Aufgaben:Exercise 1.8: Variable Edge Steepness - Revision history

Guenter at 14:19, 7 October 2021

2021-10-07T14:19:41Z

2021-09-08T15:26:24Z

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 {{quiz-Header|Buchseite=Linear_and_Time_Invariant_Systems/Some_Low-Pass_Functions_in_Systems_Theory}}
-[[File:P_ID867__LZI_A_1_8.png|right|frame|Trapezoidal low-pass filter (red) and raised-cosine low-pass filter (green)]]
+[[File:EN_LZI_A_1_8.png|right|frame|Trapezoidal low-pass filter (red) and raised-cosine low-pass filter (green)]]
 Two low-pass filters with variable edge steepnesses are compared with each other. For frequencies&nbsp;$|f| ≤ f_1$&nbsp;, &nbsp;$H(f) = 1$&nbsp; holds in both cases. In contrast, all frequencies&nbsp;$|f| ≥ f_2$&nbsp; are suppressed entirely.
 In the range&nbsp;$f_1 ≤ |f| ≤ f_2$&nbsp; the frequency responses are defined by the following equations:
-*Trapezoidal low-pass filter (TTP):
+*Trapezoidal low-pass filter&nbsp; $\rm (TLP)$:
 :$$H(f) = \frac{f_2 - |f|}{f_2 - f_1} ,$$
-*Raised-cosine low-pass filter (CRTP):
+*Raised-cosine low-pass filter&nbsp; $\rm (RCLP)$:
 :$$H(f) = \cos^2 \left(\frac{|f|- f_1}{f_2 - f_1} \cdot\frac{\pi}{2} \right).$$

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 *The exercise belongs to the chapter&nbsp; [[Linear_and_Time_Invariant_Systems/Some_Low-Pass_Functions_in_Systems_Theory|Some Low-Pass Functions in Systems Theory]].
 *In particular, reference is made to the pages&nbsp; [[Linear_and_Time_Invariant_Systems/Some_Low-Pass_Functions_in_Systems_Theory#Trapezoidal_low-pass_filter|Trapezoidal low-pass filter]]&nbsp; and&nbsp; [[Linear_and_Time_Invariant_Systems/Some_Low-Pass_Functions_in_Systems_Theory#Raised-cosine_low-pass_filter|Raised-cosine low-pass filter]].
-*You can check your results with the interactive applet &nbsp;[[Applets:Frequenzgang_und_Impulsantwort|Frequency response and impulse response]]&nbsp;.
+*You can check your results with the interactive applet &nbsp;[[Applets:Frequenzgang_und_Impulsantwort|Frequency response and impulse response]].

@@ Line 34: / Line 34: @@
 *The exercise belongs to the chapter&nbsp; [[Linear_and_Time_Invariant_Systems/Some_Low-Pass_Functions_in_Systems_Theory|Some Low-Pass Functions in Systems Theory]].
 *In particular, reference is made to the pages&nbsp; [[Linear_and_Time_Invariant_Systems/Some_Low-Pass_Functions_in_Systems_Theory#Trapezoidal_low-pass_filter|Trapezoidal low-pass filter]]&nbsp; and&nbsp; [[Linear_and_Time_Invariant_Systems/Some_Low-Pass_Functions_in_Systems_Theory#Raised-cosine_low-pass_filter|Raised-cosine low-pass filter]].
-*You can check your results with the interactive applet &nbsp;[[Applets:Frequenzgang_und_Impulsantwort|Frequency response and impulse respone]]&nbsp;.
+*You can check your results with the interactive applet &nbsp;[[Applets:Frequenzgang_und_Impulsantwort|Frequency response and impulse response]]&nbsp;.

@@ Line 34: / Line 34: @@
 *The exercise belongs to the chapter&nbsp; [[Linear_and_Time_Invariant_Systems/Some_Low-Pass_Functions_in_Systems_Theory|Some Low-Pass Functions in Systems Theory]].
 *In particular, reference is made to the pages&nbsp; [[Linear_and_Time_Invariant_Systems/Some_Low-Pass_Functions_in_Systems_Theory#Trapezoidal_low-pass_filter|Trapezoidal low-pass filter]]&nbsp; and&nbsp; [[Linear_and_Time_Invariant_Systems/Some_Low-Pass_Functions_in_Systems_Theory#Raised-cosine_low-pass_filter|Raised-cosine low-pass filter]].
-*You can check your results with the interactive applet &nbsp;[[Applets:Frequenzgang_und_Impulsantwort|Frequency response and impulse respond]]&nbsp;.
+*You can check your results with the interactive applet &nbsp;[[Applets:Frequenzgang_und_Impulsantwort|Frequency response and impulse respone]]&nbsp;.

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 *However, the zero of the numerator at&nbsp;$t / \Delta t = 2.5$&nbsp; is nullified by the denominator which also vanishes.
 *By contrast, the other zeros at&nbsp;$7.5, 12.5,\text{...} $&nbsp; remain.
-*Here, &nbsp;$r = 0$&nbsp; results in the rectangular low-pass filter and thus in the&nbsp;$\rm sinc$&ndash;shaped impulse response.
+*Here, &nbsp;$r = 0$&nbsp; also results in the rectangular low-pass filter and thus in the&nbsp;$\rm sinc$&ndash;shaped impulse response.
 *In contrast, the impulse response of the cosine-square low-pass filter&nbsp; $($special case for &nbsp;$r = 1)$&nbsp; decays extremely fast.
 *This is studied in detail in&nbsp; [[Aufgaben:Exercise_1.8Z:_Cosine-Square_Low-Pass|Exercise 1.8Z]]&nbsp;.

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 :$$\Rightarrow  \hspace{0.3cm} {t}/{ \Delta t} = \pm
 .5, \pm 7.5, \pm 12.5, \text{...}. $$
-*However, the zero of the numerator at&nbsp;$t / \Delta t = 2.5$&nbsp; is nullified by the denominator which is also vanishing.
+*However, the zero of the numerator at&nbsp;$t / \Delta t = 2.5$&nbsp; is nullified by the denominator which also vanishes.
 *By contrast, the other zeros at&nbsp;$7.5, 12.5,\text{...} $&nbsp; remain.
 *Here, &nbsp;$r = 0$&nbsp; results in the rectangular low-pass filter and thus in the&nbsp;$\rm sinc$&ndash;shaped impulse response.

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 '''(4)'''&nbsp; <u>Suggestions 1, 2 and 4</u> are correct here:
-*The impulse response&nbsp;$h_{\rm CRTP}(t)$&nbsp; of the raised-cosine low-pass filter also has zeros at an interval of&nbsp;$\Delta t$ due to the sinc&ndash;function.
+*The impulse response&nbsp;$h_{\rm CRTP}(t)$&nbsp; of the raised-cosine low-pass filter also has zeros at an interval of&nbsp;$\Delta t$ due to the &nbsp;$\rm sinc$&ndash;function.
 *The cosine function has zeros at the following times:
 :$${\cos(\pi \cdot r \cdot {t}/{ \Delta t}  )}  =  0 \hspace{0.3cm}\Rightarrow  \hspace{0.3cm}r \cdot {t}/{ \Delta t} = \pm

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-'''(3)'''&nbsp; <u>Approaches 1 and 4</u> are correct:
+'''(3)'''&nbsp; <u>Suggestions 1 and 4</u> are correct:
 *The first&nbsp;$\rm sinc$&ndash;function of&nbsp;$h_{\rm TTP}(t)$&nbsp; causes zeros at an interval of&nbsp;$\Delta t$&nbsp; (see also the equation on the information page).
 *The second &nbsp;$\rm sinc$&ndash;function causes zeros at multiples of&nbsp;$5 &middot; \Delta t$.