# Phase Congruential White Noise Generator
## 1 Introduction  
The concept of white noise realization corresponds to randomness in appearance and distribution of signals [1, 2, 3, 4]. For audible signals, the conforming range is the band of frequencies from 20 to 20,000 Hz. The randomness of such signals in this range is usually perceived by the human ear as a hissing sound with different volume intensities. White noise also manifests itself in myriad natural phenomena. For example, noises of various intensities of the sea waves, waterfall, rain, wind, etc. Outside nature, in technical systems, white noise appears in p-p junctions of semiconductors, in roars of different engines (vehicles, aircrafts, etc. ), in the overlapping of many sounds associated with big cities and metropolitan life (traffic, construction, honking, life support systems), and so on. An analysis of the literature shows that most of the articles describe different applications of white noise, as well as its recognition in transmitted signals. This includes the areas such as theoretical and applied mathematics [5, 6, 7, 8, 9], physical research [3, 4, 10, 11, 12, 13], electronic and radio engineering [14, 15, 16, 17, 18, 19, 20, 21, 22, 23], acoustics and noise phenomena [1, 2, 3, 4, 24, 25], computer algorithms [26, 27], geological prospecting and exploration [28, 29], medical and biological research [30, 31, 32, 33, 34, 35, 36], psychology and psychiatry [37, 38], and others. It is worth noting that significant results have been achieved in those fields. On the other hand, the literature describes the methods of white noise creation, as well as its use in the artificial modeling of various situations. A fundamental feature of this direction is the condition that a white noise generator is required initially. The first approach, in this case, uses techniques of recording the physical phenomena with subsequent digitizing and postprocessing filtration [39, 40]. This method provides quasi \"natural\" or \"realistic\" white noise, however the numerical accuracy of the registration of stochastic quantities is often not sufficient. Moreover, the technical realization is quite expensive. The second approach in creating a white noise generator utilizes the techniques of computer-based algorithms. In this case, the generation accuracy is quite high. However, different algorithmic methods have varying degrees of generation quality and usuallythey are associated with diverse disadvantages. To analyze the grade of generation, the verification methods are commonly applied. As an example, in [41] the following form for white noise generation is proposed; it should be noted here that this is just one of many ways to implement a white noise generator. As a basis for forming \(n\) values \(c_{j}\) of white noise (provided that \(j\in\left[\overline{0,n-1}\right]\)) the function _rand_() from _Microsoft Visual Studio_ could be utilized. The algorithmic expression used in [41] can be represented here in the following mathematical form:  
\[c_{j}=4\left(\frac{rand()}{randmax}-0.5\right) \tag{1}\]  
By using this expression, the resulting values can be obtained in the range \([-2:+2]\). This simple formula allows creation of the elements in the required diapason. The next mandatory step is to check the condition of how much the obtained values \(c_{j}\) really are the elements of a process with white noise properties. For such verification it is necessary to consider how the white noise process and the properties of its elements are formed, and here it all depends on the quality of the function _rand_() and its maximum value of _random_. Let us consider this in more details. From a general point of view, the numerical sequences form a stationary random process \(\mathbb{P}\) if the characteristics of this process are not changing during its implementation. One such characteristic is the consistent observation of the presence of numerical values over the time period. Commonly, the term observation is replaced by the equivalent concept of counting.