# Phase Congruential White Noise Generator
## 2 White Noise Autocorrelation Matrix
The simulation results in the above presented programs _P070101_, _P070201_ and _P070202_ show that the _Random_ generator creates the countings of signals with low quality for white noise process. Their autocorrelation matrix \(A\) and the corresponding matrix of autocorrelation coefficients \(R\) are far from the desirable zero values. In line with all of this, in this article we offer a new congruential generator which has better quality of generating the uniform white noise. ## 3 Theory

Consider a model, in which the values of countings of white noise random process are present at a finite observation interval \(T.\) This process starts at point \(t_{0}\). In the interval \([t_{0},\ t_{1}]=[t_{0},\ t_{0}+T]\) there are \(N_{S}\) countings of the initial signal \(S_{0}\); in the interval \([t_{1},\ t_{2}]=[t_{1},\ t_{1}+T]\) there are \(N_{S}\) countings of the signal \(S_{1}\), and so on. The designation \(N_{S}\) for the number of countings in the signal exactly corresponds to \(n\) in the previous sections above, i.e., \(N_{S}=n\). Suppose that in all signals the countings are located with a constant step:

\[\tau=\frac{T_{S}}{N_{S}-1} \tag{15}\]

In each signal \(S_{0},\ S_{1},\ \cdots,S_{n-1}\) a certain spectrum of frequencies is fixed. If among these frequencies there is one with a period \(T_{S}\), then it is denoted as the initial frequency \(\omega_{1}\). Typically, in trigonometric studies the interval \([-\pi,\ +\pi]\) is used. An isomorphic transition of an observation point \(t\in[0,T_{S}]\) to an isomorphic point \(x\in[-\pi,+\pi]\) is performed as follows:

\[x=-\pi+2\pi\frac{t}{T_{S}} \tag{16}\]

On the interval \([-\pi,+\pi]\) the frequency \(\omega_{1}\) is equal to one, i.e., \(\omega_{1}=1\). Frequencies \(\omega_{k}=k\cdot\omega_{1}\), integral multiples of the initial frequency \(\omega_{1}\), refer to the discrete Fourier spectrum, in which \(N_{F}\) is the quantity of frequencies. These frequencies make it possible to organize an orthonormal system of sine-cosine coordinates in Euclidean space \(E_{0}[-\pi,+\pi]\):

\[\frac{1}{\sqrt{2\pi}},\ \ \frac{\cos\omega_{1}x}{\sqrt{\pi}},\ \ \frac{\sin\omega_{1}x}{\sqrt{\pi}},\ \ \frac{\cos 2\omega_{1}x}{\sqrt{\pi}},\ \ \frac{\sin 2\omega_{1}x}{\sqrt{\pi}},\ \ \frac{\cos N_{F}\omega_{1}x}{\sqrt{\pi}},\ \ \frac{\sin N_{F}\omega_{1}x}{\sqrt{\pi}} \tag{17}\]

In the trigonometric space with the coordinate system (17) the countings are observed at the points \(x_{i}\in[-\pi,+\pi]\), \(i\in[\ 0:N_{S}-1]\). By the property of this space (17), the values of countings \(f(x_{i})\) could be determined by the following Fourier polynom:

\[f(x_{i})=a_{0}+\sum_{k=1}^{N_{S}}(a_{k}\cos k\omega_{1}x_{i}+b_{k}\sin k\omega _{1}x_{i}) \tag{18}\]

It is assumed that if the information signal is closely approximated to the white noise signal, then it should have the same amplitudes at all frequencies \(\omega_{k}=k\omega_{1}\). However, in Expression (18), each frequency is accompanied by two amplitudes \(a_{k}\) and \(b_{k}\) with possibly separate distributions. This approach leads to a significant complication of the algorithm for their calculations. To get around this obstacle, let us replace the calculation of each pair of intensities \(a_{k},\ b_{k}\) with the generation of uniform intensity \(A_{k}\) at multiple frequencies \(k\omega_{1}\) with a phase shift. Let us analyze how this could be done in more detail. It is known from trigonometric transformations that the spectral sine of the sum of two angles \(\gamma_{k}\) and \(\psi_{k}\) can be calculated using the following expression:

\[A_{k}\sin(\gamma_{k}+\psi_{k})=A_{k}\sin\gamma_{k}\cdot\cos\psi_{k}+A_{k}\cos \gamma_{k}\cdot\sin\psi_{k} \tag{19}\]

Considering Expressions (18) and (19) together, the following estimates are obtained:

\[ \begin{split}&\gamma_{k}=k\omega_{1}x_{i},\\ & a_{k}=A_{k}\sin\psi_{k},\\ & b_{k}=A_{k}\cos\psi_{k}\end{split}  \tag{20}\]

Using the Formulas (20), the relationship between the values \(A_{k}\) and the Fourier coefficients \(a_{k},\ b_{k}\) can be established as follows:

\[