Electrical computers: arithmetic processing and calculating – Electrical digital calculating computer – Particular function performed
Reexamination Certificate
2000-08-28
2004-09-21
Ngo, Chuong Dinh (Department: 2124)
Electrical computers: arithmetic processing and calculating
Electrical digital calculating computer
Particular function performed
Reexamination Certificate
active
06795840
ABSTRACT:
BACKGROUND OF THE INVENTION
Field of the Invention
The invention relates to a method for generating sequences of random numbers of a 1/f noise.
Random numbers of a 1/f noise can be used, for example, in a transient circuit simulation which takes account of noise, influences. A 1/f noise is understood as a stochastic process with a specific frequency spectrum which can be described by the equation
S
(
f
)
∝
1
f
β
,
β
∈
]
0
,
1
[
1/f noise sources are suitable for modeling noise influences in a multiplicity of technical and physical systems, and for systems for estimating and predicting events on the financial markets. In particular, many electronic components such as, for example, pn diodes and MOS field effect transistors have 1/f noise sources.
It is possible to approximate 1/f noise sources by carrying out a summation of the effects of many noise sources which respectively have a Lorentz spectrum as frequency spectrum. Such noise sources can be modeled, for example, by the system response of a linear time-invariant system which can also be denoted as an LTI system, at whose system input a white noise is applied. It is disadvantageous in this procedure that the dimension of the system of differential equations to be solved numerically is inflated beyond bounds. This results in long computing times and a high storage requirement for a computer system which is used to simulate a system which is subject to the influence of a 1/f noise.
SUMMARY OF THE INVENTION
It is the object of the invention to specify a method for generating a sequence of random numbers of a 1/f noise which can be carried out quickly and with a low computational outlay. It is also the object of the invention to specify an improved method for simulating a technical system which is subject to a 1/f noise. Finally, the aim is also to specify a computer program for determining sequences of random numbers of a 1/f noise which can be carried out quickly and which requires only few resources of a computer system.
This object is achieved by means of the subject matters of the independent patent claims. Improvements follow from the respective subclaims.
In accordance with an idea on which the invention is based, the numerical integration of a system path of a system which is subject to 1/f noise influences is reduced in the case of a constant integration step size to the simulation of sequences of suitably correlated random numbers. A batch method is used in order to generate sequences of such random numbers. In this context, the term “batch method” is understood to mean that the length of the sequence of the random numbers may be arbitrary. However, the random numbers must be prescribed when the method is started, the overall random number sequence being generated in a first computing step. It is possible using the method according to the invention to avoid inflation of the dimension of the system of differential equations to be solved numerically in the case of a constant integration step size. This results in considerable gains in computing time.
In accordance with a further idea on which the invention is based, the efficiency of the numerical determination of the dynamics of the simulated system is substantially raised by adaptive step size control. Specifically, the invention specifies a method which permits an adaptive step size control and a numerical simulation of 1/f noise sources to be combined with one another, it being possible for the simulation of 1/f noise to be incorporated into a transient system simulation with adaptively controlled step size. The invention permits the known methods for adaptive step size control to be combined with a numerical simulation of 1/f noise sources without the system dimension being inflated. The procedure for this purpose is a two-stage method. In a first stage, the system dynamics are integrated numerically using an adaptive step size control, the 1/f noise sources not yet being taken into account. The adaptive step sizes determined in the process serve as the basis of a second integration pass in which account is taken of the 1/f noise sources whose proportions are determined with the aid of the information obtained in the first stage of the simulation. Thus, in the case of the transient simulation of a system the invention renders it possible to use the advantages of adaptive step size control together with a simplified numerical simulation of 1/f noise sources. This produces an increased numerical approximation quality of the simulation in conjunction with efficient use of the available computing time resources, inflation of the system dimension being avoided.
In accordance with the invention, the problem of noise simulation in modeling the system to be simulated is converted into the problem of generating a random number sequence. In accordance with the invention, the correlations of these random numbers are determined, and this is used for simple and accurate generation of the corresponding random number sequences.
The methods according to the invention for generating a sequence of random numbers of a 1/f noise provide as steps, in particular, the formation of a covariance matrix
C
from which the Cholesky decomposition
L
is subsequently formed. The covariance matrix
C
differs in this process in the case in which a constant simulation step size &dgr; is used from the case in which an adaptive step size is used.
For the case of a constant step size &dgr;, the covariance matrix is determined element by element with knowledge of a desired spectral value &bgr;, the number n of the random numbers, to be generated, of a 1/f noise and the intensity constant const, one element e(i,j) each of the covariance matrix C being determined using the following equation:
e
(
i,j
)=const.|&dgr;|
&bgr;+1
·(|
i−j
+1|
&bgr;+1
−2
|i−j|
&bgr;+1
+|i−j
−1|
&bgr;+1
)
where i,j=1, . . . , n.
The Cholesky decomposition
L
of the covariance matrix
C
is subsequently formed,
C
=
L
·
L
T
,
L
assuming the form of a triangular matrix in which only the diagonals and the lower half are occupied.
The Cholesky decomposition
L
thus found can now be used to generate arbitrarily many sequences of random numbers of a 1/f noise. A vector
x
of length n is firstly formed for this purpose from random numbers normally distributed in [0,1]. Such normally distributed random numbers have an expected value 0 and a variance 1. They can be determined in a known way by transformation from a vector, of equal length, of equally distributed random numbers.
The target vector
y
with the desired 1/f-distributed random numbers is then produced by multiplying the Cholesky decomposition
L
by the vector
x
.
In the method according to the invention, which permits an adaptive numerical solution of stochastic differential equations, the procedure is essentially as described above. However, by contrast with the mode of procedure described above it is not possible to assume a constant simulation step size &dgr;. Rather, it is necessary firstly to calculate a step size vector
&dgr;
which results from prescribed instants of consideration t
0
−t
n
in the first step of the simulation. Thereafter, the covariance matrix
C
is set up by determining the elements thereof using the following equation:
e
(
i,j
)=const·(−|
t
j
−t
i
|
&bgr;+1
+|t
j−1
−t
i
|
&bgr;+1
+|t
j
−t
i−1
|
&bgr;+1
−|t
j−1
−t
i−1
|
&bgr;−1
)
where i,j=1, . . . , n.
The concept of simulating 1/f noise is based on the following thinking. The dynamics of a system which is subject to stochastic influences is modeled adequately by a stochastic process. Individual random realizations (so-called paths) of the basic stochastic process are generally calculated numerically for the purpose of simulating such sys
Denk Georg
Hillermeier Claus
Schäffler Stefan
Greenberg Laurence A.
Infineon - Technologies AG
Mayback Gregory L.
Ngo Chuong Dinh
Stemer Werner H.
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