Signal Representation/Harmonic Oscillation - Revision history

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@@ Line 231: / Line 231: @@
 :$$X(f)=\frac{A+{\rm j} \cdot B}{2}\cdot {\rm \delta} (f+f_0)+\frac{A-{\rm j} \cdot B}{2} \cdot \delta (f-f_0).$$
-This Fourier correlation is with amplitude&nbsp; $C$&nbsp; and phase&nbsp; $\varphi$&nbsp;
+This Fourier correlation is with amplitude&nbsp; $C$&nbsp; and phase&nbsp; $\varphi$:&nbsp;
 :$$x(t)=C\cdot \cos(2\pi f_0 t-\varphi)\ \, \circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\,\ X(f)={C}/{2}\cdot {\rm e}^{{\rm j} \varphi} \cdot \delta(f+f_0) + {C}/{2} \cdot {\rm e}^{\rm{-j} \varphi} \cdot \delta (f-f_0).$$

@@ Line 200: / Line 200: @@
 ==General spectral representation==
 <br>
-For a sinusoidal signal the&nbsp; [https://en.wikipedia.org/wiki/Leonhard_Euler $\text{Euler}$]  theorem  applies in a similar way:
+For a sinusoidal signal the&nbsp; &raquo;[https://en.wikipedia.org/wiki/Leonhard_Euler $\text{Euler}$]  '''theorem'''&laquo;&nbsp;  applies in a similar way:
 :$$x(t)=B\cdot \sin(2 \pi f_0 t)= \frac{\it B}{2  \rm j} \cdot \big [{\rm e}^{+{\rm j} 2 \pi \it f_{\rm 0} t}-{\rm e}^{-\rm j2 \pi \it f_{\rm 0} t}\big ]=\rm j\cdot  {\it B}/{2} \cdot \big  [{\rm e}^{-j2 \pi \it f_{\rm 0} t}-{\rm e}^{+\rm j2 \pi \it f_{\rm 0} t} \big ] .$$
@@ Line 231: / Line 231: @@
 :$$X(f)=\frac{A+{\rm j} \cdot B}{2}\cdot {\rm \delta} (f+f_0)+\frac{A-{\rm j} \cdot B}{2} \cdot \delta (f-f_0).$$
-With the amplitude&nbsp; $C$&nbsp; and the phase&nbsp; $\varphi$&nbsp; this Fourier correlation is
+This Fourier correlation is with amplitude&nbsp; $C$&nbsp; and phase&nbsp; $\varphi$&nbsp;
 :$$x(t)=C\cdot \cos(2\pi f_0 t-\varphi)\ \, \circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\,\ X(f)={C}/{2}\cdot {\rm e}^{{\rm j} \varphi} \cdot \delta(f+f_0) + {C}/{2} \cdot {\rm e}^{\rm{-j} \varphi} \cdot \delta (f-f_0).$$
@@ Line 258: / Line 258: @@
-The following (German language) learning video illustrates the properties of harmonic oscillations using scales: <br>
+The following&nbsp; (German-language)&nbsp; learning video illustrates the properties of harmonic oscillations using scales: <br>
-&nbsp; &nbsp; [[Harmonische_Schwingungen_(Lernvideo)|"Harmonische Schwingungen"]] &nbsp; &rArr; &nbsp; "Harmonic Oscillations".
+&nbsp; &nbsp; &nbsp; [[Harmonische_Schwingungen_(Lernvideo)|&raquo;Harmonische Schwingungen&raquo;]] &nbsp; &rArr; &nbsp; "Harmonic Oscillations".

@@ Line 146: / Line 146: @@
 ==Spectral representation of a cosine signal==
 <br>
-In order to derive the spectral function, we first restrict ourselves to a cosine signal, which also can be written with the&nbsp; complex exponential function&nbsp; and the &nbsp;[[Signal_Representation/Calculating_with_Complex_Numbers#Representation_by_magnitude_and_phase|"Euler theorem"]] &nbsp; in the following manner:
+In order to derive the spectral function,&nbsp; we first restrict ourselves to a cosine signal,&nbsp; which also can be written with the&nbsp; complex exponential function&nbsp; and the&nbsp; [[Signal_Representation/Calculating_with_Complex_Numbers#Representation_by_magnitude_and_phase|&raquo;Euler theorem&laquo;]] &nbsp; in the following manner:
 :$$x(t)=A \cdot \cos(2\pi f_0 t)={A}/{2}\cdot \big [{\rm e}^{\rm -j2 \pi \it f_{\rm 0} t} + {\rm e}^{\rm j2\pi \it f_{\rm 0} t} \big ].$$
-From this time domain representation it can already be seen that the cosine signal contains  &ndash; spectrally seen &ndash; only one single (physical) frequency, namely the frequency $f_0$.
+From this time domain representation it can be seen that the cosine signal contains  &ndash; spectrally seen &ndash; only one single&nbsp; $($physical$)$&nbsp; frequency,&nbsp; namely the frequency&nbsp; $f_0$.
 {{BlaueBox|TEXT=
 $\text{Proof:}$&nbsp; For the mathematical derivation of the spectral function we use the following relations:
-*the equation from the section&nbsp; [[Signal_Representation/Direct_Current_Signal_-_Limit_Case_of_a_Periodic_Signal#Dirac_.28delta.29_function_in_frequency_domain|"Dirac delta function in frequency domain"]]:&nbsp;
+*The equation from the section&nbsp; [[Signal_Representation/Direct_Current_Signal_-_Limit_Case_of_a_Periodic_Signal#Dirac_.28delta.29_function_in_frequency_domain|&raquo;Dirac delta function in frequency domain&laquo;]]:&nbsp;
-:$$x(t)=A \ \circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\, \ X(f)=A \cdot \rm \delta (\it f).$$
+:$$x(t)=A \ \circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\, \ X(f)=A \cdot \rm \delta (\it f),$$
-*the&nbsp; [[Signal_Representation/Fourier_Transform_Theorems#Shifting_Theorem|"shifting theorem"]]&nbsp; (for the frequency range)&nbsp; in anticipation of a later chapter:
+*the&nbsp; [[Signal_Representation/Fourier_Transform_Theorems#Shifting_Theorem|&raquo;shifting theorem&laquo;]]&nbsp; $($for the frequency range$)$&nbsp; in anticipation of a later chapter:
 :$$x(t) \cdot {\rm e}^{\rm j2\pi\it f_{\rm 0} t}\ \circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\, \ X(f-f_0). $$
@@ Line 165: / Line 165: @@
 This means:
-*The spectral function&nbsp; $X(f)$&nbsp; of a cosine signal with the frequency $f_0$&nbsp; is composed of two Dirac delta functions at&nbsp; $\pm f_0$.
+*The spectral function&nbsp; $X(f)$&nbsp; of a cosine signal with frequency $f_0$&nbsp; is composed of two Dirac delta functions at&nbsp; $\pm f_0$.
 * The impulse weights are each equal to half the signal amplitude.}}
@@ Line 173: / Line 173: @@
 [[File:P_ID1365__Sig_T_2_3_S4_neu.png|right|frame|Spectrum of a cosine signal]]
 $\text{Example 3:}$&nbsp;
-The diagram shows the spectrum of a cosine oscillation
+The diagram shows the spectrum of a cosine oscillation with
-*with amplitude&nbsp; $A = 4 \;{\rm V}$&nbsp; and
+*amplitude&nbsp; $A = 4 \;{\rm V}$&nbsp; and
-*the frequency&nbsp; $f_0 = 5 \;{\rm kHz}$  &nbsp; ⇒ &nbsp;  $T_0 = 200 \;{\rm &micro; s}$.
+*frequency&nbsp; $f_0 = 5 \;{\rm kHz}$  &nbsp; ⇒ &nbsp;  $T_0 = 200 \;{\rm &micro; s}$.
@@ Line 188: / Line 188: @@
 $\text{Note:}$&nbsp;
 The spectral function of any real time function except the DC signal has components at both positive and negative frequencies.
-*This fact, which often causes problems for first-year students, is formally derived from the&nbsp; [[Signal_Representation/Calculating_with_Complex_Numbers#Representation_by_magnitude_and_phase|"Euler theorem"]].
+*This fact,&nbsp; which often causes problems for first-year students,&nbsp; is formally derived from the&nbsp; [[Signal_Representation/Calculating_with_Complex_Numbers#Representation_by_magnitude_and_phase|&raquo;Euler theorem&laquo;]].
-*By extending the frequency range from&nbsp; $f \ge 0$&nbsp; to the set of real numbers you get from the physical to the "mathematical frequency".
+*By extending the frequency range from&nbsp; $f \ge 0$&nbsp; to the set of real numbers you get from the physical to the&nbsp; &raquo;mathematical frequency&laquo;.
-*However, for a negative frequency the&nbsp; [[Signal_Representation/Harmonic_Oscillation#Definition_and_properties|$\text{definition}$]]&nbsp;  is not applicable anymore:  &nbsp;  You cannot interpret&nbsp; $\rm -5 kHz$&nbsp; as&nbsp; <br>"minus 5000 oscillations per second".
+*However,&nbsp; for a negative frequency the&nbsp; [[Signal_Representation/Harmonic_Oscillation#Definition_and_properties|$\text{definition}$]]&nbsp;  is not applicable anymore:  &nbsp;  You cannot interpret&nbsp; $\rm -5\ kHz$&nbsp; as&nbsp; "minus 5000 oscillations per second".
-In the process of this course you will find that&nbsp; '''by complicating a simple subject, more complex matters can later be described very elegantly and simply'''.}}
+In the process of this course you will find that&nbsp; '''by complicating a simple subject,&nbsp; more complex matters can later be described very elegantly and simply'''.}}

@@ Line 10: / Line 10: @@
 [[File:Sig_T_2_3_S1_Version3.png|right|frame|Example of a harmonic oscillation]]
-Harmonic oscillations are of particular importance for communications engineering as well as in many natural sciences.&nbsp; The diagram shows an exemplary signal waveform.
+Harmonic oscillations are of particular importance for Communications Engineering as well as in many natural sciences.&nbsp; The diagram shows an exemplary signal waveform.
 Its importance is also related to the fact that the harmonic oscillation represents the solution of a&nbsp; differential equation&nbsp; which is found in many disciplines and reads as follows:
@@ Line 26: / Line 26: @@
 The following signal parameters are used:
-*the&nbsp; &raquo;'''amplitude}'''&laquo;nbsp; $C$&nbsp; &ndash; simultaneously the maximum value of the signal,
+*the&nbsp; &raquo;'''amplitude'''&laquo;&nbsp; $C$&nbsp; &ndash; simultaneously the maximum value of the signal,
 *the&nbsp; &raquo;'''signal frequency'''&laquo;&nbsp; $f_{0}$ &nbsp; &rArr; &nbsp; the reciprocal of the period duration&nbsp; $T_{0}$, and
-*the&nbsp; &raquo;'''zero phase angle'''&laquo;&nbsp; (or briefly the "phase") &nbsp; $\varphi$&nbsp; of the oscillation.}}
+*the&nbsp; &raquo;'''zero phase angle'''&laquo;&nbsp; $($or briefly the&nbsp; &raquo;phase&laquo;$)$ &nbsp; $\varphi$&nbsp; of the oscillation.}}
-The following (German language) learning video illustrates the properties of harmonic oscillations using scales: <br>
+The following&nbsp; $($German-language$)$&nbsp; learning video illustrates the properties of harmonic oscillations using scales: <br>
-&nbsp; &nbsp; [[Harmonische_Schwingungen_(Lernvideo)|"Harmonische Schwingungen"]] &nbsp; &rArr; &nbsp; "Harmonic Oscillations".
+&nbsp; &nbsp; &nbsp; [[Harmonische_Schwingungen_(Lernvideo)|&raquo;Harmonische Schwingungen&laquo;]] &nbsp; &rArr; &nbsp; "Harmonic Oscillations".
 {{BlaueBox|TEXT=
 $\text{Comments on nomenclature:}$&nbsp;
-In this tutorial - as usual in other literature - when describing harmonic oscillations, Fourier series and Fourier integral, the phase is entered into the equations with negative sign, whereas in connection with all modulation methods the phase is entered with a plus sign.
+In this tutorial &ndash; as usual in other literature when describing harmonic oscillations,&nbsp; Fourier series and Fourier integral &ndash;  the phase is entered into the equations with negative sign,&nbsp; whereas in connection with all modulation methods the phase is entered with a plus sign.
 *To distinguish between the two variants we use in our tutorial &nbsp; $\varphi$&nbsp; and&nbsp; $\phi$.&nbsp; Both symbols denote the small Greek "phi".&nbsp; The spelling&nbsp; $\varphi$&nbsp; is mainly used in the German and&nbsp; $\phi$&nbsp; in the anglo&ndash;american language area.
@@ Line 49: / Line 49: @@
 ==Time domain representation==
 <br>
-The amplitude&nbsp; $C$&nbsp; can be read directly from the adjacent graphic.&nbsp; The signal frequency $f_0$&nbsp; is the reciprocal of the period duration&nbsp; $T_0$.&nbsp;
+The amplitude&nbsp; $C$&nbsp; can be read directly from the adjacent graph.&nbsp; The signal frequency $f_0$&nbsp; is the reciprocal of the period duration&nbsp; $T_0$.&nbsp;
 [[File:P_ID129__Sig_T_2_3_S2a.png|right|frame|Signal parameters of a harmonic oscillation]]
@@ Line 62: / Line 62: @@
 *For a cosine signal both the parameters&nbsp; $\tau$&nbsp; and&nbsp; $\varphi$&nbsp; are zero.
-*In contrast, a sinusoidal signal&nbsp; is shifted by&nbsp; $\tau = T_0/4$&nbsp; and accordingly applies to the zero phase angle&nbsp; $\varphi = \pi/2$&nbsp; (in radians) or &nbsp; $90^{\circ}$.
+*In contrast,&nbsp; a sinusoidal signal&nbsp; is shifted by&nbsp; $\tau = T_0/4$&nbsp; and accordingly applies to the zero phase angle&nbsp; $\varphi = \pi/2$&nbsp; $($in radians$)$&nbsp; or &nbsp; $90^{\circ}$.
-So it can be stated, that - as assumed for the above sketch - at a positive value of&nbsp; $\tau$&nbsp; resp. &nbsp; $\varphi$&nbsp; the $($referring&nbsp; $t = 0)$&nbsp; nearest signal maximum comes later than at the cosine signal and at negative values earlier.
+*If a cosine signal is present at the system input and the output signal is delayed by a value&nbsp; $\tau$&nbsp; then&nbsp; $\tau$&nbsp; is also called the&nbsp; &raquo;runtime&laquo;&nbsp; of the system.
-If a cosine signal is present at the system input and the output signal is delayed by a value&nbsp; $\tau$&nbsp; then&nbsp; $\tau$&nbsp; is also called the&nbsp; "runtime"&nbsp; of the system.
+*Since a harmonic oscillation is clearly defined by three parameters,&nbsp; the entire time course from&nbsp; $-\infty$&nbsp; to&nbsp; $+\infty$&nbsp; can be calculated  analytically from only three signal values&nbsp; $x_1=x(t_1)$,&nbsp; $x_2=x(t_2)$,&nbsp; $x_3=x(t_3)$&nbsp; if the times&nbsp; $t_1$,&nbsp; $t_2$&nbsp; and&nbsp; $t_3$&nbsp; have been determined appropriately.
 {{GraueBox|TEXT=
@@ Line 85: / Line 85: @@
 :$$C \cdot \cos(2\pi  \hspace{0.05 cm} f_0  \hspace{0.05 cm} t_3 - \varphi) = +1.227\hspace{0.05 cm}.$$
-After solving this non-linear system of equations the following signal parameters are obtained:
+After solving this nonlinear system of equations the following signal parameters are obtained:
 *Signal amplitude:&nbsp; $C = 2$,
@@ Line 93: / Line 93: @@
-If you set all sampling times&nbsp; $t_1$,&nbsp; $t_2$,&nbsp; $t_3$&nbsp; in maxima, minima and/or zeros, there is no unique solution for the non-linear equation system.}}
+If you set all sampling times&nbsp; $t_1$,&nbsp; $t_2$,&nbsp; $t_3$&nbsp; in maxima, minima and/or zeros, there is no unique solution for the nonlinear equation system.}}
@@ Line 102: / Line 102: @@
 :$$x(t)=A\cdot\cos(2\pi  f_0 t)+ B\cdot\sin(2\pi f_0 t).$$
-*The terms&nbsp; $A$&nbsp; and&nbsp; $B$&nbsp; for the amplitudes of the cosine and sine components are chosen to match the nomenclature of the following chapter&nbsp; [[Signal_Representation/Fourier_Series|"Fourier Series"]].
+*The terms&nbsp; $A$&nbsp; and&nbsp; $B$&nbsp; for the amplitudes of the cosine and sine components are chosen to match the nomenclature of the following chapter&nbsp; [[Signal_Representation/Fourier_Series|&raquo;Fourier Series&laquo;]].
 *By applying trigonometric transformations we obtain from the illustration in the last section:
@@ Line 110: / Line 110: @@
 *From this follows directly by equating the coefficients:
-:$$A=C\cdot\cos(\varphi),\hspace{0.5cm}B=C\cdot\sin(\varphi).$$
+:$$A=C\cdot\cos(\varphi),$$
 *The magnitude and the zero phase angle of the harmonic oscillation can be calculated from the parameters&nbsp; $A$&nbsp; and&nbsp; $B$&nbsp; also according to simple trigonometric considerations:
-:$$C=\sqrt{A^2+B^2}, \hspace{0.5 cm}\varphi = \arctan\left({-B}/{A}\right).$$
+:$$C=\sqrt{A^2+B^2},$$
 {{BlaueBox|TEXT=
 $\text{Please note:}$&nbsp;
-The minus-sign at the calculation of the zero phase angle&nbsp; $\varphi$&nbsp; is related to the fact, that&nbsp; $\varphi$&nbsp; enters the argument of the cosine function with negative sign.&nbsp; If one would use the notation&nbsp; $\cos(2\pi f_0 t  +\phi)$&nbsp; instead of&nbsp; $\cos(2\pi f_0 t - \varphi)$&nbsp;, then&nbsp; $\phi= \arctan(B/A)$. Note the following here:
+The minus&ndash;sign at the calculation of the zero phase angle&nbsp; $\varphi$&nbsp; is related to the fact&nbsp; that&nbsp; $\varphi$&nbsp; enters the argument of the cosine function with negative sign.&nbsp; If one would use the notation&nbsp; $\cos(2\pi f_0 t  +\phi)$&nbsp; instead of&nbsp; $\cos(2\pi f_0 t - \varphi)$,&nbsp; then&nbsp; $\phi= \arctan(B/A)$.&nbsp; Note the following here:
-*For Fourier series and Fourier integral, the&nbsp; $\varphi$&nbsp; representation is common in literature.
+#For Fourier series and Fourier integral, the&nbsp; $\varphi$&nbsp; representation is common in literature.
-*For the description of the modulation methods, however, the&nbsp; $\phi$&nbsp; representation is almost always used.}}
+#For description of the modulation methods, however, the&nbsp; $\phi$&nbsp; representation is almost always used.}}
 {{GraueBox|TEXT=
-[[File:Sig_T_2_3_S3_version2.png|right|frame|Harmonic oscillation. On the right: Represented in the complex plane]]
+$\text{Example 2:}$&nbsp; The oscillation shown in the left graphic as time course is characterized by the parameters[[File:Sig_T_2_3_S3_version2.png|right|frame|Harmonic oscillation. On the right: Represented in the complex plane]]
+*$C=2,$
-$\text{Example 2:}$&nbsp;
+*$f_0 = 125 \;{\rm Hz}$,
 *$\varphi = +135^{\circ}$  &nbsp; ⇒  &nbsp; $\phi = -135^{\circ}$.

@@ Line 208: / Line 208: @@
 :$$x(t)=B\cdot \sin(2\pi f_0 t)\;\ \circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\; \ X(f)={\rm j} \cdot \big  [ {B}/{2} \cdot \delta (f+f_0)- {B}/{2} \cdot \delta (f-f_0) \big  ].$$
 [[File: P_ID2719__Sig_T_2_3_S5.png |right|frame|Spectrum of a sinusoidal signal]]
 {{GraueBox|TEXT=
 $\text{Example 4:}$&nbsp;
 The figure shows the imaginary spectrum of a sine wave&nbsp; $x(t)$&nbsp; with
 *amplitude&nbsp; $B = 3 \;{\rm V},$
 *frequency&nbsp; $f_0 = 5\;{\rm kHz},$
 *zero phase angle&nbsp;  $\varphi=90^{\circ}$  &nbsp; ⇒  &nbsp; $\phi = -90^{\circ}$.
 <br>$\text{Please note}$:
-*For the positive frequency&nbsp; $(f = +f_0)$&nbsp; the imaginary part is negative.
+#For the positive frequency&nbsp; $(f = +f_0)$&nbsp; the imaginary part is negative.<br>
+#The negative frequency&nbsp; $(f = -f_0)$&nbsp; results in a positive imaginary part.}}
@@ Line 237: / Line 239: @@
 $\text{One recognizes:}$&nbsp; The spectral function&nbsp; $X(f)$
 *is not only defined for positive and negative frequencies,
 *but it is also complex-valued.}}
-[[File:Sig_T_2_3_S6_version2.png|right|frame|Spectrum of the&nbsp; $\varphi=30^{\circ}$&nbsp; oscillation]]
+{{GraueBox|TEXT=
-{{GraueBox|TEXT=
+[[File:Sig_T_2_3_S6_version2.png|right|frame|Spectrum of the&nbsp; $\varphi=30^{\circ}$&nbsp; oscillation]]
 $\text{Example 5:}$&nbsp;
 With the parameters&nbsp; $C = 5 \;{\rm V}$,&nbsp; $f_0 = 5\;{\rm kHz}$&nbsp; and&nbsp; $\varphi=30^{\circ}$&nbsp; $($in radian measure $\pi/6)$&nbsp; results
@@ Line 252: / Line 255: @@
 Because of:
-:$$2.5 · \cos(30^{\circ}) = 2.165,  \hspace{0.3cm} 2.5 · \sin(30^{\circ}) = 1.25.$$}}
+:$$2.5 · \cos(30^{\circ}) = 2.165,$$

@@ Line 171: / Line 171: @@
 Then the following relation to the above equation applies:
-*The Dirac '''KORREKTUR: delta?''' function at&nbsp; $-f_0$&nbsp; belongs to the first term <br>$($derivable from the condition&nbsp; $f + f_0 = 0)$.
+*The Dirac delta function at&nbsp; $-f_0$&nbsp; belongs to the first term <br>$($derivable from the condition&nbsp; $f + f_0 = 0)$.
-*The Dirac '''KORREKTUR: delta?''' function for&nbsp; $+f_0$&nbsp; belongs to the second term <br>$($derivable from the condition $f - f_0 = 0)$.
+*The Dirac delta function for&nbsp; $+f_0$&nbsp; belongs to the second term <br>$($derivable from the condition $f - f_0 = 0)$.
 *Both impulse weights are&nbsp; $A/2 = 2 \;{\rm V}$.}}