Difference between revisions of "Theory of Stochastic Signals/Uniformly Distributed Random Variables"

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{{Header
 
{{Header
|Untermenü=Kontinuierliche Zufallsgrößen
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|Untermenü=Continuous Random Variables
|Vorherige Seite=Erwartungswerte und Momente
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|Vorherige Seite=Expected Values and Moments
|Nächste Seite=Gaußverteilte Zufallsgröße
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|Nächste Seite=Gaussian Distributed Random Variables
 
}}
 
}}
==Allgemeine Beschreibung und Definition==
+
==General description and definition==
{{Definition}}
+
<br>
Eine Zufallsgröße $x$ bezeichnet man als gleichverteilt, wenn sie nur Werte im Bereich von $x_{\rm min}$ bis $x_{\rm max}$ annehmen kann, und zwar mit gleicher Wahrscheinlichkeit.  
+
[[File:P_ID45__Sto_T_3_4_S1_neu100.png |frame|PDF and CDF of uniform distribution]]
{{end}}
+
{{BlaueBox|TEXT= 
 +
$\text{Definition:}$&nbsp; A random variable&nbsp; $x$&nbsp; is said to be&nbsp; '''uniformly distributed'''&nbsp; if it can only take values in the range of&nbsp; $x_{\rm min}$&nbsp; to&nbsp; $x_{\rm max}$&nbsp; with equal probability.  
  
 +
The graph shows
 +
*on the left the probability density function&nbsp; $f_{x}(x)$,
 +
*right the cumulative distribution function&nbsp; $F_{x}(r)$
  
Die Grafik zeigt links die Wahrscheinlichkeitsdichtefunktion (abgekürzt WDF) und rechts die Verteilungsfunktion (kurz VTF) einer gleichverteilten Zufallsgröße $x$.
 
  
[[File:P_ID45__Sto_T_3_4_S1_neu100.png | WDF und VTF der Gleichverteilung]]
+
of such an equally distributed random variable&nbsp; $x$.}}
  
  
Daraus können folgende Eigenschaften abgeleitet werden:  
+
From the graph and this definition,&nbsp; the following properties can be derived:  
*Die WDF $f_{\rm x}(x)$ besitzt im Bereich von $x_{\rm min}$ bis $x_{\rm max}$ den konstanten Wert $1/(x_{\rm max} - x_{\rm min})$, wobei an den beiden Bereichsgrenzen für $f_{\rm x}(x)$ jeweils nur der halbe Wert – also der Mittelwert zwischen links- und rechtsseitigem Grenzwert – zu setzen ist.  
+
*The probability density function&nbsp; $\rm (PDF)$&nbsp; has in the range from&nbsp; $x_{\rm min}$&nbsp; to&nbsp; $x_{\rm max}$&nbsp; the constant value&nbsp; $1/(x_{\rm max} - x_{\rm min})$.&nbsp;
*Die Verteilungsfunktion $F_{\rm x}(r)$ steigt im Bereich von $x_{\rm min}$ bis $x_{\rm max}$ linear von 0 auf 1 an.   
+
*On the range limits,&nbsp; only half the value - that is,&nbsp; the average value between the left&ndash;hand and right&ndash;hand limits - is to be set for&nbsp; $f_{x}(x)$&nbsp; in each case.  
*Mittelwert und Streuung haben bei der Gleichverteilung die folgenden Werte:
+
*The distribution function&nbsp; $\rm (CDF)$&nbsp; increases linearly from&nbsp; $x_{\rm min}$&nbsp; to&nbsp; $x_{\rm max}$&nbsp; in the range from&nbsp; $0$&nbsp; to&nbsp; $1$&nbsp;.   
$$m_{\rm 1} = \frac{\it x_ {\rm max} + \it x_{\rm min}}{2},\hspace{0.5cm}
+
*Mean, variance and standard deviation&nbsp; (standard deviation)&nbsp;  of the uniform distribution have the following values:
 +
:$$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 = \frac{\it x_{\rm max} - \it x_{\rm min}}{2 \sqrt{3}}.$$
 
\sigma = \frac{\it x_{\rm max} - \it x_{\rm min}}{2 \sqrt{3}}.$$
*Bei symmetrischer WDF $(x_{\rm min} = –x_{\rm max})$ erhält man als Sonderfall $m_1 =$ 0 und $σ^2 = x_{\rm max}^2/3.$
+
*For symmetric PDF &nbsp; &rArr; &nbsp; $x_{\rm min} = -x_{\rm max}$&nbsp; we obtain as a special case the mean&nbsp; $m_1 = 0$&nbsp; and the variance&nbsp; $σ^2 = x_{\rm max}^2/3.$
  
  
{{Beispiel}}  
+
{{GraueBox|TEXT= 
Hier sehen Sie zwei Signalverläufe mit gleichförmiger Amplitudenverteilung.  
+
$\text{Example 1:}$&nbsp;
 +
The graph shows two signal waveforms with uniform amplitude distribution.
  
[[File:P_ID618__Sto_T_3_4_S2_neu100.png | Beispiele gleichverteilter Signale]]
+
[[File:P_ID618__Sto_T_3_4_S2_neu100.png |right|frame|Examples of uniformly distributed signals]]
  
*Links ist statistische Unabhängigkeit der einzelnen Abtastwerte vorausgesetzt, das heißt, $x_ν$ kann alle Werte zwischen $x_{\rm min}$ und $x_{\rm max}$ mit gleicher Wahrscheinlichkeit annehmen, und zwar unabhängig von der Vergangenheit $(x_{ν–1}, x_{ν–2}, ...).$  
+
*Beim rechten Signal $y(t)$ ist diese Unabhängigkeit aufeinanderfolgender Signalwerte nicht mehr gegeben. Vielmehr stellt dieses Sägezahnsignal ein deterministisches Signal dar.
+
*On the left,&nbsp; statistical independence of the individual samples is assumed,&nbsp; that is,&nbsp; the random variable&nbsp; $x_ν$&nbsp; can take all values between&nbsp; $x_{\rm min}$&nbsp; and&nbsp; $x_{\rm max}$&nbsp; with equal probability,&nbsp; and independently of the past&nbsp; $(x_{ν-1}, x_{ν-2}, \hspace{0.1cm}\text{...}).$  
  
  
{{end}}
+
*For the right signal&nbsp; $y(t)$&nbsp; this independence of successive signal values is no longer given.&nbsp; Rather,&nbsp; this sawtooth signal represents a deterministic signal.}}
  
==Bedeutung der Gleichverteilung für die Nachrichtentechnik==
+
==Importance of the uniform distribution for communications engineering==
Die Bedeutung gleichverteilter Zufallsgrößen für die Informations- und Kommunikationstechnik ist darauf zurückzuführen, dass diese WDF–Form aus Sicht der Informationstheorie unter der Nebenbedingung „Spitzenwertbegrenzung” ein Optimum darstellt. Mit keiner anderen Verteilung als der Gleichverteilung erreicht man unter dieser Voraussetzung eine größere differentielle Entropie. Mit dieser Thematik beschäftigt sich das Kapitel 4.1 im Buch „Einführung in die Informationstheorie”.  
+
<br>
 +
The importance of uniformly distributed random variables for information&nbsp; and communication technology is due to the fact that,&nbsp; from the point of view of information theory,&nbsp; this PDF form represents an optimum under the constraint&nbsp; [[Digital_Signal_Transmission/Optimization_of_Baseband_Transmission_Systems#Power.E2.80.93_and_peak_limiting|peak limiting]]:
 +
*With no distribution other than the uniform distribution does one achieve greater&nbsp; [[Information_Theory/Differential_Entropy#Differential_entropy_of_some_peak-constrained_random_variables|differential entropy]]&nbsp; under this condition.   
 +
*This topic is dealt with in the chapter&nbsp; [[Information_Theory/Differential_Entropy|"Differential Entropy"]]&nbsp; in the book&nbsp; "Information Theory".  
  
Daneben sind unter Anderem noch folgende Punkte zu nennen:
 
*Die Bedeutung der Gleichverteiltung für die Simulation nachrichtentechnischer Systeme ist darauf zurückzuführen, dass man entsprechende ''Pseudo–Zufallsgeneratoren'' relativ einfach realisieren kann, und dass sich daraus andere Verteilungen (zum Beispiel die Gauß–, die Laplace– und die Exponentialverteilung) leicht ableiten lassen (vgl. Kapitel 3.5 bis 3.7).
 
*''In Bildverarbeitung & Bildcodierung'' wird häufig vereinfachend mit der Gleichverteilung anstelle der tatsächlichen, meist sehr viel komplizierteren Verteilung des Originalbildes gerechnet, da der Unterschied des Informationsgehaltes zwischen einem ''natürlichen Bild'' und dem auf der Gleichverteilung basierenden Modell relativ gering ist.
 
*Für die Modellierung übertragungstechnischer Systeme sind gleichverteilte Zufallsgrößen dagegen die Ausnahme. Ein Beispiel für eine tatsächlich (nahezu) gleichverteilte Zufallsgröße ist die Phase bei kreissymmetrischen Störungen, wie sie beispielsweise bei ''Quadraturmodulationsverfahren'' auftreten.
 
  
 +
{{BlaueBox|TEXT=
 +
$\text{In addition,&nbsp; the following points should be mentioned,&nbsp; among others:}$
  
Das folgende Tool berechnet unter Anderem die Kenngrößen der Gleichverteilung für beliebige Parameter $x_{\rm min}$ und $x_{\rm max}$:
+
'''(1)''' &nbsp; The importance of the uniform distribution for the simulation of communication systems is due to the fact that one can realize corresponding&nbsp; "pseudo-random generators"&nbsp; relatively easily,&nbsp; and that other distributions,&nbsp; such as the&nbsp; [[Theory_of_Stochastic_Signals/Gaussian_Distributed_Random_Variables#Probability_density_function_.26_cumulative_density_function|Gaussian distribution]]&nbsp; and the&nbsp; [[Theory_of_Stochastic_Signals/Exponentially_Distributed_Random_Variables#One-sided_exponential_distribution|exponential distribution]]&nbsp; can be easily derived.
WDF, VTF und Momente spezieller Verteilungen 
 
  
''Hinweis:'' In dieser Multimedia–Anwendung wird die Gleichverteilung als „Rechteck” bezeichnet.  
+
'''(2)''' &nbsp; In&nbsp; "Image Processing & Coding",&nbsp; simplifying calculations are often made using the uniform distribution instead of the actual distribution of the original image,&nbsp; which is usually much more complicated,&nbsp; since the difference in information content between a&nbsp; "natural image"&nbsp; and the model based on the uniform distribution is relatively small.  
  
 +
'''(3)''' &nbsp; For modeling transmission systems,&nbsp; on the other hand,&nbsp; uniformly distributed random variables are the exception.&nbsp; An example of an actually&nbsp; (nearly)&nbsp; uniformly distributed random variable is the phase in the presence of circularly symmetric interference,&nbsp; such as occurs in&nbsp; "quadrature amplitude modulation techniques"&nbsp; (QAM).}}
 +
  
 +
The interactive applet&nbsp; [[Applets:PDF,_CDF_and_Moments_of_Special_Distributions|"PDF, CDF and moments of special distributions"]]&nbsp; calculates all characteristics of the uniform distribution for any parameters&nbsp; $x_{\rm min}$&nbsp; and&nbsp; $x_{\rm max}$.
  
  
 +
==Generating a uniform distribution with pseudo&ndash;noise generators==
 +
<br>
 +
{{BlaueBox|TEXT=
 +
$\text{Definition}$&nbsp; The random generators used today are mostly&nbsp; '''pseudo&ndash;random'''.&nbsp; This means,
 +
*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&nbsp; $P$.
  
 +
 +
More on this in the chapter&nbsp; [[Theory_of_Stochastic_Signals/Generation_of_Discrete_Random_Variables|Generation of discrete random variables]]. }}
 +
 +
 +
For system simulation,&nbsp; pseudo&ndash;noise&nbsp; $\rm (PN)$&nbsp; 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,&nbsp; and
 +
*also makes troubleshooting much easier.
 +
 +
 +
{{BlaueBox|TEXT=
 +
$\text{A random sequence generator should meet the following criteria:}$
 +
 +
'''(1)''' &nbsp; The random variables&nbsp; $x_ν$&nbsp; of a generated sequence should be uniformly distributed with very good approximation.&nbsp; For discrete-value representation on a computer,&nbsp; this requires,&nbsp; among other things,&nbsp; a sufficiently&nbsp; high bit resolution,&nbsp; for example, with&nbsp; $32$&nbsp; or&nbsp; $64$&nbsp; bits per sample.
 +
 +
'''(2)''' &nbsp; If one forms from the sequential random sequence&nbsp; $〈x_ν〉$&nbsp; respectively non-overlapping pairs of random variables,&nbsp; for example&nbsp; $(x_ν, x_{ν+1})$,&nbsp; $(x_{ν+2}$,&nbsp; $x_{ν+3})$, ... , then these&nbsp; "tuples"&nbsp; should also be equally distributed in a two-dimensional representation within a square.
 +
 +
'''(3)''' &nbsp; If one forms from the sequential series&nbsp; $〈x_ν〉$&nbsp; non-overlapping&nbsp; $n$&ndash;tuples &nbsp; of random variables &nbsp; ⇒ &nbsp; $(x_ν$, . ... , $x_{ν+n-1})$,&nbsp; $(x_{ν+n}$, ... , $x_{ν+2n-1})$&nbsp; etc., then these should also yield the uniform distribution within a&nbsp; $n$&ndash;dimensional cube,&nbsp; if possible.}}
 +
 +
 +
 +
Note:
 +
*The first requirement refers exclusively to the&nbsp; "amplitude distribution"&nbsp; $\rm (PDF)$&nbsp; and is generally easier to satisfy.
 +
*The other requirements ensure&nbsp; "sufficient randomness"&nbsp; of the sequence.&nbsp; They concern the statistical independence of successive random values.
 +
 +
 +
==Multiplicative Congruental Generator==
 +
<br>
 +
$\text{Multiplicative Congruental Generator}$&nbsp; is the best known method for generating a sequence&nbsp; $〈 x_\nu 〉$&nbsp; with equally distributed values&nbsp; $ x_\nu$&nbsp; between&nbsp; $0$&nbsp; and&nbsp; $1$.&nbsp; This method is given here in a bullet-point fashion:
 +
 +
'''(1)''' &nbsp; These random generators are based on the successive manipulation of an integer variable&nbsp; $k$.&nbsp; If the number representation in the computer happens with&nbsp; $L$&nbsp; bit,&nbsp; this variable takes all values between&nbsp; $1$&nbsp; and&nbsp; $2^{L - 1}$&nbsp; exactly once each,&nbsp; if the sign bit is handled appropriately.
 +
 +
'''(2)''' &nbsp; The random variable derived from this&nbsp; $x={k}/{\rm 2^{\it L - \rm 1}}$&nbsp; is also discrete&nbsp; $($with level number&nbsp; $M = 2^{L- 1})$:
 +
:$$x={k}/{\rm 2^{\it L - \rm 1}} = k\cdot \delta x \in \{\delta x, \hspace{0.05cm}2\cdot \delta x,\hspace{0.05cm}\text{ ...}\hspace{0.05cm} , \hspace{0.05cm}1-\delta x,\hspace{0.05cm} 1\}.$$
 +
:If the bit number&nbsp; $L$&nbsp; is sufficiently large,&nbsp; the distance&nbsp; $Δx = 1/2^{L- 1}$&nbsp; between two possible values is very small,&nbsp; and one may well interpret&nbsp; $x$&nbsp; as a continuous-valued random variable in the context of simulation accuracy.
 +
 +
'''(3)''' &nbsp; The recursive generation rule of such&nbsp; "multiplicative congruential generators"&nbsp; is:
 +
:$$k_\nu=(a\cdot k_{\nu-1})\hspace{0.1cm} \rm mod \hspace{0.1cm} \it m.$$
 +
 +
'''(4)''' &nbsp; The statistical properties of the sequence depend crucially on the parameters&nbsp; $a$&nbsp; and&nbsp; $m$.&nbsp; The initial value&nbsp; $k_0$&nbsp;  has a minor importance for the statistics.
 +
 +
'''(5)''' &nbsp; The best results are obtained with the base&nbsp; $m =2\hspace{0.05cm}^l-1$,&nbsp; where&nbsp; $l$&nbsp; denotes any natural number.&nbsp; Widely used in computers with 32-bit architecture and one sign bit is the base&nbsp; $m = 2^{31} - 1 = 2\hspace{0.08cm}147\hspace{0.08cm}483\hspace{0.08cm}647$.&nbsp; A corresponding algorithm is:
 +
:$$k_\nu=(16807\cdot k_{\nu-1})\hspace{0.1cm} \rm mod\hspace{0.1cm}(2^{31}-1).$$
 +
 +
'''(6)''' &nbsp; For such a generator, only the initial value&nbsp; $k_0 = 0$&nbsp; is not allowed.&nbsp; For&nbsp; $k_0 \ne 0$&nbsp; the period duration&nbsp; $P = 2^{31} - 2.$
 +
 +
[[File:EN_Sto_T_3_4_S4.png |right|frame| Multiplicative Congruental Generator&nbsp; (C program)]]
 +
{{GraueBox|TEXT=
 +
$\text{Example 2:}$&nbsp; We analyze the&nbsp; "Multiplicative Congruental Generator"&nbsp; in more detail:
 +
 +
 +
*The algorithm,&nbsp; however,&nbsp; cannot be implemented directly on a 32 bit computer,&nbsp; 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&ndash;integer number range is exceeded.
 +
 +
 +
*The C program&nbsp; $\text{uniform( )}$&nbsp;  thus modified is given on the right.}}
 +
 +
==Exercises for the chapter==
 +
<br>
 +
[[Aufgaben:Exercise_3.5:_Triangular_and_Trapezoidal_Signal|Exercise 3.5: Triangular and Trapezoidal Signal]]
 +
 +
[[Aufgaben:Exercise_3.5Z:_Antenna_Areas|Exercise 3.5Z: Antenna Areas]]
  
  
 
{{Display}}
 
{{Display}}

Revision as of 16:45, 15 February 2022

General description and definition


PDF and CDF of uniform distribution

$\text{Definition:}$  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.

The graph shows

  • on the left the probability density function  $f_{x}(x)$,
  • 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.
  • 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:
$$m_{\rm 1} = \frac{\it x_ {\rm max} \rm + \it x_{\rm min}}{2},\hspace{0.5cm} \sigma^2 = \frac{[x_{\rm max} - x_{\rm min}]^2}{12},\hspace{0.5cm} \sigma = \frac{\it x_{\rm max} - \it x_{\rm min}}{2 \sqrt{3}}.$$
  • For symmetric PDF   ⇒   $x_{\rm min} = -x_{\rm max}$  we obtain as a special case the mean  $m_1 = 0$  and the variance  $σ^2 = x_{\rm max}^2/3.$


$\text{Example 1:}$  The graph shows two signal waveforms with uniform amplitude distribution.

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  $(x_{ν-1}, x_{ν-2}, \hspace{0.1cm}\text{...}).$


  • 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


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  peak limiting:

  • With no distribution other than the uniform distribution does one achieve greater  differential entropy  under this condition.
  • This topic is dealt with in the chapter  "Differential Entropy"  in the book  "Information Theory".


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

(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  Gaussian distribution  and the  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.

(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).


The interactive applet  "PDF, CDF and moments of special distributions"  calculates all characteristics of the uniform distribution for any parameters  $x_{\rm min}$  and  $x_{\rm max}$.


Generating a uniform distribution with pseudo–noise generators


$\text{Definition}$  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,
  • but appears to the user as stochastic due to the large period length  $P$.


More on this in the chapter  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.


$\text{A random sequence generator should meet the following criteria:}$

(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.

(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.


Note:

  • 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.


Multiplicative Congruental Generator


$\text{Multiplicative Congruental Generator}$  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:

(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  $M = 2^{L- 1})$:

$$x={k}/{\rm 2^{\it L - \rm 1}} = k\cdot \delta x \in \{\delta x, \hspace{0.05cm}2\cdot \delta x,\hspace{0.05cm}\text{ ...}\hspace{0.05cm} , \hspace{0.05cm}1-\delta x,\hspace{0.05cm} 1\}.$$
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.

(3)   The recursive generation rule of such  "multiplicative congruential generators"  is:

$$k_\nu=(a\cdot k_{\nu-1})\hspace{0.1cm} \rm mod \hspace{0.1cm} \it m.$$

(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:

$$k_\nu=(16807\cdot k_{\nu-1})\hspace{0.1cm} \rm mod\hspace{0.1cm}(2^{31}-1).$$

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

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.


  • 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.

Exercises for the chapter


Exercise 3.5: Triangular and Trapezoidal Signal

Exercise 3.5Z: Antenna Areas