Method for on-demand generation of individual random numbers of a sequence of random numbers of a 1/f noise

1. A computer implemented method for numeric simulation of an electric circuit influenced by noise, comprising the steps of:

numerically simulating the electric circuit using a model including input channels, noise input channels and output channels, the behavior of the input channels and of the output channels being described by a system of differential equations or a by a system of differential algebraic equations;

in the numerical simulation step, calculating an output vector for an input vector present on the input channels and a noise vector y of 1/f distributed random numbers present on the noise input channel, wherein generation of the noise vector y includes the steps of:

determining a desired spectral value β;

determining a number of the random numbers of the 1/f noise to be generated;

determining an intensity constant const;

setting a starting value for a running variable n;

performing a loop-type repetition until a desired number of elements y(n) of a vector y of length n is calculated from 1/f-distributed random numbers, by the steps of:

increasing a current value of the running variable n by 1;

setting a simulation time step [t n−1 ; t n ];

determining elements C ij of a covariance matrix C of dimension (n×n) according to:

C ij :=const·(−| t j −t i | β+1 +|t j−1 −t i | β+1 +|t j −t i−1 | β+1 −|t j−1 −t i−1 | β+1 ) i,j =1 , . . . ,n

determining an inverted covariance matrix C −1 by inverting the covariance matrix C;

determining a variable σ in accordance with

σ=sqrt (1 /e ( n,n ) ),

where sqrt denotes a square root function, and e(n,n) denotes an element of the inverted covariance matrix C −1 indexed by (n,n);

determining a (0,1)−normally distributed random number which forms an nth component of a vector x of length n;

forming a variable μ from first (n−1) components of an nth row of the inverted covariance matrix C −1 and (n−1) elements of the vector y calculated for a preceding (n−1) simulation time step, according to:

μ

:=

-

y

(

n

-

1

)

T

·

C

=

•

⁢

,

n

-

1

C

=

n

,

n

-

1

where y (n−1) denotes first (n−1) elements of the vector y, C ·n −1 denotes the first (n−1) components of the nth row of the inverted covariance matrix C −1 , and C n,n −1 denotes a component of the inverted covariance matrix C −1 indexed by (n,n); and

calculating an element y(n) of the vector y of length n from the 1/f-distributed random numbers, according to:

y ( n ) =x ( n )*σ+μ;

outputting the at least one sequence of random numbers of 1/f noise.

2. A method for simulating a technical system subject to 1/f noise, which comprises the steps of:

determining random numbers according to claim 1 ; and

using the random numbers for modeling variables present on input channels of the technical system.

3. A computer system, comprising:

processor programmed for executing the method for determining the sequences of random numbers of the 1/f noise according to claim 2 .

4. A computer running a computer program, comprising:

computer-executable instructions for carrying out the method according to claim 1 for determining the sequences of random numbers of the 1/f noise.

5. A computer running the instructions stored on a computer-readable data medium having the computer-executable instructions according to claim 4 .

6. A downloading method, which comprises the step of:

downloading the computer program according to claim 4 from an electronic data network onto a computer connected to the electronic data network.

7. The method according to claim 6 , which further comprises using the Internet as the electronic data network.

8. A computer system, comprising:

processor programmed for executing the method for determining the sequences of random numbers of the 1/f noise according to claim 1 .

9. A method for simulating a technical system subject to 1/f noise, which comprises the steps of:

determining random numbers according to claim 1 ; and

using the random numbers for fixing variables present on input channels of the technical system; and

simulating the technical system and outputting the result of the simulation.

10. A computer implemented method for numeric simulation of an electric circuit influenced by noise, comprising the steps of:

numerically simulating the electric circuit using a model including input channels, noise input channels and output channels, the behavior of the input channels and of the output channels being described by a system of differential equations or a by a system of differential algebraic equations;

in the numerical simulation step, calculating an output vector for an input vector present on the input channels and a noise vector y of 1/f distributed random numbers present on the noise input channel, wherein generation of the noise vector y includes the steps of:

determining a desired spectral value β;

determining a number of the random numbers of the 1/f noise to be generated;

determining an intensity constant const;

setting a starting value for a running variable n;

calculating q secpiences of the random numbers of the 1/f noise simultaneously, by performing looptype repetitions until a desired number of elements y k,n of a vector y of length n is calculated from 1/f-distributed random numbers, by the steps of:

increasing a current value of the running variable n by 1;

setting a simulation time step [t n−1 ; t n ];

determining elements C ij of a covariance matrix C of dimension (n×n) according to:

C ij :=const·(−| t j −t i | β+1 +|t j−1 −t i | β+1 +|t j −t i−1 | β+1 −|t j−1 −t i−1 | β+1 ). i,j= 1, . . . ,n

determining an inverted covariance matrix C −1 by inverting the covariance matrix C;

determining a variable σ in accordance with

σ=sqrt (1/ e ( n,n ) ),

where sqrt denotes a square root function, and e(n,n) denotes an element of the inverted covariance matrix C −1 indexed by (n,n);

determining a quantity q of (0,1)−normally distributed random numbers x k,n which form a respective last component of vectors x k of length n, where k=1, . . . , q,

forming q variables μ k according to:

μ

k

:=

-

y

(

n

-

1

)

,

k

T

·

C

=

•

⁢

,

n

-

1

C

=

n

,

n

-

1

where y (n−1),k denotes first (n−1) elements of the vector y k that were calculated for a preceding simulation time step, C ·n −1 denotes the first (n−1)

components of the nth row of the inverted covariance matrix C −1 , and C n,n −1 denotes the element of the inverted covariance matrix C −1 indexed by (n,n), where k=1, . . . , q; and

calculating q elements y k,n which form a respective nth component of the vector y k of length n from 1/f-distributed random numbers, according to:

y k,n =x k,n *σ+μ k

where k=1, . . . , q;

outputting at least one of the q sequences of random numbers of 1/f noise.

11. A method for simulating a technical system subject to 1/f noise, which comprises the steps of:

determining random numbers according to claim 10 ; and

using the random numbers for modeling variables present on input channels of the technical system.

12. A computer system, comprising:

processor programmed for executing the method for determining the sequences of random numbers of the 1/f noise according to claim 11 .

13. A computer running a computer program, comprising:

computer-executable instructions for carrying out the method according to claim 10 for determining the sequences of random numbers of the 1/f noise.

14. A computer running the instructions stored on a computer-readable data medium having the computer-executable instructions according to claim 13 .

15. A downloading method, which comprises the step of:

downloading the computer program according to claim 13 from an electronic data network onto a computer connected to the electronic data network.

16. The method according to claim 15 , which further comprises using the Internet as the electronic data network.

17. A computer system, comprising:

processor programmed for executing the method for determining the sequences of random numbers of the 1/f noise according to claim 10 .

18. A method for simulating a technical system subject to 1/f noise, which comprises the steps of:

determining random numbers according to claim 10 ; and

using the random numbers for fixing variables present on input channels of the technical system; and

simulating the technical system and outputting the result of the simulation.