Degrees of Freedom of Y Channel with Single-Antenna Users: Transmission Scheme and Beamforming Optimization

Abstract

In this paper, we investigate the degrees of freedom (DOF) of the Y channel consisting of three single-antenna users and a two-antenna common access relay, where each user intends to exchange independent messages with the other two users with the assistance of the relay. We show that the DOF of this particular scenario is 1.5. In order to prove this result, we firstly derive a DOF upper bound based on cut-set bound by allowing cooperation among users, which shows that the total DOF is upper bounded by 1.5. Then we propose a novel transmission scheme based on asymmetric signal space alignment (ASSA) to demonstrate the achievability of the upper bound. Theoretical evaluation and numerical results confirm that the upper bound can be achieved by utilizing ASSA, which also proves the optimality of the ASSA-based scheme in terms of DOF. Combining the upper bound and achievability, we conclude that the exact DOF is 1.5. Moreover, we present a novel iterative joint beamforming optimization (I-JBO) algorithm to further improve the sum rate. Numerical simulations have been provided to demonstrate the convergence speed and performance advantage of the I-JBO algorithm.

1. Introduction

The rapidly increasing demands for wireless multimedia services have promoted the development of novel signaling techniques with high spectrum efficiency [1, 2, 3]. Multiple-input multiple-output (MIMO) and relay are two outstanding representatives of these techniques [4, 5]. By combining MIMO and relay, MIMO two-way relaying channel (TWRC) has been an attractive topic for the reason that it remarkably increases the spectral efficiency of relaying system [6, 7, 8, 9]. Furthermore, the TWRC has been extended into multi-way relay channel (MWRC), which has been considered as a fundamental building block for future cooperative communication systems [10]. In particular, a novel MWRC system model termed MIMO Y channel has been introduced in [11], where each of the three users intends to exchange independent messages with the other two users via an intermediate relay. The authors also proposed a so called signal space alignment (SSA) scheme, which exploits the idea of interference alignment (IA) [12] and network coding (NC) [13, 14]. The information exchange is implemented within only two phases, namely the multiple access (MAC) phase and the broadcast (BC) phase. During the MAC phase, the users cooperatively design their precoding vectors to align the paired signals into the same spatial dimension at the relay, and then all the users transmit to the relay simultaneously. During the BC phase, the relay broadcasts the composite signal back to the users, and then the users extract the useful signals by self-interference cancellation.

As a typical representative of the MWRC, MIMO Y channel can model various communication scenarios and may play an important role in practical wireless systems [10]. Therefore, MIMO Y channel has attracted significantly growing interests. The basic MIMO Y channel has been further extended to the four-user case and the K-user case in [15] and [16] respectively, where the achievable degrees of freedom (DOF) are investigated. Moreover, some specific problems on MIMO Y channel, such as beamforming design, constellation mapping, diversity techniques, and user scheduling, have been studied in [17, 18, 19, 20]. On the other hand, a special case of the basic Y channel where all the three nodes have a single antenna has been studied in [16] and [21], where insightful results regarding the achievable DOF and the system capacity are obtained. More specifically, the algorithm in [16] is tailored for the single-input single-output (SISO) case. Moreover, the authors have not proved that the algorithm can achieve the DOF upper bound. Therefore, the exact DOF of the basic Y channel with single-antenna users remains an open problem.

In this paper, we study the DOF of the Y channel composed of three single-antenna users and a two-antenna common access relay as shown in Fig. 1. The channel model is practical since it can find many interesting applications in various ad-hoc systems. For example, in a clustered wireless sensor networks (WSN) with three users in each cluster, each sensor may want to share information with its two partners in the same cluster, while the data fusion center can play the role of the relay. Usually, the sensors are constrained by limited size and short battery life, hence multiple antennas may be unrealistic for them. Although the relay usually has greater capability, it is still restricted by the energy limited nature of WSN [22, 23]. Therefore, two antennas may be a reasonable assumption for the relay. For the considered scenario, we first derive the DOF upper bound. Then we study the transmission scheme to achieve the upper bound. According to the literatures [11, 16], the SSA scheme has been proved to significantly improve the achievable DOF of Y channel, therefore it has the potential to achieve the upper bound. However, according to [11], the traditional SSA is not applicable in this channel model for the lack of spatial dimensions. Symbol extension enables the application of SSA into the scenario by using the dimensions in time domain. Nevertheless, the symbol extension will cause delay to the network. It is noted that the asymmetric complex signaling (ACS) has been proposed in [24] to provide extra dimensions, which enables IA in the interference channel with single-antenna users. Enlightened by this, we investigate the application of ACS in Y channel and propose a novel asymmetric signal space alignment (ASSA) based scheme tailored for this special scenario, which achieves the DOF upper bound without symbol extension.

Fig. 1.A system diagram for the addressed communication scenario

Beamforming optimization is another hot topic in Y channel, because the performance of the transmission scheme is closely related with the beamforming design. In [19], a random beamforming vector selection approach is proposed to obtain the spatial diversity. Noting that the random selection method can hardly guarantee the optimality, [17] has proposed a deterministic iterative beamforming optimization as well as power allocation scheme to maximize the effective signal to noise ratio (SNR) for each channel realization. Furthermore, [18] has extended the algorithm in [17] into the Y channel with multiple data streams per message. However, the aforementioned algorithms are effective only when the dimension of the intersection space of each user pair is more than one, which doesn’t hold in the considered scenario. In order to tackle this difficulty, we propose an efficient beamforming optimization algorithm without the restriction on spatial dimensions.

This paper is an extension of the conference version in [25]. We investigate the DOF and beamforming optimization problems for the Y channel with single-antenna users. To be more specific, the main contributions of this paper are summarized as follows.

The remainder of this paper is organized as follows. Section 2 introduces the system and signal models for the addressed communication scenario. In section 3, we derive a DOF upper bound of this channel. In section 4, we present the ASSA scheme in detail to prove the achievability of the upper bound. In section 5, the I-JBO algorithm is introduced to boost the sum rate. The simulation result is given in section 6, followed by some concluding remarks in section 7.

Notations: We use bold upper and lower case letters to represent matrices and vectors, respectively. For any matrix A, ║A║F , tr(A), AH , AT , and A† denote the Frobenius norm, trace, conjugate transpose, transpose, and pseudo inverse of A , respectively. For any complex number a , Re(a), Im(a) and |a| denote the real part, imaginary part and modulus of a , respectively. E{·} is the expectation operator. ║·║ and <·> indicate the 2-norm operator and normalization operator for vectors, respectively. diag(a,b,⋯) denotes a diagonal matrix with {a,b,⋯} as its diagonal elements. R and C indicate the set of real and complex numbers, respectively.

 

2. System Model

As shown in Fig. 1, we consider a three-user Y channel where each user Si, i = 1,2,3, has a single antenna, the amplified-and-forward (AF) relay R is equipped with two antennas, and all the nodes operate in half-duplex mode1. Every user intends to send independent messages to the other two users via the relay. The direct links between users are ignored on account of severe fading and path loss. For ease of exposition, we adopt time division duplexing (TDD) system where channel reciprocity holds, therefore the channels for links Si → R and R → Si are expressed as hi ∈ C2×1 and hiT ∈ C1×2 , respectively. We also assume flat Rayleigh fading, then the entries of hi are independent and identically distributed (i.i.d.) as CN (0,1). Furthermore, perfect global channel state information (CSI) is assumed at all the nodes. The exchange of network information flow completes in two phases, namely the MAC phase and the BC phase.

During the MAC phase, all the users transmit to R simultaneously. The received signal at R is given by

where xi ∈ C is the transmit complex symbol from Si , and nR ∈ C2×1 denotes the additive white Gaussian noise (AWGN) vector with i.i.d. elements following . xi is subject to an average power constraint as E{xi2} ≤ Pi , where Pi is the transmit power of Si .

During the BC phase, the signal transmitted by R is

where G ∈ C2×2 is the processing matrix with the power restriction as . Consequently, Si observes the received signal as

where ni ∈ C denotes the AWGN following CN(0, σi2), and Si tries to extract the useful message from yi . For simplicity, we assume equal power allocation and set Pi = PR = P and .

DOF is often utilized to characterize the capacity behavior in high SNR regimes. We define the total DOF of the above Y channel as

where ρ denotes the transmit SNR and is defined as ρ = P/N0 , R(ρ) stands for the sum rate with respect to ρ, Rj,i(ρ) is the rate of the link from Si to Sj , and denotes the DOF of the link from Si to Sj .

 

3. An Upper Bound on DOF

In this section, we derive an upper bound on the DOF of the considered scenario, which serves as a guideline for the design of transmission scheme.

Theorem 1: For the Y channel composed of three single-antenna users and a two-antenna common access relay, the total DOF is upper bouned by 1.5, i.e.,

Proof: We first consider the network information flow from {S2,S3} to S1 as illustrated in Fig. 2. Using the fact that allowing cooperation of any two users among the three users does not reduce the DOF [11], we assume full cooperation between S2 and S3 to obtain an upper bound.

Fig. 2.The network information flow from {S2,S3} to S1

During the MAC phase (cut 2), S2 and S3 transmit signals to R simultaneously. We can treat the channel from S2 and S3 to R as a 2 x 2 MIMO channel, whose DOF is well-known as min {2,2} = 2. Similarly, we can obtain the DOF of the channel in the BC phase (cut 1) as min {1,2} = 1. Applying the cut-set theorem in [27] to the two phases, the cut-set bound on DOF for the network information flow from {S2,S3} to S1 is obtained as

where the pre-log 1/2 is due to the fact that the transmission is completed in two time slots.

For the other two directions of the network information flow, i.e., messages from {S1,S3} to S2 and from {S1,S2} to S3, we can get similar results as

Combining (6), (7), and (8), we obtain the total cut-set bound on DOF as

which completes the proof of the theorem.

 

4. Achieving the Upper Bound: Asymmetric Signal Space Alignment

In this section, we describe the ASSA-based scheme in detail to prove the achievability of the DOF upper bound.

4.1 Asymmetric Signal Space Alignment

According to [16], the relay and each user should have at least three and two antennas respectively to make SSA feasible for the considered scenario. In other words, traditional SSA scheme is not straightforwardly applicable when the system is lack of spatial dimension, i.e., each user is only equipped with one antenna. To this end, we first use ACS to convert the complex system into an equivalent real system with double dimensions. By employing the real-valued representation, we can rewrite (1) as

where the equivalent real-valued received signal is

yRk is the kth element of yR , the equivalent transmit signal is

the equivalent noise is

and nRk is the kth element of nR , k ∈ {1,2}. It is noted that the elements of are i.i.d. real Gaussian random variables following . The real version of the channel matrix is expressed as

where hik is the kth element of hi , k ∈ {1,2}. Consequently, the 2 × 1 complex channel hi for the link Si → R is transformed into the 4 × 2 real matrix . In other words, the number of dimensions at all the nodes is doubled. In fact, the system always works as equation (1) in actual transmission. And the real-valued equation (10) is just an equivalent representation of the system, which is used to facilitate the beamforming design.

By employing the extra spatial dimensions introduced by the ACS, we design the transmit signal as

where dj,i ∈ R is the real scalar data from Si to Sj with unit variance, and vj,i ∈ R2×1 denotes the corresponding precoding vector. To align each pair of signals into the same dimension, the paired precoding vectors vj,i and vi,j are designed to satisfy

This is equivalent to solving

where the subscript l ∈ {1,2,3} is defined as the index of user pair {Si , Sj} and l = π(i,j) = π(j,i). More specifically, we define 1 = π(1,2), 2 = π(1,3), and 3 = π(2,3). However, since is a square matrix with full rank, its nullspace, which is also the intersection subspace of span () and span (), does not exist. As a result, we still cannot align each pair of signals.

In order to tackle the difficulty above, we carefully design a projection matrix at the relay. More specifically, we employ a projection matrix Q ∈ R4×3 at the relay to change the size of the equivalent channel matrix. The columns of Q are assumed to be orthonormal, i.e., QTQ = I . Specifically, we multiply the received signal with QT . As a result, the problem in (17) is turned into

Then we can obtain from the nullspace of . We assume equal power allocation for the two data streams of each user. As a result, in order to constrain the transmit power of each pair, should satisfy . Consequently, we set , where is a normalized vector in the nullspace of . We note that there is potential for optimizing Q to improve the sum rate performance. The design of Q will be discussed in section 5.

We define the equivalent MAC channel for each user pair as

As a result, the received signal processed by QT at the relay can be presented as

where dl+ = dj,i + di,j denotes the AF network coded message for the lth pair. It is indicated in (20) that the six signals are aligned into three dimensions. Consequently, linear receiver can be used at the relay to decouple the composite signals. In the BC phase, we also need to design each user's decoding vector to align the signal space of each pair. Only then can the relay perform linear precoding. Thanks to channel reciprocity, we can simply set the decoding vectors as wi,j = vTj,i , which simplifies the computational complexity. For simplicity, we adopt zero-forcing (ZF) linear transceiver at the relay, which is designed as

where U = [u1 u2 u3], ul is given in (19), F = diag(ƒ1,ƒ2,ƒ3) is the power constraint matrix, and the diagonal elements are given as

By utilizing F , the long-term transmission power at the relay can be constrained as PR [28].

Remark: In actual transmission, the relay first transforms the complex channel vector hi into real channel matrix as equation (14) and designs the beamforming vectors for each user as equation (18). Then the relay feeds back the beamforming vectors to the users. Finally, the users design the real signal according to equation (15) and transmit the signal in complex form as , where , k = 1,2 is the kth element of , and j is imaginary unit.

4.2 Brief DOF Evaluation

After applying decoding vector wi,j = vTj,i and self-interference cancellation, Si receives the signal as

By substituting (19), (20), and (21) into (23), we can rewrite (23) as

where Fl is the lth row of F . The end-to-end sum rate is calculated as

where the received SNR for the data stream from Sj to Si is derived from (24) as follows

and the pre-log 1/2 in (25) is due to the real signal. We can see that, after all the inter-pair and intra-pair interferences are eliminated, the desired signal is only contaminated by Gaussian noise. In other words, six real data streams are transmitted without interference within two time slots. According to [29], the DOF of one real symbol is 0.5. Consequently, the DOF achieved by the proposed ASSA scheme is

which is a lower bound of the DOF. Hence we obtain

So far, we have obtained the DOF of the addressed Y channel. The main result is summarized in the following theorem.

Theorem 2: For the Y channel composed of three single-antenna users and a two-antenna common access relay, the total DOF is 1.5, i.e.,

Proof: Combining the upper bound in (5) and the lower bound in (28), we prove that the exact DOF of the channel model we investigate is 1.5.

Remark: The ASSA based scheme is not straightforwardly applicable to the scenario with more than three users, where the beamforming design is rather challenging due to the insufficient dimensions of signal space. However, we can still make the proposed algorithm applicable in such cases by combining it with the user scheduling method proposed in [20].

 

5. Beamforming Optimization for ASSA

5.1 Iterative Joint Beamforming Optimization Algorithm

It is shown in (19), (22), (25), and (26) that the sum rate is a function of Q and vj,i . This observation motivates us to study the beamforming optimization to further enhance the sum rate.

Because of the symmetrical beamforming design in MAC and BC phases, we follow [17] and only consider the MAC phase for simplicity. First, a mathematically equivalent expression of the ZF processing at the relay is presented. Specifically, instead of the pseudoinverse in (21), we use a unit norm vector pl ∈ R3×1 to extract the signal of the lth pair. According to [30], the optimal design of pl is given as pl = , where

and is a submatrix of U by removing its lth column ul . After the processing of pl and Q, the signal of the lth pair is acquired as

After some derivations, the SNR for the lth pair is obtained as

In order to maximize the sum rate, we formulate the beamforming design as the following joint optimization problem

It is shown in (34) that Q and vj,i are relevant, therefore we need to design them jointly. It is difficult to find the optimal solution because of the extremely high complexity. Therefore, the key issue is how to decouple these variables. We carefully study the coupling part of all the variables and introduce a unit-norm vector q to express all the variables. More specifically, we assume QTq = 0. As a result, we can reformulate (34) as

where αl is used to meet the power constraint in (35). From (34), we can obtain

where Aj,i ∈ R2×4 and Ai,j ∈ R2×4 contain the first and last two rows of , respectively. Consequently, the precoding vectors can be written as

In order to control the transmit power, we let

which results in . Similarly, we have . Hence, we assume

Since the column space of q is the orthogonal complement of that of Q, we arrive at

By substituting (19), (30), (39), and (43) into (43), we get the SNR expression with q as the only variable vector as follows

where the subscript a,b,c,d ∈{1,2,3}, m = π(a,b), n = π(c,d), m ≠ n, l ≠ m , and l ≠ n . And the optimization problem is simplified as

However, the above optimization problem is non-convex with respect to q [26], so it is still difficult to find the optimal solution. We obtain a local solution iteratively by using the combination of penalty function method and quasi-Newton method, which is effective to solve the equality constrained optimization problem [26]. Initially, we employ penalty function method to convert the problem in (45) into an unconstrained problem as

where

is the original objective function, h(q) = (qTq - 1)2 is the penalty function and μ > 0 is the penalty coefficient. Next, the quasi-Newton method is used to solve the problem in (47). Once q is obtained, we can calculate vj,i and Q according to (39), (40), and (43). The detail of I-JBO algorithm is summarized in Table 1. Although we cannot guarantee that the proposed iterative algorithm converges to the global optimal solution due to the nonconvexity of the objective function, we can still demonstrate the sum rate enhancement via numerical results. (See section 6 for details.)

Table 1.Iterative joint beamforming optimization algorithm

5.2 Evaluation of Computation Complexity

In this subsection, we discuss the computation complexity of the proposed beamforming optimization scheme along with the random beamforming design scheme as follows. We measure the complexity of the algorithms by counting the number of floating-point operations (flops).

We note that the calculation of ▽g(q) takes most of the flops, so we focus on it and omit the other operations. ▽g(q) can be calculated with a great number of applications of the product and quotient rules. However, the calculation process is tediously long, so we omit the calculation details here due to space limitation. Totally the flops of the proposed I-JBO algorithm is about K × I × 2.11 × 105 , where K and I are the number of iterations in the outer loop and the inner loop respectively. The number of iterations in the inner loop may vary in every outer loop, so I is an average value. For comparison, we also count the flops of the random beam design, and the result is about 3.85 × 103 . Although the proposed algorithm takes more computations, it can improve the sum rate significantly.

 

6. Numerical Results

In this section, we provide the numerical results to show the end to end sum rate performance of the proposed schemes, namely the ASSA transmission scheme, and the affiliated I-JBO beam design algorithm. In all the simulations, the i.i.d. Rayleigh flat fading channels are assumed. For fairness, we set Pi - PR = P and for all the test cases, so that the system transmit SNR becomes ρ = P/N0 . The sum rate in one channel realization is calculated according to (25). After that it is averaged from 104 channel realizations to get the ergodic sum rate. The convergence property of the I-JBO algorithm is also verified. More specifically, mean squared error (MSE) of the variable vector q and the average value of the objective function are verified for both of the inner loop and outer loop.

At first, we compare the sum rate between the naive ZF in [31] and ASSA with different beam designs. Based on the definition of DOF, we set the x-axis as log(ρ), then the slope of the sum rate curve at high SNR represents the achievable DOF. Moreover, we also plot two straight lines with slope of 1 and 1.5 as references to verify the DOF results. As shown in Fig. 3, the curves that denote the ASSA with different beamforming designs, i.e., the red, green, and blue ones, all become parallel with the straight line with slope of 1.5 as SNR increases, which means the DOF of 1.5 is achieved. Similarly, we can easily read from the figure that the achieved DOF of the naive ZF scheme is 1. It is further illustrated that the performance of the iterative scheme in [17] is the same with the random beam design. The main reason is that the dimension of the intersection space of each user pair is one in the channel model we consider, consequently there is no room for the strategy proposed in [17] to optimize the beamforming. Especially at low-to-moderate SNR regime, the ASSA with iterative and random beam designs even perform worse than the naive ZF. This is because the transmission schemes based on IA or SSA share one disadvantage, i.e., performing poorly at low SNR regime. Hence, although achieving higher DOF, it performs more poorly than the naive ZF scheme at low SNR regime. On the contrary, it is demonstrated that the I-JBO algorithm remarkably boosts the sum rate and achieves the best performance at almost all SNR regimes.

Fig. 3.Sum rate comparison between naive ZF and ASSA with different beamforming designs.

As shown in Fig. 4, the MSE of q in the inner loop, i.e., quasi-Newton method, decreases to zero as the number of iterations increases. The MSE denotes the average distance between the variable obtained in each iteration and the final value. Therefore, the results mean the inner loop is capable of converging after a certain number of iterations. By comparing the black and green lines (or the blue and red ones), we can see that more iterations are required as the penalty coefficient μ gets larger. On the contrary, fewer iterations are required as the SNR increases. Similar results can be seen in Fig. 5, which prove the convergence of the outer loop.

Fig. 4.The MSE of q versus the number of iterations in the inner loop.

Fig. 5.The MSE of q versus the number of iterations in the outer loop.

We also plot the average value of the objective function versus the number of iterations. The objective function for the inner loop is in equation (47). As shown in Fig. 6, the average value of the objective function decreases as the number of iterations increases and gets saturated after a certain number of iterations. Moreover, the inner loop converges more slowly as increases. On the contrary, higher SNR makes the loop converge faster. These results are quite consistent with the ones in Fig. 4. Meanwhile, we plot Fig. 7 to prove the convergence of the outer loop from the perspective of objective function value, where the objective function is shown in equation (45). We can see from Fig. 7 that the average value of the objective function also decreases as the number of iterations increases and converges after a few iterations. It is noted that our target is to maximize the objective function, i.e., sum rate, in equation (45). However, it is shown in Fig. 7 that the average value of the objective function decreases as the number of iterations increases. The reason is that although the vector q obtained in the first few iterations can make the value of the objective function large, it may not satisfy the constraint ║q║ = 1, which makes the termination condition of the outer loop μkh(qk+1) < εp false. As a result, the vector q is returned back to the inner loop to get a new value until the termination condition holds.

Fig. 6.The objective function in (47) versus the number of iterations in the inner loop.

Fig. 7.The objective function in (45) versus the number of iterations in the outer loop.

 

7. Conclusion

In this paper, we investigate the DOF of the Y channel composed of three single-antenna users and a two-antenna common access relay, where each user intends to exchange independent messages with the other two users with the help of the relay. Initially, a DOF upper bound for this particular scenario is derived based on cut-set bound by allowing user cooperation, which shows that the total DOF is upper bounded by 1.5. To achieve the cut-set bound, we introduce the ASSA-based scheme. By using ACS to convert the complex system into an equivalent real system, more dimensions can be utilized for beamforming. Moreover, a projection matrix is utilized at the relay to enable the ASSA. Brief analysis and simulation results demonstrate that the proposed signaling technique achieves the DOF cut-set bound. Combining the upper bound and achievability, we conclude that the DOF of the channel model we study is 1.5. Furthermore, we also come up with the I-JBO beam design algorithm to further enhance the sum rate of the ASSA scheme. It is an interesting future topic to investigate the extension of our work into a general Y channel where there are arbitrary number of users and antennas.