Architectures for Hybrid Precoding and Combining Techniques in Massive MIMO Systems Operating in the mmWave Band

Abstract

Hybrid precoding and combining techniques in millimeter-wave (mmWave) multiple-input multiple-output (MIMO) systems with various array architectures have attracted significant interest as a promising technology for the development of 6G wireless communication systems. This approach presents numerous advantages, including reduced complexity, cost, and power consumption, when compared to traditional analog precoding methods. In this chapter, we investigate hybrid precoding and combining techniques for massive MIMO systems operating in the millimeter-wave (mmWave) band, with a focus on different architectures, such as full array (FA), subarray (SA), and hybrid array (HA) architectures. We discuss the system model of each architecture. Additionally, we solve the hybrid precoding and combining optimization problem to maximize the spectral efficiency of each architecture. We then propose iterative hybrid precoding and combining algorithms for all architectures, as well as compare their performance to that of traditional hybrid design methods to demonstrate that the proposed algorithms achieve superior performance with lower complexity and hardware requirements.

1. Introduction

The use of millimeter-wave (mmWave) communications has become increasingly popular as a potential solution for current and upcoming cellular systems, mainly due to the extensive yet underutilized mmWave frequency range [1]. To ensure an adequate link margin and achieve array gain, massive multiple-input multiple-output (MIMO) antenna arrays are required for mmWave systems [2]. The use of traditional analog precoding and combining schemes in mmWave MIMO systems is not practical due to the high hardware cost and power consumption of the radio frequency (RF) chains. In light of this, hybrid precoding and combining schemes are viewed as a promising technology that can strike a balance between system performance and hardware complexity. There are two primary hybrid precoding and combining architectures used in millimeter-wave systems: full array (FA) [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15] and subarray (SA) [16, 17, 18, 19, 20, 21, 22, 23] architectures. The FA architecture is commonly employed in hybrid precoding and combining systems. With this architecture, phase shifters (PSs) connect each RF chain to each antenna, leading to a linear increase in the number of PSs with the number of antennas. In contrast, the SA architecture connects each RF chain to a subset of antennas, requiring fewer PSs than the FA architecture.

In the literature, FA hybrid architecture for mmWave systems has received significant attention. The authors of [2] proposed a hybrid precoding/combining algorithm based on simultaneous orthogonal matching pursuit, achieving performance comparable to that of optimal digital beamforming with high complexity. In [4], we introduced an iterative low-complexity hybrid design algorithm based on gradient descent. The work in [5] proposed hybrid designs for Mini-Mental State Examination (MMSE)-based rate balancing in mmWave multiuser MIMO systems and the work in [7] proposed joint hybrid precoding and combining for massive MIMO systems. In [8], a greedy approach is introduced without assumptions about channel structure or array geometry. The work in [9] presented the hybrid design by alternating minimization (HD-AM) algorithm, which achieves high spectral efficiency but is limited to equal numbers of data streams and RF chains. Manifold optimization-based hybrid precoding algorithm in [10] achieves high spectral efficiency but with high computation complexity. Sohrabi and Yu in [11] proposed a heuristic hybrid beamforming algorithm, while the authors of [12, 13] developed gradient projection algorithms for hybrid beamforming design. In multiuser scenarios [14, 15], digital beamforming removes interuser interference, and the analog precoders and combiners maximize user signal power.

Although FA hybrid architecture led to the lower complexity of hybrid precoding and combining algorithms compared to the analog one, the high cost, power consumption, and hardware complexity of this architecture persist due to the need for a phase shifter (PS) to connect each RF chain to every antenna [16, 17]. To address these challenges, the SA architecture has gained popularity as a practical solution for hybrid precoding and combining designs that offer a balance between performance, complexity, and cost. SA architectures for hybrid precoding can be classified as fixed SA [16, 17, 18], adaptive SA [19], and dynamic SA [20, 21]. In fixed SA, each RF chain is connected to a subarray of antennas, while switches are used in dynamic SA. Dynamic SA achieves similar performance as FA with high complexity as compared to the fixed SA. Overlapped SA architecture with hybrid precoding can improve the spectral efficiency of the SA architecture and still lower the complexity compared to the FA architecture [18]. A study in [16] presented an energy-efficient hybrid precoding technique for the fixed SA architecture. The technique utilized successive interference cancelation and assumed a diagonal digital precoder with real elements. Two low-complexity hybrid precoding algorithms for mmWave MIMO systems with fixed SA architecture were proposed and studied in [17]. In [18], we proposed and highlighted the use of overlapped SA architecture for improved spectral efficiency. An adaptive hybrid precoding approach for SA architecture was studied in [19]. Dynamic SA architectures in [20, 21] provided higher spectral efficiency but with increased complexity. In [20, 21], it is found that the dynamic SA architectures perform better than fixed SA architectures, but with higher hardware complexity and power consumption due to the linear increase in the switches with the number of transmit antennas. To reduce the complexity of the dynamic SA, the authors of [22, 23] proposed partially SA structures. However, the partially dynamic precoders in [22, 23] still result in greater computational and hardware complexities, as well as higher power consumption, compared to fixed SA precoders. Recently, deep learning-based hybrid designs have been explored in [24, 25, 26]. A new hybrid design approach for SA was studied in [27]. In [27], an iterative algorithm that begins by designing a hybrid precoding and combining matrix for the FA structure and then converts it into a SA matrix by setting certain entries to zero while achieving better performance was proposed and studied.

While the cost and hardware complexity of hybrid precoding and combining for SA architecture are lower than those for the FA architecture, the spectral efficiency achieved through SA architectures is still inferior to that of optimal digital precoding and combining [17]. Therefore, proposing a new hybrid array architecture that balances spectral efficiency, cost, and power consumption is an essential topic. In this chapter, we introduce a new HA architecture for mmWave MIMO systems that aims to achieve a balance between spectral efficiency, cost, and power consumption. Initially, the antennas at the transmitter/receiver are partitioned into subarrays, each containing the same number of antennas as the number of RF chains at the transmitter/receiver, and then divided into nonoverlapping subsets called groups. Finally, the antennas in each group are connected to a group of RF chains in a way similar to the connections in the FA architecture.

The main contributions of this chapter are summarized as follows:

• The FA, SA, and HA architectures’ system models for mmWave MIMO communication systems are derived and explained. The FA architecture employs PSs to connect each RF chain to all antennas. The SA architecture links each RF chain only to a subset of antennas in a subarray. In contrast, the HA architecture divides the antennas at both the transmitter and receiver into a set of subarrays, which is equivalent to the number of RF chains. The resulting subarrays are divided into groups that do not overlap, and each group’s antennas are linked to a group of RF chains in the same manner as in the FA architecture.

• The optimization problems for hybrid precoding in the FA, SA, and HA architectures are formulated and solved. In the FA architecture, the hybrid precoding optimization problem for the entire system is solved. For the SA architecture, the hybrid precoding optimization problem for each subarray is independently solved. In the HA architecture, each group’s optimization problem is independently solved.

• New iterative algorithms for hybrid precoding and combining are proposed and derived for the FA, SA, and HA architectures. The design derivation takes into account the block structure of the analog precoding/combining matrices in each architecture without relying on any other assumptions or the antenna array geometry. The proposed iterative FA (IFA) algorithm for the FA architecture iteratively determines the hybrid precoding and combining for the entire system. However, for the SA architecture, the proposed iterative SA (ISA) algorithm determines the hybrid precoding and combining for each subarray independently and then for the entire system. In the HA architecture, the proposed iterative HA (IHA) algorithm determines the hybrid precoding and combining for each group independently and then for the entire system.

• The complexities of the proposed algorithms are derived, discussed, and compared to show the simplicity of the proposed algorithms as compared with other existing algorithms.

• Simulations were used to evaluate the proposed algorithms for mmWave MIMO systems with FA, SA, and HA architectures. The results indicate that these algorithms can enhance the mmWave MIMO system’s performance and provide a high level of spectral efficiency.

2. The mmWave channel model

In this section, the mmWave channel model is discussed. $\mathbf{H}$ can be written as [2, 4, 17].

$$\mathbf{H}=\sqrt{N_t{N}_r/N_{\mathit{cl}}{N}_{\mathit{ray}}}\times \sum _i^{N_{\mathit{cl}}}\sum _l^{N_{\mathit{ray}}}\left[{\alpha }_{\mathit{il}}{\mathbf{\Lambda }}_r\left({\varphi }_{\mathit{il}}^r,{\theta }_{\mathit{il}}^r\right)\times {\mathbf{\Lambda }}_t\left({\varphi }_{\mathit{il}}^t,{\theta }_{\mathit{il}}^t\right){\mathbf{a}}_r\left({\varphi }_{\mathit{il}}^r,{\theta }_{\mathit{il}}^r\right){\mathbf{a}}_t{\left({\varphi }_{\mathit{il}}^t,{\theta }_{\mathit{il}}^t\right)}^{\ast }\right]$$E1

where $N_t$ is the number of antennas at the transmitter and $N_r$ is the number of antennas at the receiver. The numbers of clusters and paths are denoted by $N_{\mathit{cl}}$ and $N_{\mathit{ray}}$, respectively. ${\alpha }_{\mathit{il}}$ is the complex gain of the $\mathit{lth}$ path in the $\mathit{ith}$ cluster. ${\varphi }_{\mathit{il}}^t\left({\varphi }_{\mathit{il}}^r\right)$ and ${\theta }_{\mathit{il}}^t\left({\theta }_{\mathit{il}}^r\right)$ are the azimuth (elevation) angles of departure and arrival of the $\mathit{lth}$ path in the $\mathit{ith}$ cluster, respectively. The transmitter and receiver antenna element gains at their departure and arrival angles are denoted by ${\mathbf{\Lambda }}_t\left({\varphi }_{\mathit{il}}^t,{\theta }_{\mathit{il}}^t\right)$ and ${\mathbf{\Lambda }}_r\left({\varphi }_{\mathit{il}}^r,{\theta }_{\mathit{il}}^r\right)$, respectively. ${\mathbf{a}}_t\left({\varphi }_{\mathit{il}}^t,{\theta }_{\mathit{il}}^t\right)$ and ${\mathbf{a}}_r\left({\varphi }_{\mathit{il}}^r,{\theta }_{\mathit{il}}^r\right)$ are the antenna array responses at the transmitter and receiver, respectively. The array response vector in a uniform planar array can be defined as [2, 4, 17].

$${\mathbf{a}}_{\mathit{UPA}\left(\varphi ,\theta \right)}=\frac{1}{\sqrt{{\mathrm{N}}_{\mathrm{t}}}}{\left[1,\dots ,e^{\mathit{jkd}\left(\mathit{\text{xsin}}\left(\varphi \right)\mathit{sin}\left(\theta \right)+\mathit{\text{ycos}}\left(\theta \right)\right)},\dots ,e^{\mathit{jkd}(\left(w-1\right)\mathit{sin}\left(\varphi \right)\mathit{sin}\left(\theta \right)+\left(h-1\right)\mathit{cos}\left(\theta \right)}\right]}^{\mathit{T}}$$E2

where $k=\frac{2\pi }{\lambda }$, $1\le x\le \left(w-1\right),$ and $1\le y\le \left(h-1\right)$. $d=\frac{\lambda }{2}$, $w,$ and $h$ are the interantenna spacing, width, and height of the antenna array, respectively. The transmitter’s array size is $N_t=\mathit{wh}$. In this chapter, we assume perfect channel estimation at the transmitter and receiver.

3. Full array architecture system model

3.1 FA architecture

This subsection presents a discussion on the system model of the FA architecture. First, the baseband digital precoder ${\mathbf{P}}_{\mathbf{D}}$ is applied to the signal at the transmitter, after which it is precoded by the FA analog precoder ${\mathbf{P}}_{\mathbf{AFA}}$. At the receiver’s end, the FA analog combiner ${\mathbf{W}}_{\mathbf{AFA}}$ and the digital combiner ${\mathbf{W}}_{\mathbf{D}}$ are both applied. The structure of the hybrid precoding for FA architecture is depicted in Figure 1(a). The received signal in the FA architecture can be expressed as [4].

Figure 1.

Hybrid precoding at the base station (BS). (a) Full array (FA) architecture. (b) Subarray (SA) architecture.

$$\mathbf{y}=\sqrt{\mathrm{\rho }}{{\mathbf{W}}_{\mathbf{D}}}^{\mathbf{H}}{{\mathbf{W}}_{\mathbf{AFA}}}^{\mathbf{H}}\mathbf{H}{\mathbf{P}}_{\mathbf{AFA}}{\mathbf{P}}_{\mathbf{D}}\mathbf{s}+\mathbf{n}=\sqrt{\mathrm{\rho }}{{\mathbf{W}}_{\mathbf{FA}}}^{\mathbf{H}}\mathbf{H}{\mathbf{P}}_{\mathbf{FA}}\mathbf{s}+\mathbf{n}$$E3

The channel matrix is represented by $\mathbf{H}\mathbf{\in }{\mathbb{C}}^{{\mathit{N}}_{\mathit{r}}\mathbf{\times }{\mathit{N}}_{\mathit{t}}}$, and ρ denotes the average power of the received signal. ${\mathbf{P}}_{\mathbf{AFA}}$ and ${\mathbf{P}}_{\mathbf{D}}$ are $N_t\times N_{\mathit{tRF}}$ and the $N_{\mathit{tRF}}\times N_s$ matrices of analog and digital precoding matrices,$\text{respectively}$. Similarly, ${\mathbf{W}}_{\mathbf{AFA}}$ and ${\mathbf{W}}_{\mathbf{D}}$ are $N_r\times N_{\mathit{rRF}}$ and $N_{\mathit{rRF}}\times N_s$ matrices, respectively, and represent the FA analog and digital combiner matrices. The transmitted signal is represented by the $N_s\times 1$ vector $\mathbf{s}$, with $\mathbb{E}\left[\mathbf{s}{\mathbf{s}}^{\ast }\right]\mathbf{=}\frac{1}{N_s}{\mathbf{I}}_{{\mathit{N}}_{\mathit{s}}}$. The additive white Gaussian noise n is represented by the $N_s\times 1$ vector of independent and identical distribution. The matrices ${\mathbf{P}}_{\mathbf{FA}}\mathbf{=}{\mathbf{P}}_{\mathbf{AFA}}{\mathbf{P}}_{\mathbf{D}}$ and ${\mathbf{W}}_{\mathbf{FA}}\mathbf{=}{\mathbf{W}}_{\mathbf{AFA}}{\mathbf{W}}_{\mathbf{D}}$. All the elements in ${\mathbf{P}}_{\mathbf{AFA}}$ and ${\mathbf{W}}_{\mathbf{AFA}}$ have a constant amplitude, which is equal to $1/\sqrt{N_t}$ and $1/\sqrt{N_r}$, respectively [4]. The digital precoder and combiner satisfy the total power constraint and are normalized as ${‖{\mathbf{P}}_{\mathbf{AFA}}{\mathbf{P}}_{\mathbf{D}}‖}_F^2=N_s$ and ${‖{\mathbf{W}}_{\mathbf{AFA}}{\mathbf{W}}_{\mathbf{D}}‖}_F^2=N_s$. The spectral efficiency of the FA architecture can be written as [4]

$$R={log}_2\left(\left|{\mathbf{I}}_{{\mathbf{N}}_{\mathbf{r}}}+\frac{\rho }{N_s}{{\mathbf{Q}}_{\mathbf{k}}^{-\mathbf{1}}{\mathbf{W}}_{\mathbf{B}}^{\mathbf{k}}}^{\mathbf{H}}{{\mathbf{W}}_{\mathbf{AFA}}^{\mathbf{k}}}^{\mathbf{H}}\mathbf{H}{\mathbf{F}}_{\mathbf{AFA}}{\mathbf{F}}_{\mathbf{B}}{\mathbf{P}}_{\mathbf{B}}^{\mathbf{H}}{\mathbf{P}}_{\mathbf{AFA}}^{\mathbf{H}}{\mathbf{H}}_{\mathbf{k}}^{\mathbf{H}}{\mathbf{W}}_{\mathbf{AFA}}^{\mathbf{k}}{\mathbf{W}}_{\mathbf{B}}^{\mathbf{k}}\right|\right)$$E4

where ${\mathbf{Q}}_{\mathit{k}}\mathit{=}{\sigma }_n^2{{\mathbf{W}}_{\mathbf{B}}^{\mathbf{k}}}^{\mathbf{H}}{{\mathbf{W}}_{\mathbf{AFA}}^{\mathbf{k}}}^{\mathbf{H}}{\mathbf{W}}_{\mathbf{AFA}}^{\mathbf{k}}{\mathbf{W}}_{\mathbf{B}}^{\mathbf{k}}$. To optimize the spectral efficiency in (4), it is important to take into account both the total transmitted power constraint and the constraints on ${\mathbf{F}}_{\mathbf{AFA}}$ and ${\mathbf{W}}_{\mathbf{AFA}}$ during the hybrid precoder/combiner design process.

$$\begin{array}{c}\begin{array}{cc}max& R\\ {\mathbf{P}}_{\mathbf{AFA}},{\mathbf{P}}_{\mathbf{D}},{\mathbf{W}}_{\mathbf{AFA}},{\mathbf{W}}_{\mathbf{D}}& \end{array}\\ \mathrm{st}.{\mathbf{P}}_{\mathbf{AFA}}\in {\mathcal{F}}_{\mathbf{AFA}}\text{and}{\mathbf{W}}_{\mathbf{AFA}}\in {\mathfrak{I}}_{\mathbf{AFA}}\\ {‖{\mathbf{P}}_{\mathbf{AFA}}{\mathbf{P}}_{\mathbf{D}}‖}_{\mathrm{F}}^2=N_s\end{array}$$E5

where ${\mathcal{F}}_{\mathrm{AFA}}$ and ${\mathfrak{I}}_{\mathrm{AFA}}$ contain all possible precoding and combining matrices, respectively, that fulfill the amplitude constraint. The maximization of the objective function of the precoding in Eq. (5) can be expressed in a more concise manner as [4].

$$\begin{array}{c}\left({\mathbf{P}}_{\mathbf{AFA}}^{\mathbf{opt}},{\mathbf{P}}_{\mathbf{D}}^{\mathbf{opt}}\right)=\begin{array}{c}\\ arg\\ {\mathbf{P}}_{\mathbf{AFA}},\end{array}\begin{array}{c}\\ min\\ {\mathbf{P}}_{\mathbf{D}}\end{array}{‖{\mathbf{P}}_{\mathbf{FA}}^{\mathbf{opt}}-{\mathbf{P}}_{\mathbf{AFA}}{\mathbf{P}}_{\mathbf{D}}‖}_{\mathrm{F}}^2\\ \mathrm{st}.{\mathbf{P}}_{\mathbf{AFA}}\in {\mathcal{F}}_{\mathbf{AFA}},\\ {‖{\mathbf{P}}_{\mathbf{AFA}}{\mathbf{P}}_{\mathbf{D}}‖}_{\mathrm{F}}^2=N_s\end{array}$$E6

Clearly, the optimization problem in Eq. (6) is non-convex and finding its optimal solution is challenging. Nonetheless, the optimal unconstrained hybrid precoding can be determined by setting ${\mathbf{P}}_{\mathbf{FA}}^{\mathbf{opt}}$ equal to ${\mathbf{V}}_{\mathbf{1}}$, which represents the first $N_s$ column of the matrix $\mathbf{V}$ obtained through singular value decomposition (SVD) of $\mathbf{H}$, i.e., $\mathbf{H}=\mathbf{U\Sigma }{\mathbf{V}}^{\mathbf{H}}$. Similarly, the optimal unconstrained hybrid combiner can be obtained by setting optimal precoding equal to ${\mathbf{U}}_{\mathbf{1}}$, which represents the first $N_s$ column of the matrix $\mathbf{U}$ [2, 4].

3.2 Subarray architecture system model

This subsection presents a system model of the SA architecture. The structure of the hybrid precoding for SA architecture is depicted in Figure 1(b). The received signal of the SA is given by

$$\mathbf{y}=\sqrt{\rho }{{\mathbf{W}}_{\mathbf{D}}}^{\mathbf{H}}{{\mathbf{W}}_{\mathbf{ASA}}}^{\mathbf{H}}\mathbf{H}{\mathbf{P}}_{\mathbf{ASA}}{\mathbf{P}}_{\mathbf{D}}\mathbf{s}+\mathbf{n}=\sqrt{\mathrm{\rho }}{{\mathbf{W}}_{\mathbf{SA}}}^{\mathrm{H}}\mathbf{H}{\mathbf{P}}_{\mathbf{SA}}\mathbf{s}+\mathbf{n}$$E7

where ${\mathbf{P}}_{\mathbf{ASA}}$ and ${\mathbf{P}}_{\mathbf{D}}$ are the analog and the digital precoding matrices of the SA architecture, respectively, and ${\mathbf{W}}_{\mathbf{ASA}}$ and ${\mathbf{W}}_{\mathbf{D}}$ are the analog and the digital combining matrices, respectively. ${\mathbf{P}}_{\mathbf{ASA}}$ and ${\mathbf{W}}_{\mathbf{ASA}}$ can be expressed as

$${\mathbf{P}}_{\mathbf{ASA}}\mathbf{=}\left[\begin{array}{ccc}{\mathbf{p}}_{\mathrm{A}1}& {\mathbf{0}}_{{\mathrm{N}}_{\mathrm{tSA}}\times 1}& \dots {\mathbf{0}}_{{\mathrm{N}}_{\mathrm{tSA}}\times 1}\\ {\mathbf{0}}_{{\mathrm{N}}_{\mathrm{tSA}}\times 1}& {\mathbf{p}}_{\mathrm{A}2}& \begin{array}{cc}\dots & ⋮\end{array}\\ \begin{array}{c}⋮\\ {\mathbf{0}}_{{\mathrm{N}}_{\mathrm{tSA}}\times 1}\end{array}& \begin{array}{c}⋮\\ \dots \end{array}& \begin{array}{cc}\begin{array}{c}\ddots \\ {\mathbf{0}}_{{\mathrm{N}}_{\mathrm{tSA}}\times 1}\end{array}& \begin{array}{c}{\mathbf{0}}_{{\mathrm{N}}_{\mathrm{tSA}}\times 1}\\ {\mathbf{p}}_{{\text{AN}}_{\mathrm{tSA}}}\end{array}\end{array}\end{array}\right]$$E8

and

$${\mathbf{W}}_{\mathbf{ASA}}\mathbf{=}\left[\begin{array}{ccc}{\mathbf{w}}_{\mathrm{A}1}& {\mathbf{0}}_{{\mathrm{N}}_{\mathrm{rSA}}\times 1}& \begin{array}{cc}\dots & {\mathbf{0}}_{{\mathrm{N}}_{\mathrm{rSA}}\times 1}\end{array}\\ {\mathbf{0}}_{{\mathrm{N}}_{\mathrm{rSA}}\times 1}& {\mathbf{w}}_{\mathrm{A}2}& \begin{array}{cc}\dots & ⋮\end{array}\\ \begin{array}{c}⋮\\ {\mathbf{0}}_{{\mathrm{N}}_{\mathrm{rSA}}\times 1}\end{array}& \begin{array}{c}⋮\\ \dots \end{array}& \begin{array}{cc}\begin{array}{c}\ddots \\ {\mathbf{0}}_{{\mathrm{N}}_{\mathrm{rSA}}\times 1}\end{array}& \begin{array}{c}{\mathbf{0}}_{{\mathrm{N}}_{\mathrm{rSA}}\times 1}\\ {\mathbf{w}}_{{\text{AN}}_{\mathrm{rSA}}}\end{array}\end{array}\end{array}\right]$$E9

In Eq. (8), the $N_{\mathit{tSA}}\times 1$ analog precoding vector for the lth subarray ($l=1,2,\dots ..,N_{\mathit{tRF}}$) is denoted as ${\mathbf{p}}_{\mathrm{Al}}$. Its elements have equal amplitude of $1/\sqrt{N_{\mathit{tSA}}}$, but varying phases. In Eq. (9), the $N_{\mathit{rSA}}\times 1$ analog combining vector for the lth subarray ($l=1,2,\dots ..,N_{\mathit{rRF}}$) is represented by ${\mathbf{w}}_{\mathrm{Al}}$. Its elements have equal amplitude of $1/\sqrt{N_{\mathit{rSA}}}$, but varying phases. Here, $N_{\mathit{tSA}}=N_t/N_{\mathit{tRF}}$ and $N_{\mathit{rSA}}=N_r/N_{\mathit{rRF}}$ denote the number of elements in each subarray at the transmitter and receiver, respectively. The optimization problem of the $\mathit{lth}$subarray can be written as [17].

$$\begin{array}{c}\left({\mathbf{p}}_{\mathbf{Al}}^{\mathbf{opt}},{\mathbf{p}}_{\mathbf{Dl}}^{\mathbf{opt}}\right)=\begin{array}{c}\mathbf{arg}\\ {\mathbf{p}}_{\mathbf{Al}},\end{array}\begin{array}{c}\mathbf{min}\\ {\mathbf{p}}_{\mathbf{Dl}}\end{array}{‖{\mathbf{P}}_{\mathbf{l}}^{\mathbf{opt}}-{\mathbf{p}}_{\mathbf{Al}}{\mathbf{p}}_{\mathbf{Dl}}‖}_{\mathbf{F}}^{\mathbf{2}}\\ \mathbf{st}.{\mathbf{p}}_{\mathbf{Al}}\in {\overline{\mathcal{F}}}_{\mathbf{A}},\\ {‖{\mathbf{P}}_{\mathbf{ASA}}{\mathbf{P}}_{\mathbf{D}}‖}_{\mathbf{F}}^{\mathbf{2}}={\mathrm{N}}_{\mathrm{s}}\end{array}$$E10

where ${\mathbf{P}}_{\mathit{l}}^{\mathit{opt}}={\mathbf{V}}_{\mathbf{1}}\left(\left((l-1\right)N_{\mathit{tSA}}+1:l{N}_{\mathit{tSA}}\right),:)$ denotes the optimum unconstrained hybrid precoding solution of the lth subarray. ${\mathbf{p}}_{\mathbf{Dl}},$ the lth row of the ${\mathbf{P}}_{\mathbf{D}}$. ${\overline{\mathcal{F}}}_{\mathbf{A}},$ and includes all possible $N_{\mathit{tSA}}\times 1$ vectors satisfying the amplitude constraint.

3.3 Hybrid architecture system model

In this subsection, we discuss the system model for hybrid precoding and combining in mmWave MIMO systems with HA architecture. The structure of the hybrid precoding for HA is depicted in Figure 2. Antennas are divided into subarrays and grouped with RF chains. $N_{\mathit{tg}}$ and $N_{\mathit{rg}}$ groups are assumed for transmitter and receiver, respectively, with $N_{\mathit{\text{tSAg}}}=N_{\mathit{tSA}}/N_{\mathit{tg}}$ and $N_{\mathit{\text{rSAg}}}=N_{\mathit{rSA}}/N_{\mathit{rg}}$ being the number of subarrays in each group. The chapter assumes the same number of RF chains and subarrays in all groups and the number of groups must not exceed the number of RF chains but acknowledges that future work may explore cases with different numbers. In HA, the received signal can be expressed as

Figure 2.

The proposed hybrid array (HA) architecture at the base station (BS). (a) Block diagram of the hybrid precoding in the HA architecture. (b) The structure of analog precoding in the $\mathit{\text{ngth}}$ group.

$$\mathbf{y}=\sqrt{\mathrm{\rho }}{{\mathbf{W}}_{\mathrm{D}}}^{\mathrm{H}}{{\mathbf{W}}_{\mathrm{AHA}}}^{\mathrm{H}}\mathbf{H}{\mathbf{P}}_{\mathrm{AHA}}{\mathbf{P}}_{\mathrm{D}}\mathbf{s}+\mathbf{n}\mathbf{=}\sqrt{\mathrm{\rho }}{{\mathbf{W}}_{\mathrm{HA}}}^{\mathrm{H}}\mathbf{H}{\mathbf{P}}_{\mathbf{HA}}\mathbf{s}+\mathbf{n}$$E11

where ${\mathbf{P}}_{\mathrm{AHA}}$ is the matrix of the HA analog precoding, with dimension $N_t\times N_{\mathit{tRF}}$. ${\mathbf{W}}_{\mathrm{AHA}}$ is the matrix of the HA analog combining and has dimensions $N_r\times N_{\mathit{rRF}}$. The amplitudes of all elements in ${\mathbf{P}}_{\mathrm{AHA}}$ and ${\mathbf{W}}_{\mathrm{AHA}}$ are $1/\sqrt{N_t/N_{\mathit{tg}}}$ and $1/\sqrt{N_r/N_{\mathit{rg}}}$, respectively. Note that ${\mathbf{P}}_{\mathrm{HA}}$ and ${\mathbf{W}}_{\mathrm{HA}}$ must satisfy ${‖{\mathbf{P}}_{\mathrm{HA}}‖}_F^2=N_s$ and ${‖{\mathbf{W}}_{\mathrm{HA}}‖}_F^2=N_s$, where ${\mathbf{P}}_{\mathrm{HA}}\mathbf{=}{\mathbf{P}}_{\mathrm{AHA}}{\mathbf{P}}_{\mathrm{D}}$ and ${\mathbf{W}}_{\mathrm{HA}}\mathbf{=}{\mathbf{W}}_{\mathrm{AHA}}{\mathbf{W}}_{\mathrm{D}}$. The general structure of ${\mathbf{P}}_{\mathrm{AHA}}$ in the HA architecture can be expressed as

$${\mathbf{P}}_{\mathrm{AHA}}\mathbf{=}\left[\begin{array}{ccc}\begin{array}{cc}{\mathbf{P}}_{{\mathrm{G}}_1}& 0\end{array}& \dots & 0\\ \begin{array}{cc}0& {\mathbf{P}}_{{\mathrm{G}}_2}\end{array}& \dots & 0\\ \begin{array}{cc}\begin{array}{c}⋮\\ 0\end{array}& \begin{array}{c}⋮\\ 0\end{array}\end{array}& \begin{array}{c}⋮\\ \dots \end{array}& \begin{array}{c}⋮\\ {\mathbf{P}}_{{\mathrm{G}}_{{\mathrm{N}}_{\mathrm{tg}}}}\end{array}\end{array}\right]$$E12

where ${\mathbf{P}}_{{\mathrm{G}}_{\mathrm{ng}}}$ is an $\left(N_t/N_{\mathit{tg}}\right)\times \left(N_{\mathit{tRF}}/N_{\mathit{tg}}\right)$ matrix representing the analog precoding matrix of the $\mathit{\text{ngth}}$ group, where $1\le \mathit{ng}\le N_{\mathit{tg}}$. $N_{\mathit{tg}}=2^n$, where $n=0,1,\dots ,{log}_2{N}_{\mathit{tSA}}$. Note that, $N_{\mathit{tg}}=1$ when $n=0$, resulting in an FA structure [4], and $N_{\mathit{tg}}=N_{\mathit{tSA}}$ when $n={log}_2{N}_{\mathit{tSA}}$, resulting in a conventional SA structure [17]. For the HA architecture, $1\le n\le \left({log}_2{N}_{\mathit{tSA}}\right)-1$. Similarly, ${\mathbf{W}}_{\mathrm{AHA}}$ can be expressed as

$${\mathbf{W}}_{\mathrm{AHA}}\mathbf{=}\left[\begin{array}{ccc}\begin{array}{cc}{\mathbf{W}}_{{\mathrm{G}}_1}& \mathbf{0}\end{array}& \dots & \mathbf{0}\\ \begin{array}{cc}\mathbf{0}& {\mathbf{W}}_{{\mathrm{G}}_2}\end{array}& \dots & \mathbf{0}\\ \begin{array}{cc}\begin{array}{c}⋮\\ \mathbf{0}\end{array}& \begin{array}{c}⋮\\ \mathbf{0}\end{array}\end{array}& \begin{array}{c}⋮\\ \dots \end{array}& \begin{array}{c}⋮\\ {\mathbf{W}}_{{\mathrm{G}}_{{\mathrm{N}}_{\mathrm{rg}}}}\end{array}\end{array}\right]$$E13

where ${\mathbf{W}}_{{\mathrm{G}}_{\mathrm{ng}}}$ is an $\left(N_r/N_{\mathit{rg}}\right)\times \left(N_{\mathit{rRF}}/N_{\mathit{rg}}\right)$ matrix representing the analog combining matrix of the $\mathit{\text{ngth}}$ group, $1\le \mathit{ng}\le N_{\mathit{rg}}$. $N_{\mathit{rg}}$ can be computed by the same method as $N_{\mathit{tg}}$, by only replacing $N_{\mathit{tSA}}$ by $N_{\mathit{rSA}}$. The hybrid precoding optimization problem of the HA architecture can be written as

$$\begin{array}{c}\left({\mathbf{P}}_{\mathrm{AHA}}^{\mathrm{opt}},{\mathbf{P}}_{\mathrm{D}}^{\mathrm{opt}}\right)=\begin{array}{c}\\ arg\\ {\mathbf{P}}_{\mathrm{AHA}},\end{array}\begin{array}{c}\\ min\\ {\mathbf{P}}_{\mathrm{D}}\end{array}{‖{\mathbf{P}}^{\mathrm{opt}}-{\mathbf{P}}_{\mathrm{AHA}}{\mathbf{P}}_{\mathrm{D}}‖}_{\mathrm{F}}^2\\ \mathrm{st}.{\mathbf{P}}_{\mathrm{AHA}}\in {\mathcal{F}}_{\mathrm{AHA}},\\ {‖{\mathbf{P}}_{\mathrm{AHA}}{\mathbf{P}}_{\mathrm{D}}‖}_{\mathrm{F}}^2={\mathrm{N}}_{\mathrm{s}}\end{array}$$E14

where ${\mathcal{F}}_{\mathrm{AHA}}$ includes all possible precoding matrices that satisfy the amplitude constraint of the HA structure.${\mathbf{P}}^{\mathrm{opt}}={\mathbf{V}}_1$ is the optimal solution of the unconstrained hybrid precoding. Similarly, the hybrid combining optimization problem of the HA architecture can be expressed as that given in Eq. (14). However, the problem in Eq. (14) is non-convex with a difficult optimal solution. Due to the block nature of the hybrid precoding matrix in the HA architecture, ${\mathbf{P}}_{\mathrm{HA}}$ can be written as

$$\begin{array}{c}{\mathbf{P}}_{\mathrm{HA}}\mathbf{=}{\mathbf{P}}_{\mathrm{AHA}}{\mathbf{P}}_{\mathrm{D}}\mathbf{=}\left[\begin{array}{ccc}\begin{array}{cc}{\mathbf{P}}_{{\mathrm{AG}}_1}& 0\end{array}& \dots & 0\\ \begin{array}{cc}0& {\mathbf{P}}_{\mathrm{A}{\mathrm{G}}_2}\end{array}& \dots & 0\\ \begin{array}{cc}\begin{array}{c}⋮\\ 0\end{array}& \begin{array}{c}⋮\\ 0\end{array}\end{array}& \begin{array}{c}⋮\\ \dots \end{array}& \begin{array}{c}⋮\\ {\mathbf{P}}_{\mathrm{A}{\mathrm{G}}_{{\mathrm{N}}_{\mathrm{tg}}}}\end{array}\end{array}\right]\left[\begin{array}{c}{\mathbf{P}}_{{\mathrm{DG}}_1}\\ {\mathbf{P}}_{{\mathrm{DG}}_2}\\ \begin{array}{c}\mathbf{⋮}\\ {\mathbf{P}}_{{\mathrm{DG}}_{{\mathrm{N}}_{\mathrm{tg}}}}\end{array}\end{array}\right]\\ =\left[\begin{array}{c}{\mathbf{P}}_{{\mathrm{AG}}_1}{\mathbf{P}}_{{\mathrm{DG}}_1}\\ {\mathbf{P}}_{{\mathrm{AG}}_2}{\mathbf{P}}_{{\mathrm{DG}}_2}\\ \begin{array}{c}⋮\\ {\mathbf{P}}_{{\mathrm{AG}}_{{\mathrm{N}}_{\mathrm{tg}}}}{\mathbf{P}}_{{\mathrm{DG}}_{{\mathrm{N}}_{\mathrm{tg}}}}\end{array}\end{array}\right]\mathbf{=}\left[\begin{array}{c}{\mathbf{P}}_{{\mathrm{HA}}_1}\\ {\mathbf{P}}_{{\mathrm{HA}}_2}\\ \begin{array}{c}\mathbf{⋮}\\ {\mathbf{P}}_{{\mathrm{HA}}_{{\mathrm{N}}_{\mathrm{tg}}}}\end{array}\end{array}\right]\end{array}$$E15

where ${\mathbf{P}}_{{\mathrm{DG}}_{\mathrm{ng}}}$ is an $\left(N_{\mathit{tRF}}/N_{\mathit{tg}}\right)\times N_s$ matrix representing the digital precoder of the $\mathit{\text{ngth}}$ group and ${\mathbf{P}}_{{\mathrm{HA}}_{\mathrm{ng}}}$ is an $\left(N_t/N_{\mathit{tg}}\right)\times N_s$ matrix denoting the hybrid precoding of the $\mathit{\text{ngth}}$ group. Furthermore, the optimal hybrid precoding can be decomposed according to the HA architecture as

$${\mathbf{P}}^{\mathrm{opt}}\mathbf{=}\left[\begin{array}{c}{\mathbf{P}}_{{\mathrm{G}}_1}^{\mathrm{opt}}\\ {\mathbf{P}}_{{\mathrm{G}}_2}^{\mathrm{opt}}\\ \begin{array}{c}\mathbf{⋮}\\ {\mathbf{P}}_{{\mathrm{G}}_{{\mathrm{N}}_{\mathrm{tg}}}}^{\mathrm{opt}}\end{array}\end{array}\right]$$E16

where ${\mathbf{P}}_{{\mathrm{G}}_{\mathrm{ng}}}^{\mathrm{opt}}$ is the optimal digital precoding of the $\mathit{\text{ngth}}$ group in the HA architecture. Based on Eqs. (15) and (16), the problem in Eq. (8) can be decomposed into a series of $N_{\mathit{tg}}$ independent subproblems as

$${‖{\mathbf{P}}^{\mathrm{opt}}-{\mathbf{P}}_{\mathrm{AHA}}{\mathbf{P}}_{\mathrm{D}}‖}_{\mathrm{F}}^2={\sum }_{\mathit{ng}=1}^{N_{\mathit{tg}}}{‖{\mathbf{P}}_{{\mathrm{G}}_{\mathrm{ng}}}^{\mathrm{opt}}-{\mathbf{P}}_{{\mathrm{AG}}_{\mathrm{ng}}}{\mathbf{P}}_{{\mathrm{DG}}_{\mathrm{ng}}}‖}_{\mathrm{F}}^2$$E17

Now, minimizing the objective function in Eq. (8) can be performed by optimizing the $N_{\mathit{tg}}$ subproblems as

$$\begin{array}{c}\left({\mathbf{P}}_{{\mathrm{AG}}_{\mathrm{ng}}}^{\mathrm{opt}},{\mathbf{P}}_{{\mathrm{DG}}_{\mathrm{ng}}}^{\mathrm{opt}}\right)=\begin{array}{c}\\ arg\\ {\mathbf{P}}_{{\mathrm{AG}}_{\mathrm{ng}}},\end{array}\begin{array}{c}\\ min\\ {\mathbf{P}}_{{\mathrm{DG}}_{\mathrm{ng}}}\end{array}{‖{\mathbf{P}}_{{\mathrm{G}}_{\mathrm{ng}}}^{\mathrm{opt}}-{\mathbf{P}}_{{\mathrm{AG}}_{\mathrm{ng}}}{\mathbf{P}}_{{\mathrm{DG}}_{\mathrm{ng}}}‖}_{\mathrm{F}}^2\\ \mathrm{st}.{\mathbf{P}}_{\mathrm{AGng}}\in {\mathcal{F}}_{\mathrm{AHA}},\\ {‖{\mathbf{P}}_{{\mathrm{AG}}_{\mathrm{ng}}}{\mathbf{P}}_{{\mathrm{DG}}_{\mathrm{ng}}}‖}_{\mathrm{F}}^2=N_s/N_{\mathit{tg}}\end{array}$$E18

The optimal combining matrices can be achieved by optimizing the $N_{\mathit{rg}}$ subproblems in a similar fashion.

4. Proposed hybrid precoding and combining algorithms

In this section, the proposed IFA, ISA, and IHA hybrid precoding and combining algorithms for mmWave MIMO system will be derived and discussed. We only derive the equations that relate to the precoder since the derivation of the combiner is similar.

4.1 Iterative full array (IFA) algorithm

In this subsection, we propose the low-complexity IFA hybrid precoding algorithm with equal power allocation per stream. In addition, we do not assume any constraint on the optimization problem, which is related to Eq. (6). The derivation of the combiner is similar. The optimization problem in Eq. (6) is not convex and its solution is NP-hard. Therefore, we propose an iterative solution to solve the problem in Eq. (6). Specifically, we solve the following optimization problem iteratively, which is related to Eq. (6):

$$\left({\mathbf{P}}_{\mathbf{D}}^{\mathbf{opt}}\right)=arg\underset{{\mathbf{P}}_{\mathbf{D}}}{min}{‖{\mathbf{P}}_{\mathbf{FA}}^{\mathbf{opt}}-{\mathbf{P}}_{\mathbf{\text{AIFA}}}{\mathbf{P}}_{\mathbf{D}}‖}_F^2$$E19

where ${\mathbf{P}}_{\mathbf{\text{AIFA}}}$ is the proposed IFA analog precoder. The objective function can be expanded as

$$\begin{array}{rcl}{‖{\mathbf{P}}_{\mathbf{FA}}^{\mathbf{opt}}-{\mathbf{P}}_{\mathbf{\text{AIFA}}}{\mathbf{P}}_{\mathbf{D}}‖}_F^2& =& \mathrm{tr}\left({{\mathbf{P}}_{\mathbf{FA}}^{\mathbf{opt}}}^{\mathbf{H}}{\mathbf{P}}_{\mathbf{FA}}^{\mathbf{opt}}\right)-2\mathrm{tr}\left({{\mathbf{P}}_{\mathbf{FA}}^{\mathbf{opt}}}^{\mathbf{H}}{\mathbf{P}}_{\mathbf{\text{AIFA}}}{\mathbf{P}}_{\mathbf{D}}\right)\\ & & +\mathrm{tr}\left({\mathbf{P}}_{\mathbf{D}}^{\mathbf{H}}{\mathbf{P}}_{\mathbf{\text{AIFA}}}^{\mathbf{H}}{\mathbf{P}}_{\mathbf{\text{AIFA}}}{\mathbf{P}}_{\mathbf{D}}\right)\\ & =& N_S-2\mathrm{tr}\left({{\mathbf{P}}_{\mathbf{FA}}^{\mathbf{opt}}}^{\mathbf{H}}{\mathbf{P}}_{\mathbf{\text{AIFA}}}{\mathbf{P}}_{\mathbf{D}}\right)+\mathrm{tr}\left({\mathbf{P}}_{\mathbf{D}}^{\mathbf{H}}{\mathbf{P}}_{\mathbf{\text{AIFA}}}^{\mathbf{H}}{\mathbf{P}}_{\mathbf{\text{AIFA}}}{\mathbf{P}}_{\mathbf{D}}\right)\end{array}$$E20

To minimize over ${\mathbf{P}}_{\mathbf{D}}$, we set the derivative of Eq. (20) with respect to ${\mathbf{P}}_{\mathbf{D}}$ equal to zero, which yields the following minimized proposed baseband precoder ${\mathbf{P}}_{\mathbf{D}}$ (least-squares solution)

$${\mathbf{P}}_{\mathbf{D}}={\left({\mathbf{P}}_{\mathbf{\text{AIFA}}}^{\mathbf{H}}{\mathbf{P}}_{\mathbf{\text{AIFA}}}\right)}^{-\mathbf{1}}{\mathbf{P}}_{\mathbf{\text{AIFA}}}^{\mathbf{H}}{\mathbf{P}}_{\mathbf{FA}}^{\mathbf{opt}}$$E21

Then, we keep ${\mathbf{P}}_{\mathbf{D}}$ fixed and solve the same optimization problem but now minimizing over ${\mathbf{P}}_{\mathbf{\text{AIFA}}}$

$$\left({\mathbf{P}}_{\mathbf{\text{AIFA}}}^{\mathbf{opt}}\right)=arg\underset{{\mathbf{P}}_{\mathbf{\text{AIFA}}}}{min}{‖{\mathbf{P}}_{\mathbf{FA}}^{\mathbf{opt}}-{\mathbf{P}}_{\mathbf{\text{AIFA}}}{\mathbf{P}}_{\mathbf{D}}‖}_F^2$$E22

Similar to Eq. (20), expanding the objective function yields:

$${‖{\mathbf{P}}_{\mathbf{FA}}^{\mathbf{opt}}-{\mathbf{P}}_{\mathbf{\text{AIFA}}}{\mathbf{P}}_{\mathbf{D}}‖}_F^2=N_S-2\mathrm{tr}\left({{\mathbf{P}}_{\mathbf{FA}}^{\mathbf{opt}}}^{\mathbf{H}}{\mathbf{P}}_{\mathbf{\text{AIFA}}}{\mathbf{P}}_{\mathbf{D}}\right)+\mathrm{tr}\left({\mathbf{P}}_{\mathbf{D}}^{\mathbf{H}}{\mathbf{P}}_{\mathbf{\text{AIFA}}}^{\mathbf{H}}{\mathbf{P}}_{\mathbf{\text{AIFA}}}{\mathbf{P}}_{\mathbf{D}}\right)$$E23

We again set the derivative of (23) with respect to ${\mathbf{P}}_{\mathbf{\text{AIFA}}}$ as equal to zero, which yields the following equation:

$${\nabla }_f\left({\mathbf{P}}_{\mathbf{\text{AIFA}}}\right)=-{\mathbf{P}}_{\mathbf{FA}}^{\mathbf{opt}}{\mathbf{P}}_{\mathbf{D}}^{\mathbf{H}}+{\mathbf{P}}_{\mathbf{\text{AIFA}}}{\mathbf{P}}_{\mathbf{D}}{\mathbf{P}}_{\mathbf{D}}^{\mathbf{H}}=0$$E24

Since ${\mathbf{P}}_{\mathbf{D}}{\mathbf{P}}_{\mathbf{D}}^{\mathbf{H}}$ cannot be inverted when $N_S<N_{\mathit{tRF}}$, we used the gradient descent method to obtain:

$$\begin{array}{l}{P}_{AIFA}{}^{k+1}=P_{AIFA}{}^k-\alpha {\nabla }_f\left(P_{AIFA}{}^k\right)\\ P_{AIFA}{}^{k+1}=P_{AIFA}{}^k+\alpha (P_{FA}^{opt}-P_{AIFA}{}^k{P}_D)P_D^H\\ P_{AIFA}{}^{k+1}=P_{AIFA}{}^k+\alpha P_{res}{P}_D^H\end{array}$$E25

where the residual precoding matrix, denoted as ${\mathbf{P}}_{\mathbf{res}},$ is obtained by subtracting the product of ${\mathbf{P}}_{\mathbf{\text{AIFA}}}$ and ${\mathbf{P}}_{\mathbf{D}}$ from the optimized ${\mathbf{P}}_{\mathbf{FA}}^{\mathbf{opt}}$, and $\alpha$ is the step size. This approach is valid even when $N_S$ equals $N_{\mathit{tRF}}$. However, if $N_S$ is less than $N_{\mathit{tRF}}$, the ${\mathbf{P}}_{\mathbf{\text{AIFA}}}$ matrix with dimensions of $N_t\mathrm{x}{N}_{\mathit{tRF}}$needs to be completed after initialization. In each iteration, the column that results in the largest reduction of the residual is added to ${\mathbf{P}}_{\mathbf{\text{AIFA}}}$. This column is chosen from the basis of the range of the residual, which is obtained by normalizing the first singular vector of the residual element-wise. Algorithm 1 provides the pseudo-code for the proposed IFA hybrid precoder, denoted as ${\mathbf{P}}_{\mathbf{IFA}}$. In a mmWave system that employs hybrid precoding, the base station (BS) or mobile station (MS) can support up to $min(N_{\mathit{tRF}}$, $N_{\mathit{rRF}})$ [2]. The inputs to the algorithm 1 are ${\mathbf{P}}_{\mathbf{FA}}^{\mathbf{opt}}\in C^{N_t\mathrm{x}{N}_S}$ and the maximum number of iterations K. When $N_{\mathrm{S}}<N_{\mathit{tRF}}$, $K$ should be greater than or equal to $N_{\mathit{tRF}}$ - $N_S$ to compute the $N_{\mathrm{t}}\mathrm{x}{N}_{\mathit{tRF}}{\mathbf{P}}_{\mathbf{\text{AIFA}}}$ matrix. When $N_{\mathit{tRF}}$ = $N_S$, K should be greater than or equal to 1.

In the general case where $N_S\ge 1$ and $N_S\le N_{\mathit{tRF}}$, the algorithm initializes ${\mathbf{P}}_{\mathbf{\text{AIFA}}}$ by normalizing the first $N_{\mathrm{s}}$ columns of ${\mathbf{P}}_{\mathbf{FA}}^{\mathbf{opt}}$ element-wise, i.e., ${\mathbf{P}}_{\mathbf{\text{AIFA}}}={\mathbf{P}}_{\mathbf{FA}}^{\mathbf{opt}}\oslash \left(\left|{\mathbf{P}}_{\mathbf{FA}}^{\mathbf{opt}}\right|\sqrt{N_t}\right)$. Next, step 2 calculates ${\mathbf{P}}_{\mathbf{D}}$ using least squares. Steps 3 and 4 compute the residual precoding matrix ${\mathbf{P}}_{\mathbf{res}}$ and the proposed IFA analog precoder ${\mathbf{P}}_{\mathbf{\text{AIFA}}}$, respectively. Step 5 ensures that the constant-magnitude entries in ${\mathbf{P}}_{\mathbf{\text{AIFA}}}$ can be applied at radio frequency (RF) using analog phase shifters.

In steps 7 and 8, when $N_S<N_{\mathit{tRF}}$, the $N_t\mathrm{x}{N}_{\mathit{tRF}}{\mathbf{P}}_{\mathbf{\text{AIFA}}}$ needs to be completed by adding the element-wise normalization of the first singular vector of ${\mathbf{P}}_{\mathbf{res}}$ to ${\mathbf{P}}_{\mathbf{\text{AIFA}}}$. After K iterations, the algorithm finds the $N_t\mathrm{x}{N}_{\mathit{tRF}}$ proposed IFA analog precoding matrix ${\mathbf{P}}_{\mathbf{\text{AIFA}}}$ and the $N_{\mathit{tRF}}\mathrm{x}{N}_S$ baseband precoder ${\mathbf{P}}_{\mathbf{D}},$ such that ${‖{\mathbf{P}}_{\mathbf{FA}}^{\mathbf{opt}}-{\mathbf{P}}_{\mathbf{\text{AIFA}}}{\mathbf{P}}_{\mathbf{D}}‖}_F$ is minimized. In steps 12 and 13, the algorithm ensures that the transmit power constraint is satisfied and returns the proposed IFA hybrid precoder ${\mathbf{P}}_{\mathbf{IFA}}={\mathbf{P}}_{\mathbf{\text{AIFA}}}{\mathbf{P}}_{\mathbf{D}}$. The proposed hybrid combiner ${\mathbf{W}}_{\mathbf{IFA}}$ can be calculated in the same manner.

Remark 1 - Convergence of the proposed IFA hybrid precoder to local minimum points: Note that when $N_S=N_{\mathit{tRF}}$ or $N_S<N_{\mathit{tRF}}$, ${\mathbf{P}}_{\mathbf{D}}$ is a square matrix that is approximately unitary ${\mathbf{P}}_{\mathbf{D}}^{\mathbf{H}}{\mathbf{P}}_{\mathbf{D}}\approx {\mathbf{P}}_{\mathbf{D}}{\mathbf{P}}_{\mathbf{D}}^{\mathbf{H}}\approx {\mathbf{I}}_{{\mathbf{N}}_{\mathbf{S}}}$ or a non-square matrix that is approximately semi-unitary ${\mathbf{P}}_{\mathbf{D}}^{\mathbf{H}}{\mathbf{P}}_{\mathbf{D}}\approx {\mathbf{I}}_{{\mathbf{N}}_{\mathbf{s}}},$ respectively [2].

Algorithm 1. Proposed IFA Hybrid Precoding.

Input: The optimum unconstrained solution ${\mathbf{P}}_{\mathbf{FA}}^{\mathbf{opt}}\in C^{N_{\mathrm{t}}\mathrm{x}{N}_S}.$ and the maximum number of iterations$K$

Output: Analog ${\mathbf{P}}_{\mathbf{\text{AIFA}}}\in C^{N_{\mathrm{t}}\mathrm{x}{N}_{\mathrm{tRF}}}.$ with the element-wise normalization and baseband ${\mathbf{P}}_{\mathbf{D}}\in C^{N_{\mathrm{tRF}}\mathrm{x}{N}_S}$ such that ${‖{\mathbf{P}}_{\mathbf{FA}}^{\mathbf{opt}}-{\mathbf{P}}_{\mathbf{IFA}}‖}_F$ is reduced and ${‖{\mathbf{P}}_{\mathbf{IFA}}‖}_F^2=N_S$, where ${\mathbf{P}}_{\mathbf{IFA}}={\mathbf{P}}_{\mathbf{\text{AIFA}}}{\mathbf{P}}_{\mathbf{D}}$

Initialization: analog precoder ${{\mathbf{P}}_{\mathbf{\text{AIFA}}}}^1={\mathbf{P}}_{\mathbf{FA}}^{\mathbf{opt}}\oslash \left(\left|{\mathbf{P}}_{\mathbf{FA}}^{\mathbf{opt}}\right|\sqrt{N_t}\right).$

1: for $k=1:K$ do

2: Update: ${\mathbf{P}}_{\mathbf{D}}={\left({{\mathbf{P}}_{\mathbf{\text{AIFA}}}^{\mathbf{H}}}^k{{\mathbf{P}}_{\mathbf{\text{AIFA}}}}^k\right)}^{-\mathbf{1}}{{\mathbf{P}}_{\mathbf{\text{AIFA}}}^{\mathbf{H}}}^k{\mathbf{P}}_{\mathbf{FA}}^{\mathbf{opt}\mathbf{.}}$

3: Update the residual: ${\mathbf{P}}_{\mathbf{res}}={\mathbf{P}}_{\mathbf{FA}}^{\mathbf{opt}}$ - ${{\mathbf{P}}_{\mathbf{\text{AIFA}}}}^k{\mathbf{P}}_{\mathbf{D}}$

4: Update: ${{\mathbf{P}}_{\mathbf{\text{AIFA}}}}^{k+1}={{\mathbf{P}}_{\mathbf{\text{AIFA}}}}^k+\alpha {\mathbf{P}}_{\mathbf{res}}{\mathbf{P}}_{\mathbf{D}}^{\mathbf{H}}$

5: Element-Wise Normalization:

$${{\mathbf{P}}_{\mathbf{\text{AIFA}}}}^{k+1}={{\mathbf{P}}_{\mathbf{\text{AIFA}}}}^{k+1}\oslash \left(\left|{{\mathbf{P}}_{\mathbf{\text{AIFA}}}}^{k+1}\right|\sqrt{N_t}\right)$$

6: If $i\le N_{\mathit{tRF}}$ - $N_S$

7: ${\mathbf{F}}_{\mathbf{res}}=\mathbf{U\Sigma }{\mathbf{V}}^{\mathbf{H}}$

8: Append the element-wise normalization of the first vector of $\mathbf{U}$ as a new column to P_AIFA:

$${{\mathbf{P}}_{\mathbf{\text{AIFA}}}}^{k+1}=\left[{{\mathbf{P}}_{\mathbf{\text{AIFA}}}}^{k+1}{\left(\mathbf{U}\right)}_1\oslash \left(\left|{\left(\mathbf{U}\right)}_1\right|\sqrt{N_t}\right)\right]$$

9: end if

10: end for

11: ${\mathbf{P}}_{\mathbf{D}}={\left({\mathbf{P}}_{\mathbf{\text{AIFA}}}^{\mathbf{H}}{\mathbf{P}}_{\mathbf{\text{AIFA}}}\right)}^{-\mathbf{1}}{\mathbf{P}}_{\mathbf{\text{AIFA}}}^{\mathbf{H}}{\mathbf{P}}_{\mathbf{FA}}^{\mathbf{opt}}$

12: ${\mathbf{P}}_{\mathbf{D}}=\sqrt{N_S}\frac{{\mathbf{P}}_{\mathbf{D}}}{{‖{\mathbf{P}}_{\mathbf{\text{AIFA}}}{\mathbf{P}}_{\mathbf{D}}‖}_{\mathit{F}}}$

13: return ${\mathbf{P}}_{\mathbf{IFA}}={\mathbf{P}}_{\mathbf{\text{AIFA}}}{\mathbf{P}}_{\mathbf{D}}$

Thus, each iteration in Algorithm 1 minimizes the objective function ${‖{\mathbf{P}}_{\mathbf{FA}}^{\mathbf{opt}}-{\mathbf{P}}_{\mathbf{\text{AIFA}}}{\mathbf{P}}_{\mathbf{D}}‖}_F$ and the error term decreases monotonically with each iteration. Since the objective function has a lower bound, the proposed method must converge to local optimum points.

4.2 Iterative subarray (ISA) algorithm

In this subsection, the proposed ISA hybrid precoding is derived and discussed. The optimization problem of the lth subarray in Eq. (10) can be solved for each subarray in a similar way as in Eq. (22), and the following iterative solution can be obtained:

$${{\mathbf{p}}_{\mathit{Al}}}^{k+1}={{\mathbf{p}}_{\mathit{Al}}}^k+{\mathbf{P}}_{\mathit{res}}{\mathbf{p}}_{\mathit{Dl}}^H$$E26

The residual precoding matrix for the lth subarray, ${\mathbf{P}}_{\mathit{res}}$, is calculated as ${\mathbf{P}}_{\mathit{res}}={\mathbf{P}}_l^{\mathit{opt}}-{\mathbf{p}}_{\mathit{Al}}{\mathbf{p}}_{\mathit{Dl}}$. Eq. (26) shows that the updated value of ${{\mathbf{p}}_{\mathit{Al}}}^{k+1}$ for the lth subarray can be obtained by adding ${\mathbf{P}}_{\mathit{res}}{\mathbf{p}}_{\mathit{Dl}}^H$ to the value of ${{\mathbf{p}}_{\mathit{Al}}}^k$ from the previous iteration. Once initial values of ${\mathbf{p}}_{\mathit{Al}}$ and ${\mathbf{p}}_{\mathit{Dl}}$ have been obtained, these can be used to iteratively solve the optimization problem given in Eq. (26). After convergence, the resulting ${\mathbf{p}}_{\mathit{Al}}$ of the lth subarray must be normalized to satisfy the constraint in Eq. (10).

Algorithm 2. Proposed ISA Hybrid Precoding scheme

1. Input ${\mathbf{V}}_1,$K

2. Decompose ${\mathbf{V}}_{\mathbf{1}}$ as ${\mathbf{V}}_1={\left[{\stackrel{\sim }{\mathbf{V}}}_1{\stackrel{\sim }{\mathbf{V}}}_2\dots {\stackrel{\sim }{\mathbf{V}}}_{{\mathrm{L}}_{\mathrm{t}}}\right]}^{\mathrm{T}}$

3. For $1\le l\le L_t$

4. Find ${\mathbf{p}}_{\mathit{Al}}^{\mathit{\text{initial}}}=\frac{1}{\sqrt{N_{\mathit{tSA}}}}{e}^{\mathit{\text{jangle}}\left({\stackrel{\sim }{\mathbf{V}}}_l\right)}$

5. Find ${\mathbf{p}}_{\mathit{Dl}}^{\mathit{\text{initial}}}={\left({\mathbf{p}}_{\mathit{Al}}\right)}^H\ast {\stackrel{\sim }{\mathbf{V}}}_l$

6. Initial ${{\mathbf{p}}_{\mathit{Al}}}^k={\mathbf{p}}_{\mathit{Al}}^{\mathit{\text{initial}}}$

7. Initial ${\mathbf{p}}_{\mathit{Dl}}={\mathbf{p}}_{\mathit{Dl}}^{\mathit{\text{initial}}}$

8. For $1\le k\le K$

9. Compute the residual ${\mathbf{P}}_{\mathit{res}}={\stackrel{\sim }{\mathbf{V}}}_l-{{\mathbf{p}}_{\mathit{Al}}}^k{\mathbf{p}}_{\mathit{Dl}}$

10. Update ${{\mathbf{p}}_{\mathit{Al}}}^{k+1}={{\mathbf{p}}_{\mathit{Al}}}^k+{\mathbf{P}}_{\mathit{res}}{\mathbf{p}}_{\mathit{Dl}}^H$

11. Normalize ${{\mathbf{p}}_{\mathit{Al}}}^{k+1}={{\mathbf{p}}_{\mathit{Al}}}^{k+1}/\left(\left|{{\mathbf{p}}_{\mathit{Al}}}^{k+1}\right|\ast \sqrt{N_{\mathit{tSA}}}\right)$

12. ${\mathbf{p}}_{\mathit{Dl}}={\left({{\mathbf{p}}_{\mathit{Al}}}^{k+1}\right)}^H\ast {\stackrel{\sim }{\mathbf{V}}}_l$

13. end for

14. end for

15. Construct ${\mathbf{P}}_D$ and ${\mathbf{P}}_{\mathit{\text{AISA}}}$

16. Normalize ${\mathbf{P}}_D$ as ${\mathbf{P}}_D=\frac{\sqrt{N_s}}{{‖{\mathbf{P}}_{\mathit{\text{AISA}}}{\mathbf{P}}_{\mathit{D}}‖}_{\mathit{F}}}{\mathbf{P}}_D$

17. Return ${\mathbf{P}}_{\mathit{ISA}}$=${\mathbf{P}}_{\mathit{\text{AISA}}}{\mathbf{P}}_D$

The normalized ${{\mathbf{p}}_{\mathit{Al}}}^{k+1}$can be expressed as

$${{\mathbf{p}}_{\mathit{Al}}}^{k+1}={{\mathbf{p}}_{\mathit{Al}}}^{k+1}/\left(\left|{{\mathbf{p}}_{\mathit{Al}}}^{k+1}\right|\ast \sqrt{N_{\mathit{tSA}}}\right)$$E27

In summary, the pseudo-code of the proposed ISA hybrid precoding can be summarized in Algorithm 2, which can be explained as follows. First, the initial values of ${\mathbf{p}}_{\mathit{Al}}$ and ${\mathbf{p}}_{\mathit{Dl}}$ of the $\mathit{lth}$subarray must be obtained. Then, the iterative solution for the $\mathit{lth}$ subarray is applied to obtain the optimal ${\mathbf{p}}_{\mathit{Al}}$ and ${\mathbf{p}}_{\mathit{Dl}}$ and this operation will be repeated for all subarrays. Finally, ${\mathbf{P}}_D$ and ${\mathbf{P}}_A$ are constructed. Note that the initial solution of ${\mathbf{p}}_{\mathit{Al}}$ and ${\mathbf{p}}_{\mathit{Dl}}$ for each subarray can be obtained as follows:

$${\mathbf{p}}_{\mathit{Al}}^{\mathit{\text{initial}}}=\frac{\mathbf{1}}{\sqrt{N_{\mathit{tSA}}}}{e}^{j\text{angle}\left({\stackrel{\sim }{\mathbf{V}}}_{\mathbf{l}}\right)}$$E28

and

$${\mathbf{p}}_{\mathit{D}l}^{\mathit{\text{initial}}}={\left({\mathbf{p}}_{\mathit{Al}}^{\mathit{\text{initial}}}\right)}^H\ast {\stackrel{\sim }{\mathbf{V}}}_1$$E29

where ${\stackrel{\sim }{\mathbf{V}}}_1$ is an $N_{\mathit{tSA}}\times N_s$ matrix that represents the optimal precoder of the $\mathit{lth}$ subarray.

The algorithm can be summarized as follows:

1. Obtain the initial values ${\mathbf{p}}_{\mathit{Al}}^{\mathit{\text{initial}}}$ and ${\mathbf{p}}_{\mathit{D}l}^{\mathit{\text{initial}}}$ for the lth subarray.

2. Apply an iterative solution for the lth subarray to acquire the optimal ${\mathbf{p}}_{\mathit{Al}}$ and ${\mathbf{p}}_{\mathit{Dl}}$.

3. Repeat the above operation for all subarrays.

4. Construct the proposed ISA analog precoder,${\mathbf{P}}_{\mathit{\text{AISA}}}$, and digital precoder, ${\mathbf{P}}_D$.

5. Construct the hybrid precoder of the proposed ISA as ${\mathbf{P}}_{\mathit{ISA}}={\mathbf{P}}_{\mathit{\text{AISA}}}{\mathbf{P}}_D$.

Eq. (26) satisfies the property of the gradient descent method as it minimizes the objective function ${‖{\mathbf{P}}_l^{\mathit{opt}}-{\mathbf{p}}_{\mathit{Al}}{\mathbf{p}}_{\mathit{Dl}}‖}_F^2$ in each iteration from step 8 to 13 of Algorithm 2. This guarantees the convergence of ${{\mathbf{p}}_{\mathit{Al}}}^{k+1}$ to a local optimal point.

Note that the proposed ISA algorithm in this subsection differs from the proposed IFA algorithm in the previous subsection in its approach. The proposed ISA algorithm independently obtains hybrid precoding for each subarray and then combines them to find the hybrid precoding for the entire system. This makes the proposed ISA algorithm simpler than the IFA algorithm, which computes the hybrid precoding directly for the entire system.

4.3 Iterative hybrid array (IHA) algorithm

In this subsection, we introduce a low-complexity IHA hybrid precoding algorithm. The combiner derivation follows a similar approach. We know that the structure of optimal precoder of the $\mathit{\text{ngth}}$ group ${\mathbf{V}}_{{\mathbf{G}}_{\mathbf{ng}}}$is non-square semi-unitary, meaning ${\mathbf{V}}_{\mathbf{Gng}}^{\mathbf{H}}{\mathbf{V}}_{{\mathbf{G}}_{\mathbf{ng}}}={\mathbf{I}}_{{\mathbf{N}}_{\mathbf{s}}}$ when $N_{\mathit{tg}}=1$ and ${\mathbf{V}}_{\mathbf{Gng}}^{\mathbf{H}}{\mathbf{V}}_{{\mathbf{G}}_{\mathbf{ng}}}\approx {\mathbf{I}}_{{\mathbf{N}}_{\mathbf{s}}}$ when $N_{\mathit{tg}}>1$. Therefore, the HA precoder design must also be non-square semi-unitary, i.e., ${\mathbf{P}}_{\mathbf{\text{DGng}}}^{\mathbf{H}}{\mathbf{P}}_{\mathbf{\text{AGng}}}^{\mathbf{H}}{\mathbf{P}}_{{\mathbf{AG}}_{\mathbf{ng}}}{\mathbf{P}}_{{\mathbf{DG}}_{\mathbf{ng}}}={\mathbf{I}}_{{\mathbf{N}}_{\mathbf{s}}}$. This structure will be used to solve the optimization problem and make the HA precoder ${\mathbf{P}}_{{\mathbf{AG}}_{\mathbf{ng}}}{\mathbf{P}}_{{\mathbf{DG}}_{\mathbf{ng}}}$ approach the optimal precoder ${\mathbf{V}}_{{\mathbf{G}}_{\mathbf{ng}}}$ as closely as possible. Thus, by assuming the structure of the HA hybrid precoder ${\mathbf{P}}_{{\mathbf{AG}}_{\mathbf{ng}}}{\mathbf{P}}_{{\mathbf{DG}}_{\mathbf{ng}}}$ as a semi-unitary matrix, we need to solve the following optimization problem:

$$\begin{array}{c}\left({\mathbf{P}}_{{\mathbf{AG}}_{\mathbf{ng}}}^{\mathbf{opt}},{\mathbf{P}}_{{\mathbf{DG}}_{\mathbf{ng}}}^{\mathbf{opt}}\right)=\begin{array}{c}\\ arg\\ {\mathbf{P}}_{{\mathbf{AG}}_{\mathbf{ng}}},\end{array}\begin{array}{c}\\ min\\ {\mathbf{P}}_{{\mathbf{DG}}_{\mathbf{ng}}}\end{array}{‖{\mathbf{P}}_{{\mathbf{G}}_{\mathbf{ng}}}^{\mathbf{opt}}-{\mathbf{P}}_{{\mathbf{AG}}_{\mathbf{ng}}}{\mathbf{P}}_{{\mathbf{DG}}_{\mathbf{ng}}}‖}_{\mathrm{F}}^2\\ \mathit{st}.{\mathbf{P}}_{{\mathbf{AG}}_{\mathbf{ng}}}\in {\mathcal{F}}_{\mathit{AHA}},\\ {\mathbf{P}}_{{\mathbf{AG}}_{\mathbf{ng}}}^{\mathbf{H}}{\mathbf{P}}_{{\mathbf{AG}}_{\mathbf{ng}}}={\mathbf{I}}_{{\mathbf{N}}_{\mathbf{tRF}}}\text{and}{\mathbf{P}}_{{\mathbf{DG}}_{\mathbf{ng}}}^{\mathbf{H}}{\mathbf{P}}_{{\mathbf{DG}}_{\mathbf{ng}}}={\mathbf{I}}_{{\mathbf{N}}_{\mathbf{s}}}\\ {‖{\mathbf{P}}_{{\mathbf{AG}}_{\mathbf{ng}}}{\mathbf{P}}_{{\mathbf{DG}}_{\mathbf{ng}}}‖}_F^2=N_s/N_{\mathit{tg}}\end{array}$$E30

The problem in Eq. (30) is a non-convex optimization problem whose solution is mathematically intractable. However, in this subsection, we will use iterative algorithms to solve (30):

$$\left({\mathbf{P}}_{{\mathbf{AG}}_{\mathbf{ng}}}^{\mathbf{opt}},{\mathbf{P}}_{{\mathbf{DG}}_{\mathbf{ng}}}^{\mathbf{opt}}\right)=\begin{array}{c}\\ arg\\ {\mathbf{P}}_{{\mathbf{AG}}_{\mathbf{ng}}},\end{array}\begin{array}{c}\\ min\\ {\mathbf{P}}_{{\mathbf{DG}}_{\mathbf{ng}}}\end{array}{‖{\mathbf{P}}_{{\mathbf{G}}_{\mathbf{ng}}}^{\mathbf{opt}}-{\mathbf{P}}_{{\mathbf{AG}}_{\mathbf{ng}}}{\mathbf{P}}_{{\mathbf{DG}}_{\mathbf{ng}}}‖}_{\mathrm{F}}^2$$E31

We first need to find the baseband precoder ${\mathbf{P}}_{{\mathbf{DG}}_{\mathbf{ng}}}$ of the $\mathit{\text{ngth}}$ group in the HA architecture that minimizes the Euclidean distance using the initialization of the proposed HA precoder ${\mathbf{P}}_{{\mathbf{AG}}_{\mathbf{ng}}}$ of the $\mathit{\text{ngth}}$ group in the HA architecture, which is calculated by taking the first $N_{\mathit{tRF}}/N_{\mathit{tg}}$ columns from ${\mathbf{P}}_{{\mathbf{G}}_{\mathbf{ng}}}^{\mathbf{opt}}$ and then normalizing them such that each entry has constant magnitude, i.e., ${\mathbf{P}}_{{\mathbf{AG}}_{\mathbf{ng}}}=({\mathbf{P}}_{{\mathbf{G}}_{\mathbf{ng}}}^{\mathbf{opt}}\oslash \left(\left|{\mathbf{P}}_{{\mathbf{G}}_{\mathbf{ng}}}^{\mathbf{opt}}\right|\sqrt{\left(N_t/N_{\mathit{tg}}\right)}\right)$. We then find the RF precoder ${\mathbf{P}}_{{\mathbf{AG}}_{\mathbf{ng}}}$ such that the IHA hybrid precoder ${\mathbf{P}}_{{\mathbf{AG}}_{\mathbf{ng}}}{\mathbf{P}}_{{\mathbf{DG}}_{\mathbf{ng}}}$ of the $\mathit{\text{ngth}}$ group is sufficiently “close” to the optimal unconstrained digital precoder ${\mathbf{P}}_{{\mathbf{G}}_{\mathbf{ng}}}^{\mathbf{opt}}$ of the $\mathit{\text{ngth}}$ group in the HA architecture. Specifically, we would like to solve the following optimization problem first, which is related to (31):

$$\left({\mathbf{P}}_{\mathbf{\text{DGng}}}^{\mathbf{opt}}\right)=arg\underset{{\mathbf{P}}_{\mathbf{\text{DGng}}}}{min}{‖{\mathbf{P}}_{{\mathbf{G}}_{\mathbf{ng}}}^{\mathbf{opt}}-{\mathbf{P}}_{{\mathbf{AG}}_{\mathbf{ng}}}{\mathbf{P}}_{{\mathbf{DG}}_{\mathbf{ng}}}‖}_F^2$$E32

The objective function can be expanded as

$$\begin{array}{c}{‖{\mathbf{P}}_{{\mathbf{G}}_{\mathbf{ng}}}^{\mathbf{opt}}-{\mathbf{P}}_{{\mathbf{AG}}_{\mathbf{ng}}}{\mathbf{P}}_{{\mathbf{DG}}_{\mathbf{ng}}}‖}_F^2\\ =\mathrm{tr}\left({{\mathbf{P}}_{\mathbf{Gng}}^{\mathbf{opt}}}^{\mathbf{H}}{\mathbf{P}}_{\mathbf{Gng}}^{\mathbf{opt}}\right)-2\mathrm{tr}\left({{\mathbf{P}}_{\mathbf{Gng}}^{\mathbf{opt}}}^{\mathbf{H}}{\mathbf{P}}_{{\mathbf{AG}}_{\mathbf{ng}}}{\mathbf{P}}_{{\mathbf{DG}}_{\mathbf{ng}}}\right)+{‖{\mathbf{P}}_{{\mathbf{AG}}_{\mathbf{ng}}}{\mathbf{P}}_{{\mathbf{DG}}_{\mathbf{ng}}}‖}_F^2\\ =2N_S-2\mathrm{tr}\left({{\mathbf{P}}_{\mathbf{Gng}}^{\mathbf{opt}}}^{\mathbf{H}}{\mathbf{P}}_{{\mathbf{AG}}_{\mathbf{ng}}}{\mathbf{P}}_{{\mathbf{DG}}_{\mathbf{ng}}}\right)\end{array}$$E33

The solution of this problem, which is to find the maximization of ${{\mathbf{P}}_{\mathbf{Gng}}^{\mathbf{opt}}}^{\mathbf{H}}{\mathbf{P}}_{{\mathbf{AG}}_{\mathbf{ng}}}{\mathbf{P}}_{{\mathbf{DG}}_{\mathbf{ng}}},$ is solved by what is called the orthonormal Procrustes problem [28] as follows:

$${\mathbf{P}}_{{\mathbf{DG}}_{\mathbf{ng}}}=\mathbf{V}{\mathbf{U}}^{\mathbf{H}}$$E34

where ${{\mathbf{P}}_{\mathbf{Gng}}^{\mathbf{opt}}}^{\mathbf{H}}{\mathbf{P}}_{{\mathbf{AG}}_{\mathbf{ng}}}=\mathbf{U\Sigma }{\mathbf{V}}^{\mathbf{H}}$. Then, we keep ${\mathbf{P}}_{{\mathbf{DG}}_{\mathbf{ng}}}$ fixed and solve the same optimization problem but now minimizing over ${\mathbf{P}}_{{\mathbf{AG}}_{\mathbf{ng}}}$ as follows:

$$\left({\mathbf{P}}_{\mathbf{\text{AGng}}}^{\mathbf{opt}}\right)=arg\underset{{\mathbf{P}}_{\mathbf{\text{AGng}}}}{min}{‖{\mathbf{P}}_{{\mathbf{G}}_{\mathbf{ng}}}^{\mathbf{opt}}-{\mathbf{P}}_{{\mathbf{AG}}_{\mathbf{ng}}}{\mathbf{P}}_{{\mathbf{DG}}_{\mathbf{ng}}}‖}_F^2$$E35

Similar to (33), expanding the objective function yields:

$${‖{\mathbf{P}}_{{\mathbf{G}}_{\mathbf{ng}}}^{\mathbf{opt}}-{\mathbf{P}}_{{\mathbf{AG}}_{\mathbf{ng}}}{\mathbf{P}}_{{\mathbf{DG}}_{\mathbf{ng}}}‖}_F^2=2N_S-2\mathrm{tr}\left({{\mathbf{P}}_{\mathbf{Gng}}^{\mathbf{opt}}}^{\mathbf{H}}{\mathbf{P}}_{{\mathbf{AG}}_{\mathbf{ng}}}{\mathbf{P}}_{{\mathbf{DG}}_{\mathbf{ng}}}\right)$$E36

Assuming that the IHA analog precoder ${\mathbf{P}}_{{\mathbf{AG}}_{\mathbf{ng}}}$ semi-unitary matrix, the solution that maximizes ${{\mathbf{P}}_{\mathbf{Gng}}^{\mathbf{opt}}}^{\mathbf{H}}{\mathbf{P}}_{{\mathbf{AG}}_{\mathbf{ng}}}{\mathbf{P}}_{{\mathbf{DG}}_{\mathbf{ng}}}$ in (35), is also solved by the orthonormal Procrustes problem as follows:

$${\mathbf{P}}_{{\mathbf{AG}}_{\mathbf{ng}}}=\mathbf{V}{\mathbf{U}}^{\mathbf{H}}$$E37

where ${\mathbf{P}}_{{\mathbf{DG}}_{\mathbf{ng}}}{{\mathbf{P}}_{\mathbf{Gng}}^{\mathbf{opt}}}^{\mathbf{H}}=\mathbf{U\Sigma }{\mathbf{V}}^{\mathbf{H}}$. Also, there is another way to maximize ${{\mathbf{P}}_{\mathbf{Gng}}^{\mathbf{opt}}}^{\mathbf{H}}{\mathbf{P}}_{{\mathbf{AG}}_{\mathbf{ng}}}{\mathbf{P}}_{{\mathbf{DG}}_{\mathbf{ng}}}$ in (35) as follows:

$${\mathbf{P}}_{{\mathbf{AG}}_{\mathbf{ng}}}={\mathbf{P}}_{\mathbf{Gng}}^{\mathbf{opt}}{\mathbf{P}}_{\mathbf{\text{DGng}}}^{\mathbf{H}}$$E38

Both solutions in (37) and (38) are almost the same because we assume that ${\mathbf{P}}_{{\mathbf{DG}}_{\mathbf{ng}}}$ is a non-square semi-unitary or square unitary matrix in (36) and the singular values of ${\mathbf{P}}_{{\mathbf{DG}}_{\mathbf{ng}}}{{\mathbf{P}}_{\mathbf{Gng}}^{\mathbf{opt}}}^{\mathbf{H}}$ are almost unity when $N_{\mathit{tg}}=1$ and close to unity when $N_{\mathit{tg}}>1$.

The main difference between the hybrid design in this chapter and that in [29] is that our design assumes that ${\mathbf{P}}_{{\mathbf{AG}}_{\mathbf{ng}}}$ is non-square semi-unitary matrix, and ${\mathbf{P}}_{{\mathbf{DG}}_{\mathbf{ng}}}$ is non-square semi-unitary or square unitary matrix. Our design is more versatile and applicable in various scenarios, including when the number of data streams is equal to or less than the number of RF chains. In contrast, the HD-AM (hybrid design by alternating minimization) technique can only be used when the number of data streams is equal to the number of RF chains [29]. Additionally, our derivation is based on the HA architecture, while HD-AM is only applicable to the FA architecture. Our proposed algorithm is straightforward since it calculates the hybrid precoding for each group in the HA architecture independently before using it to determine that of the entire system. Conversely, the method presented in [29] computes the hybrid precoding directly for the entire system, resulting in higher computational complexity.

Algorithm 3: Proposed IHA Hybrid Precoding

Input: The optimum unconstrained solution ${\mathbf{P}}_{{\mathbf{G}}_{\mathbf{ng}}}^{\mathbf{opt}}\in C^{\left(N_t/N_{\mathit{tg}}\right)\mathrm{x}{N}_{\mathrm{S}}}$, initialized analog procoder ${\mathbf{P}}_{{\mathbf{AG}}_{\mathbf{ng}}}\in C^{\left(N_t/N_{\mathit{tg}}\right)\mathrm{x}\left(N_{\mathit{tRF}}/N_{\mathit{tg}}\right)}$ with the element-wise normalization, and the maximum number of iterations $K$.

Output: Analog ${\mathbf{P}}_{{\mathbf{HA}}_{\mathbf{ng}}}\in C^{N_t\mathrm{x}{\mathbf{N}}_{\mathbf{tRF}}}$ such that ${‖{\mathbf{P}}_{{\mathbf{G}}_{\mathbf{ng}}}^{\mathbf{opt}}-{\mathbf{P}}_{{\mathbf{HA}}_{\mathbf{ng}}}‖}_F$ is reduced and ${‖{\mathbf{P}}_{{\mathbf{HA}}_{\mathbf{ng}}}‖}_F^2=N_S/N_{\mathit{tg}}$, where ${\mathbf{P}}_{{\mathbf{HA}}_{\mathbf{ng}}}={\mathbf{P}}_{{\mathbf{AG}}_{\mathbf{ng}}}{\mathbf{P}}_{{\mathbf{DG}}_{\mathbf{ng}}}.$

1: for $i=1:K$ do

2: Update: ${\mathbf{P}}_{{\mathbf{DG}}_{\mathbf{ng}}}=\mathbf{V}{\mathbf{U}}^{\mathbf{H}},\text{where}{{\mathbf{P}}_{\mathbf{Gng}}^{\mathbf{opt}}}^{\mathbf{H}}{\mathbf{P}}_{{\mathbf{AG}}_{\mathbf{ng}}}=\mathbf{U\Sigma }{\mathbf{V}}^{\mathbf{H}}$

3: Update: ${\mathbf{P}}_{{\mathbf{AG}}_{\mathbf{ng}}}={\mathbf{P}}_{\mathbf{Gng}}^{\mathbf{opt}}{\mathbf{P}}_{\mathbf{\text{DGng}}}^{\mathbf{H}}$

4: Element-Wise Normalization: ${\mathbf{P}}_{{\mathbf{AG}}_{\mathbf{ng}}}={\mathbf{P}}_{{\mathbf{AG}}_{\mathbf{ng}}}\oslash \left(\left|{\mathbf{P}}_{{\mathbf{AG}}_{\mathbf{ng}}}\right|\sqrt{\left(N_t/N_{\mathit{tg}}\right)}\right)$

5: end for

7: ${\mathbf{P}}_{{\mathbf{DG}}_{\mathbf{ng}}}={\mathbf{P}}_{{\mathbf{AG}}_{\mathbf{ng}}}^{\mathbf{H}}{\mathbf{P}}_{\mathbf{Gng}}^{\mathbf{opt}}$

8: ${\mathbf{P}}_{\mathbf{D}}=\sqrt{N_S/N_{\mathit{tg}}}\frac{{\mathbf{P}}_{{\mathbf{DG}}_{\mathbf{ng}}}}{{‖{\mathbf{P}}_{{\mathbf{AG}}_{\mathbf{ng}}}{\mathbf{P}}_{{\mathbf{DG}}_{\mathbf{ng}}}‖}_{\mathit{F}}}$

9: Return ${\mathbf{P}}_{{\mathbf{HA}}_{\mathbf{ng}}}={\mathbf{P}}_{{\mathbf{AG}}_{\mathbf{ng}}}{\mathbf{P}}_{{\mathbf{DG}}_{\mathbf{ng}}}$.

Algorithm 3 provides the pseudo-code for the proposed IHA precoder. The inputs of the algorithm are ${\mathbf{P}}_{{\mathbf{G}}_{\mathbf{ng}}}^{\mathbf{opt}}\in C^{\left(N_t/N_{\mathit{tg}}\right)\mathrm{x}{N}_{\mathrm{S}}}$, initialized analog procoder ${\mathbf{P}}_{{\mathbf{AG}}_{\mathbf{ng}}}\in C^{\left(N_t/N_{\mathit{tg}}\right)\mathrm{x}\left(N_{\mathit{tRF}}/N_{\mathit{tg}}\right)},$ i.e., ${\mathbf{P}}_{{\mathbf{AG}}_{\mathbf{ng}}}=\left({\mathbf{P}}_{{\mathbf{G}}_{\mathbf{ng}}}^{\mathbf{opt}}\oslash \left(\left|{\mathbf{P}}_{{\mathbf{G}}_{\mathbf{ng}}}^{\mathbf{opt}}\right|\sqrt{\left(N_t/N_{\mathit{tg}}\right)}\right)\right)$, and the maximum number of iterations $K$, where $K\ge 1$ for $N_{\mathrm{S}}<\left(N_{\mathit{tRF}}/N_{\mathit{tg}}\right)$ or $\left(N_{\mathit{tRF}}/N_{\mathit{tg}}\right)\le N_S$. In the general case of $N_S\ge 1$, the algorithm starts by computing the $\mathit{\text{ngth}}$ group ${\mathbf{P}}_{{\mathbf{DG}}_{\mathbf{ng}}}$ using the orthonormal Procrustes solution in step 2. After that, the algorithm proceeds to update the $\mathit{\text{ngth}}$ group RF precoder ${\mathbf{P}}_{{\mathbf{AG}}_{\mathbf{ng}}}$ in step 3. Step 4 ensures that the proposed RF precoder ${\mathbf{P}}_{{\mathbf{AG}}_{\mathbf{ng}}}$is satisfied exactly with constant-magnitude entries, which can be applied at RF using analog phase shifters. After the last iteration of the algorithm, ${\mathbf{P}}_{{\mathbf{DG}}_{\mathbf{ng}}}$ is updated via the maximal ratio combining (MRC), instead of the least solution, which has an impact on the Frobenius norm objective function ${‖{\mathbf{P}}_{{\mathbf{G}}_{\mathbf{ng}}}^{\mathbf{opt}}-{\mathbf{P}}_{{\mathbf{AG}}_{\mathbf{ng}}}{\mathbf{P}}_{{\mathbf{DG}}_{\mathbf{ng}}}‖}_F^2;$the least solution becomes MRC after implementing the semi-unitary analog precoder, i.e., ${\mathbf{P}}_{{\mathbf{AG}}_{\mathbf{ng}}}^{\mathbf{H}}{\mathbf{P}}_{{\mathbf{AG}}_{\mathbf{ng}}}={\mathbf{I}}_{\left({\mathit{N}}_{\mathit{tRF}}/{\mathit{N}}_{\mathit{tg}}\right)}$. After $K$ iterations, the process is completed and the algorithm finds the $\left(N_t/N_{\mathit{tg}}\right)\mathrm{x}\left(N_{\mathit{tRF}}/N_{\mathit{tg}}\right)$ proposed RF precoding matrix ${\mathbf{P}}_{{\mathbf{AG}}_{\mathbf{ng}}}$ and the $\left(N_{\mathit{tRF}}/N_{\mathit{tg}}\right)\mathrm{x}{N}_S$ baseband precoder ${\mathbf{P}}_{{\mathbf{DG}}_{\mathbf{ng}}}$ such that ${‖{\mathbf{P}}_{{\mathbf{G}}_{\mathbf{ng}}}^{\mathbf{opt}}-{\mathbf{P}}_{{\mathbf{AG}}_{\mathbf{ng}}}{\mathbf{P}}_{{\mathbf{DG}}_{\mathbf{ng}}}‖}_F^2$ is minimized. In steps 8 and 9, we ensure that the transmit power constraint is satisfied for each $\mathit{\text{ngth}}$ group and return the proposed $\mathit{\text{ngth}}$ group IHA precoder ${\mathbf{P}}_{{\mathbf{HA}}_{\mathbf{ng}}}={\mathbf{P}}_{{\mathbf{AG}}_{\mathbf{ng}}}{\mathbf{P}}_{{\mathbf{DG}}_{\mathbf{ng}}}$. The proposed $\mathit{\text{ngth}}$ group hybrid combiner ${\mathbf{W}}_{{\mathbf{HA}}_{\mathbf{ng}}}$ can be calculated in the same way.

5. Complexity analysis

This section aims to examine the implementation complexities of the proposed hybrid precoding and combining algorithms for various architectures. To simplify the analysis,

we use the following notations: $N=max\left\{N_t,N_r\right\}$ represents the maximum number of antennas, $N_{\mathrm{RF}}=max\left\{N_{\mathit{tRF}},N_{\mathit{rRF}}\right\}$ represents the maximum number of RF chains, $N_{\mathrm{g}}=max\left\{N_{\mathit{tg}},N_{\mathit{rg}}\right\}$ represents the maximum number of RF groups, and K denotes the maximum number of iterations for the proposed IFA hybrid design, ISA hybrid design, and IHA hybrid design algorithms. Moreover, we denote the number of antennas for each subarray in the SA design as $N_{\mathrm{SA}}$. Our analysis is based on the total number of floating-point operations (flops) for each hybrid precoding and combining method. Table 1 shows that the computational complexities of the proposed IFA hybrid design, ISA hybrid design, and IHA hybrid design algorithms are much lower compared to that of the FA sparse hybrid precoding method, which has a complexity of O($N^2{N}_{\mathit{RF}}{N}_S$). Furthermore, the computational complexities of the proposed IHA hybrid design and ISA hybrid design algorithms are lower than that of the IFA Hybrid design, particularly for larger numbers of groups, $N_{\mathrm{g}}$. When $N_{\mathrm{g}}>1$, the proposed IHA hybrid design and ISA hybrid design algorithms have lower hardware costs than the sparse hybrid design and the proposed IFA hybrid design. To summarize, the proposed IHA hybrid design has lower computational and hardware complexities than the proposed IFA hybrid design and is comparable to that of the proposed ISA hybrid design when $N_{\mathrm{RF}}=N_{\mathrm{g}}$.

Method	Constraints	Phase Shifters Number	Complexity
Sparse hybrid design [2]	RF precoding/combining codebooks	$N{N}_{\mathrm{RF}}$	O($N^2{N}_{\mathit{RF}}{N}_S$)
Proposed IFA hybrid design	None	$N{N}_{\mathrm{RF}}$	O($N{N}_{\mathit{RF}}^{2}K$)
Proposed ISA hybrid design	None	$N$	O($N_{\mathrm{SA}}{N}_{\mathrm{RF}}{N}_{S}K$)
Proposed IHA hybrid design	None	$\left(N_{\mathrm{RF}}/N_{\mathrm{g}}\right)N$	O($N{N}_{\mathit{RF}}^{2}K/N_g^2$)

Table 1.

Complexity of the proposed algorithms.

6. Simulation results

This section presents the numerical results to show the performance advantages of the proposed IFA, ISA, and IHA hybrid precoding/combining algorithms. We consider the case where there are only one BS and one MS at a distance of 100 m. The spacing between antenna elements is equal to $\lambda /2$. The system is assumed to operate at a 28 GHz carrier frequency in an outdoor scenario, and with a path loss exponent $n=3.4$. The channel model is described in (1), with $\overline{{\mathrm{P}}_{\mathrm{\alpha },\mathrm{i}}}=1$ for all clusters. The azimuth and elevation angles of arrival and departure (AoAs/AoDs) of the rays within a cluster are assumed to be randomly Laplacian distributed. The AoAs/AoDs azimuths and elevations of the cluster means are assumed to be uniformly distributed. We use the AoD/AoA beamforming codebooks (the exact array response of the mmWave channel) at the BSs and MSs, respectively, for the sparse hybrid design [2]. The signal-to-noise ratio (SNR) in all the plots is defined as $\mathit{SNR}=\mathrm{\rho }/{\sigma }^2$. We assume perfect channel estimation at the BS and MS. For fairness, the same total power constraint is enforced on all precoding/combining solutions. The maximum number of iterations $K$ for the proposed IHA hybrid precoder/combiner, the IFA hybrid precoder/combiner, and the ISA hybrid precoder/combiner is equal to $10$ for all data cases.

In this section, we show the spectral efficiencies achieved by the proposed IFA, ISA, and IHA hybrid precoding/combining algorithms, FA sparse hybrid design [2], and the optimal unconstrained digital method at both the BS and the MS.

Figure 3 shows the spectral efficiencies achieved by the proposed IHA hybrid precoding/combining, the FA sparse hybrid precoding/combining [2], the optimal unconstrained digital design, the proposed IFA hybrid precoding/combining, and the proposed ISA hybrid precoding/combining in a $256\mathrm{x}64$ uniform planar arrays (UPAs) mmWave system for different SNR values with $N_S∊\left\{2,8\right\},N_{\mathit{tRF}}=N_{\mathit{rRF}}∊\left\{4,16\right\},\text{and}{N}_g∊\left\{1,2,4,8,16\right\}$. The spectral efficiency performance of the proposed IFA hybrid precoder/combiner is close to that of the unconstrained digital one and better than those of other methods for all cases. The proposed IHA hybrid precoding/combining method outperforms the ISA hybrid precoder, regardless of the number of data streams $N_S$ and the number of groups $N_g$. Also, the proposed IHA hybrid precoding/combining design outperforms the FA sparse hybrid design when $N_g∊\left\{1,2\right\}$ for $N_S=2$ and 8. The performance of the proposed IHA hybrid precoding/combining is degraded with the increase of $N_g$, which is equivalent to the decrease of phase shifters, leading to an increase of the interference between data streams. However, when $N_g∊\left\{4,16\right\}$, the proposed IHA hybrid precoding/combining becomes similar to the SA architecture with better performance compared to the proposed ISA hybrid precoding/combining. Also, when $N_g=1$, the performance of IHA hybrid precoding/combining is close to that of the proposed IFA hybrid precoding/combining.

Figure 3.

Average spectral efficiency achieved by the proposed iterative hybrid array (IHA) precoding/combining with K = 10, compared to the full array (FA) sparse hybrid precoding/combining design [2], the optimal unconstrained digital precoding/combining, iterative full array (IFA) hybrid precoding/combining design, and the iterative subarray (ISA) hybrid precoding/combining, for a $256\mathrm{x}64$ uniform planar arrays (UPAs) mmWave system for different signal-to-noise ratio (SNR) values with $N_S∊\left\{2,8\right\},\text{and}{N}_{\mathit{tRF}}=N_{\mathit{rRF}}∊\left\{4,16\right\}$.

In Figure 4, we use the same methods as they were used in Figure 3 in a $64\mathrm{x}16$ UPAs mmWave system for different SNR values, with $N_S∊\left\{2,4\right\},N_{\mathit{tRF}}=N_{\mathit{rRF}}∊\left\{4,8\right\},\text{and}{N}_g∊\left\{1,2,4,8\right\}$. We obtain the same results as in Figure 3. However, the proposed IHA hybrid precoding/combining method overlaps with the proposed ISA hybrid precoding/combining when $N_g=8$and $4$ for $N_S=4$and 2, respectively, where the numbers of BS and MS antennas are reduced compared to Figure 3.

Figure 4.

Average spectral efficiency achieved by the proposed iterative hybrid array (IHA) precoding/combining with K = 10, compared to the full array (FA) sparse hybrid precoding/combining design [2], the optimal unconstrained digital precoding/combining, the iterative full array (IFA) hybrid precoding/combining design, and the iterative subarray (ISA) hybrid precoding/combining, for a $64\mathrm{x}16$ UPAs mmWave system for different signal-to-noise ratio (SNR) values with $N_S∊\left\{2,4\right\},\text{and}{N}_{\mathit{tRF}}=N_{\mathit{rRF}}∊\left\{4,8\right\}$.

Figure 5 shows the performance when the number of RF chains $N_{\mathit{tRF}}=N_{\mathit{rRF}}$ is greater than the number of data streams, where $N_S∊\left\{2,4\right\}$, $N_g∊\left\{1,2,4,8\right\}$, and the SNR is fixed to 0 dB over the whole range of RF chains in a $256\mathrm{x}64$ UPAs mmWave system. The spectral efficiency of the proposed IFA hybrid precoding/combining is close to that of the unconstrained digital one with the increase of the RF chains. The performance of the IHA hybrid precoding/combining becomes worse with the increase of $N_g$, where the interference of data streams increases. The performance of the IHA hybrid precoding/combining is much better than that of the proposed ISA hybrid precoding/combining, regardless of the number of $N_g$. Also, the proposed IHA hybrid precoding/combining outperforms the FA sparse hybrid design when $N_g∊\left\{1,2\right\}$for any data stream $N_S$; however, the FA sparse hybrid design outperforms the proposed IHA hybrid precoding/combining when $N_g∊\left\{4,8\right\}$, but the performance gap between them reduces with the increase of the RF chains. Also, when $N_g=1$, the performance of IHA hybrid precoding/combining is close to that of the proposed IFA hybrid precoding/combining.

Figure 5.

Average spectral efficiency achieved by the proposed iterative hybrid array (IHA) precoding/combining with $K=10$ compared to the full array (FA) sparse hybrid precoding/combining [2], the optimal unconstrained digital precoding/combining, the iterative full array (IFA) hybrid precoding/combining with $K$ =10, and the iterative subarray (ISA) hybrid precoding/combining with $K=10$ for $256\mathrm{x}64$ uniform planar arrays (UPA) mmWave systems for signal-to-noise ratio (SNR) = 0 dB with $N_S∊\left\{2,4\right\}$ and different radio frequency (RF) chains.

Figure 6 shows the spectral efficiency achieved by the same methods when the number of RF chains equals the number of data streams, varying from 2 to 16, in a $256\mathrm{x}64$ UPAs mmWave system with $N_g∊\left\{1,2,4,8\right\}$. The SNR is fixed to 0 dB for any number of RF chains. When $N_g=1$, the performance of IHA hybrid precoder overlaps with that of the proposed IFA hybrid precoding/combining, and both are close to the unconstrained digital one. The performance of the proposed IHA hybrid precoding/combining becomes worse with the increase of $N_g$, where the interference of data streams becomes higher. As seen in Figure 6, the proposed IHA hybrid precoding/combining outperforms the FA sparse hybrid precoding/combining and the proposed ISA hybrid precoding/combining, especially for a large number of $N_g$, and$N_S=N_{\mathit{tRF}}=N_{\mathit{rRF}}$.

Figure 6.

Average spectral efficiency achieved by the proposed hybrid array (HA) precoding/combining using Algorithm 2 with $K=10$ compared to the full array (FA) sparse hybrid precoding/combining [2], the optimal unconstrained digital precoding/combining, iterative full array (IFA) hybrid precoding/combining with $K=10$, and the iterative subarray (ISA) hybrid precoding/combining with $K=10$for $256\mathrm{x}64$ uniform planar arrays (UPAs) mmWave systems for signal-to-noise ratio (SNR) = 0 dB with $N_S=N_{\mathit{tRF}}=N_{\mathit{tRF}}$.

In conclusion, although we use the proposed IHA hybrid design in both transmitter and receiver, its performance is acceptable, especially for $2\le N_g<N_{\mathit{tRF}}$ and $2\le N_g<N_{\mathit{rRF}}$, when compared to the higher hardware complexity of FA hybrid designs, such as the FA sparse hybrid design and the proposed IFA hybrid design. All FA hybrid designs require a higher hardware complexity in the BS and MS, with a higher number of phase shifters in the BS and MS, which is equal to $N_t{N}_{\mathit{tRF}}+N_r{N}_{\mathit{rRF}}$, whereas the number of phase shifters for the IHA hybrid precoder/combiner is equal to $\left(\frac{N_t{N}_{\mathit{tRF}}}{N_g}\right)+\left(\frac{N_r{N}_{\mathit{rRF}}}{N_g}\right)$. The constraint of the analog and baseband precoding/combining matrices helps to build the structure of block diagonal matrices in the proposed IHA hybrid precoder/combiner, yielding higher gains compared to the other methods. When $N_g=N_{\mathit{tRF}}$, the proposed HA structure becomes similar to the SA one; the performance of the proposed IHA hybrid precoding/combining design gives higher gains compared to the proposed ISA hybrid precoding/combining design, especially for a large number of BS antennas.

Also, when $N_g=1$, the proposed HA structure becomes similar to the FA one, and the performance of the proposed IHA hybrid precoding/combining design is comparable to that of the proposed IFA hybrid precoding/combining design. The number of iterations should be 10 or less because the gain after that will be very small, which is confirmed by our results that we did not include in this chapter.

7. Conclusion

In this chapter, we have studied and discussed the issue of hybrid precoding and combining techniques in mmWave MIMO systems for different array architectures. We presented the system models of FA, SA, and HA and solved the optimization problem of hybrid precoding and combining to maximize the spectral efficiency of each architecture. Additionally, we proposed iterative hybrid precoding and combining algorithms for all architectures. The simulation results showed that the proposed algorithms can enhance the spectral efficiency performance of mmWave MIMO systems with lower complexity and hardware requirements than traditional hybrid design methods. The findings of this chapter are expected to be of significant interest to researchers, engineers, and students working in the field of mmWave communications and MIMO systems, as they provide insights into improving the spectral efficiency and performance of wireless communication systems. Overall, this work contributes to the development of efficient and cost-effective solutions for next-generation wireless communication systems.