Abstract
This article deals with modelling the WiFi wave field, which is not visible to the naked eye, into a mathematical structure for assigning colours to observe the wave strength change compared to the source layout’s location correlation of obstacles in the survey environment. The shape function is used as a tool in modelling combined with finite sample measurement data to determine the basic parameters of the shape function. These will allow the wave field to be modelled using two techniques: the theoretical shape function and the experimental shape function. The measuring tool used in the article is the WiFi Analyzer App that runs on a smartphone. The colourassigned representation of an invisible field such as a wave will help select the signal maximal or minimization positions most beneficial to the user. The proposal is easily applied in the wave field and other quantities such as temperature, light, and concentration with similar techniques.
1 Introduction
Many physical fields exist around us but cannot be seen by the naked eye, exploiting these fields to maximize or minimize them needs to be based on understanding the spatial distribution as to how that is to observe them. Much research focused on showing the distribution of physical fields, such as Shimada et al. [1] studied the visualization of the spatial dispersion of optical fields in the gold nanoparticles’ spacetime linear array structures using scanning nearfield optical microscopy and electron microscopy. Le Kien et al. [2] studied the characterization of the guided normal field in parallelly coupled optical nanofibres. Research has shown the relationship between the spatial distributions of the field components of nanofibres and the electric field components. Sha et al. [3] used the corner transport upwind and constrained transport algorithm to study the digital interaction between the plane shock wave and the gas column in a magnetic field.
One of the popular ideas for exploiting the fields is to represent the field with mathematical modelling and assign a colour to the intensity variable to represent the field in a rainbow spectrum or digitizing graphically. Some recent studies are based on this idea. For example, Erturk et al. [4] used fractional calculus theory to study the movement of a beam on an internally bent nanowire. Then, Jajarmi et al. [5] used the classical Lagrangian method, establishing the Euler–Lagrange equations to study the complex displacement and charge of the condenser microdynamic system. Most recently, Baleanu et al. [6] used the structure of fractional derivatives and integrals to investigate the generalized fractional model of COVID19, considering the effects of isolation and quarantine. Also, by using the generalized fractional model, Jajarmi et al. [7] researched the asymptotic behaviour of immunogen tumour dynamics.
Some physical area fields have specialized equipment that makes them visual. For example, the temperature field can use thermal imaging cameras to examine its distribution directly, but the cost of the equipment is prohibitive. In addition, the approach using the interpolated function in combination with sensors to study physical fields such as temperature is also mentioned by many research studies. Oktavia et al. [8] used spatial interpolation in combination with five sensors to control temperature and humidity in data centre rooms. Wang et al. [9] used surface spline interpolation to investigate and control the temperature of integrated circuits. Bullo et al. [10] calculated the thermal field distribution of complex biological structures using the quadratic interpolation method. Long et al. [11] also studied the calculation of the thermal field distribution but used a shape function to interpolate.
Moreover, many other physical areas are not visible to the naked eye and require a solution to visualize them, so a method for this problem is issued [12]. For the WiFi wave field, its intensity satisfies the continuity in space. Therefore, they can be interpolated by approximation and continuous functions, precisely the shape function of the form [13,14]. Many authors have interpolated quantities, such as wave fields, error, or material with positive results, as long as they satisfy the continuity condition and they are approximated with a polynomial function. Gilewski and Pelczynski [15] used shape functions to solve material orientation problems for a moderately thick plate of functionally graded materials. Liu and Chang [16] used a concept called the boundary shape function to calculate and automate the quadratic nonlinear boundaryvalue problem satisfying the constraints. Xia et al. [17] compared the study with the Lasserre algorithm and showed that the shape function derivative is more intuitive and can be programmed conveniently. Silva et al. [18] compared the use of finiteelement mesh and shape functions to determine the degrees of freedom of the mesh at the data point coordinates. The research results have shown that the interpolation using the element shape functions is more than correct. Hou and Shi [19] used a shape function that generalizes datadriven timefrequency analysis to describe the frequency modulation of an intrawave. Research has also shown that it is an efficient and powerful adaptive timefrequency analysis method for intrawave frequency signals.
Because in the same environment or office using the same WiFi transmitter, users in different locations catch signals with different strengths and weaknesses. Furthermore, it is often impossible to determine the exact location of the most robust signal to use other than signal testing. So, using a finite number of measurements with a simple device combined with shape function interpolation, which is a robust and efficient method to analyse digital waveform signals, as studies [18,19] have shown, displaying the WiFi wave field graphically or numerically to help choose the most profitable, maximum, or minimum position signal for users is the driving force behind this study. With some data measured by the WiFi Analyzer App on a smartphone, it is possible to determine the mathematical function that describes the WiFi wave field in the survey space and draw the corresponding distribution. This approach allows for selecting suitable areas for the requirements of using space in parallel with using WiFi. This idea applies to the WiFi wave field and can also be applied to any field quantity that satisfies the continuum and cannot be observed with the naked eye. The article’s success opens up a direction for modelling fieldtype quantities with a lower cost of time and equipment than using specialized equipment, sometimes without specialized observation equipment. This is the most effective way to fix this problem up to now, which is the main contribution of the research team.
The article is organized as follows. Section 2 introduces the concept of the shape function, the theoretical shape function, and the form of the experimental shape function. Then, the coordinate survey space, the device to measure the WiFi signal using the WiFi Analyzer App, the calculation method to simulate the WiFi wave function in the form of the colour spectrum, and numerical graphs are presented in Section 3. Finally, we conclude in Section 4.
2 Theoretical shape function and experimental shape function form
2.1 Concept of a shape function
The shape function is commonly used in interpolation [13,14] and has many different forms suitable for different situations. For example, an interpolated space was considered, as shown in Figure 1.
In Figure 1, the intensity influence of scalar sources
where the coefficients
2.2 Theoretical shape functions
By considering a survey area in a rectangular box, as shown in Figure 2, the theoretical shape function for the eight key points in Figure 2 is determined as follows:
The reference system
in which
Thus, these values
2.3 Experimental shape functions
Like the theoretical shape functions, the
The function
From here, the stationary value of experimental shape functions is determined for the point
A unique set of stationary values according to equation (7) is not enough to determine a function of general form, it is necessary to continue investigating other points, such as examining points
Thus, the regression law allows onto determine the experimental shape functions at the
3 Simulation and experiment
3.1 Space, frame of reference, and survey object
Figure 3 is a model of a room with the WiFi transmitter placed outside, assuming that walls with different textures and distances from WiFi transmitters will cause different effects on wave strength. The WiFi wave strength at the eight corners of the room are measured, which are the eight points of the model:
The coordinate system
3.2 Measuring device
WiFi Analyzer App (Android or IOS) is a popular application for mobile devices running Google or Apple’s mobile operating systems, which includes the following features and characteristics:
Optimize WiFi for interference problems,
Channel Analyzer for nearby APs,
Realtime data and distance calculation,
History of signal strength,
Support 2.4 GHz/5 GHz,
View Hidden WiFi,
Copy MAC address,
Channel optimizer + more.
We see here the strength of the WiFi signal in dBm (decibel milliwatts). These numbers are expressed as negative integers – the larger the negative integer, the stronger the signal. If the streaming video is needed,
Figure 4 shows the results of measurements at a location with various wireless transmitters that the app recorded. When measuring, it is necessary to place the machine at exact coordinates according to the global reference system as shown in Figure 3.
3.3 Calculation according to the theoretical shape function
The coordinates of the centroid
The variables are changed as follows:
The shape functions
The interpolation function
with constraints
The survey results are obtained and calculated according to the theoretical shape function, as shown in Table 1. Note that the measurement results show a negative sign with the survey wave strength, so when assigning colours, the values have a smaller absolute value corresponding to the hot (red) colour spectrum. Larger absolute values correspond to the cool (blue) colour spectrum. These are because the colour spectrums used for the graphics must be continued in the rainbow spectrum to represent spatially varying wave intensities’ transitions. In general, the graphics should represent the relative and absolute values of the wave strength at that point compared to the surrounding points.
Number of measurements  Point

Point

Point

Point

Point

Point

Point

Point


1 








2 








3 








4 








5 








6 








7 








8 








The average value 








Behind the wall where the WiFi transmitter is placed with the most robust wave strength, in front of the wall, on the far side is the area with the slightest wave intensity (Figure 5). The fact that one of the two vertical walls has two doors like the room model also clearly affects the distribution of wave strength at the two longitudinal boundaries of the survey space. In particular, the side of the wall with the door opening WiFi signal strength is stronger than the side without the door (Figure 6), showing that obstacles such as walls significantly affect the wave propagation and the propagation distance also affects the wave strength.
According to the second observation direction, the wave is most substantial in the wall corner near the WiFi transmitter, adjacent to the room’s door. The worst wave area is the wall intersection opposite to the transmitter on the far side and the vertical wall with no doors, which is the model’s most distant and most hidden angle. The simulation results are entirely consistent with the retest results with the actual measuring device.
4 Conclusion
Determining which location has the most substantial WiFi waves to use in an area; how to arrange WiFi transmitters to ensure that the entire area has the most robust wave strength and the lowest prices are the issues that users and WiFi network design engineers are interested in. Unfortunately, there is no specialized device to display WiFi wave images for an area, such as a thermal imaging camera to display the temperature field. Moreover, measurement, testing, or arranging sensors to locate the strongest or weakest WiFi waves is very timeconsuming and labourintensive, requiring many measurements or many sensors to be arranged to obatin accurate data. Spatial shape function interpolation has been identified as an ideal solution for predicting WiFi wave field data, a field data type with spatial continuity. By using the WiFi Analyzer App, a simple application that runs on smartphones and has high reliability, to measure the value of WiFi waves at specified points in the survey space. Using spatial shape function interpolation after calculating the display of all WiFi wave images in the survey area has been performed.
Figures 5 and 6 show the results consistent with the actual distribution when the WiFi Analyzer App is retested. Thus, as long as the interpolated quantity is continuous and has suitable measuring equipment, it is possible to represent the magnitude distribution of any physical field graphically. Moreover, this technique is suitable for solving the problem in both directions, that is to find the wave intensity distribution and vice versa to find the wave strength to find the most satisfactory coordinate domain. The interpolation results are similar to Figures 5 and 6 in layout in determining experimental shape functions. However, the wave field has higher precision than the theoretical shape function due to many sample measurements.
The research team will continue to deepen the research and practical application of the proposed method by helping engineers design networks, especially in public places with high urbanization rates and high user demand for free WiFi waves is very large. The problem is calculating the location and the smallest number of WiFi transmitters so that the area has the highest number of points to catch the most substantial WiFi waves. In addition, as mentioned in Section 1, this technique can also be applied to solve problems related to temperature fields, such as calculating the number of points of heat generation needed to warm cattle on a farm in winter or the minimum number of heat points for a sanitary ware ceramic kiln with the requirement that all surfaces of the product be heated evenly to a temperature of

Funding information: The authors would like to acknowledge the Postdoctoral Science Foundation of China (Grant No. 2021M703780).

Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.

Conflict of interest: The authors state no conflict of interest.

Data availability statement: All data generated or analysed during this study are included in this published article.
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