Aufgaben:Exercise 4.2: Low-Pass for Signal Reconstruction - Revision history

Hwang at 14:31, 18 January 2023

2023-01-18T14:31:19Z

Guenter at 14:04, 11 April 2022

2022-04-11T14:04:56Z

Guenter at 16:03, 7 April 2022

2022-04-07T16:03:28Z

Guenter at 15:39, 7 April 2022

2022-04-07T15:39:21Z

Noah at 20:52, 4 April 2022

2022-04-04T20:52:52Z

Noah at 15:57, 19 March 2022

2022-03-19T15:57:42Z

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Guenter at 13:49, 16 March 2022

2022-03-16T13:49:51Z

Guenter: Guenter moved page Aufgaben:Aufgabe 4.2: Tiefpass zur Signalrekonstruktion to Aufgaben:Exercise 4.2: Low-Pass for Signal Reconstruction

2022-03-16T13:41:27Z

Guenter moved page Aufgabe 4.2: Tiefpass zur Signalrekonstruktion to Exercise 4.2: Low-Pass for Signal Reconstruction

Javier: Text replacement - "Category:Aufgaben zu Modulationsverfahren" to "Category:Modulation Methods: Exercises"

2021-03-23T12:46:34Z

Text replacement - "Category:Aufgaben zu Modulationsverfahren" to "Category:Modulation Methods: Exercises"

Javier: Text replacement - "[[Modulationsverfahren" to "[[Modulation_Methods"

2020-07-09T12:53:11Z

Text replacement - "[[Modulationsverfahren" to "[[Modulation_Methods"

@@ Line 65: / Line 65: @@
 {{ML-Kopf}}
 '''(1)'''&nbsp; Only the&nbsp; <u>first statement</u>&nbsp; is correct:
-*Sampling&nbsp; $q_{\rm dis}(t)$&nbsp; with sampling frequency&nbsp; $f_{\rm A} = 8 \ \rm kHz$&nbsp; leads to an irreversible error,&nbsp; since&nbsp; $Q_{\rm dis}(f)$&nbsp; involves a discrete spectral component&nbsp; (Dirac line)&nbsp; at&nbsp; $f_4 = 4\ \rm kHz$&nbsp; and the phase value is&nbsp; $φ_4 ≠ 0$.
+*Sampling&nbsp; $q_{\rm dis}(t)$&nbsp; with sampling frequency&nbsp; $f_{\rm A} = 8 \ \rm kHz$&nbsp; leads to an irreversible error,&nbsp; since&nbsp; $Q_{\rm dis}(f)$&nbsp; involves a discrete spectral component&nbsp; (Dirac delta line)&nbsp; at&nbsp; $f_4 = 4\ \rm kHz$&nbsp; and the phase value is&nbsp; $φ_4 ≠ 0$.
 *With the phase value&nbsp; $φ_4 = 90^\circ$&nbsp; $(4 \ \rm kHz$&nbsp; sinusoidal component$)$&nbsp;  given here holds&nbsp; $ε_{\rm dis}(t) = v_{\rm dis}(t) - q_{\rm dis}(t) = -0. 4 \ \rm V - \sin(2π \cdot f_4 \cdot t)$.&nbsp; See also solution to Exercise 4.2Z.
 *On the other hand,&nbsp; the signal&nbsp; $q_{\rm con}(t)$&nbsp; with the continuous spectrum&nbsp; $Q_{\rm con}(f)$&nbsp; can also then be measured with a rectangular low-pass filter&nbsp; $($with cutoff frequency&nbsp; $f_{\rm G} = 4\ \rm kHz)$&nbsp; be completely reconstructed if sampling frequency&nbsp; $f_{\rm A} = 8\ \rm kHz$&nbsp; was used. &nbsp; For all frequencies not equal to&nbsp; $f_4$&nbsp; the sampling theorem is satisfied.
-*The contribution of the&nbsp; $f_4$ component to the total spectrum&nbsp; $Q_{\rm con}(f)$&nbsp; is only vanishingly small &nbsp; ⇒ &nbsp; ${\rm Pr}(f_4) → 0$&nbsp; as long as the spectrum has no Dirac line  at&nbsp; $f_4$.
+*The contribution of the&nbsp; $f_4$ component to the total spectrum&nbsp; $Q_{\rm con}(f)$&nbsp; is only vanishingly small &nbsp; ⇒ &nbsp; ${\rm Pr}(f_4) → 0$&nbsp; as long as the spectrum has no Dirac delta line  at&nbsp; $f_4$.

@@ Line 3: / Line 3: @@
 }}
-[[File:P_ID1608__Mod_A_4_2.png|right|frame|Examples of continuous and discrete spectra   '''Korrektur''']]
+[[File:EN_Mod_A_4_2.png|right|frame|Examples of continuous and discrete spectra]]
 We consider in this exercise two different source signals&nbsp; $q_{\rm con}(t)$&nbsp; and&nbsp; $q_{\rm dis}(t)$ whose magnitude spectra&nbsp; $|Q_{\rm con}(f)|$&nbsp; and&nbsp; $|Q_{\rm dis}(f)|$&nbsp; are plotted. &nbsp; The highest frequency occurring in the signals is in each case&nbsp; $4 \rm kHz$.
 * Nothing more is known of the spectral function&nbsp; $Q_{\rm con}(f)$&nbsp; than that it is a continuous spectrum,&nbsp; where:
@@ Line 85: / Line 85: @@
-[[File:P_ID1609__Mod_A_4_2d.png|P_ID1609__Mod_A_4_2d.png|right|frame|Signal reconstruction with too large cutoff frequency '''Korrektur''']]
+[[File:EN_Mod_A_4_2d_neu.png|P_ID1609__Mod_A_4_2d.png|right|frame|Signal reconstruction with too large cutoff frequency]]
 '''(4)'''&nbsp; The correct solution here is&nbsp; <u>suggested solution 3</u>:
 *Sampling with&nbsp; $f_{\rm A} = 10\ \rm kHz$&nbsp; yields the periodic spectrum sketched on the right.

@@ Line 64: / Line 64: @@
 ===Solution===
 {{ML-Kopf}}
-'''(1)'''&nbsp; Only the <u>first statement</u> is correct:
+'''(1)'''&nbsp; Only the&nbsp; <u>first statement</u>&nbsp; is correct:
-*Sampling&nbsp; $q_{\rm dis}(t)$&nbsp; with sampling frequency&nbsp; $f_{\rm A} = 8 \ \rm kHz$&nbsp; leads to an irreversible error, since&nbsp; $Q_{\rm dis}(f)$&nbsp; involves a discrete spectral component (diracline) at&nbsp; $f_4 = 4\ \rm kHz$&nbsp; and the phase value&nbsp; $φ_4 ≠ 0$&nbsp; is.
+*Sampling&nbsp; $q_{\rm dis}(t)$&nbsp; with sampling frequency&nbsp; $f_{\rm A} = 8 \ \rm kHz$&nbsp; leads to an irreversible error,&nbsp; since&nbsp; $Q_{\rm dis}(f)$&nbsp; involves a discrete spectral component&nbsp; (Dirac line)&nbsp; at&nbsp; $f_4 = 4\ \rm kHz$&nbsp; and the phase value is&nbsp; $φ_4 ≠ 0$.
-*With the phase value given here&nbsp; $φ_4 = 90^\circ$&nbsp; $(4 \ \rm kHz$- sinusoidal component$)$&nbsp; holds&nbsp; $ε_{\rm dis}(t) = v_{\rm dis}(t) - q_{\rm dis}(t) = -0. 4 \ \rm V - \sin(2π \cdot f_4 \cdot t)$.&nbsp; See also sample solution to exercise 4.2Z.
+*With the phase value&nbsp; $φ_4 = 90^\circ$&nbsp; $(4 \ \rm kHz$&nbsp; sinusoidal component$)$&nbsp;  given here holds&nbsp; $ε_{\rm dis}(t) = v_{\rm dis}(t) - q_{\rm dis}(t) = -0. 4 \ \rm V - \sin(2π \cdot f_4 \cdot t)$.&nbsp; See also solution to Exercise 4.2Z.
-*On the other hand, the signal&nbsp; $q_{\rm kon}(t)$&nbsp; with the continuous spectrum&nbsp; $Q_{\rm con}(f)$&nbsp; can also then be measured with a square-wave low-pass filter&nbsp; $($with cutoff frequency&nbsp; $f_{\rm G} = 4\ \rm kHz)$&nbsp; be completely reconstructed if sampling frequency&nbsp; $f_{\rm A} = 8\ \rm kHz$&nbsp; was used. &nbsp; For all frequencies not equal to&nbsp; $f_4$&nbsp; the sampling theorem is satisfied.
+*On the other hand,&nbsp; the signal&nbsp; $q_{\rm con}(t)$&nbsp; with the continuous spectrum&nbsp; $Q_{\rm con}(f)$&nbsp; can also then be measured with a rectangular low-pass filter&nbsp; $($with cutoff frequency&nbsp; $f_{\rm G} = 4\ \rm kHz)$&nbsp; be completely reconstructed if sampling frequency&nbsp; $f_{\rm A} = 8\ \rm kHz$&nbsp; was used. &nbsp; For all frequencies not equal to&nbsp; $f_4$&nbsp; the sampling theorem is satisfied.
-*But the contribution of the&nbsp; $f_4$ component to the total spectrum&nbsp; $Q_{\rm con}(f)$&nbsp; is only vanishingly small &nbsp; ⇒ &nbsp; ${\rm Pr}(f_4) → 0$ as long as the spectrum at&nbsp; $f_4$&nbsp; has no diracline.
+*The contribution of the&nbsp; $f_4$ component to the total spectrum&nbsp; $Q_{\rm con}(f)$&nbsp; is only vanishingly small &nbsp; ⇒ &nbsp; ${\rm Pr}(f_4) → 0$&nbsp; as long as the spectrum has no Dirac line  at&nbsp; $f_4$.
-'''(2)'''&nbsp; Only the <u>proposed solution 1</u> is correct:
+'''(2)'''&nbsp; Only the&nbsp; <u>proposed solution 1</u>&nbsp; is correct:
 *With&nbsp; $f_{\rm A} = 10\ \rm kHz$&nbsp; the sampling theorem is satisfied in both cases.
-*With&nbsp; $f_{\rm G} = f_{\rm A} /2$&nbsp; both error signals&nbsp; $ε_{\rm con}(t)$ and $ε_{\rm dis}(t)$ are identically zero.
+*With&nbsp; $f_{\rm G} = f_{\rm A} /2$&nbsp; both error signals&nbsp; $ε_{\rm con}(t)$&nbsp; and&nbsp; $ε_{\rm dis}(t)$&nbsp; are identically zero.
-*In addition, the signal reconstruction also works as long as&nbsp; $f_{\rm G} > 4 \ \rm kHz$&nbsp; and&nbsp; $f_{\rm G} < 6 \ \rm kHz$&nbsp; holds.
+*In addition,&nbsp; the signal reconstruction also works as long as&nbsp; $f_{\rm G} > 4 \ \rm kHz$&nbsp; and&nbsp; $f_{\rm G} < 6 \ \rm kHz$&nbsp; holds.
-'''(3)'''&nbsp; The correct solution here is <u>suggested solution 2</u>:
+'''(3)'''&nbsp; The correct solution here is&nbsp; <u>suggested solution 2</u>:
-*With&nbsp; $f_{\rm G} = 3.5 \ \rm kHz$&nbsp; the lowpass incorrectly removes the&nbsp; $4\ \rm kHz$ component, that is, then holds:
+*With&nbsp; $f_{\rm G} = 3.5 \ \rm kHz$&nbsp; the low-pass incorrectly removes the&nbsp; $4\ \rm kHz$ component,&nbsp; that is,&nbsp; then holds:
 :$$ v_{\rm dis}(t) = q_{\rm dis}(t) - 0.4\,{\rm V} \cdot \sin (2 \pi \cdot f_{\rm 4} \cdot t)\hspace{0.3cm}\Rightarrow \hspace{0.3cm} \varepsilon_{\rm dis}(t) = - 0.4\,{\rm V} \cdot \sin (2 \pi \cdot f_{\rm 4} \cdot t)\hspace{0.05cm}.$$
-[[File:P_ID1609__Mod_A_4_2d.png|P_ID1609__Mod_A_4_2d.png|right|frame|Signal reconstruction with too large cutoff frequency]]
+[[File:P_ID1609__Mod_A_4_2d.png|P_ID1609__Mod_A_4_2d.png|right|frame|Signal reconstruction with too large cutoff frequency '''Korrektur''']]
-'''(4)'''&nbsp; The correct solution here is <u>suggested solution 3</u>:
+'''(4)'''&nbsp; The correct solution here is&nbsp; <u>suggested solution 3</u>:
 *Sampling with&nbsp; $f_{\rm A} = 10\ \rm kHz$&nbsp; yields the periodic spectrum sketched on the right.
-*The low pass with&nbsp; $f_{\rm G} = 6.5 \ \rm kHz$&nbsp; removes all discrete frequency components with&nbsp; $|f| ≥ 7\ \rm kHz$, but not the&nbsp; $6\ \rm kHz$ component.
+*The low-pass with&nbsp; $f_{\rm G} = 6.5 \ \rm kHz$&nbsp; removes all discrete frequency components with&nbsp; $|f| ≥ 7\ \rm kHz$,&nbsp; but not the&nbsp; $6\ \rm kHz$ component.
@@ Line 94: / Line 94: @@
 * the frequency&nbsp; $f_6 = f_{\rm A} - f_4 = 6\ \rm kHz$,
 * the amplitude&nbsp; $A_4$ of the&nbsp; $f_4$ component,
-* the phase&nbsp; $φ_{-4} = -φ_4$&nbsp; of the&nbsp; $Q(f)$ component at&nbsp; $f = -f_4$.
+* the phase&nbsp; $φ_{-4} = -φ_4$&nbsp; of the&nbsp; $Q(f)$&nbsp; component at&nbsp; $f = -f_4$.
 {{ML-Fuß}}

@@ Line 3: / Line 3: @@
 }}
-[[File:P_ID1608__Mod_A_4_2.png|right|frame|Continuous and discrete spectrum&nbsp; (examples)]]
+[[File:P_ID1608__Mod_A_4_2.png|right|frame|Examples of continuous and discrete spectra   '''Korrektur''']]
-We consider in this exercise two different source signals&nbsp; $q_{\rm kon}(t)$&nbsp; and&nbsp; $q_{\rm dis}(t)$ whose magnitude spectra&nbsp; $|Q_{\rm kon}(f)|$&nbsp; and&nbsp; $|Q_{\rm dis}(f)|$&nbsp; are plotted. &nbsp; The highest frequency occurring in the signals is in each case&nbsp; $4 \rm kHz$.
+We consider in this exercise two different source signals&nbsp; $q_{\rm con}(t)$&nbsp; and&nbsp; $q_{\rm dis}(t)$ whose magnitude spectra&nbsp; $|Q_{\rm con}(f)|$&nbsp; and&nbsp; $|Q_{\rm dis}(f)|$&nbsp; are plotted. &nbsp; The highest frequency occurring in the signals is in each case&nbsp; $4 \rm kHz$.
-* Nothing more is known of the spectral function&nbsp; $Q_{\rm kon}(f)$&nbsp; than that it is a continuous spectrum, where:
+* Nothing more is known of the spectral function&nbsp; $Q_{\rm con}(f)$&nbsp; than that it is a continuous spectrum,&nbsp; where:
-:$$Q_{\rm kon}(|f| \le 4\,{\rm kHz}) \ne 0 \hspace{0.05cm}.$$
+:$$Q_{\rm con}(|f| \le 4\,{\rm kHz}) \ne 0 \hspace{0.05cm}.$$
 * The spectrum&nbsp; $Q_{\rm dis}(f)$&nbsp; contains spectral lines at&nbsp; $±1 \ \rm kHz$,&nbsp; $±2 \ \rm kHz$,&nbsp; $±3 \ \rm kHz$&nbsp; and&nbsp; $±4 \ \rm kHz$.&nbsp; Thus:
 :$$q_{\rm dis}(t) = \sum_{i=1}^{4}C_i \cdot \cos (2 \pi \cdot f_i \cdot t - \varphi_i),$$
 :Amplitude values: &nbsp; $C_1 = 1.0 \ \rm V$, $C_2 = 1.8 \ \rm V$, $C_3 = 0.8 \ \rm V$, $C_4 = 0.4 \ \rm V.$
-:The phase values&nbsp; $φ_1$,&nbsp; $φ_2$&nbsp; and&nbsp; $φ_3$&nbsp; are respectively in the range&nbsp; $±18^\circ$&nbsp; and it holds&nbsp; $φ_4 = 90^\circ$.
+:The phase values&nbsp; $φ_1$,&nbsp; $φ_2$&nbsp; and&nbsp; $φ_3$&nbsp; are respectively in the range&nbsp; $±180^\circ$&nbsp; and it holds&nbsp; $φ_4 = 90^\circ$.
-The signals are each sampled at frequency&nbsp; $f_{\rm A}$&nbsp; and immediately fed to an ideal rectangular lowpass filter with cutoff frequency&nbsp; $f_{\rm G}$&nbsp; This scenario applies, for example, to.
+The signals are each sampled at frequency&nbsp; $f_{\rm A}$&nbsp; and immediately fed to an ideal rectangular low-pass filter with cutoff frequency&nbsp; $f_{\rm G}$&nbsp; This scenario applies,&nbsp; for example,&nbsp; to
-* the interference-free pulse amplitude modulation (PAM) and
+* the interference-free pulse amplitude modulation&nbsp; $\rm (PAM)$&nbsp; and
-* the interference-free pulse code modulation (PCM) at infinitely large quantization stage number&nbsp; $M$.
+* the interference-free pulse code modulation&nbsp; $\rm (PCM)$&nbsp; at infinitely large quantization stage number&nbsp; $M$.
-The output signal of the (rectangular) low-pass filter is called the sink signal&nbsp; $v(t)$&nbsp; and for the error signal&nbsp;
+The output signal of the&nbsp; (rectangular)&nbsp; low-pass filter is called the sink signal&nbsp; $v(t)$&nbsp; and for the error signal:&nbsp;
 :$$ε(t) = v(t) - q(t).$$
 This is different from zero only if the parameters of the sampling&nbsp; $($sampling frequency $f_{\rm A})$&nbsp; and/or the signal reconstruction&nbsp; $($cutoff frequency $f_{\rm G})$&nbsp; are not dimensioned in the best possible way.
@@ Line 31: / Line 28: @@
 Hints:
-*The exercise belongs to the chapter&nbsp; [[Modulation_Methods/Pulse_Code_Modulation|Pulse Code Modulation]].
+*The exercise belongs to the chapter&nbsp; [[Modulation_Methods/Pulse_Code_Modulation|"Pulse Code Modulation"]].
-*Reference is made in particular to the page&nbsp; [[Modulation_Methods/Pulse_Code_Modulation#Sampling_and_signal_reconstruction|Sampling and Signal Reconstruction]].
+*Reference is made in particular to the page&nbsp; [[Modulation_Methods/Pulse_Code_Modulation#Sampling_and_signal_reconstruction|"Sampling and Signal Reconstruction"]].
@@ Line 41: / Line 38: @@
 {Which statements are true for &nbsp;$f_{\rm A} = 8\ \rm kHz$&nbsp; and &nbsp;$f_{\rm G} = 4\ \rm kHz$&nbsp;?
 |type="[]"}
-+ The signal &nbsp;$q_{\rm kon}(t)$&nbsp; can be completely reconstructed: &nbsp; $ε_{\rm kon}(t) = 0$.
++ The signal &nbsp;$q_{\rm con}(t)$&nbsp; can be completely reconstructed: &nbsp; $ε_{\rm con}(t) = 0$.
 - The signal &nbsp;$q_{\rm dis}(t)$&nbsp; can be completely reconstructed: &nbsp; $ε_{\rm dis}(t) = 0$.
@@ Line 70: / Line 67: @@
 *Sampling&nbsp; $q_{\rm dis}(t)$&nbsp; with sampling frequency&nbsp; $f_{\rm A} = 8 \ \rm kHz$&nbsp; leads to an irreversible error, since&nbsp; $Q_{\rm dis}(f)$&nbsp; involves a discrete spectral component (diracline) at&nbsp; $f_4 = 4\ \rm kHz$&nbsp; and the phase value&nbsp; $φ_4 ≠ 0$&nbsp; is.
 *With the phase value given here&nbsp; $φ_4 = 90^\circ$&nbsp; $(4 \ \rm kHz$- sinusoidal component$)$&nbsp; holds&nbsp; $ε_{\rm dis}(t) = v_{\rm dis}(t) - q_{\rm dis}(t) = -0. 4 \ \rm V - \sin(2π \cdot f_4 \cdot t)$.&nbsp; See also sample solution to exercise 4.2Z.
-*On the other hand, the signal&nbsp; $q_{\rm kon}(t)$&nbsp; with the continuous spectrum&nbsp; $Q_{\rm kon}(f)$&nbsp; can also then be measured with a square-wave low-pass filter&nbsp; $($with cutoff frequency&nbsp; $f_{\rm G} = 4\ \rm kHz)$&nbsp; be completely reconstructed if sampling frequency&nbsp; $f_{\rm A} = 8\ \rm kHz$&nbsp; was used. &nbsp; For all frequencies not equal to&nbsp; $f_4$&nbsp; the sampling theorem is satisfied.
+*On the other hand, the signal&nbsp; $q_{\rm kon}(t)$&nbsp; with the continuous spectrum&nbsp; $Q_{\rm con}(f)$&nbsp; can also then be measured with a square-wave low-pass filter&nbsp; $($with cutoff frequency&nbsp; $f_{\rm G} = 4\ \rm kHz)$&nbsp; be completely reconstructed if sampling frequency&nbsp; $f_{\rm A} = 8\ \rm kHz$&nbsp; was used. &nbsp; For all frequencies not equal to&nbsp; $f_4$&nbsp; the sampling theorem is satisfied.
-*But the contribution of the&nbsp; $f_4$ component to the total spectrum&nbsp; $Q_{\rm kon}(f)$&nbsp; is only vanishingly small &nbsp; ⇒ &nbsp; ${\rm Pr}(f_4) → 0$ as long as the spectrum at&nbsp; $f_4$&nbsp; has no diracline.
+*But the contribution of the&nbsp; $f_4$ component to the total spectrum&nbsp; $Q_{\rm con}(f)$&nbsp; is only vanishingly small &nbsp; ⇒ &nbsp; ${\rm Pr}(f_4) → 0$ as long as the spectrum at&nbsp; $f_4$&nbsp; has no diracline.
@@ Line 77: / Line 74: @@
 '''(2)'''&nbsp; Only the <u>proposed solution 1</u> is correct:
 *With&nbsp; $f_{\rm A} = 10\ \rm kHz$&nbsp; the sampling theorem is satisfied in both cases.
-*With&nbsp; $f_{\rm G} = f_{\rm A} /2$&nbsp; both error signals&nbsp; $ε_{\rm kon}(t)$ and $ε_{\rm dis}(t)$ are identically zero.
+*With&nbsp; $f_{\rm G} = f_{\rm A} /2$&nbsp; both error signals&nbsp; $ε_{\rm con}(t)$ and $ε_{\rm dis}(t)$ are identically zero.
 *In addition, the signal reconstruction also works as long as&nbsp; $f_{\rm G} > 4 \ \rm kHz$&nbsp; and&nbsp; $f_{\rm G} < 6 \ \rm kHz$&nbsp; holds.

@@ Line 69: / Line 69: @@
 '''(1)'''&nbsp; Only the <u>first statement</u> is correct:
 *Sampling&nbsp; $q_{\rm dis}(t)$&nbsp; with sampling frequency&nbsp; $f_{\rm A} = 8 \ \rm kHz$&nbsp; leads to an irreversible error, since&nbsp; $Q_{\rm dis}(f)$&nbsp; involves a discrete spectral component (diracline) at&nbsp; $f_4 = 4\ \rm kHz$&nbsp; and the phase value&nbsp; $φ_4 ≠ 0$&nbsp; is.
-*With the phase value given here&nbsp; $φ_4 = 90^\circ$&nbsp; $(4 \ \rm kHz$- sinusoidal component$)$&nbsp; holds&nbsp; $ε_{\rm dis}(t) = v_{\rm dis}(t) - q_{\rm dis}(t) = -0. 4 \ \rm V - \sin(2π - f_4 - t)$.&nbsp; See also sample solution to exercise 4.2Z.
+*With the phase value given here&nbsp; $φ_4 = 90^\circ$&nbsp; $(4 \ \rm kHz$- sinusoidal component$)$&nbsp; holds&nbsp; $ε_{\rm dis}(t) = v_{\rm dis}(t) - q_{\rm dis}(t) = -0. 4 \ \rm V - \sin(2π \cdot f_4 \cdot t)$.&nbsp; See also sample solution to exercise 4.2Z.
 *On the other hand, the signal&nbsp; $q_{\rm kon}(t)$&nbsp; with the continuous spectrum&nbsp; $Q_{\rm kon}(f)$&nbsp; can also then be measured with a square-wave low-pass filter&nbsp; $($with cutoff frequency&nbsp; $f_{\rm G} = 4\ \rm kHz)$&nbsp; be completely reconstructed if sampling frequency&nbsp; $f_{\rm A} = 8\ \rm kHz$&nbsp; was used. &nbsp; For all frequencies not equal to&nbsp; $f_4$&nbsp; the sampling theorem is satisfied.
 *But the contribution of the&nbsp; $f_4$ component to the total spectrum&nbsp; $Q_{\rm kon}(f)$&nbsp; is only vanishingly small &nbsp; ⇒ &nbsp; ${\rm Pr}(f_4) → 0$ as long as the spectrum at&nbsp; $f_4$&nbsp; has no diracline.

← Older revision		Revision as of 13:49, 16 March 2022
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−	[[Category:Modulation Methods: Exercises\|^4.1 ~~Pulscodemodulation~~^]]	+	[[Category:Modulation Methods: Exercises\|^4.1 Pulse Code Modulation^]]

← Older revision		Revision as of 12:46, 23 March 2021
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−	[[Category:~~Aufgaben zu Modulationsverfahren~~\|^4.1 Pulscodemodulation^]]	+	[[Category:Modulation Methods: Exercises\|^4.1 Pulscodemodulation^]]

@@ Line 32: / Line 32: @@
 ''Hinweise:''
-*Die Aufgabe gehört zum  Kapitel&nbsp; [[Modulationsverfahren/Pulscodemodulation|Pulscodemodulation]].
+*Die Aufgabe gehört zum  Kapitel&nbsp; [[Modulation_Methods/Pulscodemodulation|Pulscodemodulation]].
-*Bezug genommen wird insbesondere auf die Seite&nbsp;  [[Modulationsverfahren/Pulscodemodulation#Abtastung_und_Signalrekonstruktion|Abtastung und Signalrekonstruktion]].
+*Bezug genommen wird insbesondere auf die Seite&nbsp;  [[Modulation_Methods/Pulscodemodulation#Abtastung_und_Signalrekonstruktion|Abtastung und Signalrekonstruktion]].