An Efficient Method for Channel State Information Feedback in the Relay X-Channel Mohsen Rezaee School of Electrical and Computer Engineering University of Tehran Tehran, Iran [email protected]

Abstract—The two-user relay-assisted X channel is considered when no channel state information (CSI) is available at the transmitter side. It is known that with a 2M-antenna relay and is achieved in this channel if CSI M-antenna users a DoF of 4M 3 is available at the relay. This is equal to the maximum achievable DoF in the ordinary X-channel (with no relay) with perfect CSI at the transmitters. An efficient CSI feedback method is proposed which reduces the amount of feedback necessary for the relay in order to maintain the maximum DoF. In addition, a quantization method is proposed for the case of limited feedback. The proposed method outperforms the conventional methods in terms of the number of feedback bits. Simulation results show performance improvement in the finite-SNR regime. Index Terms—degrees of freedom, relay X channel, decodeand-forward relay.

I. I NTRODUCTION In multi-user wireless networks, interference management has always been a challenging problem. Advanced interference management techniques are developed recently which can achieve higher spectral efficiencies than conventional methods. Specifically, interference alignment (IA) divides the signaling subspace into two spaces, one being reserved for communicating the desired signal and the other being allocated to the interfering signals [1] and [2]. Performing IA, however, requires global and perfect channel state information (CSI) at the transmitters (CSIT), which is difficult to achieve in practice. In order to cope with the lack of CSIT, relaying techniques can be leveraged to facilitate interference management. For instance, in the X channel with perfect CSIT including a relay does not improve the achievable degrees of freedom (DoF) [3], [4]. However, for the same X channel without CSIT, including relays provides DoF gains. More specifically, the work in [5] studied the effect of including a half-duplex amplify-and-forward (AF) relay in the two-user X channel without CSIT, where it was shown that for a single-antenna X channel without CSIT, the DoF is 4 3 . This provides larger DoF compared to the same X channel without relay, for which the DoF is 1 [6], and establishes the gain of deploying relays in X channel without CSIT. In this paper, we investigate the impact a relaying technique in the X channel with no CSIT. We focus on the scheme provided in [7] which uses the decode-and-forward (DF) strategy. It is shown that using a 2M -antenna half-duplex

Behrouz Maham Department of Electrical and Electronic Engineering School of Engineering, Nazarbayev University Astana, Kazakhstan [email protected] relay a DoF of 4M 3 can be achieved when the users have M antennas. We then propose an efficient feedback method in order to reduce the amount of feedback to the relay. As the relay is the only node that requires CSI feedback, it is important to relax its feedback requirement while maintaining the maximum achievable DoF. Furthermore, a quantization method is designed for the case of having limited feedback to the relay. Simulation results are provided which demonstrate finite-SNR performance improvement of the quantization method compared to the conventional approaches. The rest of the paper is organized as follows: In Section II, the system model of X channel with both DF and cognitive relays are presented. A DoF analysis of the DF relaying technique is presented in Sections III followed by the CSI feedback method in Section IV. Finally, the simulation results are provided in Section V. Notations: We use lower case bold face letters to denote vectors and upper bold case letters to denote matrices. Non T bold capital letters denote scalars and (.) denotes transpose. Nc (a, b) denotes a complex Gaussian distribution with mean a and variance b. The row space of a matrix is represented by R(·). II. S YSTEM M ODEL Consider a two-user Gaussian X channel, consisting of two transmitters and two receivers, where each transmitter sends an independent message to each receiver. The communication between the transmitters and receivers is assisted by a relay in DF mode. We assume that the transmitters have no CSI while the relay and receivers have full CSI. Throughout this paper, it is assumed that the entries of the channel matrices are generated from a continuous distribution, so that the channels are non-degenerate with probability one. We assume that the transmitters and receivers are equipped with M antennas and the relay has MR antennas. We define xk (t) ∈ CM and xR (t) ∈ CMR as the signals transmitted by transmitter TXk for k ∈ {1, 2}, and the relay, respectively, at time t, and accordingly define y k (t) ∈ CM and y R (t) ∈ CMR as the signals received by receiver RXk for k ∈ {1, 2}, and the relay, respectively, at time t. We denote the average power of all transmitters by P , i.e., E(kxk k2 ) ≤ P We define H kl ∈ CM ×M as the channel between transmitter l and receiver k for l, k ∈ {1, 2}, and H kR ∈ CM ×MR and

By invoking the signal models in (1) and noting that the transmitters and receivers are equipped with M antennas, the channel outputs in the nth phase for n ∈ {1, 2} are given by y 1 (n) = H 11 (n)dn1 + H 12 (n)dn2 + n1 (n), y 2 (n) = H 21 (n)dn1 + H 22 (n)dn2 + n2 (n), y R (n) = H R1 (n)dn1 + H R2 (n)dn2 + nR (n),

Fig. 1. X Channel with a DF Relay H Rl ∈ CMR ×M as the channel between the relay to receiver k, and the channel between transmitter l to the relay, respectively. In the half-duplex relaying mode, the transmissions are accomplished in two phases. At time t, if the relay is in the receiving mode (i.e., silent and listening to the transmitters), then the signals received by the relay and the receivers are y i (t) = H i1 x1 (t) + H i2 x2 (t) + ni (t), i ∈ {1, 2, R} (1) where ni (t) at the ith user and nR (t) at the relay account for the additive white Gaussian noise (AWGN) distributed as Nc (0, σ 2 ). Similarly, if relay is in the transmission mode at time t, the received signal at receiver i ∈ {1, 2} gives y i (t) = H i1 x1 (t) + H i2 x2 (t) + H iR xR (t) + ni (t) III. ACHIEVABLE DoF OF THE X-C HANNEL WITH DF-R ELAY In this section, we present an overview of the scheme of [7] in order to provide the necessary background for our proposed quantization method. The relay-aided X channel where the relay operates in the DF mode is considered. It is shown in [7] that when the transmitters do not have access to the CSI, IA can be facilitated by the relay. In this Section we assume MR = 2M . Theorem 1 ( [7]). In a two-user relay X channel with M antennas at the transmitters and receivers and 2M antennas at the half-duplex DF relay, with no CSIT and full CSI at the relay and receivers, the achieved DoF is 4M 3 . Proof. Consider a transmission scheme that is composed of three phases (time slots). Let dkl denote the M × 1 symbol vector of transmitter l intended to receiver k. In the first phase, transmitters 1 and 2 send symbols d11 and d12 , respectively, x1 (1) = d11

and x2 (1) = d12 .

(2)

In the second phase, the transmitted signals are x1 (2) = d21

and x2 (2) = d22 .

(3)

Based on expressions stated above, receiver 1 receives a linear combination of its desired messages in the first phase, while the signal received at receiver 2 is all interference. In the second phase, receiver 2 receives a linear combination of its desired messages while receiver 1 receives interference. It is noteworthy that since the relay has 2M antennas, it can decode all messages during phase 1 and 2. Finally in the third phase, the transmitters are silent and the relay constructs and sends a linear combination of the decoded messages as follows   U 1 d11 + U 2 d22 xR (3) = , (4) U 3 d21 + U 4 d12 where each U m for m ∈ {1, 2, 3, 4} is an M × M combining matrix at the relay. These matrices are chosen such that the received signal at each receiver provides excessive freedom that enables successful decoding of the desired messages. By denoting the channel between the first M antennas of the relay and receiver 1 by H 1R,1 and the channel between the second M antennas of the relay and receiver 1 by H 1R,2 , we can show the signals received at receiver 1 in three phases as (16) which is shown at the top of page 3. The whole signal space is 3M -dimensional. We need to ensure that interference is confined whithin an M -dimensional subspace, so that the desired signals can be recovered. Thus, we enforce the following condition which is based on temporal IA     H 1R,2 (3)U 3 , H 1R,1 (3)U 2 ∈ R( H 11 (2), H 12 (2) ), (17) where R(·) denotes the row space of its argument matrix. Determining U 2 and U 3 as U 2 = H −1 1R,1 (3)H 12 (2),

(18)

H −1 1R,2 (3)H 11 (2),

(19)

U3 =

satisfies this condition. Similarly U 1 and U 4 can be chosen U 1 = H −1 2R,1 (3)H 21 (1),

(20)

H −1 2R,2 (3)H 22 (1).

(21)

U4 =

Therefore, interference terms at the receivers occupy only M dimensions. Since the channels at different time slots are independent and every channel matrix has full rank almost surely, one can simply show that the desired signal space has full rank of 2M and is linearly indepent of the interference space. Therefore the intended messages can be decoded correctly and total DoF = 4M 3 can be achieved Remark 1. This result shows that achievable DoF in the MIMO X channel without CSIT assisted by 2M-antenna DF relay is 4M 3 , which is the maximum DoF for X channel with a relay, while its all nodes have full CSI as shown in [4].

   y 1 (1) y 1 (2) =  y 1 (3)

H 11 (1) 0 H 1R,1 (3)U 1

H 12 (1) 0 H 1R,2 (3)U 4

IV. CSI F EEDBACK According to the previous sections it can be said that the CSI available at the relay is equally useful as the CSI at the transmitter side from a DoF perspective (in this particular setting). This is a considerable improvement especially when the relay is closer to the receivers which reduces the feedback delay. However providing accurate CSI for the relay is still an important issue in order to achieve the desired DoF. In this section, we focus on the problem of CSI feedback to the relay and we develop an efficient quantization scheme in terms of the amount of CSI feedback. In order to design the beamforming matrices at the relay, four M × M channel matrices are required to be fed back to the relay from each receiver. For example, to design U2 and U3 , the channels H1R,1 (3), H1R,2 (3), H11 (2) and H12 (2) are required to satisfy (17). A naive approach is to feedback all these matrices to the relay in order to design U2 and U3 at the relay. This however is not very efficient in terms of the amount of information feedback. Intuitively, one can see that designing U2 and U3 according to our particular solution in (18) at the receiver and sending these matrices to the relay is more efficient in terms of the number of values to be fed back. Clearly in this case, the number of scaler values that need to be fed back is reduced by half. This intuitive observation is interesting while it has other practical issues. Despite the original channel matrices which have Gaussian elements, the elements of U2 and U3 do not have a simple distribution and the optimal quantizer for such elements is not known. Furthermore, random vector quantization (RVQ) method which is favorable for quantizing matrices with Gaussian elements is not applicable to U2 and U3 . To deal with this problem, inspired by [8], we propose another approach which relies on the invariances involved in (17) to reduce the amount of CSI feedback while using practical quantizers. First, we consider the perfect feedback situation, and then, we introduce our quantization method when limited feedback is considered. A. Perfect CSI feedback We start by introducing the following lemma. Lemma 1. In order to design U2 and U3 which satisfy (17) at the relay, it is sufficient to feedback two matrices M ×2M F1R  , F11 ∈ C which span the same row spaces as H 1R,2 (3), H 1R,1 (3) and H 11 (2), H 12 (2) respectively from receiver 1.

0 H 11 (2) H 1R,2 (3)U 3

   d11 0 d12   H 12 (2)   d21  H 1R,1 (3)U 2 d22

(16)

Proof. Assume that F1R , F11 are available at the relay. We show that the following solution satisfies (17): U 2 = F −1 1R,2 F 11,2 ,

and

U 3 = F −1 1R,1 F 11,1 ,

(22)

in which F 1R = [F 1R,1 , F 1R,2 ] and F 11 = [F 11,1 , F 11,2 ]. From (22) we have   U3 0 F 1R = F 11 . (23) 0 U2 the same  Since F 1R spans  H 1R,2 (3), H 1R,1 (3) , we can write

row

  H 1R,2 (3), H 1R,1 (3) = C 1R F 1R

space

as

(24)

for a M × M invertible matrix C 1R . Similarly, we have   H 11 (2), H 12 (2) = C 11 F 11 (25) for some M × M invertible matrix C 11 . Thus, we have   H 1R,2 (3)U 3 , H 1R,1 (3)U 2 (26)     U3 0 = H 1R,2 (3), H 1R,1 (3) (27) 0 U2   U3 0 = C 1R F 1R (28) 0 U2 = C 1R F 11

(29)

C 1R C −1 11

(30)

  H 11 (2), H 12 (2) =   ∈ R( H 11 (2), H 12 (2) ).

(31)

therefore, the expression in (17) is satisfied which completes the proof. The same argument holds for the feedback from receiver 2 in order to design U 1 and U 4 at the relay. The row space corresponding to our matrices can be represented by points on the Grassmann manifold G2M,M . The Grassmann manifold G2M,M is the set of all M -dimensional subspaces in the 2M -dimensional vector space. A point on the Grassmann manifold can be represented by any matrix X whose rows span the subspace defined by X, i.e., span(X). The real dimension of G2M,M is 2M 2 which is half of the real dimension of the original M ×2M matrices. Therefore feeding back the points on the Grassmann manifold significantly reduces the amount of feedback. B. Quantized CSI feedback Here, we assume that every receiver i can feedback a limited number of bits Ni to the relay. According to the proposed scheme, receiver i quantizes the subspace spanned by the rows of Fii and FiR each using N2i bits and feeds the index

ˆ ii = arg min dc (S, Fii ), F ˆ iR = arg min dc (S, FiR ), F S∈S S∈S in which dc (X, Y) = √12 XXH − YYH F is the chordal distance between X and Y in G2M,M . After sending the index of the quantized points, the relay finds the beamforming matrices according to (22) using the quantized matrices. Clearly this solution results in interference for the receivers where the power of interference depends on the accuracy of the quantization (the number of bits).

60 Perfect CSI Quantized CSI, Conventional, B=20

50

Quantized CSI, Proposed, B=20

Sum-Rate (bits/sec/Hz)

of the quantized codewords back to the relay. For simplicity, we assume equal quantization bits at the receivers, i.e., B , N21 = N22 . We further assume that the receivers and the relay share a predefined codebook S = {S1 , . . . , S2B } which is composed of 2B truncated unitary matrices of size M ×2M and is designed via Grassmannian subspace packing [9]. The quantized codewords at receiver i are the points in S closest to Fii and FiR , i.e.

40

30

20

10

0

0

5

10

15

20

25

SNR (dB)

Fig. 2. Sum rate comparison between quantization methods

V. S IMULATION R ESULTS In this section, we evaluate the performance of the quantization scheme of Section IV-B with RVQ codebooks. The performance metric is the sum rate evaluated through MonteCarlo simulations. M = 2 antennas are considered at each transmitter and each receiver and the relay is equipped with MR = 4 antennas. Each transmitter sends a 2-dimensional complex symbol vector to each of the receivers. Entries of the channel matrices are generated according to Nc (0, 1) and the performance results are averaged over the channel realizations. The method proposed in Section IV-B is compared to a conventional quantization method. In the conventional method the channel matrices are vectorized and normalized and then quantized using RVQ. For the proposed method, the codebook entries are independent 2 × 4 random truncated unitary matrices generated from the Haar distribution. For the conventional method, random unit norm vectors are used in the codebook construction. Figure 2 shows the achievable sum rate versus transmit SNR for B = 20 feedback bits when the precoders are designed based on the quantized feedback. Clearly the proposed scheme outperforms the conventional quantization method for the same number of feedback bits. For reference, the perfect CSI case is also plotted from which the achievability of the DoF can be verified. VI. C ONCLUSION DoF for the relay X channel with no CSIT has been investigated. It is known that in this channel, DF relay makes it possible to achieve full DoF. For M -antennas users, it provides 4M 3 DoF when the relay is equipped with 2M antennas. In this work, an efficient feedback method is provided

to reduce information feedback to the relay. Furthermore, simulations are provided that show performance improvement compared to the conventional methods at finite-SNR values. R EFERENCES [1] S. A. Jafar and S. Shamai, “Degrees of freedom region of the MIMO X channel,” IEEE Trans. Inf. Theory, vol. 54, no. 1, pp. 151–170, Jan. 2008. [2] M. A. Maddah-Ali, A. S. Motahari, and A. K. Khandani, “Communication over MIMO X channels: Interference alignment, decomposition, and performance analysis,” IEEE Trans. Inf. Theory, vol. 54, no. 8, pp. 3457– 3470, Aug. 2008. [3] B. Nourani, S. A. Motahari, and A. K. Khandani, “Relay-aided interference alignment for the quasi-static X channel,” in IEEE Int. Symp. Inf. Theory (ISIT), Seoul, Korea, Jun.-Jul. 2009, pp. 1764–1768. [4] V. R. Cadambe and S. A. Jafar, “Degrees of freedom of wireless networks with relays, feedback, cooperation, and full duplex operation,” IEEE Trans. Inf. Theory, vol. 55, no. 5, pp. 2334–2344, May 2009. [5] Y. Tian and A. Yener, “Guiding blind transmitters: degrees of freedom optimal interference alignment using relays,” IEEE Trans. Inf. Theory, vol. 59, no. 8, pp. 4819–4832, 2013. [6] C. S. Vaze and M. K. Varanasi, “The degree-of-freedom regions of MIMO broadcast, interference, and cognitive radio channels with no CSIT,” IEEE Trans. Inf. Theory, vol. 58, no. 8, pp. 5354–5374, Aug. 2012. [7] H. Zebardast, A. Tajer, B. Maham, and M. Rezaee, “Relay X channels without channel state information at the transmit sides: Degrees of freedom,” in Proc. IEEE Wireless Communications and Networking Conference (WCNC), 2015. [8] M. Rezaee and M. Guillaud, “Limited feedback for interference alignment in the K-user MIMO interference channel,” in Proc. IEEE Inf. Theory. Workshop (ITW), Lausanne, Switzerland, 2012. [9] J. H. Conway, R. H. Hardin, and N. J. Sloane, “Packing lines, planes, etc.: Packings in grassmannian spaces.” Experimental Mathematics, vol. 5, no. 2, pp. 139–159, 1996.