## 1 Introduction
In Reference [28], the sum rate maximization problem for the AF FD relay-assisted MIMO one-way system was investigated, and with the consideration of self-interference aware FD relaying, an alternating optimization (AO) method was devised. In Reference [29], the authors designed the source and relay precoders for a MIMO FD OWRN with SWIPT-enabled destination to optimize the end-to-end performance in residual loop-interference environments. In Reference [30], the hardware impairments of the FD AF OWRN was considered and an optimization problem was established to maximize the signal to a distortion-plus-noise ratio under relay and source transmit power constraints. Nonetheless, motivated by the benefit of reducing the waste of extra-channel resources and achieving a higher spectral efficiency than the one-way communication [33; 34; 35], the bidirectional communication has attracted considerable interest, and much more researches have tended to adopt bidirectional communication in the FD. In the literature [36], the joint optimization of transmit and receive beamforming for relays to maximize the achievable sum-rate in the FD BRN system with a PS scheme was considered. However, to the best of our knowledge, a joint source-relay design based on a total-MSE minimization in MIMO FD BWSN with SWIPT has not yet been studied. In this paper, a MIMO FD BWSN with PS is presented. With the consideration of processing self-interference, different from References [15; 27; 28; 29; 30], we choose to use the one presented in Reference [37]. The merit of the proposed network lies in the considerably high spectral efficiency, providing a cost-effective and perpetual power supply for WSNs and an uninterrupted transmission of information. The contributions are summarized as follows. First, for the sensor system model, contrary to Reference [36], the two source nodes are also equipped with multiple transmitter-receiver antennas for signal transmission and reception, and the multiple data streams transmitting scenarios are considered. Second, a joint optimization problem for source and relay beamforming and source receiving based on the total-MSE minimization is formulated. Third, to cope with the primal nonconvex problem, a feasible point pursuit-successive convex approximation (FPP-SCA)-basediterative algorithm is exploited. Finally, to reduce the computational complexity, a low-complexity diagonalizing method-based algorithm is introduced to simplify each non-convex subproblem into convex problems directly. In terms of the existing approach [25], the generalized singular value decomposition (GSVD) is discussed, and the scenarios in which source and relay nodes equipped with the same and different number of antennas are both discussed. The numerical results show a good performance and validate our analysis. The remainder of the paper is organized as follows. Section 2 proposes the system model, including the sensor nodes deployment and optimization model. Section 3 focuses on the scheme design. The numerical results are presented and discussed in Section 4. Finally, the conclusions are presented in Section 5. _Notation:_ Throughout this paper, scalar variables are expressed by lowercase italic letters, vectors are represented by boldface lowercase letters, and matrices are denoted by boldface uppercase letters. \(\mathbb{C}^{M\times N}\) denotes an \(M\times N\) matrix with complex entries. \(Tr(\cdot)\), \((\cdot)^{T}\), \((\cdot)^{H}\), \((\cdot)^{-1}\), \((\cdot)^{*}\), and \(\|\cdot\|\) are the trace, transpose operation, conjugate transpose operation, inverse operation, conjugate transpose operation, and the bound norm of a vector. \(\sum_{i=1}^{M}(\cdot)\) stands for the sum from 1 to \(M\). \(\sim CN\left(x,\sigma^{2}\right)\) represents a complex Gaussian distributed variable with a mean \(x\) and covariance \(\sigma\). \(vec(\cdot)\) and \(mat(\cdot)\) are the matrix vectorization operator and the corresponding inverse operation, respectively. \(\mathbb{E}[\cdot]\) signifies the expectation of the random variables in the bracket.