Spatial Multiplexing for MIMO/Massive MIMO

*Haonan Wang and Ang Li*

### **Abstract**

In this chapter, we will discuss how to achieve spatial multiplexing in multipleinput multiple-output (MIMO) communications through precoding design, for both traditional small-scale MIMO systems and massive MIMO systems. The mathematical description for MIMO communications will first be introduced, based on which we discuss both block-level precoding and the emerging symbol-level precoding techniques. We begin with simple and closed-form block-level precoders such as maximum ratio transmission (MRT), zero-forcing (ZF), and regularized ZF (RZF), followed by the classic symbol-level precoding schemes such as Tomlinson-Harashima precoder (THP) and vector perturbation (VP) precoder. Subsequently, we introduce optimization-based precoding solutions, including power minimization, SINR balancing, symbol-level interference exploitation, etc. We extend our discussion to massive MIMO systems and particularly focus on precoding designs for hardwareefficient massive MIMO systems, such as hybrid analog-digital precoding, low-bit precoding, nonlinearity-aware precoding, etc.

**Keywords:** MIMO, massive MIMO, spatial multiplexing, precoding, beamforming

### **1. Introduction**

In recent years, the demand for high-speed wireless communication has grown exponentially, driven by the proliferation of smart devices, the Internet of Things (IoT), and the increasing need for reliable and efficient data transmission [1]. To meet these demands, multiple-input multiple-output (MIMO) technology has emerged as a promising solution, offering significant improvements in spectral efficiency, capacity, and reliability. In this chapter, we will explore the concept of spatial multiplexing in MIMO communications, focusing on precoding design for both traditional small-scale MIMO systems and massive MIMO systems.

MIMO communication systems employ multiple antennas at both the transmitter and receiver ends to exploit the spatial domain, enabling the simultaneous transmission of multiple data streams over the same frequency band [2]. This spatial multiplexing capability is the key factor in achieving the high data rates and improved link reliability that MIMO systems offer. Precoding is a crucial technique in MIMO communications, as it allows the transmitter to pre-process the signals before

transmission, effectively mitigating inter-stream interference and optimizing the received signal quality. We will begin our discussion with a mathematical description of MIMO communications, providing a solid foundation for understanding the principles and techniques involved in precoding design. Based on this mathematical framework, we will dive deep into both block-level precoding and the emerging symbol-level precoding technique.

Block-level precoding techniques, such as maximum ratio transmission (MRT), zero-forcing (ZF), and regularized ZF (RZF), offer simple and closed-form solutions for mitigating inter-stream interference. These methods have been widely adopted in small-scale MIMO systems due to their ease of implementation and relatively low computational complexity. We will also discuss classic symbol-level precoding schemes, including the Tomlinson-Harashima precoder (THP) and vector perturbation (VP) precoder, which offer improved performance by exploiting the inherent structure of the transmitted symbols. As we move beyond these basic precoding techniques, we will introduce optimization-based precoding solutions that aim to further enhance the performance of MIMO systems. These approaches include power minimization, SINR balancing, and symbol-level interference exploitation, among others. By optimizing various performance metrics, these advanced precoding techniques can achieve significant gains in spectral efficiency and link reliability.

In the latter part of the chapter, we will extend our discussion to massive MIMO systems, which employ a large number of antennas at the transmitter and receiver to achieve even greater spatial multiplexing gains. While the basic principles of precoding design remain applicable to massive MIMO systems, the increased scale and complexity of these systems introduce new challenges and opportunities for precoding optimization. In particular, we will focus on precoding designs for hardware-efficient massive MIMO systems, such as hybrid analog-digital precoding, low-bit precoding, and nonlinearity-aware precoding. These techniques aim to address the practical limitations of massive MIMO systems, including hardware constraints, power consumption, and implementation complexity, while still achieving desired performance gains.

In conclusion, this chapter will provide a comprehensive overview of spatial multiplexing in MIMO communications, with a focus on precoding design for both small-scale and massive MIMO systems. By exploring a wide range of precoding techniques, from simple closed-form solutions to advanced optimization-based approaches, we aim to offer the reader a deep understanding of the principles and methods involved in achieving high-performance MIMO communications.
