The cost function that expresses the minimization of MSE is a nonlinear function of the two precoding matrices at the source and the relay, resulting in a nonconvex optimization problem

Source-precoder subproblem is solved by applying Karush-Kuhn-Tucker (KKT) conditions to single relay-precoder optimization

MRR-MRT: proportional to MF-based receive and transmit beamforming to maximize the total signal power ZFR-ZFT: proportional to ZF-based receive and transmit beamforming to remove the interference

Relay BF matrix calculations based on: (i) Maximal-Ratio Reception and Maximal-Ratio Transmission (MRR-MRT) (ii) Zero-Forcing Reception and Zero-Forcing Transmission (ZFR-ZFT)

S: single-antenna, single source R: single AF two-way single-pair MIMO D: single-antenna, single user

The sum-rate maximization problem in this two-way AF single-relay network is not convex and an approximate solution can be derived through decomposition

Joint optimization of the transceivers at both sources and relay in terms of sum-rate maximization and based on KKT conditions

S: single MIMO R: single AF two-way single-pair MIMO D: single MIMO

The global optimum regarding the maximization of the distance of network-coded symbols is complicated to be found, as it depends on the symbol constellation and the corresponding mapping rule. Moreover, for general MIMO channels between the two sources and the relay, a closed-form solution has not been derived

Design of a hybrid precoder combining three different classes of suboptimal precoders, with additional constraints of subspace alignment, subspace separation, and the maximal ratio transmission Define the optimal precoding vectors within each class in terms of maximizing the minimum distance between different network coding symbols

S: single MIMO R: single AF two-way single-pair MIMO D: Single MIMO

For each relay: : iid symbols N: number of antennas at the relay

(i) frequency domain (FD) based processing at the relays (ii) equal power allocation (EPA) across all frequencies (iii) equal power allocation (EPA) across all frequencies and relays

S: single-antenna, single source R: multiple AF MIMO D: single-antenna, single user

The joint source, relay and receive matrices optimization problem that aims at two-way MSE minimization is non-convex. The global optimum cannot be achieved with reasonable complexity (nonexhaustive searching)

Iterative algorithm for joint source, relay, and receive matrices optimization for two-way sum MSE minimization

S: single MIMO R: multiple AF two-way single-pair MIMO D: single MIMO

Depending on the imposed power constraints, the optimization problems for each optimal case induce different complexity. When multiple relays are employed, the optimization is nonconvex for the case of joint relay power constraints and joint source-relay power constraints

Max-min optimization of the source BF vector under joint relay and jointed source-relay power constraints: (i) transformation method (ii) gradient method (iii) relaxation method

S: single MIMO R: multiple AF MIMO D: single-antenna, single user

Proportional to the beamforming algorithm for the fully loaded or overloaded uplink

Linear MMSE criterion for both downlink/uplink utilizing iterative beamforming algorithm: (i) equalizer design at the user/BS (ii) forwarding matrix design at the relay station (iii) precoder design at the BS/user

S: single MIMO R: single AF MIMO D: multiple MIMO users

N: number of BS antenna M: number of relay antenna K: number of MS single antenna : iteration number in Algorithms 1 and 2 : the complexity of randomization

(i) Iterative algorithm for RS precoding design with the BS precoder fixed (ii) Design of joint BS-RS precoding by solving the BS and RS precoding alternately

S: single MIMO R: single AF two-way multi-pair MIMO D: single-antenna, multiple users

(i) The complexity of the centralized adaptive BF is per iteration J: is the number of sources and destination nodes : is the number of antennas at the kth relay (ii) For the decentralized algorithm, the complexity per iteration is equal to at the ith relay

(i) Centralized adaptive BF algorithm with the existence of a local processing center connected to all the relays and minimizing a cost function using state-space modeling approach (ii) Decentralized adaptive BF algorithm allowing each relay terminal to compute its beamforming matrix locally with limited amount of data exchange with the other relays, employing Kalman filtering to estimate its beamforming coefficients iteratively

S: single-antenna, Multiple sources R: multiple AF MIMO D: single-antenna, Multiple users

The optimization problem of meeting the QoS constraint with minimal relay power expenditure is non-convex

ZF-BF is used in order to reduce complexity by projecting the BF vector to a low dimensional space thus reducing the number of variables that are used for optimization

S: single-antenna, multiple sources R: multiple AF MIMO D: single-antenna, multiple users

As the problem of sum-rate maximization is NP-hard, the process of checking whether a set of SINR values are achievable in order to obtain the optimal solution is highly complex

(i) Sum-rate maximization through an iterative algorithm subject to a sum-power constraint of the relay BF matrices (ii) Interference neutralization beamforming scheme subject to a linear constraint on the desired signals

S: single-antenna, multiple sources R: Multiple AF MIMO D: single-antenna, multiple users

The sum-rate optimization problem is NP-hard and the global optimal solution cannot be derived with realistic computation complexity

Distributed two-hop interference pricing algorithm for relay beamforming design for maximizing end-to-end sum rates

S: multiple MIMO R: multiple AF MIMO D: multiple MIMO users

S: source, R: relay, D: destination.

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