Modulation Methods/Quadrature Amplitude Modulation - Revision history

Guenter at 13:50, 13 January 2023

2023-01-13T13:50:11Z

Hwang at 12:40, 11 January 2023

2023-01-11T12:40:40Z

Hwang at 16:54, 9 January 2023

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Hwang at 23:55, 12 November 2022

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Guenter at 14:45, 16 April 2022

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Hwang at 09:36, 16 April 2022

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Hwang at 09:22, 16 April 2022

2022-04-16T09:22:33Z

@@ Line 49: / Line 49: @@
 :$$s(t)   = {\rm Re}[s_{\rm +}(t)] = {\rm Re}[s_{\rm TP}(t) \cdot{\rm e}^{{\rm j}2\pi f_{\rm T} t}] $$
 :$$\Rightarrow \hspace{0.3cm} s(t)  = {\rm Re} \left[\left ( \sum (a_{\rm I\hspace{0.03cm}\it \nu} + {\rm j} \cdot a_{\rm Q\hspace{0.03cm}\it \nu} ) \cdot g_s (t - \nu \cdot T)\right )\left ( \cos(2 \pi f_{\rm T} t) + {\rm j} \cdot \sin(2 \pi f_{\rm T} t) \right )\right]= s_{\rm I}(t) \cdot \cos(2\pi f_{\rm T} t) -  s_{\rm Q}(t) \cdot \sin(2 \pi f_{\rm T} t) \hspace{0.05cm}.$$
-*The influence of the bandpass '''KORREKTUR: band-pass''' &nbsp;$H_{\rm BP}(f)$,&nbsp; which in practice often has to be considered at the output of the QAM modulator,&nbsp; can be assigned to the pulse shape filter &nbsp;$g_s(t)$.&nbsp;  If the passband of the bandpass '''KORREKTUR: band-pass''' filter is symmetric about&nbsp;$f_{\rm T}$,&nbsp; its low-pass equivalent&nbsp; (in the time domain) &nbsp; $h_{\rm BP\ →\ TP}(t)$ &nbsp; is purely real and one can replace &nbsp;$g_s(t)$&nbsp; with &nbsp;$g_s(t) \star h_{\rm BP\ →\ TP}(t)$&nbsp; in the model.
+*The influence of the band-pass  &nbsp;$H_{\rm BP}(f)$,&nbsp; which in practice often has to be considered at the output of the QAM modulator,&nbsp; can be assigned to the pulse shape filter &nbsp;$g_s(t)$.&nbsp;  If the passband of the band-pass  filter is symmetric about&nbsp;$f_{\rm T}$,&nbsp; its low-pass equivalent&nbsp; (in the time domain) &nbsp; $h_{\rm BP\ →\ TP}(t)$ &nbsp; is purely real and one can replace &nbsp;$g_s(t)$&nbsp; with &nbsp;$g_s(t) \star h_{\rm BP\ →\ TP}(t)$&nbsp; in the model.
 ==Power and energy of QAM signals==
@@ Line 172: / Line 172: @@
-*$\rm 4–QAM$&nbsp; (left diagram)&nbsp; can also be understood as a four-stage '''KORREKTUR: four-level''' phase modulation&nbsp; $($positions &nbsp;$±45^\circ$,&nbsp; $±135^\circ)$.&nbsp; Compared to QPSK,&nbsp; there is a rotation of &nbsp;$±45^\circ$&nbsp; $(π/4)$ &nbsp; &rArr; &nbsp; 4–QAM is also called &nbsp;${\rm π/4}\text{–QPSK}$.
+*$\rm 4–QAM$&nbsp; (left diagram)&nbsp; can also be understood as a four-level phase modulation&nbsp; $($positions &nbsp;$±45^\circ$,&nbsp; $±135^\circ)$.&nbsp; Compared to QPSK,&nbsp; there is a rotation of &nbsp;$±45^\circ$&nbsp; $(π/4)$ &nbsp; &rArr; &nbsp; 4–QAM is also called &nbsp;${\rm π/4}\text{–QPSK}$.

@@ Line 239: / Line 239: @@
 [[Aufgaben:Aufgabe_4.11Z:_Fehlerwahrscheinlichkeit_bei_QAM|Exercise 4.11Z: Error Probability with QAM]]
-[[Aufgaben:Aufgabe_4.12:_Wurzel–Nyquist–Systeme|Exercise 4.12: Root Nyquist Systems]]
+[[Aufgaben:Exercise_4.12:_Root-Nyquist_Systems|Exercise 4.12: Root-Nyquist Systems]]
 [[Aufgaben:Exercise_4.12Z:_4-QAM_Systems_again|Exercise 4.12Z: 4-QAM Systems again]]

@@ Line 49: / Line 49: @@
 :$$s(t)   = {\rm Re}[s_{\rm +}(t)] = {\rm Re}[s_{\rm TP}(t) \cdot{\rm e}^{{\rm j}2\pi f_{\rm T} t}] $$
 :$$\Rightarrow \hspace{0.3cm} s(t)  = {\rm Re} \left[\left ( \sum (a_{\rm I\hspace{0.03cm}\it \nu} + {\rm j} \cdot a_{\rm Q\hspace{0.03cm}\it \nu} ) \cdot g_s (t - \nu \cdot T)\right )\left ( \cos(2 \pi f_{\rm T} t) + {\rm j} \cdot \sin(2 \pi f_{\rm T} t) \right )\right]= s_{\rm I}(t) \cdot \cos(2\pi f_{\rm T} t) -  s_{\rm Q}(t) \cdot \sin(2 \pi f_{\rm T} t) \hspace{0.05cm}.$$
-*The influence of the bandpass &nbsp;$H_{\rm BP}(f)$,&nbsp; which in practice often has to be considered at the output of the QAM modulator,&nbsp; can be assigned to the pulse shape filter &nbsp;$g_s(t)$.&nbsp;  If the passband of the bandpass filter is symmetric about&nbsp;$f_{\rm T}$,&nbsp; its low-pass equivalent&nbsp; (in the time domain) &nbsp; $h_{\rm BP\ →\ TP}(t)$ &nbsp; is purely real and one can replace &nbsp;$g_s(t)$&nbsp; with &nbsp;$g_s(t) \star h_{\rm BP\ →\ TP}(t)$&nbsp; in the model.
+*The influence of the bandpass '''KORREKTUR: band-pass''' &nbsp;$H_{\rm BP}(f)$,&nbsp; which in practice often has to be considered at the output of the QAM modulator,&nbsp; can be assigned to the pulse shape filter &nbsp;$g_s(t)$.&nbsp;  If the passband of the bandpass '''KORREKTUR: band-pass''' filter is symmetric about&nbsp;$f_{\rm T}$,&nbsp; its low-pass equivalent&nbsp; (in the time domain) &nbsp; $h_{\rm BP\ →\ TP}(t)$ &nbsp; is purely real and one can replace &nbsp;$g_s(t)$&nbsp; with &nbsp;$g_s(t) \star h_{\rm BP\ →\ TP}(t)$&nbsp; in the model.
 ==Power and energy of QAM signals==

@@ Line 27: / Line 27: @@
 {{BlaueBox|TEXT=
 $\text{Conclusion:}$&nbsp; These statements can be summarized as follows:
-*The two transmission branches &nbsp; $\rm (I,\ Q)$&nbsp; can be thought of as two completely separate  &nbsp;$M$-step ASK systems <br>that do not interfere with each other as long as all components are optimally designed.
+*The two transmission branches &nbsp; $\rm (I,\ Q)$&nbsp; can be thought of as two completely separate  &nbsp;$M$-level ASK systems <br>that do not interfere with each other as long as all components are optimally designed.
 *Quadrature amplitude modulation thus makes it&nbsp; (ideally)&nbsp; possible to double the data rate while maintaining the same quality. }}
@@ Line 172: / Line 172: @@
-*$\rm 4–QAM$&nbsp; (left diagram)&nbsp; can also be understood as a four-stage phase modulation&nbsp; $($positions &nbsp;$±45^\circ$,&nbsp; $±135^\circ)$.&nbsp; Compared to QPSK,&nbsp; there is a rotation of &nbsp;$±45^\circ$&nbsp; $(π/4)$ &nbsp; &rArr; &nbsp; 4–QAM is also called &nbsp;${\rm π/4}\text{–QPSK}$.
+*$\rm 4–QAM$&nbsp; (left diagram)&nbsp; can also be understood as a four-stage '''KORREKTUR: four-level''' phase modulation&nbsp; $($positions &nbsp;$±45^\circ$,&nbsp; $±135^\circ)$.&nbsp; Compared to QPSK,&nbsp; there is a rotation of &nbsp;$±45^\circ$&nbsp; $(π/4)$ &nbsp; &rArr; &nbsp; 4–QAM is also called &nbsp;${\rm π/4}\text{–QPSK}$.

@@ Line 12: / Line 12: @@
 This very general model can be described as follows:
 *The binary source symbol sequence&nbsp;$〈q_k〉$&nbsp; with bit rate &nbsp;$R_{\rm B}$&nbsp;is applied to the input.&nbsp;  Thus,&nbsp; the time interval between two symbols is &nbsp;$T_{\rm B} = 1/R_{\rm B}$.
-*Two multilevel amplitude coefficients &nbsp;$a_{{\rm I}ν}$&nbsp; and &nbsp;$a_{{\rm Q}ν}$&nbsp; are derived from each &nbsp;$b$&nbsp; binary input symbols &nbsp;$q_k$,&nbsp; where &nbsp;$\rm I$&nbsp; stands for&nbsp; "inphase  component"&nbsp; and &nbsp;$\rm Q$&nbsp; stands for&nbsp; "quadrature component".
+*Two multi-level amplitude coefficients &nbsp;$a_{{\rm I}ν}$&nbsp; and &nbsp;$a_{{\rm Q}ν}$&nbsp; are derived from each &nbsp;$b$&nbsp; binary input symbols &nbsp;$q_k$,&nbsp; where &nbsp;$\rm I$&nbsp; stands for&nbsp; "inphase  component"&nbsp; and &nbsp;$\rm Q$&nbsp; stands for&nbsp; "quadrature component".
 *If &nbsp;$b$&nbsp; is even and the signal space allocation is quadratic, then the coefficients &nbsp;$a_{{\rm I}ν}$&nbsp; and &nbsp;$a_{{\rm Q}ν}$&nbsp; can each take on one of the &nbsp;$M = 2^{b/2}$&nbsp; amplitude values with equal probability.&nbsp;  This is then referred to as&nbsp; &raquo;'''quadrature amplitude modulation'''&laquo;&nbsp; $\rm (QAM)$.
 *The example considered in the graph is for the &nbsp; $\text{16-QAM}$&nbsp; with &nbsp;$b = M = 4$&nbsp; and correspondingly &nbsp;$M^2 =16$&nbsp; signal space points.&nbsp;  For a &nbsp; $\text{256-QAM}$,&nbsp; $b = 8$&nbsp; and &nbsp;$M = 16$&nbsp; would apply: &nbsp; $2^b = M^2 = 256$.
@@ Line 161: / Line 161: @@
 #$\text{16-QAM}$&nbsp; is thus nothing more than the parallel transmission of two digital signals with &nbsp;$M = 4$&nbsp; amplitude levels each.
 #For&nbsp; $\text{64-QAM}$ &nbsp; &rArr; &nbsp; $M = 8$&nbsp; would apply analogously.&nbsp; For&nbsp; $\text{256-QAM}$:&nbsp; the "one-dimensional" number of levels is &nbsp;$M = 16$.
-#All the properties for redundancy-free multilevel signals mentioned in the later chapter &nbsp;[[Digital_Signal_Transmission/Redundancy-Free_Coding|"Redundancy-free Coding"]]&nbsp; also apply here.
+#All the properties for redundancy-free multi-level signals mentioned in the later chapter &nbsp;[[Digital_Signal_Transmission/Redundancy-Free_Coding|"Redundancy-free Coding"]]&nbsp; also apply here.

@@ Line 13: / Line 13: @@
 *The binary source symbol sequence&nbsp;$〈q_k〉$&nbsp; with bit rate &nbsp;$R_{\rm B}$&nbsp;is applied to the input.&nbsp;  Thus,&nbsp; the time interval between two symbols is &nbsp;$T_{\rm B} = 1/R_{\rm B}$.
 *Two multilevel amplitude coefficients &nbsp;$a_{{\rm I}ν}$&nbsp; and &nbsp;$a_{{\rm Q}ν}$&nbsp; are derived from each &nbsp;$b$&nbsp; binary input symbols &nbsp;$q_k$,&nbsp; where &nbsp;$\rm I$&nbsp; stands for&nbsp; "inphase  component"&nbsp; and &nbsp;$\rm Q$&nbsp; stands for&nbsp; "quadrature component".
-*If &nbsp;$b$&nbsp; is even and the signal space allocation is quadratic, then the coefficients &nbsp;$a_{{\rm I}ν}$&nbsp; and &nbsp;$a_{{\rm Q}ν}$&nbsp; can each take on one of the &nbsp;$M = 2^{b/2}$&nbsp; amplitude values with equal probability.&nbsp;  This is then referred to as&nbsp; '''quadrature amplitude modulation'''&nbsp; $\rm (QAM)$.
+*If &nbsp;$b$&nbsp; is even and the signal space allocation is quadratic, then the coefficients &nbsp;$a_{{\rm I}ν}$&nbsp; and &nbsp;$a_{{\rm Q}ν}$&nbsp; can each take on one of the &nbsp;$M = 2^{b/2}$&nbsp; amplitude values with equal probability.&nbsp;  This is then referred to as&nbsp; &raquo;'''quadrature amplitude modulation'''&laquo;&nbsp; $\rm (QAM)$.
 *The example considered in the graph is for the &nbsp; $\text{16-QAM}$&nbsp; with &nbsp;$b = M = 4$&nbsp; and correspondingly &nbsp;$M^2 =16$&nbsp; signal space points.&nbsp;  For a &nbsp; $\text{256-QAM}$,&nbsp; $b = 8$&nbsp; and &nbsp;$M = 16$&nbsp; would apply: &nbsp; $2^b = M^2 = 256$.
-*Next,&nbsp; the coefficients &nbsp;$a_{{\rm I}ν}$&nbsp; and &nbsp;$a_{{\rm Q}ν}$&nbsp; are each applied to a [[Signal_Representation/Discrete-Time_Signal_Representation#Dirac_comb_in_time_and_frequency_domain|"Dirac comb"]]&nbsp; as pulse weights.&nbsp;  Thus,&nbsp; after pulse shaping with the basic transmission pulse &nbsp;$g_s(t)$,&nbsp; the following holds for both branches of the circuit diagram:
+*Next,&nbsp; the coefficients &nbsp;$a_{{\rm I}ν}$&nbsp; and &nbsp;$a_{{\rm Q}ν}$&nbsp; are each applied to a [[Signal_Representation/Discrete-Time_Signal_Representation#Dirac_comb_in_time_and_frequency_domain|$\text{Dirac comb}$]]&nbsp; as pulse weights.&nbsp;  Thus,&nbsp; after pulse shaping with the basic transmission pulse &nbsp;$g_s(t)$,&nbsp; the following holds for both branches of the circuit diagram:
 :$$s_{\rm I}(t)  = \sum_{\nu = - \infty}^{+\infty}a_{\rm I\hspace{0.03cm}\it \nu} \cdot g_s (t - \nu \cdot T)\hspace{0.05cm},\hspace{1cm}s_{\rm Q}(t) = \sum_{\nu = - \infty}^{+\infty}a_{\rm
 Q\hspace{0.03cm}\it \nu} \cdot g_s (t - \nu \cdot T)\hspace{0.05cm}.$$
@@ Line 36: / Line 36: @@
 [[File:P_ID1708__Mod_T_4_3_S2_Ganz_neu.png |right|frame| Linear modulator&nbsp; $(\rm I$ and&nbsp; $\rm Q$ components$)$&nbsp; in the equivalent low-pass range]]
 #The graph shows the simplified model in the baseband.&nbsp;  This is equivalent to the block diagram considered so far.
-#The serial-parallel conversion and the signal space allocation drawn in red in &nbsp;[[Modulation_Methods/Quadrature_Amplitude_Modulation#General_description_and_signal_space_allocation|the block diagram in the last section]]&nbsp;are retained.&nbsp; This block is no longer drawn here.
+#The serial-parallel conversion and the signal space allocation drawn in red in &nbsp;[[Modulation_Methods/Quadrature_Amplitude_Modulation#General_description_and_signal_space_allocation|"the block diagram in the last section"]]&nbsp;are retained.&nbsp; This block is no longer drawn here.
 #We also initially disregard the band-pass &nbsp;$H_{\rm BP}(f)$,&nbsp; which is often introduced for technical reasons.
 <br clear=all>
@@ Line 53: / Line 53: @@
 ==Power and energy of QAM signals==
 <br>
-As shown in the chapter &nbsp;[[Signal_Representation/Equivalent_Low-Pass_Signal_and_its_Spectral_Function|Equivalent Low-Pass Signal and its Spectral Function]]&nbsp; in the book&nbsp; "Signal Representation",&nbsp; the &nbsp; '''power'''&nbsp; of the transmitted QAM signal &nbsp;$s(t)$&nbsp; can also be calculated from the equivalent low-pass signal&nbsp; $s_{\rm TP}(t)$,&nbsp; which is always complex.&nbsp;  Thus,&nbsp; it is equally valid to write:
+As shown in the chapter &nbsp;[[Signal_Representation/Equivalent_Low-Pass_Signal_and_its_Spectral_Function|"Equivalent Low-Pass Signal and its Spectral Function"]]&nbsp; in the book&nbsp; "Signal Representation",&nbsp; the &nbsp; &raquo;'''power'''&laquo;&nbsp; of the transmitted QAM signal &nbsp;$s(t)$&nbsp; can also be calculated from the equivalent low-pass signal&nbsp; $s_{\rm TP}(t)$,&nbsp; which is always complex.&nbsp;  Thus,&nbsp; it is equally valid to write:
 :$$P  = \lim_{T_{\rm M} \rightarrow \infty} \frac{\rm 1}{T_{\rm M}}\cdot \int_{-T_{\rm M}/2}^{+T_{\rm M}/2} s^2(t)\,{\rm d}  t = {\rm 1}/{2} \cdot \lim_{T_{\rm M} \rightarrow \infty} \frac{\rm 1}{T_{\rm M}}\cdot \int_{-T_{\rm M}/2}^{+T_{\rm M}/2} |s_{\rm TP}(t)|^2\,{\rm d}  t \hspace{0.05cm}.$$
 In contrast,&nbsp; the energy of the unbounded signals&nbsp; $s(t)$&nbsp; and &nbsp; $s_{\rm TP}(t)$&nbsp; is infinite.
-However,&nbsp; if we restrict ourselves to a symbol duration&nbsp;$T$,&nbsp; we obtain the&nbsp; '''energy per symbol''':
+However,&nbsp; if we restrict ourselves to a symbol duration&nbsp;$T$,&nbsp; we obtain the&nbsp; &raquo;'''energy per symbol'''&laquo;:
 :$$E_{\rm S}  = \frac{{\rm E}[\hspace{0.05cm}|a_{\nu} |^2\hspace{0.05cm}]}{2}\cdot \int_{-\infty}^{+\infty} |g_s(t)|^2\,{\rm d}  t = \frac{{\rm E}[|a_{\nu} |^2]}{2}\cdot \int_{-\infty}^{+\infty} |G_s(f)|^2\,{\rm d}  f  \hspace{0.05cm}.$$
-On the other hand, &nbsp; $E_{\rm B} = E_{\rm S}/b$ &nbsp; gives the&nbsp; '''energy per bit'''&nbsp; if according to the signal space assignment,&nbsp; $b$&nbsp; binary symbols are combined to the complex coefficient&nbsp; $a_\nu$.
+On the other hand, &nbsp; $E_{\rm B} = E_{\rm S}/b$ &nbsp; gives the&nbsp; &raquo;'''energy per bit'''&laquo;&nbsp; if according to the signal space assignment,&nbsp; $b$&nbsp; binary symbols are combined to the complex coefficient&nbsp; $a_\nu$.
 {{GraueBox|TEXT=
@@ Line 119: / Line 119: @@
 Using the additional equations
 :$$p_{\rm B} = {\rm Q}\left ( {s_0}/{\sigma_d } \right ), \hspace{0.2cm} E_{\rm B}  =  {1}/{2} \cdot s_0^2 \cdot T_{\rm B} ,\hspace{0.2cm} \sigma_d^2 = {N_0}/{T_{\rm B} }$$
-the&nbsp; '''BPSK error probability''' is given by:
+the&nbsp; &raquo;'''BPSK error probability'''&laquo; is given by:
 $$p_{\rm B, \hspace{0.1cm}BPSK} = {\rm Q}\left ( \sqrt{{2 \cdot E_{\rm B}}/{N_0 }} \hspace{0.1cm}\right ) = {1}/{2}\cdot {\rm erfc}\left ( \sqrt{{E_{\rm B}}/{N_0 }} \hspace{0.1cm}\right ).$$
@@ Line 132: / Line 132: @@
 :$$p_{\rm B} = {\rm Q}\left ( {g_0}/{\sigma_d } \right ), \hspace{0.2cm}g_{0}  =  {s_0}/{\sqrt{2}}, \hspace{0.2cm}E_{\rm B}  =  {1}/{2} \cdot s_0^2 \cdot T_{\rm B} ,\hspace{0.2cm}
   \sigma_d^2 = {N_0}(2 \cdot T_{\rm B} )$$
-one obtains the exact same result for the&nbsp; '''4-QAM error probability'''&nbsp; as for BPSK:
+one obtains the exact same result for the&nbsp; &raquo;'''4-QAM error probability'''&laquo;&nbsp; as for BPSK:
 :$$p_{\rm B, \hspace{0.1cm}4-QAM} = {\rm Q}\left ( \sqrt{{2 \cdot E_{\rm B}}/{N_0 }} \hspace{0.1cm}\right ) = {1}/{2}\cdot {\rm erfc}\left ( \sqrt{{E_{\rm B}}/{N_0 }} \hspace{0.1cm}\right ).$$
@@ Line 152: / Line 152: @@
 The figures  apply to all Nyquist systems
 *such as the&nbsp; "rectangular-rectangular"&nbsp; configuration&nbsp; (basic transmission pulse and receiver filter impulse response are both rectangular)
-*or a &nbsp;[[Digital_Signal_Transmission/Optimization_of_Baseband_Transmission_Systems#Root_Nyquist_systems|Root cosine rolloff Nyquist system]].
+*or a &nbsp;[[Digital_Signal_Transmission/Optimization_of_Baseband_Transmission_Systems#Root_Nyquist_systems|$\text{root cosine rolloff Nyquist system}$]].
@@ Line 184: / Line 184: @@
 ==Nyquist and Root-Nyquist QAM systems==
 <br>
-So far in this chapter,&nbsp; for reasons of easy representation,&nbsp; the rectangular basic transmission pulse has always been assumed.&nbsp;   In practice,&nbsp; however,&nbsp; a&nbsp; '''root-Nyquist characteristic'''&nbsp; is usually used according to the [[Digital_Signal_Transmission/Optimization_of_Baseband_Transmission_Systems#Root_Nyquist_systems|description]]&nbsp; in the book&nbsp; "Digital Signal Transmission".&nbsp;  Briefly,&nbsp; these systems can be characterized as follows:
+So far in this chapter,&nbsp; for reasons of easy representation,&nbsp; the rectangular basic transmission pulse has always been assumed.&nbsp;   In practice,&nbsp; however,&nbsp; a&nbsp; &raquo;'''root-Nyquist characteristic'''&laquo;&nbsp; is usually used according to the [[Digital_Signal_Transmission/Optimization_of_Baseband_Transmission_Systems#Root_Nyquist_systems|$\text{description}$]]&nbsp; in the book&nbsp; "Digital Signal Transmission".&nbsp;  Briefly,&nbsp; these systems can be characterized as follows:
 #The receiver&nbsp; (German:&nbsp; "Empfänger" &nbsp; &rArr; &nbsp; subscript;&nbsp; "E")&nbsp; frequency response &nbsp;$H_{\rm E}(f)$&nbsp; is chosen here to be equal in shape to the (normalized) transmission pulse&nbsp; (German:&nbsp; "Sendeimpuls" &nbsp; &rArr; &nbsp; subscript;&nbsp; "S")&nbsp; spectrum &nbsp;$H_{\rm S}(f)$,&nbsp; which leads to the smallest possible error probability under the constraints of power limitation.
-#The overall frequency response &nbsp;$H_{\rm Nyq}(f) = H_{\rm S}(f) · H_{\rm E}(f)$&nbsp; satisfies the [[Digital_Signal_Transmission/Properties_of_Nyquist_Systems#First_Nyquist_criterion_in_the_time_domain|first Nyquist criterion]],&nbsp; so there is no intersymbol interference&nbsp; $\text{(ISI)}$&nbsp; at the receiver.&nbsp; Thus,&nbsp; $H_{\rm S}(f)$&nbsp; and &nbsp;$H_{\rm E}(f)$&nbsp; each have a root-Nyquist characteristic.
+#The overall frequency response &nbsp;$H_{\rm Nyq}(f) = H_{\rm S}(f) · H_{\rm E}(f)$&nbsp; satisfies the [[Digital_Signal_Transmission/Properties_of_Nyquist_Systems#First_Nyquist_criterion_in_the_time_domain|$\text{first Nyquist criterion}$]],&nbsp; so there is no intersymbol interference&nbsp; $\text{(ISI)}$&nbsp; at the receiver.&nbsp; Thus,&nbsp; $H_{\rm S}(f)$&nbsp; and &nbsp;$H_{\rm E}(f)$&nbsp; each have a root-Nyquist characteristic.
-#For the frequency response &nbsp;$H_{\rm Nyq}(f)$,&nbsp; one often uses a&nbsp; [[Linear_and_Time_Invariant_Systems/Some_Low-Pass_Functions_in_Systems_Theory#Cosine.E2.80.93square_low-pass_filter|cosine rolloff low-pass]]&nbsp; $H_{\rm CRO}(f)$&nbsp; with equivalent bandwidth &nbsp;$Δf_{\rm CRO} = 1/T$&nbsp; and freely chosen rolloff factor&nbsp;$ (0 ≤ r ≤ 1)$.&nbsp;  The interactive applet &nbsp;[[Applets:Frequency_%26_Impulse_Responses|"Frequency and Impulse Responses"]]&nbsp; illustrates frequency response and impulse response of this low-pass filter.
+#For the frequency response &nbsp;$H_{\rm Nyq}(f)$,&nbsp; one often uses a&nbsp; [[Linear_and_Time_Invariant_Systems/Some_Low-Pass_Functions_in_Systems_Theory#Cosine.E2.80.93square_low-pass_filter|$\text{cosine rolloff low-pass}$]]&nbsp; $H_{\rm CRO}(f)$&nbsp; with equivalent bandwidth &nbsp;$Δf_{\rm CRO} = 1/T$&nbsp; and freely chosen rolloff factor&nbsp;$ (0 ≤ r ≤ 1)$.&nbsp;  The interactive applet &nbsp;[[Applets:Frequency_%26_Impulse_Responses|"Frequency and Impulse Responses"]]&nbsp; illustrates frequency response and impulse response of this low-pass filter.
 #The advantage of these root-Nyquist systems is the much smaller bandwidth &nbsp;$(1 + r)/T$&nbsp; compared to the previously considered configuration with rectangular&nbsp; $g_s(t)$&nbsp; and rectangular&nbsp; $h_{\rm E}(t)$,&nbsp; whose spectrum is&nbsp; (theoretically)&nbsp; infinitely extended.
 #In terms of error probability,&nbsp; nothing changes compared to the&nbsp; "rectangular-rectangular"&nbsp; configuration because the basic detection pulse &nbsp;$g_d(t)$&nbsp;  before the decision has equidistant zero crossings,&nbsp; thus avoiding intersymbol interference.

@@ Line 94: / Line 94: @@
 The following graph shows the signal waveforms of&nbsp; $\rm 4-QAM$,&nbsp; where the coloring corresponds to the signal space allocation defined above.
-[[File:P_ID1711__Mod_T_4_3_S4_neu.png|right|frame| Signal waveforms for&nbsp; $\rm 4-QAM$]]
+[[File:P_ID1711__Mod_T_4_3_S4_neu.png|right|frame| Signal waveforms for&nbsp; $\text{4-QAM}$]]
 One can see from these plots:
@@ Line 204: / Line 204: @@
 ==Offset–Quadrature amplitude modulation==
 <br>
-Taking the equations for the&nbsp; "$\text{4–QAM}$" &nbsp; $($or &nbsp;"$π/4\text{–QPSK}$"$)$&nbsp; as our starting point,&nbsp; we arrive at the &nbsp; "$\text{Offset–4–QAM}$",&nbsp; which we simplify by calling &nbsp; "$\text{Offset–QPSK}$".&nbsp;  Here the transmit signal '''KORREKTUR: transmitted signal''' is given by:
+Taking the equations for the&nbsp; "$\text{4–QAM}$" &nbsp; $($or &nbsp;"$π/4\text{–QPSK}$"$)$&nbsp; as our starting point,&nbsp; we arrive at the &nbsp; "$\text{Offset–4–QAM}$",&nbsp; which we simplify by calling &nbsp; "$\text{Offset–QPSK}$".&nbsp;  Here the transmitted signal is given by:
 :$$s(t)  =s_{\rm I}(t) \cdot \cos(2 \pi f_{\rm T} t) -  s_{\rm Q}(t)
 \cdot \sin(2 \pi f_{\rm T} t),$$

@@ Line 103: / Line 103: @@
 *The QAM signal &nbsp;$s(t) = s_{\rm cos}(t) \ – \ s_{\rm –sin}(t)$&nbsp; with phase jumps by multiples of &nbsp;$±π/2$;&nbsp; their envelope is larger than the two component signals by a factor of &nbsp;$\sqrt{2}$:
 :$$s_0 = \sqrt{2} \cdot  g_0 \hspace{0.05cm}.$$
-*Here,&nbsp; the basicl transmission pulse &nbsp;$g_s(t)$&nbsp; is assumed to be rectangular between  &nbsp;$0$&nbsp; and &nbsp;$T$,&nbsp; i.e.,&nbsp; asymmetric with respect to&nbsp; $t = 0$,&nbsp; for simplicity of representation.&nbsp; The associated spectral function &nbsp;$G_s(f)$&nbsp;of this causal pulse &nbsp; $g_s(t)$&nbsp; is complex,&nbsp; though this has no implications in this context.
+*Here,&nbsp; the basic transmission pulse &nbsp;$g_s(t)$&nbsp; is assumed to be rectangular between  &nbsp;$0$&nbsp; and &nbsp;$T$,&nbsp; i.e.,&nbsp; asymmetric with respect to&nbsp; $t = 0$,&nbsp; for simplicity of representation.&nbsp; The associated spectral function &nbsp;$G_s(f)$&nbsp;of this causal pulse &nbsp; $g_s(t)$&nbsp; is complex,&nbsp; though this has no implications in this context.
 <br clear=all>