SU981999A1 - Random number generator - Google Patents

Description

(54) GENERATOR OF RANDOM NUMBERS

The invention relates to computing technology and, in particular, to devices and systems of probabilistic digital modeling of processes and dynamic systems, and allows obtaining Gaussian random numbers with a given energy spectrum.

In solving many problems of analysis and synthesis of dynamic systems, digital probabilistic modeling is widely used. In this case, the object under study is not only being modeled on the computing device, but an external disturbance is also being generated on this object. As a rule, the external disturbance is a stationary random process with a known energy spectrum.

A device for generating Gaussian random numbers with a given energy spectrum (or with a given autocorrelation function) fl is known.

However, this device cannot generate random Gaussian numbers with an arbitrarily specified shape of the spectrum. To form a given energy spectrum of a sequence of Gaussian random numbers

use the method of canonical decomposition.

However, this period requires a very large computational cost. C23 proposes an efficient method for generating Gaussian random numbers with a given energy spectrum, based on the application of the Fast Fourier Transform method and calculating the canonical decomposition of a Gaussian random process. This method consists in calculating the discrete Fourier transform of a sequence of uniformly distributed random numbers taken with weights determined by the frequency response of the shaping filter.

The closest technical solution to the invention is a device built on the basis of the fast Fourier transform method, which contains a clock-oscillator, the first output of which is connected in series

25 a generator of uniformly distributed numbers, a multiplier, a buffer block IMATED and a fast Fourier transform block, 4 to the second output are connected to the control input of the sampling unit, at 5 the first input of which is connected to the second block of memory resistors and the output is connected to the second input of the multiplier 2.

When the prototype operates, a uniformly distributed number (and the general case is complex) is generated from the clock generator signals and the first value of the weighting factor (also in the general case, the complex one) is sampled from the first block of memory registers. In a multiplier, these two numbers are multiplied together, and the result is recorded in a buffer memory. At the next clock cycles, this procedure is repeated until the entire buffer memory containing the register No. is full. After filling the buffer memory, the fast Fourier transform unit is activated, at the output of which a sequence of Gaussian random numbers with a given energy spectrum is formed. For the prototype to work successfully, the weights should be proportional to the square root of the energy spectrum values at the corresponding frequencies.

Despite the fact that the device prototype is the fastest of the known devices for generating Gaussian random numbers with an arbitrarily specified spectrum, the computational costs in it are quite large. These costs form, for a sequence of N rayccoBbiix numbers, approximately complex arithmetic operations of the addition of N (l + l / 2 og N) complex multiplication operations. The disadvantage of the prototype,. thus, there is a large amount of computational effort (low speed) in generating Gaussian random numbers with a given energy spectrum.

The purpose of the invention is to increase the speed of the device.

The goal is achieved by the fact that in a known random number generator, which contains a clock pulse generator, the first output of which is connected to the input of a generator of uniformly distributed random numbers, the output of which is connected to the first input of the multiplication unit, the second input of which is connected to the output of the key, the output of the multiplication unit is connected with the input of the first memory block, the second memory block, the weighting factor formation block, the Walsh fast transform block and the delay element whose input is connected to the second output are entered A clock pulse generator, the third output of which is connected to the input of the weight factor generation unit.

the input group of which is connected to the output group of the second storage unit, respectively, the output of the weight factor generation unit is connected to the information input of a key whose control input is connected to the output of the delay element, the output of the first memory unit is connected to the input of the fast Walsh transform unit, the output of which is generator output.

In addition, the weight shaping unit contains two memory blocks, two switches, a quad, an adder, a nonlinear converter and a multiplier, the output group of the first memory block is connected to the input group of the first switch, respectively, whose input is combined with the input of the second switch and is input The blast whose input group is the input switch group of the second switch, the output of which is connected to the first input of the multiplier, the output of which is connected to the input of the second memory block, the output of which is Exit unit, an output of the first switch via series connected squarer, an adder i.nelineyny converter connected to the second input of the multiplier.

Figure 1 shows the block diagram of the proposed random number generator with a given energy spectrum; Fig. 2 is a block diagram of the formation of weights.

The device contains a clock pulse generator 1, to the first output of which are connected serially connected generator of uniformly distributed random numbers 2, multiplier 3, first memory block 4 and fast Walsh transform block 5. Second generator output 1 is connected to the control input of the weighting factors 6, to the informational inputs of which the outputs of the second memory unit 7 are connected. The third output of the generator 1 is connected via a delay element 8 to the key 9. The weight shaping unit b consists of the first memory unit 10 connected in series, the first switch 11, the quad 12, the adder 13, the nonlinear converter 14 performing square root extraction from the input value, the multiplier 15 and the second memory block 16. The second input of the second multiplier 15 is turned on the switch 17. The second output of the generator 1 is connected to the control inputs of the switches 11 and 17, and the inputs of the second switch 17 include the outputs of the memory block 7. The output of the memory block 16 is connected to the input of the key 9. The generator works as follows. Signals from generator 1 to generator 2 generate random independent evenly distributed numbers that are fed to the first input of multiplier 3. The signals from the clock generator also go to the control input of the weight shaping unit b. The next weighting factor is formed on this signal in block b, to which key 9 sends a signal from clock generator 1 to the second input of multiplier 3. Delay element 8 is inserted so that clock 1 signals are received on key 9 only after in block b, the formation of the next weighting factor was completed. To clarify the operation of the device, we give a mathematical justification for its operation. Let F (Z) be a generalized discrete transformation. Furbe is a suspended sequence of N complex random limited and independent values of Z, K 0.1, ..., N-1 -f N-1. (1 f 2 Ph-VX-YR..K .5, l is the sequence of weighting factors; K is complete, orthonormal the system of basis functions when decomposing the sequence f into a generalized Fourier series. 2) shows that if i ( n, k) exp (- -: r system of discrete exponential functions-, n ,, l, .. N-1, the T0 distribution function of random variables f, n 0,1, ..., N -1 converges to the Gaussian function of the law, as O (I / YN). The autocorrelation function of the sequence I is equal to: orM.g-Tlf. (2 -PuPwi- N (uo,, 1, ..., Y-1) The values of L have the physical meaning readings of the energy spectrum. In the proposed device then the discrete Fourier transform is used discrete transformation of the Wall. At the same time, the Avits and solves the problem of obtaining a sequence of Gaussian random variables with the energy spectrum j LJ K 0,1, ..., N-1 or, moreover, with the correlation function R (C We decompose the sequence p according to the basis of discrete exponential functions on the interval fO, N-lJ and the basis of discrete Walsh functions), where the decomposition coefficients are represented as p (K) and f,), respectively. It is known that the spectral decomposition coefficients are related to each other by the ratio ω () 15 (and) VFS, 1c), I-0 FTsYu-- g1 9 (.)) - (5) Vrt O Func- tion called the Fourier core. The Fourier kernels constitute an orthonormal system of functions, K) Φ (P1, K) - | j if mn если if m fn From (4) we have. / Sa (K) i 1Ч, (р) Ф (p, K) ts) f (е) ф (е, к) Г (р) 5 (е) Ф (р, к), --- |; 15dr), k) / Averaging the last equality over the set we get p, (. K 11СкТ | - | hOO | (k) (f (P, K) | exponential functions and discrete Walsh functions are obtained in the form (b (K) WlV (K) | 0Cp.K) f For practical work, it is more convenient to set L values of the frequency response of the shaping filter in an engineer-based exponential function. These values are stored in memory block 7. The calculation of the weighting factors f) (k) must be performed to one time. Therefore, they can be regarded as given and neglect the computational cost of obtaining them. Let us estimate the gain in the number of arithmetic operations given by the proposed device when generating .N Gaussian random numbers with the energy spectrum values L (K). As stated above , the computational cost in the prototype is approximately N complex addition operations and N (N) complex multiplication operations. More conveniently, go to arithmetic operations on real numbers. During the operation of the prototype, it is necessary to realize approximately 2N + 3Nfog2 N arithmetic operations of addition and 2N (2 + gog N) arithmetic operations of multiplication (taking into account the fact that

L I- sequence

real numbers ClZ -X | ... sequence of complex numbers). In the proposed device, a complex sequence of random numbers Z, K 0, lf ..., N-l is multiplied, respectively, by real numbers. This requires 2N multiply operations.

To perform a discrete walsh transform on the comp; lex sequence

ci. . 0.1, .., (N-1) —It is necessary to perform approximately 2N8og N addition operations.

Let us designate the time for performing the operation of addition, t; , and the operation time of multiplication of real numbers Tsch Ctj,. The constant C for modern computing devices is in the range of 1-10.

The winning T given by the proposed device is

B-bc ogгN V (gOggN) NA-aJ t

 H-SLSG - - m For computing devices with fixed arithmetic :, t x5t. In this case

-r-.ibNCoe-xN llN NEog- NvlON

The most typical values of N are in the range of 128-1024. For such N values, the T gain in computational effort is approximately 34 times.

The block formation - weight coefficients 6 works as follows (see Fig.2). The generator signal is fed to the control inputs of the first and second switches 11 and 17. The first switch from the memory unit 10 sequentially selects the first Fourier values F (n, k) for different values of k and a fixed value n, equal to the number generated evenly distributed number. These values of Φ () are squared with quad squared 12 and summed by adder 13. After completing the definition (Φ (n, K)) for

all n O, 1, 2,. . .N-1, the square root of the nonlinear transducer is extracted from the resulting sum. The resulting value is fed to the first input of the multiplier 15. The second input of the multiplier 15 receives the value (K) selected by the second switch, numerically equal to the square root of the required value energy spectrum of a sequence of Gaussian random numbers. The resulting weights are accumulated in memory block 16.

Accumulated in the memory block 4 values of random evenly

 distributed numbers multiplied by weights are subjected to a fast Walsh transform in block 5. At the output of this block, a sequence of Gaussian

0 random numbers with a given energy spectrum. .

In the practical use of the proposed device for generating a large number of sequences from N Gaussian numbers with a given energy spectrum, the formation of weighting factors is necessary to be performed only once, after which they are sampled

0 from the memory block 16. The formation of the weighting factors is carried out according to the expression

, btKb (K) 1F (.m, k) | ,

where) - samples of the required energy spectrum of a sequence of Gaussian numbers and Φ (n, K) n values of the der Fourier function (ciCK)) and Φ (n, K) are considered given.

For the most coherent sequences with a duration of 128–1024, the winnings in computational expenditures by the proposed device will be 3–4 times as compared with the prototype.

Thus, the main advantage of the claimed device is the reduction of computational costs when generating Gaussian v random numbers with a given energy spectrum from equally distributed ones.

Claims (2)

All units included in the proposed device 5 are well known. Their implementation is not difficult. The proposed device will find application in digital systems for modeling dynamic objects. The economic effect of using the proposed device will be about 20-30 thousand rubles per year. Claim 1. A random number generator, 5 containing a clock pulse generator, the first output of which is connected to the input of a generator of uniformly distributed random numbers, the output of which is connected to the first input of the multiplication unit, the second input of which is connected to the key output, the output of the multiplication unit is connected to the input the first memory block, the second memory block, characterized in that, in order to improve speed, it contains a weight shaping unit, a Walsh fast conversion unit and a delay element, the stroke of which is connected to the second output of the clock pulse generator, the third output of which is connected to the input of the blocking weights, the group of inputs of which is connected to the main outputs of the second memory block, respectively, the output of the block of formation of the weights are connected to the information input of the key, the control input of which connected to the output of the delay element; the output of the first memory block is connected to the input of the fast Walsh transform unit, the output of which is the generator input. 2. The generator pop. 1, characterized in that the unit forming weights contains two memory blocks, two switches, a quad, an adder, a non-linear converter and a multiplier, the output group of the first memory block is connected to the input group of the first switch, respectively, whose input is combined with the input of the second KOMMyTaiTopa and is the input of the block whose input group is 0 group of inputs of the second switch,; the output of which is connected to the first input of the multiplier, the output of which is connected to the input of the second memory block whose output is the output of the block, the output of the first switch through the series-connected quadrant, c5 -1mator and nonlinear converter connected to the second input multiplier. 0 Sources of information taken into account in the examination 1. Korn G. Modeling of O-light processes on analog and analog-digital machines. M., World, 5 1968. 2. Mirkin L.I., Rabinovich M.A. and Yaroslavsky L.P. The method of generating correlated Gaussian pseudo-random numbers on a computer. ЖВМ and МФ 0 t.12 5, 1972 (prototype). From the second fuxoda tokkopo to itffpamopa From nepSoiB S / ioKa ptiucmpoB panty