# Interference Exploitation Precoding for Multi-Level Modulations: Closed-Form Solutions

In this paper, we study closed-form interference-exploitation precoding for multi-level modulations in the downlink of multi-user multiple-input single-output (MU-MISO) systems. We consider two distinct cases: first, for the case where the number of served users is not larger than the number of transmit antennas at the base station (BS), we mathematically derive the optimal precoding structure based on the Karush-Kuhn-Tucker (KKT) conditions. By formulating the dual problem, the precoding problem for multi-level modulations is transformed into a pre-scaling operation using quadratic programming (QP) optimization. We further consider the case where the number of served users is larger than the number of transmit antennas at the BS. By employing the pseudo inverse, we show that the optimal solution of the pre-scaling vector is equivalent to a linear combination of the right singular vectors corresponding to zero singular values, and derive the equivalent QP formulation. We also present the condition under which multiplexing more streams than the number of transmit antennas is achievable. For both considered scenarios, we propose a modified iterative algorithm to obtain the optimal precoding matrix, as well as a sub-optimal closed-form precoder. Numerical results validate our derivations on the optimal precoding structures for multi-level modulations, and demonstrate the superiority of interference-exploitation precoding for both scenarios.

## Authors

• 58 publications
• 12 publications
• 68 publications
• 54 publications
• 16 publications
• ### Interference Exploitation Precoding Made Practical: Closed-Form Solutions with Optimal Performance

In this paper, we propose closed-form precoding schemes with optimal per...
12/21/2017 ∙ by Ang Li, et al. ∙ 0

• ### Power Minimizer Symbol-Level Precoding: A Closed-Form Sub-Optimal Solution

In this letter, we study the optimal solution of the multiuser symbol-le...
07/27/2018 ∙ by A. Haqiqatnejad, et al. ∙ 0

• ### Symbol-Level Precoding Made Practical for Multi-Level Modulations via Block-Level Rescaling

In this letter, we propose an interference exploitation symbol-level pre...
06/27/2020 ∙ by Ang Li, et al. ∙ 0

• ### On the Throughput of Large-but-Finite MIMO Networks using Schedulers

This paper studies the sum throughput of the multi-user multiple-input-s...
11/13/2018 ∙ by Behrooz Makki, et al. ∙ 0

• ### Asymptotic Analysis of RZF over Double Scattering Channels with MMSE Estimation

This paper studies the ergodic rate performance of regularized zero-forc...
04/27/2019 ∙ by Qurrat-Ul-Ain Nadeem, et al. ∙ 0

• ### Construction of One-Bit Transmit-Signal Vectors for Downlink MU-MISO Systems with PSK Signaling

We study a downlink multi-user multiple-input single-output (MU-MISO) sy...
01/23/2019 ∙ by Gyu-Jeong Park, et al. ∙ 0

• ### The Capacity and Optimal Signaling for Gaussian MIMO Channels Under Interference Constraints (full version)

Gaussian MIMO channel under total transmit and interference power constr...
02/18/2020 ∙ by Sergey Loyka, et al. ∙ 0

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## I Introduction

PRECODING has been widely studied in multi-antenna wireless communication systems to simultaneously support data transmission to multiple users [1]. When the channel state information (CSI) is known at the transmitter side, dirty paper coding (DPC) that subtracts the interference prior to transmission achieves the channel capacity [2]. Despite its promising performance, DPC is generally difficult to implement in practical wireless systems, due to its impractical assumption of an infinite source alphabet and prohibitive complexity. Therefore, sub-optimal approximations of DPC in the form of Tomlinson-Harashima precoding (THP) and vector perturbation (VP) precoding have been proposed in [3] and [4], respectively. While offering near-optimal performance, both THP and VP are still non-linear precoding methods and include a sphere-search process, which makes their complexity still unfavorable, especially when the number of data streams is large. Accordingly, low-complexity linear precoding methods such as zero-forcing (ZF) [5] and regularized ZF (RZF) [6] have become popular. On the other hand, downlink precoding based on optimization has also received increasing research attention [7]-[13]. Among optimization-based precoding methods, the two most well-known designs are referred to as signal-to-noise-plus-interference ratio (SINR) balancing [7]-[9] and power minimization [10]-[12], where SINR balancing aims to maximize the minimum received SINR subject to a total transmit power constraint [7], [8] or a per-antenna power constraint [9], and power minimization targets minimizing the power consumption at the transmitter side while guaranteeing a minimum SINR at each receiver [11].

For both the closed-form precoding schemes and the optimization-based precoding approaches described above, the CSI at the base station (BS) is exploited to design the precoding strategy that eliminates, avoids or limits interference. The above approaches ignore the fact that the information in the transmitted data symbols themselves can also be exploited in the downlink precoding design on a symbol-by-symbol basis for further performance improvements. With information about the data symbols and their corresponding constellations, the instantaneous interference can be divided into constructive interference (CI) and destructive interference [14]. More specifically, CI is defined as interference that pushes the received signals away from the detection thresholds [15], [16], which provides further benefits for signal detection. A modified ZF precoding method was proposed in [17] to exploit the constructive part of the interference while eliminating the destructive part. A more advanced two-stage interference exploitation precoding was proposed in [18], where the phase of the destructive interference was controlled and further rotated such that the destructive interference becomes constructive. Optimization-based interference-exploitation precoding for PSK modulations has also been proposed in [19] in the context of vector perturbation precoding, where CI in the form of symbol scaling is proposed. In [20]-[22], CI precoding based on the phase-rotation metric is studied, where it is shown that a relaxed non-strict phase rotation metric is more advantageous compared to the strict phase rotation in [17], [18]. For multi-level modulations such as QAM, CI can be exploited for the outer constellation points, although all the interference for the inner constellation points is considered to be destructive, as discussed in [23]-[25] where a symbol-scaling metric is introduced. Due to the above benefits, CI has been extended to the area of low-resolution digital-to-analog converters (DACs) with PSK signaling in [26], as well as quantized constant envelope precoding with PSK and QAM signaling in [27]. More recently, it has been revealed in [28] that there exists an optimal structure for the CI precoding for PSK modulations. Nevertheless, it is still unclear whether a similar result exists for multi-level modulations such as QAM, since CI precoding for PSK modulations is based on the phase-rotation metric, while the symbol-scaling metric has to be employed for QAM constellations.

In this paper, we study closed-form interference exploitation precoding for multi-level modulations, where QAM modulation is considered as a representative example. Due to the fact that the conventional phase-rotation CI formulation is not applicable to QAM constellations, the more general symbol-scaling metric is employed. We reveal the geometric connection between the phase-rotation and symbol-scaling metrics in the CI formulation, based on which we propose the optimization problem that maximizes the CI effect for the outer constellation symbols while constraining the inner constellation symbols for multi-level modulations. We first study the case where the number of users simultaneously served by the BS is not larger than the number of BS transmit antennas. Using the Lagrangian and KKT conditions, we analyze the formulated problem and mathematically derive the structure of the optimal precoding matrix, which leads to an equivalent simplified optimization problem. By further formulating the dual problem of this equivalent optimization, we show that, similar to the case of PSK modulations, interference-exploitation precoding for multi-level modulations is equivalent to a quadratic programming (QP) optimization, and the optimal precoding matrix can be expressed as a function of the dual variables in closed form.

We further extend our analysis to the case where the number of users simultaneously served by the BS is larger than the number of BS transmit antennas, in which case conventional precoding becomes infeasible and the exact inverse included in the above analysis becomes inapplicable. In this scenario, we show that interference-exploitation precoding may still be feasible. To this end, the more generic pseudo inverse of the channel matrix is employed instead, and we derive the optimal structure of the precoding matrix. Due to the inclusion of the pseudo inverse, an additional constraint is further introduced in the equivalent optimization. Built upon this, the scaling vector for the constellation symbols is shown to be the non-zero solution of a linear equation set, which is equivalent to a linear combination of the singular vectors corresponding to the zero singular values of the coefficient matrix. Accordingly, the optimization can be transformed into an optimization on the weights for each singular vector, which is further shown to be equivalent to a QP optimization as well. Based on the equivalent QP formulation, we discuss the condition under which multiplexing more streams than the number of transmit antennas is possible with interference exploitation precoding.

For both of the scenarios considered above, we also present a generic iterative algorithm to efficiently obtain the optimal precoding matrix for multi-level modulations, where a closed-form update is included in each iteration. Based on the above transformation and algorithm, we further develop a sub-optimal closed-form non-iterative CI precoder. Our analysis for multi-level modulations in this paper complements the study on closed-form symbol-level interference-exploitation precoding in [28], which is not applicable to multi-level modulations. Simulation results validate our mathematical derivations and the optimality of the proposed algorithm. Moreover, the superiority of interference-exploitation precoding over conventional precoding methods for multi-level modulations is also revealed, especially for the case where the BS simultaneously serves a larger number of users than it has the number of transmit antennas.

We summarize the contributions of this paper below:

1. We present a geometric connection between symbol-scaling and phase-rotation metrics for interference-exploitation precoding, based on which we construct the optimization that maximizes the CI effect of the outer constellation symbols while maintaining the performance of the inner constellation symbols for multi-level modulations.

2. We perform mathematical analysis on interference-exploitation precoding for multi-level modulations. We show that CI precoding for multi-level modulations can ultimately be simplified into a QP optimization as well. Compared to CI precoding for PSK modulations where the optimization is over a simplex, it is shown that only part of the dual variables need to be constrained as non-negative in the QP formulation for multi-level modulations.

3. We further extend our analysis on CI to the case where the number of served users is larger than the number of transmit antennas at the BS. Our transformations show that the optimization for CI precoding in such scenarios is similar to the conventional case where the number of users is smaller than or equal to the number of antennas at the BS, also resulting in a QP optimization. We also present the condition under which multiplexing more streams than the number of transmit antennas based on CI is achievable.

4. We propose an iterative algorithm that is able to obtain the optimal solution of a generic QP optimization problem subject to specific constraints within only a few iterations. Based on this algorithm, the optimal precoding matrix can be efficiently obtained, for both scenarios considered in this paper. A sub-optimal closed-form non-iterative precoder is also presented.

The remainder of this paper is organized as follows: Section II introduces the system model and illustrates the connection between the two CI metrics. Section III includes the CI-based optimization problems for multi-level modulations when the number of users is smaller than or equal to the number of BS transmit antennas, and the extension to the scenario when the number of users is larger than the number of BS transmit antennas is studied in Section IV. The modified iterative algorithm and sub-optimal closed-form precoder are presented in Section V. Numerical results are provided in Section VI, and Section VII concludes the paper.

Notation: , , and denote scalar, column vector and matrix, respectively. , , , , and denote conjugate, transposition, conjugate transposition, inverse, pseudo inverse, and rank of a matrix, respectively. is the transformation of a column vector into a diagonal matrix, 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 -norm. and represent the sets of complex- and real-valued matrices, respectively. and respectively denote the real and imaginary part of a complex scalar, vector or matrix. denotes the cardinality of a set, and represents the Kronecker product. denotes the imaginary unit, denotes the identity matrix, and represents the -th column of the identity matrix.

## Ii System Model and Constructive Interference

### Ii-a System Model

We study a downlink MU-MISO system, where the BS with transmit antennas is simultaneously communicating with single-antenna users in the same time-frequency resource. We separately consider the scenarios of both and . We focus on the downlink precoding designs, and perfect CSI is assumed throughout the paper. The data symbol vector is assumed to be from a normalized multi-level modulation constellation [20], denoted as , and the received signal at the -th user can then be expressed as

 rk=hTkWs+nk, (1)

where denotes the flat-fading Rayleigh channel vector from user

to the BS with each entry following a standard complex Gaussian distribution,

is the precoding matrix, and

is the additive Gaussian noise at the receiver with zero mean and variance

.

### Ii-B Connection between Two CI Metrics for PSK Modulation

In this section, we illustrate the connection between symbol-scaling and phase-rotation metrics for CI precoding based on Fig. 1, where we employ QPSK (4QAM) as an example.

Phase Rotation Metric: As discussed in [28], we denote and , where is the objective to be optimized. We further denote as the received signal for user excluding noise, which leads to

 →OB=hTkWs=λksk, (2)

where is a complex scalar that represents the effect of interference on the data symbol for user . For -PSK constellations, the CI constraint is then constructed as [28]

 (λRk−t)tanθt≥∣∣λIk∣∣, (3)

where , , and for -PSK constellations. Accordingly, the optimization problem that maxmizes the distance of the constructive region to the detection thresholds subject to the total available transmit power based on the phase-rotation CI metric can be formulated as [28]

 P1:maxW,tt (4) s.t.hTkWs=λksk,∀k∈K (λRk−t)tanθt≥∣∣λIk∣∣,∀k∈K ∥Ws∥22≤p0

where . We have enforced a symbol-level power constraint on the precoder, since the exploitation of CI is dependent on the data symbol , which will also be shown mathematically in the following.

Symbol Scaling Metric: Following the coordinate transformation approach in [26], we first decompose the data symbol along the detection thresholds for each user , expressed as

 →OS=→OF+→OG⇒sk=sAk+sBk, (5)

where and are the bases that are parallel to the detection thresholds for each specific constellation symbol, as shown in Fig. 1. We refer the interested readers to [26] for a detailed derivation of the expressions for and for generic PSK constellations. Specifically for QPSK modulation considered in Fig. 1 as well as QAM modulations in the following part of the paper, we can obtain

 sAk=R{sk}=sRk,sBk=j⋅I{sk}=j⋅sIk. (6)

Following a similar approach to (5), we also decompose the noiseless received signal for each user along the same detection thresholds, and further introduce two real scalars and for and , respectively, which leads to

 →OB=→OD+→OE⇒hTkWs=αAksAk+αBksBk. (7)

It is then observed that the values of these two scalars directly indicate the effect of the CI. Subsequently, the corresponding optimization based on the symbol-scaling metric can be constructed as

 (8) s.t.hTkWs=αAksAk+αBksBk,∀k∈K ∥Ws∥22≤p0 U∈{A,B}

Both of the above optimization problems are convex and can be directly solved with convex optimization tools. Subsequently, based on Fig. 1 and the formulation of the above two optimizations, an important geometrical observation is given, which demonstrates the connection between the symbol-scaling and phase-rotation metric.

Observation 1: Since the noiseless received signal is located on the boundary of its constructive region, the relationship between the minimum value of in and the optimal value of in is expressed as

 (9)

where without loss of generality we have assumed user has the minimum value of . Eq. (9) is derived by considering the isosceles triangle ‘DOA’, where we can obtain

 |→OA|=2|→DO|cos∠DOA. (10)

Based on the fact that , , and , (10) leads to the expression for in (9).

It’s worth noting that while the above discussion only focuses on QPSK constellations, (9) is in fact generic to any -PSK modulation for the connection between the two CI metrics, and the only difference lies in the expression for . In the following section, the symbol-scaling CI metric is employed in the derivation of the optimal precoding matrix for multi-level modulations.

## Iii CI Precoding for the Case of K≤Nt

In this section, we focus on the common case where , and we consider 16QAM modulation as an example of multi-level modulations. For other multi-level constellations, the problem formulation and the corresponding analysis for the symbol-scaling metric readily follows our derivations in this section in a similar way.

For a generic QAM constellation, we employ the symbol-scaling metric for CI precoding since there does not exist a generic expression for the phase-rotation CI metric for QAM modulations, as shown in Fig. 2 where a 16QAM constellation is depicted as the example. The symbol-scaling metric in (7) can be further expressed in vector form as

 hTkWs=ΩTksk, (11)

where we have introduced two column vectors

 Ωk=[αAk,αBk]T,sk=[sAk,sBk]T. (12)

For QAM constellations, and are also given by (6). In this work, we consider the interference on the inner constellation points as only destructive, since the interference is less likely to be beneficial for these points. To be more specific, in Fig.2 CI exists for the real part of the constellation point type ‘B’ and imaginary part of type ‘C’, while both the real and imaginary part of the constellation point type ‘D’ can be exploited. Accordingly, we propose to construct the optimization problem that maximizes the CI effect for the outer constellation points while maintaining the performance for the inner constellation points, given by

 P3:maxW,tt (13) s.t.hTkWs=ΩTksk,∀k∈K t≤αOm,∀αOm∈O t=αIn,∀αIn∈I ∥Ws∥22≤p0

where the set consists of the real scalars corresponding to the real or imaginary part of the outer constellation points that can be scaled, and consists of the real scalars corresponding to the real or imaginary part of the constellation points that cannot exploit CI. Accordingly, we obtain

 O∪I={αA1,αB1,αA2,αB2,⋯,αAK,αBK}, (14)

and

 card{O}+card{I}=2K. (15)

is a second-order-cone programming (SOCP) problem, which can be solved via convex optimization tools such as CVX. Specifically, the optimization objective is equal to the value of in the above optimization, which can also be viewed as a scaling factor for the constellation. Moreover, if we further constrain instead of in the above optimization, the solution of the above optimization problem will become a ZF precoder.

Before we present the subsequent analysis, we first transform the power constraint included in the above optimization problem, which greatly simplifies the subsequent derivations. To be specific, we decompose the precoded signals into

 Ws=K∑i=1wisi, (16)

and similar to the case of PSK [28], we observe that the distribution of the power among each does not affect the solution of the above optimization problem, as can be viewed as a single vector for both constraints that include in . Therefore, without loss of generality and to be consistent with our problem formulation for PSK modulation in [28], we assume that the norm of each term is identical, and we obtain

 ∥Ws∥22=K2∥wisi∥22=K2s∗iwHiwisi, (17) K∑i=1s∗iwHiwisi=Ks∗iwHiwisi,

which further leads to the equivalent power constraint as

 K∑i=1s∗iwHiwisi≤p0K. (18)

We then rewrite the above optimization problem in standard minimization form as

 P4:minW,t−t (19) t−αOm≤0,∀αOm∈O t−αIn=0,∀αIn∈I K∑i=1s∗iwHiwisi≤p0K

and we express the Lagrangian of as [29]

 L(wi,t,δk,μm,νn,δ0)=−t (20) +K∑k=1δk(hTkK∑i=1wisi−ΩTksk)+card{O}∑m=1μm(t−αOm) +card{I}∑n=1νn(t−αIn)+δ0(K∑i=1s∗iwHiwisi−p0K),

where , , , and are the introduced dual variables, and , . Each and can be complex since they correspond to the equality constraints.

Based on the Lagrangian in (20), the KKT conditions for optimality can be expressed as

 ∂L∂t=−1+card{O}∑m=1μm+card{I}∑n=1νn=0 (21a) ∂L∂wi=(K∑k=1δk⋅hTk)si+δ0sis∗i⋅wHi=0,∀i∈K (21b) hTkK∑i=1wisi−ΩTksk=0,∀k∈K (21c) μm(t−αOm)=0,∀αOm∈O (21d) t−αIn=0,∀αIn∈I (21e) δ0(K∑i=1s∗iwHiwisi−p0K)=0 (21f)

Based on (21b), it is first observed that , and with the premise that we obtain , which further means that the power constraint is met with equality when optimality is achieved. Then, we can express in (21b) as

 wHi=−siδ0sis∗i(K∑k=1δk⋅hTk)=−1s∗i(K∑k=1δkδ0⋅hTk). (22)

By introducing an auxiliary variable

 ϑk=−δHkδ0,∀k∈K, (23)

we can express as

 wi=(K∑k=1ϑk⋅h∗k)1si,∀i∈K. (24)

The above expression further leads to

 wisi=(K∑k=1ϑk⋅h∗k),∀i∈K, (25)

which is constant for any and consistent with our assumption in (17).

With the obtained expression for each , we further express the precoding matrix as

 W =[w1,w2,⋯,wK] (26) =(K∑k=1ϑk⋅h∗k)[1s1,1s2,⋯,1sK] =[h∗1,h∗2,⋯,h∗K][ϑ1,ϑ2,⋯,ϑK]T[1s1,1s2,⋯,1sK] =HHΥ^sT,

where we have introduced two column vectors

 Υ=[ϑ1,ϑ2,⋯,ϑK]T,^s=[1s1,1s2,⋯,1sK]T. (27)

We express (11) in matrix form as

 HWs =[ΩT1s1,ΩT2s2,⋯,ΩTKsK]T (28) =Udiag(Ω)sE,

where and are expressed as

 Ω =[ΩT1,ΩT2,⋯,ΩTK]T (29) =[αA1,αB1,αA2,αB2,⋯,αAK,αBK]T =[αE1,αE2,⋯,αE2K−1,αE2K]T, sE =[sT1,sT2,⋯,sTK]T =[sA1,sB1,sA2,sB2,⋯,sAK,sBK]T =[sE1,sE2,⋯,sE2K−1,sE2K]T,

and the matrix is constructed as

 U=⎡⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢⎣1100⋯000011⋱⋮⋮⋮⋮⋱⋱⋱⋮⋮⋮⋮⋱⋱⋱0000⋯0011⎤⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥⎦=I⊗[1,1]. (30)

By substituting the expression for in (26) into (28), we obtain

 HHHΥ^sTs=Udiag(Ω)sE. (31)

With the premise that in this section, is invertible, and accordingly we obtain as

 Υ=1K⋅(HHH)−1Udiag(Ω)sE, (32)

which further leads to the expression for the precoding matrix as

 W=1K⋅HH(HHH)−1Udiag(Ω)sE^sT. (33)

We then substitute in (33) into the power constraint, and we obtain

 ∥Ws∥22=p0 (34) ⇒ sHWHWs=p0 ⇒ sHEdiag(Ω)UH(HHH)−1Udiag(Ω)sE=p0 ⇒ ΩTdiag(sHE)UH(HHH)−1Udiag(sE)TΩ=p0 ⇒ ΩTTΩ=p0.

Since is Hermitian and positive semi-definite, and since each entry in is real, (34) can be further transformed into

 ΩTTΩ=ΩTR{T}Ω=ΩTVΩ=p0, (35)

where is symmetric. With the expression for in (33) and the updated power constraint, we are able to construct an equivalent optimization on , given by

 P5:minΩ,t−t (36) s.t.ΩTVΩ−p0=0 t−αOm≤0,∀αOm∈O t−αIn=0,∀αIn∈I

The optimal precoding matrix for the original optimization is then obtained by substituting the solution of into (33). In the following, we analyze and derive the closed-form optimal precoding matrix as a function of the dual variables of .

The Lagrangian of is formulated as

 L(Ω,t,δ0,μm,νn)=−t+δ0(ΩTVΩ−p0) (37) +card{O}∑m=1μm(t−αOm)+card{I}∑n=1νn(t−αIn),

where , . To simplify the subsequent KKT conditions, we propose to reorder the columns and rows of the matrices and vectors included in the Lagrangian expression in (37). Specifically, we reorder the expanded symbol vector into

 sE⇒~sE=[~sTO,~sTI]T, (38)

where and are given by

 ~sO=[~s1,~s2,⋯,~scard{O}]T, (39) ~sI=[~scard{O}+1,~scard{O}+2,⋯,~s2K]T,

such that the entries in correspond to the real or imaginary part of the outer constellation points that can exploit CI, and the entries in correspond to the real and imaginary part of the inner constellation points that cannot be scaled. The corresponding scaling vector is accordingly transformed into

 Ω⇒~Ω=[~ΩTO,~ΩTI]T, (40)

where and are given by

 ~ΩO=[~α1,~α2,⋯,~αcard{O}]T, (41) ~ΩI=[~αcard{O}+1,~αcard{O}+2,⋯,~α2K]T.

We further introduce a ‘Locater’ function that returns the index of in the original expanded symbol vector , given by

 L(~sm)=k,if~sm=sEk. (42)

We can then express and as

 ~sE=FsE,~Ω=FΩ, (43)

where the transformation matrix that transforms the original and into their reordered forms is given by

 F=[eL(~s1),eL(~s2),⋯,eL(~s2K)]T, (44)

and we note that is invertible. Similarly, the corresponding reordered matrix can be obtained as

 ~V=FVFT, (45)

where the multiplication of at the left side and at the right side correspond to the row and column reordering, respectively. Using the above expressions for , and , the Lagrangian of in (37) can be further transformed into a simple form, given by

 L(~Ω,t,δ0,u1)=(1Tu1−1)t+δ0⋅~ΩT~V~Ω−uT1~Ω−δ0p0, (46)

where , and is the dual vector corresponding to the reordered , given by

 (47)

Subsequently, the KKT conditions for can be formulated as

 ∂L∂t=1Tu1−1=0 (48a) ∂L∂~Ω=δ0⋅2~V~Ω−u1=0 (48b) ~ΩT~V~Ω−p0=0 (48c) μm(t−~αm)=0,∀m∈{1,2,⋯,card{O}} (48d) t−~αn=0,∀n∈{card{I}+1,⋯,2K} (48e)

Based on (48b), we obtain an expression for as a function of , given by

 ~Ω=12δ0⋅~V−1u1, (49)

where we note that is symmetric and invertible. By substituting the expression for in (49) into the power constraint, we further obtain as

 (12δ0⋅~V−1u1)T~V(12δ0⋅~V−1u1)=p0 (50) ⇒ 14δ20⋅uT1~V−1~V~V−1u1=p0 ⇒ δ0= ⎷uT1~V−1u14p0.

For the convex optimization , it is easy to verify that Slater’s condition is met [29], which means that the dual gap is zero. Accordingly, can also be optimally solved via its dual problem, given by

 P6:D=maxu1,δ0min~Ω,tL(u1,δ0,~Ω,t). (51)

For the above dual problem, the inner minimization is achieved by (48a) and (49), and we can further simplify the dual problem into

 D =maxu1,δ0δ0⋅~ΩT~V~Ω+uT1~Ω−δ0p0 (52) =maxu1,δ0δ04δ20⋅uT1~V−1u1−12δ0uT1~V−1u1−δ0p0 =maxu1−uT1~V−1u14√uT1~V−1u14p0− ⎷uT1~V−1u14p0⋅p0 =maxu1−√p0⋅uT1~V−1u1

Based on the fact that is a monotonic function, the above dual problem is equivalent to the following minimization problem:

 P7:minu1uT1~V−1u1 (53) s.t.1Tu1−1=0 μm≥0,∀m∈{1,2,⋯,card{O}}

which is a QP optimization and can be more efficiently solved than the SOCP formulation. Moreover, based on the expression for in (49) and in (50), we finally obtain the optimal closed-form precoding matrix as a function of the dual vector in the case of as

 W= (54) 1KHH(HHH)−1Udiag⎛⎜⎝√p0uT1~V−1u1F−1~V−1u1⎞⎟⎠sE^sT,

where is to order the obtained into the original , with given in (44).

Compared to the final QP formulation for PSK modulation in [28] that is optimized over a simplex, a key difference for the case of QAM constellations is that the variable vector is no longer on a simplex, and only the dual variables that correspond to the real and imaginary part of the constellation points that can exploit CI are constrained to be non-negative, as observed in . We note that both QP formulations for PSK and QAM modulations can be solved by convex optimization tools. However, for the reasons given above, the more efficient simplex method that is generally used for solving QP problems over a simplex and the proposed iterative algorithm in [28] are not directly applicable to such multi-level modulations.

## Iv CI Precoding for the Case of K>Nt

In this section, we further extend our study to the case where the BS simultaneously serves a number of users larger than the number of the transmit antennas at the BS, i.e., . Specifically, our derivations in this section and the corresponding numerical results show that, by exploiting the information of the channel as well as the data symbols and by judiciously constructing the precoding matrix, CI precoding is able to spatially multiplex more data streams than the number of transmit antennas. Similar to the case of , the subsequent analysis is generic and can be further extended to other multi-level constellations.

When , the direct inverse included in (32) becomes infeasible, as the product is rank-deficient. In this case, the more general pseudo inverse instead of the direct matrix inverse is employed [30]. Based on (31), we can now express in the case of as

 Υ=1K⋅(HHH)+Udiag(Ω)sE, (55)

and the obtained precoding matrix as

 W=1K⋅HH(HHH)+Udiag(Ω)sE^sT. (56)

By substituting the expression for the obtained precoding matrix into the power constraint, we can similarly obtain

 ΩTdiag(sHE)UH(HHH)+Udiag(sE)Ω=p0. (57)

Then, one can easily follow a similar approach to that in Section III to obtain a QP optimization and the corresponding solution. However, we note that the solution obtained by following the above procedure is not a valid one for the original problem, since the inclusion of the pseudo inverse does not guarantee the equality of the original constraint. To be more specific, if we consider and substitute the obtained precoding matrix in (33) into (28), we obtain

 HWs=Udiag(Ω)sE (58) ⇒ H[1KHH(HHH)−1Udiag(Ω)sE^sT]s=Udiag(Ω)sE ⇒