Aufgaben:Exercise 3.4: Trapezoidal Spectrum and Pulse - Revision history

Maintenance script: Fix interlanguage link: resolve redirect chain

2026-03-16T15:56:42Z

Fix interlanguage link: resolve redirect chain

← Older revision		Revision as of 17:56, 16 March 2026
Line 8:		Line 8:

	Instead of the corner frequencies  $f_1$  and  $f_2$ , the following two descriptive variables can also be used:		Instead of the corner frequencies  $f_1$  and  $f_2$ , the following two descriptive variables can also be used:
	*the  [[Signal_Representation/~~Fourier_Transform_Theorems~~#Reciprocity_Theorem_of_time_duration_and_bandwidth\|equivalent bandwidth]]:		*the  [[Signal_Representation/The_Fourier_Transform_Theorems#Reciprocity_Theorem_of_time_duration_and_bandwidth\|equivalent bandwidth]]:
	:$$\Delta f = f_1 + f_2,$$		:$$\Delta f = f_1 + f_2,$$
	*the so-called  [[Linear_and_Time_Invariant_Systems/~~Einige_systemtheoretische_Tiefpassfunktionen~~#~~Trapez.E2.80.93Tiefpass~~\|rolloff factor]]  (in the frequency domain):		*the so-called  [[Linear_and_Time_Invariant_Systems/Some_Low-Pass_Functions_in_Systems_Theory#Trapezoidal_low-pass_filter\|rolloff factor]]  (in the frequency domain):
	:$$r_{\hspace{-0.05cm}f} = \frac{ {f_2 - f_1 }}{ {f_2 + f_1 }}.$$		:$$r_{\hspace{-0.05cm}f} = \frac{ {f_2 - f_1 }}{ {f_2 + f_1 }}.$$

Line 25:		Line 25:
	The following can be used here as descriptive variables:		The following can be used here as descriptive variables:
	*the pulse amplitude  $y_0 = y(t = 0)$,		*the pulse amplitude  $y_0 = y(t = 0)$,
	*the  [[Signal_Representation/~~Fourier_Transform_Theorems~~#Reciprocity_Theorem_of_time_duration_and_bandwidth\|equivalent pulse duration]]  (defined via the rectangle–in–time with the same area):		*the  [[Signal_Representation/The_Fourier_Transform_Theorems#Reciprocity_Theorem_of_time_duration_and_bandwidth\|equivalent pulse duration]]  (defined via the rectangle–in–time with the same area):

	:$$\Delta t = t_1 + t_2,$$		:$$\Delta t = t_1 + t_2,$$
Line 42:		Line 42:
	''Hints:''		''Hints:''
	*This exercise belongs to the chapter  [[Signal_Representation/Fourier_Transform_Theorems\|Fourier Transform Theorems]].		*This exercise belongs to the chapter  [[Signal_Representation/Fourier_Transform_Theorems\|Fourier Transform Theorems]].
	*Use the  [[Signal_Representation/~~Fourier_Transform_Theorems~~#Duality_Theorem\|Duality Theorem]]  and the  [[Signal_Representation/~~Fourier_Transform_Theorems~~#Similarity_Theorem\|Similarity Theorem]].		*Use the  [[Signal_Representation/The_Fourier_Transform_Theorems#Duality_Theorem\|Duality Theorem]]  and the  [[Signal_Representation/The_Fourier_Transform_Theorems#Similarity_Theorem\|Similarity Theorem]].

	*You can check your results using the two interactive applets		*You can check your results using the two interactive applets
Line 133:		Line 133:
	__NOEDITSECTION__		__NOEDITSECTION__
	[[Category:Signal Representation: Exercises\|^3.3 Fourier Transform Theorems^]]		[[Category:Signal Representation: Exercises\|^3.3 Fourier Transform Theorems^]]
	[[de:Aufgaben:3.~~4_Trapezspektrum_bzw~~._-impuls]]		[[de:Aufgaben:Aufgabe 3.4: Trapezspektrum bzw. -impuls]]

Maintenance script: Add German interlanguage link

2026-03-16T12:32:26Z

Add German interlanguage link

← Older revision		Revision as of 14:32, 16 March 2026
Line 133:		Line 133:
	__NOEDITSECTION__		__NOEDITSECTION__
	[[Category:Signal Representation: Exercises\|^3.3 Fourier Transform Theorems^]]		[[Category:Signal Representation: Exercises\|^3.3 Fourier Transform Theorems^]]
			[[de:Aufgaben:3.4_Trapezspektrum_bzw._-impuls]]

Guenter at 13:17, 24 May 2021

2021-05-24T13:17:21Z

← Older revision		Revision as of 15:17, 24 May 2021
Line 3:		Line 3:
	}}		}}

	[[File:P_ID508__Sig_A_3_4.png\|250px\|right\|frame\|~~Trapezspektrum~~ & ~~Trapezimpuls~~]]		[[File:P_ID508__Sig_A_3_4.png\|250px\|right\|frame\|Trapezoidal spectrum & trapezoidal pulse]]

	We consider here a trapezoidal spectral function  $X(f)$  according to the upper graph, which is completely described by the three parameters  $X_0$,  $f_1$  and  $f_2$.  For the two corner frequencies,  $f_2 > 0$  and  $0 \leq f_1 \leq f_2$ always apply.		We consider here a trapezoidal spectral function  $X(f)$  according to the upper graph, which is completely described by the three parameters  $X_0$,  $f_1$  and  $f_2$.  For the two corner frequencies,  $f_2 > 0$  and  $0 \leq f_1 \leq f_2$ always apply.

Guenter at 14:27, 26 April 2021

2021-04-26T14:27:36Z

Guenter at 14:19, 26 April 2021

2021-04-26T14:19:35Z

← Older revision		Revision as of 16:19, 26 April 2021
Line 11:		Line 11:
	:$$\Delta f = f_1 + f_2,$$		:$$\Delta f = f_1 + f_2,$$
	*the so-called  [[Linear_and_Time_Invariant_Systems/Einige_systemtheoretische_Tiefpassfunktionen#Trapez.E2.80.93Tiefpass\|rolloff factor]]  (in the frequency domain):		*the so-called  [[Linear_and_Time_Invariant_Systems/Einige_systemtheoretische_Tiefpassfunktionen#Trapez.E2.80.93Tiefpass\|rolloff factor]]  (in the frequency domain):
	:$$~~r_f~~ = \frac{ {f_2 - f_1 }}{ {f_2 + f_1 }}.$$		:$$r_{\hspace{-0.05cm}f} = \frac{ {f_2 - f_1 }}{ {f_2 + f_1 }}.$$

	With these quantities, the associated time function (see middle graph) is:		With these quantities, the associated time function (see middle graph) is:

	:$$x( t ) = X_0 \cdot \Delta f \cdot {\mathop{\rm si}\nolimits} ( { {\rm \pi} \cdot \Delta f \cdot t} ) \cdot {\mathop{\rm si}\nolimits} ( { {\rm \pi} \cdot ~~r_f~~ \cdot \Delta f\cdot t} ).$$		:$$x( t ) = X_0 \cdot \Delta f \cdot {\mathop{\rm si}\nolimits} ( { {\rm \pi} \cdot \Delta f \cdot t} ) \cdot {\mathop{\rm si}\nolimits} ( { {\rm \pi} \cdot r_{\hspace{-0.05cm}f} \cdot \Delta f\cdot t} ).$$

	Here  $\text{si}(x) = \text{sin}(x)/x$  is the so-called  "splitting function".		Here  $\text{si}(x) = \text{sin}(x)/x$  is the so-called  "splitting function".
Line 31:		Line 31:
	*the rolloff factor (in the time domain) with comparable definition as  $r_f$:		*the rolloff factor (in the time domain) with comparable definition as  $r_f$:

	:$$~~r_t~~ = \frac{ {t_2 - t_1 }}{ {t_2 + t_1 }}.$$		:$$r_{\hspace{-0.05cm}t} = \frac{ {t_2 - t_1 }}{ {t_2 + t_1 }}.$$

	Let be  $y_0 = 4\,\text{V}$,  $\Delta t = 1\,\text{ms}$  and  $r_t = 0.5$.		Let be  $y_0 = 4\,\text{V}$,  $\Delta t = 1\,\text{ms}$  and  $r_t = 0.5$.
Line 59:		Line 59:
	\|type="{}"}		\|type="{}"}
	$\Delta f \ = \ $ { 4 3% }  $\text{kHz}$		$\Delta f \ = \ $ { 4 3% }  $\text{kHz}$
	$~~r_f~~ \hspace{0.35cm} = \ $ { 0.5 3% }		$r_{\hspace{-0.05cm}f} \hspace{0.35cm} = \ $ { 0.5 3% }

	{What are the signal values of  $x(t)$  at  $t = 0$,  $t = T$  and  $t = T/2$?		{What are the signal values of  $x(t)$  at  $t = 0$,  $t = T$  and  $t = T/2$?

Guenter at 14:15, 26 April 2021

2021-04-26T14:15:35Z

← Older revision		Revision as of 16:15, 26 April 2021
Line 45:		Line 45:

	*You can check your results using the two interactive applets		*You can check your results using the two interactive applets
	::[[Applets:Pulses_and_Spectra\|Pulses and Spectra]],		:[[Applets:Pulses_and_Spectra\|Pulses and Spectra]],
	::[[Applets:Frequency_%26_Impulse_Responses\|Frequency & Impulse Responses]].		:[[Applets:Frequency_%26_Impulse_Responses\|Frequency & Impulse Responses]].


Line 56:		Line 56:
	<quiz display=simple>		<quiz display=simple>

	{~~For the given parameters, what is~~ the equivalent bandwidth and rolloff factor of the spectrum  $X(f)$?		{What are the equivalent bandwidth and the rolloff factor of the spectrum  $X(f)$  for the given parameters?
	\|type="{}"}		\|type="{}"}
	$\Delta f \ = \ $ { 4 3% }  $\text{kHz}$		$\Delta f \ = \ $ { 4 3% }  $\text{kHz}$
Line 67:		Line 67:
	$x(t=T/2)\ = \ $ { 2.293 3% }  $\text{V}$		$x(t=T/2)\ = \ $ { 2.293 3% }  $\text{V}$

	{What is the spectrum  $Y(f)$  of the trapezoidal pulse ~~witht~~  $y_0 = 4\,\text{V}$,  $\Delta t = 1\,\text{ms}$  and  $r_t = 0.5$? <br>What are the spectral values at the given frequencies?		{What is the spectrum  $Y(f)$  of the trapezoidal pulse with  $y_0 = 4\,\text{V}$,  $\Delta t = 1\,\text{ms}$  and  $r_t = 0.5$? <br>What are the spectral values at the given frequencies?
	\|type="{}"}		\|type="{}"}
	$Y(f = 0)\hspace{0.2cm} = \ $ { 4 3% }  $\text{mV/Hz}$		$Y(f = 0)\hspace{0.2cm} = \ $ { 4 3% }  $\text{mV/Hz}$

Guenter at 13:37, 26 April 2021

2021-04-26T13:37:40Z

← Older revision		Revision as of 15:37, 26 April 2021
Line 45:		Line 45:

	*You can check your results using the two interactive applets		*You can check your results using the two interactive applets
	::[[Applets:~~Impulse_und_Spektren~~\|~~Impulses~~ and Spectra]],		::[[Applets:Pulses_and_Spectra\|Pulses and Spectra]],
	::[[Applets:Frequency_%26_Impulse_Responses\|Frequency & Impulse Responses]].		::[[Applets:Frequency_%26_Impulse_Responses\|Frequency & Impulse Responses]].

Guenter at 13:34, 26 April 2021

2021-04-26T13:34:31Z

← Older revision		Revision as of 15:34, 26 April 2021
Line 5:		Line 5:
	[[File:P_ID508__Sig_A_3_4.png\|250px\|right\|frame\|Trapezspektrum & Trapezimpuls]]		[[File:P_ID508__Sig_A_3_4.png\|250px\|right\|frame\|Trapezspektrum & Trapezimpuls]]

	We consider here a trapezoidal spectral function  $X(f)$  according to the upper graph, which is completely described by the three parameters  $X_0$,  $f_1$  and  $f_2$ . For the two corner frequencies,  $f_2 > 0$  and  $0 \leq f_1 \leq f_2$ always apply.		We consider here a trapezoidal spectral function  $X(f)$  according to the upper graph, which is completely described by the three parameters  $X_0$,  $f_1$  and  $f_2$.  For the two corner frequencies,  $f_2 > 0$  and  $0 \leq f_1 \leq f_2$ always apply.

	Instead of the corner frequencies  $f_1$  and  $f_2$ , the following two descriptive variables can also be used:		Instead of the corner frequencies  $f_1$  and  $f_2$ , the following two descriptive variables can also be used:
	*the  [[Signal_Representation/~~Fourier_Transform_Laws~~#~~Reciprocity_Theorem_of_Time_duration_and_Bandwidth~~\|equivalent bandwidth]]:		*the  [[Signal_Representation/Fourier_Transform_Theorems#Reciprocity_Theorem_of_time_duration_and_bandwidth\|equivalent bandwidth]]:
	:$$\Delta f = f_1 + f_2,$$		:$$\Delta f = f_1 + f_2,$$
	*the so-called  [[Linear_and_Time_Invariant_Systems/Einige_systemtheoretische_Tiefpassfunktionen#Trapez.E2.80.93Tiefpass\|rolloff factor]]  (in the frequency domain):		*the so-called  [[Linear_and_Time_Invariant_Systems/Einige_systemtheoretische_Tiefpassfunktionen#Trapez.E2.80.93Tiefpass\|rolloff factor]]  (in the frequency domain):
	:$$r_f = \frac{ {f_2 - f_1 }}{ {f_2 + f_1 }}.$$		:$$r_f = \frac{ {f_2 - f_1 }}{ {f_2 + f_1 }}.$$

	With these quantities, the associated time function (see middle graph) is:-		With these quantities, the associated time function (see middle graph) is:

	:$$x( t ) = X_0 \cdot \Delta f \cdot {\mathop{\rm si}\nolimits} ( { {\rm \pi} \cdot \Delta f \cdot t} ) \cdot {\mathop{\rm si}\nolimits} ( { {\rm \pi} \cdot r_f \cdot \Delta f\cdot t} ).$$		:$$x( t ) = X_0 \cdot \Delta f \cdot {\mathop{\rm si}\nolimits} ( { {\rm \pi} \cdot \Delta f \cdot t} ) \cdot {\mathop{\rm si}\nolimits} ( { {\rm \pi} \cdot r_f \cdot \Delta f\cdot t} ).$$

	Here  $\text{si}(x) = \text{sin}(x)/x$  is the so-called splitting function.		Here  $\text{si}(x) = \text{sin}(x)/x$  is the so-called  "splitting function".

	In this example, the numerical values  $X_0 = 10^{–3}\,\text{V/Hz}$,  $f_1 = 1\,\text{kHz}$  and  $f_2 = 3\,\text{kHz}$  are to be used. The time  $T = 1/\Delta f$  is only used for ~~standardisation purposes~~.		In this example, the numerical values  $X_0 = 10^{–3}\,\text{V/Hz}$,  $f_1 = 1\,\text{kHz}$  and  $f_2 = 3\,\text{kHz}$  are to be used.  The time  $T = 1/\Delta f$  is only used for normative purpose.
	~~From~~ subtask  '''(3)'''  a trapezoidal signal  $y(t)$  is considered, which is identical in shape to the spectrum  $X(f)$~~ ~~.
			In the subtask  '''(3)'''  a trapezoidal signal  $y(t)$  is considered, which is identical in shape to the spectrum  $X(f)$.

	The following can be used here as descriptive variables:		The following can be used here as descriptive variables:
	*the pulse amplitude  $y_0 = y(t = 0)$,		*the pulse amplitude  $y_0 = y(t = 0)$,
	*the  [[Signal_Representation/~~Fourier_Transform_Laws~~#~~Reciprocity_Theorem_of_Time_duration_and_Bandwidth~~\|equivalent pulse duration]]  (defined via the rectangle with the same area):		*the  [[Signal_Representation/Fourier_Transform_Theorems#Reciprocity_Theorem_of_time_duration_and_bandwidth\|equivalent pulse duration]]  (defined via the rectangle–in–time with the same area):

	:$$\Delta t = t_1 + t_2,$$		:$$\Delta t = t_1 + t_2,$$
Line 32:		Line 33:
	:$$r_t = \frac{ {t_2 - t_1 }}{ {t_2 + t_1 }}.$$		:$$r_t = \frac{ {t_2 - t_1 }}{ {t_2 + t_1 }}.$$

	Let  $y_0 = 4\,\text{V}$,  $\Delta t = 1\,\text{ms}$  and  $r_t = 0.5$.		Let be  $y_0 = 4\,\text{V}$,  $\Delta t = 1\,\text{ms}$  and  $r_t = 0.5$.




Line 42:		Line 41:

	''Hints:''		''Hints:''
	*This exercise belongs to the chapter  [[Signal_Representation/~~Fourier_Transform_Laws~~\|Fourier Transform ~~Laws~~]].		*This exercise belongs to the chapter  [[Signal_Representation/Fourier_Transform_Theorems\|Fourier Transform Theorems]].
	*Use the  [[Signal_Representation/~~Fourier_Transform_Laws~~#~~Vertauschungssatz~~\|~~Vertauschungssatz~~]]  and the  [[Signal_Representation/~~Fourier_Transform_Laws~~#Similarity_Theorem\|Similarity Theorem]].		*Use the  [[Signal_Representation/Fourier_Transform_Theorems#Duality_Theorem\|Duality Theorem]]  and the  [[Signal_Representation/Fourier_Transform_Theorems#Similarity_Theorem\|Similarity Theorem]].

	*You can check your results using the two interactive applets   [[Applets:Impulse_und_Spektren\|Impulses and Spectra]]~~  sowie ~~ [[Applets:~~Frequenzgang_und_Impulsantwort~~\|~~Frequenzgang und Impulsantwort~~]]~~ ~~.		*You can check your results using the two interactive applets
			::[[Applets:Impulse_und_Spektren\|Impulses and Spectra]],
			::[[Applets:Frequency_%26_Impulse_Responses\|Frequency & Impulse Responses]].

Javier: Text replacement - "Category:Exercises for Signal Representation" to "Category:Signal Representation: Exercises"

2021-03-23T12:38:46Z

Text replacement - "Category:Exercises for Signal Representation" to "Category:Signal Representation: Exercises"

← Older revision		Revision as of 14:38, 23 March 2021
Line 131:		Line 131:

	__NOEDITSECTION__		__NOEDITSECTION__
	[[Category:~~Exercises for~~ Signal Representation\|^3.3 Fourier Transform Theorems^]]		[[Category:Signal Representation: Exercises\|^3.3 Fourier Transform Theorems^]]

Noah at 19:11, 23 January 2021

2021-01-23T19:11:58Z

← Older revision		Revision as of 21:11, 23 January 2021
Line 127:		Line 127:
	:$$Y( {f = 1\;{\rm{kHz}}} ) = Y_0 \cdot {\mathop{\rm si}\nolimits} ( {{ {\rm{\pi }}}/{2}} ) \cdot {\mathop{\rm si}\nolimits} ( {{ {\rm{\pi }}}/{4}} ) \hspace{0.15 cm}\underline{= 2.293\,{\rm{mV/Hz}}}{\rm{.}}$$		:$$Y( {f = 1\;{\rm{kHz}}} ) = Y_0 \cdot {\mathop{\rm si}\nolimits} ( {{ {\rm{\pi }}}/{2}} ) \cdot {\mathop{\rm si}\nolimits} ( {{ {\rm{\pi }}}/{4}} ) \hspace{0.15 cm}\underline{= 2.293\,{\rm{mV/Hz}}}{\rm{.}}$$

	*In subtask  '''(3)'''  this spectral value occurred at the frequency  $f = 0.5\,\rm{kHz}$  ~~aufgetreten~~.		*In subtask  '''(3)'''  this spectral value occurred at the frequency  $f = 0.5\,\rm{kHz}$ .
	{{ML-Fuß}}		{{ML-Fuß}}

	__NOEDITSECTION__		__NOEDITSECTION__
	[[Category:Exercises for Signal Representation\|^3.3 Fourier Transform Theorems^]]		[[Category:Exercises for Signal Representation\|^3.3 Fourier Transform Theorems^]]

← Older revision		Revision as of 16:27, 26 April 2021
Line 83:		Line 83:

	{{ML-Kopf}}		{{ML-Kopf}}
	'''(1)'''  The equivalent bandwidth is by definition equal to the width of the equal-area rectangle:		'''(1)'''  The equivalent bandwidth is (by definition) equal to the width of the equal-area rectangle:

	:$$\Delta f = f_1 + f_2 \hspace{0.15 cm}\underline{= 4\;{\rm{kHz}}}{\rm{.}}$$		:$$\Delta f = f_1 + f_2 \hspace{0.15 cm}\underline{= 4\;{\rm{kHz}}}{\rm{.}}$$
	*For the rolloff factor holds:		*For the rolloff factor holds:

	:$${ {~~r_f~~ = }}\frac{ {f_2 - f_1 }}{ {f_2 + f_1 }}\hspace{0.15 cm}\underline{ = 0.5}.$$		:$${ {r_{\hspace{-0.05cm}f} = }}\frac{ {f_2 - f_1 }}{ {f_2 + f_1 }}\hspace{0.15 cm}\underline{ = 0.5}.$$


Line 99:		Line 99:
	:$$x( {t = T} ) = x_0 \cdot {\mathop{\rm si}\nolimits} ( {\rm{\pi }} ) \cdot {\mathop{\rm si}\nolimits} ( { { {\rm{\pi }}}/{2}} )\hspace{0.15 cm}\underline{ = 0}.$$		:$$x( {t = T} ) = x_0 \cdot {\mathop{\rm si}\nolimits} ( {\rm{\pi }} ) \cdot {\mathop{\rm si}\nolimits} ( { { {\rm{\pi }}}/{2}} )\hspace{0.15 cm}\underline{ = 0}.$$

	*Also at all multiples of  $T$    $x(t)$  exhibits zero crossings. At time  $t = T/2$  holds:		*Also at all multiples of  $T$:    $x(t)$  exhibits zero crossings.  At time  $t = T/2$  holds:

	:$$x( {t = T/2} ) = x_0 \cdot {\mathop{\rm si}\nolimits} ( { { {\rm{\pi }}}/{2}} ) \cdot {\mathop{\rm si}\nolimits}( { { {\rm{\pi }}}/{4}} ) = x_0 \cdot \frac{ { 1 \cdot \sqrt 2 /2}}{ { {\rm{\pi /}}2 \cdot {\rm{\pi /4}}}} = x_0 \cdot \frac{ {4 \cdot \sqrt 2 }}{ { {\rm{\pi }}^{\rm{2}} }} \hspace{0.15 cm}\underline{= 2.293\;{\rm{V}}}{\rm{.}}$$		:$$x( {t = T/2} ) = x_0 \cdot {\mathop{\rm si}\nolimits} ( { { {\rm{\pi }}}/{2}} ) \cdot {\mathop{\rm si}\nolimits}( { { {\rm{\pi }}}/{4}} ) = x_0 \cdot \frac{ { 1 \cdot \sqrt 2 /2}}{ { {\rm{\pi /}}2 \cdot {\rm{\pi /4}}}} = x_0 \cdot \frac{ {4 \cdot \sqrt 2 }}{ { {\rm{\pi }}^{\rm{2}} }} \hspace{0.15 cm}\underline{= 2.293\;{\rm{V}}}{\rm{.}}$$
Line 108:		Line 108:
	:$$x( t ) = X_0 \cdot \Delta f \cdot {\mathop{\rm si}\nolimits} ( { {\rm{\pi }} \cdot \Delta f \cdot t} ) \cdot {\mathop{\rm si}\nolimits} ( { {\rm{\pi }} \cdot r_f \cdot \Delta f \cdot t} ).$$		:$$x( t ) = X_0 \cdot \Delta f \cdot {\mathop{\rm si}\nolimits} ( { {\rm{\pi }} \cdot \Delta f \cdot t} ) \cdot {\mathop{\rm si}\nolimits} ( { {\rm{\pi }} \cdot r_f \cdot \Delta f \cdot t} ).$$

	*Since both  $X(f)$  and  $x(t)$  are real and, moreover,  $y(t)$  is of the same form as  $X(f)$  , we obtain, taking into account all equivalences for the spectral function of the trapezoidal pulse:		*Since both  $X(f)$  and  $x(t)$  are real and, moreover,  $y(t)$  is of the same form as  $X(f),$  we obtain, taking into account all equivalences for the spectral function of the trapezoidal pulse:

	:$$Y( f ) = y_0 \cdot \Delta t \cdot {\mathop{\rm si}\nolimits} ( { {\rm{\pi }} \cdot \Delta t \cdot f} ) \cdot {\mathop{\rm si}\nolimits} ( { {\rm{\pi }} \cdot r_t \cdot \Delta t \cdot f} ).$$		:$$Y( f ) = y_0 \cdot \Delta t \cdot {\mathop{\rm si}\nolimits} ( { {\rm{\pi }} \cdot \Delta t \cdot f} ) \cdot {\mathop{\rm si}\nolimits} ( { {\rm{\pi }} \cdot r_t \cdot \Delta t \cdot f} ).$$