I Introduction
MULTIPLEinput multipleoutput (MIMO) systems have been widely acknowledged as a promising technology in the field of wireless communications, due to the significant gains over singleantenna systems [1]. When the channel knowledge is known at the base station (BS), the capacityachieving dirtypaper coding (DPC) scheme is proposed in [2] by presubtracting the interference prior to transmission. However, DPC is difficult to implement in practical systems due to the impractical assumption of infinite length of codewords and its high computational cost. To achieve a compromise between performance and complexity, its nonlinear counterparts in the form of TomlinsonHarashima precoding (THP) [3]
and vector perturbation (VP)
[4] have been proposed, which however are still too complicated for practice due to the inclusion of the sophisticated spheresearch algorithms. Therefore, lowcomplexity linear precoding schemes based on zeroforcing (ZF) have received increasing research attention [5], and a regularized ZF (RZF) scheme is proposed in [6] to further improve the performance of ZF. On the other hand, transmit beamforming schemes based on optimization have also been a popular research topic [7][13]. Among the optimizationbased schemes, one form of the optimization known as signaltointerferenceplusnoise ratio (SINR) balancing is to maximize the minimum SINR subject to a total power constraint [7], [8] or a perantenna power constraint [9]. An alternative downlink beamforming targets at minimizing the total transmit power at the BS subject to a minimum SINR requirement [10][12]. It has been shown that the power minimization problems can be formulated either as a virtual uplink problem with power control or as a semidefinite programming (SDP) and solved via semidefinite relaxation (SDR) [11]. As for the SINR balancing problem, it is proven to be an inverse problem to the power minimization optimization, based on which schemes via bisection search [7] and iterative algorithms [10] have been proposed.Nevertheless, both the above precoding schemes and the optimizationbased transmit beamforming designs mentioned above have ignored the fact that interference can be beneficial and further exploited on an instantaneous basis [14], [15]. The concept of constructive interference (CI) was firstly introduced in [16], where it is shown that the instantaneous interference can be categorized into constructive and destructive. A modified ZF precoding scheme is then proposed in [17], where the constructive interference is exploited while the destructive interference is cancelled. A correlation rotation scheme has been further proposed in [18], where it is shown that the destructive interference can be manipulated and rotated such that all the interference becomes constructive. Symbollevel transmit beamforming schemes based on convex optimization for CI has been proposed in [19], [20], where the concept of constructive region is introduced to relax the strict phase rotation constraint in [18] and achieve an improved performance. Further studies on the optimizationbased CI beamforming schemes can be found in [20][23]. Due to the performance benefits over conventional schemes, the concept of CI has been extended to many wireless application scenarios, including cognitive radio [24], [25], constant envelope precoding [26], wireless information and power transfer [27] and mutual coupling exploitation [28]. The above studies show that MIMO systems can benefit from the CI with a symbollevel beamforming. Nevertheless, while the performance of CIbased beamforming approaches is superior, they need to solve a convex optimization problem, which can be computationally inefficient, especially when executed on a symbolbysymbol basis.
In this paper, we design lowcomplexity optimal and suboptimal solutions for CI precoding, culminating in closedform iterative precoders. We consider an optimization problem where we maximize the distance between the constructive region and the detection thresholds such that the effect of CI is maximized. We firstly consider the optimization for strict phase rotation, where the phases of the interfering signals are rotated such that they are strictly aligned to the symbol of interest. By analyzing the formulated secondorder cone programming (SOCP) optimization with Lagrangian and KKT conditions, we derive the structure of the optimal beamforming matrix, which leads to an equivalent optimization and further simplifies the beamforming design. By formulating the dual problem of the equivalent optimization problem, it is mathematically shown that the optimization for CI beamforming is equivalent to a quadratic programming (QP) optimization over a simplex, which finally leads to a closedform expression. We extend our analysis to the case of nonstrict phase rotation, where the phases of the interfering signals are rotated such that the resulting interfered signal is located within the constructive region. By following a similar approach for the case of strict phase rotation, we analytically show that the optimal beamforming matrix for theses two scenarios shares a similar closedform expression, and a dimensional optimization for nonstrict phase rotation is equivalent to a dimensional optimization for strict phase rotation in terms of the problem formulation. Our above analysis also provides some insights on the connection between the CI beamforming and the generic ZF precoding.
Moreover, our efforts to facilitate the symbollevel CI beamforming in practice culminate in an iterative closedform scheme to efficiently obtain the optimal beamforming matrix, where a closedform solution is obtained within each iteration. We show that only in a few iterations, the closedform approach obtains optimal performance. Numerical results validate our above analysis and the optimality of the proposed iterative closedform method for both strict and nonstrict phase rotation. Moreover, it is numerically shown that the proposed iterative approach is more timeefficient compared to the conventional QP algorithms based on interiorpoint methods, especially when the number of users is small. By constraining the maximum number of iterations, we further obtain a flexible performancecomplexity tradeoff for the proposed iterative method, based on its connection with conventional ZF precoding. Both of the above motivate the use of CIbased beamforming in practical wireless systems.
For reasons of clarity, we summarize the contributions of this paper as:

We formulate the optimization problem for CIbased beamforming, where we maximize the distance between the constructive region and the detection thresholds. We derive the optimal beamforming matrix for strict phase rotation and further formulate an equivalent and simplified optimization problem.

The optimization for strict phase rotation is transformed and further shown to be equivalent to a QP problem over a simplex, which can be more efficiently solved than the originally formulated problem.

We extend our analysis to the case of nonstrict phase rotation, where the closedform expression is similar to the case of strict phase rotation. It is further shown that a dimensional optimization for nonstrict phase rotation is equivalent to a dimensional optimization for strict phase rotation in terms of the problem formulation.

We analytically study the connection between the CI beamforming and the ZF precoding, where it is shown that ZF precoding can be regarded as a special case of CIbased beamforming with all the dual variables being zero.

We further propose an iterative closedform scheme to obtain the optimal beamforming matrix for both the strict and nonstrict phase rotation cases, where within each iteration a closedform solution can be derived. We show that the closedform precoder obtains an optimal performance in only a few iterations.
The remainder of this paper is organized as follows. Section II introduces the system model and briefly reviews CI. Section III includes the analysis for the optimization problems with both strict and nonstrict phase rotation constraints. The connection between the CI beamforming and conventional ZF precoding is discussed in Section IV. The proposed iterative closedform scheme is introduced in Section V. The numerical results are shown in Section VI, and Section VII concludes the paper.
: , , and denote scalar, vector and matrix, respectively. , and denote transposition, conjugate transposition and trace of a matrix, respectively. denotes the imaginary unit, and denotes the vectorization operation. denotes the entry in the row and th column of . denotes the absolute value of a real number or the modulus of a complex number, and denotes the Frobenius norm. represents an matrix in the complex set, and
denotes the identity matrix.
and denote the real and imaginary part of a complex number, respectively. denotes the cardinality of a set.Ii System Model and Constructive Interference
In this section, the system model that we consider is firstly introduced, followed by a brief review of CI and the constructive region.
Iia System Model
We consider a multiuser MISO system in the downlink, where the BS structure with a symbollevel precoding is depicted in Fig. 1, where the iterative closedform algorithm will be introduced in Section V. The BS with transmit antennas is simultaneously communicating with singleantenna users in the same timefrequency resource, where . We focus on the transmit beamforming designs and perfect CSI is assumed throughout the paper. The data symbol vector is assumed to be from a normalized PSK modulation constellation [19], denoted as . Then, the received signal at the th user can be expressed as
(1) 
where denotes the flatfading Rayleigh channel vector from user to the BS, and each entry in
follows a standard complex Gaussian distribution.
is the beamforming matrix andis the additive Gaussian noise with zero mean and variance
at the receiver.IiB Constructive Interference
CI is defined as the interference that pushes the received signals away from the detection thresholds [14][18]. CI for strict phase rotation refers to the cases where the phases of the interfering signals are controlled and rotated, such that they are strictly aligned to those of the data symbols of interest [18]. The constructive region has been further introduced in [19], where it is shown that the phases of the interfering signals may not be necessarily strictly aligned to that of the data symbols of interest, known as the nonstrict phase rotation. It is demonstrated that, as long as the resulting interfered signals are located in the constructive region, this increases the distance to the detection thresholds and returns an improved performance. To show this intuitively, in Fig. 2 and Fig. 3 we depict the case for strict phase rotation and nonstrict phase rotation respectively, where the constellation point from a normalized QPSK constellation is employed as the example to illustrate these two cases. We can observe that for both strict phase rotation and nonstrict phase rotation, the distance of the received signals to the detection thresholds is increased, which will improve the detection performance.
Iii Constructive Interference Beamforming
In this section, we firstly focus on the CI beamforming for strict phase rotation, and we further extend our analysis to the case of nonstrict phase rotation.
Iiia Strict Phase Rotation
Before formulating the optimization problem, based on the geometry of the modulation constellation we firstly construct the conditions that the beamformer should satisfy to achieve the strict phase rotation. In Fig. 2, without loss of generality we denote and is the object to be maximized. We further assume that the node ‘B’ denotes the noiseless received signal for user that is colinear to for strict phase rotation, which leads to
(2) 
Then, by introducing a realvalued scaling factor , we further express as
(3) 
where based on the geometry we can obtain that is a real number, and the condition on to achieve CI for strict phase rotation is given by
(4) 
where . With the above formulation, we can construct the optimization problem for strict phase rotation as
(5)  
where denotes the total available transmit power. A symbollevel power constraint is employed, as the exploitation of CI is related to the transmit symbol vector, which will also be shown mathematically in the following. belongs to the SOCP and can be solved with convex optimization tools such as CVX [19]. We decompose the beamforming matrix into vectors
(6) 
and based on the virtual multicast formulation in [19] we obtain that each is identical. This leads to the equivalent transformation of the power constraint, given by
(7) 
We further transform in (5) into a standard minimization problem, expressed as
(8)  
In the following we analyze with Lagrangian and KKT conditions. The Lagrangian of is expressed as [29]
(9)  
where , and are the dual variables, and we have and , . Based on the Lagrangian in (9), the KKT conditions for optimality can be obtained as
(10a)  
(10b)  
(10c)  
(10d)  
(10e) 
Based on (10b), it is firstly obtained that , and with the fact that we can further obtain . Then, in (10b) can be expressed as
(11) 
By denoting
(12) 
where we note that can be complex, the expression of is obtained as
(13) 
Based on (13), we further obtain that
(14) 
which is a constant for any . This mathematically verifies that the beamforming vector for one symbol is a phaserotated version of the beamforming vector for another symbol. Then, with each obtained, the beamforming matrix can be obtained and further expressed in a matrix form as
(15)  
We further express (3) in a compact form as
(16) 
where is the channel matrix and . By substituting (15) into (16), we can further obtain
(17)  
With (17), we can obtain the structure of the optimal beamforming matrix as a function of scaling vector as
(18) 
It is easy to observe from (18) that the CI beamforming is a symbollevel beamforming scheme since the beamforming matrix includes the expression of the symbol vector . Moreover, with (18) the original optimization problem on is transformed into an optimization on the realvalued scaling vector . With the fact that , based on (10e) we can obtain that the power constraint is strictly active, which leads to
(19)  
where we note that as each is real, and is defined as
(20) 
It is easy to obtain that is Hermitian and positive semidefinite, which further leads to
(21) 
where is a symmetric and positive semidefinite matrix. With (21) obtained, we can formulate a new convex optimization problem on that is equivalent to the original optimization , expressed as
(22)  
The optimal beamforming matrix for the original problem in (5) can be obtained with (18) based on the obtained by solving . In the following, we analyze the convex optimization with the Lagrangian approach, where the Lagrangian of is formulated as
(23)  
where and are the dual variables and , . is a column vector that consists of the dual variables and the vector . Based on (23), the KKT conditions of for optimality are expressed as
(24a)  
(24b)  
(24c)  
(24d) 
Based on (24b), firstly we have , and we can further obtain the expression of , given by
(25) 
where we note that is symmetric. With , based on (24c) it is obtained that the power constraint is strictly active, and by substituting (25) into (24c), we can express as a function of the dual vector , given by
(26)  
For the convex optimization in (22), it is easy to verify that the Slater’s condition is satisfied [29], which means that the dual gap is zero. Therefore, we can solve by solving its corresponding dual problem, which is given by
(27) 
For the dual problem , the inner minimization is achieved with (24a) and the obtained in (25), and therefore the dual problem can be further transformed into
(28)  
Due to the fact that is a monotonic function, therefore the dual problem is equivalent to the following optimization problem
(29)  
where the first constraint comes from (24a).
Based on our analysis and transformations above, we have transformed and simplified the original problem, and shown that the original optimization can be solved by solving . To be more specific, through (26), (25) and (18), we arrive at a final closedform expression of the optimal beamforming matrix as a function of , given by
(30) 
Moreover, it is observed that is a typical QP optimization problem over a simplex, which can be more efficiently solved with the simplex method [30] or interiorpoint methods [31], compared to the original CI beamforming formulation which is a SOCP optimization.
IiiB NonStrict Phase Rotation
We extend our analysis to the case of nonstrict phase rotation. Similarly, before formulating the optimization problem, we firstly construct the condition that the beamforming designs should satisfy such that the received signals are located in the constructive region. Based on Fig. 3, for consistency we denote and is the objective to be maximized. Following (2), we denote the received signal for user as , which is expressed as
(31) 
In the case of nonstrict phase rotation, each can be a complex value, which mathematically represents that a phase rotation is included for the received signal compared to the data symbol , as shown in Fig. 3. This is different from the case of strict phase rotation where each is strictly real. Then, based on the fact that and are perpendicular, we can obtain the expression of and , given by
(32) 
where based on Fig. 3 the imaginary unit ‘’ denotes a phase rotation of geometrically. For simplicity of denotation, we denote and , respectively. Due to the fact that the nodes ‘O’, ‘A’ and ‘C’ are colinear, we can further obtain the expression of as
(33) 
In Fig. 3, we can observe that to have the received signal located in the constructive region is equivalent to the following condition:
(34)  
In the case of , , (34) is identical to (4), and the nonstrict phase rotation reduces to the strict phase rotation. For PSK modulation, it is observed from the modulation constellation that the threshold angle can be expressed as
(35) 
With the above formulation, we can construct the optimization problem of CI for nonstrict phase rotation as
(36)  
To further analyze the optimization problem for nonstrict phase rotation, we first transform in (36) into a standard minimization form, given by
(37)  
Then, by following a similar step in (9)(17) with the Lagrangian approach, we can obtain that the optimal beamforming structure for nonstrict phase rotation is the same as that for strict phase rotation, which is given in (18). With the power constraint strictly active, we can further obtain that
(38)  
where is given by (20). However, we note that, different from the case of strict phase rotation, for the case of nonstrict phase rotation (38) is not in a quadratic form since each can be complex. By decomposing
(39) 
we can expand (38) with its real and imaginary components and further transform the power constraint into a quadratic form, given by
(40)  
Similar to the optimization in (22) for strict phase rotation, we can formulate an optimization problem on for nonstrict phase rotation, expressed as
(41)  
where we have transformed the CI constraint with the absolute value on into two separate constraints. We then analyze with Lagrangian and KKT conditions, where the Lagrangian of is constructed as
(42)  
where , and are the dual variables, and , , . By introducing
(43)  
where and , the Lagrangian for can be further simplified into
(44) 
Based on (44), we express the KKT conditions for optimality of in the following:
(45a)  
(45b)  
(45c)  
(45d)  
(45e) 
Based on (45b) we can obtain and the expression of , given by
(46) 
where we note that is symmetric. Moreover, from (45c) we obtain that the power constraint is strictly active with , and we can further obtain the expression of as
(47)  
where for simplicity and consistency we introduce
(48) 
Similar to the case for strict phase rotation, it is easy to observe that the Slater’s condition is satisfied for , and therefore by following a similar approach in (27) and (28), the dual problem for can be formulated into
(49) 
which further leads to the following equivalent optimization for nonstrict phase rotation
(50)  
where we denote as the th entry in , and we obtain based on (48). is also a QP optimization over a simplex, which can be efficiently solved. The final optimal beamforming matrix for nonstrict phase rotation can be similarly obtained in a closed form as a function of , given by
(51) 
where is a transformation matrix that transform the realvalued vector into its complex equivalence.
Based on the formulated equivalent optimization problems in (29) and in (50), we note the similarity of the optimization problem for strict phase rotation and nonstrict phase rotation. We observe that the objective function of for strict phase rotation and for nonstrict phase rotation is identical, and both optimization problems share the same constraints. It is further observed that the only difference between and is the problem size. It is then concluded that a dimensional optimization problem for nonstrict phase rotation and a dimensional optimization for strict phase rotation share the same problem formulation, and therefore they can be solved in a similar way.
Iv CI as a Generalization of ZF Precoding
In this section, we discuss the connection between the CI beamforming for strict phase rotation and the conventional ZF precoding. For the CI beamforming with nonstrict phase rotation, the connection can be obtained in a similar way. To compare the CI beamforming and the conventional ZF precoding, as a reference we first present the precoded signal vector of ZF, given by
(52) 
where is the scaling factor to meet the transmit power constraint. For fairness of comparison, we employ a symbollevel normalization for such that as for the considered CI beamforming, which leads to the expression of as
(53) 
By denoting , the expression of can be further transformed into
(54)  
Subsequently, we perform the mathematical analysis on the optimization problem on for strict phase rotation. By applying the Lagrangian approach, we can obtain the Lagrangian of , given by
(55)  
where the vector consists of each nonnegative dual variable of . Based on (55), we express the KKT conditions of as
(56a)  
(56b)  
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