Modulation Methods/General Description of OFDM - Revision history

Guenter at 16:31, 23 January 2023

2023-01-23T16:31:46Z

Guenter at 15:22, 19 January 2023

2023-01-19T15:22:25Z

Hwang at 18:27, 18 January 2023

2023-01-18T18:27:47Z

Hwang at 18:27, 18 January 2023

2023-01-18T18:27:10Z

Hwang at 18:25, 18 January 2023

2023-01-18T18:25:20Z

Hwang at 18:22, 18 January 2023

2023-01-18T18:22:24Z

Hwang at 18:20, 18 January 2023

2023-01-18T18:20:58Z

Hwang at 17:13, 18 January 2023

2023-01-18T17:13:01Z

Hwang at 12:45, 18 January 2023

2023-01-18T12:45:53Z

Reed at 14:01, 9 April 2022

2022-04-09T14:01:13Z

@@ Line 91: / Line 91: @@
 The diagram shows an example of the magnitude spectrum of an OFDM signal with five carriers.
-[[File:ENneu_Mod_T_5_5_S4.png|right|frame| Spectrum of an OFDM signal '''KORREKTUR: Note?''']]
+[[File:ENneu_Mod_T_5_5_S4_v2.png|right|frame| Spectrum of an OFDM signal]]
 *It is remarkable that the maximum of each subcarrier coincides with the zeros of all other carriers.&nbsp; This corresponds to the first Nyquist condition in the frequency domain.

@@ Line 111: / Line 111: @@
 Here,&nbsp; the &nbsp;$k$–th transmitted pulse
 *is the sum of the basic pulses &nbsp;$g_{\mu}(t)$ shifted by&nbsp;$k · T$,&nbsp;
-*each of which is weighted by the &nbsp;$\mu$–th amplitude coefficients of the QAM coder '''KORREKTUR: encoder''' at time &nbsp;$k$.&nbsp;
+*each of which is weighted by the &nbsp;$\mu$–th amplitude coefficients of the QAM encoder at time &nbsp;$k$.&nbsp;

@@ Line 111: / Line 111: @@
 Here,&nbsp; the &nbsp;$k$–th transmitted pulse
 *is the sum of the basic pulses &nbsp;$g_{\mu}(t)$ shifted by&nbsp;$k · T$,&nbsp;
-*each of which is weighted by the &nbsp;$\mu$–th amplitude coefficients of the QAM coder at time &nbsp;$k$.&nbsp;
+*each of which is weighted by the &nbsp;$\mu$–th amplitude coefficients of the QAM coder '''KORREKTUR: encoder''' at time &nbsp;$k$.&nbsp;

@@ Line 101: / Line 101: @@
 {{BlaueBox|TEXT=
 $\text{Conclusion:}$&nbsp; An OFDM signal under the condition of a rectangular pulse shaping and a subcarrier spacing of &nbsp;$f_0$&nbsp;
-*fulfills the &nbsp;[[Digital_Signal_Transmission/Eigenschaften_von_Nyquistsystemen#Erstes_Nyquistkriterium_im_Zeitbereich|$\text{first Nyquist condition in the time domain}$]]&nbsp; and thus, of course,
+*fulfills the &nbsp;[[Digital_Signal_Transmission/Properties_of_Nyquist_Systems#First_Nyquist_criterion_in_the_time_domain|$\text{first Nyquist condition in the time domain}$]]&nbsp; and thus, of course,
-*also the &nbsp;[[Digital_Signal_Transmission/Eigenschaften_von_Nyquistsystemen#Erstes_Nyquistkriterium_im_Frequenzbereich |$\text{first Nyquist condition in the frequency domain}$]].}}
+*also the &nbsp;[[Digital_Signal_Transmission/Properties_of_Nyquist_Systems#First_Nyquist_criterion_in_the_frequency_domain |$\text{first Nyquist condition in the frequency domain}$]].}}

@@ Line 91: / Line 91: @@
 The diagram shows an example of the magnitude spectrum of an OFDM signal with five carriers.
-[[File:ENneu_Mod_T_5_5_S4.png|right|frame| Spectrum of an OFDM signal]]
+[[File:ENneu_Mod_T_5_5_S4.png|right|frame| Spectrum of an OFDM signal '''KORREKTUR: Note?''']]
 *It is remarkable that the maximum of each subcarrier coincides with the zeros of all other carriers.&nbsp; This corresponds to the first Nyquist condition in the frequency domain.

@@ Line 67: / Line 67: @@
 For simplicity,&nbsp; we assume the basic pulse  &nbsp;$g_{\mu}(t)$&nbsp; to be symmetric at &nbsp;$t = 0$.&nbsp;&nbsp; Then it is valid with &nbsp;$f_0 = 1/T$:
 :$$g_\mu (t) = \left\{ \begin{array}{l} s_0 \cdot {\rm{e}}^{ {\kern 1pt} {\rm{j2 \pi}} {\kern 1pt} \mu f_0 t} \quad \quad - T/2 < t < T/2, \\ 0 \quad \quad \quad \quad \quad \quad \; {\rm otherwise.} \\ \end{array} \right.$$
-In the spectral domain,&nbsp; such an acausal rectangular function modulated by a&nbsp; (complex)&nbsp; exponential function of frequency &nbsp;$\mu · f_0$&nbsp; corresponds to an&nbsp; $\rm sinc$ function shifted by&nbsp;$\mu · f_0$:
+In the spectral domain,&nbsp; such an acausal rectangular function modulated by a&nbsp; (complex)&nbsp; exponential function of frequency &nbsp;$\mu · f_0$&nbsp; corresponds to a&nbsp; $\rm sinc$ function shifted by&nbsp;$\mu · f_0$:
 :$$G_\mu (f) = s_0 \cdot T \cdot {\rm{sinc}} \big((f - \mu  f_0 )\cdot T \big ).$$
 On the right,&nbsp;  this spectral function&nbsp; $($normalized to the maximum &nbsp;$s_0 · T)$&nbsp; is shown for &nbsp;$\mu = 5$. &nbsp; &ndash; &nbsp; The arrow is to suggest that if the basic pulse is unconstrained in time,&nbsp; the &nbsp;$\rm sinc$ function shown would have to be replaced by a Dirac delta function at &nbsp;$\mu · f_0$.

@@ Line 48: / Line 48: @@
 *The symbol duration in the single carrier&nbsp; $($ $\rm SC)$&nbsp; case is then:
 :$$T_\text{SC} = 1 \cdot {\rm{log}_2}(4) \cdot 1.3 \,{\rm{&micro; s} } \approx 2.6 \,{\rm{&micro; s} }.$$
-*Assuming that for a multi carrier&nbsp; $($ $\rm MC)$&nbsp; system with &nbsp;$N = 32$&nbsp; carriers the modulation method&nbsp; $\text{16–QAM}$&nbsp; is used,&nbsp; the symbol duration is larger by a factor of&nbsp; $64$:&nbsp;
+*Assuming that for a multicarrier&nbsp; $($ $\rm MC)$&nbsp; system with &nbsp;$N = 32$&nbsp; carriers the modulation method&nbsp; $\text{16–QAM}$&nbsp; is used,&nbsp; the symbol duration is larger by a factor of&nbsp; $64$:&nbsp;
 :$$T_\text{MC} = 32 \cdot  {\rm{log}_2}(16) \cdot 1.3 \,{\rm{&micro; s} } \approx 0.167\, {\rm{ms} }.$$ }}

@@ Line 13: / Line 13: @@
-*In OFDM,&nbsp; the subcarriers themselves are usually modulated by conventional &nbsp;[[Modulation_Methods/Quadratur–Amplitudenmodulation|quadrature amplitude modulation]]&nbsp; $\rm (QAM)$&nbsp; or by &nbsp;[[Modulation_Methods/Lineare_digitale_Modulation#BPSK_.E2.80.93_Binary_Phase_Shift_Keying|binary phase modulation]]&nbsp; $\rm (BPSK)$,&nbsp; although the individual carriers may well differ in terms of the modulation type.
+*In OFDM,&nbsp; the subcarriers themselves are usually modulated by conventional &nbsp;[[Modulation_Methods/Quadratur–Amplitudenmodulation|$\text{quadrature amplitude modulation}$]]&nbsp; $\rm (QAM)$&nbsp; or by &nbsp;[[Modulation_Methods/Lineare_digitale_Modulation#BPSK_.E2.80.93_Binary_Phase_Shift_Keying|$\text{binary phase modulation}$]]&nbsp; $\rm (BPSK)$,&nbsp; although the individual carriers may well differ in terms of the modulation type.
@@ Line 22: / Line 22: @@
 *The &nbsp;$N–1$&nbsp; parts of the data stream &nbsp;$〈q_{\mu,k}〉$&nbsp; present at time&nbsp;$k$&nbsp; are first 4-QAM encoded by combining two bits at a time.&nbsp;
 *Then the generally complex amplitude&nbsp;$a_{\mu,\hspace{0.08cm}k}$&nbsp; $($with control variables &nbsp;$\mu = 1$, ... , $N–1)$&nbsp; is pulse-shaped and modulated with the&nbsp; $\mu$–th multiple of the basic frequency &nbsp;$f_0$.&nbsp;
-*The transmitted signal is now the additive superposition of the individual partial signals.&nbsp; The consideration takes place here and also in the following in the&nbsp;[[Signal_Representation/Equivalent_Low-Pass_Signal_and_its_Spectral_Function|equivalent low-pass range]],&nbsp; whereby the index&nbsp; "TP"&nbsp; (German:&nbsp; "Tiefpassbereich")&nbsp; is renounced.
+*The transmitted signal is now the additive superposition of the individual partial signals.&nbsp; The consideration takes place here and also in the following in the&nbsp;[[Signal_Representation/Equivalent_Low-Pass_Signal_and_its_Spectral_Function|$\text{equivalent low-pass range}$]],&nbsp; whereby the index&nbsp; "TP"&nbsp; (German:&nbsp; "Tiefpassbereich")&nbsp; is renounced.
 *The pulse shape filter &nbsp;$g_s(t)$&nbsp; is a rectangle of height &nbsp;$s_0$&nbsp; limited to the range &nbsp;$0 ≤ t < T$.&nbsp;
-*We call &nbsp;$T$&nbsp; the&nbsp; '''symbol duration'''&nbsp; and denote the reciprocal &nbsp;$f_0 = 1/T$&nbsp; as the&nbsp; '''basic frequency'''.
+*We call &nbsp;$T$&nbsp; the&nbsp; &raquo;'''symbol duration'''&laquo;&nbsp; and denote the reciprocal &nbsp;$f_0 = 1/T$&nbsp; as the&nbsp; &raquo;'''basic frequency'''&laquo;.
 If we now summarize this filter with the respective modulation to
 :$$g_\mu (t) = \left\{ \begin{array}{l} s_0  \cdot {\rm{e}}^{ {\kern 1pt} {\rm{j2 \pi}} {\kern 1pt} \mu f_0 t} \quad 0 \le t < T, \\ 0 \quad \quad \quad \quad \quad {\rm otherwise} \\ \end{array} \right.$$
-with&nbsp; $\mu ∈ \{0, \ \text{...}\ , N–1\}$,&nbsp; we obtain the&nbsp; '''OFDM transmitted signal&nbsp; $s_k(t)$&nbsp; in the &nbsp;$k$–th time interval''':
+with&nbsp; $\mu ∈ \{0, \ \text{...}\ , N–1\}$,&nbsp; we obtain the&nbsp; &raquo;'''OFDM transmitted signal&nbsp; $s_k(t)$&nbsp; in the &nbsp;$k$–th time interval'''&laquo;:
 :$$s_k (t) = \sum\limits_{\mu = 0}^{N - 1} {a_{\mu ,\hspace{0.08cm}k} \cdot g_\mu (t - k \cdot T_{\rm{R}} )}.$$
 {{BlaueBox|TEXT=
-The total&nbsp; '''OFDM transmitted signal considering all time intervals'''&nbsp; is then:
+The total&nbsp; &raquo;'''OFDM transmitted signal considering all time intervals'''&laquo;&nbsp; is then:
 :$$s(t) = \sum\limits_{k = - \infty }^{+\infty} {\sum\limits_{\mu = 0}^{N - 1} {a_{\mu ,\hspace{0.08cm}k} \cdot g_\mu (t - k \cdot T_{\rm{R} } )} }.$$
 *$T_{\rm R}$&nbsp; denotes the frame&nbsp; (German:&nbsp; "Rahmen" &nbsp; &rArr; &nbsp; subscript "R")&nbsp; duration.&nbsp; Within this time the same data is present at the input and after&nbsp; $T_{\rm R}$&nbsp; the next frame follows.
@@ Line 56: / Line 56: @@
 *The duration of a symbol increases significantly for a multicarrier system compared to a single-carrier system,&nbsp; reducing the spurious influence of the channel impulse response and decreasing intersymbol interference.
-*The ability to use different robust modulation schemes for different subbands is one of the&nbsp; '''major advantages of OFDM'''.&nbsp; This will be discussed in more detail in the following sections&nbsp; [[Modulation_Methods/OFDM_für_4G–Netze|OFDM for 4G networks]]&nbsp; and&nbsp; [[Examples_of_Communication_Systems/Allgemeine_Beschreibung_von_DSL|Digital Subscriber Line]]&nbsp; $\rm (DSL)$.&nbsp; }}
+*The ability to use different robust modulation schemes for different subbands is one of the&nbsp; &raquo;'''major advantages of OFDM'''&laquo;.&nbsp; This will be discussed in more detail in the following sections&nbsp; [[Modulation_Methods/OFDM_für_4G–Netze|"OFDM for 4G networks"]]&nbsp; and&nbsp; [[Examples_of_Communication_Systems/Allgemeine_Beschreibung_von_DSL|"Digital Subscriber Line"]]&nbsp; $\rm (DSL)$.&nbsp; }}
@@ Line 101: / Line 101: @@
 {{BlaueBox|TEXT=
 $\text{Conclusion:}$&nbsp; An OFDM signal under the condition of a rectangular pulse shaping and a subcarrier spacing of &nbsp;$f_0$&nbsp;
-*fulfills the &nbsp;[[Digital_Signal_Transmission/Eigenschaften_von_Nyquistsystemen#Erstes_Nyquistkriterium_im_Zeitbereich|first Nyquist condition in the time domain]]&nbsp; and thus, of course,
+*fulfills the &nbsp;[[Digital_Signal_Transmission/Eigenschaften_von_Nyquistsystemen#Erstes_Nyquistkriterium_im_Zeitbereich|$\text{first Nyquist condition in the time domain}$]]&nbsp; and thus, of course,
-*also the &nbsp;[[Digital_Signal_Transmission/Eigenschaften_von_Nyquistsystemen#Erstes_Nyquistkriterium_im_Frequenzbereich |first Nyquist condition in the frequency domain]].}}
+*also the &nbsp;[[Digital_Signal_Transmission/Eigenschaften_von_Nyquistsystemen#Erstes_Nyquistkriterium_im_Frequenzbereich |$\text{first Nyquist condition in the frequency domain}$]].}}
@@ Line 161: / Line 161: @@
 {{BlaueBox|TEXT=
-$\text{Conclusion:}$&nbsp; &nbsp; The&nbsp; '''orthogonality of the OFDM signal'''&nbsp; &nbsp;$s(t)$&nbsp;
+$\text{Conclusion:}$&nbsp; &nbsp; The&nbsp; &raquo;'''orthogonality of the OFDM signal'''&laquo;&nbsp; &nbsp;$s(t)$&nbsp;
 *is given for the control variable &nbsp;$k$&nbsp; $\rm (time)$&nbsp;
 *as well as for the control variable &nbsp;$\mu$&nbsp; $\text{(carrier frequencies)}$&nbsp;!}}

@@ Line 95: / Line 95: @@
 *It is remarkable that the maximum of each subcarrier coincides with the zeros of all other carriers.&nbsp; This corresponds to the first Nyquist condition in the frequency domain.
 *This property allows&nbsp; "ICI"&ndash;free sampling&nbsp; (that is: &nbsp; without intercarrier interference)&nbsp; of the spectrum at multiples of &nbsp;$f_0$.&nbsp; Orthogonality is therefore guaranteed.
-*If one were to dispense with the time limit in pulse shaping, the displayed&nbsp; $\rm sinc$&nbsp; functions would each become Dirac lines at the distance&nbsp; $f_0$&nbsp;  (drawn in gray in the diagram).
+*If one were to dispense with the time limit in pulse shaping, the displayed&nbsp; $\rm sinc$&nbsp; functions would each become Dirac delta lines at the distance&nbsp; $f_0$&nbsp;  (drawn in gray in the diagram).
 *This idealizing simplification is unfortunately not realizable in practice.&nbsp; Indeed,&nbsp; the requirement &nbsp;$T → ∞$&nbsp; means at the same time that only one frame could be transmitted in an infinitely long time.
 <br clear=all>

@@ Line 2: / Line 2: @@
 {{Header
 |Untermenü= Multiple Access Methods
-|Vorherige Seite=Fehlerwahrscheinlichkeit der PN–Modulation
+|Vorherige Seite=Error Probability of Direct-Sequence Spread Spectrum Modulation
-|Nächste Seite=Realisierung von OFDM-Systemen
+|Nächste Seite=Implementation of OFDM Systems
 }}
 ==The principle of OFDM - system consideration in the time domain==