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

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	*Mean, variance and standard deviation have the following values:		*Mean, variance and standard deviation have the following values:
	:$$m_{\rm 1} = \frac{x_ {\rm max} + x_{\rm min} }{2},\hspace{0.5cm}		:$$m_{\rm 1} = \frac{x_ {\rm max} + x_{\rm min} }{2},\hspace{0.5cm}\sigma^2 = \frac{[x_{\rm max} - x_{\rm min}]^2}{12},\hspace{0.5cm}\sigma = \frac{x_{\rm max} - x_{\rm min} }{2 \sqrt{3} }.$$
	\sigma^2 = \frac{[x_{\rm max} - x_{\rm min}]^2}{12},\hspace{0.5cm}
	\sigma = \frac{x_{\rm max} - x_{\rm min} }{2 \sqrt{3} }.$$

	*For symmetric PDF   ⇒   $x_{\rm min} = -x_{\rm max}$  we obtain as a special case the mean  $m_1 = 0$  and the variance  $σ^2 = x_{\rm max}^2/3.$}}		*For symmetric PDF   ⇒   $x_{\rm min} = -x_{\rm max}$  we obtain as a special case the mean  $m_1 = 0$  and the variance  $σ^2 = x_{\rm max}^2/3.$}}

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	{{Display}}		{{Display}}
			[[de:Stochastische_Signaltheorie/Gleichverteilte_Zufallsgrößen]]

Guenter at 17:01, 20 February 2024

2024-02-20T17:01:13Z

← Older revision		Revision as of 19:01, 20 February 2024
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	The ~~interactive~~ applet  [[Applets:PDF,_CDF_and_Moments_of_Special_Distributions\|"PDF, CDF and moments of special distributions"]]  calculates ~~all~~ characteristics of the uniform distribution for any parameters  $x_{\rm min}$  and  $x_{\rm max}$.		The HTML5/JavaScript applet  [[Applets:PDF,_CDF_and_Moments_of_Special_Distributions\|»PDF, CDF and moments of special distributions«]]  calculates the characteristics of the uniform distribution for any parameters  $x_{\rm min}$  and  $x_{\rm max}$.


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	{{BlaueBox\|TEXT=		{{BlaueBox\|TEXT=
	$\text{Definition}$  The random generators used today are mostly  »'''pseudo–random'''«.  This means,		$\text{Definition}$  The random generators used today are mostly  »'''pseudo–random'''«.  This means,
	*that the sequence generated is actually deterministic as the result of a fixed algorithm,		*that the sequence generated is actually deterministic as the result of a fixed algorithm,

	*but appears to the user as stochastic due to the large period length  $P$.		*but appears to the user as stochastic due to the large period length  $P$.


	More on this in the chapter  [[Theory_of_Stochastic_Signals/Generation_of_Discrete_Random_Variables\|"Generation of discrete random variables"]]. }}		More on this in the chapter  [[Theory_of_Stochastic_Signals/Generation_of_Discrete_Random_Variables\|»Generation of discrete random variables«]]. }}


	For system simulation,  pseudo–noise  $\rm (PN)$  generators have the distinct advantage over true random generators that the generated random sequences can be reproduced without storage, which
	*allows the comparison of different system models,  and
	*also makes troubleshooting much easier.

			For system simulation,  pseudo–noise  $\rm (PN)$  generators have the distinct advantage over true random generators that the generated random sequences can be reproduced without storage, which allows the comparison of different system models,  and also makes troubleshooting much easier.

	{{BlaueBox\|TEXT=		{{BlaueBox\|TEXT=
	$\text{A random sequence generator should meet the following criteria:}$		$\text{A random sequence generator should meet the following criteria:}$

	'''(1)'''   The random variables  $x_ν$  of a generated sequence should be uniformly distributed with very good approximation.  For discrete-value representation on a computer,  this requires,  among other things,  a sufficiently  high bit resolution,  for example, with  $32$  or  $64$  bits per sample.		'''(1)'''   The random variables  $x_ν$  of a generated sequence should be uniformly distributed with very good approximation.  For the discrete-value representation on a computer,  this requires,  among other things,  a sufficiently  high bit resolution,  for example, with  $32$  or  $64$  bits per sample.

	'''(2)'''   If one forms from the sequential random sequence  $〈x_ν〉$  respectively non-overlapping pairs of random variables,  for example  $(x_ν, x_{ν+1})$,  $(x_{ν+2}$,  $x_{ν+3})$, ... , then these  "tuples"  should also be equally distributed in a two-dimensional representation within a square.		'''(2)'''   If one forms from the sequential random sequence  $〈x_ν〉$  respectively non-overlapping pairs of random variables,  for example  $(x_ν, x_{ν+1})$,  $(x_{ν+2}$,  $x_{ν+3})$, ... , then these  »tuples«  should also be equally distributed in a two-dimensional representation within a square.

	'''(3)'''   If one forms from the sequential series  $〈x_ν〉$  non-overlapping  $n$–tuples   of random variables   ⇒   $(x_ν$, . ... , $x_{ν+n-1})$,  $(x_{ν+n}$, ... , $x_{ν+2n-1})$  ~~etc.~~, then these should also yield the uniform distribution within a  $n$–dimensional cube~~,  if possible~~.}}		'''(3)'''   If one forms from the sequential series  $〈x_ν〉$  non-overlapping  $n$–tuples   of random variables   ⇒   $(x_ν$, . ... , $x_{ν+n-1})$,  $(x_{ν+n}$, ... , $x_{ν+2n-1})$  and so on, <br>then these should also yield the uniform distribution within a  $n$–dimensional cube.}}



	Note:		$\text{Note:}$
	*The first requirement refers exclusively to the  "amplitude distribution"  $\rm (PDF)$  and is generally easier to satisfy.		*The first requirement refers exclusively to the  »amplitude distribution«  $\rm (PDF)$  and is generally easier to satisfy.
	*The other requirements ensure  "sufficient randomness"  of the sequence.  They concern the statistical independence of successive random values.
			*The other requirements ensure  »sufficient randomness«  of the sequence.  They concern the statistical independence of successive random values.


	==Multiplicative Congruental Generator==		==Multiplicative Congruental Generator==
	<br>		<br>
	~~$\text{Multiplicative Congruental Generator}$~~  is the best known method for generating a sequence  $〈 x_\nu 〉$  with equally distributed values  $ x_\nu$  between  $0$  and  $1$.  This method is given here in a bullet-point fashion:		The  »'''multiplicative congruental generator'''«  is the best known method for generating a sequence  $〈 x_\nu 〉$  with equally distributed values  $ x_\nu$  between  $0$  and  $1$.  This method is given here in a bullet-point fashion:

	'''(1)'''   These random generators are based on the successive manipulation of an integer variable  $k$.  If the number representation in the computer happens with  $L$  bit,  this variable takes all values between  $1$  and  $2^{L - 1}$  exactly once each,  if the sign bit is handled appropriately.		'''(1)'''   These random generators are based on the successive manipulation of an integer variable  $k$.  If the number representation in the computer happens with  $L$  bit,  this variable takes all values between  $1$  and  $2^{L - 1}$  exactly once each,  if the sign bit is handled appropriately.

	'''(2)'''   The random variable ~~derived from this~~  $x={k}/{\rm 2^{\it L - \rm 1}}$  is also discrete  $($with level number  $M = 2^{L- 1})$:		'''(2)'''   The random variable  $x={k}/{\rm 2^{\it L - \rm 1}}$  derived from this is also discrete-valued  $($with level number  $M = 2^{L- 1})$:
	:$$x={k}/{\rm 2^{\it L - \rm 1}} = k\cdot \~~delta~~ x \in \{\~~delta~~ x, \hspace{0.05cm}2\cdot \~~delta~~ x,\hspace{0.05cm}\text{ ...}\hspace{0.05cm} , \hspace{0.05cm}1-\~~delta~~ x,\hspace{0.05cm} 1\}.$$		::$$x={k}/{\rm 2^{\it L - \rm 1}} = k\cdot \Delta x \in \{\Delta x, \hspace{0.05cm}2\cdot \Delta x,\hspace{0.05cm}\text{ ...}\hspace{0.05cm} , \hspace{0.05cm}1-\Delta x,\hspace{0.05cm} 1\}.$$
	:If the bit number  $L$  is sufficiently large,  the distance  $Δx = 1/2^{L- 1}$  between two possible values is very small,  and one may well interpret  $x$  as a continuous-valued random variable in the context of simulation accuracy.		:If the bit number  $L$  is sufficiently large,  the distance  $Δx = 1/2^{L- 1}$  between two possible values is very small,  and one may well interpret  $x$  as a continuous-valued random variable in the context of simulation accuracy.

	'''(3)'''   The recursive generation rule of such  "multiplicative congruential generators"  is:		'''(3)'''   The recursive generation rule of such  »multiplicative congruential generators«  is:
	:$$k_\nu=(a\cdot k_{\nu-1})\hspace{0.1cm} \rm mod \hspace{0.1cm} \it m.$$		::$$k_\nu=(a\cdot k_{\nu-1})\hspace{0.1cm} \rm mod \hspace{0.1cm} \it m.$$

	'''(4)'''   The statistical properties of the sequence depend crucially on the parameters  $a$  and  $m$.  The initial value  $k_0$  has a minor importance for the statistics.		'''(4)'''   The statistical properties of the sequence depend crucially on the parameters  $a$  and  $m$.  The initial value  $k_0$  has a minor importance for the statistics.

	'''(5)'''   The best results are obtained with the base  $m =2\hspace{0.05cm}^l-1$,  where  $l$  denotes any natural number.  Widely used in computers with 32-bit architecture and one sign bit is the base  $m = 2^{31} - 1 = 2\hspace{0.08cm}147\hspace{0.08cm}483\hspace{0.08cm}647$.  A corresponding algorithm is:		'''(5)'''   The best results are obtained with the base  $m =2\hspace{0.05cm}^l-1$,  where  $l$  denotes any natural number.  Widely used in computers with 32-bit architecture and one sign bit is the base  $m = 2^{31} - 1 = 2\hspace{0.08cm}147\hspace{0.08cm}483\hspace{0.08cm}647$.  A corresponding algorithm is:
	:$$k_\nu=(16807\cdot k_{\nu-1})\hspace{0.1cm} \rm mod\hspace{0.1cm}(2^{31}-1).$$		::$$k_\nu=(16807\cdot k_{\nu-1})\hspace{0.1cm} \rm mod\hspace{0.1cm}(2^{31}-1).$$

	'''(6)'''   For such a generator, only the initial value  $k_0 = 0$  is not allowed.  For  $k_0 \ne 0$  the period duration  $P = 2^{31} - 2.$		'''(6)'''   For such a generator, only the initial value  $k_0 = 0$  is not allowed.  For  $k_0 \ne 0$  the period duration  $P = 2^{31} - 2.$

	~~[[File:EN_Sto_T_3_4_S4.png \|right\|frame\| Multiplicative Congruental Generator  (C program)]]~~
	{{GraueBox\|TEXT=		{{GraueBox\|TEXT=
	~~$\text{Example 2~~:~~}$  We analyze the  "~~Multiplicative Congruental Generator"  ~~in more detail:~~		[[File:EN_Sto_T_3_4_S4.png \|right\|frame\| Multiplicative Congruental Generator  $($C program$)$]]

			$\text{Example 2:}$  We analyze the  »multiplicative congruental generator«  in more detail:

	*The algorithm,  however,  cannot be implemented directly on a 32–bit computer,  since the multiplication result requires up to 46 bits.


			*The algorithm cannot be implemented directly on a 32–bit computer,  since the multiplication result requires up to 46 bits.

	*But it can be modified in such a way that at no time during the calculation the 32–bit integer number range is exceeded.		*But it can be modified in such a way that at no time during the calculation the 32–bit integer number range is exceeded.


	*The C program  $\text{uniform( )}$  thus modified is given on the right.}}		*The C program  $\text{uniform( )}$  thus modified is given on the right.}}

Guenter at 16:27, 20 February 2024

2024-02-20T16:27:09Z

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	==General description and definition==		==General description and definition==
	<br>		<br>
	~~[[File:P_ID45__Sto_T_3_4_S1_neu100.png \|frame\|PDF and CDF of uniform distribution]]~~
	{{BlaueBox\|TEXT=		{{BlaueBox\|TEXT=
	$\text{Definition:}$  A random variable  $x$  is said to be  »'''uniformly distributed'''«  if it can only take values in the range of  $x_{\rm min}$  to  $x_{\rm max}$  with equal probability.		$\text{Definition:}$  A random variable  $x$  is said to be  »'''uniformly distributed'''«  if it can only take values in the range of  $x_{\rm min}$  to  $x_{\rm max}$  with equal probability.
			[[File:P_ID45__Sto_T_3_4_S1_neu100.png \|frame\|PDF and CDF of uniform distribution]]

	The graph shows		The graph shows of such an equally distributed random variable  $x$
	*on the left the probability density function  $f_{x}(x)$,		#on the left the probability density function  $f_{x}(x)$,
	*right the cumulative distribution function  $F_{x}(r)$		#on the right the cumulative distribution function  $F_{x}(r)$.


	~~of such an equally distributed random variable~~  $x$.}}		From the graph and this definition,  the following properties can be derived:
			*The probability density function  $\rm (PDF)$  has in the range from  $x_{\rm min}$  to  $x_{\rm max}$  the constant value  $1/(x_{\rm max} - x_{\rm min})$.

			*On the range limits,  only half the value – that is,  the average value between the left–hand and right–hand limits – is to be set for  $f_{x}(x)$  in each case.

	~~From the graph and this definition,  the following properties can be derived:~~		*The cumulative distribution function  $\rm (CDF)$  increases linearly from  $x_{\rm min}$  to  $x_{\rm max}$  in the range from  $0$  to  $1$ .
	*The probability density function  $\rm (PDF)$  has in the range from  $x_{\rm min}$  to  $x_{\rm max}$  the constant value  $1/(x_{\rm max} - x_{\rm min})$.
	*On the range limits,  only half the value - that is,  the average value between the left–hand and right–hand limits - is to be set for  $f_{x}(x)$  in each case.		*Mean, variance and standard deviation have the following values:
	*The distribution function  $\rm (CDF)$  increases linearly from  $x_{\rm min}$  to  $x_{\rm max}$  in the range from  $0$  to  $1$ .		:$$m_{\rm 1} = \frac{x_ {\rm max} + x_{\rm min} }{2},\hspace{0.5cm}
	*Mean, variance and standard deviation ~~of the uniform distribution~~ have the following values:
	:$$m_{\rm 1} = \frac{~~\it~~ x_ {\rm max} ~~\rm~~ + ~~\it~~ x_{\rm min}}{2},\hspace{0.5cm}
	\sigma^2 = \frac{[x_{\rm max} - x_{\rm min}]^2}{12},\hspace{0.5cm}		\sigma^2 = \frac{[x_{\rm max} - x_{\rm min}]^2}{12},\hspace{0.5cm}
	\sigma = \frac{~~\it~~ x_{\rm max} - ~~\it~~ x_{\rm min}}{2 \sqrt{3}}.$$		\sigma = \frac{x_{\rm max} - x_{\rm min} }{2 \sqrt{3} }.$$
	*For symmetric PDF   ⇒   $x_{\rm min} = -x_{\rm max}$  we obtain as a special case the mean  $m_1 = 0$  and the variance  $σ^2 = x_{\rm max}^2/3.$
			*For symmetric PDF   ⇒   $x_{\rm min} = -x_{\rm max}$  we obtain as a special case the mean  $m_1 = 0$  and the variance  $σ^2 = x_{\rm max}^2/3.$}}


	{{GraueBox\|TEXT=		{{GraueBox\|TEXT=
	$\text{Example 1:}$		$\text{Example 1:}$  The graph shows two signal waveforms with uniform amplitude distribution.
	The graph shows two signal waveforms with uniform amplitude distribution.		[[File:P_ID618__Sto_T_3_4_S2_neu100.png \|right\|frame\|Examples of uniformly distributed signals]]

	~~[[File:P_ID618__Sto_T_3_4_S2_neu100.png \|right\|frame\|Examples of uniformly distributed signals]]~~


	*On the left,  statistical independence of the individual samples is assumed,  that is,  the random variable  $x_ν$  can take all values between  $x_{\rm min}$  and  $x_{\rm max}$  with equal probability,  and independently of the past  $(x_{ν-1}, x_{ν-2}, \hspace{0.1cm}\text{...}).$		$\rm (A)$  On the left,  statistical independence of the individual samples is assumed,  that is,  the random variable  $x_ν$  can take all values between  $x_{\rm min}$  and  $x_{\rm max}$  with equal probability,  and independently of the past  $(x_{ν-1}, x_{ν-2}, \hspace{0.1cm}\text{...}).$


	*For the right signal  $y(t)$  this independence of successive signal values is no longer given.  Rather,  this sawtooth signal represents a deterministic signal.}}		$\rm (B)$  For the right signal  $y(t)$  this independence of successive signal values is no longer given.  Rather,  this sawtooth signal represents a deterministic signal.}}

	==Importance of the uniform distribution for ~~communications engineering~~==		==Importance of the uniform distribution for Communications Engineering==
	<br>		<br>
	The importance of uniformly distributed random variables for information  and communication technology is due to the fact that,  from the point of view of information theory,  this PDF form represents an optimum under the constraint  [[Digital_Signal_Transmission/Optimization_of_Baseband_Transmission_Systems#Power_and_peak_limitation\|~~$\text{~~peak limitation}$]]:		The importance of uniformly distributed random variables for information  and communication technology is due to the fact that,  from the point of view of information theory,  this PDF form represents an optimum under the constraint  [[Digital_Signal_Transmission/Optimization_of_Baseband_Transmission_Systems#Power_and_peak_limitation\|»peak limitation«]]:
	*With no distribution other than the uniform distribution one achieves greater  [[Information_Theory/Differential_Entropy#Differential_entropy_of_some_peak-constrained_random_variables\|~~$\text{~~differential entropy}$]]  under this condition.		*With no distribution other than the uniform distribution one achieves greater  [[Information_Theory/Differential_Entropy#Differential_entropy_of_some_peak-constrained_random_variables\|»differential entropy«]]  under this condition.
	*This topic is dealt with in the chapter  [[Information_Theory/Differential_Entropy\|"Differential Entropy"]]  in the book  "Information Theory".
			*This topic is dealt with in the chapter  [[Information_Theory/Differential_Entropy\|»Differential Entropy«]]  in the book  »Information Theory«.


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	$\text{In addition,  the following points should be mentioned,  among others:}$		$\text{In addition,  the following points should be mentioned,  among others:}$

	'''(1)'''   The importance of the uniform distribution for the simulation of communication systems is due to the fact that one can realize corresponding  "pseudo-random generators"  relatively easily,  and that other distributions,  such as the  [[Theory_of_Stochastic_Signals/Gaussian_Distributed_Random_Variables#Probability_density_function_.E2.80.93_Cumulative_density_function\|~~$\text{~~Gaussian distribution}$]]  and the  [[Theory_of_Stochastic_Signals/Exponentially_Distributed_Random_Variables#One-sided_exponential_distribution\|~~$\text{~~exponential distribution}$]]  can be easily derived.		'''(1)'''   The importance of the uniform distribution for the simulation of communication systems is due to the fact that one can realize corresponding  »pseudo-random generators«  relatively easily,  and that other distributions,  such as the  [[Theory_of_Stochastic_Signals/Gaussian_Distributed_Random_Variables#Probability_density_function_.E2.80.93_Cumulative_density_function\|»Gaussian distribution«]]  and the  [[Theory_of_Stochastic_Signals/Exponentially_Distributed_Random_Variables#One-sided_exponential_distribution\|»exponential distribution«]]  can be easily derived.

	'''(2)'''   In  "Image Processing & ~~Coding"~~,  simplifying calculations are often made using the uniform distribution instead of the actual distribution of the original image,  which is ~~usually~~ much more complicated,  since the difference in information content between a  "natural image"  and the model based on the uniform distribution is relatively small.		'''(2)'''   In  »Image Processing«,  simplifying calculations are often made using the uniform distribution instead of the actual distribution of the original image,  which is often much more complicated,  since the difference in information content between a  »natural image«  and the model based on the uniform distribution is relatively small.

	'''(3)'''   For modeling transmission systems,  on the other hand,  uniformly distributed random variables are the exception.  An example of an ~~actually~~  (nearly)  uniformly distributed random variable is the phase in the presence of circularly symmetric interference,  such as occurs in  "quadrature amplitude modulation techniques"  (QAM).}}		'''(3)'''   For modeling transmission systems,  on the other hand,  uniformly distributed random variables are the exception.  An example of an  $($nearly$)$  uniformly distributed random variable is the phase in the presence of circularly symmetric interference,  such as occurs in  »quadrature amplitude modulation techniques«  $\rm (QAM)$.}}

Guenter at 08:59, 22 December 2022

2022-12-22T08:59:25Z

← Older revision		Revision as of 10:59, 22 December 2022
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	*On the range limits,  only half the value - that is,  the average value between the left–hand and right–hand limits - is to be set for  $f_{x}(x)$  in each case.		*On the range limits,  only half the value - that is,  the average value between the left–hand and right–hand limits - is to be set for  $f_{x}(x)$  in each case.
	*The distribution function  $\rm (CDF)$  increases linearly from  $x_{\rm min}$  to  $x_{\rm max}$  in the range from  $0$  to  $1$ .		*The distribution function  $\rm (CDF)$  increases linearly from  $x_{\rm min}$  to  $x_{\rm max}$  in the range from  $0$  to  $1$ .
	*Mean, variance and standard deviation~~  (standard deviation)  '''KORREKTUR: 2x standard deviation?'''~~ of the uniform distribution have the following values:		*Mean, variance and standard deviation of the uniform distribution have the following values:
	:$$m_{\rm 1} = \frac{\it x_ {\rm max} \rm + \it x_{\rm min}}{2},\hspace{0.5cm}		:$$m_{\rm 1} = \frac{\it x_ {\rm max} \rm + \it x_{\rm min}}{2},\hspace{0.5cm}
	\sigma^2 = \frac{[x_{\rm max} - x_{\rm min}]^2}{12},\hspace{0.5cm}		\sigma^2 = \frac{[x_{\rm max} - x_{\rm min}]^2}{12},\hspace{0.5cm}

Hwang at 19:19, 20 December 2022

2022-12-20T19:19:20Z

← Older revision		Revision as of 21:19, 20 December 2022
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	*The algorithm,  however,  cannot be implemented directly on a 32 bit computer,  since the multiplication result requires up to 46 bits.		*The algorithm,  however,  cannot be implemented directly on a 32–bit computer,  since the multiplication result requires up to 46 bits.


	*But it can be modified in such a way that at no time during the calculation the 32 ~~bit~~–integer number range is exceeded.		*But it can be modified in such a way that at no time during the calculation the 32–bit integer number range is exceeded.

Hwang at 20:27, 13 December 2022

2022-12-13T20:27:37Z

← Older revision		Revision as of 22:27, 13 December 2022
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	<br>		<br>
	The importance of uniformly distributed random variables for information  and communication technology is due to the fact that,  from the point of view of information theory,  this PDF form represents an optimum under the constraint  [[Digital_Signal_Transmission/Optimization_of_Baseband_Transmission_Systems#Power_and_peak_limitation\|$\text{peak limitation}$]]:		The importance of uniformly distributed random variables for information  and communication technology is due to the fact that,  from the point of view of information theory,  this PDF form represents an optimum under the constraint  [[Digital_Signal_Transmission/Optimization_of_Baseband_Transmission_Systems#Power_and_peak_limitation\|$\text{peak limitation}$]]:
	*With no distribution other than the uniform distribution ~~does~~ one ~~achieve~~ greater  [[Information_Theory/Differential_Entropy#Differential_entropy_of_some_peak-constrained_random_variables\|$\text{differential entropy}$]]  under this condition.		*With no distribution other than the uniform distribution one achieves greater  [[Information_Theory/Differential_Entropy#Differential_entropy_of_some_peak-constrained_random_variables\|$\text{differential entropy}$]]  under this condition.
	*This topic is dealt with in the chapter  [[Information_Theory/Differential_Entropy\|"Differential Entropy"]]  in the book  "Information Theory".		*This topic is dealt with in the chapter  [[Information_Theory/Differential_Entropy\|"Differential Entropy"]]  in the book  "Information Theory".

Hwang at 20:25, 13 December 2022

2022-12-13T20:25:54Z

← Older revision		Revision as of 22:25, 13 December 2022
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	$\text{In addition,  the following points should be mentioned,  among others:}$		$\text{In addition,  the following points should be mentioned,  among others:}$

	'''(1)'''   The importance of the uniform distribution for the simulation of communication systems is due to the fact that one can realize corresponding  "pseudo-random generators"  relatively easily,  and that other distributions,  such as the  [[Theory_of_Stochastic_Signals/Gaussian_Distributed_Random_Variables#Probability_density_function_.~~26_cumulative_density_function~~\|$\text{Gaussian distribution}$]]  and the  [[Theory_of_Stochastic_Signals/Exponentially_Distributed_Random_Variables#One-sided_exponential_distribution\|$\text{exponential distribution}$]]  can be easily derived.		'''(1)'''   The importance of the uniform distribution for the simulation of communication systems is due to the fact that one can realize corresponding  "pseudo-random generators"  relatively easily,  and that other distributions,  such as the  [[Theory_of_Stochastic_Signals/Gaussian_Distributed_Random_Variables#Probability_density_function_.E2.80.93_Cumulative_density_function\|$\text{Gaussian distribution}$]]  and the  [[Theory_of_Stochastic_Signals/Exponentially_Distributed_Random_Variables#One-sided_exponential_distribution\|$\text{exponential distribution}$]]  can be easily derived.

	'''(2)'''   In  "Image Processing & Coding",  simplifying calculations are often made using the uniform distribution instead of the actual distribution of the original image,  which is usually much more complicated,  since the difference in information content between a  "natural image"  and the model based on the uniform distribution is relatively small.		'''(2)'''   In  "Image Processing & Coding",  simplifying calculations are often made using the uniform distribution instead of the actual distribution of the original image,  which is usually much more complicated,  since the difference in information content between a  "natural image"  and the model based on the uniform distribution is relatively small.

Hwang at 20:25, 13 December 2022

2022-12-13T20:25:17Z

← Older revision		Revision as of 22:25, 13 December 2022
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	==Importance of the uniform distribution for communications engineering==		==Importance of the uniform distribution for communications engineering==
	<br>		<br>
	The importance of uniformly distributed random variables for information  and communication technology is due to the fact that,  from the point of view of information theory,  this PDF form represents an optimum under the constraint  [[Digital_Signal_Transmission/Optimization_of_Baseband_Transmission_Systems#~~Power.E2.80.93_and_peak_limiting~~\|$\text{peak ~~limiting~~}$]]:		The importance of uniformly distributed random variables for information  and communication technology is due to the fact that,  from the point of view of information theory,  this PDF form represents an optimum under the constraint  [[Digital_Signal_Transmission/Optimization_of_Baseband_Transmission_Systems#Power_and_peak_limitation\|$\text{peak limitation}$]]:
	*With no distribution other than the uniform distribution does one achieve greater  [[Information_Theory/Differential_Entropy#Differential_entropy_of_some_peak-constrained_random_variables\|$\text{differential entropy}$]]  under this condition.		*With no distribution other than the uniform distribution does one achieve greater  [[Information_Theory/Differential_Entropy#Differential_entropy_of_some_peak-constrained_random_variables\|$\text{differential entropy}$]]  under this condition.
	*This topic is dealt with in the chapter  [[Information_Theory/Differential_Entropy\|"Differential Entropy"]]  in the book  "Information Theory".		*This topic is dealt with in the chapter  [[Information_Theory/Differential_Entropy\|"Differential Entropy"]]  in the book  "Information Theory".

Hwang at 20:23, 13 December 2022

2022-12-13T20:23:36Z

← Older revision		Revision as of 22:23, 13 December 2022
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	*On the range limits,  only half the value - that is,  the average value between the left–hand and right–hand limits - is to be set for  $f_{x}(x)$  in each case.		*On the range limits,  only half the value - that is,  the average value between the left–hand and right–hand limits - is to be set for  $f_{x}(x)$  in each case.
	*The distribution function  $\rm (CDF)$  increases linearly from  $x_{\rm min}$  to  $x_{\rm max}$  in the range from  $0$  to  $1$ .		*The distribution function  $\rm (CDF)$  increases linearly from  $x_{\rm min}$  to  $x_{\rm max}$  in the range from  $0$  to  $1$ .
	*Mean, variance and standard deviation  (standard deviation)  of the uniform distribution have the following values:		*Mean, variance and standard deviation  (standard deviation)  '''KORREKTUR: 2x standard deviation?''' of the uniform distribution have the following values:
	:$$m_{\rm 1} = \frac{\it x_ {\rm max} \rm + \it x_{\rm min}}{2},\hspace{0.5cm}		:$$m_{\rm 1} = \frac{\it x_ {\rm max} \rm + \it x_{\rm min}}{2},\hspace{0.5cm}
	\sigma^2 = \frac{[x_{\rm max} - x_{\rm min}]^2}{12},\hspace{0.5cm}		\sigma^2 = \frac{[x_{\rm max} - x_{\rm min}]^2}{12},\hspace{0.5cm}