Modulation Methods/General Description of OFDM - Revision history

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	*In OFDM,  the subcarriers themselves are usually modulated by conventional  [[Modulation_Methods/Quadratur–Amplitudenmodulation\|$\text{quadrature amplitude modulation}$]]  $\rm (QAM)$  or by  [[Modulation_Methods/~~Lineare_digitale_Modulation~~#~~BPSK_.E2.80.93_Binary_Phase_Shift_Keying~~\|$\text{binary phase modulation}$]]  $\rm (BPSK)$,  although the individual carriers may well differ in terms of the modulation type.		*In OFDM,  the subcarriers themselves are usually modulated by conventional  [[Modulation_Methods/Quadratur–Amplitudenmodulation\|$\text{quadrature amplitude modulation}$]]  $\rm (QAM)$  or by  [[Modulation_Methods/Linear_Digital_Modulation#BPSK_–_Binary_Phase_Shift_Keying\|$\text{binary phase modulation}$]]  $\rm (BPSK)$,  although the individual carriers may well differ in terms of the modulation type.


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	:$$S(\mu \cdot f_0 ) = S_0 (\mu \cdot f_0 ) + \ \text{...} \ + S_\mu (\mu \cdot f_0 ) + \ \text{...} \ + S_{N - 1} (\mu \cdot f_0 ) = S_\mu (\mu \cdot f_0 ).$$		:$$S(\mu \cdot f_0 ) = S_0 (\mu \cdot f_0 ) + \ \text{...} \ + S_\mu (\mu \cdot f_0 ) + \ \text{...} \ + S_{N - 1} (\mu \cdot f_0 ) = S_\mu (\mu \cdot f_0 ).$$
	*Here and in the following,  we omit the index  $k$  of the frame number. From		*Here and in the following,  we omit the index  $k$  of the frame number. From
	:$$s_\mu (t) = s_0 \cdot a_\mu \cdot {\rm{e} }^{{\rm j \hspace{0.03cm}\cdot\hspace{0.03cm}2\pi } \hspace{0.03cm}\cdot \hspace{0.03cm} \mu \hspace{0.03cm}\cdot \hspace{0.03cm} f_0 \hspace{0.03cm}\cdot \hspace{0.03cm} t} \cdot {\rm{rect} } \left( {\frac{ {t - T/2} }{T} } \right) \hspace{0.45cm} {\rm{and }} \hspace{0.45cm} S_\mu (f) = \int_{ - \infty }^{+\infty} {s_\mu (t) \cdot {\rm{e} }^{ - {\rm j \hspace{0.03cm}\cdot \hspace{0.03cm}2\pi } \hspace{0.03cm}\cdot \hspace{0.03cm} f \hspace{0.03cm}\cdot		:$$s_\mu (t) = s_0 \cdot a_\mu \cdot {\rm{e} }^{{\rm j \hspace{0.03cm}\cdot\hspace{0.03cm}2\pi } \hspace{0.03cm}\cdot \hspace{0.03cm} \mu \hspace{0.03cm}\cdot \hspace{0.03cm} f_0 \hspace{0.03cm}\cdot \hspace{0.03cm} t} \cdot {\rm{rect} } \left( {\frac{ {t - T/2} }{T} } \right) \hspace{0.45cm} {\rm{and }} \hspace{0.45cm} S_\mu (f) = \int_{ - \infty }^{+\infty} {s_\mu (t) \cdot {\rm{e} }^{ - {\rm j \hspace{0.03cm}\cdot \hspace{0.03cm}2\pi } \hspace{0.03cm}\cdot \hspace{0.03cm} f \hspace{0.03cm}\cdot\hspace{0.03cm} t} \hspace{0.06cm}{\rm d}t}$$
	\hspace{0.03cm} t} \hspace{0.06cm}{\rm d}t}$$

	:the spectrum  $S(f)$  results in general to:		:the spectrum  $S(f)$  results in general to:
	:$$S(f) = \left( {s_0 \cdot a_0 \cdot T \cdot {\rm{sinc} }(f T ) \cdot {\rm{e} }^{ - {\rm j \hspace{0.03cm}\cdot \hspace{0.03cm}2\pi }\hspace{0.03cm}\cdot \hspace{0.03cm} {T}/{2}\hspace{0.03cm}\cdot \hspace{0.03cm} f} } \right) * \int_{ - \infty }^{+\infty} { {\rm{e} }^{\rm{0} } \cdot {\rm{e} }^{ - {\rm{j\hspace{0.03cm}\cdot \hspace{0.03cm}2\pi } } \hspace{0.03cm}\cdot \hspace{0.03cm} f \hspace{0.03cm}\cdot \hspace{0.03cm} t} \hspace{0.06cm}{\rm d}t} \hspace{0.08cm}+ \text{...} $$		:$$S(f) = \left( {s_0 \cdot a_0 \cdot T \cdot {\rm{sinc} }(f T ) \cdot {\rm{e} }^{ - {\rm j \hspace{0.03cm}\cdot \hspace{0.03cm}2\pi }\hspace{0.03cm}\cdot \hspace{0.03cm} {T}/{2}\hspace{0.03cm}\cdot \hspace{0.03cm} f} } \right) * \int_{ - \infty }^{+\infty} { {\rm{e} }^{\rm{0} } \cdot {\rm{e} }^{ - {\rm{j\hspace{0.03cm}\cdot \hspace{0.03cm}2\pi } } \hspace{0.03cm}\cdot \hspace{0.03cm} f \hspace{0.03cm}\cdot \hspace{0.03cm} t} \hspace{0.06cm}{\rm d}t} \hspace{0.08cm}+ \text{...} $$
	:$$\hspace{0.5cm}\text{...} + \left( {s_0 \cdot a_\mu \cdot T \cdot {\rm{sinc} } ( f T ) \cdot {\rm{e} }^{ - {\rm j \hspace{0.03cm}\cdot \hspace{0.03cm}2\pi }\hspace{0.03cm}\cdot \hspace{0.03cm}{T}/{2}\hspace{0.03cm}\cdot \hspace{0.03cm} f} } \right) * \int_{ - \infty }^{+\infty} { {\rm{e} }^{ {\rm{j\hspace{0.03cm}\cdot \hspace{0.03cm}2\pi } } \hspace{0.03cm}\cdot \hspace{0.03cm}\mu \hspace{0.03cm}\cdot \hspace{0.03cm} f_0 \hspace{0.03cm}\cdot \hspace{0.03cm} t} \cdot {\rm{e} }^{ - {\rm{j\hspace{0.03cm}\cdot \hspace{0.03cm}2\pi } } \hspace{0.03cm}\cdot \hspace{0.03cm} f \hspace{0.03cm}\cdot		:$$\hspace{0.5cm}\text{...} + \left( {s_0 \cdot a_\mu \cdot T \cdot {\rm{sinc} } ( f T ) \cdot {\rm{e} }^{ - {\rm j \hspace{0.03cm}\cdot \hspace{0.03cm}2\pi }\hspace{0.03cm}\cdot \hspace{0.03cm}{T}/{2}\hspace{0.03cm}\cdot \hspace{0.03cm} f} } \right) * \int_{ - \infty }^{+\infty} { {\rm{e} }^{ {\rm{j\hspace{0.03cm}\cdot \hspace{0.03cm}2\pi } } \hspace{0.03cm}\cdot \hspace{0.03cm}\mu \hspace{0.03cm}\cdot \hspace{0.03cm} f_0 \hspace{0.03cm}\cdot \hspace{0.03cm} t} \cdot {\rm{e} }^{ - {\rm{j\hspace{0.03cm}\cdot \hspace{0.03cm}2\pi } } \hspace{0.03cm}\cdot \hspace{0.03cm} f \hspace{0.03cm}\cdot\hspace{0.03cm} t} \hspace{0.06cm}{\rm d}t} \hspace{0.08cm}+ \text{...} $$
	\hspace{0.03cm} t} \hspace{0.06cm}{\rm d}t} \hspace{0.08cm}+ \text{...} $$		:$$ \hspace{0.5cm}\text{...} +\left( {s_0 \cdot a_{N - 1} \cdot T \cdot {\rm{sinc} } ( f T ) \cdot {\rm{e} }^{ - {\rm j \hspace{0.03cm}\cdot \hspace{0.03cm}2\pi }\hspace{0.03cm}\cdot \hspace{0.03cm} {T}/{2}\hspace{0.03cm}\cdot \hspace{0.03cm} f} }\right)* \int_{ - \infty }^{+\infty} { {\rm{e} }^{ {\rm{j\hspace{0.03cm}\cdot \hspace{0.03cm}2\pi } } \hspace{0.03cm}\cdot \hspace{0.03cm}(N-1) \hspace{0.03cm}\cdot \hspace{0.03cm} f_0 \hspace{0.03cm}\cdot \hspace{0.03cm} t} \cdot {\rm{e} }^{ - {\rm{j\hspace{0.03cm}\cdot \hspace{0.03cm}2\pi } } \hspace{0.03cm}\cdot \hspace{0.03cm} f \hspace{0.03cm}\cdot \hspace{0.03cm} t} \hspace{0.06cm}{\rm d}t} .$$
	:$$ \hspace{0.5cm}\text{...} +\left( {s_0 \cdot a_{N - 1} \cdot T \cdot {\rm{sinc} } ( f T ) \cdot {\rm{e} }^{ - {\rm j \hspace{0.03cm}\cdot \hspace{0.03cm}2\pi }\hspace{0.03cm}\cdot \hspace{0.03cm} {T}/{2}\hspace{0.03cm}\cdot \hspace{0.03cm} f} }\right)
	* \int_{ - \infty }^{+\infty} { {\rm{e} }^{ {\rm{j\hspace{0.03cm}\cdot \hspace{0.03cm}2\pi } } \hspace{0.03cm}\cdot \hspace{0.03cm}(N-1) \hspace{0.03cm}\cdot \hspace{0.03cm} f_0 \hspace{0.03cm}\cdot \hspace{0.03cm} t} \cdot {\rm{e} }^{ - {\rm{j\hspace{0.03cm}\cdot \hspace{0.03cm}2\pi } } \hspace{0.03cm}\cdot \hspace{0.03cm} f \hspace{0.03cm}\cdot \hspace{0.03cm} t} \hspace{0.06cm}{\rm d}t} .$$

	*With distributions, this equation can be expressed as follows:		*With distributions, this equation can be expressed as follows:
	:$$S(f) = \left( {s_0 \cdot a_0 \cdot T \cdot {\rm{sinc} }(f T ) \cdot {\rm{e} }^{ - {\rm j \hspace{0.03cm}\cdot \hspace{0.03cm}\pi }\hspace{0.03cm}\cdot \hspace{0.03cm} {T}\hspace{0.03cm}\cdot \hspace{0.03cm} f} } \right) * \delta (f) \hspace{0.08cm}+ \text{...} $$		:$$S(f) = \left( {s_0 \cdot a_0 \cdot T \cdot {\rm{sinc} }(f T ) \cdot {\rm{e} }^{ - {\rm j \hspace{0.03cm}\cdot \hspace{0.03cm}\pi }\hspace{0.03cm}\cdot \hspace{0.03cm} {T}\hspace{0.03cm}\cdot \hspace{0.03cm} f} } \right) * \delta (f) \hspace{0.08cm}+ \text{...} $$
	:$$\hspace{0.5cm} \text{...} + \left( {s_0 \cdot a_\mu \cdot T \cdot {\rm{sinc} } ( fT )\cdot {\rm{e} }^{ - {\rm j \hspace{0.03cm}\cdot		:$$\hspace{0.5cm} \text{...} + \left( {s_0 \cdot a_\mu \cdot T \cdot {\rm{sinc} } ( fT )\cdot {\rm{e} }^{ - {\rm j \hspace{0.03cm}\cdot\hspace{0.03cm}\pi }\hspace{0.03cm}\cdot \hspace{0.03cm} {T}\hspace{0.03cm}\cdot\hspace{0.03cm} f} } \right) * \delta (f - \mu \cdot f_0 )\hspace{0.08cm}+ \text{...} $$
	\hspace{0.03cm}\pi }\hspace{0.03cm}\cdot \hspace{0.03cm} {T}\hspace{0.03cm}\cdot		:$$\hspace{0.5cm} \text{...}+ \left( {s_0 \cdot a_{N - 1} \cdot T \cdot {\rm{sinc} } ( f T ) \cdot {\rm{e} }^{ - {\rm j \hspace{0.03cm}\cdot \hspace{0.03cm}\pi }\hspace{0.03cm}\cdot \hspace{0.03cm} {T}\hspace{0.03cm}\cdot\hspace{0.03cm} f} }\right) * \delta (f-(N - 1) \cdot f_0 ) .$$
	\hspace{0.03cm} f} } \right) * \delta (f - \mu \cdot f_0 )\hspace{0.08cm}+ \text{...} $$
	:$$\hspace{0.5cm} \text{...}
	+ \left( {s_0 \cdot a_{N - 1} \cdot T \cdot {\rm{sinc} } ( f T ) \cdot {\rm{e} }^{ - {\rm j \hspace{0.03cm}\cdot \hspace{0.03cm}\pi }\hspace{0.03cm}\cdot \hspace{0.03cm} {T}\hspace{0.03cm}\cdot
	\hspace{0.03cm} f} }\right) * \delta (f-(N - 1) \cdot f_0 ) .$$
	:$$\Rightarrow \hspace{0.3cm}S(f) = {s_0 \cdot a_0 \cdot T \cdot {\rm{sinc} }( T \cdot f) \cdot {\rm{e} }^{ - {\rm j \hspace{0.03cm}\cdot \hspace{0.03cm}\pi }\hspace{0.03cm}\cdot \hspace{0.03cm} {T}\hspace{0.03cm}\cdot \hspace{0.03cm} f} } \hspace{0.08cm}+\hspace{0.08cm} \text{...} $$		:$$\Rightarrow \hspace{0.3cm}S(f) = {s_0 \cdot a_0 \cdot T \cdot {\rm{sinc} }( T \cdot f) \cdot {\rm{e} }^{ - {\rm j \hspace{0.03cm}\cdot \hspace{0.03cm}\pi }\hspace{0.03cm}\cdot \hspace{0.03cm} {T}\hspace{0.03cm}\cdot \hspace{0.03cm} f} } \hspace{0.08cm}+\hspace{0.08cm} \text{...} $$
	:$$\hspace{1.4cm}\text{...} + {s_0 \cdot a_\mu \cdot T \cdot {\rm{sinc} } ( T \cdot (f - \mu \cdot f_0 ))\cdot {\rm{e} }^{ - {\rm j \hspace{0.03cm}\cdot \hspace{0.03cm}\pi }\hspace{0.03cm}\cdot		:$$\hspace{1.4cm}\text{...} + {s_0 \cdot a_\mu \cdot T \cdot {\rm{sinc} } ( T \cdot (f - \mu \cdot f_0 ))\cdot {\rm{e} }^{ - {\rm j \hspace{0.03cm}\cdot \hspace{0.03cm}\pi }\hspace{0.03cm}\cdot\hspace{0.03cm} {T}\hspace{0.03cm}\cdot \hspace{0.03cm} (f - \mu \hspace{0.03cm}\cdot \hspace{0.03cm}f_0 )} } \hspace{0.08cm}+ \hspace{0.08cm}\text{...}$$
	\hspace{0.03cm} {T}\hspace{0.03cm}\cdot \hspace{0.03cm} (f - \mu \hspace{0.03cm}\cdot \hspace{0.03cm}f_0 )} } \hspace{0.08cm}+ \hspace{0.08cm}\text{...}$$		:$$ \hspace{1.4cm}\text{...} + s_0 \cdot a_{N - 1} \cdot T \cdot {\rm sinc } ( T \cdot \big [f-(N - 1) \cdot f_0 ) \big ] \cdot {\rm e}^{ - {\rm j \hspace{0.03cm}\cdot\hspace{0.03cm}\pi }\hspace{0.03cm}\cdot \hspace{0.03cm} {T}\hspace{0.03cm}\hspace{0.03cm}\cdot \hspace{0.03cm} \big [f-(N - 1) \hspace{0.03cm}\cdot \hspace{0.03cm}f_0 \big ]}.$$
	:$$ \hspace{1.4cm}\text{...} + s_0 \cdot a_{N - 1} \cdot T \cdot {\rm sinc } ( T \cdot \big [f-(N - 1) \cdot f_0 ) \big ] \cdot {\rm e}^{ - {\rm j \hspace{0.03cm}\cdot
	\hspace{0.03cm}\pi }\hspace{0.03cm}\cdot \hspace{0.03cm} {T}\hspace{0.03cm}\hspace{0.03cm}\cdot \hspace{0.03cm} \big [f-(N - 1) \hspace{0.03cm}\cdot \hspace{0.03cm}f_0 \big ]}.$$
	*Now setting  $f = \mu · f_0$,  we obtain:		*Now setting  $f = \mu · f_0$,  we obtain:
	:$$S (\mu \cdot f_0) = 0 \hspace{0.08cm}+ \hspace{0.08cm} \text{...} \hspace{0.08cm}+\hspace{0.08cm} s_0 \cdot a_\mu \cdot T \cdot {\rm{sinc} } (0) \cdot {\rm{e} }^0 \hspace{0.08cm}+\hspace{0.08cm} \text{...}+ 0 = s_0 \cdot a_\mu \cdot T = S_\mu ( \mu \cdot f_0 ).$$		:$$S (\mu \cdot f_0) = 0 \hspace{0.08cm}+ \hspace{0.08cm} \text{...} \hspace{0.08cm}+\hspace{0.08cm} s_0 \cdot a_\mu \cdot T \cdot {\rm{sinc} } (0) \cdot {\rm{e} }^0 \hspace{0.08cm}+\hspace{0.08cm} \text{...}+ 0 = s_0 \cdot a_\mu \cdot T = S_\mu ( \mu \cdot f_0 ).$$

Maintenance script: Add German interlanguage link

2026-03-16T12:28:09Z

Add German interlanguage link

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	{{Display}}		{{Display}}
			[[de:Modulationsverfahren/Allgemeine_Beschreibung_von_OFDM]]

Guenter at 16:31, 23 January 2023

2023-01-23T16:31:46Z

← Older revision		Revision as of 18:31, 23 January 2023
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	The diagram shows an example of the magnitude spectrum of an OFDM signal with five carriers.		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.  This corresponds to the first Nyquist condition in the frequency domain.		*It is remarkable that the maximum of each subcarrier coincides with the zeros of all other carriers.  This corresponds to the first Nyquist condition in the frequency domain.

Guenter at 15:22, 19 January 2023

2023-01-19T15:22:25Z

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

Hwang at 18:27, 18 January 2023

2023-01-18T18:27:47Z

← Older revision		Revision as of 20:27, 18 January 2023
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	Here,  the  $k$–th transmitted pulse		Here,  the  $k$–th transmitted pulse
	*is the sum of the basic pulses  $g_{\mu}(t)$ shifted by $k · T$,		*is the sum of the basic pulses  $g_{\mu}(t)$ shifted by $k · T$,
	*each of which is weighted by the  $\mu$–th amplitude coefficients of the QAM coder at time  $k$.		*each of which is weighted by the  $\mu$–th amplitude coefficients of the QAM coder '''KORREKTUR: encoder''' at time  $k$.

Hwang at 18:27, 18 January 2023

2023-01-18T18:27:10Z

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

Hwang at 18:25, 18 January 2023

2023-01-18T18:25:20Z

← Older revision		Revision as of 20:25, 18 January 2023
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	The diagram shows an example of the magnitude spectrum of an OFDM signal with five carriers.		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.  This corresponds to the first Nyquist condition in the frequency domain.		*It is remarkable that the maximum of each subcarrier coincides with the zeros of all other carriers.  This corresponds to the first Nyquist condition in the frequency domain.

Hwang at 18:22, 18 January 2023

2023-01-18T18:22:24Z

← Older revision		Revision as of 20:22, 18 January 2023
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	For simplicity,  we assume the basic pulse  $g_{\mu}(t)$  to be symmetric at  $t = 0$.   Then it is valid with  $f_0 = 1/T$:		For simplicity,  we assume the basic pulse  $g_{\mu}(t)$  to be symmetric at  $t = 0$.   Then it is valid with  $f_0 = 1/T$:
	:$$g_\mu (t) = \left\{ \begin{array}{l} s_0 \cdot {\rm{e}}^{ {\kern 1pt} {\rm{j2 \pi}} {\kern 1pt} \mu f_0 t} \quad \quad - T/2 < t < T/2, \\ 0 \quad \quad \quad \quad \quad \quad \; {\rm otherwise.} \\ \end{array} \right.$$		:$$g_\mu (t) = \left\{ \begin{array}{l} s_0 \cdot {\rm{e}}^{ {\kern 1pt} {\rm{j2 \pi}} {\kern 1pt} \mu f_0 t} \quad \quad - T/2 < t < T/2, \\ 0 \quad \quad \quad \quad \quad \quad \; {\rm otherwise.} \\ \end{array} \right.$$
	In the spectral domain,  such an acausal rectangular function modulated by a  (complex)  exponential function of frequency  $\mu · f_0$  corresponds to an  $\rm sinc$ function shifted by $\mu · f_0$:		In the spectral domain,  such an acausal rectangular function modulated by a  (complex)  exponential function of frequency  $\mu · f_0$  corresponds to a  $\rm sinc$ function shifted by $\mu · f_0$:
	:$$G_\mu (f) = s_0 \cdot T \cdot {\rm{sinc}} \big((f - \mu f_0 )\cdot T \big ).$$		:$$G_\mu (f) = s_0 \cdot T \cdot {\rm{sinc}} \big((f - \mu f_0 )\cdot T \big ).$$
	On the right,  this spectral function  $($normalized to the maximum  $s_0 · T)$  is shown for  $\mu = 5$.   –   The arrow is to suggest that if the basic pulse is unconstrained in time,  the  $\rm sinc$ function shown would have to be replaced by a Dirac delta function at  $\mu · f_0$.		On the right,  this spectral function  $($normalized to the maximum  $s_0 · T)$  is shown for  $\mu = 5$.   –   The arrow is to suggest that if the basic pulse is unconstrained in time,  the  $\rm sinc$ function shown would have to be replaced by a Dirac delta function at  $\mu · f_0$.

Hwang at 18:20, 18 January 2023

2023-01-18T18:20:58Z

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

Hwang at 17:13, 18 January 2023

2023-01-18T17:13:01Z

← Older revision		Revision as of 19:13, 18 January 2023
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	*In OFDM,  the subcarriers themselves are usually modulated by conventional  [[Modulation_Methods/Quadratur–Amplitudenmodulation\|quadrature amplitude modulation]]  $\rm (QAM)$  or by  [[Modulation_Methods/Lineare_digitale_Modulation#BPSK_.E2.80.93_Binary_Phase_Shift_Keying\|binary phase modulation]]  $\rm (BPSK)$,  although the individual carriers may well differ in terms of the modulation type.		*In OFDM,  the subcarriers themselves are usually modulated by conventional  [[Modulation_Methods/Quadratur–Amplitudenmodulation\|$\text{quadrature amplitude modulation}$]]  $\rm (QAM)$  or by  [[Modulation_Methods/Lineare_digitale_Modulation#BPSK_.E2.80.93_Binary_Phase_Shift_Keying\|$\text{binary phase modulation}$]]  $\rm (BPSK)$,  although the individual carriers may well differ in terms of the modulation type.


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	*The  $N–1$  parts of the data stream  $〈q_{\mu,k}〉$  present at time $k$  are first 4-QAM encoded by combining two bits at a time.		*The  $N–1$  parts of the data stream  $〈q_{\mu,k}〉$  present at time $k$  are first 4-QAM encoded by combining two bits at a time.
	*Then the generally complex amplitude $a_{\mu,\hspace{0.08cm}k}$  $($with control variables  $\mu = 1$, ... , $N–1)$  is pulse-shaped and modulated with the  $\mu$–th multiple of the basic frequency  $f_0$.		*Then the generally complex amplitude $a_{\mu,\hspace{0.08cm}k}$  $($with control variables  $\mu = 1$, ... , $N–1)$  is pulse-shaped and modulated with the  $\mu$–th multiple of the basic frequency  $f_0$.
	*The transmitted signal is now the additive superposition of the individual partial signals.  The consideration takes place here and also in the following in the [[Signal_Representation/Equivalent_Low-Pass_Signal_and_its_Spectral_Function\|equivalent low-pass range]],  whereby the index  "TP"  (German:  "Tiefpassbereich")  is renounced.		*The transmitted signal is now the additive superposition of the individual partial signals.  The consideration takes place here and also in the following in the [[Signal_Representation/Equivalent_Low-Pass_Signal_and_its_Spectral_Function\|$\text{equivalent low-pass range}$]],  whereby the index  "TP"  (German:  "Tiefpassbereich")  is renounced.
	*The pulse shape filter  $g_s(t)$  is a rectangle of height  $s_0$  limited to the range  $0 ≤ t < T$.		*The pulse shape filter  $g_s(t)$  is a rectangle of height  $s_0$  limited to the range  $0 ≤ t < T$.
	*We call  $T$  the  '''symbol duration'''  and denote the reciprocal  $f_0 = 1/T$  as the  '''basic frequency'''.		*We call  $T$  the  »'''symbol duration'''«  and denote the reciprocal  $f_0 = 1/T$  as the  »'''basic frequency'''«.


	If we now summarize this filter with the respective modulation to		If we now summarize this filter with the respective modulation to
	:$$g_\mu (t) = \left\{ \begin{array}{l} s_0 \cdot {\rm{e}}^{ {\kern 1pt} {\rm{j2 \pi}} {\kern 1pt} \mu f_0 t} \quad 0 \le t < T, \\ 0 \quad \quad \quad \quad \quad {\rm otherwise} \\ \end{array} \right.$$		:$$g_\mu (t) = \left\{ \begin{array}{l} s_0 \cdot {\rm{e}}^{ {\kern 1pt} {\rm{j2 \pi}} {\kern 1pt} \mu f_0 t} \quad 0 \le t < T, \\ 0 \quad \quad \quad \quad \quad {\rm otherwise} \\ \end{array} \right.$$
	with  $\mu ∈ \{0, \ \text{...}\ , N–1\}$,  we obtain the  '''OFDM transmitted signal  $s_k(t)$  in the  $k$–th time interval''':		with  $\mu ∈ \{0, \ \text{...}\ , N–1\}$,  we obtain the  »'''OFDM transmitted signal  $s_k(t)$  in the  $k$–th time interval'''«:
	:$$s_k (t) = \sum\limits_{\mu = 0}^{N - 1} {a_{\mu ,\hspace{0.08cm}k} \cdot g_\mu (t - k \cdot T_{\rm{R}} )}.$$		:$$s_k (t) = \sum\limits_{\mu = 0}^{N - 1} {a_{\mu ,\hspace{0.08cm}k} \cdot g_\mu (t - k \cdot T_{\rm{R}} )}.$$

	{{BlaueBox\|TEXT=		{{BlaueBox\|TEXT=
	The total  '''OFDM transmitted signal considering all time intervals'''  is then:		The total  »'''OFDM transmitted signal considering all time intervals'''«  is then:
	:$$s(t) = \sum\limits_{k = - \infty }^{+\infty} {\sum\limits_{\mu = 0}^{N - 1} {a_{\mu ,\hspace{0.08cm}k} \cdot g_\mu (t - k \cdot T_{\rm{R} } )} }.$$		:$$s(t) = \sum\limits_{k = - \infty }^{+\infty} {\sum\limits_{\mu = 0}^{N - 1} {a_{\mu ,\hspace{0.08cm}k} \cdot g_\mu (t - k \cdot T_{\rm{R} } )} }.$$
	*$T_{\rm R}$  denotes the frame  (German:  "Rahmen"   ⇒   subscript "R")  duration.  Within this time the same data is present at the input and after  $T_{\rm R}$  the next frame follows.		*$T_{\rm R}$  denotes the frame  (German:  "Rahmen"   ⇒   subscript "R")  duration.  Within this time the same data is present at the input and after  $T_{\rm R}$  the next frame follows.
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	*The duration of a symbol increases significantly for a multicarrier system compared to a single-carrier system,  reducing the spurious influence of the channel impulse response and decreasing intersymbol interference.		*The duration of a symbol increases significantly for a multicarrier system compared to a single-carrier system,  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  '''major advantages of OFDM'''.  This will be discussed in more detail in the following sections  [[Modulation_Methods/OFDM_für_4G–Netze\|OFDM for 4G networks]]  and  [[Examples_of_Communication_Systems/Allgemeine_Beschreibung_von_DSL\|Digital Subscriber Line]]  $\rm (DSL)$.  }}		*The ability to use different robust modulation schemes for different subbands is one of the  »'''major advantages of OFDM'''«.  This will be discussed in more detail in the following sections  [[Modulation_Methods/OFDM_für_4G–Netze\|"OFDM for 4G networks"]]  and  [[Examples_of_Communication_Systems/Allgemeine_Beschreibung_von_DSL\|"Digital Subscriber Line"]]  $\rm (DSL)$.  }}


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


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	{{BlaueBox\|TEXT=		{{BlaueBox\|TEXT=
	$\text{Conclusion:}$    The  '''orthogonality of the OFDM signal'''   $s(t)$		$\text{Conclusion:}$    The  »'''orthogonality of the OFDM signal'''«   $s(t)$
	*is given for the control variable  $k$  $\rm (time)$		*is given for the control variable  $k$  $\rm (time)$
	*as well as for the control variable  $\mu$  $\text{(carrier frequencies)}$ !}}		*as well as for the control variable  $\mu$  $\text{(carrier frequencies)}$ !}}