MULTIPLE-input multiple-output (MIMO) systems have been widely acknowledged as a promising technology in the field of wireless communications, due to the significant gains over single-antenna systems . When the channel knowledge is known at the base station (BS), the capacity-achieving dirty-paper coding (DPC) scheme is proposed in  by pre-subtracting 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 non-linear counterparts in the form of Tomlinson-Harashima precoding (THP) 
and vector perturbation (VP) have been proposed, which however are still too complicated for practice due to the inclusion of the sophisticated sphere-search algorithms. Therefore, low-complexity linear precoding schemes based on zero-forcing (ZF) have received increasing research attention , and a regularized ZF (RZF) scheme is proposed in  to further improve the performance of ZF. On the other hand, transmit beamforming schemes based on optimization have also been a popular research topic -. Among the optimization-based schemes, one form of the optimization known as signal-to-interference-plus-noise ratio (SINR) balancing is to maximize the minimum SINR subject to a total power constraint ,  or a per-antenna power constraint . An alternative downlink beamforming targets at minimizing the total transmit power at the BS subject to a minimum SINR requirement -. It has been shown that the power minimization problems can be formulated either as a virtual uplink problem with power control or as a semi-definite programming (SDP) and solved via semi-definite relaxation (SDR) . 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  and iterative algorithms  have been proposed.
Nevertheless, both the above precoding schemes and the optimization-based transmit beamforming designs mentioned above have ignored the fact that interference can be beneficial and further exploited on an instantaneous basis , . The concept of constructive interference (CI) was firstly introduced in , where it is shown that the instantaneous interference can be categorized into constructive and destructive. A modified ZF precoding scheme is then proposed in , where the constructive interference is exploited while the destructive interference is cancelled. A correlation rotation scheme has been further proposed in , where it is shown that the destructive interference can be manipulated and rotated such that all the interference becomes constructive. Symbol-level transmit beamforming schemes based on convex optimization for CI has been proposed in , , where the concept of constructive region is introduced to relax the strict phase rotation constraint in  and achieve an improved performance. Further studies on the optimization-based CI beamforming schemes can be found in -. Due to the performance benefits over conventional schemes, the concept of CI has been extended to many wireless application scenarios, including cognitive radio , , constant envelope precoding , wireless information and power transfer  and mutual coupling exploitation . The above studies show that MIMO systems can benefit from the CI with a symbol-level beamforming. Nevertheless, while the performance of CI-based beamforming approaches is superior, they need to solve a convex optimization problem, which can be computationally inefficient, especially when executed on a symbol-by-symbol basis.
In this paper, we design low-complexity optimal and sub-optimal solutions for CI precoding, culminating in closed-form 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 second-order 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 closed-form expression. We extend our analysis to the case of non-strict 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 closed-form expression, and a -dimensional optimization for non-strict 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 symbol-level CI beamforming in practice culminate in an iterative closed-form scheme to efficiently obtain the optimal beamforming matrix, where a closed-form solution is obtained within each iteration. We show that only in a few iterations, the closed-form approach obtains optimal performance. Numerical results validate our above analysis and the optimality of the proposed iterative closed-form method for both strict and non-strict phase rotation. Moreover, it is numerically shown that the proposed iterative approach is more time-efficient compared to the conventional QP algorithms based on interior-point methods, especially when the number of users is small. By constraining the maximum number of iterations, we further obtain a flexible performance-complexity tradeoff for the proposed iterative method, based on its connection with conventional ZF precoding. Both of the above motivate the use of CI-based beamforming in practical wireless systems.
For reasons of clarity, we summarize the contributions of this paper as:
We formulate the optimization problem for CI-based 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 non-strict phase rotation, where the closed-form expression is similar to the case of strict phase rotation. It is further shown that a -dimensional optimization for non-strict 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 CI-based beamforming with all the dual variables being zero.
We further propose an iterative closed-form scheme to obtain the optimal beamforming matrix for both the strict and non-strict phase rotation cases, where within each iteration a closed-form solution can be derived. We show that the closed-form 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 non-strict phase rotation constraints. The connection between the CI beamforming and conventional ZF precoding is discussed in Section IV. The proposed iterative closed-form 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.
Ii-a System Model
We consider a multiuser MISO system in the downlink, where the BS structure with a symbol-level precoding is depicted in Fig. 1, where the iterative closed-form algorithm will be introduced in Section V. The BS with transmit antennas is simultaneously communicating with single-antenna users in the same time-frequency 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 , denoted as . Then, the received signal at the -th user can be expressed as
where denotes the flat-fading Rayleigh channel vector from user to the BS, and each entry in
follows a standard complex Gaussian distribution.is the beamforming matrix and
is the additive Gaussian noise with zero mean and varianceat the receiver.
Ii-B Constructive Interference
CI is defined as the interference that pushes the received signals away from the detection thresholds -. 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 . The constructive region has been further introduced in , 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 non-strict 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 non-strict 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 non-strict 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 non-strict phase rotation.
Iii-a 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 co-linear to for strict phase rotation, which leads to
Then, by introducing a real-valued scaling factor , we further express as
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
where . With the above formulation, we can construct the optimization problem for strict phase rotation as
where denotes the total available transmit power. A symbol-level 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 . We decompose the beamforming matrix into vectors
and based on the virtual multicast formulation in  we obtain that each is identical. This leads to the equivalent transformation of the power constraint, given by
We further transform in (5) into a standard minimization problem, expressed as
In the following we analyze with Lagrangian and KKT conditions. The Lagrangian of is expressed as 
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
where we note that can be complex, the expression of is obtained as
Based on (13), we further obtain that
which is a constant for any . This mathematically verifies that the beamforming vector for one symbol is a phase-rotated 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
We further express (3) in a compact form as
With (17), we can obtain the structure of the optimal beamforming matrix as a function of scaling vector as
It is easy to observe from (18) that the CI beamforming is a symbol-level 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 real-valued scaling vector . With the fact that , based on (10e) we can obtain that the power constraint is strictly active, which leads to
where we note that as each is real, and is defined as
It is easy to obtain that is Hermitian and positive semi-definite, which further leads to
where is a symmetric and positive semi-definite matrix. With (21) obtained, we can formulate a new convex optimization problem on that is equivalent to the original optimization , expressed as
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
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
Based on (24b), firstly we have , and we can further obtain the expression of , given by
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
For the convex optimization in (22), it is easy to verify that the Slater’s condition is satisfied , which means that the dual gap is zero. Therefore, we can solve by solving its corresponding dual problem, which is given by
Due to the fact that is a monotonic function, therefore the dual problem is equivalent to the following optimization problem
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 closed-form expression of the optimal beamforming matrix as a function of , given by
Moreover, it is observed that is a typical QP optimization problem over a simplex, which can be more efficiently solved with the simplex method  or interior-point methods , compared to the original CI beamforming formulation which is a SOCP optimization.
Iii-B Non-Strict Phase Rotation
We extend our analysis to the case of non-strict 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
In the case of non-strict 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
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 co-linear, we can further obtain the expression of as
In Fig. 3, we can observe that to have the received signal located in the constructive region is equivalent to the following condition:
In the case of , , (34) is identical to (4), and the non-strict 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
With the above formulation, we can construct the optimization problem of CI for non-strict phase rotation as
To further analyze the optimization problem for non-strict phase rotation, we first transform in (36) into a standard minimization form, given by
Then, by following a similar step in (9)-(17) with the Lagrangian approach, we can obtain that the optimal beamforming structure for non-strict 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
where is given by (20). However, we note that, different from the case of strict phase rotation, for the case of non-strict phase rotation (38) is not in a quadratic form since each can be complex. By decomposing
we can expand (38) with its real and imaginary components and further transform the power constraint into a quadratic form, given by
Similar to the optimization in (22) for strict phase rotation, we can formulate an optimization problem on for non-strict phase rotation, expressed as
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
where , and are the dual variables, and , , . By introducing
where and , the Lagrangian for can be further simplified into
Based on (44), we express the KKT conditions for optimality of in the following:
Based on (45b) we can obtain and the expression of , given by
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
where for simplicity and consistency we introduce
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
which further leads to the following equivalent optimization for non-strict phase rotation
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 non-strict phase rotation can be similarly obtained in a closed form as a function of , given by
where is a transformation matrix that transform the real-valued 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 non-strict phase rotation. We observe that the objective function of for strict phase rotation and for non-strict 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 non-strict 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 non-strict 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
where is the scaling factor to meet the transmit power constraint. For fairness of comparison, we employ a symbol-level normalization for such that as for the considered CI beamforming, which leads to the expression of as
By denoting , the expression of can be further transformed into
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
where the vector consists of each non-negative dual variable of . Based on (55), we express the KKT conditions of as