Aufgaben:Exercise 4.11Z: C Program "acf2" - Revision history

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Guenter at 08:59, 28 January 2022

2022-01-28T08:59:27Z

Guenter: Guenter moved page Aufgaben:Aufgabe 4.11Z: C-Programm „akf2” to Aufgaben:Exercise 4.11Z: C Program "acf2"

2022-01-28T08:54:50Z

Guenter moved page Aufgabe 4.11Z: C-Programm „akf2” to Exercise 4.11Z: C Program "acf2"

@@ Line 3: / Line 3: @@
 }}
-[[File:EN Sto Z 4 11.png|right|frame|C program&nbsp; $2$&nbsp; for ACF&ndash;calculation '''correction''']]
+[[File:EN_Sto_Z_4_11_neu.png|right|frame|C program&nbsp; $2$&nbsp; for ACF&ndash;calculation]]
 You can see on the right the C program&nbsp; "acf2"&nbsp; for calculating the discrete ACF values&nbsp; $\varphi_x(k)$&nbsp; with index&nbsp; $k = 0$, ... , $l$.

@@ Line 3: / Line 3: @@
 }}
-[[File:P_ID395__Sto_Z_4_11.png|right|frame|C program&nbsp; $2$&nbsp; for ACF&ndash;calculation '''correction''']]
+[[File:EN Sto Z 4 11.png|right|frame|C program&nbsp; $2$&nbsp; for ACF&ndash;calculation '''correction''']]
 You can see on the right the C program&nbsp; "acf2"&nbsp; for calculating the discrete ACF values&nbsp; $\varphi_x(k)$&nbsp; with index&nbsp; $k = 0$, ... , $l$.

@@ Line 53: / Line 53: @@
 ===Solution===
 {{ML-Kopf}}
-'''(1)'''&nbsp; To calculate the ACF value&nbsp; $\varphi_x(0)$&nbsp; is averaged over&nbsp; $\underline{N =10000}$&nbsp; summands, for&nbsp; $\varphi_x(10)$&nbsp; only over&nbsp; $\underline{N = 9990}$.
+'''(1)'''&nbsp; To calculate the ACF value&nbsp; $\varphi_x(0)$&nbsp; you have to average over&nbsp; $\underline{N =10000}$&nbsp; summands, for&nbsp; $\varphi_x(10)$&nbsp; only over&nbsp; $\underline{N = 9990}$.
-'''(2)'''&nbsp; Correct are <u>the proposed solutions 1, 3 and 4</u>:
+'''(2)'''&nbsp; Correct are&nbsp; <u>the proposed solutions 1, 3 and 4</u>:
 *The computation time increases approximately linearly with&nbsp; $N$&nbsp; and&nbsp; $l + 1$&nbsp; as can be seen from the ACF calculation highlighted in red.
-* In contrast, the computation time for the other parts of the program can be neglected.
+* In contrast,&nbsp; the computation time for the other parts of the program can be neglected.
-*Naturally, the calculation also becomes more accurate with increasing&nbsp; $N$&nbsp;.
+*Naturally,&nbsp; the calculation also becomes more accurate with increasing&nbsp; $N$.
-*This goes here &ndash; in contrast to the program "acf1" of exercise 4.11 &ndash; however at the expense of the necessary memory requirement.
+*This goes here &ndash; in contrast to the program&nbsp; "acf1"&nbsp; of Exercise 4.11 &ndash; however at the expense of the necessary memory requirement.
 *Since each float variable takes up exactly four bytes, the auxiliary field&nbsp; ${\rm H}[10000 ]$&nbsp; alone requires a memory of 40 kByte.
-'''(3)'''&nbsp; Correct are <u>suggested solutions 1 and 3</u>:
+'''(3)'''&nbsp; Correct are the&nbsp; <u>suggested solutions 1 and 3</u>:
-*The stronger the statistical bindings within the random process, the less accurate is the ACF calculation for a given&nbsp; $N$&nbsp;.
+*The stronger the statistical bindings within the random process,&nbsp; the less accurate is the ACF calculation for a given&nbsp; $N$.
 *This fact can be illustrated for example by the power calculation&nbsp; $($ACF value at&nbsp; $k=0)$&nbsp;:
-**If all&nbsp; $N$&nbsp; samples are statistically independent, then all contributions provide the maximum information about the ACF&ndash;value&nbsp; $\varphi_x(k=0)$.
+**If all&nbsp; $N$&nbsp; samples are statistically independent,&nbsp; then all contributions provide the maximum information about the ACF value&nbsp; $\varphi_x(k=0)$.
-**However, if there are statistical bindings between&nbsp; $x_\nu$&nbsp; and&nbsp; $x_{\nu+1}$, but not between&nbsp; $x_\nu$&nbsp; and&nbsp; $x_{\nu+2}$, only half of all samples provide the full information&nbsp; about&nbsp; $\varphi_x(k=0)$&nbsp; and the others provide only limited information.
+**However,&nbsp; if there are statistical bindings between&nbsp; $x_\nu$&nbsp; and&nbsp; $x_{\nu+1}$,&nbsp; but not between&nbsp; $x_\nu$&nbsp; and&nbsp; $x_{\nu+2}$,&nbsp; only half of all samples provide the full information&nbsp; about&nbsp; $\varphi_x(k=0)$&nbsp; and the others provide only limited information.
 *This loss of information based on correlations can only be compensated in this example by doubling&nbsp; $N$&nbsp;.
-*The last statement is also true, as explained in detail on the page "Accuracy of the numerical ACF calculation" in the theory section.
+*The last statement is also true,&nbsp; as explained in detail on the page&nbsp; "Accuracy of the numerical ACF calculation"&nbsp; in the theory section.
 {{ML-Fuß}}

@@ Line 45: / Line 45: @@
 + The stronger the internal statistical bindings of the process are,&nbsp; the more&nbsp; <u>inexact</u>&nbsp; is the ACF result for a given&nbsp; $N$.
 - The stronger the internal statistical bindings of the process are,&nbsp; the more&nbsp; <u>accurate</u>&nbsp; is the ACF result for a given&nbsp; $N$.
-+ If the process has statistical bindings,&nbsp; the errors of the numerical ACF calculation are also correlated. <br>&nbsp; &nbsp; <i>Example:</i> &nbsp; &nbsp; If the value&nbsp; $\varphi_x(k=5)$&nbsp; is too large,&nbsp; it is very likely that&nbsp; $\varphi_x(k=4)$&nbsp; and&nbsp; $\varphi_x(k=6)$&nbsp; will also be too large.
++ If the process has statistical bindings,&nbsp; the errors of the numerical ACF calculation are also correlated. <br>&nbsp; &nbsp; <u>Example:</u> &nbsp; &nbsp; If the value&nbsp; $\varphi_x(k=5)$&nbsp; is too large,&nbsp; it is very likely that&nbsp; $\varphi_x(k=4)$&nbsp; and&nbsp; $\varphi_x(k=6)$&nbsp; will also be too large.

@@ Line 4: / Line 4: @@
 [[File:P_ID395__Sto_Z_4_11.png|right|frame|C program&nbsp; $2$&nbsp; for ACF&ndash;calculation '''correction''']]
-You can see on the right the C program "acf2" for calculating the discrete ACF values&nbsp; $\varphi_x(k)$&nbsp; with index&nbsp; $k = 0$, ... , $l$.
+You can see on the right the C program&nbsp; "acf2"&nbsp; for calculating the discrete ACF values&nbsp; $\varphi_x(k)$&nbsp; with index&nbsp; $k = 0$, ... , $l$.
-In contrast to the program "acf1" from&nbsp; [[Aufgaben:Exercise_4.11:_C_Program_"acf1"|Exercise 4.11]]&nbsp; here the algorithm described in the theory part is applied directly.&nbsp; It should be noted:
+In contrast to the program&nbsp; "acf1"&nbsp; from&nbsp; [[Aufgaben:Exercise_4.11:_C_Program_"acf1"|Exercise 4.11]],&nbsp; here the algorithm described in the theory part is applied directly.&nbsp; It should be noted:
-*Let the long value passed to the program be here&nbsp; $l=10$.
+*Let the long value passed to the program be&nbsp; $l=10$.
-*The calculated ACF values&nbsp; $\varphi_x(0)$, ... , $\varphi_x(10)$&nbsp; are returned to the main program with the float field&nbsp; $\rm ACF[ \ ]$&nbsp; In lines 7 and 8, this field is prefilled with zeros.
+*The calculated ACF values&nbsp; $\varphi_x(0)$, ... , $\varphi_x(10)$&nbsp; are returned to the main program with the float field&nbsp; $\rm ACF[ \ ]$.&nbsp; In lines 7 and 8,&nbsp; this field is prefilled with zeros.
-*The random variable&nbsp; $x( \ )$&nbsp; is defined as a float function in line 4, as is an auxiliary field&nbsp; ${\rm H}[10000 ]$, into which the&nbsp; $N = 10000$&nbsp; samples&nbsp; $x_\nu$&nbsp; are entered (lines 9 and 10).
+*The random variable&nbsp; $x( \ )$&nbsp; is defined as a float function in line 4,&nbsp; as is an auxiliary field&nbsp; ${\rm H}[10000 ]$,&nbsp; into which the&nbsp; $N = 10000$&nbsp; samples&nbsp; $x_\nu$&nbsp; are entered&nbsp; (lines 9 and 10).
-*The names of the run variables in line 6 are adapted to the given algorithm.
+*The names of the control variables in line 6 are adapted to the given algorithm.
 *The actual ACF calculation is done from line 11.&nbsp; This program part is marked red in the program code.
@@ Line 39: / Line 35: @@
 {Which of the following statements are correct?
 |type="[]"}
-+ The calculation time increases linearly with&nbsp; $l + 1$, i.e. with the number of ACF values to be calculated.
++ The calculation time increases linearly with&nbsp; $l + 1$,&nbsp; i.e. with the number of ACF values to be calculated.
 - The calculation time increases quadratically with the number&nbsp; $N$&nbsp; of samples considered.
 + The calculation becomes more accurate as&nbsp; $N$&nbsp; increases.
-+ If a float variable is represented with&nbsp; $\rm 4 \ bytes$&nbsp;, "acf2" needs at least&nbsp; $4 \cdot N$&nbsp; bytes of memory.
++ If a float variable is represented with&nbsp; $\rm 4 \ bytes$,&nbsp; "acf2"&nbsp; needs at least&nbsp; $4 \cdot N$&nbsp; bytes of memory.
 {Which of the following statements are true?
 |type="[]"}
-+ The stronger the internal statistical bindings of the process are, the more <u>inexact</u> is the ACF result for a given&nbsp; $N$&nbsp;.
++ The stronger the internal statistical bindings of the process are,&nbsp; the more&nbsp; <u>inexact</u>&nbsp; is the ACF result for a given&nbsp; $N$.
-- The stronger the internal statistical bindings of the process are, the <u>more accurate</u>, for a given&nbsp; $N$&nbsp; is the ACF result.
+- The stronger the internal statistical bindings of the process are,&nbsp; the more&nbsp; <u>accurate</u>&nbsp; is the ACF result for a given&nbsp; $N$.
-+ If the process has statistical bindings, the errors of the numerical ACF calculation are also correlated. <br>&nbsp; &nbsp; <i>Example:</i> &nbsp; &nbsp; If the value&nbsp; $\varphi_x(k=5)$&nbsp; is too large, it is very likely that&nbsp; $\varphi_x(k=4)$&nbsp; and&nbsp; $\varphi_x(k=6)$&nbsp; will also be too large.
++ If the process has statistical bindings,&nbsp; the errors of the numerical ACF calculation are also correlated. <br>&nbsp; &nbsp; <i>Example:</i> &nbsp; &nbsp; If the value&nbsp; $\varphi_x(k=5)$&nbsp; is too large,&nbsp; it is very likely that&nbsp; $\varphi_x(k=4)$&nbsp; and&nbsp; $\varphi_x(k=6)$&nbsp; will also be too large.

@@ Line 3: / Line 3: @@
 }}
-[[File:P_ID395__Sto_Z_4_11.png|right|frame|C-Programm&nbsp; $2$&nbsp; zur AKF&ndash;Berechnung]]
+[[File:P_ID395__Sto_Z_4_11.png|right|frame|C-Programm&nbsp; $2$&nbsp; zur AKF&ndash;Berechnung '''Korrektur''']]
 Sie sehen rechts das C-Programm "akf2" zur Berechnung der diskreten AKF-Werte&nbsp; $\varphi_x(k)$&nbsp; mit Index&nbsp; $k = 0$, ... , $l$.
@@ Line 81: / Line 81: @@
-[[Category:Theory of Stochastic Signals: Exercises|^4.4 Autokorrelationsfunktion (AKF)^]]
+[[Category:Theory of Stochastic Signals: Exercises|^4.4 Auto-Correlation Function^]]

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