Distributed Massive MIMO for LEO Satellite Networks

Several approaches have been proposed to address challenges associated with the high mobility of LEO satellites. In [13], the authors proposed a software-defined networking (SDN) architecture to control LEO satellites using a controller on the ground that connects to the LEOs via a GEO satellite. Along similar lines, the authors in [14] proposed an extensible architecture utilizing several layers of terrestrial relays (TRs), HAPSs, LEOs, and GEOs for relay purposes, and they studied the handover procedure among these different systems. However, using GEOs in the network, as investigated in these studies, entails a long-delay segment in the communication cycle that directly impacts the QoS of users and requires coordination between different satellite operators. In addition, these two studies, and other current approaches (e.g., [15, 16]), focused on the connectivity of UTs with a single satellite in their visibility. Therefore, the UT’s service time (i.e., connection time without handover interruption) is limited by the visibility of a single LEO satellite, which is only a few minutes (about 10 minutes in the Iridium system [14], for example). The handover rate is inevitably high, regardless of the adopted handover management technique. Moreover, most of these works are based on old LEO constellations (e.g., Iridium) and do not exploit the features of the new LEO mega-constellations, such as SpaceX’s Starlink and Telesat’s Lightspeed.

To overcome the challenges of single-satellite connectivity, the use of multiple-input multiple-output (MIMO) techniques in satellite communications, and in LEO SatNets in particular, has been investigated in a handful of works. Former studies, such as [17, 18, 19], investigated the connection of ground terminals to two GEO satellites, or to two antennas deployed on a single GEO satellite, for diversity purposes to address fading issues (e.g., in rainy and foggy weathers). As for LEO satellites, the authors in [20] focused on feeder links by studying the connection to multiple ground stations via optical links to realize site diversity in future LEO SatNets. In [21], the authors adopted a bipartite graph model for the ground gateway stations and the multiple LEO satellites visible to them. They solved this multi-connectivity problem by using maximum matching techniques. However, they utilized basic MIMO techniques (i.e., considering the general case that each ground station node can be connected to multiple satellite nodes and vice versa) and assumed that the ground stations had accurate information about the motion of the LEO satellites in their visibility. In [22], the authors considered a similar MIMO model and analyzed the capacity of the LEO-MIMO links by taking into consideration the Doppler shift due to the motion of the satellites. Nevertheless, these studies considered classical MIMO models to describe the connectivity of ground UTs with multiple satellites without investigating the details of the network architecture, channel estimation, precoding, and interference between users.

On another front, massive MIMO in LEO SatNets was studied in [23, 24]. In these works, the authors assumed that the LEO satellites use arrays of uniform planar antennas that can realize massive MIMO. However, due to the line-of-sight (LoS) connection to the ground terminals, the collocated satellite massive MIMO system cannot achieve the desired benefits as in terrestrial networks if the ground users are not sufficiently separated due to the so-called “unfavourable propagation” environment [25] that results in a “keyhole channel” matrix [26]. Therefore, collocated massive MIMO would not be suitable for single-user MIMO scenarios (sending multiple layers to the same user), for instance.

One significant difference between future LEO SatNets and old LEO constellations is ultra-dense deployment. That is, future LEO mega-constellations will include thousands of satellites. For example, SpaceX is planning to deploy $30,000$ LEO satellites for their second-generation constellation in addition to the current plan of around $12,000$ satellites [27]. This means that multiple LEO satellites will be visible to ground UTs simultaneously, which will open the door for advanced distributed MIMO techniques, such as cell-free massive MIMO (CF-mMIMO).

CF-mMIMO was recently proposed for next-generation terrestrial networks that build on coordinated multi-point (CoMP) and network MIMO techniques for large spectral efficiency, power efficiency, and network flexibility gains [28]. In terrestrial CF-mMIMO, multiple access points can be used to cooperatively communicate with users in a cell-free manner. This technique can be utilized in future LEO satellite networks exploiting an ultra-dense deployment, very high-speed inter-satellite links (ISLs), and LoS connections with ground UTs. Moreover, the CF-mMIMO architecture enables the cross-layer design to jointly optimize the upper and lower layers of networking. This cross-layer design is of utmost importance to LEO SatNets given that the network nodes are moving (i.e., the network topology is dynamic) and all links (with UTs, gateways, and other satellites) are wireless (radio frequency (RF) or free-space optical (FSO)). This means that the design of the lower layers has a significant impact on the performance of the upper ones. Therefore, CF-mMIMO can be leveraged for an efficient, resilient satellite network.

Noting that CF-mMIMO is a terrestrial technology that belongs to the distributed MIMO techniques, in this paper, we focus on the generalization of CF-mMIMO and propose a distributed massive MIMO (DM-MIMO) LEO satellite network architecture. Besides, we investigate the cross-layer design and an artificial intelligence (AI)-based implementation. To the best of our knowledge, this is the first work to introduce and study a DM-MIMO approach in LEO constellations.

The major contributions of this paper can be summarized as follows:

• •

  We propose a LEO SatNet architecture based on DM-MIMO techniques. More specifically, we investigate the network topology, required ISLs, duplexing mode, beamforming, power control, frequency reuse, and handover management strategies. Moreover, we highlight the benefits of using this architecture and investigate the associated challenges and potential solutions to realize it.

• •

  We develop an optimized cross-layer design framework based on the proposed DM-MIMO-based architecture, such that the power allocation and handover management processes are jointly optimized. For this purpose, we describe the channel model, uplink training and channel estimation, downlink data transmission, and formulate a novel multi-objective optimization problem. It should be noted that it is not straightforward to establish a combined channel and data transmission models based on those from cell-free massive MIMO and satellite communications. In the optimization problem, the aggregate throughput is maximized while minimizing the handover rate. We refer to this optimized cross-layer design as the distributed joint power allocation and handover management (D-JPAHM) technique.

• •

  We introduce an AI-based implementation for the developed cross-layer control framework, which can be used in practical satellite systems. For this purpose, we leverage deep learning to provide accurate predictions for the solution of the formulated multi-objective optimization problem without actually solving the problem. Deep learning provides several benefits compared to traditional optimization-based and heuristic approaches. For instance, it exploits offline computations to reduce the computational complexity of the online operation. It also adapts to changing environments, which is vital for the dynamic satellite network. Furthermore, the scalability and support of distributed data processing and storage provided by deep learning techniques are crucial for LEO satellites that are less capable compared to GEO satellites.

• •

  We conduct extensive simulations to evaluate the performance of the proposed architecture, cross-layer design, and AI-based implementation. In addition, we compare the performance with that of conventional approaches and architectures from the literature. The simulation results show the superiority of the proposed DM-MIMO-based architecture and solutions compared to the traditional single-connectivity approach.

The remainder of this paper is organized as follows. In Section II, the proposed DM-MIMO-based LEO SatNet architecture is discussed, and several network design aspects are investigated. Then, in Section III, we highlight the benefits of using this architecture, the associated challenges and potential solutions. In Section IV, the optimized cross-layer design is detailed. In Section V, we discuss the AI-driven approach for implementing the proposed cross-layer optimization framework. In Section VI, we present and discuss the results of the simulations to evaluate the performance of the proposed DM-MIMO-based architecture and solutions in comparison with that of traditional single satellite connectivity. Finally, we conclude the paper in Section VII.

Fig. 1 shows the proposed DM-MIMO-based LEO SatNet architecture. As we can see, the satellites are divided into clusters. To be consistent with the terrestrial CF-mMIMO terminology, each satellite in the cluster is called a satellite access point (SAP). In the event that a satellite has multiple antennas, each antenna would be considered a separate SAP. These SAPs are connected to a central processing unit (CPU) through ISLs for fronthaul purposes. This CPU can be deployed on a central satellite with more advanced computing capabilities. These central satellites are called super satellite nodes (SSNs).

To exploit the reciprocity of the uplink (UL) and downlink (DL) channels, time division duplexing (TDD) is the adopted duplexing mode in CF-mMIMO systems [28]. For the purpose of channel estimation, both UL and DL pilots can be used [29]. However, in most CF-mMIMO studies (e.g., [28, 30, 31]), only UL pilots are considered. This is also suitable for SatNets because the users do not need to estimate their effective channel gain and to use most of the TDD frame for data transmission. It is important to mention that the propagation delay and Doppler shift can be compensated in the time and frequency synchronization processes, given that the satellites move in a predetermined orbit, and their motion information can be known a priori [32]. Therefore, TDD was adopted in several satellite LEO communication studies [33] and systems [34, 35]. Accordingly, the TDD frame can be structured, as shown in Fig. 2.

The channel coherence interval is defined as the time-frequency interval during which the channel characteristics can be considered static. This coherence interval depends on the channel condition, the mobility of the satellite and the UT, and the carrier frequency. The coherence interval samples–or channel uses–designated $\tau_{c}$, are divided into three parts: the initial $\tau^{p}_{u}$ samples are used for UL pilot transmission, the next $\tau^{d}_{u}$ samples are used for UL data transmission, and the last $\tau^{d}_{d}$ samples are reserved for DL data transmission. It is worth mentioning that the guard intervals are excluded from this coherence time interval. Utilizing the UL pilots, all the UL channels are estimated at the SAPs locally without forwarding them to the CPU. This supports the scalability of the network, since the signalling overhead is independent of the number of SAPs. Due to reciprocity, these channel estimates are valid for the DL direction as well. Therefore, the estimated channels are used for DL data precoding and UL data detection.

Efficient radio resource allocation techniques can exploit the advantages of the proposed architecture (including transmit and receive diversity) to achieve higher throughput for the connected terminals, ensuring their QoS satisfaction, minimizing interference, and minimizing the handover rate. In this subsection, we discuss different aspects of resource allocation for the proposed DM-MIMO-based SatNets architecture.

The traditional satellite handover process (i.e., based on single satellite connectivity) is depicted in Fig. 3a. In this case, when the signal level is below a certain threshold, the link is switched to the next LEO satellite in the cone visibility of the UT. This can be accomplished by using the satellite reference signals that are broadcast by the satellite. In the network layer, handover is required to forward the data arriving for the old address to the new home network, since the UT is given a new IP address in the new home network. As discussed above, the service time in this case is limited by the satellite visibility, which is a few minutes in LEO SatNets. However, using the proposed DM-MIMO architecture, the UT is connected to a cluster of satellites or SAPs. Consequently, the service time is limited by the visibility of the target cluster, which is longer than that of a single satellite. In addition, the resource allocation process can be optimized such that the service time is maximized. This minimizes the handover rate, associated losses, processing delays, and signalling overhead. In what follows, we discuss the link-layer and network-layer handover processes based on the proposed DM-MIMO architecture.

The major benefits of using DM-MIMO in LEO SatNets can be summarized as follows:

• •

  Improved throughput: As indicated earlier, the dominant LoS communication between the ground UTs and LEO satellites imposes limitations on the use of MIMO techniques in LEO SatNets. This is due to the unfavourable propagation environment that lowers the rank of the channel matrix between the UTs and satellites [25]. Therefore, this limits the MIMO gain to the keyhole capacity as investigated in [26], except if proper user grouping and spacing are employed to alleviate this effect. The use of DM-MIMO realizes this spatial distribution among the serving satellites and could address the collocation issues. This is in addition to the full frequency reuse that enhances spectral efficiency. Therefore, the use of DM-MIMO could significantly improve the satellite network throughput.

• •

  Lower handover rate: As discussed in Section II-D, the use of DM-MIMO-based architecture enables the UTs to be served by a cluster of LEO satellites. Therefore, this extends the UT’s service time to the visibility period of a cluster of satellites instead of a single one. This directly reduces the handover rate of UTs and the issues associated with a high handover rate, such as signalling overhead, processing delay, data forwarding, tunnelling, IP addressing issues, and location update.

• •

  Flexible operation: The concept of CPUs and SAPs is similar to centralized units (CUs) and distributed units (DUs) that are used in the context of open radio access networks (O-RAN), as discussed in [41]. This opens the door to the advanced flexible O-RAN techniques [42] to be utilized in LEO SatNets.

• •

  Cross-layer design: As indicated in Section I, the nodes of the satellite network are moving and have wireless links among them and to the users on the ground. Therefore, the performance of lower layers highly impacts the upper ones. The use of DM-MIMO enables the cross-layer design of SatNets to jointly optimize upper and lower layers, which provides several benefits to SatNets [43]. In this regard, a cross-layer framework is proposed in Section IV to show the benefits of using this design.

Although the DM-MIMO-based architecture provides several benefits to LEO SatNets, several challenges need to be addressed to leverage this technology in SatNets. The major challenges and potential solutions can be summarized as follows:

• •

  Dynamic topology: Due to the dynamic environment of LEO satellites (due to movement at different directions and at different altitudes), the topology of network nodes changes with time. This entails issues with defining the cluster that serves the UTs (i.e., the number of SAPs and which SAPs serve each UT). However, this cluster formation process can be optimized by the SSN on the basis of different performance objectives and associated deployment costs. For instance, this depends on the amount of traffic per spot (e.g., hotspots should be served with a larger number of SAPs compared to cold spots), the capabilities of the serving SAPs (e.g., the power and computational resources), and the kind of traffic (e.g., narrowband versus broadband). An example of such a dynamic clustering approach is proposed in [44] for terrestrial cell-free massive MIMO networks. This concept can be adapted to DM-MIMO-based SatNets. Moreover, user grouping (see [45] for instance) can be employed to assign groups of ground UTs to dynamic clusters of SAPs to optimize user-cluster association.

• •

  Synchronization: To achieve such cooperative transmission between a cluster of satellites, synchronization in time, frequency, and phase is required. Several time synchronization techniques have been proposed in the literature to align the signals at the symbol level and achieve coherent transmission (see [46, 47] for instance). Besides, frequency and phase synchronization has been widely studied in the literature to achieve cooperative transmission or sensing as discussed in [48]. These synchronization techniques can be utilized and adapted to be applied in the considered scenario of DM-MIMO-based LEO SatNets.

• •

  Outdated channel information: As discussed in Section II, the channel knowledge (which is vital to several network management functionalities) is based on the estimated channel utilizing the UL pilots. However, due to the long propagation delay (around $7.9$ ms for LEOs), this channel information could be outdated. To overcome this issue, intelligent machine learning techniques can be utilized to estimate the channel while taking into account the propagation delay. An example of this approach is studied in [49], where the authors propose a deep learning model that is based on long short term with memory (LSTM) to predict the channels while exploiting their correlation to overcome channel aging issues in LEO SatNets.

• •

  Complexity of SSNs: The SSNs are used to control the SAPs in their clusters. Therefore, this requires additional computing and power resources for these satellites to be able to perform such control tasks. To overcome this issue, distributed processing can be utilized as much as possible. For example, as discussed in Section II, several network management functions (e.g., beamforming and pilot assignment) can be implemented in a distributed manner to reduce the computations on the SSNs. In addition, software operation can be exploited to flexibly implement the functions of the CPU without requiring extra hardware at the SSNs.

• •

  Signalling overhead: To implement the cooperative operation of SAPs, additional signalling between the CPU and SAPs is required. To overcome this signalling overhead, the exchange of the information between the SAPs and the CPU should be minimized; for example, by restricting it to payload data and power control coefficients as discussed in [28]. Besides, the use of high-speed FSO communications [50], which enables the information to be exchanged at very high rate with low latency, can be utilized to overcome this issue.

We consider a cluster of LEO satellites that includes a set of $M$ SAPs indexed by $\mathcal{M}=\{1,~{}2,\cdots,~{}m,\cdots,~{}M\}$. This cluster serves a set of single-antenna ground UTs set, indexed by $\mathcal{K}=\{1,~{}2,\cdots,~{}k,\cdots,~{}K\}$. Assume that the channel conditions are static in a coherence time interval of $\tau_{c}$ samples. Due to the strong LoS component between the UTs and SAPs, the channel between the $k$th UT and the $m$th SAP is modelled as Rician and can be calculated by [51]

where $\kappa_{m,k}$ is the Rician K-factor, $h^{\prime}_{m,k}$ and $h^{\prime\prime}_{m,k}$ represent the LoS and non-LoS (NLoS) components, respectively. The large scale fading and losses are represented by $L_{m,k}=\newline 10^{-(L^{dist}_{m,k}+L^{shad}_{m,k}+L^{angl}_{m,k})/10}$, where $L^{dist}_{m,k}$ is the power loss (in dB) due to distance between UT $k$ and SAP $m$, $L^{shad}_{m,k}\sim\mathcal{N}(0,\sigma_{sh}^{2})$ is the shadowing attenuation (in dB) with variance $\sigma_{sh}^{2}$, and $L^{angl}_{m,k}$ is the loss due to the boresight angle and can be calculated (in dB) by [52]

where $\theta_{m,k}$ is the boresight angle between the $k$th UT and the $m$th SAP, and $\eta$ is the antenna factor determining the coverage radius, assuming that the aperture efficiency is unity.

Suppose that the NLoS component, $h^{\prime\prime}_{m,k}$, is a Rayleigh random variable, i.e., $h^{\prime\prime}_{m,k}\sim\mathcal{C}\mathcal{N}(0,1)$. The LoS component is given by $h^{\prime}_{m,k}=e^{j\phi_{m,k}}$, where $\phi_{m,k}\sim\mathcal{U}[-\pi,\pi]$ is a uniform random variable that represents the phase shift due to the mobility of the SAP and UT and propagation delay.

For simplicity, we rewrite (1) as follows:

where

In (3), $\tilde{h}_{m,k}\sim\mathcal{C}\mathcal{N}(0,\lambda_{m,k})$, $\lambda_{m,k}=L_{m,k}/(\kappa_{m,k}+1)$, and $\beta_{m,k}\in\mathbb{R}$. Therefore, $\beta_{m,k}$ and $\lambda_{m,k}$ are changing slowly [53] compared to small-scale fading that changes instantaneously. Since they mainly depend on the UT’s position, they can be calculated a priori. In addition, we assume that the propagation delay and Doppler shift are compensated in the time and frequency synchronization processes. This is a reasonable assumption since they primarily depend on the satellite and user location and velocity, which can be determined given the satellite orbital information [32]. In addition, other advanced techniques can be utilized to compensate for Doppler offset in real-time and achieve symbol synchronization without a priori satellite information (e.g., [32, 54]).

As discussed in Section II, TDD is used with the frame structure shown in Fig. 2. Therefore, to estimate the UL channels at the SAPs, every UT transmits a pilot on the initial $\tau_{u}^{p}$ samples of the coherence block. Since we assume that the number of UTs is larger than the number of mutual orthogonal pilots (i.e., $K>\tau_{u}^{p}$), every subset of UTs is assigned the same pilot. The subset of UTs that are assigned the same pilot as UT $k$ is denoted by $\mathcal{C}_{k}$. Define $\sqrt{q_{k}}\psi_{k}\in\mathbb{C}^{\tau_{u}^{p}\times 1}$ as the $\tau_{u}^{p}$-length pilot sequence transmitted by the $k$th UT, where $q_{k}$ is the pilot power and $\psi_{k}^{H}\psi_{k}=||\psi_{k}||^{2}=\tau_{u}^{p}$. Therefore, the received signal vector at the $m$th SAP, $\textbf{y}_{m}^{p}\in\mathbb{C}^{\tau_{u}^{p}\times 1}$, from all $K$ UTs’ pilot transmissions is given thus:

where $\textbf{n}_{m}^{p}\sim\mathcal{CN}(\textbf{0}_{\tau_{u}^{p}},\sigma_{n^{p}}^{2}\textbf{I}_{\tau_{u}^{p}})$ is the additive white Gaussian noise (AWGN) vector.

To estimate the UL channel of UT $k$, sufficient statistics are derived from the received signal by calculating the inner product between the received signal vector, $\textbf{y}_{m}^{p}$, and $\psi_{k}$, as follows:

This is because

This statistic can be used to estimate the UL channel, $h_{m,k}$, at the $m$th SAP using techniques such as MMSE and linear MMSE (LMMSE) estimators. We assume that a phase-aware MMSE channel estimator is used. Therefore, the estimated UL channel can be given accordingly [36]:

with the following statistics

where $\mathbb{E}\{\cdot\},~{}\text{and}~{}\text{Var}\{\cdot\}$ are the expectation and variance operators, respectively.

Given that most of the traffic is in the DL direction, we consider the DL power allocation and manage the handover process. In the DL, the SAPs transmit the same symbol to the UT in a cooperative manner. Assume that the symbol to be sent to UT $k$ is $s_{k}\in\mathbb{C}$. Every symbol is precoded by a precoding vector $\textbf{v}_{k}=[v_{1,k},v_{2,k},\cdots,v_{M,k}]^{T}$, where $v_{m,k}\in\mathbb{C}$. Therefore, if the signal vector to be sent to the $K$ UTs is $\textbf{s}=[s_{1},s_{2},\cdots,s_{K}]^{T}$, then the signal vector to be transmitted by the $M$ SAPs is given thus:

where $\textbf{V}=[\textbf{v}_{1},\textbf{v}_{2},\cdots,\textbf{v}_{K}]$ is an $M\times K$ matrix.

Therefore, the signal received by the $k$th UT can be calculated by

where $\textbf{h}_{k}=[h_{1,k},h_{2,k},\cdots,h_{M,k}]^{T}$ and $n_{k}\sim\mathcal{CN}(0,\sigma^{2}_{n})$ is the AWGN noise. Assuming that the UT approximates the precoded channel by the average value $\mathbb{E}\{\textbf{v}_{k}^{H}\textbf{h}_{k}\}$, the signal-to-interference and noise (SINR) can be calculated by [36]

In this study, we adopt coherent beamforming as the technique used to minimize the interference between the UTs. Therefore, the precoding coefficient for the $k$th UT and $m$th SAP is $v_{m,k}=\sqrt{p_{m,k}}\hat{h}_{m,k}$, where $p_{m,k}$ is a power scaling factor and $\hat{h}_{m,k}$ is the estimated UL channel, which is valid for the DL direction by virtue of channel reciprocity. That is, the precoding vector for the $k$th UT is given by

where $\textbf{P}_{k}=\text{diag}\left(\frac{p_{1,k}}{\mathbb{E}\{|\hat{h}_{1,k}|^{2}\}},\frac{p_{2,k}}{\mathbb{E}\{|\hat{h}_{2,k}|^{2}\}},\cdots,\frac{p_{M,k}}{\mathbb{E}\{|\hat{h}_{M,k}|^{2}\}}\right)$ and $\hat{\textbf{h}}_{k}=[\hat{h}_{1,k},\hat{h}_{2,k},\cdots,\hat{h}_{M,k}]^{T}$ .

Accordingly, by using coherent beamforming as in (18), phase-aware MMSE channel estimation as in (9), and the SINR in (17), the SINR can be derived as in [36] as follows:

where

Therefore, the achievable DL data rate (in bps/Hz) of the $k$th UT served by this cluster of SAPs can be calculated as

To optimize the power allocation and the handover decisions such that the cluster throughput is maximized and the handover rate is minimized, we formulate the power allocation and handover processes as a multi-objective optimization problem, where the objective functions to be maximized are the UTs’ aggregate data rate and their service time before being switched to another cluster. For the latter, we maximize the number of served UTs with a guaranteed minimum data rate based on their link conditions. When the link condition does not allow optimized power allocation to serve the UT with the minimum required data rate, a handover request is issued. Then, a handover decision is taken when this repeats, and the visibility of the UT is confirmed by the next serving cluster, as discussed in Section II. That is, the two objectives of the optimization problem at the $t$th time slot are given as follows:

where $R_{k}[t]$ is the data rate of the $k$th UT during the $t$th time slot based on its channel conditions and power allocation during that time slot, as given in (27). $I_{k}[t]$ is an indicator variable that indicates whether the $k$th UT can be served by the cluster during the $t$th time slot with an acceptable data rate, through optimizing the power allocation, or it is infeasible and a handover might be considered. If a handover is triggered, then the implementation procedure would be as shown in Fig. 4. By introducing this indicator variable, we can jointly optimize the data rate of the UTs and their service time.

To deal with this multi-objective optimization problem, we construct a weighted sum of the two objectives to combine the two conflicting objectives into a single function. Thus, the complete handover-aware power allocation problem at time slot $t$ is formulated as follows:

where $\alpha$ is a weighting coefficient that combines the two competing objectives and can be used to prioritize them. That is, by setting $\alpha=0$, we target maximizing the aggregate UTs’ data rate only, and by setting $\alpha=1$, we target minimizing the handover rate only. Constraint (29a) is used to ensure that the served UTs satisfy their minimum rate level, where $R_{k}^{min}$ is the required minimum rate of UT $k$. Constraint (29c) is expressed to ensure that the total power scaling factors of every SAP are within the required range, where $P_{m}^{max}$ is the maximum total value of transmit power of the $m$th SAP. The binary value of the indicator variable, $I_{k}[t]$, and the non-negative value of $p_{u,k}$ are imposed by constraints (29b) and (29d), respectively.

Since the decision variables contain continuous (i.e., the power coefficient) and discrete (i.e., the handover indicator) ones and observing its structure, this optimization problem is modelled as a mixed-integer non-linear program (MINLP). This kind of optimization problem is known to be generally NP-hard due to its combinatorial behaviour [55]. Therefore, exponential computational complexity is required to solve the problem in (29). This means that solving this problem optimally cannot be done in real-time. In the next section, we leverage deep learning techniques to build a model that can predict efficient solutions to this problem in a computationally efficient manner. This is one of several benefits provided by AI techniques, as discussed in the next section.

During online operation, the DNN model calculates predictions for the desired output vector for the input values based on the experience gained in the training phase. For this purpose, the output of each layer is calculated as in (33). Therefore, the computational complexity to calculate $\textbf{z}^{(e)}$ is due to the multiplication operation and the activation function. Using the big-O notation, this computational complexity is in the order of $\mathcal{O}(U_{e}U_{e-1}+U_{e})=\mathcal{O}(U_{e}U_{e-1})$. Therefore, the complexity of calculating the output of the DNN model is $\sum_{e=1}^{E}\mathcal{O}(U_{e}U_{e-1})$. For example, if two hidden layers are used, and each layer has the same number of units as the input, then the computational complexity can be calculated as follows:

This is much lower than the exponential complexity of solving an NP-hard problem, such as (29). In addition, other benefits of using machine learning-based techniques are utilized, such as adaptability to the dynamics of the satellite link and topology, scalability, and support of distributed data processing and storage.