**2. Body of the manuscript**

In Section 3, we will provide an introduction to the MIMO communication system, which will include a mathematical description of the MIMO system, performance metrics of MIMO communications, and emerging massive MIMO techniques. In Section 4, we will explain traditional precoding design, which will include preliminaries on precoding and classical precoding schemes. Subsequently, in Section 5, we will discuss optimization-based precoding to demonstrate the use of convex optimization in precoding design. Finally, in recognition of the wide application of massive MIMO, Section 6 will introduce hardware-efficient precoding as a means of achieving a favorable balance between communication performance and power consumption.

### **3. MIMO communication systems**

Due to the increasing demand for higher data rates and reliability for wireless networks, MIMO techniques have appeared and received extensive research attention. To support spatial multiplexing, parallel data streams can be transmitted simultaneously with multiple antennas deployed at the BS. To improve reliability, space-time coding techniques can be employed by sending copies of the same information across the antenna array. In this section, we present an overview of the fundamental concepts of multi-antenna technology, which serves as a foundation for the subsequent discussion on precoding. Given that spatial multiplexing is the primary focus of this chapter, our attention is primarily directed toward multi-user multi-input singleoutput (MU-MISO) systems.

#### **3.1 Mathematical description for MIMO communications**

In a wireless multi-user MISO (MU-MISO) system, as depicted in **Figure 1**, the data symbol vector is denoted as **s**, and one BS with *Nt* antennas transmits wireless signals to *K* single-antenna receivers. Mathematically, the signal vector at the receiver can be expressed as.where *hi*,*<sup>j</sup>* denotes the complex channel gain between the *i*-th receiver and the *j*-th transmit antenna, *xj* denotes the transmit signal on the *j*-th transmit antenna, *yi* denotes the received signal of the *j*-th receiver, and *ni* denotes the additive Gaussian noise corresponding to the *i*-th receiver. Based on that, the *k*-th user's received signal can be expressed as

$$
\begin{bmatrix} \mathcal{Y}\_1 \\ \mathcal{Y}\_2 \\ \vdots \\ \mathcal{Y}\_K \end{bmatrix} = \begin{bmatrix} h\_{1,1} & h\_{1,2} & \cdots & h\_{1,N\_t} \\ h\_{2,1} & h\_{2,2} & \cdots & h\_{2,N\_t} \\ \vdots & \vdots & \ddots & \vdots \\ h\_{K,1} & h\_{K,2} & \cdots & h\_{K,N\_t} \end{bmatrix} \begin{bmatrix} \mathcal{X}\_1 \\ \mathcal{X}\_2 \\ \vdots \\ \mathcal{X}\_N \end{bmatrix} + \begin{bmatrix} n\_1 \\ n\_2 \\ \vdots \\ n\_K \end{bmatrix}, \tag{1}
$$

$$
\mathcal{Y}\_k = \mathbf{h}\_k^T \mathbf{x} + n\_k,\tag{2}
$$

where *yk* denotes the *k*-th user's received signal, **h***<sup>k</sup>* ∈ *Nt*�<sup>1</sup> denotes the *k*-th user's channel vector, **x**∈ *Nt*�<sup>1</sup> denotes the transmit signal vector, and *nk* denotes the additive noise vector which follows the complex Gaussian distribution ℂℕ 0, *σ*<sup>2</sup> *<sup>k</sup>***<sup>I</sup>** � � with the zero mean and *σ*<sup>2</sup> *<sup>k</sup>* noise power. The combining process is eliminated at the receiver side, for the single-antenna configuration. Based on (2), the transmission process in MU-MISO can be reorganized into a matrix form, as shown below:

$$\mathbf{y} = \mathbf{H}\mathbf{x} + \mathbf{n},\tag{3}$$

**Figure 1.** *A block diagram of MU-MISO systems.*

with **y** ¼ *y*1, *y*2, … , *yK* <sup>T</sup> , **H** ¼ ½ � **h**1, **h**2, … , **h***<sup>K</sup>* T, and **<sup>n</sup>** <sup>¼</sup> ½ � *<sup>n</sup>*1, *<sup>n</sup>*2, … , *nK* T.

To mitigate the detrimental impact of channel fading, the transmitter performs precoding on the symbol vector to obtain the transmitted signal, expressed as **x** ¼ **Ws***:* Precoding is achieved using a matrix **W**∈ *Nt*�*K:* The design of the precoding matrix **W** is the crucial signal processing procedure in MIMO downlink transmission, as it enables each receiver to achieve a received signal *yk* that closely approximates the original symbol *sk*.

#### **3.2 Performance metrics for MIMO communications**

In order to measure the communication performance of MIMO systems, bit error rate (BER) and channel capacity are the two performance metrics that are usually employed, as explained below.

#### *3.2.1 BER*

Bit Error Rate (BER) refers to the proportion of erroneously transmitted bits to the total number of transmitted bits during the transmission process and is the most commonly used performance metric to evaluate the reliability of digital communication systems. Its mathematical definition can be given as

$$P\_b = \frac{N\_e}{N\_b},\tag{4}$$

where *Ne* denotes the erroneous transmitted bits, and *Nb* denotes the total transmitted bits.

#### *3.2.2 Channel capacity*

The channel capacity represents the maximum rate of information transmission that can be sustained by a communication system when the bit error rate approaches zero. Its mathematical definition is given as the maximum mutual information between the input and output signals of the channel, which represents the extent to which the received signal preserves information about the transmitted signal after the channel. More specifically, the channel capacity is determined by identifying the input distribution that maximizes the mutual information, subject to the constraints of the channel's physical properties and the power limitations of the system. Therefore, it serves as a fundamental limit on the data transmission rate and is a crucial performance metric for evaluating the effectiveness of communication systems. The definition of channel capacity can be expressed as

$$\mathbf{C} = \max \mathbf{I}(\text{input}; \text{output}), \tag{5}$$

where C denotes the channel capacity, and I (*x*; *y*) denotes the mutual information between *x* and *y*. For SISO systems, when both the transmitter and receiver have perfect Channel State Information (CSI), the channel capacity can be obtained as

$$\mathbf{C} = B \log\_2(\mathbf{1} + \boldsymbol{\chi}),\tag{6}$$

where *B* denotes the system bandwidth, and *γ* denotes the receive SNR. The physical interpretation of (8) has been discussed in ref. [2].

In the context of MIMO systems, it is feasible to decompose the channel into a sum of multiple SISO channels via singular value decomposition (SVD) [2]. Subsequently, utilizing "water-filling" power allocation strategy [2], it is possible to harness the full potential of the system and achieve channel capacity. In an ideal scenario where both the transmitter and receiver possess perfect CSI, the channel capacity of an *Nr* � *Nt* MIMO channel can be captured precisely using the following equation:

$$\mathbf{C} = \log\_2 \det \left( \mathbf{I}\_{N\_r} + \frac{\rho}{N\_r} \mathbf{H} \mathbf{H}^\mathrm{H} \right), \tag{7}$$

where *ρ* denotes the transmit SNR.

#### **3.3 Massive MIMO**

As mobile communication technologies continue to evolve, wireless network capacity and communication quality have become increasingly critical. Traditional wireless communication systems face limitations that prevent them from satisfying the modern industry's demands for high-speed, high-capacity, and high-quality communication. Massive MIMO technology has emerged as a promising solution to these challenges.

Massive MIMO is an extension of conventional MIMO technology [3, 4]. In contrast to the typical tens-of-antenna configuration in traditional MIMO systems for signal transmission and reception, Massive MIMO employs significantly more antennas, for example, hundreds or even thousands of antennas.

Massive MIMO technology enjoys wide applications in various fields of wireless communications, such as 5G and IoT [5]. It has several notable features: channel hardening, favorable propagation, power concentration, capacity enhancement, interference reduction, and spectral efficiency improvement. In particular, channel hardening refers to the property that as the antenna array size increases, the relative fluctuations of channel coefficients decrease [5]. Although randomness still exists, its impact on communication approximates that of non-fading channels. Favorable propagation is a phenomenon in which the channels of different users become nearly orthogonal in the spatial domain as the number of antennas at the base station increases significantly. This leads to a substantial reduction in inter-user interference and further improved spectral efficiency, making massive MIMO a promising technology for future wireless communication systems. Power concentration refers to Massive MIMO's ability to focus transmitted power more efficiently through finer beamforming techniques, especially for millimeter-wave communication where channel gain drops off precipitously with distance [6]. Capacity enhancement is achieved by processing more data streams than traditional MIMO systems, leading to improved network capacity. Interference reduction is accomplished through spatial multiplexing and beamforming, which minimize inter-signal interference and enhance signal quality and reliability. Last, spectral efficiency improvement results from more efficient utilization of bandwidth resources, which enhances data transmission speeds.

However, Massive MIMO technology still faces certain challenges in engineering applications, such as high power consumption [7] and hardware costs. To be more

specific, traditional MIMO systems equip each antenna with radio frequency (RF) chains and high-resolution digital-to-analog converters (DACs), causing significant power loss when the antenna array is large. In such a scenario, the advanced signal processing mechanisms required to handle a large number of antennas for signal transmission and reception are generally more complex, necessitating much more energy consumption than traditional wireless communication systems. From this perspective, hardware-efficient precoding techniques hold significant research value and promising application prospects.
