Aufgaben:Exercise 4.3Z: Hilbert Transformator - Revision history

Guenter at 14:13, 24 May 2021

2021-05-24T14:13:04Z

Guenter at 14:40, 6 May 2021

2021-05-06T14:40:13Z

Guenter at 14:29, 6 May 2021

2021-05-06T14:29:24Z

Guenter at 14:10, 6 May 2021

2021-05-06T14:10:16Z

Javier: Text replacement - "bandpass" to "band-pass"

2021-05-03T15:12:59Z

Text replacement - "bandpass" to "band-pass"

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

2021-03-23T12:39:50Z

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

Noah at 20:11, 5 February 2021

2021-02-05T20:11:02Z

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Oezdemir: Oezdemir moved page Aufgaben:Aufgabe 4.3Z: Hilbert-Transformator to Aufgaben:Exercise 4.3Z: Hilbert Transformator

2020-11-24T19:41:34Z

Oezdemir moved page Aufgabe 4.3Z: Hilbert-Transformator to Exercise 4.3Z: Hilbert Transformator

Oezdemir at 19:39, 24 November 2020

2020-11-24T19:39:28Z

Javier: Text replacement - "Signaldarstellung/Analytisches Signal und zugehörige Spektralfunktion" to "Signal_Representation/Analytical_Signal_and_Its_Spectral_Function"

2020-09-01T10:55:40Z

Text replacement - "Signaldarstellung/Analytisches Signal und zugehörige Spektralfunktion" to "Signal_Representation/Analytical_Signal_and_Its_Spectral_Function"

@@ Line 3: / Line 3: @@
 }}
-[[File:P_ID717__Sig_Z_4_3_neu.png|right|frame|Hilbert transformer ]]
+[[File:P_ID717__Sig_Z_4_3_neu.png|right|frame|Hilbert transformator ]]
 The diagram describes a model of how, at least mentally,
 *the analytical signal&nbsp; $x_{+}(t)$&nbsp; can be generated,

@@ Line 71: / Line 71: @@
 :$$X_{\rm +}(f)= \left(1 + {\rm
 sign}(f)\right) \cdot  X(f)$$
-:shows that&nbsp; $H_{\rm HT}(f) = - {\rm j} \cdot \sign(f)$&nbsp; ist.
+:shows that&nbsp; $H_{\rm HT}(f) = - {\rm j} \cdot \sign(f)$.
 *Thus, the real part we are looking for is&nbsp; ${\rm Re}[X_{\rm +}(f)]\hspace{0.15cm}\underline{=0}$&nbsp; and the imaginary part is equal to&nbsp; ${\rm Im}[X_{\rm +}(f)]\hspace{0.15cm}\underline{=-1}$.
@@ Line 93: / Line 93: @@
 :$$Y_2(f) = -{A}/{2}\cdot\delta (f + f_{0})-
 {A}/{2}\cdot\delta (f - f_{0}).$$
-*It follows that $y_2(t) = - A \cdot \cos(2\pi f_0 t)$ und $y_2(t = 0)\;  \underline{= -\hspace{-0.08cm}1 \,\text{V}}$.
+*It follows that&nbsp; $y_2(t) = - A \cdot \cos(2\pi f_0 t)$&nbsp; and&nbsp; $y_2(t = 0)\;  \underline{= -\hspace{-0.08cm}1 \,\text{V}}$.
@@ Line 104: / Line 104: @@
 \Rightarrow \hspace{0.3cm}y_3(t) = A \cdot  {\cos} ( 2 \pi f_0 t - 3\pi/4).$$
 *The signal phase is thus&nbsp; $\varphi = \pi /4$.
-*The Hilbert transformer delays this by&nbsp; $\varphi_{\rm HT} \;  \underline{= 90^\circ} \; (\pi /2)$&nbsp; verzögert.
+*The Hilbert transformer delays this by&nbsp; $\varphi_{\rm HT} \;  \underline{= 90^\circ} \; (\pi /2)$.
 *Therefore, the output signal&nbsp; $y_3(t) = A \cdot \cos(2\pi f_0 t -3 \pi /4)$&nbsp; and the signal value at time&nbsp; $t = 0$&nbsp; is&nbsp; $A \cdot \cos(135^\circ) \; \underline{= -0.707  \,\text{V}}$.
@@ Line 117: / Line 117: @@
 :$$X_{3+}(f) =  {A_0} \cdot {\rm e}^{-{\rm j} \varphi}\cdot\delta (f
 - f_{\rm 0})  .$$
-*By applying the&nbsp; [[Signal_Representation/Fourier_Transform_Laws#Verschiebungssatz|Verschiebungssatzes]]&nbsp;, the associated time function with&nbsp; $\varphi = \pi /4$ is thus::
+*By applying the&nbsp; "Shifting Theorem"&nbsp;, the associated time function with&nbsp; $\varphi = \pi /4$ is:
 :$$x_{3+}(t) = A_0 \cdot {\rm e}^{{\rm j}( 2 \pi f_{\rm 0} t
 \hspace{0.05cm}-\hspace{0.05cm} \varphi)}.$$
@@ Line 125: / Line 125: @@
 \cdot{\sin} ( 45^\circ)= \hspace{0.15 cm}\underline{{\rm 0.707 \hspace{0.05cm} V}-{\rm
 j}\cdot {\rm 0.707 \hspace{0.05cm} V}}.$$
 ''Hint'': &nbsp;
-*To get from&nbsp; $x(t)$&nbsp; to&nbsp; $x_+(t)$&nbsp; , just replace the cosine function with the complex exponential function.
+*To get from&nbsp; $x(t)$&nbsp; to&nbsp; $x_+(t)$,&nbsp; just replace the cosine function with the complex exponential function.
 *For example, the following applies to a harmonic oscillation:
 :$$x(t) = A \cdot  {\cos} ( 2 \pi f_0 t

@@ Line 4: / Line 4: @@
 [[File:P_ID717__Sig_Z_4_3_neu.png|right|frame|Hilbert transformer ]]
-The diagram describes a model of how, at least mentally
+The diagram describes a model of how, at least mentally,
-*from the real band-pass signal&nbsp; $x(t)$
+*the analytical signal&nbsp; $x_{+}(t)$&nbsp; can be generated,
-*the analytical signal&nbsp; $x_{+}(t)$
+*from the real band-pass signal&nbsp; $x(t)$.
-can be generated.
+Its output signal&nbsp; $y(t)$&nbsp;  is multiplied by the imaginary unit&nbsp; $\rm j$&nbsp; and added to the signal&nbsp; $x(t)$&nbsp;:
 :$$x_{\rm +}(t)= x(t) + {\rm j}\cdot y(t)  .$$
 As test signals are used, each with&nbsp; $A = 1 \, \text{V}$&nbsp; and&nbsp; $f_0 = 10 \, \text{kHz}$:
@@ Line 17: / Line 17: @@
 :$$x_2(t) = A \cdot  {\sin} ( 2 \pi f_0 t ),$$
 :$$x_3(t) = A \cdot  {\cos} \big( 2 \pi f_0 (t - \tau) \big) \hspace{0.3cm}{\rm with}\hspace{0.3cm}\tau = 12.5 \hspace{0.1cm}{\rm &micro; s}.$$
@@ Line 26: / Line 23: @@
 ''Hints:''
-*This exercise belongs to the task&nbsp; [[Signal_Representation/Analytical_Signal_and_Its_Spectral_Function|Analytical Signal and Its Spectral Function]].
+*This exercise belongs to the chapter&nbsp; [[Signal_Representation/Analytical_Signal_and_Its_Spectral_Function|Analytical Signal and its Spectral Function]].
 *The following applies to the spectral function of the analytical signal:
@@ Line 35: / Line 32: @@
 <quiz display=simple>
-{Calculate the frequency response&nbsp; $H_{HT}(f)$&nbsp; of the Hilbert transformer. Which value is valid for the frequency&nbsp; $f_0 = 10 \text{ kHz}$?
+{Calculate the frequency response&nbsp; $H_{\rm HT}(f)$&nbsp; of the Hilbert transformer.&nbsp; Which value is valid for the frequency&nbsp; $f_0 = 10 \text{ kHz}$?
 |type="{}"}
 $\text{Re}[H_{\rm HT}(f = f_0)]\ = \ $  { 0. }
@@ Line 41: / Line 38: @@
-{What is the Hilbert transform&nbsp; $y_1(t)$&nbsp; for the input signal&nbsp; $x_1(t)$? In particular, what value results at&nbsp; $t = 0$?
+{What is the Hilbert transform&nbsp; $y_1(t)$&nbsp; for the input signal&nbsp; $x_1(t)$?&nbsp; In particular, what value results at&nbsp; $t = 0$?
 |type="{}"}
 $y_1(t = 0)\ = \ $ { 0. } &nbsp;$\rm V$
-{What is the Hilbert transform&nbsp; $y_2(t)$&nbsp; for the input signal&nbsp; $x_2(t)$? Which value results in particular at&nbsp; $t = 0$?
+{What is the Hilbert transform&nbsp; $y_2(t)$&nbsp; for the input signal&nbsp; $x_2(t)$?&nbsp; Which value results in particular at&nbsp; $t = 0$?
 |type="{}"}
 $y_2(t = 0)\ = \ $ { -1.03--0.97 } &nbsp;$\rm V$
-{What is the Hilbert transform&nbsp; $y_3(t)$&nbsp; for the input signal&nbsp; $x_3(t)$? What value results for&nbsp; $t=0$? <br>What is the phase delay&nbsp; $\varphi_{\rm HT}$&nbsp; of the Hilbert transformer?
+{What is the Hilbert transform&nbsp; $y_3(t)$&nbsp; for the input signal&nbsp; $x_3(t)$?&nbsp; What value results for&nbsp; $t=0$?&nbsp; What is the phase delay&nbsp; $\varphi_{\rm HT}$&nbsp; of the Hilbert transformer?
 |type="{}"}
 $\varphi_{\rm HT}\ = \ $ { 90 3% } &nbsp;$\text{Grad}$
@@ Line 57: / Line 54: @@
-{What is the analytical signal associated with&nbsp; $x_3(t)$&nbsp; gehörige analytische Signal? What are the values of the real and imaginary parts of this complex signal at time&nbsp; $t = 0$?
+{What is the analytical signal associated with&nbsp; $x_3(t)$?&nbsp; What are the values of the real and imaginary parts of this complex signal at time&nbsp; $t = 0$?
 |type="{}"}
 $\text{Re}[x_{3+}(t = 0)]\ = \ $ { 0.707 3% } &nbsp;$\text{V}$

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@@ Line 5: / Line 5: @@
 [[File:P_ID717__Sig_Z_4_3_neu.png|right|frame|Hilbert transformer ]]
 The diagram describes a model of how, at least mentally
-*from the real bandpass signal&nbsp; $x(t)$
+*from the real band-pass signal&nbsp; $x(t)$
 *the analytical signal&nbsp; $x_{+}(t)$

← Older revision		Revision as of 12:39, 23 March 2021
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← Older revision	Revision as of 19:41, 24 November 2020
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