Modulation Methods/Single-Sideband Modulation - Revision history

Hwang at 12:43, 18 January 2023

2023-01-18T12:43:19Z

Guenter at 09:50, 13 January 2023

2023-01-13T09:50:15Z

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Hwang at 21:58, 2 January 2023

2023-01-02T21:58:24Z

Hwang at 21:48, 2 January 2023

2023-01-02T21:48:52Z

Reed at 14:07, 9 April 2022

2022-04-09T14:07:24Z

Reed at 14:05, 9 April 2022

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Reed at 14:03, 9 April 2022

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Reed at 15:51, 31 March 2022

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Reed at 15:49, 31 March 2022

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Reed at 15:48, 31 March 2022

2022-03-31T15:48:33Z

@@ Line 65: / Line 65: @@
 *For SSB–AM the convolution products overlap with the left and the right Dirac delta function at every frequency.&nbsp; These are denoted by&nbsp; "+"&nbsp; and&nbsp; "&ndash;"&nbsp; respectively in the [[Modulation_Methods/Synchronous_Demodulation#Description_in_the_frequency_domain|$\text{corresponding DSB graph}$]].
-*In contrast,&nbsp; for USB,&nbsp; only the convolution with the Dirac line at &nbsp;$ -f_{\rm T}$&nbsp; yields the &nbsp;$V(f)$&nbsp; component for positive frequencies,&nbsp; and for LSB modulation it is the convolution with the Dirac delta function &nbsp;$δ(f - f_{\rm T})$.
+*In contrast,&nbsp; for USB,&nbsp; only the convolution with the Dirac delta line at &nbsp;$ -f_{\rm T}$&nbsp; yields the &nbsp;$V(f)$&nbsp; component for positive frequencies,&nbsp; and for LSB modulation it is the convolution with the Dirac delta function &nbsp;$δ(f - f_{\rm T})$.
 *In the case of DSB–AM,&nbsp; $v(t) = q(t)$&nbsp; is reached with the receiver-side carrier signal &nbsp;$z_{\rm E}(t) =  2 · \cos(ω_{\rm T} · t)$.&nbsp; In contrast,&nbsp; the carrier amplitude must be increased to &nbsp;$A_{\rm T} = 4$ &nbsp; for SSB-AM.

@@ Line 125: / Line 125: @@
 This graph makes the following clear:
-#There are more similarities between the upper and lower signal than between the first two,&nbsp; because
+#There are more similarities between the upper and lower signal than between the first two,&nbsp; because '''KORREKTUR: because?'''
 #The comparison of a DSB-AM and a SSB-AM should be made for the same numerical value of &nbsp;$m$&nbsp; and &nbsp;$\mu$&nbsp; if possible.
 #For each value&nbsp;  $\mu$&nbsp; of the sideband-to-carrier ratio,&nbsp;  the  envelope demodulation with SSB-AM leads to severe distortions.

@@ Line 12: / Line 12: @@
-The so-called&nbsp; '''single-sideband amplitude modulation'''&nbsp; $\rm (SSB–AM)$&nbsp; takes advantage of this property by transmitting only one of these sidebands,&nbsp; either the upper sideband (USB) or the lower sideband (LSB).&nbsp; This reduces the required bandwidth by half compared to DSB-AM.
+The so-called&nbsp; &raquo;'''single-sideband amplitude modulation'''&laquo;&nbsp; $\rm (SSB–AM)$&nbsp; takes advantage of this property by transmitting only one of these sidebands,&nbsp; either the upper sideband (USB) or the lower sideband (LSB).&nbsp; This reduces the required bandwidth by half compared to DSB-AM.
 The graph illustrates single-sideband amplitude modulation in the frequency domain and simultaneously presents a realization of a SSB modulator.&nbsp;
@@ Line 35: / Line 35: @@
 *In accordance with a hierarchical structure,&nbsp; three telephone channels&nbsp;  – each band-limited to the range from &nbsp;$300\ \rm Hz$&nbsp; to &nbsp;$3.4 \ \rm kHz$&nbsp; –&nbsp; were initially combined to form a preliminary grouping with a bandwidth of &nbsp;$12 \ \rm  kHz$&nbsp;.
 *In order to be able to accommodate three channels in &nbsp;$12 \ \rm  kHz$&nbsp; with a safety margin,&nbsp; only one sideband&nbsp; (LSB or USB)&nbsp; of each telephone channel was considered.
-*Using further combinations,&nbsp; the long distance traffic system&nbsp; '''V10800'''&nbsp; with up to&nbsp; $10800$&nbsp; voice channels and a total bandwidth of &nbsp;$60 \ \rm  MHz$&nbsp; was thus realized.}}
+*Using further combinations,&nbsp; the long distance traffic system&nbsp; &raquo;'''V10800'''&laquo;&nbsp; with up to&nbsp; $10800$&nbsp; voice channels and a total bandwidth of &nbsp;$60 \ \rm  MHz$&nbsp; was thus realized.}}
 {{BlaueBox|TEXT=
 $\text{Conclusion:}$&nbsp;
-*The great advantage of a&nbsp; $\rm SSB-AM$&nbsp; is that it has '''half the bandwidth''' compared to&nbsp; $\rm DSB-AM$.
+*The great advantage of a&nbsp; $\rm SSB-AM$&nbsp; is that it has &raquo;'''half the bandwidth'''&laquo; compared to&nbsp; $\rm DSB-AM$.
 *The trade-off with disadvantages required for this advantage is explained in the next sections.}}
@@ Line 46: / Line 46: @@
 ==Synchronous demodulation of a SSB-AM signal==
 <br>
-Let us now consider a SSB-AM modulated signal and a&nbsp; [[Modulation_Methods/Synchronous_Demodulation|synchronous demodulator]]&nbsp; at the receiver.
+Let us now consider a SSB-AM modulated signal and a&nbsp; [[Modulation_Methods/Synchronous_Demodulation|$\text{synchronous demodulator}$]]&nbsp; at the receiver.
 *Perfect frequency and phase synchronization will be assumed.
 *Without affecting generality,&nbsp; in the rest of this section we will always let &nbsp;${ ϕ}_{\rm T} = 0$&nbsp; &rArr; &nbsp; cosine carrier.
@@ Line 52: / Line 52: @@
 [[File:EN_Mod_T_2_4_S2_neu.png |right|frame| Synchronous demodulation of a SSB-AM signal]]
 <br>
-A comparison with the&nbsp; [[Modulation_Methods/Synchronous_Demodulation|characteristics of the synchronous demodulator for DSB-AM]]&nbsp;  shows the follows similarities and differences:
+A comparison with the&nbsp; [[Modulation_Methods/Synchronous_Demodulation|$\text{characteristics of the synchronous demodulator for DSB-AM}$]]&nbsp;  shows the follows similarities and differences:
 *The spectrum &nbsp;$V(f)$&nbsp; of the sink signal results in both cases from the convolution of the spectra &nbsp;$R(f)$&nbsp; and &nbsp;$Z_{\rm E}(f)$,&nbsp; the latter being composed of two Dirac delta functions at&nbsp;$±f_{\rm T}$.
-*For SSB–AM the convolution products overlap with the left and the right Dirac delta function at every frequency.&nbsp; These are denoted by&nbsp; "+"&nbsp; and&nbsp; "&ndash;"&nbsp; respectively in the [[Modulation_Methods/Synchronous_Demodulation#Description_in_the_frequency_domain|corresponding DSB graph]].
+*For SSB–AM the convolution products overlap with the left and the right Dirac delta function at every frequency.&nbsp; These are denoted by&nbsp; "+"&nbsp; and&nbsp; "&ndash;"&nbsp; respectively in the [[Modulation_Methods/Synchronous_Demodulation#Description_in_the_frequency_domain|$\text{corresponding DSB graph}$]].
-*In contrast,&nbsp; for USB,&nbsp; only the convolution with the Dirac line at &nbsp;$ -f_{\rm T}$&nbsp; yields the &nbsp;$V(f)$&nbsp; component for positive frequencies,&nbsp; and for LSB modulation it is the convolution with the Dirac function &nbsp;$δ(f - f_{\rm T})$.
+*In contrast,&nbsp; for USB,&nbsp; only the convolution with the Dirac line at &nbsp;$ -f_{\rm T}$&nbsp; yields the &nbsp;$V(f)$&nbsp; component for positive frequencies,&nbsp; and for LSB modulation it is the convolution with the Dirac delta function &nbsp;$δ(f - f_{\rm T})$.
@@ Line 66: / Line 66: @@
 <br clear=all>
 {{BlaueBox|TEXT=
-$\text{Conclusion:}$&nbsp; If there is a frequency offset between the carrier signals &nbsp;$z(t)$&nbsp; and &nbsp;$z_{\rm E}(t)$,&nbsp; strong nonlinear distortion always occurs, i.e.,&nbsp; regardless of whether DSB-AM or SSB-AM is present.&nbsp; For the &nbsp; '''implementation of a synchronous demodulator'''&nbsp; a &nbsp; '''perfect frequency synchronization'''&nbsp; is essential. }}
+$\text{Conclusion:}$&nbsp; If there is a frequency offset between the carrier signals &nbsp;$z(t)$&nbsp; and &nbsp;$z_{\rm E}(t)$,&nbsp; strong nonlinear distortion always occurs, i.e.,&nbsp; regardless of whether DSB-AM or SSB-AM is present.&nbsp; For the &nbsp; &raquo;'''implementation of a synchronous demodulator'''&laquo;&nbsp; a &nbsp; &raquo;'''perfect frequency synchronization'''&laquo;&nbsp; is essential. }}
@@ Line 84: / Line 84: @@
 $\text{From this equation we can see:}$
 *The two delay times &nbsp;$τ_1 = Δ{\mathbf ϕ}_{\rm T}/ω_1$&nbsp; and &nbsp;$τ_2 = Δ{\mathbf ϕ}_{\rm T}/ω_2$&nbsp; are different.
-*This means that a phase offset in single sideband amplitude modulation&nbsp; (USB–AM or LSB–AM)&nbsp; leads to&nbsp; '''phase distortions'''&nbsp; (i.e., to linear distortions).
+*This means that a phase offset in single sideband amplitude modulation&nbsp; (USB–AM or LSB–AM)&nbsp; leads to&nbsp; &raquo;'''phase distortions'''&laquo;&nbsp; (i.e., to linear distortions).
 *A positive value of&nbsp; $Δ{\mathbf ϕ}_{\rm T}$&nbsp; will cause
 **positive values of &nbsp;$τ_1$&nbsp; and &nbsp;$τ_2$&nbsp; (i.e., lagging signals with respect to the cosine)&nbsp; for USB,&nbsp; and
@@ Line 96: / Line 96: @@
 ==Sideband-to-carrier ratio==
 <br>
-An important parameter for DSB–AM is the&nbsp; [[Modulation_Methods/Double-Sideband_Amplitude_Modulation#Double-Sideband_Amplitude_Modulation_with_carrier|modulation depth]] &nbsp;$m = q_{\rm max}/A_{\rm T}$.&nbsp; In the special case of a harmonic oscillation, &nbsp;$m = A_{\rm N}/A_{\rm T}$&nbsp; holds and one obtains the spectrum &nbsp;$S_+(f)$&nbsp; of the analytical signal corresponding to the upper graph.&nbsp; Please note the normalization with respect to  &nbsp;$A_{\rm T}$.
+An important parameter for DSB–AM is the&nbsp; [[Modulation_Methods/Double-Sideband_Amplitude_Modulation#Double-Sideband_Amplitude_Modulation_with_carrier|$\text{modulation depth}$]] &nbsp;$m = q_{\rm max}/A_{\rm T}$.&nbsp; In the special case of a harmonic oscillation, &nbsp;$m = A_{\rm N}/A_{\rm T}$&nbsp; holds and one obtains the spectrum &nbsp;$S_+(f)$&nbsp; of the analytical signal corresponding to the upper graph.&nbsp; Please note the normalization with respect to  &nbsp;$A_{\rm T}$.
 [[File:EN_Mod_T_2_4_S4a.png|right|frame|Spectra for DSB-AM,&nbsp; USB-AM]]
@@ Line 106: / Line 106: @@
 In a similar way,&nbsp; this can be written as
 :$$s_+(t) = A_{\rm T} \cdot \left({\rm e}^{{\rm j}\hspace{0.03cm}\cdot \hspace{0.05cm} \omega_{\rm T} \hspace{0.03cm}\cdot \hspace{0.05cm}t }   + \mu \cdot {\rm e}^{{\rm j}\hspace{0.03cm}\cdot \hspace{0.03cm} (\omega_{\rm T} + \omega_{\rm N}) \hspace{0.03cm}\cdot \hspace{0.05cm}t }\right)$$
-now using the &nbsp; '''sideband–to–carrier ratio'''&nbsp;:
+now using the &nbsp; &raquo;'''sideband–to–carrier ratio'''&laquo;&nbsp;:
 :$$\mu = \frac{A_{\rm N}}{2 \cdot A_{\rm T}} \hspace{0.05cm}.$$
 If the source signal is not a harmonic oscillation,&nbsp; it is difficult to specify this quantity.
@@ Line 133: / Line 133: @@
 <br>
 The key advantage of&nbsp;  $\text{SSB&ndash;AM}$&nbsp; compared to&nbsp;  $\text{DSB&ndash;AM}$&nbsp; is the bandwidth requirement,&nbsp; smaller by a factor of &nbsp;$2$.&nbsp; However,&nbsp; for the half bandwidth in SSB-AM,&nbsp; some disadvantages have to be accepted,&nbsp; which will be investigated in the exercises for this section:
-*The information about the source signal &nbsp;$q(t)$&nbsp; is,&nbsp; in contrast to DSB-AM,&nbsp; no longer exclusively in the amplitude,&nbsp; but equally in the phase&nbsp; (see [[Aufgaben:Exercise_2.11:_Envelope_Demodulation_of_an_SSB_Signal| Exercise 2.11]]).
+*The information about the source signal &nbsp;$q(t)$&nbsp; is,&nbsp; in contrast to DSB-AM,&nbsp; no longer exclusively in the amplitude,&nbsp; but equally in the phase&nbsp; (see [[Aufgaben:Exercise_2.11:_Envelope_Demodulation_of_an_SSB_Signal|"Exercise 2.11"]]).
 *Synchronous demodulation of a SSB–AM signal leads to phase distortions if there is a phase offset between the carrier signals at the transmitter and the receiver.
-*The application of&nbsp; $\text{envelope demodulation}$&nbsp; with&nbsp; "USB&ndash;AM"&nbsp; or&nbsp;   "LSB&ndash;AM"&nbsp; thus always leads to strong nonlinear distortions&nbsp; (see [[Aufgaben:Exercise_2.11Z:_Once_again_SSB-AM_and_Envelope_Demodulator|Exercise 2.11Z]]).
+*The application of&nbsp; $\text{envelope demodulation}$&nbsp; with&nbsp; "USB&ndash;AM"&nbsp; or&nbsp;   "LSB&ndash;AM"&nbsp; thus always leads to strong nonlinear distortions&nbsp; (see [[Aufgaben:Exercise_2.11Z:_Once_again_SSB-AM_and_Envelope_Demodulator|"Exercise 2.11Z"]]).
 As with DSB–AM and synchronous demodulation,&nbsp; the following statements also apply here:
-*Attenuation distortions of the channel also lead only to (linear)&nbsp; attenuation distortions with respect to the sink signal.&nbsp; No nonlinear distortions arise&nbsp; (see&nbsp; [[Aufgaben:Exercise_2.10:_SSB-AM_with_Channel_Distortions|Exercise A2.10]]).
+*Attenuation distortions of the channel also lead only to (linear)&nbsp; attenuation distortions with respect to the sink signal.&nbsp; No nonlinear distortions arise&nbsp; (see&nbsp; [[Aufgaben:Exercise_2.10:_SSB-AM_with_Channel_Distortions|"Exercise 2.10"]]).
 *SSB–AM without a carrier shows the exact same noise behaviour as DSB-AM without a carrier.&nbsp; The advantage of the smaller RF bandwidth is cancelled out by the necessary level matching.
-*A SSB–AM with a sideband-to-carrier ratio &nbsp;$\mu$&nbsp; shows similar noise behaviour as in DSB-AM with modulation depth &nbsp;$m = \sqrt{2} · \mu$&nbsp; (see&nbsp; [[Aufgaben:Exercise_2.10Z:_Noise_with_DSB-AM_and_SSB-AM|Exercise 2.10Z]]).
+*A SSB–AM with a sideband-to-carrier ratio &nbsp;$\mu$&nbsp; shows similar noise behaviour as in DSB-AM with modulation depth &nbsp;$m = \sqrt{2} · \mu$&nbsp; (see&nbsp; [[Aufgaben:Exercise_2.10Z:_Noise_with_DSB-AM_and_SSB-AM|"Exercise 2.10Z"]]).
 *In any case,&nbsp; it should be noted that SSB-AM with carrier is not very useful due to the nonlinear distortions brought about with envelope demodulation.

@@ Line 134: / Line 134: @@
 The key advantage of&nbsp;  $\text{SSB&ndash;AM}$&nbsp; compared to&nbsp;  $\text{DSB&ndash;AM}$&nbsp; is the bandwidth requirement,&nbsp; smaller by a factor of &nbsp;$2$.&nbsp; However,&nbsp; for the half bandwidth in SSB-AM,&nbsp; some disadvantages have to be accepted,&nbsp; which will be investigated in the exercises for this section:
 *The information about the source signal &nbsp;$q(t)$&nbsp; is,&nbsp; in contrast to DSB-AM,&nbsp; no longer exclusively in the amplitude,&nbsp; but equally in the phase&nbsp; (see [[Aufgaben:Exercise_2.11:_Envelope_Demodulation_of_an_SSB_Signal| Exercise 2.11]]).
-*Synchronous demodulation of a SSB–AM signal leds to phase distoritions if there is a phase offset between the carrier signals at the transmitter and the receiver.
+*Synchronous demodulation of a SSB–AM signal leads to phase distortions if there is a phase offset between the carrier signals at the transmitter and the receiver.
 *The application of&nbsp; $\text{envelope demodulation}$&nbsp; with&nbsp; "USB&ndash;AM"&nbsp; or&nbsp;   "LSB&ndash;AM"&nbsp; thus always leads to strong nonlinear distortions&nbsp; (see [[Aufgaben:Exercise_2.11Z:_Once_again_SSB-AM_and_Envelope_Demodulator|Exercise 2.11Z]]).

@@ Line 111: / Line 111: @@
 *Here one can use the following approximation:
 :$$\mu \approx \frac{q_{\rm max}}{2 \cdot A_{\rm T}} \hspace{0.3cm}{\rm with}\hspace{0.3cm} q_{\rm max} = \max_{t} \hspace{0.05cm} |q(t)|  \hspace{0.05cm}.$$
-*Thus, &nbsp;$\mu \approx m/2.$&nbsp; However,&nbsp; a comparison of DSB-AM to SSB-AM should always be made for the same numerical value of &nbsp;$m$&nbsp;and &nbsp;$,\mu$&nbsp; resp..
+*Thus, &nbsp;$\mu \approx m/2.$&nbsp; However,&nbsp; a comparison of DSB-AM to SSB-AM should always be made for the same numerical value of &nbsp;$m$&nbsp;and &nbsp;$,\mu$&nbsp; resp.

@@ Line 26: / Line 26: @@
 <br clear=all>
 {{GraueBox|TEXT=
-$\text{Example 1:}$&nbsp; For &nbsp;$f_{\rm T} = 100 \ \rm kHz$,&nbsp; $B_{\rm NF} = 3 \ \rm kHz$&nbsp; we obtain  from the DSB an&nbsp; "USB-AM with carrier"&nbsp; if the filter cuts out all frequencies below  &nbsp;$99.999\text{...} \ \rm kHz$&nbsp;.
+$\text{Example 1:}$&nbsp; For &nbsp;$f_{\rm T} = 100 \ \rm kHz$,&nbsp; $B_{\rm NF} = 3 \ \rm kHz$&nbsp;, DSB gives us an&nbsp; "USB-AM with carrier"&nbsp; if the filter cuts out all frequencies below  &nbsp;$99.999\text{...} \ \rm kHz$&nbsp;.
 *If the lower cut-off frequency &nbsp;$f_{\rm U}$&nbsp; is larger than &nbsp;$f_{\rm T}$&nbsp; by an&nbsp; (arbitrarily)&nbsp; small&nbsp; "epsilon",&nbsp;  we get an&nbsp; "USB-AM without carrier".
 *A&nbsp; "LSB with/without carrier"&nbsp; can be realised accordingly with &nbsp;$f_{\rm O} =100 \ \rm kHz$&nbsp; and &nbsp;$f_{\rm O} = 99.999\text{...} \ \rm kHz$,&nbsp; resp. }}

@@ Line 2: / Line 2: @@
 {{Header
 |Untermenü=Amplitude Modulation and Demodulation
-|Vorherige Seite=Hüllkurvendemodulation
+|Vorherige Seite=Envelope Demodulation
-|Nächste Seite=Weitere AM–Varianten
+|Nächste Seite=Further AM Variants
 }}
 ==Description in the frequency domain==

@@ Line 133: / Line 133: @@
 <br>
 The key advantage of&nbsp;  $\text{SSB&ndash;AM}$&nbsp; compared to&nbsp;  $\text{DSB&ndash;AM}$&nbsp; is the bandwidth requirement,&nbsp; smaller by a factor of &nbsp;$2$.&nbsp; However,&nbsp; for the half bandwidth in SSB-AM,&nbsp; some disadvantages have to be accepted,&nbsp; which will be investigated in the exercises for this section:
-*The information about the source signal &nbsp;$q(t)$&nbsp; is,&nbsp; in contrast to DSB-AM,&nbsp; no longer exclusively in the amplitude,&nbsp; but equally in the phase&nbsp; (see [[Aufgaben:Exercise_2.11:_Envelope_Demodulation_of_an_ESB_Signal| Exercise 2.11]]).
+*The information about the source signal &nbsp;$q(t)$&nbsp; is,&nbsp; in contrast to DSB-AM,&nbsp; no longer exclusively in the amplitude,&nbsp; but equally in the phase&nbsp; (see [[Aufgaben:Exercise_2.11:_Envelope_Demodulation_of_an_SSB_Signal| Exercise 2.11]]).
 *Synchronous demodulation of a SSB–AM signal leds to phase distoritions if there is a phase offset between the carrier signals at the transmitter and the receiver.
-*The application of&nbsp; $\text{envelope demodulation}$&nbsp; with&nbsp; "USB&ndash;AM"&nbsp; or&nbsp;   "LSB&ndash;AM"&nbsp; thus always leads to strong nonlinear distortions&nbsp; (see [[Aufgaben:Exercise_2.11Z:_Once_again_ESB-AM_and_Envelope_Demodulator|Exercise 2.11Z]]).
+*The application of&nbsp; $\text{envelope demodulation}$&nbsp; with&nbsp; "USB&ndash;AM"&nbsp; or&nbsp;   "LSB&ndash;AM"&nbsp; thus always leads to strong nonlinear distortions&nbsp; (see [[Aufgaben:Exercise_2.11Z:_Once_again_SSB-AM_and_Envelope_Demodulator|Exercise 2.11Z]]).
@@ Line 152: / Line 152: @@
 [[Aufgaben:Exercise_2.10Z:_Noise_with_DSB-AM_and_SSB-AM|Exercise 2.10Z: Noise with DSB-AM and SSB-AM]]
-[[Aufgaben:Exercise_2.11:_Envelope_Demodulation_of_an_ESB_Signal|Exercise 2.11: Envelope Demodulation of an SSB signal]]
+[[Aufgaben:Exercise_2.11:_Envelope_Demodulation_of_an_SSB_Signal|Exercise 2.11: Envelope Demodulation of an SSB signal]]
-[[Aufgaben:Exercise_2.11Z:_Once_again_ESB-AM_and_Envelope_Demodulator|Exercise 2.11Z: Once again SSB-AM and Envelope Demodulator]]
+[[Aufgaben:Exercise_2.11Z:_Once_again_SSB-AM_and_Envelope_Demodulator|Exercise 2.11Z: Once again SSB-AM and Envelope Demodulator]]
 {{Display}}