Full-Duplex Relay based on Zero-Forcing Beamforming

In this paper, we propose a full-duplex relay (FDR) based on a zero-forcing beamforming (ZFBF) for a multiuser MIMO relay system. The ZFBF is employed at the base station to suppress both the self-interference of the relay and the multiuser interference at the same time. Numerical results show that the proposed FDR can enhance the sum rate performance as compared to the half-duplex relay (HDR), if sufficient isolation between the transmit and receive antennas is ensured at the relay.

MIMO. Moreover, the need for multiple antennas at the RS is eliminated by letting the BS perform the self-interference cancellation. The performance of the proposed FDR will be compared with that of an HDR with ZFBF and an ordinary multiuser MIMO without RS.

II. SYSTEM MODEL
We consider a downlink scenario where the BS has L transmit antennas and a RS has one receive antenna and one transmit antenna. The RS is assumed to employ the decodeand-forward protocol [9], where the RS decodes all the signals from the BS and forwards them to the MS. We categorize mobile stations (MS's), each assumed to be equipped with one receive antenna, into two groups according to the reachability of the signals from the BS: 1) MS's in the first group can receive the signals from both the BS and RS. 2) MS's in the second group can receive only the signal from the RS. For notational convenience, we will refer to the set of N i MS's in the i-th group as MS-i, where N i denotes the number of MS's in the MS-i. Moreover, we assume that the BS selects the MS's from each MS group so that the sum rate is maximized [8].
The received signal y i,j at the j-th MS in the MS-i, i = 1, 2, is respectively given as y 1,j = h BS 1,j s BS + h RS 1,j s RS + z 1,j , y 2,j = h RS 2,j s RS + z 2,j , where h BS 1,j denotes a 1×L channel vector between the BS and the j-th MS in MS-1. Moreover, h RS i,j is a channel coefficient between the RS and the j-th MS in MS-i, s BS denotes the L × 1 transmit signal vector from the BS, s RS denotes the transmit signal from the RS, and z i,j denotes the zero-mean additive white Gaussian noise (AWGN) with unit variance. Note that s BS and s RS are determined according to the MS selection. When generating s BS at the BS and s RS at the RS, the total available transmit power is assumed to be P BS T and P RS T , respectively. The signal transmission at the BS and RS is assumed to occur every time slot, the duration of which is assumed to be normalized to the unity.

III. RELAY TRANSMISSION SCHEMES
Although our main interest is in the FDR with ZFBF, the HDR transmission can also accommodate a ZFBF. For fair comparison between the HDR and FDR, we first develop a ZFBF for the HDR and the derived ZFBF will be incorporated into the FDR. In this section, we assume that all the exact channel state informations (CSI's) are available at the BS as arXiv:1006.1690v1 [cs.IT] 9 Jun 2010 in [10]. The impact of CSI feedback errors will be discussed in Section IV.

A. Half-Duplex Relay (HDR)
An HDR transmission scheme can be realized by time sharing of each slot between the BS and RS. Specifically, in the first (1−t) fraction of each time slot, the BS can serve the RS as well as MS's in the MS-1 using a ZFBF scheme. The received signal at the RS during this period can be expressed as where z RS also has unit variance. Let Γ ⊂ {1, 2, · · · , N 1 } with |Γ| < L be the set of MS's served by the BS, which is a subset of indices for the MS's in the MS-1. With Γ = {γ 1 , γ 2 , · · · , γ |Γ| }, the received signals at the MS's associated with Γ, y Γ = y 1,γ1 y 1,γ2 · · · y 1,γ |Γ| T , can be given as where the |Γ| × L channel matrix H (Γ) is given as From (2) and (3), we have where the (|Γ| + 1) × L channel matrix H Γ is defined as and the L × 1 BS transmit signal vector s BS Γ can be expressed as In (7), the |Γ| × 1 transmit symbol vector x Γ is defined as x Γ = x 1,γ1 x 1,γ2 · · · x 1,γ |Γ| T , where x 1,γm is the data symbol destined for the γ m -th MS in the MS-1. x RS is the transmit symbol destined for the RS, which the RS decodes and forwards to an MS in the MS-2. The L × (|Γ| + 1) ZFBF precoder W Γ and the (|Γ| + 1) × (|Γ| + 1) power allocation matrix P Γ are defined as and P Γ = diag P 1,γ1 , P 1,γ2 , · · · , P 1,γ |Γ| , P RS (9) with (·) † denoting the conjugate transpose. The L × 1 vectors w BS 1,γm and w BS RS in (8) denote the ZFBF vectors at the BS corresponding to the γ m -th MS in the MS-1 and to the RS, respectively. Then, the sum rate for each link during the (1−t) fraction of a time slot is given as where the transmit power P 1,γm allocated to the γ m -th MS in the MS-1 and P RS allocated to the RS should satisfy the following power constraint, |Γ| m=1 ||w BS 1,γm || 2 P 1,γm + ||w BS RS || 2 P RS,Γ = P BS T .
Here, we define τ 1,γm = 1 ||w BS 1,γm || 2 and τ RS = 1 ||w BS RS || 2 , and perform a water-filling as in [8]. Then, P 1,γm and P RS that maximize the sum rate R Γ + R RS are given as where (x) + denotes max{x, 0} and In the remaining t fraction of the time slot, we consider two possible scenarios: 1) the BS serves MS's in the MS-1 and the RS forwards the decoded data to an MS in the MS-2 at the same time. 2) the BS serves MS's in the MS-1 and the RS forwards the decoded data to another MS in the MS-1 at the same time. For the first scenario, the RS's signal destined for an MS in the MS-2 becomes the multiuser interference to the MS's in the MS-1. In the following, we cancel this interference by using a ZFBF, which will be incorporated into the FDR in the next subsection. The received signals at the MS's in the MS-1 and an MS in the MS-2 can be expressed as The (|Γ| + 1) × (L + 1) channel matrix HΓ is given as where H Γ can be defined as in (4) and h Γ = The (L + 1) × 1 transmit signal vector can be expressed in a matrix form and |Γ| ≤ L. In (16), the (L + 1) × (|Γ| + 1) ZFBF precoder and the (|Γ| + 1) × (|Γ| + 1) power allocation matrix are given as and respectively. Note thatx 2,j denotes a delayed version of x 2,j destined for the j-th MS in the MS-2 with a delay corresponding to the decoding delay at the RS. WΓ in (17) can be rewritten as [11] where w RS 2,j is the beamforming weight at the RS and w BS 2,j denotes the L × 1 ZFBF vectors at the BS corresponding to the j-th MS in the MS-2. The L × |Γ| beamforming matrix W BŠ Γ is defined as Then, the transmit signals in (16) can be rewritten as Note that the BS transmit signal s BS contains not only the beamformed signal of xΓ but also the beamformed signal ofx 2,j . It should be noted that the BS can transmit the beamformed signal ofx 2,j , sincex 2,j is a delayed version of x 2,j that is originated from the BS. It is also seen that the RS transmits only the beamformed signal ofx 2,j , due to the form of the precoding matrix WΓ ,j in (19). It is reasonable because the RS can know onlyx 2,j after decoding x 2,j . The sum rate served to MS's is given as From (21), it is obvious that the BS power constraint is given as As in (12), the water-filling provides the available powerP BS 1,γm andP BS 2,j at the BS. From the RS power constraint, the available power at the RS can be also computed as Now, we determine final transmit power P 1,γm and P 2,j usinḡ P BS 1,γm ,P BS 2,j andP RS 2,j . In the case ofP RS 2,j >P BS 2,j , we simply set P 1,γm =P BS 1,γm and P 2,j =P BS 2,j . Otherwise, we set P 2,j = P RS 2,j . Then, we have and perform the water-filling again to determine P 1,γm . Then, P 1,γm and P 2,j are used to compute (22). Assuming that all the received signals in the (1 − t) time fraction are decoded and forwarded by the RS during the remaining t time fraction, t is chosen such that (1 − t)R RS = tR 2,j [12]. This implies that the RS can forward only the information that has been received from the BS. Then, the overall sum rate in the HDR is given as For the second scenario, the RS receives x 1,j from the BS and forwardsx 1,j to the j-th MS in the MS-1. HΓ ,j can be given as Here, the MS-1 receives the signals from both the BS and RS such that h BS 1,j is not a zero vector. The pseudo-inverse of (27) yields where the 1 × |Γ| vector w RS 1,j is also not a zero vector unlike in (19). From (16), we obtain the RS transmit signal shown as Note that the RS has no information on xΓ since it received and decoded only x 1,j in the (1 − t) time fraction. Therefore, it is obvious that the RS cannot compose (29), and thus we do not consider this transmission scenario. Due to the same reason, the scenario cannot be realized in the FDR as well, which will be described in the next subsection.

B. Full-Duplex Relay (FDR)
In the FDR transmission, the received signal at the RS and at an MS in each group can be expressed as where the second term in y RS represents the self-interference due to simultaneous transmission and reception. During an entire time slot, the BS serves MS's in the MS-1 as well as the RS, while the RS serves an MS in the MS-2 at the same time. In the proposed FDR, h RS RS should be available at the BS. Here, we assume that the RS transmits h RS RS to the BS by using an appropriate method, such as the analog linear modulation in [10].
The ZFBF precoder in (34) can be rewritten as [11] where w BS RS and w BS 2,j denote the L × 1 ZFBF vectors at the BS, w RS 2,j is the beamforming weight at the RS and Then, the transmit signals in (33) can be rewritten as As described in Section III-A, it is reasonable that s BS contains not only the beamformed signal of x Γ and x 2,j but also the beamformed signal ofx 2,j . Moreover, the RS transmits only the beamformed signal ofx 2,j . Substituting (33)-(35) into (31), we get y 1,γm = P 1,γm x 1,γm + z 1,γm with m = 1, 2, · · · , |Γ| and In (40), it is seen that the ZFBF precoder W Γ,j makes y RS free from the self-interference. Thus, the RS can decode x 2,j and forwardx 2,j to the j-th MS in the MS-2 without interference. It is also obvious that we can set P RS = P 2,j , since x 2,j and x 2,j are associated with the same information rate. The sum rate served to MS's is found as After water-filling with (41) and (42) as in (12), the available powerP 1,γm andP RS at the BS can be obtained. Similarly, the available powerP 2,j at the RS can be computed as UsingP 1,γm ,P RS , andP 2,j obtained from (42) and (43), we determine final transmit power P 1,γm and P RS of the BS. In the case ofP RS <P 2,j , we simply get P 1,γm =P 1,γm and P RS =P RS . Otherwise, we set P RS =P 2,j . Then we have |Γ| m=1 ||w BS 1,γm || 2 P 1,γm = P BS T − ||w BS RS || 2 + ||w BS 2,j || 2 P RS (44) and P 1,γm is computed by the water-filling with (44). Then, P 1,γm and P RS are used to compute the sum rate of the proposed FDR in (41). In order to realize the FDR with ZFBF, the CSI between the RS's transmit antenna and receive antenna, h RS RS , is required at the BS. Moreover, the BS should inform the RS of w RS 2,j P RS after constructing W Γ,j , P 1,γm and P RS .
We can establish an interesting connection between the proposed FDR and multiuser MIMO transmission. The signal model for the proposed FDR is illustrated in Fig. 1(a), where the number of BS transmit antennas is assumed to be L = 2. If we regard the RS transmit antenna as a virtual BS transmit antenna in Fig. 1(a), the FDR signal model can equivalently be translated into a multiuser MIMO transmission model in Fig. 1(b), where the restriction on the received signals at the MS-2 is due to our assumption of the signal reachability in Section II. Comparing Figs. 1(a) and 1(b), we see that the selfinterference in the FDR corresponds to multiuser interference in the equivalent multiuser MIMO model. This justifies our approach of eliminating both the self-interference and the multiuser interference simultaneously, since the ZFBF was adopted to suppress the multiuser interference in the multiuser MIMO system [8].
It is worth discussing a practical issue to be considered in deploying the FDR. The self-interference signal might be much stronger than the desired signal at the receiver front-end of the RS. In order to avoid possible saturation in this case, it is important to make sure that the received signals at the RS fall within the dynamic range of the receiver. This can be assured through the use of an analog-to-digital converter (ADC) with high resolution and/or sufficient isolation between the transmit and receive antennas [3], [4].

IV. IMPACT OF CSI FEEDBACK ERRORS
In order to investigate the impact of CSI feedback errors on the sum rate performance, we adopt the CSI error model in [13]. According to the model, the CSI's at the BS can be represented asĥ where e BS 1,j , e RS 1,j , e RS 2,j , e BS RS , and e RS RS denote the feedback errors, which are assumed to follow independent and identically distributed (i.i.d.) complex Gaussian distribution with zero mean and variance of σ 2 E . As in (32), we can composeĤ Γ,j using (45) and the FDR ZFBF precoder with the feedback errors,Ŵ Γ,j = , can be written aŝ  where Note that we have assumed that P RS = P 2,j in (40), since the incoming information rate and outgoing information rate are the same at the RS. However, it is obvious that log 2 (1 + SINR RS ) = log 2 (1 + SINR 2,j ) with CSI feedback errors, even though the actual transmit power satisfies P RS = P 2,j . Therefore, in this case, it may be reasonable to modify the sum rate in (41) For the HDR with ZFBF,R HDR can also be computed with the CSI feedback errors similarly to the procedure described above.

V. NUMERICAL RESULTS
We evaluate the ergodic sum rate performance of the HDR and FDR schemes through Monte Carlo simulations; 10,000 independent channel realizations are used to obtain the ergodic sum rate of each scheme. We also present the performance of an ordinary multiuser MIMO system without RS [8] as a baseline. We assume P BS T = 100 and P RS T = 50. Moreover, for every MS in the first or second group, channel coefficient of the BS-to-MS link is assumed to follow i.i.d. complex Gaussian distribution with zero mean and unit variance. Similarly, channel coefficient of the RS-to-MS link is assumed to follow the same distribution with zero mean and the variance of Q dB. These assumptions imply that the signal-to-noise  ratio (SNR) of the BS-to-MS link and RS-to-MS link is set to 20dB and (Q + 17)dB, respectively. It is also assumed that the average channel gain of each entry in h BS RS and that of h RS RS are, respectively, G-dB and I-dB larger than that of the BS-to-MS link. Assuming that the number of MS's in each group is the same, we denote the number of MS's in each group as N (i.e., N i = N with i = 1, 2).
In Fig. 2, we presnet the variation of E R HDR and E R FDR with σ 2 E , when L = 2, N = 4, G = 20, I = 10, and Q = 6. The ergodic sum rate of ZFBF without RS [8] is also presented. The sum rate performance for all transmission schemes is shown to decrease steeply when σ 2 E > 10 −3 . Moreover, it is seen that the proposed FDR with ZFBF outperforms the HDR with ZFBF and the ordinary ZFBF without RS, even though the CSI's at the BS are erroneous. Fig. 3 shows the variation of the ergodic sum rate performance with Q, when L = 2, N = 4, G = 20, I = 10, and σ 2 E = 0.01. As shown in Fig. 3, even though the RS-to-MS link becomes more reliable, the HDR sum rate is almost constant due to the time sharing. However, the sum rate of the proposed FDR increases steeply compared with the HDR. Fig. 4 presents how the ergodic sum rate performance varies with G, when L = 2, N = 4, I = 10, and Q = 6. In Fig.  4, σ 2 E = 0 indicates that the perfect CSI's are available at the BS. Note that the larger value of G makes the BS-to-RS link more reliable. The FDR with ZFBF is shown to outperform the HDR when G > 8. Fig. 5 shows the variation of the ergodic sum rates with I, when L = 2, N = 4, G = 20, Q = 6, and σ 2 E = 0.01. Note that the larger I causes the stronger self-interference to the RS receiver in the case of the FDR with ZFBF. The HDR scheme must not depend on I. For the FDR to outperform the HDR, I must be less than 21. This suggests that sufficient isolation between the transmit and receive antennas at the RS is necessary for the proposed FDR.

VI. CONCLUSION
We have proposed an FDR transmission scheme for the downlink of a cellular system. The FDR has been realized through the use of a ZFBF scheme based on multiple antennas at the base station. Numerical results have verified that the proposed FDR provides the performance improvement over the HDR when the antenna isolation is sufficient for the proposed FDR to operate within a tolerable self-interference range. We have also investigated the impact of CSI errors on the sum rate performance.