içgörü - Pseudorandom number generation - # Gaussian random number generation from binary sequences

Generating Gaussian Pseudorandom Noise from Binary Sequences with Guaranteed Statistical Properties

Q: How can the proposed GRNG be extended to generate Gaussian random vectors with desired correlation structure?

To extend the proposed Gaussian Random Number Generator (GRNG) to generate Gaussian random vectors with a desired correlation structure, we can leverage the concept of multivariate Gaussian distributions. By considering the correlation matrix of the desired random vectors, we can modify the generation process to ensure that the generated samples exhibit the specified correlation structure. One approach is to introduce a transformation step after generating the individual Gaussian random variables. This transformation can be based on techniques such as Cholesky decomposition or eigen decomposition of the correlation matrix. By applying this transformation to the generated random variables, we can achieve the desired correlation structure while preserving the Gaussian distribution properties. Additionally, we can incorporate the correlation information directly into the generation process by modifying the algorithm to generate correlated Gaussian random vectors directly. This can involve adjusting the weights or parameters of the underlying binary sequences to reflect the desired correlations between the random variables.

Q: What are the implications of the correlation peaks in the input binary sequences on the performance of Monte-Carlo and Quasi Monte-Carlo algorithms that rely on the Gaussian multivariate properties of the generated samples?

The presence of correlation peaks in the input binary sequences used for generating Gaussian random numbers can have significant implications for Monte-Carlo and Quasi Monte-Carlo algorithms that rely on the Gaussian multivariate properties of the samples. Bias in Estimation: Correlation peaks can introduce bias in the estimation of integrals or functions in Monte-Carlo simulations. The non-randomness introduced by the peaks can lead to systematic errors in the estimation process, affecting the accuracy of the results. Convergence Rate: Correlation peaks can impact the convergence rate of Monte-Carlo algorithms. Higher-order correlations can slow down the convergence of the algorithms, requiring more samples to achieve the desired level of accuracy. Variance of Estimates: The presence of correlation peaks can increase the variance of the estimates obtained from Monte-Carlo simulations. This can result in wider confidence intervals and less precise estimates of the quantities of interest. Efficiency of Sampling: In Quasi Monte-Carlo algorithms, correlation peaks can disrupt the uniformity of the sample points, leading to suboptimal coverage of the integration domain. This can reduce the efficiency of the sampling process and affect the accuracy of the results.

Q: Can the search for characteristic polynomials of the input binary sequences be formulated as an optimization problem to balance memory requirements and statistical performance of the GRNG?

Yes, the search for characteristic polynomials of the input binary sequences can be formulated as an optimization problem to balance memory requirements and statistical performance of the Gaussian Random Number Generator (GRNG). By defining appropriate objective functions and constraints, we can optimize the selection of characteristic polynomials to achieve a trade-off between memory efficiency and statistical properties. Objective Function: The objective function can be designed to maximize the statistical properties of the generated Gaussian random numbers while minimizing the memory requirements of the GRNG. This can involve optimizing the correlation measures, linear complexity, or other relevant statistical metrics. Constraints: Constraints can be imposed to ensure that the selected characteristic polynomials meet specific criteria related to memory usage, computational efficiency, or desired statistical properties. For example, constraints on the degree of the polynomials or the number of terms can be included to control memory requirements. Optimization Algorithms: Various optimization algorithms such as genetic algorithms, simulated annealing, or gradient-based methods can be employed to search for the optimal characteristic polynomials. These algorithms can iteratively adjust the parameters of the polynomials to find the best balance between memory efficiency and statistical performance. By formulating the search for characteristic polynomials as an optimization problem, we can systematically explore the space of possible polynomials to identify the configurations that offer the most favorable trade-offs for the GRNG.

Temel Kavramlar

This work studies the theoretical framework to apply the Central Limit Theorem to generate Gaussian pseudorandom sequences from sums of binary sequences with good correlation properties, providing a relationship between the pseudorandomness of the input binary sequences and the statistical moments of the output Gaussian sequences.

Özet

The paper focuses on generating Gaussian pseudorandom noise using binary sequences, which provides a simpler hardware implementation compared to other methods like the Box-Muller algorithm.

Key highlights:

Theorem 1 and Theorem 2 establish bounds on the product moments of the generated Gaussian sequences in terms of the correlation measures of the input binary sequences.
The authors analyze the correlation properties of Gold codes, showing that they do not exhibit full peaks in the third and fourth correlation measures, making them a good candidate for the proposed Gaussian random number generator (GRNG).
Computational experiments compare the performance of GRNGs using m-sequences and Gold codes, demonstrating that Gold codes offer better statistical and pseudorandom properties.
The paper discusses the influence of the correlation peaks in the input binary sequences on the Gaussian multivariate properties of the output, and proposes further research directions to study the Tausworthe model and the search for characteristic polynomials that balance memory requirements and statistical performance.

Customize Summary

Rewrite with AI

Generate Citations

Translate Source

To Another Language

Generate MindMap

from source content

Visit Source

arxiv.org

İstatistikler

None.

Alıntılar

None.

Önemli Bilgiler Şuradan Elde Edildi

Generating gaussian pseudorandom noise with binary sequences

by Fran... : arxiv.org 04-04-2024

https://arxiv.org/pdf/2404.02251.pdf

Generating gaussian pseudorandom noise with binary sequences

Daha Derin Sorular

How can the proposed GRNG be extended to generate Gaussian random vectors with desired correlation structure?

To extend the proposed Gaussian Random Number Generator (GRNG) to generate Gaussian random vectors with a desired correlation structure, we can leverage the concept of multivariate Gaussian distributions. By considering the correlation matrix of the desired random vectors, we can modify the generation process to ensure that the generated samples exhibit the specified correlation structure.
One approach is to introduce a transformation step after generating the individual Gaussian random variables. This transformation can be based on techniques such as Cholesky decomposition or eigen decomposition of the correlation matrix. By applying this transformation to the generated random variables, we can achieve the desired correlation structure while preserving the Gaussian distribution properties.
Additionally, we can incorporate the correlation information directly into the generation process by modifying the algorithm to generate correlated Gaussian random vectors directly. This can involve adjusting the weights or parameters of the underlying binary sequences to reflect the desired correlations between the random variables.

What are the implications of the correlation peaks in the input binary sequences on the performance of Monte-Carlo and Quasi Monte-Carlo algorithms that rely on the Gaussian multivariate properties of the generated samples?

The presence of correlation peaks in the input binary sequences used for generating Gaussian random numbers can have significant implications for Monte-Carlo and Quasi Monte-Carlo algorithms that rely on the Gaussian multivariate properties of the samples.

Bias in Estimation: Correlation peaks can introduce bias in the estimation of integrals or functions in Monte-Carlo simulations. The non-randomness introduced by the peaks can lead to systematic errors in the estimation process, affecting the accuracy of the results.

Convergence Rate: Correlation peaks can impact the convergence rate of Monte-Carlo algorithms. Higher-order correlations can slow down the convergence of the algorithms, requiring more samples to achieve the desired level of accuracy.

Variance of Estimates: The presence of correlation peaks can increase the variance of the estimates obtained from Monte-Carlo simulations. This can result in wider confidence intervals and less precise estimates of the quantities of interest.

Efficiency of Sampling: In Quasi Monte-Carlo algorithms, correlation peaks can disrupt the uniformity of the sample points, leading to suboptimal coverage of the integration domain. This can reduce the efficiency of the sampling process and affect the accuracy of the results.

Can the search for characteristic polynomials of the input binary sequences be formulated as an optimization problem to balance memory requirements and statistical performance of the GRNG?

Yes, the search for characteristic polynomials of the input binary sequences can be formulated as an optimization problem to balance memory requirements and statistical performance of the Gaussian Random Number Generator (GRNG). By defining appropriate objective functions and constraints, we can optimize the selection of characteristic polynomials to achieve a trade-off between memory efficiency and statistical properties.

Objective Function: The objective function can be designed to maximize the statistical properties of the generated Gaussian random numbers while minimizing the memory requirements of the GRNG. This can involve optimizing the correlation measures, linear complexity, or other relevant statistical metrics.

Constraints: Constraints can be imposed to ensure that the selected characteristic polynomials meet specific criteria related to memory usage, computational efficiency, or desired statistical properties. For example, constraints on the degree of the polynomials or the number of terms can be included to control memory requirements.

Optimization Algorithms: Various optimization algorithms such as genetic algorithms, simulated annealing, or gradient-based methods can be employed to search for the optimal characteristic polynomials. These algorithms can iteratively adjust the parameters of the polynomials to find the best balance between memory efficiency and statistical performance.

By formulating the search for characteristic polynomials as an optimization problem, we can systematically explore the space of possible polynomials to identify the configurations that offer the most favorable trade-offs for the GRNG.