Modulation Methods/Synchronous Demodulation - Revision history

Guenter at 09:32, 13 January 2023

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Guenter at 09:24, 13 January 2023

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Hwang at 20:56, 2 January 2023

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Hwang at 20:39, 2 January 2023

2023-01-02T20:39:28Z

@@ Line 34: / Line 34: @@
 :$$B(f) = R(f) \star Z_{\rm E}(f)\hspace{0.05cm}.$$
-*Convolution of the Dirac delta function &nbsp;$ - {\rm j} \cdot δ(f – f_{\rm T})$&nbsp; with the wholly '''KORREKTUR: purely''' imaginary spectrum &nbsp;$R(f)$&nbsp; leads to completely real spectral components around &nbsp;$f = 0$&nbsp; and &nbsp;$f = 2f_{\rm T}$.&nbsp; These components are marked with a&nbsp;  "plus"&nbsp; on the lower graph.
+*Convolution of the Dirac delta function &nbsp;$ - {\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}$.&nbsp; These spectral components are marked with&nbsp; "minus".
@@ Line 182: / Line 182: @@
 In particular,&nbsp; we make the following assumptions:
 *Only double-sideband amplitude modulation with modulation depth &nbsp;$m$&nbsp; and an ideal synchronous demodulator without phase and frequency offsets are considered.
-*According to the extended AWGN channel model,&nbsp; where &nbsp;$α_{\rm K}$&nbsp; is a frequency-independent transmission factor of the channel and the interference '''KORREKTUR: noise''' signal &nbsp;$n(t)$&nbsp; models white noise with two-sided noise power density &nbsp;$N_0/2$,&nbsp; the following holds for the received signal:
+*According to the extended AWGN channel model,&nbsp; where &nbsp;$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:
 :$$r(t) = \alpha_{\rm K} \cdot s(t) + n(t) \hspace{0.05cm}.$$
 *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:
@@ Line 230: / Line 230: @@
 \vert f \vert < B_{\rm NF} \hspace{0.05cm}, \\ {\rm otherwise} \hspace{0.05cm}. \\ \end{array}$$
-*By integration,&nbsp; we obtain the power &nbsp;$P_ε = 2N_0 · B_{\rm NF}$. &nbsp;  Thus,&nbsp; with this intermediate result,&nbsp; the sinking '''KORREKTUR: sink''' SNR can be given as:
+*By integration,&nbsp; we obtain the power &nbsp;$P_ε = 2N_0 · B_{\rm NF}$. &nbsp;  Thus,&nbsp; with this intermediate result,&nbsp; the&nbsp; "sink SNR"&nbsp; can be given as:
 :$$\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}.$$
-In the next section,&nbsp; we still establish the relationship between the power&nbsp; $P_q$&nbsp; of the source signal and the transmission '''KORREKTUR: transmit?''' power&nbsp; $P_{\rm S}$.}}
+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}$.}}
 ==Relationship between the powers from source signal and transmitted signal==
 <br>
-To characterise the relationship between the sink&ndash;SNR  &nbsp;$\rho_v$&nbsp; and the transmission '''KORREKTUR: transmit?''' power &nbsp;$P_{\rm S}$,&nbsp; we first need the relationships between
+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
 *source signal &nbsp;$q(t)$ &nbsp;    ⇒   &nbsp; power &nbsp;$P_q$,&nbsp; and
-*transmitted signal &nbsp;$s(t)$ &nbsp;      ⇒   &nbsp; transmission '''KORREKTUR: transmit?''' power &nbsp;$P_{\rm S}$.
+*transmitted signal &nbsp;$s(t)$ &nbsp;      ⇒   &nbsp; transmit power &nbsp;$P_{\rm S}$.
@@ Line 288: / Line 288: @@
 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.
-*A greater transmission '''KORREKTUR: 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.
+*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.
 *For&nbsp; &raquo;DSB–AM with carrier&laquo;&nbsp; with a modulation depth &nbsp;$m$:
 :$$\rho_v = \xi \cdot \frac{1}{1 + {2}/{m^2}}\hspace{0.3cm}\Rightarrow \hspace{0.3cm}

@@ Line 303: / Line 303: @@
 [[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|Exercise 2.5: DSB-AM via a Gaussian Channel]]
+[[Aufgaben:Exercise_2.5:_DSB-AM_via_a_Gaussian_channel|Exercise 2.5: DSB-AM via a Gaussian Channel]]
 [[Aufgaben:Exercise_2.5Z:_Linear_Distortions_with_DSB-AM|Exercise 2.5Z: Linear Distortions with DSB-AM]]

@@ Line 297: / Line 297: @@
 *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==
+==Exercises for the chapter==
 <br>
 [[Aufgaben:Aufgabe_2.4:_Frequenz–_und_Phasenversatz|Exercise 2.4: Frequency and Phase Offset]]

@@ Line 295: / Line 295: @@
 *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 insignifican deterioration of less than one dB compared to&nbsp; &raquo;DSB–AM without carrier&laquo;.
+*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==

@@ Line 237: / Line 237: @@
 ==Relationship between the powers from source signal and transmitted signal==
 <br>
-To characterise the relationship between the sink&ndash;SNR  &nbsp;$\rho_v$&nbsp; and the transmission power &nbsp;$P_{\rm S}$,&nbsp; we first need the relationships between
+To characterise the relationship between the sink&ndash;SNR  &nbsp;$\rho_v$&nbsp; and the transmission '''KORREKTUR: transmit?''' power &nbsp;$P_{\rm S}$,&nbsp; we first need the relationships between
 *source signal &nbsp;$q(t)$ &nbsp;    ⇒   &nbsp; power &nbsp;$P_q$,&nbsp; and
-*transmitted signal &nbsp;$s(t)$ &nbsp;      ⇒   &nbsp; transmission power &nbsp;$P_{\rm S}$.
+*transmitted signal &nbsp;$s(t)$ &nbsp;      ⇒   &nbsp; transmission '''KORREKTUR: transmit?''' power &nbsp;$P_{\rm S}$.
@@ Line 288: / Line 288: @@
 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.
-*A greater transmission 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.
+*A greater transmission '''KORREKTUR: 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.
 *For&nbsp; &raquo;DSB–AM with carrier&laquo;&nbsp; with a modulation depth &nbsp;$m$:
 :$$\rho_v = \xi \cdot \frac{1}{1 + {2}/{m^2}}\hspace{0.3cm}\Rightarrow \hspace{0.3cm}