View source for Modulation Methods/General Description of OFDM

 
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|Untermenü= Multiple Access Methods
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|Nächste Seite=Implementation of OFDM Systems
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==The principle of OFDM - system consideration in the time domain==
<br>
Orthogonal Frequency Division Multiplex&nbsp; $\rm (OFDM)$&nbsp; is a digital multicarrier modulation method with the following characteristics:
[[File:P_ID1635__Mod_T_5_5_S1a_neu.png|right|frame| Principle of an&nbsp; $\rm OFDM$ transmitter based on&nbsp; $\text{4-QAM}$]]

*Instead of a broadband,&nbsp; strongly modulated signal,&nbsp; a large number of narrowband,&nbsp; mutually orthogonal subcarriers are used for data transmission.&nbsp; Among other things,&nbsp; this enables adaptation to a frequency-selective channel.


*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.


*Differences in the degree of modulation result in different data rates for the subcarriers.&nbsp; This means that a high-rate source signal must be split into several signals of significantly lower symbol rate for transmission. 
<br clear=all>
The diagram shows the principle of an OFDM transmitter based on the modulation method&nbsp; $\text{4-QAM}$.&nbsp;  
*The representation of the&nbsp; "zeroth"&nbsp; branch &nbsp;$(\mu = 0)$,&nbsp; which represents the DC part,&nbsp; has been deliberately omitted here because it is often set to zero &nbsp; ⇒ &nbsp; for all frames &nbsp;$k$ &nbsp; &rArr; &nbsp; $a_{0,\hspace{0.08cm} k} =0 $&nbsp; holds. 
*The &nbsp;$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|$\text{equivalent low-pass range}$]],&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; &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; &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; &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.

*For a multicarrier system with&nbsp; $M$&nbsp; QAM signal space points and the bit duration&nbsp; $T_{\rm B}$&nbsp; of the binary source symbols,&nbsp; the symbol duration&nbsp; $T$&nbsp; is generally given by
:$$T = N \cdot {\rm{log}_2}(M) \cdot T_{\rm{B} } ,$$
:where&nbsp; $N$&nbsp; is again the number of subcarriers.&nbsp; 
*For the frame duration, &nbsp; $T_{\rm R} \ge T$&nbsp; must hold.&nbsp; Initially,&nbsp; let&nbsp; $T_{\rm R} = T$.}} 


{{GraueBox|TEXT=
$\text{Example 1:}$&nbsp; 
*We first assume a single-carrier system with data rate &nbsp;$R_{\rm B} = 768 \ \rm kbit/s$  &nbsp; ⇒  &nbsp; $T_{\rm B} ≈ 1.3 \ \rm &micro; s$ &nbsp; and a mapping with &nbsp;$M = 4$&nbsp; signal space points&nbsp; $\text{(4–QAM)}$.&nbsp;&nbsp; 
*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 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;
:$$T_\text{MC} = 32 \cdot  {\rm{log}_2}(16) \cdot 1.3 \,{\rm{&micro; s} } \approx 0.167\, {\rm{ms} }.$$ }}


{{BlaueBox|TEXT=
$\text{Conclusions:}$&nbsp; 
*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; &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; }}



==System consideration in the frequency domain with acausal basic pulse==
<br>
We consider again the OFDM transmitted signal in the &nbsp;$k$–th time interval with setting &nbsp;$T_{\rm R} = T$: 
:$$s_k (t) = \sum\limits_{\mu = 0}^{N - 1} {a_{\mu ,\hspace{0.08cm}k} \cdot g_\mu (t - k \cdot T)}.$$
[[File:EN_Mod_T_5_5_S2_a.png|right|frame| Spectrum of a non-causal basic pulse ]]
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 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;$\mu · f_0$.

{{BlaueBox|TEXT=
$\text{Conclusions:}$ &nbsp;  If all amplitude coefficients &nbsp;$a_{μ,\hspace{0.08cm}k} ≠ 0$,&nbsp; the spectrum &nbsp;$S_k(f)$&nbsp; of the transmitted signal in the &nbsp;$k$–th time domain interval 
*is composed of&nbsp; $N$ &nbsp; $\rm sinc$&nbsp; functions &nbsp; &rArr; &nbsp; ${\rm sinc}(x) = \sin(\pi x)/(\pi x),$
*each shifted by a multiple of the basic frequency &nbsp;$f_0$.}}


==System consideration in the frequency domain with causal basic pulse==
<br>
{{BlaueBox|TEXT=
$\text{Important result:}$ &nbsp; 

*If one also takes into account that in reality a causal basic pulse
:$$g_\mu (t) = \left\{ \begin{array}{l} s_0  \cdot {\rm{e} }^{ {\kern 1pt} {\rm{j{\kern 1pt}\cdot {\kern 1pt}2 \pi} } {\kern 1pt}\cdot {\kern 1pt} \mu f_0 {\kern 1pt}\cdot {\kern 1pt}t} \quad 0 \le t < T, \\ 0\quad \quad \quad \quad \quad \quad {\rm otherwise}, \\ \end{array} \right.$$
:has to be assumed,&nbsp; the spectrum is given by
:$$S_k (f) = s_0 \cdot T \cdot \sum\limits_{\mu = 0}^{N - 1} {a_{\mu ,\hspace{0.08cm}k} \cdot \,} {\rm{sinc} }\big( (f - \mu \cdot f_0 )\cdot T\big) \cdot {\rm{e} }^{ - {\rm{j{\kern 1pt}\cdot {\kern 1pt}2\pi} }\hspace{0.05cm}\cdot \hspace{0.05cm} {T}/{2} \hspace{0.05cm}\cdot \hspace{0.05cm} (f - \mu \hspace{0.05cm}\cdot \hspace{0.05cm}f_0 )} .$$
*The complex exponential function comes from the limits of the rectangle used here for pulse shaping in the time domain &nbsp;$0$ ... $T$ &nbsp;  $($shift by &nbsp;$T/2)$. 
*The purely real&nbsp; $\rm sinc$&nbsp; function shown before,&nbsp; on the other hand,&nbsp; would correspond to the non-causal rectangle of &nbsp;$ -T/2$ ... $+T/2$.}}


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?''']]  

*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 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.
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{{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/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/Properties_of_Nyquist_Systems#First_Nyquist_criterion_in_the_frequency_domain |$\text{first Nyquist condition in the frequency domain}$]].}} 


==Orthogonality properties of the carriers==
<br>
The time limit of the basic pulse allows the separate consideration of the two sums in the equation of the OFDM transmitted signal:
:$$s(t) = \sum\limits_{k = - \infty }^{+\infty} {s_k (t)} \quad {\rm{with}} \quad s_k (t) = \sum\limits_{\mu = 0}^{N - 1} {a_{\mu ,\hspace{0.08cm}k} \cdot g_\mu (t - k \cdot T )}.$$
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; 


{{BlaueBox|TEXT=
$\text{Another important result:}$&nbsp;
 
This gives for the spectrum &nbsp;$S_{\mu,\hspace{0.08cm}k}(f)$&nbsp; of the &nbsp;$\mu$–th carrier in the &nbsp;$k$–th interval: 
:$$S_{\mu ,\hspace{0.08cm}k} (f) = s_0 \cdot a_{\mu ,\hspace{0.08cm}k} \cdot T \cdot {\rm{sinc} }\big( (f - \mu \cdot f_0 )\cdot T\big) \cdot {\rm{e} }^{ - {\rm{j \hspace{0.05cm} \cdot \hspace{0.05cm} \pi} } \hspace{0.05cm} \cdot \hspace{0.05cm}T \hspace{0.05cm} \cdot \hspace{0.05cm} (f - \mu \hspace{0.05cm} \cdot \hspace{0.05cm} f_0)}.$$
The following properties, which are important for the OFDM principle, apply:
*The  &nbsp;$s_k(t)$&nbsp; are orthogonal to each other in time $($control variable &nbsp;$k)$,&nbsp; since they do not overlap in time due to the time limitation of the pulse shape filter &nbsp;$g_s(t)$.
*Although the time limitation of the pulses leads to a spectral overlap,&nbsp; there is nevertheless also orthogonality with respect to the carriers $($control variable $\mu)$,&nbsp; since: 
:$$S_k (\mu \cdot f_0 ) = S_{\mu ,\hspace{0.08cm}k} (\mu \cdot f_0 ) = s_0 \cdot a_{\mu ,\hspace{0.08cm}k} \cdot T.$$}}


{{BlaueBox|TEXT=
$\text{Proof:}$&nbsp; 
For orthogonality at the frequency support points &nbsp;$\mu · f_0$&nbsp; must hold:
:$$S(\mu \cdot f_0 ) = S_0 (\mu \cdot f_0 ) + \ \text{...} \ + S_\mu (\mu \cdot f_0 ) + \ \text{...} \  + S_{N - 1} (\mu \cdot f_0 ) = S_\mu (\mu \cdot f_0 ).$$
*Here and in the following,&nbsp; we omit the index &nbsp;$k$&nbsp; of the frame number. From 
:$$s_\mu (t) = s_0 \cdot a_\mu \cdot {\rm{e} }^{{\rm j \hspace{0.03cm}\cdot\hspace{0.03cm}2\pi } \hspace{0.03cm}\cdot \hspace{0.03cm} \mu \hspace{0.03cm}\cdot \hspace{0.03cm} f_0 \hspace{0.03cm}\cdot \hspace{0.03cm} t} \cdot {\rm{rect} } \left( {\frac{ {t - T/2} }{T} } \right) \hspace{0.45cm} {\rm{and }} \hspace{0.45cm} S_\mu (f) = \int_{ - \infty }^{+\infty} {s_\mu (t) \cdot {\rm{e} }^{ - {\rm j \hspace{0.03cm}\cdot \hspace{0.03cm}2\pi } \hspace{0.03cm}\cdot \hspace{0.03cm} f \hspace{0.03cm}\cdot
 \hspace{0.03cm} t} \hspace{0.06cm}{\rm d}t}$$

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

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

*Thus,&nbsp; the spectrum at &nbsp;$f = \mu · f_0$&nbsp; is composed only of components of the &nbsp;$\mu$–th carrier,&nbsp; with all other carriers becoming identically zero.
*Orthogonality is guaranteed. <div align="right">'''q.e.d.'''</div> }}                     


{{BlaueBox|TEXT=
$\text{Conclusion:}$&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;!}} 

==Exercises for the chapter==
<br>
[[Aufgaben:Exercise_5.6:_OFDM_Spectrum|Exercise 5.6: OFDM Spectrum]]

[[Aufgaben:Exercise_5.6Z:_Single_and_Multiple_Beam_System|Exercise 5.6Z: Single and Multiple Beam System]]


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