Signal Representation/The Fourier Transform Theorems - Revision history

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

Guenter at 16:26, 14 June 2023

2023-06-14T16:26:50Z

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

	*With the further renaming  $u \to f$  one gets to the [[Signal_Representation/~~Fourier_Transform_and_Its_Inverse~~#The_second_Fourier_integral\|»second Fourier integral«]]:		*With the further renaming  $u \to f$  one gets to the [[Signal_Representation/The_Fourier_Transform_and_its_Inverse#The_second_Fourier_integral\|»second Fourier integral«]]:

	:$$X^{\star}(t ) = \int_{ - \infty }^{ + \infty } {x^{\star}( f )} \cdot {\rm e}^{ {\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f\hspace{0.05cm}\cdot \hspace{0.05cm}t}\hspace{0.1cm} {\rm d}f.\hspace{7.9cm}$$		:$$X^{\star}(t ) = \int_{ - \infty }^{ + \infty } {x^{\star}( f )} \cdot {\rm e}^{ {\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f\hspace{0.05cm}\cdot \hspace{0.05cm}t}\hspace{0.1cm} {\rm d}f.\hspace{7.9cm}$$

Guenter at 16:24, 14 June 2023

2023-06-14T16:24:08Z

← Older revision		Revision as of 18:24, 14 June 2023
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	{{BlaueBox\|TEXT=		{{BlaueBox\|TEXT=
	$\text{Proof of Equation (1):}$ This equation results from differentiation of the  [[Signal_Representation/~~Fourier_Transform_and_Its_Inverse~~#The_second_Fourier_integral\|»second Fourier integral«]]:		$\text{Proof of Equation (1):}$ This equation results from differentiation of the  [[Signal_Representation/The_Fourier_Transform_and_its_Inverse#The_second_Fourier_integral\|»second Fourier integral«]]:

	:$$y(t) = \frac{\text{d}x(t)}{\text{d}t} = \frac{\text{d} }{\text{d}t}\int_{ - \infty }^{ + \infty } X( f ) \cdot {\rm e}^{ {\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}2\hspace{0.03cm}\pi\hspace{0.05cm}\cdot \hspace{0.05cm} f\hspace{0.05cm}\cdot \hspace{0.05cm}t}\hspace{0.1cm} {\rm d}f = \int_{ - \infty }^{ + \infty } {X( f )} \cdot {\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}2\pi f \cdot {\rm e}^{ {\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}2\pi \hspace{0.05cm}\cdot \hspace{0.05cm} f\hspace{0.05cm}\cdot \hspace{0.05cm}t} {\rm d}f.$$		:$$y(t) = \frac{\text{d}x(t)}{\text{d}t} = \frac{\text{d} }{\text{d}t}\int_{ - \infty }^{ + \infty } X( f ) \cdot {\rm e}^{ {\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}2\hspace{0.03cm}\pi\hspace{0.05cm}\cdot \hspace{0.05cm} f\hspace{0.05cm}\cdot \hspace{0.05cm}t}\hspace{0.1cm} {\rm d}f = \int_{ - \infty }^{ + \infty } {X( f )} \cdot {\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}2\pi f \cdot {\rm e}^{ {\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}2\pi \hspace{0.05cm}\cdot \hspace{0.05cm} f\hspace{0.05cm}\cdot \hspace{0.05cm}t} {\rm d}f.$$
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	#By comparing the integrands,  the variation  $\mathbf{(1)}$  of the differentiation theorem is obtained.		#By comparing the integrands,  the variation  $\mathbf{(1)}$  of the differentiation theorem is obtained.
	#To derive the second variant one proceeds from the  [[Signal_Representation/~~Fourier_Transform_and_Its_Inverse~~#The_first_Fourier_integral\|»first Fourier integral«]]  in an analogous manner.		#To derive the second variant one proceeds from the  [[Signal_Representation/The_Fourier_Transform_and_its_Inverse#The_first_Fourier_integral\|»first Fourier integral«]]  in an analogous manner.
	#The negative exponent in the first Fourier integral leads to the minus sign in the time function. <div align="right">q.e.d.</div>}}		#The negative exponent in the first Fourier integral leads to the minus sign in the time function. <div align="right">q.e.d.</div>}}

Guenter at 16:20, 14 June 2023

2023-06-14T16:20:44Z

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	{{BlaueBox\|TEXT=		{{BlaueBox\|TEXT=
	$\text{Proof of Equation (1):}$		$\text{Proof of Equation (1):}$
	The  [[Signal_Representation/~~Fourier_Transform_and_Its_Inverse~~#The_first_Fourier_integral\|»first Fourier integral«]]  for signal   $x_{\rm V}(t) = x(t-t_0)$  shifted to the right by  $t_0$  is defined with the substitution  $\tau = t - t_0$:		The  [[Signal_Representation/The_Fourier_Transform_and_its_Inverse#The_first_Fourier_integral\|»first Fourier integral«]]  for signal   $x_{\rm V}(t) = x(t-t_0)$  shifted to the right by  $t_0$  is defined with the substitution  $\tau = t - t_0$:

	:$$X_{\rm V} ( f ) = \int_{ - \infty }^{ + \infty } {x( {t - t_0 } ) \cdot {\rm e}^{ - {\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f\hspace{0.05cm}\cdot \hspace{0.05cm}t}\hspace{0.1cm}{\rm d}t}		:$$X_{\rm V} ( f ) = \int_{ - \infty }^{ + \infty } {x( {t - t_0 } ) \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}

Guenter at 16:19, 14 June 2023

2023-06-14T16:19:08Z

← Older revision		Revision as of 18:19, 14 June 2023
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	If we restrict ourselves to real time functions,  the signs for  »conjugated complex»  can be omitted on both sides of the Fourier correspondence.}}		If we restrict ourselves to real time functions,  the signs for  »conjugated complex»  can be omitted on both sides of the Fourier correspondence.}}


	{{BlaueBox\|TEXT=		{{BlaueBox\|TEXT=
	$\text{Proof:}$  The  [[Signal_Representation/~~The_Fourier_Transform_and_Its_Inverse~~#The_first_Fourier_integral\|»first Fourier integral«]]  is after successive renaming  $t \to u$,  $f \to t$:		$\text{Proof:}$  The  [[Signal_Representation/The_Fourier_Transform_and_its_Inverse#The_first_Fourier_integral\|»first Fourier integral«]]  is after successive renaming  $t \to u$,  $f \to t$:

	:$$X( f ) = \int_{ - \infty }^{ + \infty } {x( u )} \cdot {\rm e}^{ - {\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f\hspace{0.05cm}\cdot \hspace{0.05cm}u} \hspace{0.1cm}{\rm d}u, \hspace{1cm}		:$$X( f ) = \int_{ - \infty }^{ + \infty } {x( u )} \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}u} \hspace{0.1cm}{\rm d}u, \hspace{1cm}

Guenter at 16:08, 14 June 2023

2023-06-14T16:08:05Z

← Older revision		Revision as of 18:08, 14 June 2023
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	If we restrict ourselves to real time functions,  the signs for  »conjugated complex»  can be omitted on both sides of the Fourier correspondence.}}		If we restrict ourselves to real time functions,  the signs for  »conjugated complex»  can be omitted on both sides of the Fourier correspondence.}}


	{{BlaueBox\|TEXT=		{{BlaueBox\|TEXT=
	$\text{Proof:}$  The  [[Signal_Representation/~~Fourier_Transform_and_Its_Inverse~~#The_first_Fourier_integral\|»first Fourier integral«]]  is after successive renaming  $t \to u$,  $f \to t$:		$\text{Proof:}$  The  [[Signal_Representation/The_Fourier_Transform_and_Its_Inverse#The_first_Fourier_integral\|»first Fourier integral«]]  is after successive renaming  $t \to u$,  $f \to t$:

	:$$X( f ) = \int_{ - \infty }^{ + \infty } {x( u )} \cdot {\rm e}^{ - {\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f\hspace{0.05cm}\cdot \hspace{0.05cm}u} \hspace{0.1cm}{\rm d}u, \hspace{1cm}		:$$X( f ) = \int_{ - \infty }^{ + \infty } {x( u )} \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}u} \hspace{0.1cm}{\rm d}u, \hspace{1cm}

Guenter at 16:05, 14 June 2023

2023-06-14T16:05:34Z

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	We consider two rectangles of equal height, where  $T_2 = T_1/2$  holds.		We consider two rectangles of equal height, where  $T_2 = T_1/2$  holds.

	*The spectral function of  $x_1(t)$  results after the  [[Signal_Representation/~~Fourier_Transform_and_Its_Inverse~~#The_first_Fourier_integral\|»first Fourier Integral«]]  to		*The spectral function of  $x_1(t)$  results after the  [[Signal_Representation/The_Fourier_Transform_and_its_Inverse#The_first_Fourier_integral\|»first Fourier Integral«]]  to

	:$$X_1 (f) = A \cdot \frac{ {1 - {\rm e}^{ - {\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}fT_1 } } }{ { {\rm j}2\pi f} } .$$		:$$X_1 (f) = A \cdot \frac{ {1 - {\rm e}^{ - {\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}fT_1 } } }{ { {\rm j}2\pi f} } .$$

Guenter at 16:00, 14 June 2023

2023-06-14T16:00:38Z

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	:This relation can be used for simplification by omitting the constant  $k$  $($which can be a gain,  an attenuation or a unit factor$)$  and adding it to the result later.		:This relation can be used for simplification by omitting the constant  $k$  $($which can be a gain,  an attenuation or a unit factor$)$  and adding it to the result later.

	:The above sentence follows directly from the definition of the  [[Signal_Representation/~~Fourier_Transform_and_Its_Inverse~~#The_first_Fourier_integral\|»first Fourier integral«]],  as well as from the  »addition theorem«,  which formulates the foundation of the  »superposition principle«.		:The above sentence follows directly from the definition of the  [[Signal_Representation/The_Fourier_Transform_and_its_Inverse#The_first_Fourier_integral\|»first Fourier integral«]],  as well as from the  »addition theorem«,  which formulates the foundation of the  »superposition principle«.

	{{BlaueBox\|TEXT=		{{BlaueBox\|TEXT=

Guenter at 15:58, 14 June 2023

2023-06-14T15:58:45Z

Show changes

← Older revision		Revision as of 15:47, 16 March 2026
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	Therefore the following applies:		Therefore the following applies:
	#The even  $($blue$)$  signal part  $x_{\rm g} (t) = {1}/{2}$  is a constant with the real spectral function  $X_{\rm R}(f) = {1}/{2} \cdot \delta(f)$.		#The even  $($blue$)$  signal part  $x_{\rm g} (t) = {1}/{2}$  is a constant with the real spectral function  $X_{\rm R}(f) = {1}/{2} \cdot \delta(f)$.
	#The spectrum  ${\rm j} \cdot X_{\rm I}(f)$  of the odd  $($green$)$  signum function  $x_{\rm u} (t)$  was already calculated in the earlier  [[Signal_Representation/~~Fourier_Transform_and_Its_Inverse~~#Fourier_transform\|$\text{Example 3}$]]  in the section  »Fourier transform«.		#The spectrum  ${\rm j} \cdot X_{\rm I}(f)$  of the odd  $($green$)$  signum function  $x_{\rm u} (t)$  was already calculated in the earlier  [[Signal_Representation/The_Fourier_Transform_and_its_Inverse#Fourier_transform\|$\text{Example 3}$]]  in the section  »Fourier transform«.
	#This results for the spectrum of the  $($red$)$  sketched  jump function:		#This results for the spectrum of the  $($red$)$  sketched  jump function:
	::$$X(f) = X_{\rm R}(f) + {\rm j}\cdot X_{\rm I}(f)= {1}/{2} \cdot \delta (f) - {\rm j}\cdot \frac{1}{2\pi f}.$$}}		::$$X(f) = X_{\rm R}(f) + {\rm j}\cdot X_{\rm I}(f)= {1}/{2} \cdot \delta (f) - {\rm j}\cdot \frac{1}{2\pi f}.$$}}
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	==Reciprocity Theorem of time duration and bandwidth==		==Reciprocity Theorem of time duration and bandwidth==
	<br>		<br>
	This law follows directly from the  [[Signal_Representation/~~Fourier_Transform_Theorems~~#Similarity_Theorem\|»similarity theorem«]]:   The wider a pulse is in its extension,  the narrower and higher is the corresponding spectrum and vice versa.		This law follows directly from the  [[Signal_Representation/The_Fourier_Transform_Theorems#Similarity_Theorem\|»similarity theorem«]]:   The wider a pulse is in its extension,  the narrower and higher is the corresponding spectrum and vice versa.

	:To be able to make quantitative statements,  we define two parameters for energy-limited signals.  Both quantities are shown in the diagram in  $\text{Example 4}$  for a Gaussian pulse and its likewise Gaussian spectrum.		:To be able to make quantitative statements,  we define two parameters for energy-limited signals.  Both quantities are shown in the diagram in  $\text{Example 4}$  for a Gaussian pulse and its likewise Gaussian spectrum.
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	Note that  $\Delta f$  is defined over the actual spectrum  $X(f)$  and not over  $\|X(f)\|$.		Note that  $\Delta f$  is defined over the actual spectrum  $X(f)$  and not over  $\|X(f)\|$.
	*For real functions the integration over the even function part is sufficient,  since the integral over the odd part is always zero due to the  [[Signal_Representation/~~Fourier_Transform_Theorems~~#Assignment_Theorem\|»assignment theorem«]].		*For real functions the integration over the even function part is sufficient,  since the integral over the odd part is always zero due to the  [[Signal_Representation/The_Fourier_Transform_Theorems#Assignment_Theorem\|»assignment theorem«]].

	*For odd time functions and thus purely imaginary spectra,  the two definitions of  $\Delta t$  and  $\Delta f$  fail.		*For odd time functions and thus purely imaginary spectra,  the two definitions of  $\Delta t$  and  $\Delta f$  fail.
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	$\text{Proof:}$  The  [[Signal_Representation/The_Fourier_Transform_and_its_Inverse#The_first_Fourier_integral\|»first Fourier integral«]]  is after successive renaming  $t \to u$,  $f \to t$:		$\text{Proof:}$  The  [[Signal_Representation/The_Fourier_Transform_and_its_Inverse#The_first_Fourier_integral\|»first Fourier integral«]]  is after successive renaming  $t \to u$,  $f \to t$:

	:$$X( f ) = \int_{ - \infty }^{ + \infty } {x( u )} \cdot {\rm e}^{ - {\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f\hspace{0.05cm}\cdot \hspace{0.05cm}u} \hspace{0.1cm}{\rm d}u, \hspace{1cm}		:$$X( f ) = \int_{ - \infty }^{ + \infty } {x( u )} \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}u} \hspace{0.1cm}{\rm d}u, \hspace{1cm}X(t ) = \int_{ - \infty }^{ + \infty } {x( u )} \cdot {\rm e}^{ - {\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}t\hspace{0.05cm}\cdot \hspace{0.05cm}u}\hspace{0.1cm} {\rm d}u.$$
	X(t ) = \int_{ - \infty }^{ + \infty } {x( u )} \cdot {\rm e}^{ - {\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}t\hspace{0.05cm}\cdot \hspace{0.05cm}u}\hspace{0.1cm} {\rm d}u.$$

	If you change the sign in the exponent,  you have to replace  $X(t)$  by  $X^(t)$  and  $x(u)$  by  $x^*(u)$ :		If you change the sign in the exponent,  you have to replace  $X(t)$  by  $X^(t)$  and  $x(u)$  by  $x^*(u)$ :
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	The  [[Signal_Representation/The_Fourier_Transform_and_its_Inverse#The_first_Fourier_integral\|»first Fourier integral«]]  for signal   $x_{\rm V}(t) = x(t-t_0)$  shifted to the right by  $t_0$  is defined with the substitution  $\tau = t - t_0$:		The  [[Signal_Representation/The_Fourier_Transform_and_its_Inverse#The_first_Fourier_integral\|»first Fourier integral«]]  for signal   $x_{\rm V}(t) = x(t-t_0)$  shifted to the right by  $t_0$  is defined with the substitution  $\tau = t - t_0$:

	:$$X_{\rm V} ( f ) = \int_{ - \infty }^{ + \infty } {x( {t - t_0 } ) \cdot {\rm e}^{ - {\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f\hspace{0.05cm}\cdot \hspace{0.05cm}t}\hspace{0.1cm}{\rm d}t}		:$$X_{\rm V} ( f ) = \int_{ - \infty }^{ + \infty } {x( {t - t_0 } ) \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}= \int_{ - \infty }^{ + \infty } {x( \tau ) \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}( {\tau + t_0 } )}\hspace{0.1cm} {\rm d}\tau .}$$
	= \int_{ - \infty }^{ + \infty } {x( \tau ) \cdot {\rm e}^{ - {\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f\hspace{0.05cm}\cdot \hspace{0.05cm}( {\tau + t_0 } )}\hspace{0.1cm} {\rm d}\tau .}$$

	*The term independent from the integration variable  $\tau$  can be dragged in front of the integral.		*The term independent from the integration variable  $\tau$  can be dragged in front of the integral.
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	:$$X_2( f ) = \frac{A }{2\pi f} \cdot \sin ( 2\pi fT) + {\rm j}\cdot \frac{A }{2\pi f} \cdot \big[ {\cos ( {2\pi fT} ) - 1} \big] .$$		:$$X_2( f ) = \frac{A }{2\pi f} \cdot \sin ( 2\pi fT) + {\rm j}\cdot \frac{A }{2\pi f} \cdot \big[ {\cos ( {2\pi fT} ) - 1} \big] .$$

	*The same result can be obtained with the  [[Signal_Representation/~~Fourier_Transform_Theorems~~#Assignment_Theorem\|»assignment theorem«]];   <br>⇒   The real part of the spectrum belongs to the even signal part  $x_{\rm g}(t)$,  the imaginary part to the odd signal part  $x_{\rm u}(t)$.}}		*The same result can be obtained with the  [[Signal_Representation/The_Fourier_Transform_Theorems#Assignment_Theorem\|»assignment theorem«]];   <br>⇒   The real part of the spectrum belongs to the even signal part  $x_{\rm g}(t)$,  the imaginary part to the odd signal part  $x_{\rm u}(t)$.}}


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	:$$X_1( f ) = \frac{1 }{ {\rm j\cdot \pi} f}, $$		:$$X_1( f ) = \frac{1 }{ {\rm j\cdot \pi} f}, $$
	:$$X_2( f ) = 2 = {\rm const.}\hspace{0.3cm}		:$$X_2( f ) = 2 = {\rm const.}\hspace{0.3cm}\Rightarrow \hspace{0.3cm} X_2(f) = X_1(f) \cdot {\rm j}\cdot 2\pi f.$$
	\Rightarrow \hspace{0.3cm} X_2(f) = X_1(f) \cdot {\rm j}\cdot 2\pi f.$$
	#From the differentiation theorem it follows that  $x_2(t)$  is equal to the time-derivative of  $x_1(t)$ .		#From the differentiation theorem it follows that  $x_2(t)$  is equal to the time-derivative of  $x_1(t)$ .
	#This is actually correct:  For  $t \neq 0$  ⇒   $x_1(t)$  is constant, i.e. the derivative is zero.		#This is actually correct:  For  $t \neq 0$  ⇒   $x_1(t)$  is constant, i.e. the derivative is zero.
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	$\text{Illustration – not an exact proof:}$		$\text{Illustration – not an exact proof:}$

	The integration theorem represents exactly the inversion of the  [[Signal_Representation/~~Fourier_Transform_Theorems~~#Differentiation_Theorem\|»differentiation theorem«]].  If one applies the differentiation theorem to the equation  $\mathbf{(1)}$  one obtains		The integration theorem represents exactly the inversion of the  [[Signal_Representation/The_Fourier_Transform_Theorems#Differentiation_Theorem\|»differentiation theorem«]].  If one applies the differentiation theorem to the equation  $\mathbf{(1)}$  one obtains

	:$$\frac{ {\rm d} }{ {\rm d}t} \int_{ - \infty }^{\hspace{0.05cm}t} {x( \tau )} \hspace{0.1cm}{\rm d}\tau\ \ \circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\,\ \ X( f )\cdot \left( {\frac{1}{ { {\rm j}\cdot 2\pi f} } + \frac{1}{2}\cdot \delta ( f )} \right) \cdot {\rm j}\cdot 2\pi f.$$		:$$\frac{ {\rm d} }{ {\rm d}t} \int_{ - \infty }^{\hspace{0.05cm}t} {x( \tau )} \hspace{0.1cm}{\rm d}\tau\ \ \circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\,\ \ X( f )\cdot \left( {\frac{1}{ { {\rm j}\cdot 2\pi f} } + \frac{1}{2}\cdot \delta ( f )} \right) \cdot {\rm j}\cdot 2\pi f.$$