T})\hspace{0.05cm}.$$
 
:$$z_{\rm E}(t) = 2 \cdot \cos(\omega_{\rm T} \cdot t + \phi_{\rm T})\hspace{0.05cm}.$$
*DThe result of the multiplication is the signal &nbsp;$b(t)$.&nbsp;. Considering the trigonometric transformation &nbsp;$\cos^2(α) = 1/2 · \big [1 + \cos(2α)\big ]$&nbsp; we get
+
 
 +
*The result of the multiplication is the signal &nbsp;$b(t)$. &nbsp;  Considering the trigonometric transformation &nbsp; $\cos^2(α) = 1/2 · \big [1 + \cos(2α)\big ]$ &nbsp; we get
 
:$$b(t)  = r(t) \cdot z_{\rm E}(t) = 2 \cdot q(t) \cdot \cos^2(\omega_{\rm T} \cdot t + \phi_{\rm T}) = q(t) + q(t) \cdot \cos(2 \cdot \omega_{\rm T} \cdot t + 2\cdot \phi_{\rm T})\hspace{0.05cm}.$$
 
:$$b(t)  = r(t) \cdot z_{\rm E}(t) = 2 \cdot q(t) \cdot \cos^2(\omega_{\rm T} \cdot t + \phi_{\rm T}) = q(t) + q(t) \cdot \cos(2 \cdot \omega_{\rm T} \cdot t + 2\cdot \phi_{\rm T})\hspace{0.05cm}.$$
*The second term is in the range around twice the carrier frequency.  If the signal bandwidth is &nbsp;$B_{\rm NF} < f_{\rm T}$, which is always true in practice, this component can be suppressed by a low-pass filter &nbsp;$H_{\rm E}(f)$&nbsp; with suitable dimensions, and we obtain &nbsp;$v(t) = q(t)$.
+
 
 +
*The second term is in the range around twice the carrier frequency&nbsp; $f_{\rm T}$.&nbsp; If the signal bandwidth is &nbsp;$B_{\rm NF} < f_{\rm T}$,&nbsp; which is always true in practice,&nbsp; this component can be suppressed by a low-pass filter &nbsp;$H_{\rm E}(f)$&nbsp; with suitable dimensions,&nbsp; and we obtain &nbsp;
 +
:$$v(t) = q(t).$$
  
 
==Description in the frequency domain==
 
==Description in the frequency domain==
 
<br>
 
<br>
Assuming an even source signal &nbsp;$q(t)$  &nbsp; ⇒ &nbsp;  with a real spectrum &nbsp;$Q(f)$&nbsp; and a sinusoidal carrier &nbsp;$z(t)$&nbsp;, the imaginary transmitted spectrum &nbsp;$S(f)$&nbsp; is obtained according to the second plot, where &nbsp;$A_{\rm T} ≠ 0$&nbsp; also takes into account the DSB-AM with a carrier (shown by the red Dirac function). Because the channel is ideal, &nbsp;$R(f) = S(f)$.
+
Assuming an even source signal &nbsp;$q(t)$  &nbsp; ⇒ &nbsp;  with a real spectrum &nbsp;$Q(f)$&nbsp; and a sinusoidal carrier &nbsp;$z(t)$,&nbsp; the imaginary transmitted spectrum &nbsp;$S(f)$&nbsp; is obtained according to the second plot,&nbsp; where &nbsp;$A_{\rm T} ≠ 0$&nbsp; also takes into account the DSB-AM with a carrier&nbsp; (shown by the red Dirac delta function).&nbsp; Because the channel is ideal,&nbsp; $R(f) = S(f)$.
  
[[File:EN_Mod_T_2_2_S2.png|right|frame| Illustration of synchronous demodulation in the frequency domain]]
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[[File:EN_Mod_T_2_2_S2.png|right|frame| Illustration of synchronous demodulation in the frequency domain<br>$B(f)$&nbsp; according to the sketch below,&nbsp; but with shaded portions]]
  
 
The operation of the synchronous demodulator can be explained in the frequency domain as follows:  
 
The operation of the synchronous demodulator can be explained in the frequency domain as follows:  
*The receiver-side carrier signal&nbsp;$z_{\rm E}(t) = 2 · z(t) = 2 · \sin(ω_{\rm T} · t)$&nbsp; leads to two Dirac functions in the spectral domain at &nbsp;$\pm f_{\rm T}$&nbsp; with weights &nbsp;$\pm \rm j$.&nbsp; The negative imaginary part occurs at &nbsp;$f = +f_{\rm T}$&nbsp;.  
+
*The receiver-side carrier signal&nbsp;$z_{\rm E}(t) = 2 · z(t) = 2 · \sin(ω_{\rm T} · t)$&nbsp; leads to two Dirac delta functions in the spectral domain at &nbsp;$\pm f_{\rm T}$&nbsp; with weights &nbsp;$\pm \rm j$.&nbsp; The negative imaginary part occurs at &nbsp;$f = +f_{\rm T}$.  
  
*The multiplication &nbsp;$b(t) = r(t) · z_{\rm E}(t)$&nbsp; corresponds to the convolution of the associated spectral functions:  
+
*The multiplication &nbsp;$b(t) = r(t) · z_{\rm E}(t)$&nbsp; corresponds to the convolution of the associated spectral functions.&nbsp; The result is shown in the sketch below,&nbsp; if the influence of the subsequent low-pass&nbsp; (hatching)&nbsp; is disregarded:
 
:$$B(f) = R(f) \star Z_{\rm E}(f)\hspace{0.05cm}.$$
 
:$$B(f) = R(f) \star Z_{\rm E}(f)\hspace{0.05cm}.$$
  
*Convolution of the Dirac function &nbsp;$ - {\rm j} \cdot δ(f – f_{\rm T})$&nbsp; with the wholly imaginary spectrum &nbsp;$R(f)$&nbsp; leads to completely real spectral components around &nbsp;$f = 0$&nbsp; and &nbsp;$f = 2f_{\rm T}$.&nbsp; These components are marked with a "+" on the graph.
+
*Convolution of the Dirac delta function &nbsp;$ - {\rm j} \cdot δ(f – f_{\rm T})$&nbsp; with the purely imaginary spectrum &nbsp;$R(f)$&nbsp; leads to completely real spectral components around &nbsp;$f = 0$&nbsp; and &nbsp;$f = 2f_{\rm T}$.&nbsp; These components are marked with a&nbsp;  "plus"&nbsp; on the lower graph.
  
*The second convolution product &nbsp;${\rm j} · δ(f + f_{\rm T}) \star R(f)$&nbsp; yields a low-frequency spectral component around&nbsp;$f = 0$.&nbsp; in addition to a component at &nbsp;$–2f_{\rm T}$&nbsp;. These spectral components are marked with "".
+
*The second convolution product &nbsp;${\rm j} · δ(f + f_{\rm T}) \star R(f)$&nbsp; yields a low-frequency spectral component around&nbsp;$f = 0$&nbsp; in addition to a component at &nbsp;$–2f_{\rm T}$.&nbsp; These spectral components are marked with&nbsp; "minus".
  
*The spectrum after the low-pass &nbsp;$H_{\rm E}(f)$&nbsp; is &nbsp;$V(f) = Q(f) + A_{\rm T} · δ(f)$.&nbsp; For DSB-AM with a carrier, the interfering DC component can be removed by a band limit on the low end, i.e. &nbsp;$H_{\rm E}(f = 0) = 0$.
+
*The spectrum after the low-pass &nbsp;$H_{\rm E}(f)$&nbsp; is &nbsp;$V(f) = Q(f) + A_{\rm T} · δ(f)$.&nbsp; For DSB-AM with a carrier, the interfering DC component can be removed by a lower band limit:&nbsp;  
 +
:$$H_{\rm E}(f = 0) = 0.$$
  
 
+
*The color assignment in the diagram&nbsp; (USB in blue, LSB in green, carrier in red) shows that the synchronous demodulator uses both the&nbsp; $\rm USB$&nbsp; and the&nbsp; $\rm LSB$&nbsp; for signal reconstruction.  
*The color assignment in the diagram (USB blue, LSB green, carrier red) shows that the synchronous demodulator uses both the USB and the LSB for signal reconstruction.  
 
 
<br clear=all>
 
<br clear=all>
 
==Prerequisites for the application of a synchronous demodulator==
 
==Prerequisites for the application of a synchronous demodulator==
 
<br>
 
<br>
The output signal &nbsp;$v(t)$&nbsp; is identical to the source signal  &nbsp;$q(t)$ if the following conditions are satisfied:
+
The output signal &nbsp;$v(t)$&nbsp; is identical to the source signal  &nbsp;$q(t)$&nbsp; if the following conditions are satisfied:
 
*The bandwidth &nbsp;$B_{\rm NF}$&nbsp; of the source signal is smaller than the carrier frequency &nbsp;$f_{\rm T}$.&nbsp; This restriction is not particularly drastic and is not relevant for practical applications.
 
*The bandwidth &nbsp;$B_{\rm NF}$&nbsp; of the source signal is smaller than the carrier frequency &nbsp;$f_{\rm T}$.&nbsp; This restriction is not particularly drastic and is not relevant for practical applications.
  
*The carrier frequencies of the transmitter and receiver are exactly matched.  This requires carrier recovery at the receiver and thus involves some "cost".  
+
*The carrier frequencies of the transmitter and receiver are exactly matched.&nbsp; This requires carrier recovery at the receiver and thus involves some&nbsp; "cost".  
  
*There is also perfect phase synchrony between the the carrier signals &nbsp;$z(t)$&nbsp; and &nbsp;$z_{\rm E}(t)$&nbsp; at the transmitter and receiver end, respectively.
+
*There is also perfect phase synchronicity between the the carrier signals &nbsp;$z(t)$&nbsp; and &nbsp;$z_{\rm E}(t)$&nbsp; at the transmitter and receiver end, respectively.
  
*The channel frequency response &nbsp;$H_{\rm K}(f)$&nbsp; is ideally equal to &nbsp; $1$.&nbsp; in the passband &nbsp;$f_{\rm T} - B_{\rm NF} ≤ |f| ≤ f_{\rm T} + B_{\rm NF}$&nbsp;. Frequency-independent attenuation or frequency-linear phase (delay) are usually tolerated.
+
*The channel frequency response &nbsp;$H_{\rm K}(f)$&nbsp; is ideally equal to &nbsp; $1$&nbsp; in the passband &nbsp;$f_{\rm T} - B_{\rm NF} ≤ |f| ≤ f_{\rm T} + B_{\rm NF}$.&nbsp; Frequency-independent attenuation or frequency-linear phase&nbsp; (delay)&nbsp; are usually tolerated.&nbsp; Note:&nbsp; The subscript&nbsp; "K"&nbsp; stands for German&nbsp; "Kanal"&nbsp; ⇒ &nbsp; channel).
  
*The influence of noise and external disturbances is assumed to be negligible in this model. However, even with non-negligible noise, the synchronous demodulator is superior to other demodulators.  
+
*The influence of noise and external disturbances is assumed to be negligible in this model.&nbsp; However,&nbsp; even with non-negligible noise,&nbsp; the synchronous demodulator is superior to other demodulators.  
  
*The receiver filter &nbsp;$H_{\rm E}(f)$&nbsp; (subscript E stands for German "Empfangsfilter" ⇒ receiver filter) is equal to "one" for &nbsp;$|f| ≤ B_{\rm NF}$&nbsp; and equal to "zero" for $|f| ≥ 2f_{\rm T} - B_{\rm NF}$.&nbsp; The process in between is not relevant (see graph in the&nbsp; [[Modulation_Methods/Synchronous_Demodulation#Description_in_the_frequency_domain|previous section]]).  
+
*The receiver filter &nbsp;$H_{\rm E}(f)$&nbsp;   is equal to&nbsp; "one"&nbsp; for &nbsp;$|f| ≤ B_{\rm NF}$&nbsp; and equal to&nbsp; "zero"&nbsp; for $|f| ≥ 2f_{\rm T} - B_{\rm NF}$.&nbsp; The range in between is not relevant&nbsp; (see graph in the&nbsp; [[Modulation_Methods/Synchronous_Demodulation#Description_in_the_frequency_domain|"previous section"]]).&nbsp; Note:&nbsp; The subscript&nbsp; "E"&nbsp; stands for German&nbsp; "Empfänger"&nbsp; ⇒ &nbsp; "receiver").
  
*BFor the modulation method "DSB-AM with carrier", &nbsp;$H_{\rm E}(f = 0) ≡ 0$&nbsp; must additionally ensure that the carrier added at the transmitter is no longer included in the sink signal.
+
*For the modulation method&nbsp; "DSB-AM with carrier",&nbsp; the condition&nbsp; $H_{\rm E}(f = 0) ≡ 0$&nbsp; must additionally ensure that the carrier added at the transmitter is no longer included in the sink signal.
  
The following sections describe the effects if some of the above conditions are not met.
+
*The following four sections describe the effects if some of the above conditions are not met.
  
==Influence of a frequency offset
+
==Influence of a frequency offset==
==
 
 
<br>
 
<br>
As the name&nbsp; "synchronous demodulator”&nbsp;  it only works when there is complete synchrony between the carrier signals of the transmitter and receiver.  On the other hand, if the carrier frequencies differ by a &nbsp; '''frequency offset '''&nbsp; $Δ\hspace{-0.05cm}f_{\rm T}$, for example
+
As the name&nbsp; "synchronous demodulator”&nbsp;  it only works when there is complete synchronization between the carrier signals of the transmitter and receiver.&nbsp;  
 +
 
 +
On the other hand,&nbsp; if the carrier frequencies differ by a &nbsp; &raquo;'''frequency offset '''&laquo;&nbsp; $Δ\hspace{-0.05cm}f_{\rm T}$,&nbsp; for example
 
:$$\begin{align*}z(t) & = 1 \cdot \cos(2 \pi f_{\rm T} \cdot t + \phi_{\rm T})\hspace{0.05cm},  \\ z_{\rm E}(t) & = 2 \cdot \cos(2 \pi (f_{\rm T} + \Delta\hspace{-0.05cm} f_{\rm T}) \cdot  t + \phi_{\rm T})\hspace{0.05cm},\end{align*}$$
 
:$$\begin{align*}z(t) & = 1 \cdot \cos(2 \pi f_{\rm T} \cdot t + \phi_{\rm T})\hspace{0.05cm},  \\ z_{\rm E}(t) & = 2 \cdot \cos(2 \pi (f_{\rm T} + \Delta\hspace{-0.05cm} f_{\rm T}) \cdot  t + \phi_{\rm T})\hspace{0.05cm},\end{align*}$$
then for the sink signal spectrum we obtain:  
+
then we obtain for the sink signal spectrum:  
 
:$$V(f)  = {1}/{2}\cdot Q(f + \Delta \hspace{-0.05cm}f_{\rm T}) + {1}/{2}\cdot Q(f - \Delta\hspace{-0.05cm} f_{\rm T}) =  Q(f) \star \big[ {1}/{2}\cdot \delta(f + \Delta\hspace{-0.05cm} f_{\rm T}) + {1}/{2}\cdot \delta (f - \Delta \hspace{-0.05cm}f_{\rm T}) \big] \hspace{0.05cm}.$$
 
:$$V(f)  = {1}/{2}\cdot Q(f + \Delta \hspace{-0.05cm}f_{\rm T}) + {1}/{2}\cdot Q(f - \Delta\hspace{-0.05cm} f_{\rm T}) =  Q(f) \star \big[ {1}/{2}\cdot \delta(f + \Delta\hspace{-0.05cm} f_{\rm T}) + {1}/{2}\cdot \delta (f - \Delta \hspace{-0.05cm}f_{\rm T}) \big] \hspace{0.05cm}.$$
  
This result can be verified in the frequency domain using the sketch on the &nbsp;[[Modulation_Methods/Synchronous_Demodulation#Description_in_the_frequency_domain|Description]]&nbsp; page. After transforming the equation to the time domain, we get:  
+
This result can be verified in the frequency domain using the sketch in the &nbsp;[[Modulation_Methods/Synchronous_Demodulation#Description_in_the_frequency_domain|$\text{frequency description}$]]&nbsp; section.&nbsp; After transforming the equation to the time domain,&nbsp; we get:  
 
:$$v(t) = q(t) \cdot \cos(2 \pi \cdot \Delta \hspace{-0.05cm}f_{\rm T} \cdot t )\hspace{0.05cm}.$$
 
:$$v(t) = q(t) \cdot \cos(2 \pi \cdot \Delta \hspace{-0.05cm}f_{\rm T} \cdot t )\hspace{0.05cm}.$$
  
 
{{BlaueBox|TEXT=
 
{{BlaueBox|TEXT=
$\text{Conclusion:}$&nbsp;  for DSB-AM (with or without carrier), '''synchronous demodulation with frequency offset'''&nbsp; $ \Delta \hspace{-0.05cm}f_{\rm T} \cdot t )$ leads to &nbsp; [[Linear_and_Time_Invariant_Systems/Linear_and_Time_Invariant_Systems/Linear_Distortions#Attenuation_distortions|attenuation distortions]], characterized by the ''time-dependent factor'' &nbsp; $\cos(2 \pi \cdot \Delta \hspace{-0.05cm}f_{\rm T} \cdot t )$.  
+
$\text{Conclusions:}$&nbsp;   
 +
*For DSB-AM&nbsp; (with or without carrier),&nbsp; &raquo;'''synchronous demodulation with frequency offset'''&laquo;&nbsp; $ \Delta \hspace{-0.05cm}f_{\rm T}$ leads to&nbsp; [[Linear_and_Time_Invariant_Systems/Linear_Distortions#Attenuation_distortions|$\text{attenuation distortions}$]],&nbsp; characterized by the&nbsp; "time-dependent factor" &nbsp; $\cos(2 \pi \cdot \Delta \hspace{-0.05cm}f_{\rm T} \cdot t )$.  
  
Der Frequenzversatz&nbsp; $Δ\hspace{-0.05cm}f_{\rm T}$, der auf Realisierungsungenauigkeiten der Trägerrückgewinnung zurückgeht, ist meist sehr klein und bewegt sich im Bereich von einigen Hertz bis etwa&nbsp; $100\text{ Hz}$.&nbsp; In diesem Zusammenhang spricht man dann von einer&nbsp; "Schwebung". }}  
+
*The frequency offset &nbsp; $Δ\hspace{-0.05cm}f_{\rm T}$,&nbsp; which is due to&nbsp; &raquo;'''inaccuracies in the realization of carrier recovery'''&laquo;,&nbsp; is usually very small and ranges from a few Hertz to about &nbsp; $100\text{ Hz}$.&nbsp; In this context,&nbsp; one usually speaks of a&nbsp; "beat".}}  
  
  
[[File: P_ID1002__Mod_T_2_2_S4_neu.png |right|frame| Beeinträchtigung der Synchrondemodulation durch einen Frequenzversatz]]  
+
[[File: P_ID1002__Mod_T_2_2_S4_neu.png |right|frame| Impairment of synchronous demodulation due to a frequency offset]]  
  
 
{{GraueBox|TEXT=
 
{{GraueBox|TEXT=
$\text{Beispiel 1:}$&nbsp;  
+
$\text{Example 1:}$&nbsp;  
Die Grafik zeigt
+
The graph shows
*ein cosinusförmiges Quellensignal mit der Frequenz &nbsp;$f_{\rm N} = 1\ \rm  kHz$  &nbsp; <br>⇒  &nbsp; blaue Schwingungund
+
*a cosine-shaped source signal with frequency  &nbsp;$f_{\rm N} = 1\ \rm  kHz$  &nbsp; <br>⇒  &nbsp; blue oscillationand
*das mit einem Synchrondemodulator gewonnene Sinkensignal &nbsp;$v(t)$ &nbsp; <br>⇒  &nbsp;  rote Kurve.
+
 +
*the sink signal&nbsp; $v(t)$&nbsp; obtained with a synchronous demodulator <br>⇒  &nbsp;  red curve.
  
  
Hierbei wurde ein Frequenzversatz von &nbsp;$Δ\hspace{-0.05cm}f_{\rm T} = 100\ \rm  Hz$&nbsp; zugrundegelegt.&nbsp; Damit ergibt sich:  
+
Here,&nbsp; a frequency offset of  &nbsp;$Δ\hspace{-0.05cm}f_{\rm T} = 100\ \rm  Hz$&nbsp; was used as a basis.&nbsp; This results in:
 
:$$\begin{align*}v(t ) & = 1\,{\rm V} \cdot \cos (2 \pi \cdot 1\,{\rm kHz} \cdot t) \cdot \cos (2 \pi \cdot 0.1\,{\rm kHz} \cdot t)  =\\ &= 0.5\,{\rm V} \cdot \cos (2 \pi \cdot 0.9\,{\rm kHz} \cdot t) + 0.5\,{\rm V} \cdot \cos (2 \pi \cdot 1.1\,{\rm kHz} \cdot t) \hspace{0.05cm}.\end{align*}$$
 
:$$\begin{align*}v(t ) & = 1\,{\rm V} \cdot \cos (2 \pi \cdot 1\,{\rm kHz} \cdot t) \cdot \cos (2 \pi \cdot 0.1\,{\rm kHz} \cdot t)  =\\ &= 0.5\,{\rm V} \cdot \cos (2 \pi \cdot 0.9\,{\rm kHz} \cdot t) + 0.5\,{\rm V} \cdot \cos (2 \pi \cdot 1.1\,{\rm kHz} \cdot t) \hspace{0.05cm}.\end{align*}$$
  
Spektral gesehen werden aus der &nbsp;$1\ \rm  kHz$–Schwingung zwei überlagerte Schwingungen mit den Frequenzen &nbsp;$0.9\ \rm  kHz$&nbsp; und &nbsp;$1.1\ \rm  kHz$&nbsp; halber Amplitude.
 
*Es entstehen neue Frequenzen – also&nbsp; [[Linear_and_Time_Invariant_Systems/Nichtlineare_Verzerrungen|nichtlineare Verzerrungen]].&nbsp;
 
*Die gesendete Frequenz&nbsp; $(1\ \rm  kHz)$&nbsp; ist dagegen in &nbsp;$v(t)$&nbsp; nicht mehr enthalten. }}
 
  
==Einfluss eines Phasenversatzes==
+
From a spectral point of view, the &nbsp;$1\ \rm  kHz$ oscillation becomes two superimposed oscillations with frequencies &nbsp;$0.9\ \rm  kHz$&nbsp; and &nbsp;$1.1\ \rm  kHz$&nbsp; with half the amplitude.
 +
*New frequencies occur as a result &nbsp; &rArr; &nbsp; [[Linear_and_Time_Invariant_Systems/Nonlinear_Distortions|$\text{nonlinear distortions}$]].&nbsp;
 +
 +
*In contrast,&nbsp; the transmitted frequency&nbsp; $(1\ \rm  kHz)$&nbsp; is no longer included in &nbsp;$v(t)$&nbsp;. }}
 +
 
 +
==Influence of a phase offset==
 
<br>
 
<br>
Nun gelte für das sende– und für das empfängerseitige Trägersignal:
+
Now,&nbsp; for the transmitter-side and receiver-side carrier signal,&nbsp; it holds that
 
:$$\begin{align*}z(t) & = 1 \cdot \cos(2 \pi f_{\rm T} t + \phi_{\rm T})\hspace{0.05cm},  \\ z_{\rm E}(t) & = 2 \cdot \cos(2 \pi f_{\rm T}  t + \phi_{\rm E})\hspace{0.05cm}.\end{align*}$$
 
:$$\begin{align*}z(t) & = 1 \cdot \cos(2 \pi f_{\rm T} t + \phi_{\rm T})\hspace{0.05cm},  \\ z_{\rm E}(t) & = 2 \cdot \cos(2 \pi f_{\rm T}  t + \phi_{\rm E})\hspace{0.05cm}.\end{align*}$$
Damit erhält man für das Signal direkt nach der Multiplikation mit dem&nbsp; '''Phasenversatz''' &nbsp;$Δ \mathbf{ϕ_{\rm T} = ϕ_{\rm E} - ϕ_{\rm T} }$:  
+
For the signal immediately after multiplication by the&nbsp; &raquo;'''phase offset'''&laquo; &nbsp;$Δ \mathbf{ϕ_{\rm T} = ϕ_{\rm E} - ϕ_{\rm T} }$,&nbsp; this gives:
 
:$$b(t)  = q(t) \cdot \cos(\omega_{\rm T} \cdot t + \phi_{\rm T})  \cdot 2 \cdot \cos(\omega_{\rm T} \cdot t + \phi_{\rm E})=  q(t)\cdot \cos(\Delta \phi_{\rm T}) + q(t) \cdot \cos(2 \cdot \omega_{\rm T} \cdot t + \phi_{\rm E}+ \phi_{\rm T})\hspace{0.05cm}.$$
 
:$$b(t)  = q(t) \cdot \cos(\omega_{\rm T} \cdot t + \phi_{\rm T})  \cdot 2 \cdot \cos(\omega_{\rm T} \cdot t + \phi_{\rm E})=  q(t)\cdot \cos(\Delta \phi_{\rm T}) + q(t) \cdot \cos(2 \cdot \omega_{\rm T} \cdot t + \phi_{\rm E}+ \phi_{\rm T})\hspace{0.05cm}.$$
  
Unter Berücksichtigung des Tiefpassfilters ergibt sich somit für das Sinkensignal:
+
Thus,&nbsp; taking into account the low-pass filter,&nbsp; the result for the sink signal is:
 
:$$v(t) = q(t)\cdot \cos(\Delta \phi_{\rm T}) \hspace{0.05cm}.$$
 
:$$v(t) = q(t)\cdot \cos(\Delta \phi_{\rm T}) \hspace{0.05cm}.$$
  
 
{{BlaueBox|TEXT=
 
{{BlaueBox|TEXT=
$\text{Fazit:}$&nbsp; Bei ZSB–AM&nbsp; (mit oder ohne Träger)&nbsp; führt die&nbsp; '''Synchrondemodulation mit Phasenversatz'''&nbsp; &nbsp;$Δ\mathbf{ϕ_{\rm T} }$&nbsp; nicht zu Verzerrungen, sondern lediglich zu einer frequenzunabhängigen Dämpfung um den&nbsp; ''zeitunabhängigen Faktor'' &nbsp;$\cos(Δ\mathbf{ϕ_{\rm T} })$.&nbsp;  
+
$\text{Conclusions:}$&nbsp;  
  
Der Grund für diese weniger gravierende Signalveränderung als im Falle eines  Frequenzversatzes ist, dass hier die Zeit&nbsp; $t$&nbsp; im Argument der Cosinusfunktion fehlt. }}  
+
*For DSB-AM&nbsp; (with or without carrier), &nbsp; &raquo;'''synchronous demodulation with phase offset'''&laquo;&nbsp; &nbsp;$Δ\mathbf{ϕ_{\rm T} }$&nbsp; does not lead to distortions,
 +
:but only to a frequency independent attenuation by the time independent factor &nbsp;$\cos(Δ\mathbf{ϕ_{\rm T} })$.&nbsp;
  
 +
*The reason for this less serious signal change than in the case of a frequency offset is that in this case the time&nbsp; $t$&nbsp; is missing in the argument of the cosine function. }}
  
  
 +
[[File: P_ID1003__Mod_T_2_2_S5_neu.png |right|frame| Impairment of synchronous demodulation due to phase offset]]
 
{{GraueBox|TEXT=
 
{{GraueBox|TEXT=
$\text{Beispiel 2:}$&nbsp;  
+
$\text{Example 2:}$&nbsp;  
Die Grafik zeigt oben die Signale &nbsp;$q(t)$&nbsp; und &nbsp;$s(t)$&nbsp; am Sender und unten die empfängerseitigen Signale &nbsp;$b(t)$&nbsp; und &nbsp;$v(t)$.
+
The diagram shows
 +
*the signals &nbsp;$q(t)$&nbsp; and &nbsp;$s(t)$&nbsp; at the transmitter above,
 +
 +
*and the receiver-side signals &nbsp;$b(t)$&nbsp; and &nbsp;$v(t)$&nbsp; below.
 +
 
  
[[File: P_ID1003__Mod_T_2_2_S5_neu.png |right|frame| Beeinträchtigung der Synchrondemodulation durch Phasenversatz]]
 
  
*Aufgrund des Phasenversatzes um &nbsp;$Δ \mathbf{ϕ_{\rm T} } = π/3\  (60^\circ)$&nbsp; ist das Sinkensignal &nbsp;$v(t)$&nbsp; nur halb so groß wie das Quellensignal &nbsp;$q(t)$.  
+
Due to the phase offset of &nbsp;$Δ \mathbf{ϕ_{\rm T} } = π/3\  (60^\circ)$,&nbsp; the sink signal &nbsp;$v(t)$&nbsp; is only half the size of the source signal &nbsp;$q(t)$.  
  
  
*Die Signalform von &nbsp;$q(t)$&nbsp; bleibt jedoch in &nbsp;$v(t)$&nbsp; erhalten. }}
+
However,&nbsp; the waveform of &nbsp;$q(t)$&nbsp; is preserved in the output signal &nbsp;$v(t)$. }}
 
<br clear=all>
 
<br clear=all>
==Einfluss linearer Kanalverzerrungen==
+
==Influence of linear channel distortions==
 
<br>
 
<br>
Im Abschnitt &nbsp;[[Linear_and_Time_Invariant_Systems/Lineare_Verzerrungen#D.C3.A4mpfungsverzerrungen|Dämpfungsverzerrungen]]&nbsp; des Buches "Lineare zeitinvariante Systemewurde bereits angedeutet, dass das gesamte Übertragungssystem – bestehend aus&nbsp; '''M'''odulator,&nbsp; '''K'''anal und&nbsp; '''D'''emodulator – durch den resultierenden Frequenzgang &nbsp;$H_{\rm MKD}(f)$&nbsp; vollständig beschrieben werden kann, wenn
+
In the section &nbsp;[[Linear_and_Time_Invariant_Systems/Linear_Distortions#Attenuation_distortions|"Attenuation distortions"]]&nbsp; of the book&nbsp; "Linear and Time Invariant Systems"&nbsp; it was already suggested that the entire communication system &nbsp; &ndash; consisting of the&nbsp; $\rm M$odulator,&nbsp; the channel &nbsp; $($&rArr; &nbsp; German:&nbsp; $\rm K$anal$)$ &nbsp; and the&nbsp; $\rm D$emodulator &ndash;&nbsp;  can be completely described by the resulting frequency response &nbsp;$H_{\rm MKD}(f)$&nbsp;
[[File:EN_Mod_T_2_2_S6_v2.png|right|frame| Amplitudenmodulation und Synchrondemodulation]]  
+
[[File:EN_Mod_T_2_2_S6_v2.png|right|frame| Amplitude modulation and Synchronous demodulation]]  
* entweder das System verzerrungsfrei ist, oder
+
*if either the system is free of distortion,&nbsp; or
*lediglich lineare Verzerrungen hinsichtlich der Signale &nbsp;$q(t)$&nbsp; und &nbsp;$v(t)$&nbsp; entstehen.  
+
 
 +
*only linear distortions arise with respect to the signals &nbsp;$q(t)$&nbsp; and &nbsp;$v(t)$.  
 +
 
  
 +
In contrast,&nbsp; [[Linear_and_Time_Invariant_Systems/Nonlinear_Distortions|$\text{nonlinear distortions}$]]&nbsp; are not captured by this diagram
 +
#because the multiplicative relationship &nbsp;$V(f) = Q(f)\cdot H_{\rm MKD}(f)$&nbsp; does not allow for the emergence of new frequencies;
 +
#if &nbsp;$Q(f_0) = 0$,&nbsp; then &nbsp;$V(f_0) = 0$&nbsp; will always hold as well.
  
Dagegen werden &nbsp;[[Linear_and_Time_Invariant_Systems/Nichtlineare_Verzerrungen|nichtlineare Verzerrungen]]&nbsp; durch dieses Ersatzschaltbild nicht erfasst, da aufgrund des multiplikativen Zusammenhangs &nbsp;$V(f) = Q(f)\cdot H_{\rm MKD}(f)$&nbsp; das Entstehen neuer Frequenzen nicht möglich ist.&nbsp; Ist &nbsp;$Q(f_0) = 0$, so wird stets auch &nbsp;$V(f_0) = 0$&nbsp; gelten.
 
  
 +
The above conditions are fulfilled for the following system variant:
 +
*The modulator generates a&nbsp;"DSB-AM"&nbsp; (with or without carrier)&nbsp; around the carrier frequency &nbsp;$f_{\rm T}$.
  
Obige Voraussetzungen sind bei folgender Systemvariante erfüllt:
+
*The&nbsp; "channel"&nbsp; is describable by the frequency response  &nbsp;$H_{\rm K}(f)$&nbsp; with band-pass characteristics and its bandwidth is sufficient.
*Der Modulator erzeugt eine ZSB–AM (mit oder ohne Träger) um die Trägerfrequenz &nbsp;$f_{\rm T}$.
+
*Der Kanal ist durch den Frequenzgang &nbsp;$H_{\rm K}(f)$&nbsp; mit Bandpass–Charakter beschreibbar und dessen Bandbreite ausreichend.  
+
*The&nbsp; "synchronous demodulator"&nbsp; is synchronous in frequency and phase and the receiver filter &nbsp;$H_{\rm E}(f)$&nbsp; is ideal (rectangular).
*Der Synchrondemodulator ist frequenz– und phasensynchron und das Filter &nbsp;$H_{\rm E}(f)$&nbsp; ideal (rechteckförmig).  
 
  
  
 
{{BlaueBox|TEXT=
 
{{BlaueBox|TEXT=
$\text{Definition:}$&nbsp; Damit lautet der&nbsp; '''resultierende Frequenzgang'''&nbsp; von Modulator–Kanal–Demodulator:  
+
$\text{Definition:}$&nbsp; With these favorable conditions,&nbsp; the&nbsp; &raquo;'''resulting frequency response'''&laquo;&nbsp; of&nbsp; "modulator-channel-demodulator"&nbsp; is:  
 
:$$H_{\rm MKD}(f) = {1}/{2} \cdot \big[ H_{\rm K}(f + f_{\rm T}) + H_{\rm K}(f - f_{\rm T})\big] \hspace{0.05cm}.$$}}
 
:$$H_{\rm MKD}(f) = {1}/{2} \cdot \big[ H_{\rm K}(f + f_{\rm T}) + H_{\rm K}(f - f_{\rm T})\big] \hspace{0.05cm}.$$}}
  
  
*Ist  &nbsp;$\vert H_{\rm MKD}(f) \vert $&nbsp; im Bereich der Signalbandbreite nicht konstant, so werden die verschiedenen Spektralanteile des Quellensignals &nbsp;$q(t)$&nbsp; auch unterschiedlich übertragen &nbsp; &rArr; &nbsp; ''Dämpfungsverzerrungen''.  
+
*If  &nbsp;$\vert H_{\rm MKD}(f) \vert $&nbsp; is not constant in the range of the signal bandwidth, the different spectral components of the source signal  &nbsp;$q(t)$&nbsp; are also transmitted differently &nbsp; &rArr; &nbsp; "attenuation distortions".
*Ebenso kann es zu&nbsp; ''Phasenverzerrungen''&nbsp; kommen, wenn die Phasenfunktion &nbsp;$\text{arc} \ H_{\rm MKD}(f) $&nbsp; nichtlinear in &nbsp;$f$&nbsp; ist.
+
 +
*Similarly, "phase distortions" may occur if the phase function &nbsp;$\text{arc} \ H_{\rm MKD}(f) $&nbsp; is nonlinear in &nbsp;$f$.
 +
 
  
  
 
{{GraueBox|TEXT=
 
{{GraueBox|TEXT=
$\text{Beispiel 3:}$&nbsp; Die Grafik verdeutlicht die obige Berechnungsvorschrift für die resultierende Systemfunktion.  
+
$\text{Example 3:}$&nbsp; The diagram illustrates the above calculation rule for the resulting system function.  
  
[[File: P_ID1005__Mod_T_2_2_S6b_neu.png|right|frame| Einfluss linearer Kanalverzerrungen]]
+
[[File: P_ID1005__Mod_T_2_2_S6b_neu.png|right|frame| Influence of linear channel distortions]]
  
*Aus dem unsymmetrischen Bandpass &nbsp;$H_{\rm K}(f)$&nbsp; –&nbsp; bezogen auf die Trägerfrequenz &nbsp;$f_{\rm T}$&nbsp; – wird die im NF–Bereich &nbsp;$($um $f = 0)$&nbsp; symmetrische Funktion &nbsp;$H_{\rm MKD}(f)$.
+
*From the asymmetric band-pass  &nbsp;$H_{\rm K}(f)$&nbsp; – related to the carrier frequency &nbsp;$f_{\rm T}$&nbsp; – we get the symmetric function &nbsp;$H_{\rm MKD}(f)$&nbsp; in the low-frequency range&nbsp; $($around&nbsp; $f = 0)$.
  
  
*Besteht das Quellensignal aus zwei Frequenzanteilen – in der Grafik an den roten Markierungspfeilen zu erkennen so wird die Spektrallinie bei &nbsp;$f_2$&nbsp; stärker gedämpft als die Frequenz &nbsp;$f_1$ &nbsp; <br>&rArr; &nbsp; es entstehen lineare Dämpfungsverzerrungen.
+
*If the source signal consists of two frequency components indicated in the graph by the red marker arrows the spectral line at &nbsp;$f_2$&nbsp; is more attenuated than the frequency  &nbsp;$f_1$&nbsp; &rArr; &nbsp;linear attenuation distortions arise.
 +
  
+
*The fact that &nbsp;$H_{\rm MKD}(f)$&nbsp; also includes components around &nbsp;$±2f_{\rm T}$&nbsp; is not a major concern. These do not affect low-pass considerations.}}  
*Dass &nbsp;$H_{\rm MKD}(f)$&nbsp; auch Anteile um &nbsp;$±2f_{\rm T}$&nbsp; beinhaltet, ist nicht weiter störend.&nbsp; Diese beeinträchtigen die Tiefpass-Betrachtung nicht.}}  
 
  
  
==Einfluss von Rauschstörungen==
+
==Influence of noise interference==
 
<br>
 
<br>
Nun soll die Frage geklärt werden, in wie weit die Übertragungsqualität durch ein stochastisches Stör- bzw. Rauschsignal &nbsp;$n(t)$&nbsp; beeinträchtigt wird.&nbsp; Wir gehen dabei von folgendem Szenario aus, das bereits auf der Seite &nbsp;[[Modulation_Methods/Qualitätskriterien#Untersuchungen_beim_AWGN.E2.80.93Kanal|Untersuchungen beim AWGN–Kanal]]&nbsp; vorgestellt wurde.
+
Now we wish to clarify the extent to which the transmission quality is affected by a stochastic interference or a noise signal&nbsp;$n(t)$.&nbsp; We will be assuming the following scenario,&nbsp; which has already been presented in the [[Modulation_Methods/Quality_Criteria#Investigations_at_the_AWGN_channel|"Investigations at the AWGN channel"]]&nbsp; section.
  
[[File:EN_Mod_T_2_2_S7.png|center|frame| Untersuchungen zum AWGN–Kanal]]
+
[[File:EN_Mod_T_2_2_S7.png|right|frame| Investigating the AWGN channel]]
  
Insbesondere werden folgende Annahmen getroffen:  
+
In particular,&nbsp; we make the following assumptions:  
*Betrachtet wird eine Zweiseitenband–Amplitudenmodulation mit Modulationsgrad &nbsp;$m$&nbsp; sowie ein idealer Synchrondemodulator ohne Phasen- und Frequenzversatz.  
+
*Only double-sideband amplitude modulation with modulation depth &nbsp;$m$&nbsp; and an ideal synchronous demodulator without phase and frequency offsets are considered.
*Entsprechend dem erweiterten AWGN–Kanalmodell gilt für das Empfangssignal, wobei &nbsp;$α_{\rm K}$&nbsp; ein frequenzunabhängiger Übertragungsfaktor ist und das Störsignal &nbsp;$n(t)$&nbsp; weißes Rauschen mit der zweiseitigen Rauschleistungsdichte &nbsp;$N_0/2$&nbsp; modelliert:  
+
 
 +
*According to the extended AWGN channel model,&nbsp; where &nbsp;$α_{\rm K}$&nbsp; is a frequency-independent transmission factor of the channel and the interference&nbsp; $($noise$)$&nbsp; signal &nbsp;$n(t)$&nbsp; models white noise with two-sided noise power density &nbsp;$N_0/2$,&nbsp; the following holds for the received signal:
 
:$$r(t) = \alpha_{\rm K} \cdot s(t) + n(t) \hspace{0.05cm}.$$
 
:$$r(t) = \alpha_{\rm K} \cdot s(t) + n(t) \hspace{0.05cm}.$$
*Stellvertretend für ein Quellensignal &nbsp;$q(t)$&nbsp; der Bandbreite &nbsp;$B_{\rm NF}$&nbsp; wird hier von einem cosinusförmigen Nachrichtensignal der Frequenz &nbsp;$B_{\rm NF}$&nbsp; ausgegangen:  
+
 
 +
*Representing a source signal &nbsp;$q(t)$&nbsp; of bandwidth &nbsp;$B_{\rm NF}$&nbsp;, a cosine-shaped message signal of frequency &nbsp;$B_{\rm NF}$&nbsp; is assumed here:
 
:$$q(t) = A_{\rm N} \cdot \cos(2 \pi \cdot B_{\rm NF} \cdot t ) \hspace{0.05cm}.$$
 
:$$q(t) = A_{\rm N} \cdot \cos(2 \pi \cdot B_{\rm NF} \cdot t ) \hspace{0.05cm}.$$
  
Mit diesen Annahmen gilt für das Sinkensignal &nbsp;$v(t) = \alpha_{\rm K} \cdot q(t) + \varepsilon(t) \hspace{0.05cm}$, wobei die Ursache der stochastischen Komponente &nbsp;$ε(t)$&nbsp; das Bandpass–Rauschen &nbsp;$n(t)$&nbsp; am Eingang des Synchrondemodulators ist.
+
With these assumptions,&nbsp; the sink signal is &nbsp;$v(t) = \alpha_{\rm K} \cdot q(t) + \varepsilon(t) \hspace{0.05cm}$,&nbsp; where the cause of the stochastic component &nbsp;$ε(t)$&nbsp; at the output is the band-pass noise &nbsp;$n(t)$&nbsp; at the input of the synchronous demodulator.
  
 
{{BlaueBox|TEXT=
 
{{BlaueBox|TEXT=
$\text{Definition:}$&nbsp; Als quantitatives Maß für die Übertragungsqualität wird das&nbsp; '''Signal–zu–Rausch–Leistungsverhältnis'''&nbsp; an der Sinke verwendet, das hier mit den Leistungen von &nbsp;$q(t)$&nbsp; und &nbsp;$ε(t)$&nbsp; wie folgt lautet:
+
$\text{Definition:}$&nbsp;  
 +
 
 +
*The&nbsp; &raquo;'''signal-to-noise power ratio'''&laquo;&nbsp; at the sink is used as a quantitative measure of transmission quality,&nbsp; and is expressed here in terms of the power of &nbsp;$q(t)$&nbsp; and &nbsp;$ε(t)$, respectively:
 
:$$\rho_v = \frac{\alpha_{\rm K}^2 \cdot P_q}{P_\varepsilon} \hspace{0.05cm}.$$
 
:$$\rho_v = \frac{\alpha_{\rm K}^2 \cdot P_q}{P_\varepsilon} \hspace{0.05cm}.$$
  
Dieses Verhältnis bezeichnen wir im Folgenden kurz als das&nbsp; '''Sinken–SNR'''&nbsp; &nbsp;$ρ_v$&nbsp; und die logarithmische Darstellung &nbsp;$10 · \lg ρ_v$&nbsp; als den&nbsp; ''Sinken-Störabstand''&nbsp; in dB.}}
+
*In the following, we will refer to this ratio as the &laquo;'''sink SNR'''&laquo; &nbsp;$ρ_v$&nbsp; and the logarithmic representation &nbsp;$10 · \lg ρ_v$&nbsp; as the &raquo;'''sink SNR'''&laquo; in dB. }}
  
  
==Berechnung der Rauschleistung==
+
==Calculating the noise power==
 
<br>
 
<br>
Wir berechnen zunächst die Leistung &nbsp;$P_ε$&nbsp; des Fehlersignals &nbsp;$ε(t)$, die wir der Einfachheit halber als „Rauschleistung” bezeichnen.&nbsp; Das Fehlersignal &nbsp;$ε(t)$&nbsp; erhält man aus dem Störsignal &nbsp;$n(t)$&nbsp; am Eingang durch
+
We first calculate the power &nbsp;$P_ε$&nbsp; of the error signal &nbsp;$ε(t)$,&nbsp; which we refer to as the&nbsp; "noise power" for simplicity. 
*Multiplikation mit&nbsp; $z_{\rm E}(t) = 2 · \cos(ω_{\rm T} · t + \mathbf{ϕ_{\rm T} })$&nbsp; und
+
 
*eine anschließende (ideale) Tiefpassfilterung auf den Frequenzbereich&nbsp; $\pm B_{\rm NF}$.
+
The error signal &nbsp;$ε(t)$&nbsp; at the output is obtained from the noise signal&nbsp; $n(t)$&nbsp; at the input by
 +
*multiplying by&nbsp; $z_{\rm E}(t) = 2 · \cos(ω_{\rm T} · t + \mathbf{ϕ_{\rm T} })$&nbsp; and
 +
 
 +
*a subsequent&nbsp; (ideal)&nbsp; low-pass filtering to the low-frequency range &nbsp; $\pm B_{\rm NF}$.
  
  
Für das Leistungsdichtespektrum &nbsp;${\it Φ_ε}'(f)$&nbsp; ohne Berücksichtigung des Tiefpasses gilt mit &nbsp;${\it Φ}_n(f) = N_0/2$:
+
For the power-spectral density &nbsp; $\rm (PSD)$ &nbsp; ${\it Φ_ε}'(f)$&nbsp; without considering the low-pass filter,&nbsp; it holds with &nbsp;${\it Φ}_n(f) = N_0/2$:
 
:$${\it \Phi}_\varepsilon \hspace{-0.10cm} '(f) = {\it \Phi}_n (f) \star {\it \Phi}_{z {\rm E}}(f)  \hspace{0.05cm}.$$
 
:$${\it \Phi}_\varepsilon \hspace{-0.10cm} '(f) = {\it \Phi}_n (f) \star {\it \Phi}_{z {\rm E}}(f)  \hspace{0.05cm}.$$
  
In den Büchern „Signaldarstellung” und „Stochastische Signaltheorie” wurde gezeigt, dass das &nbsp;[[Signal_Representation/Fourier_Transform_and_Its_Inverse#Das_erste_Fourierintegral|Spektrum]]&nbsp; und das&nbsp; [[Theory_of_Stochastic_Signals/Leistungsdichtespektrum_(LDS)|Leistungsdichtespektrum]]&nbsp; eines Cosinussignals &nbsp;$x(t) = A · \cos(2πf_{\rm T}t)$&nbsp; wie folgt gegeben sind:
+
In the books&nbsp; "Signal Representation"&nbsp; and&nbsp; "Theory of Stochastic Signals”,&nbsp; it was shown that
 +
*the&nbsp;[[Signal_Representation/Fourier_Transform_and_its_Inverse#The_first_Fourier_integral|$\text{spectrum}$]]&nbsp; of a cosine signal &nbsp;$x(t) = A · \cos(2πf_{\rm T}t)$&nbsp; is given by:
 
:$$X(f) = \frac{A}{2}\cdot \delta(f + f_{\rm T})  + \frac{A}{2}\cdot \delta(f - f_{\rm T})  \hspace{0.05cm}, $$
 
:$$X(f) = \frac{A}{2}\cdot \delta(f + f_{\rm T})  + \frac{A}{2}\cdot \delta(f - f_{\rm T})  \hspace{0.05cm}, $$
 +
 +
*whereas for the&nbsp; [[Theory_of_Stochastic_Signals/Power-Spectral_Density|$\text{power-spectral density}$]]&nbsp; of a cosine signal &nbsp;$x(t) = A · \cos(2πf_{\rm T}t)$:
 
:$$ {\it \Phi}_x (f) = \frac{A^2}{4}\cdot \delta(f + f_{\rm T}) + \frac{A^2}{4}\cdot \delta(f - f_{\rm T}) \hspace{0.05cm}.$$
 
:$$ {\it \Phi}_x (f) = \frac{A^2}{4}\cdot \delta(f + f_{\rm T}) + \frac{A^2}{4}\cdot \delta(f - f_{\rm T}) \hspace{0.05cm}.$$
  
Angewandt auf das empfangsseitige Trägersignal &nbsp;$z_{\rm E}(t)$&nbsp; lautet die zweite Gleichung mit &nbsp;$A = 2$, und zwar unabhängig von der Phase&nbsp; (da im Leistungsdichtespektrum alle Phasenbeziehungen verloren gehen):
+
Applied to the receiver-side carrier signal &nbsp;$z_{\rm E}(t)$,&nbsp; with &nbsp;$A = 2$:  
:$${\it \Phi}_{z {\rm E}}(f)= \delta(f + f_{\rm T})  + \delta(f - f_{\rm T})  \hspace{0.05cm}.$$
+
:$${\it \Phi}_{z {\rm E}}(f)= \delta(f + f_{\rm T})  + \delta(f - f_{\rm T})  \hspace{0.05cm},$$
 +
and this is independent of phase&nbsp; (since all phase relationships are lost in the power-spectral density).
  
Unter Berücksichtigung, dass &nbsp;${\it Φ}_n(f)$&nbsp; für alle Frequenzen konstant ist  &nbsp; ⇒  &nbsp; „Weißes Rauschen” , ergibt sich:
+
{{GraueBox|TEXT=
 +
$\text{Example 4:}$ &nbsp;
 +
Considering that &nbsp;${\it Φ}_n(f)$&nbsp; is constant for all frequencies &nbsp; ⇒  &nbsp; "White Noise",&nbsp; we get:
 
:$${\it \Phi}_\varepsilon \hspace{-0.10cm} '(f) = {\it \Phi}_n (f + f_{\rm T}) + {\it \Phi}_n (f - f_{\rm T}) = 2 {\it \Phi}_n (f) = N_0  \hspace{0.05cm}.$$
 
:$${\it \Phi}_\varepsilon \hspace{-0.10cm} '(f) = {\it \Phi}_n (f + f_{\rm T}) + {\it \Phi}_n (f - f_{\rm T}) = 2 {\it \Phi}_n (f) = N_0  \hspace{0.05cm}.$$
  
Das Leistungsdichtespektrum (LDS) nach dem Tiefpassfilter ist für &nbsp;$|f| < B_{\rm NF}$&nbsp; genau so groß und außerhalb Null:
+
*The power-spectral density&nbsp; $\rm (PSD)$&nbsp; after the low-pass filter is exactly as large for &nbsp;$ \vert f \vert < B_{\rm NF}$&nbsp; and zero beyond this range.
 
:$${\it \Phi}_\varepsilon (f)  = \left\{ \begin{array}{c} N_0 \\ 0 \\  \end{array} \right.\quad
 
:$${\it \Phi}_\varepsilon (f)  = \left\{ \begin{array}{c} N_0 \\ 0 \\  \end{array} \right.\quad
\begin{array}{*{10}c}    {\rm{f\ddot{u}r}}\\    \\ \end{array}\begin{array}{*{20}c}
+
\begin{array}{*{10}c}    {\rm{for} }\\    \\ \end{array}\begin{array}{*{20}c}
|f|< B_{\rm NF} \hspace{0.05cm}, \\ {\rm sonst} \hspace{0.05cm}. \\ \end{array}$$
+
\vert f \vert < B_{\rm NF} \hspace{0.05cm}, \\ {\rm otherwise} \hspace{0.05cm}. \\ \end{array}$$
  
Durch Integration erhält man die Leistung &nbsp;$P_ε = 2N_0 · B_{\rm NF}$.&nbsp; Mit diesem Zwischenergebnis kann somit für das Sinken–SNR geschrieben werden:
+
*By integration,&nbsp; we obtain the power &nbsp;$P_ε = 2N_0 · B_{\rm NF}$. &nbsp; Thus,&nbsp; with this intermediate result,&nbsp; the&nbsp; "sink SNR"&nbsp; can be given as:
:$$\rho_v = \frac{\alpha_{\rm K}^2 \cdot P_q}{P_\varepsilon} = \frac{\alpha_{\rm K}^2 \cdot  P_q}{N_0 \cdot B_{\rm NF}}  \hspace{0.05cm}.$$
+
:$$\rho_v = \frac{\alpha_{\rm K}^2 \cdot P_q}{P_\varepsilon} = \frac{\alpha_{\rm K}^2 \cdot  P_q}{N_0 \cdot B_{\rm NF} }  \hspace{0.05cm}.$$
  
Im nächsten Abschnitt wird noch der Zusammenhang zwischen der Leistung &nbsp;$P_q$&nbsp; des Quellensignals und der Sendeleistung &nbsp;$P_{\rm S}$&nbsp; hergestellt.
+
In the next section,&nbsp; we still establish the relationship between the power&nbsp; $P_q$&nbsp; of the source signal and the transmit power&nbsp; $P_{\rm S}$.}}
  
==Zusammenhang zwischen $P_q$ und $P_{\rm S}$==
+
==Relationship between the powers of source signal and transmitted signal==
 
<br>
 
<br>
Um den Zusammenhang zwischen Sinken&ndash;SNR  &nbsp;$\rho_v$&nbsp; und Sendeleistung &nbsp;$P_{\rm S}$&nbsp; angeben zu können, benötigen wir noch den Zusammenhang zwischen den Leistungen von
+
To characterise the relationship between the sink&ndash;SNR  &nbsp;$\rho_v$&nbsp; and the transmit power &nbsp;$P_{\rm S}$,&nbsp; we first need the relationships between
*Quellensignal &nbsp;$q(t)$&nbsp;    ⇒  &nbsp; Leistung &nbsp;$P_q$, und
+
*source signal &nbsp;$q(t)$ &nbsp;    ⇒  &nbsp; power &nbsp;$P_q$,&nbsp; and
*Sendesignal &nbsp;$s(t)$&nbsp;      ⇒  &nbsp; Sendeleistung &nbsp;$P_{\rm S}$.
+
 
 +
*transmitted signal &nbsp;$s(t)$ &nbsp;      ⇒  &nbsp; transmit power &nbsp;$P_{\rm S}$.
  
  
 
{{BlaueBox|TEXT=
 
{{BlaueBox|TEXT=
$\text{Vorweg genommenes Ergebnis:}$&nbsp;  
+
$\text{Anticipated result:}$&nbsp;  
Im Falle der „ZSB–AM mit Träger” gilt dabei mit dem Modulationsgrad &nbsp;$m$:
+
In the case of&nbsp; &raquo;DSB–AM with carrier&laquo;&nbsp; the following holds true for a modulation depth &nbsp;$m$:
 
:$$P_{\rm S} = { P_q}/{2} \cdot \hspace{0.05cm} \left(  1 + {2}/{m^2}\right)\hspace{0.05cm}.$$
 
:$$P_{\rm S} = { P_q}/{2} \cdot \hspace{0.05cm} \left(  1 + {2}/{m^2}\right)\hspace{0.05cm}.$$
  
*Anzumerken ist, dass diese Gleichung nur dann anwendbar ist, wenn &nbsp;$q(t)$&nbsp; eine harmonische Schwingung beschreibt.&nbsp;  
+
*Note that this equation is only applicable when &nbsp;$q(t)$&nbsp; describes a harmonic oscillation.
*Die „ZSB–AM ohne Träger” ist in der Gleichung als Sonderfall für &nbsp;$m → ∞$&nbsp; mit enthalten.}}
+
 +
*The&nbsp; &raquo;DSB–AM without carrier&laquo;&nbsp; is included in the equation as a special case for&nbsp; $m \to \infty$.}}
  
  
 
{{BlaueBox|TEXT=
 
{{BlaueBox|TEXT=
$\text{Beweis:}$&nbsp;  
+
$\text{Proof:}$&nbsp;  
Ausgegangen wird jeweils von Cosinusschwingungen, also den folgenden Gleichungen:
+
Cosine oscillations are assumed in each case,&nbsp; i.e. the following equations:
 
:$$q(t) = A_{\rm N} \cdot \cos(\omega_{\rm N} \cdot t ) \hspace{0.05cm},$$
 
:$$q(t) = A_{\rm N} \cdot \cos(\omega_{\rm N} \cdot t ) \hspace{0.05cm},$$
 
:$$s(t)  =  \big[ q(t) + A_{\rm T}\big] \cdot \cos(\omega_{\rm T} \cdot t ) = A_{\rm T} \cdot \cos(\omega_{\rm T} \cdot t ) + {A_{\rm N} }/{2}\cdot \cos\big [(\omega_{\rm T}+ \omega_{\rm N}) \cdot t \big] + {A_{\rm N} }/{2}\cdot \cos\big[(\omega_{\rm T}- \omega_{\rm N}) \cdot t \big]\hspace{0.05cm}.$$
 
:$$s(t)  =  \big[ q(t) + A_{\rm T}\big] \cdot \cos(\omega_{\rm T} \cdot t ) = A_{\rm T} \cdot \cos(\omega_{\rm T} \cdot t ) + {A_{\rm N} }/{2}\cdot \cos\big [(\omega_{\rm T}+ \omega_{\rm N}) \cdot t \big] + {A_{\rm N} }/{2}\cdot \cos\big[(\omega_{\rm T}- \omega_{\rm N}) \cdot t \big]\hspace{0.05cm}.$$
*Die Leistung des Quellensignals, bezogen auf den Widerstand &nbsp;$1 \ \rm Ω$, beträgt mit der Periodendauer &nbsp;$T_{\rm N}$:
+
*The power of the source signal,&nbsp; in relation to a resistance of &nbsp;$1 \ \rm Ω$,&nbsp; with the period &nbsp;$T_{\rm N}$,&nbsp; is:
 
:$$P_{q}  =  \frac{1}{T_{\rm N} }\cdot\int_{0}^{  T_{\rm N} }
 
:$$P_{q}  =  \frac{1}{T_{\rm N} }\cdot\int_{0}^{  T_{\rm N} }
 
  {q^2(t)}\hspace{0.1cm}{\rm d}t =  \frac{A_{\rm N}^2}{T_{\rm N} }\cdot\int_{0}^{T_{\rm N} }
 
  {q^2(t)}\hspace{0.1cm}{\rm d}t =  \frac{A_{\rm N}^2}{T_{\rm N} }\cdot\int_{0}^{T_{\rm N} }
 
  {\cos^2(2 \pi\cdot{t}/{T_{\rm N} })}\hspace{0.1cm}{\rm d}t = \frac{A_{\rm N}^2}{2}\hspace{0.05cm}.$$
 
  {\cos^2(2 \pi\cdot{t}/{T_{\rm N} })}\hspace{0.1cm}{\rm d}t = \frac{A_{\rm N}^2}{2}\hspace{0.05cm}.$$
*Entsprechend erhält man für die Leistung des Sendesignals:
+
 
 +
*Accordingly,&nbsp; for the power of the transmitted signal,&nbsp; one obtains:
 
:$$P_{\rm S}  =  \frac{A_{\rm T}^2}{2} + \frac{(A_{\rm N}/2)^2}{2} + \frac{(A_{\rm N}/2)^2}{2} =
 
:$$P_{\rm S}  =  \frac{A_{\rm T}^2}{2} + \frac{(A_{\rm N}/2)^2}{2} + \frac{(A_{\rm N}/2)^2}{2} =
 
  \frac{A_{\rm T}^2}{2} + \frac{A_{\rm N}^2}{4}
 
  \frac{A_{\rm T}^2}{2} + \frac{A_{\rm N}^2}{4}
Line 245: Line 284:
 
  \hspace{0.05cm}.$$
 
  \hspace{0.05cm}.$$
  
*Diese Gleichung gilt sowohl für ZSB–AM ohne Träger &nbsp;$(A_{\rm T} = 0)$&nbsp; als auch für ZSB–AM mit Träger.&nbsp; Da &nbsp;$q(t)$&nbsp; als eine harmonische Schwingung vorausgesetzt wurde, kann mit dem Modulationsgrad &nbsp;$m = A_{\rm N}/A_{\rm T}$&nbsp; hierfür auch geschrieben werden:
+
*This equation is valid for both&nbsp; &raquo;DSB–AM without carrier&laquo; &nbsp;$(A_{\rm T} = 0)$&nbsp; and&nbsp; &raquo;DSB–AM with carrier&laquo;.&nbsp; Since &nbsp;$q(t)$&nbsp; was assumed to be a harmonic oscillation,&nbsp; &raquo;DSB–AM without carrier&laquo;&nbsp; given a modulation depth &nbsp;$m = A_{\rm N}/A_{\rm T}$,&nbsp; this can also be written as:
 
:$$P_{\rm S}  =  {A_{\rm N}^2}/{4} \cdot \left( 1  +{2A_{\rm T}^2}/{A_{\rm N}^2} \right)= P_q/2 \cdot \left( 1  +{2}/{m^2} \right)\hspace{0.05cm}.\hspace{5.4cm}{\rm q.e.d.}$$}}
 
:$$P_{\rm S}  =  {A_{\rm N}^2}/{4} \cdot \left( 1  +{2A_{\rm T}^2}/{A_{\rm N}^2} \right)= P_q/2 \cdot \left( 1  +{2}/{m^2} \right)\hspace{0.05cm}.\hspace{5.4cm}{\rm q.e.d.}$$}}
  
  
==Sinken-SNR und Leistungskenngröße==
+
==Sink SNR and the performance parameter==
 
<br>
 
<br>
{{BlaueBox|TEXT=Mit den Ergebnissen der letzten drei Abschnitte kann deshalb für das&nbsp; '''Sinken–SNR'''&nbsp; der&nbsp; '''ZSB-AM'''&nbsp; geschrieben werden:
+
{{BlaueBox|TEXT=With the results of the last three sections, the&nbsp; &raquo;'''sink SNR of double-sideband amplitude modulation'''&laquo;&nbsp; can be written as:  
 
:$$\rho_v = \frac{\alpha_{\rm K}^2 \cdot P_q}{P_\varepsilon} = \frac{\alpha_{\rm K}^2 \cdot P_{\rm S} }{N_0 \cdot B_{\rm NF} } \cdot \frac{1}{1 + {2}/{m^2} } \hspace{0.05cm}.$$}}
 
:$$\rho_v = \frac{\alpha_{\rm K}^2 \cdot P_q}{P_\varepsilon} = \frac{\alpha_{\rm K}^2 \cdot P_{\rm S} }{N_0 \cdot B_{\rm NF} } \cdot \frac{1}{1 + {2}/{m^2} } \hspace{0.05cm}.$$}}
  
  
Nachfolgend wird diese Gleichung ausführlich diskutiert.&nbsp; Bereits im Kapitel &nbsp;[[Modulation_Methods/Qualitätskriterien#Untersuchungen_beim_AWGN.E2.80.93Kanal|Untersuchungen beim AWGN-Kanal]]&nbsp; wurde begründet, warum es Sinn macht, das Sinken–SNR &nbsp;$ρ_v$&nbsp; in Abhängigkeit der nachfolgend benannten Leistungskenngröße &nbsp;$ξ$&nbsp; anzugeben:
+
[[File: EN_Mod_T_2_2_S7b.png|right|frame|Linear und double–logarithmic representation of sink SNR]]
 +
The following is a detailed discussion of this equation.&nbsp; Already in the chapter &nbsp;[[Modulation_Methods/Quality_Criteria#Investigations_at_the_AWGN_channel|"Investigations at the AWGN channel"]]&nbsp; it was justified why it makes sense to state the sink SNR  &nbsp;$ρ_v$&nbsp; as a function of the performance parameter &nbsp;$ξ$&nbsp; named below:
 
:$$\xi = \frac{\alpha_{\rm K}^2 \cdot P_{\rm S}}{N_0 \cdot B_{\rm NF}}
 
:$$\xi = \frac{\alpha_{\rm K}^2 \cdot P_{\rm S}}{N_0 \cdot B_{\rm NF}}
 
\hspace{0.5cm}\Rightarrow \hspace{0.5cm} \rho_v = \frac{\xi}{1 + {2}/{m^2}}  
 
\hspace{0.5cm}\Rightarrow \hspace{0.5cm} \rho_v = \frac{\xi}{1 + {2}/{m^2}}  
 
\hspace{0.05cm}.$$
 
\hspace{0.05cm}.$$
  
Die beiden Grafiken zeigen die entsprechenden Kurven – links linear und rechts in doppelt–logarithmischer Darstellung.  
+
The two graphs show the corresponding curves
 +
*linear on the left,
 +
 
 +
*double-logarithmic on the right.
 +
<br clear=all>
 +
The curves are to be interpreted as follows:
 +
*For the system variant&nbsp; &raquo;DSB–AM without carrier&laquo;&nbsp; with &nbsp;$m → ∞$&nbsp; the above equation yields the simple relationship &nbsp;$ρ_v = ξ$.&nbsp; This gives the bisector of the angle for both the linear and double logarithmic plots.
  
[[File: EN_Mod_T_2_2_S7b.png|center|frame|Sinken–SNR in linearer und doppelt–logarithmischer Darstellung]]
+
*A greater transmit power &nbsp;$P_{\rm S}$&nbsp;  leads to a better sink SNR,&nbsp; as does a larger transmission factor  &nbsp; $α_{\rm K}$&nbsp; (⇒ lower attenuation).&nbsp; However, &nbsp; $10 · \lg ρ_v$&nbsp; is also increased by a smaller noise power density &nbsp;$N_0$&nbsp; and a smaller bandwidth &nbsp;$B_{\rm NF}$,&nbsp; other things being equal.
  
Die Kurven sind wie folgt zu interpretieren:
+
*For&nbsp; &raquo;DSB–AM with carrier&laquo;&nbsp; with a modulation depth &nbsp;$m$:
*Für die Systemvariante „ZSB–AM ohne Träger” erhält man mit &nbsp;$m → ∞$&nbsp; aus der oberen Gleichung den einfachen Zusammenhang &nbsp;$ρ_v = ξ$. Dies ergibt sowohl bei der linearen als auch bei der doppelt–logarithmischen Darstellung die Winkelhalbierende.
 
*Eine größere Sendeleistung &nbsp;$P_{\rm S}$&nbsp; führt ebenso wie ein größerer Übertragungsfaktor &nbsp;$α_{\rm K}$&nbsp; (⇒ geringere Dämpfung) zu einem besseren Sinken–SNR.&nbsp; $10 · \lg ρ_v$&nbsp; wird aber auch durch eine kleinere Rauschleistungsdichte &nbsp;$N_0$&nbsp; und eine kleinere Bandbreite &nbsp;$B_{\rm NF}$&nbsp; bei sonst gleichen Bedingungen vergrößert.
 
*Bei einer „ZSB–AM mit Träger” gilt mit dem Modulationsgrad &nbsp;$m$:
 
 
:$$\rho_v = \xi \cdot \frac{1}{1 + {2}/{m^2}}\hspace{0.3cm}\Rightarrow \hspace{0.3cm}
 
:$$\rho_v = \xi \cdot \frac{1}{1 + {2}/{m^2}}\hspace{0.3cm}\Rightarrow \hspace{0.3cm}
 
  10 \cdot {\rm lg }\hspace{0.1cm}\rho_v  = 10 \cdot {\rm lg }\hspace{0.1cm}\xi - 10 \cdot {\rm lg }\hspace{0.1cm} \left({1 + {2}/{m^2}}\right)\hspace{0.05cm}.$$
 
  10 \cdot {\rm lg }\hspace{0.1cm}\rho_v  = 10 \cdot {\rm lg }\hspace{0.1cm}\xi - 10 \cdot {\rm lg }\hspace{0.1cm} \left({1 + {2}/{m^2}}\right)\hspace{0.05cm}.$$
*In der doppelt–logarithmischen Darstellung&nbsp; (siehe rechte Grafik)&nbsp; führt dies zu einer Parallelverschiebung der Kurven nach unten, zum Beispiel  bei &nbsp;$m = 1$&nbsp; um &nbsp;$4.77$ dB&nbsp; und bei &nbsp;$m = 0.5$&nbsp; um &nbsp;$9.54$ dB.
 
*Alle Aussagen gelten unter der Voraussetzung eines idealen Synchrondemodulators.&nbsp; In diesem Fall macht das Verfahren „ZSB–AM mit Träger” eigentlich keinen Sinn.&nbsp; Der zugesetzte Träger führt hier nur zu einer unnötig großen Sendeleistung und kann zur Demodulation nicht genutzt werden.
 
*Die Kurven gelten für perfekte Frequenz– und Phasensynchronisation.&nbsp; Um die Parameter &nbsp;$f_{\rm T}$&nbsp; und &nbsp;$\mathbf{ϕ_{\rm T} }$&nbsp; mit weniger Aufwand aus dem Empfangssignal &nbsp;$r(t)$&nbsp; ermitteln zu können, macht allerdings ein kleiner Trägeranteil im Sendesignal durchaus Sinn.
 
*Mit &nbsp;$m = 3$&nbsp; ergibt sich dann nur eine unwesentliche Verschlechterung gegenüber „ZSB–AM ohne Träger” von weniger als einem dB.
 
  
==Aufgaben zum Kapitel==
+
*In the double logarithmic representation&nbsp;  (see right graph),&nbsp; this leads to a downward parallel shift of the curves,&nbsp; <br>e.g. at &nbsp;$m = 1$&nbsp; by &nbsp;$4.77$ dB&nbsp; and at &nbsp;$m = 0.5$&nbsp; by &nbsp;$9.54$ dB.
 +
 
 +
*All statements are valid under the assumption of an ideal synchronous demodulator.&nbsp;  In this case,&nbsp; the&nbsp; &raquo;DSB–AM with carrier&laquo;&nbsp; method actually makes no sense.&nbsp;  The added carrier would only lead to an unnecessarily large transmit power and could not be used for demodulation.
 +
 
 +
*The curves are valid for perfect frequency and phase synchronization.&nbsp;  However,&nbsp; in order to be able to determine the parameters &nbsp;$f_{\rm T}$&nbsp; and &nbsp;$\mathbf{ϕ_{\rm T} }$&nbsp; from the received signal &nbsp;$r(t)$&nbsp; with less effort,&nbsp; a small carrier component in the transmitted signal makes sense.
 +
 
 +
*When &nbsp;$m = 3$,&nbsp; it is only an insignificant deterioration of less than one dB compared to&nbsp; &raquo;DSB–AM without carrier&laquo;.
 +
 
 +
==Exercises for the chapter==
 
<br>
 
<br>
[[Aufgaben:Aufgabe_2.4:_Frequenz–_und_Phasenversatz|Aufgabe 2.4: Frequenz– und Phasenversatz]]
+
[[Aufgaben:Aufgabe_2.4:_Frequenz–_und_Phasenversatz|Exercise 2.4: Frequency and Phase Offset]]
  
[[Aufgaben:Aufgabe_2.4Z:_Tiefpass-Einfluss_bei_Synchrondemodulation|Aufgabe 2.4Z: Tiefpass-Einfluss bei Synchrondemodulation]]
+
[[Aufgaben:Aufgabe_2.4Z:_Tiefpass-Einfluss_bei_Synchrondemodulation|Exercise 2.4Z: Low-pass influence with Synchronous Demodulation]]
  
[[Aufgaben:Aufgabe_2.5:_ZSB–AM_über_einen_Gaußkanal|Aufgabe 2.5: ZSB–AM über einen Gaußkanal]]
+
[[Aufgaben:Exercise_2.5:_DSB-AM_via_a_Gaussian_channel|Exercise 2.5: DSB-AM via a Gaussian Channel]]
  
[[Aufgaben:Aufgabe_2.5Z:_Nochmals_Verzerrungen_bei_ZSB-AM|Aufgabe 2.5Z: Nochmals Verzerrungen bei ZSB-AM]]
+
[[Aufgaben:Exercise_2.5Z:_Linear_Distortions_with_DSB-AM|Exercise 2.5Z: Linear Distortions with DSB-AM]]
  
[[Aufgaben:Aufgabe_2.6:_Freiraumdämpfung|Aufgabe 2.6: Freiraumdämpfung]]
+
[[Aufgaben:Aufgabe_2.6:_Freiraumdämpfung|Exercise 2.6: Free Space Attenuation]]
  
[[Aufgaben:Aufgabe_2.6Z:_Signal-to-Noise-Ratio_(SNR)|Aufgabe 2.6Z: Signal-to-Noise-Ratio (SNR)]]
+
[[Aufgaben:Aufgabe_2.6Z:_Signal-to-Noise-Ratio_(SNR)|Exercise 2.6Z: Signal-to-Noise-Ratio (SNR)]]
  
  
 
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Latest revision as of 10:32, 13 January 2023

Block diagram and time domain representation


Modulation at the transmitter only makes sense,  if it is possible to reverse this signal conversion at the receiver end,  ideally without loss of information.  In any form of amplitude modulation  - be it double sideband (DSB) or single sideband (SSB),  with or without a carrier signal -  the so-called  »synchronous demodulator«  fulfils this task.

DSB amplitude modulation and synchronous demodulation

The following should be noted with regard to the block diagram on the right:

  • For modulation,  we will focus on  "DSB–AM without a carrier"  $($modulation depth  $m → ∞)$.  However,  synchronous demodulation is also applicable in  "DSB–AM with carrier".
  • Let the channel be ideal and the distortions negligible,  that the received signal  $r(t)$  is identical to the transmitted signal  $s(t)$:
$$r(t) = s(t) = q(t) \cdot \cos(\omega_{\rm T} \cdot t + \phi_{\rm T})\hspace{0.05cm}.$$
  • At the receiver,  $r(t)$  is first multiplied by the receiver-side carrier  $z_{\rm E}(t)$,  which is identical shaped to the transmitter-side carrier  $z(t)$  but larger by a factor of  $2$:
$$z_{\rm E}(t) = 2 \cdot \cos(\omega_{\rm T} \cdot t + \phi_{\rm T})\hspace{0.05cm}.$$
  • The result of the multiplication is the signal  $b(t)$.   Considering the trigonometric transformation   $\cos^2(α) = 1/2 · \big [1 + \cos(2α)\big ]$   we get
$$b(t) = r(t) \cdot z_{\rm E}(t) = 2 \cdot q(t) \cdot \cos^2(\omega_{\rm T} \cdot t + \phi_{\rm T}) = q(t) + q(t) \cdot \cos(2 \cdot \omega_{\rm T} \cdot t + 2\cdot \phi_{\rm T})\hspace{0.05cm}.$$
  • The second term is in the range around twice the carrier frequency  $f_{\rm T}$.  If the signal bandwidth is  $B_{\rm NF} < f_{\rm T}$,  which is always true in practice,  this component can be suppressed by a low-pass filter  $H_{\rm E}(f)$  with suitable dimensions,  and we obtain  
$$v(t) = q(t).$$

Description in the frequency domain


Assuming an even source signal  $q(t)$   ⇒   with a real spectrum  $Q(f)$  and a sinusoidal carrier  $z(t)$,  the imaginary transmitted spectrum  $S(f)$  is obtained according to the second plot,  where  $A_{\rm T} ≠ 0$  also takes into account the DSB-AM with a carrier  (shown by the red Dirac delta function).  Because the channel is ideal,  $R(f) = S(f)$.

Illustration of synchronous demodulation in the frequency domain
$B(f)$  according to the sketch below,  but with shaded portions

The operation of the synchronous demodulator can be explained in the frequency domain as follows:

  • The receiver-side carrier signal $z_{\rm E}(t) = 2 · z(t) = 2 · \sin(ω_{\rm T} · t)$  leads to two Dirac delta functions in the spectral domain at  $\pm f_{\rm T}$  with weights  $\pm \rm j$.  The negative imaginary part occurs at  $f = +f_{\rm T}$.
  • The multiplication  $b(t) = r(t) · z_{\rm E}(t)$  corresponds to the convolution of the associated spectral functions.  The result is shown in the sketch below,  if the influence of the subsequent low-pass  (hatching)  is disregarded:
$$B(f) = R(f) \star Z_{\rm E}(f)\hspace{0.05cm}.$$
  • Convolution of the Dirac delta function  $ - {\rm j} \cdot δ(f – f_{\rm T})$  with the purely imaginary spectrum  $R(f)$  leads to completely real spectral components around  $f = 0$  and  $f = 2f_{\rm T}$.  These components are marked with a  "plus"  on the lower graph.
  • The second convolution product  ${\rm j} · δ(f + f_{\rm T}) \star R(f)$  yields a low-frequency spectral component around $f = 0$  in addition to a component at  $–2f_{\rm T}$.  These spectral components are marked with  "minus".
  • The spectrum after the low-pass  $H_{\rm E}(f)$  is  $V(f) = Q(f) + A_{\rm T} · δ(f)$.  For DSB-AM with a carrier, the interfering DC component can be removed by a lower band limit: 
$$H_{\rm E}(f = 0) = 0.$$
  • The color assignment in the diagram  (USB in blue, LSB in green, carrier in red) shows that the synchronous demodulator uses both the  $\rm USB$  and the  $\rm LSB$  for signal reconstruction.


Prerequisites for the application of a synchronous demodulator


The output signal  $v(t)$  is identical to the source signal  $q(t)$  if the following conditions are satisfied:

  • The bandwidth  $B_{\rm NF}$  of the source signal is smaller than the carrier frequency  $f_{\rm T}$.  This restriction is not particularly drastic and is not relevant for practical applications.
  • The carrier frequencies of the transmitter and receiver are exactly matched.  This requires carrier recovery at the receiver and thus involves some  "cost".
  • There is also perfect phase synchronicity between the the carrier signals  $z(t)$  and  $z_{\rm E}(t)$  at the transmitter and receiver end, respectively.
  • The channel frequency response  $H_{\rm K}(f)$  is ideally equal to   $1$  in the passband  $f_{\rm T} - B_{\rm NF} ≤ |f| ≤ f_{\rm T} + B_{\rm NF}$.  Frequency-independent attenuation or frequency-linear phase  (delay)  are usually tolerated.  Note:  The subscript  "K"  stands for German  "Kanal"  ⇒   channel).
  • The influence of noise and external disturbances is assumed to be negligible in this model.  However,  even with non-negligible noise,  the synchronous demodulator is superior to other demodulators.
  • The receiver filter  $H_{\rm E}(f)$  is equal to  "one"  for  $|f| ≤ B_{\rm NF}$  and equal to  "zero"  for $|f| ≥ 2f_{\rm T} - B_{\rm NF}$.  The range in between is not relevant  (see graph in the  "previous section").  Note:  The subscript  "E"  stands for German  "Empfänger"  ⇒   "receiver").
  • For the modulation method  "DSB-AM with carrier",  the condition  $H_{\rm E}(f = 0) ≡ 0$  must additionally ensure that the carrier added at the transmitter is no longer included in the sink signal.
  • The following four sections describe the effects if some of the above conditions are not met.

Influence of a frequency offset


As the name  "synchronous demodulator”  it only works when there is complete synchronization between the carrier signals of the transmitter and receiver. 

On the other hand,  if the carrier frequencies differ by a   »frequency offset «  $Δ\hspace{-0.05cm}f_{\rm T}$,  for example

$$\begin{align*}z(t) & = 1 \cdot \cos(2 \pi f_{\rm T} \cdot t + \phi_{\rm T})\hspace{0.05cm}, \\ z_{\rm E}(t) & = 2 \cdot \cos(2 \pi (f_{\rm T} + \Delta\hspace{-0.05cm} f_{\rm T}) \cdot t + \phi_{\rm T})\hspace{0.05cm},\end{align*}$$

then we obtain for the sink signal spectrum:

$$V(f) = {1}/{2}\cdot Q(f + \Delta \hspace{-0.05cm}f_{\rm T}) + {1}/{2}\cdot Q(f - \Delta\hspace{-0.05cm} f_{\rm T}) = Q(f) \star \big[ {1}/{2}\cdot \delta(f + \Delta\hspace{-0.05cm} f_{\rm T}) + {1}/{2}\cdot \delta (f - \Delta \hspace{-0.05cm}f_{\rm T}) \big] \hspace{0.05cm}.$$

This result can be verified in the frequency domain using the sketch in the  $\text{frequency description}$  section.  After transforming the equation to the time domain,  we get:

$$v(t) = q(t) \cdot \cos(2 \pi \cdot \Delta \hspace{-0.05cm}f_{\rm T} \cdot t )\hspace{0.05cm}.$$

$\text{Conclusions:}$ 

  • For DSB-AM  (with or without carrier),  »synchronous demodulation with frequency offset«  $ \Delta \hspace{-0.05cm}f_{\rm T}$ leads to  $\text{attenuation distortions}$,  characterized by the  "time-dependent factor"   $\cos(2 \pi \cdot \Delta \hspace{-0.05cm}f_{\rm T} \cdot t )$.
  • The frequency offset   $Δ\hspace{-0.05cm}f_{\rm T}$,  which is due to  »inaccuracies in the realization of carrier recovery«,  is usually very small and ranges from a few Hertz to about   $100\text{ Hz}$.  In this context,  one usually speaks of a  "beat".


Impairment of synchronous demodulation due to a frequency offset

$\text{Example 1:}$  The graph shows

  • a cosine-shaped source signal with frequency  $f_{\rm N} = 1\ \rm kHz$  
    ⇒   blue oscillation, and
  • the sink signal  $v(t)$  obtained with a synchronous demodulator
    ⇒   red curve.


Here,  a frequency offset of  $Δ\hspace{-0.05cm}f_{\rm T} = 100\ \rm Hz$  was used as a basis.  This results in:

$$\begin{align*}v(t ) & = 1\,{\rm V} \cdot \cos (2 \pi \cdot 1\,{\rm kHz} \cdot t) \cdot \cos (2 \pi \cdot 0.1\,{\rm kHz} \cdot t) =\\ &= 0.5\,{\rm V} \cdot \cos (2 \pi \cdot 0.9\,{\rm kHz} \cdot t) + 0.5\,{\rm V} \cdot \cos (2 \pi \cdot 1.1\,{\rm kHz} \cdot t) \hspace{0.05cm}.\end{align*}$$


From a spectral point of view, the  $1\ \rm kHz$ oscillation becomes two superimposed oscillations with frequencies  $0.9\ \rm kHz$  and  $1.1\ \rm kHz$  with half the amplitude.

  • In contrast,  the transmitted frequency  $(1\ \rm kHz)$  is no longer included in  $v(t)$ .

Influence of a phase offset


Now,  for the transmitter-side and receiver-side carrier signal,  it holds that

$$\begin{align*}z(t) & = 1 \cdot \cos(2 \pi f_{\rm T} t + \phi_{\rm T})\hspace{0.05cm}, \\ z_{\rm E}(t) & = 2 \cdot \cos(2 \pi f_{\rm T} t + \phi_{\rm E})\hspace{0.05cm}.\end{align*}$$

For the signal immediately after multiplication by the  »phase offset«  $Δ \mathbf{ϕ_{\rm T} = ϕ_{\rm E} - ϕ_{\rm T} }$,  this gives:

$$b(t) = q(t) \cdot \cos(\omega_{\rm T} \cdot t + \phi_{\rm T}) \cdot 2 \cdot \cos(\omega_{\rm T} \cdot t + \phi_{\rm E})= q(t)\cdot \cos(\Delta \phi_{\rm T}) + q(t) \cdot \cos(2 \cdot \omega_{\rm T} \cdot t + \phi_{\rm E}+ \phi_{\rm T})\hspace{0.05cm}.$$

Thus,  taking into account the low-pass filter,  the result for the sink signal is:

$$v(t) = q(t)\cdot \cos(\Delta \phi_{\rm T}) \hspace{0.05cm}.$$

$\text{Conclusions:}$ 

  • For DSB-AM  (with or without carrier),   »synchronous demodulation with phase offset«   $Δ\mathbf{ϕ_{\rm T} }$  does not lead to distortions,
but only to a frequency independent attenuation by the time independent factor  $\cos(Δ\mathbf{ϕ_{\rm T} })$. 
  • The reason for this less serious signal change than in the case of a frequency offset is that in this case the time  $t$  is missing in the argument of the cosine function.


Impairment of synchronous demodulation due to phase offset

$\text{Example 2:}$  The diagram shows

  • the signals  $q(t)$  and  $s(t)$  at the transmitter above,
  • and the receiver-side signals  $b(t)$  and  $v(t)$  below.


Due to the phase offset of  $Δ \mathbf{ϕ_{\rm T} } = π/3\ (60^\circ)$,  the sink signal  $v(t)$  is only half the size of the source signal  $q(t)$.


However,  the waveform of  $q(t)$  is preserved in the output signal  $v(t)$.


Influence of linear channel distortions


In the section  "Attenuation distortions"  of the book  "Linear and Time Invariant Systems"  it was already suggested that the entire communication system   – consisting of the  $\rm M$odulator,  the channel   $($⇒   German:  $\rm K$anal$)$   and the  $\rm D$emodulator –  can be completely described by the resulting frequency response  $H_{\rm MKD}(f)$ 

Amplitude modulation and Synchronous demodulation
  • if either the system is free of distortion,  or
  • only linear distortions arise with respect to the signals  $q(t)$  and  $v(t)$.


In contrast,  $\text{nonlinear distortions}$  are not captured by this diagram

  1. because the multiplicative relationship  $V(f) = Q(f)\cdot H_{\rm MKD}(f)$  does not allow for the emergence of new frequencies;
  2. if  $Q(f_0) = 0$,  then  $V(f_0) = 0$  will always hold as well.


The above conditions are fulfilled for the following system variant:

  • The modulator generates a "DSB-AM"  (with or without carrier)  around the carrier frequency  $f_{\rm T}$.
  • The  "channel"  is describable by the frequency response  $H_{\rm K}(f)$  with band-pass characteristics and its bandwidth is sufficient.
  • The  "synchronous demodulator"  is synchronous in frequency and phase and the receiver filter  $H_{\rm E}(f)$  is ideal (rectangular).


$\text{Definition:}$  With these favorable conditions,  the  »resulting frequency response«  of  "modulator-channel-demodulator"  is:

$$H_{\rm MKD}(f) = {1}/{2} \cdot \big[ H_{\rm K}(f + f_{\rm T}) + H_{\rm K}(f - f_{\rm T})\big] \hspace{0.05cm}.$$


  • If  $\vert H_{\rm MKD}(f) \vert $  is not constant in the range of the signal bandwidth, the different spectral components of the source signal  $q(t)$  are also transmitted differently   ⇒   "attenuation distortions".
  • Similarly, "phase distortions" may occur if the phase function  $\text{arc} \ H_{\rm MKD}(f) $  is nonlinear in  $f$.


$\text{Example 3:}$  The diagram illustrates the above calculation rule for the resulting system function.

Influence of linear channel distortions
  • From the asymmetric band-pass  $H_{\rm K}(f)$  – related to the carrier frequency  $f_{\rm T}$  – we get the symmetric function  $H_{\rm MKD}(f)$  in the low-frequency range  $($around  $f = 0)$.


  • If the source signal consists of two frequency components – indicated in the graph by the red marker arrows – the spectral line at  $f_2$  is more attenuated than the frequency  $f_1$  ⇒  linear attenuation distortions arise.


  • The fact that  $H_{\rm MKD}(f)$  also includes components around  $±2f_{\rm T}$  is not a major concern. These do not affect low-pass considerations.


Influence of noise interference


Now we wish to clarify the extent to which the transmission quality is affected by a stochastic interference or a noise signal $n(t)$.  We will be assuming the following scenario,  which has already been presented in the "Investigations at the AWGN channel"  section.

Investigating the AWGN channel

In particular,  we make the following assumptions:

  • Only double-sideband amplitude modulation with modulation depth  $m$  and an ideal synchronous demodulator without phase and frequency offsets are considered.
  • According to the extended AWGN channel model,  where  $α_{\rm K}$  is a frequency-independent transmission factor of the channel and the interference  $($noise$)$  signal  $n(t)$  models white noise with two-sided noise power density  $N_0/2$,  the following holds for the received signal:
$$r(t) = \alpha_{\rm K} \cdot s(t) + n(t) \hspace{0.05cm}.$$
  • Representing a source signal  $q(t)$  of bandwidth  $B_{\rm NF}$ , a cosine-shaped message signal of frequency  $B_{\rm NF}$  is assumed here:
$$q(t) = A_{\rm N} \cdot \cos(2 \pi \cdot B_{\rm NF} \cdot t ) \hspace{0.05cm}.$$

With these assumptions,  the sink signal is  $v(t) = \alpha_{\rm K} \cdot q(t) + \varepsilon(t) \hspace{0.05cm}$,  where the cause of the stochastic component  $ε(t)$  at the output is the band-pass noise  $n(t)$  at the input of the synchronous demodulator.

$\text{Definition:}$ 

  • The  »signal-to-noise power ratio«  at the sink is used as a quantitative measure of transmission quality,  and is expressed here in terms of the power of  $q(t)$  and  $ε(t)$, respectively:
$$\rho_v = \frac{\alpha_{\rm K}^2 \cdot P_q}{P_\varepsilon} \hspace{0.05cm}.$$
  • In the following, we will refer to this ratio as the «sink SNR«  $ρ_v$  and the logarithmic representation  $10 · \lg ρ_v$  as the »sink SNR« in dB.


Calculating the noise power


We first calculate the power  $P_ε$  of the error signal  $ε(t)$,  which we refer to as the  "noise power" for simplicity.

The error signal  $ε(t)$  at the output is obtained from the noise signal  $n(t)$  at the input by

  • multiplying by  $z_{\rm E}(t) = 2 · \cos(ω_{\rm T} · t + \mathbf{ϕ_{\rm T} })$  and
  • a subsequent  (ideal)  low-pass filtering to the low-frequency range   $\pm B_{\rm NF}$.


For the power-spectral density   $\rm (PSD)$   ${\it Φ_ε}'(f)$  without considering the low-pass filter,  it holds with  ${\it Φ}_n(f) = N_0/2$:

$${\it \Phi}_\varepsilon \hspace{-0.10cm} '(f) = {\it \Phi}_n (f) \star {\it \Phi}_{z {\rm E}}(f) \hspace{0.05cm}.$$

In the books  "Signal Representation"  and  "Theory of Stochastic Signals”,  it was shown that

  • the $\text{spectrum}$  of a cosine signal  $x(t) = A · \cos(2πf_{\rm T}t)$  is given by:
$$X(f) = \frac{A}{2}\cdot \delta(f + f_{\rm T}) + \frac{A}{2}\cdot \delta(f - f_{\rm T}) \hspace{0.05cm}, $$
$$ {\it \Phi}_x (f) = \frac{A^2}{4}\cdot \delta(f + f_{\rm T}) + \frac{A^2}{4}\cdot \delta(f - f_{\rm T}) \hspace{0.05cm}.$$

Applied to the receiver-side carrier signal  $z_{\rm E}(t)$,  with  $A = 2$:

$${\it \Phi}_{z {\rm E}}(f)= \delta(f + f_{\rm T}) + \delta(f - f_{\rm T}) \hspace{0.05cm},$$

and this is independent of phase  (since all phase relationships are lost in the power-spectral density).

$\text{Example 4:}$   Considering that  ${\it Φ}_n(f)$  is constant for all frequencies   ⇒   "White Noise",  we get:

$${\it \Phi}_\varepsilon \hspace{-0.10cm} '(f) = {\it \Phi}_n (f + f_{\rm T}) + {\it \Phi}_n (f - f_{\rm T}) = 2 {\it \Phi}_n (f) = N_0 \hspace{0.05cm}.$$
  • The power-spectral density  $\rm (PSD)$  after the low-pass filter is exactly as large for  $ \vert f \vert < B_{\rm NF}$  and zero beyond this range.
$${\it \Phi}_\varepsilon (f) = \left\{ \begin{array}{c} N_0 \\ 0 \\ \end{array} \right.\quad \begin{array}{*{10}c} {\rm{for} }\\ \\ \end{array}\begin{array}{*{20}c} \vert f \vert < B_{\rm NF} \hspace{0.05cm}, \\ {\rm otherwise} \hspace{0.05cm}. \\ \end{array}$$
  • By integration,  we obtain the power  $P_ε = 2N_0 · B_{\rm NF}$.   Thus,  with this intermediate result,  the  "sink SNR"  can be given as:
$$\rho_v = \frac{\alpha_{\rm K}^2 \cdot P_q}{P_\varepsilon} = \frac{\alpha_{\rm K}^2 \cdot P_q}{N_0 \cdot B_{\rm NF} } \hspace{0.05cm}.$$

In the next section,  we still establish the relationship between the power  $P_q$  of the source signal and the transmit power  $P_{\rm S}$.

Relationship between the powers of source signal and transmitted signal


To characterise the relationship between the sink–SNR  $\rho_v$  and the transmit power  $P_{\rm S}$,  we first need the relationships between

  • source signal  $q(t)$   ⇒   power  $P_q$,  and
  • transmitted signal  $s(t)$   ⇒   transmit power  $P_{\rm S}$.


$\text{Anticipated result:}$  In the case of  »DSB–AM with carrier«  the following holds true for a modulation depth  $m$:

$$P_{\rm S} = { P_q}/{2} \cdot \hspace{0.05cm} \left( 1 + {2}/{m^2}\right)\hspace{0.05cm}.$$
  • Note that this equation is only applicable when  $q(t)$  describes a harmonic oscillation.
  • The  »DSB–AM without carrier«  is included in the equation as a special case for  $m \to \infty$.


$\text{Proof:}$  Cosine oscillations are assumed in each case,  i.e. the following equations:

$$q(t) = A_{\rm N} \cdot \cos(\omega_{\rm N} \cdot t ) \hspace{0.05cm},$$
$$s(t) = \big[ q(t) + A_{\rm T}\big] \cdot \cos(\omega_{\rm T} \cdot t ) = A_{\rm T} \cdot \cos(\omega_{\rm T} \cdot t ) + {A_{\rm N} }/{2}\cdot \cos\big [(\omega_{\rm T}+ \omega_{\rm N}) \cdot t \big] + {A_{\rm N} }/{2}\cdot \cos\big[(\omega_{\rm T}- \omega_{\rm N}) \cdot t \big]\hspace{0.05cm}.$$
  • The power of the source signal,  in relation to a resistance of  $1 \ \rm Ω$,  with the period  $T_{\rm N}$,  is:
$$P_{q} = \frac{1}{T_{\rm N} }\cdot\int_{0}^{ T_{\rm N} } {q^2(t)}\hspace{0.1cm}{\rm d}t = \frac{A_{\rm N}^2}{T_{\rm N} }\cdot\int_{0}^{T_{\rm N} } {\cos^2(2 \pi\cdot{t}/{T_{\rm N} })}\hspace{0.1cm}{\rm d}t = \frac{A_{\rm N}^2}{2}\hspace{0.05cm}.$$
  • Accordingly,  for the power of the transmitted signal,  one obtains:
$$P_{\rm S} = \frac{A_{\rm T}^2}{2} + \frac{(A_{\rm N}/2)^2}{2} + \frac{(A_{\rm N}/2)^2}{2} = \frac{A_{\rm T}^2}{2} + \frac{A_{\rm N}^2}{4} \hspace{0.3cm}\Rightarrow \hspace{0.3cm} P_{\rm S} = {1}/{2} \cdot \left( P_q + A_{\rm T}^2 \right) \hspace{0.05cm}.$$
  • This equation is valid for both  »DSB–AM without carrier«  $(A_{\rm T} = 0)$  and  »DSB–AM with carrier«.  Since  $q(t)$  was assumed to be a harmonic oscillation,  »DSB–AM without carrier«  given a modulation depth  $m = A_{\rm N}/A_{\rm T}$,  this can also be written as:
$$P_{\rm S} = {A_{\rm N}^2}/{4} \cdot \left( 1 +{2A_{\rm T}^2}/{A_{\rm N}^2} \right)= P_q/2 \cdot \left( 1 +{2}/{m^2} \right)\hspace{0.05cm}.\hspace{5.4cm}{\rm q.e.d.}$$


Sink SNR and the performance parameter


With the results of the last three sections, the  »sink SNR of double-sideband amplitude modulation«  can be written as:

$$\rho_v = \frac{\alpha_{\rm K}^2 \cdot P_q}{P_\varepsilon} = \frac{\alpha_{\rm K}^2 \cdot P_{\rm S} }{N_0 \cdot B_{\rm NF} } \cdot \frac{1}{1 + {2}/{m^2} } \hspace{0.05cm}.$$


Linear und double–logarithmic representation of sink SNR

The following is a detailed discussion of this equation.  Already in the chapter  "Investigations at the AWGN channel"  it was justified why it makes sense to state the sink SNR  $ρ_v$  as a function of the performance parameter  $ξ$  named below:

$$\xi = \frac{\alpha_{\rm K}^2 \cdot P_{\rm S}}{N_0 \cdot B_{\rm NF}} \hspace{0.5cm}\Rightarrow \hspace{0.5cm} \rho_v = \frac{\xi}{1 + {2}/{m^2}} \hspace{0.05cm}.$$

The two graphs show the corresponding curves

  • linear on the left,
  • double-logarithmic on the right.


The curves are to be interpreted as follows:

  • For the system variant  »DSB–AM without carrier«  with  $m → ∞$  the above equation yields the simple relationship  $ρ_v = ξ$.  This gives the bisector of the angle for both the linear and double logarithmic plots.
  • A greater transmit power  $P_{\rm S}$  leads to a better sink SNR,  as does a larger transmission factor   $α_{\rm K}$  (⇒ lower attenuation).  However,   $10 · \lg ρ_v$  is also increased by a smaller noise power density  $N_0$  and a smaller bandwidth  $B_{\rm NF}$,  other things being equal.
  • For  »DSB–AM with carrier«  with a modulation depth  $m$:
$$\rho_v = \xi \cdot \frac{1}{1 + {2}/{m^2}}\hspace{0.3cm}\Rightarrow \hspace{0.3cm} 10 \cdot {\rm lg }\hspace{0.1cm}\rho_v = 10 \cdot {\rm lg }\hspace{0.1cm}\xi - 10 \cdot {\rm lg }\hspace{0.1cm} \left({1 + {2}/{m^2}}\right)\hspace{0.05cm}.$$
  • In the double logarithmic representation  (see right graph),  this leads to a downward parallel shift of the curves, 
    e.g. at  $m = 1$  by  $4.77$ dB  and at  $m = 0.5$  by  $9.54$ dB.
  • All statements are valid under the assumption of an ideal synchronous demodulator.  In this case,  the  »DSB–AM with carrier«  method actually makes no sense.  The added carrier would only lead to an unnecessarily large transmit power and could not be used for demodulation.
  • The curves are valid for perfect frequency and phase synchronization.  However,  in order to be able to determine the parameters  $f_{\rm T}$  and  $\mathbf{ϕ_{\rm T} }$  from the received signal  $r(t)$  with less effort,  a small carrier component in the transmitted signal makes sense.
  • When  $m = 3$,  it is only an insignificant deterioration of less than one dB compared to  »DSB–AM without carrier«.

Exercises for the chapter


Exercise 2.4: Frequency and Phase Offset

Exercise 2.4Z: Low-pass influence with Synchronous Demodulation

Exercise 2.5: DSB-AM via a Gaussian Channel

Exercise 2.5Z: Linear Distortions with DSB-AM

Exercise 2.6: Free Space Attenuation

Exercise 2.6Z: Signal-to-Noise-Ratio (SNR)