Evolutionary game-based cooperative strategy for effective capacity of multiple-input-multiple-output communications

Abstract

Cooperative-relaying networks have great potential for deployment in next-generation wireless communication networks. However, a cooperative-relaying network using two-dimensional multiple-input-multiple-output technology can further enhance the network performance. In this article, we propose a game theoretical framework for mobile two-dimensional multiple-input-multiple-output communication networks to achieve optimal relay selection and cooperative control. A source node, relay, and destination node can make service-selection decisions dynamically in a two-level network, based on the mobile-channel satisfaction parameters (e.g. spatial-temporal correlation and relay survivability). To model this dynamic interactive decision problem, we propose a hierarchical dynamic game framework. At the outer level, we formulate an evolutionary game to model and analyze the process of adaptive selection of relays for maximizing the multiple-input-multiple-output capacity by relay selection and power allocation. At the inner level, by aligning relays, we formulate an evolutionary game to model a self-organizing network structure for relays, to increase the capacity. A closed-loop evolutionary game equilibrium is considered to solve the dynamic game. Numerical results show that the proposed algorithm can effectively improve the quality of service for mobile two-dimensional multiple-input-multiple-output communication networks.

Keywords

Multiple-input-multiple-output communications capacity game theory relay selection survivability

Introduction

A cooperative communication network based on relays consists of transmitters, relays, and receivers, in which the relay nodes provide assistance for video data delivery in a multiple-input-multiple-output (MIMO) channel. Recently, a heightened attention has been focused on the fifth generation (5G) wireless communication networks, in which MIMO systems^1–3 are equipped with hundreds of antennas to serve tens of or more users simultaneously. Using multiuser MIMO and by concentrating power in a specific direction, the capacity of a MIMO system can be increased linearly with minimum number of transmitting and receiving antennas. The MIMO systems are more robust than the conventional MIMO systems as they offer extra degrees of freedom to update the working of the antenna arrays.

The capacity can be increased by allowing two or more antenna users to transmit data independently in the same sub-channel,⁴ which is achieved using the same frequency bands and time slots. If the number of receiving antennas on the relay is equal to or larger than the number of antennas on the receiver, the multiple data flowing from different antennas can be decoded at the receiver. This kind of cooperative transmission is called the relay-multiple-input-multiple-output (R-MIMO) transmission. Existing relay transmission schemes are designed to maximize the sum rate. The relay assignment and power allocation for the amplify-and-forward (AF) relay system can maximize the sum rates of all users.⁵ A joint power allocation and channel design conveys information from a source to a destination through multiple orthogonal sub-channels and maximizes the sum rate via optimal power allocation to the source and to the active channel.⁶ A joint cell, channel, and power allocation problem is formulated as an overall optimization problem, where the objective is to maximize the minimum user throughput, and a feasible cell allocation is obtained via either greedy allocation or exhaustive search.⁷ The relay transmits the combined signals that were received from both sources, with the aim of achieving better spectral efficiency. A relay cooperation scheme was proposed to investigate the trade-off between spectral efficiency and energy efficiency in multicell MIMO cellular networks.⁸ The relay node may be responsible for the control signals of the device-to-device (D2D) link (i.e. relay-aided D2D communication).⁹ The multiantenna cellular base station (BS) acts as a cooperative relay, helping the D2D nodes forward packets to improve the throughput of the network.¹⁰

The game theory framework formulates the interactions among the source nodes, relay nodes, and destination nodes. Game theory has been used to model routing, flow control, power control, and resource allocations in cooperative communications.^11–14 In Zappone and Jorswieck,¹¹ the AF cooperative communication scheme was modeled using the Stackelberg market framework, wherein a relay is willing to sell its resources, power, and bandwidth, to multiple users to maximize its revenue. In Liu et al.,¹³ a model of secure and dependable virtual service in a sensor cloud was proposed to improve the virtual capacity of sensor cloud services using a stochastic evolutionary coalition game. In Zhang et al.,¹⁴ two efficient competitive auction mechanisms were presented to optimize the traffic handling capacity gain through offloading with budget constraints. The existence and uniqueness of the Nash equilibrium (NE) were investigated using the concavity of the utility function and the exact potential game associated with the proposed utility function.¹⁵ The performance in cooperative communication depends on careful resource allocation such as relay selection and power control; a two-level Stackelberg game was employed to consider the benefits of the source node and the relay nodes, to help the source find the relays.¹⁶ Moreover, addressing the dynamic cooperative partner-selection problem with incomplete private information and an energy-aware perspective was a challenging issue; hence, a cooperative partner-selection evolutionary algorithm based on the Q-learning approach was proposed.¹⁷ A theoretical non-stationary three-dimensional (3D) wideband twin-cluster channel model for massive MIMO systems with carrier frequencies in the order of GHz was proposed. This model investigated the impact of cluster elevation angles on the correlation properties of the proposed massive MIMO channel models.¹⁸ Relaying is a promising technique in cellular networks for improving the system coverage capacity; hence, a selection criterion with a metric called survival probability for an energy-harvesting relay was proposed to support data transmissions.¹⁹

Since the number of antennas are to increase in next-generation wireless communication networks, the far-field assumptions in conventional MIMO channels and relay selection may no longer be appropriate, a maximum-achievable-capacity model was proposed for 3D massive MIMO systems.²⁰ The MIMO capacity converges to its lower bound when the distance between the transmitting and receiving antenna arrays goes to infinity. However, the mobile MIMO channels may be highly correlated; thus, Su et al.²⁰ may not offer the real time and survival capacity increase provided by the conventional ground/near-ground MIMO wireless communications. Michailidis et al.²¹ proposed 3D scattering modeling of MIMO mobile-to-mobile via stratospheric relay (MMSR) fading channels AF networks to evaluate the channel capacity, and they have no further consideration to power allocation under the condition of relay selection. Compared with this article, we consider capacity increase, power budget constraints, and investigate the cooperative mechanism of relays for MIMO communications between multiple transmitting and receiving antennas. In Shen et al.,²² the authors only investigated a relay-assisted switching scheme of MIMO system based on the joint consideration of the received signal-to-noise ratio (SNR) at the user equipment (UE) and 3D distances. However, they do not investigate the power and angle allocation among the transmitter, relay, and receiver antennas, which is an effective technique to maximize the capacity and mitigate interference in the MIMO communication. Chen et al.²³ proposed a new theoretical 3D geometrical scattering reference model for MIMO base station to relay station (BS2RS) backhaul channels. A 3D geometry-based stochastic scattering model (GBSSM) for wideband MIMO vehicle-to-vehicle (V2V) relay-based cooperative fading channel is proposed.²⁴ B Talha and M Patzold²⁵ dealt with the modeling and analysis of narrowband MIMO mobile-to-mobile (M2M) fading channels in relay-based cooperative networks, and general analytical solutions are presented for the four-dimensional (4D) space–time cross-correlation function (CCF), the 3D spatial CCF, the two-dimensional (2D) transmit (relay and receive) correlation function (CF), and the temporal autocorrelation function (ACF). However, they failed to consider survival capacity in the dynamic environments. In this article, we address the aforementioned limitations of past works on the power allocation in dynamic relay-based cooperative 3D MIMO communications by utilizing the one-ring scattering model in a 2D plane. Therefore, without loss of generality, we assume that mobile relays with multiple antennas are deployed between the multiple transmitter and receiver antennas to improve the capacity of MIMO communications.

The contributions of this article are summarized as follows:

We propose a hierarchical (i.e. two-level) dynamic relay-selecting approach based on the evolutionary game theory to investigate the cooperative mechanism of relays for MIMO communications between multiple transmitting and receiving antennas. Unlike the existing traditional relay selections and 3D MIMO models in the literature, the proposed framework considers the dynamic (i.e. alignment-dependent) and multiple transmitting antennas’ decisions. It also accounts for the relays’ survivability.

We show the advantages of dynamic strategies (i.e. dynamic alignment angle and dynamic power) over static strategies in terms of higher capacity. In addition, the replicator dynamics of the evolutionary game at the outer level converges fast to the evolutionarily stable state (ESS) of the relay selection combining power allocation with angle-alignment feedback.

We show that the replicator dynamics of the evolutionary game at the inner level for the non-stationary relay evolution of MIMO globally converges to the ESS, even for a small change in the transmission state. In addition, the stability of the relay network is ensured.

We extend the proposed evolutionary game framework to analyze the impact of the relays’ survivability on the equilibrium solutions, which improves the adaptive transmission capacity of MIMO communications.

The rest of the article is organized as follows: In section “System model,” we develop the general model for a dynamic relay-assisted MIMO communication network with arbitrary antenna array alignment. In section “The problem formulation for effective capacity,” we first consider the MIMO transmission decision-making problems involving the transmitter, relay, and receiver antennas. Then, we develop a two-level evolutionary game for the power control and angle allocation that enables capacity maximization for each transmitter antenna in the MIMO communication network. We also propose algorithms for selecting the relay using the evolutionary game. In section “Adaptive capacity based on survival evolutionary update of a relay,” we first describe the survival model for dynamic evolutionary relays; then, we design algorithms for adaptive capacity adjustment based on the survival evolutionary update when the transmitter antenna array and receiver encounter jamming attacks from malicious nodes or disconnection of links. Extensive numerical studies are presented in section “Numerical results.” Finally, a few conclusions are drawn in section “Conclusion.”

System model

In this section, we discuss the system model. Notations used in this article are summarized in Table 1.

Table 1.

Notations.

Notation	Meaning
$m$	The number of transmitter antennas
$k$	The number of relay antennas
$n$	The number of receiver antennas
$y$	The received signal vector
$x$	The transmitted signal vector
$P_{s, d}^{i}$	The normalized power
$P_{s, r}^{i}$	The normalized power
$H_{s, d}^{[l]}$	The channel coefficient matrices
$H_{s, r}^{[l]}$	The channel coefficient matrices
$D_{th}$	The threshold of the distance
$D_{s, d}$	The distance between s and d
$σ_{s, r}$ , $σ_{r, d}$ , $σ_{s, d}$	The noise power for different links
$η_{s, r}$ , $η_{s, d}$	The additive noise vectors
$\tilde{R}$	A ring with radius
$α_{s, r}^{[l]}$	Narrow angular spread
$D_{s, r}^{[l]}$	The average distance
$θ_{s, r}^{[l]}$	The angle of departure (AoD)
${\bar{θ}}_{s, r}^{[l]}$	The mean angle
$J_{0}$	The zeroth-order modified Bessel function
$I_{m}, I_{k}$	An identity matrix
$l$	An index of the antennas
$H$	The Hermitian operator
$H_{r, d}^{[l]}$	The spatial correlation matrix
$λ_{s, r}$	The spatial-temporal correlation values
$λ_{r, d}$	The spatial-temporal correlation values
$D_{r, d}^{[l]}$	The average distance
$θ$	The set of strategies of alignment angles
$p_{t}$	The set of strategies of power allocations
$P_{s, r}^{[i]} (θ_{s, r}^{[i]})$	The power azimuth spectrum
$p_{s, r}^{[l]}$	The allocated power
$γ^{a}$	The vector of an antenna array state
$θ_{r, s}^{[i]}$	The alignment angle of each relay antenna
$θ_{r, s}^{[l]}$	The alignment angle of the feedback
$θ_{s, r}^{[l]}$	The alignment angle of the transmitter
$p_{r, s}^{[l]}$	The power allocation for the feedback
${\bar{U}}_{j}^{b} (t)$	The average utility of the relay antenna
$\tilde{G}$	The gain of a MIMO system
$Z$	Different levels of output gain
$ϒ$	Independent protection groups
$λ_{hij}$	Different levels of STC value protections
$ε_{sr}$ , $ε_{rs}$	The rates of change of the feedback
$ρ$	The average SNR at each receive antenna
$U_{l}^{a}$	The utility of the transmitter antenna array
$U_{j}^{b}$	The MIMO utility in the relay
$P_{h} (θ_{hi}, θ_{hj})$	The allocated power for the angles
$Φ_{h}$	The partition of the set
$C_{h}^{a}$	The cost to select alignment angles
$φ (h, θ_{hi}, θ_{hj})$	The cardinality of each subset $Φ_{hij}$
$C_{h}^{as}$	The total cost
$C_{mu} (θ, λ)$	The mutation total cost
$C_{ge} (θ, λ)$	The genetic total cost
$Δ (x_{i}, y_{i})$	The expected changes in $x_{i}$ and $y_{i}$
$C_{1}$ , $C_{2}$ . $C_{3}$ , $C_{4}$	Temporary variables
$C_{rs}^{(1)}$ , $C_{rs}^{(2)}$	The average capacity of each relay

MIMO: multiple-input-multiple-output; STC: spatial-temporal correlation; SNR: signal-to-noise ratio.

We consider a dynamic relay-assisted MIMO communication network, where mobile BSs are treated as potential relay nodes and ad hoc techniques are exploited to assist the self-organization of the relays. Zhu and Zhang²⁶ proposed resource allocation scheme and used the simple half-duplex relay protocols with AF to maximize the capacity for relay networks. AF relay links were also applied to 3D scattering MIMO model.²¹ Therefore, we employ the AF cooperation protocol as relaying strategies to trigger the adaption of the power and alignment angles among the transmitter, relay, and receiver antennas. Figure 1 shows an example of a relay-assisted MIMO communication network, including three transmitters equipped with three antennas, each of three relays equipped with three antennas and three receivers equipped with three antennas, in which the distance between the transmitter and receiver antenna array reaches the threshold $D_{th}$ . By allowing cooperation via relays, it is possible to increase the service coverage capacity, balance traffic, and improve energy awareness.

Figure 1.

Example of a relay-assisted MIMO communication network.

We concentrate on a distributed system where transmitters with different quality-of-service (QoS) requirements play repeated games for the relay. The transmitters make decisions to select relay antenna in the games with the consideration of their capacities. We consider the scenario that there are $m$ transmission antennas, $k$ potential relay antennas, and $n$ receiver antennas arrays denoted by $s = {s_{1}, s_{2}, \dots, s_{m}}$ , $r = {r_{1}, r_{2}, \dots, r_{k}}$ , and $d = {d_{1}, d_{2}, \dots, d_{n}}$ , respectively. We focus on the simple half-duplex relay protocols with AF.

MIMO channel model

The received signals of the mobile MIMO communication system can be modeled as

y_{s, d} = \sqrt{P_{s, d}} H_{s, d}^{[l]} x + η_{s, d}

(1)

y_{s, r} = \sqrt{P_{s, r}} H_{s, r}^{[l]} x + η_{s, r}

(2)

where $y = [y_{1}, \dots, y_{k}]^{T}$ is the received signal vector; $P_{s, d}$ and $P_{s, r}$ are the transmission power at the transmitters. $x = [x_{1}, \dots, x_{m}]^{T}$ is the transmitted signal vector with a normalized average power constraint $\sum_{i = 1}^{m} P_{s, d}^{i} x_{i} \leq 1$ and $\sum_{i = 1}^{m} P_{s, r}^{i} x_{i} \leq 1$ ; $H_{s, d}^{[l]}$ and $H_{s, r}^{[l]}$ are the channel coefficient matrices of antennas. $η_{s, r}$ and $η_{s, d}$ are additive noise vectors whose elements are assumed to be independent and ideally distributed (i.i.d.) complex Gaussian random variables with zero mean. Without loss of generality, we assume that the noise power is denoted by $σ_{s, r}$ , $σ_{r, d}$ , and $σ_{s, d}$ for different links. In the MIMO systems, if there are not enough local scattering surrounding the relay, then relays that have large numbers of antennas must be deployed to improve local scattering in mobile environments, correspondingly, channel measurements have shown that most of the signal energy is localized in the azimuth direction.²⁷ Therefore, we consider the one-ring local scattering model^28,29 to characterize the MIMO channel. The local scattering surrounding the relay is modeled by a ring with radius $\tilde{R}$ ; whereas, the transmission signal from the transmitter has a narrow angular spread (AS) denoted by $α_{s, r}^{[l]} = 2 \tan^{- 1} (R / D_{s, r}^{[l]})$ , where $D_{s, r}^{[l]}$ is the average distance between the transmitter $s$ and the relay $r$ , $l$ is an index of antennas. Because of the transmitter antennas align within the power azimuth, $P_{s, r}$ is only function of $θ_{s, r}$ . We use the von Mises model to characterize the power azimuth spectrum (PAS) with respect to $θ_{s, r}^{[l]}$ ²⁸ in the relay $l$ as follows

P_{s, r}^{[l]} (θ_{s, r}^{[l]}) = \frac{\exp [α_{s, r}^{[l]} \cos (θ_{s, r}^{[l]} - {\bar{θ}}_{s, r}^{[l]})]}{2 π J_{0} (α_{s, r}^{[l]})}

(3)

where $θ_{s, r}^{[l]}$ is the angle of departure (AoD) of a path from the transmitter $s$ to the relay $r$ . ${\bar{θ}}_{s, r}^{[l]}$ is the mean angle of the AoD. $J_{0}$ is the zeroth-order modified Bessel function. $H_{s, r}^{[l]}$ describes the spatial correlations between $p th$ and $q th$ antenna elements between the transmitter antenna array and the potential relay antenna array and is defined as

\begin{matrix} {[H_{s, r}^{[l]}]}_{(p, q)} = \int_{- π}^{π} e^{j [ϕ_{p}^{[l]} (θ_{xz}) - ϕ_{q}^{[l]} (θ_{xz})]} P_{s, r}^{[l]} (θ_{xz}) d θ_{xz} \\ + \int_{0}^{2 π} e^{j [ϕ_{p}^{[l]} (θ_{xy}) - ϕ_{q}^{[l]} (θ_{xy})]} P_{s, r}^{[l]} (θ_{xy}) d θ_{xy} \end{matrix}

(4)

where we $ϕ_{p}^{[l]} (θ_{xz}) - ϕ_{q}^{[l]} (θ_{xz})$ and $ϕ_{p}^{[l]} (θ_{xy}) - ϕ_{q}^{[l]} (θ_{xy})$ account for the phase difference between $p th$ and $q th$ antenna elements in the azimuth direction in the $(x, z) - plane$ and $(x, y) - plane$ , respectively, at the transmitter antenna array.

The capacity of the relay-assisted MIMO communication without alignment angles

The capacity of the mobile MIMO communication network between the transmitter $s$ and the relay $r$ without alignment angles is

C_{s, r} = lo g_{2} [\det (I_{m} + \sum_{l = 1}^{m} (\frac{P_{s, r}^{[l]}}{m σ_{s, r}} H_{s, r}^{[l]} \overset{H}{(H_{s, r}^{[l]})}))]

(5)

where $I_{m}$ is an identity matrix of size $m$ , $H$ stands for the Hermitian operator at the transmitter antenna array. The MIMO capacity may reach its lower bound when the distance $D_{s, d} \to \infty$ .²⁰ The relay helps the transmitter antennas forward packets along multiple paths to improve the throughput of the network. The multipath complex gains between the transmitter antenna and the receiver antenna can be presented as

H_{s, r, d}^{[l]} = H_{s, r}^{[l]} + H_{r, d}^{[l]}

(6)

where $H_{r, d}^{[l]}$ is the channel coefficient matrices at the relay antenna. The capacity of the relay-assisted MIMO communication network between the transmitter $s$ and the receiver $d$ is calculated from equation (7), where $I_{k}$ is an identity matrix of size $k$

\begin{array}{l} C_{s, r, d} = \\ l o g_{2} [d e t (I_{m} + \sum_{l = 1}^{m} (\frac{P_{s, r}^{[l]}}{m σ_{s, r}} H_{s, r}^{[l]} \overset{H}{(H_{s, r}^{[l]})})) + d e t (I_{k} + \sum_{l = 1}^{k} (\frac{P_{r, d}^{[l]}}{k σ_{r, d}} H_{r, d}^{[l]} \overset{H}{(H_{r, d}^{[l]})}))] \end{array}

(7)

The capacity of the relay-assisted MIMO communication with alignment angles

If the transmitter antennas work with the relay cooperatively using alignment angle, the achieved cooperative capacity of the mobile MIMO communication network is

\begin{array}{l} C (P_{s, r, d}^{[l]} (θ_{s, r, d}^{[l]})) \\ = l o g_{2} (1 + \frac{P_{s, r}^{[l]} (θ_{s, r}^{[l]}) λ_{s, r}^{2}}{m σ_{s, r} \overset{2}{(D_{s, r}^{[l]})}} + \frac{P_{r, d}^{[l]} (θ_{r, d}^{[l]}) λ_{r, d}^{2}}{k σ_{r, d} \overset{2}{(D_{r, d}^{[l]})}}) \end{array}

(8)

where $λ_{s, r}$ and $λ_{r, d}$ are the spatial-temporal correlation values (STC values). $D_{r, d}^{[l]}$ is the average distance between the relay and receiver antennas.

The problem formulation for effective capacity

To optimize capacity in equation (8), we proposed a hierarchical dynamic game framework. It is composed of two levels: an outer-level evolutionary game for modeling the dynamic alignment strategy selection of transmitter antenna arrays and an inner-level evolutionary game for modeling the dynamic feedback strategy of relay antenna arrays (Figure 2).

Outer-level evolutionary game: the transmitter antenna could dynamically adjust alignment angle $θ_{s, r}^{[l]}$ according to capacity $C_{j}$ and feedback alignment angle $θ_{s, r}^{[l]}$ . The received capacity and feedback alignment angle are time-varying, which depends on the power allocation strategy controlled by the transmitter antenna array. Without complete information, all transmitter antennas gradually learn during the capacity and feedback alignment angle adaptation process. The evolutionary game with transmitter antennas having bounded rationality is a suitable tool for modeling this learning process. With an evolutionary game model using replicator dynamics, at the equilibrium, all the transmitter antennas in the same population receive the same individual utility as the average.

Inner-level evolutionary game: the relay antennas dynamically adjusts its $θ_{r, s}^{[l]}$ according to dynamic capacity $C_{i}$ to maximize their own payoffs. We use an evolutionary game to analyze the dynamic feedback strategy of the relay antenna. The main motivation behind using a game is that the outer-level players (i.e. the transmitter antenna) need to make the optimal decision dynamically according to the time-varying capacity and feedback alignment angle of the inner-level relay antennas, while conventional non-cooperative game theory can be used mostly for static analysis of the capacity and feedback alignment angle. The optimization problem to maximize the capacity with alignment angles of the MIMO channel is formulated as follows

\begin{matrix} max_{θ, p_{t}} C_{l} (θ, p_{t}) \\ s . t . p_{ti, min} \leq p_{ti} \leq p_{ti, max} \\ θ_{i, min} \leq θ_{i} \leq θ_{i, max} \end{matrix}

(9)

Figure 2.

Hierarchical dynamic game framework for the power allocations corresponding to the alignment angles.

From equation (9), we can show that once all the transmitter antennas have made their individual decisions, their chosen strategies form a profile $(θ, p_{t})$ , where $θ = [θ_{1}, θ_{2}, \dots, θ_{m}]$ is the set of strategies for the alignment angles and $p_{t} = [p_{t 1}, p_{t 2}, \dots, p_{m}]$ is the set of strategies for the power allocations corresponding to the alignment angles.

Evolutionary game among transmitter antennas at the outer level

The objective of the power and angle allocation is to enable each transmitter antenna in the MIMO communication network to maximize the capacity. Therefore, it is a typical two-stage leader–follower game that can be analyzed using an evolutionary game. We first study the evolutionary behavior among the transmitter antennas, which selects the relay antennas for channel sharing. Multiple transmitter antennas may allocate the same alignment angle to the same relay antenna, which may reduce the utility of the relay antenna. Hence, the relay antenna will send a new feedback angle to the transmitter antennas in order to achieve higher utility. Consequently, these transmitter antennas change their alignment angles and powers and switch to a different relay antenna. This process is repeated several times until all transmitter antennas in the same array achieve identical capacities. As indicated, each transmitter antenna cooperatively chooses a relay antenna according to its own capacity. Therefore, to analyze the behavior of the transmitter antennas, we employ an evolutionary game framework.

We define the basic components of the evolutionary game framework for the transmitter antennas as follows:

Players: each transmitter and relay antenna in the network shares the transmitter and relay antenna information, as well as the network status, in this evolutionary game.

Population: refers to the antenna arrays of transmitters and relays in the network; each antenna array forms an independent population.

Strategy: the set of strategies refers to the transmitter antennas available to alignment angles and the corresponding power allocations for them.

Utility: the utility of each player is defined as the capacity that can be achieved by proper power allocation and the use of correct alignment angles.

A transmitter antenna is always willing to align with a relay antenna at an angle such that a higher capacity is attained. The transmitter antennas within an antenna array can communicate with each other and exchange information about their strategies. If one antenna observes that another antenna has chosen a different relay antenna that has a higher capacity, it may learn the strategy of the observed antenna and gradually changes its alignment angle with the hope of achieving a higher capacity. We assume that all transmitter antennas choosing the same relay are allocated identical powers and alignment angles. Then, the utility of the transmitter antenna array $a$ aligned with the relay antenna $l$ is defined as

U_{l}^{a} = C (P_{s, r}^{[l]} (θ_{s, r}^{[l]})) - p_{s, r}^{[l]} (θ_{s, r}^{[l]})

(10)

So, the utility function can be written as

U_{l}^{a} = \log_{2} (1 + \frac{\sum_{i = 1}^{m} P_{s, r}^{[i]} (θ_{s, r}^{[i]}) λ_{s, r}^{2}}{m σ_{s, r} {(D_{s, r}^{[l]})}^{2}}) - p_{s, r}^{[l]} (θ_{s, r}^{[l]})

(11)

where $P_{s, r}^{[l]} (θ_{s, r}^{[l]}) = \sum_{i = 1}^{m} P_{s, r}^{[i]} (θ_{s, r}^{[i]})$ , $P_{s, r}^{[i]} (θ_{s, r}^{[i]})$ is the PAS of each transmitter antenna, $i$ is an index of the power allocations corresponding to the alignment angles, and $p_{s, r}^{[l]} (θ_{s, r}^{[l]})$ is the allocated power for the changed alignment angle $θ_{s, r}^{[l]}$ . As we have described, the transmitter antennas in the same antenna array can learn from each other’s strategy; hence, the strategy of one player in an antenna array can be replicated by other players in the same antenna array. Here, we introduce replicator dynamics to describe the evolution in the antenna array. As per replicator dynamics, the sharing of a strategy in an antenna array grows at a rate equal to the difference between the utility of that strategy and the average utility of the antenna array. For the antenna array $a$ , let $γ^{a}$ denote the vector of an antenna array state. $γ_{i}^{a}$ is the antenna array share of the strategy $i$ . Then, $γ^{a}$ can be expressed as

γ^{a} = [γ_{1}^{a}, γ_{2}^{a}, \dots, γ_{m}^{a}]

(12)

In the evolutionary game of relay-alignment selection, the decisions of the transmitter antennas on the relay-alignment selection at time $t$ are made according to the antenna array state at $t - 1$ . In this case, the replicator dynamics³⁰ with delayed information is given by

{\overset{\cdot}{γ}}_{i}^{a} (t) = γ_{i}^{a} (t - 1) (π_{i}^{a} (t - 1) - {\bar{π}}_{i}^{a} (t - 1))

(13)

where $γ_{i}^{a} (t - 1)$ denotes the proportion of antennas choosing new alignment angle at time $t - 1$ . ${\bar{π}}_{i}^{a} (t - 1)$ is the average capacity of the entire antenna array. The average capacity is given by

{\bar{π}}^{a} = \frac{1}{m} \sum_{i = 1}^{m} π_{i}^{a} γ_{i}^{a}

(14)

This equation shows that the transmitters sharing antenna’s strategy can provide higher capacity and their capacities will increase with time. The relay with multiple antennas share the alignment angle information. Each antenna in the same relay can get information about other antennas’ allocation of the alignment angle and adjust new alignment angle to resist jamming attack for improving transmission capacity according to attained information. Based on this replicator dynamics of the antennas in relay, the number of antennas as player to choose the probability of alignment angle increases if their capacity is above the average capacity. It is impossible for an antenna to choose new alignment angle, which provides a lower capacity than the current capacity. This replicator dynamics satisfies the condition of $\sum_{i = 1}^{m} {\overset{\cdot}{γ}}_{i}^{a} (t) = 0$ , and it indicates that the summation of all transmitter antenna shares does not vary with time. An evolutionary equilibrium is a fixed point of the replicator dynamics. At this fixed point, which can be obtained numerically, capacities of all antennas in the relay are identical. In other words, since the rate of strategy adaptation is zero (i.e. ${\overset{\cdot}{γ}}_{i}^{a} (t) = 0$ ), there is no antenna that deviates angle to gain a higher capacity. Hence, an evolutionary equilibrium can be attained. Since the rate of change of the alignment angle is zero, no transmitter antenna has an incentive to change its chosen alignment angle under the evolutionary equilibrium condition.

Cooperation among relay antennas at the inner level

Multiple antennas in a relay can exchange information about their load capacities. When the load capacity of an antenna is equal to the average capacity of the entire relay, the load capacity of this antenna is the maximum possible value in the relay. For incoming link requests from a transmitter antenna array, the antenna in the relay sends a feedback with a new alignment angle. Furthermore, each antenna in the relay must send a proper feedback with a new alignment angle in order to maximize its utility. Hence, the cooperative game among these relay antennas can be defined as a tuple $Φ = < V, θ_{j}, U_{j}^{b} >$ , where $V$ is the set of relay players and $θ_{j}$ is the feedback strategy set of relay player $j$ . Let $P (θ) = [p_{r, s}^{[1]} (θ_{r, s}^{[1]}), p_{r, s}^{[2]} (θ_{r, s}^{[2]}), \dots, p_{r, s}^{[k]} (θ_{r, s}^{[k]})]$ be the feedback strategy profile. $U_{j}^{b}$ is the utility function of player $j$ in the relay antenna array $b$ . Then, the MIMO utility about capacity in the relay can be calculated as

U_{j}^{b} = \log_{2} (\sum_{i = 1}^{k} θ_{r, s}^{[i]} + k \frac{P_{s, r}^{[l]} (θ_{s, r}^{[l]}) λ_{s, r}^{2}}{σ_{s, r} {(D_{s, r}^{[l]})}^{2}}) - p_{r, s}^{[l]} (θ_{r, s}^{[l]})

(15)

where $θ_{r, s}^{[l]} = \sum_{i = 1}^{k} θ_{r, s}^{[i]}$ is the total alignment angle of the relay antenna $l$ sent in the feedback between the relay $r$ and the transmitter $s$ . $θ_{r, s}^{[i]}$ is the alignment angle of each relay antenna, $i$ is an index of the alignment angle of each relay antenna, and $p_{r, s}^{[l]} (θ_{r, s}^{[l]})$ is the power allocation for the alignment feedback angle $θ_{r, s}^{[l]}$ .

Replicator dynamics of transmitter antenna array

The evolutionary game converges to its evolutionary equilibrium when $\sum_{i = 1}^{m} {\overset{\cdot}{γ}}_{i}^{a} (t) = 0$ . The evolution will stop when the transmitter antennas choosing different relay antennas obtain identical utility $U_{1}^{a} = U_{2}^{a} = \cdot \cdot \cdot = U_{m}^{a}$ , as no transmitter antenna in the antenna array will be able to find another antenna with a higher utility. Therefore, the evolutionary equilibrium can be obtained by solving $U_{1}^{a} = U_{2}^{a}$ and the following equation

\begin{matrix} \log_{2} (1 + \frac{P_{s, r}^{[1]} (θ_{s, r}^{[1]}) λ_{s, r}^{2}}{m σ_{s, r} {(D_{s, r}^{[1]})}^{2}}) - p_{s, r}^{[1]} (θ_{s, r}^{[1]}) \\ = \log_{2} (1 + \frac{P_{s, r}^{[2]} (θ_{s, r}^{[2]}) λ_{s, r}^{2}}{m σ_{s, r} {(D_{s, r}^{[2]})}^{2}}) - p_{s, r}^{[2]} (θ_{s, r}^{[2]}) \end{matrix}

(16)

We can get

\begin{matrix} λ_{s, r} = D_{s, r}^{[1]} D_{s, r}^{[2]} \sqrt{\frac{(A - 1) m σ_{s, r}}{(\overset{2}{(D_{s, r}^{[2]})} P_{s, r}^{[1]} (θ_{s, r}^{[1]}) - A \cdot \overset{2}{(D_{s, r}^{[1]})} P_{s, r}^{[2]} (θ_{s, r}^{[2]}))}} \\ = \sqrt{\frac{(A - 1) m σ_{s, r}}{(\frac{P_{s, r}^{[1]} (θ_{s, r}^{[1]})}{\overset{2}{(D_{s, r}^{[1]})}} - A \cdot \frac{P_{s, r}^{[2]} (θ_{s, r}^{[2]})}{\overset{2}{(D_{s, r}^{[2]})}})}} \end{matrix}

(17)

where $A = e^{\ln 2 \times (p_{s, r}^{[1]} (θ_{s, r}^{[1]}) - p_{s, r}^{[2]} (θ_{s, r}^{[2]}))}$ .

The stability of the evolutionary game between relay antennas

During the evolution of the STC between the transmitter and relay antennas, each relay antenna will adjust its feedback strategy to achieve a higher utility. After substituting equation (17) into equation (15), we can obtain equation (18)

U_{j}^{b} = \log_{2} (\sum_{i = 1}^{k} θ_{r, s}^{[i]} + \frac{{mkP}_{s, r}^{[l]} (θ_{s, r}^{[l]}) \cdot {(D_{s, r}^{[1]} D_{s, r}^{[2]})}^{2} (A - 1)}{{(D_{s, r}^{[l]})}^{2} \cdot ({(D_{s, r}^{[2]})}^{2} P_{s, r}^{[1]} (θ_{s, r}^{[1]}) - A \cdot {(D_{s, r}^{[1]})}^{2} P_{s, r}^{[2]} (θ_{s, r}^{[2]}))}) - p_{r, s}^{[l]} (θ_{r, s}^{[l]})

(18)

\begin{array}{l} \frac{\partial U_{j}^{b}}{\partial θ_{r, s}^{[l]}} = (\frac{{(D_{s, r}^{[l]})}^{2} \cdot ({(D_{s, r}^{[2]})}^{2} P_{s, r}^{[1]} (θ_{s, r}^{[1]}) - A \cdot {(D_{s, r}^{[1]})}^{2} P_{s, r}^{[2]} (θ_{s, r}^{[2]}))}{\ln 2 \cdot [θ_{r, s}^{[l]} \cdot {(D_{s, r}^{[l]})}^{2} \cdot ({(D_{s, r}^{[2]})}^{2} P_{s, r}^{[1]} (θ_{s, r}^{[1]}) - A \cdot {(D_{s, r}^{[1]})}^{2} P_{s, r}^{[2]} (θ_{s, r}^{[2]})) + m k P_{s, r}^{[l]} (θ_{s, r}^{[l]}) \cdot {(D_{s, r}^{[1]} D_{s, r}^{[2]})}^{2} (A - 1)]}) \\ - {(p_{r, s}^{[l]} (θ_{r, s}^{[l]}))}^{'} \end{array}

(19)

The first-order derivative of $U_{j}^{b}$ with respect to $θ_{r, s}^{[l]}$ is calculated in equation (19) to reach a stable point of NE of the evolutionary. Afterward, the optimal power of each relay antenna and can be computed in equation (20) by setting the resulting $\partial U_{j}^{b} / \partial θ_{r, s}^{[l]} = 0$ , and there also exists the optimal power allocation strategy $p_{r, s}^{* [l]} (θ_{r, s}^{[l]})$ for the relay to maximize $U_{j}^{b}$ .

(p_{r, s}^{[l]} (θ_{r, s}^{[l]}))' = \underset{p_{r, s}^{* [l]} (θ_{r, s}^{[l]})}{\underset{︸}{\frac{\exp [α_{r, s}^{[l]} \cos (θ_{r, s}^{[l]} - {\bar{θ}}_{r, s}^{[l]})]}{2 π J_{0} (α_{r, s}^{[l]})}}} \cdot (α_{r, s}^{[l]}) \cdot (- \sin (θ_{r, s}^{[l]} - {\bar{θ}}_{r, s}^{[l]}))

(20)

p_{r, s}^{* [l]} (θ_{r, s}^{[l]}) = \frac{{(D_{s, r}^{[l]})}^{2} \cdot (A \cdot {(D_{s, r}^{[1]})}^{2} P_{s, r}^{[2]} (θ_{s, r}^{[2]}) - {(D_{s, r}^{[2]})}^{2} P_{s, r}^{[1]} (θ_{s, r}^{[1]}))}{\ln 2 \cdot (α_{r, s}^{[l]}) \cdot \sin (θ_{r, s}^{[l]} - {\bar{θ}}_{r, s}^{[l]}) [θ_{r, s}^{[l]} \cdot {(D_{s, r}^{[l]})}^{2} \cdot ({(D_{s, r}^{[2]})}^{2} P_{s, r}^{[1]} (θ_{s, r}^{[1]}) - A \cdot {(D_{s, r}^{[1]})}^{2} P_{s, r}^{[2]} (θ_{s, r}^{[2]})) + {mkP}_{s, r}^{[l]} (θ_{s, r}^{[l]}) \cdot {(D_{s, r}^{[1]} D_{s, r}^{[2]})}^{2} (A - 1)]}

(21)

{(p_{r, s}^{* [l]} (θ_{r, s}^{* [l]}))}_{\min} = \frac{{(D_{s, r}^{[l]})}^{2} \cdot (A \cdot {(D_{s, r}^{[1]})}^{2} P_{s, r}^{[2]} (θ_{s, r}^{[2]}) - {(D_{s, r}^{[2]})}^{2} P_{s, r}^{[1]} (θ_{s, r}^{[1]}))}{\ln 2 \cdot (α_{r, s}^{[l]}) [π \cdot {(D_{s, r}^{[l]})}^{2} \cdot ({(D_{s, r}^{[2]})}^{2} P_{s, r}^{[1]} (θ_{s, r}^{[1]}) - A \cdot {(D_{s, r}^{[1]})}^{2} P_{s, r}^{[2]} (θ_{s, r}^{[2]})) + {mkP}_{s, r}^{[l]} (θ_{s, r}^{[l]}) \cdot {(D_{s, r}^{[1]} D_{s, r}^{[2]})}^{2} (A - 1)]}

(22)

According to equation (3), the power is the $θ$ function in the STC, the optimal power corresponding to the alignment angle in the relay $l$ can be calculated with equation (21). From equation (21), we can see that for $θ_{r, s}^{[l]}$ and ${\bar{θ}}_{r, s}^{[l]}$ in the relay $l$ , if $θ_{r, s}^{[l]} = π$ and ${\bar{θ}}_{r, s}^{[l]} = \frac{π}{2}$ , then the optimal power corresponding to the alignment angle in the relay $l$ achieves the minimum in (22). By combining equations (15) and (22), the MIMO maximum capacity in the relay antenna $l$ can be calculated as

C_{\max}^{b} = \log_{2} ({\bar{θ}}_{r, s}^{[l]} + k \frac{P_{s, r}^{[l]} (θ_{s, r}^{[l]}) λ_{s, r}^{2}}{σ_{s, r} {(D_{s, r}^{[l]})}^{2}}) - {(p_{r, s}^{* [l]} (θ_{r, s}^{* [l]}))}_{\min}

(23)

Transmitter antennas to select relay using evolutionary game

We present an iterative algorithm for the transmitter antennas to converge to the evolutionary equilibrium. Based on replicator dynamics, each transmitter antenna changes its strategy to maximize its capacity. Since the controller of a transmitter antenna array is responsible for collecting feedback from a relay, the transmitter antennas in the same array can exchange information; the current choice and capacity of one transmitter antenna will be available to all other antennas in the same array, but they will not be available to the antennas in other arrays. A transmitter antenna can also receive the average capacity of its own array. The transmitter antenna changes its alignment angle with a certain power if it finds another transmitter antenna having a higher capacity in the same array. When all transmitter antennas in the same array obtain the same capacity, the evolution is complete. Algorithms 1 and 2 describe the specific procedures for selecting a relay. As indicated by the evolution game framework for the transmitter antenna array in the MIMO communication network, all transmitter antennas in the same array can obtain equal capacity at equilibrium. Algorithm 1 shows the process executed by each transmitter antenna (Table 2). Algorithm 2 shows the process executed by each relay antenna. In Table 3, ${\bar{U}}_{j}^{b} (t)$ is the average utility of the relay antenna array.

Table 2.

The specific procedure of selecting the relay for each transmitter antenna at the outer level.

Algorithm 1. Executed by each transmitter antenna.
1. Initially, each transmitter antenna links to an available relay antenna randomly using an alignment angle with a certain power. 2. Each transmitter antenna computes its utility using the allocated alignment angle with the certain power according to equation (11). 3. After communicating with the other antennas in the same array, each transmitter antenna gets the choices and utilities of the other antennas in the array. 4. If $U_{1}^{a} \neq U_{2}^{a} \neq \cdot \cdot \cdot \neq U_{m}^{a}$ , then the transmitter antenna changes its alignment angle with some power to that of another relay antenna that offers a higher capacity. Otherwise, the transmitter antenna keeps the current alignment angle. 5. Repeat procedures 2 to 4. 6. Computes the spatial-temporal correlation $λ$ according to equation (17).

Algorithm 1. Executed by each transmitter antenna.

1. Initially, each transmitter antenna links to an available relay antenna randomly using an alignment angle with a certain power.
2. Each transmitter antenna computes its utility using the allocated alignment angle with the certain power according to equation (11).
3. After communicating with the other antennas in the same array, each transmitter antenna gets the choices and utilities of the other antennas in the array.
4. If

U_{1}^{a} \neq U_{2}^{a} \neq \cdot \cdot \cdot \neq U_{m}^{a}

, then the transmitter antenna changes its alignment angle with some power to that of another relay antenna that offers a higher capacity. Otherwise, the transmitter antenna keeps the current alignment angle.
5. Repeat procedures 2 to 4.
6. Computes the spatial-temporal correlation

λ

according to equation (17).

Table 3.

The specific procedure of selecting the transmitter for each relay antenna at the inner level.

Algorithm 2. Executed by each relay antenna
1. For each relay antenna array do 2. Set the initial alignment angle with the certain power for the relay antenna. 3. Repeat 4. Wait for a period $T$ for the alignment angles from transmitter antennas. 5. Each relay computes its utility according to equation (18). 6. If $θ_{r, s}^{[l]} \neq π$ and ${\bar{θ}}_{r, s}^{[l]} \neq \frac{π}{2}$ then 7. Adjust alignment angle according to equation (21). 8. Else Allocate alignment angle $θ_{r, s}^{[l]}$ . 9. Send feedback information ${(p_{r, s}^{* [l]} (θ_{r, s}^{* [l]}))}_{\min}$ to the controller of the transmitter antenna array. 10. Until ${\bar{U}}_{j}^{b} (t) = U_{j}^{b} (t)$ . 11. End for

Algorithm 2. Executed by each relay antenna

1. For each relay antenna array do
2. Set the initial alignment angle with the certain power for the relay antenna.
3. Repeat
4. Wait for a period

T

for the alignment angles from transmitter antennas.
5. Each relay computes its utility according to equation (18).
6. If

θ_{r, s}^{[l]} \neq π

and

{\bar{θ}}_{r, s}^{[l]} \neq \frac{π}{2}

then
7. Adjust alignment angle according to equation (21).
8. Else Allocate alignment angle

θ_{r, s}^{[l]}

.
9. Send feedback information

{(p_{r, s}^{* [l]} (θ_{r, s}^{* [l]}))}_{\min}

to the controller of the transmitter antenna array.
10. Until

{\bar{U}}_{j}^{b} (t) = U_{j}^{b} (t)

.
11. End for

Adaptive capacity based on survival evolutionary update of a relay

Model description of survival evolution

For MIMO communications, we define relay-assisted success as its ability to meet a demanded STC value $λ$ and measure the system survivability³¹ as

S_{MIMO} = \Pr {G \geq λ}

(24)

where $G$ is the gain of a MIMO system. For MIMO, which has a finite number of states, there can be $Z$ different levels of output gain performance: $G_{MIMO} \in G = {G_{z}, 1 \leq z \leq Z}$ , and the system output performance distribution can be defined by two finite vectors $G$ and $q = {q_{z}}$ , where $q_{z} = \Pr {G = G_{z}}, 1 \leq z \leq Z$ . We can define relay-assisted MIMO system survivability as the probability that a system remains in those states in which $G_{z} \geq λ$ and

S_{MIMO} (λ) = \sum_{G_{z} \geq λ} q_{z}

(25)

Antenna switching, based on reconfigurable antennas, and opportunistic communication were introduced in the interference-alignment networks in Zhao et al.^32,33 to reduce the sizes of the receivers and to improve the signal-to-interference-plus-noise ratio (SINR) performance of the antenna-selection scheme. Hence, we consider a MIMO system consisting of linked transmitter antennas. For each antenna, different levels of an alignment protection can be chosen. It is assumed that the antenna alignment, for an antenna belonging to the same array, can be destroyed by the impact of jamming attacks or mobility. The alignment angles belonging to antenna array $h$ can be separated into $ϒ$ independent protection groups, where $ϒ$ can vary from 1 to $E_{h}$ . For the protection groups $i$ and $j$ , within the antenna array $h$ , different levels of STC value protections can be chosen through $λ_{hij}$ , and there exists a pair of vectors of different alignment angles at available powers. A vector contains a pair $(θ_{hj}, θ_{hi})$ of alignment angles for the powers chosen for the transmitter antenna group $i$ and receiver antenna group $j$ . When $G_{z} < λ_{hij}, θ_{hi} \neq θ_{hj}$ , the alignment angles of the transmitter antenna group $i$ and receiver antenna group $j$ are alignment-separation; while $G_{z} \geq λ_{hij}, θ_{hi} = θ_{hj}$ , their alignment angles keep each other cooperatively. The total cost of the alignment angles chosen for the $i th$ and $j th$ protection groups in the antenna array $h$ is

C_{h}^{a} = \sum_{i, j = 1}^{E_{h}} P_{h} (θ_{hi}, θ_{hj})

(26)

where $P_{h} (θ_{hi}, θ_{hj})$ is the allocated power for the alignment angles of the $i th$ and $j th$ protection groups in the antenna array $h$ . The alignment-separation problem for each antenna array can be considered as a problem of partitioning a set $Φ_{h}$ , where each set can contain 0 to $E_{h}$ elements. The partition of the set $Φ_{h}$ can be represented by the vector $λ_{hij}$ . One can easily obtain the cardinality of each subset $Φ_{hij}$ as

φ (h, θ_{hi}, θ_{hj}) = | Φ_{hij} | = Π_{i, j = 1}^{E_{h}} f (λ_{hij}, θ_{hi}, θ_{hj})

(27)

where $f (λ_{hij}, θ_{hi}, θ_{hj}) = {\begin{matrix} 1, G_{z} \geq λ_{hij}, θ_{hi} = θ_{hj} \\ 0, G_{z} < λ_{hij}, θ_{hi} \neq θ_{hj} \end{matrix}$

From equation (17), when $D_{s, r}^{[2]}$ does not change, we can observe that when the average distance $D_{s, r}^{[1]}$ between the transmitter and relay antennas increases, the STC value $λ$ increases and achieves $G_{z} < λ_{hij}$ , they change alignment angles and powers and $θ_{hi} \neq θ_{hj}$ . When the average distance between the transmitter and receiver antennas decreases, the STC value $λ$ decreases and achieves $G_{z} > λ_{hij}$ , and they keep alignment angles and powers and $θ_{hi} = θ_{hj}$ .

The total cost of alignment-separation and protection for the antenna array $h$ , determined by the pair of vectors, can be calculated as

C_{h}^{as} = \sum_{h = 1}^{m} c_{h} (φ (h, θ_{hi}, θ_{hj}), λ_{hij})

(28)

where $c_{h} (φ (h, θ_{hi}, θ_{hj}), λ_{hij}) = {\begin{matrix} 1, φ (h, θ_{hi}, θ_{hj}) = 1 \\ 0, φ (h, θ_{hi}, θ_{hj}) = 0 \end{matrix}$

When the average distance between the transmitter and receiver antennas increases and exceeds the threshold value $D_{th}$ , they change alignment angles and powers, and we call this changed cost as the mutation cost while unchanged cost as the genetic cost.³⁴ The mutation total cost of MIMO survivability can be determined as

\begin{array}{l} C_{m u} (θ, λ) = \\ \sum_{h = 1}^{m} \sum_{i, j = 1}^{E_{h}} [P_{h} (θ_{h i}, θ_{h j}) \cdot 1_{{G_{z} < λ_{h i j}}} + c_{h} (φ (h, θ_{h i}, θ_{h j}), λ_{h i j}) \cdot 1_{{φ (h, θ_{h i}, θ_{h j}) = 0}}] \end{array}

(29)

where $1_{{G_{z} < λ_{hij}}}$ denotes the indicator function, its indicator is equal to 1 if alignment angles are reallocated and 0 otherwise. $1_{{φ (h, θ_{hi}, θ_{hj}) = 0}}$ also denotes the indicator function, its indicator is equal to 1 if alignment angles are separation and 0 otherwise. The genetic total cost of MIMO survivability can be determined as

\begin{array}{l} C_{g e} (θ, λ) = \\ \sum_{h = 1}^{m} \sum_{i, j = 1}^{E_{h}} [P_{h} (θ_{h i}, θ_{h j}) \cdot 1_{{G_{z} \geq λ_{h i j}}} + c_{h} (φ (h, θ_{h i}, θ_{h j}), λ_{h i j}) \cdot 1_{{φ (h, θ_{h i}, θ_{h j}) = 1}}] \end{array}

(30)

where $1_{{G_{z} \geq λ_{hij}}}$ denotes the indicator function, its indicator is equal to 1 if alignment angles are unchanged and 0 otherwise. $1_{{φ (h, θ_{hi}, θ_{hj}) = 1}}$ also denotes the indicator function, its indicator is equal to 1 if alignment angles are non-separation and 0 otherwise.

The stability of survival evolutionary update for adaptive capacity

In an elementary step of survival update, when the controller of transmitter antenna array senses the feedback alignment angle of the relay antenna array increasing by $(1 / k) θ_{j}$ , the transmitter antennas increase the capacity by selecting an alignment strategy with probability

\begin{matrix} \Pr (ε_{sr} < (1 / k) θ_{j}) = \overset{the number of alignment}{\overset{︷}{\frac{m}{m + k} (1 - x_{i})}} \\ \times [\underset{genetic alignment}{\underset{︸}{(1 - u) \frac{x_{i} \cdot C_{ge} (θ, λ)}{C_{mu} (θ, λ) + C_{ge} (θ, λ)}}} + \underset{mutation alignment}{\underset{︸}{u \frac{1}{C_{mu} (θ, λ)}}}] \end{matrix}

(31)

In this situation, some transmitter antennas adopt a new strategy with probability $u$ to adjust the alignment angle while other transmitter antennas inherit an old strategy with probability $1 - u$ to keep the alignment angle. $x_{i}$ is the probability of the alignment angle change of the transmitter antenna. $1 - x_{i}$ is the probability of the alignment angle remaining the same. $ε_{sr}$ represents the rates of change of the feedback alignment angles of the relay antenna array.

When the controller of the transmitter antenna array senses the feedback alignment angle from the relay antenna array decreasing by $(1 / k) θ_{j}$ , the transmitter antennas decrease the capacity by selecting the alignment strategy with probability

\begin{matrix} \Pr (ε_{sr} > (1 / k) θ_{j}) = \overset{the number of non - alignment}{\overset{︷}{\frac{m}{m + k} x_{i}}} \\ \times [\underset{non - genetic alignment}{\underset{︸}{(1 - u) (1 - \frac{x_{i} \cdot C_{ge} (θ, λ)}{C_{mu} (θ, λ) + C_{ge} (θ, λ)})}} + \underset{non - mutation alignment}{\underset{︸}{u (1 - \frac{1}{C_{mu} (θ, λ)})}}] \end{matrix}

(32)

When the controller of the relay antenna array senses that the allocated alignment angle from the transmitter antenna array has increased by $(1 / m) θ_{i}$ , the relay antennas increase the capacity by selecting the alignment strategy with probability

\begin{matrix} \Pr (ε_{rs} < (1 / m) θ_{i}) = \overset{the number of alignment}{\overset{︷}{\frac{m}{m + k} (1 - y_{i})}} \\ \times [\underset{genetic alignment}{\underset{︸}{(1 - v) \frac{y_{i} \cdot C_{ge} (θ, λ)}{C_{mu} (θ, λ) + C_{ge} (θ, λ)}}} + \underset{mutation alignment}{\underset{︸}{v \frac{1}{C_{mu} (θ, λ)}}}] \end{matrix}

(33)

\begin{matrix} \Pr (ε_{rs} > (1 / m) θ_{i}) = \overset{the number of non - alignment}{\overset{︷}{\frac{m}{m + k} y_{i}}} \\ \times [\underset{non - genetic alignment}{\underset{︸}{(1 - v) (1 - \frac{y_{i} \cdot C_{ge} (θ, λ)}{C_{mu} (θ, λ) + C_{ge} (θ, λ)})}} + \underset{non - mutation alignment}{\underset{︸}{v (1 - \frac{1}{C_{mu} (θ, λ)})}}] \end{matrix}

(34)

In this situation, some relay antennas adopt a new strategy with probability $v$ to send a feedback alignment angle while other relay antennas inherit an old strategy with probability $1 - v$ to keep the alignment angle. $y_{i}$ is the probability of the alignment angle change of the relay antenna. $1 - y_{i}$ is the probability of the alignment angle remaining the same. $ε_{rs}$ represents the rates of change of the alignment angles of the transmitter antenna array.

When the controller of the relay antenna array senses the allocated alignment angle from the transmitter antenna array decreasing by $(1 / m) θ_{i}$ , the relay antennas decrease the capacity by selecting the alignment strategy with probability in (34). Therefore, the expected changes in $x_{i}$ and $y_{i}$ , in the step of update state of the alignment angle, is calculated as

\begin{matrix} Δ (x_{i}, y_{i}) = \\ \frac{x_{i}}{k} (\Pr (ε_{sr} < (1 / k) θ_{j}) - \Pr (ε_{sr} > (1 / k) θ_{j})) \\ + \frac{y_{i}}{m} (\Pr (ε_{rs} < (1 / m) θ_{i}) - \Pr (ε_{rs} > (1 / m) θ_{i})) \end{matrix}

(35)

and $0 < Δ (x_{i}, y_{i}) \leq 1$ , which shows that the evolutionary update state is converging and bounded. When the transmitter and relay antenna array make the decision of selecting the alignment strategy, current information at a certain time $t$ may not be available. Therefore, the transmitter and relay antenna array must rely on historical information, which again may be delayed for a certain period. This delay can occur due to the information exchange latency between the transmitter antenna array and relay antenna array. Thus, we assume that the transmitter and relay antenna array make an alignment strategy selection at time $t$ based on the information from time $t - 1$ . In this case, the dynamics of the evolutionary update state can be modified in equation (36).

In Tables 4 and 5, $C_{sr}^{(1)}$ and $C_{sr}^{(2)}$ are the average capacity of each transmitter antenna for different $Δ C$ values. $Γ_{sr}^{(1)}$ and $Γ_{sr}^{(2)}$ are the probability of the alignment strategy selected by each transmitter antenna for different $Δ C$ values. $C_{1}$ , $C_{2}$ . $C_{3}$ , and $C_{4}$ are some temporary variables.

Table 4.

The specific procedure of the adaptive capacity update for each transmitter antenna at the outer level.

Algorithm 3. Adaptive capacity update executed by each transmitter antenna.
1. Initial values $u$ , $v$ , $x_{i}$ , $C_{1} = 0$ , $C_{2} = 0$ . $C_{3} = 0$ , $C_{4} = 0$ . 2. Label the alignment antenna in the set of $a 1$ and not the alignment antenna in the set of $b 1$ . 3. Repeat 4. The transmitter antenna listens to the relay antenna and computes the capacity $C_{j} = C (P_{s, r}^{[l]} ((1 / k) θ_{j})) - p_{s, r}^{[l]} (θ_{s, r}^{[l]})$ , and its difference $Δ C = C_{j} - C_{j - 1}$ . 5. If $Δ C > 0$ then 6. Compute $Γ_{sr}^{(1)} = \Pr (ε_{sr} < (1 / k) θ_{j})$ . 7. For each antenna in the set $a 1$ do 8. Adjust alignment angle $θ_{s, r}^{[l]}$ with probability $(1 - u) \frac{x_{i} \cdot C_{ge} (θ, λ)}{C_{mu} (θ, λ) + C_{ge} (θ, λ)}$ . 9. Set $C_{1} = C_{1} + (C_{j} - C_{j - 1})$ . 10. End for 11. For each antenna in the set $b 1$ do 12. Adjust alignment angle $θ_{s, r}^{[l]}$ with probability $u \frac{1}{C_{mu} (θ, λ)}$ . 13. Set $C_{2} = C_{2} + (C_{j} - C_{j - 1})$ . 14. Set $C_{sr}^{(1)} = Γ_{sr}^{(1)} \cdot (C_{1} + C_{2})$ . 15. End for 16. If $Δ C < 0$ then 17. Compute $Γ_{sr}^{(2)} = \Pr (ε_{sr} > (1 / k) θ_{j})$ . 18. For each antenna in the set $a 1$ do 19. Adjust alignment angle $θ_{s, r}^{[l]}$ with probability $1 - (1 - u) (1 - \frac{x_{i} \cdot C_{ge} (θ, λ)}{C_{mu} (θ, λ) + C_{ge} (θ, λ)})$ . 20. Set $C_{3} = C_{3} + (C_{j} - C_{j - 1})$ . 21. For each antenna in the set $b 1$ do 22. Adjust alignment angle $θ_{s, r}^{[l]}$ with probability $1 - u \cdot (1 - \frac{1}{C_{mu} (θ, λ)})$ . 23. Set $C_{4} = C_{4} + (C_{j} - C_{j - 1})$ . 24. Set $C_{sr}^{(2)} = (1 - Γ_{sr}^{(2)}) \cdot (C_{3} + C_{4})$ . 25. End for 26. Compute $Δ (x_{i}) = \frac{x_{i}}{k} (Γ_{sr}^{(1)} - Γ_{sr}^{(2)})$ . 27. Until $Δ (x_{i}) < 0$ or $Δ (x_{i}) > 1$ .

Algorithm 3. Adaptive capacity update executed by each transmitter antenna.

1. Initial values

u

v

x_{i}

C_{1} = 0

C_{2} = 0

C_{3} = 0

C_{4} = 0

.
2. Label the alignment antenna in the set of

a 1

and not the alignment antenna in the set of

b 1

.
3. Repeat
4. The transmitter antenna listens to the relay antenna and computes the capacity

C_{j} = C (P_{s, r}^{[l]} ((1 / k) θ_{j})) - p_{s, r}^{[l]} (θ_{s, r}^{[l]})

, and its difference

Δ C = C_{j} - C_{j - 1}

.
5. If

Δ C > 0

then
6. Compute

Γ_{sr}^{(1)} = \Pr (ε_{sr} < (1 / k) θ_{j})

.
7. For each antenna in the set

a 1

do
8. Adjust alignment angle

θ_{s, r}^{[l]}

with probability

(1 - u) \frac{x_{i} \cdot C_{ge} (θ, λ)}{C_{mu} (θ, λ) + C_{ge} (θ, λ)}

.
9. Set

C_{1} = C_{1} + (C_{j} - C_{j - 1})

.
10. End for
11. For each antenna in the set

b 1

do
12. Adjust alignment angle

θ_{s, r}^{[l]}

with probability

u \frac{1}{C_{mu} (θ, λ)}

.
13. Set

C_{2} = C_{2} + (C_{j} - C_{j - 1})

.
14. Set

C_{sr}^{(1)} = Γ_{sr}^{(1)} \cdot (C_{1} + C_{2})

.
15. End for
16. If

Δ C < 0

then
17. Compute

Γ_{sr}^{(2)} = \Pr (ε_{sr} > (1 / k) θ_{j})

.
18. For each antenna in the set

a 1

do
19. Adjust alignment angle

θ_{s, r}^{[l]}

with probability

1 - (1 - u) (1 - \frac{x_{i} \cdot C_{ge} (θ, λ)}{C_{mu} (θ, λ) + C_{ge} (θ, λ)})

.
20. Set

C_{3} = C_{3} + (C_{j} - C_{j - 1})

.
21. For each antenna in the set

b 1

do
22. Adjust alignment angle

θ_{s, r}^{[l]}

with probability

1 - u \cdot (1 - \frac{1}{C_{mu} (θ, λ)})

.
23. Set

C_{4} = C_{4} + (C_{j} - C_{j - 1})

.
24. Set

C_{sr}^{(2)} = (1 - Γ_{sr}^{(2)}) \cdot (C_{3} + C_{4})

.
25. End for
26. Compute

Δ (x_{i}) = \frac{x_{i}}{k} (Γ_{sr}^{(1)} - Γ_{sr}^{(2)})

.
27. Until

Δ (x_{i}) < 0

Δ (x_{i}) > 1

Table 5.

The specific procedure of the adaptive capacity update for each relay antenna at the inner level.

Algorithm 4. Adaptive capacity update executed by each relay antenna.
1. Initial values: $u$ , $v$ , $y_{i}$ , $C_{1} = 0$ , $C_{2} = 0$ . $C_{3} = 0$ , $C_{4} = 0$ . 2. Label the alignment antennas in the set $a 2$ and not the alignment antennas in the set $b 2$ . 3. Repeat 4. The relay antenna listens to the transmitter antenna and computes the capacity $C_{i} = \log_{2} (\sum_{i = 1}^{k} (1 / m) θ_{i}) + k \frac{P_{s, r}^{[l]} (θ_{s, r}^{[l]}) λ_{s, r}^{2}}{σ_{s, r} {(D_{s, r}^{[l]})}^{2}}) - p_{r, s}^{[l]} (θ_{r, s}^{[l]})$ and its difference $Δ C = C_{i} - C_{i - 1}$ . 5. If $Δ C > 0$ then 6. Compute $Γ_{rs}^{(1)} = \Pr (ε_{rs} < (1 / m) θ_{i})$ . 7. For each antenna in the set $a 2$ do 8. Feedback new alignment angle $θ_{r, s}^{[l]}$ with probability $(1 - v) \frac{y_{i} \cdot C_{ge} (θ, λ)}{C_{mu} (θ, λ) + C_{ge} (θ, λ)}$ . 9. Set $C_{1} = C_{1} + (C_{i} - C_{i - 1})$ . 10. End for 11. For each antenna in the set $b 2$ do 12. Feedback alignment angle $θ_{r, s}^{[l]}$ with probability $v \frac{1}{C_{mu} (θ, λ)}$ . 13. Set $C_{2} = C_{2} + (C_{i} - C_{i - 1})$ . 14. Set $C_{rs}^{(1)} = Γ_{rs}^{(1)} \cdot (C_{1} + C_{2})$ . 15. End for 16. If $Δ C < 0$ then 17. Compute $Γ_{rs}^{(2)} = \Pr (ε_{rs} > (1 / m) θ_{i})$ . 18. For each antenna in the set $a 2$ do 19. Feedback alignment angle $θ_{r, s}^{[l]}$ with probability $1 - (1 - v) (1 - \frac{y_{i} \cdot C_{ge} (θ, λ)}{C_{mu} (θ, λ) + C_{ge} (θ, λ)})$ . 20. Set $C_{3} = C_{3} + (C_{i} - C_{i - 1})$ . 21. For each antenna in the set $b 2$ do 22. Adjust alignment angle $θ_{r, s}^{[l]}$ with probability $1 - v (1 - \frac{1}{C_{mu} (θ, λ)})$ . 23. Set $C_{4} = C_{4} + (C_{i} - C_{i - 1})$ . 24. Set $C_{rs}^{(2)} = (1 - Γ_{rs}^{(2)}) \cdot (C_{3} + C_{4})$ . 25. End for 26. Compute $Δ (y_{i}) = \frac{y_{i}}{m} (Γ_{rs}^{(1)} - Γ_{rs}^{(2)})$ . 27. Until $Δ (y_{i}) < 0$ or $Δ (y_{i}) > 1$ .

Algorithm 4. Adaptive capacity update executed by each relay antenna.

1. Initial values:

u

v

y_{i}

C_{1} = 0

C_{2} = 0

C_{3} = 0

C_{4} = 0

.
2. Label the alignment antennas in the set

a 2

and not the alignment antennas in the set

b 2

.
3. Repeat
4. The relay antenna listens to the transmitter antenna and computes the capacity

C_{i} = \log_{2} (\sum_{i = 1}^{k} (1 / m) θ_{i}) + k \frac{P_{s, r}^{[l]} (θ_{s, r}^{[l]}) λ_{s, r}^{2}}{σ_{s, r} {(D_{s, r}^{[l]})}^{2}}) - p_{r, s}^{[l]} (θ_{r, s}^{[l]})

and its difference

Δ C = C_{i} - C_{i - 1}

.
5. If

Δ C > 0

then
6. Compute

Γ_{rs}^{(1)} = \Pr (ε_{rs} < (1 / m) θ_{i})

.
7. For each antenna in the set

a 2

do
8. Feedback new alignment angle

θ_{r, s}^{[l]}

with probability

(1 - v) \frac{y_{i} \cdot C_{ge} (θ, λ)}{C_{mu} (θ, λ) + C_{ge} (θ, λ)}

.
9. Set

C_{1} = C_{1} + (C_{i} - C_{i - 1})

.
10. End for
11. For each antenna in the set

b 2

do
12. Feedback alignment angle

θ_{r, s}^{[l]}

with probability

v \frac{1}{C_{mu} (θ, λ)}

.
13. Set

C_{2} = C_{2} + (C_{i} - C_{i - 1})

.
14. Set

C_{rs}^{(1)} = Γ_{rs}^{(1)} \cdot (C_{1} + C_{2})

.
15. End for
16. If

Δ C < 0

then
17. Compute

Γ_{rs}^{(2)} = \Pr (ε_{rs} > (1 / m) θ_{i})

.
18. For each antenna in the set

a 2

do
19. Feedback alignment angle

θ_{r, s}^{[l]}

with probability

1 - (1 - v) (1 - \frac{y_{i} \cdot C_{ge} (θ, λ)}{C_{mu} (θ, λ) + C_{ge} (θ, λ)})

.
20. Set

C_{3} = C_{3} + (C_{i} - C_{i - 1})

.
21. For each antenna in the set

b 2

do
22. Adjust alignment angle

θ_{r, s}^{[l]}

with probability

1 - v (1 - \frac{1}{C_{mu} (θ, λ)})

.
23. Set

C_{4} = C_{4} + (C_{i} - C_{i - 1})

.
24. Set

C_{rs}^{(2)} = (1 - Γ_{rs}^{(2)}) \cdot (C_{3} + C_{4})

.
25. End for
26. Compute

Δ (y_{i}) = \frac{y_{i}}{m} (Γ_{rs}^{(1)} - Γ_{rs}^{(2)})

.
27. Until

Δ (y_{i}) < 0

Δ (y_{i}) > 1

$C_{rs}^{(1)}$ and $C_{rs}^{(2)}$ are the average capacity of each relay antenna for different $Δ C$ values. $Γ_{rs}^{(1)}$ and $Γ_{rs}^{(2)}$ are the probability of the alignment strategy selected by each relay antenna for different $Δ C$ values.

Adaptive capacity applications and comparison with conventional Rayleigh fading

The MIMO communication system reaches its adaptive capacity using the game strategy and relay selection in the following two examples.

Example 1

When the distance between a transmitter antenna array and a receiver antenna array is greater than the threshold, that is, $D_{s, d} > D_{th}$ , the average capacity decreases as the distance increases. Each antenna in the transmitter antenna array begins to select a relay antenna in a relay and reaches NE according to STC value. The adaptive capacity that can be attained in mobile situation by the transmitter antenna array is calculated from equation (37)

\begin{matrix} Δ (x_{i}, y_{i}) (t) \\ = \frac{x_{i}}{k} (\Pr (ε_{sr} < (1 / k) \cdot θ_{j} (t - 1)) - \Pr (ε_{sr} > (1 / k) \cdot θ_{j} (t - 1))) \\ + \frac{y_{i}}{m} (\Pr (ε_{rs} < (1 / m) \cdot θ_{i} (t - 1)) - \Pr (ε_{rs} > (1 / m) \cdot θ_{i} (t - 1))) \end{matrix}

(36)

\begin{matrix} C_{D_{s, d} > D_{th}} = (1 - \Pr (ε_{sr} > (1 / k) θ_{j})) \\ \times (C (P_{s, r}^{[l]} ((1 / k) θ_{j})) - θ_{s, r}^{[l]} p_{s, r}^{[l]}) \\ = (C (P_{s, r}^{[l]} ((1 / k) θ_{j})) - θ_{s, r}^{[l]} p_{s, r}^{[l]}) \frac{m}{m + k} (1 - x_{i}) \\ [\frac{x_{i} \cdot C_{ge} (θ, λ)}{C_{mu} (θ, λ) + C_{ge} (θ, λ)} + \frac{u + C_{mu} (θ, λ)}{C_{mu} (θ, λ)}] \end{matrix}

(37)

Example 2

When a transmitter antenna array and a receiver encounter jamming attacks or disconnected links from malicious nodes, the average capacity of the transmitter antenna array decreases after the jamming attack or disconnection of the links. Each antenna in the transmitter antenna array begins to adjust a new alignment angle to resist the jamming attack either according to the feedback of a new alignment angle from the relay or by adjusting the alignment angle on its own, which makes $\sum_{i = 1}^{m} P_{s, r}^{[i]} (θ_{s, r}^{[i]}) = p_{r, s}^{* [l]} (θ_{r, s}^{[l]})$ . After substituting equation (21) into equation (11), the adaptive capacity that can be achieved by the transmitter antenna array during jamming attacks or after disconnection of links is obtained as

C_{Jamming - attack} = \log_{2} (1 + \frac{p_{r, s}^{* [l]} (θ_{r, s}^{[l]}) λ_{s, r}^{2}}{m σ_{s, r} {(D_{s, r}^{[l]})}^{2}}) - p_{s, r}^{[l]} (θ_{s, r}^{[l]})

(38)

The above capacity $C_{Jamming - attack}$ is ensured through the controller of the transmitter antenna array adjusting alignment angle $θ_{r, s}^{[l]}$ to avoid jamming attacks or disconnected links caused by the malicious nodes. The average capacity of a Rayleigh fading MIMO communication system with $m$ transmitter antennas and $n$ receiver antennas is given by Su et al.¹⁹

C_{Rayleigh} = m \log_{2} (1 + ρ) - \frac{m}{\ln 2}

(39)

where $ρ$ is the average SNR at each receive antenna. The transmitter antennas work with the relay cooperatively. The achieved cooperative average capacity of the mobile MIMO communication network is

C_{Assisted - relay} = \log_{2} (1 + ρ_{s, d} (θ) + ρ_{s, r} (θ) + ρ_{r, d} (θ))

(40)

where $ρ (θ)$ is the SNR related to the alignment angles. The cooperative average capacity of the relay-assisted mobile MIMO communication is greater than that of the conventional Rayleigh fading on-ground MIMO network scenario, $C_{Assisted - relay} > C_{Rayleigh}$ . After substituting equation (17) into equation (11), the average capacity of the transmitter antenna array is

\begin{matrix} C_{Trans} = \log_{2} [1 + P_{s, r}^{[l]} (θ_{s, r}^{[l]}) (\frac{{(D_{s, r}^{[1]} D_{s, r}^{[2]})}^{2} (A - 1)}{{(D_{s, r}^{[l]})}^{2} (B - A \cdot Q)})] \\ = \log_{2} [1 + \frac{\exp [α_{s, r}^{[l]} \cos (θ_{s, r}^{[l]} - {\bar{θ}}_{s, r}^{[l]})] {(D_{s, r}^{[1]} D_{s, r}^{[2]})}^{2} (A - 1)}{2 π J_{0} (α_{s, r}^{[l]}) [{(D_{s, r}^{[l]})}^{2} (B - A \cdot Q)]}] \end{matrix}

(41)

where $B = (D_{s, r}^{[2]})^{2} \cdot P_{s, r}^{[1]} (θ_{s, r}^{[1]})$ and $Q = (D_{s, r}^{[1]})^{2} \cdot P_{s, r}^{[2]} (θ_{s, r}^{[2]})$ . When $θ_{s, r}^{[l]}$ is equal to ${\bar{θ}}_{s, r}^{[l]}$ by adjusting the alignment angles, transmitter antenna array achieves the maximum capacity

C_{Trans}^{Max} = \log_{2} [1 + \frac{\exp (α_{s, r}^{[l]}) {(D_{s, r}^{[1]} D_{s, r}^{[2]})}^{2} (A - 1)}{2 π J_{0} (α_{s, r}^{[l]}) [{(D_{s, r}^{[l]})}^{2} (B - A \cdot Q)]}]

(42)

Thus, $C_{Trans}^{Max} > C_{Rayleigh}$ , the cooperative average capacity of the transmitter antennas in the mobile MIMO communication using game strategy and relay selection is greater than that of the conventional Rayleigh fading on-ground MIMO network scenario in the case of mobility, jamming attack, or disconnected-link situations.

Numerical results

In this section, we demonstrate numerical results to illustrate the capacities of the MIMO communication systems with different transmitters, relays, and receiver antenna arrays in dynamic environments; in particular, we will show the average capacity when the distance between a transmitter antenna array and a receiver antenna array is greater than the threshold and when it suffers from jamming attacks. We set the number of the transmitter, relay, and receiver to 3, respectively. We consider three different settings for each group. In the numerical studies, setting the number of the antennas to 8, 4, and 3 in the transmitter group, the number of the antennas to 4, 3, and 4 in the relay group, the number of antennas to 8, 2, and 5 in the receiver group. Therefore, we gain the number of antennas be $8 \times 4 \times 8$ , $4 \times 3 \times 2$ , and $3 \times 4 \times 5$ , respectively. We adjust the transmission power $P$ with a random alignment angle such that the average SNR increases from 0 to 20 dB, and in Figure 3, we show the average capacity response where $N_{s}$ , $N_{r}$ , and $N_{d}$ are the number of the transmitter, relay, and receiver antennas, respectively.

Figure 3.

Average capacity response of different antenna arrays.

For a given distance, which is greater than the threshold, the transmitter and receiver antenna arrays select the relay antennas; they adjust the transmission power. When the number of the transmitter antennas is equal to the number of the receiver antennas and there is a $8 \times 4 \times 8$ array in the MIMO channel, a highest average capacity response of 50 bps/Hz is achieved. We also observe that the average capacity of a $3 \times 4 \times 5$ antenna array is slightly greater than that of a $4 \times 3 \times 2$ antenna array. This is because the numbers of the transmitter and receiver antennas with adjustment powers are equal and the number of relay antennas are greater than those of the $4 \times 3 \times 2$ antenna array.

In Figure 4, we observe that the average capacity with the feedback update channels is 28 bps/Hz when the distance between the transmitter and receiver antenna arrays is less than the threshold. However, with increase in the distance, the maximum average capacity will decrease.

Figure 4.

Average capacity of the transmitter and receiver antennas for $D < 50 m$ .

The effect of different distances on the average capacities of the transmitter and receiver antenna arrays is shown in Figures 5 and 6. When the distance between the transmitter and receiver antenna array is $100 m$ , the transmitter and receiver antenna arrays have to select a relay antenna to complete the transmission. The average capacity of transmitter and receiver antennas for $D = 100 m$ is shown in Figure 5.

Figure 5.

Average capacity of the transmitter, relay, and receiver antennas for $D = 100 m$ .

Figure 6.

Average capacity of the transmitter, relay, and receiver antennas for $D = 500 m$ .

When we increase the SNR to 5 dB, the average capacity begins to increase quickly. The maximum average capacity of the transmitter, relay, and receiver antennas with the feedback update channels can reach up to 17.5 bps/Hz. The maximum average capacity of the transmitter, relay, and receiver antennas with the random channels can reach up to 16.8 bps/Hz. This is because, in MIMO transmission, the transmitter and receiver antennas select the relay antenna by adjusting the transmission power with the alignment angle according to the change in distance, so that they can update the channels. In Figure 6, when the distance between the transmitter and receiver antenna arrays is $500 m$ and the SNR is 10 dB, the average capacity begins to increase quickly. The maximum average capacity of the transmitter, relay, and receiver antennas with the feedback update channels can go up to 14 bps/Hz. The maximum average capacity of the transmitter, relay, and receiver antennas with the random channels can go up to 13 bps/Hz.

In Figure 7, we plot the capacity of MIMO communication systems for $D = 100 m$ and bandwidth $bw = 20 Gbps$ suffering from jamming attacks. The maximum average capacity of the transmitter, relay, and receiver antennas with the adjustment parameters is 2.25 bps/Hz.

Figure 7.

Average capacity of the transmitter, relay, and receiver antennas for $D = 100 m$ , $bw = 20 Gbps$ , suffering from jamming attacks.

This is because the transmitter and receiver antennas select a relay antenna by dynamically adjusting the transmission power with alignment angle parameters (Algorithms 3 and 4) according to the changes in distance and jamming, so that they can avoid the interference of attack angles. We can observe from comparisons that, with increasing the distance and the jamming attacks from malicious relays, the average capacity of MIMO communication systems increases, but it is less than the maximum average capacity.

Figure 8 shows the average capacity of the transmitter and relay antennas of a MIMO communication system. In this figure, we can see that as the number of the transmitter and antennas increases, the performance of both the adaptive capacity $C_{Jamming - attack}$ of the transmitter antenna array during jamming attacks and the average capacity $C_{Trans}$ of the transmitter antenna array increase. The maximum average capacity $C_{Jamming - attack}$ of the transmitter antenna array can reach up to $7.1212 \times 10^{- 7} bps / Hz$ . The maximum average capacity $C_{Trans}$ of the transmitter antenna array can reach up to $16.6253 bps / Hz$ . This is mainly due to the fact that the transmitter antenna array is able to adaptively update capacity through Algorithms 3 and 4, adjusting the alignment angle to the changes due to mobility or jamming attacks. Figure 8 shows that as the number of the transmitter and antennas increases, the average capacity $C_{Rayleigh}$ of the transmitter antenna array decreases for setting the SNR of $C_{Rayleigh}$ less than 1 dB.

Figure 8.

Average capacity of the transmitter and relay antennas of the MIMO communication system for setting the SNR of $C_{Rayleigh}$ less than 1 dB and $D = 10 m$ .

Figure 9 shows that as the number of the transmitter and antennas increases, the average capacity $C_{Rayleigh}$ of the transmitter antenna array increases for setting the SNR of $C_{Rayleigh}$ to 3 dB. The maximum average capacity $C_{Rayleigh}$ of the transmitter antenna array can reach up to $11.0145 bps / Hz$ and is less than the maximum average capacity $C_{Trans}$ of the transmitter antenna. The increase in the performance advantage highlights the ability of the transmitter antenna array to improve effectively the capacity of MIMO communications through the proposed adaptive capacity update algorithms.

Figure 9.

Average capacity of the transmitter and relay antennas of the MIMO communication system for setting the SNR of $C_{Rayleigh}$ to 3 dB and $D = 10 m$ .

Figure 10 compares the average capacity of the transmitter, relay, and receiver antennas between cooperative relay antennas and non-cooperative relay antennas with varying SNRs. When the SNR increase, we can see that the average capacity of the proposed scheme is greater than that of the non-cooperative relay antennas in a MIMO communications. The non-cooperative relay antennas do not consider the feedback alignment angle, while the feedbacks have significant impact on capacity.

Figure 10.

Average capacity of the transmitter, relay, and receiver antennas of the MIMO communication system.

We vary the probability $ξ_{j}$ of the alignment strategy taken by the transmitter antennas in the range of 0.4–0.73. Figure 11 shows that the relay antennas can adjust their alignment angle using the game strategy. The game of the alignment strategy between the transmitter antennas and the relay antennas eventually converges to the stable state.

Figure 11.

The stability of survival evolutionary update for adaptive capacity.

Conclusion

In this article, we analyzed the capacity of MIMO wireless communication networks with relay antenna array alignments and proposed a two-level dynamic game framework based on the evolutionary game theory to investigate the cooperative mechanism of relay for MIMO communications to improve the possible capacity. With an evolutionary equilibrium equation and a first-order derivative of the game utility, we obtained an STC value for the MIMO capacity equilibrium, optimal power, and alignment angle of each relay antenna, which enabled the controller of the antenna array to adjust them adaptively to increase the game utility. Then, we proposed a model for the survival evolutionary update of relays and developed algorithms for the adaptive capacity update of the transmitter and relay antennas when the transmitter and receiver antenna arrays suffer from jamming attacks from malicious nodes or disconnection of links. Finally, we discussed two MIMO communication scenarios that improve the capacity. Extensive numerical studies validated our theoretical developments. When the distance between the transmitter and receiver antenna arrays was large or when the arrays were subjected to jamming attacks, we were able to design adaptive capacity update systems for the relay antennas such that their capacities exceeded the average capacity of conventional MIMO communications over Rayleigh fading channels.

Footnotes

Handling Editor: Olivier Berder

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was partly supported by National Natural Science Foundation of China under grant nos 61572014, 61772338, and 61772018 and by Zhejiang Provincial Natural Science Foundation of China under grant no. LY16F020028.

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