Difference between revisions of "Signal Representation/Special Cases of Pulses"

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{{Header
 
{{Header
 
|Untermenü=Aperiodic Signals - Impulses
 
|Untermenü=Aperiodic Signals - Impulses
|Vorherige Seite=Fourier Transform and Its Inverse
+
|Vorherige Seite=The Fourier Transform and its Inverse
 
|Nächste Seite=Fourier Transform Theorems
 
|Nächste Seite=Fourier Transform Theorems
 
}}
 
}}
  
==Rectangular Impulse==
+
==Rectangular pulse==
 
<br>
 
<br>
[[File:Sig_T_3_2_S1_version3.png|right|frame|Rectangle Pulse and Its Spectrum]]
+
[[File:Sig_T_3_2_S1_version3.png|right|frame|Rectangular pulse and its spectrum]]
One speaks of a&nbsp; '''rectangle-pulse'', if the following applies for the time domain:
+
One speaks of a&nbsp; &raquo;'''rectangular pulse'''&laquo;,&nbsp; if the following applies for the time domain:
  
 
:$$x(t) = \left\{ \begin{array}{l} \hspace{0.25cm}A  \\  A /2 \\ \hspace{0.25cm} 0 \\  \end{array} \right.\quad \quad \begin{array}{*{20}c}  {\rm{for}}  \\  {\rm{for}}  \\  {\rm{for}}  \\ \end{array}\begin{array}{*{20}c}  {\left| \hspace{0.05cm} t\hspace{0.05cm} \right| < T/2,}  \\  {\left| \hspace{0.05cm}t\hspace{0.05cm} \right| = T/2,}  \\  {\left|\hspace{0.05cm} t \hspace{0.05cm} \right| > T/2.}  \\ \end{array}$$
 
:$$x(t) = \left\{ \begin{array}{l} \hspace{0.25cm}A  \\  A /2 \\ \hspace{0.25cm} 0 \\  \end{array} \right.\quad \quad \begin{array}{*{20}c}  {\rm{for}}  \\  {\rm{for}}  \\  {\rm{for}}  \\ \end{array}\begin{array}{*{20}c}  {\left| \hspace{0.05cm} t\hspace{0.05cm} \right| < T/2,}  \\  {\left| \hspace{0.05cm}t\hspace{0.05cm} \right| = T/2,}  \\  {\left|\hspace{0.05cm} t \hspace{0.05cm} \right| > T/2.}  \\ \end{array}$$
 
   
 
   
&nbsp; $A$&nbsp; denotes the amplitude of the pulse and&nbsp; $T$&nbsp; the pulse duration.
+
$A$&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#The_first_Fourier_integral|&raquo;first Fourier integral&laquo;]]:
 +
:$$X(f) = \int_{ - T/2}^{+T/2} A  \cdot {\rm e}^{ -{\rm j}2\pi  ft}\, {\rm d}t ,$$
 +
:$$\Rightarrow \hspace{0.3cm} X(f) =  A \cdot  \int_{ - T/2}^{+T/2} {\cos ( {2\pi}ft )\,{\rm d}t - {\rm j} \cdot A} \int_{ - T/2}^{+T/2} {\sin ( {2\pi ft} )}\,{\rm d}t .$$
  
The corresponding spectral function&nbsp; $X(f)$&nbsp; is obtained by using the&nbsp; [[Signal_Representation/Fourier_Transform_and_Its_Inverse#Das_erste_Fourierintegral|first Fourier integral]]:
+
*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; $+T/2$.
 
:$$X(f) = \int_{ - T/2}^{+T/2} {A  \cdot {\rm e}^{ -{\rm j}2\pi  ft}\, {\rm d}t = A }\cdot  \int_{ - T/2}^{+T/2} {\cos ( {2\pi}ft )\,{\rm d}t - {\rm j} \cdot A} \int_{ - T/2}^{+T/2} {\sin ( {2\pi ft} )}\,{\rm d}t .$$
 
  
*Here the integration limits&nbsp; $\pm T/2$ take into account that&nbsp; $x(t)$&nbsp; is identical to zero outside the interval from&nbsp; $-T/2$&nbsp; to&nbsp; $+T/2$&nbsp; .
 
 
*The second integral disappears due to the odd integrand and you get
 
*The second integral disappears due to the odd integrand and you get
 
   
 
   
 
:$$X(f) = \frac{A  \cdot \sin \left( {\pi fT} \right)}{\pi f}.$$
 
:$$X(f) = \frac{A  \cdot \sin \left( {\pi fT} \right)}{\pi f}.$$
 +
 +
*By extending the numerator and denominator each with&nbsp; $T$&nbsp; one can also write for the spectral function of the rectangular pulse:
 +
 +
:$$X( f ) = A \cdot T \cdot {\rm si}\hspace{-0.08cm}\left( {\pi fT} \right)\hspace{1.3cm}\text{resp.}\hspace{1.3cm}\ X( f ) = A \cdot T \cdot {\rm sinc}\hspace{-0.08cm}\left( {fT} \right).$$
  
 
{{BlaueBox|TEXT=   
 
{{BlaueBox|TEXT=   
$\text{Definition:}$&nbsp;
+
$\text{Definitions:}$&nbsp; For abbreviation we define the following functions:
For abbreviation we define the following function and call it&nbsp; '''si-function'''&nbsp; or also as&nbsp; '''splitting function''':
+
*&raquo;'''sinc&ndash;function&laquo;'''&nbsp; $($predominantly used in Anglo-American literature$)$
 +
:$${\rm sinc}( x ) =  {\sin  (\pi  x) }/(\pi  x ),$$
 +
 
 +
*&raquo;'''si&ndash;function'''&laquo;&nbsp; or&nbsp; &raquo;$\text{splitting function}$&laquo; &nbsp;$($predominantly used in German literature$)$
 +
:$${\rm si}\left( x \right) = \sin \left( x \right)/x = {\rm sinc}(x/\pi ).$$}}
  
:$${\rm si}\hspace{-0.08cm}\left( x \right) = \sin \left( x \right)/x.$$}}
 
  
 +
<u>Note:</u> &nbsp; In our&nbsp; $\rm LNTwww$&nbsp; (because of the German original)&nbsp; we mostly use the function&nbsp; ${\rm si}(x)$,&nbsp; but important results are also given in the&nbsp; ${\rm sinc}(x)$ form.
  
By extending the numerator and denominator each with&nbsp; $T$&nbsp; one can also write for the spectral function of the rectangular pulse:
+
&rArr; &nbsp; As the right graph on the top 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; &raquo;area of the rectangle&laquo;.
:$$X( f ) = A \cdot T \cdot {\rm si}\hspace{-0.08cm}\left( {\pi fT} \right).$$
 
  
As the graphic shows,&nbsp; $X(f)$&nbsp; has the following properties:
+
*For the frequencies&nbsp; $f_n = n/T$&nbsp; with&nbsp; $n = ±1, ±2, ±3,\text{ ...} $&nbsp; the spectral function has zeroes:
*The maximum is at frequency&nbsp; $f=0$&nbsp; and has the value&nbsp; $A \cdot T$&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:
 
  
 
:$$X( {f = f_n } ) = 0.$$
 
:$$X( {f = f_n } ) = 0.$$
Line 47: Line 52:
  
  
==Gaussian Impulse==
+
==Gaussian pulse==
 
<br>
 
<br>
Another example of an aperiodic signal is the&nbsp; '''Gaussian pulse'''&nbsp; with the time
+
Another example of an aperiodic signal is the&nbsp; &raquo;'''Gaussian pulse'''&laquo;:
 
   
 
   
 
:$$x(t) = A  \cdot {\rm e}^{ - \pi \left( {t/\Delta t} \right)^2 } .$$
 
:$$x(t) = A  \cdot {\rm e}^{ - \pi \left( {t/\Delta t} \right)^2 } .$$
  
This pulse is described by two parameters, namely
+
This pulse is described by two parameters,&nbsp; namely
*the pulse amplitude&nbsp; $A$, and
+
*the pulse amplitude&nbsp; $A$,&nbsp; and
 
*the equivalent pulse duration&nbsp; $\Delta t$.
 
*the equivalent pulse duration&nbsp; $\Delta t$.
  
  
 
{{BlaueBox|TEXT=   
 
{{BlaueBox|TEXT=   
$\text{Definition:}$&nbsp; The term&nbsp; '''equivalent pulse duration'''&nbsp; is generally used to describe the duration of a rectangular pulse with the same amplitude and area as the given pulse-shaped signal&nbsp; $x(t)$:
+
$\text{Definition:}$&nbsp; The term&nbsp; &raquo;'''equivalent pulse duration'''&laquo;&nbsp; is generally used to describe the duration of a rectangular pulse with the same amplitude and area as the given pulse-like signal&nbsp; $x(t)$:
  
 
:$$\Delta t = \frac{1}{A }\cdot \hspace{-0.15cm} \int_{ - \infty }^{ + \infty } {x( t )\, {\rm d}t.}$$}}
 
:$$\Delta t = \frac{1}{A }\cdot \hspace{-0.15cm} \int_{ - \infty }^{ + \infty } {x( t )\, {\rm d}t.}$$}}
  
  
The Gaussian pulse&nbsp; $x(t)$&nbsp; has the following properties $($see graphic in&nbsp; $\text{Example 1})$:
+
The Gaussian pulse&nbsp; $x(t)$&nbsp; has the following properties&nbsp; $($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.  
+
#The time function is for all times from&nbsp; $-\infty$&nbsp; to&nbsp; $+\infty$&nbsp; existent and positive.
*This means simultaneously:&nbsp; The absolute impulse duration is infinite.
+
#This means simultaneously:&nbsp; The absolute pulse duration is infinite.
*The pulse maximum&nbsp; $A$&nbsp; is $t = 0$.
+
#With the above definition the pulse maximum&nbsp; $A$&nbsp; is at&nbsp; $t = 0$.
*At&nbsp; $t = \pm \Delta t/2$&nbsp;, the impulse is decayed to &nbsp; $\text{e}^{-\pi/4} \approx 0.456$&nbsp; of the Maximum. And at&nbsp; $t = \pm \Delta t$&nbsp;, the signal value is less than&nbsp; $3.5 \cdot 10^{-6} \cdot A$.
+
#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; $x(t)$:
+
#The  spectral function&nbsp; $X(f)$&nbsp; is also Gaussian and has similar characteristics as the Gaussian pulse&nbsp; $x(t)$:
  
:$$X(f) = A  \cdot \Delta t \cdot {\rm e}^{ - \pi \left( {f \cdot \Delta t} \right)^2 }.$$
+
::$$X(f) = A  \cdot \Delta t \cdot {\rm e}^{ - \pi \left( {f \hspace{0.05cm} \cdot \hspace{0.05cm} \Delta t} \right)^2 }.$$
 
   
 
   
On the page&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|&raquo;Reciprocity Theorem&laquo;]]&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; $X(f)$
 
  
 +
The following example illustrates the similarities and differences between the Gaussian pulse&nbsp; $x(t)$&nbsp; and its spectrum&nbsp; $X(f)$.
  
[[File:P_ID559__Sig_T_3_2_S2_b_neu.png|right|frame|Gaussian Pulse and Its Spectrum (Numerical Values)]]
+
{{GraueBox|TEXT=
{{GraueBox|TEXT=  
+
[[File:P_ID559__Sig_T_3_2_S2_b_neu.png|right|frame|Gaussian pulse and its spectrum&nbsp; $($with numerical values$)$]]   
 
$\text{Example 1:}$&nbsp;
 
$\text{Example 1:}$&nbsp;
The output power pulse&nbsp; $x(t)$&nbsp; of a laser for digital optical transmission can be assumed to be Gaussian in the equivalent low pass range with good approximation.
+
The output power pulse&nbsp; $x(t)$&nbsp; of a laser for digital optical transmission can be assumed to be Gaussian in the equivalent low-pass range with good approximation.
  
 
Let the signal parameters be&nbsp; $A = 1 \,\text{mW}$&nbsp; and&nbsp; $\Delta t =1 \,\text{ns}$.  
 
Let the signal parameters be&nbsp; $A = 1 \,\text{mW}$&nbsp; and&nbsp; $\Delta t =1 \,\text{ns}$.  
  
Damit erhält man im Spektralbereich die vergleichbaren Kenngrößen:
+
This gives the following comparable parameters in the spectral range:
* das Maximum&nbsp; $X_0 = X(f=0) = A \cdot \Delta t = 10^{-12} \,\text{W/Hz}$,
+
* The maximum&nbsp; $X_0 = X(f=0) = A \cdot \Delta t = 10^{-12} \,\text{W/Hz}$,
 +
 
 
*the equivalent bandwidth&nbsp; $\Delta f = 1/\Delta t = 1 \,\text{GHz}$.  
 
*the equivalent bandwidth&nbsp; $\Delta f = 1/\Delta t = 1 \,\text{GHz}$.  
<br clear=all>
 
Theoretically, the absolute frequency band extends to infinity. However, at&nbsp; $f = 2 \cdot \delta f = 2\,\text{GHz}$&nbsp; the spectral function is already reduced by the factor&nbsp; $3.5 \cdot 10^{-6}$&nbsp; compared to its maximum.}
 
  
  
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 LZI&ndash;system parameterized:
+
Theoretically,&nbsp; the absolute frequency band extends to infinity.&nbsp; However,&nbsp; at&nbsp; $f = 2 \cdot \Delta f = 2\,\text{GHz}$&nbsp; the spectral function is already reduced by the factor&nbsp; $3.5 \cdot 10^{-6}$&nbsp; compared to its maximum.}}
*[[Applets:Impulse_und_Spektren|Pulses and Spectra]]
+
 
*[[Applets:Frequenzgang_und_Impulsantwort|Frequency Response and Impulse Response]]
 
  
 +
&rArr; &nbsp; 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|&raquo;Pulses and Spectra&laquo;]],
  
It is also possible to display the so-called "dual correspondences".
+
*[[Applets:Frequenzgang_und_Impulsantwort|&raquo;Frequency Response and Impulse Response&laquo;]].
  
  
==Dirac Delta Impulse==
+
==Dirac delta or impulse==
 
<br>
 
<br>
In the chapter&nbsp; [[Signal_Representation/General_Description|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|&raquo;Periodic Signals&laquo;]]&nbsp; the&nbsp; &raquo;Dirac delta function&laquo;&nbsp; was already used to describe the spectrum of a direct current&nbsp; $\rm (DC)$&nbsp; signal or a harmonic oscillation.&nbsp; However,&nbsp; 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 impulse-like processes with the help of this mathematical function in the time domain.
 
  
[[File:Sig_T_3_2_S3_version3.png|right|frame|Dirac Pulse and Corresponding Spectrum]]
+
{{BlaueBox|TEXT=
{{BlaueBox|TEXT= 
+
[[File:Sig_T_3_2_S3_version3.png|right|frame|Dirac delta and corresponding spectrum]]  
 
$\text{Definition:}$&nbsp;
 
$\text{Definition:}$&nbsp;
One calls as&nbsp; '''Diracimpuls'''&nbsp; the time response
+
A &nbsp; &raquo;'''Dirac delta'''&laquo;,&nbsp; also called&raquo;'''impulse'''&laquo;,&nbsp; is denoted as
 
:$$x(t) = X_0 \cdot \delta (t),$$
 
:$$x(t) = X_0 \cdot \delta (t),$$
 
   
 
   
which can be characterised as follows (see plot):
+
and can be characterised as follows&nbsp; $($see plot$)$:
*The Dirac-pulse is infinitely narrow &nbsp; &rArr; &nbsp; it holds&nbsp; $x(t)$ = 0&nbsp; for&nbsp; $t \neq 0$&nbsp; and at time $t = 0$ the Dirac-pulse is infinitely high.
+
*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.
*I f&nbsp; $x(t)$&nbsp;describes a voltage curve, so the pulse weight&nbsp; $X_0$&nbsp; has the unit "Vs" (i.e. the unit "V/Hz" of a spectral function), since&nbsp; $\delta (t)$&nbsp; itself has the unit "1/s".
+
 
*The spectral function of the Dirac impulse includes all frequencies&nbsp; $f$&nbsp; equally: &nbsp;  
+
*If&nbsp; $x(t)$&nbsp;describes a voltage curve,&nbsp; 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$)$,&nbsp; 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;  
 
:$$X(f) = X_0 = \rm const.$$}}
 
:$$X(f) = X_0 = \rm const.$$}}
  
  
The properties mentioned here are shown in the german learning video&nbsp; [[Herleitung_und_Visualisierung_der_Diracfunktion_(Lernvideo)|Derivation and Visualisation of the Dirac Function]]&nbsp;.
+
The properties mentioned here are shown in the&nbsp; $($German language$)$&nbsp; learning video:&nbsp; <br> &nbsp; &nbsp; &nbsp;[[Herleitung_und_Visualisierung_der_Diracfunktion_(Lernvideo)|&raquo;Herleitung und Visualisierung der Diracfunktion&laquo;]] &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 Pulse]]
+
{{GraueBox|TEXT=
{{GraueBox|TEXT=
+
$\text{Example 2:}$&nbsp; We consider a network with low-pass characteristic and very low cutoff frequency&nbsp; $f_{\rm G} = 10\,\text{ kHz}$.&nbsp; The output signal&nbsp; $y(t)$&nbsp; does not change significantly when one of the sketched signals&nbsp; $x_1(t)$,&nbsp; $x_2(t)$&nbsp; or&nbsp; $x_3(t)$&nbsp; is applied to the input.&nbsp; This result can be interpreted as follows:
$\text{Example 2:}$&nbsp; We consider a  
+
[[File:P_ID561__Sig_T_3_2_S3b_neu.png|right|frame|On the significance of the Dirac delta]] 
Network with low pass&ndash;characteristic and very low cutoff frequency&nbsp; $f_{\rm G} = 10\,\text{ kHz}$. The output signal&nbsp; $y(t)$&nbsp; does not (almost) change when one of the sketched signals&nbsp; $x_1(t)$,&nbsp; $x_2(t)$&nbsp; or&nbsp; $x_3(t)$&nbsp; is applied to the input.
+
 
+
#The&nbsp; &raquo;equivalent pulse durations&laquo;&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;
This result can be interpreted as follows:
+
#Both the rectangle&nbsp; $x_1(t)$&nbsp; and the triangle&nbsp; $x_2(t)$&nbsp; can be approximated by the Dirac delta&nbsp; $x_3(t)$.&nbsp; The impulse weight&nbsp; $X_0 = 6 \cdot 10^{-6}\, \text{Vs}$&nbsp; must be equal to the pulse areas of&nbsp; $x_1(t)$&nbsp; resp.&nbsp; $x_2(t)$.
equivalent pulse durations are the same in each case&nbsp; $(\Delta t = 1\, &micro;\text{s})$&nbsp; and this is much smaller than&nbsp; $1/f_{\rm G} = 100 \, &micro;\text{s}$, the actual pulse shape (rectangle or triangle) has only a minor influence on the output signal&nbsp; $y(t)$.
+
#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; $1/f_{\rm G} = 100 \, \text{ns}$.
*Both the rectangle&nbsp; $x_1(t)$&nbsp; and the triangle&nbsp; $x_2(t)$ - can be approximated by the diraculse&nbsp; $x_3(t)$&nbsp;.
+
#Even if the Dirac delta is drawn with the same height as the other two pulses,&nbsp; it still has an infinite value at the time&nbsp; $t = 0$.&nbsp;  
*The impulse weight&nbsp; $X_0 = 6 - 10^{-6}\, \text{Vs}$&nbsp; must be equal to the impulse areas of&nbsp; $x_1(t)$&nbsp; and&nbsp; $x_2(t)$&nbsp;. However, a sufficiently small cutoff frequency is required for this approximation. With&nbsp; $f_{\rm G} = 10 \, \text{MHz}$ &nbsp; &rArr; &nbsp; $1/f_{\rm G} = 100 \, \text{ns}$&nbsp; this simplification would not be permitted.
+
#For the Dirac delta the impulse area&nbsp; $($&raquo;impulse weight&laquo;$)$&nbsp; 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"$)$.}}
*Even if the Dirac-pulse is drawn with the same height as the other two pulses, it still has an infinite value at the time&nbsp; $t = 0$&nbsp;.
 
*The pulse area ("pulse weight") is always specified for the Dirac impulse. This differs from the other pulse amplitudes already in the unit&nbsp; (for example "Vs" instead of "V").}}
 
  
  
==Exercises for the Chapter==
+
==Exercises for the chapter==
  
[[Aufgaben:Exercise 3.3: From The Signal to the Spectrum|Exercise 3.3: From The Signal to the Spectrum]]
+
[[Aufgaben:Exercise 3.3: From The Signal to the Spectrum|Exercise 3.3: From the Signal to the Spectrum]]
  
[[Aufgaben:Exercise 3.3Z: Rectangular- and Dirac Pulse|Exercise 3.3Z: Rectangular- and Dirac Pulse]]
+
[[Aufgaben:Exercise_3.3Z:_Rectangular_Pulse_and_Dirac_Delta|Exercise 3.3Z: Rectangular Pulse and Dirac Delta]]
  
  
  
 
{{Display}}
 
{{Display}}

Latest revision as of 17:26, 15 November 2023

Rectangular pulse


Rectangular pulse and its spectrum

One speaks of a  »rectangular pulse«,  if the following applies for the time domain:

$$x(t) = \left\{ \begin{array}{l} \hspace{0.25cm}A \\ A /2 \\ \hspace{0.25cm} 0 \\ \end{array} \right.\quad \quad \begin{array}{*{20}c} {\rm{for}} \\ {\rm{for}} \\ {\rm{for}} \\ \end{array}\begin{array}{*{20}c} {\left| \hspace{0.05cm} t\hspace{0.05cm} \right| < T/2,} \\ {\left| \hspace{0.05cm}t\hspace{0.05cm} \right| = T/2,} \\ {\left|\hspace{0.05cm} t \hspace{0.05cm} \right| > T/2.} \\ \end{array}$$

$A$  denotes the amplitude of the pulse and  $T$  the pulse duration.

The corresponding spectral function  $X(f)$  is obtained by using the  »first Fourier integral«:

$$X(f) = \int_{ - T/2}^{+T/2} A \cdot {\rm e}^{ -{\rm j}2\pi ft}\, {\rm d}t ,$$
$$\Rightarrow \hspace{0.3cm} X(f) = A \cdot \int_{ - T/2}^{+T/2} {\cos ( {2\pi}ft )\,{\rm d}t - {\rm j} \cdot A} \int_{ - T/2}^{+T/2} {\sin ( {2\pi ft} )}\,{\rm d}t .$$
  • Here the integration limits  $\pm T/2$  take into account that  $x(t)$  is identical to zero outside the interval from  $-T/2$  to  $+T/2$.
  • The second integral disappears due to the odd integrand and you get
$$X(f) = \frac{A \cdot \sin \left( {\pi fT} \right)}{\pi f}.$$
  • By extending the numerator and denominator each with  $T$  one can also write for the spectral function of the rectangular pulse:
$$X( f ) = A \cdot T \cdot {\rm si}\hspace{-0.08cm}\left( {\pi fT} \right)\hspace{1.3cm}\text{resp.}\hspace{1.3cm}\ X( f ) = A \cdot T \cdot {\rm sinc}\hspace{-0.08cm}\left( {fT} \right).$$

$\text{Definitions:}$  For abbreviation we define the following functions:

  • »sinc–function«  $($predominantly used in Anglo-American literature$)$
$${\rm sinc}( x ) = {\sin (\pi x) }/(\pi x ),$$
  • »si–function«  or  »$\text{splitting function}$«  $($predominantly used in German literature$)$
$${\rm si}\left( x \right) = \sin \left( x \right)/x = {\rm sinc}(x/\pi ).$$


Note:   In our  $\rm LNTwww$  (because of the German original)  we mostly use the function  ${\rm si}(x)$,  but important results are also given in the  ${\rm sinc}(x)$ form.

⇒   As the right graph on the top shows,  $X(f)$  has the following properties:

  • The maximum is at frequency  $f=0$  and has the value  $A \cdot T$  ⇒   »area of the rectangle«.
  • For the frequencies  $f_n = n/T$  with  $n = ±1, ±2, ±3,\text{ ...} $  the spectral function has zeroes:
$$X( {f = f_n } ) = 0.$$
  • The following constraint applies to the magnitude spectrum:
$$\left| {X( f )} \right| \le \frac{A}{\pi \cdot \left| f \right|}.$$


Gaussian pulse


Another example of an aperiodic signal is the  »Gaussian pulse«:

$$x(t) = A \cdot {\rm e}^{ - \pi \left( {t/\Delta t} \right)^2 } .$$

This pulse is described by two parameters,  namely

  • the pulse amplitude  $A$,  and
  • the equivalent pulse duration  $\Delta t$.


$\text{Definition:}$  The term  »equivalent pulse duration«  is generally used to describe the duration of a rectangular pulse with the same amplitude and area as the given pulse-like signal  $x(t)$:

$$\Delta t = \frac{1}{A }\cdot \hspace{-0.15cm} \int_{ - \infty }^{ + \infty } {x( t )\, {\rm d}t.}$$


The Gaussian pulse  $x(t)$  has the following properties  $($see graphic in  $\text{Example 1})$:

  1. The time function is for all times from  $-\infty$  to  $+\infty$  existent and positive.
  2. This means simultaneously:  The absolute pulse duration is infinite.
  3. With the above definition the pulse maximum  $A$  is at  $t = 0$.
  4. At  $t = \pm \Delta t/2$ , the pulse is decayed to   $\text{e}^{-\pi/4} \approx 0.456$  of the maximum.  And at  $t = \pm \Delta t$,  the signal value is less than  $3.5 \cdot 10^{-6} \cdot A$.
  5. The spectral function  $X(f)$  is also Gaussian and has similar characteristics as the Gaussian pulse  $x(t)$:
$$X(f) = A \cdot \Delta t \cdot {\rm e}^{ - \pi \left( {f \hspace{0.05cm} \cdot \hspace{0.05cm} \Delta t} \right)^2 }.$$

In the section  »Reciprocity Theorem«  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  $x(t)$  and its spectrum  $X(f)$.

Gaussian pulse and its spectrum  $($with numerical values$)$

$\text{Example 1:}$  The output power pulse  $x(t)$  of a laser for digital optical transmission can be assumed to be Gaussian in the equivalent low-pass range with good approximation.

Let the signal parameters be  $A = 1 \,\text{mW}$  and  $\Delta t =1 \,\text{ns}$.

This gives the following comparable parameters in the spectral range:

  • The maximum  $X_0 = X(f=0) = A \cdot \Delta t = 10^{-12} \,\text{W/Hz}$,
  • the equivalent bandwidth  $\Delta f = 1/\Delta t = 1 \,\text{GHz}$.


Theoretically,  the absolute frequency band extends to infinity.  However,  at  $f = 2 \cdot \Delta f = 2\,\text{GHz}$  the spectral function is already reduced by the factor  $3.5 \cdot 10^{-6}$  compared to its maximum.


⇒   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:


Dirac delta or impulse


In the chapter  »Periodic Signals«  the  »Dirac delta function«  was already used to describe the spectrum of a direct current  $\rm (DC)$  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.


Dirac delta and corresponding spectrum

$\text{Definition:}$  A   »Dirac delta«,  also called»impulse«,  is denoted as

$$x(t) = X_0 \cdot \delta (t),$$

and can be characterised as follows  $($see plot$)$:

  • The Dirac delta is infinitely narrow   ⇒   it holds  $x(t)\equiv 0$  for  $t \neq 0$  and at time  $t = 0$  the Dirac delta is infinitely high.
  • If  $x(t)$ describes a voltage curve,  then the impulse weight  $X_0$  has the unit  "$\textrm{V} \cdot \textrm{s}$"  $($i.e. the unit  "$\textrm{V}/\textrm{Hz}$"  of a spectral function$)$,  since  $\delta (t)$  itself has the unit  "$1/\textrm{s}$".
  • The Fourier transform of the Dirac delta includes all frequencies $f$  equally:  
$$X(f) = X_0 = \rm const.$$


The properties mentioned here are shown in the  $($German language$)$  learning video: 
     »Herleitung und Visualisierung der Diracfunktion«   ⇒   "Derivation and visualisation of the Dirac delta function".

$\text{Example 2:}$  We consider a network with low-pass characteristic and very low cutoff frequency  $f_{\rm G} = 10\,\text{ kHz}$.  The output signal  $y(t)$  does not change significantly when one of the sketched signals  $x_1(t)$,  $x_2(t)$  or  $x_3(t)$  is applied to the input.  This result can be interpreted as follows:

On the significance of the Dirac delta
  1. The  »equivalent pulse durations«  are the same in each case  $(\Delta t = 1\, µ\text{s})$  and this is much smaller than  $1/f_{\rm G} = 100 \, µ\text{s}$.  The actual pulse shape  $($rectangle or triangle$)$  has only a minor influence on the output signal  $y(t)$. 
  2. Both the rectangle  $x_1(t)$  and the triangle  $x_2(t)$  can be approximated by the Dirac delta  $x_3(t)$.  The impulse weight  $X_0 = 6 \cdot 10^{-6}\, \text{Vs}$  must be equal to the pulse areas of  $x_1(t)$  resp.  $x_2(t)$.
  3. However, a sufficiently small cutoff frequency is required for this approximation.  This simplification would not be permitted with  $f_{\rm G} = 10 \, \text{MHz}$   ⇒   $1/f_{\rm G} = 100 \, \text{ns}$.
  4. 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  $t = 0$. 
  5. For the Dirac delta the impulse area  $($»impulse weight«$)$  is always specified.  This differs from the other pulse amplitudes already in the unit  $($e.g.  "Vs"  instead of  "V"$)$.


Exercises for the chapter

Exercise 3.3: From the Signal to the Spectrum

Exercise 3.3Z: Rectangular Pulse and Dirac Delta