Difference between revisions of "Signal Representation/Discrete Fourier Transform (DFT)"

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{{Header
 
{{Header
|Untermenü=Zeit- und frequenzdiskrete Signaldarstellung
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|Untermenü=Time and Frequency-Discrete Signal Representation
|Vorherige Seite=Zeitdiskrete Signaldarstellung
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|Vorherige Seite=Discrete-Time Signal Representation
|Nächste Seite=Fehlermöglichkeiten bei Anwendung der DFT
+
|Nächste Seite=Possible Errors When Using DFT
 
}}
 
}}
  
==Argumente für die diskrete Realisierung der Fouriertransformation==
+
==Arguments for the discrete implementation of the Fourier transform==
 
<br>
 
<br>
Die&nbsp; '''Fouriertransformation'''&nbsp; gemäß der bisherigen Beschreibung im Kapitel&nbsp; [[Signaldarstellung/Fouriertransformation_und_-rücktransformation|Aperiodische Signale &ndash; Impulse]]&nbsp; weist aufgrund der unbegrenzten Ausdehnung des Integrationsintervalls eine unendlich hohe Selektivität auf und ist deshalb ein ideales theoretisches Hilfsmittel der Spektralanalyse.
+
The&nbsp; &raquo;'''Fourier transform'''&laquo;&nbsp; according to the previous description in chapter&nbsp; [[Signal_Representation/The_Fourier_Transform_and_its_Inverse#The_first_Fourier_integral|&raquo;Aperiodic Signals &ndash; Pulses&laquo;]]&nbsp; has an infinitely high selectivity due to the unlimited extension of the integration interval and is therefore an ideal theoretical tool of spectral analysis.
  
Sollen die Spektralanteile&nbsp; $X(f)$&nbsp; einer Zeitfunktion&nbsp; $x(t)$&nbsp; numerisch ermittelt werden, so sind die allgemeinen Transformationsgleichungen
+
If the spectral components&nbsp; $X(f)$&nbsp; of a time function&nbsp; $x(t)$&nbsp; are to be determined numerically,&nbsp; the general transformation equations
 
   
 
   
 
:$$\begin{align*}X(f) & =  \int_{-\infty
 
:$$\begin{align*}X(f) & =  \int_{-\infty
  }^{+\infty}x(t) \cdot {\rm e}^{-{\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm} 2 \pi f t}\hspace{0.1cm} {\rm d}t\hspace{0.5cm} \Rightarrow\hspace{0.5cm} \text{Hintransformation}\hspace{0.7cm} \Rightarrow\hspace{0.5cm} \text{Erstes Fourierintegral}
+
  }^{+\infty}x(t) \cdot {\rm e}^{-{\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm} 2 \pi f t}\hspace{0.1cm} {\rm d}t\hspace{0.5cm} \Rightarrow\hspace{0.5cm} \text{Transform}\hspace{0.7cm} \Rightarrow\hspace{0.5cm} \text{first  Fourier integral}
 
  \hspace{0.05cm},\\
 
  \hspace{0.05cm},\\
 
x(t) & =  \int_{-\infty
 
x(t) & =  \int_{-\infty
 
  }^{+\infty}\hspace{-0.15cm}X(f) \cdot {\rm e}^{\hspace{0.05cm}+{\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm} 2 \pi f t}\hspace{0.1cm} {\rm d}f\hspace{0.35cm} \Rightarrow\hspace{0.5cm}
 
  }^{+\infty}\hspace{-0.15cm}X(f) \cdot {\rm e}^{\hspace{0.05cm}+{\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm} 2 \pi f t}\hspace{0.1cm} {\rm d}f\hspace{0.35cm} \Rightarrow\hspace{0.5cm}
\text{Rücktransformation}\hspace{0.4cm} \Rightarrow\hspace{0.5cm} \text{Zweites Fourierintegral}
+
\text{Inverse Transform}\hspace{0.4cm} \Rightarrow\hspace{0.5cm} \text{second Fourier integral}
 
  \hspace{0.05cm}\end{align*}$$
 
  \hspace{0.05cm}\end{align*}$$
  
aus zwei Gründen ungeeignet:
+
are unsuitable for two reasons:
*Die Gleichungen gelten ausschließlich für zeitkontinuierliche Signale. Mit Digitalrechnern oder Signalprozessoren kann man jedoch nur zeitdiskrete Signale verarbeiten.
+
#The equations apply exclusively to continuous-time signals.&nbsp; With digital computers or signal processors,&nbsp; however,&nbsp; one can only process discrete-time signals.
*Für eine numerische Auswertung der beiden Fourierintegrale ist es erforderlich, das jeweilige Integrationsintervall auf einen endlichen Wert zu begrenzen.
+
#For a numerical evaluation of the two Fourier integrals it is necessary to limit the respective integration interval to a finite value.
  
  
 
{{BlaueBox|TEXT=
 
{{BlaueBox|TEXT=
$\text{Daraus ergibt sich folgende Konsequenz:}$&nbsp;
+
$\text{This leads to the following consequence:}$&nbsp;
  
Ein&nbsp; '''kontinuierliches Signal'''&nbsp; muss vor der numerischen Bestimmung seiner Spektraleigenschaften zwei Prozesse durchlaufen, nämlich
+
A&nbsp; &raquo;'''continuous-valued signal'''&laquo;&nbsp; must undergo two processes before the numerical determination of its spectral properties,&nbsp; viz.
*den der&nbsp; '''Abtastung'''&nbsp; zur Diskretisierung, und
+
#that of&nbsp; &raquo;'''sampling'''&laquo;&nbsp; for discretization,&nbsp; and<br>
*den der&nbsp; '''Fensterung'''&nbsp; zur Begrenzung des Integrationsintervalls.}}
+
#that of&nbsp; &raquo;'''windowing'''&laquo;&nbsp; to limit the integration interval.}}
  
  
Im Folgenden wird ausgehend von einer aperiodischen Zeitfunktion&nbsp; $x(t)$&nbsp; und dem dazugehörigen Fourierspektrum&nbsp; $X(f)$&nbsp; eine für die Rechnerverarbeitung geeignete zeit– und frequenzdiskrete Beschreibung schrittweise entwickelt.
+
In the following,&nbsp; starting from an aperiodic time function&nbsp; $x(t)$&nbsp; and the corresponding Fourier spectrum&nbsp; $X(f)$&nbsp; a time and frequency-discrete description suitable for computer processing is developed step by step.
  
  
==Zeitdiskretisierung &ndash; Periodifizierung im Frequenzbereich==
+
 
 +
==Time discretization &ndash; Periodification in the frequency domain==
 
<br>
 
<br>
Die folgenden Grafiken zeigen einheitlich links den Zeitbereich und rechts den Frequenzbereich. Ohne Einschränkung der Allgemeingültigkeit sind&nbsp; $x(t)$&nbsp; und&nbsp; $X(f)$&nbsp; jeweils reell und gaußförmig.
+
The following graphs show uniformly the time domain on the left and the frequency domain on the right.&nbsp;
 +
*Without limiting generality,&nbsp; $x(t)$&nbsp; and&nbsp; $X(f)$&nbsp; are each real and Gaussian.
 +
 
 +
*According to the chapter&nbsp; [[Signal_Representation/Time_Discrete_Signal_Representation|&raquo;Discrete-Time Signal Representation&laquo;]]&nbsp; one can describe the sampling of the time signal&nbsp; $x(t)$&nbsp; by multiplying it by a Dirac delta train &nbsp; &rArr; &nbsp; <br>&raquo;Dirac comb&nbsp; in the time domain&laquo;
 +
[[File:P_ID1132__Sig_T_5_1_S2_neu.png|right|frame| Time discretization &nbsp; &rArr; &nbsp; Periodification in the frequency domain.&nbsp; <u>Notation:</u><br> &nbsp; &nbsp; ${\rm A}\{x(t)\}$:&nbsp; Signal&nbsp; $x(t)$&nbsp; after&nbsp; &raquo;sampling&laquo;&nbsp; $($German:&nbsp; "Abtastung"$)$ &nbsp; &rArr; &nbsp; $\rm A\{\text{...}\}$ <br> &nbsp; &nbsp; ${\rm P}\{X(f)\}$:&nbsp; Spectrum&nbsp; $X(f)$&nbsp; after&nbsp; &raquo;periodification&laquo;&nbsp;  &nbsp; &rArr; &nbsp; $\rm P\{\text{...}\}$]]
  
[[File:P_ID1132__Sig_T_5_1_S2_neu.png|center|frame|Diskretisierung im Zeitbereich – Periodifizierung im Frequenzbereich]]
+
:$$p_{\delta}(t) =  \sum_{\nu = - \infty }^{+\infty} T_{\rm A} \cdot
 +
\delta (t- \nu \cdot T_{\rm A}
 +
)\hspace{0.05cm}.$$
  
Entsprechend dem Kapitel&nbsp; [[Signaldarstellung/Zeitdiskrete_Signaldarstellung|Zeitdiskrete Signaldarstellung]]&nbsp; kann man die Abtastung des Zeitsignals&nbsp; $x(t)$&nbsp; durch die Multiplikation mit einem Diracpuls&nbsp; $p_{\delta}(t)$&nbsp; beschreiben. Es ergibt sich das im Abstand&nbsp; $T_{\rm A}$&nbsp; abgetastete Zeitsignal
+
The result is the time signal sampled at a distance&nbsp; $T_{\rm A}$:&nbsp;
 
   
 
   
 
:$${\rm A}\{x(t)\} =  \sum_{\nu = - \infty }^{+\infty} T_{\rm A} \cdot x(\nu \cdot T_{\rm A})\cdot
 
:$${\rm A}\{x(t)\} =  \sum_{\nu = - \infty }^{+\infty} T_{\rm A} \cdot x(\nu \cdot T_{\rm A})\cdot
Line 48: Line 55:
 
  )\hspace{0.05cm}.$$
 
  )\hspace{0.05cm}.$$
  
Dieses abgetastete Signal&nbsp; $\text{A}\{ x(t)\}$&nbsp; transformieren wir nun in den Frequenzbereich. Der Multiplikation des Diracpulses&nbsp; $p_{\delta}(t)$&nbsp; mit&nbsp; $x(t)$&nbsp; entspricht im Frequenzbereich die Faltung von&nbsp; $P_{\delta}(f)$&nbsp; mit&nbsp; $X(f)$. Es ergibt sich das periodifizierte Spektrum&nbsp; $\text{P}\{ X(f)\}$, wobei&nbsp; $f_{\rm P}$&nbsp; die Frequenzperiode der Funktion&nbsp; $\text{P}\{ X(f)\}$&nbsp; angibt:
+
We transform this sampled signal&nbsp; $\text{A}\{ x(t)\}$&nbsp; into the frequency domain:
 +
*The multiplication of the Dirac comb&nbsp; $p_{\delta}(t)$&nbsp; with&nbsp; $x(t)$&nbsp; corresponds in the frequency domain to the convolution of&nbsp; $P_{\delta}(f)$&nbsp; with&nbsp; $X(f)$.
 +
 
 +
*The result is the periodified spectrum&nbsp; $\text{P}\{ X(f)\}$,&nbsp; where&nbsp; $f_{\rm
 +
P}$&nbsp; indicates the frequency period of the function&nbsp; $\text{P}\{ X(f)\}$&nbsp;:
 
   
 
   
 
:$${\rm A}\{x(t)\} \hspace{0.2cm}\circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\, \hspace{0.2cm} {\rm P}\{X(f)\} =  \sum_{\mu = - \infty }^{+\infty}
 
:$${\rm A}\{x(t)\} \hspace{0.2cm}\circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\, \hspace{0.2cm} {\rm P}\{X(f)\} =  \sum_{\mu = - \infty }^{+\infty}
  X (f- \mu \cdot f_{\rm P} )\hspace{0.5cm} {\rm mit }\hspace{0.5cm}f_{\rm
+
  X (f- \mu \cdot f_{\rm P} ).$$
P}= {1}/{T_{\rm A}}\hspace{0.05cm}.$$
 
  
Dieser Zusammenhang wurde ebenfalls bereits im Kapitel&nbsp; [[Signaldarstellung/Zeitdiskrete_Signaldarstellung|Zeitdiskrete Signaldarstellung]]&nbsp;  hergeleitet, jedoch mit etwas anderer Nomenklatur:
+
This relation was already derived in the chapter&nbsp; [[Signal_Representation/Time_Discrete_Signal_Representation|"Discrete-Time Signal Representation"]]&nbsp;  but with slightly different nomenclature:
*Das abgetastete Signal nennen wir nun&nbsp; $\text{A}\{ x(t)\}$&nbsp; anstelle von&nbsp; $x_{\rm A}(t)$.
+
*We now denote the sampled signal by&nbsp; $\text{A}\{ x(t)\}$&nbsp; instead of&nbsp; $x_{\rm A}(t)$.
* Die&nbsp; '''Frequenzperiode'''&nbsp; wird nun mit&nbsp; $f_{\rm P}$ = $1/T_{\rm A}$&nbsp; anstelle von&nbsp; $f_{\rm A} = 1/T_{\rm A}$&nbsp; bezeichnet.
 
  
 +
* The&nbsp; &raquo;frequency period&laquo;&nbsp; is now denoted by&nbsp; $f_{\rm P} = 1/T_{\rm A}$&nbsp; instead of&nbsp; $f_{\rm A} = 1/T_{\rm A}$.
  
Diese Nomenklaturänderung wird auf den folgenden Seiten begründet.
 
  
Die obige Grafik zeigt den hier beschriebenen Funktionalzusammenhang. Es ist anzumerken:
+
These nomenclature changes are justified in the following sections.
*Die Frequenzperiode&nbsp; $f_{\rm P}$&nbsp; wurde hier bewusst klein gewählt, so dass die Überlappung der zu summierenden Spektren deutlich zu erkennen ist.
 
*In der Praxis sollte&nbsp; $f_{\rm P}$&nbsp; aufgrund des Abtasttheorems mindestens doppelt so groß sein wie die größte im Signal&nbsp; $x(t)$&nbsp; enthaltene Frequenz.
 
*Ist dies nicht erfüllt, so muss mit&nbsp; '''Aliasing'''&nbsp; gerechnet werden – siehe Kapitel&nbsp; [[Signaldarstellung/Fehlermöglichkeiten_bei_Anwendung_der_DFT|Fehlermöglichkeiten bei Anwendung der DFT]].
 
  
 +
The graph above shows the functional relationship described here.&nbsp; It should be noted:
 +
#The frequency period&nbsp; $f_{\rm P}$&nbsp; has been deliberately chosen to be small here so that the overlap of the spectra to be summed can be clearly seen.
 +
#In practice&nbsp; $f_{\rm P}$&nbsp; should be at least twice as large as the largest frequency contained in the signal&nbsp; $x(t)$&nbsp; due to the sampling theorem.
 +
#If this is not fulfilled,&nbsp; then&nbsp; &raquo;'''aliasing'''&laquo;&nbsp; must be expected - see chapter&nbsp; [[Signal_Representation/Possible_Errors_When_Using_DFT|&raquo;Possible Errors when using DFT&laquo;]].
  
==Frequenzdiskretisierung &ndash; Periodifizierung im Zeitbereich==
+
 
 +
==Frequency discretization &ndash; Periodification in the time domain ==
 
<br>
 
<br>
Die Diskretisierung von&nbsp; $X(f)$&nbsp; lässt sich ebenfalls durch eine Multiplikation mit einem Diracpuls beschreiben. Es ergibt sich das im Abstand&nbsp; $f_{\rm A}$&nbsp; abgetastete Spektrum:
+
The discretization of&nbsp; $X(f)$&nbsp; can also be described by a multiplication with a Dirac comb in the frequency domain.&nbsp; The result is the sampled spectrum with distance&nbsp; $f_{\rm A}$:  
 
 
:$${\rm A}\{X(f)\} =  X(f) \cdot  \sum_{\mu = - \infty }^{+\infty}
 
:$${\rm A}\{X(f)\} =  X(f) \cdot  \sum_{\mu = - \infty }^{+\infty}
 
  f_{\rm A} \cdot \delta (f- \mu \cdot f_{\rm A } ) =  \sum_{\mu = - \infty }^{+\infty}
 
  f_{\rm A} \cdot \delta (f- \mu \cdot f_{\rm A } ) =  \sum_{\mu = - \infty }^{+\infty}
 
  f_{\rm A} \cdot X(\mu \cdot f_{\rm A } ) \cdot\delta (f- \mu \cdot f_{\rm A } )\hspace{0.05cm}.$$
 
  f_{\rm A} \cdot X(\mu \cdot f_{\rm A } ) \cdot\delta (f- \mu \cdot f_{\rm A } )\hspace{0.05cm}.$$
  
Transformiert man den hier verwendeten Frequenz–Diracpuls $($mit Impulsgewichten&nbsp; $f_{\rm A})$&nbsp; in den Zeitbereich, so erhält man mit&nbsp; $T_{\rm P} = 1/f_{\rm A}$:
+
*If one transforms the Dirac comb $($with impulse weights&nbsp; $f_{\rm A})$&nbsp; into the time domain,&nbsp; one obtains with&nbsp; $T_{\rm P} = 1/f_{\rm A}$:
 
   
 
   
 
:$$\sum_{\mu = - \infty }^{+\infty}
 
:$$\sum_{\mu = - \infty }^{+\infty}
Line 82: Line 92:
 
   \delta (t- \nu \cdot T_{\rm P } ) \hspace{0.05cm}.$$
 
   \delta (t- \nu \cdot T_{\rm P } ) \hspace{0.05cm}.$$
  
Die Multiplikation mit&nbsp; $X(f)$&nbsp; entspricht im Zeitbereich der Faltung mit&nbsp; $x(t)$. Man erhält das im Abstand&nbsp; $T_{\rm P}$&nbsp; periodifizierte Signal&nbsp; $\text{P}\{ x(t)\}$:
+
*The multiplication with&nbsp; $X(f)$&nbsp; corresponds in the time domain to the convolution with&nbsp; $x(t)$.&nbsp; One obtains the signal&nbsp; $\text{P}\{ x(t)\}$&nbsp; periodified with distance&nbsp; $T_{\rm P}$:
 
   
 
   
 
:$${\rm A}\{X(f)\} \hspace{0.2cm}\bullet\!\!-\!\!\!-\!\!\!-\!\!\circ\, \hspace{0.2cm}
 
:$${\rm A}\{X(f)\} \hspace{0.2cm}\bullet\!\!-\!\!\!-\!\!\!-\!\!\circ\, \hspace{0.2cm}
Line 89: Line 99:
 
   x (t- \nu \cdot T_{\rm P } ) \hspace{0.05cm}.$$
 
   x (t- \nu \cdot T_{\rm P } ) \hspace{0.05cm}.$$
  
[[File:P_ID1134__Sig_T_5_1_S3_neu.png|right|frame|Diskretisierung im Frequenzbereich – Periodifizierung im Zeitbereich]]
+
 
 
{{GraueBox|TEXT=
 
{{GraueBox|TEXT=
$\text{Beispiel 1:}$&nbsp;
+
[[File:P_ID1134__Sig_T_5_1_S3_neu.png|right|frame|Frequency discretization &ndash; Periodization in the time domain]]
Dieser Zusammenhang ist in der Grafik veranschaulicht:  
+
$\text{Example 1:}$&nbsp;
*Aufgrund der groben Frequenzrasterung ergibt sich in diesem Beispiel für die Zeitperiode&nbsp; $T_{\rm P}$&nbsp; ein relativ kleiner Wert.
+
This correlation is illustrated in the graph:  
 +
*Due to the coarse frequency rastering,&nbsp; this example results in a relatively small value for the time period&nbsp; $T_{\rm P}$&nbsp;.
 +
 
  
 +
* Therefore due to overlaps,&nbsp; the&nbsp; $($blue$)$&nbsp; periodified time signal&nbsp; $\text{P}\{ x(t)\}$&nbsp; differs significantly from&nbsp; $x(t)$.
  
* Deshalb  unterscheidet sich das (blaue) periodifizierte Zeitsignal&nbsp; $\text{P}\{ x(t)\}$&nbsp; aufgrund von Überlappungen deutlich von&nbsp; $x(t)$.}}
 
  
 +
*If one wants to achieve&nbsp; $\text{P}\{ x(t)\} \approx x(t)$&nbsp; then&nbsp; $T_{\rm P}$&nbsp; must be chosen much larger than in this example.}}
  
==Finite Signaldarstellung==
+
==Finite signal representation==
 
<br>
 
<br>
[[File:P_ID1135__Sig_T_5_1_S4_neu.png|right|frame|Finite Signale der Diskreten Fouriertransformation (DFT)]]
+
[[File:P_ID1135__Sig_T_5_1_S4_neu.png|right|frame|Finite signals of the Discrete Fourier Transform&nbsp; $\rm (DFT)$]]
Zur so genannten&nbsp; ''finiten Signaldarstellung''&nbsp; kommt man,
+
One arrives at the so-called&nbsp; &raquo;finite signal representation&laquo;,&nbsp; if both
*wenn sowohl die Zeitfunktion&nbsp; $x(t)$  
+
*the time function&nbsp; $x(t)$,&nbsp;
*als auch die Spektralfunktion&nbsp; $X(f)$  
+
 +
*the spectral function&nbsp; $X(f)$  
 +
 
  
 +
are specified exclusively by their sample values.&nbsp; This graph is to be interpreted as follows:
 +
*In the left graph,&nbsp; the function&nbsp; $\text{A}\{ \text{P}\{ x(t)\}\}$&nbsp; is drawn in blue.&nbsp; This results from sampling the periodified time function&nbsp; $\text{P}\{ x(t)\}$&nbsp; with equidistant Dirac deltas with distance&nbsp; $T_{\rm A} = 1/f_{\rm P}$.
  
ausschließlich durch ihre Abtastwerte angegeben werden.
+
*In the right graph,&nbsp; the function&nbsp; $\text{P}\{ \text{A}\{ X(f)\}\}$&nbsp; is drawn in green.&nbsp; This results from the periodification&nbsp; $($with&nbsp; $f_{\rm P})$&nbsp; of the sampled spectral function&nbsp; $\{ \text{A}\{ X(f)\}\}$.
<br clear=all>
+
Die Grafik ist wie folgt zu interpretieren:
+
*There is a Fourier correspondence between the blue finite signal&nbsp; $($in the left sketch$)$&nbsp; and the green finite signal&nbsp; $($in the right sketch$)$,&nbsp; as follows:
*Im linken Bild blau eingezeichnet ist die Funktion&nbsp; $\text{A}\{ \text{P}\{ x(t)\}\}$. Diese ergibt sich durch Abtastung der periodifizierten Zeitfunktion&nbsp; $\text{P}\{ x(t)\}$&nbsp; mit äquidistanten Diracimpulsen im Abstand&nbsp; $T_{\rm A} = 1/f_{\rm P}$.
 
*Im rechten Bild grün eingezeichnet ist die Funktion&nbsp; $\text{P}\{ \text{A}\{ X(f)\}\}$. Diese ergibt sich durch Periodifizierung $($mit&nbsp; $f_{\rm P})$&nbsp; der abgetasteten Spektralfunktion&nbsp; $\{ \text{A}\{ X(f)\}\}$.
 
*Zwischen dem blauen finiten Signal und dem grünen finiten Signal besteht eine Fourierkorrespondenz, und zwar folgende:
 
 
   
 
   
 
:$${\rm A}\{{\rm P}\{x(t)\}\} \hspace{0.2cm}\circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\, \hspace{0.2cm} {\rm P}\{{\rm A}\{X(f)\}\} \hspace{0.05cm}.$$
 
:$${\rm A}\{{\rm P}\{x(t)\}\} \hspace{0.2cm}\circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\, \hspace{0.2cm} {\rm P}\{{\rm A}\{X(f)\}\} \hspace{0.05cm}.$$
  
*Die Diraclinien der periodischen Fortsetzung&nbsp; $\text{P}\{ \text{A}\{ X(f)\}\}$&nbsp; der abgetasteten Spektralfunktion fallen allerdings nur dann in das gleiche Frequenzraster wie diejenigen von&nbsp; $\text{A}\{ X(f)\}$, wenn die Frequenzperiode&nbsp; $f_{\rm P}$&nbsp; ein ganzzahliges Vielfaches&nbsp; $(N)$&nbsp; des Frequenzabtastabstandes&nbsp; $f_{\rm A}$&nbsp; ist.
+
*The Dirac delta lines of the periodic continuation&nbsp; $\text{P}\{ \text{A}\{ X(f)\}\}$&nbsp; of the sampled spectral function,&nbsp; however,&nbsp; only fall into the same frequency grid as those of&nbsp; $\text{A}\{ X(f)\}$&nbsp; if the frequency period&nbsp; $f_{\rm P}$&nbsp; is an integer multiple&nbsp; $(N)$&nbsp; of the frequency sampling interval&nbsp; $f_{\rm A}$.
*Deshalb muss bei Anwendung der finiten Signaldarstellung stets die folgende Bedingung erfüllt sein, wobei für die natürliche Zahl&nbsp; $N$&nbsp; in der Praxis meist eine Zweierpotenz verwendet wird&nbsp; (der obigen Grafik liegt der Wert&nbsp; $N = 8$&nbsp; zugrunde):
+
 
 +
*Therefore,&nbsp; when using the finite signal representation,&nbsp; the following condition must always be fulfilled,&nbsp; where in practice the natural number&nbsp; $N$&nbsp; is usually a power of two&nbsp; $($the above graph is based on the value&nbsp; $N = 8)$:
 
   
 
   
 
:$$f_{\rm P} = N \cdot f_{\rm A} \hspace{0.5cm} \Rightarrow\hspace{0.5cm} {1}/{T_{\rm A}}= N \cdot f_{\rm A} \hspace{0.5cm} \Rightarrow\hspace{0.5cm}
 
:$$f_{\rm P} = N \cdot f_{\rm A} \hspace{0.5cm} \Rightarrow\hspace{0.5cm} {1}/{T_{\rm A}}= N \cdot f_{\rm A} \hspace{0.5cm} \Rightarrow\hspace{0.5cm}
 
  N \cdot f_{\rm A}\cdot T_{\rm A} = 1\hspace{0.05cm}.$$
 
  N \cdot f_{\rm A}\cdot T_{\rm A} = 1\hspace{0.05cm}.$$
  
*Bei Einhaltung der Bedingung&nbsp; $N \cdot f_{\rm A} \cdot T_{\rm A} = 1$&nbsp; ist die Reihenfolge von Periodifizierung und Abtastung vertauschbar. Somit gilt:
+
*If the condition&nbsp; $N \cdot f_{\rm A} \cdot T_{\rm A} = 1$&nbsp; is fulfilled then the order of periodization and sampling is interchangeable.&nbsp; Thus:
 
   
 
   
 
:$${\rm A}\{{\rm P}\{x(t)\}\} = {\rm P}\{{\rm A}\{x(t)\}\}\hspace{0.2cm}\circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\, \hspace{0.2cm}
 
:$${\rm A}\{{\rm P}\{x(t)\}\} = {\rm P}\{{\rm A}\{x(t)\}\}\hspace{0.2cm}\circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\, \hspace{0.2cm}
Line 128: Line 143:
  
 
{{BlaueBox|TEXT=
 
{{BlaueBox|TEXT=
$\text{Fazit:}$&nbsp;
+
$\text{Conclusions:}$&nbsp;
*Die Zeitfunktion&nbsp; $\text{P}\{ \text{A}\{ x(t)\}\}$&nbsp; besitzt die Periode&nbsp; $T_{\rm P} = N \cdot T_{\rm A}$.  
+
#The time function&nbsp; $\text{P}\{ \text{A}\{ x(t)\}\}$&nbsp; has the period&nbsp; $T_{\rm P} = N \cdot T_{\rm A}$.  
*Die Periode im Frequenzbereich ist&nbsp; $f_{\rm P} = N \cdot f_{\rm A}$.  
+
#The period in the frequency domain is&nbsp; $f_{\rm P} = N \cdot f_{\rm A}$.  
*Zur Beschreibung des diskretisierten Zeit– und Frequenzverlaufs reichen somit jeweils&nbsp; $N$&nbsp; '''komplexe Zahlenwerte''' in Form von Impulsgewichten aus.}}
+
#For the description of the discretized time and frequency response in each case&nbsp; $N$&nbsp; '''complex numerical values'''&nbsp; in the form of impulse weights are sufficient.}}
  
  
 
{{GraueBox|TEXT=
 
{{GraueBox|TEXT=
$\text{Beispiel 2:}$&nbsp;
+
$\text{Example 2:}$&nbsp;
Es liegt ein zeitbegrenztes (impulsartiges) Signal&nbsp; $x(t)$&nbsp; in abgetasteter Form vor, wobei  der Abstand zweier Abtastwerte&nbsp; $T_{\rm A} = 1\, {\rm &micro; s}$&nbsp; beträgt:  
+
A time-limited&nbsp; $($pulse-like$)$&nbsp; signal&nbsp; $x(t)$&nbsp; is present in sampled form,&nbsp; where the distance between two samples is&nbsp; $T_{\rm A} = 1\, {\rm &micro; s}$:  
*Nach einer diskreten Fouriertransformation mit&nbsp; $N = 512$&nbsp; liegt das Spektrum&nbsp; $X(f)$&nbsp; in Form von Abtastwerten im Abstand&nbsp; $f_{\rm A} = (N \cdot T_{\rm A})^{–1} \approx 1.953\,\text{kHz} $&nbsp; vor.  
+
*After a discrete Fourier transform with&nbsp; $N = 512$&nbsp; the spectrum&nbsp; $X(f)$&nbsp; is available in form of frequency-samples at spacing&nbsp; $f_{\rm A} = (N \cdot T_{\rm A})^{-1} \approx 1.953\,\text{kHz} $.  
*Vergrößert man den DFT&ndash;Parameter auf&nbsp;  $N= 2048$, so ergibt sich ein feineres Frequenzraster mit&nbsp; $f_{\rm A} \approx 488\,\text{Hz}$.}}
 
  
 +
*Increasing the DFT parameter to&nbsp; $N= 2048$ results in a&nbsp; $($four times$)$&nbsp; finer frequency grid with&nbsp; $f_{\rm A} \approx 488\,\text{Hz}$.}}
  
==Von der kontinuierlichen zur diskreten Fouriertransformation==
+
 
 +
==From the continuous to the discrete Fourier transform==
 
<br>
 
<br>
Aus dem herkömmlichen&nbsp; [[Signaldarstellung/Fouriertransformation_und_-rücktransformation#Das_erste_Fourierintegral|ersten Fourierintegral]]
+
From the conventional&nbsp; [[Signal_Representation/Fourier_Transform_and_its_Inverse#The_first_Fourier_integral|&raquo;first Fourier integral&laquo;]]
 
   
 
   
 
:$$X(f) =\int_{-\infty
 
:$$X(f) =\int_{-\infty
 
  }^{+\infty}x(t) \cdot {\rm e}^{-{\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm} 2 \pi \hspace{0.05cm}\cdot \hspace{0.05cm} f  \hspace{0.05cm}\cdot \hspace{0.05cm}t}\hspace{0.1cm} {\rm d}t$$
 
  }^{+\infty}x(t) \cdot {\rm e}^{-{\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm} 2 \pi \hspace{0.05cm}\cdot \hspace{0.05cm} f  \hspace{0.05cm}\cdot \hspace{0.05cm}t}\hspace{0.1cm} {\rm d}t$$
  
entsteht durch Diskretisierung&nbsp; $(\text{d}t \to T_{\rm A}$,&nbsp; $t \to \nu \cdot T_{\rm A}$,&nbsp; $f \to \mu \cdot f_{\rm A}$,&nbsp; $T_{\rm A} \cdot f_{\rm A} = 1/N)$&nbsp; die abgetastete und periodifizierte Spektralfunktion
+
arises from discretization&nbsp; $(\text{d}t \to T_{\rm A}$,&nbsp; $t \to \nu \cdot T_{\rm A}$,&nbsp; $f \to \mu \cdot f_{\rm A}$,&nbsp; $T_{\rm A} \cdot f_{\rm A} = 1/N)$&nbsp; the sampled and periodified spectral function
 
   
 
   
 
:$${\rm P}\{X(\mu \cdot f_{\rm A})\} = T_{\rm A} \cdot \sum_{\nu = 0 }^{N-1}
 
:$${\rm P}\{X(\mu \cdot f_{\rm A})\} = T_{\rm A} \cdot \sum_{\nu = 0 }^{N-1}
Line 154: Line 170:
 
  \cdot \hspace{0.05cm}\mu /N} \hspace{0.05cm}.$$
 
  \cdot \hspace{0.05cm}\mu /N} \hspace{0.05cm}.$$
  
Es ist berücksichtigt, dass aufgrund der Diskretisierung jeweils die periodifizierten Funktionen einzusetzen sind.  
+
It is taken into account that due to the discretization,&nbsp; the periodified functions are to be used in each case.  
  
Aus Gründen einer vereinfachten Schreibweise nehmen wir nun die folgenden Substitutionen vor:
+
For reasons of simplified notation, we now make the following substitutions:
*Die&nbsp; $N$&nbsp; '''Zeitbereichskoeffizienten'''&nbsp; seien mit der Laufvariablen&nbsp; $\nu = 0$, ... , $N - 1$:
+
*The&nbsp; $N$&nbsp; &raquo;time-domain coefficients&laquo;&nbsp; are with the variable&nbsp; $\nu = 0$, ... , $N - 1$:
 
:$$d(\nu) =
 
:$$d(\nu) =
 
   {\rm P}\left\{x(t)\right\}{\big|}_{t \hspace{0.05cm}= \hspace{0.05cm}\nu \hspace{0.05cm}\cdot \hspace{0.05cm}T_{\rm A}}\hspace{0.05cm}.$$
 
   {\rm P}\left\{x(t)\right\}{\big|}_{t \hspace{0.05cm}= \hspace{0.05cm}\nu \hspace{0.05cm}\cdot \hspace{0.05cm}T_{\rm A}}\hspace{0.05cm}.$$
*Die&nbsp; $N$&nbsp; '''Frequenzbereichskoeffizienten'''&nbsp; seien mit der Laufvariablen&nbsp; $\mu = 0,$ ... , $N$ – 1:
+
*Let&nbsp; $N$&nbsp; &raquo;frequency domain coefficients&laquo;&nbsp; be associated with the variable&nbsp; $\mu = 0,$ ... , $N-1$:
 
:$$D(\mu) = f_{\rm A} \cdot
 
:$$D(\mu) = f_{\rm A} \cdot
 
   {\rm P}\left\{X(f)\right\}{\big|}_{f \hspace{0.05cm}= \hspace{0.05cm}\mu \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm A}}\hspace{0.05cm}.$$
 
   {\rm P}\left\{X(f)\right\}{\big|}_{f \hspace{0.05cm}= \hspace{0.05cm}\mu \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm A}}\hspace{0.05cm}.$$
*Abkürzend wird für den von&nbsp; $N$&nbsp; abhängigen&nbsp; '''komplexen Drehfaktor'''&nbsp; geschrieben:
+
*Abbreviation is written for the from&nbsp; $N$&nbsp; dependent&nbsp; &raquo;complex rotation factor&laquo;&nbsp;:
:$$w = {\rm e}^{-{\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm} 2 \pi /N}
+
:$$w = {\rm e}^{-{\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm} 2 \pi /N}
  = \cos \left( {2 \pi}/{N}\right)-{\rm j} \cdot \sin \left( {2 \pi}/{N}\right)
+
  = \cos \left( {2 \pi}/{N}\right)-{\rm j} \cdot \sin \left( {2 \pi}/{N}\right)
 
  \hspace{0.05cm}.$$  
 
  \hspace{0.05cm}.$$  
  
[[File:P_ID2730__Sig_T_5_1_S5_neu.png|right|frame|Zur Definition der Diskreten Fouriertransformation (DFT) mit&nbsp; $N=8$]]
 
 
{{BlaueBox|TEXT=
 
{{BlaueBox|TEXT=
 
$\text{Definition:}$&nbsp;
 
$\text{Definition:}$&nbsp;
 
+
The term&nbsp; &raquo;Discrete Fourier Transform&laquo;&nbsp; $\rm  (DFT)$&nbsp; means the calculation of the&nbsp; $N$&nbsp; spectral coefficients&nbsp; $D(\mu)$&nbsp; from the&nbsp; $N$&nbsp; signal coefficients&nbsp; $d(\nu)$:
Unter dem Begriff&nbsp; '''Diskrete Fouriertransformation'''&nbsp; (kurz '''DFT''')&nbsp; versteht man die Berechnung der&nbsp; $N$&nbsp; Spektralkoeffizienten&nbsp; $D(\mu)$&nbsp; aus den&nbsp; $N$&nbsp; Signalkoeffizienten&nbsp; $d(\nu)$:
+
[[File:P_ID2730__Sig_T_5_1_S5_neu.png|right|frame|On defining the&nbsp; &raquo;Discrete Fourier Transform&laquo;&nbsp; $\rm (DFT)$&nbsp; with&nbsp; $N=8$]]
 
 
:$$D(\mu) = \frac{1}{N} \cdot \sum_{\nu = 0 }^{N-1}
 
:$$D(\mu) = \frac{1}{N} \cdot \sum_{\nu = 0 }^{N-1}
   d(\nu)\cdot  {w}^{\hspace{0.05cm}\nu \hspace{0.03cm} \cdot \hspace{0.05cm}\mu} \hspace{0.05cm}. $$
+
   d(\nu)\cdot  {w}^{\hspace{0.05cm}\nu \hspace{0.07cm} \cdot \hspace{0.05cm}\mu} \hspace{0.05cm}. $$
 +
:
 +
In the diagram you can see:
 +
*The&nbsp; $N = 8$&nbsp; signal coefficients&nbsp; $d(\nu)$&nbsp; by the blue filling,
  
In der Grafik erkennt man an einem Beispiel
+
*the&nbsp; $N = 8$&nbsp; spectral coefficients&nbsp; $D(\mu)$&nbsp; at the green filling.}}
*die&nbsp; $N = 8$&nbsp; Signalkoeffizienten&nbsp; $d(\nu)$&nbsp; an der blauen Füllung,
 
*die&nbsp; $N = 8$&nbsp; Spektralkoeffizienten&nbsp; $D(\mu)$&nbsp; an der grünen Füllung.}}
 
  
  
==Inverse Diskrete Fouriertransformation==
+
==Inverse discrete Fourier transform==
 
<br>
 
<br>
Die Inverse Diskrete Fouriertransformation (IDFT) beschreibt das&nbsp;  [[Signaldarstellung/Fouriertransformation_und_-rücktransformation#Das_zweite_Fourierintegral|zweite Fourierintegral]]
+
The&nbsp; &raquo;inverse discrete Fourier transform&laquo;&nbsp;  describes the&nbsp;  [[Signal_Representation/Fourier_Transform_and_its_Inverse#The_second_Fourier_integral|&raquo;second Fourier integral&laquo;]]:
 
   
 
   
 
:$$\begin{align*}x(t) & =  \int_{-\infty
 
:$$\begin{align*}x(t) & =  \int_{-\infty
Line 190: Line 205:
 
  t}\hspace{0.1cm} {\rm d}f\end{align*}$$
 
  t}\hspace{0.1cm} {\rm d}f\end{align*}$$
  
in diskretisierter Form: &nbsp; $d(\nu) =
+
in discretized form: &nbsp;  
 +
:$$d(\nu) =
 
   {\rm P}\left\{x(t)\right\}{\big|}_{t \hspace{0.05cm}= \hspace{0.05cm}\nu \hspace{0.05cm}\cdot \hspace{0.05cm}T_{\rm
 
   {\rm P}\left\{x(t)\right\}{\big|}_{t \hspace{0.05cm}= \hspace{0.05cm}\nu \hspace{0.05cm}\cdot \hspace{0.05cm}T_{\rm
   A}}\hspace{0.01cm}.$
+
   A}}\hspace{0.01cm}.$$
  
[[File:P_ID2731__Sig_T_5_1_S6_neu.png|right|frame|Zur Definition der IDFT mit&nbsp; $N=8$]]
+
{{BlaueBox|TEXT= $\text{Definition:}$&nbsp; The term&nbsp; &raquo;Inverse Discrete Fourier Transform&raquo; &nbsp; $\rm (IDFT)$&nbsp;&nbsp; means the calculation of the signal coefficients&nbsp; $d(\nu)$&nbsp; from the spectral coefficients&nbsp; $D(\mu)$:
{{BlaueBox|TEXT=
+
[[File:P_ID2731__Sig_T_5_1_S6_neu.png|right|frame|On defining the&nbsp; &raquo;Inverse Discrete Fourier Transform&laquo;&nbsp; $\rm (IDFT)$&nbsp; with&nbsp; $N=8$]]  
$\text{Definition:}$&nbsp;
+
:$$d(\nu) = \sum_{\mu = 0 }^{N-1}
 +
D(\mu) \cdot {w}^{-\nu \hspace{0.07cm} \cdot \hspace{0.05cm}\mu} \hspace{0.05cm}.$$
  
Unter dem Begriff&nbsp;  '''Inversen Diskreten Fouriertransformation'''&nbsp; (kurz '''IDFT''')&nbsp; versteht man die Berechnung der Signalkoeffizienten&nbsp; $d(\nu)$&nbsp; aus den Spektralkoeffizienten&nbsp; $D(\mu)$:
+
*With indices&nbsp; $\nu = 0, \hspace{0.05cm}\text{...} \hspace{0.05cm}, N-1, $&nbsp; and &nbsp; $\mu = 0, \hspace{0.05cm}\text{...} \hspace{0.05cm}, N-1$&nbsp; holds:
 
:$$d(\nu) =  \sum_{\mu = 0 }^{N-1}
 
D(\mu) \cdot  {w}^{-\nu \hspace{0.03cm} \cdot \hspace{0.05cm}\mu} \hspace{0.05cm}.$$
 
 
 
Mit den Laufvariablen&nbsp; $\nu = 0, \hspace{0.05cm}\text{...} \hspace{0.05cm}, N-1$&nbsp; und&nbsp; $\mu = 0, \hspace{0.05cm}\text{...} \hspace{0.05cm}, N-1$&nbsp; gilt auch hier:
 
 
:$$d(\nu) =
 
:$$d(\nu) =
 
   {\rm P}\left\{x(t)\right\}{\big \vert}_{t \hspace{0.05cm}= \hspace{0.05cm}\nu \hspace{0.05cm}\cdot \hspace{0.05cm}T_{\rm
 
   {\rm P}\left\{x(t)\right\}{\big \vert}_{t \hspace{0.05cm}= \hspace{0.05cm}\nu \hspace{0.05cm}\cdot \hspace{0.05cm}T_{\rm
Line 213: Line 225:
  
 
:$$w  = {\rm e}^{- {\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm} 2 \pi /N}
 
:$$w  = {\rm e}^{- {\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm} 2 \pi /N}
  \hspace{0.01cm}.$$}}
+
  \hspace{0.01cm}.$$
  
+
*A comparison between&nbsp; [[Signal_Representation/Discrete_Fourier_Transform_(DFT)#From_the_continuous_to_the_discrete_Fourier_transform|&raquo;DFT&laquo;]]&nbsp; and&nbsp; &raquo;IDFT&laquo;&nbsp; shows that exactly the same algorithm can be used.&nbsp;
Ein Vergleich zwischen der&nbsp; [[Signaldarstellung/Diskrete_Fouriertransformation_(DFT)#Von_der_kontinuierlichen_zur_diskreten_Fouriertransformation|DFT]]&nbsp; und IDFT zeigt, dass genau der gleiche Algorithmus verwendet werden kann. Die einzigen Unterschiede der IDFT gegenüber der DFT sind:
+
 
*Der Exponent des Drehfaktors ist mit unterschiedlichem Vorzeichen anzusetzen.
+
*The only differences between IDFT and DFT are:
*Bei der IDFT entfällt die Division durch&nbsp; $N$.
+
#The exponent of the rotation factor is to be applied with different sign.
 +
#In the IDFT,&nbsp; the division by&nbsp; $N$&nbsp; is omitted.}}
  
  
==Interpretation von DFT und IDFT==
+
==Interpretation of DFT and IDFT==
 
<br>
 
<br>
Die Grafik zeigt die diskreten Koeffizienten im Zeit– und Frequenzbereich zusammen mit den periodifizierten zeitkontinuierlichen Funktionen.
+
The graph shows the discrete coefficients in the time and frequency domain together with the periodified continuous-time functions.
  
[[File:P_ID1136__Sig_T_5_1_S7_neu.png|center|frame|Zeit&ndash; und Frequenzbereichskoeffizienten der DFT]]
+
[[File:P_ID1136__Sig_T_5_1_S7_neu.png|right|frame|Time and frequency coefficients of the DFT]]
  
Bei Anwendung von DFT bzw. IDFT ist zu beachten:
+
When using DFT or IDFT,&nbsp; please note:
*Nach obigen Definitionen besitzen die DFT–Koeffizienten&nbsp; $d(ν)$&nbsp; und&nbsp; $D(\mu)$&nbsp; stets die Einheit der Zeitfunktion.  
+
#According to the above definitions, the DFT coefficients&nbsp; $d(ν)$&nbsp; and&nbsp; $D(\mu)$&nbsp; always have the unit of the time function.  
*Dividiert man&nbsp; $D(\mu)$&nbsp; durch&nbsp; $f_{\rm A}$, so erhält man den Spektralwert&nbsp; $X(\mu \cdot f_{\rm A})$.
+
#Dividing&nbsp; $D(\mu)$&nbsp; by&nbsp; $f_{\rm A}$,&nbsp; one obtains the spectral value&nbsp; $X(\mu \cdot f_{\rm A})$.
*Die Spektralkoeffizienten&nbsp; $D(\mu)$&nbsp; müssen stets komplex angesetzt werden, um auch ungerade Zeitfunktionen berücksichtigen zu können.
+
#The spectral coefficients&nbsp; $D(\mu)$&nbsp; must always be complex in order to be able to consider odd time functions.
*Um auch Bandpass–Signale im äquivalenten Tiefpass&ndash;Bereich transformieren zu können, verwendet man meist auch komplexe Zeitkoeffizienten&nbsp; $d(\nu)$.
+
#One also uses complex time coefficients&nbsp; $d(\nu)$ &nbsp; &rArr; &nbsp; DFT and IDFT are also applicable to band-pass signals.
*Als Grundintervall für&nbsp; $\nu$&nbsp; und&nbsp; $\mu$&nbsp; definiert man meist – wie in obiger Grafik – den Bereich von&nbsp; $0$&nbsp; bis&nbsp; $N - 1$.  
+
#The basic interval for&nbsp; $\nu$&nbsp; and&nbsp; $\mu$&nbsp; is usually defined as the range from&nbsp; $0$&nbsp; to&nbsp; $N - 1$&nbsp; $($filled circles in the graph$)$.
*Mit den komplexwertigen Zahlenfolgen&nbsp; $\langle \hspace{0.1cm}d(\nu)\hspace{0.1cm}\rangle = \langle \hspace{0.1cm}d(0), \hspace{0.05cm}\text{...} \hspace{0.05cm} , d(N-1) \hspace{0.1cm}\rangle$ &nbsp; sowie &nbsp; $\langle \hspace{0.1cm}D(\mu)\hspace{0.1cm}\rangle =   \langle \hspace{0.1cm}D(0), \hspace{0.05cm}\text{...} \hspace{0.05cm} , D(N-1) \hspace{0.1cm}\rangle$&nbsp; werden DFT und IDFT ähnlich wie die herkömmliche Fouriertransformation symbolisiert:
+
<br clear=all>
:$$\langle \hspace{0.1cm} D(\mu)\hspace{0.1cm}\rangle \hspace{0.2cm}\bullet\!\!-\!\!\!-(N)\!-\!\!\!-\!\!\hspace{0.05cm}\circ\, \hspace{0.2cm} \langle \hspace{0.1cm} d(\nu) \hspace{0.1cm}\rangle \hspace{0.05cm}.$$  
+
With the complex-valued number sequences&nbsp;  
*Ist die Zeitfunktion&nbsp; $x(t)$&nbsp; bereits auf den Bereich&nbsp; $0 \le t \lt N \cdot T_{\rm A}$&nbsp; begrenzt, dann geben die von der IDFT ausgegebenen Zeitkoeffizienten direkt die Abtastwerte der Zeitfunktion an: &nbsp; $d(\nu) = x(\nu \cdot T_{\rm A}).$
+
:$$\langle \hspace{0.03cm}d(\nu)\hspace{0.03cm}\rangle = \langle \hspace{0.03cm}d(0), \hspace{0.05cm}\text{...} \hspace{0.05cm} , d(N-1) \hspace{0.03cm}\rangle,$$
*Ist&nbsp; $x(t)$&nbsp; gegenüber dem Grundintervall verschoben, so muss man die im&nbsp; $\text{Beispiel 3}$&nbsp; gezeigte Zuordnung zwischen&nbsp; $x(t)$&nbsp; und den Koeffizienten&nbsp; $d(\nu)$&nbsp; wählen.
+
:$$\langle \hspace{0.03cm}D(\mu)\hspace{0.03cm}\rangle = \langle \hspace{0.03cm}D(0), \hspace{0.05cm}\text{...} \hspace{0.05cm} , D(N-1) \hspace{0.03cm}\rangle,$$
 +
 
 +
DFT and IDFT are symbolized similarly to the conventional Fourier transform:  
 +
:$$\langle \hspace{0.03cm} D(\mu)\hspace{0.03cm}\rangle \hspace{0.2cm}\bullet\!\!-\!\!\!-(N)\!-\!\!\!-\!\!\hspace{0.05cm}\circ\, \hspace{0.2cm} \langle \hspace{0.03cm} d(\nu) \hspace{0.03cm}\rangle \hspace{0.05cm}.$$  
 +
#If the function&nbsp; $x(t)$&nbsp; is already limited to the range&nbsp; $0 \le t \lt N \cdot T_{\rm A}$&nbsp; then the IDFT output directly give the samples of the time function:   &nbsp; $d(\nu) = x(\nu \cdot T_{\rm A}).$
 +
#If&nbsp; $x(t)$&nbsp; is shifted with respect to the basic interval,&nbsp; one has to choose the association shown in&nbsp; $\text{Example 3}$&nbsp; between&nbsp; $x(t)$&nbsp; and the coefficients&nbsp; $d(\nu)$.
  
  
 
{{GraueBox|TEXT=
 
{{GraueBox|TEXT=
$\text{Beispiel 3:}$&nbsp;
+
$\text{Example 3:}$&nbsp;
Die obere Grafik zeigt den unsymmetrischen Dreieckimpuls&nbsp; $x(t)$, dessen absolute Breite kleiner ist als&nbsp; $T_{\rm P} = N \cdot T_{\rm A}$.  
+
The upper graph shows the asymmetric triangular pulse&nbsp; $x(t)$ whose absolute width is smaller than&nbsp; $T_{\rm P} = N \cdot T_{\rm A}$.  
 +
 
 +
[[File:EN_Sig_T_5_1_S7b_neu.png|right|frame|On assigning of the DFT coefficients with&nbsp; $N=8$]]
  
[[File:P_ID1139__Sig_T_5_1_S7b_neu.png|right|frame|Zur Belegung der DFT-Koeffizienten  mit&nbsp; $N=8$]]
 
  
Die untere Skizze zeigt die zugeordneten DFT–Koeffizienten gültig für&nbsp; $N = 8$
+
The sketch below shows the assigned DFT coefficients $($valid for&nbsp; $N = 8)$.
  
*Für&nbsp; $\nu = 0,\hspace{0.05cm}\text{...} \hspace{0.05cm} , N/2 = 4$&nbsp; gilt&nbsp; $d(\nu) = x(\nu \cdot T_{\rm A})$:
+
*For&nbsp; $\nu = 0,\hspace{0.05cm}\text{...} \hspace{0.05cm} , N/2 = 4$&nbsp; holds &nbsp; $d(\nu) = x(\nu \cdot T_{\rm A})$ :
  
 
:$$d(0) = x (0)\hspace{0.05cm}, \hspace{0.15cm}
 
:$$d(0) = x (0)\hspace{0.05cm}, \hspace{0.15cm}
Line 254: Line 273:
 
:$$d(3) = x (3T_{\rm A})\hspace{0.05cm}, \hspace{0.15cm}
 
:$$d(3) = x (3T_{\rm A})\hspace{0.05cm}, \hspace{0.15cm}
 
d(4) = x (4T_{\rm A})\hspace{0.05cm}.$$  
 
d(4) = x (4T_{\rm A})\hspace{0.05cm}.$$  
*Dagegen sind die Koeffizienten&nbsp; $d(5)$,&nbsp; $d(6)$&nbsp; und&nbsp; d$(7)$&nbsp; wie folgt zu setzen:
+
*The coefficients&nbsp; $d(5)$,&nbsp; $d(6)$&nbsp; and&nbsp; d$(7)$&nbsp; are to be set as follows:
  
:$$d(\nu) = x \big ((\nu\hspace{-0.05cm} - \hspace{-0.05cm} N ) \cdot T_{\rm  A}\big )  $$
+
:$$d(5) = x (-3T_{\rm A})\hspace{0.05cm}, \hspace{0.35cm}
 +
d(6) = x (-2T_{\rm A})\hspace{0.05cm}, \hspace{0.35cm}
 +
d(7) = x (-T_{\rm A})\hspace{0.05cm}$$
 +
:$$ \Rightarrow \hspace{0.2cm}d(\nu) = x \big ((\nu\hspace{-0.05cm} - \hspace{-0.05cm} N ) \cdot T_{\rm  A}\big ). $$
  
:$$ \Rightarrow \hspace{0.2cm}d(5) = x (-3T_{\rm A})\hspace{0.05cm}, \hspace{0.35cm}
+
}}
d(6) = x (-2T_{\rm A})\hspace{0.05cm}, \hspace{0.35cm}
 
d(7) = x (-T_{\rm A})\hspace{0.05cm}.$$ }}
 
  
==Aufgaben zum Kapitel==
+
==Exercises for the chapter==
 
<br>
 
<br>
[[Aufgaben:Aufgabe_5.2:_Inverse_Diskrete_Fouriertransformation|Aufgabe 5.2: Inverse Diskrete Fouriertransformation]]
+
[[Aufgaben:Exercise 5.2: Inverse Discrete Fourier Transform|Exercise 5.2: Inverse Discrete Fourier Transform]]
  
[[Aufgaben:Aufgabe_5.2Z:_DFT_eines_Dreieckimpulses|Aufgabe 5.2Z: DFT eines Dreieckimpulses]]
+
[[Aufgaben:Exercise 5.2Z: DFT of a Triangular Pulse|Exercise 5.2Z: DFT of a Triangular Pulse]]
  
 
      
 
      
 
{{Display}}
 
{{Display}}

Latest revision as of 16:57, 28 June 2023

Arguments for the discrete implementation of the Fourier transform


The  »Fourier transform«  according to the previous description in chapter  »Aperiodic Signals – Pulses«  has an infinitely high selectivity due to the unlimited extension of the integration interval and is therefore an ideal theoretical tool of spectral analysis.

If the spectral components  $X(f)$  of a time function  $x(t)$  are to be determined numerically,  the general transformation equations

$$\begin{align*}X(f) & = \int_{-\infty }^{+\infty}x(t) \cdot {\rm e}^{-{\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm} 2 \pi f t}\hspace{0.1cm} {\rm d}t\hspace{0.5cm} \Rightarrow\hspace{0.5cm} \text{Transform}\hspace{0.7cm} \Rightarrow\hspace{0.5cm} \text{first Fourier integral} \hspace{0.05cm},\\ x(t) & = \int_{-\infty }^{+\infty}\hspace{-0.15cm}X(f) \cdot {\rm e}^{\hspace{0.05cm}+{\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm} 2 \pi f t}\hspace{0.1cm} {\rm d}f\hspace{0.35cm} \Rightarrow\hspace{0.5cm} \text{Inverse Transform}\hspace{0.4cm} \Rightarrow\hspace{0.5cm} \text{second Fourier integral} \hspace{0.05cm}\end{align*}$$

are unsuitable for two reasons:

  1. The equations apply exclusively to continuous-time signals.  With digital computers or signal processors,  however,  one can only process discrete-time signals.
  2. For a numerical evaluation of the two Fourier integrals it is necessary to limit the respective integration interval to a finite value.


$\text{This leads to the following consequence:}$ 

A  »continuous-valued signal«  must undergo two processes before the numerical determination of its spectral properties,  viz.

  1. that of  »sampling«  for discretization,  and
  2. that of  »windowing«  to limit the integration interval.


In the following,  starting from an aperiodic time function  $x(t)$  and the corresponding Fourier spectrum  $X(f)$  a time and frequency-discrete description suitable for computer processing is developed step by step.


Time discretization – Periodification in the frequency domain


The following graphs show uniformly the time domain on the left and the frequency domain on the right. 

  • Without limiting generality,  $x(t)$  and  $X(f)$  are each real and Gaussian.
  • According to the chapter  »Discrete-Time Signal Representation«  one can describe the sampling of the time signal  $x(t)$  by multiplying it by a Dirac delta train   ⇒  
    »Dirac comb  in the time domain«
Time discretization   ⇒   Periodification in the frequency domain.  Notation:
    ${\rm A}\{x(t)\}$:  Signal  $x(t)$  after  »sampling«  $($German:  "Abtastung"$)$   ⇒   $\rm A\{\text{...}\}$
    ${\rm P}\{X(f)\}$:  Spectrum  $X(f)$  after  »periodification«    ⇒   $\rm P\{\text{...}\}$
$$p_{\delta}(t) = \sum_{\nu = - \infty }^{+\infty} T_{\rm A} \cdot \delta (t- \nu \cdot T_{\rm A} )\hspace{0.05cm}.$$

The result is the time signal sampled at a distance  $T_{\rm A}$: 

$${\rm A}\{x(t)\} = \sum_{\nu = - \infty }^{+\infty} T_{\rm A} \cdot x(\nu \cdot T_{\rm A})\cdot \delta (t- \nu \cdot T_{\rm A} )\hspace{0.05cm}.$$

We transform this sampled signal  $\text{A}\{ x(t)\}$  into the frequency domain:

  • The multiplication of the Dirac comb  $p_{\delta}(t)$  with  $x(t)$  corresponds in the frequency domain to the convolution of  $P_{\delta}(f)$  with  $X(f)$.
  • The result is the periodified spectrum  $\text{P}\{ X(f)\}$,  where  $f_{\rm P}$  indicates the frequency period of the function  $\text{P}\{ X(f)\}$ :
$${\rm A}\{x(t)\} \hspace{0.2cm}\circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\, \hspace{0.2cm} {\rm P}\{X(f)\} = \sum_{\mu = - \infty }^{+\infty} X (f- \mu \cdot f_{\rm P} ).$$

This relation was already derived in the chapter  "Discrete-Time Signal Representation"  but with slightly different nomenclature:

  • We now denote the sampled signal by  $\text{A}\{ x(t)\}$  instead of  $x_{\rm A}(t)$.
  • The  »frequency period«  is now denoted by  $f_{\rm P} = 1/T_{\rm A}$  instead of  $f_{\rm A} = 1/T_{\rm A}$.


These nomenclature changes are justified in the following sections.

The graph above shows the functional relationship described here.  It should be noted:

  1. The frequency period  $f_{\rm P}$  has been deliberately chosen to be small here so that the overlap of the spectra to be summed can be clearly seen.
  2. In practice  $f_{\rm P}$  should be at least twice as large as the largest frequency contained in the signal  $x(t)$  due to the sampling theorem.
  3. If this is not fulfilled,  then  »aliasing«  must be expected - see chapter  »Possible Errors when using DFT«.


Frequency discretization – Periodification in the time domain


The discretization of  $X(f)$  can also be described by a multiplication with a Dirac comb in the frequency domain.  The result is the sampled spectrum with distance  $f_{\rm A}$:

$${\rm A}\{X(f)\} = X(f) \cdot \sum_{\mu = - \infty }^{+\infty} f_{\rm A} \cdot \delta (f- \mu \cdot f_{\rm A } ) = \sum_{\mu = - \infty }^{+\infty} f_{\rm A} \cdot X(\mu \cdot f_{\rm A } ) \cdot\delta (f- \mu \cdot f_{\rm A } )\hspace{0.05cm}.$$
  • If one transforms the Dirac comb $($with impulse weights  $f_{\rm A})$  into the time domain,  one obtains with  $T_{\rm P} = 1/f_{\rm A}$:
$$\sum_{\mu = - \infty }^{+\infty} f_{\rm A} \cdot \delta (f- \mu \cdot f_{\rm A } ) \hspace{0.2cm}\bullet\!\!-\!\!\!-\!\!\!-\!\!\circ\, \hspace{0.2cm} \sum_{\nu = - \infty }^{+\infty} \delta (t- \nu \cdot T_{\rm P } ) \hspace{0.05cm}.$$
  • The multiplication with  $X(f)$  corresponds in the time domain to the convolution with  $x(t)$.  One obtains the signal  $\text{P}\{ x(t)\}$  periodified with distance  $T_{\rm P}$:
$${\rm A}\{X(f)\} \hspace{0.2cm}\bullet\!\!-\!\!\!-\!\!\!-\!\!\circ\, \hspace{0.2cm} {\rm P}\{x(t)\} = x(t) \star \sum_{\nu = - \infty }^{+\infty} \delta (t- \nu \cdot T_{\rm P } )= \sum_{\nu = - \infty }^{+\infty} x (t- \nu \cdot T_{\rm P } ) \hspace{0.05cm}.$$


Frequency discretization – Periodization in the time domain

$\text{Example 1:}$  This correlation is illustrated in the graph:

  • Due to the coarse frequency rastering,  this example results in a relatively small value for the time period  $T_{\rm P}$ .


  • Therefore due to overlaps,  the  $($blue$)$  periodified time signal  $\text{P}\{ x(t)\}$  differs significantly from  $x(t)$.


  • If one wants to achieve  $\text{P}\{ x(t)\} \approx x(t)$  then  $T_{\rm P}$  must be chosen much larger than in this example.

Finite signal representation


Finite signals of the Discrete Fourier Transform  $\rm (DFT)$

One arrives at the so-called  »finite signal representation«,  if both

  • the time function  $x(t)$, 
  • the spectral function  $X(f)$


are specified exclusively by their sample values.  This graph is to be interpreted as follows:

  • In the left graph,  the function  $\text{A}\{ \text{P}\{ x(t)\}\}$  is drawn in blue.  This results from sampling the periodified time function  $\text{P}\{ x(t)\}$  with equidistant Dirac deltas with distance  $T_{\rm A} = 1/f_{\rm P}$.
  • In the right graph,  the function  $\text{P}\{ \text{A}\{ X(f)\}\}$  is drawn in green.  This results from the periodification  $($with  $f_{\rm P})$  of the sampled spectral function  $\{ \text{A}\{ X(f)\}\}$.
  • There is a Fourier correspondence between the blue finite signal  $($in the left sketch$)$  and the green finite signal  $($in the right sketch$)$,  as follows:
$${\rm A}\{{\rm P}\{x(t)\}\} \hspace{0.2cm}\circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\, \hspace{0.2cm} {\rm P}\{{\rm A}\{X(f)\}\} \hspace{0.05cm}.$$
  • The Dirac delta lines of the periodic continuation  $\text{P}\{ \text{A}\{ X(f)\}\}$  of the sampled spectral function,  however,  only fall into the same frequency grid as those of  $\text{A}\{ X(f)\}$  if the frequency period  $f_{\rm P}$  is an integer multiple  $(N)$  of the frequency sampling interval  $f_{\rm A}$.
  • Therefore,  when using the finite signal representation,  the following condition must always be fulfilled,  where in practice the natural number  $N$  is usually a power of two  $($the above graph is based on the value  $N = 8)$:
$$f_{\rm P} = N \cdot f_{\rm A} \hspace{0.5cm} \Rightarrow\hspace{0.5cm} {1}/{T_{\rm A}}= N \cdot f_{\rm A} \hspace{0.5cm} \Rightarrow\hspace{0.5cm} N \cdot f_{\rm A}\cdot T_{\rm A} = 1\hspace{0.05cm}.$$
  • If the condition  $N \cdot f_{\rm A} \cdot T_{\rm A} = 1$  is fulfilled then the order of periodization and sampling is interchangeable.  Thus:
$${\rm A}\{{\rm P}\{x(t)\}\} = {\rm P}\{{\rm A}\{x(t)\}\}\hspace{0.2cm}\circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\, \hspace{0.2cm} {\rm P}\{{\rm A}\{X(f)\}\} = {\rm A}\{{\rm P}\{X(f)\}\}\hspace{0.05cm}.$$

$\text{Conclusions:}$ 

  1. The time function  $\text{P}\{ \text{A}\{ x(t)\}\}$  has the period  $T_{\rm P} = N \cdot T_{\rm A}$.
  2. The period in the frequency domain is  $f_{\rm P} = N \cdot f_{\rm A}$.
  3. For the description of the discretized time and frequency response in each case  $N$  complex numerical values  in the form of impulse weights are sufficient.


$\text{Example 2:}$  A time-limited  $($pulse-like$)$  signal  $x(t)$  is present in sampled form,  where the distance between two samples is  $T_{\rm A} = 1\, {\rm µ s}$:

  • After a discrete Fourier transform with  $N = 512$  the spectrum  $X(f)$  is available in form of frequency-samples at spacing  $f_{\rm A} = (N \cdot T_{\rm A})^{-1} \approx 1.953\,\text{kHz} $.
  • Increasing the DFT parameter to  $N= 2048$ results in a  $($four times$)$  finer frequency grid with  $f_{\rm A} \approx 488\,\text{Hz}$.


From the continuous to the discrete Fourier transform


From the conventional  »first Fourier integral«

$$X(f) =\int_{-\infty }^{+\infty}x(t) \cdot {\rm e}^{-{\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm} 2 \pi \hspace{0.05cm}\cdot \hspace{0.05cm} f \hspace{0.05cm}\cdot \hspace{0.05cm}t}\hspace{0.1cm} {\rm d}t$$

arises from discretization  $(\text{d}t \to T_{\rm A}$,  $t \to \nu \cdot T_{\rm A}$,  $f \to \mu \cdot f_{\rm A}$,  $T_{\rm A} \cdot f_{\rm A} = 1/N)$  the sampled and periodified spectral function

$${\rm P}\{X(\mu \cdot f_{\rm A})\} = T_{\rm A} \cdot \sum_{\nu = 0 }^{N-1} {\rm P}\{x(\nu \cdot T_{\rm A})\}\cdot {\rm e}^{-{\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm} 2 \pi \hspace{0.05cm} \cdot \hspace{0.05cm}\nu \hspace{0.05cm} \cdot \hspace{0.05cm}\mu /N} \hspace{0.05cm}.$$

It is taken into account that due to the discretization,  the periodified functions are to be used in each case.

For reasons of simplified notation, we now make the following substitutions:

  • The  $N$  »time-domain coefficients«  are with the variable  $\nu = 0$, ... , $N - 1$:
$$d(\nu) = {\rm P}\left\{x(t)\right\}{\big|}_{t \hspace{0.05cm}= \hspace{0.05cm}\nu \hspace{0.05cm}\cdot \hspace{0.05cm}T_{\rm A}}\hspace{0.05cm}.$$
  • Let  $N$  »frequency domain coefficients«  be associated with the variable  $\mu = 0,$ ... , $N-1$:
$$D(\mu) = f_{\rm A} \cdot {\rm P}\left\{X(f)\right\}{\big|}_{f \hspace{0.05cm}= \hspace{0.05cm}\mu \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm A}}\hspace{0.05cm}.$$
  • Abbreviation is written for the from  $N$  dependent  »complex rotation factor« :
$$w = {\rm e}^{-{\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm} 2 \pi /N} = \cos \left( {2 \pi}/{N}\right)-{\rm j} \cdot \sin \left( {2 \pi}/{N}\right) \hspace{0.05cm}.$$

$\text{Definition:}$  The term  »Discrete Fourier Transform«  $\rm (DFT)$  means the calculation of the  $N$  spectral coefficients  $D(\mu)$  from the  $N$  signal coefficients  $d(\nu)$:

On defining the  »Discrete Fourier Transform«  $\rm (DFT)$  with  $N=8$
$$D(\mu) = \frac{1}{N} \cdot \sum_{\nu = 0 }^{N-1} d(\nu)\cdot {w}^{\hspace{0.05cm}\nu \hspace{0.07cm} \cdot \hspace{0.05cm}\mu} \hspace{0.05cm}. $$

In the diagram you can see:

  • The  $N = 8$  signal coefficients  $d(\nu)$  by the blue filling,
  • the  $N = 8$  spectral coefficients  $D(\mu)$  at the green filling.


Inverse discrete Fourier transform


The  »inverse discrete Fourier transform«  describes the  »second Fourier integral«:

$$\begin{align*}x(t) & = \int_{-\infty }^{+\infty}X(f) \cdot {\rm e}^{\hspace{0.05cm}{\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm} 2 \pi \hspace{0.05cm}\cdot \hspace{0.05cm} f \hspace{0.05cm}\cdot \hspace{0.05cm} t}\hspace{0.1cm} {\rm d}f\end{align*}$$

in discretized form:  

$$d(\nu) = {\rm P}\left\{x(t)\right\}{\big|}_{t \hspace{0.05cm}= \hspace{0.05cm}\nu \hspace{0.05cm}\cdot \hspace{0.05cm}T_{\rm A}}\hspace{0.01cm}.$$

$\text{Definition:}$  The term  »Inverse Discrete Fourier Transform»   $\rm (IDFT)$   means the calculation of the signal coefficients  $d(\nu)$  from the spectral coefficients  $D(\mu)$:

On defining the  »Inverse Discrete Fourier Transform«  $\rm (IDFT)$  with  $N=8$
$$d(\nu) = \sum_{\mu = 0 }^{N-1} D(\mu) \cdot {w}^{-\nu \hspace{0.07cm} \cdot \hspace{0.05cm}\mu} \hspace{0.05cm}.$$
  • With indices  $\nu = 0, \hspace{0.05cm}\text{...} \hspace{0.05cm}, N-1, $  and   $\mu = 0, \hspace{0.05cm}\text{...} \hspace{0.05cm}, N-1$  holds:
$$d(\nu) = {\rm P}\left\{x(t)\right\}{\big \vert}_{t \hspace{0.05cm}= \hspace{0.05cm}\nu \hspace{0.05cm}\cdot \hspace{0.05cm}T_{\rm A} }\hspace{0.01cm},$$
$$D(\mu) = f_{\rm A} \cdot {\rm P}\left\{X(f)\right\}{\big \vert}_{f \hspace{0.05cm}= \hspace{0.05cm}\mu \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm A} } \hspace{0.01cm},$$
$$w = {\rm e}^{- {\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm} 2 \pi /N} \hspace{0.01cm}.$$
  • A comparison between  »DFT«  and  »IDFT«  shows that exactly the same algorithm can be used. 
  • The only differences between IDFT and DFT are:
  1. The exponent of the rotation factor is to be applied with different sign.
  2. In the IDFT,  the division by  $N$  is omitted.


Interpretation of DFT and IDFT


The graph shows the discrete coefficients in the time and frequency domain together with the periodified continuous-time functions.

Time and frequency coefficients of the DFT

When using DFT or IDFT,  please note:

  1. According to the above definitions, the DFT coefficients  $d(ν)$  and  $D(\mu)$  always have the unit of the time function.
  2. Dividing  $D(\mu)$  by  $f_{\rm A}$,  one obtains the spectral value  $X(\mu \cdot f_{\rm A})$.
  3. The spectral coefficients  $D(\mu)$  must always be complex in order to be able to consider odd time functions.
  4. One also uses complex time coefficients  $d(\nu)$   ⇒   DFT and IDFT are also applicable to band-pass signals.
  5. The basic interval for  $\nu$  and  $\mu$  is usually defined as the range from  $0$  to  $N - 1$  $($filled circles in the graph$)$.


With the complex-valued number sequences 

$$\langle \hspace{0.03cm}d(\nu)\hspace{0.03cm}\rangle = \langle \hspace{0.03cm}d(0), \hspace{0.05cm}\text{...} \hspace{0.05cm} , d(N-1) \hspace{0.03cm}\rangle,$$
$$\langle \hspace{0.03cm}D(\mu)\hspace{0.03cm}\rangle = \langle \hspace{0.03cm}D(0), \hspace{0.05cm}\text{...} \hspace{0.05cm} , D(N-1) \hspace{0.03cm}\rangle,$$

DFT and IDFT are symbolized similarly to the conventional Fourier transform:

$$\langle \hspace{0.03cm} D(\mu)\hspace{0.03cm}\rangle \hspace{0.2cm}\bullet\!\!-\!\!\!-(N)\!-\!\!\!-\!\!\hspace{0.05cm}\circ\, \hspace{0.2cm} \langle \hspace{0.03cm} d(\nu) \hspace{0.03cm}\rangle \hspace{0.05cm}.$$
  1. If the function  $x(t)$  is already limited to the range  $0 \le t \lt N \cdot T_{\rm A}$  then the IDFT output directly give the samples of the time function:   $d(\nu) = x(\nu \cdot T_{\rm A}).$
  2. If  $x(t)$  is shifted with respect to the basic interval,  one has to choose the association shown in  $\text{Example 3}$  between  $x(t)$  and the coefficients  $d(\nu)$.


$\text{Example 3:}$  The upper graph shows the asymmetric triangular pulse  $x(t)$ whose absolute width is smaller than  $T_{\rm P} = N \cdot T_{\rm A}$.

On assigning of the DFT coefficients with  $N=8$


The sketch below shows the assigned DFT coefficients $($valid for  $N = 8)$.

  • For  $\nu = 0,\hspace{0.05cm}\text{...} \hspace{0.05cm} , N/2 = 4$  holds   $d(\nu) = x(\nu \cdot T_{\rm A})$ :
$$d(0) = x (0)\hspace{0.05cm}, \hspace{0.15cm} d(1) = x (T_{\rm A})\hspace{0.05cm}, \hspace{0.15cm} d(2) = x (2T_{\rm A})\hspace{0.05cm}, $$
$$d(3) = x (3T_{\rm A})\hspace{0.05cm}, \hspace{0.15cm} d(4) = x (4T_{\rm A})\hspace{0.05cm}.$$
  • The coefficients  $d(5)$,  $d(6)$  and  d$(7)$  are to be set as follows:
$$d(5) = x (-3T_{\rm A})\hspace{0.05cm}, \hspace{0.35cm} d(6) = x (-2T_{\rm A})\hspace{0.05cm}, \hspace{0.35cm} d(7) = x (-T_{\rm A})\hspace{0.05cm}$$
$$ \Rightarrow \hspace{0.2cm}d(\nu) = x \big ((\nu\hspace{-0.05cm} - \hspace{-0.05cm} N ) \cdot T_{\rm A}\big ). $$

Exercises for the chapter


Exercise 5.2: Inverse Discrete Fourier Transform

Exercise 5.2Z: DFT of a Triangular Pulse