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Full-Duplex MIMO Precoding for Sum-Rate Maximization with Sequential Convex Programming Sean Huberman and Tho Le-Ngoc

Abstract—This paper focuses on precoding design for sumrate maximization while considering the effects of residual selfinterference for Multi-User MIMO (MU-MIMO) Full-Duplex (FD) systems. The problem formulation leads to a non-convex matrix-variable optimization problem, where we develop two efficient sum-rate maximization algorithms using Sequential Convex Programming (SCP), namely the Difference of Convex functions (DC)-based and the Sequential Convex Approximations for Matrix-variable Programming (SCAMP) algorithms. In addition, we derive the achievable sum-rate under the effect of residual self-interference. Simulation results show that even in cases of high self-interference and high estimation error, the SCAMP algorithm provides approximately 20-30% sum-rate improvements over both conventional optimized HD transmission and the existing state-of-the-art FD algorithm in a wide range of scenarios. Finally, the convergence results indicate that the DC-based algorithm tends to initially give the best performance, however, at convergence the SCAMP algorithm tends to significantly outperform the other algorithms. Index Terms—Full-duplex, MIMO, precoding, sequential convex programming, non-convex optimization.

I. I NTRODUCTION The recent attraction of Full-Duplex (FD) systems stems from the fact that they can provide approximately double sumrate improvements over Half-Duplex (HD) systems [1], [2]. However, while FD systems do not require separate time or frequency slots, the systems suffer from self-interference in addition to the co-channel interference present in HD systems. Self-interference is the interference seen by a receiver due to its own transmission. The self-interference is typically significantly larger than the inter-user co-channel interference and can prevent the potential “double” sum-rate gains of FD transmission. Hence, effective self-interference management is crucial to significantly improve the spectral efficiency of FD systems with respect to HD systems. Precoding has been applied in the context of FD relay networks (e.g., [3]–[8]), where most approaches focus on selfinterference suppression. For FD Single-User Multiple-InputMultiple-Output (SU-MIMO) networks, [9] applies beamforming to minimize the transmit power subject to total Signal Sean Huberman and Tho Le-Ngoc are with the Department of Electrical and Computer Engineering, McGill University, 3480 University Street, Montreal, Quebec, Canada, H3A 0E9 (e-mails: [email protected] and [email protected]). The work presented in this paper is partly supported by the R&D Contract Advanced Transmission Multiple-Access Schemes for c Future Radio Access from Huawei Technologies Canada. Copyright 2013 IEEE. Personal use of this material is permitted. However, permission to use this material for any other purposes must be obtained from the IEEE by sending a request to [email protected].

to Interference plus Noise Ratio (SINR) constraints and selfinterference constraints. Beamforming while taking explicitly modeling the dynamic-range limitations is proposed in [10] while applying pilot-aided channel estimation. The SelfInterference Pricing (SIP) algorithm was presented in [11] to replace the direct non-convex optimization problem by four pricing-based sub-problems. A FD Multi-User MIMO (MU-MIMO) system where BSs operate in FD-mode while Mobile Stations (MSs) operate in HD-mode was considered in [12], [13]. The interference caused by uplink transmission to downlink MSs was ignored in the approaches taken by [12] and [13], in order to simplify the mathematics. As well, perfect channel knowledge was assumed for both the forward and self-interference channels. Based on these assumptions, a non-convex optimization problem was formulated. Due to FD transmission, the uplink and downlink channels are coupled. Sequential and joint precoding schemes were presented in [12] to approximate the solution of the non-convex optimization problem. The sequential precoding scheme first selected a downlink precoding scheme and then optimized the uplink precoding scheme assuming the fixed downlink precoding. The joint precoding scheme has the form of a Difference of Convex functions (DC), where a firstorder approximation to the objective function was applied in order to convexify the optimization problem. The work in [12] was extended in [13] by taking into account both spectral and energy efficient designs. For FD MU-MIMO with HD MS, the interference caused by uplink transmission to downlink MSs is considered in [14], where the effect of large-scale MIMO systems on FD is investigated by selecting the ratio between the number of transmit and receive antennas at the BS. This paper focuses on precoding design for sum-rate maximization while considering the effects of residual selfinterference for MU-MIMO systems. The FD MU-MIMO system presented in this paper is valid for the cases of FD MSs, HD MSs, and mixed FD/HD MSs. As well, it generalizes the results from our previous work, [15], for the FD SU-MIMO system. Moreover, the FD MU-MIMO with HD MSs system in this work generalizes the system model used for precoding design in [12]. In particular, while the configuration is similar, the environment is different and more realistic since we include the interference from uplink transmission to downlink MSs. Furthermore, we consider the effects of imperfect channel knowledge. The FD MU-MIMO problem formulation leads to a non-convex matrix-variable optimization problem, where we consider two Sequential Convex Programming (SCP) approaches to develop efficient sum-rate maximization algo-

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rithms. The first algorithm takes advantage of the DC structure of the non-convex optimization problem by directly looking at the objective function decomposition. The form of the original objective function naturally lends itself to this choice of DC decomposition. In particular, the MU-MIMO DC-based algorithm derived in this paper generalizes the joint precoding scheme presented in [12] and the FD SU-MIMO DC-based algorithm presented in our previous work [15], which can be recovered as special cases. The second algorithm called Sequential Convex Approximations for Matrix-variable Programming (SCAMP) is developed for solving general non-convex matrix-variable optimization problems with logarithmic objective functions. In this paper, we apply the SCAMP algorithm to the MU-MIMO FD systems. The SCAMP algorithm approximates the non-convex objective function by first applying a lower-bound and then applying an upper-bound on one of the terms in the lowerbound. The result is a non-convex approximation of the original non-convex objective function. First-order approximations are applied to the non-convex terms of the approximation function in order to ensure that the approximate objective function satisfies the Disciplined Convex Programming (DCP) ruleset [16]. Finally, we derive an analytical expression for the loss in sum-rate incurred due to the effect of the residual selfinterference. The residual self-interference results due to the effects of imperfect channel knowledge. The remainder of this paper is organized as follows: Section II describes the system models. Section III presents the proposed DC-based and SCAMP algorithms. Section IV derives an analytical expression for the effect of channel estimation error on the achievable sum-rate. Section V provides some illustrative results, and Section VI provides some concluding remarks. Notation: In this paper, non-bold variables denote scalars, lower-case bold variables denote vectors, and upper-case bold variables denote matrices. E{·} refers to the expected value operation. A† refers to the conjugate transpose of matrix A. Tr{A} refers to the trace of matrix A. |A| refers to the determinant of matrix A. Id and 0d refer to the d × d identity and all-zero matrices, respectively. Finally, A < 0 implies that A is a positive semi-definite matrix. II. S YSTEM M ODEL A. Full-Duplex MU-MIMO with HD MSs The FD MU-MIMO with HD MSs system model is shown in Fig. 1. Let the number of downlink MS be KDL and the number of uplink MS be KUL . It is assumed that each of the MS are equipped with NMS,T transmit and NMS,R receive antennas, respectively. Hence, it is assumed that the BS is equipped with NBS,T = KDL NMS,R transmit and NBS,R = KUL NMS,T receive antennas, respectively. The received signal at the BS can be written as: X X yBS = GBS VBS,k xBS,k + zBS , [HUL ]u VMS,u xMS,u + u∈IU

k∈ID

(1)

[HDL]1

MS

DL

[HDL]K MS BS (Tx / Rx)

GBS

GMS [HUL]1

UL MS

[HUL]k

MS

Fig. 1. Full-duplex MU-MIMO with HD MSs system model.

where IU denotes the set of uplink MSs, ID denotes the set of downlink MSs, [HUL ]u ∈ CNBS,R ×NMS,T is the channel matrix from the u-th MS to the BS, VMS,u ∈ CNMS,T ×NMS,T is the precoding matrix for the u-th MS, xMS,u ∈ CNMS,T ×1 vector of uplink data symbols for the u-th MS, GBS ∈ CNBS,R ×NBS,T is the BS self-interference matrix, VBS,k ∈ CNBS,T ×NBS,T is the precoding matrix for the k-th user at the BS, xBS,k ∈ CNBS,T ×1 vector of transmitted symbols from the BS to the k-th MS, and zBS ∈ CNBS,R ×1 is the additive white Gaussian noise. Similarly, the received signal at the k-th MS (k ∈ ID ) can be written as: X [HDL ]k VBS,l xBS,l yk = [HDL ]k VBS,k xBS,k + l∈ID \{k}

+

X

[GMS ]k,u VMS,u xMS,u + zk ,

(2)

u∈IU

where [HDL ]k ∈ CNMS,R ×NBS,T is the channel matrix from the BS to the k-th MS, [GMS ]k,u ∈ CNMS,R ×NMS,T is the interference matrix from the u-th uplink MS to the k-th downlink MS, and zk ∈ CNMS,R ×1 is the additive white Gaussian noise. Note that it is assumed that k ∈ ID . Hence, the downlink sum-rate can be written as: X
˜ DL,k −1 CDL,k , RDL = log2 INMS ,R + Σk + C k∈ID

˜ DL,k and CDL,k are the downlink where Σk = E{zk z†k }, C receive covariance matrices for the interference and direct channels, respectively, defined as:

CDL,k = [HDL ]k QBS,k [HDL ]†k , X X ˜ DL,k = [HDL ]k QBS,l [HDL ]†k+ [GMS ]k,u QMS,u [GMS ]†k,u , C l∈ID \{k}

u∈IU

where the transmit covariance matrices QBS,k and QMS,u are defined as: † QBS,k = VBS,k SBS,k VBS,k , for k ∈ ID , † QMS,u = VMS,u SMS,u VMS,u , for u ∈ IU ,

where SBS,k = E{xBS,k x†BS,k } and SMS,u = E{xMS,u x†MS,u }. Similarly, the uplink sum-rate can be written as:
˜ UL −1 CUL , RUL = log2 INBS,R + ΣBS + C

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˜ UL and CUL are the uplink receive where ΣBS = E{zBS z†BS }, C covariance matrices for the interference and direct channels, respectively, defined as: X CUL = [HUL ]u QMS,u [HUL ]†u ,

[HUL]1 MS (Tx/Rx)

[HDL]1

u∈IU

˜ UL = C

X

GBS QBS,k G†BS .

k∈ID

BS (Tx / Rx)

GBS

The non-convex sum-rate maximization problem can be written as: max

VBS,k ,VMS,u , k∈ID , u∈IU

[HDL]k

RDL + RUL

† subject to: Tr{VMS,u SMS,u VMS,u } ≤ Pmax,u , u ∈ IU (3) X † Tr{VBS,k SBS,k VBS,k } ≤ Pmax,BS .

GMS

[HUL]k

MS (Tx/Rx)

Fig. 2. Full-duplex MU-MIMO system model.

k∈ID

The self-interference channels, GBS and GMS , are assumed to be estimated, while the forward channels, HDL and HUL , are assumed to be known perfectly, in order to more easily compare with HD transmission. More specifically, it is assumed that: ˆ BS + ∆GBS , GBS = G ˆ MS + ∆GMS , GMS = G ˆ BS where GBS and GMS are the true channel matrices, G ˆ and GMS are the estimated channel matrices, and ∆GBS and ∆GMS are the channel estimation error matrices, with zero 2 . mean and variance σerr ˆ ˆ UL are the estimates of RDL and RUL Similarly, RDL and R and are defined as: X
ˆ DL,k −1 CDL,k , ˆ DL = R log2 INMS,R + Σk + C k∈ID

ˆ UL R


ˆ UL −1 CUL , = log2 INBS,R + ΣBS + C

III. MU-MIMO: S EQUENTIAL C ONVEX P ROGRAMMING This section presents the DC-based and SCAMP SCP algorithms for the MU-MIMO system. As discussed in Section II-B, the derivation applies to the cases of MU-MIMO systems with HD MSs, FD MSs, and mixed FD/HD MSs. As well, the algorithms also generalize the DC-based and SCAMP algorithms for the SU-MIMO system [15], which can be recovered as a special case when ID = IU = {1}. Optimization problem (3) can be re-written as: max

QBS,k ,QMS,u , k∈ID , u∈IU

RDL + RUL

(4a)

subject to: Tr{QMS,u } ≤ Pmax,u , u ∈ IU , X Tr{QBS,k } ≤ Pmax,BS ,

(4b) (4c)

k∈ID

QBS,k < 0, k ∈ ID ,

(4d)

QMS,u < 0, u ∈ IU ,

(4e)

ˆ DL,k and C ˆ UL are the estimates of the covariance where the positive semi-definite constraints ensure the solved where C for covariance matrices are feasible. Once optimization prob˜ DL,k and C ˜ UL , respectively, and are given by: matrices C lem (4) is solved, the corresponding precoding matrices can X X ˆ MS ]k,u QMS,u [G ˆ MS ]† , be recovered using the Cholesky decomposition. In particular, ˆ DL,k = [HDL ]k QBS,l [HDL ]†k+ [G C k,u QMS,u = LMS,u L†MS,u , QBS,k = LBS,k L†BS,k , and hence, the u∈IU l∈ID \{k} X precoding matrices can be computed as: ˆ† . ˆ BS QBS,k G ˆ UL = G C BS −1/2 k∈ID (5) VMS,u = LMS,u SMS,u , −1/2

B. Full-Duplex MU-MIMO The FD MU-MIMO system model (with FD MS) is shown in Fig. 2. In this system, a FD BS services K FD MSs. The received signals at the BS and MS are identical to that of (1) and (2), respectively, when IU = ID = {1, . . . , K}. As such, the FD MU-MIMO with FD MSs system can be recovered as a special case of the FD MU-MIMO with HD MSs derivations provided in this paper. Furthermore, a system where the BS operates in FD-mode while some MSs operate in HD-mode and some MSs operate in FD-mode (i.e., a mixed HD/FD MS scenario) can also be recovered as a special case of the FD MU-MIMO with HD MSs derivations provided in this paper where each FD MS is an element of the set IU ∩ ID .

VBS,k = LBS,k SBS,k .

(6)

A. DC-Based Algorithm The DC-based approach is a SCP algorithm that writes the non-convex optimization problem as a DC (i.e., f = g − h, where g and h are convex) and applies a first-order approximation to h to make the objective function convex. The objective function lends itself naturally to a particular choice of DC decomposition. First, we re-write the objective function as a minimization problem: min

QBS,k ,QMS,u , k∈ID , u∈IU

f

subject to: (4b) − (4e)

(7)

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ˆ DL − R ˆ UL and where R ˆ DL and R ˆ UL are the where f = −R estimates of RDL and RUL and are defined in Section II-A. The non-convex function, f , can be written as a difference ˆ DL and −R ˆ UL as a differof convex functions by writing −R ˆ ence of convex functions. −RDL can be re-written as: X ˆ DL,k ˆ DL = log2 Σk + C −R k∈ID

−

X

k∈ID

ˆ DL,k + CDL,k log2 Σk + C

k∈ID

= gDL − hDL

˜ UL= − log |ΩUL |− Tr h 2

where gDL and hDL are defined as follows: X X gDL = − log2 Σk + [HDL ]k QBS,l [HDL ]†k k∈ID

l∈ID

X † ˆ MS ]k,u QMS,u [G ˆ MS ] , + [G k,u u∈I

hDL

U X X [HDL ]k QBS,l [HDL ]†k log2 Σk + =−

k∈ID

X ˆ MS ]k,u QMS,u [G ˆ MS ]† , [G + k,u u∈IU

where both gDL and hDL are convex (since log |X| is concave ˆ DL is not. for X < 0, |X| 6= 0); however, their difference, −R ˆ Similarly, −RUL can be re-written as: ˆ UL = gUL − hUL , −R

u∈IU X ˆ † , ˆ = − log2 ΣBS + GBS QBS,k G BS k∈ID

where both gUL and hUL are convex; however, their difference, ˆ UL is not. −R Hence, f can be written as a difference of convex functions, f = g − h, where (8) (9)

˜ The DC-based algorithm approximates f by f˜MU-DC = g−h, where ˜h is the first-order Taylor’s series approximation [17, p. 69] of h (i.e., (9)). The first-order Taylor’s series approximation for the function log2 |A + X| is given by (10) h
i
−1 1 log2 A+X ≥ log2 A+X0 + Tr A+X0 X−X0 . ln(2) (10) ˜ can be written as: Hence, h ˜=˜ h hDL + ˜ hUL ,

and where ΩDL,k and ΩUL are given by: X (n) [HDL ]k QBS,l [HDL ]†k ΩDL,k = Σk + l∈ID \{k}

+

X

ˆ MS ]† , ˆ MS ]k,u Q(n) [G [G MS,u k,u

u∈IU

X

ˆ† ˆ BS Q(n) G G BS,l BS .

l∈ID

The function, f˜MU-DC , is convex since it is the difference between a convex function, g, (i.e., (8)) and an affine function, ˜ (i.e., (11)). As such, the non-convex optimization problem h, (7) can be locally approximated by the convex optimization problem (12): f˜MU-DC (12)

subject to: (4b) − (4e).

[HUL ]u QMS,u [HUL ]†u

g = gDL + gUL h = hDL + hUL .

k∈ID

min

l∈ID

hUL

#   X Ω−1 (n) † UL ˆ ˆ GBS QBS,k −QBS,k GBS , ln(2)

QBS,k ,QMS,u , k∈ID , u∈IU

where gUL and hUL are defined as follows: X ˆ† ˆ BS QBS,l G gUL = − log2 ΣBS + G BS X

"

ΩUL = ΣBS +

l∈ID \{k}

+

to the covariance matrix associated with the n-th iteration, are given by: # " −1   X ΩDL,k X (n) † ˜ DL= ˆ ˆ MS ]k,u QMS,u − Q [G Tr h MS,u [GMS ]k,u ln(2) u∈IU k∈ID   −1   X ΩDL,k X (n) Tr  + [HDL ]k QBS,l − QBS,l [HDL ]†k  ln(2) k∈ID l∈ID \{k} X log2 |ΩDL,k | , −

(11)

˜ ˜ wherethe expressions  for hDL and hUL centered around the (n) (n) point QBS,k , QMS,u for k ∈ ID and u ∈ IU , where n refers

The DC-based algorithm is described in Algorithm 1. A sequence of convex semi-definite programming optimization problems are solved where the objective function is updated for each iteration to locally approximate the original nonconvex optimization problem. Note that each convex subproblem can be solved using cvx, a package for solving disciplined convex programs in Matlab [18], [19]. Convergence to a local optimum is guaranteed since after each iteration the objective function is non-increasing and the optimization problem is bounded below (i.e., the negative of the total system sum-rate). B. SCAMP Algorithm This section presents the scheme we developed for solving general non-convex matrix-variable optimization problems with logarithmic objective functions applied to the FD MUMIMO scenario. The SCAMP algorithm results in an alternate objective function approximation to the DC-based algorithm. The SCAMP approximation is derived by combining the first-order Taylor’s approximations [17, p. 69] of the functions log2 |A + X| (given by 10) and log2 |X|, which can be expressed as: h
i 1 X − X0 . (13) Tr X−1 log2 X ≥ log2 X0 + 0 ln(2)

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Algorithm 1: DC-based algorithm. (0)

Randomly initialize QBS,k , k ∈ ID . (0) Randomly initialize QMS,u , u ∈ IU . Initialize n = 0. repeat ˜ using (11), centered around the point Update h   (n) (n) QBS,k , QMS,u , k ∈ ID , u ∈ IU . ˜ Update f˜MU-DC = g − h.  ⋆ Solve (12) for QBS,k , Q⋆MS,u , k ∈ ID , u ∈ IU . n = n + 1. (n) Update QMS,u = Q⋆MS,u , u ∈ IU . (n) Update QBS,k = Q⋆BS,k , k ∈ ID . until f˜MU−DC converges; Apply Cholesky decomposition: Q⋆MS,u = LMS,u L†MS,u . Apply Cholesky decomposition: Q⋆BS,k = LBS,k L†BS,k . Solve for VMS,u using (5), u ∈ IU . Solve for VBS,k using (6), k ∈ ID .

The inverse of a sum of two matrices can be written as [20]:
−1 −1 (A + X0 )−1 = X−1 I + AX−1 AX−1 0 − X0 0 0 . (14)

Hence, substituting (14) into (10) gives:   Tr X0−1 (X − X0 ) (15) log2 A + X ≥ log2 A + X0 + ln(2) h i
−1 −1 −1 Tr X−1 I + AX AX (X − X ) 0 0 0 0 − . ln(2) Next, (13) and (15) can be combined by substituting for the  common term Tr X−1 (X − X ) , which gives: 0 0 log2 A + X ≈ log2 |X| −

1 Tr [ΦX] + β, ln(2)

(16)

where Φ and β are given by:
−1 AX−1 (17) I + AX−1 Φ = X−1 0 , 0 0 h i
−1 Tr X−1 I + AX−1 A 0 0 − log2 |X0 | . β = log2 |A + X0 |+ ln(2) (18) The following definitions will assist in the derivation of the MU-MIMO SCAMP algorithm. ˆ DL,k , ΥDL,k = Σk + C ˆ UL , ΥUL = ΣBS + C

(19) (20)

ˆ DL,k and C ˆ UL are defined as in Section II-A to be the where C ˜ ˜ UL , respectively. estimates of CDL,k and C (n) (n) (n) (n) (n) ˆ (n) ˆ As well, ΥDL,k , C DL,k , CDL,k , ΥUL , CUL , and CUL refer to each respective expression evaluated at the point  (n) (n) QBS,k , QMS,u for k ∈ ID and u ∈ IU , where n refers to the covariance matrix associated with the n-th iteration.

ˆ DL gives Applying the SCAMP approximation, (16), to R the following equation:  X  1 ˆ tDL,k − βDL,k , ηDL,k + νDL,k + −RDL ≈ ln(2) k∈ID

where βDL,k is defined as in (18) with A = I and X0 =  −1 (n) (n) ΥDL,k CDL,k . As well, ηDL,k , νDL,k , and tDL,k are defined as follows: ηDL,k = − log2 |CDL,k | , νDL,k = log2 |ΥDL,k | ,   tDL,k = Tr ΦDL,k Υ−1 C DL,k DL,k ,

(21) (22) (23)

where ΦDL,k is given by (17) with A = I and X0 =  −1 (n) (n) ΥDL,k CDL,k . ˆ DL satisfies the DCP ruleset [16], In order to ensure that −R νDL,k and tDL,k were replaced by their respective first-order Taylor’s series approximations. The Taylor’s series approximations applied are given by (10) for νDL,k , and by (24) for tDL,k . h i    
−1 † Tr AX−1 Y ≈ Tr AX−1 + Tr Y (Y−Y ) AX 0 0 0 0 h i
−1 −1 † − Tr X0 Y0 AX0 (X−X0 ) , (24)

which can be computed using the concept of matrix differentials [21], [22]. Hence, the Taylor’s series approximations of νDL,k and  (n) (n) tDL,k , centered around the point QBS,k , QMS,u , for k ∈ ID and u ∈ IU , are denoted by ν˜DL,k and t˜DL,k and are given by (25) and (26), respectively.   −1 (n)   Υ (n)  DL,k ˆ DL,k − C ˆ (n)  ν˜DL,k = log2 ΥDL,k +Tr C DL,k , (25) ln(2) t˜DL,k =

(n) tDL,k

# "   −1 † (n) (n) CDL,k − CDL,k + Tr ΦDL,k ΥDL,k

− Tr

" 

(n)

ΥDL,k

−1

 −1 † (n) (n) × CDL,k ΦDL,k ΥDL,k

#   (n) ˆ DL,k − C ˆ C , DL,k

(26)

(n)

where tDL,k refers to (23) evaluated at the point  (n) (n) QBS,k , QMS,u , for k ∈ ID and u ∈ IU . ˆ DL , denoted by R ˜ DL , is Hence, the DCP approximation to R given by:  X 1 ˜ ˜ DL = ηDL,k + ν˜DL,k + −R tDL,k − βDL,k , (27) ln(2) k∈ID

where ηDL,k is given by (21), ν˜DL,k is given by (25), t˜DL,k is given by (26), and βDL,k is given by (18) with A = I and  −1 (n) (n) X0 = ΥDL,k CDL,k .

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Similar to the downlink case, applying the SCAMP approxˆ UL gives the following equation: imation, (16), to R ˆ UL ≈ ηUL + νUL + −R

1 tUL − βUL , ln(2)

where βUL is defined as in (18) with A = I and X0 =  −1 (n) (n) ΥUL CUL . As well, ηUL , νUL , and tUL are defined as follows: ηUL = − log2 |CUL | , νUL = log2 |ΥUL | ,   tUL = Tr ΦUL Υ−1 UL CUL ,

(28) (29) (30)

where ΦUL is given by (17) with A = I and X0 =  −1 (n) (n) CUL . ΥUL ˆ UL Similar to the downlink case, in order to ensure that R satisfies the DCP ruleset, νUL and tUL were replaced by their respective first-order Taylor’s series approximations.   −1 1 (n) (n) ˆ UL − C ˆ (n) , C Tr ΥUL ν˜UL = log2 ΥUL + UL ln(2) (31) # "  †  −1   (n) (n) (n) (32) CUL − CUL t˜UL = tUL + Tr ΦUL ΥUL

Algorithm 2: SCAMP algorithm. (n)

Randomly initialize QBS,k , k ∈ ID . (n) Randomly initialize QMS,u , u ∈ IU . Initialize n = 0. repeat (n) ˆ (n) , and C(n) ˆ (n) , C(n) , Υ(n) , C Update ΥDL,k , C UL UL UL DL,k DL,k  (n) (n) centered around the point QBS,k , QMS,u , k ∈ ID , u ∈ IU . Update ν˜DL,k using (25), t˜DL,k using (26), ν˜UL using (31), and t˜UL using (32). ˜ DL using (27) and −R ˜ Update −R  UL using (33).  ⋆ ⋆ Solve (34) for QBS,k , QMS,u , k ∈ ID , u ∈ IU . n = n + 1. (n) Update QMS,u = Q⋆MS,u , u ∈ IU . (n) Update QBS,k = Q⋆BS,k , k ∈ ID . ˜ DL − R ˜ UL converges; until −R Apply Cholesky decomposition: Q⋆MS,u = LMS,u L†MS,u . Apply Cholesky decomposition: Q⋆BS,k = LBS,k L†BS,k . Solve for VMS,u using (5), u ∈ IU . Solve for VBS,k using (6), k ∈ ID .

# analytical results to express the loss incurred due to inaccurate −1 †   (n) (n) ˆ UL − C ˆ , channel knowledge resulting in residual self-interference. ΥUL C − Tr UL ⋆ ˆ ⋆ (respectively, R⋆ and R ˆ ⋆ ) be the downLet RDL and R DL UL UL link (respectively, uplink) true and expectedsum-rates corre(n)  where tUL is given by (30) evaluated at the point  ⋆ ⋆ sponding to the transmit covariance matrices Q , Q (n) (n) MS,u , BS,k QBS,k , QMS,u , k ∈ ID , u ∈ IU . for k ∈ ID and u ∈ IU . ˆ UL , denoted by R ˜ UL , is Hence, the DCP approximation to R ⋆ ˜⋆ ˜⋆ Similarly, let C⋆DL,k and C DL,k (respectively, CUL and CUL ) given by: be the downlink (respectively, uplink) transmit covariance 1 matrices corresponding to the transmit covariance matrices  ˜ UL = ηUL + ν˜UL + −R t˜UL − βUL , (33)  ⋆ ln(2) QBS,k , Q⋆MS,u , for k ∈ ID and u ∈ IU . " 

(n) ΥUL

−1

(n) CUL ΦUL

where ηUL is given by (28), ν˜UL is given by (31), t˜UL is given by (32),  and βUL is given by (18) with A = I and X0 =  (n)

−1

˜⋆ ˜⋆ Hence, C DL,k and CUL are given by:

(n)

CUL . ΥUL Note that the SCAMP optimization problem can be written as: ˜ DL − R ˜ UL −R min QBS,k ,QMS,u , k∈ID , u∈IU

(34)

˜⋆ ˜⋆ ˆ⋆ C DL,k = CDL,k + ∆CDL,k , ˜⋆ = C ˆ ⋆ + ∆C ˜⋆ , C UL UL UL ˆ⋆ ˜⋆ where C DL,k and ∆CDL,k are given by:

subject to: (4b) − (4e). By construction, optimization problem (34) is convex and satisfies the DCP ruleset. The SCAMP algorithm is described in Algorithm 2. As with the DC-based algorithm, optimization problem (34) can be solved using cvx [18], [19]. As well, it can be seen that the SCAMP algorithm converges to a locally optimal point of the original non-convex optimization problem by using a similar argument to that of Section III-A. IV. E FFECT OF E STIMATION E RROR

ON THE

S UM -R ATE

This section investigates the effect of channel estimation error on the achievable sum-rate. In particular, it provides

ˆ ⋆DL,k = C

X

[HDL ]k Q⋆BS,l [HDL ]†k

l∈ID \{k}

+

X

ˆ MS ]k,u Q⋆MS,u [G ˆ MS ]† , [G k,u

u∈IU

˜ ⋆DL,k = ∆C

X

u∈IU

(

[∆GMS ]k,u Q⋆MS,u [∆GMS ]†k,u ˆ MS ]k,u Q⋆MS,u [∆GMS ]† + [G k,u +

ˆ MS ]† [∆GMS ]k,u Q⋆MS,u [G k,u

)

,

7

k∈ID

˜ ⋆UL ∆C

=

X

(

ˆ† ∆GBS Q⋆BS,k ∆G†BS + ∆GBS Q⋆BS,k G BS

k∈ID

) † ⋆ ˆ + GBS QBS,k ∆GBS .

Hence, the uplink sum-rate can be written as: −1  ˆ ⋆ + ∆C ˜⋆ C RUL = log2 I + ΣBS + C UL . UL UL

−1  ˆ ⋆ + ∆C ˜⋆ can be re-written as: Using (14), ΣBS + C UL UL



˜⋆ ΥUL + ∆C UL

−1 −1  −1 ˜ ⋆ −1 ˜ ⋆ −1 = Υ−1 UL − ΥUL I + ∆CUL ΥUL ∆CULΥUL ,

ˆ⋆ . where ΥUL = ΣBS + C UL Hence, RUL can be re-written as:

ˆ UL + ∆RUL , RUL = R where ∆RUL is the uplink estimation error term given by: −1  ˜ ⋆UL Υ−1 ˜ ⋆UL Υ−1 CUL , I + ∆ C ∆ C ∆RUL = log2 I − Ψ−1 UL UL UL
where ΨUL = ΥUL I + Υ−1 UL CUL . Note that ∆RUL is derived using the fact that log2 |A − B| = log2 |A| + log2 I − A−1 B . Similarly, the estimation error term for transmission to the k-th downlink MS (k ∈ ID ), ∆RDL,k , is given by:  −1 −1 ˜⋆ ∆RDL,k = log2 I − Ψ−1 × DL,k I + ∆CDL,k ΥDL,k ˜ ⋆ Υ−1 CDL,k , ∆C DL,k DL,k

  where ΨDL,k = ΥDL,k I + Υ−1 DL,k CDL,k and ΥDL,k = Σk + ˆ⋆ . C

Similarly, [GMS ]k,u for k ∈ ID , u ∈ IU was generated as a zero-mean complex Gaussian random variable with a variance equal to the co-channel Interference-to-Noise Ratio at the Mobile Station (INRMS,in ), where SNR/INRMS,in represents the Signal-to-co-channel-Interference Ratio at the MS receiver input (SIRMS,in ) before interference cancellation. Note that since the BS antennas are co-located, they suffer from very strong self-interference, whereas since the downlink and uplink MSs are geographically separated, they suffer from a weaker interference. Hence, INRBS,in > INRMS,in . The simulations assume Kdl = Kul = 3, NMS,T = NMS,R = 2, NBS,R = NBS,T = 6, Pmax,u = 23 dBm for all u ∈ IU , and Pmax,BS = 30 dBm in accordance with the 3GPP LTE (Release 9) simulation baseline parameters for a picocell deployment [23, p.59]. Fig. 3 shows the FD-to-HD sum-rate ratio vs. SNR with 2 SIRBS,in = −15 dB, SIRMS,in = −5 dB, and σerr = 1.1 The results show that the SCAMP algorithm can offer 15–25% sum-rate improvements over optimized HD for a wide-range of realistic SNR values. In particular, it is interesting to note that the SCAMP algorithm outperforms the DC-based algorithm for all SNR values. 1.4 SCAMP DC Scheme [12] Half−Duplex

1.3 1.2 FD−to−HD sum−rate ratio

ˆ ⋆ and ∆C ˜ ⋆ are given by: and C UL UL X ˆ⋆ = ˆ BS Q⋆ G ˆ† C G UL BS,k BS ,

1.1 1 0.9 0.8 0.7

0

5

10 SNR (dB)

15

20

DL,k

V. I LLUSTRATIVE R ESULTS

Fig. 3. FD-to-HD sum-rate ratio vs. SNR with SIRBS,in = −15 dB, 2 = 1. SIRMS,in = −5 dB, and σerr

A. FD MU-MIMO: Sum-Rate Comparison This sub-section compares the performance of the DCbased and SCAMP algorithms, the optimized HD and the joint design proposed in [12] for the MU-MIMO with HD MSs system shown in Fig. 1. Note that for the MU-MIMO with HD MSs system, the HD optimization problem is non-convex and can be solved by using SCP. The noise was normalized such that E{zBS z†BS } = INR,BS and E{zk z†k } = INR,MS , for k ∈ ID , and HDL and HUL were generated as zero-mean complex Gaussian random variables with a variance equal to the SNR. Conversely, the self-interference channel, GBS , was generated as a zero-mean complex Gaussian random variable with a variance equal to the self-Interference-to-Noise Ratio at the Base Station (INRBS,in ), where SNR/INRBS,in represents the Signal-to-self-Interference Ratio at the BS receiver input (SIRBS,in ) before interference cancellation.

Fig. 4 shows the FD-to-HD sum-rate ratio vs. SIRBS,in 2 with SNR = 5 dB, SIRMS,in = −5 dB, and σerr = 1. The results show that the SCAMP algorithm provides about 20– 30% sum-rate improvements over optimized HD for a widerange of SIRBS,in values. In particular, for SIRBS,in > −25 dB, the SCAMP algorithm provides significantly better sumrate than the other candidate SCP algorithms. Furthermore, with SIRBS,in = −40 dB, the SCAMP algorithm still provides performance improvements over the optimized HD approach. 2 Fig. 5 shows the FD-to-HD sum-rate ratio vs. σerr with SNR = 5 dB, SIRBS,in = −20 dB, and SIRMS,in = −5 dB. The 1 Setting σ 2 = 1 is a reasonable choice for practical FD systems since it err implies the estimation error of the self-interference channel is on the same order of magnitude as the background noise.

8

perfect approximation of the forward channels. Conversely, the SCAMP approach approximates each of the forward, interference, and noise terms but it spreads out the approximations over multiple terms. As such, for FD SU-MIMO systems, where there is fewer sources of interference (self or otherwise), an accurate forward channel is more important, leading to superior performance for the DC-based approach (i.e., the DCbased algorithm better approximates the FD SU-MIMO system than the SCAMP algorithm). However, for FD MU-MIMO systems, there are many forms of interference (e.g., many selfinterference terms and uplink co-channel interference) which results in the SCAMP algorithm better-approximating the FD MU-MIMO environment than the DC-based algorithm.

1.3

FD−to−HD sum−rate ratio

1.2

1.1

1

0.9

0.8 SCAMP DC Scheme [12] Half−Duplex

0.7 −40

−35

−30

−25

−20 −15 SIRBS,in (dB)

−10

−5

0

C. FD MU-MIMO: Convergence Comparison

Fig. 4. FD-to-HD sum-rate ratio vs. SIRBS,in with SNR = 5 dB, SIRMS,in = 2 = 1. −5 dB, and σerr

results show that the SCAMP algorithm is capable of achieving more than 20% sum-rate improvements over optimized HD for 2 2 some reasonable values of σerr . In addition, when σerr = 10, corresponding to large inaccuracies in the channel knowledge, the SCAMP algorithm was still capable of providing 5% performance improvements over optimized HD.

1.35 SCAMP DC Scheme [12] Half−Duplex

1.3

FD−to−HD sum−rate ratio

1.25 1.2 1.15 1.1 1.05 1 0.95 0.9 0.85 −2 10

−1

0

10

10

1

10

2

σerr 2 with SNR = 5 dB, SIR Fig. 5. FD-to-HD sum-rate ratio vs. σerr BS,in = −20 dB, and SIRMS,in = −5 dB.

This sub-section provides a convergence comparison of the various MU-MIMO SCP algorithms for particular realizations. Hence, although SCAMP outperforms the DC-based approach on average for the FD MU-MIMO simulations, on a perrealization basis, it is not guaranteed to outperform the other approaches. As discussed in Section V-B, the SCAMP and DCbased algorithms apply different convex approximations to the non-convex objective function, and as such, some realizations may be better suited for the SCAMP approximation, while others may be better suited for the DC-based approximation. Fig. 6 shows the convergence of the SCP algorithms compared to the converged optimized HD value for two different realizations with SNR = 5 dB, SIRBS,in = −20 dB, SIRMS,in = 2 −5 dB, and σerr = 1.2 It is important to note that the optimized HD also requires multiple iterations; however, here we only show the final converged value of the optimized HD approach to highlight the convergence of the FD SCP-based approaches. The results show that the DC-based approach tends to initially outperform the SCAMP approach but, in general, the SCAMP approach outperforms the DC-based approach at convergence. As well, it was observed that the SCP algorithms periodically have a step-like convergence where the algorithm will level-off for some iterations before increasing significantly and beginning to level-off again. This is due to the internal convex optimization solver, since occasionally the convex optimization solver (i.e., cvx) will fail to find an optimal point or will return a point which is considered an inaccurate solution; however, after slightly tuning the initial points to the convex solver (i.e., after several more iterations), an optimal point can be found, resulting in a sudden sum-rate increase and the step-like convergence.

B. Comparison with FD SU-MIMO It is interesting to note that in Section V-A, for each FD MUMIMO scenario tested, the SCAMP algorithm outperformed the DC-based algorithm. However, results in [15] indicate that the DC-based algorithm offers a better sum-rate than the SCAMP algorithm for the FD SU-MIMO scenario. This is due to the method by which the DC-based and SCAMP algorithms approximate their respective objective functions. In particular, the DC-based algorithm only approximates the interference-plus-noise terms, and hence, has a

VI. C ONCLUDING R EMARKS This paper presented two SCP-based algorithms for solving the non-convex sum-rate maximization problem with matrix variables for MU-MIMO systems. The DC-based algorithm is based on the fact that the structure of the problem naturally lends itself to a particular DC decomposition. The SCAMP 2 The

n-th data-point corresponds to the sum-rate (b/s/Hz) associated with (n) and QMS,u of the respective algorithm (i.e., the BS and MS transmit covariance matrices associated with n-th iteration).

(n) QBS,k

9

of high self-interference, the SCAMP algorithm is capable of providing significant sum-rate improvements over existing optimized HD techniques for FD MU-MIMO systems.

16 15 14

R EFERENCES

Sum−rate (b/s/Hz)

13 12 11 10 9 8

SCAMP DC Scheme [12] Half−Duplex

7 6 5

0

200

400 600 Number of Iterations

800

1000

(a) Realization 1 13 12

Sum−rate (b/s/Hz)

11 10 9 8 7

SCAMP DC Scheme [12] Half−Duplex

6 5

0

200

400 600 Number of Iterations

800

1000

(b) Realization 2 Fig. 6. Convergence comparison for SNR = 5 dB, SIRBS,in = −20 dB, 2 = 1. SIRMS,in = −5 dB, and σerr

algorithm is a general technique proposed for solving nonconvex optimization problems with a logarithmic objective function, which we apply to the FD MU-MIMO system. The SCAMP algorithm is a multi-step procedure that begins by applying a non-convex approximation to the non-convex objective function and then proceeds to convexify the nonconvex approximation. The two SCP-based algorithms result in two different convex approximations to the non-convex objective function. An analytical expression for the estimation error of the sum-rate was derived, which provides an analytical result to express the loss incurred due to inaccurate channel knowledge resulting in residual self-interference. Illustrative results show that the SCAMP algorithm provides approximately 20-30% sum-rate improvements over optimized HD in a wide-range of FD MU-MIMO system scenarios. The results indicated that initially, the DC-based algorithm tends to provide better performance but at convergence the SCAMP algorithm tends to provide significantly better performance, on average. Overall, the results indicate that even in the presence

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Sean Huberman received his B.Sc. Engineering (first class honours) from Queen’s University (Kingston, Canada) in 2008. As of November 2014, he completed his Ph.D. in Electrical Engineering at McGill University (Montreal, Canada). His research interests include techniques of dynamic resource allocation and mathematical optimization with applications to wireless and wireline half- and full-duplex systems. Mr. Huberman was the recipient of the Hydro Quebec Engineering Doctoral Award in 2009. In 2010, he was the recipient of the Vadasz Doctoral Fellowship in Engineering. He was also the recipient of a three-year doctoral National Science and Engineering Research Council (NSERC) award and a three-year McGill Engineering Doctoral Award (MEDA).

Tho Le-Ngoc obtained his B.Eng. (with Distinction) in Electrical Engineering in 1976, his M.Eng. in 1978 from McGill University, Montreal, and his Ph.D. in Digital Communications in 1983 from the University of Ottawa, Canada. During 1977-1982, he was with Spar Aerospace Limited and involved in the development and design of satellite communications systems. During 1982-1985, he was an Engineering Manager of the Radio Group in the Department of Development Engineering of SRTelecom Inc., where he developed the new point-to-multipoint DA-TDMA/TDM Subscriber Radio System SR500. During 1985-2000, he was a Professor at the Department of Electrical and Computer Engineering of Concordia University. Since 2000, he has been with the Department of Electrical and Computer Engineering of McGill University. His research interest is in the area of broadband digital communications. He is a fellow of the Institute of Electrical and Electronics Engineers (IEEE), the Engineering Institute of Canada (EIC), the Canadian Academy of Engineering (CAE) and the Royal Society of Canada (RSC). He is the recipient of the 2004 Canadian Award in Telecommunications Research, and recipient of the IEEE Canada Fessenden Award 2005. He holds a Canada Research Chair (Tier I) on Broadband Access Communications, and a Bell Canada/NSERC Industrial Research Chair on Performance & Resource Management in Broadband xDSL Access Networks.