Modulation Methods/Linear Digital Modulation - Revision history

Guenter at 13:47, 13 January 2023

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Guenter at 13:46, 13 January 2023

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Hwang at 16:02, 9 January 2023

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Guenter at 14:19, 11 April 2022

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Guenter at 09:36, 8 April 2022

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Hwang at 11:13, 7 April 2022

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Guenter at 11:47, 4 April 2022

2022-04-04T11:47:39Z

@@ Line 214: / Line 214: @@
 <br>
 The graphic shows the block diagram of a transmission system with DPSK modulation&nbsp; ("Differential Phase Shift Keying")&nbsp; and differential coherent demodulation.&nbsp; In bullet point form,&nbsp; the given system can be described as follows:
-[[File:EN_Mod_T_4_2_S7c_neu.png|right|frame| DPSK modulation and differential coherent demodulation '''KORREKTUR''': Takt]]
+[[File:EN_Mod_T_4_2_S7c_neu.png|right|frame| DPSK modulation and differential coherent demodulation]]
 *Ignoring the modulation with the carrier signals &nbsp;$z(t)$&nbsp; and &nbsp;$2 · z(t)$&nbsp; resp.,&nbsp; the symbol &nbsp;$m_ν = m_{ν-1} · q_ν$&nbsp; is present in the interval &nbsp;$ν$&nbsp; at input&nbsp; '''(1)'''&nbsp; of the multiplier highlighted in yellow,&nbsp; and the symbol&nbsp; $m_{ν-1}$&nbsp; is present at input &nbsp; '''(2)'''.
 *The multiplication of&nbsp; '''(1)'''&nbsp; and&nbsp; '''(2)'''&nbsp; gives the desired result:

@@ Line 49: / Line 49: @@
 <br>
 The diagram shows the block diagram of an ASK system including the receiver components.&nbsp;
-[[File:EN_Mod_T_4_2_S3a_neu.png|right|frame|Block diagram of an ASK system including receiver components '''KORREKTUR''': Takt]]
+[[File:EN_Mod_T_4_2_S3a_neu.png|right|frame|Block diagram of an ASK system including receiver components]]
 *Let the source signal &nbsp;$q(t)$&nbsp; be NRZ-rectangular and unipolar:
 :$$a_ν ∈ \{0, \ 1\}.$$

@@ Line 49: / Line 49: @@
 <br>
 The diagram shows the block diagram of an ASK system including the receiver components.&nbsp;
-[[File:EN_Mod_T_4_2_S3a.png|right|frame|Block diagram of an ASK system including receiver components '''KORREKTUR''': Takt]]
+[[File:EN_Mod_T_4_2_S3a_neu.png|right|frame|Block diagram of an ASK system including receiver components '''KORREKTUR''': Takt]]
 *Let the source signal &nbsp;$q(t)$&nbsp; be NRZ-rectangular and unipolar:
 :$$a_ν ∈ \{0, \ 1\}.$$
@@ Line 90: / Line 90: @@
 <br>
 We further assume ASK modulation as well as an ideal transmission channel, i.e.
-[[File:P_ID1691__Mod_T_4_2_S4a_neu.png|right|frame|Block diagram of an incoherent ASK demodulator '''KORREKTUR''': Takt]]
+[[File:EN_Mod_T_4_2_S4a_neu.png|right|frame|Block diagram of an incoherent ASK demodulator]]
 #distortion-free,
 #attenuation-free and
@@ Line 214: / Line 214: @@
 <br>
 The graphic shows the block diagram of a transmission system with DPSK modulation&nbsp; ("Differential Phase Shift Keying")&nbsp; and differential coherent demodulation.&nbsp; In bullet point form,&nbsp; the given system can be described as follows:
-[[File:EN_Mod_T_4_2_S7c.png |right|frame| DPSK modulation and differential coherent demodulation '''KORREKTUR''': Takt]]
+[[File:EN_Mod_T_4_2_S7c_neu.png|right|frame| DPSK modulation and differential coherent demodulation '''KORREKTUR''': Takt]]
 *Ignoring the modulation with the carrier signals &nbsp;$z(t)$&nbsp; and &nbsp;$2 · z(t)$&nbsp; resp.,&nbsp; the symbol &nbsp;$m_ν = m_{ν-1} · q_ν$&nbsp; is present in the interval &nbsp;$ν$&nbsp; at input&nbsp; '''(1)'''&nbsp; of the multiplier highlighted in yellow,&nbsp; and the symbol&nbsp; $m_{ν-1}$&nbsp; is present at input &nbsp; '''(2)'''.
 *The multiplication of&nbsp; '''(1)'''&nbsp; and&nbsp; '''(2)'''&nbsp; gives the desired result:

@@ Line 2: / Line 2: @@
 {{Header
 |Untermenü=Digital Modulation Methods
-|Vorherige Seite=Pulscodemodulation
+|Vorherige Seite=Pulse Code Modulation
-|Nächste Seite=Quadratur–Amplitudenmodulation
+|Nächste Seite=Quadrature Amplitude Modulation
 }}
 ==Differences between analog and digital modulation methods==

@@ Line 49: / Line 49: @@
 <br>
 The diagram shows the block diagram of an ASK system including the receiver components.&nbsp;
-[[File:EN_Mod_T_4_2_S3a.png|right|frame|Block diagram of an ASK system including receiver components]]
+[[File:EN_Mod_T_4_2_S3a.png|right|frame|Block diagram of an ASK system including receiver components '''KORREKTUR''': Takt]]
 *Let the source signal &nbsp;$q(t)$&nbsp; be NRZ-rectangular and unipolar:
 :$$a_ν ∈ \{0, \ 1\}.$$
@@ Line 90: / Line 90: @@
 <br>
 We further assume ASK modulation as well as an ideal transmission channel, i.e.
-[[File:P_ID1691__Mod_T_4_2_S4a_neu.png|right|frame|Block diagram of an incoherent ASK demodulator]]
+[[File:P_ID1691__Mod_T_4_2_S4a_neu.png|right|frame|Block diagram of an incoherent ASK demodulator '''KORREKTUR''': Takt]]
 #distortion-free,
 #attenuation-free and

@@ Line 207: / Line 207: @@
 Below the phase values &nbsp;$ϕ_{\rm S}$&nbsp; are given for the first six symbols:
-*Additional phase rotation on the channel,&nbsp; e.g. by &nbsp;$70.3π$, does change the absolute phase values to &nbsp;$69.8π$, &nbsp;$69.8π$, &nbsp;$70.8π$, &nbsp;$70.8π$, &nbsp;$69.8π$&nbsp; and &nbsp;$70.8π$.
+*Additional phase rotation on the channel,&nbsp; e.g. by &nbsp;$70.3π$,&nbsp; does change the absolute phase values to &nbsp;$69.8π$, &nbsp;$69.8π$, &nbsp;$70.8π$, &nbsp;$70.8π$, &nbsp;$69.8π$&nbsp; and &nbsp;$70.8π$.
 *However,&nbsp; the phase difference of adjacent symbols is preserved, so the differential coherent demodulation still works.&nbsp;
 *A corresponding demodulator is presented in the following section. }}

@@ Line 229: / Line 229: @@
 ==Error probabilities - a brief overview==
 <br>
-The error probabilities of the discussed digital modulation methods&nbsp; (ASK, BPSK, DPSK)&nbsp; are calculated in chapter &nbsp;[[Digital_Signal_Transmission/Lineare_digitale_Modulation_–_Kohärente_Demodulation|Linear Digital Modulation - Coherent Demodulation]]&nbsp; of the book "Digital Signal Transmission" under different boundary conditions.&nbsp;
+The error probabilities of the discussed digital modulation methods&nbsp; (ASK,&nbsp; BPSK,&nbsp; DPSK)&nbsp; are calculated in chapter&nbsp; [[Digital_Signal_Transmission/Lineare_digitale_Modulation_–_Kohärente_Demodulation|Linear Digital Modulation - Coherent Demodulation]]&nbsp; of the book&nbsp; "Digital Signal Transmission"&nbsp; under different boundary conditions.&nbsp; Here only some results are given without proof,&nbsp; valid for
+#a transmitted signal with the average energy &nbsp;$E_{\rm B}$&nbsp; per bit,
-Here only some results are given without proof, valid for
+#AWGN noise with the&nbsp; (one-sided)&nbsp; noise power density &nbsp;$N_0$,&nbsp; and
-*a transmitted signal with the average energy &nbsp;$E_{\rm B}$&nbsp; per bit,
+#best possible receiver realization using the matched filter principle.
 {{BlaueBox|TEXT=
-Let us first consider the bit error probability of &nbsp;[[Modulation_Methods/Linear_Digital_Modulation#Demodulation_and_detection_of_BPSK_signals|Binary Phase Shift Keying]] &nbsp; $\rm (BPSK)$&nbsp; assuming a coherent receiver:
+$\text{Here without proof:}$&nbsp;
-:$$p_{\rm B} = {\rm Q}\left ( \sqrt{ {2 \cdot E_{\rm B} }/{N_0 } } \hspace{0.1cm}\right
+Let us first consider the&nbsp; '''bit error probability'''&nbsp; of &nbsp;[[Modulation_Methods/Linear_Digital_Modulation#Demodulation_and_detection_of_BPSK_signals|"Binary Phase Shift Keying"]] &nbsp; $\rm (BPSK)$&nbsp; assuming a coherent receiver:
   ) = {1}/{2}\cdot {\rm erfc}\left ( \sqrt{ {E_{\rm B} }/{N_0 } } \hspace{0.1cm}\right ).$$
-In contrast, for &nbsp;[[Modulation_Methods/Linear_Digital_Modulation#Coherent_demodulation_of_ASK_signals|Amplitude Shift Keying]]&nbsp; $\rm (ASK)$&nbsp; with coherent demodulation:
+*[[Applets:Komplementäre_Gaußsche_Fehlerfunktionen|Two variants of the complementary Gaussian error function]]&nbsp; were used in the formulas:
-:$$p_{\rm B} = {\rm Q}\left ( \sqrt{{E_{\rm B} }/{N_0 } } \hspace{0.1cm}\right
+::$${\rm Q} ({\it x}) = \frac{\rm 1}{\sqrt{\rm 2\pi} }\int_{\it
 x}^{+\infty}{\rm e}^{ {\it -u}^{\rm 2}/\rm 2}\,{\rm d} {\it u}
 \hspace{0.05cm},\hspace{0.3cm} {\rm erfc} ( {\it x} ) = \frac{\rm
 }{\sqrt{\rm \pi} }\int_{\it x}^{+\infty}{\rm e}^{ {\it -u}^{\rm
-} }\,{\rm d} {\it u} \hspace{0.05cm}.$$}}
+} }\,{\rm d} {\it u} \hspace{0.05cm}.$$
-If we plot the bit error probability &nbsp;$p_{\rm B}$&nbsp; over the quotient &nbsp;$E_{\rm B}/N_0$&nbsp; in double logarithmic scale, the ASK curve is always &nbsp;$3 \ \rm dB$&nbsp; to the right of the BPSK curve.&nbsp; This degradation is also a reason why ASK is rarely used in practice.
+The equations given here are to be evaluated in&nbsp; [[Aufgaben:Exercise_4.8:_Different_Error_Probabilities|Exercise 4.8]].&nbsp;
-Incoherent demodulation of a BPSK signal, on the other hand, is not possible.&nbsp; For the &nbsp;[[Modulation_Methods/Linear_Digital_Modulation#Incoherent_demodulation_of_ASK_signals|incoherent ASK demodulation]]&nbsp; one obtains:
+{{GraueBox|TEXT=
-:$$p_{\rm B} = {1}/{2}\cdot {\rm e}^{-{E_{\rm B}}/{(2N_0) }} .$$
+$\text{Example 4:}$&nbsp;
 ==Exercises for the chapter==

@@ Line 149: / Line 149: @@
 ==Demodulation and detection of BPSK signals==
 <br>
-Due to the constant envelope of the BPSK signal, the demodulation must always be coherent here.&nbsp; The &nbsp;[[Modulation_Methods/Linear_Digital_Modulation#Coherent_demodulation_of_ASK_signals|same block diagram]]&nbsp; can be assumed as for coherent ASK demodulation.
+Due to the constant envelope of the BPSK signal,&nbsp; the demodulation must always be coherent here.&nbsp; The &nbsp;[[Modulation_Methods/Linear_Digital_Modulation#Coherent_demodulation_of_ASK_signals|same block diagram]]&nbsp; can be assumed as for coherent ASK demodulation.
 [[File:EN_Mod_T_4_2_S4b.png|right|frame|Signals with BPSK modulation and coherent demodulation]]
 {{GraueBox|TEXT=
 $\text{Example 2:}$&nbsp; The diagram shows from top to bottom
-*the source signal &nbsp;$q(t)$,
+#the bipolar source signal &nbsp;$q(t)$,
-*the received signal &nbsp;$r(t) = s(t)$&nbsp; with ideal channel,
+#the received signal &nbsp;$r(t) = s(t)$&nbsp; with ideal channel,
-*the signal &nbsp;$b(t)$&nbsp; after multiplication by the carrier signal &nbsp;$z_{\rm E}(t) = 2 \cdot z(t)$ at the receiver end,
+#$b(t)$&nbsp; after multiplication by the receiver-side carrier &nbsp;$z_{\rm E}(t) = 2 \cdot z(t)$,
-*the detection signal &nbsp;$d(t)$&nbsp; after "integration" by the matched filter,
+#the detection signal &nbsp;$d(t)$&nbsp; after&nbsp; "integration"&nbsp; by the matched filter,
-*the sink signal &nbsp;$v(t)$.
+#the sink signal &nbsp;$v(t)$.
 A comparison with the corresponding&nbsp; [[Modulation_Methods/Linear_Digital_Modulation#Coherent_demodulation_of_ASK_signals|signals]]&nbsp; in the coherent demodulation of the ASK shows:
-*The square-wave signals &nbsp;$q(t)$&nbsp; and &nbsp;$v(t)$&nbsp; are now bipolar.
+*The rectangular signals &nbsp;$q(t)$&nbsp; and &nbsp;$v(t)$&nbsp; are now bipolar.
 *For the detection signal at BPSK, compared to ASK, the following applies:
 :$$d_{\rm BPSK}(t) =
-  \cdot d_{\rm ASK}(t)-s_0.$$
+*For the detection signal at BPSK,&nbsp; compared to ASK,&nbsp; the following applies:
-*In the considered attenuation-, distortion- and noise-free case all detection samples are &nbsp;$d(ν · T) = ±s_0$.&nbsp; Therefore, the decision threshold &nbsp;$E = 0$&nbsp; should be used here.
+:$$d_{\rm BPSK}(t) =2  \cdot d_{\rm ASK}(t)-s_0.$$
-*One can see the double distance of the BPSK detection samples (circle marks) from the threshold, which decisively improves the &nbsp;[[Modulation_Methods/Linear_Digital_Modulation#Error_probabilities_-_a_brief_overview|error probability]].&nbsp; }}
+*One can see the double distance of the BPSK detection samples&nbsp; (circle marks)&nbsp; from the threshold,&nbsp; which decisively improves &nbsp;[[Modulation_Methods/Linear_Digital_Modulation#Error_probabilities_-_a_brief_overview|error probability]].&nbsp; }}
@@ Line 174: / Line 177: @@
 <br>
 [[File:EN_Mod_T_4_2_S7.png|right|frame| DPSK transmitter]]
-The diagram on the right shows the block diagram of the modulator for &nbsp;''Differential Phase Shift Keying''&nbsp; $\rm (DPSK)$.
+The diagram on the right shows the block diagram of the modulator for &nbsp;"Differential Phase Shift Keying"&nbsp; $\rm (DPSK)$.
-*The (bipolar) source signal &nbsp;$q(t)$&nbsp; with the amplitude coefficients &nbsp;$q_ν ∈ \{-1, +1\}$&nbsp; is transformed according to this mapping into the signal &nbsp;$m(t)$&nbsp; with the amplitude coefficients
+*The bipolar source signal &nbsp;$q(t)$&nbsp; with the amplitude coefficients &nbsp;$q_ν ∈ \{-1, +1\}$&nbsp; is transformed according to this mapping into the signal &nbsp;$m(t)$&nbsp; with the amplitude coefficients
 :$$m_{\nu} = m_{\nu -1} \cdot q_{\nu} \in \{ -1, +1\}$$
-before it is fed to the BPSK modulator.
+:before it is fed to the BPSK modulator.
-Although the exact carrier frequency &nbsp;$f_{\rm T}$&nbsp; must be known to the receiver, it is still referred to as an incoherent PSK demodulator, and sometimes as a differential coherent PSK demodulator.
+*An essential advantage of differential binary phase modulation is that the resulting signal &nbsp;$s(t)$&nbsp; can be demodulated even without knowledge of the carrier phase &nbsp;$ϕ_{\rm T}$,&nbsp; see next section.&nbsp;  Although the exact carrier frequency &nbsp;$f_{\rm T}$&nbsp; must be known to the receiver,&nbsp; it is still referred to as an&nbsp; "incoherent PSK demodulator",&nbsp; and sometimes as a&nbsp; "differential coherent PSK demodulator".
 <br clear=all>
 [[File:EN_Mod_T_4_2_S7b.png|right|frame|Signals at the DPSK transmitter]]
@@ Line 192: / Line 193: @@
 $\text{Example 3:}$&nbsp;
 The diagram on the right shows
-*the source signal &nbsp;$q(t)$
+#the bipolar source signal &nbsp;$q(t)$,
-*the mapping signal &nbsp;$m(t)$&nbsp; and
+#the mapping signal &nbsp;$m(t)$,&nbsp; and
-*the DPSK transmitted signal &nbsp;$s(t)$.
+#the DPSK transmitted signal &nbsp;$s(t)$.
-In the following, &nbsp;$v_ν$&nbsp; denotes the coefficients after the decision, which should coincide with the transmission-side amplitude coefficients &nbsp;$q_ν$.&nbsp; The following principle can be seen:
+The following principle can be seen:
-*Whenever the receiver detects a phase jump, it decides for itself &nbsp;$v_ν = -1$.&nbsp; It holds:
+*Whenever the receiver detects a phase jump,&nbsp; it decides&nbsp; $v_ν = -1$:
 :$$v_3 = v_5 =v_6 =-1.$$
 *If no phase jump is recognizable, &nbsp;$v_ν = +1$&nbsp; is set:
 :$$v_1 = v_2 =v_4 =+1.$$
-<br clear=all>
-Below the transmitted signal, the phase values &nbsp;$ϕ_{\rm S}$&nbsp; are given for the first six symbols.
+Below the phase values &nbsp;$ϕ_{\rm S}$&nbsp; are given for the first six symbols:
-*Additional phase rotation on the channel, for example by &nbsp;$70.3π$, does change the absolute phase values to &nbsp;$69.8π$, &nbsp;$69.8π$, &nbsp;$70.8π$, &nbsp;$70.8π$, &nbsp;$69.8π$&nbsp; and &nbsp;$70.8π$.
+*Additional phase rotation on the channel,&nbsp; e.g. by &nbsp;$70.3π$, does change the absolute phase values to &nbsp;$69.8π$, &nbsp;$69.8π$, &nbsp;$70.8π$, &nbsp;$70.8π$, &nbsp;$69.8π$&nbsp; and &nbsp;$70.8π$.
-*However, the phase difference of adjacent symbols is preserved, so the differential coherent demodulation still works.&nbsp; A corresponding demodulator is presented in the following section. }}
+*However,&nbsp; the phase difference of adjacent symbols is preserved, so the differential coherent demodulation still works.&nbsp;
 ==Differential coherent demodulation of the DPSK signal==
 <br>
-On the page end shown is the block diagram of a transmission system with DPSK modulation&nbsp; (''Differential Phase Shift Keying'')&nbsp; and differential coherent demodulation.&nbsp; In bullet point form, the mode of operation can be described as follows:
+The graphic shows the block diagram of a transmission system with DPSK modulation&nbsp; ("Differential Phase Shift Keying")&nbsp; and differential coherent demodulation.&nbsp; In bullet point form,&nbsp; the given system can be described as follows:
-*Ignoring the modulation with the carrier signals &nbsp;$z(t)$&nbsp; and &nbsp;$2 · z(t)$&nbsp; respectively, the symbol &nbsp;$m_ν = m_{ν-1} · q_ν$&nbsp; is present in the interval &nbsp;$ν$&nbsp; at the input&nbsp; '''(1)'''&nbsp; of the multiplier highlighted in yellow, and the symbol&nbsp;$m_{ν-1}$&nbsp; is present at the input &nbsp; '''(2)'''.
+[[File:EN_Mod_T_4_2_S7c.png |right|frame| DPSK modulation and differential coherent demodulation]]
-*The multiplication of&nbsp; '''(1)'''&nbsp; and&nbsp; '''(2)'''&nbsp; gives the desired result, namely &nbsp;$v_ν = m_{ν-1} · q_ν · m_{ν-1} = q_ν$.&nbsp; It is taken into account here that &nbsp;$m_{ν-1} ∈ \{+1, –1\}$&nbsp; holds.
+*Ignoring the modulation with the carrier signals &nbsp;$z(t)$&nbsp; and &nbsp;$2 · z(t)$&nbsp; resp.,&nbsp; the symbol &nbsp;$m_ν = m_{ν-1} · q_ν$&nbsp; is present in the interval &nbsp;$ν$&nbsp; at input&nbsp; '''(1)'''&nbsp; of the multiplier highlighted in yellow,&nbsp; and the symbol&nbsp; $m_{ν-1}$&nbsp; is present at input &nbsp; '''(2)'''.
-*The matched filter with frequency response &nbsp;$H_{\rm E}(f)$&nbsp; eliminates the unwanted components around the doubled carrier frequency which result from the twofold multiplication by &nbsp;$z(t)$&nbsp; and &nbsp;$2 · z(t)$.&nbsp; If the basic pulse &nbsp;$g_q(t)$&nbsp; is rectangular, the frequency response &nbsp;$H_{\rm E}(f)$&nbsp; can also be realized very easily by an integrator.
+*The multiplication of&nbsp; '''(1)'''&nbsp; and&nbsp; '''(2)'''&nbsp; gives the desired result:
-*We assume that the channel causes a phase rotation by &nbsp;$ϕ$,&nbsp; which the receiver does not know (red block).&nbsp; For example, assuming the transmitting side carrier &nbsp;$z(t) = \cos (2π · f_{\rm T} · t)$,&nbsp; the received signal &nbsp;$r(t)$&nbsp; contains a multiplicative component with &nbsp;$\cos (2π · f_{\rm T} · t + ϕ)$.&nbsp; Thus, the addition of the receiving carrier &nbsp;$2 · z(t)$&nbsp; is not phase synchronous.
+:$$v_ν = m_{ν-1} · q_ν · m_{ν-1} = q_ν.$$
-*The signal &nbsp;$r(t – T)$&nbsp; delayed  by a symbol duration &nbsp;$T$&nbsp; has the same phase &nbsp;$ϕ$.&nbsp; The correlation between &nbsp;$2 · r(t) · z(t)$&nbsp; and &nbsp;$2 · r(t – T) · z(t – T)$&nbsp; makes the decision result independent of the random phase &nbsp;$ϕ$.&nbsp; This type of demodulation is called ''differential-coherent''.
+:It is taken into account here that &nbsp;$m_{ν-1} ∈ \{+1, –1\}$&nbsp; holds.
 ==Error probabilities - a brief overview==