Modulation Methods/Single-Sideband Modulation - Revision history

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	*In contrast,  for USB-AM we get:		*In contrast,  for USB-AM we get:
	:$$v(t) = A_1 \cdot \cos(\omega_1 \cdot t - \Delta \phi_{\rm T}) + A_2		:$$v(t) = A_1 \cdot \cos(\omega_1 \cdot t - \Delta \phi_{\rm T}) + A_2\cdot \cos(\omega_2 \cdot t - \Delta \phi_{\rm T})= A_1 \cdot \cos(\omega_1 \cdot (t - \tau_1)) + A_2 \cdot\cos(\omega_2 \cdot (t - \tau_2))\hspace{0.05cm}.$$
	\cdot \cos(\omega_2 \cdot t - \Delta \phi_{\rm T})= A_1 \cdot \cos(\omega_1 \cdot (t - \tau_1)) + A_2 \cdot
	\cos(\omega_2 \cdot (t - \tau_2))\hspace{0.05cm}.$$

	{{BlaueBox\|TEXT=		{{BlaueBox\|TEXT=

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	{{Display}}		{{Display}}
			[[de:Modulationsverfahren/Einseitenbandmodulation]]

Hwang at 12:43, 18 January 2023

2023-01-18T12:43:19Z

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	*For SSB–AM the convolution products overlap with the left and the right Dirac delta function at every frequency.  These are denoted by  "+"  and  "–"  respectively in the [[Modulation_Methods/Synchronous_Demodulation#Description_in_the_frequency_domain\|$\text{corresponding DSB graph}$]].		*For SSB–AM the convolution products overlap with the left and the right Dirac delta function at every frequency.  These are denoted by  "+"  and  "–"  respectively in the [[Modulation_Methods/Synchronous_Demodulation#Description_in_the_frequency_domain\|$\text{corresponding DSB graph}$]].

	*In contrast,  for USB,  only the convolution with the Dirac line at  $ -f_{\rm T}$  yields the  $V(f)$  component for positive frequencies,  and for LSB modulation it is the convolution with the Dirac delta function  $δ(f - f_{\rm T})$.		*In contrast,  for USB,  only the convolution with the Dirac delta line at  $ -f_{\rm T}$  yields the  $V(f)$  component for positive frequencies,  and for LSB modulation it is the convolution with the Dirac delta function  $δ(f - f_{\rm T})$.

	*In the case of DSB–AM,  $v(t) = q(t)$  is reached with the receiver-side carrier signal  $z_{\rm E}(t) = 2 · \cos(ω_{\rm T} · t)$.  In contrast,  the carrier amplitude must be increased to  $A_{\rm T} = 4$   for SSB-AM.		*In the case of DSB–AM,  $v(t) = q(t)$  is reached with the receiver-side carrier signal  $z_{\rm E}(t) = 2 · \cos(ω_{\rm T} · t)$.  In contrast,  the carrier amplitude must be increased to  $A_{\rm T} = 4$   for SSB-AM.

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

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

Hwang at 21:48, 2 January 2023

2023-01-02T21:48:52Z

Reed at 14:07, 9 April 2022

2022-04-09T14:07:24Z

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	The key advantage of  $\text{SSB–AM}$  compared to  $\text{DSB–AM}$  is the bandwidth requirement,  smaller by a factor of  $2$.  However,  for the half bandwidth in SSB-AM,  some disadvantages have to be accepted,  which will be investigated in the exercises for this section:		The key advantage of  $\text{SSB–AM}$  compared to  $\text{DSB–AM}$  is the bandwidth requirement,  smaller by a factor of  $2$.  However,  for the half bandwidth in SSB-AM,  some disadvantages have to be accepted,  which will be investigated in the exercises for this section:
	*The information about the source signal  $q(t)$  is,  in contrast to DSB-AM,  no longer exclusively in the amplitude,  but equally in the phase  (see [[Aufgaben:Exercise_2.11:_Envelope_Demodulation_of_an_SSB_Signal\| Exercise 2.11]]).		*The information about the source signal  $q(t)$  is,  in contrast to DSB-AM,  no longer exclusively in the amplitude,  but equally in the phase  (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  $\text{envelope demodulation}$  with  "USB–AM"  or  "LSB–AM"  thus always leads to strong nonlinear distortions  (see [[Aufgaben:Exercise_2.11Z:_Once_again_SSB-AM_and_Envelope_Demodulator\|Exercise 2.11Z]]).		*The application of  $\text{envelope demodulation}$  with  "USB–AM"  or  "LSB–AM"  thus always leads to strong nonlinear distortions  (see [[Aufgaben:Exercise_2.11Z:_Once_again_SSB-AM_and_Envelope_Demodulator\|Exercise 2.11Z]]).

Reed at 14:05, 9 April 2022

2022-04-09T14:05:37Z

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

Reed at 14:03, 9 April 2022

2022-04-09T14:03:54Z

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

Reed at 15:51, 31 March 2022

2022-03-31T15:51:09Z

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	{{Header		{{Header
	\|Untermenü=Amplitude Modulation and Demodulation		\|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==		==Description in the frequency domain==

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	The so-called  '''single-sideband amplitude modulation'''  $\rm (SSB–AM)$  takes advantage of this property by transmitting only one of these sidebands,  either the upper sideband (USB) or the lower sideband (LSB).  This reduces the required bandwidth by half compared to DSB-AM.		The so-called  »'''single-sideband amplitude modulation'''«  $\rm (SSB–AM)$  takes advantage of this property by transmitting only one of these sidebands,  either the upper sideband (USB) or the lower sideband (LSB).  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.		The graph illustrates single-sideband amplitude modulation in the frequency domain and simultaneously presents a realization of a SSB modulator.
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	*In accordance with a hierarchical structure,  three telephone channels  – each band-limited to the range from  $300\ \rm Hz$  to  $3.4 \ \rm kHz$  –  were initially combined to form a preliminary grouping with a bandwidth of  $12 \ \rm kHz$ .		*In accordance with a hierarchical structure,  three telephone channels  – each band-limited to the range from  $300\ \rm Hz$  to  $3.4 \ \rm kHz$  –  were initially combined to form a preliminary grouping with a bandwidth of  $12 \ \rm kHz$ .
	*In order to be able to accommodate three channels in  $12 \ \rm kHz$  with a safety margin,  only one sideband  (LSB or USB)  of each telephone channel was considered.		*In order to be able to accommodate three channels in  $12 \ \rm kHz$  with a safety margin,  only one sideband  (LSB or USB)  of each telephone channel was considered.
	*Using further combinations,  the long distance traffic system  '''V10800'''  with up to  $10800$  voice channels and a total bandwidth of  $60 \ \rm MHz$  was thus realized.}}		*Using further combinations,  the long distance traffic system  »'''V10800'''«  with up to  $10800$  voice channels and a total bandwidth of  $60 \ \rm MHz$  was thus realized.}}


	{{BlaueBox\|TEXT=		{{BlaueBox\|TEXT=
	$\text{Conclusion:}$		$\text{Conclusion:}$
	*The great advantage of a  $\rm SSB-AM$  is that it has '''half the bandwidth''' compared to  $\rm DSB-AM$.		*The great advantage of a  $\rm SSB-AM$  is that it has »'''half the bandwidth'''« compared to  $\rm DSB-AM$.
	*The trade-off with disadvantages required for this advantage is explained in the next sections.}}		*The trade-off with disadvantages required for this advantage is explained in the next sections.}}

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	==Synchronous demodulation of a SSB-AM signal==		==Synchronous demodulation of a SSB-AM signal==
	<br>		<br>
	Let us now consider a SSB-AM modulated signal and a  [[Modulation_Methods/Synchronous_Demodulation\|synchronous demodulator]]  at the receiver.		Let us now consider a SSB-AM modulated signal and a  [[Modulation_Methods/Synchronous_Demodulation\|$\text{synchronous demodulator}$]]  at the receiver.
	*Perfect frequency and phase synchronization will be assumed.		*Perfect frequency and phase synchronization will be assumed.
	*Without affecting generality,  in the rest of this section we will always let  ${ ϕ}_{\rm T} = 0$  ⇒   cosine carrier.		*Without affecting generality,  in the rest of this section we will always let  ${ ϕ}_{\rm T} = 0$  ⇒   cosine carrier.
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	[[File:EN_Mod_T_2_4_S2_neu.png \|right\|frame\| Synchronous demodulation of a SSB-AM signal]]		[[File:EN_Mod_T_2_4_S2_neu.png \|right\|frame\| Synchronous demodulation of a SSB-AM signal]]
	<br>		<br>
	A comparison with the  [[Modulation_Methods/Synchronous_Demodulation\|characteristics of the synchronous demodulator for DSB-AM]]  shows the follows similarities and differences:		A comparison with the  [[Modulation_Methods/Synchronous_Demodulation\|$\text{characteristics of the synchronous demodulator for DSB-AM}$]]  shows the follows similarities and differences:

	*The spectrum  $V(f)$  of the sink signal results in both cases from the convolution of the spectra  $R(f)$  and  $Z_{\rm E}(f)$,  the latter being composed of two Dirac delta functions at $±f_{\rm T}$.		*The spectrum  $V(f)$  of the sink signal results in both cases from the convolution of the spectra  $R(f)$  and  $Z_{\rm E}(f)$,  the latter being composed of two Dirac delta functions at $±f_{\rm T}$.


	*For SSB–AM the convolution products overlap with the left and the right Dirac delta function at every frequency.  These are denoted by  "+"  and  "–"  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.  These are denoted by  "+"  and  "–"  respectively in the [[Modulation_Methods/Synchronous_Demodulation#Description_in_the_frequency_domain\|$\text{corresponding DSB graph}$]].


	*In contrast,  for USB,  only the convolution with the Dirac line at  $ -f_{\rm T}$  yields the  $V(f)$  component for positive frequencies,  and for LSB modulation it is the convolution with the Dirac function  $δ(f - f_{\rm T})$.		*In contrast,  for USB,  only the convolution with the Dirac line at  $ -f_{\rm T}$  yields the  $V(f)$  component for positive frequencies,  and for LSB modulation it is the convolution with the Dirac delta function  $δ(f - f_{\rm T})$.


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	<br clear=all>		<br clear=all>
	{{BlaueBox\|TEXT=		{{BlaueBox\|TEXT=
	$\text{Conclusion:}$  If there is a frequency offset between the carrier signals  $z(t)$  and  $z_{\rm E}(t)$,  strong nonlinear distortion always occurs, i.e.,  regardless of whether DSB-AM or SSB-AM is present.  For the   '''implementation of a synchronous demodulator'''  a   '''perfect frequency synchronization'''  is essential. }}		$\text{Conclusion:}$  If there is a frequency offset between the carrier signals  $z(t)$  and  $z_{\rm E}(t)$,  strong nonlinear distortion always occurs, i.e.,  regardless of whether DSB-AM or SSB-AM is present.  For the   »'''implementation of a synchronous demodulator'''«  a   »'''perfect frequency synchronization'''«  is essential. }}


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	$\text{From this equation we can see:}$		$\text{From this equation we can see:}$
	*The two delay times  $τ_1 = Δ{\mathbf ϕ}_{\rm T}/ω_1$  and  $τ_2 = Δ{\mathbf ϕ}_{\rm T}/ω_2$  are different.		*The two delay times  $τ_1 = Δ{\mathbf ϕ}_{\rm T}/ω_1$  and  $τ_2 = Δ{\mathbf ϕ}_{\rm T}/ω_2$  are different.
	*This means that a phase offset in single sideband amplitude modulation  (USB–AM or LSB–AM)  leads to  '''phase distortions'''  (i.e., to linear distortions).		*This means that a phase offset in single sideband amplitude modulation  (USB–AM or LSB–AM)  leads to  »'''phase distortions'''«  (i.e., to linear distortions).
	*A positive value of  $Δ{\mathbf ϕ}_{\rm T}$  will cause		*A positive value of  $Δ{\mathbf ϕ}_{\rm T}$  will cause
	**positive values of  $τ_1$  and  $τ_2$  (i.e., lagging signals with respect to the cosine)  for USB,  and		**positive values of  $τ_1$  and  $τ_2$  (i.e., lagging signals with respect to the cosine)  for USB,  and
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	==Sideband-to-carrier ratio==		==Sideband-to-carrier ratio==
	<br>		<br>
	An important parameter for DSB–AM is the  [[Modulation_Methods/Double-Sideband_Amplitude_Modulation#Double-Sideband_Amplitude_Modulation_with_carrier\|modulation depth]]  $m = q_{\rm max}/A_{\rm T}$.  In the special case of a harmonic oscillation,  $m = A_{\rm N}/A_{\rm T}$  holds and one obtains the spectrum  $S_+(f)$  of the analytical signal corresponding to the upper graph.  Please note the normalization with respect to  $A_{\rm T}$.		An important parameter for DSB–AM is the  [[Modulation_Methods/Double-Sideband_Amplitude_Modulation#Double-Sideband_Amplitude_Modulation_with_carrier\|$\text{modulation depth}$]]  $m = q_{\rm max}/A_{\rm T}$.  In the special case of a harmonic oscillation,  $m = A_{\rm N}/A_{\rm T}$  holds and one obtains the spectrum  $S_+(f)$  of the analytical signal corresponding to the upper graph.  Please note the normalization with respect to  $A_{\rm T}$.

	[[File:EN_Mod_T_2_4_S4a.png\|right\|frame\|Spectra for DSB-AM,  USB-AM]]		[[File:EN_Mod_T_2_4_S4a.png\|right\|frame\|Spectra for DSB-AM,  USB-AM]]
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	In a similar way,  this can be written as		In a similar way,  this can be written as
	:$$s_+(t) = A_{\rm T} \cdot \left({\rm e}^{{\rm j}\hspace{0.03cm}\cdot \hspace{0.05cm} \omega_{\rm T} \hspace{0.03cm}\cdot \hspace{0.05cm}t } + \mu \cdot {\rm e}^{{\rm j}\hspace{0.03cm}\cdot \hspace{0.03cm} (\omega_{\rm T} + \omega_{\rm N}) \hspace{0.03cm}\cdot \hspace{0.05cm}t }\right)$$		:$$s_+(t) = A_{\rm T} \cdot \left({\rm e}^{{\rm j}\hspace{0.03cm}\cdot \hspace{0.05cm} \omega_{\rm T} \hspace{0.03cm}\cdot \hspace{0.05cm}t } + \mu \cdot {\rm e}^{{\rm j}\hspace{0.03cm}\cdot \hspace{0.03cm} (\omega_{\rm T} + \omega_{\rm N}) \hspace{0.03cm}\cdot \hspace{0.05cm}t }\right)$$
	now using the   '''sideband–to–carrier ratio''' :		now using the   »'''sideband–to–carrier ratio'''« :
	:$$\mu = \frac{A_{\rm N}}{2 \cdot A_{\rm T}} \hspace{0.05cm}.$$		:$$\mu = \frac{A_{\rm N}}{2 \cdot A_{\rm T}} \hspace{0.05cm}.$$
	If the source signal is not a harmonic oscillation,  it is difficult to specify this quantity.		If the source signal is not a harmonic oscillation,  it is difficult to specify this quantity.
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	<br>		<br>
	The key advantage of  $\text{SSB–AM}$  compared to  $\text{DSB–AM}$  is the bandwidth requirement,  smaller by a factor of  $2$.  However,  for the half bandwidth in SSB-AM,  some disadvantages have to be accepted,  which will be investigated in the exercises for this section:		The key advantage of  $\text{SSB–AM}$  compared to  $\text{DSB–AM}$  is the bandwidth requirement,  smaller by a factor of  $2$.  However,  for the half bandwidth in SSB-AM,  some disadvantages have to be accepted,  which will be investigated in the exercises for this section:
	*The information about the source signal  $q(t)$  is,  in contrast to DSB-AM,  no longer exclusively in the amplitude,  but equally in the phase  (see [[Aufgaben:Exercise_2.11:_Envelope_Demodulation_of_an_SSB_Signal\| Exercise 2.11]]).		*The information about the source signal  $q(t)$  is,  in contrast to DSB-AM,  no longer exclusively in the amplitude,  but equally in the phase  (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.		*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  $\text{envelope demodulation}$  with  "USB–AM"  or  "LSB–AM"  thus always leads to strong nonlinear distortions  (see [[Aufgaben:Exercise_2.11Z:_Once_again_SSB-AM_and_Envelope_Demodulator\|Exercise 2.11Z]]).		*The application of  $\text{envelope demodulation}$  with  "USB–AM"  or  "LSB–AM"  thus always leads to strong nonlinear distortions  (see [[Aufgaben:Exercise_2.11Z:_Once_again_SSB-AM_and_Envelope_Demodulator\|"Exercise 2.11Z"]]).



	As with DSB–AM and synchronous demodulation,  the following statements also apply here:		As with DSB–AM and synchronous demodulation,  the following statements also apply here:
	*Attenuation distortions of the channel also lead only to (linear)  attenuation distortions with respect to the sink signal.  No nonlinear distortions arise  (see  [[Aufgaben:Exercise_2.10:_SSB-AM_with_Channel_Distortions\|Exercise A2.10]]).		*Attenuation distortions of the channel also lead only to (linear)  attenuation distortions with respect to the sink signal.  No nonlinear distortions arise  (see  [[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.  The advantage of the smaller RF bandwidth is cancelled out by the necessary level matching.		*SSB–AM without a carrier shows the exact same noise behaviour as DSB-AM without a carrier.  The advantage of the smaller RF bandwidth is cancelled out by the necessary level matching.
	*A SSB–AM with a sideband-to-carrier ratio  $\mu$  shows similar noise behaviour as in DSB-AM with modulation depth  $m = \sqrt{2} · \mu$  (see  [[Aufgaben:Exercise_2.10Z:_Noise_with_DSB-AM_and_SSB-AM\|Exercise 2.10Z]]).		*A SSB–AM with a sideband-to-carrier ratio  $\mu$  shows similar noise behaviour as in DSB-AM with modulation depth  $m = \sqrt{2} · \mu$  (see  [[Aufgaben:Exercise_2.10Z:_Noise_with_DSB-AM_and_SSB-AM\|"Exercise 2.10Z"]]).
	*In any case,  it should be noted that SSB-AM with carrier is not very useful due to the nonlinear distortions brought about with envelope demodulation.		*In any case,  it should be noted that SSB-AM with carrier is not very useful due to the nonlinear distortions brought about with envelope demodulation.