Theory of Stochastic Signals/Gaussian Distributed Random Variables - Revision history

Guenter at 09:00, 22 December 2022

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Hwang at 12:56, 21 December 2022

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Hwang at 12:47, 21 December 2022

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Hwang at 19:56, 20 December 2022

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Hwang at 19:33, 20 December 2022

2022-12-20T19:33:45Z

@@ Line 14: / Line 14: @@
 :$$x=\sum\limits_{i=\rm 1}^{\it I}x_i .$$
-*According to the&nbsp; [https://en.wikipedia.org/wiki/Central_limit_theorem $\text{central limit theorem of statistics}$]&nbsp; this sum has a Gaussian PDF in the limiting case&nbsp; $(I → ∞)$&nbsp; as long as the individual components&nbsp; $x_i$&nbsp; have no statistical ties '''KORREKTUR: ties or bindings?'''.&nbsp; This holds&nbsp; (almost)&nbsp; for all density functions of the individual summands.
+*According to the&nbsp; [https://en.wikipedia.org/wiki/Central_limit_theorem $\text{central limit theorem of statistics}$]&nbsp; this sum has a Gaussian PDF in the limiting case&nbsp; $(I → ∞)$&nbsp; as long as the individual components&nbsp; $x_i$&nbsp; have no statistical  bindings.&nbsp; This holds&nbsp; (almost)&nbsp; for all density functions of the individual summands.
 *Many&nbsp; "noise processes"&nbsp; fulfill exactly this condition,&nbsp; that is,&nbsp; they are additively composed of a large number of independent individual contributions,&nbsp; so that their pattern functions&nbsp; ("noise signals")&nbsp; exhibit a Gaussian amplitude distribution.
-*If one applies a Gaussian distributed signal to a linear filter for spectral shaping,&nbsp; the output signal is also Gaussian distributed. &nbsp; Only the distribution parameters such as mean and standard deviation change,&nbsp; as well as the internal statistical ties '''KORREKTUR: ties or bindings?''' of the samples.}}.
+*If one applies a Gaussian distributed signal to a linear filter for spectral shaping,&nbsp; the output signal is also Gaussian distributed. &nbsp; Only the distribution parameters such as mean and standard deviation change,&nbsp; as well as the internal statistical bindings of the samples.}}.
 [[File:P_ID68__Sto_T_3_5_S1_neu.png |right|frame|Gaussian distributed and uniformly distributed random signal]]

@@ Line 164: / Line 164: @@
 The Box and Muller method&nbsp; &ndash; hereafter abbreviated to&nbsp; "BM"&nbsp; &ndash; can be characterized as follows:
 *The theoretical background for the validity of above generation rules is based on the regularities for&nbsp; [[Theory_of_Stochastic_Signals/Two-Dimensional_Random_Variables|$\text{two-dimensional random variables}$]].
-*Obvious equations successively yield two Gaussian values without statistical bindings '''KORREKTUR: ties or bindings?'''.&nbsp; This fact can be used to reduce simulation time by generating a tuple&nbsp; $(x, \ y)$&nbsp; of Gaussian values at each function call.
+*Obvious equations successively yield two Gaussian values without statistical bindings.&nbsp; This fact can be used to reduce simulation time by generating a tuple&nbsp; $(x, \ y)$&nbsp; of Gaussian values at each function call.
 *A comparison of the computation times shows that&nbsp; &ndash; with the best possible implementation in each case&nbsp; &ndash; the BM method is superior to the addition method with&nbsp; $I =12$&nbsp; by&nbsp;  (approximately)&nbsp; a factor of&nbsp; $3$.
 *The range of values is less limited in the BM method than in the addition method,&nbsp; so that even small probabilities are simulated more accurately.&nbsp; But even with the BM method,&nbsp; it is not possible to simulate arbitrarily small error probabilities.

@@ Line 14: / Line 14: @@
 :$$x=\sum\limits_{i=\rm 1}^{\it I}x_i .$$
-*According to the&nbsp; [https://en.wikipedia.org/wiki/Central_limit_theorem $\text{central limit theorem of statistics}$]&nbsp; this sum has a Gaussian PDF in the limiting case&nbsp; $(I → ∞)$&nbsp; as long as the individual components&nbsp; $x_i$&nbsp; have no statistical ties.&nbsp; This holds&nbsp; (almost)&nbsp; for all density functions of the individual summands.
+*According to the&nbsp; [https://en.wikipedia.org/wiki/Central_limit_theorem $\text{central limit theorem of statistics}$]&nbsp; this sum has a Gaussian PDF in the limiting case&nbsp; $(I → ∞)$&nbsp; as long as the individual components&nbsp; $x_i$&nbsp; have no statistical ties '''KORREKTUR: ties or bindings?'''.&nbsp; This holds&nbsp; (almost)&nbsp; for all density functions of the individual summands.
 *Many&nbsp; "noise processes"&nbsp; fulfill exactly this condition,&nbsp; that is,&nbsp; they are additively composed of a large number of independent individual contributions,&nbsp; so that their pattern functions&nbsp; ("noise signals")&nbsp; exhibit a Gaussian amplitude distribution.
-*If one applies a Gaussian distributed signal to a linear filter for spectral shaping,&nbsp; the output signal is also Gaussian distributed. &nbsp; Only the distribution parameters such as mean and standard deviation change,&nbsp; as well as the internal statistical ties of the samples.}}.
+*If one applies a Gaussian distributed signal to a linear filter for spectral shaping,&nbsp; the output signal is also Gaussian distributed. &nbsp; Only the distribution parameters such as mean and standard deviation change,&nbsp; as well as the internal statistical ties '''KORREKTUR: ties or bindings?''' of the samples.}}.
 [[File:P_ID68__Sto_T_3_5_S1_neu.png |right|frame|Gaussian distributed and uniformly distributed random signal]]
@@ Line 164: / Line 164: @@
 The Box and Muller method&nbsp; &ndash; hereafter abbreviated to&nbsp; "BM"&nbsp; &ndash; can be characterized as follows:
 *The theoretical background for the validity of above generation rules is based on the regularities for&nbsp; [[Theory_of_Stochastic_Signals/Two-Dimensional_Random_Variables|$\text{two-dimensional random variables}$]].
-*Obvious equations successively yield two Gaussian values without statistical bindings '''KORREKTUR: ties?'''.&nbsp; This fact can be used to reduce simulation time by generating a tuple&nbsp; $(x, \ y)$&nbsp; of Gaussian values at each function call.
+*Obvious equations successively yield two Gaussian values without statistical bindings '''KORREKTUR: ties or bindings?'''.&nbsp; This fact can be used to reduce simulation time by generating a tuple&nbsp; $(x, \ y)$&nbsp; of Gaussian values at each function call.
 *A comparison of the computation times shows that&nbsp; &ndash; with the best possible implementation in each case&nbsp; &ndash; the BM method is superior to the addition method with&nbsp; $I =12$&nbsp; by&nbsp;  (approximately)&nbsp; a factor of&nbsp; $3$.
 *The range of values is less limited in the BM method than in the addition method,&nbsp; so that even small probabilities are simulated more accurately.&nbsp; But even with the BM method,&nbsp; it is not possible to simulate arbitrarily small error probabilities.

@@ Line 197: / Line 197: @@
 $\text{Example 4:}$&nbsp;
 The sketch shows the PDF splitting for&nbsp; $J = 16$&nbsp; by the boundaries&nbsp; $I_{-7}$, ... , $ I_7$.
-*These interval boundaries were chosen so that each interval has the same area&nbsp; $p_j = 1/J = 1/16$&nbsp; .
+*These interval boundaries were chosen so that each interval has the same area&nbsp; $p_j = 1/J = 1/16$.&nbsp;
 *The characteristic value&nbsp; $C_j$&nbsp; of each interval lies exactly midway between&nbsp; $I_{j-1}$&nbsp; and&nbsp; $I_j$.

@@ Line 178: / Line 178: @@
 However,&nbsp; a simulation documented in&nbsp; [ES96]<ref name='ES96'>Eck, P.; Söder, G.:&nbsp; Tabulated Inversion, a Fast Method for White Gaussian Noise Simulation.&nbsp; In:&nbsp; AEÜ Int. J. Electron. Commun. 50 (1996), pp. 41-48.</ref>&nbsp; over&nbsp; $10^{9}$&nbsp; samples has shown that the BM method approximates the Q function very well only up to error probabilities of&nbsp; $10^{-5}$&nbsp; but then the curve shape breaks off steeply.
 *The maximum occurring value of the root expression was not&nbsp; $6.55$,&nbsp; but due to the current random variables&nbsp; $u$&nbsp; and&nbsp; $v$&nbsp; only about&nbsp; $4.6$,&nbsp; which explains the abrupt drop from about&nbsp; $10^{-5}$&nbsp; on.
-*Of course,&nbsp; this method works much better with 64 bit arithmetic operations}}.
+*Of course,&nbsp; this method works much better with 64 bit arithmetic operations.}}
 ==Gaussian generation with the "Tabulated Inversion" method==

@@ Line 158: / Line 158: @@
 ==Gaussian generation with the Box/Muller method==
 <br>
-In this method,&nbsp; two statistically independent Gaussian distributed random variables&nbsp; $x$&nbsp; and&nbsp; $y$&nbsp; are generated&nbsp; (approximately)&nbsp; from the two&nbsp; $($between&nbsp; $0$&nbsp; and&nbsp; $1$&nbsp; uniformlyly distributed and statistically independent random variables&nbsp; $u$&nbsp; and&nbsp; $v)$&nbsp; by&nbsp; [[Theory_of_Stochastic_Signals/Exponentially_Distributed_Random_Variables#Transformation_of_random_variables|$\text{nonlinear transformation}$]]:
+In this method,&nbsp; two statistically independent Gaussian distributed random variables&nbsp; $x$&nbsp; and&nbsp; $y$&nbsp; are generated&nbsp; (approximately)&nbsp; from the two&nbsp; $($between&nbsp; $0$&nbsp; and&nbsp; $1$&nbsp; uniformly distributed and statistically independent random variables&nbsp; $u$&nbsp; and&nbsp; $v)$&nbsp; by&nbsp; [[Theory_of_Stochastic_Signals/Exponentially_Distributed_Random_Variables#Transformation_of_random_variables|$\text{nonlinear transformation}$]]:
 :$$x=m_x+\sigma_{x}\cdot \cos(2 \pi u)\cdot\sqrt{-2\cdot \ln(v)},$$
 :$$y=m_y+\sigma_{y}\cdot \sin(2 \pi u)\cdot\sqrt{-2\cdot \ln(v)}.$$

@@ Line 71: / Line 71: @@
 [[File:P_ID621__Sto_T_3_5_S3neu.png |right|frame| Complementary Gaussian error integral&nbsp; ${\rm Q}(x)$]]
 :$${\rm Pr}(x > x_{\rm 0})={\rm Q}({x_{\rm 0} }/{\sigma}).$$
-*Here,&nbsp; ${\rm Q}(x) = 1 - {\rm ϕ}(x)$&nbsp; denotes the complementary function to&nbsp; $ {\rm ϕ}(x)$.&nbsp; This function is called the&nbsp; &raquo;'''Complementary Gaussian error integral'''&laquo;&nbsp; and the following calculation rule applies:
+*Here,&nbsp; ${\rm Q}(x) = 1 - {\rm ϕ}(x)$&nbsp; denotes the complementary function to&nbsp; $ {\rm ϕ}(x)$.&nbsp; This function is called the&nbsp; &raquo;'''complementary Gaussian error integral'''&laquo;&nbsp; and the following calculation rule applies:
 :$$\rm Q (\it x\rm ) = \rm 1- \phi (\it x)$$

@@ Line 42: / Line 42: @@
 [[File:EN_Sto_T_3_5_S2.png |right|frame| PDF and CDF of a Gaussian distributed random variable]]
 $$\hspace{0.4cm}f_x(x) = \frac{1}{\sqrt{2\pi}\cdot\sigma}\cdot {\rm e}^{-(x-m_1)^2 /(2\sigma^2) }.$$
-The parameters of such a Gaussian PDF are.
+The parameters of such a Gaussian PDF are
 *$m_1$&nbsp; ("mean"&nbsp; or&nbsp; "DC component"),
 *$σ$&nbsp; ("standard deviation").