# Phase Congruential White Noise Generator
## 1 Introduction
Therefore, the summation of all countings makes up the process with elements \(c_{j}\in\mathbb{P}\). A number of \(n\) consecutive countings can be collected together as an information signal \(S\). Formally, there can be many signals, and each of them is a subset of the process \(S\subset\mathbb{P}\). Let us denote the initial counting in signal \(S\) as \(c_{0}\in S\); the next one should have the following index \(c_{1}\in S\). Carrying on with the numbering of countings, the last one in this case would have the designation \(c_{n-1}\in S\). This means that only the first initial \(n\) countings are considered in signal \(S\). Let us assign this initial signal as \(S_{0}\subset\mathbb{P}\). Considering the random process further, it becomes obvious that \(c_{n}\
otin S_{0}\) since there are only \(n\) countings in that signal, starting from index \(0\) to \(n-1\). This means that the next signal \(S_{1}\) starts from the counting \(c_{n}\). However, the initial counting in each signal (\(S_{0}\), \(S_{1}\), etc.) should have a zero index relative to the beginning of the signal. To overcome this issue, the dual indexing, such as \(c_{ij}\), is used. The first index \(i\) indicates the signal number \(S_{i}\), while the second one \(j\) specifies the assigned number of the counting in the corresponding signal. In this case the next counting \(c_{n}\in\mathbb{P}\) in a random process is denoted in a random signal \(S_{1}\subset\mathbb{P}\) by a double indexing \(c_{n}=c_{10}\in S_{1}\). This approach allows consideration of the countings of a random process from the point of view related to their sorting by signals. Usually, the sequential signals are considered, and each of them contains an equal number of \(n\) countings with equal time intervals \(\tau\) between them. In this case, the time \(T_{S}\) of each information signal \(S\) could be designated as \(T_{S}=(n-1)\tau\). If the time \(\tau\) between countings admits a continual interpretation, then continuous stationary random processes are considered. If the interval time \(\tau\) and the number of countings \(n\) are finite in the signal \(S\), then such processes are discrete. Generally, the mathematical transition from continuous signals to discrete ones is carried out using the Dirac delta function. In a random discrete process, the countings are following one by one sequentially. Dividing a process into \(n\) sequential countings can be interpreted as observing the sequential discrete signal. Thus, the initial \(n\) countings provide forming the signal \(S_{0}\); the next \(n\) countings organize the subsequent signal \(S_{1}\), and so on. For the next step in signal analysis, it is necessary to check whether the countings in each signal are independent. The consistent observation of countings in a period of time is not exhaustive evidence of their statistical independence. Therefore, it is necessary to analyze their interaction among other countings within the signal. For this purpose, the adjacent pairs of signals are evaluated using the mathematical correlation method. In this case, it is convenient to introduce the formal autvectors for each pair of signals. The first pair consists of the initial signal \(S_{0}\) and the adjacent one \(S_{1}\); the next pair is formed from signals \(S_{1}\) and \(S_{2}\), and so on. Thus, the correlation approach uses adjacent signals \(S_{k}\) and \(S_{k+1}\). To any pair of adjacent signals, for example, \(S_{0}\) and \(S_{1}\), there is correspondence of \(n\) autvectors \({}^{0}Z_{0},\ {}^{0}Z_{1},\cdots,{}^{0}Z_{n-1}\). The upper left index 0 underlines the origin of countings, which are taken from signal \(S_{0}\). Countings of the original signal \(c_{0j}\in S_{0}\) are initially located in the autvector \({}^{0}z_{0j}\in{}^{0}Z_{0}\), which means that countings in the signal \(S_{0}\) are transferred to the initial autvector \({}^{0}Z_{0}\) as follows:  
\[ \begin{split}{}^{0}z_{0,0}&=c_{0,0},\\ {}^{0}z_{0,1}&=c_{0,1},\ldots\\ {}^{0}z_{0,n-1}&=c_{0,n-1}\end{split}