Joint Source and Relay Design for Multi-user MIMO Non-regenerative Relay Networks with Direct Links

Recently, MIMO relay network has attracted considerable interest from both academic and industrial communities. It has been verified that wireless relay can increase coverage and capacity of the wireless networks [1]. Meanwhile, MIMO techniques can provide significant improvement for the spectral efficiency and link reliability in scattered environments because of its multiplexing and diversity gains [2]. A MIMO relay network, combining the relaying and MIMO techniques, can make use of both advantages to increase the data rate in the network edge and extend the network coverage. It is a promising technique for the next generation’s wireless communications.

The capacity of MIMO relay network has been extensively investigated in the literature [3, 4, 5, 6, 7]. Recent works on MIMO non-regenerative relay are focusing on how to design the source precoding matrix and relay processing matrix. For a single-user MIMO relay network, an optimal relay processing matrix which maximizes the end-to-end mutual information is designed in [8] and [9] independently, and the optimal structures of jointly designed source precoding matrix and relay processing matrix are derived in [10]. In [11] and [12], the relay processing matrix to minimize the mean-squared error (MSE) at the destination is developed. A unified framework to jointly optimize the source precoding matrix and the relay processing matrix is established in [13]. For a multi-user single-antenna relay network, the optimal relay processing is designed to maximize the system capacity [14, 15, 16]. In [17], the optimal source precoding matrices and relay processing matrix are developed in the downlink and uplink scenarios of an MU-MIMO relay network without considering S-D links. There are only a few works considering the direct S-D links. In [18] and [19], the optimal relay processing matrix is designed based on MSE criterion with and without the optimal source precoding matrix in the presence of direct links, respectively. However, for a relay network with direct S-D links, jointly optimizing the source precoding matrix and the relay processing matrix based on capacity or MSE is much difficult, especially for an MU-MIMO relay network.

In this paper, we consider an MU-MIMO non-regenerative relay network where each node is equipped with multiple antennas. We take the effect of S-D link into the joint optimization of the source precoding matrices and relay processing matrix, which is more complicated than the relatively simple case without considering S-D links [17]. To our best knowledge, there is no such work in the literature on the joint optimization of source precoding and relay processing for MU-MIMO non-regenerative relay networks with direct S-D links. Two major contributions of this paper over the conventional works are as follows:

• •

  We first introduce a general strategy to the joint design of source precoding matrices and relay processing matrix by transforming the network into a set of parallel scalar sub-systems just as a point-to-point MIMO channel under a relay modified power constraint, and show that the capacity based source precoding matrices and relay processing matrix respectively share the same structures with the MSE based ones.

• •

  A nested iterative algorithm is presented to solve the joint optimization of sources precoding and relay processing based on capacity and MSE respectively. Simulation results show that the proposed algorithm outperforms the existing methods.

The rest of this paper is organized as follows. Section II illustrates the system model. Section III presents the optimal structures of source precoding and relay processing, and a nested iterative algorithm to solve the joint optimization of sources precoding and relay processing. Section IV devotes to the simulation results. Finally, Section V concludes the paper.

Notations: Lower-case letter, boldface lower-case letter, and boldface upper-case letter denote scalar, vector, and matrix, respectively. $\textsf{E}(\cdot)$, $\mathrm{tr}(\cdot)$, $(\cdot)^{-1}$, $(\cdot)^{{\dagger}}$, $|\cdot|$, and $\|\cdot\|_{F}$ denote expectation, trace, inverse, conjugate transpose, determinant, and Frobenius norm of a matrix, respectively. $\mathbf{I}_{N}$ stands for the identity matrix of order $N$. $\mathrm{diag}(a_{1},\ldots,a_{N})$ is a diagonal matrix with the $i$th diagonal entry $a_{i}$. $\log$ is of base $2$. $\mathcal{C}^{M\times N}$ represents the set of $M\times N$ matrices over complex field, and $\sim\mathcal{CN}(x,y)$ means satisfying a circularly symmetric complex Gaussian distribution with mean $x$ and covariance $y$. $[x]^{+}$ denotes $\max\{0,x\}$.

We consider a multiple access MIMO relay network with two source nodes (SNs), one relay node (RN) and one destination node (DN) as illustrated in Fig. 1, where the channel matrices have been shown. The numbers of antennas equipped at the SNs, RN and DN are $N_{s},N_{r}$, and $N_{d}$, respectively. We assume that there is only two SNs and both SNs have the same number of antennas for simplicity. However, it is easy to be generalized to the scenario of multiple SNs with different numbers of antennas at each SN. In this paper, we consider a non-regenerative half-duplex relaying strategy applied at the RN to process the received signals. Thus, the transmission will take place in two phases. Suppose that perfect synchronization has been established between SN₁ and SN₂ prior to transmission, and both SN₁ and SN₂ transmit their independent messages to the RN and DN simultaneously during the first phase. Then the RN processes the received signals and forwards them to the DN during the second phase.

Let $\mathbf{H}_{ri}\in\mathcal{C}^{N_{r}\times N_{s}},\mathbf{H}_{di}\in\mathcal{C}^{N_{d}\times N_{s}},$ and $\mathbf{H}_{dr}\in\mathcal{C}^{N_{d}\times N_{r}}$ denote the channel matrices of the $i$th SN to RN, to DN, and RN to DN, respectively. Each entry of the channel matrices is assumed to be complex Gaussian variable with zero-mean and variance $\sigma^{2}_{h}$. Furthermore, all the channels involved are assumed to be quasi-static i.i.d. Rayleigh fading combining with large scale fading over a common narrow-band. Let $\mathbf{F}_{1}\in\mathcal{C}^{N_{s}\times N_{s}}$ and $\mathbf{F}_{2}\in\mathcal{C}^{N_{s}\times N_{s}}$ denote the precoding matrices for SN₁ and SN₂, respectively, which satisfy the power constraint $\textsf{E}[\mathbf{F}_{i}\mathbf{s}_{i}\mathbf{s}^{{\dagger}}_{i}\mathbf{F}^{{\dagger}}_{i}]=\mathrm{tr}(\mathbf{F}_{i}\mathbf{F}^{{\dagger}}_{i})\leq P_{i}$. Let $\mathbf{G}\in\mathcal{C}^{N_{r}\times N_{r}}$ denote the relay processing matrix. Suppose that $\mathbf{n}_{r}\in\mathcal{C}^{N_{r}\times 1}$ and $\mathbf{n}_{i}\in\mathcal{C}^{N_{d}\times 1}$ are the noise vectors at RN and DN, respectively, and all noise are independent and identically distributed additive white Gaussian noise (AWGN) with zero-mean and unit variance. Then, the baseband signal vectors $\mathbf{y}_{1}$ and $\mathbf{y}_{2}$ received at the DN during the two consecutive phases can be expressed as follows:

where $\mathbf{s}_{i}\in\mathcal{C}^{N_{s}\times 1}$ is assumed to be a zero-mean circularly symmetric complex Gaussian signal vector transmitted by the $i$th SN and satisfies $\textsf{E}(\mathbf{s}_{i}\mathbf{s}^{{\dagger}}_{i})=\mathbf{I}_{N_{s}}$. Let Y, $\textbf{H}_{i}$ ($i=1,2,3$), and N, shown in (13), denote the effective receive signal, effective channels and effective noise respectively. Then $\mathbf{H}_{3}\textsf{E}[\mathbf{NN}^{{\dagger}}]\mathbf{H}_{3}^{{\dagger}}=\mathbf{H}_{3}\mathbf{H}_{3}^{{\dagger}}=\mathrm{diag}(\mathbf{I}_{N_{d}},\mathbf{R})$, where $\mathbf{R}=\mathbf{I}_{N_{d}}+\mathbf{H}_{dr}\mathbf{GG}^{{\dagger}}\mathbf{H}^{{\dagger}}_{dr}$ is the covariance matrix of the effective noise at the DN during the second phase.

In this section, the capacity and MSE for the MMSE detector with successive interference cancelation (SIC) at DN are analyzed. Then, we will exploit the optimal structures of source precoding and relay processing based on capacity and MSE respectively. Then a new algorithm on how to jointly optimize the sources precoding matrices and the relay processing matrix is proposed to maximize the capacity or minimize MSE of the entire network.

In this section, simulation results are carried out to verify the performance superiority of the proposed joint source-relay design scheme (JDS) for MU-MIMO relay network with direct links. We first compare the proposed scheme with other three schemes in terms of the ergodic capacity and the Cumulative Distribution Function (CDF) of instantaneous capacity of the MIMO relaying networks, and then compare the sum-MSE of the networks. The alternative schemes are:

1. (1)

   Naive scheme (NAS): The source covariances are fixed to be scaled by the identity matrices $\frac{P_{1}}{N_{S}}\mathbf{I}$ and $\frac{P_{2}}{N_{S}}\mathbf{I}$ at SN₁ and SN₂, respectively, and the relay processing matrix is $\mathbf{G}=\eta\mathbf{I}$, where $\eta=\sqrt{\frac{P_{r}}{\mathrm{tr}(\mathbf{I}+\sum^{2}_{i=1}\mathbf{H}_{ri}\mathbf{F}_{i}\mathbf{F}^{{\dagger}}_{i}\mathbf{H}^{{\dagger}}_{ri})}}$ is a power control factor. The S-D links contribution is included.

2. (2)

   Suboptimal scheme (SOS): This scheme is proposed in [17] for MU-MIMO relay network without considering S-D links in design. But, the S-D links contribution of capacity is included in the simulation for fair comparison. Note that this scheme is optimal for the scenario without considering the S-D links.

3. (3)

   No-direct links scheme (NOD): This scheme is like SOS, but, without S-D links contribution.

Noting that both SOS and NOD have different power control polices to accommodate the capacity and MSE criterions. In the simulations, we consider a linear two-dimensional symmetric network geometry as depicted in Fig. 1, where both SNs are deployed at the same position, and the distance between SNs (or RN) and DN is set to be $\ell_{sd}$ (or $\ell_{rd}$), and $\ell_{sd}=\ell_{sr}+\ell_{rd}$. The channel gains are modeled as the combination of large scale fading (related to distance) and small scale fading (Rayleigh fading), and all channel matrices have i.i.d. $\mathcal{CN}(0,\frac{1}{\ell^{\tau}})$ entries, where $\ell$ is the distance between two nodes, and $\tau=3$ is the path loss exponent.

Fig. 2-4 are based on capacity criterion. Fig. 2 shows the CDF of instantaneous capacity for different power constraints, when all nodes positions are fixed. Fig. 3 shows the capacity of the network versus the power constraints, when all nodes positions are fixed. These two figures show that capacity offered by the proposed relaying scheme is better than both SOS and NOD schemes at all SNR regime, especially at high SNR regime. The naive scheme surpasses both SOS and NOD schemes at high SNR regime, which demonstrates that the direct S-D link should not be ignored in design. Fig. 4 shows the capacity of the network versus the distance ($\ell_{sr}$) between SNs and RN, for fixed $\ell_{sd}$. It is clear that the capacity offered by the proposed scheme is better than those by the SOS, NAS and NOD schemes. NOD scheme is the worst performance scheme at any relay position at moderate and high SNR regimes.

Fig. 5 and Fig. 6, are based on MSE criterion, the similar conclusions can be drawn.

In this paper, we propose a optimal structure of the source precoding matrices and relay processing matrix for MU-MIMO non-regenerative relay network with direct S-D links based on capacity and MSE respectively. We show that the capacity based optimal source precoding matrices share the same structures with the MSE based ones. So does relay processing matrix. A nested iterative algorithm jointly optimizing the source precoding and relay processing is proposed. Simulation results show that the proposed algorithm provides better performance than the existing methods.