Constructive Multiuser Interference in Symbol Level Precoding for the MISO Downlink Channel
Abstract
This paper investigates the problem of interference among the simultaneous
multiuser
transmissions in the downlink of multiple
antennas systems. Using symbol level precoding, a new approach towards the multiuser interference is discussed along
this paper. The concept of exploiting the interference between
the spatial multiuser transmissions by jointly utilizing the data information
(DI)
and channel state information (CSI), in order to design symbollevel precoders,
is proposed. In this direction, the interference among the data streams
is transformed under certain conditions to useful signal that can improve the signal to interference
noise ratio (SINR) of the downlink transmissions. We propose a maximum ratio
transmission (MRT) based algorithm that jointly exploits DI and CSI to glean the benefits from constructive multiuser interference. Subsequently, a relation between the constructive interference downlink transmission and physical
layer multicasting is established. In this context, novel constructive interference precoding techniques that tackle the transmit power minimization (min power) with
individual SINR constraints at each user’s receivers is proposed. Furthermore, fairness through maximizing the weighted minimum SINR (max min SINR) of the users is addressed by finding the link between the min power and max min SINR problems. Moreover, heuristic precoding techniques are proposed
to tackle the weighted sum rate problem.
Finally, extensive numerical results
show that the proposed schemes outperform other state of the art techniques.
I Introduction
Interference is one of the crucial and limiting factors in wireless networks. The idea of utilizing the time and frequency resources has been proposed in the literature to allow different users to share the resouces without inducing harmful interference. The concept of exploiting the users’ spatial separation has been a fertile research domain for more than one decade. This can be implemented by adding multiple antennas at one or both communication sides. Multiantenna transceivers empower the communication systems with more degrees of freedom that can boost the performance if the multiuser interference is mitigated properly. Exploiting the space dimension, to serve different users simultaneously in the same time slot and the same frequency band through spatial division multiplexing (SDMA), has been investigated in [1][8].
The applications of SDMA, in which a single multiple antennas transmitter wants to communicate with multiple receivers, vary according to the requested service. The first service type is known as a broadcast in which a transmitter has a common message to be sent to multiple receivers. In physical layer research, this service has been studied under the term of physical layer multicasting (i.e. PHY multicasting) [14][15]. Since a single data stream is sent to all receivers, there is no multiuser interference. In the remainder of this paper, this case will be referred to as multicast transmission. The second service type is known as unicast, in which a transmitter has an individual message for each receiver. Due to the nature of the wireless medium and the use of multiple antennas, multiple simultaneous unicast transmissions are possible in the downlink of a base station (BS). In these cases, multiple streams are simultaneously sent, which motivates precoding techniques that mitigate the multiuser interference. In information theory terms, this service type has been studied using the broadcast channel [8]. In the remainder of this paper, this case will be referred to as downlink transmission.
In the literature, the precoding techniques for downlink tranmission can be further classified as:

Userlevel precoding in which multiple codewords are transmitted simultaneously but each codeword is addressed to a single user. This case is also known as multiantenna broadcast channel precoding [6][13] and the precoder design is dependant on the channels of the individual users. This is a special case of group level precoding where each group consists of a single user.
In the last category, the main idea is to constructively correlate the interference among the spatial streams rather than fully decorrelate them as in the conventional schemes [9]. In [22], the interference in the scenario of BPSK and QPSK is classified into types: constructive and desctructive. Based on this classification, a selective channel inversion scheme is proposed to eliminate the destructive interference while it keeps the constructive one to be received at the users’ terminal. A more advanced scheme is proposed in [23], which rotates the destructive interference to be received as useful signal with the constructive one. These schemes outperform the conventional precodings [9] and show considerable gains. However, the anticipated gains come at the expense of additional complexity at the system design level. Assuming that the channel coherence time is , and the symbol period is , with for slow fading channels, the user precoder has to be recalculated with a frequency of in comparison with the symbol based precoder . Therefore, faster precoder calculation and switching is needed in the symbollevel precoding which can be translated to more expensive hardware. The contributions of this paper can be summarized in the following points:

A generalized characterization of the constructive interference for any MPSK is described. Based on this characterization, we propose a new constructive interference precoding scheme, called constructive interference maximum ratio transmissions (CIMRT). This technique exploits the weakness points of constructive interference zero forcing precoding (CIZF) in [23].

We find the relation between the constructive interference precoding problem and PHY layer multicasting and verify it for any MPSK modulation scenario.

We propose different symbol based precoding schemes that aim at optimizing different performance metrics such as minimizing the transmit power while acheiving certain SNR targets, maximizing the minimum SNR among the user while keeping the power constraint in the system satisfied and finally maximizing the sum rate of all users without exceeding the permissible amount of power in the system.
The rest of the paper is organized as follows: the channel and the system model is explained in section (II), while section (III) discusses how the conventional downlink precoding techniques tackle interference. Symbol level precoding is described in (IV). Moreover, techniques that exploit the multiuser interference in symbolbased precoding are described (V). The relation to PHYlayer multicasting and the solution to the power min problem are investigated in (VI). The problem of maximizing the minimum SINR is tackled in section (VII). Heuristic sum rate maximization techniques are discussed (VIII). Finally, the performance of the proposed algorithms is evaluated in section (X).
Notation: We use boldface upper and lower case letters for matrices and column vectors, respectively. , stand for Hermitian transpose and conjugate of . and denote the statistical expectation and the Euclidean norm, is used to indicate the positive semidefinite matrix. , are the angle and magnitude of respectively. , are the real and the imaginary part of . Finally, the vector of all zeros with length of is defined as .
Ii System and Signal Models
We consider a singlecell multipleantenna downlink scenario, where a single BS is equipped with transmit antennas that serves user terminals, each one of them equipped with a single receiving antenna. The adopted modulation technique is MPSK. We assume a quasi static block fading channel between the BS antennas and the user, where the received signal at j user is written as
(1) 
is the transmitted signal vector from the multiple antennas transmitter and denotes the noise at receiver, which is assumed i.d.d complex Gaussian distributed variable . A compact formulation of the received signal at all users’ receivers can be written as
(2) 
Let be written as , where is the unit power precoding vector for the user . The received signal at user in symbol period is given by
(3) 
where is the allocated power to the user. A more detailed compact system formulation is obtained by stacking the received signals and the noise components for the set of K selected users as
(4) 
with , as the compact channel and precoding matrices. Notice that the transmitted signal includes the uncorrelated data symbols for all users with , is the power allocation matrix . It should be noted that CSI and DI are available at the transmitter side.
Iii Conventional Multiuser Precoding Techniques
The main goal of transmit beamforming is to increase the signal power at the intended user and mitigate the interference to nonintended users. This can be obtained by precoding the transmitted symbols in a way that optimizes the spatial directions of the simultaneous transmissions by means of beamforming. This can be mathematically translated to a design problem that targets beamforming vectors to have maximal inner products with the intended channels and minimal inner products with the nonintended user channels. There are several proposed beamforming techniques in the literature. One of the simplest approaches is to encode the transmitted signal by premultiplying it with the pseudo inverse of the multiuser matrix channel. Several approaches have been proposed including minimizing the sum power while satisfying a set of SINR constraints[6] and maximizing the jointly achievable SINR margin under a power constraint[7]. In any scenario, the generic received signal can be formulated as
(5)  
The corresponding SINR can be expressed as
(6) 
This paper tries to go beyond this conventional look at the interference by employing symbollevel precoding. This approach can under certain conditions convert the inner product with the nonintended channels into useful power by maximizing them but with the specific directions to which constructively addup at each user receivers. Taking into account the I/Q plane of the symbol detection, the constructive interference is achieved by using the interfering signal vector to move the received point deeper into the correct detection region. Considering that each user receives a constructive interference from other users’ streams, the received signal can be written as
(7) 
This yields the SINR expression for MPSK symbols as
(8) 
Different precoding techniques that redesign the terms to constructively correlate them with are proposed in the next sections (V)(VIII).
Iiia Power constraints for user based and symbol based precodings
In the conventional user based precoding, the transmitter needs to precode every which means that the power constraint has to be satisfied along the coherence time . Taking the expectation of , and since is fixed along , the previous expression can be reformulated as , where due to uncorrelated symbols over .
However, in symbol level precoding the power constraint should be guaranteed for each symbol vector transmission namely for each . In this case the power constraint equals to . In the next sections, we characterize the constructive interference and show how to exploit it in the multiuser downlink transmissions^{1}^{1}1From now on, we assume that the transmssion changes at each symbol and we drop the time index for the ease of notation .
Iv Constructive Interference
The interference is a random deviation which can move the desired constellation point in any direction. To address this problem, the power of the interference has been used in the past to regulate its effect on the desired signal point. The interference among the multiuser spatial streams leads to deviation of the received symbols outside of their detection region. However, in symbol level precoding (e.g. MPSK) this interference pushes the received symbols further into the correct detection region and, as a consequence it enhances the system performance. Therefore, the interference can be classified into constructive or destructive based on whether it facilitates or deteriorates the correct detection of the received symbol. For BPSK and QPSK scenarios, a detailed classification of interference is discussed thoroughly in [22]. In this section, we describe the required conditions to have constructive interference for any MPSK modulation.
Iva Constructive Interference Definition
Assuming both DI and CSI are available at the transmitter, the unitpower created interference from the data stream on user can be formulated as:
(9) 
Since the adopted modulations are MPSK ones, a definition for constructive interference can be stated as
Lemma 1.
For any MPSK modulated symbol , it is said to receive constructive interference from another simultaneously transmitted symbol which is associated with if and only if the following inequalities hold
Proof.
For any MPSK modulated symbol, the region of correct detection lies in , where is the angle of the detected symbols. In order for the interference to be constructive, the received interfering signal should lie in the region of the target symbol. For the first condition, the function checks whether the received interfering signal originating from the transmit symbol is located in the detection region of the target symbol. However, the trigonometric functions are not onetoone functions. This means that it manages to check the two quadrants which the interfering symbol may lie in. To find which one of these quadrants is the correct one, an additional constraint is added to check the sign compatibility of the target and received interfering signals. ∎
Corollary 1.
The constructive interference is mutual.
If the symbol constructively interferes with , then
the interference from transmitting the symbol
is constructive to .
For constructively interfering symbols, the value of the received signal can be bounded as
(10) 
The inequality (a) holds when all simultaneous users are orthogonal (i.e. ), while (b) holds when all created interference is aligned with the transmitted symbol as and , . Eq. (10) indicates that in the case of constructive interference, having fully correlated signals is beneficial as they contribute to received signal power. In conventional precoding techniques, the previous inequality can be reformulated as
(11) 
The worst case scenario can occur when all users are colinear . The channel cannot be inverted and thus the interference cannot be mitigated. The optimal scenario takes place when all users have physically orthogonal channels which entails no multiuser interference. Therefore, utilizing the CSI and DI leads to higher performance in comparison with conventional techniques.
V Constructive Interference Precoding for MISO Downlink Channels
In the remainder of this paper, it is assumed that the transmitter is capable of designing precoding on symbol level utilizing both CSI and DI^{2}^{2}2From this section, we combine the the precoding design with power allocation.
Va Correlation Rotation Zero Forcing Precoding (CIZF)
The precoder aims at minimizing the mean square error while it takes into the account the rotated constructive interference [23]. The optimization problem can be formulated as
where in this scenario. The solution can be easily expressed as
(12) 
where ensures the power normalization. The cross correlation factor between the user’s channel and transmitted data stream can be expressed as
(13) 
The relative phase that grants the constructive simultaneous transmissions can be expressed as
(14) 
The corresponding rotation matrix can be implemented as:
(15) 
and the received signal at user can be expressed as
(16) 
where has the same magnitude as but with different phase, and . By taking a look at (16b), it has a multicast formulation since it seems for each user that BS sends the same symbol for all users by applying a userdependent rotation..
Remark 1.
It can be noted that this solution includes a zero forcing step and a correlation step . The correlation step aims at making the transmit signals constructively received at each user. Unfortunately, this design fails when we deal with colinear users . However, intuitively having colinear users should create more constructive interference and higher gain should be anticipated. It can be easily concluded that the source of this contradiction is the zero forcing step. In an effort to overcome the problem, we propose a new precoding technique in the next section.
VB Proposed Constructive Interference Maximum Ratio Transmission (CIMRT)
The maximum ratio transmission (MRT) is not suitable for multiuser downlink transmissions in MISO system due to the intolerable amount of the created interference. On the other hand, this feature makes it a good candidate for constructive interference. The naive maximum ratio transmission (nMRT) can be formulated as
(17) 
A new look at the received signal can be viewed by exploiting the singular value decomposition of , and as follows
(18)  
where
is a unitary matrix that contains the leftsingular vectors of , the matrix is an diagonal matrix with nonnegative real numbers on the diagonal, and contains rightsingular vectors of . is the power scaled of to normalize each column in to unit. The received signal can be as
(19) 
Utilizing the reformulation of in (18), the received signal can be written as
(20) 
where is the row of the matrix , . Since is a unitary matrix, it can have uncoupled rotations which can grant the constructivity of interference. Let be the rotation matrix in the plane, which performs an orthogonal rotation of the and columns of a unitary matrix while keeping the others fixed, thus preserving unitarity. Assume without loss of generality that . Givens rotation matrix in the plane can be defined as
where the non trivial entries appear at the intersections of and rows and columns. Hence, any unitary matrix can be expressed using the following parameterization
(21) 
It can be seen from the structure of the matrix in (VB) that rotation in the (,)plane does not change the directions of the remaining beamforming vectors. Therefore, it just modifies the value of , and the precoder reads as
(22) 
To grant constructive interference, we need to rotate the (,)plane by formulating the rotation as a set of nonlinear equations as
(23) 
Remark 2.
The rotation of plane is independent and decoupled from any other
plane. This means that any implemented rotation on this plane only affects the pair.
Since the set of nonlinear equations can have different roots, the function needs to be evaluated at the obtained root in order to find the optimal ones. The optimal solution can be found when solving for , . Sometimes it is not feasible to solve for and , and their values need to be reduced correspondingly. The proposed algorithm can be illustrated in the following table
A1: Constructive Interference Rotation for CIMRT Algorithm 

Vi Constructive Interference for Power Minimization
Via Constructive Interference Power Minimization Precoding (CIPM)
From the definition of constructive interference, we should design the constructive interference precoders by granting that the sum of the precoders and data symbols in the expression forces the received signal to the detection region of the desired symbol for each user. Therefore, the optimization that minimizes the transmit power and grants the constructive reception of the transmitted data symbols can be written as
where is the SNR target for the user that should be granted by the transmitter, and is the vector that contains all the SNR targets. The set of constraints guarantees that each user receives its corresponding data symbol . A reformulation for the previous problem (VIA) using can be expressed as
The replaced variables ’s indicate that it is not necessary to send the exact symbols ; they can be included in precoding design as long as they are received correctly at users’ terminals. Then, we design the final output vector instead of designing the whole with the assumption that is fixed.This means that the proposed methods move away from the classic approach of linear beamforming, where the precoding matrix is multiplied with the symbol vector. Instead, we adopt an approach where the transmit signal vector is designed directly based on an optimization problem.
Lemma 2.
Assuming a conventional linear precoder , the transmitted signal vector which minimizes the transmit power can be calculated using a unitrank precoding matrix .
Proof.
This can be proved by using the auxiliary variable and substituting it in the optimization problem (VIA). The optimization can be rewritten as
is a vector, which makes the solution a unit rank as , and the virtual input vector . ∎
Based on Lemma 2, the differentiation between the conventional and constructive interference precoding techniques is illustrated in Fig. (1)(2). Fig. (1) shows how the conventional precoding depends only on the CSI information to optimize that carry the data symbols and without any design dependency between them. Therefore, the transmitted output vector can be formulated as . The final output vector only depends on the DI and CSI and this dependence is a linear one. On the other hand, in constructive interference precoding schemes, the precoding directly depends on both the CSI and DI information to exploit the interference through skipping the intermediate step (i.e. optimizing ) and optimizing directly the vector . In constructive interference schemes (2), the relation between the data symbols in and the final output vector cannot be explicitly described as in linear conventional precoding scheme. This can be explained by the fact that the DI is used to design the output vector but is not necessarily physically transmitted as in conventional linear precoding. An implicit set of virtual data is used instead which is explained later in this paper.
ViB The Relation Between Constructive Interference Precoding and Constrained Constellation Multicast
By taking a look at Lemma 2, the solution of the optimization problem resembles
the solution of multicast problem in which the transmitter sends a single message to
multiple users[14][17]. However in our problem,
we have an additional constraint in (VIA)(2) which guarantees that each user detects correctly its symbol based on the received signal.
Theorem 1.
Proof.
We assume that we have the following equivalent channel as
(28) 
The power minimization can be rewritten by replacing by its equivalent channel in (26) as
(29)  
where is the row of the . Rewriting the first constraints in (29) as
(30)  
shows the equivalence between the constrained constellation multicast channel and constructive interference downlink channel. ∎
By taking a look at (2), the objective function is unit rank and thereby it is a convex. The convexity holds for , however, the phase constraints in are not convex. Therefore a formulation for is required. We can reformulate the constraint as
The minimum transmit power in (VIA)(2) occurs when the inequality constraints are replaced by equality (i.e. all users should achieve their target threshold SNR). A final formulation can be expressed as
It can be viewed that the constraints in (VIB) are turned from inequality constraints to equality constraint (VIB) due to signal aligning requirements. The Lagrangian function can be derived as follows
where and are the Lagrangian dual variables. The derivative for the Lagrangian function can be written as
(34) 
By equating this term to zero, can be written as
(35)  
where . The optimal values of the Lagrangian variables and can be found by substituting in the constraints (VIB) which result in solving the set of equations (36). The final constrained constellation multicast precoder can be found by substituting all and in (35).
(36) 
Corollary 2.
The CI precoding for power minimization as well as constrained constellation multicast precoding must span the subspaces of each user’s channel.
It can be noted from the formulation of that BS should use the same precoder for all users. This result resembles the multicast approach in which the BS wants to deliver the same message to all users[14][15]. However in multicast systems, a different symbol should be detected correctly at each user.
Using (35), we can rewrite the received signal at receiver as
From (VIB), the constellation constrained multicast can be formulated as a constructive interference downlink channel with set of precoders , where each one of these precoders is allocated with power and associated with the symbol .
Corollary 3.
The solution of problem with uniformly scaled SINR constraints is given simply by scaling the output vector of the original problem as follows:
where .
Proof.
We define the normalized precoder equals to . For any , . Therefore, all users can receive their target data symbols scaled to a certain SNR value. This implies that scaling uniformly all users’ SNR targets does change . Using the simultaneous set of equations (36), we can replace each by . This multiplies each value of , by . As a consequence, a scaling factor of is multiplied with the original output vector which proves the corollary. ∎
ViC Constructive Interference Power minimization bounds
In order to assess the performance of the proposed algorithm, we mention two theoretical upper bound as follows
ViC1 Genie aided upper bound
This bound occurs when all multiuser transmissions are constructively interfering by nature and without the need to optimize the output vector. The minimum transmit power for a system that exploits the constructive interference on symbol basis can be found by the following theoretical bound
Theorem 2.
The genieaided minimum transmit power in the downlink of multiuser MISO system can be found by solving the following optimization
Proof.
ViC2 Optimal Multicast
Based on theorem (2), a theoretical upperbound can be characterized. This bound occurs if we drop the phase alignment constraint . The intuition of using this technique is the complete correlation among the information that needs to be communicated (i.e. same symbol for all users). The optimal input covariance for power minimization in multicast system can be found as a solution of the following optimization
(38) 
This problem is thoroughly solved in [14]. A tighter upperbound can be found by imposing a unit rank constraint on [15], to allow the comparison with the unit rank transmit power minimization constructive interference precoding
(39) 
Eq. (39) presents a tighter upper bound in comparison (38). It assumes a unit rank approximation of (38).
Vii Weighted Max Min SINR Algorithm for Constructive Interference Precoding (CIMM)
The weighted maxmin SINR beamforming aims at improving the relative fairness in the system by maximizing the worst user SINR. This problem has been studied in different frameworks such as multicast [14], and downlink transmissions[10]. In [14], the authors have solved this problem by finding the relation between the minpwr problem and maxmin problem and formulating both problem as convex optimization ones. On the other hand, the authors of [10] have solved the problem using the bisection technique. In this work, we exploit the constructive interference to enhance the user fairness in terms of weighted SNR. The challenging aspect is the additional constraints which guarantee the data have been detected correctly at the receivers. The constructive interference maxmin problem can be formulated as
where denotes the requested SNR target for the user. If we denote , the previous optimization can be formulated as
where is the vector that contains all the weights . In the following, it is shown that the optimal output vector is a scaled version of the minpwr solution in (VIA)[14]. The weighted maximum minimum SINR problem has been solved using bisection method over [10].
Viia Maxmin SINR and minpwr relation
Lemma 3.
The relationship between min power and maxmin problem can be described as .
Proof.
The problem (VII) can be formulated
The optimal value of denoted by can be found by solving the minpwr. ∎
Thus, the maxmin SINR solution is a scaled version of min power solution, which means that the system designer needs to find the optimal value of to solve the maxmin problem. In the next section, we propose a simple method that can find this parameter influenced by the literature [10].
ViiB Maxmin SINR Constructive Interference Precoding
In comparison with (VII), we have additional constraints that limit the system performance. The problem can be formulated as
A solution for (VIIB) can be found in the same fashion by using
the bisection method as [10] and can be summarized as
A2: Bisection for maxmin SINR for CI precoding (CIMM) 

Repeat 
set 
set 
if 
then 
else 
Until 
Return 
Viii Weighted Sum Rate Maximization Algorithms for constructive interference Precoding (CISR)
The sum rate problem of the multiuser downlink of multiple antennas for userlevel precoding has been investigated in the literature [3][4]. The authors in [3] prove that the sum rate problem is NP hard. However, a simpler solution for the sum rate problem is characterized in [4] by rotating the MRTs of each user’s channel to reduce the amount of the created interference on other users’ transmissions. On the other hand, the weighted sum rate optimization in single group multicast scenarios is studied [16], which tries to design closed form precoders at different high SNR scenarios and proposes an iterative algorithm with low computation complexity for general SNR case. Furthermore, heuristic solutions for sum rate maximization of group multicast precoding with perantenna power constraint are proposed in [19].
In this work, we take into the account that the interference can be exploited among the different multiuser data streams. This requires that the sum rate problem should be formulated to take into consideration this new feature. The weighted sum rate maximization with a unit rank assumption for the precoding matrix can be written as^{3}^{3}3For the sum rate problrm, it should be noted that the optimal solution is not necessarily unit rank, but we employ this assumption to enable tractable heuristic solutions
(44)  