# The Effect of Blue Noise on the Optimization Ability of Hopfield Neural Network
## 3 Materials and Methods
### Constructing Colored Noise Generators  
Noise signals are signals generated by random processes and can be generally classified as white noise and colored noise. White noise is a noise signal whose power spectral density (PSD, power spectral density) is constant in the whole frequency domain and has been widely used in the noise correlation test of various models; however, white noise as a purely theoretical construct will inevitably have deviations when simulating realistic application scenarios [30], so taking various types of spectral situations of colored noise becomes relevant. Some of the typical colored noise signals are shown in Table 1.  
There are many colored noise generation methods, but in the face of the complex N-dimensional vector situation of neural networks, most of them cannot be directly applied to neural networks, so the article improves the typical \(PSD\propto f^{\alpha}\) (\(\alpha\in\)R) colored noise sequence generator [31] based on these typical colored noise types, the original noise generator can only generate colored noise sequences satisfying the one-dimensional case. After the improvement, colored noise networks matching the ranks of neural networks will be generated when the size of the ranks of neural networks is obtained and will meet the requirements of Gaussian noise distribution, based on the fusion scheme proposed in the paper, and the colored noise network will directly perturb the neural network and generate the output. Since the improved noise generator can freely choose the colored noise network dimension according to the rank value size of the neural network, the noise generator also supports the output of N-dimensional noise sequences with a single point, which is convenient for the subsequent fusion work with the neural network. Based on the original generator [32], the dimension reconstruction function is added to this generator, which is composed of fast Fourier transform (FFT), inverse fast Fourier transform (IFFT), divisor, and multipliers. The digital input signal is input continuously point by point, and the FFT transform is made first, and according to the desired colored noise, the frequency characteristic \(H(n)\) is divided in the frequency domain; then, the IFFT transform is made, and the output is reconstructed according to the desired dimension, the output of the colored noise generator is obtained cumulatively, and the colored noise after reconstruction still obeys the Gaussian distribution, so it does not cause the probability density to change, and its difference is mainly reflected in the sequence power spectral density. The difference is mainly reflected in the sequence power spectral density and its own numerical distribution. Taking blue noise as an example, the constructed noise sequence of its own characteristics and white noise is shown in Figure 1, the generated blue noise still meets the requirements of Gaussian noise distribution. However, it can be obviously seen in the gray figure that the gray value of the blue noise after spectrum transformation is higher than that of the original white noise, while the autocorrelation function and power spectral density are a Fourier transform pair, and it can be seen that the noise amplitude after re-construction is significantly reduced. \begin{table}
\begin{tabular}{c c c} \hline
**Noise Type** & **PSD** & **Examples of Applications** \\ \hline Red noise & \(\propto\frac{1}{f^{2}}\) & Identifying steady-state transitions in the integrated analysis of climate records \\ Pink noise & \(\propto\frac{1}{f}\) & Promotes neural oscillatory activity \\ Blue noise & \(\propto f\) & Multi-class blue noise sampling \\ Violet noise & \(\propto f^{2}\) & Acoustic thermal noise signal of water \\ \hline \end{tabular}
\end{table} 
Table 1: Example of typical colored noise.