**5. Optimization-based precoding**

With the deepening of research on precoding technology, an increasing number of mathematical tools, such as convex optimization, have been introduced into the precoding design process to improve precoding performance as much as possible. In addition, optimization-based precoding can flexibly serve various communication targets, and therefore has a wide range of applications in practical engineering systems.

#### **5.1 Block-level precoding**

#### *5.1.1 Preliminary*

Based on the analysis above, due to the linear relationship between the transmitted signal vector **x**, the symbol vector **s**, and the precoding matrix **W**, the transmitted signal **x** can be regarded as a linear weighted combination of the precoding matrix **W**, where the weighting coefficients are given by the symbol vector **s**. Therefore, the wireless transmission process of (7) and (8) can be reformulated in the following form:

$$\mathbf{y}\_k = \mathbf{h}\_k \sum\_{i=1}^K \mathbf{w}\_i \mathbf{s}\_i + n\_k = \mathbf{h}\_k \mathbf{w}\_k \mathbf{s}\_k + \mathbf{h}\_k \sum\_{i \neq k}^K \mathbf{w}\_i \mathbf{s}\_i + n\_k,\tag{24}$$

where the first component denotes the expected received signal of the *k*-th user, the second component denotes the interference, and the third component denotes the additive noise. Based on that, the received SINR of the *k*-th user can be given as

*Spatial Multiplexing for MIMO/Massive MIMO DOI: http://dx.doi.org/10.5772/intechopen.112041*

$$\gamma\_k = \frac{\left|\mathbf{h}\_k \mathbf{w}\_k\right|^2}{\sum\_{i \neq k}^K \left|\mathbf{h}\_k \mathbf{w}\_i\right|^2 + \sigma^2}. \tag{25}$$

Based on the analysis above, there are two main schemes for optimization-based block-level precoding, as discussed in the following.

#### *5.1.2 Power minimization (PM) scheme*

Power minimization precoding, also known as minimum power beamforming<sup>1</sup> , is a technique used to minimize the total transmitted power subject to a set of quality of service (QoS) constraints. The goal of this technique is to transmit the signal with the minimum possible power while ensuring that the received signal quality meets the desired level. This technique is particularly useful in situations where power consumption is a critical issue or in large-scale MIMO systems where the number of antennas is much larger than the number of users.

The PM design problem can be formulated as below [13]:

$$\begin{aligned} \mathcal{P}\_1 &: \min\_{\mathbf{w}\_i} \sum\_{i=1}^K ||\mathbf{w}\_i||\_F^2 \\ \text{s.t.} & \quad \frac{|\mathbf{h}\_k \mathbf{w}\_k|^2}{\sum\_{i \neq k}^K \left| \mathbf{h}\_k \mathbf{w}\_i \right|^2 + \sigma^2} \ge \Gamma\_k, \forall k \in \{1, 2, \dots, K\} \end{aligned} \tag{26}$$

where Γ*<sup>k</sup>* denotes the SINR threshold for the *k*-th user. It is proved that P<sup>1</sup> is convex which can be solved via convex optimization algorithms efficiently. In addition to conventional convex optimization algorithms, literature has revealed an uplink-downlink duality in ref. [14], which has led to the development of an efficient iterative algorithm for solving downlink precoding optimization. Meanwhile, after transforming PM optimization into a semi-definite programming (SDP) problem, the semi-definite relaxation (SDR) approach [15–17] can be used to design the precoding matrix efficiently.

#### *5.1.3 SINR balancing (SB) scheme*

SINR balancing precoding is a technique used to balance the signal-to- interference-plus-noise ratio (SINR) across all users in a multi-user system. The goal of this technique is to allocate the transmit power among the users such that each user experiences an equal SINR. This technique is particularly useful in situations where there are multiple users with different channel conditions, as it ensures that each user receives an equal quality of service. To be more specific, the SB design problem can be formulated as below [18]:

<sup>1</sup> It is noted that in this chapter the term 'beamforming' and 'precoding' are interchangeable.

$$\begin{aligned} \mathcal{P}\_2: \quad \max\_{\mathbf{w}\_i} \min\_k & \gamma\_k\\ \text{s.t.} \quad \gamma\_k &= \frac{|\mathbf{h}\_k \mathbf{w}\_k|^2}{\sum\_{i \neq k}^K |\mathbf{h}\_k \mathbf{w}\_i|^2 + \sigma^2}, \forall k \in \{1, 2, \dots, K\} \\ & \sum\_{i=1}^K ||\mathbf{w}\_i||\_F^2 \le P\_0 \end{aligned} \tag{27}$$

where *P*<sup>0</sup> is the maximum transmit power. Unlike the PM design problem, P<sup>2</sup> is non-convex, which brings difficulties to the optimal precoding design. However, SB precoding can be efficiently designed through the bisection search method in ref. [16], or via an iterative algorithm in [14].

#### **5.2 Symbol-level precoding**

Block-level precoding is a precoding design based on CSI and is generally independent of the transmitted symbols. These algorithms tend to eliminate inter-user interference. In recent years, symbol-level precoding has received increasing attention [19]. Compared with block-level precoding, symbol-level precoding accomplishes precoding design based on both CSI and transmitted symbols, which gives it the ability to manipulate interference vectors more wisely compared with block-level precoding. With symbol-level precoding, the system can manage and utilize inter-user interference, which offers an additional power gain to improve system performance. In this subsection, we first introduce the concept of constructive interference (CI) to reveal the main idea of interference exploitation and then discuss the design problem of symbol-level precoding in different scenarios.

#### *5.2.1 Concept for interference exploitation*

Interference is commonly considered a factor that limits performance in wireless communication systems. It arises due to the superimposition of transmit signals for different users in the wireless channel during multi-user transmission. Precoding strategies capitalize on the availability of CSI at the base station, along with data symbol information, to predict interference before transmission. Information theory analysis reveals that known interference will not affect the broadcast channel's capacity when CSI is available at the transmitter. However, most existing linear precoding schemes aim to eliminate, avoid or limit interference, and operate on a block level. Recent studies suggest that constructive interference (CI) precoding via Symbol-Level Precoding (SLP) can control both the power and direction of interfering signals, allowing interference to contribute to error-less signal detection and improve system performance [20]. Interference exploitation techniques are most useful in systems where interference can be predicted. In this subsection, we will give an illustrative example to demonstrate the division of instantaneous interference into CI and destructive interference (DI) [20].

Let us consider a scenario where the desired symbol *u* is from a nominal BPSK constellation, with the assumption that *u* ¼ 1. We use *i* to denote the interfering signal and discuss two cases: (i) *i*> 0 and (ii) *i*<0.

In the first case, when *i* >0, as shown in **Figure 3(a)**, the received signal can be expressed as <sup>~</sup>*<sup>y</sup>* <sup>¼</sup> *huu* <sup>þ</sup> <sup>~</sup> *hii* þ *n* ¼ ~*r* þ *n*, where ~*r* represents the received signal excluding noise, and *n* denotes the additive noise at the receiver side. **Figure 3(a)** shows that ProjOE<sup>~</sup> ð Þ~*r* >ProjOE~ð Þ *huu* , which means that the interference has pushed *r* further away

*Spatial Multiplexing for MIMO/Massive MIMO DOI: http://dx.doi.org/10.5772/intechopen.112041*

**Figure 3.** *The geometrical representation of CI and DI.*

from the detection threshold of BPSK when compared to the original data symbol *u*. Here Proj**<sup>d</sup>** ð Þ **x** denotes the projection of vector **x** on the direction of **d**. In this situation, the interfering signal is actually constructive and contributes to the useful signal power. Given a fixed noise power, ~*y* ¼ ~*r* þ *n* is more likely to be detected correctly than the interference-free case *y*<sup>0</sup> ¼ *huu* þ *n:* Thus, we can expect improved performance.

On the other hand, in the second case, when *i*<0, as shown in **Figure 3(b)**, the interfering signal causes the received signal *r* to move closer to the detection threshold. In this case, the interfering signal reduces the useful signal power and is therefore destructive. The noiseless received signal *r* ¼ *huu* þ *hii* is more susceptible to noise than *r*<sup>0</sup> ¼ *u* in this scenario.

In summary, symbol-level precoding offers more precise interference management and control, with the added benefit of improved performance through beneficial interference. This makes it a better communication performance option compared to traditional block-level precoding. Next, we will introduce the design principles of symbol-level precoding by discussing classical CI-SLP precoding methods.

#### *5.2.2 Phase rotation metric*

As depicted in **Figure 4**, CI-SLP is a technique that manipulates inter-user interference to ensure that the noise-free receive signal falls within the constructive region.

**Figure 4.** *CI-SLP, 'phase-rotation' metric, 8-PSK.*

The SLP matrix **W** is designed to maximize the distance between the worst user's constructive region and the detection threshold, thereby improving the transmission performance. Masouros [21] first proposed the "phase rotation" metric for PSK modulated systems. Based on this metric, the noise-free receive signal can be expressed as follows [22]:

$$
\overrightarrow{OA} = \mathbf{h}\_k^T \mathbf{W} \mathbf{s} = \lambda\_k \mathbf{s}\_k. \tag{28}
$$

The constructive factor *λ<sup>k</sup>* quantifies the constructive effect of interference exploitation for that user. Based on this factor, the constructive region can be described as follows:

$$\begin{aligned} \theta\_{AB} \leq \theta\_t &\Rightarrow \tan \theta\_{AB} \leq \tan \theta\_t\\ \Rightarrow \frac{\left| j \cdot \lambda\_k^{\mathcal{T}} s\_k \right|}{\left| \left[ \lambda\_k^{\mathcal{R}} - \sqrt{\Gamma\_k \sigma^2} \right] s\_k \right|} &\leq \tan \theta\_t\\ \Rightarrow \left[ \lambda\_k^{\mathcal{R}} - \sqrt{\Gamma\_k \sigma^2} \right] \tan \theta\_t &\geq \left| \lambda\_k^{\mathcal{T}} \right| \end{aligned} \tag{29}$$

According to the transmit power minimization criterion, the CI-SLP design problem is shown below

$$\begin{aligned} \mathcal{P}\_3: \min\_{\mathbf{w}} \|\mathbf{W}\mathbf{s}\|\_{\mathrm{F}}^2\\ \text{s.t.} &\mathbf{h}\_k \mathbf{W} \mathbf{s} = \lambda\_k s\_k, \forall k \in \{1, 2, \dots, K\} \\ &\left[\lambda\_k^{\otimes} - \sqrt{\Gamma\_k \sigma^2}\right] \tan \theta\_t \ge \left|\Gamma\_k^{\mathrm{T}}\right|, \forall k \in \{1, 2, \dots, K\}, \end{aligned} \tag{30}$$

where Γ<sup>I</sup> *<sup>k</sup>* denotes the Quality of Serves (QoS) threshold of the *k*-th user.

The convexity of P<sup>3</sup> can be proven, similar to the traditional PM problem, enabling the use of several convex optimization algorithms to solve this problem conveniently. Similarly, the CI-SLP design problem based on the SB criterion can be formulated as

$$\begin{array}{lcl}\mathcal{P}\_{4}: & \max\_{\mathbf{W},t} t\\ \text{s.t.} & \mathbf{h}\_{k}\mathbf{W}\mathbf{s} = \boldsymbol{\lambda}\_{k}\boldsymbol{s}\_{k}, \forall k \in \{1, 2, \cdots, K\} \\ & \left[\boldsymbol{\lambda}\_{k}^{\otimes} - t\right] \tan \theta\_{t} \ge \left|\boldsymbol{\lambda}\_{k}^{\mathbb{Z}}\right|, \forall k \in \{1, 2, \cdots, K\} \\ & \|\mathbf{W}\mathbf{s}\|\_{\mathrm{F}}^{2} \le P\_{0}. \end{array} \tag{31}$$

It is worth noting that the convexity of the equation shown above can also be proven, which distinguishes it from the traditional SB problem and renders it more mathematically tractable.

#### *5.2.3 Symbol scaling metric*

In QAM modulation, the interference exploitation is conditional, unlike PSK modulation. The constellation signal points of QAM modulation can be classified into four groups based on their interference exploitation characteristics, as shown in **Figure 5**. Group A' represents signal points that do not exploit any interference, while Group B<sup>0</sup> and Group C<sup>0</sup> represent signal points that exploit interference in the real and

*Spatial Multiplexing for MIMO/Massive MIMO DOI: http://dx.doi.org/10.5772/intechopen.112041*

**Figure 5.** *CI-SLP, 'symbol-scaling' metric, 16-QAM.*

imaginary parts, respectively. Group D<sup>0</sup> represents signal points that exploit interference in both the real and imaginary parts, resulting in full interference exploitation.

The interference exploitation procedure via the "symbol-scaling" [23] metric and decomposition of the noiseless receive signal of the *k*-th user can be described as follows:

$$\mathbf{h}\_k^T \mathbf{W} \mathbf{s} = \mathbf{a}\_k^T \mathbf{s}\_k,\tag{32}$$

where

$$\mathbf{a}\_{k} = \begin{bmatrix} a\_{k}^{\mathcal{A}}, a\_{k}^{\mathcal{A}} \end{bmatrix}^{\mathrm{T}}, \mathbf{s}\_{k} = \begin{bmatrix} s\_{k}^{\mathcal{A}}, s\_{k}^{\mathcal{A}} \end{bmatrix}^{\mathrm{T}} \tag{33}$$

with

$$s\_k^{\mathcal{A}} = \Re(s\_k), s\_k^{\mathcal{B}} = \ (s\_k), \ k = \mathbf{1}, \mathbf{2}, \dots, \mathbf{K}. \tag{34}$$

Based on that, the CI-SLP design problem in QAM-modulated systems can be described as follows

$$\begin{array}{c} \mathcal{P}\_{\mathbb{S}} : \, \max \, \mathbf{w}, \boldsymbol{\alpha}\_{k,t} \quad t \\ \text{s.t.} \qquad \mathbf{h}\_{k}^{T} \mathbf{W} \mathbf{s} = \boldsymbol{\alpha}\_{k}^{\mathrm{T}} \mathbf{s}\_{k}, \forall k \in \mathcal{K} \\ \qquad t \le \boldsymbol{\alpha}\_{m}^{\mathcal{O}}, \forall \boldsymbol{\alpha}\_{m}^{\mathcal{O}} \in \mathcal{O} \\ \qquad t = \boldsymbol{\alpha}\_{n}^{\mathcal{T}}, \forall \boldsymbol{\alpha}\_{n}^{\mathcal{T}} \in \mathcal{T} \\ \|\mathbf{W} \mathbf{s}\|\_{2}^{2} \le p\_{0}. \end{array} \tag{35}$$

The set O comprises the indices of successful interference exploitation corresponding to the real part of the symbol in group B<sup>0</sup> , the imaginary part of the symbol in group C<sup>0</sup> , and both the real and imaginary parts of the symbol in group D<sup>0</sup> . Conversely, the set I comprises the indices of unsuccessful interference exploitation corresponding to the imaginary part of the symbol in group B<sup>0</sup> , the real part of the

symbol in group C0 , and both the real and imaginary parts of the symbol in group A'. It follows that O and I satisfy the following relationship:

$$\begin{aligned} \mathcal{O}\cup\mathcal{T} &= \mathcal{K}, \mathcal{O}\cap\mathcal{Z} = \mathcal{Q},\\ \mathsf{card}\{\mathcal{O}\} + \mathsf{card}\{\mathcal{T}\} &= \mathsf{Z}K. \end{aligned} \tag{36}$$

The definitions of the sets O and I reveal the difference between the phase rotation criterion and the symbol scaling criterion. The former exploits interference unconditionally, i.e., all constellation points participate in interference exploitation, while the latter exploits interference conditionally. For QAM modulation systems, the inner constellation points do not participate in interference exploitation, and beneficial interference only results in performance gains for the outer constellation points. This difference arises from the inherent properties of QAM and PSK modulation schemes. In PSK modulation, the amplitude of the constellation points does not carry any information, and therefore, any constellation point can be exploited for interference without adversely affecting the detection of other constellation points. However, for the inner constellation points in QAM modulation, interference vectors that push the noiseless receive signal points in any direction will adversely affect the error decision of other constellation points. It is worth noting that these two design criteria only differ in their description of the interference exploitation process and are essentially equivalent. Li et al. [23] has proven that under PSK modulation, the symbol scaling criterion and the phase rotation criterion are equivalent, as depicted in **Figure 4**, where the symbol-scaling metric is also applicable. Therefore, the symbol scaling criterion is more universal in this sense.

### **6. Hardware-efficient precoding**

The use of technologies such as General Artificial Intelligence (AI), has led to a surge in users' demand for mobile data traffic. One way to address this issue is to utilize massive MIMO systems, which employ a large number of antennas at the base station to improve data rate and link reliability. This approach allows signals to be dynamically adjusted in both horizontal and vertical directions, reducing interference between small areas and enabling more accurate pointing toward specific users. However, directly applying Massive MIMO technology to traditional communication system architectures can result in new problems [3]. To be more specific, traditional MIMO systems equip each antenna with RF chains and high-resolution DACs, causing significant power loss when the antenna array is large. To solve this issue, there are three general approaches: reducing the number of RF chains, lowering the resolution of the DACs, or employing power-efficient nonlinear power amplifiers. However, these hardware-efficient architectures introduce new challenges to precoding designs, which will be explained in more detail in the following.

#### **6.1 Hybrid analog-digital (HAD) precoding**

Fully-digital precoders can be used in traditional sub-6 GHz bands, but for millimeter wave (mmWave) communications, the cost and power consumption of hardware components make this approach impractical. To solve this issue, researchers

*Spatial Multiplexing for MIMO/Massive MIMO DOI: http://dx.doi.org/10.5772/intechopen.112041*

have developed the hybrid analog-digital structure, which provides a promising tradeoff between the cost, complexity, and capacity of the mmWave network. This structure reduces hardware complexity and power consumption by reducing the total number of RF chains. Specifically, the mmWave transceivers first process data streams with a low-dimension digital precoder, followed by high-dimension analog precoding using low-cost phase shifters, switches [24], or lens [25]. While the performance of the hybrid precoder is usually inferior to that of a fully-digital precoder, it offers a cost-efficient and energy-efficient solution for mmWave communication.

In an MU-MIMO system illustrated in **Figure 6**, *Nt* transmit antennas are utilized by the BS to serve *K* single-antenna users simultaneously. The transmitter has *N<sup>t</sup>* RF RF chains, where *N<sup>t</sup>* RF ≪ *Nt*. In this subsection, we use phase shifter-based hybrid architecture as an illustrative example, without loss of generality.

Based on that, the transmit symbol vector **x** can be expressed as

$$\mathbf{x} = \mathbf{F}\_{\text{RF}} \mathbf{F}\_{\text{BB}} \mathbf{s},\tag{37}$$

where **F**RF ∈ *N<sup>t</sup>* RF�*Nt* denotes the hybrid precoding matrix, **F**BB ∈ *<sup>K</sup>*�*N<sup>t</sup>* RF denotes the digital baseband precoding matrix, and **s** ∈ *<sup>K</sup>*�<sup>1</sup> denotes the data symbol vector with **ss**<sup>H</sup> <sup>¼</sup> <sup>1</sup> *<sup>K</sup>* **I***K*, respectively. Considering that the hybrid precoding matrix is the mathematical description of phase shifters, we have the constant-module constraint for the hybrid precoder, as shown below:

$$\left|\mathbf{F}\_{\rm RF}(i,j)\right| = \mathbf{1}, \ \mathbf{1} \le i \le N\_{\rm RF}^t, \quad \mathbf{1} \le j \le N\_t. \tag{38}$$

Meanwhile, the power constraint at the transmit side can be expressed as

$$\left\|\mathbf{F\_{BB}F\_{RF}}\right\|\_{F}^{2} = P\_{\mathbf{0}},\tag{39}$$

where *P*<sup>0</sup> is the maximum transmit power.

Based on that, the *k*-th user's received signal can be expressed as

$$\mathcal{Y}\_k = \mathbf{h}\_k^H \mathbf{F}\_{\text{RF}} \mathbf{F}\_{\text{BB}} \mathbf{s} + n\_k,\tag{40}$$

where **h***<sup>k</sup>* ∈ *Nt*�<sup>1</sup> denotes the complex channel matrix for the *k*-th user, and *nk* � CN 0, *<sup>σ</sup>*<sup>2</sup> *k* denotes the additive Gaussian noise vector for the *k*-th user with the zero-mean and *σ*<sup>2</sup> *<sup>k</sup>* noise power.

Aimed at maximizing the spectral efficiency, a common HAD precoding design problem can be formulated as [26].

**Figure 6.** *The HAD MIMO system.*

$$\begin{aligned} \mathcal{P}\_{6} & \coloneqq \max\_{\mathbf{F}\_{\text{RF}}, \mathbf{f}\_{k}^{\text{BB}}} \sum\_{k=1}^{K} \log\_{2} \left( 1 + \frac{\left| \mathbf{h}\_{k}^{\text{H}} \mathbf{F}\_{\text{RF}} \mathbf{f}\_{k}^{\text{BB}} \right|^{2}}{\sum\_{i \neq k} \left| \mathbf{h}\_{k}^{\text{H}} \mathbf{F}\_{\text{RF}} \mathbf{f}\_{i}^{\text{BB}} \right|^{2} + \sigma\_{k}^{2}} \right) \\ & \text{s.t. } \mathbf{F}\_{\text{RF}} \in \mathcal{F}, \forall 1 \le k \le K, \\ & \left\| \mathbf{F}\_{\text{RF}} \left[ \mathbf{f}\_{1}^{\text{BB}}, \mathbf{f}\_{2}^{\text{BB}}, \dots, \mathbf{f}\_{K}^{\text{BB}} \right] \right\|\_{F}^{2} = P\_{0}, \end{aligned} \tag{41}$$

where ℱ denotes the available region of **F**RF, as defined below:

$$\mathcal{F} = \left\{ \mathbf{F}\_{\rm RF} \left| \left| \mathbf{F}\_{\rm RF}(i, j) \right| = 1, \ 1 \le i \le N\_{\rm RF}^t, \ 1 \le j \le N\_t \right\} \right. \tag{42}$$

The non-convexity of P<sup>6</sup> is due to the constant-module constraint of **F**RF, making it difficult to solve. To address this issue, a two-stage hybrid precoding algorithm was proposed in ref. [27] where the analog precoder maximizes the effective channel gain and the digital precoder mitigates multi-user interference based on the ZF principle. In ref. [28], it was demonstrated that hybrid precoding can achieve any fully-digital precoding when the number of RF chains is twice the number of data streams, and a near-optimal hybrid precoding design was proposed for single-user and multi-user transmissions with fewer RF chains. Reference [29] focused specifically on partiallyconnected structures in multi-user scenarios and proposed hybrid precoding designs based on successive interference cancelation (SIC). This approach decomposes the total spectral efficiency optimization problem into a series of sub-rate optimization problems that can be solved efficiently using the power iteration algorithm. Other works on hybrid precoding include low-complexity designs based on MRT [30], virtual path selection [31], and SVD [32].

#### **6.2 Low-bit precoding**

Using low-resolution DACs instead of high-resolution DACs in massive MIMO architecture can be an effective way to reduce the power consumption of BS. This approach reduces the power consumption per RF chain, as depicted in **Figure 7**, instead of reducing the number of RF chains like in the hybrid architecture.

High-resolution DACs are required for each transmit signal to avoid signal distortion, but they consume significant power due to their linear relationship with bandwidth and exponential relationship with resolution [33]. Large-scale antenna arrays, with hundreds of antenna elements, require a significantly large number of DACs, posing practical challenges. To address this issue, low-resolution DACs, particularly 1 bit DACs, can substantially simplify hardware and reduce the corresponding power

**Figure 7.** *The architecture of low-bit MIMO system.*

consumption at the BS. Furthermore, 1-bit DACs generate CE signals, which facilitate the use of power-efficient amplifiers, further reducing hardware complexity. The common low-bit precoding design problem can be formulated as [34].

$$\begin{aligned} \mathcal{P}\_{\mathcal{T}} &: \min\_{\mathbf{x}} \|\mathbf{s} - \boldsymbol{\beta}\_{\text{DAC}} \cdot \mathbf{H} \mathbf{x} \|\_{2}^{2} + \mathbf{K} \boldsymbol{\beta}\_{\text{DAC}}^{2} \sigma^{2} \\ &\text{s.t.} \mathbf{x} \in \mathcal{X}\_{\text{DAC}} \\ &\boldsymbol{\beta}\_{\text{DAC}} > \mathbf{0}. \end{aligned} \tag{43}$$

The optimization problem P<sup>7</sup> seeks to minimize the MSE between transmitted and received symbols using low-resolution DACs. For 1-bit DACs, the set of output signals is denoted as X DAC ¼ � ffiffiffiffiffiffi *P*0 2*Nt* q � ffiffiffiffiffi *P*0 2*Nt* q � *j* n o. In ref. [35], a non-linear precoding method based on a biconvex relaxation framework achieved promising performance with a low computational cost. Its corresponding VLSI design architectures were illustrated in refs. [36]. Alternatively, Jacobsson et al. [37] proposed several 1-bit precoding schemes based on SDR, sphere encoding, and squared *l*∞-norm relaxation, while Landau and de Lamare [38] described a 1-bit precoding method based on the branch-and-bound framework that can theoretically achieve optimal performance. Other downlink precoding designs for low-resolution DACs include SER minimization in refs. [39, 40] and alternating minimization in ref. [34]. Nonlinear precoding designs tend to outperform linear methods when low-resolution DACs are used at the transmitter. For example, CI-based symbol-level precoding design has been discussed in low-resolution DACs systems [41–43]. Several efficient solutions [43–45] have been proposed for the NP-hard optimization problem, both for 1-bit and few-bit DACs systems.

#### **6.3 Nonlinearity-aware precoding**

In a massive multiple-input-multiple-output (MIMO) system, the integration of power-efficient nonlinear power amplifiers (PAs) can reduce the power consumption of each RF chain, similar to the architecture of low-bit digital-to-analog converters. Consequently, this leads to an improved energy efficiency of the system. However, in traditional multi-antenna systems, the limited linear region of nonlinear PAs causes significant signal distortions when transmitting signals with high peak-to-average power ratios (PAPRs). This consequently negatively impacts system performance.

To resolve the issue of PAPR, traditional research falls into two categories: (a) constant envelope precoding (CEP) schemes that maintain signal power at a constant value, commonly known as SLP schemes; and (b) frame-level precoding matrix optimization aimed at reducing the PAPR of the transmit signal. CEP eliminates the performance loss introduced by nonlinear PAs by limiting the amplitude of the transmit signal to a constant value, while the low-PAPR precoding relaxes the strict CE constraint by allowing the maximum PAPR to a certain value. In recent years, there has been a growing body of literature that explores the precoding design based on the knowledge of the nonlinear response characteristics of PAs. This approach represents a departure from the traditional emphasis solely on reducing the peak-to-average power ratio (PAPR) of transmitted signals. To be more specific, nonlinearity-aware precoding utilizes a clipping function to model the response characteristics of nonlinear PAs and developed a precoder that can resist both interference and PA nonlinearity by describing the modeled response characteristics [46].

**Figure 8.** *The nonlinearity-aware precoding system.*

The nonlinearity-aware precoding system can be shown in **Figure 8**. Considering a multi-user MISO system, the *k*-th user's received signal can be expressed as

$$\mathcal{Y}\_k = \mathbf{h}\_k^T \mathcal{F}(\mathbf{W}\mathbf{s}) + n\_k,\tag{44}$$

where ℱð Þ� : ! is the nonlinearity function that delineates the input-output response properties of nonlinear power amplifiers [47]. Based on that, the nonlinearity-aware precoding design problem aimed at maximizing the sum rate can be expressed as

$$\begin{aligned} \mathcal{P}\_{\mathsf{S}} &: \max\_{\mathbf{W} \in \mathsf{C}^{K \times N\_{\mathsf{t}}}} R\_{\text{sum}}(\mathbf{W}) \\ \text{s.t.} & \quad \mathbb{F}\left[\left\|\phi(\mathbf{W}\mathbf{s})\right\|^{2}\right] = P\_{\mathsf{t}}, \end{aligned} \tag{45}$$

where *Pt* denotes the maximum transmit power constraint. The problem has been addressed through the introduction of a distortion-aware beamforming (DAB) algorithm as proposed by [48]. This method adopts an iterative approach to optimize data rate while minimizing the effect of distortions. In addition, several other precoding strategies have been developed with a focus on accounting for nonlinearity in the system. Specifically, Aghdam et al. [49] studied a precoding scheme that incorporates power amplifier effects in massive MU-MIMO downlink systems and put forth a robust algorithm to mitigate interference and nonlinearity resulting from power amplifiers. Moreover, Zayani et al. [50] presented a power control mechanism and a precoding scheme for SU-MISO communication systems that utilize nonlinear power amplifiers at the base station. The proposed method maximizes the received SINR while utilizing an iterative precoding algorithm. Finally, Jee et al. [51] optimized both precoding and power allocation strategies jointly to maximize the achievable sum rate of MU-MIMO systems.

### **7. Conclusions**

In this chapter, we have provided a comprehensive overview of precoding design for achieving spatial multiplexing in MIMO communications.

We began in Section 3 by introducing the fundamental concepts of MIMO systems, including the mathematical description of MIMO communications, performance metrics, and the increasingly important and widely used massive MIMO technology in 5G. These concepts laid a solid foundation for the subsequent discussions on the precoding design.

### *Spatial Multiplexing for MIMO/Massive MIMO DOI: http://dx.doi.org/10.5772/intechopen.112041*

In Section 4, we discussed traditional precoding design methods, including closedform linear block-level precoding techniques such as MRT, ZF, and RZF, as well as traditional nonlinear symbol-level precoding techniques such as THP and VP. Through these algorithms, we introduced the basic principles and guidelines of precoding design.

In Section 5, we discussed more complex precoding design methods based on convex optimization, including power minimization, SINR balancing, and the emerging CI-SLP precoding. These methods provide more flexibility and adaptability in precoding design and can achieve better performance in practical communication systems.

In Section 6, we focused on the hardware-efficient precoding design for massive MIMO systems in 5G. We discussed hybrid analog-digital precoding, low-bit precoding, and nonlinearity-aware precoding, which are essential for reducing power consumption and computational complexity while maintaining high communication performance.

Overall, this chapter highlights the importance of efficient precoding design for achieving efficient and reliable wireless transmission. Precoding design is a critical component of MIMO technology, and it requires a careful balance between communication performance, power consumption, and computational complexity. The discussions in this chapter provide a comprehensive understanding of the various precoding techniques that can be employed to achieve spatial multiplexing in MIMO communications and underscore the significance of efficient precoding design for realizing the full potential of MIMO technology in wireless communication systems.
