Signal Representation/Discrete Fourier Transform (DFT) - Revision history

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

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

	are unsuitable for two reasons:		are unsuitable for two reasons:
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	==From the continuous to the discrete Fourier transform==		==From the continuous to the discrete Fourier transform==
	<br>		<br>
	From the conventional  [[Signal_Representation/~~Fourier_Transform_and_its_Inverse~~#The_first_Fourier_integral\|»first Fourier integral«]]		From the conventional  [[Signal_Representation/The_Fourier_Transform_and_its_Inverse#The_first_Fourier_integral\|»first Fourier integral«]]

	:$$X(f) =\int_{-\infty}^{+\infty}x(t) \cdot {\rm e}^{-{\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm} 2 \pi \hspace{0.05cm}\cdot \hspace{0.05cm} f \hspace{0.05cm}\cdot \hspace{0.05cm}t}\hspace{0.1cm} {\rm d}t$$		:$$X(f) =\int_{-\infty}^{+\infty}x(t) \cdot {\rm e}^{-{\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm} 2 \pi \hspace{0.05cm}\cdot \hspace{0.05cm} f \hspace{0.05cm}\cdot \hspace{0.05cm}t}\hspace{0.1cm} {\rm d}t$$
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	==Inverse discrete Fourier transform==		==Inverse discrete Fourier transform==
	<br>		<br>
	The  »inverse discrete Fourier transform«  describes the  [[Signal_Representation/~~Fourier_Transform_and_its_Inverse~~#The_second_Fourier_integral\|»second Fourier integral«]]:		The  »inverse discrete Fourier transform«  describes the  [[Signal_Representation/The_Fourier_Transform_and_its_Inverse#The_second_Fourier_integral\|»second Fourier integral«]]:

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

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

	{{BlaueBox\|TEXT= $\text{Definition:}$  The term  »Inverse Discrete Fourier Transform»   $\rm (IDFT)$   means the calculation of the signal coefficients  $d(\nu)$  from the spectral coefficients  $D(\mu)$:		{{BlaueBox\|TEXT= $\text{Definition:}$  The term  »Inverse Discrete Fourier Transform»   $\rm (IDFT)$   means the calculation of the signal coefficients  $d(\nu)$  from the spectral coefficients  $D(\mu)$:
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	*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:		*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:
	:$$d(\nu) ={\rm P}\left\{x(t)\right\}{\big \vert}_{t \hspace{0.05cm}= \hspace{0.05cm}\nu \hspace{0.05cm}\cdot \hspace{0.05cm}T_{\~~rmA~~} }\hspace{0.01cm},$$		:$$d(\nu) ={\rm P}\left\{x(t)\right\}{\big \vert}_{t \hspace{0.05cm}= \hspace{0.05cm}\nu \hspace{0.05cm}\cdot \hspace{0.05cm}T_{\rm A} }\hspace{0.01cm},$$

	:$$D(\mu) = f_{\rm A} \cdot{\rm P}\left\{X(f)\right\}{\big \vert}_{f \hspace{0.05cm}= \hspace{0.05cm}\mu \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm A} }\hspace{0.01cm},$$		:$$D(\mu) = f_{\rm A} \cdot{\rm P}\left\{X(f)\right\}{\big \vert}_{f \hspace{0.05cm}= \hspace{0.05cm}\mu \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm A} }\hspace{0.01cm},$$

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← Older revision		Revision as of 14:28, 16 March 2026
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	{{Display}}		{{Display}}
			[[de:Signaldarstellung/Diskrete_Fouriertransformation_(DFT)]]

Guenter at 15:57, 28 June 2023

2023-06-28T15:57:57Z

← Older revision		Revision as of 17:57, 28 June 2023
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	<br clear=all>		<br clear=all>
	With the complex-valued number sequences		With the complex-valued number sequences
	:$$\langle \hspace{0.~~1cm~~}d(\nu)\hspace{0.~~1cm~~}\rangle = \langle \hspace{0.~~1cm~~}d(0), \hspace{0.05cm}\text{...} \hspace{0.05cm} , d(N-1) \hspace{0.~~1cm~~}\rangle,$$		:$$\langle \hspace{0.03cm}d(\nu)\hspace{0.03cm}\rangle = \langle \hspace{0.03cm}d(0), \hspace{0.05cm}\text{...} \hspace{0.05cm} , d(N-1) \hspace{0.03cm}\rangle,$$
	:$$\langle \hspace{0.~~1cm~~}D(\mu)\hspace{0.~~1cm~~}\rangle = \langle \hspace{0.~~1cm~~}D(0), \hspace{0.05cm}\text{...} \hspace{0.05cm} , D(N-1) \hspace{0.~~1cm~~}\rangle,$$		:$$\langle \hspace{0.03cm}D(\mu)\hspace{0.03cm}\rangle = \langle \hspace{0.03cm}D(0), \hspace{0.05cm}\text{...} \hspace{0.05cm} , D(N-1) \hspace{0.03cm}\rangle,$$

	DFT and IDFT are symbolized similarly to the conventional Fourier transform:		DFT and IDFT are symbolized similarly to the conventional Fourier transform:
	:$$\langle \hspace{0.~~1cm~~} D(\mu)\hspace{0.~~1cm~~}\rangle \hspace{0.2cm}\bullet\!\!-\!\!\!-(N)\!-\!\!\!-\!\!\hspace{0.05cm}\circ\, \hspace{0.2cm} \langle \hspace{0.~~1cm~~} d(\nu) \hspace{0.~~1cm~~}\rangle \hspace{0.05cm}.$$		:$$\langle \hspace{0.03cm} D(\mu)\hspace{0.03cm}\rangle \hspace{0.2cm}\bullet\!\!-\!\!\!-(N)\!-\!\!\!-\!\!\hspace{0.05cm}\circ\, \hspace{0.2cm} \langle \hspace{0.03cm} d(\nu) \hspace{0.03cm}\rangle \hspace{0.05cm}.$$
	#If the function  $x(t)$  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:   $d(\nu) = x(\nu \cdot T_{\rm A}).$		#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:   $d(\nu) = x(\nu \cdot T_{\rm A}).$
	#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  $d(\nu)$.		#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  $d(\nu)$.

Guenter at 17:03, 23 June 2023

2023-06-23T17:03:34Z

← Older revision		Revision as of 19:03, 23 June 2023
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	For reasons of simplified notation, we now make the following substitutions:		For reasons of simplified notation, we now make the following substitutions:
	*The  $N$  »time-domain coefficients«  are with the variable  $\nu = 0$, ... , $N - 1$:		*The  $N$  »time-domain coefficients«  are with the variable  $\nu = 0$, ... , $N - 1$:
	:$$d(\nu) =.		:$$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}.$$		{\rm P}\left\{x(t)\right\}{\big\|}_{t \hspace{0.05cm}= \hspace{0.05cm}\nu \hspace{0.05cm}\cdot \hspace{0.05cm}T_{\rm A}}\hspace{0.05cm}.$$
	*Let  $N$  »frequency domain coefficients«  be associated with the variable  $\mu = 0,$ ... , $N-1$:		*Let  $N$  »frequency domain coefficients«  be associated with the variable  $\mu = 0,$ ... , $N-1$:

Guenter at 16:05, 23 June 2023

2023-06-23T16:05:47Z

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	*Without limiting generality,  $x(t)$  and  $X(f)$  are each real and Gaussian.		*Without limiting generality,  $x(t)$  and  $X(f)$  are each real and Gaussian.

	*According to the chapter  [[Signal_Representation/Time_Discrete_Signal_Representation\|»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«		*According to the chapter  [[Signal_Representation/Time_Discrete_Signal_Representation\|»Discrete-Time Signal Representation«]]  one can describe the sampling of the time signal  $x(t)$  by multiplying it by a Dirac delta train   ⇒   <br>»Dirac comb  in the time domain«
	[[File:P_ID1132__Sig_T_5_1_S2_neu.png\|right\|frame\| Time discretization   ⇒   Periodification in the frequency domain.  <u>Notation:</u><br>     ${\rm A}\{x(t)\}$:  Signal  $x(t)$  after  »sampling«  $($German:  "Abtastung"$)$   ⇒   $\rm A\{\text{...}\}$ <br>     ${\rm P}\{X(f)\}$:  Spectrum  $X(f)$  after  »periodification«    ⇒   $\rm P\{\text{...}\}$]]		[[File:P_ID1132__Sig_T_5_1_S2_neu.png\|right\|frame\| Time discretization   ⇒   Periodification in the frequency domain.  <u>Notation:</u><br>     ${\rm A}\{x(t)\}$:  Signal  $x(t)$  after  »sampling«  $($German:  "Abtastung"$)$   ⇒   $\rm A\{\text{...}\}$ <br>     ${\rm P}\{X(f)\}$:  Spectrum  $X(f)$  after  »periodification«    ⇒   $\rm P\{\text{...}\}$]]

Guenter at 16:04, 23 June 2023

2023-06-23T16:04:08Z

← Older revision		Revision as of 18:04, 23 June 2023
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	==Arguments for the discrete implementation of the Fourier transform==		==Arguments for the discrete implementation of the Fourier transform==
	<br>		<br>
	The  »'''Fourier transform'''«  according to the previous description in chapter  [[Signal_Representation/The_Fourier_Transform_and_its_Inverse\|»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.		The  »'''Fourier transform'''«  according to the previous description in chapter  [[Signal_Representation/The_Fourier_Transform_and_its_Inverse#The_first_Fourier_integral\|»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		If the spectral components  $X(f)$  of a time function  $x(t)$  are to be determined numerically,  the general transformation equations

Guenter at 16:02, 23 June 2023

2023-06-23T16:02:36Z

← Older revision		Revision as of 18:02, 23 June 2023
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	==Arguments for the discrete implementation of the Fourier transform==		==Arguments for the discrete implementation of the Fourier transform==
	<br>		<br>
	The  »'''Fourier transform'''«  according to the previous description in chapter  [[Signal_Representation/~~Fourier_Transform_and_Its_Inverse~~\|»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.		The  »'''Fourier transform'''«  according to the previous description in chapter  [[Signal_Representation/The_Fourier_Transform_and_its_Inverse\|»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		If the spectral components  $X(f)$  of a time function  $x(t)$  are to be determined numerically,  the general transformation equations

Guenter at 16:01, 23 June 2023

2023-06-23T16:01:47Z

← Older revision		Revision as of 18:01, 23 June 2023
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	==Arguments for the discrete implementation of the Fourier transform==		==Arguments for the discrete implementation of the Fourier transform==
	<br>		<br>
	The  »'''Fourier transform'''«  according to the previous description in chapter  [[Signal_Representation/~~The_Fourier_Transform_and_Its_Inverse~~\|»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.		The  »'''Fourier transform'''«  according to the previous description in chapter  [[Signal_Representation/Fourier_Transform_and_Its_Inverse\|»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		If the spectral components  $X(f)$  of a time function  $x(t)$  are to be determined numerically,  the general transformation equations

Guenter at 16:01, 23 June 2023

2023-06-23T16:01:18Z

← Older revision		Revision as of 18:01, 23 June 2023
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	==Arguments for the discrete implementation of the Fourier transform==		==Arguments for the discrete implementation of the Fourier transform==
	<br>		<br>
	The  »'''Fourier transform'''«  according to the previous description in chapter  [[Signal_Representation/~~Fourier_Transform_and_Its_Inverse~~\|»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.		The  »'''Fourier transform'''«  according to the previous description in chapter  [[Signal_Representation/The_Fourier_Transform_and_Its_Inverse\|»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		If the spectral components  $X(f)$  of a time function  $x(t)$  are to be determined numerically,  the general transformation equations