Modulation Methods/Quality Criteria - Revision history

Guenter at 17:01, 12 January 2023

2023-01-12T17:01:24Z

Hwang at 16:24, 2 January 2023

2023-01-02T16:24:38Z

Hwang at 16:24, 2 January 2023

2023-01-02T16:24:17Z

Hwang at 16:23, 2 January 2023

2023-01-02T16:23:57Z

Hwang at 16:23, 2 January 2023

2023-01-02T16:23:42Z

Hwang at 16:20, 2 January 2023

2023-01-02T16:20:19Z

Hwang at 16:14, 2 January 2023

2023-01-02T16:14:22Z

Hwang at 16:06, 2 January 2023

2023-01-02T16:06:46Z

Hwang at 16:03, 2 January 2023

2023-01-02T16:03:57Z

Hwang at 16:00, 2 January 2023

2023-01-02T16:00:52Z

@@ Line 11: / Line 11: @@
 The task of any message transmission system is
 *to provide a sink signal&nbsp; $v(t)$&nbsp; at a spatially distant sink
 *that differs as little as possible from the source signal &nbsp; $q(t)$&nbsp;.
 <br clear=all>
@@ Line 23: / Line 24: @@
 *Non-ideal realization of the modulator and the demodulator,
 *linear attenuation distortions and phase distortions,&nbsp; as well as nonlinearities,
 *external disturbances and additional stochastic noise processes,
-*frequency-independent damping '''KORREKTUR: attenuation''' and delay.
@@ Line 33: / Line 37: @@
 *Due to the attenuation factor&nbsp; $α$,&nbsp; the sink signal &nbsp;$v(t)$ is only&nbsp; "quieter"&nbsp; compared to the source signal&nbsp; $q(t)$.
 *Even a delay&nbsp; $τ$&nbsp; is often tolerable,&nbsp; at least for a unidirectional transmission.
 *In contrast,&nbsp; in bidirectional communications – such as a telephone call – a delay of&nbsp; $300$&nbsp; milliseconds is already perceived as a significant disturbance.
@@ Line 48: / Line 54: @@
 {{BlaueBox|TEXT=
-$\text{Definition:}$&nbsp; As a measure of the quality of the communication system,&nbsp; the&nbsp; &raquo;'''signal-to-noise (power) ratio'''&laquo;&nbsp; $\rm (SNR)$&nbsp; $ρ_v$&nbsp; at the sink is defined as the quotient of the signal power (variance) of the useful component &nbsp;$v(t) - ε(t)$&nbsp; and the disturbance '''KORREKTUR: noise?''' component &nbsp;$ε(t)$,&nbsp; respectively:
+$\text{Definition:}$&nbsp; As a measure of the quality of the communication system,&nbsp; the&nbsp; &raquo;'''signal-to-noise (power) ratio'''&laquo;&nbsp; $\rm (SNR)$&nbsp; $ρ_v$&nbsp; at the sink is defined as the quotient of the signal power (variance) of the useful component &nbsp;$v(t) - ε(t)$&nbsp; and the disturbing component &nbsp;$ε(t)$,&nbsp; respectively:
 :$$\rho_{v} = \frac{  P_{v -\varepsilon} }{P_{\varepsilon} } \hspace{0.05cm},\hspace{0.7cm}\text{with}\hspace{0.7cm} P_{v -\varepsilon}  = \overline{[v(t)-\varepsilon(t)]^2} = \lim_{T_{\rm M} \rightarrow \infty}\hspace{0.1cm}\frac{1}{T_{\rm M} } \cdot \int_{0}^{  T_{\rm M} }
   {\big[v(t)-\varepsilon(t)\big]^2 }\hspace{0.1cm}{\rm d}t,\hspace{0.5cm}
@@ Line 63: / Line 69: @@
 :$$\rho_{v} = \frac{\alpha^2 \cdot  P_{q} }{P_{\varepsilon} }  \hspace{0.3cm}\Rightarrow \hspace{0.3cm} 10 \cdot {\rm lg}\hspace{0.15cm}\rho_{v} =
 \cdot {\rm lg} \hspace{0.15cm} \frac{\alpha^2 \cdot  P_{q} }{P_{\varepsilon} } \hspace{0.05cm}.$$
-*In the following,&nbsp; we will refer to &nbsp;$ρ_v$&nbsp; as the&nbsp; &raquo;'''sink signal–to–noise ratio'''&laquo; &nbsp; (or short:&nbsp; &raquo;'''sink SNR'''&laquo;).
+*In the following,&nbsp; we will refer to &nbsp;$ρ_v$&nbsp; as the&nbsp; &raquo;'''sink signal–to–noise ratio'''&laquo; &nbsp; or short:&nbsp; &raquo;'''sink SNR'''&laquo;.
 *One often uses the logarithmic form &nbsp; &rArr; &nbsp; $10 · \lg \ ρ_v$&nbsp; which is expressed in&nbsp; $\rm dB$&nbsp; when using the logarithm of base ten &nbsp; $(\lg)$&nbsp;.}}
@@ Line 71: / Line 77: @@
 $\text{Example 1:}$&nbsp; On the right, you can see an exemplary section of the&nbsp; (blue)&nbsp; source signal &nbsp;$q(t)$&nbsp; and the&nbsp; (red)&nbsp; sink signal &nbsp;$v(t)$, which are noticeably different.
-However, the middle graph makes it clear that the main difference between  &nbsp;$q(t)$&nbsp; and &nbsp;$v(t)$&nbsp; is due to the damping '''KORREKTUR: attenuation''' factor &nbsp;$α = 0.7$&nbsp; and the transmission delay &nbsp;$τ = 0.1\text{ ms}$.
+However, the middle graph makes it clear that the main difference between  &nbsp;$q(t)$&nbsp; and &nbsp;$v(t)$&nbsp; is due to the  attenuation factor &nbsp;$α = 0.7$&nbsp; and the transmission delay &nbsp;$τ = 0.1\text{ ms}$.
 The bottom sketch shows the remaining error signal &nbsp;$ε(t) = v(t) - α · q(t - τ)$&nbsp; after correcting for attenuation and delay.&nbsp; We refer to the mean square ⇒ "variance" of this signal as the noise power &nbsp;$P_ε$.
@@ Line 88: / Line 94: @@
 [[File:EN_Mod_T_1_2_S3.png|right|frame| Simplified model of a communication system]]
 :$$v(t) ≠ α · q(t - τ),$$
-which differs from &nbsp;$q(t)$&nbsp; by more than just damping '''KORREKTUR: attenuation''' and delay.&nbsp; For the study of these signal distortions, we always assume the following model and premises:
+which differs from &nbsp;$q(t)$&nbsp; by more than just attenuation and delay.&nbsp; For the study of these signal distortions, we always assume the following model and premises:
 *The additive noise signal &nbsp;$n(t)$&nbsp; at the channel output&nbsp; (demodulator input)&nbsp; is negligible and ignored.

@@ Line 157: / Line 157: @@
 {{BlaueBox|TEXT=
 $\text{Conclusion:}$&nbsp;  Considering these four assumptions, we conclude that it makes sense to express the sink SNR in normalized form as
-:$$\rho_{v } = \rho_{v }(\xi) \hspace{0.5cm} {\rm with} \hspace{0.5cm}\xi = \frac{ {\alpha_{\rm K} }^2 \cdot P_{\rm S} }{N_0 \cdot B_{\rm NF}. }$$
+:$$\rho_{v } = \rho_{v }(\xi) \hspace{0.5cm} {\rm with} \hspace{0.5cm}\xi = \frac{ {\alpha_{\rm K} }^2 \cdot P_{\rm S} }{N_0 \cdot B_{\rm NF} }.$$
 &nbsp; In the following,&nbsp; we refer to &nbsp; $ξ$&nbsp; as the&nbsp; &raquo;'''performance parameter'''&laquo;.}}

@@ Line 157: / Line 157: @@
 {{BlaueBox|TEXT=
 $\text{Conclusion:}$&nbsp;  Considering these four assumptions, we conclude that it makes sense to express the sink SNR in normalized form as
-:$$\rho_{v } = \rho_{v }(\xi) \hspace{0.5cm} {\rm with} \hspace{0.5cm}\xi = \frac{ {\alpha_{\rm K} }^2 \cdot P_{\rm S} }{N_0 \cdot B_{\rm NF} }$.$
+:$$\rho_{v } = \rho_{v }(\xi) \hspace{0.5cm} {\rm with} \hspace{0.5cm}\xi = \frac{ {\alpha_{\rm K} }^2 \cdot P_{\rm S} }{N_0 \cdot B_{\rm NF}. }$$
 &nbsp; In the following,&nbsp; we refer to &nbsp; $ξ$&nbsp; as the&nbsp; &raquo;'''performance parameter'''&laquo;.}}

@@ Line 157: / Line 157: @@
 {{BlaueBox|TEXT=
 $\text{Conclusion:}$&nbsp;  Considering these four assumptions, we conclude that it makes sense to express the sink SNR in normalized form as
-:$$\rho_{v } = \rho_{v }(\xi) \hspace{0.5cm} {\rm with} \hspace{0.5cm}\xi = \frac{ {\alpha_{\rm K} }^2 \cdot P_{\rm S} }{N_0 \cdot B_{\rm NF} }$$.
+:$$\rho_{v } = \rho_{v }(\xi) \hspace{0.5cm} {\rm with} \hspace{0.5cm}\xi = \frac{ {\alpha_{\rm K} }^2 \cdot P_{\rm S} }{N_0 \cdot B_{\rm NF} }$.$
 &nbsp; In the following,&nbsp; we refer to &nbsp; $ξ$&nbsp; as the&nbsp; &raquo;'''performance parameter'''&laquo;.}}

@@ Line 157: / Line 157: @@
 {{BlaueBox|TEXT=
 $\text{Conclusion:}$&nbsp;  Considering these four assumptions, we conclude that it makes sense to express the sink SNR in normalized form as
-:$$\rho_{v } = \rho_{v }(\xi) \hspace{0.5cm} {\rm with} \hspace{0.5cm}\xi = \frac{ {\alpha_{\rm K} }^2 \cdot P_{\rm S} }{N_0 \cdot B_{\rm NF} }$$
+:$$\rho_{v } = \rho_{v }(\xi) \hspace{0.5cm} {\rm with} \hspace{0.5cm}\xi = \frac{ {\alpha_{\rm K} }^2 \cdot P_{\rm S} }{N_0 \cdot B_{\rm NF} }$$.
 &nbsp; In the following,&nbsp; we refer to &nbsp; $ξ$&nbsp; as the&nbsp; &raquo;'''performance parameter'''&laquo;.}}

@@ Line 91: / Line 91: @@
 *The additive noise signal &nbsp;$n(t)$&nbsp; at the channel output&nbsp; (demodulator input)&nbsp; is negligible and ignored.
-*All components of modulator and demodulator are treated as linear,
+*All components of modulator and demodulator are treated as linear.
 *Similarly,&nbsp; the channel is assumed to be linear,&nbsp; and is thus completely characterized by its frequency response &nbsp;$H_{\rm K}(f)$&nbsp;.
 <br clear=all>

@@ Line 15: / Line 15: @@
 <br clear=all>
 {{BlaueBox|TEXT=
-$\text{Definition:}$&nbsp;  An&nbsp; '''ideal system'''&nbsp; is achieved when the following conditions hold:
+$\text{Definition:}$&nbsp;  An&nbsp; &raquo;'''ideal system'''&laquo;&nbsp; is achieved when the following conditions hold:
 :$$v(t) = q(t) + n(t), \hspace{1cm}n(t)  \to 0.$$
-This takes into account that &nbsp;$n(t) \equiv 0$&nbsp; is physically impossible due to&nbsp;  [[Aufgaben:Exercise_1.3Z:_Thermal_Noise|Thermal Noise]].}}
+This takes into account that &nbsp;$n(t) \equiv 0$&nbsp; is physically impossible due to&nbsp;  [[Aufgaben:Exercise_1.3Z:_Thermal_Noise|$\text{thermal noise}$]].}}
@@ Line 29: / Line 29: @@
 {{BlaueBox|TEXT=
-$\text{Definition:}$&nbsp;  A&nbsp; '''distortionless system'''&nbsp; is achieved,&nbsp; if from the above list only the last restriction is effective:
+$\text{Definition:}$&nbsp;  A&nbsp; &raquo;'''distortionless system'''&laquo;&nbsp; is achieved,&nbsp; if from the above list only the last restriction is effective:
 :$$v(t) = \alpha \cdot q(t- \tau) + n(t), \hspace{1cm}n(t)  \to 0.$$}}
@@ Line 48: / Line 48: @@
 {{BlaueBox|TEXT=
-$\text{Definition:}$&nbsp; As a measure of the quality of the communication system,&nbsp; the&nbsp; '''signal-to-noise (power) ratio'''&nbsp; $\rm (SNR)$&nbsp; $ρ_v$&nbsp; at the sink is defined as the quotient of the signal power (variance) of the useful component &nbsp;$v(t) - ε(t)$&nbsp; and the disturbance component &nbsp;$ε(t)$,&nbsp; respectively:
+$\text{Definition:}$&nbsp; As a measure of the quality of the communication system,&nbsp; the&nbsp; &raquo;'''signal-to-noise (power) ratio'''&laquo;&nbsp; $\rm (SNR)$&nbsp; $ρ_v$&nbsp; at the sink is defined as the quotient of the signal power (variance) of the useful component &nbsp;$v(t) - ε(t)$&nbsp; and the disturbance component &nbsp;$ε(t)$,&nbsp; respectively:
 :$$\rho_{v} = \frac{  P_{v -\varepsilon} }{P_{\varepsilon} } \hspace{0.05cm},\hspace{0.7cm}\text{with}\hspace{0.7cm} P_{v -\varepsilon}  = \overline{[v(t)-\varepsilon(t)]^2} = \lim_{T_{\rm M} \rightarrow \infty}\hspace{0.1cm}\frac{1}{T_{\rm M} } \cdot \int_{0}^{  T_{\rm M} }
   {\big[v(t)-\varepsilon(t)\big]^2 }\hspace{0.1cm}{\rm d}t,\hspace{0.5cm}
@@ Line 63: / Line 63: @@
 :$$\rho_{v} = \frac{\alpha^2 \cdot  P_{q} }{P_{\varepsilon} }  \hspace{0.3cm}\Rightarrow \hspace{0.3cm} 10 \cdot {\rm lg}\hspace{0.15cm}\rho_{v} =
 \cdot {\rm lg} \hspace{0.15cm} \frac{\alpha^2 \cdot  P_{q} }{P_{\varepsilon} } \hspace{0.05cm}.$$
-*In the following,&nbsp; we will refer to &nbsp;$ρ_v$&nbsp; as the&nbsp; '''sink signal–to–noise ratio''' &nbsp; (or short:&nbsp; '''sink SNR''').
+*In the following,&nbsp; we will refer to &nbsp;$ρ_v$&nbsp; as the&nbsp; &raquo;'''sink signal–to–noise ratio'''&laquo; &nbsp; (or short:&nbsp; &raquo;'''sink SNR'''&laquo;).
 *One often uses the logarithmic form &nbsp; &rArr; &nbsp; $10 · \lg \ ρ_v$&nbsp; which is expressed in&nbsp; $\rm dB$&nbsp; when using the logarithm of base ten &nbsp; $(\lg)$&nbsp;.}}
@@ Line 158: / Line 158: @@
 $\text{Conclusion:}$&nbsp;  Considering these four assumptions, we conclude that it makes sense to express the sink SNR in normalized form as
 :$$\rho_{v } = \rho_{v }(\xi) \hspace{0.5cm} {\rm with} \hspace{0.5cm}\xi = \frac{ {\alpha_{\rm K} }^2 \cdot P_{\rm S} }{N_0 \cdot B_{\rm NF} }$$
-&nbsp; In the following,&nbsp; we refer to &nbsp; $ξ$&nbsp; as the&nbsp; '''performance parameter'''.}}
+&nbsp; In the following,&nbsp; we refer to &nbsp; $ξ$&nbsp; as the&nbsp; &raquo;'''performance parameter'''&laquo;.}}
@@ Line 186: / Line 186: @@
 B_{\rm NF}= 5\; {\rm kHz}\hspace{0.05cm}.$$
 *In &nbsp;$\text{System B}$,&nbsp; there is also a linear relationship of &nbsp;$ρ_v = ξ/3$.&nbsp; The line also passes through the origin.&nbsp;  However, the slope is only  &nbsp;$1/3$.
-*It should be noted that the noise behavior corresponding to &nbsp;$\text{System A}$&nbsp; is observed for  &nbsp; [[Modulation_Methods/Double-Sideband_Amplitude_Modulation#Description_in_the_frequency_domain|Double-sideband suppressed-carrier amplitude modulation]]&nbsp; $($modulation depth &nbsp;$m → ∞)$,&nbsp; while &nbsp;$\text{System B}$&nbsp; describes &nbsp;  [[Modulation_Methods/Double-Sideband_Amplitude_Modulation#Double-Sideband_Amplitude_Modulation_with_carrier|Double-sideband amplitude modulation with carrier]]&nbsp; $(m ≈ 0.5)$.
+*It should be noted that the noise behavior corresponding to &nbsp;$\text{System A}$&nbsp; is observed for &nbsp; [[Modulation_Methods/Double-Sideband_Amplitude_Modulation#Description_in_the_frequency_domain|$\text{double-sideband suppressed-carrier amplitude modulation}$]]&nbsp; $($modulation depth &nbsp;$m → ∞)$,&nbsp; while &nbsp;$\text{System B}$&nbsp; describes &nbsp;  [[Modulation_Methods/Double-Sideband_Amplitude_Modulation#Double-Sideband_Amplitude_Modulation_with_carrier|$\text{double-sideband amplitude modulation with carrier}$]]&nbsp; $(m ≈ 0.5)$.
 *$\text{System C}$&nbsp; shows a completely different noise behavior.&nbsp;  For small &nbsp;$ξ$&ndash;values,&nbsp; this system is superior to &nbsp;$\text{System A}$,&nbsp; though the quality of both systems is the same at &nbsp;$ξ = 10000$.

@@ Line 5: / Line 5: @@
 |Nächste Seite=General Model of Modulation
 }}
-==Ideal and Distortionless System==
+==Ideal and distortionless system==
 <br>
 [[File:EN_Mod_T_1_2_S1.png |right|frame| Block diagram describing modulation and demodulation]]