Theory of Stochastic Signals/Generation of Discrete Random Variables - Revision history

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← Older revision		Revision as of 17:39, 8 February 2024
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−	For reasons of comprehension,  a cumbersome programming was chosen here.  The given C program part could also be written more compactly:	+	For reasons of comprehension,  a cumbersome programming was chosen here.  The given C–program part could also be written more compactly:

	:$$\text{\{ float random(); return((long) (random()*M)); \} }$$}}		:$$\text{\{ float random(); return((long) (random()*M)); \} }$$}}

@@ Line 124: / Line 124: @@
 ==Generation of multi-level random variables==
 <br>
-Many high-level programming languages provide pseudo-random generators that return real random numbers&nbsp; $x$&nbsp; equally distributed between&nbsp; $0$&nbsp; and&nbsp; $1$.&nbsp;  For example,&nbsp; a corresponding C&nbsp; function call is:
+Many high-level programming languages provide pseudo-random generators that return real random numbers&nbsp; $x$&nbsp; equally distributed between&nbsp; $0$&nbsp; and&nbsp; $1$.&nbsp;  For example,&nbsp; a corresponding C&ndash;function call is:
 :$$x = \text{random()}.$$
-By successively calling this&nbsp; "random function",&nbsp; a periodically repeating sequence of real numbers is created,&nbsp; where all values&nbsp; $0 \le x < 1$&nbsp; are equally likely &nbsp; &rArr; &nbsp; see chapter&nbsp; [[Theory_of_Stochastic_Signals/Uniformly_Distributed_Random_Variables|"Uniformly Distributed Random Variables"]].
+By successively calling this&nbsp; random function,&nbsp; a periodically repeating sequence of real numbers is created,&nbsp; where all values&nbsp; $0 \le x < 1$&nbsp; are equally likely <br>(see chapter&nbsp; [[Theory_of_Stochastic_Signals/Uniformly_Distributed_Random_Variables|&raquo;Uniformly Distributed Random Variables&raquo;$)$]].
-*However,&nbsp; since the period&nbsp; $P$&nbsp; is very large,&nbsp; this sequence can be considered as&nbsp; "pseudo-random".
+#However,&nbsp; since the period&nbsp; $P$&nbsp; is very large,&nbsp; this sequence can be considered as&nbsp; &raquo;pseudo-random&laquo;.
-*By specifying a starting value,&nbsp; the pseudo-random sequence is started at certain points.
+#By specifying a starting value,&nbsp; the pseudo-random sequence is started at certain points.
@@ Line 137: / Line 137: @@
 {{GraueBox|TEXT=
 $\text{Example 5:}$&nbsp;
-The graphic shows the principle for the special case&nbsp; $M = 3$ &nbsp; &rArr; &nbsp; ternary random sequence&nbsp; $〈z_{\rm ν}〉$&nbsp; with&nbsp; $z_{\rm ν} ∈ \{0,\ 1,\ 2\}$.&nbsp; Assumed here is the between&nbsp; $0$&nbsp; and&nbsp; $1$&nbsp; uniformly distributed random variable&nbsp; $x$.
+The graphic shows the principle for the special case&nbsp; $M = 3$ &nbsp; &rArr; &nbsp; ternary random sequence&nbsp; $〈z_{\rm ν}〉$&nbsp; with&nbsp; $z_{\rm ν} ∈ \{0,\ 1,\ 2\}$.&nbsp;
 [[File:P_ID457__Sto_T_2_5_S6neu.png|right|frame|For generating multi-level  random variables]]
-The desired probabilities are designated as follows:
+*Assumed is the between&nbsp; $0$&nbsp; and&nbsp; $1$&nbsp; uniformly distributed random variable&nbsp; $x$.
-*$p_0 = {\rm Pr}(z_{\rm ν} =0)$,
-*$p_1 = {\rm Pr}(z_{\rm ν} =1)$,
+*The desired probabilities are designated as follows:
-*$p_2 = {\rm Pr}(z_{\rm ν} =2)$.
+:$$p_0 = {\rm Pr}(z_{\rm ν} =0),\hspace{0.5cm} p_1 = {\rm Pr}(z_{\rm ν} =1),\hspace{0.5cm} p_2 = {\rm Pr}(z_{\rm ν} =2). $$
-Then holds:
+In the listed C&ndash;program,
-*If the current&nbsp; $x_{\rm ν} <p_0$,&nbsp; then the ternary random variable is set to&nbsp; $z_{\rm ν} = 0$.
+*we get for&nbsp; $M = 3$&nbsp; and for the current random value&nbsp; $x = 0.57$&nbsp;
-*In the range&nbsp; $p_0 ≤ x_{\rm ν} < p_0 + p_1$&nbsp; the output is&nbsp; $z_{\rm ν} = 1$&nbsp;.
-*For&nbsp; $x_{\rm ν} \ge p_0 + p_1$&nbsp; the ternary random variable becomes&nbsp; $z_{\rm ν} =2$.
+*the product&nbsp; $x · M = 0.57 · 3 = 1.71$&nbsp; and thus the ternary random variable&nbsp; $z = 1$.
-<br clear=all>
-In the C program listed, for&nbsp; $M = 3$&nbsp; and for the current random value&nbsp; $x = 0.57$&nbsp; we get the product&nbsp; $x · M = 0.57 · 3 = 1.71$&nbsp; and thus the ternary random variable&nbsp; $z = 1$. For an other random value&nbsp; $x = 0.95$&nbsp; on the other hand,&nbsp; the function would return the result&nbsp; $z = 2$.
+*For an other random value&nbsp; $x = 0.95$&nbsp; on the other hand,&nbsp; the function would return the result&nbsp; $z = 2$.
 For reasons of comprehension,&nbsp; a cumbersome programming was chosen here.&nbsp; The given C program part could also be written more compactly:

@@ Line 62: / Line 62: @@
 If not all&nbsp; $L$&nbsp; memory cells of the shift register are preallocated with zeros,&nbsp; a periodic random sequence&nbsp; $〈z_ν\rangle$ always results:
 [[File:EN_Sto_T_2_5_S5_neu.png|right|frame|Composition of some favorable generator polynomials]]
-*The period length&nbsp; $P$&nbsp; of this sequence depends to a strong degree on the feedback coefficients.
+*The period length&nbsp; $P$&nbsp; of the pseudo-noise sequence depends to a strong degree on the feedback coefficients.
 *For each degree&nbsp; $L$&nbsp; there is at least one configuration with&nbsp; &raquo;'''maximum period length'''&laquo;
 :$$P_{\rm max} = \rm 2^{\it L}-\rm 1.$$
-*Such a PN sequence is also often referred to as&nbsp; &raquo;'''M-sequence'''&laquo;,&nbsp; where the&nbsp; "M"&nbsp; stands for&nbsp; "Maximum".
+*Such a pseudo-noise sequence is also often referred to as&nbsp; &raquo;'''M-sequence'''&laquo;,&nbsp; where the&nbsp; &raquo;$\rm M$&laquo;&nbsp; stands for&nbsp; &raquo;maximum&laquo;.
-<br><br><br><br><br><br><br>
+<br><br>
-The table on the right lists PN generators of maximum length up to degree&nbsp; $L = 31$&nbsp;.
+The table lists pseudo-noise generators of maximum length up to degree&nbsp; $L = 31$.
-*The selection is limited to configurations with only one tap &ndash; that is, with two returns.
-*This means that the associated polynomials consist of only three summands.
+<u>Note:</u>
-*For applications that require high speed, such generators are very useful.
+#The selection is limited to configurations with only one tap&nbsp; $($two returns$)$.
 <br clear=all>
 {{BlueBox|TEXT=
-$\text{For now, without proof:}$&nbsp;
+$\text{For now without proof:}$&nbsp;
-#&nbsp; A&nbsp; &raquo;'''M-sequence'''&laquo;&nbsp; can be recognized by the fact that the generator polynomial&nbsp; $G(D)$&nbsp; is primitive.&nbsp;
-#&nbsp; As detailed in the book&nbsp; [[Channel_Coding|"Channel Coding"]],&nbsp; a polynomial&nbsp; $G(D)$&nbsp; of degree&nbsp; $L$&nbsp; is called&nbsp; &raquo;'''primitive'''&laquo; if the following condition is satisfied:
+$(1)$&nbsp; An&nbsp; &raquo;'''M-sequence'''&laquo;&nbsp; can be recognized by the fact that the generator polynomial&nbsp; $G(D)$&nbsp; is&nbsp; &raquo;'''primitive'''&laquo;.&nbsp;
-::$$\frac{D^n+ 1}{G(D)} \neq 0\hspace{0.5cm} {\rm for}\hspace{0.5cm}\it n<P_{\rm max} \rm = \rm 2^{\it L}-\rm 1.$$}}
 <br clear=all>
 {{GraueBox|TEXT=
 $\text{Example 3:}$&nbsp; A shift register of degree&nbsp; $L = 4$&nbsp; with octal identifier&nbsp; $(31)$&nbsp; and generator polynomial&nbsp; $G(D) = D^4 + D^3 + 1$&nbsp; leads to a sequence of maximum length:
 :$$P_{\rm max} = 2^4 - 1 = 15.$$
-The mathematical proof for this is complex:
+The mathematical proof for this equation is rathercomplex:
-*Using the above polynomial division for&nbsp; $n = 1$, &nbsp;...&nbsp; , $14$&nbsp; one must show that the quotient is always nonzero.
+#Using the above polynomial division for&nbsp; $n = 1$, &nbsp;...&nbsp; , $14$&nbsp; one must show that the quotient is always non&ndash;zero.
-*First the division&nbsp; $(D^{15} + 1)/G(D)$&nbsp; may give a result without remainder.
+#First the division&nbsp; $(D^{15} + 1)/G(D)$&nbsp; may give a result without remainder.
-*Here it is to be considered that in the modulo-2 algebra&nbsp; $+1$&nbsp; and&nbsp; $-1$&nbsp; are identical. }}
+#Here it is to be considered that in the modulo-2 algebra&nbsp; $+1$&nbsp; and&nbsp; $-1$&nbsp; are identical. }}
 ==Reciprocal polynomials==
@@ Line 94: / Line 98: @@
-Between the two shift registers with polynomials&nbsp; $G(D)$&nbsp; and&nbsp; $G_{\rm R}(D)$&nbsp; respectively,&nbsp; there are the following relations:
+There are the following relations between the two shift registers with polynomials&nbsp; $G(D)$&nbsp; resp.&nbsp; $G_{\rm R}(D)$:
-*If&nbsp; $G(D)$&nbsp; provides a sequence of maximum length &nbsp; ⇒ &nbsp; $P_{\rm max} = 2^L - 1$,&nbsp; then the period length of the reciprocal polynomial&nbsp; $G_{\rm R}(D)$&nbsp; is also maximum.
+#If&nbsp; $G(D)$&nbsp; provides a sequence of maximum length &nbsp; ⇒ &nbsp; $P_{\rm max} = 2^L - 1$,&nbsp; then the period length of the reciprocal polynomial&nbsp; $G_{\rm R}(D)$&nbsp; is also maximum.
-*The output sequences of reciprocal configurations are inverses of each other.&nbsp; That is:
+#The output sequences of reciprocal configurations are&nbsp; &raquo;inverses of each other&laquo;. &nbsp; That is:<br> &nbsp; &nbsp; &nbsp; The sequence of&nbsp; $G(D)$ &ndash; read from right to left &ndash; gives the sequence of the reciprocal configuration&nbsp; $G_{\rm R}(D)$.
@@ Line 104: / Line 107: @@
 {{GraueBox|TEXT=
 $\text{Example 4:}$&nbsp;
-We consider again the degree&nbsp; $L = 4$.
+We consider again a pseudo-noise generator with degree&nbsp; $L = 4$&nbsp; and shift register structure&nbsp; $(31)_{\rm oct}$&nbsp;:
+:$$G(D) =  1+D^{3} + D^{4}.$$
-:and thus the configuration with the octal identifier&nbsp; $(23)$.
+* For the reciprocal polynomial we obtain the configuration with the octal identifier&nbsp; $(23)$:
 *The corresponding output sequences each have the maximum period length&nbsp; $P_{\rm max} = 15$&nbsp; and are inverses of each other:
 :* ... $0 \ 0 \ 0 \ 1 \ 0 \ 0 \ 1 \ 1 \ 0 \ 1 \ 0 \ 1 \ 1 \ 1 \ 1$ ...
@@ Line 115: / Line 118: @@
 *This means that the upper initial sequence of&nbsp; $(31)$,&nbsp; read from right to left,&nbsp; yields the sequence of reciprocal order&nbsp; $(23)$&nbsp; given below.
 *However,&nbsp; a cyclic phase shift of four binary digits can be seen. }}

@@ Line 7: / Line 7: @@
 ==Pseudo-random variables==
 <br>
-One way to generate a binary random sequence&nbsp; $〈z_{\rm ν}〉 ∈ \{0, 1\}$&nbsp; with good statistical properties is offered by the so-called&nbsp; &raquo;'''pseudo-random generators'''&laquo;,&nbsp; also known as&nbsp; &raquo;'''PN generators'''&laquo;,&nbsp; where&nbsp; "PN"&nbsp; stands for&nbsp; "pseudo-noise".
+One way to generate a binary random sequence&nbsp; $〈z_{\rm ν}〉 ∈ \{0, 1\}$&nbsp; with good statistical properties is offered by the so-called&nbsp; &raquo;'''pseudo-random generators'''&laquo;,&nbsp; also known as&nbsp; &raquo;'''PN generators'''&laquo;,&nbsp; where&nbsp; &raquo;PN&laquo;&nbsp; stands for&nbsp; &raquo;pseudo-noise&laquo;.
 These have the following properties:
-*The binary sequence generated by such a generator is not stochastic in a strict sense,&nbsp; but exhibits periodic and thus deterministic properties.
+#The binary sequence generated by such a generator is not stochastic in a strict sense,&nbsp; but exhibits periodic and thus deterministic properties.
-*If the period length&nbsp; $P$&nbsp; is sufficiently large,&nbsp; the sequence appears to an observer as random with sufficiently good statistical properties for many applications.
+#If the period length&nbsp; $P$&nbsp; is sufficiently large,&nbsp; the sequence appears to an observer as random with sufficiently good statistical properties for many applications.
-*The advantage of a pseudo-random generator over a&nbsp; "real"&nbsp; stochastic source is that the random sequence is reproducible if a few parameters are known.
+#The advantage of a pseudo-random generator over a&nbsp; real&nbsp; stochastic source is that the random sequence is reproducible if a few parameters are known.
 {{GraueBox|TEXT=
-$\text{Example 1:}$&nbsp; The latter property also gives rise to the main applications of pseudo-noise generators:
+$\text{Example 1:}$&nbsp; The latter property also gives rise to the main applications of pseudo-noise generators,
-*first, the&nbsp; "error frequency measurement"&nbsp; in digital signal transmission,
+# the&nbsp; &raquo;error rate measurement&laquo;&nbsp; in digital signal transmission,
-*secondly,&nbsp; for band spreading in CDMA&nbsp; ("Code Division Multiple Access").
+# for&nbsp; &raquo;band spreading&laquo;&nbsp; in CDMA&nbsp; $($Code Division Multiple Access$)$.
-In such a&nbsp; "Spread Spectrum System"&nbsp; the transmitted signal is modulated with a binary random sequence,&nbsp; whose symbol repetition frequency is significantly higher than the bit frequency.&nbsp; This offers the possibility of multiple channel utilization.&nbsp; Since the same sequence must be added in phase at the receiver,&nbsp; the use of reproducible PN sequences is also common here.
+In such a&nbsp; &raquo;Spread Spectrum System&laquo;&nbsp; the transmitted signal is modulated with a binary random sequence,&nbsp; whose symbol repetition frequency is significantly higher than the bit frequency.&nbsp; This offers the possibility of multiple channel utilization.&nbsp; Since the same sequence must be added in phase at the receiver,&nbsp; the use of reproducible pseudo-noise sequences is also common here.
-Detailed information on bandspreading methods can be found in the chapter&nbsp; [[Examples_of_Communication_Systems/General_Description_of_UMTS|"UMTS - Universal Mobile Telecommunications System"]]&nbsp; of the book&nbsp; "Examples of Communication Systems".}}
+Detailed information on bandspreading methods can be found in the chapter&nbsp; [[Examples_of_Communication_Systems/General_Description_of_UMTS|&raquo;UMTS - Universal Mobile Telecommunications System&laquo;]]&nbsp; of the book&nbsp; &raquo;Examples of Communication Systems&laquo;.}}
 ==Realization of PN generators==
 <br>
-Pseudo-random generators are usually realized by feedback shift registers,&nbsp; where at each clock instant the content of the register is shifted one place to the right&nbsp; (see diagram).&nbsp; For the currently generated symbol,&nbsp; with&nbsp; $g_l ∈ \{0, 1\}$&nbsp; and&nbsp; $l = 1$, ... , $L-1$:
+Pseudo-random generators are usually realized by &raquo;feedback shift registers&laquo;,&nbsp; where at each clock instant the content of the register is shifted one place to the right&nbsp; $($see diagram$)$.&nbsp; For the currently generated symbol,&nbsp; with&nbsp; $g_l ∈ \{0, 1\}$&nbsp; and&nbsp; $l = 1$, ... , $L-1$:
 :$$z_\nu = (g_1\cdot z_{\nu-1}+g_2\cdot z_{\nu-2}+\hspace{0.1cm}\text{...}\hspace{0.1cm}+g_l\cdot z_{\nu-l}+\hspace{0.1cm}\text{...}\hspace{0.1cm}+g_{L-1}\cdot z_{\nu-L+1}+ z_{\nu-L})\hspace{0.1cm} \rm mod \hspace{0.2cm}2.$$
-*The binary values&nbsp; $z_{ν-1}$&nbsp; to&nbsp; $z_{ν-L}$&nbsp; generated at previous time points are stored in the memory cells of the shift register.
+#The binary values&nbsp; $z_{ν-1}$&nbsp; to&nbsp; $z_{ν-L}$&nbsp; generated at previous time points are stored in the memory cells of the shift register.
-*The coefficients&nbsp; $g_1$, ... ,&nbsp; $g_{L-1}$&nbsp; are also binary values,&nbsp; where a&nbsp; "one"&nbsp; indicates a feedback at the corresponding location and a&nbsp; "zero"&nbsp; indicates no feedback.
+#The coefficients&nbsp; $g_1$, ... ,&nbsp; $g_{L-1}$&nbsp; are also binary values,&nbsp; where&nbsp; $1$&nbsp; indicates a feedback at the corresponding location and&nbsp; $0$&nbsp; indicates no feedback.
-*Modulo-2 addition can be implemented,&nbsp; for example,&nbsp; using an XOR operation:
+#Modulo-2 addition can be implemented,&nbsp; for example,&nbsp; using an XOR operation:
 [[File:EN_Sto_T_2_5_S2.png |right|frame| Pseudo-noise generator]]
 :$$(x + y)\hspace{0.15cm} \rm mod\hspace{0.15cm}2 = \it x\hspace{0.15cm}\rm XOR\hspace{0.15cm} \it y = \left\{\begin{array}{*{2}{c}} \rm 0 & \rm if\hspace{0.15cm} \it x= y,\\ 1 & \rm if\hspace{0.15cm} \it x\neq y. \\ \end{array} \right.$$
@@ Line 39: / Line 39: @@
 The statistical properties of the generated pseudo-noise random sequence&nbsp; $〈z_{\rm ν}〉$&nbsp; are essentially determined by
 *the&nbsp; &raquo;'''degree'''&laquo;&nbsp; $L$,&nbsp; and
 *the&nbsp; &raquo;'''feedback coefficients'''&laquo;&nbsp; $g_{\hspace{0.05cm}l}$&nbsp; $($with&nbsp; $l = 1$, ... , $L-1)$.
-A prerequisite for the generation of a PN sequence is that not all memory elements are preallocated with zeros, otherwise the modulo-2 addition would always generate the symbol&nbsp; "$0$".
+A prerequisite for the generation of a pseudo-noise sequence is that not all memory elements are preallocated with zeros,&nbsp; otherwise the modulo-2 addition would always generate the symbol&nbsp; $0$.
-To identify different PN generators one uses in the literature alternatively:
+To identify different pseudo-noise generators one uses in the literature alternatively:
-*The&nbsp; &raquo;'''generator polynomials'''&laquo;&nbsp; of the type
+*The&nbsp; &raquo;'''generator polynomial'''&laquo;&nbsp;
 :$$G(D) = g_L\cdot D^L+g_{L-1}\cdot D^{L-1}+ \ \text{...} \ +g_1\cdot D+g_0 .$$
 :Here,&nbsp;  $g_0 = g_L = 1$&nbsp; is always to be set;&nbsp; $D$&nbsp; is a formal parameter indicating a delay by one clock&nbsp; $(D^L$&nbsp; indicates a delay by&nbsp; $L$&nbsp; clocks$)$.
-*The&nbsp; &raquo;'''octal representation'''&laquo;&nbsp; of the binary number&nbsp; $(g_L\ \text{ ...} \ g_2 \ g_1 \ g_0)$.&nbsp; It is important that here the feedback coefficients &ndash; starting from the right with&nbsp; $g_0$&nbsp; &ndash; are combined into triples and these are written octal&nbsp; $(0 \ \text{...}\ 7)$.
+*The&nbsp; &raquo;'''octal representation'''&laquo;&nbsp; of the binary number&nbsp; $(g_L\ \text{ ...} \ g_2 \ g_1 \ g_0)$.&nbsp; <br>It is important that the feedback coefficients &ndash; starting from the right with&nbsp; $g_0$&nbsp; &ndash; are combined into triples and these are written octal&nbsp; $(0 \ \text{...}\ 7)$.
@@ Line 55: / Line 56: @@
 $\text{Example 2:}$&nbsp;
 The generator polynomial&nbsp; $D^4 + D^3 + 1$&nbsp; belongs to a shift register of degree&nbsp; $L = 4$&nbsp; with the following octal representation.
-:$$(g_4 \ g_3 \ g_2 \ g_1 \ g_0) = (11001)_{\rm bin} = (31)_{\rm oct}. $$}}
+:$$(g_4 \ g_3 \ g_2 \ g_1 \ g_0) = (11\hspace{0.05cm} 001)_{\rm bin} = (31)_{\rm oct}. $$}}
 ==Sequences of maximum length (M-sequences)==

@@ Line 60: / Line 60: @@
 <br>
 If not all&nbsp; $L$&nbsp; memory cells of the shift register are preallocated with zeros,&nbsp; a periodic random sequence&nbsp; $〈z_ν\rangle$ always results:
-[[File: EN_Sto_T_2_5_S5.png|right|frame|Composition of some favorable generator polynomials '''Korrektur:''' identisch, Polynom]]
+[[File:EN_Sto_T_2_5_S5_neu.png|right|frame|Composition of some favorable generator polynomials]]
 *The period length&nbsp; $P$&nbsp; of this sequence depends to a strong degree on the feedback coefficients.
 *For each degree&nbsp; $L$&nbsp; there is at least one configuration with&nbsp; &raquo;'''maximum period length'''&laquo;

← Older revision		Revision as of 14:40, 13 December 2022
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−	:The topic of this chapter is illustrated with examples in the  (German language)  learning video: <br>     [[Erläuterung_der_PN–Generatoren_an_einem_Beispiel_(Lernvideo)\|"Erläuterung der PN-Generatoren an einem Beispiel"]]  $\Rightarrow$ Explanation of PN generators using an example.	+	⇒   The topic of this chapter is illustrated with examples in the  (German language)  learning video: <br>     [[Erläuterung_der_PN–Generatoren_an_einem_Beispiel_(Lernvideo)\|"Erläuterung der PN-Generatoren an einem Beispiel"]]  $\Rightarrow$ Explanation of PN generators using an example.

@@ Line 151: / Line 151: @@
 In the C program listed, for&nbsp; $M = 3$&nbsp; and for the current random value&nbsp; $x = 0.57$&nbsp; we get the product&nbsp; $x · M = 0.57 · 3 = 1.71$&nbsp; and thus the ternary random variable&nbsp; $z = 1$. For an other random value&nbsp; $x = 0.95$&nbsp; on the other hand,&nbsp; the function would return the result&nbsp; $z = 2$.
-For reasons of comprehension,&nbsp; a cumbersome programming was chosen here&nbsp; The given C program part could also be written more compactly:
+For reasons of comprehension,&nbsp; a cumbersome programming was chosen here.&nbsp; The given C program part could also be written more compactly:
 :$$\text{\{ float random(); return((long) (random()*M)); \} }$$}}

@@ Line 120: / Line 120: @@
-==Generation of multilevel random variables=='''KORREKTUR: multi-level'''
+==Generation of multilevel random variables==
 <br>
-Many high-level programming languages provide pseudo-random generators that return real random numbers&nbsp; $x$&nbsp; equally distributed between&nbsp; $0$&nbsp; and&nbsp; $1$.&nbsp;  For example,&nbsp; a corresponding C&nbsp; function call is:
+'''KORREKTUR: multi-level''' Many high-level programming languages provide pseudo-random generators that return real random numbers&nbsp; $x$&nbsp; equally distributed between&nbsp; $0$&nbsp; and&nbsp; $1$.&nbsp;  For example,&nbsp; a corresponding C&nbsp; function call is:
 :$$x = \text{random()}.$$