Signal Representation/General Description - Revision history

Guenter at 14:13, 8 June 2023

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Guenter at 15:48, 8 February 2023

2023-02-08T15:48:42Z

Guenter at 14:09, 5 December 2022

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Guenter at 17:00, 2 December 2022

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Hwang at 19:10, 25 November 2022

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Guenter at 16:21, 17 May 2021

2021-05-17T16:21:23Z

Guenter at 11:26, 14 April 2021

2021-04-14T11:26:45Z

@@ Line 21: / Line 21: @@
 ==Features and applications==
 <br>
-Periodic signals are of great importance for communications engineering:
+Periodic signals are of great importance for Communications Engineering:
-*They belong to the class of&nbsp;[[Signal_Representation/Signal_classification#Deterministic_and_stochastic_signals|"deterministic signals"]],&nbsp; whose time function can be specified in analytical form.
+*They belong to the class of&nbsp; [[Signal_Representation/Signal_classification#Deterministic_and_stochastic_signals|&raquo;deterministic signals&laquo;]],&nbsp; whose time function can be specified in analytical form.
 *Their signal path is thus known for all times&nbsp; $t$&nbsp; and can be clearly predicted for the future.
@@ Line 29: / Line 29: @@
-Nevertheless, periodic signals are often also required in communications engineering, for example
+Nevertheless,&nbsp; periodic signals are often also required in Communications Engineering,&nbsp; for example
 *for modulation and demodulation in carrier frequency systems,
@@ Line 46: / Line 46: @@
-As can be seen from the displayed settings, the period duration of both signals is one millisecond and the amplitude one volt.}}
+As can be seen from the displayed settings,&nbsp; the period duration of both signals is one millisecond and the amplitude one volt.}}
 ==Definition and parameters==
 <br>
-Before we turn to the signal parameters of a periodic signal, the term&nbsp; "periodicity"&nbsp; shall be clearly defined:
+Before we turn to the signal parameters of a periodic signal,&nbsp; the term&nbsp; &raquo;periodicity&laquo;&nbsp; shall be clearly defined:
 {{BlaueBox|TEXT=
@@ Line 60: / Line 60: @@
 This results in the following parameters:
-*The&nbsp; &raquo;'''period duration'''&laquo;&nbsp; $T_{0}$&nbsp; indicates the smallest possible value, which satisfies the above equation.
+*The&nbsp; &raquo;'''period duration'''&laquo;&nbsp; $T_{0}$&nbsp; indicates the smallest possible value,&nbsp; which satisfies the above equation.
-*The&nbsp; &raquo;'''basic frequency'''&laquo;&nbsp; $f_{0} = 1/T_{0}$&nbsp; describes the number of periods per time unit (mostly per second).
+*The&nbsp; &raquo;'''basic frequency'''&laquo;&nbsp; $f_{0} = 1/T_{0}$&nbsp; describes the number of periods per time unit&nbsp; $($mostly per second$)$.
-*The unit&nbsp; "1/s"&nbsp; is also called&nbsp; "Hz", named after the German physicist &nbsp; [https://en.wikipedia.org/wiki/Heinrich_Hertz $\text{Heinrich Hertz}$].
+*The unit&nbsp; "1/s"&nbsp; is also called&nbsp; "Hz",&nbsp; named after the German physicist &nbsp; [https://en.wikipedia.org/wiki/Heinrich_Hertz $\text{Heinrich Hertz}$].
-*The&nbsp; &raquo;'''basic circular frequency'''&laquo;&nbsp; $\omega_{0}$&nbsp; represents the angular rotation per second, usually given in radians.
+*The&nbsp; &raquo;'''basic circular frequency'''&laquo;&nbsp; $\omega_{0}$&nbsp; represents the angular rotation per second,&nbsp; usually given in radians.
-*In contrast to the basic frequency, the unit&nbsp; "Hz"&nbsp; is not common here, but&nbsp; "1/s".&nbsp; The following equation applies:
+*In contrast to the basic frequency,&nbsp; the unit&nbsp; "Hz"&nbsp; is not common here, but&nbsp; "1/s".&nbsp; The following equation applies:
 :$$\omega_{0}=2\pi f_{0} = {2\pi}/{T_{0}}.$$
@@ Line 78: / Line 78: @@
 *The period duration is&nbsp; $T_{0} = 2.5 \ \rm ms$.
-*From this the basic frequency &nbsp; $f_0 =  400  \ \rm  Hz$ is calculated.
+*From this the basic frequency &nbsp; $f_0 =  400  \ \rm  Hz$&nbsp; is calculated.
 *The basic circular frequency results to &nbsp;$\omega_{0}=2513 \ \rm  1/s.$}}
@@ Line 86: / Line 86: @@
 ==Resulting period duration==
 <br>
-If a signal&nbsp; $x(t)$&nbsp; consists of the sum of two periodic signals&nbsp; $x_{1}(t)$&nbsp; and&nbsp; $x_{2}(t)$&nbsp; with the period durations&nbsp; $T_{1}$&nbsp; or &nbsp; $T_{2}$, the resulting period duration of the sum signal is the smallest common multiple of&nbsp; $T_{1}$&nbsp; and&nbsp; $T_{2}$.
+If a signal&nbsp; $x(t)$&nbsp; consists of the sum of two periodic signals&nbsp; $x_{1}(t)$&nbsp; and&nbsp; $x_{2}(t)$&nbsp; with period durations&nbsp; $T_{1}$&nbsp; or &nbsp; $T_{2}$,&nbsp; the resulting period duration of the sum signal is the smallest common multiple of&nbsp; $T_{1}$&nbsp; and&nbsp; $T_{2}$.
 *This statement applies independently of the amplitude and phase relations.
-*On the other hand, if &nbsp; $T_{1}$&nbsp; and&nbsp; $T_{2}$&nbsp; don't  have a  rational common multiple&nbsp; $($Example: &nbsp; $T_{2} = \pi \cdot T_{1})$, <br>then the sum signal&nbsp; $x(t)$&nbsp; is in contrast to its two components&nbsp; $x_{1}(t)$&nbsp; and&nbsp; $x_{2}(t)$&nbsp; not periodic.
+*On the other hand,&nbsp; if &nbsp; $T_{1}$&nbsp; and&nbsp; $T_{2}$&nbsp; don't  have a  rational common multiple&nbsp; $($Example: &nbsp; $T_{2} = \pi \cdot T_{1})$,&nbsp; then the sum signal&nbsp; $x(t)$&nbsp; is in contrast to its two components&nbsp; $x_{1}(t)$&nbsp; and&nbsp; $x_{2}(t)$&nbsp; not periodic.
 {{GraueBox|TEXT=
 $\text{Example 3:}$&nbsp;
-Here, a cosinusoidal signal&nbsp; $x_{1}(t)$&nbsp; with period duration&nbsp; $T_{1} = 2\; {\rm ms}$&nbsp; (blue signal course)&nbsp;is added with  a sinusoidal signal&nbsp; $x_{2}(t)$&nbsp; with period duration&nbsp; $T_{2} = 5\; {\rm ms}$&nbsp; and twice the amplitude (green curve).
+Here,&nbsp; a cosinusoidal signal&nbsp; $x_{1}(t)$&nbsp; with period duration&nbsp; $T_{1} = 2\; {\rm ms}$&nbsp; $($blue signal course$)$&nbsp;is added with  a sinusoidal signal&nbsp; $x_{2}(t)$&nbsp; with period duration&nbsp; $T_{2} = 5\; {\rm ms}$&nbsp; and twice the amplitude&nbsp; $($green curve).
 [[File:P_ID247__Sig_T_2_1_S3_neu.png|frame|Resulting period duration of the sum of cosine and sine signal]]
-*The (red) sum signal&nbsp; $x(t) = x_{1}(t) + x_{2}(t)$&nbsp; then shows the resulting period duration&nbsp; $T_{0} = 10\; {\rm ms}$ &nbsp; &rArr; &nbsp; basic frequency&nbsp; $f_{0} = 100\; {\rm Hz}$.
+*The&nbsp; $($red$)$&nbsp; sum signal&nbsp; $x(t) = x_{1}(t) + x_{2}(t)$&nbsp; then shows the resulting period duration&nbsp; $T_{0} = 10\; {\rm ms}$ &nbsp; &rArr; &nbsp; basic frequency&nbsp; $f_{0} = 100\; {\rm Hz}$.
-*The frequency&nbsp; $f_{0}$&nbsp; itself is not contained in&nbsp; $x(t)$&nbsp; only integer multiples of it, namely&nbsp;
+*The frequency&nbsp; $f_{0}$&nbsp; itself is not contained in&nbsp; $x(t)$&nbsp; only integer multiples of it,&nbsp; namely&nbsp;
-::$f_{1} = 500\; {\rm Hz}$&nbsp; and&nbsp; $f_{2} = 200\; {\rm Hz}$. }}
+::$f_{1} = 500\; {\rm Hz}$&nbsp; and&nbsp; $f_{2} = 200\; {\rm Hz}$.
-&rArr; &nbsp; With the interactive applet&nbsp; [[Applets:Period_Duration_of_Periodic_Signals|"Period Duration of Periodic Signals"]]&nbsp; the resulting period of two harmonic oscillations can be determined.

@@ Line 7: / Line 7: @@
 == # OVERVIEW OF THE SECOND MAIN CHAPTER # ==
 <br>
-In this chapter,&nbsp; &raquo;periodic signals&laquo;&nbsp; are considered and described mathematically &raquo;in the time and frequency domain&laquo;.
+In this chapter,&nbsp; &raquo;'''periodic signals'''&laquo;&nbsp; are considered and described mathematically &raquo;'''in the time and frequency domain'''&laquo;.
 This chapter contains in detail:
@@ Line 22: / Line 22: @@
 <br>
 Periodic signals are of great importance for communications engineering:
-*They belong to the class of&nbsp;[[Signal_Representation/Signal_classification#Deterministic_and_stochastic_signals|$\text{deterministic signals}$]], whose time function can be specified in analytical form.
+*They belong to the class of&nbsp;[[Signal_Representation/Signal_classification#Deterministic_and_stochastic_signals|"deterministic signals"]],&nbsp; whose time function can be specified in analytical form.
 *Their signal path is thus known for all times&nbsp; $t$&nbsp; and can be clearly predicted for the future.
 *They are therefore never information-carrying signals.
@@ Line 29: / Line 31: @@
 Nevertheless, periodic signals are often also required in communications engineering, for example
 *for modulation and demodulation in carrier frequency systems,
 *for synchronization and clock regeneration in digital systems,
 *as test and verification signals during system implementation.
-[[File:P_ID161__Sig_T_2_1_S1.png|right|frame|Oscilloscope image of cosine and triangular signals]]
+{{GraueBox|TEXT=
-{{GraueBox|TEXT=
+[[File:P_ID161__Sig_T_2_1_S1.png|right|frame|Oscilloscope image of cosine and triangular signals]]
 $\text{Example 1:}$&nbsp;
 The oscilloscope image shows two typical representatives of periodic signals:
 *above a cosine signal,
 *below a triangular signal.
@@ Line 50: / Line 55: @@
 {{BlaueBox|TEXT=
 $\text{Definition:}$&nbsp;
-A&nbsp; $\text{periodic signal}$&nbsp; $x(t)$&nbsp; is present if for all arbitrary values of&nbsp; $t$&nbsp; and all integer values of&nbsp; $i$&nbsp; with an appropriate&nbsp; $T_{0}$&nbsp; applies:
+A&nbsp; &raquo;'''periodic signal'''&laquo;&nbsp; $x(t)$&nbsp; is present if for all arbitrary values of&nbsp; $t$&nbsp; and all integer values of&nbsp; $i$&nbsp; with an appropriate&nbsp; $T_{0}$&nbsp; applies:
 :$$x(t+i\cdot T_{0}) = x(t).$$}}
 This results in the following parameters:
-*The&nbsp; $\text{period duration}$&nbsp; $T_{0}$&nbsp; indicates the smallest possible value, which satisfies the above equation.
+*The&nbsp; &raquo;'''period duration'''&laquo;&nbsp; $T_{0}$&nbsp; indicates the smallest possible value, which satisfies the above equation.
-*The&nbsp; $\text{basic frequency}$&nbsp; $f_{0} = 1/T_{0}$&nbsp; describes the number of periods per time unit (mostly per second).
 *The unit&nbsp; "1/s"&nbsp; is also called&nbsp; "Hz", named after the German physicist &nbsp; [https://en.wikipedia.org/wiki/Heinrich_Hertz $\text{Heinrich Hertz}$].
-*The&nbsp; $\text{basic circular frequency}$&nbsp; $\omega_{0}$&nbsp; represents the angular rotation per second, usually given in radians.
 *In contrast to the basic frequency, the unit&nbsp; "Hz"&nbsp; is not common here, but&nbsp; "1/s".&nbsp; The following equation applies:
 :$$\omega_{0}=2\pi f_{0} = {2\pi}/{T_{0}}.$$
-[[File:P_ID211__Sig_T_2_1_S2_neu.png|right|frame|Given signal and period duration]]
+{{GraueBox|TEXT=
-{{GraueBox|TEXT=
+[[File:P_ID211__Sig_T_2_1_S2_neu.png|right|frame|Given signal and period duration]]
 $\text{Example 2:}$&nbsp;
 Here, a periodic time signal is shown:
 *The period duration is&nbsp; $T_{0} = 2.5 \ \rm ms$.
 *From this the basic frequency &nbsp; $f_0 =  400  \ \rm  Hz$ is calculated.
 *The basic circular frequency results to &nbsp;$\omega_{0}=2513 \ \rm  1/s.$}}
@@ Line 77: / Line 88: @@
 If a signal&nbsp; $x(t)$&nbsp; consists of the sum of two periodic signals&nbsp; $x_{1}(t)$&nbsp; and&nbsp; $x_{2}(t)$&nbsp; with the period durations&nbsp; $T_{1}$&nbsp; or &nbsp; $T_{2}$, the resulting period duration of the sum signal is the smallest common multiple of&nbsp; $T_{1}$&nbsp; and&nbsp; $T_{2}$.
 *This statement applies independently of the amplitude and phase relations.
 *On the other hand, if &nbsp; $T_{1}$&nbsp; and&nbsp; $T_{2}$&nbsp; don't  have a  rational common multiple&nbsp; $($Example: &nbsp; $T_{2} = \pi \cdot T_{1})$, <br>then the sum signal&nbsp; $x(t)$&nbsp; is in contrast to its two components&nbsp; $x_{1}(t)$&nbsp; and&nbsp; $x_{2}(t)$&nbsp; not periodic.
@@ Line 87: / Line 99: @@
 *The (red) sum signal&nbsp; $x(t) = x_{1}(t) + x_{2}(t)$&nbsp; then shows the resulting period duration&nbsp; $T_{0} = 10\; {\rm ms}$ &nbsp; &rArr; &nbsp; basic frequency&nbsp; $f_{0} = 100\; {\rm Hz}$.
 *The frequency&nbsp; $f_{0}$&nbsp; itself is not contained in&nbsp; $x(t)$&nbsp; only integer multiples of it, namely&nbsp;
 ::$f_{1} = 500\; {\rm Hz}$&nbsp; and&nbsp; $f_{2} = 200\; {\rm Hz}$. }}
-With the interactive applet&nbsp; [[Applets:Period_Duration_of_Periodic_Signals|"Period Duration of Periodic Signals"]]&nbsp; the resulting period of two harmonic oscillations can be determined.
+&rArr; &nbsp; With the interactive applet&nbsp; [[Applets:Period_Duration_of_Periodic_Signals|"Period Duration of Periodic Signals"]]&nbsp; the resulting period of two harmonic oscillations can be determined.

@@ Line 7: / Line 7: @@
 == # OVERVIEW OF THE SECOND MAIN CHAPTER # ==
 <br>
-In this chapter,&nbsp; periodic signals&nbsp; are considered and described mathematically in the time and frequency domain.
+In this chapter,&nbsp; &raquo;periodic signals&laquo;&nbsp; are considered and described mathematically &raquo;in the time and frequency domain&laquo;.
 This chapter contains in detail:
-* Some basic terms like&nbsp; "period duration", "basic frequency"&nbsp; and&nbsp; "circular frequency",
+# Some basic terms like&nbsp; &raquo;period duration&laquo;,&nbsp; &raquo;basic frequency&laquo;&nbsp; and&nbsp; &raquo;circular frequency&laquo;,
-* the properties of a&nbsp; "DC signal"&nbsp; as a boundary case of a periodic signal,
+# the properties of a&nbsp; &raquo;DC signal&laquo;&nbsp; as a limiting case of a periodic signal,
-* the definition and interpretation of the Dirac delta function",
+# the definition and interpretation of the&nbsp; &raquo;Dirac delta function&laquo;,
-* the spectral representation of a DC signal or a DC signal component,
+# the&nbsp; &raquo;spectral representation&laquo;&nbsp; of a DC signal or a DC signal component,
-* the time  and frequency representation of&nbsp; "harmonic oscillations", and finally
+# the time  and frequency representation of&nbsp; &raquo;harmonic oscillations&laquo;,&nbsp; and finally
-* the application of&nbsp; "Fourier series"&nbsp; for spectral analysis of periodic signals.
+# the application of&nbsp; &raquo;Fourier series&laquo;&nbsp; for spectral analysis of periodic signals.

@@ Line 12: / Line 12: @@
 * Some basic terms like&nbsp; "period duration", "basic frequency"&nbsp; and&nbsp; "circular frequency",
 * the properties of a&nbsp; "DC signal"&nbsp; as a boundary case of a periodic signal,
-* the definition and interpretation of the&nbsp; "Dirac '''KORREKTUR: Dirac delta?''' function",
+* the definition and interpretation of the Dirac delta function",
 * the spectral representation of a DC signal or a DC signal component,
 * the time  and frequency representation of&nbsp; "harmonic oscillations", and finally

@@ Line 22: / Line 22: @@
 <br>
 Periodic signals are of great importance for communications engineering:
-*They belong to the class of&nbsp;[[Signal_Representation/Signal_classification#Deterministische_und_stochastische_Signale|$\text{deterministic signals}$]], whose time function can be specified in analytical form.
+*They belong to the class of&nbsp;[[Signal_Representation/Signal_classification#Deterministic_and_stochastic_signals|$\text{deterministic signals}$]], whose time function can be specified in analytical form.
 *Their signal path is thus known for all times&nbsp; $t$&nbsp; and can be clearly predicted for the future.
 *They are therefore never information-carrying signals.

@@ Line 94: / Line 94: @@
-==Exercises for the Chapter==
+==Exercises for the chapter==
 <br>
 [[Aufgaben:Exercise_2.1:_Rectifying|Exercise 2.1: Rectification]]