Performance analysis of satellite link using Gaussian mixture model under rain

The evolution of communication systems to the next generation, for example, B5G and 6G, demands an ultrareliable performance regardless of weather conditions. Such ultrareliable system design will require that the effects of adverse weather events on the communication system have to be computed more accurately so that physical layer compensation should be optimally and dynamically adaptive to such events. The performance of satellite links is severely affected by dynamic rain attenuation, and thus, accurate and reliable modeling of performance parameters is essential for dynamic fade countermeasures, especially above 10 GHz. In this work, we model the energy per bit to noise spectral density ratio ( Eb/N0 ) using Gaussian mixture (GM) model during rainy events. The developed mathematical expression is used to accurately model the average Eb/N0 , bit error rate (BER), outage probability, and ergodic channel capacity of the link. The average BER, upper bound on BER, and average ergodic capacity of an M‐ary phase shift keying scheme (MPSK) using the GM model of Eb/N0 are derived to evaluate the performance of the link under such weather impairments. We then show the numerical results and analysis using the GM model of the measured Eb/N0 data obtained with the AMoS‐7 satellite at a site located in Israel.


| INTRODUCTION
The future generation of satellites will include the link design above 10 GHz to support higher data rates.Such a shift to higher-frequency bands is needed to enhance the data rate over the communication networks. 1The B5G and 6G systems will be integrating the satellite and the terrestrial networks to form a heterogeneous communication network.As the B5G and 6G will be designed at mm-wave and sub-THz bands, several challenges are posed due to the physical aspects of the channel at such frequencies.At such high frequencies, the communication link suffers due to the various long-term and dynamic atmospheric channel effects. 2 Especially rain is one of the major causes of dynamic fading to the transmitted signal at such high frequencies.Due to increased attenuation over the atmospheric channel under rain, there is an increase in antenna temperature at the ground station in addition to the dynamic signal attenuation. 3,4Besides, the multipath and scintillation effects also show enhanced activity during rain. 5The scintillation fading shows stronger and more rapid variations under rain over clear weather conditions due to the wet scintillation effects and the intensity of scintillation fading increases depending on the rain attenuation.The signal-to-noise ratio (SNR) at the ground station will thus be changing rapidly under rain. 6][9][10] Thus, a more accurate and ultrareliable model for the satellite communication link is required such that a high quality of service and better link availability can be guaranteed under all weather conditions with more energy-efficient system designs. 11As the next-generation system design requires ultrareliable performance, it would be better to design compensation techniques adaptive to the short-term weather forecast, for example, hourly basis. 124][15] Hence, the FMT would require more accurate and reliable modeling of channel effects on the received signal at the ground station under the rain in order to guarantee a high quality of service with optimal usage of resources. 16 the rain intensity changes rapidly during a rainy event, the signal attenuation would vary strongly and dynamically.These events are marked by many such short-duration rains followed by no-rain events.Thus, SNR will show stronger and more rapid variations over a very short duration of time and hence will show many clusters around different mean SNR values.Therefore, the SNR time-series signal under rainy events can be statistically modeled as distributions centered around different mean values with different spreading (variances).Considering a received signal data with a duration, for example, short duration of 1 h, 24 h, or long term interspersed with many rainy and no rain events, it will be more appropriate to model the SNR distribution as a mixture of many Gaussian distributions.The received SNR could thus be modeled as a mixture of Gaussian models with different mean and variances that would provide a more accurate, reliable, and energy-efficient model for evaluating the link performance parameters under rapidly changing dynamics of atmospheric channels.
8][19] In Boulanger et al, 20 the authors proposed a rain attenuation time-series synthesizer with many rain and no-rain periods using the mixed Dirac-log-normal modeling of the absolute rain attenuation.An inverse Gaussian model is used to analyze the rain attenuation. 21A Weibull distribution-based rain attenuation time-series synthesizer is developed in Kanellopoulos et al. 22 Assuming a log-normal distribution of rain rate in Kupferman and Arnon 23 have derived the error probability for a satellite communication system.Arnau et al 24 have investigated the effects of spatial correlation of rain attenuation assuming a log-normal distribution and derived the outage capacity of the link at low and high SNR regimes.Kritikos et al 25 have assumed a log-normal distribution for the point rainfall rate and the slant path attenuation to derive the expression for signal-to-noise-plus-totalinterference ratio.However, the above analyses are based on the assumption that the long-term statistical model of attenuation follows the given probability distribution, which is univariate.On the other hand, the next-generation communication system design should adapt optimally to the dynamic weather conditions that impair the link performance. 12The univariate model is a special case of the multivariate model with a single component.
The multivariate model offers more flexibility and allows for a more comprehensive representation of the underlying distribution and dependencies in the SNR data when dealing with multiple dimensions or features.In the univariate case, the SNR data are represented by a single dimension or feature, whereas in the multivariate case, there are multiple dimensions or features.By using a multivariate model with multiple components, we can potentially capture more complex patterns and structures in the SNR data, which may not be fully captured by the univariate model.7][28][29] It allows for capturing correlations and dependencies between different dimensions or features of the SNR data.This can provide a more comprehensive understanding of the underlying structure in the SNR and potentially improve the modeling accuracy of the communication system.Moreover, mixture models can handle situations where the data points do not strictly belong to a single cluster or group.Instead, they can have a probabilistic assignment to multiple clusters, reflecting the uncertainty in the data.This flexibility is particularly valuable when dealing with complex and dynamic scenarios such as rapidly changing atmospheric channel conditions.This will ensure an ultrareliable and more energyefficient link design under dynamic weather impairments for the adaptive link design. 30 this work, we first develop the expression of the probability density function (PDF) of E b =N 0 using a Gaussian mixture (GM) model and derive the outage probability of the communication link based on the GM model of E b =N 0 (dB) under rain.We then consider an M-ary phase shift keying scheme (MPSK) signaling scheme that is used for the satellite link under consideration to derive the average bit error rate (BER) performance, its upper bound, and average channel capacity to evaluate the link performance under rainy events.We then compute the link performance parameters, that is, outage probability, average BER, upper bound on BER, and average channel capacity utilizing the developed expressions with the measured data obtained at a site in Israel.The parameters of the multivariate Gaussian or GM model are computed using the expectation-maximization (EM) algorithm.The comparison of results is done with the univariate Gaussian model of E b =N 0 (dB).The analysis has been done for short-term duration of 1 h, 24 h, and long term to show the broad applicability and generalizability of the GM model.The satellite under consideration is AMoS-7 satellite which uses an 8-PSK signaling scheme.
The article is organized as follows: In Section 2, we develop the GM model of SNR of a satellite link, and the expression for the outage probability of the link is derived.Using the developed expression of the GM model of SNR, the expression of average BER, upper bound on BER, and average channel capacity have been derived.Section 3 presents the numerical results obtained with the measured SNR data with AMoS-7 satellite using an 8-PSK signaling scheme, and a comparison of results with univariate Gaussian model is shown.Section 4 concludes the paper.
In this section, we model the normalized SNR received at the ground station using the GM model.Based on the GM model of SNR, we develop the mathematical expressions to compute the average BER, its upper bound, and average ergodic channel capacity under rainy events.

| GMM modeling of SNR
The GM model is a soft clustering method, unlike the K-means algorithm which applies hard clustering to the data distribution.The multivariate model is composed of several Gaussian distribution (i ¼ 1,2,…, N) each defined by its mean (μ , and a mixing probabilities (π i ¼ π 1 , π 2 , …, π N ).The mixing probabilities would satisfy the following condition Given the data points x, the PDF for a GM of models can be expressed as 31 where we use random variable x to represent the normalized signal-to-noise ratio (dB) or energy per bit to noise spectral density ratio in dB denoted as E b =N 0 (dB).
The average of the random variable E b =N 0 will then be expressed as In order to evaluate the parameters of the GM model Θ ¼ fμ, σ, πg, an iterative algorithm, namely, expectation-maximization (EM) is used.The EM algorithm is based on the maximization (M-step) of expectation of log-likelihood (E-step) and is discussed elaborately in previous studies. 32,33e outage probability of the satellite link using the GM model of PDF of E b =N 0 (dB), that is, probability that E b =N 0 falls below a threshold value γ (dB) can now be written as where erf is the error function given by erfðxÞ

| BER and channel capacity
The quadrature amplitude modulation (QAM) signaling scheme is more widely used in modern communication systems.Though, phase shift keying (PSK) would be preferred especially under adverse channel conditions and for power-efficient communication.5][36][37] In our case, the MPSK signaling is chosen, as the transmitting satellite with which we have obtained the data uses the 8-PSK modulation.The expressions developed in this study can be extended to other modulation schemes including QAM and frequency shift keying (FSK).
The BER of a satellite link is modeled with the GM model of random variable E b =N 0 (dB).We use M-PSK signaling scheme for the computation of the BER at the receiver as the signaling scheme used by the satellite in consideration, that is, AMoS-7 is PSK.The average probability of symbol error for MPSK can be expressed as where M is the number of symbols and Qð:Þ denotes the Q-function that is written in terms of error function as For the case of equiprobable symbol error and number of symbols given as M ¼ 2 K with K denoting the number of binary symbols, we can express the BER as 38 where x denotes the random variable E b =N 0 (dB).
As the x ¼ E b =N 0 (dB) is a dynamically and strongly varying random variable during a rainy event, we can find the average and worst-case scenario performance of the link.To evaluate the system performance under the worst-case scenario, we find the upper bounds on BER as a function of the random variable E b =N 0 .For the MPSK modulation scheme, we can write the upper bounds 38 on the BER as follows: The expression for average BER can be now computed using ( 2) and ( 7) as The average BER in (9) has to be evaluated numerically once the parameters of GM model Θ ¼ fμ, σ, πg of normalized SNR are found using the EM algorithm.Additionally, it would be better to evaluate the link average performance under the worst-case scenario; we could find the average of the worst-case BER, that is, the upper bound BER as expressed in (8).We can compute the average performance of the upper bound of BER using ( 2) and (8) given by The above expression for average BER performance in (10) under the worst-case scenario can be evaluated numerically given the parameters of the GM model Θ ¼ fμ, σ, πg.
Another parameter to evaluate the quality of the fading channel is the ergodic channel capacity.The ergodic capacity of a fading channel is defined as the maximum rate at which the information can be transmitted over the channel with negligible error probability.The ergodic capacity (bps/Hz) will be expressed as 39,40 where B is the bandwidth and R b is the data rate.For an MPSK signaling scheme, the bandwidth efficiency (R b =B) is given by R b B ¼ 1 2 log 2 M. The average channel capacity will now be expressed using the GM model of For larger values of E b =N 0 > > 1, the above expression of average ergodic capacity would be approximated as Now, we can manipulate above expression to write the average capacity for the case of

| NUMERICAL RESULTS AND DISCUSSION
In this section, we first provide the measurement data and its preprocessing.

| Measurement data
The symbol energy to noise spectral density E S =N 0 (dB) measurement is done at Petah Tikva in Israel.The link under consideration is a feeder link.
The diameter of the receiver antenna is 1.2 m.The low noise block (LNB) converter has a gain of 60 dB, an input level of À2 to 8 dBm at the reference of 10 MHz, and a noise figure of 0.8 dB at 25 C.The sampling time is 240 ms, and the E S =N 0 data are processed by a median filter of order 5 to remove the transients.The modulation scheme is 8-PSK with the AMoS-7 satellite at 3.9 W and a downlink frequency of 11.113 GHz.From the E S =N 0 (dB) data, we obtain the bit energy to noise spectral density ratio as E b =N 0 ðdBÞ ¼ E S =N 0 ðdBÞ À 10log 10 ðlog 2 MÞ with M ¼ 8.
The data have been collected from Dec 2019 to May 2020.From the obtained signal time series, we divide the data into signal segments with a period of 1 and 24 h as required for the analysis to be done in Section 3.3.We analyze the data when there are occurrences of rain on a particular day in the data to model it as a mixture of the Gaussian model for the period of time duration in consideration.There has been a total of 30 such signal segments having a duration of 24 h marked with many rainy and no rain events.The plot of time-series signal F I G U R E 2 Bayesian information criterion (BIC) plot as the number of components is varied to find the optimum Gaussian mixture (GM) model for the signal with a duration of 24 h.
F I G U R E 3 Time-series E b =N 0 signal recorded on January 8, 2020 on a rainy day with a duration of 24 h.
E b =N 0 ðdBÞ is shown in Figure 1A on January 4, 2020 with various rainy and no rain events for a time duration of 24 h.To illustrate our point that it would be more appropriate to model the E b =N 0 ðdBÞ as a mixture of Gaussian rather than a univariate Gaussian, we show the histogram in Figure 1B.

| Methodology
The number of components in the GM model is determined using the Bayesian information criterion (BIC). 41The BIC penalizes models with more parameters, thereby encouraging model selection that avoids overfitting.It strikes a balance between goodness of fit (as reflected in the log-likelihood) and model complexity (as reflected in the number of parameters).In the context of GM models, the log-likelihood has been computed using rain, where there may be additional noise and signal attenuation variations, the optimal number of components in the GM model may be different compared with signals obtained under different conditions.The BIC criteria are defined as follows: where N is the number of parameters in the GM model, which is calculated based on the number of mixture components and the dimensionality of the data, p denotes the number of observations in the data points, and L is the maximized likelihood function.
In order to illustrate this point, we consider the data with the span of 24 h (as shown in Figure 2) and show its BIC and the corresponding number of components.In Figure 3, we show the BIC plot as the number of components are changed.The GM model having the lowest BIC is then automatically selected as the optimum model.A regularization parameter of 0.001 is used to avoid the ill-conditioned covariance matrix.In this case, the number of components for the optimum GM model is found to be 14.The values of the parameters of these 14 clusters include the following: mean μ i are   In Figure 4, we show the plot of average E b =N 0 for each of the signals in consideration using the GM model as shown in (3) with the mixture components Θ ¼ fμ, σ, πg evaluated using EM algorithm and the number of components as determined by BIC for each of these signal segments.

F I G U R E 8
The number of components in the Gaussian mixture (GM) model for the signals in consideration with a duration of 1 h.
T A B L E 1 GM model parameters.We also compare the results with univariate Gaussian modeling of average E b =N 0 .It can be observed that the average E b =N 0 (dB) shows similar values except for some signal samples indexed as 3, 8, and 10.
In Figure 5, we show the plot of outage probability for the signals each having a duration of 1 h (indexed as 1 to 24) using the GM model and univariate model.The threshold value is taken to be 10 dB.It is observed that models show quite a difference in the values of outage probability obtained by the two models.For the signal indexed as 4, 18, 19, 21, 22, 23, and 24, the GM model shows zero outage probability while the univariate Gaussian model overestimates.
In Figure 6, we show the plots of average BER and average upper bound on the BER performance for each of the signals in consideration using the GM model and univariate Gaussian.The average BER with a univariate model for many signals in consideration underestimates the performance parameters compared with the GM model.This is also true for the worst-case performance, that is, average upper bound on BER.We observe that for the univariate Gaussian, the average and the worst-case performance show significant differences with the GM model.Also, the average BER and its upper bounds show similar trends for the two models.Though for many signal samples, both the models show similar average BER and its upper bound.However, there would be significant underestimation of these parameters by univariate model.
F I G U R E 9 Outage probability evaluated using Gaussian mixture (GM) and univariate Gaussian models shown along with the actual outage probability.
F I G U R E 1 0 Average E b =N 0 for the signals in consideration with a duration of 24 h using (GM) and univariate Gaussian model.
Figure 7 shows the plot of average channel capacity using the multivariate and univariate Gaussian models.In most of the cases, both models provide similar average capacity.Though, for some signals, their values differ as shown for the signal indices 3 and 8.
In Figure 8, the plots of the number of components for the GM model are shown for the signals.The number of components have been obtained by using the BIC criterion and that shown in the plots corresponds to the lowest BIC obtained.The parameters for these signals have been shown in In Figure 9, we show the outage probability as a function of threshold SNR for the GM and the univariate model.The actual outage probability using the measured E b =N 0 data has also been shown.It is observed that the GM model will more closely approximate the actual outage probability thereby proving to be a better model.

F I G U R E 1 1
Outage probability of the signals in consideration with a duration of 24 h using Gaussian mixture (GM) and univariate Gaussian model.
F I G U R E 1 2 Bit error rate (BER) performance for the signals in consideration with a duration of 24 h using Gaussian mixture (GM) and univariate Gaussian model.

| Analysis on 24 h basis
Now, we show the performance analysis for the signal with a duration of 24 h.In Figure 10, we show the plot of outage probability using the mixture of Gaussian and univariate Gaussian models.We can observe that both the models give approximately similar average E b =N 0 on 24 h basis.
The plot of outage probability is shown in Figure 11.Notably, the evaluated value of outage probability using the GM model shows quite a difference from the values obtained with the univariate model.The plots of outage probability are provided assuming that the threshold value γ Therefore, it is crucial to prioritize the analysis of the link using the GM model for optimal design and compensation.By leveraging the GM model, more accurate estimations of the outage probability can be achieved, enabling better resource allocation and facilitating informed decisions regarding adaptive coding and modulation strategies.This approach ensures the link's optimal performance and minimizes the risk of improper resource allocation.
F I G U R E 1 3 Average channel capacity (bps/Hz) for the signals in consideration with a duration of 24 h using Gaussian mixture (GM) and univariate Gaussian model.
The number of components in the Gaussian mixture (GM) model for the signals in consideration with a duration of 24 h.
In Figure 12, the plots of average BER and average upper bound on the BER performance are shown for the 30 rainy days without any forward error correction codes.It is evident that the average BER calculated using the univariate model often underestimates the performance parameters compared to the GM model.This underestimation is also observed in the worst-case scenario, where the average upper bound on BER is considered.Significant differences between the univariate Gaussian model and the GM model can be observed in terms of both the average and worst-case performance.However, the average BER and its upper bounds generally exhibit similar trends when comparing the two models.Although there are instances where the average BER and its upper bounds align closely between the two models, it should be noted that the univariate models tend to severely underestimate these parameters on other days.Overall, the results suggest that the GM model provides more accurate estimations of the average BER and its upper bounds, while the univariate models may lead to serious underestimation of these parameters on certain days.
In Figure 13, we show the average channel capacity using the multivariate and univariate Gaussian models.The channel capacity using both models shows almost similar values as seen in Figure 13.The number of components in the GM model using the BIC criterion is shown in Figure 14.The parameters for the signal are presented in Table 2, utilizing the EM algorithm.For ease of reference, values of the parameters are displayed only for the first 10 signals.In the case of the univariate model, the obtained means are as follows: In Figure 15, we show the outage probability as the threshold E b =N 0 is varied for the GM model and univariate Gaussian model.The actual outage probability is also shown in the figure.It can be observed that the univariate Gaussian significantly underestimates the outage probability.
As the threshold normalized SNR γ (dB) goes lower, more is the inaccurate estimation done by the univariate model.For example, for the outage probability of 10 À5 , the difference between the threshold SNR between the two models is about 2.6 dB and thus univariate model will not result in an energy-efficient design of the communication system.It is also notable that the GM model follows the actual probability very closely.A univariate model will provide inaccurate results for a system designed to communicate at lower values of SNR which usually happens during adverse weather events.

| Long-term analysis
The performance analysis for the signal with long-term duration is shown in this section.The total duration of the signal under analysis is approximately 720 h by combining the data obtained on 30 rainy days each having a duration of 24 h.The number of components obtained using the BIC criterion in the GM model turned out to be 15.The corresponding values of the parameters include the weights π i given by 0. Gaussian model are 10.9616 and 1.3560, respectively.In Table 3, the performance parameters have been compared for the two models.The F I G U R E 1 5 Outage probability evaluated using Gaussian mixture (GM) and univariate Gaussian models shown along with the actual outage probability.
T A B L E 3 Performance parameters.threshold SNR for the calculation of outage probability is taken to be 10 dB.Significant differences in the performance parameters can be noted in the Table, especially for the outage probability, average BER, and average upper bound on BER.
In Figure 16, the plots of outage probability are shown using the GM and univariate Gaussian model.The actual outage probability using the measured E b =N 0 data is also shown in the plot.Again, we note the point that the GM model can provide more accurate measure of the outage probability.
Thus, the mixture of Gaussian models would be a preferred for the estimation of performance parameters of the satellite link under rainy conditions.Such a model will thus help in designing the compensation schemes to provide an ultrareliable and more accurate communication system, especially under adverse weather events with the optimal use of resources.To show its applicability, we have also shown the results and analysis at a higher frequency in Ku-Bands in Appendix A. This will prove useful model design as the satellite communication is moving to higherfrequency bands due to congestion at lower bands.

| CONCLUSION
In this work, we have used mixture of Gaussian to model the SNR at the ground station receiver under rain.The expression for outage probability, BER performance, and average ergodic capacity with a MPSK scheme is developed to evaluate the link performance using the GM model.The parameters of the mixture and the number of components are evaluated using the EM algorithm and BIC criterion, respectively, for the received signal marked by various rainy and nonrainy events.The model will provide a more accurate and reliable computation of outage probability, average BER, upper bound on the BER, and average channel capacity performance of the satellite link under rain.Experimental results using the SNR data obtained with AMoS-7 satellite using an 8 PSK signaling scheme are presented to compute the performance parameters of the link using the proposed GM model.The study done in this work can be extended to other modulation schemes, for example, QAM.
As we move to higher-frequency bands for the design of future generation of satellite communication systems, the other factors that affect the signal fading are clouds and fog besides rain.The rain including convective rain events, clouds, and fog will be major causes of random and rapid deep fading at higher frequencies.Under such a scenario, it will be extremely challenging to ensure an ultrareliable link with a highly energyefficient system design.To ensure this, it would be extremely useful to model the short-term channel effects due to rain, fog, and clouds as a mixture of Gaussian models for more accurate and ultrareliable communication link design.
The analysis of E b =N 0 using the GM model can provide group structure in the underlying data.By identifying different groups or clusters within the data, adaptive coding and modulation schemes can be optimized accordingly.The identified groups can indicate variations in channel quality, such as different levels of attenuation.The code rate, modulation symbol size, transmitter power, and beamforming can then be optimally adapted depending on such group structure in the data and according to the specifics of each group.Such a group-wise mitigation strategy will ensure that resources are efficiently allocated based on the varying channel conditions, resulting in improved overall system performance and energy efficiency.
F I G U R E 1 6 Outage probability evaluated using Gaussian mixture (GM) and univariate Gaussian models shown along with the actual outage probability.
assume that the value of E b =N 0 (dB) under clear weather is 25 dB.Using the excess rain attenuation after scaling, we thus compute the E b =N 0 (dB) time-series data.The E b =N 0 (dB) time-series data are then modeled using the GM model, and the parameters of the model are evaluated using the EM algorithm.The number of components is found using the BIC criterion.
The value of parameters using the univariate Gaussian model is given as follows: mean of 12.3585 and a standard deviation of 4.9123.The number of components for the GM model is found to be 12.The parameters of the GM model are as follows: weights given by 0. In Figure A1, we show the outage probability versus the threshold SNR plot using the GM model and univariate model along with the actual values.We can observe the significant differences in the outage probability as the threshold SNR is varied for both models.It can be noted that for the low outage probability, the univariate model gives an underestimated value of outage probability for the same threshold SNR that will lead to a nonefficient fade countermeasure.In Table A1, we show the respective values of performance parameters using the GM and univariate model.The threshold SNR for calculating the outage probability is taken to be the same as previously, that is, 10 dB.Significant differences in the link performance parameters can be noted.
Following this, we provide the criteria used to find the number of components in the multivariate model.Thereafter, we analyze the satellite link using the developed expressions of link performance parameters with the GM model and compare the results with a univariate Gaussian model of E b =N 0 (dB).The analysis has been done for the short term and long term as well.A univariate Gaussian model is the most common way to model the normalized SNR under rainy events.F I G U R E 1 E b =N 0 ðdBÞ data recorded on January 4, 2020 for the signal with a duration of 24 h interspersed with many rain and no-rain events.(A) Time-series plot.(B) Histogram plot.
the EM algorithm.The EM algorithm estimates the parameters of the model (e.g., mean, variances, and mixture weights) that maximize the likelihood of the observed data.By comparing the BIC values of different GM models trained with varying numbers of components, we have selected the model with the lowest BIC as the best-fitting model for the given data.The number of components would vary depending on the signal variation and its clustering around a mean value and will be different for different signals obtained under rain.In the case of signals obtained under F I G U R E 4 Average E b =N 0 for the signals in consideration with a duration of 1 h using Gaussian mixture (GM) and univariate Gaussian model.F I G U R E 5 Outage probability of the signals in consideration with a duration of 1 h using Gaussian mixture (GM) and univariate Gaussian model.

F I G U R E 6
Bit error rate (BER) performance for the signals in consideration with a duration of 1 h using Gaussian mixture (GM) and univariate Gaussian model.F I G U R E 7 Average channel capacity (bps/Hz) for the signals in consideration with a duration of 1 h using Gaussian mixture (GM) and univariate Gaussian model.

3. 3 |
Performance analysis3.3.1 | Short-term analysis: 1 hFirst, we analyze the performance of the GM model on a short-term duration that is, hourly basis.The comparison of the GM model has been made with the univariate Gaussian model which is most commonly used.The time-series E b =N 0 signal for which the hourly analysis will be done is shown in Figure3.The data have been divided into the duration of 1 h, and we have indexed these signal samples as numbered from 1 to 24.
of 10 dB without any coding scheme.Except on days like February 7, February 18, February 19, February 29, March 5, March 12, March 13, March 20, and April 4, the predicted outage probability with multivariate Gaussian shows quite a difference with the univariate model.The univariate model has the tendency to either overestimate or underestimate the performance indicator.Such inaccurate estimations often result in suboptimal utilization of resources when employing adaptive coding and modulation techniques for the physical layer to compensate for the link.

F I G U R E A 1
Outage probability evaluated using Gaussian mixture (GM) model and univariate Gaussian model for the E b =N 0 data obtained by scaling of rain attenuation at a higher frequency of 26 GHz shown along with the actual outage probability.T A B L E A 1 Performance parameters.