# Phase Congruential White Noise Generator
## 2 White Noise Autocorrelation Matrix  
The white noise autocorrelation matrix plays a fundamental role in analyzing the statistical independence of white noise signals [1; 2; 3; 4; 42; 43; 44]. Let us demonstrate this utilization the example of the matrix of autvectors \({}^{0}Z\) from the previous section. Using the matrix \({}^{0}Z\), it is possible to calculate the paired scalar multiplications of autvectors \({}^{0}Z_{0}\), \({}^{0}Z_{1}\),\(\cdots\)\({}^{0}Z_{n-1}\). To simplify the notation of these autvectors, let us omit the upper left index \(0\), i.e., instead of \({}^{0}Z_{i}\) the indication \(Z_{i}\) is considered further. For autvectors \(Z_{i}\) and \(Z_{k}\) the scalar multiplication \((Z_{i},Z_{k})\) is determined by the sum of multiplications of countings as follows:  
\[(Z_{i},Z_{k})=\sum_{m=0}^{n-1}z_{im}\cdot z_{km} \tag{5}\]  
In Formula (5), the sum of multiplications of the corresponding countings is calculated, provided that these autvectors are in an orthogonal coordinate system. Formally, the geometric length or norm \(\|Z_{i}\|\) of the autovector \(Z_{i}\) in an \(n\)-dimensional orthogonal coordinate system is calculated by the following scalar multiplication:  
\[\|Z_{i}\|=\sqrt{\sum_{m=0}^{n-1}x_{im}\cdot x_{im}}=\sqrt{\sum_{m=0}^{n-1}x_{ im}^{2}} \tag{6}\]  
Formula (6) formally coincides with the Pythagorean Theorem in a multidimensional linear geometric space. From analytical geometry it is known (or it could be counted directly) that cosine of the angle between two linear vectors is determined by the scalar multiplication (5) between them and their norms (6). So, using Expressions (5) and (6), the following Formula (7) for the angle between the autvectors \(Z_{i}\) and \(Z_{k}\) can be obtained:  
\[\cos\varphi_{ik}=\frac{(Z_{i},Z_{k})}{\|Z_{i}\|\cdot\|Z_{k}\|} \tag{7}\]  
In applied signal analysis the statistical estimates are very important and could be easily obtained by means of computer technologies. In this sense, it is of significant interest to evaluate the linear connections of autvectors, which for \(\cos\varphi_{ik}=0\) are usually calledstatistically independent. In this case such autvectors have a correlation coefficient equal to zero. For autvectors \(Z_{i}\) and \(Z_{k}\) the correlation coefficient is calculated as follows:  
\[r_{ik}==\frac{\sum_{m=0}^{n-1}(z_{im}-E_{1}(Z_{i}))\cdot(z_{km}-E_{1}(Z_{k}))}{ \sqrt{\sum_{m=0}^{n-1}(z_{im}-E_{1}(Z_{i}))^{2}}\sqrt{\sum_{k=0}^{n-1}(z_{km}-E _{1}(Z_{k}))^{2}}} \tag{8}\]  
In Expression (8) the correlation coefficient \(r_{ik}\) deals with centered autvectors relative to the first statistical moments \(E_{1}(Z_{i})\) and \(E_{1}(Z_{k})\), which are the following:  
\[E_{1}(Z_{i})=\frac{1}{n}\sum_{m=0}^{n-1}Z_{im},\qquad E_{1}(Z_{k})=\frac{1}{n} \sum_{m=0}^{n-1}Z_{km} \tag{9}\]  
Further, using the Expressions (9) let us determine the autvector \(V_{i}\), which is statistically centered for the autovector \(Z_{i}\) after corresponding adjustment of each its counting \(z_{im}\):  
\[v_{im}=z_{im}-E_{1}(Z_{i}),\qquad m\in\big{[}\overline{0:n-1}\big{]} \tag{10}\]  
Finally, the angle \(\varphi_{ik}\) between the autvectors \(V_{i}\) and \(V_{k}\) is defined by using Expressions (5)-(10) in the following form:  
\[\cos\varphi_{ik}=r_{ki}=\frac{(V_{i},V_{k})}{\|V_{i}\|\cdot\|V_{k}\|} \tag{11}\]  
From Expressions (8) and (11) it follows that the autvectors \(Z_{i},\ Z_{k}\) and the autvectors \(V_{i},\ V_{k}\) have equal angles to each other. Thus, the initial signal \(S_{0}\) is statistically independent, since all pairs of its autvectors are orthogonal. The centered autvectors \(V_{i}\) could be located in the matrix of autvectors \(V\) having the size \(n\times n\). Below is a program _P070102_, in which the matrix \(V\) contains the autvectors \(V_{i}\) obtained from the autvectors \(Z_{i}\) of the matrix \(Z\) after using the transformation (10).