Signal Representation/The Fourier Transform Theorems - Revision history

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@@ Line 233: / Line 233: @@
 :$$X^{\star}(t ) = \int_{ - \infty }^{ + \infty } {x^{\star}( u )}  \cdot {\rm e}^{ {\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}2\pi\hspace{0.05cm}\cdot \hspace{0.05cm} t\hspace{0.05cm}\cdot \hspace{0.05cm}u}\hspace{0.1cm} {\rm d}u.$$
-*With the further renaming &nbsp;$u \to f$&nbsp; one gets to the [[Signal_Representation/Fourier_Transform_and_Its_Inverse#The_second_Fourier_integral|&raquo;second Fourier integral&laquo;]]:
+*With the further renaming &nbsp;$u \to f$&nbsp; one gets to the [[Signal_Representation/The_Fourier_Transform_and_its_Inverse#The_second_Fourier_integral|&raquo;second Fourier integral&laquo;]]:
 :$$X^{\star}(t ) = \int_{ - \infty }^{ + \infty } {x^{\star}( f )}  \cdot {\rm e}^{ {\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f\hspace{0.05cm}\cdot \hspace{0.05cm}t}\hspace{0.1cm} {\rm d}f.\hspace{7.9cm}$$

@@ Line 336: / Line 336: @@
 {{BlaueBox|TEXT=
-$\text{Proof of Equation (1):}$&nbsp;This equation results from differentiation of the&nbsp; [[Signal_Representation/Fourier_Transform_and_Its_Inverse#The_second_Fourier_integral|&raquo;second Fourier integral&laquo;]]:
+$\text{Proof of Equation (1):}$&nbsp;This equation results from differentiation of the&nbsp; [[Signal_Representation/The_Fourier_Transform_and_its_Inverse#The_second_Fourier_integral|&raquo;second Fourier integral&laquo;]]:
 :$$y(t) = \frac{\text{d}x(t)}{\text{d}t} = \frac{\text{d} }{\text{d}t}\int_{ - \infty }^{ + \infty } X( f )  \cdot {\rm e}^{ {\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}2\hspace{0.03cm}\pi\hspace{0.05cm}\cdot \hspace{0.05cm} f\hspace{0.05cm}\cdot \hspace{0.05cm}t}\hspace{0.1cm} {\rm d}f = \int_{ - \infty }^{ + \infty } {X( f )}  \cdot {\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}2\pi f \cdot {\rm e}^{ {\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}2\pi \hspace{0.05cm}\cdot \hspace{0.05cm} f\hspace{0.05cm}\cdot \hspace{0.05cm}t} {\rm d}f.$$
@@ Line 345: / Line 345: @@
 #By comparing the integrands,&nbsp; the variation&nbsp; $\mathbf{(1)}$&nbsp; of the differentiation theorem is obtained.
-#To derive the second variant one proceeds from the &nbsp;[[Signal_Representation/Fourier_Transform_and_Its_Inverse#The_first_Fourier_integral|&raquo;first Fourier integral&laquo;]]&nbsp; in an analogous manner.
+#To derive the second variant one proceeds from the &nbsp;[[Signal_Representation/The_Fourier_Transform_and_its_Inverse#The_first_Fourier_integral|&raquo;first Fourier integral&laquo;]]&nbsp; 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>}}

@@ Line 287: / Line 287: @@
 {{BlaueBox|TEXT=
 $\text{Proof of Equation (1):}$&nbsp;
-The&nbsp;  [[Signal_Representation/Fourier_Transform_and_Its_Inverse#The_first_Fourier_integral|&raquo;first Fourier integral&laquo;]]&nbsp;  for signal &nbsp; $x_{\rm V}(t) = x(t-t_0)$&nbsp; shifted to the right by&nbsp; $t_0$&nbsp; is defined with the substitution&nbsp; $\tau = t - t_0$:
+The&nbsp;  [[Signal_Representation/The_Fourier_Transform_and_its_Inverse#The_first_Fourier_integral|&raquo;first Fourier integral&laquo;]]&nbsp;  for signal &nbsp; $x_{\rm V}(t) = x(t-t_0)$&nbsp; shifted to the right by&nbsp; $t_0$&nbsp; is defined with the substitution&nbsp; $\tau = t - t_0$:
 :$$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}

@@ Line 221: / Line 221: @@
 If we restrict ourselves to real time functions,&nbsp; the signs for&nbsp; &raquo;conjugated complex&raquo;&nbsp; can be omitted on both sides of the Fourier correspondence.}}
 {{BlaueBox|TEXT=
-$\text{Proof:}$&nbsp; The&nbsp; [[Signal_Representation/The_Fourier_Transform_and_Its_Inverse#The_first_Fourier_integral|&raquo;first Fourier integral&laquo;]]&nbsp; is after successive renaming  &nbsp;$t \to u$,&nbsp;  $f \to t$:
+$\text{Proof:}$&nbsp; The&nbsp; [[Signal_Representation/The_Fourier_Transform_and_its_Inverse#The_first_Fourier_integral|&raquo;first Fourier integral&laquo;]]&nbsp; is after successive renaming  &nbsp;$t \to u$,&nbsp;  $f \to t$:
 :$$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}

@@ Line 221: / Line 221: @@
 If we restrict ourselves to real time functions,&nbsp; the signs for&nbsp; &raquo;conjugated complex&raquo;&nbsp; can be omitted on both sides of the Fourier correspondence.}}
 {{BlaueBox|TEXT=
-$\text{Proof:}$&nbsp; The&nbsp; [[Signal_Representation/Fourier_Transform_and_Its_Inverse#The_first_Fourier_integral|&raquo;first Fourier integral&laquo;]]&nbsp; is after successive renaming  &nbsp;$t \to u$,&nbsp;  $f \to t$:
+$\text{Proof:}$&nbsp; The&nbsp; [[Signal_Representation/The_Fourier_Transform_and_Its_Inverse#The_first_Fourier_integral|&raquo;first Fourier integral&laquo;]]&nbsp; is after successive renaming  &nbsp;$t \to u$,&nbsp;  $f \to t$:
 :$$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}

@@ Line 132: / Line 132: @@
 We consider two rectangles of equal height, where&nbsp; $T_2 = T_1/2$&nbsp; holds.
-*The spectral function of&nbsp; $x_1(t)$&nbsp; results after the&nbsp; [[Signal_Representation/Fourier_Transform_and_Its_Inverse#The_first_Fourier_integral|&raquo;first Fourier Integral&laquo;]]&nbsp; to
+*The spectral function of&nbsp; $x_1(t)$&nbsp; results after the&nbsp; [[Signal_Representation/The_Fourier_Transform_and_its_Inverse#The_first_Fourier_integral|&raquo;first Fourier Integral&laquo;]]&nbsp; to
 :$$X_1 (f) = A  \cdot \frac{ {1 - {\rm e}^{ - {\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}fT_1 } } }{ { {\rm j}2\pi f} } .$$

@@ Line 22: / Line 22: @@
 :This relation can be used for simplification by omitting the constant&nbsp; $k$&nbsp; $($which can be a gain,&nbsp; an attenuation  or a unit factor$)$&nbsp; and adding it to the result later.
-:The above sentence follows directly from the definition of the&nbsp; [[Signal_Representation/Fourier_Transform_and_Its_Inverse#The_first_Fourier_integral|&raquo;first Fourier integral&laquo;]],&nbsp; as well as from the&nbsp; &raquo;addition theorem&laquo;,&nbsp; which formulates the foundation of the&nbsp; &raquo;superposition principle&laquo;.
+:The above sentence follows directly from the definition of the&nbsp; [[Signal_Representation/The_Fourier_Transform_and_its_Inverse#The_first_Fourier_integral|&raquo;first Fourier integral&laquo;]],&nbsp; as well as from the&nbsp; &raquo;addition theorem&laquo;,&nbsp; which formulates the foundation of the&nbsp; &raquo;superposition principle&laquo;.
 {{BlaueBox|TEXT=

@@ Line 269: / Line 269: @@
 ==Shifting Theorem==
 <br>
-We now consider a shift of the time function &ndash; e.g. caused by a delay &ndash; or a frequency shift, as it occurs for example with (analog)&nbsp; [[Modulation_Methods/Double-Sideband_Amplitude_Modulation#Description_in_the_frequency_domain|"double-sideband amplitude modulation"]].
+We now consider
 {{BlaueBox|TEXT=
-$\text{Shifting Theorem:}$&nbsp; If&nbsp; $X(f)$&nbsp; is the Fourier transform of &nbsp; $x(t)$, the following correlations also apply:
+$\text{Shifting Theorem:}$&nbsp; If&nbsp; $X(f)$&nbsp; is the Fourier transform of&nbsp; $x(t)$,&nbsp; the following correspondences also apply:
 $$\mathbf{(1)}\hspace{0.5cm}x( {t - t_0 } )\hspace{0.15cm}\circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\hspace{0.15cm}X( f ) \cdot {\rm e}^{ - {\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}2\pi \hspace{0.05cm}\cdot \hspace{0.05cm} f\hspace{0.05cm}\cdot \hspace{0.05cm}t_0 },$$
@@ Line 278: / Line 282: @@
 $$\mathbf{(2)}\hspace{0.5cm}x( t ) \cdot {\rm e}^{  {\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_0 \hspace{0.05cm}\cdot \hspace{0.05cm}t}\hspace{0.15cm}\circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\hspace{0.15cm}X( {f - f_0 } ).$$
-Here&nbsp; $t_0$&nbsp; and&nbsp; $f_0$&nbsp; are any time or frequency values.}}
+Here,&nbsp; $t_0$&nbsp; and&nbsp; $f_0$&nbsp; are any time or frequency values.}}
 {{BlaueBox|TEXT=
 $\text{Proof of Equation (1):}$&nbsp;
-The&nbsp;  [[Signal_Representation/Fourier_Transform_and_Its_Inverse#The_first_Fourier_integral|$\text{first Fourier integral}$]]&nbsp;  for signal &nbsp; $x_{\rm V}(t) = x(t-t_0)$&nbsp; shifted to the right by &nbsp; $t_0$&nbsp; is defined with the substitution &nbsp; $\tau = t - t_0$:
+The&nbsp;  [[Signal_Representation/Fourier_Transform_and_Its_Inverse#The_first_Fourier_integral|&raquo;first Fourier integral&laquo;]]&nbsp;  for signal &nbsp; $x_{\rm V}(t) = x(t-t_0)$&nbsp; shifted to the right by&nbsp; $t_0$&nbsp; is defined with the substitution&nbsp; $\tau = t - t_0$:
 :$$X_{\rm V} ( f ) = \int_{ - \infty }^{ + \infty } {x( {t - t_0 } ) \cdot {\rm e}^{ - {\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f\hspace{0.05cm}\cdot \hspace{0.05cm}t}\hspace{0.1cm}{\rm d}t}
 = \int_{ - \infty }^{ + \infty } {x( \tau  ) \cdot {\rm e}^{ - {\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f\hspace{0.05cm}\cdot \hspace{0.05cm}( {\tau  + t_0 } )}\hspace{0.1cm} {\rm d}\tau .}$$
-The term independent from the integration variable&nbsp; $\tau$&nbsp; can be dragged in front of the integral.&nbsp; With the renaming&nbsp; $\tau \to t$&nbsp; one then obtains
+*The term independent from the integration variable&nbsp; $\tau$&nbsp; can be dragged in front of the integral.&nbsp;
 :$$X_{\rm V}( f ) = {\rm e}^{ - {\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f\hspace{0.05cm}\cdot \hspace{0.05cm}t_0 }  \cdot \int_{ - \infty }^{ + \infty } {x( t )}  \cdot {\rm e}^{ - {\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f\hspace{0.05cm}\cdot \hspace{0.05cm}t}\hspace{0.1cm} {\rm d}t  = {\rm e}^{ - {\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}2 \pi \hspace{0.05cm}\cdot \hspace{0.05cm}f\hspace{0.05cm}\cdot \hspace{0.05cm}t_0 }  \cdot X( f).$$
@@ Line 294: / Line 300: @@
-[[File:P_ID478__Sig_T_3_3_S6_neu.png|right|frame|Example for Shifting Theorem]]
+{{GraueBox|TEXT=
-{{GraueBox|TEXT=
+[[File:P_ID478__Sig_T_3_3_S6_neu.png|right|frame|Shifting theorem example]]
 $\text{Example 6:}$&nbsp; As already mentioned, the symmetrical rectangular pulse&nbsp; $x_1(t)$&nbsp; has the spectrum
 :$$X_1 ( f ) = A  \cdot T \cdot {\rm si}( {\pi fT} )= A  \cdot T \cdot {\rm sinc}( {fT} )$$
-:$$\hspace{0.5cm} \text{with} \hspace{0.5cm} {\rm si}(x)= \sin(x)/x,\hspace{0.5cm}{\rm sinc}(x)= \sin(\pi x)/(\pi x)={\rm si}(x/\pi).$$
+:$$\hspace{0.9cm} \text{with} \hspace{0.5cm} {\rm si}(x)= \sin(x)/x,\hspace{0.5cm}{\rm sinc}(x)= \sin(\pi x)/(\pi x)={\rm si}(x/\pi).$$
-*The rectangular pulse&nbsp; $x_2(t)$&nbsp; displayed below is shifted to the right with respect to&nbsp; $x_1(t)$&nbsp; by&nbsp; $T/2$&nbsp;: &nbsp;
+*The rectangular pulse&nbsp; $x_2(t)$&nbsp; displayed below is shifted to the right with respect to&nbsp; $x_1(t)$&nbsp; by&nbsp; $T/2$: &nbsp;
 :$$x_2(t) = x_1(t-T/2).$$
 *Thus its spectrum is:
@@ Line 307: / Line 313: @@
 :$$X_2( f ) = A  \cdot T \cdot {\rm sinc}( {fT} ) \cdot {\rm e}^{ - {\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f\hspace{0.05cm}\cdot \hspace{0.05cm}T} .$$
-*This spectral function can also be written as follows with the&nbsp; [[Signal_Representation/Calculating_with_Complex_Numbers#Representation_by_magnitude_and_phase|$\text{theorem of Euler}$]]&nbsp; and some trigonometric conversions:
+*This spectral function can also be written as follows with the&nbsp; [[Signal_Representation/Calculating_with_Complex_Numbers#Representation_by_magnitude_and_phase|&raquo;theorem of Euler&laquo;]]&nbsp; and some trigonometric conversions:
 :$$X_2( f ) =  \frac{A }{2\pi f} \cdot \sin ( 2\pi fT) +  {\rm j}\cdot  \frac{A }{2\pi f} \cdot \big[ {\cos ( {2\pi fT} ) - 1} \big] .$$
-*The same result can be obtained with the&nbsp; [[Signal_Representation/Fourier_Transform_Theorems#Assignment_Theorem|$\text{assignment theorem}$]] &nbsp; <br>&rArr; &nbsp; The real part of the spectrum belongs to the even signal part&nbsp; $x_{\rm g}(t)$, the imaginary part to the odd part&nbsp; $x_{\rm u}(t)$.}}
+*The same result can be obtained with the&nbsp; [[Signal_Representation/Fourier_Transform_Theorems#Assignment_Theorem|&raquo;assignment theorem&laquo;]]; &nbsp; <br>&rArr; &nbsp; The real part of the spectrum belongs to the even signal part&nbsp; $x_{\rm g}(t)$,&nbsp; the imaginary part to the odd signal part&nbsp; $x_{\rm u}(t)$.}}

@@ Line 216: / Line 216: @@
 {{BlaueBox|TEXT=
-$\text{Duality Theorem:}$&nbsp; If&nbsp; $X(f)$&nbsp; is the Fourier transform of&nbsp; $x(t)$, then:
+$\text{Duality Theorem:}$&nbsp; If&nbsp; $X(f)$&nbsp; is the Fourier transform of&nbsp; $x(t)$,&nbsp; then:
 :$$X^{\star}(t)\hspace{0.15cm}\circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\hspace{0.15cm}x^{\star}( f ).$$
-If we restrict ourselves to real time functions, the signs for&nbsp; "conjugated complex"&nbsp; can be omitted on both sides of the Fourier correspondence.}}
+If we restrict ourselves to real time functions,&nbsp; the signs for&nbsp; &raquo;conjugated complex&raquo;&nbsp; can be omitted on both sides of the Fourier correspondence.}}
 {{BlaueBox|TEXT=
-$\text{Proof:}$&nbsp; The&nbsp; [[Signal_Representation/Fourier_Transform_and_Its_Inverse#The_first_Fourier_integral|$\text{first Fourier integral}$]]&nbsp; is after successive renaming  &nbsp;$t \to u$,&nbsp; respectively&nbsp; $f \to t$:
+$\text{Proof:}$&nbsp; The&nbsp; [[Signal_Representation/Fourier_Transform_and_Its_Inverse#The_first_Fourier_integral|&raquo;first Fourier integral&laquo;]]&nbsp; is after successive renaming  &nbsp;$t \to u$,&nbsp;  $f \to t$:
 :$$X( f ) = \int_{ - \infty }^{ + \infty } {x( u )}  \cdot {\rm e}^{ - {\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f\hspace{0.05cm}\cdot \hspace{0.05cm}u} \hspace{0.1cm}{\rm d}u, \hspace{1cm}
 X(t ) = \int_{ - \infty }^{ + \infty } {x( u )}  \cdot {\rm e}^{ - {\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}t\hspace{0.05cm}\cdot \hspace{0.05cm}u}\hspace{0.1cm} {\rm d}u.$$
-*If you change the sign in the exponent, you have to replace&nbsp; $X(t)$&nbsp; by&nbsp; $X^*(t)$&nbsp; and&nbsp; $x(u)$&nbsp; by&nbsp; $x^*(u)$&nbsp;:
+*If you change the sign in the exponent,&nbsp; you have to replace&nbsp; $X(t)$&nbsp; by&nbsp; $X^*(t)$&nbsp; and&nbsp; $x(u)$&nbsp; by&nbsp; $x^*(u)$&nbsp;:
 :$$X^{\star}(t ) = \int_{ - \infty }^{ + \infty } {x^{\star}( u )}  \cdot {\rm e}^{ {\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}2\pi\hspace{0.05cm}\cdot \hspace{0.05cm} t\hspace{0.05cm}\cdot \hspace{0.05cm}u}\hspace{0.1cm} {\rm d}u.$$
-*With the further renaming &nbsp;$u \to f$&nbsp; one gets to the [[Signal_Representation/Fourier_Transform_and_Its_Inverse#The_second_Fourier_integral|$\text{second Fourier Integral}$]]:
+*With the further renaming &nbsp;$u \to f$&nbsp; one gets to the [[Signal_Representation/Fourier_Transform_and_Its_Inverse#The_second_Fourier_integral|&raquo;second Fourier integral&laquo;]]:
 :$$X^{\star}(t ) = \int_{ - \infty }^{ + \infty } {x^{\star}( f )}  \cdot {\rm e}^{ {\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f\hspace{0.05cm}\cdot \hspace{0.05cm}t}\hspace{0.1cm} {\rm d}f.\hspace{7.9cm}$$
@@ Line 240: / Line 240: @@
 [[File:P_ID475__Sig_T_3_3_S5_neu.png|right|frame|$\text{Rectangular-in-time}$ &nbsp; &rArr;  $\text{sinc-in-frequency};$ <br>$\text{sinc-in-time}$ &nbsp; &rArr;  $\text{rectangular-in-frequency}$]]
 {{GraueBox|TEXT=
 $\text{Example 5:}$&nbsp;
-According to the&nbsp; "duality theorem", the spectral function of the Dirac delta  is therefore:
+The figure on the right shows an application of the duality theorem,&nbsp; namely the functional relations between
 :$$ x(t) = \delta(t)\hspace{0.15cm}\circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\hspace{0.15cm} X(f)= 1.$$
@@ Line 251: / Line 264: @@
-The figure on the right shows another application of the duality theorem, namely the functional relations between
+}}