Signal Representation/Special Cases of Pulses - Revision history

Guenter at 16:26, 15 November 2023

2023-11-15T16:26:34Z

Guenter at 16:22, 15 November 2023

2023-11-15T16:22:35Z

Guenter at 12:52, 14 June 2023

2023-06-14T12:52:31Z

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Hwang at 11:47, 23 January 2023

2023-01-23T11:47:30Z

Guenter at 14:56, 10 December 2022

2022-12-10T14:56:33Z

Hwang at 14:47, 1 December 2022

2022-12-01T14:47:12Z

Hwang at 14:44, 1 December 2022

2022-12-01T14:44:36Z

Hwang at 14:43, 1 December 2022

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Hwang at 14:42, 1 December 2022

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Hwang at 12:25, 1 December 2022

2022-12-01T12:25:36Z

@@ Line 2: / Line 2: @@
 {{Header
 |Untermenü=Aperiodic Signals - Impulses
-|Vorherige Seite=The Fourier Transform and Its Inverse
+|Vorherige Seite=The Fourier Transform and its Inverse
 |Nächste Seite=Fourier Transform Theorems
 }}

@@ Line 2: / Line 2: @@
 {{Header
 |Untermenü=Aperiodic Signals - Impulses
-|Vorherige Seite=Fourier Transform and Its Inverse
+|Vorherige Seite=The Fourier Transform and Its Inverse
 |Nächste Seite=Fourier Transform Theorems
 }}

@@ Line 109: / Line 109: @@
 ==Dirac delta or impulse==
 <br>
-In the chapter&nbsp; [[Signal_Representation/Direct_Current_Signal_-_Limit_Case_of_a_Periodic_Signal#Dirac_.28delta.29_function_in_frequency_domain|"Periodic Signals"]]&nbsp; the&nbsp; "Dirac function"&nbsp; was already used to describe the spectrum of a DC signal or a harmonic oscillation.
+In the chapter&nbsp; [[Signal_Representation/Direct_Current_Signal_-_Limit_Case_of_a_Periodic_Signal#Dirac_.28delta.29_function_in_frequency_domain|"Periodic Signals"]]&nbsp; the&nbsp; "Dirac delta function"&nbsp; was already used to describe the spectrum of a DC signal or a harmonic oscillation.
 However, in communications engineering it is also common and extremely advantageous to describe and analyze short-term processes with the help of this mathematical function in the time domain.
@@ Line 128: / Line 128: @@
-The properties mentioned here are shown in the (German language) learning video:&nbsp; <br> &nbsp; &nbsp; &nbsp;[[Herleitung_und_Visualisierung_der_Diracfunktion_(Lernvideo)|"Herleitung und Visualisierung der Diracfunktion"]] &nbsp; &rArr; &nbsp; "Derivation and visualisation of the Dirac function".
+The properties mentioned here are shown in the (German language) learning video:&nbsp; <br> &nbsp; &nbsp; &nbsp;[[Herleitung_und_Visualisierung_der_Diracfunktion_(Lernvideo)|"Herleitung und Visualisierung der Diracfunktion"]] &nbsp; &rArr; &nbsp; "Derivation and visualisation of the Dirac delta function".
 [[File:P_ID561__Sig_T_3_2_S3b_neu.png|right|frame|On the significance of the Dirac delta]]

@@ Line 20: / Line 20: @@
 *Here the integration limits&nbsp; $\pm T/2$&nbsp; take into account that&nbsp; $x(t)$&nbsp; is identical to zero outside the interval from&nbsp; $-T/2$&nbsp; to&nbsp; $+T/2$.
 *The second integral disappears due to the odd integrand and you get
@@ Line 31: / Line 32: @@
 $\text{Definitions:}$&nbsp; For abbreviation we define the following functions:
 *$\text{sinc&ndash;function}$&nbsp; (predominantly used in Anglo-American literature) &nbsp; &nbsp; &rArr; &nbsp; &nbsp; ${\rm sinc}( x ) =  {\sin  (\pi  x) }/(\pi  x ),$
 *$\text{si&ndash;function}$&nbsp; or $\text{splitting function}$ &nbsp;(predominantly used in German literature) &nbsp; &nbsp; &rArr; &nbsp; &nbsp; ${\rm si}\left( x \right) = \sin \left( x \right)/x = {\rm sinc}(x/\pi ).$}}
@@ Line 38: / Line 40: @@
 As the graphic on the right shows,&nbsp; $X(f)$&nbsp; has the following properties:
 *The maximum is at frequency&nbsp; $f=0$&nbsp; and has the value&nbsp; $A \cdot T$&nbsp; &rArr; &nbsp; "area of the rectangle".
 *For the frequencies&nbsp; $f_n = n/T$&nbsp; with&nbsp; $n = ±1, ±2, ±3,\text{ ...} $&nbsp; has the spectrum zeroes:
@@ Line 55: / Line 58: @@
 This pulse is described by two parameters, namely
 *the pulse amplitude&nbsp; $A$,&nbsp; and
 *the equivalent pulse duration&nbsp; $\Delta t$.
@@ Line 65: / Line 69: @@
 The Gaussian pulse&nbsp; $x(t)$&nbsp; has the following properties $($see graphic in&nbsp; $\text{Example 1})$:
 *The time function is for all times from&nbsp; $-\infty$&nbsp; to&nbsp; $+\infty$&nbsp; existent and positive.
 *This means simultaneously:&nbsp; The absolute pulse duration is infinite.
 *With the above definition the pulse maximum&nbsp; $A$&nbsp; is at&nbsp; $t = 0$.
 *At&nbsp; $t = \pm \Delta t/2$&nbsp;, the pulse is decayed to &nbsp; $\text{e}^{-\pi/4} \approx 0.456$&nbsp; of the maximum.&nbsp; And at&nbsp; $t = \pm \Delta t$&nbsp;, the signal value is less than&nbsp; $3.5 \cdot 10^{-6} \cdot A$.
 *The  spectral function&nbsp; $X(f)$&nbsp; is also Gaussian and has the same properties as the Gaussian pulse&nbsp; $x(t)$:
@@ Line 86: / Line 94: @@
 This gives the following comparable parameters in the spectral range:
 * The maximum&nbsp; $X_0 = X(f=0) = A \cdot \Delta t = 10^{-12} \,\text{W/Hz}$,
 *the equivalent bandwidth&nbsp; $\Delta f = 1/\Delta t = 1 \,\text{GHz}$.
@@ Line 94: / Line 103: @@
 We would like to draw your attention to two interactive applets on this topic with which you can display the time and frequency domain representations of the Gaussian pulse, rectangular pulse, triangular pulse, trapezoidal pulse and cosine rolloff pulse or the comparable quantities of an LTI system parameterized:
 *[[Applets:Pulses_and_Spectra|"Pulses and Spectra"]],
 *[[Applets:Frequenzgang_und_Impulsantwort|"Frequency Response and Impulse Response"]].
@@ Line 111: / Line 121: @@
 and can be characterised as follows (see plot):
 *The Dirac delta is infinitely narrow &nbsp; &rArr; &nbsp; it holds&nbsp; $x(t)\equiv 0$&nbsp; for&nbsp; $t \neq 0$&nbsp; and at time&nbsp; $t = 0$&nbsp; the Dirac delta is infinitely high.
 *If&nbsp; $x(t)$&nbsp;describes a voltage curve, then the impulse weight&nbsp; $X_0$&nbsp; has the unit&nbsp; $\textrm{V} \cdot \textrm{s}$&nbsp; (i.e. the unit&nbsp; $\textrm{V}/\textrm{Hz}$&nbsp; of a spectral function), since&nbsp; $\delta (t)$&nbsp; itself has the unit&nbsp; $1/\textrm{s}$.
 *The Fourier transform of the Dirac delta includes all frequencies $f$&nbsp; equally: &nbsp;
 :$$X(f) = X_0 = \rm const.$$}}
@@ Line 124: / Line 136: @@
 This result can be interpreted as follows:&nbsp;
 *The&nbsp; "equivalent pulse durations"&nbsp; are the same in each case&nbsp; $(\Delta t = 1\, &micro;\text{s})$&nbsp; and this is much smaller than&nbsp; $1/f_{\rm G} = 100 \, &micro;\text{s}$.&nbsp; The actual pulse shape&nbsp; (rectangle or triangle)&nbsp; has only a minor influence on the output signal&nbsp; $y(t)$.&nbsp;
 *Both the rectangle&nbsp; $x_1(t)$&nbsp; and the triangle&nbsp; $x_2(t)$ - can be approximated by the Dirac delta&nbsp; $x_3(t)$.&nbsp; The impulse weight&nbsp; $X_0 = 6 - 10^{-6}\, \text{Vs}$&nbsp; must be equal to the pulse areas of&nbsp; $x_1(t)$&nbsp; resp.&nbsp; $x_2(t)$.
 *However, a sufficiently small cutoff frequency is required for this approximation.&nbsp;  This simplification would not be permitted with&nbsp; $f_{\rm G} = 10 \, \text{MHz}$ &nbsp; &rArr; &nbsp; $1/f_{\rm G} = 100 \, \text{ns}$.
 *Even if the Dirac delta is drawn with the same height as the other two pulses, it still has an infinite value at the time&nbsp; $t = 0$.&nbsp; For the Dirac delta the impulse area ("impulse weight") is always specified.&nbsp; This differs from the other pulse amplitudes already in the unit&nbsp; (e.g.&nbsp; "Vs"&nbsp; instead of&nbsp; "V").}}

@@ Line 101: / Line 101: @@
 In the chapter&nbsp; [[Signal_Representation/Direct_Current_Signal_-_Limit_Case_of_a_Periodic_Signal#Dirac_.28delta.29_function_in_frequency_domain|"Periodic Signals"]]&nbsp; the&nbsp; "Dirac function"&nbsp; was already used to describe the spectrum of a DC signal or a harmonic oscillation.
-However, in Communications Engineering it is also common and extremely advantageous to describe and analyze short-term processes with the help of this mathematical function in the time domain.
+However, in communications engineering it is also common and extremely advantageous to describe and analyze short-term processes with the help of this mathematical function in the time domain.
 [[File:Sig_T_3_2_S3_version3.png|right|frame|Dirac delta and corresponding spectrum]]

@@ Line 73: / Line 73: @@
 :$$X(f) = A  \cdot \Delta t \cdot {\rm e}^{ - \pi \left( {f \hspace{0.05cm} \cdot \hspace{0.05cm} \Delta t} \right)^2 }.$$
-In the section&nbsp; [[Signal_Representation/Fourier_Transform_Laws#Reziprozit.C3.A4tsgesetz_von_Zeitdauer_und_Bandbreite|"Reciprocity Theorem"]]&nbsp;  the analogies of time domain and frequency domain of the Gaussian pulse are discussed in detail.
+In the section&nbsp; [[Signal_Representation/Fourier_Transform_Theorems#Reciprocity_Theorem_of_time_duration_and_bandwidth|"Reciprocity Theorem"]]&nbsp;  the analogies of time domain and frequency domain of the Gaussian pulse are discussed in detail.
 The following example illustrates the similarities and differences between the Gaussian pulse&nbsp; $x(t)$&nbsp; and its spectrum&nbsp; $X(f)$.

@@ Line 15: / Line 15: @@
 $A$&nbsp; denotes the amplitude of the pulse and&nbsp; $T$&nbsp; the pulse duration.
-The corresponding spectral function&nbsp; $X(f)$&nbsp; is obtained by using the&nbsp; [[Signal_Representation/Fourier_Transform_and_Its_Inverse#Das_erste_Fourierintegral|$\text{first Fourier integral}$]]:
+The corresponding spectral function&nbsp; $X(f)$&nbsp; is obtained by using the&nbsp; [[Signal_Representation/Fourier_Transform_and_its_Inverse#The_first_Fourier_integral|$\text{first Fourier integral}$]]:
 :$$X(f) = \int_{ - T/2}^{+T/2} A  \cdot {\rm e}^{ -{\rm j}2\pi  ft}\, {\rm d}t ,$$
 :$$\Rightarrow \hspace{0.3cm} X(f) =  A \cdot  \int_{ - T/2}^{+T/2} {\cos ( {2\pi}ft )\,{\rm d}t - {\rm j} \cdot A} \int_{ - T/2}^{+T/2} {\sin ( {2\pi ft} )}\,{\rm d}t .$$