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Wireless MIMO switching: distributed zeroforcing and MMSE relaying using network coding
EURASIP Journal on Advances in Signal Processing volume 2013, Article number: 130 (2013)
Abstract
In a switching problem, a onetoone mapping from the inputs to the outputs is conducted according to a switch pattern, i.e., a permutation matrix. In this paper, we investigate a wireless switching problem, in which a group of singleantenna relays acts together as a multipleinputmultipleoutput (MIMO) switch to carry out distributed precodeandforward. All users transmit simultaneously to the MIMO switch in the uplink and then the MIMO switch precodes the received signals and broadcasts in the downlink. Ideally, each user could receive its desired signal from one other user with no or little interference from other users. Selfinterference is allowed in the received signals, as it can be canceled when each user has the channel gain of its selfinterference. We propose two distributed relaying schemes based on two widely adopted criteria, i.e., zeroforcing relaying and minimum mean square error (MMSE) relaying. For the distributed zeroforcing relaying, we further propose a message passing approach, with which the proposed zeroforcing relaying achieves significant throughput gain with little attendant overhead. We also claim that the proposed MMSE relaying achieves even larger throughput at the expense of larger amount of message passing. Simulation results validate the throughput gains of the proposed relaying schemes.
1 Introduction
Twoway relaying has attracted tremendous attention due to its potential of significantly improving the spectral efficiency [1–4]. By applying physicallayer network coding (PNC) [5], two halfduplex nodes can accomplish a bidirectional data exchange in two phases with the assist of a halfduplex relay.
Recently, much of the interest has been generalized to multiway relaying [6–17]. The major challenge of designing a multiway relaying system lies in the cochannel interference induced by multiple users, which could dramatically degrade the system performance if not handled properly. So far, most related works focus on two traffic patterns, i.e., pairwise data exchange where the users form pairs to exchange data within each pair, and full data exchange, where each user and full data exchange, where each user broadcasts to all other users [8–11]. In contrast, the traffic pattern studied in our earlier papers [12, 13, 16, 17] is an arbitrary unicast, where the mapping from transmitters to receivers can be an arbitrary permutation and thus is more general. Such a framework is called wireless multipleinputmultipleoutput (MIMO) switching [12], and it is realized via a multiantenna relay. In this paper, we follow the same framework and study the relaying schemes of the distributed relays.
Specifically, we investigate distributed relaying schemes for multiple singleantenna relays in the wireless MIMO switching network. Most of the existing works on multiway relaying including our previous works exploited a multiantenna relay to improve relaying efficiency [6, 7, 10–13, 16, 17]. However, due to limited size and power, etc., it is natural to deploy multiple singleantenna relays in the network. A group of such relays forms a MIMO switch where distributed beamforming is carried out. Since data exchange among the distributed relays is expensive, decodeandforward schemes which involve joint coding and decoding at the relays are not practical. Thus, nonregenerative relaying is used instead in the network. In addition, the precoding matrix at the relays is diagonal as each relay can only multiply its received signal with a complex gain. The schemes for a multiple antenna relay fail for distributed relays. In [14], the authors proposed an orthogonalizeandforward protocol to make multiple pairs of twoway transmissions independent with the aid of multiple singleantenna relays. In this paper, we propose a scheme to improve the system sum rate which chooses a precoding vector in the interference null space under the criterion of maximizing the ratio of overall signal power to noise power. In [15], it was assumed that the users do not have knowledge of the channel state information. In order to achieve interferencefree transmission at the users, the equivalent channel gains of each desired signal and its corresponding selfinterference signal are designed to be equal to 1, and the gains of interuser to 0 with proper precoding at the relays citemolisch11. We will show that by limited attendant message limited attendant message passing, the constraint that the channel gains of desired signals and selfinterferences are equal can be relaxed. By doing so, the minimum required number of relays is reduced, we could achieve higher throughput. Furthermore, both citeBor07 and citemolisch11 assume that the [15] assume that the uplink and downlink channels are reciprocal. We will indicate that the relaying schemes in [14, 15] fail for nonpairwise traffic pattern and solve this problem later in this paper.
As we know, zeroforcing schemes could boost noise at low signaltonoise ratio (SNR) or for an illconditioned channel. Minimum mean square error (MMSE) schemes strike a balance between interference and noise. MMSE precoder and receiver were investigated in [18] for a singlehop transmission. Similar MMSE schemes with a multiantenna relay were proposed to minimize sum mean square error (MSE) of all the users in [7, 11] for multiway relaying. In contrast, due to the practical issues aforementioned, we propose a distributed MMSE relaying scheme to circumvent the noiseelevating issue of zeroforcing schemes.
Specifically, in this paper, we first propose a distributed zeroforcing relaying scheme, which nulls all interuser interferences and maximizes the ratio of overall signal power to noise power. For this relaying scheme, we also propose a messagepassing approach with low overhead. After that, we propose a distributed MMSE relaying which minimizes the sum MSE and improves the throughput further by jointly optimizing the precoders at the relays and the receive filters at the users. We aim to deploy both relaying schemes for pairwise/nonpairwise switching under reciprocal/nonreciprocal channels. For nonpairwise switching in reciprocal channels, a desired signal and the interference signal of its reverse link belong to the same signal space, which could make the proposed schemes fail. A trick by using the conjugate of the received signals at each hop is provided to deal with this problem.
Notation. The operators (·)^{T}, (·)^{∗}, (·)^{H}, and (·)^{−1} denote the transposition, the complex conjugation, the Hermitian and the inverse of a matrix, respectively. ⊙ denotes the Hadamard product, i.e., elementbyelement product. Tr{W} and rank(W) denote the trace and the rank of W, respectively. $\mathbb{E}[\phantom{\rule{0.3em}{0ex}}\xb7]$ is the expectation of a random variable. W=diag{w} denotes a diagonal matrix with the elements of w on the diagonal. w=diag{W} denotes a column vector that contains the diagonal elements of the matrix W. [W]_{diag} denotes a diagonal matrix with the same diagonal elements as W.
The remainder of this paper is organized as follows. The system setup of a distributed MIMO switching is introduced in Section 2. Two distributed relaying schemes based on zeroforcing and MMSE criteria are proposed in Sections 3 and 4, respectively. In Section 5, special cases of the two relaying schemes are further investigated in reciprocal channels. The simulation results are presented in Section 6. The paper is concluded in Section 7.
2 System model
Consider a wireless MIMO switching network with K singleantenna users and N distributed singleantenna relay nodes which form a virtual MIMO switch as shown in Figure 1. The K users communicate with each other via the MIMO switch, assuming that direct links are not available between any two users. In this paper, we focus on a pure unicast traffic pattern where each user transmits to one other user only. We assume that each user and relay is halfduplex, and then each transmission consists of two phases. In the uplink phase, all users transmit their signals simultaneously to the relays. After that, the relays broadcast their precoded signals back to the users in the downlink phase.
Let x=[x _{1},⋯,x _{ K }]^{T} be the signal vector of the users satisfying $\mathbb{E}\left[\phantom{\rule{0.3em}{0ex}}\mathbf{x}{\mathbf{x}}^{\mathrm{H}}\right]={\mathbf{I}}_{K}$. The received signals at the relays can be expressed as
where H=(h _{1},⋯,h _{ K }) is the uplink channel matrix, and ${\mathbf{h}}_{k}\in {\u2102}^{N\times 1}$ is the channel vector from user k to the N relays; $\mathbf{u}\in {\u2102}^{N\times 1}$ is the noise vector at the relays, with independent and identically distributed (i.i.d.) noise following a circularly symmetric complex Gaussian (CSCG) distribution, i.e., ${u}_{n}\sim \mathcal{C}\mathcal{N}(0,{\sigma}^{2})$. Upon receiving the uplink signal, each relay scales its received signal by a complex gain before forwarding to the users, that is,
where G=diag{g}, and the n th element of g is the complex gain factor at relay n. G is designed to realize a particular switch pattern among the users, which is specified by a mapping π(·). That is to say, user i sends data to user π(i) through the relays in a transmission. The total transmission power of the relays is bounded by P _{R}, i.e.,
where $\mathbf{\Phi}\triangleq \phantom{\rule{2.77626pt}{0ex}}{[\phantom{\rule{0.3em}{0ex}}\mathbf{H}{\mathbf{H}}^{\mathrm{H}}+{\gamma}^{2}{\mathbf{I}}_{N}]}_{\mathrm{d}\mathit{\text{iag}}}$. The received signals at the users can be written as
where $\mathbf{w}\in {\u2102}^{K\times 1}$ is the noise vector at the users, with i.i.d. noise samples following a CSCG distribution, i.e., ${w}_{k}\sim \mathcal{C}\mathcal{N}(0,{\sigma}^{2})$. Equivalently, the received signal of user π(i) is explicitly rewritten as
where the downlink channel is written as F = (f _{1},⋯,f _{ K })^{T}, and ${\mathbf{f}}_{k}\in {\u2102}^{N\times 1}$ is the channel vector from the relays to user k. The first term of the right hand side of (6) is the desired signal; the second term is the selfinterference which can be canceled by user π(i) as long as the equivalent channel gain ${\mathbf{f}}_{\pi \left(i\right)}^{T}\mathbf{G}{\mathbf{h}}_{\pi \left(i\right)}$ is known to this user; the third term is the interuser interference induced by other users; the last two terms are noises from the relays and the user end. Assuming that the upper and downlink phase both consume half of the total intervals, we obtain the sum rate as
In the following discussion, we optimize the sum rate based on two different criteria in the distributed MIMO switching network.
3 Distributed zeroforcing relaying
In this section, we propose a distributed zeroforcing relaying scheme that eliminates all interferences at the users. Specifically, we use a vector in the null space of the total interuser interferences as the precoder. By properly choosing the vector, we can maximize the ratio of the overall signal power to noise as well. In what follows, we elaborate the details of this scheme.
The total power of interuser interference at user π(i) is written as
Let
To null the interuser interference of user π(i), the precoder vector g satisfies
Then, all interuser interferences of the users being zero can be equivalently expressed as
where $\mathbf{W}=\sum _{i=1}^{K}{\mathbf{W}}_{i}.$ Since each user has K−2 independent interuser interferences, the total number of independent interuser interferences is K(K−2) when the uplink and downlink channels are nonreciprocal.^{a} In this case, the number of relays is required to be larger than K(K−2) to guarantee each interuser interference to be zero, i.e., N>K(K−2).
From (14), it is obvious that g is in the null space of W. Since W is the sum of K(K−2) different interference subspaces, the rank of W is K(K−2) with probability of 1. The orthonormal basis of the null space of W denoted as ${\mathbf{N}}_{\mathbf{W}}\in {\u2102}^{N\times (NK(K2\left)\right)}$ here can be obtained by the singular value decomposition:
The solution to (14) is thus a linear combination of the columns of N _{ W } and can be written as
where α= [α _{1},⋯,α _{ N−K(K−2)}]^{T} is the linear combination vector. We then determine the weights α by maximizing the ratio of the total power of the desired signals to the noise power. The total desired signal power is written as
where $\mathbf{\Omega}={\mathbf{N}}_{\mathbf{W}}\left[\phantom{\rule{0.3em}{0ex}}\sum _{i=1}^{K}{({\mathbf{f}}_{\pi \left(i\right)}\odot {\mathbf{h}}_{i})}^{\ast}{({\mathbf{f}}_{\pi \left(i\right)}\odot {\mathbf{h}}_{i})}^{\mathrm{T}}\right]{\mathbf{N}}_{\mathbf{W}}^{\mathrm{H}}$.The total noise power is
where $\mathbf{\Gamma}={\mathbf{N}}_{\mathbf{W}}^{\mathrm{H}}{\left[{\mathbf{F}}^{\mathrm{H}}\mathbf{F}\right]}_{\text{diag}}{\mathbf{N}}_{\mathbf{W}}$. The ratio of the total power of desired signal to the total noise power η can be obtained as
By inserting (15) into (4), the total power constraint at the relays can be rewritten as
where ${\mathbf{\Phi}}^{\prime}={\mathbf{N}}_{\mathbf{W}}^{\mathrm{H}}\mathbf{\Phi}{\mathbf{N}}_{\mathbf{W}}$. We claim that the relays need to transmit with total power P _{R} to maximize η. It can be proved as follows: suppose α _{opt} is the optimal combination vector and ${\mathbf{\alpha}}_{\text{opt}}^{\mathrm{H}}{\mathbf{\Phi}}^{\prime}{\mathbf{\alpha}}_{\text{opt}}<{P}_{\mathrm{R}}$, we can always find a α=ρ α _{opt} to make α ^{H} Φ ^{′} α=P _{R} with ρ>1. For α, the corresponding η _{ α } is
which contradicts with that α _{opt} is optimal. Thus, the optimal α satisfies the equation $\frac{{\mathbf{\alpha}}^{\mathrm{H}}{\mathbf{\Phi}}^{\prime}\mathbf{\alpha}}{{P}_{\mathrm{R}}}=1$. Denote that ${\mathbf{\Gamma}}^{\prime}=\mathbf{\Gamma}+\frac{K}{{P}_{\mathrm{R}}}{\mathbf{\Phi}}^{\prime}$. We can rewrite η as
which is a generalized Rayleigh ratio problem [19]. The optimal α to maximize η is $\mathbf{\alpha}=\kappa \stackrel{\u0304}{\mathbf{\alpha}}$ where $\stackrel{\u0304}{\mathbf{\alpha}}$ is the eigenvector of (Γ ^{′})^{−1} Ω corresponding to its largest eigenvalue, and κ is a scaling factor to guarantee that the relays transmit with maximum power in (4), i.e.,
Remark 1
To perform selfinterference cancelation, the equivalent channel response of selfinterference ${\mathbf{f}}_{\pi \left(i\right)}^{T}\mathbf{G}{\mathbf{h}}_{\pi \left(i\right)}$ in (6) is needed at user π(i). In addition, the equivalent channel response of desired signal ${\mathbf{f}}_{\pi \left(i\right)}^{T}\mathbf{G}{\mathbf{h}}_{i}$ in (6) is also required at user π(i) to estimate the desired signal. We claim that the equivalent channel gains can be delivered within at most three steps for an arbitrary unicast pattern. We provide an example to show how the users get the required channel responses. Consider seven users with traffic demands as shown in Figure 2. If only user i (e.g., user 1) sends a training pilot to the relays and the relays broadcast the precoded signal back to the users, in such a transmission, user i (e.g., user 1) obtains the gain of its selfinterference and user π(i) (e.g., user 2) gets the channel gain of the desired signal link. Since zeroforcing relaying nulls interuser interference completely, multiple users could send training pilots simultaneously without interfering each others, as long as users i, π ^{−1}(i), and π(i) (e.g., users 1, 3, and 2) do not transmit together. Based on this requirement, the users in a unicast cycle with even users can be scheduled in two steps, i.e., adjacent users are scheduled in different steps. The users in a unicast cycle with odd users can be scheduled in three steps. Note that the users from different cycles can be scheduled together because different cycles do not interfere with each other. As illustrated in Figure 2, the required channel information needed can be obtained in three steps shown in Figure 2b for the switching pattern in Figure 2a. In the first transmission, users 1, 4, and 6 estimate the equivalent channel gains of selfinterferences, and users 2, 5, and 7 estimate the gains of the desired signals. In the second transmission, users 2, 5, and 7 estimate the equivalent channel gains of selfinterferences, and users 3, 4, and 6 estimate the gains of the desired signals. In the last transmission, user 3 estimates the equivalent channel gain of selfinterference, and user 1 estimates the gain of its desired signal. Thus, we conclude that at most three steps are required for message passing before data transmission of the network.
Note that a central node is required to compute the precoding matrix at the relays following the assumption in [16, 17]. In addition, channel information H and F are required at the relays. The channel estimation at the relays is out of the scope of this paper. More details can be found in [20, 21].
4 Distributed MMSE relaying
In this section, we propose a distributed MMSE relaying scheme that minimizes the sum MSE by jointly optimizing the precoder at each relay and the receive filter at each user end.
Similar to [13], we use a diagonal matrix B to express the weights of selfinterference and a diagonal matrix C to denote the receive filters. The receivers first filter the signal with C and then subtract the selfinterferences B x from the filtered signal C r. The sum MSE can be written as $\mathbb{E}\left\{\parallel \mathbf{P}\mathbf{x}(\mathbf{C}\mathbf{r}\mathbf{B}\mathbf{x}){\parallel}^{2}\right\}$, where P is a Kdimensional permutation matrix realizing the required switch pattern, i.e., the mapping defined by π(·) as mentioned in Section 2; that is, column i of P is equal to e _{ π(i)}, i.e., p _{ i }=e _{ π(i)}, where e _{ i } is column i of an identity matrix. Together with the total transmission power constraint of the relays in (4), the sum MSE minimization problem is formulated as follows:
Unfortunately, the optimization problem (33a) is nonconvex and thus is generally difficult to solve in an optimal way. We then propose an iterative algorithm to optimize G, B, and C. A similar method was used in [7, 18], which considered a singlehop problem or a multiway relaying without network coding. Here we show that PNC can be easily incorporated and optimized in the iterative method.
To solve the problem (33), we introduce a cofactor β and define
The Lagrangian function is written as
where $\stackrel{\u0304}{\mathbf{g}}=\text{diag}\left\{\stackrel{\u0304}{\mathbf{G}}\right\}\in {\u2102}^{N\times 1}$, and the last step follows by plugging (5) and (34).
4.1 Optimal $\mathbf{(}\stackrel{\mathbf{\u0304}}{\mathbf{g}},\mathbf{\beta})$ for fixed $\mathbf{(}\mathbf{B}\mathbf{,}\stackrel{\mathbf{\u0304}}{\mathbf{C}}\mathbf{)}$
For fixed $(\mathbf{B},\stackrel{\u0304}{\mathbf{C}})$, the Lagrangian function (36) can be rewritten as
where
From the KKT conditions, we have
Substituting (41) into (42), we get
Plugging (43) to (40), we have
Note that similar KKT conditions were considered in [7]. However, the derivation provided here is much simpler.
4.2 Optimal $(\mathbf{B},\stackrel{\u0304}{\mathbf{C}})$ for fixed $\mathbf{(}\stackrel{\u0304}{g},\beta )$
Denote $\mathbf{b}=\text{diag}\left\{\mathbf{B}\right\}\in {\u2102}^{K\times 1}$, $\stackrel{\u0304}{\mathbf{c}}=\text{diag}\left\{\stackrel{\u0304}{\mathbf{C}}\right\}\in {\u2102}^{K\times 1}$. For fixed $(\stackrel{\u0304}{\mathbf{g}},\beta )$, i.e., fixed G, the Lagrangian function can be rewritten as
where
From (45), we obtain the solution to minimize the sum MSE for fixed $(\stackrel{\u0304}{\mathbf{g}},\beta )$,
Note that from (51) the interference canceled at the user end is rewritten as
which implies that selfinterference is completely canceled by the solution of B in (51).
By performing the above two steps iteratively, we present the proposed iterative approach as follows:

1:
Init: B=B _{0}, $\overline{\mathbf{C}}={\overline{\mathbf{C}}}_{0}$;

2:
while the sum MSE can be reduced by more than e do

3:
Calculate $\overline{\mathbf{g}}$ and ß using (44) and (41), respectively;

4:
Calculate B and $\overline{\mathbf{C}}$ using (51) and (52), respectively; 5: end while
4.3 Convergence of the iterative algorithm
The sum MSE decreases monotonically in the iteration. Together with the fact that the sum MSE is nonnegative, we conclude that the iterative algorithm converges. Figure 3 shows the sum rate versus the number of iteration with different numbers of random initial vectors. When multiple random initial vectors are used, we choose the vector that converges to the highest sum rate. It can be observed that the sum rate converges within 20 iterations. Higher sum rate can be achieved by setting various random initials, but the performance gain is marginal when the number of random initials exceeds 20.
Remark 2
With the proposed MMSE relaying, each user also demands the equivalent channel gains of the desired signal and the selfinterference to recover the desired signal, as that for the proposed zeroforcing relaying mentioned previously in Remark 1. However, each user needs to send its pilot individually in different steps, since there is residual interuser interference at the users for the MMSE relaying. Larger overheads are required compared to the proposed zeroforcing scheme. In the simulations, we will show that with the larger overhead, the MMSE relaying can further improve the sum rate that achieved by the proposed zeroforcing scheme.
Remark 3
Here we compare the computational complexity of our schemes with the existing schemes in [14, 15]. We count the numbers of matrix inversion and singular value decomposition (SVD) needed in each scheme as the complexity of other operations can be neglected compared to them. In the scheme proposed in [14], the major operation is an SVD operation of a N×K(K−2) interference matrix with complexity $\mathcal{O}\left(N{K}^{2}{(K2)}^{2}\right)$. For the scheme in [15], the major operations are the SVD of a N×K(K−2) matrix and a matrix inverse operation of two N×N matrices. The computational complexity is $\mathcal{O}\left(N{K}^{2}{(K2)}^{2}\right)+\mathcal{O}\left(2{N}^{3}\right)$. In our proposed zeroforcing scheme, to get the weighting vector (15) at the relays, we need to compute N _{ W } and α. Specifically, we need to compute the null space of a N×K(K−2) to obtain N _{ W }. To get α, we need to compute the inverse of a N×N matrix and the eigenvector of a N×N matrix. The computational complexity is $\mathcal{O}\left(N{K}^{2}{(K2)}^{2}\right)+\mathcal{O}\left(2{N}^{3}\right)$. With MMSE relaying, we need to compute the inverse matrix of a N×N matrix to obtain $(\stackrel{\u0304}{\mathbf{g}},\beta )$ for fixed $(\mathbf{B},\stackrel{\u0304}{\mathbf{C}})$ and the matrix inverse of a K×K matrix to obtain $(\mathbf{B},\stackrel{\u0304}{\mathbf{C}})$ for fixed $(\stackrel{\u0304}{\mathbf{g}},\beta )$ in each iterative step. Suppose M (M is less than 20 as evaluated in Figure 3) iterations are needed. The computational complexity is $\mathcal{O}\left(M{N}^{3}\right)+\mathcal{O}\left(M{K}^{3}\right)$.
5 Further discussion in reciprocal channels
We consider the case that the uplink channel and downlink channel are reciprocal, i.e., F=H ^{T}, which is a widely adopted assumption for timedivision duplex systems. In this case, the equivalent channel from user i to user j is exactly the same as that from user j to user i. In what follows, we investigate whether the proposed schemes can be used as well in the case of channel reciprocity.
5.1 Distributed zeroforcing relaying
We notice that for reciprocal channels, the situations for pairwise and nonpairwise traffic patterns are different as will be elaborated later in this subsection. We first refer to pairwise switching.
5.1.1 Pairwise switching
Recall that there are K(K−2) independent interuser interferences in nonreciprocal channels. Due to channel reciprocity, the interference from user i to user j, j≠π(i) (its equivalent channel is ${\mathbf{h}}_{j}^{T}\mathbf{G}{\mathbf{h}}_{i}$) and that from user j to user i (its equivalent channel is ${\mathbf{h}}_{i}^{T}\mathbf{G}{\mathbf{h}}_{j}$) belongs to the same space spanned by the equivalent channel vector h _{ i }⊙h _{ j }. With this result, the number of relays required to null all interuser interferences is larger than $\frac{1}{2}K(K2)$, which is half of the required number for the nonreciprocal case.
5.1.2 Nonpairwise switching
For a nonpairwise switching, if the link from user i to user j is for a desired signal transmission, i.e., j=π(i), then the reverse link from user j to user i is an interference link, i.e., i≠π(j) (or else it becomes a pairwise switching). In the case of channel reciprocity, the desired signal and the interference in its reverse link are in the same space spanned by the equivalent channel vector h _{ i }⊙h _{ j }. Thus, the interference can not be nulled by projection. We next provide a trick here to differentiate the link from its reverse link.
In the downlink, the relays send the conjugate of the precoded signal z in (2) instead, i.e.,
At the user end, all users detect r ^{′}, which is the conjugate of the received signals, i.e.,
With this trick, the equivalent channel gain of the link from user i to user j becomes ${\mathbf{h}}_{j}^{\mathrm{H}}\mathbf{G}{\mathbf{h}}_{i}$, and its reverse channel gain is ${\mathbf{h}}_{i}^{\mathrm{H}}\mathbf{G}{\mathbf{h}}_{j}$. Since the spaces spanned by ${\mathbf{h}}_{i}\odot {\mathbf{h}}_{j}^{\ast}$ and ${\mathbf{h}}_{i}^{\ast}\odot {\mathbf{h}}_{j}$ are different, the problem becomes the same as the one for the nonreciprocal case and can be solved accordingly.
Remark 4
The minimum relays required in different cases are listed in Table 1. Recall the schemes in [14, 15] that were proposed for multiuser twoway relaying, i.e., pairwise traffic pattern, assuming channel reciprocity. They can be directly extended to the case of channel nonreciprocity, while with different minimum required relays. However, the schemes in [14, 15] cannot be extended to nonpairwise switching for reciprocal channels. The proposed conjugate trick could also heal the two proposed schemes.
5.2 Distributed MMSE relaying
For the MMSE scheme, the requirement of minimum relays is not necessary. However, for the nonpairwise switching with the assumption of channel reciprocity, even with sufficient relays, it is still difficult to improve the performance because the interferences and the desired signals are coupled together. The throughput can be boosted when the spaces of signal and interference are completely decoupled with the trick of using conjugate, which will be shown later in Section 6.
6 Numerical results
In this section, we evaluate the sum MSE and the sum rates of the distributed MIMO switching network using different relaying schemes. Consider a pure unicast traffic pattern with four users, i.e., K=4. The maximum total transmission power of the relays is equal to the transmission power of each user, i.e., P _{R}=1. The noise levels at each relay and user are the same. The entries of the channel vectors are generated according to $\mathcal{C}\mathcal{N}(0,1)$ and all the results are obtained by averaging 10,000 Monte Carlo simulations. In the simulation, we use N _{1} and N _{2} to denote the required minimum relays for the proposed zeroforcing relaying and the scheme in [15], respectively. Note that the values of N _{1} and N _{2} can be different in different cases as shown in Table 1. Then, we present four observations as follows:
Observation 1
The proposed MMSE relaying scheme has better sum MSE performance than zeroforcing relaying schemes.
In Figure 4, we compare the sum MSE of our proposed schemes with the schemes in [14, 15]. The simulations show that MMSE relaying outperforms the zeroforcing schemes significantly. With N _{1} relays, i.e., the minimum required relays for the proposed zeroforcing scheme, the scheme in [15] fails with large sum MSE over the whole SNR region. The proposed zeroforcing scheme has the same sum MSE with the scheme in [14] as α is a scalar determined by the power constraint at the relays which is the same for both schemes with N _{1} relays. With MMSE relaying, sum MSE is reduced significantly as the effect of noise is considered in the design of the scheme.
Observation 2
The proposed distributed zeroforcing and MMSE relaying schemes achieve significant throughput gain over the distributed schemes in [14, 15].
In Figures 5, 6, and 7, we compare the sum rates of the two proposed approaches with the relaying schemes in [14, 15] in different cases, i.e., Figure 5 for nonpairwise switching with nonreciprocal channels, Figure 6 for pairwise switching with reciprocal channels, and Figure 7 for nonpairwise switching with reciprocal channels.^{b} It is shown that roughly the same results can be obtained from the three different cases. Both the proposed schemes outperform the schemes in [14, 15].
Specifically, from Table 1, we see that the proposed zeroforcing scheme needs less relays than the scheme in [15]. When using N _{1} relays, i.e., the minimum required relays for the proposed zeroforcing scheme, the scheme in [15] fails without efficient throughput. When the number of relays increases to N _{2}, i.e., the minimum required relays of the scheme in [15], the proposed schemes still outperform this scheme significantly. Compared to the scheme in [14], the proposed zeroforcing scheme has the same throughput when using N _{1} y relays. This is because with N _{1} relays, α is the same for both schemes as stated above. However, as the number of relays increases, the proposed zeroforcing scheme becomes better. The reason lies in that our scheme always optimizes α to maximize the ratio of overall signal power to noise power. However, the scheme in [16] fails in this.
In all the cases, the proposed MMSE scheme achieves the best throughput performance, and it does not require minimum relays. The penalty is that its overhead is larger than the other schemes.
Observation 3
For reciprocal channels, the minimum required relays can be reduced to almost half of the other cases for pairwise switching, and the proposed conjugate trick could avoid the failure of all the schemes for nonpairwise switching.
In Table 1, it can be observed that the number of relays required for pairwise switching in reciprocal channels is almost half of the other cases. The results in Figures 5, 6, and 7 further indicate that similar sum rate can be achieved in this case with almost half relays.
Figure 8 compares the performance of proposed MMSE relaying with conventional MMSE relaying not using conjugate for nonpairwise switching in reciprocal channels. In this case, the sum rate of conventional MMSE relaying is quite low and can be improved trivially by increasing the number of relays, since the desired signals and some interference signals belong to the same signal space. By using conjugate trick, the problem can be solved and the sum rate increases significantly.
Observation 4
The throughput performance improves as the number of relays increases, specially for the proposed zeroforcing relaying scheme with the number of relays from N _{1} to N _{1}+1.
Figure 8 compares the performance of the proposed zeroforcing and MMSE relaying verse different number of relays. We take the nonpairwise switching in reciprocal channels as an example, as similar results can be obtained for other cases. Note that the performance of MMSEbased scheme degrades smoothly when N decreases, which makes this scheme also suitable for smaller system configurations. As for the zeroforcing relaying, one more relay boosts the performance compared to the case with minimum number of relays. However, the throughput improvement is limited when N increases.
7 Conclusion
In this paper, two relaying schemes are proposed for the distributed MIMO switching network. We indicate that the proposed distributed zeroforcing scheme exhibits large improvement of system throughput with limited attendant overhead compared to the existing distributed zeroforcing schemes. The proposed distributed MMSEbased scheme can further improve the throughput performance over the proposed zeroforcing scheme at the cost of larger overhead. Both schemes can be applied to an arbitrary unicast pattern under reciprocal/nonreciprocal channels, with the proposed conjugate trick healing the failure of nonpairwise switching with reciprocal channels.
We assume singleantenna users and relays and perfect channel knowledge at the relays. Future research will focus on the distributed relaying schemes for multiantenna nodes and with partial channel information at the relays.
Endnotes
^{a} As for reciprocal channels, the number is different, which will be discussed later in Section 5.
^{b} For nonreciprocal channels, the minimum required relays for pairwise and nonpairwise switching is exactly the same. In addition, similar results can be concluded from both pairwise and nonpairwise switching. Thus, we only show the simulation results of nonpairwise switching in Figure 5.
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Acknowledgements
This work was partially supported by the National Natural Science Foundation (no. 61201201 and U1261109), the State Key Lab of Rail Traffic Control and Safety (no. RCS2011ZT011), the Fundamental Research Funds for the Central Universities (nos. 2011JBM203, 2010RC007, and 2010JBZ008), and the Key Grant Project of Chinese Ministry of education (no. 313006).
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Wang, M., Wang, F. & Zhong, Z. Wireless MIMO switching: distributed zeroforcing and MMSE relaying using network coding. EURASIP J. Adv. Signal Process. 2013, 130 (2013) doi:10.1186/168761802013130
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Keywords
 Minimum Mean Square Error
 Equivalent Channel
 Total Transmission Power
 Reciprocal Channel
 Minimum Mean Square Error Scheme