Comparative Study of Beamforming Algorithms for MM-wave Hybrid MIMO Systems

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Irfan Ali Awan

COMPARATIVE STUDY OF BEAMFORMING ALGORITHMS FOR MM-WAVE HYBRID MIMO SYSTEMS

Faculty of Electrical

Engineering

Master’s Thesis

October 2020

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ABSTRACT

Irfan Ali Awan: Comparative study of beamforming algorithms for mm-wave hybrid MIMO systems

Master’s Thesis Tampere University

Master’s in Electrical Engineering (Wireless Communications and RF) October 2020

In this thesis, we will perform a comparative study on beamforming algorithms for hybrid millimetre-wave MIMO systems. We will compare different hybrid beamforming algorithms based on their throughput, complexity, transmit and receive antennas. All the considered algorithms are implemented in MATLAB and according to simulation results, the algorithms are compared. In order to gain deep insights about each algorithm, different metrics will be considered. These metrics are spectral efficiency vs SNR, throughput vs transmit antennas, and throughput vs receive antennas. Other important metrics that we will use to evaluate different algorithms are complexity and energy consumption. In this thesis, we will focus on algorithms designed for partially connected hybrid architecture.

.

Keywords: MM-wave, Beamforming, Hybrid Beamforming, Partially Connected, Algorithms, Massive MIMO.

The originality of this thesis has been checked using the Turnitin Originality Check service.

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PREFACE

First of all, I would like to thank Allah for his ultimate support and for providing me such an opportunity to do my thesis at Tampere University, Finland. After that, I am very grateful to my parents for their prayers and their full support in every aspect of my life and making me able to achieve my goals.

My thesis topic is “Comparative study of beamforming algorithms for mm-wave hybrid MIMO systems” which is supervised by Lauri Anttila, who is working as a University Researcher at Tampere University. In the company, my Line Manager is Juha Liias and my thesis is being supervised by Mohammad Majidzadeh.

Being a student of Wireless Communications and RF, I am thankful to both of my supervisors at the company and University for giving me this opportunity to write this thesis under their supervision. While writing this thesis, I faced many challenges but with the help of my supervisors in university and company, I was capable to cope up with them. I was able to learn new concepts with these problems. While writing the thesis, I was able to understand the latest technology of mobile networks.

Finally, the presence of my family members, friends, and my supervisors made me able to achieve this goal, who were always supporting me in my difficult time.

Tampere, October 2020 Irfan Ali Awan

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LIST OF FIGURES:

Figure 1:MM-wave frequency range... 3

Figure 2:Basic MIMO structure ... 4

Figure 3: Single Input Ingle Output (SISO) Structure ... 5

Figure 4:Single Input Multiple Output (SIMO) Structure ... 5

Figure 5:Multiple Input Single Output (MISO) Structure ... 6

Figure 6:Multiple Input Multiple Output (MIMO) Structure. ... 6

Figure 7:Multiuser MIMO with single receive antennas ... 7

Figure 8:Multiuser MIMO with multiple receive antennas ... 7

Figure 9: Principle of Beamforming ... 8

Figure 10: Transmitted signal interferences ... 9

Figure 11: Beamforming Structure ... 10

Figure 12: Figure (a) is showing fully connected architecture and Figure (b) is showing partially connected architecture in the Analog Precoder A. ... 11

Figure 13: Figure (a) is showing full array based architecture and Figure (b) is showing subarray based architecture in the Digital Precoder D. ... 11

Figure 14: A System model. ... 12

Figure 16: Fully Digital ZF Algorithm Architecture. ... 15

Figure 17: SIC Hybrid Precoding ... 20

Figure 18: Inverters and switches-based architecture ... 23

Figure 19: Throughput versus Signal to noise Ratio with Nt=64 ... 31

Figure 24: Throughput versus number of transmitter antennas with Nu=4 ... 36

Figure 25: Throughput versus number of transmitter antennas with Nu=2 ... 38

Figure 26: Throughput versus number of receiver antennas ... 39

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LIST OF SYMBOLS AND ABBREVIATIONS

2G Second Generation

3G Third Generation

4G Fourth Generation

5G Fifth Generation

ACE Adaptive cross-entropy

ADC Analog to Digital converter Alt-min Alternative minimum

AWGN Additive white gaussian noise

BS Base station

CE Cross entropy

Ch Channel

CSI Channel state information

dB Decibels

DL Downlink

FR Frequency Range

GPP Generation Partnership Project MIMO Multiple Input Multiple Output MISO Multiple Input Multiple Output

MM MM-Wave

MRT Maximum Ratio Transmission

MU Multiuser

Nr Number of receive antennas

Nt Number of transmitters

Nu Number of users

PS Phase shifters

QCQP Quadratic constraint Quadratic programming

SDP Semidefinite Programming

SI Switch and Inverter

SIC Successive interference cancellation SIMO Single Input Multiple Output

SINR Signal to interference-plus-noise ratio SISO Single Input Single Output

SNR Signal to Noise Ratio

SVD Singular value decomposition

SW Switches

UE User Equipment

ULA Uniform Linear Array

UP Uplink

UPA Uniform Planar Array

ZF Zero Forcing

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1 INTRODUCTION

The increasing world population demands the requirement of the fastest wireless communications. To meet the requirement of the fastest wireless communications and larger bandwidth, extensive research work has been carried out to forge ahead of the communication speed. The advanced wireless communication system utilizes spatial multiplexing in which multiple transceiver antennas are used to maximize throughput and to make communications faster. In mm-wave, hybrid beamforming is considered as a cheap and efficient solution to realize spatial multiplexing [1]. Typically, this technology has been proposed in the development of fifth-generation (5G) wireless networks to offer higher data rates and low latency to a large number of users.

Digital beamforming is widely used in lower frequencies. In such systems, multiple RF chains and ADC are which ultimately increases the overall cost of the system [2]. These increased number of integrated modules require to end up consuming significant power, thus, limits their application in the industry. To make them cost-effective and comparatively simpler implementation, researchers proposed an idea of hybrid beamforming in which both digital and analog beamforming is used which consumes less power and is less complex than digital beamforming [3]. Later on, various algorithms and beamforming techniques were introduced to attain higher spectral efficiencies and lower power consumption. Therefore, the algorithms characterized by spectral efficiencies and performance are implemented following the nature of the application.

In academic and industry, research is in progress about 5G wireless networks to meet and solve multiple technical problems and challenges [4]. With the advancements in the electronics industry and the increased number of portable devices, there is a rapid increase in internet users. Additionally, the 3D video games, virtual reality (VR) video streaming, and the internet of things require higher bandwidths for the smooth opera- tions. Therefore, the need arises to develop solutions with larger bandwidth to offer higher data rates for smooth, uninterrupted, and reliable wireless communication. The purpose is to provide a larger bandwidth to accommodate such high data rates so that the consumers can enjoy uninterruptable communication. So, for this purpose, 5G will provide higher data rates and bandwidth which will be almost 100 times more efficient than the 4G system. To provide this higher bandwidth and at the same time maintaining

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the energy-efficient design, millimeter-wave plays a vital role in hybrid beamforming in the 5G wireless network.

In this thesis, we will analyze various hybrid beamforming schemes for massive MIMO transceiver systems. We will also consider a different antenna configuration along with several hybrid beamforming algorithms. Considered algorithms will be compared based on spectral efficiency, complexity, and power consumption. We will also conclude that every algorithm has its own feasibility that where it will be used, however, single-cell multi-users use cases will be considered in this study.

1.1 Notations

Notations: Lower-case boldface letters are representing vectors and upper-case boldface letters are representing matrices; (. )^𝑇 represents the transpose, (. )^𝐻represents conjugate transpose, (. )⁻¹represents conjugate transpose and |.| represents inversion a determinant of a matrix respectively; whereas ||. ||₁ denotes the 𝑙₁and ||. ||₂ is representing 𝑙₁-vector’s norm; ||. ||_𝐹 represents matrix having the Frobenius norm; Re{.} is the real part and Im{.} is the imaginary part of the complex number; E(.) is used to denote the expectation; and 𝐼_𝑁 is representing the N×N identity matrix.

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2 OVERVIEW

In this section, we will discuss the basic working principle, an overview of MM-wave, MIMO, beamforming, hybrid-beamforming, and various types of beamforming.

2.1 MM-Wave

The existing third-generation partnership project (3GPP) specifications which are used for 2G,3G and 4G mobile communications uses the frequency ranges less than 6 GHz [5]. Mm-wave are operated in the frequency ranges between 30 GHz to 300 GHz [6].

They are named as mm-wave as their respective wavelengths of these frequencies lie between 1 mm to 10 mm. 5G wireless technology, frequency ranges i.e. FR1 and FR2, are used to define 5G New Radio (NR). The frequency span in between 24.25 GHz to 52.6 GHz is known as frequency range 2 (FR2), including some of the mm-wave frequency range whereas, (FR1) comprises the frequency range from 450 MHz to 6 GHz which includes the LTE frequency range and the cm-waves [7]. Typically, the mm-wave range possesses shorter wavelengths, but their bandwidths are higher as compared to FR1 [8].

The needs for mm-wave have increased due to the enhanced mobile broadband ser- vices, playing a major role to drive the 5G to meet the increasing demand for faster and better mobile experiences [9]. The mobile networks are facing a drastic increase in the demand for mobile data as the number of internet-enabled devices is increasing day- by-day, which shares and consumes high-definition multi-media. With more advanced electronic devices, the amount of the online data generated in the form of the high definition videos, cloud computing, and virtual reality had lead to an ever-increasing demand for faster and reliable connectivity devices [10].

While using these mm-wave band we can use larger bandwidths (Hundreds of MHz) which provides extremely high data rates and a significant increase in the capacity as well. Figure 1 shows the mm-wave frequency range.

Figure 1: MM-wave frequency range

300GHz

3GHz 30GHz

cm wave band mm-wave band

300MHz 6GHz band

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Historically, mm-wave were not used due to its high propagation loss and blockage by different objects such as hand, head, body, and buildings [11]. Because of advancements in antenna techniques, these problems can be sorted out and we can tackle with higher propagation loss [12]. One of the techniques being used is to multiple-input multiple-output (MIMO), which will be discussed in the next subsection 2.2. Due to its high frequency and shorter wavelength, it is possible to achieve the data rates up to or over 10 Gbps whereas, in other frequency ranges such as microwaves frequency range, the maximum achievable data rate is 1 Gbit/s.

2.2 MIMO Basics

In this subsection, we will discuss the basics of MIMO. MIMO technology uses multiple transmitters and multiple receivers to transfer data simultaneously [13]. MIMO technology uses a radio-wave phenomenon called multipath where the transmitted signal ar- rives at the receiver using multiple paths, being reflected from various objects introduc- ing various time delays and SNR levels. Figure 2 below shows the basic MIMO structure.

Figure 2: Basic MIMO structure

By using multiple antennas in MIMO technology, it is possible to increase the channel capacity and throughput of the channel. Due to this characteristic MIMO technology has been widely deployed in the industry during recent years. There are different types of antenna configurations, some of these are briefly discussed below:

2.2.1 SISO

One type of antenna configuration is SISO-Single Input Single Output [14]. Figure 3 shows the structure of the SISO configuration, in which both the transmitter and the

Tx Rx

1 2 3 n

1 2 3 m

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receiver have one transmitting and receiving antenna, respectively. However, its spectral efficiency is very low due to the low number of antennas and is limited in its performance as interference and fading will impact the system more than a MIMO. Channel bandwidth and the Signal to noise ratio decides the throughput of the channel [15].

Figure 3: Single Input Single Output (SISO) Structure

2.2.2 SIMO

The single input and multiple output configuration is another configuration of MIMO technology, where the transmitter has a single antenna, whereas, the receiver has multiple antennas. Figure 4 shows the structure of a single input multiple outputs (SIMO) structure. This is also known as receive diversity. This type of configuration is used to enable the receiver system robust towards the fading effect and let it receive the signal from a number of independent sources. The advantage of SIMO is that it is easy to implement, and it also has some disadvantages at the receiver side while processing the received signal. SIMO can be used in many applications such as digital televisions (DTV), metropolitan area networks (MANs), and wireless local area networks (WLANs) but when the receiver is located in a mobile device like a cell phone handset the processing may be limited by size, cost, and battery drain.

Figure 4: Single Input Multiple Output (SIMO) Structure

Tx Rx

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2.2.3 MISO

Multiple Input Single User (MISO) is also known as transmit diversity. In this case, the transmitter has multiple antennas and the receiver has only one antenna. Figure 5 shows the structure of the MISO configuration. Identical data is transmitted from the multiple antennas at the transmitter end and the receiver is then able to receive the optimum signal which is then used to extract the data. The advantage of MISO is that the coding/processing is moved from the receiver to the transmitter end, lowering the size, cost, and enhancing the battery life of the mobile device receiver.

Figure 5: Multiple Input Single Output (MISO) Structure

2.2.4 MIMO

Multiple Input Multiple Output (MIMO) is the configuration in which both the transmitter and the receiver has multiple antennas, where multiple signal paths are used to carry the user data. By using this method, the data capacity of the radio link is multiplied using multiple transmit and receive antennas. The following Figure 6 shows the structure of this configuration.

Figure 6: Multiple Input Multiple Output (MIMO) Structure.

Tx Rx

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2.2.5 Multi-user MIMO

In a multi-user MIMO configuration, we have one transmitter and multiple receivers.

The receivers could have either have one antenna or multiple antennas. However, in this research work, we are considering the receivers with only one antenna. Figure 7 shows the configuration with multiuser MIMO with single receive antennas and Figure 8 illustrates multi-user MIMO with multiple receive antennas.

Figure 7: Multiuser MIMO with single receive antennas

Figure 8: Multiuser MIMO with multiple receive antennas

2.3 Beamforming

Antenna Beamforming is a technique which has widely been used in the cellular network and especially in 5G technology [16]. The antenna beamforming system consists of multiple antennas, which changes the radiation pattern of the overall system by changing the phase and amplitude of the transmitted signal [17]. Beamforming requires multiple antennas elements for the constructive and destructive combination of signals.

When two signals with equal amplitude, the same frequency are added together in phase, then the resultant signal has twice the signal level. If two signals with equal

Tx Rx

Rx

Rx Rx

Tx Rx

Rx

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amplitude, the same frequency are added with opposite phases then the two signals cancel each other. This causes constructive and destructive interference. Figure 9 shows how the beams are formed depending upon the number of elements. As we are increasing the number of radiating elements the more energy it is transmitting, and secondary lobes start appearing with the main lobe. We can also conclude that as we are increasing the radiating elements, the directivity of the signal increases.

Figure 10 shows how the signals are transmitted from the antenna element and how constructive and destructive interference is formed. There is a different kind of beamforming such as digital beamforming, analog beamforming, and hybrid beamforming, the hybrid beamforming being a combination of digital and analog beamforming. In this research work, we will only consider hybrid beamforming.

Figure 9: Principle of Beamforming

Single radiating element Two radiating elements Four radiating elements Twice the energy

Main Lobe

Secondary lobe

Four times the energy

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Figure 10: Transmitted signal interferences

Figure 11 shows the beamforming structure of the hybrid beamforming at the transmitter side. Analog circuitry comprises of the phase shifters and the amplitude tuners for each RF channel, which is driving each antenna element. The beamforming circuit being cascaded between the transmitter antennas and the analog circuit is further connected to the power amplifiers.

Tx

Rx

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Figure 11: Beamforming Structure

2.4 Hybrid Beamforming

Hybrid beamforming combines the advantages of both analog and digital beamforming architectures. The basic idea is to use the phased array antennas. In a fully digital beamforming system, a digital to analog converter (DAC), and an RF chain per antenna is needed making the overall system energy inefficient and complex structured. To make it energy-efficient and less complex, hybrid beamforming is introduced which requires less DAC’s and RF chains [1]. As hybrid beamforming uses both analog and digital beamforming so only one RF chain is required for a subset of antennas which reduces complexity and power consumption. The hybrid beamforming systems are classified according to the analog/RF beamforming architecture. There are two types of analog architectures, fully connected architecture, and partially connected architecture [18]. In partially connected architecture every single RF chain is connected to a single subarray and in fully connected architecture every single RF chain is connected to all the antennas in the Analog part, as can be seen in Figure 12 (a) and (b). The

Baseband Processing

RF

. . . . . .

RF chains

Beamforming .

.

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reason for choosing the partially connected architecture is that it consumes less power and is less complex as compared to the fully connected architecture.

When we talk about the digital beamforming structures, we have two types of structures, i.e. subarray-based and full array-based structures [2]. In full array-based structure, every data stream is transmitted to all the available RF chains through the digital precoder part. In sub-array-based structure, each data stream is transmitted to a single RF chain [3]. These types of structures can be seen in Figure 13 (a) and (b), which is representing the digital part of the hybrid beamforming.

Figure 12: Figure (a) is showing fully connected RF architecture and Figure (b) is showing partially connected RF architecture in the Analog Precoder.

Figure 13: Figure (a) is showing full array-based architecture and Figure (b) is show- ing subarray-based architecture in the Digital Precoder D.

RF chain

.

. .

. . . .

. . .

. .

. . . .

. .

Analog Precoder Analog Precoder

(a) Fully Connected (b) Partially Connected

11 12

𝑁 1 𝑁 2 𝑁 𝑁 1𝑁

𝑁 𝑁 1 11

2

𝑁

RF chain

. . . .

. .

. . .

1

𝑁

RF chain

. . .

(a) (b)

Digital Precoder D Digital Precoder D

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3 SYSTEM MODEL

Figure 13 illustrates two typical architectures of hybrid beamforming and is used in mm- wave MIMO systems [2]. The architecture of a full and partially connected MIMO system is shown in Figure 13(a) and Figure 13(b), respectively. In this section, the narrow- band single-cell Multi-user-MISO system is considered with partially connected architecture operating in downlink (DL) mode.

The system consists of a base station (BS) having N number of antenna elements serving K number of single-antenna users. Each RF is serving a specific number of antennas known as subarrays, having several small antennas. There is a total Na number of RF chains and subarrays. The total number of antennas in the subarray is n=N/Na. The total number of users considered is equal to the number of RF chains, that are K=Na. Here it is assumed that the base station (BS) has the perfect Channel State Information (CSI).

Figure 14: A system model.

The above Figure 14 shows the MU-MISO system model for the hybrid beamforming and Figure 13 (a) and (b) is showing two architectures, fully connected and partially connected, respectively. We can further divide this partially connected architecture into subarray-based architecture and full array-based architecture in the analog precoder part as can be seen in Figure 12 (a) and (b), respectively. When we discuss the full array-based system, in which RF chains are connected to all the data streams, the digital precoder 𝐅BB∈ 𝐶^𝑁 ^×𝐾 can be written as:

𝐅_BB= (

11 ⋯ _1𝑘

⋮ ⋱ ⋮

𝑁 1 ⋯ _𝑁 _𝐾) (1)

Digital Precoder

Analog Precoder

RF chain RF chain

RF chain 1

2

𝑘

User 1

User 2

User K

. . .

. . K .

1

2

𝑘

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As far as subarray-based beamforming is concerned, every RF chain is connected to a single data stream. The digital precoder can be illustrated with the diagonal matrix 𝐅_BB= 𝑖𝑎𝑔 ( ₁₁, ₂₂, … . _𝑁 _𝐾). The analog precoder becomes 𝐅_RF ∈ 𝐶^𝑁×𝑁 can be written as

𝐅_RF = (

𝑎₁ 0 ⋯ 0

0 𝑎₂ ⋯ 0

⋮ ⋮ ⋱ ⋮

0 0 ⋯ 𝑎_𝑁

) (2)

where 𝒂_𝑖 𝜖 𝐶^𝑛×1, 𝑖 = 1,2,3, … . , is the analog precoder of the ith subarray, and 0𝜖𝐶^𝑛×1 is a zero vector. The signal which is received to the kth can be written as:

𝑦_𝑘 = 𝐡_𝑘^𝐻𝐅_RF𝐯_D^k _𝑘+ 𝐡_𝑘^𝐻 ∑ 𝐅_RF𝐯_Dⁱ _𝑖+ 𝑧_𝑘

𝐾

𝑖=1,𝑖≠𝑘

(3)

where h_𝑘𝜖𝐶^𝑛×1 is known as the channel vector which is representing the channel from the transmit side to the 𝑘th user, 𝐯_D𝜖𝐶^𝑛×1 is known as 𝑘th column which is present in the digital beamforming matrix, _𝑘 is known as the user 𝑘 data symbol, and 𝑧_𝑘~𝐶 (0, ₀) is known as the additive white gaussian noise of the user 𝑘. For 𝑘th user, rate expression can be written as:

𝑅_𝑘 = log₂(1 + |𝐡_𝑘^𝐻𝐅_RF𝐯^𝑘|²

0+ ∑^𝐾_{𝑖=1,𝑖≠𝑘}|h_𝑘^𝐻𝐅_RF𝐯_Dⁱ|²) (4) Optimizing of the system is aimed to maximize the average sum rate of the users. The optimization problem in mathematical form can be expressed as:

𝑚𝑎𝑥𝑖𝑚𝑖𝑧𝑒 ∑ 𝑅_𝑘(𝐅_RF, 𝐅_BB)

𝐾

𝑘=1

. 𝑡 𝑡𝑟(𝐅_RF𝐅_BB𝐅_BB^H 𝐅_RF^H) ≤ 𝑃 (5) where digital beamforming 𝐅BB and analog beamforming 𝐅RF are the optimization variable and the maximum power denoted by P is available at the base station. As equation number (5) is non-convex, so we cannot solve optimally. To find sub-optimal solutions, convex approximation methods can be used. There are different algorithms based on their complexity and performance which are categorized according to their needs [19].

They are discussed later in this section.

The channel model used for the performance evaluation and to produce MU-MISO channel realization with geometric uniform linear array (ULA) settings is based on Saleh-Valenzuela model [20] and it is based on the stochastic approach. This model is

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representing the stochastic properties of the delays of arrival and amplitudes of resolv- able multipath components (MPCs) in indoor wideband wireless transmission system.

It basically models the arrival of MPCs in the form of cluster, where cluster arrivals and MPCs arrival within each cluster is governed by the Poisson Processes [21].

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4 ALGORITHMS

In this section, we will go through some different hybrid beamforming algorithms for partially connected MU-Multiple input single-output (MISO) systems having a large number of antenna arrays. All the algorithm employs subarray-based architecture. For all the hybrid beamforming algorithms, we will allocate equal powers, i.e., 𝑃_𝑘 = 𝑃/ , 𝑘 = 1,2, … , . Values of the power are gathered in a diagonal matrix as P=diag(𝑃₁, 𝑃₂, … , 𝑃_𝐾). In the sub-array-based structure, every user is allocated with a single subarray. In this method, every RF chain receives a single data stream. Hence, the digital precoder can be represented as a diagonal matrix as 𝐅_BB= 𝑖𝑎𝑔( ₁₁, ₂₂, … , _𝑘𝑘), in which streams are routed to the RF chains and remaining beamforming is processed at the analog part. The following are the algorithms which are compared here in this paper:

1. Fully Digital Zero Forcing algorithm 2. Fully Digital MRT algorithm

3. Hybrid subarray based Zero-Forcing matching algorithm 4. Maximum ratio Transmission (MRT) matching algorithm 5. Successive interference cancellation (SIC) algorithm 6. Adaptive Cross entropy (ACE) algorithm

7. SDR-Alt Min algorithm

4.1 Fully Digital Zero Forcing (ZF) Algorithm:

The Fully digital Zero forcing beamforming architecture [22] consists of only Digital Pre- coding, DAC’s, and RF chains as can be seen in the following Figure 16.

Digital Precoding

RF chain

RF Chain

. . .

DAC

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Figure 15: Fully Digital ZF Algorithm architecture

Here, in the case of Fully Digital beamforming ZF matrix 𝐕_𝑍𝐹 can be calculated from the propagation channel 𝐇 ∈ 𝐶^𝐾×𝑁 as:

𝐕_𝑍𝐹 = 𝐇^𝐻(𝐇𝐇^𝐻)⁻¹ (6)

4.2 Fully Digital Maximum Ratio Transmission (MRT) Algorithm:

The Fully digital Maximum Ratio Transmission (MRT) architecture consists of the same architecture as shown in Figure 16 above. In fully digital beamforming, the precoding is performed in the digital baseband due to which there is flexibility in the vector or precoding matrix. Here, we are considering the linear Maximum Ratio Transmission (MRT) precoder, which is designed in such a way that it increases the SNR for the specific user. To calculate the MRT precoder, we only need the channel vectors between the user equipment and the serving base stations. We can calculate the Fully digital MRT matrix as:

𝐕_𝑀𝑅𝑇 = 𝐇^𝐻

‖𝐇‖ (7)

4.3 Zero Forcing matching Algorithm:

The purpose of this algorithm [3] is to drive the hybrid beamformer closer to the fully digital zero-forcing solution of each user. The optimization problem can be written as:

minimize

F_RFF_BB ||𝐕_ZF− 𝐅_RF𝐅_BB||_𝐹 (8) where 𝐕_ZF represents the fully digital beamforming composed of Zero Forcing beamforming vectors representing all users. As 𝐅_RF and 𝐅_BB in the above equation (8) is considered as non-convex. Then it is difficult to solve it optimally. To tackle this, a sub- optimal solution can be used by using a technique where variables are optimized one by one while keeping other variables fixed. To start the optimization process, we first initialize the analog beamformer and after that optimize the digital beamformer which can be written as:

minimize

F_BB ||𝐕_ZF− 𝐅_RF𝐅_BB||

𝐹 (9)

Further, we can fix the digitally optimized beamformer and then optimize the analog beamformer. Then we can solve the analog beamforming part from:

minimize

F_RF ||𝐕_ZF− 𝐅_RF𝐅_BB||

𝐹 (10)

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This procedure is continued until ít reaches the maximum number of iterations or the difference between the two successive iterations crosses the threshold value. The convex nature of the optimization step helps to improve the objective function so that the sub-optimal solution can be possible. If the optimization tools are used then these optimization problems can be solved. Additionally, in step (9) a constraint of orthogonality is added in the optimization of digital beamforming so that the hybrid beams become orthogonal to each other. We can express the constraint as:

(𝐅_RF𝐅BB)_𝑖^𝐻(𝐅_BB𝐅RF)_𝑗≤ 𝜖 ∀𝑖 ≠ 𝑗 (11) In the sub-array based Zero Forcing method, each sub-array utilizes zero-forcing beamforming to its corresponding user while it nullifies the interference of the other users. If we want interference cancellation for other users, then the number of antennas in every subarray should be equal to or more than the number of users. When we stacked the vectors of the channel, then normalizing the subarray based zero-forcing beamforming and it can be formed such that:

𝐕̅ =

( [𝐕₁]₁

||[𝐕₁]₁|| 0 ⋯ 0 0 [𝐕₂]₂

||[𝐕₂]₂|| ⋯ 0

⋮ ⋮ ⋱ ⋮

0 0 ⋯ [𝐕_K]_K

||[𝐕_K]_K||)

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where 𝐕_𝐾= 𝐇_𝑘^𝐻(𝐇_𝐾𝐇_𝐾^𝐻)⁻¹. The above equation 12 is representing the analog precoding part. Here, 𝐕_K is the matrix and in [𝐕_𝐾]_𝐾 the outer K is representing the 𝑘th column. The corresponding channel can be shown with the matrix 𝐇_𝑘𝜖𝐶^𝑛×𝐾 between the 𝑘th subarray and the available K users and [. ]_𝑘 represents the 𝑘th column of the matrix argu- ment. Finally, the normalized 𝐕̅ in then used to calculate the beamforming which can be written as 𝐕 = √𝐏𝐕̅ also can be seen in Table 1.

Table 1: Zero Forcing Matching Algorithm 1: Initializing 𝐅_RF

2: Iterate

3: Solving 𝐅_BB from (7) while keeping 𝐅_RF fixed 4: Solving 𝐅_RF from (8) while keeping 𝐅_BB fixed 5: until it meets {Max iterations |Threshold}

6: V1← normalized 𝐅_RF𝐅_BB 7: V←√𝑷 𝑽̅

8: Return V

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4.4 Maximum Ratio Transmission (MRT) Matching Algorithm

The purpose of the maximum ratio transmission (MRT) algorithm [3] is to bring closer hybrid beamforming and the fully digital MRT beamforming for every user.

The optimization process follows the same steps as that of the zero-forcing (ZF) matching algorithm. The formulation of the MRT-matching algorithm can be written as:

𝑚𝑖𝑛𝑖𝑚𝑖𝑧𝑒

𝐅_RF,𝐅_BB ||𝐕_MRT− 𝐅_RF𝐅_BB||_F (13) where 𝐕_MRT stands for the fully digital beamformer which contains vectors of MRT beamforming corresponding to all the users. As the above equation (13) is non-convex, so an alternative optimization strategy needs to be developed so that we can find the sub-optimal solution. To start the optimization procedure, it is necessary to initialize the analog beamforming and after that optimize the digital beamforming as:

𝐅_BB ||𝐕_MRT− 𝐅_RF𝐅_BB||_F (14)

After that, by keeping the resultant optimized digital beamformer fixed and the analog beamformer is optimized as:

𝐅_RF ||𝐕_MRT− 𝐅_RF𝐅_BB||_F (15)

In the case of subarray-based precoding, every data stream is transferred to only a single RF chain. Due to which the digital precoder becomes a diagonal matrix as 𝐅_BB= 𝑖𝑎𝑔 ( ₁₁, ₂₂, … , _𝑁_𝑎_𝐾). The subarray-based process has been shown in Table 3 and following equation 16. This process keeps on continued until it reaches the required state of convergence. We can write the hybrid precoder as 𝐕 = √𝐏𝐕̅. This algorithm is described in the following Algorithm in Table 2.

Table 2:MRT Matching Algorithm 1: Initializing 𝐅_RF

2: Iteration

3: Solving 𝐅_BB from (12) while fixing 𝐅_RF 4: Solving 𝐅_RF from (13) while fixing 𝐅_BB

5: process continues until meeting {Max iterations | Threshold}

6: V1← normalized 𝐅_RF𝐅_BB 7: V←√𝐏 ^𝐕̅

8: Return V

In this method, every subarray used MRT based solution for its corresponding user.

We can denote the normalizing MRT beamformer as:

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𝐕̅ =

( 𝐡₁

||𝐡₁|| 0 ⋯ 0

0 𝐡2

||𝐡₂|| … 0

⋮ ⋮ ⋱ ⋮

0 0 ⋯ 𝐡K

||𝐡_K||)

(16)

where 𝐡_𝑘 represents the channel vectors from the kth subarray to its corresponding user. We can use the ̅ from the above equation 16 to calculate the resultant beamformer which can be denoted by 𝑽 = √𝑷𝑽̅.

4.5 Successive interference cancellation (SIC) Algorithm

In this section, we will discuss about Successive interference cancellation (SIC) Algo- rithm:

4.5.1 Outline of SIC algorithm based on hybrid beamforming

In the SIC algorithm, we are going to discuss the hybrid beamforming part, and we aim to amplify the total probable rate R of mm-wave MIMO systems. Along with this max- min fairness criterion [23] will also be considered. The achievable rate can be written as:

𝑅 = log₂(|𝐈_𝐾+ 𝜌

𝜎²𝐇𝐏𝐏^H𝐇^H|) (17)

We can write the hybrid precoding matrix V as

𝐕 = 𝐅_BB𝐅_RF = 𝑖𝑎𝑔{𝐚₁, … , 𝐚_𝑁}. 𝑖𝑎𝑔{ ₁, … , _𝑁}.

However, when we consider the hybrid precoding matrix P and especially to its block diagonal structure, we can notice that each sub-antenna array has its precoding and they do not depend on each other. Due to this reason, we can decompose the total achievable rate into a series of sub-rate optimization problems, and each will consider only one sub-antenna array.

Then, as a result, the maximum achievable rate R is written as:

𝑅 = ∑ 𝑙𝑜𝑔₂(1 +

𝑁

𝑛=1

𝜌

𝜎²𝒑_𝑛^𝐻𝑯^𝐻𝑻_𝑛−1⁻¹ 𝑯𝒑_𝑛), (18) here we have 𝐓_𝑛 = 𝐈_𝐾+_𝑁𝜎^𝜌₂𝐇𝐏_n𝐏_𝑛^𝐻𝐇^𝐻 and 𝐓₀= 𝐈_𝑁. From the above equation (18), we can optimize all the sub-antenna arrays one after the other. By using the algorithm of SIC for multiuser signal detection [24], we can update the corresponding matrices once

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every sub-antenna array sub-rate is optimized one after the other. This process keeps on going for all the subarray. Figure 16 illustrates the hybrid precoding based on the SIC algorithm. Now we will concentrate on how to optimize sub-rate and how that can be implemented to all the other remaining sub-antenna arrays. For nth sub-antenna array, the sub-rate optimization precoding vector p_n can be stated as:

𝐩_𝑛^𝑜𝑝𝑡= arg max log₂(1 + 𝜌

𝜎²𝐩_𝑛^𝐻𝐆_𝑛−1𝐩_𝑛 ) (19) here 𝐆_𝑛−1 is defined as 𝐆_𝑛−1= 𝐇^𝐻𝐓_𝑛−1⁻¹ 𝐇.

Figure 16: SIC Hybrid Precoding

4.5.2 Optimal solution using the Low-complexity Algorithm

The algorithm starts by using the simple power iteration algorithm, from which we can calculate the maximum eigenvalue and the respective eigenvector in a diagonal matrix.

As 𝐆_𝑛−1 is known as the Hermitian matrix, so we can conclude that: 1) 𝐆_𝑛−1 is a matrix that can be diagonalized; 2) The singular values of 𝐆_𝑛−1 are similar to eigenvalues (eigenvectors). Due to this reason, it is possible to compute the v₁ also the largest singular value Σ₁ of 𝐆_𝑛−1 which have less complexity using the power iteration algorithm. The summary of the power iteration algorithm has been shown in Table 3:

Table 3: Power iteration algorithm

Input: (1) 𝐆_𝑛−1

(2) Starting solution 𝐮⁽⁰⁾ (3) Highest iterations S for 1≤s≤

(1) 𝒛^{( )}= 𝑮_𝑛−1𝐮^{( −1)} Set 𝐓₀ = 𝐈_𝑁

Optimize the sub-rate

Update matrix 𝐓₁

Optimize the sub-rate

Update matrix 𝐓_𝑁−1

Optimize the sub-rate Sub-antenna array 1 Sub-antenna array 2 Sub-antenna array N

⋯

p₁ p₂ p_𝑁

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(2) 𝑚^{( )}= arg max |𝑧_𝑖^{( )}| (3) if 1≤s≤ 2

𝑛^{( )}= 𝑚^{( )} else

n^(s)=^𝑚^{( )}^𝑚^{( −2)}^−(𝑚^{( −1)}⁾

2

𝑚^{( )}−2𝑚( −1)+𝑚( −2)

end if (4) 𝐮^{( )} =_𝑛^𝐳^{( )}_{( )} end for

Output: (1) The largest singular value Σ₁ = 𝑛^{( )} (2) The first singular vector 𝐯₁= ^𝐮^{( )}

||𝐮^{( )}||

2

4.5.3 SIC-Based Hybrid Precoding

The pseudo-code of the SIC algorithm is summarized and can be seen in Table 4:

Table 4: SIC algorithm based on Hybrid Precoding

Input: 𝐆𝟎

for 1≤ 𝑛 ≤

(1) Computing Σ₁and 𝑣₁ of 𝐆_𝑛−1 by using Algorithm 1.

(2) 𝐚_𝑛^{−𝑜𝑝𝑡}=_√𝑀¹ 𝑒^{𝑗 𝑛𝑔𝑙𝑒(𝐯}¹⁾, _𝑛^𝑜𝑝𝑡 =^||𝐯_√𝑀¹^||¹ 𝐩_𝑛^{−𝑜𝑝𝑡}= ¹

√𝑀||𝐯₁||𝑒^{𝑗 𝑛𝑔𝑙𝑒(𝐯}¹⁾ (3) 𝐆_𝑛 = 𝐆_𝑛−1−

𝜌 𝑁𝜎2Σ₁²𝐯₁𝐯₁^𝐻

1+ ^𝜌

𝑁𝜎2Σ₁ (Proposition 2) end for

Output: (1) D= 𝑖𝑎𝑔{ ₁^𝑜𝑝𝑡, … , _𝑁^𝑜𝑝𝑡} (2) A= 𝑖𝑎𝑔{𝐚₁^{−𝑜𝑝𝑡}, … , 𝐚_𝑁^{−𝑜𝑝𝑡}} (3) 𝐕 = 𝐅_BB𝐅_RF

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4.5.4 Evaluation of the Complexity

In this sub-section, we will discuss the complexities of the SIC-based hybrid precoding algorithm which is based on multiplications and divisions which are complex. The complexity of Algorithm 2 consists of the following four steps:

First, we compute the first part which is 𝐆_𝟎= 𝐑𝐇^𝐻𝐇𝐑^𝐻. Here R is denoting

the selection matrix and the size of the H is K×NM. As a result, it includes 𝑀² multiplications and excluding divisions.

1. Another complexity comes from Algorithm 1. We can see that in every iteration it is mandatory to calculate a matrix-to-vector multiplication 𝐳^{( )} = 𝐆_𝑛−1𝐮^{( −1)} along with the Aitken acceleration method. So, the total number of multiplications is (𝑀²+ 2) − 4 and (2 − 2) times of division.

2. The third complexity comes while calculating the optimal solution 𝐩_𝑛^{−𝑜𝑝𝑡} in step 2 of Algorithm 2. This part is simple as it requires only 2 times of multiplication and has no division, as 𝐯_𝟏 is calculated and ¹

√𝑀 is a constant which cannot be changed.

3. The endmost complexity occurs while updating 𝐆_𝑛. When we look at Prop- osition 2, which implies that we utilize Σ₁ and 𝐯₁ which are obtained from the power iteration algorithm from table 3 to update 𝐆_n, which mainly in- volves vector to vector multiplication rather than a complex matrix to matrix multiplication and matrix inversion. We can see that this applies mainly to the outer product 𝐯₁𝐯₁^𝐻. In this case, it includes 𝑀² number of multiplication and have only a single division.

In a nutshell, this algorithm requires almost 𝒪(𝑀²( + )) times of multiplication and 𝒪(2 ) number of time divisions.

4.6 Adaptive Cross-Entropy (ACE) Algorithm:

In this algorithm [25], we will discuss the architecture which is based on switches and inverters (SI) which have a low cost of hardware and which reduces energy consumption. In the previous architecture, the analog part was composed of phase shifters, it is now composed of switches and inverters which are quite an energy-efficient. Using this architecture, the proposed Adaptive cross-entropy (ACE) algorithm is extracted.

(29)

Machine learning [26] usually uses this Cross-Entropy (CE) technique and using this architecture the precoding scheme known as adaptive CE (ACE) is implemented. Sev- eral random candidate hybrid precoders are generated first in this technique. Afterward, according to their sum-rates, these candidate hybrid precoders are adaptively weighted and elements in hybrid precoding are refined according to their probability distributions by minimizing the Cross-Entropy (CE). When looking at the simulation results, we can conclude that this algorithm is energy efficient than the rest of the algorithms.

4.6.1 The SI-based hybrid precoding architecture

Figure 17 below is representing the switch inverter-based architecture (SW architecture) [27].

Figure 17: Inverters and switches-based architecture

When we compare the switch inverter architecture with the finite resolution phase shifter architecture, we can conclude that the energy consumptions are different for each architecture depending upon the energy required by each component such as RF chain, finite-resolution phase shifter, and baseband. The phase-shifter based architecture requires high energy consumption as compared to the switch-based architecture [28].

As a solution, Switch inverter-based architecture is introduced and it can be a good trade-off between the Phase shifter-based architecture and the Switch-based architecture which is quite an energy-efficient. RF chains in Switch inverter-based architecture are attached to a sub-antenna array having 𝑛 = / antennas instead of all the N antenna [29]. In SI based architecture, only one inverter is used for the connection in

Digital Precoder

RF chain

. . .

Inverters and switches

Analog beamforming

⋮

(30)

the RF chain and the sub-antenna array and M number of switches rather than N phase shifters. The energy consumption in this SI-based architecture is written as:

𝑃 𝐼− 𝑟𝑐ℎ𝑖𝑡𝑒𝑐𝑡𝑢𝑟𝑒= 𝜌 + _𝑅𝐹𝑃_𝑅𝐹+ _𝑅𝐹𝑃_𝐼𝑁+ 𝑃_𝑊+ 𝑃_𝐵𝐵 (20) where 𝑃_𝐼𝑁 is the energy consumed by the inverter. The energy consumption of switches is approximately equal to the digital chip (i.e., 𝑃_𝐼𝑁≈ 𝑃_𝑊). We can also notice that the SI-based architecture consumes less energy than PS-based architecture. Moreover, in SI-based architecture, all the antennas are used so it can provide the maximum gains of the arrays of mm-wave massive MIMO which are proven later-on.

We will first discuss the constraints of the hardware which are produced by the SI- based architecture. In the first constraint, instead of the full matrix, the block diagonal matrix is preferred in the analog beamformer 𝐅_𝑅𝐹 and can be denoted as:

𝐅_𝑅𝐹 = [ 𝐟1

RF 0 ⋯ 0

0 𝐟₂^RF ⋱ 0

⋮ ⋮ ⋱ ⋮

0 0 ⋯ 𝐟_N^RF_RF]_𝑁×𝑁

𝑅𝐹

(21)

where 𝐟_n^RF is representing the analog beamformer having the order of M×1 of 𝑛th sub- antenna array. Another limitation is that this architecture only uses switches and inverters so all the non-zero elements in 𝑅_𝑅𝐹 should be:

1

√ {−1, +1}. (22)

4.6.2 Hybrid precoding scheme using ACE

In this section, we will design the digital precoder 𝐅_𝐵𝐵 as well as an analog beamformer 𝐅_𝑅𝐹 so that we can get the maximum sum-rate R, which can be represented as follows:

(𝐅_𝑅𝐹^𝑜𝑝𝑡, 𝐅_𝐵𝐵^𝑜𝑝𝑡) = arg max 𝑅,

. 𝑡 𝐅_𝑅𝐹 ∈ ℱ, (23)

||𝐅_𝑅𝐹𝐅_𝐵𝐵||_𝐹² = 𝜌,

where ℱ is representing the group having the possible analog beamformers which can satisfy these two constraints (21) and (22) as discussed above, according to that we can denote the sum-rate as:

𝑅 = ∑ log₂(1 + 𝛾_𝑘),

𝐾

𝑘=1

(24) Signal to-interference-plus-noise ratio (SINR) of the 𝑘th is represented by:

(31)

𝛾_𝑘 = |h_k^H𝐅RF𝐟_k^BB|²

∑^K_k`≠k|h_k^H𝐅_RF𝐟_k´^BB|²+ σ²

(25)

where 𝐟_k^BB is representing the kth column of 𝐅_BB.

It should be noted that we can observe the non-convex behavior of the constraints in equations (21) and (22) in the analog beamformer 𝐅_RF. Due to this reason solving (23) is quite difficult. As N non-zero elements of 𝐅_RF belongs from ¹

√𝑁{−1, +1}, so there are a finite number of possible 𝐅_RF. As a result, (23) is considered as a combining problem.

As a possible solution, we can first calculate optimal 𝐅BB by selecting a candidate 𝐅RFby following the channel matrix 𝐇𝐅RF having no non-convex constraints. Now we have the set of the possible 𝐅_RF´s so we can now get the optimized analog beamformer 𝐅_RF^opt and digital beamformer 𝐅_BB^opt. In this search scheme, it is required to search 2^𝑁 𝐅_RF´s and 𝐅_BB´s, due to this reason, the complexity is increased because N is large in mm-wave massive MIMO systems. To solve this problem, the ACE algorithm is introduced, which is known as the updated version of the CE algorithm by using machine learning [30].

First, we will give a brief introduction about the CE algorithm, which used an iterative procedure to solve the combining problem. S candidates are generated by the CE algorithm (e.g., here it is hybrid precoders) in every iteration as per the probability distribution. After that, the objective value is calculated (in our problem it is an achievable sum-rate) of every candidate, and after that, it then selects the _{𝑒𝑙𝑖𝑡𝑒} the suitable candidate as “elite”. In the end, the process of updating the probability distribution starts by minimizing the CE based on the selected elites. After the number of procedures, the close solution will be obtained by using the probability distribution and this solution will be similar to the optimal solution having quite a high probability. The CE algorithm has many disadvantages although it has been widely used in machine learning. One disad- vantage is that the contribution of all elites is the same. Intuitively, while updating the probability distribution, the elite which has better objective value will be considered as more important. As a result, if we weigh the elites adaptively by using their objective values then we can expect better performance. By using this idea, we can solve (23) by using the ACE algorithm.

The ACE algorithm starts by formulating the non-zero elements present in 𝐅_RF as × 1 vector 𝐟 = [(𝐟₁^RF)

T

, (𝐟₂^RF)

T

, … (𝐟_RF^RF)

T

]

T

, and by setting the probability parameter 𝐮 = [𝑢₁, 𝑢₂, … 𝑢_𝑁]^𝑇 as an × 1 vector, here 0 ≤ 𝑢_𝑛 ≤ 1 is representing the probability 𝑓_𝑛 =

1

√𝑁, 𝑓_𝑛 is the nth element of f. After that, by setting the 𝐮⁽⁰⁾=¹

2× 1_𝑁×1, we consider that

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all N non-zero elements of 𝐅_RF belongs to the ¹

√𝑁{−1, +1} with equal probability. After that the following steps are followed:

1. In step 1, when the 𝑖th iteration is executed, it generates S number of candidate analog beamformers {𝐅_𝐑𝐅}₌₁ which uses probability distribution (ℱ; 𝐮^(𝑖)) (i.e., generates {𝐟^s}_s=1^S as per 𝐮^(𝑖) and changing the shapes to matrices which belong to ℱ).

2. Followed with step number 2, according to the most powerful channel 𝐇_eq^s = 𝐇𝐅_RF^s , we will calculate the equivalent digital precoder 𝐅_𝐁𝐁 for 1≤ ≤ . There are a lot of advances digital precoder schemes [31], but here we will adapt the classical ZF digital precoder whose performance is quite optimal and whose complexity is low, and 𝐅_𝐁𝐁 can be calculated as:

𝐆^s= (𝐇_eq^s )^𝐻(𝐇_eq^s (𝐇_eq^s )^𝐻)⁻¹, 𝐅_BB^s = β^s𝐆^s, (26) where 𝛽 = √𝜌/||𝐅RF𝐆 ||

𝐹 is known as a power normalized factor. There is some vari- ation in this algorithm in the Digital precoder part which is composed of full array-based digital beamforming architecture as can be seen in Figure 13(a).

3. In this step, we will calculate the achievable sum-rate {𝑅(𝐅_RF)}₌₁ by putting 𝐅_RF and 𝐅_BB into equation (25).

4. In step 4, we calculate {𝑅(𝐅_RF^s )}₌₁ in descending order.

5. As per the standard CE algorithm, the upcoming step will be to update 𝐮^(𝑖+1) using elites with the help of minimizing CE, and it can be written as:

𝐮^(𝑖+1)= arg 𝑚𝑎𝑥1

∑ ln (𝐅_RF^[s]; 𝐮^(𝑖)) ,

𝑒𝑙𝑖𝑡𝑒

=1

(27)

where 𝐅_RF^[s]; 𝐮^(𝑖) is representing the probability to generate 𝐅_RF^[s]. In the above equation (27), the elite's contributions are considered as the same, due to which there is a lack of performance. To solve this problem, each of the elite will be weighted adaptively which is based on the attainable sum-rate. After that we presented an additional parameter of the elites which represents the average achievable sum-rate as:

𝑇 = ¹

𝑒𝑙𝑖𝑡𝑒∑₌₁^{𝑒𝑙𝑖𝑡𝑒}𝑅(𝐅_RF^[s]) (28)

6. In this step, the weight 𝑤 of elite 𝐅_RF^[s] is calculated as 𝑤 = 𝑅(𝐅_RF^[s])/𝑇. According to {𝑤}₌₁^{_𝑒𝑙𝑖𝑡𝑒}, equation (29) can be written as :

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𝐮^(𝑖+1)= arg max1

𝑤 ln (𝐅_RF^[s]; 𝐮^(𝑖)) . (29) Here (𝐅_RF^[s]; 𝐮^(𝑖))=(𝐟^[𝐬]; 𝐮^(𝑖)), and the nth element 𝑓_𝑛^{[ ]} of 𝒇^{[ ]} is representing the Bernoulli random variable, and whereas 𝑓_𝑛^{[ ]} = 1/√ having the probability 𝑢_𝑛^(𝑖) and 𝑓_𝑛^{[ ]} = 1/√

with probability 1 − 𝑢_𝑛^(𝑖). As a result, we have:

(𝐅_RF^[s]; 𝐮^(𝑖)) = ∏ (𝑢_𝑛^(𝑖))

1

2(1+√𝑁𝑓_𝑛^{[ ]})

(1 − 𝑢_𝑛^(𝑖))

1 2(1−√𝑁𝑓𝑛[ ]

).

𝑁

(30)

After substituting equation (30) into (29) we will get:

¹∑ 𝑤 (^{1+√𝑁𝑓}^𝑛

[ ]

2𝑢_𝑛^(𝑖)

𝑒𝑙𝑖𝑡𝑒

=1 −^{1−√𝑁𝑓}^𝑛

[ ]

2(1−𝑢_𝑛^(𝑖))). (31) by setting (31) to zero we will get step 7.

7. So, in step 7, 𝑢_𝑛^(𝑖+1) can be updated as:

𝑢_𝑛^(𝑖+1)=^∑ ^𝑤^(√𝑁𝑓^𝑛

𝑒𝑙𝑖𝑡𝑒 [ ]

=1 +1)

2 ∑^{𝑒𝑙𝑖𝑡𝑒}₌₁ 𝑤 (32) 8. In step 8, this process keeps on repeating (𝑖 = 𝑖 + 1), until it reaches the final iteration 𝐼, and then the analog beamformer is selected as 𝐅_RF^[1] and the digital precoder will be selected as 𝐅_BB^[1]. This ACE algorithm is also applicable when the users have multiple antennas.

4.6.3 Complexity Analysis

Here we will discuss the complexity of the proposed ACE-based hybrid precoding algorithm.

When we look at the algorithm, we will see that the complexity in this algorithm comes mainly from the steps 2,3,6, and 7 of the above algorithm. S effective channel matrices {𝐇eqs }₌₁ and digital precoders {𝐅_BB^s }₌₁are calculated by using equation (26) in step 2.

As a result, we can write the complexity as 𝒪( ²). In step number 3, each candidate's sum-rate is calculated. After employing digital ZF precoder, the SINR 𝛾_𝑘 of the kth user for the sth candidate can be written as 𝛾_𝑘 = (^𝛽_𝜎)². The complexity of this part is 𝒪( ). After that _{𝑒𝑙𝑖𝑡𝑒} are calculated in step 6 which are based on their weights as shown in (28), that is quite simple and complexity can be defined as 𝒪( _{𝑒𝑙𝑖𝑡𝑒}). In the last step

Comparative Study of Beamforming Algorithms for MM-wave Hybrid MIMO Systems

Irfan Ali Awan