Theory of Stochastic Signals/Creation of Predefined ACF Properties - Revision history

Hwang at 18:55, 21 December 2022

2022-12-21T18:55:12Z

Hwang at 18:52, 21 December 2022

2022-12-21T18:52:57Z

Hwang at 18:50, 21 December 2022

2022-12-21T18:50:38Z

Hwang at 18:50, 21 December 2022

2022-12-21T18:50:21Z

Hwang at 18:45, 21 December 2022

2022-12-21T18:45:40Z

Guenter at 10:07, 25 February 2022

2022-02-25T10:07:14Z

Bene: Text replacement - "rms value" to "standard deviation"

2022-02-15T15:45:35Z

Text replacement - "rms value" to "standard deviation"

Guenter at 16:36, 29 January 2022

2022-01-29T16:36:04Z

Guenter at 17:04, 28 January 2022

2022-01-28T17:04:29Z

Guenter at 17:03, 28 January 2022

2022-01-28T17:03:29Z

@@ Line 79: / Line 79: @@
 {{GraueBox|TEXT=
-$\text{Example 3:}$&nbsp; As shown in the&nbsp; [[Theory_of_Stochastic_Signals/Creation_of_Predefined_ACF_Properties#Ambiguities_in_the_determination_of_coefficients|"last section"]],&nbsp; the parameter set&nbsp;$a_0 = 0.7$, &nbsp;$a_1 = 0.3$&nbsp; is suitable to generate the ACF values&nbsp;$φ_0 = 0.58$&nbsp; and &nbsp;$φ_1 = 0.21$.&nbsp; The desired ACF of the output sequence is then in detailed notation:
+$\text{Example 3:}$&nbsp; As shown in the&nbsp; [[Theory_of_Stochastic_Signals/Creation_of_Predefined_ACF_Properties#Ambiguities_in_the_determination_of_the_coefficients|"last section"]],&nbsp; the parameter set&nbsp;$a_0 = 0.7$, &nbsp;$a_1 = 0.3$&nbsp; is suitable to generate the ACF values&nbsp;$φ_0 = 0.58$&nbsp; and &nbsp;$φ_1 = 0.21$.&nbsp; The desired ACF of the output sequence is then in detailed notation:
 [[File:P_ID557__Sto_T_5_3_S2_b_neu_100.png |frame| Example of ACF calculation|right]]
 :$$\varphi_y(\tau) = 0.58 \cdot \delta(\tau) + 0.21 \cdot \delta(\tau - T_{\rm A})

@@ Line 38: / Line 38: @@
 <br>
 Now the following question is to be answered: &nbsp; How can the coefficients&nbsp; $a_0$, ... , $a_M$&nbsp; of a&nbsp; $M$&ndash;th order non-recursive filter can be determined,
-* if the desired ACF values&nbsp; $φ_y(0)$, ... ,s&nbsp; $φ_y(M · T_{\rm A})$&nbsp; are given,s&nbsp; and
+* if the desired ACF values&nbsp; $φ_y(0)$, ... ,s&nbsp; $φ_y(M · T_{\rm A})$&nbsp; are given&nbsp; and
 * outside the range from&nbsp;  $-M · T_{\rm A}$&nbsp; to&nbsp; $+M · T_{\rm A}$&nbsp; all ACF values are to be zero.

@@ Line 7: / Line 7: @@
 ==ACF at the output of a non-recursive filter==
 <br>
-We consider a non-recursive&nbsp; $M$&ndash;th order digital&nbsp; filter according to the following diagram.
+We consider &nbsp; a non-recursive&nbsp; $M$&ndash;th order digital&nbsp; filter according to the following diagram.
 [[File:P_ID555__Sto_T_5_3_S1_neu.png |frame| $M$-th order non-recursive filter]]
 The discrete-time input variable&nbsp; $〈\hspace{0.05cm}x_ν\hspace{0.05cm}〉$&nbsp; is

@@ Line 7: / Line 7: @@
 ==ACF at the output of a non-recursive filter==
 <br>
-We consider a &nbsp; non-recursive&nbsp; $M$&ndash;th order digital&nbsp; filter according to the following diagram.
+We consider a non-recursive&nbsp; $M$&ndash;th order digital&nbsp; filter according to the following diagram.
 [[File:P_ID555__Sto_T_5_3_S1_neu.png |frame| $M$-th order non-recursive filter]]
 The discrete-time input variable&nbsp; $〈\hspace{0.05cm}x_ν\hspace{0.05cm}〉$&nbsp; is

@@ Line 79: / Line 79: @@
 {{GraueBox|TEXT=
-$\text{Example 3:}$&nbsp; As shown in the&nbsp; [[Theory_of_Stochastic_Signals/Creation_of_Predefined_ACF_Properties#Ambiguities_in_the_determination_of_coefficients|last section]],&nbsp; the parameter set&nbsp;$a_0 = 0.7$, &nbsp;$a_1 = 0.3$&nbsp; is suitable to generate the ACF values&nbsp;$φ_0 = 0.58$&nbsp; and &nbsp;$φ_1 = 0.21$.&nbsp; The desired ACF of the output sequence is then in detailed notation:
+$\text{Example 3:}$&nbsp; As shown in the&nbsp; [[Theory_of_Stochastic_Signals/Creation_of_Predefined_ACF_Properties#Ambiguities_in_the_determination_of_coefficients|"last section"]],&nbsp; the parameter set&nbsp;$a_0 = 0.7$, &nbsp;$a_1 = 0.3$&nbsp; is suitable to generate the ACF values&nbsp;$φ_0 = 0.58$&nbsp; and &nbsp;$φ_1 = 0.21$.&nbsp; The desired ACF of the output sequence is then in detailed notation:
 [[File:P_ID557__Sto_T_5_3_S2_b_neu_100.png |frame| Example of ACF calculation|right]]
 :$$\varphi_y(\tau) = 0.58 \cdot \delta(\tau) + 0.21 \cdot \delta(\tau - T_{\rm A})

@@ Line 101: / Line 101: @@
 [[Aufgaben:Exercise_5.5:_ACF-Equivalent_Filters|Exercise 5.5: ACF-equivalent Filters]]
-[[Aufgaben:Exercise_5.5Z:_ACF_after_1st_order_filter|Exercise 5.5Z: ACF after 1st Order Filter]]
+[[Aufgaben:Exercise_5.5Z:_ACF_after_1st_Order_Filter|Exercise 5.5Z: ACF after 1st Order Filter]]
 [[Aufgaben:Exercise_5.6:_Filter_Dimensioning|Exercise 5.6: Filter Dimensioning]]

@@ Line 11: / Line 11: @@
 The discrete-time input variable&nbsp; $〈\hspace{0.05cm}x_ν\hspace{0.05cm}〉$&nbsp; is
 * mean&ndash;free&nbsp; $(m_x = 0)$,
-* Gaussian distributed&nbsp; (with&nbsp; standard deviation&nbsp; &rArr; &nbsp; "rms value"&nbsp; $σ_x)$,&nbsp; and
+* Gaussian distributed&nbsp; (with&nbsp; standard deviation&nbsp; &rArr; &nbsp; "standard deviation"&nbsp; $σ_x)$,&nbsp; and
 * without memory&nbsp; ("white noise") &nbsp; &rArr; &nbsp; statistically independent samples.
@@ Line 42: / Line 42: @@
-For the rms value&nbsp; $σ_x = 1$,&nbsp; the following nonlinear system of equations is obtained,&nbsp; using&nbsp; $φ_k = φ_y(k · T_{\rm A})$&nbsp; for simplicity of notation:
+For the standard deviation&nbsp; $σ_x = 1$,&nbsp; the following nonlinear system of equations is obtained,&nbsp; using&nbsp; $φ_k = φ_y(k · T_{\rm A})$&nbsp; for simplicity of notation:
 :$$\begin{align*}\varphi _0 & = \sum\limits_{\mu  = 0}^M {a_\mu^2  ,}\\ \varphi _1 &  = \sum\limits_{\mu  = 0}^{M - 1} {a_\mu   \cdot a_{\mu  + 1} ,}  \\ & ... &\\  \varphi _{M - 1} & = a_0  \cdot a_{M - 1}  + a_1  \cdot a_M ,  \\ \varphi _M  & =  a_0  \cdot a_M .\end{align*}$$
@@ Line 54: / Line 54: @@
 $\text{Example 2:}$&nbsp; We consider the following constellation:
 #a recursive filter of first order &nbsp; ⇒  &nbsp; $M = 1$,
-#a discrete-time input sequence&nbsp; $〈\hspace{0.05cm}x_ν\hspace{0.05cm}〉$&nbsp; with mean&nbsp; $m_x =$ 0 &nbsp; and&nbsp; rms value&nbsp; $σ_x = 1$,
+#a discrete-time input sequence&nbsp; $〈\hspace{0.05cm}x_ν\hspace{0.05cm}〉$&nbsp; with mean&nbsp; $m_x =$ 0 &nbsp; and&nbsp; standard deviation&nbsp; $σ_x = 1$,
 #desired ACF values of the sequence&nbsp; $〈\hspace{0.05cm}y_ν\hspace{0.05cm}〉$: &nbsp; $ φ_y(0) = φ_0 =0.58$, &nbsp;  $φ_y(±T_{\rm A}) = φ_1 = 0.21$.

@@ Line 63: / Line 63: @@
 *This leads to an equation of degree&nbsp; $4$,&nbsp;
 :$$a_0 ^2  + \left( { { {0.21} }/{ {a_0 } } } \right)^2  = 0.58\quad  \Rightarrow \quad a_0 ^4  - 0.58 \cdot a_0 ^2  + 0.21^2  = 0.$$
-*A solution represents&nbsp; $a_0 = 0.7$.&nbsp; &nbsp; By inserting it into the second equation, we find&nbsp; $a_1 = 0.3$. }}
+*A solution represents&nbsp; $a_0 = 0.7$.&nbsp; &nbsp; By inserting it into the second equation, we find&nbsp; $a_1 = 0.3$.
-One recognizes from this example that already in the simplest case &nbsp;  ⇒ &nbsp;  $M = 1$&nbsp; for&nbsp; $a_0$&nbsp; a nonlinear determination equation of degree&nbsp; $4$&nbsp; results.
+One recognizes from this example that already in the simplest case &nbsp;  ⇒ &nbsp;  $M = 1$&nbsp; for&nbsp; $a_0$&nbsp; a nonlinear determination equation of degree&nbsp; $4$&nbsp; results.}}
 ==Ambiguities in the determination of the coefficients==

@@ Line 99: / Line 99: @@
 ==Exercises for the chapter==
 <br>
-[[Aufgaben:Exercise_5.5:_ACF-Equivalent_Filters|Exercise 5.5: ACF-Equivalent Filters]]
+[[Aufgaben:Exercise_5.5:_ACF-Equivalent_Filters|Exercise 5.5: ACF-equivalent Filters]]
 [[Aufgaben:Exercise_5.5Z:_ACF_after_1st_order_filter|Exercise 5.5Z: ACF after 1st order filter]]