# Capacity Analysis of Hybrid Satellite–Terrestrial Systems with Selection Relaying

^{*}

## Abstract

**:**

## 1. Introduction

#### 1.1. Related Work

#### 1.2. Contribution and Organization

- We propose a model of the satellite–terrestrial communication system, where the transmitted signal at the relay depends on the result of the comparison of the received SNR with the arbitrary local threshold;
- In this paper, we derive the novel analytical expressions for the outage probability for the arbitrary threshold at the relay, which can be different from the threshold at the destination, extending our previous results published in [34], which were based on more general probability density functions. Additionally, we derive novel expressions for the outage capacity and ergodic capacity, providing the corresponding numerical results;
- We derive the analytical expressions for the probability density function of the received SNR at the destination for the given threshold at the relay;
- We derive the analytical expressions for the outage capacity of the system, which is a relevant performance measure for applications with delay constraints;
- We derive the relevant analytical expressions for the ergodic capacity of the analyzed system for applications with no delay requirements;
- All analytical expressions derived in the paper are given in polynomial–exponential form, and they are valid for the general case of the shadowed Rice fading environment with the integer-valued fading parameter at the satellite–terrestrial links and Nakagami-m fading environment at the terrestrial link;
- The derived expressions are general and applicable to a transmission system with arbitrary system parameters;
- We propose a novel method for generating the time series that corresponds to the time-varying channel gains in satellite–terrestrial links as an improved version of the simulation method that we proposed in [35]. In this paper, we use an improved simulation method that includes the terrestrial component, deriving the corresponding temporal autocorrelation function of the complex channel gain in the satellite–terrestrial link;
- Analytical results are confirmed using an independent Monte Carlo simulation method, and the corresponding numerical results are presented for the various propagation scenarios and typical parameters of the active LEO satellite systems.

## 2. System and Channel Model

- In the first half of the signalization interval, the source sends a signal to the receivers at the destination and at the relay. The instantaneous received SNR in the S-D link in an arbitrary time instant t is denoted by γ
_{1}(t), and the instantaneous received SNR in the S-R link is denoted by γ_{2}(t). - After the first half of the signalization interval, the relay performs decoding. If the decoding is successful, in the second half of the signalization interval, the receiver at the destination receives the signal from the relay, with the instantaneously received SNR in the R-D link being denoted by γ
_{3}(t). Otherwise, the relay is silent (it does not send any signal). The decoding at the relay is usually considered successful in the time instant t if the received SNR is larger than the predefined threshold, denoted by γ_{th,R}[14]. The value of the threshold depends on the applied modulation and coding scheme at the corresponding communication link, as well as the receiver sensitivity. - The signals received in the first and the second half of the signalization interval are combined. In this paper, we assume that MRC is applied at the destination receiver. Therefore, the SNR at the output of the MRC combiner is obtained as the sum of the SNRs in two intervals [33] as follows:

_{i}(t) denotes the time-varying complex channel gain in the i-th communication channel. As defined before, the channel gain between the satellite and the destination (mobile user) is the first one (i = 1), the channel gain between the satellite and the relay is the second one (i = 2), and the channel gain between the relay and the destination is the third one (i = 3).

_{S}, the average transmitted power at the relay is denoted by P

_{R}, d

_{SD}denotes the distance between the satellite and the destination, d

_{SR}denotes the distance between the satellite and the relay, and d

_{RD}denotes the distance between the relay and the destination. The corresponding path loss factors are denoted by n

_{SD}, n

_{SR}, and n

_{RD}. We assume that the noise power values at the destination and at the relay are equal and are denoted by σ

^{2}.

_{R}= θ

_{D}= θ, and the corresponding distances are approximately the same, i.e., d

_{SD}≈ d

_{SR}= H/sin(θ).

- A.
- Satellite–terrestrial channel

_{i}(t). In the available literature, various channel models were developed to describe the time-varying complex channel gain in a narrowband land mobile satellite channel. It is usually considered that the fluctuations are the result of many weak scattered components (multipath fading) and the random variations of the total power of the multipath components (shadowing) [37].

_{i}(t) of the scattering component is uniformly distributed, while the corresponding amplitude a

_{i}(t) exhibits Rayleigh distribution as follows:

_{0,i}denotes the average power of the scattering component.

_{0,i}and Nakagami-m distributed envelope z

_{i}(t) as follows:

_{i}denotes the average power of the LOS component in i-th channel and m

_{i}denotes the corresponding Nakagami fading parameter.

_{i}(t) = |h

_{i}(t)|

^{2}, can be provided using the following [12]:

_{1}F

_{1}( ; · ; ) denotes the confluent hypergeometric function of the first kind.

_{1}and m

_{2}, the hypergeometric function can be represented in the form of the finite summation of polynomial–exponential terms [28], resulting in a simplified PDF expression as follows:

_{k}= t(t + 1)···(t + k − 1) denotes the Pochhammer symbol, and the following substitutions are to be applied:

_{i}(t) (i = 1, 2) are dominantly determined using the ACF of the scattering component.

_{0,i}can be successfully described using the autocorrelation function as follows:

_{D}

_{m}denotes the maximum Doppler frequency. The above expression was given in the book [41], where the second-order statistics for the isotropic scattering are presented in detail. The second-order statistics for the diversity systems were described in [42,43] for the case of Rayleigh distribution, and in [44] for the case of Nakagami-m distribution (both distributions will be used in our simulation model).

- B.
- Channel model in terrestrial links

_{3}(t) = |h

_{3}(t)| can be modeled using a random variable with a Gamma distribution as follows:

_{3}E{λ

_{3}} denotes is the average power gain, m

_{3}denotes the corresponding fading parameter, and Γ(·) denotes the Gamma function.

_{S}

^{2}) are of a certain value, then the Nakagami-m parameters m

_{3}and ${\mathsf{\Omega}}_{3}$ can be obtained as ${m}_{3}=(1+{K}^{2})/(2K+1)$ and ${{\sigma}_{S}}^{2}={\mathsf{\Omega}}_{3}\left(1-\sqrt{1-1/{m}_{3}}\right)/2$ [40].

_{2}F

_{1}(·, ·; ·; ·) denotes the hypergeometric function [46] and J

_{0}(.) denotes the Bessel function of the first kind and the zeroth order. In [47], the authors argued that the rank statistics of the Nakagami-m fading envelope are approximately the same for various values of the fading parameter m

_{3}. As the rank autocorrelation coefficient is known for Rayleigh fading (m

_{3}= 1), we will assume that the same expression is valid for higher values of m

_{3}, thus determining the second-order statistics in the terrestrial link.

## 3. Outage Capacity

_{out}(γ

_{th}).

_{th,R}. However, such an analysis could not provide complete information about the impact of the S-R link. By using the expression for the instantaneous SNR at the output of the MRC combiner, denoted by γ(t), the above expression can be rewritten in a more compact form as follows:

- In the case when γ
_{th,R}→∞, we obtain I_{2}→1, I_{3}→0, and the resulting outage probability corresponds to the case without relaying (when only S-D link is present).

- 2.
- In the case when γ
_{th,R}→0 (the reliable S-R link), the instantaneous SNR is obtained as $\mathsf{\gamma}(t)={\mathsf{\gamma}}_{1}(t)+{\mathsf{\gamma}}_{3}(t)$, we obtain I_{2}→0, I_{3}→1, and the resulting outage probability is determined with the expression of I_{4}, i.e.,$$\begin{array}{c}{P}_{out}({\mathsf{\gamma}}_{th})\approx {\displaystyle \underset{0}{\overset{{\mathsf{\gamma}}_{\mathrm{t}\mathrm{h}}}{\int}}{f}_{{\mathsf{\Gamma}}_{1}}({\mathsf{\gamma}}_{1}){\displaystyle \underset{0}{\overset{{\mathsf{\gamma}}_{\mathrm{t}\mathrm{h}}-{\mathsf{\gamma}}_{1}}{\int}}{f}_{{\mathsf{\Gamma}}_{3}}({\mathsf{\gamma}}_{3})d{\mathsf{\gamma}}_{3}}d{\mathsf{\gamma}}_{1}}=1-{e}^{-{K}_{21}{\mathsf{\gamma}}_{th}}{\displaystyle \sum _{k=0}^{{m}_{1}-1}{\displaystyle \sum _{j=0}^{k}{K}_{41}(k,j){\mathsf{\gamma}}_{th}^{j}}}\hfill \\ \hfill +{\displaystyle \sum _{k=0}^{{m}_{1}-1}{\displaystyle \sum _{p=0}^{{m}_{3}-1}{\displaystyle \sum _{q=0}^{p}{K}_{42}(k,p,q)}{\mathsf{\gamma}}_{th}^{p-q}}}{e}^{-C{\mathsf{\gamma}}_{th}}-{\displaystyle \sum _{k=0}^{{m}_{1}-1}{\displaystyle \sum _{p=0}^{{m}_{3}-1}{\displaystyle \sum _{q=0}^{p}{\displaystyle \sum _{j=0}^{k+q}{K}_{43}(k,p,q,j)}}{\mathsf{\gamma}}_{th}^{p-q+j}}}{e}^{-{K}_{21}{\mathsf{\gamma}}_{th}}.\end{array}$$

- In the case when powerful error correction codes are applied in all communication links, the above expression is a good approximation in the region of very small values for γ
_{th}.

- 3.
- In the case when γ
_{th,R}= γ_{th}, the derived outage probability reduced to the expression derived in our conference paper [34].

- 1.
- The analysis for the fixed relaying DF protocol, where it is assumed that the relay retransmits signal even in the case when the instantaneous SNR at the relay is below the threshold, i.e., the outage appears at the destination whenever ${\mathsf{\gamma}}_{2}(t)<{\mathsf{\gamma}}_{\mathrm{th},\mathrm{R}}$, i.e.,$${P}_{out,FR}({\mathsf{\gamma}}_{th}\left|{\mathsf{\gamma}}_{th,R}\right.)={\displaystyle \underset{0}{\overset{{\mathsf{\gamma}}_{\mathrm{t}\mathrm{h},\mathrm{R}}}{\int}}{f}_{{\mathsf{\Gamma}}_{2}}({\mathsf{\gamma}}_{2})d{\mathsf{\gamma}}_{2}}+{\displaystyle \underset{{\mathsf{\gamma}}_{\mathrm{t}\mathrm{h},\mathrm{R}}}{\overset{\infty}{\int}}{f}_{{\mathsf{\Gamma}}_{2}}({\mathsf{\gamma}}_{2})d{\mathsf{\gamma}}_{2}}{\displaystyle \underset{0}{\overset{{\mathsf{\gamma}}_{\mathrm{t}\mathrm{h}}}{\int}}{f}_{{\mathsf{\Gamma}}_{1}}({\mathsf{\gamma}}_{1}){\displaystyle \underset{0}{\overset{{\mathsf{\gamma}}_{\mathrm{t}\mathrm{h}}-{\mathsf{\gamma}}_{1}}{\int}}{f}_{{\mathsf{\Gamma}}_{3}}({\mathsf{\gamma}}_{3})d{\mathsf{\gamma}}_{3}}d{\mathsf{\gamma}}_{1}}.$$

- Finally, the closed form expression is easily obtained from Equation (32) if the first and third summation (the second and the fourth term) are ignored. This solution represents the closed form solution for the analysis given in [33].

- 2.
- The analysis for the simple DF protocol, where it is assumed that the S-D link is blocked [21], can be obtained if we set ${\mathsf{\gamma}}_{1}(t)$ = 0, and therefore$$\mathsf{\gamma}(t)=\left\{\begin{array}{cc}0,\hfill & {\mathsf{\gamma}}_{2}(t)<{\mathsf{\gamma}}_{\mathrm{th},\mathrm{R}},\hfill \\ {\mathsf{\gamma}}_{3}(t),\hfill & {\mathsf{\gamma}}_{2}(t)\ge {\mathsf{\gamma}}_{\mathrm{th},\mathrm{R}}.\hfill \end{array}\right.$$

- The outage probability is derived through setting ${{f}_{\mathsf{\Gamma}}}_{1}({\mathsf{\gamma}}_{1})=\mathsf{\delta}({\mathsf{\gamma}}_{1}=0)$, where $\mathsf{\delta}(\cdot )$ denotes the delta function, and it can be determined using the simplified expression as follows:$${P}_{out,DF}({\mathsf{\gamma}}_{th}\left|{\mathsf{\gamma}}_{th,R}\right.)={\displaystyle \underset{0}{\overset{{\mathsf{\gamma}}_{\mathrm{t}\mathrm{h},\mathrm{R}}}{\int}}{f}_{{\mathsf{\Gamma}}_{2}}({\mathsf{\gamma}}_{2})d{\mathsf{\gamma}}_{2}}+{\displaystyle \underset{{\mathsf{\gamma}}_{\mathrm{t}\mathrm{h},\mathrm{R}}}{\overset{\infty}{\int}}{f}_{{\mathsf{\Gamma}}_{2}}({\mathsf{\gamma}}_{2})d{\mathsf{\gamma}}_{2}}{\displaystyle \underset{0}{\overset{{\mathsf{\gamma}}_{\mathrm{t}\mathrm{h}}}{\int}}{f}_{{\mathsf{\Gamma}}_{3}}({\mathsf{\gamma}}_{3})d{\mathsf{\gamma}}_{3}}d{\mathsf{\gamma}}_{1}.$$

## 4. Ergodic Capacity

_{th,R}→∞, only the first term remains, and this corresponds to the system with the S-D link only. If γ

_{th,R}→0, the PDF can be simplified to the expression that can be easily derived by combining (34) and (41).

## 5. Numerical Results

_{1}[dB] = EIRP [dB] − n

_{SD}× 10log10(4πd

_{SD}f

_{0}/c) − L

_{A}[dB] + G [dB] − 10log10(kT

_{S}B),

_{A}denotes atmospheric losses due to oxygen and water as well as other losses (polarization mismatch, antenna misalignment), and G denotes the antenna gain at the receiver. The second term at the right side of the above equation corresponds to free-space path loss (FSPL), where d

_{SD}denotes the distance between the satellite and receiver at the destination, n

_{SD}denotes the corresponding path loss factor, f

_{0}denotes the carrier frequency at the satellite–terrestrial link, and c = 3 × 10

^{8}m/s denotes the speed of light. The last term in (50) corresponds to the noise level, where k = 1.38 × 10

^{−23}J/K denotes the Boltzmann constant, T

_{S}denotes the temperature of the system, and B denotes the channel bandwidth.

_{1}[dB] = P

_{S}[dB] − n

_{SD}× 10log

_{10}(d

_{SD}) − 10log10(σ

^{2}),

_{S}takes into account the same power-related parameters for all system users:

_{S}[dB] = EIRP [dB] − n

_{SD}× 10log

_{10}(4πf

_{0}/c) − L

_{A}[dB] + G [dB],

_{A}, f

_{0}, G, T

_{S}, and B for active LEO satellite systems can be found in [5,11]. Due to the relatively small diameter of the beam (L << H), it can be assumed that the elevation angles at both the destination and relay satisfy the condition θ

_{R}= θ

_{D}= θ, and the corresponding distances and path loss factors are approximately the same, i.e., d

_{SD}≈ d

_{SR}, n

_{SD}≈ n

_{SR}, and therefore $\overline{\mathsf{\gamma}}={\overline{\mathsf{\gamma}}}_{i},i=1,2.$ The terrestrial link operates in the urban environment, with a higher path loss factor, where n

_{RD}= 4 [52]. If it is not differently stated, we also assume that the P

_{R}is adjusted to provide approximately the same average SNR in all channels (i.e., ${\overline{\mathsf{\gamma}}}_{i}=\overline{\mathsf{\gamma}},i=1,2,3$), and, for the parameters presented in Table 2, we obtain that $\overline{\mathsf{\gamma}}=13.2\mathrm{d}\mathrm{B}.$

- We apply the method based on autoregressive models [53] to generate the time series x(n) that describes the multipath component. It corresponds to the complex Gaussian random process with a Rayleigh distributed envelope. In the case of isotropic scattering, the normalized autocorrelation function is given by ${R}_{r}(\tau )={J}_{0}(2\pi {f}_{Dm}\tau )$, where f
_{Dm}denotes the maximum Doppler shift for the multipath component (we assume f_{Dm}= 100 Hz). - The first step is repeated to generate a time series y(n), independent from x(n), with the Rayleigh distributed envelope and ACF ${R}_{s}(\tau )={J}_{0}(2\pi {f}_{Ds}\tau )$, where f
_{Ds}denotes the maximum Doppler shift for the shadowing (which is usually f_{Ds}<< f_{Dm}, while in our simulations, we chose f_{Ds}= 1 Hz [54]). - Based on the rejection/acceptance technique described in [55,56], we have generated a temporally uncorrelated time series z
_{un}_{1}(n) with Nakagami distribution. The rank matching method described in [47] is applied to reorder the samples in that process according to the previously generated reference y(n). The resulting time series z(n) corresponds to the time-varying LOS component that has an envelope with Nakagami distribution (as in z_{un}_{1}(n)) and a normalized ACF ${R}_{s}(\tau )={J}_{0}(2\pi {f}_{Ds}\tau )$(as in y(n)). - Two previous steps are repeated to generate w(n) with a Rayleigh distributed envelope and the maximum Doppler shift f
_{Dt}. It was combined with the temporally uncorrelated time series z_{un}_{2}(n), resulting in the time series h_{3}(n) with a Nakagami distributed envelope and a normalized ACF, where ${R}_{t}(\tau )={J}_{0}(2\pi {f}_{Dt}\tau )$. This corresponds to the channel gain of the terrestrial R-D link. - Channel gains for any satellite–terrestrial link (S-D or S-R) are obtained through the expression h(n) = x(n) + w(n).

_{3}= 5, Ω

_{3}= 1), and heavy or average shadowing at the S-R link.

_{th,R}= 5 dB, and an average shadowing scenario at the S-R link, the corresponding SNR will usually be above the SNR threshold, and we can notice that γ(n) ≈ γ

_{1}(n) + γ

_{3}(n). However, in the case of heavy shadowing at the S-R link, the impact of the relaying is less significant, and the output SNR values are similar to the SNR values at the S-D link.

_{th,R}. Furthermore, we can notice that every particular value of the threshold at the relay results in a different waveform of the SNR at the output of the MRC combiner. Therefore, statistics of the SNR at the destination receiver depend on the threshold applied at the relay receiver.

_{th,R}. The theoretical results are obtained using Equation (42), and the simulation results are estimated based on N = 10

^{7}samples for the case of the average shadowing at the S-D and S-R channel, and the fading parameter in the R-D channel has the value m

_{3}= 5. In accordance with the expectations, lower values of the threshold at the relay result in distributions with larger mean values. For smaller values of γ

_{th,R}, the probability that the SNR at the MRC output in the destination belongs to the range of large values (e.g., larger than 15 dB, or in absolute values γ > 31.63) increases. Therefore, we can expect that the outage probability will be reduced, and the system capacity will be increased if the SNR threshold at the relay has smaller values.

_{th,R}< 5 dB.

_{th,R.}

_{1}(t) and h

_{2}(t) are mostly determined using the ACF of the corresponding scattering component. The discrete representation of the instantaneous channel gain in the i-th channel (i = 1, 2), given by h

_{i}(k) = x

_{i}(n) + z

_{i}(n), has the following discrete autocorrelation function:

_{i}(n) and z

_{i}(n) are mutually independent, as the scattered and LOS components are assumed to be statistically independent in the shadowed Rice channel model [12]. Therefore, the second and the third expectation operators are equal to zero. The first expectation corresponds to the ACF of the complex time series x

_{i}(n), denoted by R

_{x}

_{,i}(k), and the fourth expectation corresponds to the ACF of the real-valued series z(n), denoted by R

_{z}

_{,i}(k).

_{0,i}, estimated from the generated time series x(n), should have the following discrete autocorrelation function:

_{i}, estimated from the generated time series z(n), should have the discrete ACF.

_{h}

_{,i}(k) = R

_{x}

_{,i}(k) + R

_{z}

_{,i}(k). The accuracy of the simulator is demonstrated in Figure 6 for the case of the average shadowing scenario, previously determined using Doppler shifts (f

_{Dm}= 100 Hz, f

_{Ds}= 1 Hz), and for the case where f

_{Ds}= 15 Hz. The simulation results correspond very well with the theoretical expressions for the ACF. If the LOS varies slowly, the corresponding ACF R

_{z}

_{,i}(k) is almost constant for the observed lags, and the shape of R

_{h}

_{,i}(k) is mostly determined using the shape of R

_{x}

_{,i}(k). It is easy to verify that the normalized autocovariance functions of time series h

_{i}(n) and x

_{i}(n), denoted, respectively, by C

_{h}

_{,i}(k) and C

_{x}

_{,i}(k), would satisfy the relation C

_{h}

_{,i}(k) ≈ C

_{x}

_{,i}(k).

_{th}= log

_{2}(1 + γ

_{th}) denotes a predefined capacity threshold.

_{3}= 5, Ω

_{3}= 1). It is interesting to note that the average value of the capacity is larger in the referent system, as the benefit of MRC cannot compensate for the inefficient use of the bandwidth due to the relaying. However, during the observed interval, the instantaneous capacity of the system with relaying does not decrease below the threshold C

_{th}= 1 bit/s/Hz, contrary to the instantaneous capacity of the referent system. This indicates that the ergodic capacity of the system with relaying is lower when compared to the referent system, while the outage capacity of the system with relaying can be larger when compared to the referent system for chosen system parameters and this particular capacity threshold.

_{3}= 5 at the terrestrial link. Also, we provide the comparison of three DF-based protocols as follows:

- -
- simple DF protocol, for the case where the S-D link is blocked, as a typical scenario for HSTRNs, previously analyzed in [21];
- -
- fixed relaying protocol, i.e., the DF protocol applied for the case where the S-D link is present, the relay always retransmits the signal, and MRC is applied at the destination was analyzed in [33];
- -

_{th,R}is equal to the threshold at the destination γ

_{th}. Taking into the account that the same modulation and coding scheme is used in all communication links, this assumption is reasonable in the case where the decoding is always successful if the SNR is above the threshold, and always unsuccessful if the SNR is below the threshold.

_{th,R}depends on the applied modulation and coding scheme, as well as the SNR statistics at the communication links. Although analysis is out of the scope of this paper, we present the numerical results for the outage probability for the two following simplified cases:

- -
- γ
_{th,R}= γ_{th}, as assumed in most of the papers; - -
- value γ
_{th,R}is fixed, and P_{out}is given for a typical range of γ_{D}.

^{7}samples and the basic principles of communication system simulations [58]. It can be observed that a typical DF protocol [21], applied when the S-R link is blocked (P

_{out}given in Equation (38)), exhibits inferior performance when compared to the referent system, with P

_{out}given in Equation (33). When DF is combined with MRC, a minor improvement is visible if the relay always retransmits the received signal [33]. The selection relaying has a potential to further improve the performance of the satellite–terrestrial network [33], and the closed-form expression for the outage probability was derived in our conference paper [34] for the case where γ

_{th,R}= γ

_{th}. In this paper, we have derived a more general expression for the outage probability, valid for γ

_{th,R}≠ γ

_{th}, when the satellite–terrestrial links undergo shadowed Rice fading, and propagation in terrestrial links can be described using Nakagami-m statistics. Although the scenario with the fixed γ

_{th,R}is simplified, it can be noticed that additional performance improvements are possible if γ

_{th,R}< γ

_{th}. If γ

_{th,R}→0 and γ

_{th}→0, the performances of the system with highly reliable links can be estimated from the simplified expression (34). This can be used as a guideline for the further optimization of parameter γ

_{th,R}, based on the procedure presented in [57].

_{3}in the terrestrial link R-D. It is interesting to notice that the referent system (where only the S-D link is present) outperforms the system by relaying for larger threshold values. This can be explained using Figure 6, where, for the presented samples, we estimate that P

_{out}(C

_{th}) = 1 and P

_{out,ref}(C

_{th}) < 1 if C

_{th}= 5. The effect of relaying is more visible for the light shadowing in satellite–terrestrial links, and, in such a case, the outage probability is lower when compared to the referent system, even for high threshold levels. Furthermore, it can be noticed that the outage probability of the system is more sensitive to the quality of the R-D link if there is light shadowing in the S-D link and S-R link.

_{out}for any value of the target outage probability when compared to the other propagation scenarios. Furthermore, the impact of the fading factor in the R-D link is largest for the light shadowing. The system with relaying outperforms the referent system for lower outage probabilities only. This effect is more noticeable in the case of light shadowing in satellite–terrestrial links, where the referent system provides larger C

_{out}in a wide range of practically important outage probabilities.

^{7}. The instantaneous capacity for the system with relaying and the referent system is determined using Equations (15) and (16), respectively. The outage capacity is estimated through comparing the instantaneous capacity with the corresponding threshold.

_{3}= 5 and Ω

_{3}= 1, and the transmitted power at the relay is adjusted to satisfy the equation ${P}_{R}={P}_{S}\times {d}_{RD}^{{n}_{RD}}/{d}_{SD}^{{n}_{SR}},$ and, therefore, $\overline{\mathsf{\gamma}}={\overline{\mathsf{\gamma}}}_{i},i=1,2,3.$ The presented numerical results indicate that relaying increases the outage capacity of the satellite–terrestrial system for typical values of the average SNR, although the system with relaying is spectrally less efficient. If the heavy shadowing is present both in the S-D and the S-R link, the capacity gain due to relaying is more significant for larger values of the average SNR and smaller values of the outage probability (if $\overline{\mathsf{\gamma}}=20\mathrm{dB}$, outage capacity C

_{out}= 3.48C

_{out}

_{,ref}for P

_{out}= 0.01, and C

_{out}= 1.03C

_{out}

_{,ref}for P

_{out}= 0.1).

_{out}= 0.01, it is also significant for P

_{out}= 0.1.

_{out}= 0.01, and numerical results are obtained using the time series with N = 10

^{7}samples. The two following scenarios are considered—the first one when average shadowing is present both at the S-D and the S-R link, and the second one when light shadowing is present at both links. The numerical results are presented for two values of the Nakagami-m fading parameter at the R-D link, namely m

_{3}= 1 and m

_{3}= 10. The increase in the outage capacity due to the applied selection relaying is visible in a wide range of average SNRs, both for the average and light shadowing.

_{3}= 1, the outage capacity for the system with relaying is greater than the outage capacity of the referent system (S-D only) in the range $\overline{\mathsf{\gamma}}<12\mathrm{dB}$. For the same propagation conditions at the communication links between satellite and terrestrial receivers, but for the increased fading parameter at the terrestrial link (m

_{3}= 10), the improvement due to relaying is visible in the range $\overline{\mathsf{\gamma}}<15\mathrm{dB}$.

_{3}= 1, the outage capacity for the system with relaying is greater than the outage capacity of the referent system in the range $\overline{\mathsf{\gamma}}<18.5\mathrm{dB}$. A greater value of the Nakagami-m fading parameter in the terrestrial channel leads to a higher outage capacity. If m

_{3}= 10, for the same propagation conditions at the communication links between the satellite and terrestrial receivers, the improvement due to relaying is visible in the whole analyzed range $\overline{\mathsf{\gamma}}<20\mathrm{dB}$.

_{1}(n), h

_{2}(n), h

_{3}(n), with N = 10

^{7}samples, generated using the described simulation method. The corresponding SNRs are obtained using Equations (2) and (3) for the given average SNR and system parameters, the relay operated based on Equation (1), and the time series for the instantaneous capacity for the system, with the relaying and referent system being determined using Equations (15) and (16), respectively. The ergodic capacity is estimated via averaging the instantaneous capacity, i.e., the mean value on the corresponding waveform is determined.

_{3}= 5 and Ω

_{3}= 1). The threshold at the relay was set to γ

_{th}

_{,R}= 0dB. The results for the system with relaying are compared to the numerical results obtained for the referent system, where only the S-D link is present. The theoretical results for the referent system can also be calculated using Equation (45) if we set γ

_{th}

_{,R}= ∞, while, in our experiments, we set γ

_{th}

_{,R}= 10,000.

_{SD}≈ d

_{SR}= H/sin(θ). Furthermore, it is well known that the parameters of the shadowed Rice model can be determined as a function of the elevation angle of the satellite, based on the following expressions:

## 6. Conclusions

_{out}. In general, it can be noticed that the system performances are more sensitive to the quality of the terrestrial link if the propagation in the S-D and S-R links corresponds to the light shadowing scenario. In the case when the S-R link experiences better propagation conditions, the relaying is more effective for all analyzed values of the average SNR. On the other hand, the relaying increases the ergodic capacity of the system only for the heavy propagation at the satellite–terrestrial links and smaller values of the average SNR, and the effect of relaying is more visible if the average SNR in the terrestrial link can be additionally increased.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A

_{4}can be split into two terms as follows:

_{1}is integer valued, was already derived in (24) in the following closed form:

_{3}through the use of expression (22) and the fact that $\mathsf{\Gamma}({m}_{3})=({m}_{3}-1)!$ [46]; from this, we obtain

## References

- Dahlman, E.; Parkvall, S.J.; Skold, J. 5G NR: The Next Generation Wireless Access Technology, 2nd ed.; Academic Press: Cambridge, MA, USA, 2020. [Google Scholar]
- 3rd Generation Partnership Project. TR 38.821. Solutions for NR to Support Non-Terrestrial Networks. (NTN), V16.0.0, January 2020. Available online: https://atisorg.s3.amazonaws.com/archive/3gpp-documents/Rel16/ATIS.3GPP.38.821.V1600.pdf (accessed on 15 February 2024).
- Darwish, T.; Kurt, G.K.; Yanikomeroglu, H.; Bellemare, M.; Lamontagne, G. LEO satellites in 5G and beyond networks: A review from a standardization perspective. IEEE Access
**2022**, 10, 35040–35060. [Google Scholar] [CrossRef] - Kodheli, O.; Lagunas, E.; Maturo, N.S.; Sharma, K.; Shankar, B.; Montoya, J.F.M.; Duncan, J.C.M.; Spano, D.; Chatzinotas, S.; Kisseleff, S.; et al. Satellite communications in the new space era: A survey and future challenges. IEEE Commun. Surv. Tutor.
**2021**, 23, 70–109. [Google Scholar] [CrossRef] - Del Portillo, I.; Cameron, B.G.; Crawley, E.F. A technical comparison of three low earth orbit satellite constellation systems to provide global broadband. Acta Astronaut.
**2019**, 159, 123–135. [Google Scholar] [CrossRef] - Maral, G.; Bousquet, M.; Sun, Z. Satellite Communications Systems: Systems, Techniques and Technology, 6th ed.; John Wiley & Sons Ltd.: Boca Raton, FL, USA, 2020. [Google Scholar]
- Gongora-Torres, J.M.; Vargas-Rosales, C.; Aragón-Zavala, A.; Villalpando-Hernandez, R. Elevation Angle Characterization for LEO Satellites: First and Second Order Statistics. Appl. Sci.
**2023**, 13, 4405. [Google Scholar] [CrossRef] - Zhou, H.; Liu, L.; Ma, H. Coverage and Capacity Analysis of LEO Satellite Network Supporting Internet of Things. In Proceedings of the 2019 IEEE International Conference on Communications (ICC 2019), Shanghai, China, 20–24 May 2019; pp. 1–6. [Google Scholar] [CrossRef]
- Liu, F.; Qian, G. Simulation Analysis of Network Capacity for LEO Satellite. In Proceedings of the 2020 International Conference on Computer Science and Management Technology (ICCSMT 2020), Shanghai, China, 20–22 November 2020; pp. 100–104. [Google Scholar] [CrossRef]
- Deng, R.; Di, B.; Zhang, H.; Kuang, L.; Song, L. Ultra-Dense LEO Satellite Constellations: How Many LEO Satellites Do We Need? IEEE Trans. Wirel. Commun.
**2021**, 20, 4843–4857. [Google Scholar] [CrossRef] - Rozenvasser, D.; Shulakova, K. Estimation of the Starlink Global Satellite System Capacity. In Proceedings of the 12th International Conference on Applied Innovation in IT (ICAIIT 2024), Kothen, Germany, 9 March 2023; pp. 55–59. [Google Scholar] [CrossRef]
- Abdi, A.; Lau, W.C.; Alouini, M.-S.; Kaveh, M. A new simple model for land mobile satellite channels: First- and second-order statistics. IEEE Trans. Wirel. Commun
**2003**, 2, 519–528. [Google Scholar] [CrossRef] - Sendonaris, A.; Erkip, E.; Aazhang, B. User cooperation diversity. Part, I. System description. IEEE Trans. Commun.
**2003**, 51, 1927–1938. [Google Scholar] [CrossRef] - Laneman, J.N.; Tse, D.N.C.; Wornell, G.W. Cooperative diversity in wireless networks: Efficient protocols and outage behavior. IEEE Trans. Inf. Theory
**2004**, 50, 3062–3080. [Google Scholar] [CrossRef] - Yu, M.; Li, J. Is amplify-and-forward practically better than decode-and-forward or vice versa? In Proceedings of the IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP 2005), Philadelphia, PA, USA, 23 March 2005; Volume 3, pp. iii/365–iii/368. [Google Scholar] [CrossRef]
- Siriwongpairat, W.P.; Himsoon, T.; Su, W.; Liu, K.J.R. Optimum threshold-selection relaying for decode-and-forward cooperation protocol. In Proceedings of the IEEE Wireless Communications and Networking Conference (WCNC 2006), Las Vegas, NV, USA, 3–6 April 2006; pp. 1015–1020. [Google Scholar] [CrossRef]
- Evans, B.; Werner, M.; Lutz, E.; Bousquet, M.; Corazza, G.; Maral, G.; Rumeau, R. Integration of satellite and terrestrial systems in future multimedia communications. IEEE Trans. Wirel. Commun.
**2005**, 12, 72–80. [Google Scholar] [CrossRef] - Bhatnagar, M.R.; Arti, M.K. Performance analysis of AF based hybrid satellite-terrestrial cooperative network over generalize fading channels. IEEE Commun. Lett.
**2013**, 17, 1912–1915. [Google Scholar] [CrossRef] - Arti, M.K.; Bhatnagar, M.R. Beamforming and combining in hybrid satellite-terrestrial cooperative systems. IEEE Commun. Lett.
**2014**, 18, 483–486. [Google Scholar] [CrossRef] - An, K.; Lin, M.; Liang, T.; Wang, J.B.; Wang, J.; Huang, Y.; Swindlehurst, A.L. Performance Analysis of Multi-Antenna Hybrid Satellite-Terrestrial Relay Networks in the Presence of Interference. IEEE Trans. Commun.
**2015**, 63, 4390–4404. [Google Scholar] [CrossRef] - An, K.; Liang, T. Hybrid satellite-terrestrial relay networks with adaptive transmission. Trans. Veh. Technol.
**2019**, 68, 12448–12452. [Google Scholar] [CrossRef] - Guo, K.; Lin, M.; Zhang, B.; Wang, J.-B.; Wu, Y.; Zhu, W.-P.; Cheng, J. Performance Analysis of Hybrid Satellite-Terrestrial Cooperative Networks with Relay Selection. IEEE Trans. Veh. Technol.
**2020**, 69, 9053–9067. [Google Scholar] [CrossRef] - Ding, Y.; Zou, Y.; Li, B. Ergodic Capacity and Outage Probability Analysis of RIS-Aided Hybrid Satellite-Terrestrial Networks. In Proceedings of the 14th International Conference on Wireless Communications and Signal Processing (WCSP 2022), Nanjing, China, 1–3 November 2022; pp. 837–841. [Google Scholar] [CrossRef]
- Bankey, V.; Upadhyay, P.K. Ergodic capacity of multiuser hybrid satellite-terrestrial fixed-gain AF relay networks with CCI and outdated CSI. IEEE Trans. Veh. Technol.
**2018**, 67, 4666–4671. [Google Scholar] [CrossRef] - Akhmetkaziyev, Y.; Nauryzbayev, G.; Arzykulov, S.; Rabie, K.; Eltawil, A. Ergodic Capacity of Cognitive Satellite-Terrestrial Relay Networks with Practical Limitations. In Proceedings of the International Conference on Information and Communication Technology Convergence (ICTC 2021), Jeju Island, Republic of Korea, 20–22 October 2021; pp. 555–560. [Google Scholar] [CrossRef]
- Lin, H.; Zhang, C.; Huang, Y.; Zhao, R.; Yang, L. Downlink Outage Analysis of Integrated Satellite-Terrestrial Relay Network with Relay Selection and Outdated CSI. In Proceedings of the 93rd IEEE Vehicular Technology Conference (VTC2021-Spring), Helsinki, Finland, 25–28 April 2021. [Google Scholar] [CrossRef]
- Zhang, H.; Pan, G.; Ke, S.; Wang, S.; An, J. Outage Analysis of Cooperative Satellite-Aerial-Terrestrial Networks with Spatially Random Terminals. IEEE Trans. Commun.
**2022**, 70, 4972–4987. [Google Scholar] [CrossRef] - Pan, G.; Ye, J.; An, J.; Alouini, S. Latency versus Reliability in LEO Mega-Constellations: Terrestrial, Aerial, or Space Relay. IEEE Trans. Mobile Comput.
**2023**, 22, 5330–5345. [Google Scholar] [CrossRef] - Lin, Z.; Lin, M.; Champagne, B.; Zhu, W.-P.; Al-Dhahir, N. Secrecy-Energy Efficient Hybrid Beamforming for Satellite-Terrestrial Integrated Networks. IEEE Trans. Commun.
**2021**, 69, 6345–6360. [Google Scholar] [CrossRef] - Lin, Z.; Lin, M.; de Cola, T.; Wang, J.-B.; Zhu, W.-P.; Cheng, J. Supporting IoT with Rate-Splitting Multiple Access in Satellite and Aerial-Integrated Networks. IEEE Internet Things J.
**2021**, 8, 11123–11134. [Google Scholar] [CrossRef] - Lin, Z.; Lin, M.; Wang, J.-B.; de Cola, T.; Wang, J. Joint Beamforming and Power Allocation for Satellite-Terrestrial Integrated Networks with Non-Orthogonal Multiple Access. IEEE J. Sel. Top. Signal Process.
**2019**, 13, 657–670. [Google Scholar] [CrossRef] - Lin, Z.; Lin, M.; Champagne, B.; Zhu, W.-P.; Al-Dhahir, N. Secure and Energy Efficient Transmission for RSMA-Based Cognitive Satellite-Terrestrial Networks. IEEE Wirel. Commun. Lett.
**2021**, 10, 251–255. [Google Scholar] [CrossRef] - Sakarellos, V.K.; Kourogiorgas, C.; Panagopolous, A.D. Cooperative hybrid land mobile satellite-terrestrial broadcasting systems: Outage probability evaluation and accurate simulation. Wirel. Pers. Commun.
**2014**, 79, 1471–1481. [Google Scholar] [CrossRef] - Milojković, J.; Ivaniš, P.; Blagojević, V.; Brkić, S. Performance analysis of land mobile satellite-terrestrial systems with selection relaying. In Proceedings of the 10th International Conference on Electrical, Electronic and Computing Engineering (IcETRAN 2023), TEI 1.2, East Sarajevo, Bosnia and Herzegovina, 5–8 June 2023. [Google Scholar] [CrossRef]
- Ivaniš, P.; Blagojević, V.; Đorđević, G. The method of generating shadowed Rice fading with desired statistical properties. In Proceedings of the 22nd International Symposium INFOTEH-JAHORINA (INFOTEH 2023), Jahorina, Bosnia and Herzegovina, 15–17 March 2023. [Google Scholar] [CrossRef]
- Ye, J.; Pan, G.; Alouini, M.-S. Earth Rotation-Aware Non-Stationary Satellite Communication Systems: Modeling and Analysis. IEEE Trans. Wirel. Commun.
**2021**, 20, 5942–5956. [Google Scholar] [CrossRef] - Lutz, E. Modelling of the land mobile satellite communications channel. In Proceedings of the IEEE-APS Topical Conference on Antennas and Propagation in Wireless Communications (APWC 2013), Turin, Italy, 9–13 September 2013; pp. 199–202. [Google Scholar] [CrossRef]
- Loo, C. A statistical model for a land mobile satellite link. IEEE Trans. Veh. Technol.
**1985**, 34, 122–127. [Google Scholar] [CrossRef] - Abdi, A.; Lau, W.C.; Alouini, M.-S.; Kaveh, M. On the second-order statistics of a new simple model for land mobile satellite channels. In Proceedings of the 54th Vehicular Technology Conference (VTC Fall 2001), Atlantic City, NJ, USA, 7–11 October 2001; Volume 1, pp. 301–304. [Google Scholar] [CrossRef]
- Clarke, R.H. A Statistical Theory of Mobile Radio Reception. Bell Syst. Tech. J.
**1968**, 47, 957–1000. [Google Scholar] [CrossRef] - Jakes, W.C. Microwave Mobile Communications, 2nd ed.; IEEE Press: New York, NY, USA, 1993. [Google Scholar]
- Ivanis, P.; Drajic, D.; Vucetic, B. The second order statistics of maximal ratio combining with unbalanced branches. IEEE Commun. Lett.
**2008**, 12, 508–510. [Google Scholar] [CrossRef] - Ivanis, P.; Blagojevic, V.; Drajic, D.; Vucetic, B. Closed-Form Level Crossing Rates Expressions of Orthogonalized Correlated MIMO Channels. IEEE Trans. Veh. Technol.
**2011**, 60, 1910–1916. [Google Scholar] [CrossRef] - Ivanis, P.; Blagojevic, V.; Drajic, D.; Vucetic, B. Second Order Statistics of a Maximum Ratio Combiner with Unbalanced and Unequally Distributed Nakagami Branches. IET Commun.
**2011**, 5, 1829–1835. [Google Scholar] [CrossRef] - Yacoub, M.D.; Bautista, J.E.V.; Guerra de Rezende Guedes, L. On higher order statistics of the Nakagami-m distribution. IEEE Trans. Veh. Technol.
**1999**, 48, 790–794. [Google Scholar] [CrossRef] - Abramowitz, M.; Stegun, I.A. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables; Dover: New York, NY, USA, 1972. [Google Scholar]
- Silveira Santos Filho, J.C.; Yacoub, M.D.; Fraidenraich, G. A Simple Accurate Method for Generating Autocorrelated Nakagami-m Envelope Sequences. IEEE Commun. Lett.
**2007**, 11, 231–233. [Google Scholar] [CrossRef] - Sakarellos, V.K.; Skraparlis, D.; Panagopoulos, A.D.; Kanellopoulos, J.D. Cooperative diversity performance of selection relaying over correlated shadowing. Phys. Commun.
**2011**, 4, 182–189. [Google Scholar] [CrossRef] - Goldsmith, A. Wireless Communications; Cambridge University Press: Cambridge, UK, 2005. [Google Scholar]
- Thakur, S.; Singh, A. Ergodic Secrecy Capacity in Nakagami-m Fading Channels. In Proceedings of the International Carnahan Conference on Security Technology (ICCST 2019), Chennai, India, 1–3 October 2019. [Google Scholar] [CrossRef]
- Available online: https://functions.wolfram.com/ (accessed on 20 January 2024).
- Matolak, D.W.; Zhang, Q.; Wu, Q. Path Loss in an Urban Peer-to-Peer Channel for Six Public-Safety Frequency Bands. IEEE Wirel. Commun. Lett.
**2013**, 2, 263–266. [Google Scholar] [CrossRef] - Baddour, K.E.; Beaulieu, N.C. Autoregressive modeling for fading channel simulation. IEEE Trans. Wireless Commun.
**2005**, 4, 1650–1662. [Google Scholar] [CrossRef] - Vucetic, B.; Du, J. Channel modeling and simulation in satellite mobile communication systems. IEEE J. Sel. Areas Commun.
**1992**, 10, 1209–1218. [Google Scholar] [CrossRef] - Papoulis, A.; Pillai, S.U. Probability Random Variables and Stochastic Processes, 4th ed.; McGraw-Hill: Boston, MA, USA, 2002. [Google Scholar]
- Zhu, Q.-M.; Yu, X.-B.; Wang, J.-B.; Xu, D.-Z.; Chen, X.-M. A new generation method for spatial-temporal correlated MIMO Nakagami fading channel. Int. J. Antennas Propag.
**2012**, 2012, 713594. [Google Scholar] [CrossRef] - Ikki, S.; Ahmed, M.H. Performance of Decode-and-Forward Cooperative Diversity Networks Over Nakagami-m Fading Channels. In Proceedings of the IEEE Global Telecommunications Conference (GLOBECOM 2007), Washington, DC, USA, 26–30 November 2007; pp. 4328–4333. [Google Scholar] [CrossRef]
- Jeruchim, M.C.; Balaban, P.; Shanmugan, K.S. Simulation of Communication Systems: Modeling, Methodology and Techniques; Springer Science & Business Media: Berlin, Germany, 2006. [Google Scholar]
- Milojković, J.; Brkić, S.; Ivaniš, P.; Čiča, Z. Performance of Handover Execution in Satellite Networks with Shadowed-Rician Fading. In Proceedings of the 16th International Conference on Advanced Technologies, Systems and Services in Telecommunications (TELSIKS 2023), Niš, Serbia, 25–27 October 2023; pp. 155–158. [Google Scholar] [CrossRef]
- Gradshteyn, I.S.; Ryzhik, I.M. Table of Integrals, Series and Products, 5th ed.; Academic Press Inc.: San Diego, CA, USA, 1994. [Google Scholar]

**Figure 3.**The simulator of the Nakagami-m and shadowed Rice channel gains with desired first- and second-order statistics.

**Figure 4.**The discrete waveforms of the instantaneous SNRs at the S-R link, γ

_{2}(n), at the S-D link, γ

_{1}(n), γ

_{3}(n), and at the output of the MRC receiver at the destination, γ(n).

**Figure 6.**ACF of the channel gain, average shadowing, f

_{Dm}= 100 Hz, f

_{Ds}= 1 Hz, and f

_{Ds}= 15 Hz.

**Figure 7.**Instantaneous capacity of the system with relaying C(t) and the referent system C

_{ref}(t).

Symbol | Explanation |
---|---|

P_{S} | the power of the signal from the satellite |

P_{R} | the power the signal from the relay |

h_{1} | the complex channel gain between the satellite (S) and destination (D) |

h_{2} | the complex channel gain between the satellite (S) and relay (R) |

h_{3} | the complex channel gain between the relay (R) and destination (D) |

γ_{1} | the instantaneous SNR at the output of the S-D link |

γ_{2} | the instantaneous SNR at the output of the S-R link |

γ_{3} | the instantaneous SNR at the output of the R-D link |

γ | the instantaneous SNR at the output of the MRC combiner at the destination |

γ_{th} | the threshold at the destination |

γ_{th,R} | the threshold at the relay |

d_{SD} | the distance between the satellite and destination |

d_{SR} | the distance between the satellite and relay |

d_{RD} | the distance between the relay and destination |

n_{SD} | the path loss between the satellite and destination |

n_{SR} | the path loss between the satellite and relay |

n_{RD} | the path loss between the relay and destination |

θ | elevation |

σ^{2} | the noise power at the destination |

${\mathsf{\lambda}}_{i}$ | the normalized power gain in the i-th channel |

${\overline{\mathsf{\gamma}}}_{i}$ | the average signal–noise ratio at the output of the i-th channel |

m_{i} | the Nakagami fading parameter at the i-th channel |

Ω_{i} | the average power of the LOS component in the i-th channel |

b_{0,i} | the average power of the scattering component in the satellite–terrestrial links |

f_{Dm} | the maximum Doppler shift for the multipath in the S-D and S-R links |

f_{Ds} | the maximum Doppler shift for the shadowing in the S-D and S-R links |

f_{Dt} | the maximum Doppler shift for the terrestrial link |

f_{0} | the carrier frequency at the satellite–terrestrial links |

c | the speed light |

C | the instantaneous capacity |

C_{out} | the outage capacity |

C_{e} | the ergodic capacity |

C_{th} | the capacity threshold |

P_{out} | the outage probability |

EIRP | the effective isotropic radiated power |

L_{A} | denotes the typical loss due to the atmospheric conditions |

G | the antenna gain at the receiver |

T_{S} | the temperature of the system |

B | the channel bandwidth |

k | the Boltzmann constant |

System Parameters | Value | Simulation Parameters | Value |
---|---|---|---|

EIRP | 36.7 dBW | P_{S} | 9.9 dBW |

L_{A} | 14 dB | σ^{2} | −89 dBm |

G | 10.5 dB | n_{SD}, n_{SR} | 2 |

T_{s} | 363 K | H | 550 km |

B | 250 MHz | θ | 60° |

f_{0} | 11 GHz | L | 25 km |

Propagation Scenario | b_{0} | m | Ω |
---|---|---|---|

Infrequent light shadowing | 0.158 | 19 | 1.29 |

Average shadowing | 0.126 | 10 | 0.835 |

Frequent heavy shadowing | 0.063 | 1 | 0.000897 |

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**MDPI and ACS Style**

Ivaniš, P.; Milojković, J.; Blagojević, V.; Brkić, S.
Capacity Analysis of Hybrid Satellite–Terrestrial Systems with Selection Relaying. *Entropy* **2024**, *26*, 419.
https://doi.org/10.3390/e26050419

**AMA Style**

Ivaniš P, Milojković J, Blagojević V, Brkić S.
Capacity Analysis of Hybrid Satellite–Terrestrial Systems with Selection Relaying. *Entropy*. 2024; 26(5):419.
https://doi.org/10.3390/e26050419

**Chicago/Turabian Style**

Ivaniš, Predrag, Jovan Milojković, Vesna Blagojević, and Srđan Brkić.
2024. "Capacity Analysis of Hybrid Satellite–Terrestrial Systems with Selection Relaying" *Entropy* 26, no. 5: 419.
https://doi.org/10.3390/e26050419