Aufgaben:Exercise 3.7Z: Partial Fraction Decomposition - Revision history

Guenter at 13:55, 25 January 2022

2022-01-25T13:55:58Z

Guenter at 13:45, 25 January 2022

2022-01-25T13:45:35Z

Oezer at 10:02, 26 October 2021

2021-10-26T10:02:24Z

Oezer at 10:00, 26 October 2021

2021-10-26T10:00:35Z

Oezer at 09:59, 26 October 2021

2021-10-26T09:59:33Z

Oezer at 09:57, 26 October 2021

2021-10-26T09:57:00Z

Oezer at 07:30, 26 October 2021

2021-10-26T07:30:18Z

Oezer at 07:30, 26 October 2021

2021-10-26T07:30:00Z

Oezer at 07:27, 26 October 2021

2021-10-26T07:27:59Z

Oezer at 07:27, 26 October 2021

2021-10-26T07:27:17Z

@@ Line 91: / Line 91: @@
 ===Solution===
 {{ML-Kopf}}
-'''(1)'''&nbsp; The <u> suggested solutions 1 and 2</u> are correct:
+'''(1)'''&nbsp; The&nbsp; <u> suggested solutions 1 and 2</u>&nbsp; are correct:
-*According to the criteria given in exercise 3.4Z, there is always an all-pass filter at hand if there is a corresponding zero &nbsp;$p_{\rm o} = + A + {\rm j} \cdot B$&nbsp; in the right $p$&ndash;half-plane for each pole &nbsp;$p_{\rm x} = - A + {\rm j} \cdot B$&nbsp; in the left half-plane.
+*According to the criteria given in exercise 3.4Z,&nbsp; there is always an all-pass filter at hand <br>if there is a corresponding zero &nbsp;$p_{\rm o} = + A + {\rm j} \cdot B$&nbsp; in the right $p$&ndash;half-plane for each pole &nbsp;$p_{\rm x} = - A + {\rm j} \cdot B$&nbsp; in the left half-plane.
 *Considering&nbsp; $K = 1$&nbsp; the attenuation function is then &nbsp;$a(f) = 0 \ \rm  Np$ &nbsp; &#8658; &nbsp; $|H(f)| = 1$.
 *The following can be seen from the graph on the information page: &nbsp; The configurations &nbsp;$(1)$ and &nbsp;$(2)$ satisfy exactly these symmetry properties.
@@ Line 98: / Line 98: @@
-'''(2)'''&nbsp; The <u> suggested solution 4</u> is correct:
+'''(2)'''&nbsp; The&nbsp; <u> suggested solution 4</u>&nbsp; is correct:
 *The transfer function &nbsp;$H_{\rm L}^{(5)}(p)$&nbsp; is also described by configuration &nbsp;$(4)$&nbsp; as the following calculation shows:
 :$$H_{\rm L}^{(5)}(p) \hspace{0.25cm} =  \hspace{0.2cm} \frac{p/A}{(\sqrt{p/A}+\sqrt{A/p})^2}
@@ Line 105: / Line 105: @@
   }= H_{\rm L}^{(4)}(p)
   \hspace{0.05cm}.$$
-*The double zero is at &nbsp;$p_{\rm o} = 0$ and the double pole at &nbsp;$p_{\rm x} = -A = -2$.
+*The double zero is at &nbsp;$p_{\rm o} = 0$&nbsp; and the double pole at &nbsp;$p_{\rm x} = -A = -2$.
-'''(3)'''&nbsp; The following holds for configuration &nbsp;$(1)$&nbsp;:
+'''(3)'''&nbsp; The following holds for configuration &nbsp;$(1)$:
 :$$H_{\rm L}(p)  =\frac{p-2}{p+2}=\frac{p+2-4}{p+2}= 1 - \frac{4}{p+2}=1- H_{\rm L}\hspace{-0.05cm}'(p)
 \hspace{0.3cm} \Rightarrow \hspace{0.3cm}H_{\rm L}\hspace{-0.05cm}'(p)  = \frac{4}{p+2}
@@ Line 118: / Line 118: @@
-'''(4)'''&nbsp; Similarly, the following is obtained for configuration &nbsp;$(2)$:
+'''(4)'''&nbsp; Similarly,&nbsp; the following is obtained for configuration &nbsp;$(2)$:
 :$$H_{\rm L}(p)  =\frac{(p-2 - {\rm j} \cdot 2)(p-2 + {\rm j} \cdot 2)}{(p+2 - {\rm j} \cdot 2)(p+2 + {\rm j} \cdot 2)}=
    \frac{p^2 -4\cdot p  +8 }{p^2 +4\cdot p  +8}=
@@ Line 127: / Line 127: @@
   \hspace{0.05cm}.$$
-Thus, the <u> suggested solutions 2 and 3</u> are correct in contrast to statement 1:
+Thus,&nbsp; the&nbsp; <u> suggested solutions 2 and 3</u>&nbsp; are correct in contrast to statement 1:
 * While &nbsp;$H_{\rm L}(p)$&nbsp; has two conjugate complex zeros,
-* &nbsp;$H_{\rm L}\hspace{0.01cm}'(p)$&nbsp; only has a single zero at &nbsp;$p_{\rm o}\hspace{0.01cm}' = 0$.
+* $H_{\rm L}\hspace{0.01cm}'(p)$&nbsp; only has a single zero at &nbsp;$p_{\rm o}\hspace{0.01cm}' = 0$.
@@ Line 142: / Line 142: @@
   \hspace{0.05cm}.$$
 *The zero of &nbsp;$H_{\rm L}\hspace{0.01cm}'(p)$&nbsp; is now at &nbsp;$p_{\rm o}\hspace{0.01cm}' = -2$.
-*The constant is &nbsp;$K\hspace{0.01cm}' = 4$ &nbsp; &#8658; &nbsp; only <u> suggested solution 2</u> is correct here.
+*The constant is &nbsp;$K\hspace{0.01cm}' = 4$ &nbsp; &#8658; &nbsp; only &nbsp; <u> suggested solution 2</u> &nbsp; is correct here.
-'''(6)'''&nbsp; Finally, the following holds for configuration &nbsp;$(4)$:
+'''(6)'''&nbsp; Finally,&nbsp; the following holds for configuration &nbsp;$(4)$:
 :$$H_{\rm L}(p)  =  \frac{p^2 }{(p+2)^2}=\frac{p^2 +4\cdot p  +4 -4\cdot p  -4 }{p^2 +4\cdot p  +4}
    = 1- \frac{4\cdot p  +4 }{p^2 +4\cdot p  +4}
@@ Line 152: / Line 152: @@
   \cdot \frac{p+1}{(p+2)^2}
   \hspace{0.05cm}.$$
-<u>Suggested solution 2</u> is also correct here. In general, it can be said that:
+<u>Suggested solution 2</u>&nbsp; is correct here.&nbsp; In general,&nbsp; it can be said that:
 *The partial fraction decomposition changes the number and position of the zeros.
-* On the contrary, the poles of $H_{\rm L}\hspace{0.01cm}'(p)$ are always identical to those of $H_{\rm L}(p)$.
+* On the contrary,&nbsp; the poles of&nbsp; $H_{\rm L}\hspace{0.01cm}'(p)$&nbsp; are always identical to those of&nbsp; $H_{\rm L}(p)$.
 {{ML-Fuß}}

@@ Line 4: / Line 4: @@
 [[File:P_ID1789__LZI_Z_3_7.png|right|frame|Pole-zero diagrams]]
-In the graph, four two-port networks are given by their pole&ndash;zero diagrams &nbsp;$H_{\rm L}(p)$&nbsp;.
+In the graph,&nbsp; four two-port networks are given by their pole&ndash;zero diagrams &nbsp;$H_{\rm L}(p)$.
 * They all have in common that the number &nbsp;$Z$&nbsp; of zeros is equal to the number &nbsp;$N$&nbsp; of poles.
 *The constant factor in each case is &nbsp;$K=1$.
-In the special case &nbsp;$Z = N$&nbsp; the residue theorem cannot be applied directly to compute the impulse response &nbsp;$h(t)$&nbsp;.
+In the special case &nbsp;$Z = N$&nbsp; the residue theorem cannot be applied directly to compute the impulse response &nbsp;$h(t)$.
-Rather, a partial fraction decomposition corresponding to
+Rather,&nbsp; a&nbsp; "partial fraction decomposition"&nbsp; corresponding to
 :$$H_{\rm L}(p)  =1- H_{\rm L}\hspace{0.05cm}'(p)
   \hspace{0.05cm}$$
 must be made beforehand. Then,
 :$$h(t)  = \delta(t)- h\hspace{0.03cm}'(t)
-  \hspace{0.05cm}$$ holds for the impulse response.
+  \hspace{0.05cm}$$
-&nbsp;$h\hspace{0.03cm}'(t)$&nbsp; is the inverse Laplace transform of &nbsp;$H_{\rm L}\hspace{0.05cm}'(p)$&nbsp;, where the condition &nbsp;$Z' < N'$&nbsp; is satisfied.
+holds for the impulse response.
-Two of the four configurations given are so-called ''all-pass filters''.
+Two of the four configurations given are so-called&nbsp; "all-pass filters".
-*This refers to two-port networks for which the Fourier spectral function satisfies the condition &nbsp;$|H(f)| = 1$ &nbsp; &#8658; &nbsp; $a(f) = 0$&nbsp;.
+*This refers to two-port networks for which the Fourier spectrum satisfies the condition &nbsp;$|H(f)| = 1$ &nbsp; &#8658; &nbsp; $a(f) = 0$&nbsp;.
 *In [[Aufgaben:Exercise_3.4Z:_Various_All-Pass_Filters|Exercise 3.4Z]] it is given how the poles and zeros of such an all-pass filter must be positioned.
-Furthermore, in this exercise the &nbsp;$p$&ndash;transfer function
+Furthermore,&nbsp; in this exercise the &nbsp;$p$&ndash;transfer function
 :$$H_{\rm L}^{(5)}(p) =\frac{p/A}{\left (\sqrt{p/A}+\sqrt{A/p} \right )^2}
   \hspace{0.05cm}$$
-&rArr; &nbsp; "configuration $(5)$" will be examined in more detail, which can be represented by one of the four pole&ndash;zero diagrams given in the graph if the parameter &nbsp;$A$&nbsp; is chosen correctly.
+&rArr; &nbsp; "configuration $(5)$" will be examined in more detail,&nbsp; which can be represented by one of the four pole&ndash;zero diagrams given in the graph if the parameter &nbsp;$A$&nbsp; is chosen correctly.
@@ Line 33: / Line 34: @@
-''Please note:''
+Please note:
 *The exercise belongs to the chapter&nbsp;   [[Linear_and_Time_Invariant_Systems/Inverse_Laplace_Transform|Inverse Laplace Transform]].
 *In particular, reference is made to the page&nbsp; [[Linear_and_Time_Invariant_Systems/Inverse_Laplace_Transform#Partial_fraction_decomposition|Partial fraction decomposition]].
@@ Line 59: / Line 60: @@
-{Compute the function &nbsp;$H_{\rm L}\hspace{0.01cm}'(p)$&nbsp; after a partial fraction decomposition for configuration&nbsp; '''(1)'''. <br>Enter the function value for &nbsp;$p = 0$&nbsp;.
+{Compute the function &nbsp;$H_{\rm L}\hspace{0.01cm}'(p)$&nbsp; after a partial fraction decomposition for configuration&nbsp; '''(1)'''.&nbsp; Enter the function value for &nbsp;$p = 0$.
 |type="{}"}
 $H_{\rm L}\hspace{0.01cm}'(p = 0) \ = \ $  { 2 3% }

@@ Line 133: / Line 133: @@
-'''(5)'''&nbsp; The following applies for Konfiguration &nbsp;$(3)$&nbsp;:
+'''(5)'''&nbsp; The following applies for configuration &nbsp;$(3)$&nbsp;:
 :$$H_{\rm L}(p)  =
    \frac{p^2 }{p^2 +4\cdot p  +8}=\frac{p^2 +4\cdot p  +8 -4\cdot p  -8 }{p^2 +4\cdot p  +8}

@@ Line 64: / Line 64: @@
-{Compute &nbsp;$H_{\rm L}\hspace{0.01cm}'(p)$&nbsp; for the configuration &nbsp;$(2)$.&nbsp; Which statements are true here?
+{Compute &nbsp;$H_{\rm L}\hspace{0.01cm}'(p)$&nbsp; for configuration &nbsp;$(2)$.&nbsp; Which statements are true here?
 |type="[]"}
 - $H_{\rm L}\hspace{0.01cm}'(p)$&nbsp; has the same zeros as &nbsp;$H_{\rm L}(p)$.
@@ Line 71: / Line 71: @@
-{Compute &nbsp;$H_{\rm L}\hspace{0.01cm}'(p)$&nbsp; for the configuration &nbsp;$(3)$.&nbsp; Which statements are true here?
+{Compute &nbsp;$H_{\rm L}\hspace{0.01cm}'(p)$&nbsp; for configuration &nbsp;$(3)$.&nbsp; Which statements are true here?
 |type="[]"}
 - $H_{\rm L}\hspace{0.01cm}'(p)$&nbsp; has the same zeros as &nbsp;$H_{\rm L}(p)$.
@@ Line 78: / Line 78: @@
-{Compute &nbsp;$H_{\rm L}\hspace{0.01cm}'(p)$&nbsp; for the configuration &nbsp;$(4)$.&nbsp; Which statements are true here?
+{Compute &nbsp;$H_{\rm L}\hspace{0.01cm}'(p)$&nbsp; for configuration &nbsp;$(4)$.&nbsp; Which statements are true here?
 |type="[]"}
 - $H_{\rm L}\hspace{0.01cm}'(p)$&nbsp; has the same zeros as &nbsp;$H_{\rm L}(p)$.

@@ Line 51: / Line 51: @@
-{Which quadripole has the transfer function &nbsp;$H_{\rm L}^{(5)}(p)$?
+{Which two-port network has the transfer function &nbsp;$H_{\rm L}^{(5)}(p)$?
 |type="()"}
 - Configuration &nbsp;$(1)$,