Method to produce function of distribution of probabilities of initial signal of system

FIELD: information technologies.

SUBSTANCE: analytical method is created to produce signal probability distribution function (PDF) at the system output (initial system). This method is applicable, if: the system is described by analytical function; the system depends on accidental value; PDF of accidental value at the system input is available. There is an inverse function to function that describes system. Created method solves the following formal task. To produce PDF analytically for specified function y(x), if PDF of accidental value x in the interval [a,b] is available.

EFFECT: increased accuracy of production of probability characteristics of initial signal.

1 dwg


The invention relates to the field of measuring technology. The invention can be used to obtain well-defined probabilistic characteristics of the original signal.

The objective of the invention to obtain probability distribution function (DAF) of the signal at the system output.

This object is achieved in that determines the probability of finding analytically the function inverse to y(x) (the function describing the system), and is functional transformation for a given algorithm.

The invention consists in that the method of obtaining WDP is in the implementation of functional transformation y(x) algorithm:

int{Sx(x(y))dx/dy}/{int_[a, b](Sx(x(y))dx/dy)dy},

where int is an indefinite integral of a function, int_[a,b] is the Lebesgue integral on the field values in the interval x∈[a, b], Sxthe distribution density of the test signal.

Get WDP analytically for a given function y(x) is possible if the WDP random variable x on the interval [a, b] is known and can be found analytically the inverse function x(y). Let x be a random variable with a given distribution function.

Let the original random variable ξ has a density distribution of Sx(x) on the interval [a, b]. Let the function through which the signal passes, is defined as y(x). If the inverse function x(y) on the interval [a, b] is unambiguous, continuous dy/dx vanishes, then the density distribution of the resulting random variable Sy(y) is proportional to:

Using the normalization, we find the absolute value of Sy:

where int_[a, b] is a Lebesgue integral over the area of the y values at x∈[a, b] (Not necessarily to the interval coincided c [y(a), (b)], since y(x) may have a gap. For example, y=x-1has a tear on [-1, 1].) Result (2) is the probability density function (MF). The first derivative of WDP is RO. If x(y) ambiguously defined and(or) dy/dx vanishes on [a, b], we use another method. Also for those y that are not values of x(y) in dots dy/dx=0, use the following method:

let y0- the point at which you want to find Sy.

1. Get a set of line segments [a1b1], [a2b2], ..., [anbn], ... each of which y(x) is monotonic and inseparably, dy/dx has no zeros, and there is only one solution for the equation y(xn)=y0. In the worst case, the number of such intervals is countable.

2. Take the interval [anbn], find the probability Pnthat ξ belongs to this fragment (itwhere- defined the integral of anto bn)get private density:

DG is - definite integral of y(an) to y(bn).

3. The total density is the sum of private densities

where sum_n - sum of elements n.

4. The value of the integral (3) and the probability density of Syn(y0) significantly positive values.

5. The result (4) is the RO to obtain WDP need to integrate.

This method can be implemented in a simple blocks. The circuit implementation shown in the drawing, where

- block 1 - block generate signal x(y), the corresponding inverse functions to the test signal y(x);

- block 2 - block of differentiation;

- block 3 - block density functions of the distribution of the test signal;

blocks 4 and 7 blocks the multiplication signals;

blocks 5 and 8 blocks integrators;

- unit 6 - unit exponentiation (-1);

blocks 9 and 10 oscilloscopes.

The output unit 1 is connected to the input unit 2 and unit 3. The output unit 2 is connected to the input unit 4. The output unit 3 is connected to the input unit 4 and unit 5. The output of block 4 is connected to the input unit 7. The output unit 5 is connected to the input unit 6. The output unit 6 is connected to the input unit 7. The output unit 7 is connected to the input unit 8 and unit 9. The output unit 8 is connected to the input unit 10. Signal transmission in the direction from output to input.

The system operates the trail the way. The signal output from the block generating the signal 1 is fed to the input of block 2, which is differentiated, and the input unit 3, which is converted according to the density function for the distribution of the test signal, the output signal from unit 2 to the input of the multiplier 4, which is multiplied by the signal output unit 3, the signal output unit 3 is also fed to the input of block integration 5, which integrates the signal from the output unit 5 is fed to the input of block 6 of the exponent -1, the signals from the outputs of the block 6 and 4 are sent to the input unit 7, which are multiplied together. The output unit 7 is fed to the input unit 8 is integrated, and the input unit 9, which is clearly displayed on the oscilloscope, the output unit 8 is also fed to the input of the oscilloscope 10, which displays clearly.

The technical result of the use of the invention is to obtain accurate probabilistic characteristics of the original signal that can be used, for example, to create a reference source signal systems testing of technical devices.

The method of obtaining the probability distribution function of the original signal with a given density distribution of the test signal in the interval consisting of the transmission signal described by the function inverse to the probability distribution function of the input signal through a differentiator, and the parallel through the block, converts the incoming signal randomly with a given probability density, then the signal at the output of this block is fed to the first of two multiplier units, where it is multiplied by the output signal of the differentiator, and the first of the two integrators; the signal coming from the integrator is fed to the unit exponentiation -1, the output of which the signal arrives at the second of the two multipliers, where is multiplied by the output signal from the first multiplier and the output signal from the second multiplier is passed through the second of the two integrators, the output of which is a probability distribution of the source signal that can be represented in as algorithm:
where int is an indefinite integral of a function, int_[a,b] is the Lebesgue integral on the field values in the interval x∈[a,b}, Sx - density distribution of the test signal.


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