1 Introduction

Radio wave attenuation is primarily caused by the absorption of a radio signals by some atmospheric phenomenon like rain, snow, ice, clouds, dust etc. These attenuations are more prevalent in the frequency range above 10 GHz. Attenuation caused by rain is not only limited to satellite up-link and down-links, but it can also affect the point-to-point terrestrial microwave links above 10 GHz [1,2,3,4,5]. The attenuations caused due to presence of rain is called rain fade. Some of the studies comprised of mathematical models [3, 8, 10]. Results of these models are compared with the ITU-R model. Some experimentation has been performed at different geographical locations. Based on the results obtained, empirical model was developed in order to predict attenuation due to rain. These models tries to predict the attenuation but it was not accurate in case of real situation. The recommendations ITU-R [4] rain attenuation model is the widely used method for the predictions of rain effects on satellite communications system. In order to predict the attenuations for particular frequencies the values of k and α coefficients for both horizontal and vertical polarization are required. In order to calculate the values of these coefficients complex methodologies are given in ITU-R recommendations [4]. Some simpler methodology is needed to be developed in order to find these values.

The proposed paper has contributed following points:

  • A simpler and improved model is developed in order to calculate values of K and α coefficients in order to calculate specific attenuation.

  • An algorithm is developed in order to calculate specific attenuation in dB/km caused due to rain.

  • A statistical analysis has been done in order to measure the performance of proposed model.

The present paper is organized in following manner. Second section discuss about the literature review. Section third is described about the proposed model. Fourth section discuss about the results and discussions followed by the last conclusion section.

2 Literature Review

In order to predict the attenuation caused due to rain, lot of studies has been performed. Details of these studies are shown in Table 1.

Table 1 Comparative analysis of literature review
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The recommendations ITU-R rain attenuation model is the widely used method for the predictions of rain effects on satellite communications system [4].

The attenuation is calculated for 99.99% fade depth by:

$$A_{0.01} = {\text{kR}}^{\alpha } dr dB$$
(1)

where R is given by 99.99% of rain rate for rain region in mm/h, kRα is the specific attenuation in dB/km and d is the link distance in km. the formula for r is given as:

$$r = 1/\left( {1 + d/d0} \right)$$
(2)

where

$$d0 = 35e^{ - 0.015R} {\text{km}}$$
(3)

where d0 is the effective path length and r is called the distance factor. K and α are the regression coefficient for frequencies and polarization [4].

In Fig. 1, a comparative study of different model like ITU-R, Gracia, RAL and Brazil were shown. In this implementation it is observed that brazil model shows more attenuation as compared with the ITU-R model. This is because of the different geographical conditions as brazil model was based on different experimental studies conducted in brazil.

Fig. 1
figure 1

Comparison of various rain models

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The Figs. 2, 3, 4 and 5 shows the value of k and α over frequency in GHz for horizontal and vertical polarization described in ITU-R rain model [4]. From the following figures it is clearly observed that the values of k and α are very uneven for frequencies 1 to 10 GHz.

Fig. 2
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Value of K coefficient for horizontal polarization

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Fig. 3
figure 3

Value of α coefficient for horizontal polarization

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Fig. 4
figure 4

Value of k coefficient for vertical polarization

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Fig. 5
figure 5

Value of α coefficient for vertical polarization

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In order to predict the attenuations for particular frequencies the values of coefficients are identified with the help of these graphs. Due to the unevenness of the curves, researchers find it very difficult to identify the values. Some methodology is needed to be developed in order to find these values.

Different mathematical, empirical and experimental studies have been performed in last decade. It has been concluding from the studies that rain affects significantly the deterioration of radio signals. Due to the complex nature of rain more thorough study of attenuation due to rain is required. The work done till now is based on mathematical analyses. This paper discusses about the machine learning approach. Before implementation of Machine learning it is very important to understand about Machine Learning.

Machine learning is a tool that transforms data into knowledge. Data has exploded over the past 50 years. This large amount of data is useless without analyzing and finding hidden patterns. Machine learning techniques are used to automatically detect important basic patterns in complex data. Late patterns and problematic information can be used to predict future events and make all kinds of difficult decisions.

In supervised learning, the goal is to learn the mapping (rules) between input and output sets. For example, an entrance could be a weather forecast and a production could be a beach visitor. The purpose of supervised learning is to study a map that explains the relationship between temperature and number of beach visitors.

For example, in the course of training, the model behaves by identifying data about previous input and output pairs, and “supervise” learning accordingly. In the beach example, new temperature calculation related data have been collected for long time and a machine learning algorithm used by collected data predicts temperature for future visitors.

Regression is another form of supervised learning. The difference between classification and regression is that regression outputs numbers rather than classes. Therefore, regression is useful for predicting numerical problems such as stock market prices, temperature on a specific date, or probability of events. For unsupervised learning, the examples provide input only. There are no obvious examples of results to find.

3 Proposed Model

The problem in ITU-R recommendations model is that availability of complex methodology for calculation of k and α coefficients. One of the solutions for this problem is finding out approximations. In order to get better results of approximation, machine learning algorithms come into play. When curve is drawn between the frequency and k-coefficient, it is not so smooth for getting approximation equation. One of the solutions of this problem is to break the curve into different parts and do approximation in order to get polynomial equations. The challenges are that to break down the curve in different sections. It is a very complicated task to make decision for finding the breaking point of the curve. In order to solve this problem, machine leaning algorithm have been incorporated for classification. Soft-Clustering algorithm is used to identify the breaking point. After classification different curves are drawn between K-coefficient versus frequency for each classification. Then in order to fit the curve, regression analysis techniques of machine learning have been incorporated in proposed model (Table 2).

Table 2 Clusters of frequency range from 1 to 100 GHz
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The first task is to draw the curve between K-coefficient versus frequency (GHz) then in order to get the breaking point, classification algorithms of Soft-Clusteringis performed. Soft-Clustering techniques breaks the curves in different parts based on the chanaging properties of the curve. The implementation is shown in Fig. 6. In this graph, curves breaks into seven clusters. Then in order to get approximation curve, regression analysis of machine learning algorithm is used. In that algorithm, training values are k-coefficient and frequency in GHz. Regression algorithm is applied multiple number of times in order to get fitted curve with minimum errors. From fitted curve different equation are obtained for K-values at different frequency range from Eqs. (4)–(8). Detailed description of obtained expression are as follows:

Fig. 6
figure 6

Cluster of K—Coefficients versus frequency

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From 1 to 100 GHz value of k for Vertical Polarization is given by:

$$K \, = \, ax^{3} + bx^{2} + cx + d$$
(4)

Values of coefficient a, b, c and d are taken from Table 3.

Table 3 Coefficient values for Eq. (4)
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From 1 to 100 GHz value of α for Vertical Polarization is given by:

$$\alpha \, = \, ax^{3} + bx^{2} + cx + d$$
(5)

Values of coefficient a, b, c and d are taken from Table 4.

Table 4 Coefficient values for Eq. (5)
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In the same way, model for horizontal polarization developed and related equations.

are as follows.

From 1 to 11 GHz value of k for horizontal polarization is given by:

$$k \, = \, ax^{4} + bx^{3} + cx^{2} + dx + e$$
(6)

From 12 to 100 GHz value of k for horizontal polarization is given by:

$$k \, = \, ax^{3} + bx^{2} + cx + d$$
(7)

Values of coefficient a, b, c, d and e are taken from Table 5.

Table 5 Coefficient values for Eq. (6) and (7)
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From 1 to 100 GHz value of α for horizontal polarization is given by:

$$\alpha \, = ax^{3} + bx^{2} + cx + d$$
(8)

Values of coefficient a, b, c and d are taken from Table 6.

Table 6 Coefficient values for Eq. (8)
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4 Algorithm for Proposed Model for Calculating specific attenuation due to rain in dB/km

figure e

5 Results and Discussions

The values of K-Coefficients and α-Coefficients for both horizontal and vertical polarization are described from Eqs. (4)–(8). Implementation of this model and comparative studies are described in Figs. 7 and 8 for both vertical and horizontal polarization. The implementation described the graph between frequencies in GHz and specific attenuation in dB/km. Different parameters used in the model are frequencies from 1 to 100 GHz, K and α Coefficients for both vertical and horizontal polarization calculated from Eqs. (4)–(8), rain rate in mm/h (Tables 7, 8, 9 and 10).

Fig. 7
figure 7

Comparative results of ITU-R and Proposed model for vertical polarization

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Table 7 Statistical analysis by rain rate for vertical polarization of proposed model
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Table 8 Statistical analysis by frequency for vertical polarization of proposed model
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Table 9 Statistical analysis by rain rate for horizontal polarization of proposed model
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Fig. 8
figure 8

Comparative results of ITU-R and Proposed model for Horizontal polarization

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Table 10 Error analysis for vertical polarization
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For frequency of 60 GHz, the error observed was 0.0595 which one is very low. For rain rate of 50 mm/h, error became 0.5131. Details are shown in Table 11. From Table 11, it has been observed that proposed model performs better then ITU-R model and are simpler to use and less complex in nature. It has also been observed that with the rise in frequency the attenuation is also increased for above 10 GHz frequency ranges.

The highly accurate approximation model was obtained due to the use of machine learning techniques. Two machine learning algorithms was used, one is Soft-Clustering algorithm and another is regression analysis. The clustering algorithm helps to classify the curves where exactly they change their property. In this case property was the change of flow of curves that is increase or decrease of curves. Once the continuous flow was identified then it will be easy to implement regression analysis in order to obtain polynomial equations (Fig. 9).

Fig. 9
figure 9

Comparative result of ITU-R and Proposed model at 100 mm

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Statically analysis of obtained model was performed and results are shown in Tables 7, 8, 9 and 10. The parameters used in order to analyses performance of proposed model are MAPE, MAD, MSE, and RMSE for rain rate of 10 mm/h, 20 mm/h, 30 mm/h, 50 mm/h, 100 mm/h, 150 mm/h and 200 mm/h. Same are done with frequency ranges from 10 to 100 GHz (Table 11).

Table 11 Error analysis of horizontal polarization
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The values of K-Coefficients and α-Coefficients for both horizontal and vertical polarization are described from Eq. (4)–(28). Implementation of this model and comparative studies are described in Figs. 7 and 8 for both vertical and horizontal polarization. The implementation descried the graph between frequencies in GHz and specific attenuation in dB/Km. Different parameters used in the model are frequencies from 1 to 100 GHz, K and α Coefficients for both vertical and horizontal polarization calculated from Eq. (4)–(8), rain rate in mm/h. average % error has been calculated for the implementation results obtained for proposed model and ITU-R model.

In case of higher frequencies from 80 to 100 GHz. at rain rate of 10 mm/h, 50 mm/h, 100 mm/h and 150 mm/h, proposed results are comparable with ITU-R model shown in Fig. 10.

Fig. 10
figure 10

Comparative result of ITU-R and proposed model (rain rate at a 10 mm/h, b 50 mm/h, c 100 mm/h and d 150 mm/h)

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6 Conclusions

Various researchers have proposed different models for rain attenuations but ITU-Rain model is widely acceptable model. In the ITU model different values of regression coefficients k and α are given for different frequencies. In this work approximations are calculated for those values in the form of approximation curves. From the simulation work it has been observed that approximation model closely matches with the original ITU-Rain model. Different equations are provided for different values. It helps researchers and engineers to identify coefficients for a given frequencies.