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Beamforming Techniques for mmWave HybridAnalog-Digital Transceivers

Nabil Akdim

To cite this version:Nabil Akdim. Beamforming Techniques for mmWave Hybrid Analog-Digital Transceivers. Signal andImage processing. Université Paris-Saclay, 2021. English. NNT : 2021UPASG092. tel-03595055

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Beamforming Techniques for mmWave Hybrid Analog-Digital Transceivers

Thèse de doctorat de l'université Paris-Saclay

École doctorale n°580 : sciences et technologies de l'information et de la communication (STIC)

Spécialité de doctorat : Réseaux, information et communications Unité de recherche : Université Paris-Saclay, CNRS, CentraleSupélec, Laboratoire

des signaux et systèmes, 91190, Gif-sur-Yvette, France. Référent : CentraleSupélec

Thèse présentée et soutenue à Paris-Saclay,

le 13/12/2021, par

Nabil AKDIM

Composition du Jury

Didier le Ruyet Professeur, Conservatoire National des Arts et Métiers

Président & Rapporteur

Ghaya Rekaya Professeure, Télécom Paris Rapporteure & Examinatrice

Direction de la thèse Pierre Duhamel Professeur, CentraleSupélec Directeur de thèse

Yang Sheng Professeur, CentraleSupélec Co-Directeur de thèse

Mustapha Benjillali Professeur, Institut National des Postes et Télécommunications

Invité

Carles Navarro Manchon Professeur associé, Aalborg University Invité

Thès

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doc

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NT

: 202

1UPA

SG09

2

Maison du doctorat de l’Université Paris-Saclay 2ème étage aile ouest, Ecole normale supérieure Paris-Saclay 4 avenue des Sciences, 91190 Gif sur Yvette, France

Titre : Techniques de formation de beamformers pour les émetteurs-récepteurs hybrides analogiques- numériques mmWave

Mots clés : beamformers, émetteurs-récepteurs hybrides, mmWave

Résumé : La forte augmentation des applications gourmandes en bande passante ces dernières années et la pénurie mondiale des bandes cellulaire dans le spectre traditionnel des bandes basses ont rendu le spectre vacant dans les bandes de fréquences à ondes millimétriques (mmWave) d’une importance primordiale pour tous les acteurs clés de l’industrie cellulaire. Cela étant dit, communi- quer sans fil sur les fréquences mmWave serait néanmoins une tâche difficile, principalement en rai-son de la faible réflectivité, de l’absorption élevée et des pertes de propagation en espace libre impor-tantes sur des bandes aussi élevées.

L’une des solutions très populaires pour surmonter les problèmes de propagation susmentionnés con-siste à utiliser des réseaux d’antennes massifs, une technique rendue possible grace a la proportionna-lité entre la taille physique du réseau et la longueur d’onde de la porteuse (les fréquences mmWave sont caractérisées par une petite longueur d’onde, ce qui se traduit par réseaux d’antennes compacts avec un grand nombre d’éléments). Cependant, le coût élevé, la consommation d’énergie et la comple-xité du matériel de signal mixte chez mmWave ren-dent impossible l’utilisation de grands réseaux d’an-tennes avec des éléments à commande numérique.

Le cloisonnement des traitements de signal liés aux émetteurs-récepteurs mmWave a rendu possible une mise en œuvre économique et énergétique-ment effi cace de ces derniers. Ces nouvelles archi-tectures sont connues sous le nom d’émetteurs-récepteurs hybrides analogiques/numériques de for-mation de faisceaux.

Ces architectures hybrides divisent le traitement de précodage/combinaison entre les domaines analogique et numérique, ce qui réduit considéra-blement le nombre requis de chaînes RF.

Les structures des réseaux hybrides soulevent néanmoins leurs propres défis, le faible rapport signal sur bruit (SNR) résultant des pertes de pro-pagation élevées, la grande dimensionnalité de la matrice de canaux sans fil Multiple-Input-Multiple-Output (MIMO) et la présence de traitement ana-logique compliquent l’acquisition des informations d’état du canal (CSI) et le calcul des précodeurs et combineurs MIMO.

Relever les défis susmentionnés est essentiel pour activer le cellulaire basé sur mmWave. Avec cette motivation à l’esprit, cette thèse propose de nouvelles solutions algorithmiques qui les abor-dent. Nous proposons des solutions rapides et de faible complexité qui offrent des performances ef-ficaces tout en respectant les contraintes matériel-les. Les principales contributions de la présente thèse consistent à concevoir des algorithmes rapi-des et de faible complexité qui permettent de construire des précodeurs et des combineurs ro-bustes pendant la phase d’apprentissage du be-amformer et sans avoir besoin d’estimer explicite-ment le CSI puis de l’utiliser pour dériver ces be-amformers.

Maison du doctorat de l’Université Paris-Saclay 2ème étage aile ouest, Ecole normale supérieure Paris-Saclay 4 avenue des Sciences, 91190 Gif sur Yvette, France

Title : Beamforming Techniques for mmWave Hybrid Analog-Digital Transceivers.

Keywords : Beamforming, mmWave, Hybrid Analog-Digital Transceivers.

Abstract : The high increase of bandwidth greedy applications in recent years, and the global wireless bandwidth shortage in the traditional low band spectra has made vacant spectrum in millimeter-wave (mmWave) frequency bands of a paramount importance for all key players in the cellular indus-try. This being said, wireless communications over mmWave frequencies will nevertheless be a chal-lenging task, mainly due to the diffraction capability, high absorption and large free space propagation losses on such high bands.

One of the very popular solutions to overcome the aforementioned propagation issues is to use mas-sive antenna arrays, a technique that is made pos-sible because if the proportionality between the ar-ray’s physical size and the carrier wavelength (mmWave frequencies are characterized by small wavelength, which translates to compact antenna arrays with high number of elements). However, the high cost, power consumption and complexity of the mixed signal hardware at mmWave make having large antenna arrays with digitally controlled ele-ments infeasible.

The partitioning of the signal processing operations related to the mmWave transceivers made having a cost and energy effective implementation of these latter possible. These novel architectures are known as hybrid analog/digital beamforming trans-ceivers. These hybrid architectures divide the pre-coding/combining processing between the analog and digital domains, which reduces considerably the required number of RF chains.

Hybrid array structures entail nevertheless their own challenges, the low signal-to-noise ration (SNR) resulting from high propagation losses, the large dimensionality of the Multiple-Input-Multiple-Output (MIMO) wireless channel matrix and the presence of analog processing complicate the ac-quisition of the channel state information (CSI) and the computation of the MIMO precoders and combiners.

Addressing the aforementioned challenges is key to enabling mmWave based cellular. With this mo-tivation in mind, this dissertation proposes novel algorithmic solutions that tackle them. We pro-pose fast and low complexity solutions that yield efficient performance while respecting the hardware constraints. The main contributions of the present thesis consist of devising fast and low complexity algorithms that enable building robust precoders and combiners during the beam trai-ning phase and without the need of explicitly esti-mating the CSI and then using it to derive these beamformers.

Beamforming Techniques for mmWave Hybrid Analog-Digital

Transceivers

by

Nabil Akdim

DISSERTATION

Presented to the Faculty of the Graduate School of

CentraleSupélec - Paris-Saclay University

in Partial Fulfillment of the Requirements

for the Degree of

DOCTOR OF PHILOSOPHY

CentraleSupélec - Paris-Saclay University

France

September 2021

Beamforming Techniques for mmWave Hybrid Analog-Digital

Transceivers

Nabil Akdim

CentraleSupélec - Paris-Saclay University, 2021

Abstract

The high increase of bandwidth greedy applications in recent years, and the global wireless band-

width shortage in the traditional low band spectra has made vacant spectrum in millimeter-wave

(mmWave) frequency bands of a paramount importance for all key players in the cellular industry.

This being said, wireless communications over mmWave frequencies will nevertheless be a challenging

task, mainly due to the poor diffraction capability, high absorption and large free space propagation

losses on such high bands.

One of the very popular solutions to overcome the aforementioned propagation issues is to use

massive antenna arrays, a technique that is made possible because of the proportionality between

the array’s physical size and the carrier wavelength. However, the high cost, power consumption and

complexity of the mixed signal hardware at mmWave complicates to a great extent implementing large

antenna arrays with digitally controlled elements.

Nevertheless, the partitioning of the signal processing operations related to the mmWave transceivers

has allowed cost and energy effective implementation of these latter. These novel architectures are

known as hybrid analog/digital beamforming transceivers. These hybrid architectures divide the pre-

coding/combining processing between the analog and the digital domains, which reduces considerably

ii

iii

the required number of RF chains.

Hybrid array structures entail nevertheless their own challenges, the low signal-to-noise ratio (SNR)

resulting from high propagation losses, the large dimensionality of the multiple-input-multiple-output

(MIMO) wireless channel matrix and the presence of analog processing complicate the acquisition of

the channel state information (CSI) and the computation of the MIMO precoders and combiners.

Addressing the aforementioned challenges is key to enabling mmWave based cellular. With this

motivation in mind, this dissertation proposes novel algorithmic solutions that tackle these latter.

We propose fast and low complexity solutions that yield efficient performance while respecting the

hardware constraints. The main contributions of the present thesis are deriving fast and low complexity

algorithms that enable building robust precoders and combiners during the beam training phase and

without the need of explicitly estimating the CSI and then using it to derive these beamformers.

In the first contribution, we use advanced tools from Bayesian active learning and approximate in-

ference theories, to devise a robust and fast hierarchical beam search algorithm that reduces the amount

of time or resources needed for the beam search process while guaranteeing a low beam misalignment

probability. Our second contribution proposes a novel beam training strategy based on alternating

transmissions between two hybrid mmWave transceivers. The main idea behind our proposal is to

exploit the reciprocity of the mmWave MIMO channel between the two transceivers. With appropriate

processing at each device, the alternate transmissions implicitly implement an algebraic power itera-

tion that leads to approximating the top left and right singular vectors of that MIMO channel matrix.

Mathematical analysis as well as numerical simulations illustrate the promising performance of the

proposed solutions, making them as enabling technologies for mmWave hybrid transceiver systems.

Contents

Abstract vii

List of Figures ix

List of Tables xi

1 Introduction 1

1.1 Why do Cellular Communications Need mmWave Bands ? . . . . . . . . . . . . . . . . . 2

1.2 Hybrid Beamforming as an Enabler . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.3 Hybrid Beamforming Challenges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.4 Overview of Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.5 Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2 MmWave Channel Characteristics 9

2.1 MmWave vs. Sub-6 Ghz propagation environements . . . . . . . . . . . . . . . . . . . . 9

2.2 MmWave Channel Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.2.1 Large Scale Fading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.2.2 Small Scale Fading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.3 MmWave Transceiver Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.3.1 MmWave Hybrid Digital-Analog Antenna Array Architectures . . . . . . . . . . 22

v

vi CONTENTS

3 mmWave Wireless Channels Variational Online Learning 31

3.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.3 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.4 RF Codebook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3.5 Sequential Beam Pair Search via Variational HiePM . . . . . . . . . . . . . . . . . . . . 38

3.5.1 Sequential Active Learning via the HiePM Strategy . . . . . . . . . . . . . . . . 38

3.5.2 The Variational Expectation Maximization HiePM Scheme: VEM-HiePM . . . . 40

3.5.3 The Variational Model Comparison Based HiePM Scheme : VMC-HiePM . . . . 44

3.6 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

3.7 Conclusion and Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

4 Mutli-Stream Beamforming with Hybrid Arrays 57

4.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

4.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

4.3 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

4.4 Hybrid Ping Pong Multi Beam Training : Hybrid PPMBT . . . . . . . . . . . . . . . . . 62

4.4.1 Ping-Pong Multi Beam Training with Digital Antenna Arrays: Digital PPMBT . 63

4.4.2 Analog Precoder Multi Level Codebook . . . . . . . . . . . . . . . . . . . . . . . 64

4.4.3 Ping Pong Multi Beam Training with Hybrid Antenna Arrays : Hybrid PPMBT 65

4.5 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

4.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

5 Concluding Remarks 77

5.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

5.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

CONTENTS vii

Appendices 81

A Variational Hierarchical Posterior Matching for mmWave Wireless Channels Online

Learning 83

B Ping Pong Beam Training for Multi Stream MIMO Communications with Hybrid

Antenna Arrays 89

Bibliography 97

List of Figures

1.1 Estimation of Global Mobile Subscriptions [1] . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2 Current Cellular Spectrum in the US [2] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.3 Vehicle-To-Everything (V2X) Communication . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.4 Device to Device for Wearable Devices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.1 MmWave vs. Sub-6 Ghz Propagation Differences . . . . . . . . . . . . . . . . . . . . . . . . 10

2.2 MmWave Blockage Agents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.3 Separate Modeling of LOS and NLOS Links . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.4 Two-state Markov Model for Blockage Events . . . . . . . . . . . . . . . . . . . . . . . . 15

2.5 Two-State Shadowing Event with 0dB Threshold Showing Unshadowed (Black Line)

and Shadowed (Red Line) Regions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.6 Time Domain Clusters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.7 Angular Domain Clusters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.8 28 GHz Time Domain Clusters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.9 28 GHz Angular Domain Clusters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.10 MIMO Transceiver Architecture at Frequencies below 6 GHz . . . . . . . . . . . . . . . . . . 21

2.11 MIMO Hybrid Digital-Analog MmWave Transceiver Architecture . . . . . . . . . . . . . . . . 23

2.12 Hybrid Precoding w/ Phase Shifters – Fully Connected . . . . . . . . . . . . . . . . . . . . . 24

2.13 Hybrid Precoding w/ Phase Shifters – Partially Connected . . . . . . . . . . . . . . . . . . . 24

ix

x LIST OF FIGURES

2.14 Hybrid Precoding with Lenses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.1 Hybrid Transceiver Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

3.2 Structure of a Multi-resolution Codebook with a Resolution Parameter with N = 8 . . . . . . 37

3.3 Resulting Beam Patterns of the Beamforming Vectors in the First Three Codebook Levels of

a Hierarchical Codebook. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.4 Beamforming Loss of the Different search schemes in a Channel with L = 0 Scattered

Components. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

3.5 Beamforming Loss of the Different Search Schemes in a Channel with L = 0 Scattered

Components. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

3.6 VMC-HiePM Performance in Channels with Different Power Ratios Between the Dom-

inant Path and L = 3 Scattered Components. . . . . . . . . . . . . . . . . . . . . . . . . 55

4.1 Structure of the Hybrid Transceivers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

4.2 Array Gains Obtained with the Analog Beamformers of the Proposed Multi-Level Code-

book, NA = 16, NRF = 4, LA = 3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

4.3 Spectral Efficiency When NRFA =NRF

B = 8, NS=4, L=8, ρ=30dB. . . . . . . . . . . . . . 71

4.4 Spectral Efficiency for Different NS Values. NA=NB=128, NRFA =NRF

B =8, L=8, ρ=30dB. 72

4.5 Spectral Efficiency over Different NS Values. NA=NB=64, NRFA =NRF

B =8, L=7. . . . . 75

(a) Spectral Efficiency over SNR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

(b) Spectral Efficiency over PP Iterations . . . . . . . . . . . . . . . . . . . . . . . . . 75

4.6 Spectral Efficiency When Device A is Equipped with a Hybrid Array and Device B has

a Full Digital Architecture. NA=128, NB=4, NRFA =16, NS=4, L=40. . . . . . . . . . . . 76

List of Tables

2.1 Omnidirectional Path Loss Models in the Umi Scenario . . . . . . . . . . . . . . . . . . . . . 27

2.2 LOS Probability Models in the Umi Scenario . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.3 Range for the power Consumption for the different devices in a mmWave front-end . . . . . . 29

xi

Chapter 1

Introduction

1

2 CHAPTER 1. INTRODUCTION

1.1 Why do Cellular Communications Need mmWave Bands ?

The next couple of years are expected to see a surge in the number of mobile subscriptions. According

to a study conducted by the international telecommunication union (ITU) [1], and as can be seen in

Fig. 1.1, the total number of global mobile subscribers is estimated to keep increasing by about 10%

year over year and will attain more than 17 billion by the year 2030.

Figure 1.1: Estimation of Global Mobile Subscriptions [1]

Such a high number of connected wireless devices, together with the emergence of new mobile

applications that require very high data rate communications, emanating from these latter, as well as

the wireless network’s user’s expectation to have a satisfactory end-user experience even in crowded

areas [3], will pose a big challenge ahead of the wireless service providers, who will have to come up

with new and innovative technologies in order to be able to cope with this explosive increase in the

mobile data traffic.

The wireless service providers will also have to overcome the global bandwidth shortage which is

caused by the congestion and fragmentation of the traditional low band spectra [4]. Fig 1.2 [2] shows

how the below 3Ghz cellular spectrum currently in use in the united states shows is badly packed, it

also shows how scarce is the total remaining free bandwidth that can be used for new applications

(only about 600 Mhz) and also how high is the cost to acquire such small bandwidth.

The difficulty to achieve the aforementioned demands faced by wireless service providers, using the

1.1. WHY DO CELLULAR COMMUNICATIONS NEED MMWAVE BANDS ? 3

Figure 1.2: Current Cellular Spectrum in the US [2]

congested and fragmented traditional low spectral bands, has pushed them to start exploring vacant

spectrum at the millimeter-wave (mmWave) frequency bands [4].

Mmwave carrier frequencies span the spectrum range from 30 GHz to 300 GHz, whereas the ma-

jority of to date deployed wireless systems operate in the below 6 Ghz spectra. Going up to such

high frequencies opens up the door to accessing multi-Ghz unused spectral bandwidths, which would

then translate to higher data rates. An example of such use is the WiGig [5] wireless systems that is

deployed in the 60 GHz unlicensed mmWave band and that makes use of a 2 Ghz large bandwidth and

combines with the orthogonal frequency division multiplexing (OFDM) wavefrom modulation scheme

to reach data rates up to 6 Gbps.

Interest in mmWave frequencies for wireless communications dates back to the 19th century. Bose

and Lebedev [6] started experimenting in such high bands already in the 1890s. But, this interest

started to gain momentum and materialise when academia, industry as well as governments, with the

hope of solving the above-mentioned issues related to bandwidth scarcity and fragmentation in the

the sub 6 Ghz spectrum, began backing the idea of deploying a flavor of the 5th generation of cellular

systems, known as 5G new radio (NR) [7], on mmWave bands [8–10]. All these efforts resulted in two

types of 5G NR cellular deployments, the first known as frequency range 1 (FR1) NR option, deployed

on sub 6 Ghz frequencies, and the second known as frequency range 2 NR (FR2) option [11], deployed

over the mmWave frequency bands. This latter FR2 option benefited, in its initial deployment, from


large bandwidths (of up to 3 Ghz) in the 28 Ghz and 39 Ghz frequency band [12], which, combined with

advanced multi antenna beamforming techniques, allowed achieving the astonishingly high throughput

of up to 4.3 Gbps on a cellular hand-held device [13].

Aside from cellular applications, mmWave can have many other potential applications. Being

already heavily used in the well established automotive radar business [14], adding the communication

dimension to it would enable new mmWave vehicle-to-everything (V2X) applications like cloud assisted

or fully automated driving, as shown in Fig. 1.31, and would open new possibilities for the assisted

and autonomous driving industry [15].

MmWave is also of interest for high speed wearable networks that connect cell phone, smart watch,

augmented reality glasses and virtual reality headsets [16]. Examples of such use cases are shown in

Fig. 1.3 and Fig. 1.42.

Figure 1.3: Vehicle-To-Everything (V2X) Communication

This being said, wireless communications over mmWave frequencies will nevertheless be a challeng-

ing task, mainly due to the poor diffraction capability, high absorption and large free space propagation

losses on such high bands [17].

1Figure is taken from Professor Robert W. Heath Jr’s mmWave communications tutorial given in the signal processingsummer school that was held in Chalmers University in Gothenburg, Sweden, during summer 2017.

2Similar to Fig. 1.3, this figure is taken from Professor Robert W. Heath Jr’s mmWave communications tutorial givenin the signal processing summer school that was held in Chalmers University in Gothenburg, Sweden, during summer2017.

1.2. HYBRID BEAMFORMING AS AN ENABLER 5

Figure 1.4: Device to Device for Wearable Devices

1.2 Hybrid Beamforming as an Enabler

Fortunately, this frequency range will allow for the use of compact and small antenna arrays with high

number of elements, as the physical size of the array is proportional to the carrier wavelength. The

large beamforming gains that such large-scale arrays enable will be used to compensate for the above

limitations. However, the high cost, power consumption and complexity of the mixed signal hardware

at mm-wave make having large antenna arrays with digitally controlled elements infeasible [18]. A

direct implication of these new hardware constraints is the renewed interest in partitioning the related

signal processing operations between analog and digital domains to reduce the number of required

mixed signal hardware components, like analog-to-digital converters (ADCs), or their resolution, thus

reducing their power consumption and die size footprints. This has led to the development of new and

novel transceiver architectures dubbed hybrid analog/digital beamforming architectures. These hybrid

architectures divide the precoding/combining processing between the analog and digital domains, which

reduces the required number of RF chains. reducing thus hardware implementation complexity and

related power consumption, which enables in turn having portable device being able to communicate

wirelessly over mmWave frequencies.


1.3 Hybrid Beamforming Challenges

The hardware constraints associated with the hybrid architectures such as the limitations on the RF

components and the coupling between analog and digital precoders, however, impose new constraints

on the precoding/combining and the wireless channel estimation design problems.

Accurate Channel State information is critical for efficient operation in wireless communication

systems. The task of obtaining such information at hybrid beamforming mmWave systems represents

a major challenge. In addition to the large training overhead associated with the large arrays and

the SNR that is typically low before beamforming design, the hardware constraints, that results from

RF/hybrid precoding, makes the channels at the baseband seen only through the RF lens [18]. This

has renewed the interest in beam training techniques. These techniques make use of multi-stage radio

frequency (RF) codebooks together with adaptive beamwidth beamforning algorithms to jointly design

the transmitter and receiver beamforming vectors with the goal of maximizing the effective receive

gain of the wireless link being used [18, 19]. Despite the reduced complexity of the aforementioned

algorithms, they do entail a large search overhead and are prone to errors in noisy channels. This

has motivated devising novel beam search techniques that are robust both to inter-beam interference

leakage caused by the RF codebook imperfections and to the additive and multiplicative noise that is

inherent to any wireless communication system.

1.4 Overview of Contributions

Addressing the aforementioned challenges is the key to enabling mmWave based cellular. With this

motivation, this dissertation proposes novel algorithmic solutions that tackle them. We propose low-

complexity and fast solutions that yield efficient performance while respecting the hardware constraints.

The primary contributions of this dissertation can be summarized as follows.

• Our first contribution proposes efficient single stream sequential noisy beam search techniques for

mmWave systems with hybrid architectures. We use a combination of bayesian active learning

1.5. ORGANIZATION 7

and of advanced inference techniques to benefit from the reduced search time of the classical

hierarchical beam-search technique, while at the same time reduce this latter’s inherent high

probability of beam misalignment, especially on noisy channels [20].

• Our second contribution proposes a novel beam training strategy based on alternating transmis-

sions between two hybrid mmWave transceivers. The main idea behind our proposal is to exploit

the reciprocity of the mmWave MIMO channel between the two transceivers. With appropriate

processing at each device, the alternate transmissions implicitly implement an algebraic power it-

eration that leads to approximating the top left and right singular vectors of that MIMO channel

matrix [21].

1.5 Organization

The rest of this thesis is organized as follows. In Chapter 2, we introduce mmWave channel character-

istics as well as mmWave hybrid transceiver design challenges. We then detail our first contribution,

dubbed "Variational Hierarchical Posterior Matching for mmWave Wireless Channels Online Learn-

ing" in Chapter 3. We show how the interplay of bayesian active learning and of advanced inference

techniques can devise fast and efficient beam search algorithms for single stream mmWave communi-

cation systems. In Chapter 4, we detail our second contribution, entitled "Ping Pong Beam Training

for Multi Stream MIMO Communications with Hybrid Antenna Arrays". We show how our proposal

approximates singular value decomposition (SVD) precoding with hybrid transceivers, enabling thus

robust multi-stream mmWave wireless communication systems. Concluding remarks and future work

are finally presented in Chapter 5.

Chapter 2

MmWave Channel Characteristics

Understanding the mmWave signal propagation characteristics and properties is fundamental to be

able to come up with accurate mmWave channel models. These latter are needed to help assess the

usability and also compare the different mmWave wireless communication systems. We will discuss in

this chapter the details of such propagation mechanisms.

2.1 MmWave vs. Sub-6 Ghz propagation environements

Spectral wave propagation at mmWave frequencies differ in many aspects from that of the low frequency

bands, mainly due to the very small wavelength compared to the size of most of the objects in the

environment. On one hand, diffraction effects, one of the main propagation mechanisms in sub 6 Ghz

bands [22], contributes much less to the overall mmWave signal propagation due to the reduced Fresnel

zone. Scattering, on the other hand, tends to be higher due to the increased effective roughness of

materials, but remains limited still and not as rich as in lower frequency bands. mmWave propagation

is also characterized by higher absorption and larger free space propagation losses [17,18].

All these differences affect heavily all mmWave channel’s properties. Multi-paths in mmWave tend

to be more clustered and exhibit far fewer paths than on lower frequency channels, leading to more

9

10 CHAPTER 2. MMWAVE CHANNEL CHARACTERISTICS

sparsity in the delay, angular and spatial domains. Doppler shift will have a bigger noisy effect on the

signal due to the large carrier frequency and large bandwidths on mmWave bands. Angular spread

will be smaller because of the high sensitivity to blockages (buildings, human body or even user’s

own fingers) and strong differences between line-of-sight and non-line-of-sight propagation conditions.

Fig.2.11 summarizes all these differences.

Figure 2.1: MmWave vs. Sub-6 Ghz Propagation Differences

2.2 MmWave Channel Models

mmWave Channel models are required for simulating the wireless mmWave signal propagation mecha-

nisms in a reproducible and cost-effective way. This is needed to accurately design and compare radio

air interfaces, system deployment and develop adequate signal processing algorithms for mmWave

transmitters and receivers.

MmWave channel model parameters can be split into two classes :

• The large scale parameter class, which encompasses characteristics like path loss, shadowing and

blockage (this latter translates to line of sight and non line of sight (LOS/NLOS) probability1Similar to Fig. 1.3, this figure is taken from Professor Robert W. Heath Jr’s mmWave communications tutorial given

in the signal processing summer school that was held in Chalmers University in Gothenburg, Sweden, during summer2017.

2.2. MMWAVE CHANNEL MODELS 11

models).

• The small scale parameter class, which encompasses characteristics like delay spread, Doppler

spread, angular spread and number of multi-path component clusters.

Let us discuss next the details of each of these two classes.

2.2.1 Large Scale Fading

In this section, we will detail the main large scale parameters of mmWave channel models. These

include path loss and shadowing parameters as well as large scale blockage parameters and related

modelling.

Path Loss and Shadowing Models

Path loss and Shadowing models for mmWave channels are inspired by Friis Law [19] and follow an

additive white noise linear log-distance parametric model [22,23] as shown in equation 2.1 below :

PL(d)[dB] = α+ 10β log10(d) + ξ, ξ ∼ N(0, σ2) (2.1)

d is the distance separating the transmitter and the receiver, α and β are parameter models that

depend on the wavelength λ being used, on the omnidirectional gains of the transmit and receive

antennas, Gt and Gr and on penetration losses of the material that the spectral waves might penetrate

in the surrounding environment. ξ is the log-normal term that accounts for variances in shadowing,

and which is also partially affected by the penetration losses.

Large scale Blockage Models

Blockage is a major impairment at mmWave. As shown in Fig.2.22, it can be caused by surrounding

objects like buildings, by surrounding people, or even by the user’s own body.2Similar to Fig. 1.3, this figure is taken from Professor Robert W. Heath Jr’s mmWave communications tutorial given

in the signal processing summer school that was held in Chalmers University in Gothenburg, Sweden, during summer2017.


Figure 2.2: MmWave Blockage Agents

On one hand, penetration losses that are caused by building walls can introduce attenuation up to

80 dB [24], and those that are caused by the human body can result for up to 35 dB loss [25]. On the

other hand, reflective capabilities of all these blockers allow them to be important scatterers to enable

coverage via NLOS paths for mmWave cellular systems [26]. Measurements conducted by New York

University (NYU) confirm that even in extremely dense urban environments, coverage is possible up

to 200 m from a potential cell site [4].

Blockage can be modeled in different ways. Random shape theory [27] and stochastic geometry

theory [28] are mathematical tools that can used to evaluate coverage and capacity in mmWave cellular

networks analytically. Data driven methods can also be used to quantify the effect of blockage, an

example of such methods is to model the mmwave wireless link states using a two-state model (LOS

and NLOS) or a three state model (LOS, NLOS, and signal outage), where both model’s states are

chosen to be parametric statistical functions of the distance between transmitters and receivers, and

where each state’s parameters are fit using the field sounding measurements [29], then the resulting

fitted functions are used to calculate the probability of the link being in each of these states. Fig.2.33

shows this separate modeling of LOS and NLOS links.

A widely used two state model [29–32] is described below:



Figure 2.3: Separate Modeling of LOS and NLOS Links

• LOS model i.e not blocked :

PLLOS(d)[dB] = αLOS + 10βLOS log10(d) + ξLOS (2.2)

• NLOS model i.e blocked :

PLNLOS(d)[dB] = αNLOS + 10βNLOS log10(d) + ξNLOS (2.3)

• Choice of LOS or NLOS is determined by a Bernoulli random variable p(d):

PL(d)[dB] = p(d)PLLOS(d)[dB] + (1− p(d))PLNLOS(d)[dB] (2.4)

• The stochastic blocking function p(d) is modeled as:

P(p(d) = 1) = exp−λd (2.5)

Many organizations have conducted extensive mmWave channel field sounding measurements to

help collect data and fit the above statistical model parameters (λ, αLOS, βLOS, αNLOS, βNLOS and

the shadowing variances). The four major ones are :

• The 3rd Generation Partnership Project (3GPP TR 38.901 [32]), which provides channel models

from 0.5–100 GHz based on a modification of 3GPP’s extensive effort to develop models from 6

to 100 GHz in TR 38.900 [33]. 3GPP TR documents are a continual work in progress and serve

as the international industry standard for 5G cellular.


• 5G Channel Model (5GCM) [34], an ad-hoc group of 15 companies and universities that developed

models based on extensive measurement campaigns and helped seed 3GPP understanding for TR

38.900 [33].

• Mobile and wireless communications Enablers for the Twenty–twenty Information Society (METIS) [35],

a large research project sponsored by European Union.

• Millimeter-Wave Based Mobile Radio Access Network for 5G Integrated Communications (mm-

MAGIC) [36], another large research project sponsored by the European Union.

An example of the results of the measurement campaigns conducted by the above bodies are sum-

marized, for the urban micro-cellular (UMi) propagation scenario, in Table 2.1 for the Omnidirectional

path loss model and Table 2.2 for the LOS probability model.

Blockage introduces not only LOS/NLOS large scale fading effects on the mmWave signal prop-

agation, but also small scale rapid signal variations, mainly caused by people walking between the

transmitter and the receiver. These small scale effects can be by modeled a multi-state Markov model

where transition probability rates can be determined from the field measurements [37]. An example

of such a model is the simple two-state Markov model that is used to characterize unshadowed and

shadowed states for a wireless link in the presence of pedestrian induced variations in received signal

strength [38, 39]. Fig.2.4 shows a diagram of a two-state Markov model where Punshad and Pshad in-

dicate the transition probabilities of going from a shadowed to unshadowed state and an unshadowed

to shadowed state, respectively, and to shadowed state, respectively, and Fig.2.5 depicts the charac-

terization of a typical blockage event with two-states when applying a 0 dB threshold relative to the

zero-crossings for the beginning and end of a shadowing event.

2.2.2 Small Scale Fading

In this section, we will detail the main small scale parameters of mmWave channel models. We fist

motivate the need for using large antenna arrays for mmWave communications. We then explore the


Figure 2.4: Two-state Markov Model for Blockage Events

Figure 2.5: Two-State Shadowing Event with 0dB Threshold Showing Unshadowed (Black Line) andShadowed (Red Line) Regions

clustered nature of the such high frequency channels. We discuss finally the impact of large antenna

arrays and the clustering characteristics on the mmwave channel mathematical model formulation.

Motivating large arrays for mmWave

As discussed so far, large free space propagation losses and high sensitivity to blockage make wireless

communications over mmwave channels a very challenging task. Fortunately, such high frequencies

will allow the use of compact and small antenna arrays with high number of elements, as the physical


size of the array is proportional to the carrier wavelength. The large beamforming gains and adaptive

directionality that such large-scale arrays enable will be used to compensate for the above limitations.

In fact, according to Friis Law, the far field receive power Pr and the transmit power Pt are related

by equation 2.6 in free space propagation :

Pr =Pt

4πR2

λ2

4πGtGr (2.6)

where R is the distance separating the transmitter and the receiver, λ is the wavelength and Gt

and Gr are the transmit and receive antenna gains.

Equation 2.6 can be dissected into three components :

• A first component that depends mainly on the transmit power and on the separation distance

between the transmitter and the receiver. This component is called the receive power spectral

density and is defined by Pt4πR2 .

• A second component that depends mainly on the wavelength used and on the receive antenna

characteristics. This component is called the effective receive aperture and is defined by λ2

4πGr.

• A third component that depends mainly on the transmit antenna characteristics. This component

equals the transmit antenna gain defined by Gt.

The receive antenna gain Gr increases with the number of antennas used at the receiver (effectively,

the more antennas we use for reception, the higher is the power will receive). So knowing that the

antenna size scales inversely to the wavelength (we refer to antenna size relative to wavelength. For

example : a 1/2 wave dipole antenna is approximately half a wavelength long), we can see that the

higher the frequency is (i.e the lower the wavelength), the higher is the number of antennas we can

accommodate in a given physical area, i.e the more effective power we can actually receive receive.

Therefore, the scaling of the antenna gains increases the effective receive aperture λ2

4πGr and more

than compensates for the increased free-space path-loss at mmWave frequencies.


Compensating for path loss in this manner will require then directional transmissions with high-

dimensional antenna arrays (32 antennas and above). This explains why large arrays is a defining

characteristic of mmWave communication.

A Clustered Channel Model

Extensive field measurements [40] have shown that mmwave channels usually assume clustered spatio-

temporal models, where the channel’s multipath components are clustered both in time and angular

domains, as shown in Fig.2.6 and Fig.2.74. The time cluster–spatial lobe (TCSL) [41] approach is

then used to develop statistical spatial channel models (SSCM) for mmwave channels, as the TCSL

framework is shown to faithfully reproduce the first- and second-order time and angular statistics of

these types of wireless channels [42].

The mmWave channel’s SSCM small scale parameters are similar to those of low frequency SSCM

channel models, these are the per cluster parametric distributions of delay, power, central angles of

departure (AoD) and angles of arrival (AoA), together with the angle and delay spreads within each

cluster. Field Sounding campaigns show that these small scale characteristics exhibit spatio-temporal

sparsity due to the small angular spread and the low number of clusters caused by the limited scattering

in mmWave bands. Typically, measurements in the 28 GHz band [40] show the existence of two main

clusters in the time domain (Fig.2.8) and 5 main clusters in the angular domain (Fig.2.9)

Small Scale Fading Mathematical Model

The clustered nature of the mmwwave channel models, as well as their inherent need for large antenna

arrays makes many of the statistical fading distributions used for the traditional multiple input multiple

output (MIMO) low frequency channels inaccurate for mmWave channel modeling. For this reason,

we adopt the extended Saleh-Valenzuela [43] based clustered model representation, which allows us to

accurately capture the mathematical structure present in mmWave channels.4Similar to Fig. 1.3, both of these two figures are taken from Professor Robert W. Heath Jr’s mmWave communications

tutorial given in the signal processing summer school that was held in Chalmers University in Gothenburg, Sweden, duringsummer 2017.


Figure 2.6: Time Domain Clusters

Figure 2.7: Angular Domain Clusters

Consider a mmWave MIMO system with Nt transmit and Nr receive antennas. Assuming a static

2D narrow-band mmWave channels where uniform linear arrays (ULA) are used (extensions to dynamic

3D wide-band models with different array structures are straightforward [18, 40]), then our channel

model can be described by a Nr ×Nt complex matrix H, as set by equation 2.7.

H =

L∑

l=1

αlar(φr,l)aHt (φt,l) (2.7)

We have:

• L : the number of multi-path components.

• The elements on the ULAs are separated by a distance d. Typically d = λ/2, where λ is the the

mmWave wavelength of interest.

• αl is the complex fading channel gain of the l-th multi-path component. αl is typically assumed


Figure 2.8: 28 GHz Time Domain Clusters

Figure 2.9: 28 GHz Angular Domain Clusters


to be complex Gaussian distributed.

• at(φt,l) and ar(φr,l) are the ULA array response vectors of the l-th multi-path component, at

the transmitter and receiver respectively.

• φt,l and φr,l are the incidence angles of the lth multi-path component, at the transmitter

and receiver respectively. These are modeled as at (φt,l) =[1, e−jωt,l , . . . , e−j(Nt−1)ωt,l

]Tand

ar (φr,l) =[1, e−jωr,l , . . . , e−j(Nr−1)ωr,l

]T, with ωt,l(φt,l) = 2π

λ d cos (φt,l) and ωr,l(φr,l) = 2πλ d cos (φr,l)

being the directional cosine angles of the lth multi-path component. The incidence angles φA

and φB are assumed to be sampled from the ranges [θt,1, θt,2] and [θr,1, θr,2] respectively.

2.3 MmWave Transceiver Design

As we saw above, reliable wireless communication over mmWave channels cannot be achieved without

large arrays. We will detail then in this section the challenges that come into picture when dealing

with large array transceiver design.

Signal processing for multiple antenna (MIMO) transceivers at low frequencies happens completely

in the digital baseband domain. This is made possible because in such systems, all antennas can be

digitally controlled through dedicated radio frequency (RF) chains as the number of antennas used in

low bands tend to be small (4 antennas typically), and because the mixed signal and RF components

(digital-to-analog and analog-to-digital converters, power amplifiers and low noise amplifiers) needed

for these RF chains do not consume very high power and are easy to integrate into a single system on

chip (SoC) subsystem. An example of such a transceiver system is shown in Fig. 2.105.

This is not the case for the mmWave communication systems as large antenna arrays are a defining

characteristic of these setups. This will have a big architectural impact on the mmWave transceiver

design.


2.3. MMWAVE TRANSCEIVER DESIGN 21

Figure 2.10: MIMO Transceiver Architecture at Frequencies below 6 GHz

Current hardware technology makes it very challenging to tie a separate RF chain (and all the

related baseband circuitry) for each antenna at the mmWave frequencies [44]. The array’s antenna

elements should be placed very close to each other to avoid granting lobes, this space limitation makes

it difficult to pack the RF chain needed complicated mixed signal and baseband circuitry behind each

antenna.

Controlling digitally every antenna of the array separately will drive the overall mmWave transceiver

power consumption very high : (i) mixed signal devices like PA and ADCs/DACs are power hungry

at such high frequencies [45]; (ii) benefiting from the large available bandwidth and the MIMO capa-

bilities of the massive arrays used in mmwave would require processing many parallel high throughput

data streams. This will strain the baseband digital signal processing chain and will drive the overall

transceiver’s power consumption excessively high [46].

Other aspects to take into account are the architectural challenges imposed by the mmWave analog

front end domain, where key power greedy hardware components include power amplifiers, phase

shifters, and switches.

A tremendous effort has been spent on building low power amplifiers, as these latter are an es-

sential component in the radio frequency chain, in integrated circuit (IC) design. In contrast, phase


shifters, originally utilized in radar systems, are the newly-introduced hardware components in hybrid

beamforming systems. Neverthless, the exact power consumption depends on the specifications and

technology used to implement these components. Table 2.3 shows the range of the power consumed by

different devices included in a mmWave front-end. Data were taken from a number of recent papers

proposing protoype devices for PAs [47–49], LNAs [50–53], phase shifters [54–57], VCOs [58–60] and

ADCs [61–64] at mmWave frequencies. Lt(Lr) is the number of RF chains at the TX(RX). A detailed

treatment of mmWave RF and analog devices and multi-gbps digital baseband circuits can be found

in [23].

All these hardware and power consumption constraints have motivated the wireless communication

research community to look into alternative mmWave specific MIMO transceiver architectures where

the required signal processing is split between the analog and digital domains, which are known as

hybrid digital-analog antenna array architectures [65], or where different design trade-offs are made

with respect to number of antennas or resolution of the RF chain’s components (DAC/ADCs, PAs and

phase shifters for example), these are known as low resolution transceivers [66], or some mix of both

of these solutions.

We will review in this section one of the main MIMO architectures for mmWave systems, namely

the hybrid digital-analog antenna array architectures. The reader can refer to [18] for an overview

about all such architectures.

2.3.1 MmWave Hybrid Digital-Analog Antenna Array Architectures

The hybrid digital-analog antenna array architectures, or hybrid beamformers for short, are composed

by large antenna arrays that are steered using analog phase shifters and only a few digitally modulated

radio-frequency (RF) chains. An illustration of such an architecture is shown in Fig.2.116.

The architecture shown in Fig.2.11 divides the mmWave MIMO transceiver between the digital and



Figure 2.11: MIMO Hybrid Digital-Analog MmWave Transceiver Architecture

the analog domains. The digital component in hybrid architectures can handle each of the baseband

data streams separately, similar to the conventional fully digital MIMO transceivers, allowing thus spa-

tial multiplexing and multiuser MIMO when Ns > 1. The signal processing of the analog components

is however different. Since the transmitted signals for all baseband data streams are mixed together

through digital precoding, and the number of physical antennas is bigger than the number of streams

and RF chains, then the analog network FRF ∈ CNt×lt will be a common component shared by all

these baseband streams. This would impact greatly the algorithm design for such architectures and

would make reusing traditional low frequency channel estimation and precoding/combining techniques

very challenging.

The RF precoding/combining stage can be implemented using different analog approaches like

phase shifters [67], switches [68] or lenses [69]. Two hybrid structures are possible. In the first one

(see Fig.2.12), all the antennas can connect to each RF chain. In the second one (see Fig.2.13), the

array can be divided into subarrays, where each subarray connects to its own individual transceiver.

Having multiple subarrays reduces hardware complexity at the expense of less overall array flexibility.

A complete analysis of the energy efficiency and spectrum-efficieny of both architectures is provided

in [67].

Fig.2.12 and Fig.2.137 show the example of hybrid precoding structures with fully and partially7Similar to Fig. 1.3, both of these two figures are taken from Professor Robert W. Heath Jr’s mmWave communications

tutorial given in the signal processing summer school that was held in Chalmers University in Gothenburg, Sweden, during


connected phase shifters respectively. Such structures are enabled through digitally controlled phased

shifters with quantized phases, where the digital precoder/combiner can correct for lack of precision in

the analog, for example to cancel residual multi-stream interference, allowing thus hybrid precoding to

approach the performance of the unconstrained solutions [43]. The multi-subarray partially connected

structure allows for a great reduction in hardware complexity and power consumption [67].

Figure 2.12: Hybrid Precoding w/ Phase Shifters – Fully Connected

Figure 2.13: Hybrid Precoding w/ Phase Shifters – Partially Connected

To further reduce the overall implementation and power consumption complexity of the hybrid

architecture, an alternative mmWave hybrid architecture that makes use of switching networks has

been recently proposed [68]. This architecture, illustrated in Fig.2.12, exploits the sparse nature of the

mmwave channel by implementing a compressed spatial sampling of the received signal. The analog

combiner design is performed by a subset antenna selection algorithm instead of an optimization over

summer 2017.


all quantized phase values. Every switch can be connected to all the antennas if the array size is small

or to a subset of antennas for larger arrays.

Hybrid architecture can also be realized using a lens antenna at the front-end, using the funda-

mental fact that lenses compute a spatial Fourier transform thereby enabling direct channel access in

beamspace [69]. This continuous aperture phased (CAP) MIMO transceiver architecture is illustrated

in Fig.2.148 and suggests a practical pathway for realizing high dimensional MIMO transceivers at

mmWave frequencies with significantly low hardware complexity compared to conventional approaches

based on digital beamforming. The antennas and RF precoder/combiner in Fig.2.12 are replaced by

the continuous- aperture lens antenna and mmWave beam selector in Fig.2.14. CAP-MIMO directly

samples in beamspace via an array of feed antennas arranged on the focal surface of the lens antenna.

Figure 2.14: Hybrid Precoding with Lenses

The number of ADC/DAC modules and transmit/receive chains tracks the number of data streams,

as in the phase-array-based hybrid transceiver. However, the mapping of the digitally pre-coded data

streams into corresponding beams is accomplished via the mmWave beam selector that maps the

mmWave signal for a particular data stream into a feed antenna representing the corresponding beam.

The wideband lens can be designed in a number of efficient ways, including a discrete lens array (DLA)

for lower frequencies or a dielectric lens at higher frequencies [69].

There are many implications of using a hybrid architecture for mmWave MIMO. Given channel



state information, new algorithms are needed to design the separate precoders/combiners since they

decompose into products of matrices with different constraints (analog and digital matrices as we saw

above). Learning the channel state is also harder, since training data is sent through analog precoders

and combiners. More challenges are found when going to broadband channels as the analog processing

is frequency flat while the digital processing can be frequency selective. There are many opportunities

for future research into designing cellular networks that support hybrid architectures.


PL[dB], fc is in GHz and d3D is in meters meters Shadow fading std [dB] Applicability range and Parameters

5GCM

5GCM UMi-Street CI Model with 1 m reference distance: σSF = 3.76 6 < fc < 100 GHz

Canyon LOS PL = 32.4 + 21log10(d) + 20log10(fc)

5GCM UMi-Street CI Model with 1 m reference distance:

Canyon NLOS PL = 32.4 + 31.7log10(d) + 20log10(fc) σSF = 8.09 6 < fc < 100 GHz

ABG model:

PL = 22.4 + 35.3log10(d) + 21.3log10(fc) σSF = 7.82

5GCM UMi-Open CI Model with 1 m reference distance: σSF = 4.2 6 < fc < 100 GHz

Square LOS PL = 32.4 + 18.5log10(d) + 20log10(fc)

5GCM UMi-Open CI Model with 1 m reference distance:

Square NLOS PL = 32.4 + 28.9log10(d) + 20log10(fc) σSF = 7.1 6 < fc < 100 GHz

ABG model:

PL =3.66 + 41.4log10(d) + 24.3 log10(fc) σSF = 7.0

3GPP TR 38.901

3GPP UMi-Street PLUMi−LOS =

= 32.4 + 21 log10(d) + 20log10(fc) , 10m < d < dm

= 32.4 + 40 log10(d) + 20log10(fc)− 9.5 log10(d2m + (hBS − hUE)2) , dm < d < 5 km

σSF = 4.0 0.5 < fc < 100 GHz, 1.5 m < hUE < 22.5 m

Canyon LOS , hBS = 10 m, dm is specified in 3GPP TR 38.901

3GPP UMi-Street PL = max(PLUMi−LOS(d), PLUMi−NLOS(d)) σSF = 7.52 0.5 < fc < 100 GHz, 1.5 m < hUE < 22.5 m

Canyon NLOS PLUMi−NLOS = 22.4 + 35.3 log10(d) + 21.310(fc)− 0.3(hUE − 1.5) 10 m < d < 5000 m, hBS = 10 m

METIS

METIS UMi-Street PLUMi−LOS =

= 28.0 + 22 log10(d) + 20log10(fc) + PL0 , 10m < d < dm

= 35.8 + 40 log10(d) + 22 log10(dm) + 22log10(fc)− 18 log10(hBShUE) + PL0 , dm < d < 500 mσSF = 3.1 0.5 < fc < 60 GHz, 1.5 m < hUE < 22.5 m

Canyon LOS , hBS = 10 m, dm is specified in METIS specifications

METIS UMi-Street PL = max(PLUMi−LOS(d), PLUMi−NLOS(d)) σSF = 4.0 0.45 < fc < 6 GHz, 1.5 m < hUE < 22.5 m

Canyon NLOS PLUMi−NLOS = 23.15 + 36.7 log10(d) + 2610(fc)− 0.3(hUE) 10 m < d < 2000 m, hBS = 10 m

mmMAGIC

mmMAGIC UMi-Street PL = 32.9 + 19.2log10(d) + 20.8 log 10(fc) σSF = 2.0 65 < fc < 100 GHz

Canyon LOS

mmMAGIC UMi-Street PL = 31.0 + 45.0log10(d) + 20.0 log 10(fc) σSF = 7.82 65 < fc < 100 GHz

Canyon NLOS

Note : PL is the path loss, d is the T-R Euclidean distance

Table 2.1: Omnidirectional Path Loss Models in the Umi Scenario


LOS Probability models (distances in meters) Parameters

3GPP TR 38.901 PLLOS(d) = min(d1/d, 1)(1− exp(−d/d2)) + exp(−d/d2) d1 = 18 m, d2 = 36 m

5GCM d1/d2 model: d1/d2 model:

PLLOS(d) = min(d1/d, 1)(1− exp(−d/d2)) + exp(−d/d2) d1 = 20 m, d2 = 39 m

NYU squared model: NYU squared model:

PLLOS(d) = (min(d1/d, 1)(1− exp(−d/d2)) + exp(−d/d2))2 d1 = 22 m, d2 = 100 m

METIS PLLOS(d) = min(d1/d, 1)(1− exp(−d/d2)) + exp(−d/d2) d1 = 18 m, d2 = 36 m

d >= 10 m

mmMAGIC PLLOS(d) = min(d1/d, 1)(1− exp(−d/d2)) + exp(−d/d2) d1 = 18 m, d2 = 36 m

Table 2.2: LOS Probability Models in the Umi Scenario


Device devices Power (mW)

(Per device)

PA Nt(Nr) 40-250

LNA Nt(Nr) 4-86

Phase shifter Nt(Nr)× Lt(Lr) 15-110

ADC Lt(Lr) 15-795

VCO Lt(Lr) 4-25

Table 2.3: Range for the power Consumption for the different devices in a mmWave front-end

Chapter 3

mmWave Wireless Channels

Variational Online Learning

3.1 Overview

We propose in this chapter1 two variational Bayesian acftive learning schemes that enable initial

access for hybrid digital-analog enabled devices operating in mmWave wireless channels. The proposed

schemes are devised with the goal to balance the beam search time and achieving higher beamforming

gain, while accounting for uncertainties on the unknown channel (gain and noise variance).

3.2 Introduction

As we discussed in earlier chapters, mmWave frequency bands (30− 300Ghz) is one of the most promis-

ing technologies that will make 5G and beyond cellular networks able to serve a large number of wireless

terminals with high data rates [4]. We saw how the free space propagation losses, poor diffraction ca-

1This chapter is based on the work published in the conference paper : N. Akdim, C. N. Manchón, M. Benjillali andP. Duhamel, "Variational Hierarchical Posterior Matching for mmWave Wireless Channels Online Learning," 2020 IEEE21st International Workshop on Signal Processing Advances in Wireless Communications (SPAWC), 2020, pp. 1-5, doi:10.1109/SPAWC48557.2020.9154340.

31

32 CHAPTER 3. MMWAVE WIRELESS CHANNELS VARIATIONAL ONLINE LEARNING

pability and high absorption of such high frequencies make building wireless transceivers operating

on such high frequencies a challenging task [17]. We introduced the concept of large antenna arrays

and we saw that the antenna array’s physical size proportionality to the carrier wavelength makes it

feasible to engineer small and compact antenna arrays with large number of elements at mmwave. We

also discussed how the high beamforming gains and the spatial steering capabilities of these arrays can

help greatly compensate the aforementioned limitations on these high bands.

When it comes to implementation, we discussed how the complexity of the mixed signal hardware at

mmWave as well as its high cost and power consumption make having element wise digitally controlled

large antenna arrays operating on such high frequencies infeasible [18]. We argued how such limitations

have motivated the wireless communication community to adopt a novel transceiver architecture termed

the hybrid digital-analog antenna array architecture [65]. This architecture helps bring down the cost

and power consumption of the mmwave transceivers by allowing to steer their large antenna arrays

using only few digitally modulated radio-frequency (RF) chains as shown in Fig. 4.1.

Baseband

Precoder

Baseband

Combiner

RF Chain

RF Chain

RF Chain

RF Chain

RF Precoder RF Combiner

NA N

B

NA

RF

NB

RF

H

TRANSCEIVER A TRANSCEIVER B

Figure 3.1: Hybrid Transceiver Structure

The hybrid digital-analog transceiver architecture brings its own challenges though. As already dis-

cussed, sensing the mmwave wireless channel using hybrid structure allows to access only a compressed

3.2. INTRODUCTION 33

version of it and requires running an exhaustive and time consuming search in the angular domain

to be able to estimate the CSI accurately. Also, using the acquired CSI to build the needed MIMO

precoders and combiners for such large MIMO channel, under the hybrid architecture constraint, is

not an easy task, it requires splitting the MIMO processing into two components, one digital and one

analog, a split that is challenging to properly perform due to the interconnection and interplay of the

analog domain design constraints with those of the digital domain [4, 18].

To overcome the long search time issue, the scientific community explored using the sparsity friendly

techniques for the CSI acquisition and precoding/combining algorithm design. These techniques were

believed, at least theoretically, to help bring down the number of channel measurements needed to

estimate CSI and build robust precoders and combiners. An example of such techniques are the

compressed sensing based approaches [18, 43, 70]. These latter have been shown, however, not only to

feature high computational complexity but also require long search time in general [70].

Other sparsity friendly techniques that are good alternative schemes to alleviate the aforementioned

issues are the hierarchical beam-search algorithms [21, 70]. The beam search mechanism in these

algorithms is designed based on the bisection concept. In particular, these algorithms start initially

by dividing the angular space into a number of partitions, which equivalently divides the AoAs/AoDs

range into a number of intervals, and design the multi-stage training precoding and combining set of

vectors, this group of vectors is known as a hierarchical beamforning codebook. The codebook’s design

is done in a way to let the combined angular spread of each stage, i.e the union of is vectors angular

spreads, cover entirely the AoAs/AoDs range of interest. Vectors of the first stage are used to sense the

angular space partitions, the received signal is then used to determine the partition(s) that are highly

likely to have non-zero element(s) which are further divided into smaller partitions in the later stages

until detecting the non-zero elements, the AoAs/AoDs, with the required resolution. If the number of

precoding vectors used in each stage equals K , where K is a design parameter, then the number of

adaptive stages needed to detect the AoAs/AoDs with a resolution of 2π/N is S = logK N . This shows

that this family of algorithms does reduce the beam search time. Nevertheless, it was demonstrated


that this reduction in the channel measurement overhead comes at the expense of entailing a high

probability of beam misalignment, especially on noisy channels [71]. To overcome the long beam search

overhead while still benefiting from the reduced search time of the hierarchical beam-search algorithms,

Chiu et al. proposed in [72] to use Bayesian active learning. They designed an algorithm which they

dubbed HierPM. This newly proposed scheme augments the hierarchical beam-search schemes with a

technique called posterior matching [73].

HierPM builds upon the connection between unit norm constrained hybrid beafmorning [70] and

noisy Bayesian active learning [74, 75] to, based on the wireless channel’s incidence angles posterior

distribution, sequentially choose the pair of precoder/combiner to use in subsequent measurements;

this choice of the precoder/combiner is performed in a way that is guaranteed to reduce both the

search time and the beam misalignment probability. HierPM as proposed in [72] presents two main

limitations: first, it is derived for systems in which only one of the communicating devices is equipped

with hybrid digital-analog arrays; this makes it not practical for cases when both devices require

beam steering as is common for mmWave communications; second, it requires knowledge of mmwave

wireless channel parameters such as the complex gain of the channel’s line-of-sight (LOS) component

as well as the noise variance. These parameters are assumed to be known or estimated in [72], but no

practical estimation algorithm is proposed to obtain these estimates. This absence of a good estimate

of the wireless channel CSI (channel gain and noise variance) makes the incidence angles posterior

distribution calculation intractable and hinders the proposed algorithm use in practical scenarios.

We will detail here the first contribution of the present thesis. We will discuss two novel sequential

noisy beam search techniques that build on HierPM principle but solve its above mentioned limitations.

Our proposed strategies extend HierPM to bi-directional beam alignemnt, in which both partici-

pating devices need to coordinate to find the correct transmission and reception directions. In addi-

tion, building upon the variational inference concept, namely the variational expectation-maximization

based inference framework [76] and the variational model comparison based inference framework [76],

our newly proposed schemes naturally account for the uncertainty about the channel’s gain and noise

3.3. SYSTEM MODEL 35

variance at the two communicating hybrid array enabled devices. The proposed estimation process

used together with both proposed strategies is gracefully embedded in the HierPM algorithm, and

enables its use in the usual situation in which the channel parameters are unknown. Numerical sim-

ulation results show that the proposed methods are able to effectively handle the uncertainty in the

channel parameters, resulting in beamforming gains close to these of an exhaustive search algorithm

while requiring an amount of pilot measurements comparable to those of hierarchical search algorithms.

We start by describing the system model and the RF codebook used. We next discuss the technical

details of each of our proposed schemes the rest. We finally show through numerical simulations how

effective these are in terms of their beamforming gains.

3.3 System Model

Our system is composed of two hybrid digital analog antenna array devices A and B, equipped with

uniform linear arrays (ULA) of NA and NB antenna elements respectively. The elements on the ULAs

are separated by a distance d = λ/2, where λ is the the mm-Wave wavelength of interest. Device

A (Device B respectively) digitally control its ULA with NRFA (NRF

B respectively) RF chains each.

The two devices communicate over a reciprocal LOS wireless MIMO channel. This is considered to be

static and narrowband, and is modeled according to the finite scatterer channel model with one single

dominant path [21,77] as:

H = α(φB)H(φA) (3.1)

where H ∈ CNB×NA is the wireless channel MIMO matrix, and α is the complex fading channel

gain, modeled as a standard complex Gaussian variable. (φA) and (φB) are the ULA array re-

sponse vectors at devices A and B with incidence angles φA and φB respectively, modeled as (ωA) =

[1, e−jωA , . . . , e−j(NA−1)ωA

]Tand (ωB) =

[1, e−jωB , . . . , e−j(NB−1)ωB

]T, with ωA(φA) = 2π

λ d cos (φA)

and ωB(φB) = 2πλ d cos (φB). The incidence angles φA and φB are modeled as uniformly distributed in

the range [θA,1, θA,2] and [θB,1, θB,2] respectively.


The two devices go through an initial access phase consisting of a pilot based beam alignment

procedure in order to establish the wireless link between them. We assume in this work that, during

this initial access phase, the CSI learning and beam search processes for the two devices are centralized,

i.e one of the devices, say B, is collecting measurements based on device A’s pilot transmission, uses

such measurements to learn the channel’s gain and noise variance and also devises the combiner it will

use for the next pilot reception occasion together with the precoder that device A should use in sending

that pilot. This decision is communicated to device A through an ideal, error-free control channel,

which can e.g. be established via a sub-6 GHz link in a non-stand-alone deployment. The extension

of our proposal to distributed CSI learning and beam search setups will not be discussed here and will

the object of a future work.

At time instant t, device A sends a pilot symbol to B, who after pilot removal observes the signal

yB,t =√PwH

B,tHfA,t +wHB,tnB,t (3.2)

where fA,t ∈ CNA and wB,t ∈ CNB denote the effective precoder and combiner used at time t by

transceivers A and B respectively. These effective precoder and combiner are obtained from hybrid

digital-analog codebooks detailed in the next section. In addition, nB,t ∈ CNB is a complex, circularly-

symmetric additive white Gaussian noise vector, obtained after training sequence removal and with

i.i.d elements, each with variance σ2B .√P is the average transmit power of the pilot signal.

3.4 RF Codebook

The adaptive beamforming strategy proposed herein utilizes the hierarchical beamforming codebook

in [70]. Such a codebook, noted CS hereafter, is designed to have S levels of beam patterns. The beams

in each level l (l = 1, . . . S) are optimized to leverages the digital-analog transceiver architecture of

the devices by properly setting digital and analog beamformers to approach the desired analog beams

shape. These desired beams should have the following ideal properties:

• They divide the angular region of interest, say [θ1, θ2] dyadically in a hierarchical manner,

3.4. RF CODEBOOK 37

• The angular coverage of any two different beams of them are disjoint,

• The union of all such beams is the whole region of interest.

We note Cl the collection of beams belonging to level l. Then, Cl will contain 2l beamforming vectors

that divide the sector [θ1, θ2] into 2l directions, each associated with a certain range of incidence angles

Rml , such that [θ1, θ2] = ∪2l

m=1Rml . We note each of such 2l vectors as either fA (Rml ) or wB (Rml ),

depending on the considered device.

Figure 3.2 shows the first three levels of an example codebook with N = 8, and figure 3.3 illustrates

the beam patterns of the beamforming vectors of each codebook level.

Figure 3.2: Structure of a Multi-resolution Codebook with a Resolution Parameter with N = 8

Figure 3.3: Resulting Beam Patterns of the Beamforming Vectors in the First Three Codebook Levels of aHierarchical Codebook.


3.5 Sequential Beam Pair Search via Variational HiePM

As described in the introduction, to allow devices A and B to run a fast and reliable beam alignment

process, an adaptive beamforming technique termed “Hierarchical Posterior Matching (HierPM)” [72]

is used hereafter. This technique uses the hierarchical beamforming codebook structure described

above to sequentially, i.e based on all available measurements at a certain point of time, choose the

next set of precoder/combiner pairs that shall be used to take a new measurement, so that both the

average best beam pair search time and the beam pair misalignment probability are optimally reduced.

In this section, we first review the details of this adaptive scheme and show that knowledge of the

channel gain α and the noise variance σ2 is necessary to make the strategy usable in practice. We then

detail our contribution. Thee newly proposed schemes overcome the aforementioned shortcomings

of the vanilla HierPM scheme, by either considering the channel’s parameters to be non random

latent unknowns and uses the variational expectation maximization scheme to estimate them, or by

by either considering the channel’s parameters to be random latent variables and resorts to using

a novel variational model comparison based inference framework [76] to account for that. We dub

our novel strategies as “Variational Expectation Maximization Based Hierarchical Posterior Matching

(VEM-HierPM)” and “Variational Model Comparison Based Hierarchical Posterior Matching (VMC-

HierPM)”.

3.5.1 Sequential Active Learning via the HiePM Strategy

We illustrate here the use of the vanilla HiePM scheme [72] for device A (an analogous strategy will

be used for device B). HiePM selects fA,t+1 based on the posterior at time t of the incidence angle

φA. We discretize the noisy beam search problem above by assuming that the beam search resolution

δA is an integer power of two and that the AoA φA is of the form:

φA ∈ φA,1, . . . , φA,δA, φA,i = θA,1 +(i− 1)

δA(θA,2 − θA,1) (3.3)

Note that Such discretization approaches the original problem of initial access as δA → 0 [72]. Note

3.5. SEQUENTIAL BEAM PAIR SEARCH VIA VARIATIONAL HIEPM 39

also that to support this level of resolution, the corresponding number of levels of the hierarchical

beamforming codebook at device A should be : SA = log2 (δA).

With the above setup, the posterior distribution of φA given all measurements up to time t (collected

in vector yB,1:t), can be written as a δA-dimensional vector πA (t) with entries

πA,i (t) := Pr (φA = φA,i|yB,1:t) , i = 1, . . . , δA. (3.4)

The posterior probability of φA being in a certain range, say Rmi , can be computed as

πA,Rmi (t) :=∑

φA,i∈Rmi

πA,i (t). (3.5)

The HiePM strategy examines the posterior probability πA,Rmi (t) for all i = 1, . . . , SA and m =

1, . . . , 2i and selects fA,t+1 ∈ CS to be the beamformer corresponding to the angular range that satisfies:

(i∗t+1,m∗t+1) = argmin

(i,m)

∣∣∣∣πA,Rmi (t)− 1

2

∣∣∣∣ (3.6)

Intuitively, HiePM chooses at time instant t+ 1 a narrower beam than the one being used at time

instant t only if the posterior of its parent beam is bigger than 12 (making this parent beam the most

suitable to choose from a Bayesian standpoint) and such a narrow beam itself has the highest posterior

among the children of its parent beam. Doing so, it is then guaranteed [72] that this scheme will

sequentially refine the width of the beamformer around the true incidence angle φA.

Next we describe how the posterior belief around φA is updated once a new measurement is taken

with the pair of beamformers chosen previously with HiePM. Based on the measurement model in (3.2),

the posterior update at time instant t+ 1 can be expressed using Bayes rule as

πA,i (t+ 1) ∝ πA,i (t) f (yB,t+1|φA = φA,i) ,

i = 1, . . . , δA

(3.7)

where f (yB,t+1|φA = φA,i) is the likelihood of φA from measurement yB,t+1. Unfortunately, the de-

pendency of likelihood term f (yB,t+1|φA = φA,i) on the latent parameters, namely the channel gain

α, the noise variance σ2B and the incidence angle φB , makes it difficult to be calculated in closed form,

thus hindering its practical use..


3.5.2 The Variational Expectation Maximization HiePM Scheme: VEM-

HiePM

Our first contribution is based on considering the channel gain α and the noise variance σ2B as classical

unknown parameters, i.e latent parameters that are not probabilistic. We explain first the Variational

Expectation Maximization approximate inference framework used in its most general form. We then

show how to apply it to our problem to account for the unknown parameters, α and σ2B , and to jointly

derive posterior updates of our incidence angles φA and φB .

Primer on the Expectation Maximization framework

We start by listing the different types of variables that the Expectation Maximization (EM) approxi-

mate inference framework builds upon:

• x is the observed data vector, which is in our case yB,1:t+1.

• z = (z1, z2, . . . ,zL) denotes the L-dim vector of latent unknown parameters that parameterize

the measurement model (3.2). In our case, these are the channel’s gain and the noise variance,

i.e z = (α, ν). ν is the inverse of σ2B .

• m ∈ 1, 2, . . . , δA×δB denotes the mth pair of angles (φA,im , φB,jm), with im ∈ 1, . . . , δA, and

jm ∈ 1, . . . , δB. Choosing a certain label m is equivalent to assuming that our measurement

model in (3.2) is parameterized by the the mth pair of angles. m will be our model’s latent

random variable.

The EM framework is used to find an estimate for the hidden unknown parameters of our model,

i.e z, that maximize the log likelihood L(z) = log(x; z). EM assumes that deriving a good maximum

log likelihood estimate of z is not easily solved directly, but that the corresponding problem in which

m is also observed is mathematically tractable and can be solved efficiently.

The EM algorithm starts with some initial guess for the maximum likelihood parameter z(0), and


then proceeds to iteratively generate successive estimates, z(1), z(2), . . . by repeatedly applying the

following two steps, for t = 1, 2, . . . , until convergence is reached :

• E−Step : Compute a distribution q(t) over the range of m such that q(t)(m) = p(m|x; z(t−1)).

• M − Step : Set z(t) to z that maximizes Eq(t) [log(p(x,m; z)]

The recursive operation above solves the issue of having the random variable m latent, i.e unob-

served, by representing its corresponding value by a distribution of values in the E step, and then

performing a maximum likelihood estimation for the join data obtained by combining this with the

known value of the observed variable x.

The VEM-HiePM algorithm

We detail next the VEM-HiePM scheme. Suppose we made t measurements. We have from our

measurement model (3.2)

p(x,m;x) = p(yB,1:t|m; (α, ν))p(m) (3.8)

where

• ν = σ−2 is the noise precision at device B;

• p(y1:t|m; (α, ν)) =∏t+1i=1 CN(yi;

√PαwH

B,iAmfA,i, σ2) is the likelihood of our measurement model

(we assume here that the sequential noise samples are i.i.d)2, Am = (φB,jm)H(φA,im);

• p(m) = 1δAδB

is the prior belief overm, which is assumed to be uniform to make it non informative.

As already discussed, the EM algorithm starts with some initial guess for the maximum likelihood

parameters z(0)t = (α

(0)t , ν

(0)t ), and then proceeds to iteratively generate successive estimates, z(1)

t =

(α(1)t , ν

(1)t ), z(2)

t == (α(2)t , ν

(2)t ), . . . , by repeatedly applying the expectation and maximization steps,

for s = 0, 1, 2, . . . , until convergence is reached.

2In the above, CN(·;µ, λ) denotes the complex Gaussian pdf with mean µ and precision λ, Γ(·; a, b) denotes theGamma pdf with shape and rate parameters a and b.


Let q(s)t , α(s)

t and ν(s)t be the estimates of posterior distribution over m, the channel’s gain and

noise precision, respectively, at the measurement epoch t and after s > 0 iterations. We obtain after

some algebra:

αst =Eq(s−1)t

(∑td=1

(√PwH

B,tAmfA,t

)∗yB,d

)

Eq(s−1)t

(∑td=1

∣∣∣√PwH

B,tAmfA,t

∣∣∣2) (3.9)

νst =t

Eq(s−1)t

(∑td=1

∥∥∥yd −√PwH

B,tAmfA,tα(s)t

∥∥∥2) (3.10)

Now that we know how to derive estimate for α and ν if we are given an estimate for the posterior

distribution over m, let us see how to derive the posterior distribution over m given α and ν estimates.

Let q∞t and q∞t−1 be the estimates for the posterior distribution over m at measurement epochs t − 1

and t, and α(∞)t−1 and ν

(∞)t−1 be the estimates for α and ν respectively, where all these estimates are

considered after that the EM iterations converge. We have

q(∞)t (m)

(a)∼ CN(yt;√PαwH

B,tAmfA,t, ν∞t−1)q

(∞)t−1 (m)

(b)∼t∏

i=1

CN(yi;√PαwH

B,iAmfA,i, ν(∞)t−1 )p(m)

(c)∼ exp (−ν∞t−1

t∑

d=1

∥∥∥yd −√PwH

B,tAmfA,tα(∞)t

∥∥∥2

)p(m)

(3.11)

where (a) results from applying bayes rule on q(∞)t (m) and (b) and (c) are results of direct appli-

cation of our system model assumptions.

The posteriors over φA,im and φB,jm are obtained from the posterior q(∞)t (m) as

qA,t(i) =∑

m:im=iq

(∞)t (m), i = 1, . . . , δA (3.12)

qB,t(j) =∑

m:jm=jq

(∞)t (m), j = 1, . . . , δB (3.13)

The posterior probability of the incidence angles φA and φB to be in a certain range RnA,i and RpB,j

respectively, read as:


qA,t(RnA,i) :=∑

φA,i∈RnA,i

qA,t(i), (3.14)

qB,t(RpB,j) :=∑

φB,j∈RpB,j

qB,t(i), (3.15)

The vanilla HierPM scheme is then applied separately to qA,t(i) and qB,t(j), to choose the pair of

beamformers to use for the next measurement occasion.

Algorithm 1 runs all above operations in a loop, until the measurement budget is exhausted:

device B decides which pair of beamformers devices A and B shall use to take the next measurement

by applying the HiePM scheme separately to the current posteriors qA,t and qB,t, it then takes a new

measurement yB,t+1 with those latter, and finally run variational EM inference to derive estimates of

νB = 1σ2B

and α as well as approximate of φA and φB .

Algorithm 1: VEM-HiePM1 Input : Antenna Array Size NA and NB , The search resolution δA and δB , the codebooksCSA and CSB , Search time τ

2 Output : Estimates of φA, φB ; for t = 1, 2, . . . , τ − 1 do3 #HiePM Based BF selection according to Eq.(3.6)

(fA,t+1,wB,t+1) =(fA

(RkA,t+1

A,lA,t+1

),wB

(RkB,t+1

B,lB,t+1

))

4 #Take next measurement5 yB,t+1 =

√PwH

B,t+1HfA,t+1 + wHB,t+1nB,t+1,

6 #Variational EM Posterior Update7 while (No convergence yet) do8 for m = 1 : δAδB do9 update qt(m) via (3.11)

10 end11 update αt via (3.9)12 update νt via (3.10)13 end14 #Angle and Angular Range Posterior Update15 update qA,t+1(i) via (3.21) and qB,t+1(j) via (3.22)16 update qA,t+1(RnA,i) via (3.23) and qB,t+1(RnB,j) via (3.24)

17 #Final Precoder/Combiner Vector design(lt+1,A, kt+1,A

)= (SA, arg maxk (qA,t+1(k)))

18(lt+1,B , kt+1,B

)= (SB , arg maxk (qB,t+1(k)))

19 end20 Output : φA = φA,kτ,A , φB = φB,kτ,B


3.5.3 The Variational Model Comparison Based HiePM Scheme : VMC-

HiePM

Our second contribution in this chapter is based on considering the channel gain α and the noise

variance σ2 as probabilistic latent variables. We show next, how “VMC-HiePM" is able, using the

variational model based approximate inference framework described in [76], to infer all above unknowns

and uses them efficiently to calculate the posterior update needed for HiePM, in a consistent and elegant

way. Such an inference framework lends itself naturally in the HiePM context: we make the best use

of the measurements by first estimating posteriors over the channel gain and noise variance and then

use those to robustly update the angle of incidence posterior, doing so allows VMC-HiePM to take

the channel’s gain and noise variance estimation uncertainties properly into account when deriving the

posterior of the incidence angles, thus making a robust HiePM based decision when choosing the next

precoder/combiner pair to use.

We explain first the variational model comparison based approximate inference framework used in

its most general form, then show how to apply it to our problem to derive posterior updates for our

parameters of interest.

Primer on the Variational Model Comparison based approximate inference framework

Variational Model Comparison based Posterior Update

As we did for the VEM-HiePM study, We start by listing the different types of variables that the

variational model comparison based approximate inference framework deals with:

• X is the observed data vector, in our case is yB,1:t+1.

• Z = (Z1,Z2, . . . ,ZL) denotes the L-dim vector of latent variables that parameterize the mea-

surement model (3.2). In our case, Z = (α, σ2).

• m ∈ 1, 2, . . . , δA×δB denotes the mth pair of angles (φA,im , φB,jm), with im ∈ 1, . . . , δA, and


jm ∈ 1, . . . , δB. Choosing a certain label m is equivalent to assuming that our measurement

model in (3.2) is parameterized by the the mth pair of angles.

The framework performs joint inference on the hidden variables to find a set of distributions

q(Z|m), q(m)1:m that approximate the true posterior p(Z,m|X), by minimizing the Kullback-Leibler

(KL) divergence:

KL(q(Z|m)q(m), p(Z,m|X)). (3.16)

HiePM then uses the approximate incidence angle posterior q(m) to decide which is the best

measurement model candidate fitting the observed data vector X. Algorithm 2 [76, Chapter 10.4] lists

the steps required to perform such operations.

Algorithm 2: Variational Model Comparison based Posterior Update1 for m = 1 : δAδB do2 while (No convergence yet) do3 for j = 1, 2, . . . , L do4 q(Zj |m) ∝ Ei6=j(log(p(X|Z,m)))

5 Lm =∫Zq(Z|m) log(p(Z,m|X)

q(Z|m) )

6 q(m) ∝ p(m) exp(Lm)

Posterior Update for our measurement Model and the overall V-HiePM Algorithm

From our measurement model (3.2), we have

p(X,Z,m) = p(yB,1:t|α, ν,m)p(α)p(ν)p(m) (3.17)

where

• ν = σ−2 is the noise precision at device B

• p(X|Z,m) = p(yB,1:t|α, νB ,m) =∏t+1i=1 CN(yB,i;

√PαwH

B,iAmfA,i, σ2B) is the likelihood of

our measurement model (we assume here that the sequential noise samples are i.i.d); p(α) =

CN(α;α0, β0) is the prior belief over α, considered to be Gaussian with a known initial mean α0


and initial precision β03. p(νB) = Γ(νB ; a0, b0) is the non informative prior belief over νB , with

parameters a0 = 0 and b0 = 0. Am = (φB,jm)H(φA,im)

• p(m) = 1δAδB

is the prior belief overm, which is assumed to be uniform to make it non informative

as well4.

After some algebra, the obtained approximate posteriors for α and νB , up to the measurement

iteration t, are shown to keep the form of their respective priors, but with parameters that depend on

the measurement vector yB,1:t :

qt(α|m) has the form of complex Gaussian pdf with mean αt,m and precision βt,m reading

βt,m =at,mbt,m

t∑

d=1

∣∣∣√PwH

B,tAmfA,t

∣∣∣2

+ β0 (3.18a)

αt,m =at,m

bt,mβt,m

t∑

d=1

(√PwH

B,tAmfA,t

)∗yB,d +

α0β0

βt,m(3.18b)

qt(νB |m)5follows a Gamma pdf with parameters shape and rate parameters at,m and bt,m given by

at,m = a0 + t, (3.19a)

bt,m = b0 − 2Re(∑t

d=1

(√PwH

B,tAmfA,t

)∗yB,dα

∗t,m

)+

∑td=1

[|yB,d|2 +

(1

βt,m+ |αt,m|2

) ∣∣∣√PwH

B,tAmfA,t

∣∣∣2]

(3.19b)

Note that the choice of our prior distributions is not arbitrary, the priors chosen above correspond to

the maximum entropy distributions [78] that respect constraints that need to be put on their respective

parameters, namely α being a complex variable having a known initial mean and variance, νB being a

non negative variable and m being a discrete variable). Such a choice makes our proposal assume the

least information about our measurement model’s unknowns.3The first and second order moments of α are the only assumed known values in our model.4In the above, CN(·;µ, λ) denotes the complex Gaussian pdf with mean µ and precision λ, Γ(·; a, b) denotes the

Gamma pdf with shape and rate parameters a and b.5Note that (3.18) and (3.19) can be re-written, after performing some algebra, in a recursive format w.r.t their terms

involving summation over measurements epochs. This results in a significant reduction of the algorithm’s memory andcomputation complexity footprint.


The posterior of the model, indexed by m, is then updated following Lines 5 and 6 in Algorithm 2,

where Lm reads

Lt,m = log(1

β2t,m

) + at,m (1− log(bt,m))

+ log(Γ(at,m))− b0at,mbt,m

−(

t∑

d=1

|yB,d|2at,mbt,m

− βt,m |αt,m|2)

(3.20)

The posteriors over φA,im and φB,jm are obtained from the posterior qt(m) as

qA,t(i) =∑

m:im=iqt(m), i = 1, . . . , δA (3.21)

qB,t(j) =∑

m:jm=jqt(m), j = 1, . . . , δB (3.22)

The posterior probability of the incidence angles φA and φB to be in a certain range RnA,i and RpB,j

resp, read as:

qA,t(RnA,i) :=∑

φA,i∈RnA,i

qA,t(i), (3.23)

qB,t(RpB,j) :=∑

φB,j∈RpB,j

qB,t(i), (3.24)

The vanilla HierPM scheme is then applied separately to qA,t(i) and qB,t(j), to choose the pair of

beamformers to use for the next measurement occasion.

Algorithm 3 runs all above operations in a loop, until the measurement budget is exhausted:

device B decides which pair of beamformers devices A and B shall use to take the next measurement

by applying the HiePM scheme separately to the current posteriors qA,t and qB,t, it then takes a

new measurement yB,t+1 with those latter, and finally run variational inference to derive approximate

posteriors of νB = 1σ2B, α as well as of φA and φB .


Algorithm 3: VMC-HiePM1 Input : Antenna Array Size NA and NB , The search resolution δA and δB , the codebooksCSA and CSB , Search time τ

2 Output : Estimates of φA, φB , α and νB3 for t = 1, 2, . . . , τ − 1 do4 #HierPM Based BF selection according to Eq.(3.6)

(fA,t+1,wB,t+1) =(fA

(RkA,t+1

A,lA,t+1

),wB

(RkB,t+1

B,lB,t+1

))


√PwH


7 #Variational Model Comparison Posterior Update8 for m = 1 : δAδB do9 while (No convergence yet) do

10 update qt+1(α|m) via (3.18) then qt+1(νB |m) via (3.19)11 end12 update qt+1(m) via (3.20)13 end14 #Angle and Angular Range Posterior Update15 update qA,t+1(i) via (3.21) and qB,t+1(j) via (3.22)16 update qA,t+1(RnA,i) via (3.23) and qB,t+1(RnB,j) via (3.24)

17 #Final Precoder/Combiner Vector design(lt+1,A, kt+1,A

)= (SA, arg maxk (qA,t+1(k)))

18(lt+1,B , kt+1,B

)= (SB , arg maxk (qB,t+1(k)))

19 end20 Output : φA = φA,kτ,A , φB = φB,kτ,B

3.6 Numerical Results

To assess the effectiveness of the proposed algorithms, we run Monte Carlo simulations on a setup with

two hybrid digital-analog beamforming devices A and B. The channel matrix H ∈ CNB×NA reads

H = α(φB)H(φA) +

L∑

l=1

αl(φB,l)H(φA,l) (3.25)

and contains one dominant multipath component and L scattered components. All incidence angles

are independently drawn from a uniform distribution between 0 and π. The channel gains are inde-

pendently drawn from a a set of complex Gaussian distribution with mean 0 and variances fulfilling

Varα+∑lVarαl = 1, so that the average SNR ρ between the nth element of the array at A and the

mth element of the array at B equals E|Hnm|2/E|σB |2=1/σ2B

6. In all simulations below, the two

6Hnm is the channel coefficient between device B’s nth array element and device A’s mth array element, and E isthe expectation operator.

3.6. NUMERICAL RESULTS 49

devices are equipped with identical arrays made of NA = NB = 32 elements, digitally controlled with

NRFA = NRF

B = 8 RF chains. Device A uses a codebook CA with a depth of SA = log2(δA), δA = 128.

CA is built using the orthogonal matching pursuit as described in [70]. A similar codebook, CB , is used

for device B.7

We define beamforming gains, achieved after taking t measurements, for our proposed schemes as

follows :

Gx =∣∣∣wH(φB,kt,B )Hf(φA,kt,A)

∣∣∣2

(3.26)

where x can be either em for VEM-HiePM or vh for VMC-HiePM.

We benchmark these beamforming gains with different measurement budget sizes8 and under dif-

ferent channel assumptions, against that of the different state of the art schemes listed below:

• Gph of the vanilla HiePM scheme of [72]. Here, such a scheme assumes that all of the energy in

the channel is concentrated in the path corresponding to the known gain α and all other gains

αl are null, it also assumes that σ2B is known. In such case, the posterior update is done, simply

using Bayes rule as in equation (21) in [72], on the beam pair corresponding to that main path,

and then HiePM is applied to the marginals over those angles separately, similar to what our

schemes do.

• Gbs of the noisy binary search algorithm of [70], which is achieved by 4 log2(maxNA, NB) = 28

measurements.

• Gmax the best achievable beamforming gain of the used codebook, defined as:

Gmax = maxw∈CBSB ,f∈C

ASA

∣∣wHHf∣∣2 . (3.27)

We begin by assuming that the dominant component is the only component that is present in the

channel (i.e. L = 0).

7Note that the multi-RF chain setups are used solely to help build acceptable RF codebooks [70], and are not usedfor multi-stream MIMO operations.

8note the the exhaustive search needs NANB = 16384 measurements to settle.


Fig. 3.4 shows the beamforming losses of the benchmarked algorithms with respect to the optimum

pair of beamformers, defined as Lvh = Gvh/Gmax, Lem = Gem/Gmax, and Lbs = Gbs/Gmax.


-10

-8-6

-4-2

02

46

81

0

SN

R i

n d

B

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.91

Beamforming Gain

Lv

h :

=

28

Le

m :

=

28

Lv

h :

=

56

Le

m :

=

56

Lv

h :

=

100

Le

m :

=

100

Lb

s :

=

28

Figure3.4:

Beamform

ingLo

ssof

theDifferentsearch

schemes

inaCha

nnel

withL

=0ScatteredCom

ponents.


It can be seen that both or our proposed schemes are clearly much better, in all SNR regimes, than

the binary search based beamforming scheme proposed in [70], we see that even with as little as 28

measurements, which is the measurement budget needed by this latter to settle, both of our algorithms

perform quite well in all SNR regimes, we see that even with such a small measurement budget, VMC-

HiePM can achieve at least 50% of the maximum beamforming gain that can be achieved with the

RF codebook being used across all SNR points. Also, We observe that with only 100 measurements

(compare this with the number of measurements needed for exhaustive search to achieve

We compare next the performance of our schemes to that of the vanilla HiePM with perfect CSI

and operating SNR knowledge scheme. Fig. 4.3 shows the beamforming losses of VMC-HiePM, VEM-

HiePM and of vanilla HiePM schemes with respect to the optimum pair of beamformers (these are

defined similar to the above : Lvh = Gvh/Gmax, Lem = Gem/Gmax and Lph = Gph/Gmax) under

the same simulation assumption of Fig. 3.4. The results show that VMC-HiePM can achieve similar or

even better performance compared to vanilla HiePM with perfect CSI and operating SNR knowledge,

we can see as well that VEM-HiePM is underperfoming when copared to the vanilla HiePM scheme.

Also, it can be observed that the vanilla HiePM scheme with perfect channel gain knowledge saturates

at high SNR: this is an effect of the algorithm assuming that the component’s incidence angle lies

on a discrete grid of values, whereas the actual angles are sampled from a continuous distribution.

VMC-HiePM is less sensitive to this model mismatch, due to the estimation of the channel gain and

inverse noise variance: in practice, these estimates partly account for the mismatch in the assumed

values of the angles and provide robustness to the overall procedure.


-10

-8-6

-4-2

02

46

81

0

SN

R i

n d

B

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.91

Beamforming Gain

Lv

h :

=

28

Le

m :

=

28

Lp

h :

=

28

Lv

h :

=

56

Le

m :

=

56

Lp

h :

=

56

Lv

h :

=

100

Le

m :

=

100

Lp

h :

=

100

Figure3.5:

Beamform

ingLo

ssof

theDifferentSearch

Schemes

inaCha

nnel

withL

=0ScatteredCom

ponents.


Next, we explore the robustness of our best performing method, namely VMC-HiePM, against

channels containing more than one multipath component. For this, we consider a channel with L = 3

scattered components with gains of equal variance, and with the power ratio between the dominant

and scattered components being LOSR = Eα2/(Eα2 +∑l Eα2

l ). Fig. 4.5 shows beamforming

gains achieved by our algorithm after 100 measurements compared to the maximum gains achievable

Gmax.

As it can be observed, the maximum achievable beamforming gain decreases as the power is more

evenly distributed among the channel’s components. Although VMC-HiePM assumes the existence of

a single component, it shows remarkable resilience to the presence of other components. Even when

all components in the model have comparable power, our proposed method is able to perform within

2 dB of the optimum for sufficiently high SNR.


-10

-8-6

-4-2

02

46

81

0

SN

R i

n d

B

20

21

22

23

24

25

26

27

28

29

30

Beamforming Gain in dB

Gv

h :

L=

3, L

OS

R =

0.2

Gm

ax :

L=

3, L

OS

R =

0.2

Gv

h :

L=

3, L

OS

R =

0.7

Gm

ax :

L=

3, L

OS

R =

0.7

Gv

h :

L=

0, L

OS

R =

1

Gm

ax :

L=

0, L

OS

R =

1

Figure3.6:

VMC-H

iePM

Perform

ance

inCha

nnelswithDifferentPow

erRatiosBetweentheDom

inan

tPathan

dL

=3ScatteredCom

ponents.


3.7 Conclusion and Future Work

We proposed in this work two variational Bayesian online learning schemes that enable initial access

for hybrid digital-analog enabled devices operating in mmWave wireless channels. When compared

to state of the art beam acquisition schemes, our methods shows superiority, in terms of balancing

the beam search time versus achieving higher beamforming gain, in being able to properly do so

while accounting for uncertainties on the unknown CSI (gain and noise variance) and in being very

resilient to the dominant single path assumption. Even though both of the schemes are derived based

on a discretized model of the angles of incidence of the channel’s main component, they showed great

robustness against off-grid angles as well as working with a realistic codebook implementation. Further

research will focus on adapting the proposed online learning algorithms to operating in time-varying

channels.

Chapter 4

Mutli-Stream Beamforming with

Hybrid Arrays

4.1 Overview

In this chapter1 we propose a method to derive precoders and combiners for multi-stream MIMO

transmission between two devices equipped with hybrid digital-analog antenna arrays. The method

relies on a low- complexity “multi-beam split and drop with backtracking” procedure to update the

analog precoders, while digital precoders are computed with the QR-decomposition based method. We

show numerically that for sufficiently large SNRs, our proposal can approximate well the unconstrained

SVD-based precoder design and can thus enable high throughput mmWave communication systems.

1This chapter is based on the work published in the conference paper : N. Akdim, C. N. Manchón, M. Benjillali andE. de Carvalho, "Ping Pong Beam Training for Multi Stream MIMO Communications with Hybrid Antenna Arrays,"2018 IEEE Globecom Workshops (GC Wkshps), 2018, pp. 1-7, doi: 10.1109/GLOCOMW.2018.8644444.

57

58 CHAPTER 4. MUTLI-STREAM BEAMFORMING WITH HYBRID ARRAYS

4.2 Introduction

Upcoming wireless communication networks are expected to provide service to an unprecendently large

number of wireless devices with peak data rates in the order of tens of Gbps.

As we discussed in great details in earlier chapters, the low complexity and power friendly mmWave

Hybrid Antenna array transceivers are seen as a great trasnceiver design that would solve.........

the congestion and fragmentation of the traditional spectral bands below 6GHz has pushed wireless

service providers to explore vacant spectrum at the millimeter-wave (mmWave) frequency bands (30–

300 GHz) in order to fulfill that goal [4]. Having a poor diffraction capability and high absorption and

free space propagation losses, wireless communications over such high frequencies will be a challenging

task [17]. As detailed already, this frequency range will allow for the use of compact and small

antenna arrays with high number of elements, as the physical size of the array is proportional to the

carrier wavelength. The large beamforming gains that such large-scale arrays enable will be used to

compensate for the above limitations.

However, the high cost, power consumption and complexity of the mixed signal hardware at mm-

wave make having large antenna arrays with digitally controlled elements infeasible [18]. This has

motivated the wireless communication research community to look at the hybrid digital-analog antenna

array architectures [65]. In such architectures, the large antenna array is steered using analog phase

shifters and only a few digitally modulated radio-frequency (RF) chains. An illustration of such an

architecture is shown in Fig. 4.1.

Although Hybrid Antenna arrays help making packing large antenna arrays in small devices a

feasible task by reducing the implementation complexity as well as the power consumption of the

overall baseband and RF chains, they do bring their own challenges: the low SNR resulting from high

propagation losses, the large dimensionality of the MIMO channel matrix and the presence of analog

processing complicate the acquisition of the channel state information (CSI) and the computation of

the MIMO precoders and combiners [4,18]. Luckily, channel measurement campaigns [17] have shown

4.2. INTRODUCTION 59

that mm-wave channels are sparse in the angular domain, which enables the proposal of CSI acquisition

and precoding/combining algorithms that exploit such property. An example of these are compressed

sensing based approaches such as [18,43,70], which are generally computationally complex and require

a large amount of channel measurements. An alternative are exhaustive and hierarchical beam-search

techniques, which may entail significant latency and probability of miss detection [71].

In this work, we focus on a beam training strategy based on alternating transmissions between two

transceivers, which has been coined ping pong beam training (PPBT). The main idea behind PPBT

is to exploit the reciprocity of the MIMO channel. With appropriate processing at each device, the

alternate transmissions implicitly implement an algebraic power iteration that leads to approximating

the top left and right singular vectors of the MIMO channel matrix. This idea was first applied in

the digital arrays context for single stream wireless communications in [79, 80], and was extended to

multi stream setups in [81], to large antenna array and frequency selective systems in [82] and to

noisy MIMO channels in [83]. More recently, similar approaches have been proposed in the context of

mmWave communications with hybrid digital-analog antenna arrays, which we review next. In [77], the

basic ping pong beam training method for single-stream MIMO transmission was adapted to the hybrid

array architecture with the inclusion of a “beam split-and-drop” procedure for the setting of analog

precoders. The subspace estimation and decomposition method in [84] proposes a ping pong based

algorithm that iteratively estimates the channel’s right and left eigenvectors using a Krylov subspace

estimation method. This algorithm is based on exhaustive measurements with a large set of different

analog precoders, which are then linearly combined in order to cancel the effect of the analog precoders.

It therefore requires significant amount of transmissions, which imply large signalling overhead and

latency. Lastly, the power iteration based training method introduced in [85] is a technique that

extends the solution proposed in [81] to the multi stream case, where the digital precoders are set

based on an algebraic power iteration technique, while the analog precoders update is done based on

a compressed sensing technique called simultaneous orthogonal matching pursuit [86].

Compared to the above approaches, we propose in this article a strategy that sets the digital and


analog precoders of the devices in a way to approximate the top NS left and right singular vectors

of the channel matrix, with NS being the desired number of spatial streams. Our new technique,

which we dub “hybrid ping pong multi beam training“ (Hybrid PPMBT) extends the work done in [77]

to the multi stream case. It adapts the PPBT strategy to hybrid arrays by progressively choosing

the analog precoders at each device from a predefined hierarchical codebook. After one round-trip

transmission, a novel "multi beam split and drop strategy with backtracking" is applied to focus the

analog precoders towards the spatial directions that are most likely containing the channel’s top NS

multipath components. The digital precoders are updated via an orthogonal decomposition operation

on the received signal as described in [81]. In comparison to the approaches in [84] and [85], Hybrid

PPMBT is much simpler from a computational complexity aspect and has a low training overhead as it

requires significantly fewer transmissions. Simulation results show that our proposed scheme performs

very well in retrieving the wanted NS channel’s top eigenmodes for sufficiently large signal-to-noise

ratio, both in terms of accuracy and convergence speed.

Baseband

Precoder

Baseband

Combiner

RF Chain

RF Chain

RF Chain

RF Chain


NA N

B

NA

RF

NB

RF

H

TRANSCEIVER A TRANSCEIVER B

Figure 4.1: Structure of the Hybrid Transceivers

4.3. SYSTEM MODEL 61

4.3 System Model

We consider a system in which two hybrid analog digital transceivers A and B, equipped with uniform

linear arrays (ULA) composed of NA and NB antenna elements. Such elements are separated with

a distance d = λ/2, where λ is the wavelength of interest. The two devices control digitally their

arrays with NRFA and NRF

B RF chains respectively and exchange data over a reciprocal wireless MIMO

channel using NS parallel data streams. The channel from device A to device B is considered to

be static and narrowband and is modeled according to the finite scatterer channel model with L

propagation paths [77,87], as

H =

√NANBL

L∑

l=1

αl(ΩB,l)H(ΩA,l), (4.1)

here, H ∈ CNB×NA , L is the number of multipath components (MPC), αl is the complex fading channel

gain for MPC l, ΩA,l = 2πλ d cosφA,l and ΩB,l = 2π

λ d cosφB,l are the directional cosines corresponding

to the lth MPC at arrays A and B respectively, where φA,l, φB,l are the angles of incidence of that same

path, and and are the array response vectors at device A and B respectively. The αl are modeled as

independent, standard complex gaussian variables, the φA,l and φB,l as uniformly distributed in the

range [0, 2π) radians and the array responses as (ΩA,l) = [1, e−jΩA,l , . . . , e−j(NA−1)ΩA,l ]T/√NA and

(ΩB,l) = [1, e−jΩB,l , . . . , e−j(NB−1)ΩB,l ]T/√NB .

In order to establish the wireless link with device B, device A (we assume, without loss of generality,

that device A is performing the first transmission) transmits T , an NS × NS orthogonal training

sequence i.e TTH = INS . Upon reception, device B cancels the training sequence effect by multiplying

its received digital signal by TH . The resulting signal can be expressed as:

YB = FHBHFAWA + FH

BNB , (4.2)

where FA ∈ CNA×NRFA and FB ∈ CNB×NRFB contain the states of the analog precoder and com-

biner of transceivers A and B, WA ∈ CNRFA ×NS denotes the digital precoder of transceiver A and

NB ∈ CNB×NS is a complex, circularly-symmetric additive white gaussian noise matrix, obtained

after training sequence removal and with i.i.d elements, each with variance σ2. Transmissions from


device B to device A are modeled analogously as

YA = FHAHHFBWB + FH

ANA, (4.3)

where WB ∈ CNRFB ×NS and NA ∈ CNA×NS are defined similar to the above.

4.4 Hybrid Ping Pong Multi Beam Training : Hybrid PPMBT

Given the signal model in (4.2) and (4.3), the beamforming task consists of selecting the set of analog

and digital precoders and combiners that maximize the spectral efficiency over a given channel matrix

H. For transmission from device A to B, and assuming unit transmit power equally allocated across

the NS streams, the spectral efficiency reads

R = log2 det(INS +

R−1NB

NSHeH

He

), (4.4)

where RNB= σ2WH

BFHBFBWB is the noise covariance matrix after receive combining at device B,

He = WHBFH

BHFAWA is the equivalent channel after precoding and combining at both devices. An

analogous expression applies for transmission from device B to device A.

The optimal precoders maximizing (4.4) are known to be the NS top right and left singular vectors

of H. However, the hybrid structure of the antenna array makes the computation of such precoders

challenging. On the one hand, as digital measurements of the channel are only obtained after analog

precoding and combining, estimating the full channel matrix H in order to obtain its singular value

decomposition requires a large number of measurements and hence large overhead and latency [84].

On the other hand, even if the channel matrix H can be estimated, the precoders have to be built

as the product of the analog precoding matrix FA and the digital precoding matrix WA. While the

elements of WA can take any complex value due to its digital implementation, the operation modeled

by FA is implemented via phase shifters and combiners, which restricts the values it can take. In this

work, we restrict the entries of FA to satisfy |(FA)l,i|2 ∈ 1

M(i)A

; 0, where M (i)A being the number of

activated array elements in the ith column of FA, and the option (FA)l,i = 0 accounts for the option of

2(FA)l,i is the entry of the matrix FA belonging to its lth row and ith column.

4.4. HYBRID PING PONG MULTI BEAM TRAINING : HYBRID PPMBT 63

leaving some elements of the array unused. In addition, a transmit power constraint is enforced such

that ‖FAWA‖F = 1.3 With these constraints, finding the combination of digital and analog precoders

to best approximate the channel’s singular vectors becomes a computationally intensive optimization

problem [43].

To overcome such difficulties, we propose an iterative multi beam training scheme based on alternate

transmissions between the two devices, this procedure estimates progressively and simultaneously the

top NS right and left singular vectors of H and sets the digital and analog precoders so that they

approach those singular vectors. It consists of two parts: 1. a "backtracking beam split and drop"

approach to select the analog precoders FA and FB from a predefined multi level codebook, 2. a

method to select the digital precoders WA and WB inspired by the QR decomposition algorithm

described in [81].

We will proceed by reviewing the beam training procedure for digital antenna arrays proposed

in [81], then briefly present the multi level codebook that is used for the analog precoder update, and

finally explain our multi beam training solution.

4.4.1 Ping-Pong Multi Beam Training with Digital Antenna Arrays: Dig-

ital PPMBT

We review the digital PPBT algorithm over a narrowband reciprocal channel H as described in [81].

We consider two devices A and B equipped with digitally controlled antenna arrays with NA and NB

elements respectively. At the initial (0th) iteration, the process starts with a random initialization

of the precoder at device A, W [0]A . A uses then this initial precoder to transmit a training sequence

to B. Upon reception and training sequence removal, device A gets an estimate of HW[0]A , makes a

QR-decomposition on it, and uses the Q part of that decomposition as its precoder W [0]B . It then uses

that precoder to transmit a training sequence back to device A, who will repeat the same operations.

This process is reiterated until convergence, at which, device A gets an estimate of the top NS right

3Obviously, the same constraints apply to the analog precoder of device B.


singular vectors of H and device B gets an estimate of the top NS right singular vectors of HH.

Further details on this procedure can be found in [81].

4.4.2 Analog Precoder Multi Level Codebook

We illustrate here the codebook definition for transceiver A (an analogous codebook is used for the

transceiver B). We consider a codebook CA which is composed of LA= log2(NA/NRFA )+1. levels.

Note that for the considered codebook design we constrain NA and NRFA to be both integer pow-

ers of two. For the kth level, we define a subcodebook C(k)A =ϕ(k)

A,i, i=0, 1, . . . ,M(k)A -1 consisting of

M(k)A =NRF

A 2k-1 column vectors, k=1, 2, . . . , LA. Each of the elements of the subcodebook is defined

as :

ϕ(k)A,i=

[1, e-jψ

(k)A,i , . . . , e-j(M

(k)A -1)ψ

(k)A,i ,0T

NA-M(k)A

]T/

√M

(k)A (4.5)

where ψ(k)A,i=π-π(2i+1)/M

(k)A is the directional cosine of the ith vector at the kth level (ϕ(k)

A,i steers the

array in the direction θ(k)A,i= arccosψ

(k)A,i/π, with a lobe whose width decreases with the codebook level

k), and 0N is the N -dimensional column zero vector. Further details about the codebook used here

can be found in [77].

Basically ϕ(j)A,i steers the array in the direction θ

(j)A,i = arccosψ

(j)A,i/π, with a lobe whose width

decreases with the codebook level j. Note that for a fixed level k, the directional cosines θ(k)A,i, i =

0, . . . ,M(k)A − 1 are set to uniformly sample the directional cosine range [−π, π] and as k increases this

range is sampled with larger resolution and more array elements are used for higher-level codebook

elements, resulting in more directive beamforming vectors (an illustration of this is shown in figure 4.2).

For more details about RF codebook optimization methods, we refer the reader to [88,89].


Angle (radians)

0 0.5 1 1.5 2 2.5 3 3.5

Arr

ay G

ain

(d

B)

0

5

10

15Codebook Level 1

Angle (radians)

0 0.5 1 1.5 2 2.5 3 3.5

Arr

ay G

ain

(d

B)

0

5

10

15Codebook Level 2

Angle (radians)

0 0.5 1 1.5 2 2.5 3 3.5

Arr

ay G

ain

(d

B)

0

5

10

15Codebook Level 3

Figure 4.2: Array Gains Obtained with the Analog Beamformers of the Proposed Multi-Level Code-book, NA = 16, NRF = 4, LA = 3.

4.4.3 Ping Pong Multi Beam Training with Hybrid Antenna Arrays : Hy-

brid PPMBT

The proposed algorithm for beam training with hybrid arrays is described in pseudocode. Algorithm 4

presents the overall training scheme, while Algorithm 5 describes the subroutine used to update the

analog precoding matrices.


Initialization

First, the digital and analog precoders are initialized. FA and FB are initialized to C(1)A and C(1)

B

respectively, while WA is initialized to a random unitary matrix. The columns of WA are then

normalized to fulfill the transmit power constraint of the effective precoder FAWA. Finally, a set of

empty arrays, pB ,kB , iB, are created. Those arrays will store the needed information to perform

the analog precoder updates, as explained below.

Ping Pong Iterations

After the initialization phase, a sequence of alternate pilot transmissions starts between the two devices.

The baseband precoders are updated after each reception step by means of a QR-decomposition,

followed by a normalization step. These two operations are detailed in lines 3-7, 12-16 and 20-24 in

Algorithm. 4. Immediately after updating their baseband precoders upon reception of a transmission,

the devices transmit back with the updated digital precoders and the same analog precoder as used for

reception. Only after the transmission has been made will the transmitting device update its analog

precoders (using Algorithm 5), such that next reception is done with the updated setting. This allows

for the QR based iteration to converge, as each reception-transmission cycle is performed over a static

setting of the analog precoders.

Update of Analog Precoders

The devices invoke the routine outlined in Algorithm 5 to update their analog precoder state. This

routine bases its update on the current state of the device’s RF precoder , its codebook C and on the

update history of its baseband precoder . Using all previous updates of allows for backtracking–i.e,

correcting for wrong RF precoder updates. The routine works as follows:

a) Three sequences of values are generated: kn stores the level of the codebook of the nth column

of , pn stores the squared norm of the nth row of , i.e the aggregate energy received on it, and


Algorithm 4: Ping Pong Multi Beam Training with Hybrid Arrays1 Input : Antenna Array Size NA and NB , the codebooks CSA and CSB , Search time τ

2 F[0]A ←

[ϕ

(1)A,0,ϕ

(1)A,1, . . . ,ϕ

(1)

A,M(1)A −1

]

3 F[0]B ←

[ϕ

(1)B,0,ϕ

(1)B,1, . . . ,ϕ

(1)

B,M(1)B −1

]

4 Initialize W[0]A to an orthogonal matrix of its size.

5 for s = 1, 2, . . . , NS − 1 do6 W

[0]A (:, s)←W

[0]A (:, s)/

√NS

∥∥∥F [0]A W

[0]A (:, s)

∥∥∥2

7 end8 pA,kA, iA ← [], [], []9 pB ,kB , iB ← [], [], []

10 A transmits, B receives:11 Y

[0]B =(F

[0]B )HHF

[0]A W

[0]A +(F

[0]B )HN

[0]B

12 [Q,R]← QR(Y[0]B )

13 for s = 1 : NS do14 W

[0]B (:, s)← Q(:, s)

15 W[0]B (:, s)←W

[0]B (:, s)/

√NS

∥∥∥F [0]B W

[0]B (:, s)

∥∥∥2

16 end17 for t = 1, 2, . . . , τ − 1 do18 B transmits, A receives:19 Y

[t]A =(F

[t−1]A )HHHF

[t−1]B W

[t−1]B +(F

[t−1]A )HN

[t]A

20 [Q,R]← qr(Y[t]A )

21 for s = 1 : NS do22 W

[t]A (:, s)← Q(:, s)

23 W[t]A (:, s)←W

[t]A (:, s)/

√NS

∥∥∥F [t−1]A W

[t]A (:, s)

∥∥∥2

24 end25 [F

[t]B , pB ,kB , iB]← Upd.An.Pr[F

[t−1]B ,W

[t−1]B , CB , pB ,kB , iB]

26 A transmits, B receives:27 Y

[t]B =(F

[t]B )HHF

[t−1]A W

[t]A +(F

[t]B )HN

[t]B

28 [Q,R]← qr(Y[t]B )

29 for s = 1 : NS do30 W

[t]B (:, s)← Q(:, s)

31 W[t]B (:, s)← W

[t]B (:,s)

√NS

∥∥∥F [t]B W

[t]B (:,s)

∥∥∥2

32 end33 [F

[t]A , pA,kA, iA]← Upd.An.Pr[F

[t−1]A ,W

[t]A , CA, pA,kA, iA]

34 end35 Output : [F

[τ ]A ],W

[τ ]A ,F

[τ ]B ],W

[τ ]B ]

in stores the index of the nth column of , out of the level of the codebook to which that column

belongs.


Algorithm 5: Analog Precoder Update : Upd.An.Pr1 Input : [F , W , C,p,k,i]2 Generate : kn, in, pn, n=1, . . . , NRF , where kn will store the codebook level to which the

′s

nth column belongs, in will store its index out of that level and pn will store the norm of thenth row of , i.e the aggregate energy received on it.

3 p← [p, p1, . . . , pNRF ],4 k← [k, k1, . . . , kNRF ],5 i← [i, i1, . . . , iNRF ]6 Sort p in a descending manner and store the result in pS , then store the arrangement of the

elements of p into pS in pI .7 bI ← []8 bL ← []

9 while n ≤ NRF do10 bI ← [bI , pI,n], pI,n is the nth element of pI , ppI,n ← 0, ppI,n is the pI,nth element of p,11 if kpI,n= log2( N

NRF)+1 or n=NRF -1 then

12 n← n+ 1, bL ← [bL, 1]13 else

14 n← n+ 2, bL ← [bL, 2]

15 for t = 1 : Length(bI) do16 m← bL,t17 if m = 2 then18 F ← [F,ϕ

(km+1)2im

,ϕ(km+1)2im+1 ]

19 else

20 F ← [F,ϕ(km)im

]

21 return F ; p,k, i

b) The sequences generated above will be used to update three vectors: kn will be appended to

k, with k being an array storing the codebook levels of the columns of used over consecutive

ping-pong iterations. pn will be appended to p, with p being an array storing the received energy

over the different spatial directions set by the analog beamformer . in will be appended to i in

a similar manner to the above.

c) Once p is updated, it will be sorted in a descending manner and the resulting sorted indices will

be stored in pI . This newly formed array will be used to find the entries of p that are most likely

to direct the analog precoders where the MPCs of H are.

d) Two new vectors are built: bI contains the K first indexes of pI i.e it identifies the beams that

are most aligned with the channel’s MPCs (K is the length of bI , which can be derived from


lines 9-13). bL is a vector that is made of 1’s and 2’s. The ith entry of bL is set to 2 when

the precoder corresponding to the ith entry of bI is replaced with the two precoders belonging

to one step higher level of the codebook and that have their beams covering together its same

spatial area, otherwise it is set to 1. Deciding to append 1 or 2 to bL depends on whether we

already consumed all columns of and on whether the element of pI in question belong to the last

level of the codebook or not (see lines 17-21). Line 8 of the algorithm erases the measurement

stored over a beam that is selected to be included in the analog precoding matrix, either directly

or after splitting it into two beams of the immediately higher level. As new measurements

will be obtained over that beam in the next iteration, the old measurement is deleted to avoid

unnecessarily coming back to the previous configuration corresponding to that old measurement.

e) Finally, the RF precoder columns are updated (lines 15-22).

We dub the above procedure for updating the analog precoders “beam split and drop with backtracking”,

owing to the way it operates. At each iteration, a decision is made as to whether a given beam is

split –i.e replaced by two more directive beams– or dropped –i.e. removed from the precoding matrix.

The backtracking feature refers to the fact that, via the vector p, the measurements obtained in prior

ping-pong iterations are kept in memory, allowing for returning to lower level beams in the codebook

in the cases where noise leads to erroneous decisions in the the split-and-drop procedure.

4.5 Numerical Results

In order to assess the effectiveness of the proposed algorithm, we perform Monte Carlo simulations

for multiple configurations of the hybrid arrays at devices A and B. The channel matrix H follows

the model in (4.1). The average SNR for the sth stream link between the nth element of the array at

device B and the mth element of the array at A is defined as ρ=E|Hnm|2/E|Nns|2=1/σ2, where

Nns is the nth entry of the sth column of the noise matrix N . We assume here for simplicity that

the SNR ρ is the same for all streams. Hnm is the channel coefficient between device B’s nth array


element and device A’s mth array element, and E is the expectation operator.

We measure the performance with the average spectral efficiency per bits/s/Hz, expressed with :

log2

( ∣∣∣∣INS+R−1N

NSHeH

He

∣∣∣∣)

where He=(F[n−1]B W

[n−1]B )HHF

[n−1]A W

[n]A and RN=σ2(F

[n−1]B W

[n−1]B )HF

[n−1]B W

[n−1]B for integer it-

eration n, and He=(F[n]B W

[n]B )HHF

[n−1]A W

[n]A and RN=σ2(F

[n]B W

[n]B )HF

[n]B W

[n]B for half iteration

n+0.5.

We benchmark the performance of the hybrid PPMBT against the performance of the digital ping

pong multi stream beam training method (digital PPMBT) described in [81], for which the spectral

efficiency is calculated in a similar manner to that of the hybrid PPMBT. We compare it also with

the optimal unconstrained SVD based precoder, which maximize the spectral efficiency and which is

obtained by using the top NS left and right singular vectors of H. Note that the SVD based precoder

derivation requires full channel knowledge at both devices, an information that is hard to get when

hybrid architectures are used as explained earlier, and which our proposed scheme does not need at

the start, but rather learns while setting the precoders.

We start our evaluation in the high SNR regime (ρ=30dB), in which the training process will not

be impaired too much by noise. This allows assessing how the arrays size, the number RF chains and

the number of spatial streams really affect the algorithm’s performance.

Fig. 4.3 depicts the algorithm’s performance over a channel with 8 multipath components, when the

two devices are equipped with identical arrays made of NA=NB=32, 128 or 1024 elements and a fixed

number of RF chains NRFA =NRF

B =8 and try to establish an NS=4 streams MIMO communication.

We see that the algorithm reaches, in very few iterations, for small, medium and large sized arrays, to

about 1 to 2 bits/s/Hz of the digital PPMBT and SVD based precoding schemes performances. We also

see that the convergence speed (in terms of PP iterations) scales inversely to the codebook depth for

each of the topologies: the convergence is slower when large arrays are used because, in such cases, the

codebook has more levels and its few last levels contain a high number of beamformers with very narrow

beams. The selection of such directive beams is more prone to error than those in codebooks with lower


0 2 4 6 8 10 12 14 16 18 20 22


30

40

50

60

70

80

90

100S

pe

ctr

al

Eff

icie

nc

y

Optimal Precoding (SVD)

Digital PPMBT

Hybrid PPMBT

NA

=NB

=1024

NA

=NB

=128

NA

=NB

=32

Figure 4.3: Spectral Efficiency (bits/s/Hz) Attained by the Algorithm over PP Iterations. Devices Aand B are Equipped with a Hybrid Array : NRF

A =NRFB = 8, NS=4, L=8, ρ=30dB.

resolution. Although the backtracking mechanism can correct for the errors made, this comes at the

expense of a larger number of iterations needed for convergence. Note that the number of PP iterations

needed for convergence for all topologies is still much below what is needed for exhaustive search: this

latter requires as an example, for NA=NB=1024 with NRFA =NRF

B =8, NA/NRFA =NB/NRF

B =128 PP

iterations to find the best beams, compared with only 16 for our algorithm.

Fig. 4.4 shows the algorithm’s performance over the same channel as the one used in Fig. 4.3, but

with the number of MIMO streams, NS , taking different values (4, 6 and 8) and the two devices being

equipped with identical arrays made of NA=NB=128 elements and use 8 RF chains each. The purpose

of these simulations is to investigate how does the ratio of the number of MIMO streams to the number

of RF chains NS/NRF affect the algorithm’s performance. We can clearly see that the algorithm


behaves in three different ways depending on the aforementioned ratio: 1) When NS ≤ NRF /2,

the convergence is very quick and no backtracking is performed. 2) When NRF /2 < NS < NRF , the

algorithm’s convergence is slowed down and one observes some irregularity of the convergence behavior

over iterations which is due to the backtracking mechanism. 3) When NRF = NS , the algorithm fails

to provide acceptable performance.

0 5 10 15 20 25 30 35 40


0

10

20

30

40

50

60

70

80

90

100

110

Sp

ectr

al E

ffic

ien

cy


Digital PPMBT

Hybrid PPMBT

NS=4

NS=6

NS=8

Figure 4.4: Spectral Efficiency (bits/s/Hz) Attained by the Algorithm over PP Iterations for DifferentNS Values. Devices A and B are Equipped with a Hybrid Array : NA=NB=128, NRF

A =NRFB =8, L=8,

ρ=30dB.

Next, in Fig. 4.5, we fix the size of antenna arrays and number of RF chains used at both devices to

NA=NB=64 and NRFA =NRF

B =8, and evaluate the algorithm’s performance over a channel with L = 7

multipath components at different SNR values. In Fig. 4.5a, we show the algorithm’s performance

against SNR after training convergence for different NS values. The results show that the number of

streams that the training algorithm can handle efficiently grows with SNR, as is to be expected. At


low SNR regimes (−15 to 0 dB) the algorithm works better when it attempts to estimate only the

dominant singular vector of the channel; if more singular vectors are estimated, the large estimation

error degrades the overall spectral efficiency of the system. As the SNR grows, an increasing number

of singular vectors can be accurately estimated by the algorithm and, hence, increasing the number of

streams provides significant spectral efficiency gains. For all cases, we observe that the performance

of the Hybrid PPMBT is very close to that of the fully-digital counterpart, and approaches the SVD

precoding performance as the SNR grows. In Fig. 4.5b the convergence behavior of the training

algorithm with different NS settings is evaluated at their respective SNR values of interest. We observe

that the method’s spectral efficiency tends to saturate at around 10 ping pong iterations for moderate

and high SNR. Convergence for lower SNR values is, however, slower. We attribute this effect to the

fact that the large noise power induces numerous incorrect beam selection errors in the updates of the

analog precoder; although the backtracking feature of the algorithm can help correct some of them,

the price to pay is a longer training time. In any case, we remark that the algorithm reaches about

70% of the spectral efficiency obtained at convergence within the first 6 iterations, regardless of the

SNR value and number of spatial streams.

To conclude, we evaluate our proposed training procedure in a system in which only one of the

devices is equipped with a large, hybrid array (NA=256, NRFA =16), while the other has a digitally-

controlled array of moderate size (NB=4). Such topologies can be seen as massive MIMO systems

operating at microwave frequencies. In addition, to reflect the richer scattering experienced in such

frequency bands [90], we adopt the following channel model:

H=√NANBL

ABΛAHA (4.6)

where AA contains in its columns the steering vectors (ΩA,p), p=1, 2, . . . , P , AB is defined analogously,

L is again the number of multipath components, and Λ is a L×L matrix with i.i.d. standard complex

Gaussian entries. This setting will allow to test the validity of our approach in channels with richer

scattering and its robustness against the sparse assumption of the channel.

Fig. 4.6 shows the algorithm’s performance over a rich scattering channel (L=40) when NA=128,


NRFA =16, NB=NS=4. It can be seen that the proposed scheme performs remarkably well and very

close to the full digital and optimal SVD based precoder solutions at low, mid and high SNRs. These

results show that, although the algorithm was originally designed to exploit the sparse nature of

mmWave channels, it is robust to channels with richer scattering.

4.6 Conclusion

We proposed a method to derive precoders and combiners for multi-stream MIMO transmission be-

tween two devices equipped with hybrid digital-analog antenna arrays. The method relies on a low-

complexity “multi-beam split and drop with backtracking” procedure to update the analog precoders,

while digital precoders are computed with the QR-decomposition based method in [81]. For sufficiently

large SNR, the resulting precoders approximate well the unconstrained SVD-based precoders, as our

numerical assessment shows. We envision that the proposed algorithm can be especially useful in

mmWave communication systems.

Compared to the state-of-art methods, our approach offers the advantage of computational sim-

plicity while achieving high-spectral efficiency with moderate training overhead. The numerical results

show that the method achieves convergence within NRF (log2(N/NRF )+1) ping pong iterations in the

low SNR regime and log2(N/NRF )+1 iterations in the mid and high SNR regime, assuming both

transceivers are equipped with arrays made of N elements and NRF RF chains. Although the method

was developed with sparse channels in mind, the performance assessment shows that it is robust against

this assumption and also performs well in rich scattering channels.

Also, in order to further reduce the training overhead, the proposed scheme can be interleaved with

transmission of payload with increasing data-rate. This, the extension to multi-user environments and

to time varying channels will be the subject of our future work.

4.6. CONCLUSION 75

-20 -15 -10 -5 0 5 10 15

SNR(dB)

0

5

10

15

20

25

30

35

40

45

50

Sp

ectr

al E

ffic

ien

cy


Digital PPMBT

Hybrid PPMBT

NS=2

NS=1

NS=4

(a) Spectral Efficiency over SNR

0 5 10 15 20 25


0

5

10

15

20

25

30

35

40

45

50

Sp

etc

ral E

ffic

ien

cy


Digital PPMBT

Hybrid PPMBT

ρ=0dB, NS=2

ρ=15dB, NS=4

ρ=-15dB, NS=1

(b) Spectral Efficiency over PP Iterations

Figure 4.5: Spectral Efficiency (bits/s/Hz) Attained by the Algorithm over PP Iterations for DifferentNS Values. Devices A and B are Equipped with a Hybrid Array. NA=NB=64, NRF

A =NRFB =8, L=7.


0 2 4 6 8 10 12 14 16


0

10

20

30

40

50

60

70

80

Sp

ec

tra

l E

ffic

ien

cy


Digital PPMBT

Hybrid PPMBT

ρ=-15dB

ρ=30dB

ρ=0dB

Figure 4.6: Spectral Efficiency (bits/s/Hz) Attained by the Algorithm over PP Iterations. De-vice A is Equipped with a Hybrid Array and Device B has a Full Digital Architecture.NA=128, NB=4, NRF

A =16, NS=4, L=40.

Chapter 5

Concluding Remarks

5.1 Summary

The present thesis focused on developing beam search and beamforning solutions that address key

challenges in mmWave massive MIMO systems. The algorithms proposed herein tackles problems

related to the aforementioned system’s hardware constraints, channel acquisition overhead, and the

related beamforming design complexity.

We first proposed efficient Bayesian active beam search algorithms that suit best the noisy and

sparse nature of the mmWave massive MIMO line of sight channels. We showed how the proposed

techniques alleviate the need for separate channel estimation and beamforming design, while showing

high robustness against noise in low signal to noise ratio regimes. Then, we developed a novel multi

beam search algorithm that relaxes the channel’s line of sight assumption and helps establish multi-

stream mmWave wireless links. By leveraging the iterative power iteration technique, we showed that

our proposed scheme almost approximates singular value decomposition based multi stream beam-

forming.

For the line of sight mmWave massive MIMO systems, we proposed two variational Bayesian active

learning schemes that enable initial access for hybrid digital-analog enabled devices operating in these

77

78 CHAPTER 5. CONCLUDING REMARKS

highly sparse channels. The proposed schemes are devised with the goal to balance the beam search

time and to achieve high beamforming gain, while accounting for uncertainties on the unknown channel

(gain and noise variance). They build upon an active learning algorithm called hierarchical posterior

matching and extend it with tools from Bayesian inference to devise bi-directional beam alignment

algorithms that are numerically shown to effectively handle the uncertainty in the channel parameters,

thus resulting in beamforming gains close to these of exhaustive beam search algorithms, while requiring

an amount of pilot measurements comparable to that of hierarchical search algorithms.

For the multi beam alignment problem, we proposed an algorithm that derives precoders and

combiners for multi-steam MIMO wireless communications systems that are quipped with hybrid

digital-analog arrays. The proposed method uses a novel beam search mechanism called "multi-beam

split and drop with backtracking" to update the analog precoders and combiners, and uses the QR-

decomposition based update to devise the digital ones. Numerical simulations show that our algorithm

can approximate, with low pilot overhead, quite well the unconstrained SVD-based precoder/combiner

design for sufficiently large SNRs.

5.2 Future Work

There are several possible directions for future research.

In Chapter 3, we proposed two variational Bayesian online learning schemes that enable initial

access for hybrid digital-analog enabled devices operating in mmWave wireless channels, for point to

point hybrid array transceivers, communicating over static wireless channels. Both of our schemes

showed superiority, when compared to state of the art, in balancing the beam training overhead versus

attaining high beamforming gain. Further work is still required to adapt our proposals, and especially

the HiePM search mechanism, to both time varying channels as well as multi user environments.

In Chapter 4, we proposed a hybrid scheme that marries the novel multi-beam split and drop

with backtracking search mechanism with the QR-decomposition to devise the analog and digital

5.2. FUTURE WORK 79

precoders and combiners, for point to point hybrid array transceivers, communicating over static

wireless channels.

Theoretical guarantees for ’the multi-beam split and drop with backtracking’ scheme:

For fully digital transceivers, the QR-decomposition based precoding and combining is guaranteed

to converge to the wireless channel’s main singular value decomposition (SVD) components. One

important extension of this work is to consider investigating similar theoretical guarantees for our

hybrid algorithm and see under which conditions can the multi-beam split and drop with backtracking

search algorithm, when combined with QR-decomposition or any other baseband linear subspace search

algorithm, attain the performance of the unconstrained SVD based precoding and combining.

Training Overhead reduction for the proposed Ping Pong Multi Beam Training with

Hybrid Antenna Arrays scheme: Another interesting direction for future study is to investigate

how our proposed scheme can be interleaved with transmission of data payload with increasing data

rate. This would reduce the training overhead and make better use of the wireless channel resources.

Extension to multi-user environments and to time varying channels: One pre-requisite

to have our scheme considered for a concrete implementation in a cellular context is to make it usable

in multi-user environments and robust against the channel’s time variability. Future work in these two

directions is thus of great importance.

Appendices

81

Appendix A

Variational Hierarchical Posterior

Matching for mmWave Wireless

Channels Online Learning

83

Variational Hierarchical Posterior Matching formmWave Wireless Channels Online Learning

Nabil Akdim1, Carles Navarro Manchon2, Mustapha Benjillali3 and Pierre Duhamel41 Apple, Munich, Germany

2 Department of Electronic Systems, Aalborg University, Denmark3 Communication Systems Department, INPT, Rabat, Morocco

4 Laboratoire des Signaux et Systemes (L2S), CNRS-CentraleSup, FranceEmails: [email protected], [email protected], [email protected], [email protected]

Abstract—We propose a beam alignment algorithm thatenables initial access establishment between two transceiversequipped with hybrid digital-analog antenna arrays operating inmillimeter wave wireless channels. The proposed method buildsupon an active channel learning method based on hierarchicalposterior matching that was originally proposed for single-sidedbeam alignment on single path dominant channels. We extend itto the double-sided alignment problem and propose an estimationframework based on variational Bayesian inference that accountsfor the uncertainties on the unknown channel complex gain andnoise variance. The proposed approach is numerically shownto be resilient to the single path assumption and reaches nearoptimal beamforming gains with a moderate training overhead,even at low signal-to-noise ratios.

I. INTRODUCTION

Low power consumption and implementation complexity ofhybrid digital-analog transceiver architectures have acceleratedtheir adoption as a beamforming solution that can enableefficient wireless communications over the harsh mmWave fre-quency limited-scattering and blockage-prone wireless chan-nels for 5G and beyond wireless cellular networks [1]. Also,angular sparsity of such channels [1], [2] allows for the useof adaptive sparsity-friendly techniques to ease the initialalignment and channel state information (CSI) acquisition onthem, when using such transceiver designs [3]–[6].

In this study, we focus on hierarchical posterior matching(HiePM), an initial access scheme introduced in [7] whichprovably enables fast and reliable initial access establishmentbetween two wireless transceivers over wireless mmWavechannels with a single dominant path. Chiu et al. have shownin that contribution that using posterior matching [8] togetherwith hierarchical beam search [4] can significantly reducethe initial access acquisition time while keeping the corre-sponding misalignment probability relatively low, providedthat the channel’s complex gain and operating signal-to-noiseration (SNR) are fully known to the communicating devices.These limiting constraints were relaxed in [9] by proposingto augment HiePM with extra simplifying assumptions onthe statistical properties of the channel’s CSI and then touse either a sampling scheme or a linear filtering scheme(Kalman filter) to learn it in parallel to running HiePM.Although this latter extension of the vanilla HiePM makesit robust with respect to uncertainties on the channel’s CSI,

it still presents some limitations. First, it assumes perfectknowledge of the operating SNR. Second, the assumptionsmade on the statistical distribution of the channel’s complexgain (needed to make HiePM able to run as we will seelater), are simplistic and not justified from a theoretical orpractical view. Third, the proposed methods that build onsuch statistical assumption to overcome the CSI uncertaintyissue are either very restrictive and computationally heavy(in the case of the sampling method) or show relativelymoderate to low performance (in the case of the Kalmanfilter method). Finally, the overall tweaked setup assumes thatone of the communicating transceivers has a single antenna,and the extension to the case where both communicatingdevices use the hybrid digital-analog transceiver structure isnot straightforward.

Our contribution, in this work, which we dub “VariationalHiePM (V-HiePM)” will address shortcomings of proposalsin both of the aforementioned works [7], [9]. Specifically, wewill augment HiePM with a variational approximate inferencemodel [10] that will:

• make it robust against uncertainties of both CSI andoperating SNR,

• allow for a natural and theoretically grounded parametri-sation of the statistical properties of the CSI and operatingSNR, in a way that will make HiePM run smoothly,

• make the overall setup performing very close to thevanilla HiePM scheme with perfect SNR/CSI knowledge,

• allow for both communicating devices to be equippedwith hybrid digital-analog arrays.

II. SYSTEM MODEL

Our system is composed of two hybrid digital-analog an-tenna array devices A and B, equipped with uniform lineararrays (ULAs) of NA and NB antenna elements respectively.The elements on the ULAs are separated by a distanced = /2, where is the the mmWave wavelength ofinterest. Device A (B respectively) digitally controls its ULAwith NRF

A (NRFB respectively) RF chains. The two devices

communicate over a reciprocal static and narrowband wirelessmmWave MIMO channel, that is modeled as a NB NA

complex matrix H , sampled from the finite scatterer channelmodel [5] with a single dominant path as1:

H = ↵aB(B)aHA(A) (1)

where ↵ is the complex fading channel gain. aA(A) andaB(B) are the ULA array response vectors at devices A andB with incidence angles A and B respectively, modeledas aA (!A) =

1, ej!A , . . . , ej(NA1)!A

Tand aB (!B) =

1, ej!B , . . . , ej(NB1)!BT

, with !A(A) = 2 d cos (A)

and !B(B) = 2 d cos (B). The incidence angles A and

B are assumed to be sampled from the ranges [A,1, A,2]and [B,1, B,2] respectively. 2

The two devices go through an initial access phase con-sisting of a pilot based beam alignment procedure in order toestablish the wireless link between them. We assume in thiswork that, during this initial access phase, the CSI learningand beam search processes for the two devices are centralized,i.e one of the devices, say B, is collecting measurementsbased on device A’s pilot transmission, uses them to learnthe channel’s statistics and devises the beamformer it willuse for the next pilot reception occasion together with thebeamformer that device A should use in sending that pilot,then communicates such information to device A through anideal, error-free control channel3. At time instant t, device Asends a pilot symbol to B, which observes, after pilot removal:

yB,t =p

PwHB,tHfA,t + wH

B,tnB,t (2)where fA,t 2 CNA and wB,t 2 CNB are the effectivebeamformer and combiner used at time t by transceiversA and B respectively. These are chosen from the hybriddigital-analog codebooks detailed next. nB,t 2 CNB is acomplex circularly-symmetric additive white Gaussian noisevector with i.i.d elements with an unknown variance 2

B ,obtained after training sequence removal.

pP is the average

transmit power of the pilot signal.The adaptive beamforming strategy proposed herein utilizes

the hierarchical beamforming codebook of [4]. Such a code-book, noted CS hereafter, is designed to have S levels of beampatterns. We note Cl the collection of beams belonging to levell. Then, Cl contains 2l beamforming vectors that divide thesector [1, 2] into 2l directions, each associated with a certainrange of incidence angles Rm

l , such that [1, 2] = [2l

m=1Rml .

We note each of such 2l vectors as either fA (Rml ) or

wB (Rml ), depending on the considered device.

III. SEQUENTIAL BEAM PAIR SEARCH VIA VARIATIONALHIERARCHICAL POSTERIOR MATCHING

We start this section by first reviewing the details of thevanilla HiePM scheme [7], showing that knowledge of the

1This assumption is used only for the analytical derivation of our scheme.In Section IV, we show numerically that the method is resilient to such alimitation.

2No statistical assumptions on the distribution of the gain and the anglesare made here, we will later justify the choice of distributions we will useduring the inference process.

3Such channel can e.g. be established via a sub-6 GHz link in a non-stand-alone deployment. Control feedback channel design details will not bediscussed here due to space constraints.

channel gain ↵ as well as the noise variance 2 is necessary tomake such a search strategy usable in practice. We then detailour main contribution, which consists of augmenting HiePMwith a novel variational model comparison based approximateinference framework [10] to account for the uncertainties about↵ and 2

B and thus overcome the shortcomings, as detailed inthe introduction, of the vanilla HiePM and modified HiePMschemes proposed in [7] and [9] respectively.

A. Sequential Active Learning via the HiePM Strategy

We illustrate here the use of vanilla HiePM scheme [7]for device A (an analogous strategy will be used for deviceB). HiePM selects fA,t+1 based on the posterior at time t ofthe incidence angle A. We discretize4 the noisy beam searchproblem above by assuming that the beam search resolutionA

5 is an integer power of two and that the AoA A is of theform:A 2 A,1, . . . ,A,A,A,i = A,1+

(i 1)

A(A,2 A,1) (3)

With the above setup, the posterior distribution of A givenall measurements up to time t (collected in vector yB,1:t), canbe written as a A-dimensional vector A (t) with entries

A,i (t) := Pr (A = A,i|yB,1:t) , i = 1, . . . , A. (4)

The posterior probability of A being in a certain range,say Rm

i , can be computed asA,Rm

i(t) :=

X

A,i2Rmi

A,i (t). (5)

The HiePM strategy examines the posterior probabilityA,Rm

i(t) for all i = 1, . . . , SA and m = 1, . . . , 2i and

selects fA,t+1 2 CS to be the beamformer corresponding tothe angular range that satisfies:

(it+1, mt+1) = arg min

(i,m)

A,Rmi

(t) 1

2

(6)

Doing so, it is guaranteed [7] to sequentially refine the widthof the beamformer around the true incidence angle A.

Next we describe how the posterior bielief around A isupdated once a new measurement is taken with the pair ofbeamformers chosen previously with HiePM. Based on themeasurement model in (2), the posterior update at time instantt + 1 can be expressed using Bayes rule as

A,i (t + 1) / A,i (t) f (yB,t+1|A = A,i) ,

i = 1, . . . , A(7)

where f (yB,t+1|A = A,i) is the likelihood of A frommeasurement yB,t+1. Unfortunately, the likelihood term abovecannot be calculated in closed form due to the unknownchannel gain ↵, noise variance 2

B and incidence angle B .We will show next, how “V-HiePM” is able, using the

variational model based approximate inference frameworkdescribed in [10], to infer all above unknowns and uses them

4Such discretization approaches the original problem of initial access asA ! 0 [7].

5To support this level of resolution, the corresponding number of levelsof the hierarchical beamforming codebook at device A should be : SA =log2 (A).

efficiently to calculate the posterior update needed for HiePM,in a consistent and elegant way6.

B. The V-HiePM Scheme

We explain first the variational model based approximateinference framework used in its most general form, then showhow to apply it to our problem to derive posterior updates forour parameters of interest.

1) Variational Model Comparison based Posterior Up-date: We start by listing the different types of variables thatthe variational model comparison based approximate inferenceframework deals with:

• X is the observed data vector, in our case is yB,1:t+1.• Z = (Z1, Z2, . . . , ZL) denotes the L-dim vector of latent

variables that parameterize the measurement model (2).In our case, Z = (↵,2

B).• m 2 1, 2, . . . , A B denotes the mth pair of

angles (A,im,B,jm

), with im 2 1, . . . , A, andjm 2 1, . . . , B. Choosing a certain label m is equiv-alent to assuming that our measurement model in (2) isparameterized by the the mth pair of angles.

The framework performs joint inference on the hiddenvariables to find a set of distributions q(Z|m), q(m)1:m thatapproximate the true posterior p(Z, m|X), by minimizing theKullback-Leibler (KL) divergence:

KL(q(Z|m)q(m), p(Z, m|X)). (8)HiePM then uses the approximate incidence angle posterior

q(m) to decide which is the best measurement model can-didate fitting the observed data vector X . Algorithm 1 liststhe steps required to perform such operations. We omit themathematical derivation because of space constrains and referto [10, Chapter 10.4] for such details.

Algorithm 1: Variational Model Comparison basedPosterior Update

1 for m = 1 : AB do2 while (No convergence yet) do3 for j = 1, 2, . . . , L do4 q(Zj |m) / Ei 6=j(log(p(X|Z, m)))

5 Lm =R

Zq(Z|m) log(

p(Z,m|X)q(Z|m)

)

6 q(m) / p(m) exp(Lm)

2) Posterior Update for our measurement Model andthe overall V-HiePM Algorithm: From our measurementmodel (2), we have

p(X, Z, m) = p(yB,1:t|↵, B , m)p(↵)p(B)p(m) (9)where B = 2

B is the noise precision atdevice B; p(X|Z, m) = p(yB,1:t|↵, B , m) =Qt+1

i=1 CN(yB,i;p

P↵wHB,iAmfA,i,

2B) is the likelihood

6As it will be detailed below, such an inference framework lends itselfnaturally in the HiePM context: we make the best use of the measurementsby first estimating posteriors over the channel gain and noise varianceand then use those to robustly update the angle of incidence posterior,doing so allows V-HiePM to take the channel’s gain and noise varianceestimation uncertainties properly into account when deriving the posteriorof the incidence angles, thus making a robust HiePM based decision whenchoosing the next precoder/combiner pair to use.

of our measurement model (we assume here that thesequential noise samples are i.i.d); p(↵) = CN(↵;↵0,0)is the prior belief over ↵, considered to be Gaussianwith a known initial mean ↵0 and initial precision 0

7;p(B) = (B ; a0, b0) is the non informative prior beliefover B , with parameters a0 = 0 and b0 = 0; finally,p(m) = 1

ABis the prior belief over m, which is assumed to

be uniform to make it non informative as well8. In addition,Am = aB(B,jm

)aHA(A,im

) is the assumed unfaded channelmatrix under the mth pair of incidence angles.

The obtained approximate posteriors for ↵ and B , up tothe measurement iteration t, can be shown to keep the form oftheir respective priors, but with parameters that depend on themeasurement vector yB,1:t: qt(↵|m) has the form of complexGaussian pdf with mean ↵t,m and precision t,m reading

t,m =at,m

bt,m

tX

d=1

p

PwHB,tAmfA,t

2

+ 0 (10a)

↵t,m =at,m

bt,mt,m

tX

d=1

pPwH

B,tAmfA,t

yB,d +

↵00

t,m

(10b)and qt(B |m)9follows a Gamma pdf with parameters shapeand rate parameters at,m and bt,m given by

at,m = a0 + t, (11a)

bt,m = b02RePt

d=1

pPwH

B,tAmfA,t

yB,d↵

t,m

+

Ptd=1

|yB,d|2 +

1

t,m+ |↵t,m|2

p

PwHB,tAmfA,t

2

(11b)Note that the choice of our prior distributions is not ar-

bitrary, the priors chosen above correspond to the maximumentropy distributions [11] that respect constraints that needto be put on their respective parameters, namely ↵ being acomplex variable having a known initial mean and variance,B being a non negative variable and m being a discretevariable). Such a choice makes our proposal assume the leastinformation about our measurement model’s unknowns.

The posterior of the model, indexed by m, is then updatedfollowing Lines 5 and 6 in Algorithm 1, where Lm reads

Lt,m = log(1

2t,m

) + at,m (1 log(bt,m)) + log((at,m))

b0at,m

bt,m

tX

d=1

|yB,d|2at,m

bt,m t,m |↵t,m|2

!(12)

7The first and second order moments of ↵ are the only assumed knownvalues in our model.

8In the above, CN(·; µ,) denotes the complex Gaussian pdf with meanµ and precision , (·; a, b) denotes the Gamma pdf with shape and rateparameters a and b.

9Note that (10) and (11) can be re-written, after performing some algebra,in a recursive format w.r.t their terms involving summation over measurementsepochs. This results in a significant reduction of the algorithm’s memory andcomputation complexity footprint.

The posteriors over A,imand B,jm

are obtained from theposterior qt(m) as

qA,t(i) =X

m:im=iqt(m), i = 1, . . . , A (13)

qB,t(j) =X

m:jm=jqt(m), j = 1, . . . , B (14)

The posterior probability of the incidence angles A andB to be in a certain range Rn

A,i and RpB,j resp, read as:

qA,t(RnA,i) :=

X

A,i2RnA,i

qA,t(i), (15)

qB,t(RpB,j) :=

X

B,j2RpB,j

qB,t(i), (16)

The vanilla HierPM scheme is then applied separately toqA,t(i) and qB,t(j), to choose the pair of beamformers to usefor the next measurement occasion.

The modes ↵t and Bt of the approximate posteriorsqt(↵|m

t ) and qt(B |mt ), with m

t = arg maxm(qt(m)), canbe seen as approximations of the MMSE estimates of ↵ andB respectively. These estimates are given by:

↵t = ↵t,mt, Bt = at,m

t/bt,m

t. (17)

Algorithm 2 runs all above operations in a loop, until themeasurement budget is exhausted: device B decides whichpair of beamformers devices A and B shall use to take thenext measurement by applying the HiePM scheme separatelyto the current posteriors qA,t and qB,t, it then takes a newmeasurement yB,t+1 with those latter, and finally run varia-tional inference to derive approximate posteriors of B = 1

2B

,↵ as well as of A and B .

IV. NUMERICAL RESULTS

To assess the effectiveness of V-HiePM, we run Monte Carlosimulations on a setup with two hybrid digital-analog beam-forming devices A and B. The channel matrix H 2 CNBNA

reads

H = ↵aB(B)aHA(A) +

LX

l=1

↵laB(B,l)aHA(A,l) (18)

and contains one dominant multipath component and L scat-tered components. All incidence angles are independentlydrawn from a uniform distribution between 0 and . Thechannel gains are independently drawn from a a set of complexGaussian distribution with mean 0 and variances fulfillingVar↵+

Pl Var↵l = 1, so that the average SNR between

the nth element of the array at A and the mth element ofthe array at B equals E|Hnm|2/E|B |2=1/2

B10. In all

simulations below, the two devices are equipped with identicalarrays made of NA = NB = 32 elements, digitally controlledwith NRF

A = NRFB = 8 RF chains. Device A uses a codebook

CA with a depth of SA = log2(A), A = 128. CA is built

10Hnm is the channel coefficient between device B’s nth array elementand device A’s mth array element, and E is the expectation operator.

using the orthogonal matching pursuit as described in [4]. Asimilar codebook, CB , is used for device B.11

We benchmark our algorithm’s beamforming gain after tmeasurements, defined as:

Gvh =wH(B,kt,B

)Hf(A,kt,A)2

(19)with different measurement budget sizes and under differentchannel assumptions (note the the exhaustive search needsNANB = 16384 measurements to settle), against that of thedifferent state of the art schemes listed below:

• Gph of the vanilla HiePM scheme of [7]. Here, such ascheme assumes that most of the energy in the channelis concentrated in the path corresponding to the knowngain ↵ and all other gains ↵l are null, it also assumesthat 2

B is known. In such case, the posterior update isdone, simply using Bayes rule as in equation (21) in [7],on the beam pair corresponding to that main path, andthen HiePM is applied to the marginals over those anglesseparately, similar to what V-HiePM does.

• Gbs of the noisy binary search algorithm of [4], which isachieved by 4 log2(maxNA, NB) = 28 measurements.

As a reference, we consider as well the best achievablebeamforming gain of the codebook, defined as

Gmax = maxw2CB

SB,f2CA

SA

wHHf2 . (20)

11Note that the multi-RF chain setups are used solely to help buildacceptable RF codebooks [4], and are not used for multi-stream MIMOoperations.

Algorithm 2: V-HiePM1 Input : Antenna Array Size NA and NB , The search resolution A and B ,

the codebooks CSAand CSB

, Search time 2 Output : Estimates of A, B , ↵ and B

3 for t = 1, 2, . . . , 1 do4 #HierPM Based BF selection according to Eq.(6)

(fA,t+1, wB,t+1) =

fA

RkA,t+1

A,lA,t+1

, wB

RkB,t+1

B,lB,t+1


pPwH


7 #Variational Model Comparison Posterior Update8 for m = 1 : AB do9 while (No convergence yet) do

10 update qt+1(↵|m) via (10) then qt+1(B |m) via (11)11 end12 update qt+1(m) via (12)13 end

14 #Angle and Angular Range Posterior Update15 update qA,t+1(i) via (13) and qB,t+1(j) via (14)16 update qA,t+1(Rn

A,i) via (15) and qB,t+1(RnB,j) via (16)

17 #Final Precoder/Combiner Vector designlt+1,A, kt+1,A

= (SA, arg maxk (qA,t+1(k)))

18

lt+1,B , kt+1,B

= (SB , arg maxk (qB,t+1(k)))

19 #Channel’s Gain and Noise Precision MAP Estimates update ↵t+1 andBt+1 via (17)

20 end21 Output : A = A,k,A

, B = B,k,B, ↵ = ↵ , B = B

-10 -5 0 5 10

SNR in dB

-25

-20

-15

-10

-5

0

Be

am

form

ing

Lo

ss

in

dB

Lbs

: = 28

Lvh

: = 28

Lph

: =28

Lvh

: = 56

Lph

: =56

Lvh

: = 128

Lph

: =128

Figure 1: Beamforming loss of the different search schemes in achannel with L = 0 scattered components.

We begin by assuming that only the dominant componentis present (i.e. L = 0). Fig. 1 shows the beamforming lossesof the benchmarked algorithms with respect to the optimumpair of beamformers, defined as Lvh = Gvh/Gmax, Lph =Gph/Gmax, and Lbs = Gbs/Gmax. The results show thesuperiority of our scheme compared to the binary search of[4], and that it achieves similar or even better performancecompared to vanilla HiePM with perfect CSI and operatingSNR knowledge. It can be observed that the vanilla HiePMscheme with perfect channel gain knowledge saturates at highSNR: this is an effect of the algorithm assuming that thecomponent’s incidence angle lies on a discrete grid of values,whereas the actual angles are sampled from a continuousdistribution. Our proposed method is less sensitive to thismodel mismatch, due to the estimation of the channel gainand inverse noise variance: in practice, these estimates partlyaccount for the mismatch in the assumed values of the anglesand provide robustness to the overall procedure.

Next, we explore the robustness of the proposed methodagainst channels containing more than one multipath compo-nent. For this, we consider a channel with L = 3 scatteredcomponents with gains of equal variance, and with the powerratio between the dominant and scattered components beingLOSR = E↵2/(E↵2 +

Pl E↵2

l ). Fig. 2 shows beam-forming gains achieved by our algorithm after 100 measure-ments compared to the maximum gains achievable Gmax.

As it can be observed, the maximum achievable beamform-ing gain decreases as the power is more evenly distributedamong the channel’s components. Although V-HiePM assumesthe existence of a single component, it shows remarkableresilience to the presence of other components. Even whenall components in the model have comparable power, ourproposed method is able to perform within 2 dB of theoptimum for sufficiently high SNR.

V. CONCLUSION AND FUTURE WORK

We proposed in this work a variational Bayesian onlinelearning scheme that enables initial access for hybrid digital-analog enabled devices operating in mmWave wireless chan-nels. When compared to state of the art beam acquisitionschemes, our method shows superiority, in terms of balancing

-10 -5 0 5 10

SNR in dB

20

22

24

26

28

30

Be

am

form

ing

Ga

in i

n d

B

Gvh

: L=3, LOSR

= 0.2

Gmax

: L=3, LOSR

= 0.2

Gvh

: L=3, LOSR

= 0.7

Gmax

: L=3, LOSR

= 0.7

Gvh

: L=0, LOSR

= 1

Gmax

: L=0, LOSR

= 1

Figure 2: V-HiePM performance in channels with different powerratios between the dominant path and L = 3 scattered components.

the beam search time versus achieving higher beamforminggain, in being able to properly do so while accounting foruncertainties on the unknown CSI (gain and noise variance)and in being very resilient to the dominant single path assump-tion. Even though the scheme is derived based on a discretizedmodel of the angles of incidence of the channel’s maincomponent, it showed great robustness against off-grid anglesas well as working with a realistic codebook implementation.Further research will focus on adapting the proposed onlinelearning algorithm to operating in time-varying channels.

REFERENCES

[1] R. W. Heath, N. Gonzalez-Prelcic, S. Rangan, W. Roh, and A. M.Sayeed, “An overview of signal processing techniques for millimeterwave MIMO systems,” IEEE Journal of Selected Topics in SignalProcessing, vol. 10, no. 3, pp. 436–453, Apr. 2016.

[2] S. Hur, S. Baek, B. Kim, Y. Chang, A. F. Molisch, T. S. Rappaport,K. Haneda, and J. Park, “Proposal on millimeter-wave channel modelingfor 5G cellular systems,” IEEE Journal of Selected Topics in SignalProcessing, vol. 10, no. 3, pp. 454–469, Apr. 2016.

[3] O. E. Ayach, S. Rajagopal, S. Abu-Surra, Z. Pi, and R. W. Heath,“Spatially sparse precoding in millimeter wave MIMO systems,” IEEETransactions on Wireless Communications, vol. 13, no. 3, pp. 1499–1513, Mar. 2014.

[4] A. Alkhateeb, O. E. Ayach, G. Leus, and R. W. Heath, “Channelestimation and hybrid precoding for millimeter wave cellular systems,”IEEE Journal of Selected Topics in Signal Processing, vol. 8, no. 5, pp.831–846, Oct. 2014.

[5] N. Akdim, C. N. Manchon, M. Benjillali, and E. de Carvalho, “Pingpong beam training for multi stream MIMO communications with hybridantenna arrays,” in 2018 IEEE Globecom Workshops (GC Wkshps), Dec.2018, pp. 1–7.

[6] Y. Ding, S. Chiu, and B. D. Rao, “Bayesian channel estimation algo-rithms for massive MIMO systems with hybrid analog-digital processingand low-resolution ADCs,” IEEE Journal of Selected Topics in SignalProcessing, vol. 12, no. 3, pp. 499–513, Jun. 2018.

[7] S. Chiu, N. Ronquillo, and T. Javidi, “Active learning and CSI acquisi-tion for mmwave initial alignment,” IEEE Journal on Selected Areas inCommunications, pp. 1–1, 2019.

[8] O. Shayevitz and M. Feder, “Optimal feedback communication viaposterior matching,” IEEE Transactions on Information Theory, vol. 57,no. 3, pp. 1186–1222, Mar. 2011.

[9] N. Ronquillo, S. Chiu, and T. Javidi, “Sequential learning of CSIfor mmwave initial alignment,” CoRR, vol. abs/1912.12738, 2019.[Online]. Available: http://arxiv.org/abs/1912.12738

[10] C. M. Bishop, Pattern Recognition and Machine Learning (InformationScience and Statistics). Berlin, Heidelberg: Springer-Verlag, 2006.

[11] M. U. Thomas, “A generalized maximum entropy principle,” OperationsResearch, vol. 27, no. 6, pp. 1188–1196, 1979.

Appendix B

Ping Pong Beam Training for Multi

Stream MIMO Communications with

Hybrid Antenna Arrays

89

Ping Pong Beam Training forMulti Stream MIMO Communications

with Hybrid Antenna ArraysNabil Akdim1, Carles Navarro Manchon2, Mustapha Benjillali3, and Elisabeth de Carvalho2

1 Intel Deutschland, Munich, Germany2 Department of Electronic Systems, Aalborg University, Denmark

3 Communication Systems Department, INPT, Rabat, MoroccoEmails: [email protected], [email protected], [email protected], [email protected]

Abstract—We propose an iterative training procedure that ap-proximates multi-stream MIMO eigenmode transmission betweentwo transceivers equipped with hybrid digital analog antennaarrays. The procedure is based on a series of alternate (pingpong) transmissions between the two devices in order to exploitthe reciprocity of the wireless channel. During the ping pongiterations, the update of the devices’ digital precoders/combinersis performed based on a QR decomposition of the receivedsignal matrix. Concurrently, their analog precoders/combinersare progressively updated by a novel “multi-beam split anddrop with backtracking” mechanism that tracks the channel’smain spatial components. As shown throughout the paper, theproposed algorithm converges with only few iterations, hasminimal computational complexity, and performs very closelyto optimal singular value decomposition based precoding withsufficiently large signal-to-noise ratio.

I. INTRODUCTION

Upcoming wireless communication networks are expectedto provide service to an unprecendently large number of wire-less devices with peak data rates in the order of tens of Gbps.The congestion and fragmentation of the traditional spectralbands below 6GHz has pushed wireless service providers toexplore vacant spectrum at the millimeter-wave (mmWave)frequency bands (30–300 GHz) in order to fulfill that goal [1].Nevertheless, the poor reflectivity and high absorption and freespace propagation losses make communicating wirelessly oversuch high frequencies a challenging task [2]. Fortunately, thisfrequency range will allow for the use of compact and smallantenna arrays with high number of elements, as the physicalsize of the array is proportional to the carrier wavelength. Thelarge beamforming gains that such large-scale arrays enablewill be used to compensate for the above limitations.

However, the high cost, power consumption and complexityof the mixed signal hardware at mm-wave make having largeantenna arrays with digitally controlled elements infeasible [3].This has motivated the wireless communication research com-munity to look at the hybrid digital-analog antenna arrayarchitectures [4]. In such architectures, the large antenna arrayis steered using analog phase shifters and only a few digitallymodulated radio-frequency (RF) chains. An illustration of suchan architecture is shown in Fig. 1.

In addition to their cost and implementation advantages,hybrid array structures entail their own challenges: the lowSNR resulting from high propagation losses, the large dimen-sionality of the MIMO channel matrix and the presence ofanalog processing complicate the acquisition of the channelstate information (CSI) and the computation of the MIMOprecoders and combiners [1], [3]. Luckily, channel measure-ment campaigns [2] have shown that mm-wave channels aresparse in the angular domain, which enables the proposalof CSI acquisition and precoding/combining algorithms thatexploit such property. An example of these are compressedsensing based approaches such as [3], [5], [6], which aregenerally computationally complex and require a large amountof channel measurements. An alternative are exhaustive andhierarchical beam-search techniques, which may entail signif-icant latency and probability of miss detection [7].

In this work, we focus on a beam training strategy basedon alternating transmissions between two transceivers, whichhas been coined ping pong beam training (PPBT). The mainidea behind PPBT is to exploit the reciprocity of the MIMOchannel. With appropriate processing at each device, the alter-nate transmissions implicitly implement an algebraic poweriteration that leads to approximating the top left and rightsingular vectors of the MIMO channel matrix. This idea wasfirst applied in the digital arrays context for single streamwireless communications in [8], [9], and was extended tomulti stream setups in [10], to large antenna array andfrequency selective systems in [11] and to noisy MIMOchannels in [12]. More recently, similar approaches have beenproposed in the context of mmWave communications withhybrid digital-analog antenna arrays, which we review next.In [13], the basic ping pong beam training method for single-stream MIMO transmission was adapted to the hybrid arrayarchitecture with the inclusion of a “beam split-and-drop”procedure for the setting of analog precoders. The subspaceestimation and decomposition method in [14] proposes a pingpong based algorithm that iteratively estimates the channel’sright and left eigenvectors using a Krylov subspace estimationmethod. This algorithm is based on exhaustive measurementswith a large set of different analog precoders, which are then

Baseband

Precoder

Baseband

Combiner

RF Chain

RF Chain

RF Chain

RF Chain


NA NB

NA

RF

NB

RF

H

WA

FATRANSCEIVER A TRANSCEIVER B

FB

WB

YB[1]

YB[ ]NB

RF

Fig. 1: Structure of the transceivers

linearly combined in order to cancel the effect of the analogprecoders. It therefore requires significant amount of trans-missions, which imply large signalling overhead and latency.Lastly, the power iteration based training method introducedin [15] is a technique that extends the solution proposedin [10] to the multi stream case, where the digital precodersare set based on an algebraic power iteration technique, whilethe analog precoders update is done based on a compressedsensing technique called simultaneous orthogonal matchingpursuit [16].

Compared to the above approaches, we propose in thisarticle a strategy that sets the digital and analog precodersof the devices in a way to approximate the top NS left andright singular vectors of the channel matrix, with NS beingthe desired number of spatial streams. Our new technique,which we dub “hybrid ping pong multi beam training“(Hybrid PPMBT) extends the work done in [13] to the multistream case. It adapts the PPBT strategy to hybrid arrays byprogressively choosing the analog precoders at each devicefrom a predefined hierarchical codebook. After one round-triptransmission, a novel ”multi beam split and drop strategy withbacktracking” is applied to focus the analog precoders towardsthe spatial directions that are most likely containing thechannel’s top NS multipath components. The digital precodersare updated via an orthogonal decomposition operation on thereceived signal as described in [10]. In comparison to theapproaches in [14] and [15], Hybrid PPMBT is much simplerfrom a computational complexity aspect and has a low trainingoverhead as it requires significantly fewer transmissions. Sim-ulation results show that our proposed scheme performs verywell in retrieving the wanted NS channel’s top eigenmodesfor sufficiently large signal-to-noise ratio, both in terms ofaccuracy and convergence speed.

II. SYSTEM MODEL

We consider a system in which two hybrid analog digitaltransceivers A and B, equipped with uniform linear arrays(ULA) composed of NA and NB antenna elements. Suchelements are separated with a distance d = /2, where isthe wavelength of interest. The two devices control digitallytheir arrays with NRF

A and NRFB RF chains respectively and

exchange data over a reciprocal wireless MIMO channel usingNS parallel data streams. The channel from device A to

device B is considered to be static and narrowband and ismodeled according to the finite scatterer channel model withL propagation paths [13], [17], as

H =

rNANB

L

LX

l=1

↵laB(B,l)aHA(A,l), (1)

here, H 2 CNBNA , L is the number of multipath com-ponents (MPC), ↵l is the complex fading channel gain forMPC l, A,l = 2

d cosA,l and B,l = 2 d cosB,l

are the directional cosines corresponding to the lth MPCat arrays A and B respectively, where A,l, B,l are theangles of incidence of that same path, and aA and aB

are the array response vectors at device A and B respec-tively. The ↵l are modeled as independent, standard com-plex gaussian variables, the A,l and B,l as uniformly dis-tributed in the range [0, 2) radians and the array responsesas aA(A,l) = [1, ejA,l , . . . , ej(NA1)A,l ]T/

pNA and

aB(B,l) = [1, ejB,l , . . . , ej(NB1)B,l ]T/p

NB .In order to establish the wireless link with device B, device

A (we assume, without loss of generality, that device A isperforming the first transmission) transmits T , an NS NS

orthogonal training sequence i.e TT H = INS. Upon reception,

device B cancels the training sequence effect by multiplyingits received digital signal by T H . The resulting signal can beexpressed as:

YB = F HBHFAWA + F H

BNB , (2)where FA 2 CNANRF

A and FB 2 CNBNRFB contain the

states of the analog precoder and combiner of transceiversA and B, WA 2 CNRF

A NS denotes the digital precoder oftransceiver A and NB 2 CNBNS is a complex, circularly-symmetric additive white gaussian noise matrix, obtained aftertraining sequence removal and with i.i.d elements, each withvariance 2. Transmissions from device B to device A aremodeled analogously as

YA = F HAHHFBWB + F H

ANA, (3)where WB 2 CNRF

B NS and NA 2 CNANS are definedsimilar to the above.

III. HYBRID PING PONG MULTI BEAM TRAINING :HYBRID PPMBT

Given the signal model in (2) and (3), the beamforming taskconsists of selecting the set of analog and digital precoders andcombiners that maximize the spectral efficiency over a givenchannel matrix H . For transmission from device A to B, andassuming unit transmit power equally allocated across the NS

streams, the spectral efficiency reads

R = log2 detINS

+R1

NB

NSHeH

He

, (4)

where RNB= 2W H

BF HBFBWB is the noise covari-

ance matrix after receive combining at device B, He =W H

BF HBHFAWA is the equivalent channel after precoding

and combining at both devices. An analogous expressionapplies for transmission from device B to device A.

The optimal precoders maximizing (4) are known to be theNS top right and left singular vectors of H . However, thehybrid structure of the antenna array makes the computation

of such precoders challenging. On the one hand, as digitalmeasurements of the channel are only obtained after analogprecoding and combining, estimating the full channel matrixH in order to obtain its singular value decomposition requiresa large number of measurements and hence large overheadand latency [14]. On the other hand, even if the channelmatrix H can be estimated, the precoders have to be builtas the product of the analog precoding matrix FA and thedigital precoding matrix WA. While the elements of WA cantake any complex value due to its digital implementation, theoperation modeled by FA is implemented via phase shiftersand combiners, which restricts the values it can take. Inthis work, we restrict the entries of FA to satisfy |(FA)l,i|12 1

M(i)A

; 0, where M(i)A being the number of activated

array elements in the ith column of FA, and the option(FA)l,i = 0 accounts for the option of leaving some elementsof the array unused. In addition, a transmit power constraint isenforced such that kFAWAkF = 1.2 With these constraints,finding the combination of digital and analog precoders tobest approximate the channel’s singular vectors becomes acomputationally intensive optimization problem [5].

To overcome such difficulties, we propose an iterative multibeam training scheme based on alternate transmissions be-tween the two devices, this procedure estimates progressivelyand simultaneously the top NS right and left singular vectorsof H and sets the digital and analog precoders so that theyapproach those singular vectors. It consists of two parts: 1) a”backtracking beam split and drop” approach to select theanalog precoders FA and FB from a predefined multi levelcodebook, 2) a method to select the digital precoders WA andWB inspired by the QR decomposition algorithm describedin [10].

We will proceed by reviewing the beam training procedurefor digital antenna arrays proposed in [10], then briefly presentthe multi level codebook that is used for the analog precoderupdate, and finally explain our multi beam training solution.

A. Ping-Pong Multi Beam Training with Digital AntennaArrays: Digital PPMBT

We review the digital PPBT algorithm over a narrowbandreciprocal channel H as described in [10]. We consider twodevices A and B equipped with digitally controlled antennaarrays with NA and NB elements respectively. At the initial(0th) iteration, the process starts with a random initializationof the precoder at device A, W

[0]A . A uses then this initial

precoder to transmit a training sequence to B. Upon receptionand training sequence removal, device A gets an estimate ofHW

[0]A , makes a QR-decomposition on it, and uses the Q

part of that decomposition as its precoder W[0]B . It then uses

that precoder to transmit a training sequence back to device A,who will repeat the same operations. This process is reiterateduntil convergence, at which, device A gets an estimate of the

1(FA)l,i is the entry of the matrix FA belonging to its lth row and ithcolumn.

2Obviously, the same constraints apply to the analog precoder of device B.

top NS right singular vectors of H and device B gets anestimate of the top NS right singular vectors of HH. Furtherdetails on this procedure can be found in [10].

Algorithm 1 Ping Pong Multi Beam Training with HybridArrays

1: Initialize:

F[0]A

'

(1)A,0,'

(1)A,1, . . . ,'

(1)

A,M(1)A

1

,

F[0]B

'

(1)B,0,'

(1)B,1, . . . ,'

(1)

B,M(1)B

1

,

Initialize W[0]A to an orthogonal matrix of its size.

for s = 1 : NS do

W[0]A (:, s) W

[0]A

(:,s)pNS

F[0]A

W[0]A

(:,s)2

end forpA, kA, iA [], [], [], pB , kB , iB [], [], [].

2: A transmits, B receives: Y[0]

B =(F[0]B )HHF

[0]A W

[0]A +(F

[0]B )HN

[0]B

3: [Q, R] qr(Y[0]

B )4: for s = 1 : NS do5: W

[0]B (:, s) Q(:, s)

6: W[0]B (:, s) W

[0]B

(:,s)pNS

F[0]B

W[0]B

(:,s)

27: end for8: t 19: loop

10: B transmits,11: A receives Y

[t]A =(F

[t1]A )HHHF

[t1]B W

[t1]B +(F

[t1]A )HN

[t]A

12: [Q, R] qr(Y[t]

A )13: for s = 1 : NS do14: W

[t]A (:, s) Q(:, s)

15: W[t]A (:, s) W

[t]A

(:,s)pNS

F[t1]A

W[t]A

(:,s)

216: end for17: [F

[t]B , pB , kB , iB]

UPD.AN.PR(F[t1]B , W

[t1]B , CB , pB , kB , iB)

18: A transmits,19: B receives: Y

[t]B =(F

[t]B )HHF

[t1]A W

[t]A +(F

[t]B )HN

[t]B

20: [Q, R] qr(Y[t]

B )21: for s = 1 : NS do22: W

[t]B (:, s) Q(:, s)

23: W[t]B (:, s) W

[t]B

(:,s)pNS

F[t]B

W[t]B

(:,s)

224: end for25: [F

[t]A , pA, kA, iA]

UPD.AN.PR(F[t1]A , W

[t]A , CA, pA, kA, iA)

26: t t + 127: end loop

B. Analog Precoder Multi Level Codebook

We illustrate here the codebook definition for transceiverA (an analogous codebook is used for the transceiverB). We consider a codebook CA which is composedof LA= log2 (NA/NRF

A )+13 levels. For the kth level,we define a subcodebook C(k)

A ='(k)A,i, i=0, 1, . . . , M

(k)A -1

consisting of M(k)A =NRF

A 2k-1 column vectors, k=1, 2, . . . , LA.Each of the elements of the subcodebook is defined as

3For the considered codebook design we constrain NA and NRFA to be

both integer powers of two.

'(k)A,i=

1, e-j (k)

A,i , . . . , e-j(M(k)A -1) (k)

A,i ,0TNA-M(k)

A

T

/

qM

(k)A ,

where (k)A,i=-(2i+1)/M

(k)A is the directional cosine of the

ith vector at the kth level ('(k)A,i steers the array in the direction

(k)A,i= arccos

(k)A,i/, with a lobe whose width decreases with

the codebook level k), and 0N is the N -dimensional columnzero vector. Further details about the codebook used here canbe found in [13].

C. Ping Pong Multi Beam Training with Hybrid AntennaArrays : Hybrid PPMBT

The proposed algorithm for beam training with hybridarrays is described in pseudocode. Algorithm 1 presents theoverall training scheme, while Algorithm 2 describes thesubroutine used to update the analog precoding matrices.

1) Initialization: First, the digital and analog precodersare initialized. FA and FB are initialized to C(1)

A and C(1)B

respectively, while WA is initialized to a random unitarymatrix. The columns of WA are then normalized to fulfill thetransmit power constraint of the effective precoder FAWA.Finally, a set of empty arrays, pB , kB , iB, are created.Those arrays will store the needed information to perform theanalog precoder updates, as explained below.

2) Ping Pong Iterations: After the initialization phase,a sequence of alternate pilot transmissions starts between thetwo devices. The baseband precoders are updated after eachreception step by means of a QR-decomposition, followedby a normalization step. These two operations are detailedin lines 3-7, 12-16 and 20-24 in Algorithm. 1. Immediatelyafter updating their baseband precoders upon reception ofa transmission, the devices transmit back with the updateddigital precoders and the same analog precoder as used forreception. Only after the transmission has been made willthe transmitting device update its analog precoders (usingAlgorithm 2), such that next reception is done with the updatedsetting. This allows for the QR based iteration to converge,as each reception-transmission cycle is performed over a staticsetting of the analog precoders.

3) Update of Analog Precoders: The devices invokethe routine outlined in Algorithm 2 to update their analogprecoder state. This routine bases its update on the currentstate of the device’s RF precoder F , its codebook C and on theupdate history of its baseband precoder W . Using all previousupdates of W allows for backtracking–i.e, correcting forwrong RF precoder updates. The routine works as follows:

a) Three sequences of values are generated: kn stores thelevel of the codebook of the nth column of F , pn storesthe squared norm of the nth row of W , i.e the aggregateenergy received on it, and in stores the index of the nthcolumn of F , out of the level of the codebook to whichthat column belongs.

b) The sequences generated above will be used to updatethree vectors: kn will be appended to k, with k beingan array storing the codebook levels of the columns ofF used over consecutive ping-pong iterations. pn will

be appended to p, with p being an array storing thereceived energy over the different spatial directions setby the analog beamformer F . in will be appended to iin a similar manner to the above.

c) Once p is updated, it will be sorted in a descendingmanner and the resulting sorted indices will be storedin pI . This newly formed array will be used to findthe entries of p that are most likely to direct the analogprecoders where the MPCs of H are.

d) Two new vectors are built: bI contains the K first indexesof pI i.e it identifies the beams that are most alignedwith the channel’s MPCs (K is the length of bI , whichcan be derived from lines 9-13). bL is a vector that ismade of 1’s and 2’s. The ith entry of bL is set to 2when the precoder corresponding to the ith entry of bI

is replaced with the two precoders belonging to one stephigher level of the codebook and that have their beamscovering together its same spatial area, otherwise it isset to 1. Deciding to append 1 or 2 to bL depends onwhether we already consumed all columns of F and onwhether the element of pI in question belong to the lastlevel of the codebook or not (see lines 17-21). Line 8 ofthe algorithm erases the measurement stored over a beamthat is selected to be included in the analog precodingmatrix, either directly or after splitting it into two beamsof the immediately higher level. As new measurementswill be obtained over that beam in the next iteration, theold measurement is deleted to avoid unnecessarily comingback to the previous configuration corresponding to that

Algorithm 2 Analog Precoder Update

1: function UPD.AN.PR(F , W , C, p,k,i)2: Generate : kn, in, pn, n=1, . . . , NRF , where kn will store

the codebook level to which the F0s nth column belongs, in

will store its index out of that level and pn will store the normof the nth row of W , i.e the aggregate energy received on it.

3: p [p, p1, . . . , pNRF ], k [k, k1, . . . , kNRF ], i [i, i1, . . . , iNRF ]

4: Sort p in a descending manner and store the result in pS ,then store the arrangement of the elements of p into pS in pI .

5: bI [], bL []6: while n NRF do7: bI [bI , pI,n], pI,n is the nth element of pI ,8: ppI,n 0, ppI,n is the pI,nth element of p,9: if kpI,n = log2(

NNRF )+1 or n=NRF -1 then

10: n n + 1, bL [bL, 1]11: else12: n n + 2, bL [bL, 2]13: end if14: end while15: for t = 1 : Length(bI) do16: m bL,t

17: if m = 2 then18: F [F ,'

(km+1)2im

,'(km+1)2im+1 ]

19: else20: F [F ,'

(km)im

]21: end if22: end for23: return F ; p, k, i24: end function

old measurement.e) Finally, the RF precoder columns are updated (lines

15-22).We dub the above procedure for updating the analog precoders“beam split and drop with backtracking”, owing to the way itoperates. At each iteration, a decision is made as to whethera given beam is split –i.e replaced by two more directivebeams– or dropped –i.e. removed from the precoding matrix.The backtracking feature refers to the fact that, via the vectorp, the measurements obtained in prior ping-pong iterations arekept in memory, allowing for returning to lower level beamsin the codebook in the cases where noise leads to erroneousdecisions in the the split-and-drop procedure.

IV. NUMERICAL RESULTS

In order to assess the effectiveness of the proposed al-gorithm, we perform Monte Carlo simulations for multipleconfigurations of the hybrid arrays at devices A and B. Thechannel matrix H follows the model in (1). The average SNRfor the sth stream link between the nth element of the array atdevice B and the mth element of the array at A is defined as=E|Hnm|2/E|Nns|2=1/2, where Nns is the nth entryof the sth column of the noise matrix N . We assume here forsimplicity that the SNR is the same for all streams. Hnm isthe channel coefficient between device B’s nth array elementand device A’s mth array element, and E is the expectationoperator.

We measure the performance with the av-erage spectral efficiency per bits/s/Hz, ex-

pressed with log2

INS+R1

N

NSHeH

He

, where

He=(F[n1]B W

[n1]B )HHF

[n1]A W

[n]A and

RN =2(F[n1]B W

[n1]B )HF

[n1]B W

[n1]B for integer it-

eration n, and He=(F[n]B W

[n]B )HHF

[n1]A W

[n]A and

RN =2(F[n]B W

[n]B )HF

[n]B W

[n]B for half iteration n+0.5.

We benchmark the performance of the hybrid PPMBTagainst the performance of the digital ping pong multi streambeam training method (digital PPMBT) described in [10], forwhich the spectral efficiency is calculated in a similar mannerto that of the hybrid PPMBT. We compare it also with theoptimal unconstrained SVD based precoder, which maximizethe spectral efficiency and which is obtained by using the topNS left and right singular vectors of H . Note that the SVDbased precoder derivation requires full channel knowledge atboth devices, an information that is hard to get when hybridarchitectures are used as explained earlier, and which ourproposed scheme does not need at the start, but rather learnswhile setting the precoders.

We start our evaluation in the high SNR regime (=30dB),in which the training process will not be impaired too much bynoise. This allows assessing how the arrays size, the numberRF chains and the number of spatial streams really affect thealgorithm’s performance.

Fig. 2 depicts the algorithm’s performance over a channelwith 8 multipath components, when the two devices are

0 2 4 6 8 10 12 14 16 18 20 22


30

40

50

60

70

80

90

100

Sp

ec

tra

l E

ffic

ien

cy


Digital PPMBT

Hybrid PPMBT

NA

=NB

=1024

NA

=NB

=128

NA

=NB

=32

Fig. 2: Spectral Efficiency (bits/s/Hz) attained by the algorithmover PP Iterations. Devices A and B are equipped with ahybrid array. NRF

A =NRFB = 8, NS=4, L=8, =30dB.

equipped with identical arrays made of NA=NB=32, 128 or1024 elements and a fixed number of RF chains NRF

A =NRFB =8

and try to establish an NS=4 streams MIMO communication.We see that the algorithm reaches, in very few iterations, forsmall, medium and large sized arrays, to about 1 to 2 bits/s/Hzof the digital PPMBT and SVD based precoding schemesperformances. We also see that the convergence speed (interms of PP iterations) scales inversely to the codebook depthfor each of the topologies: the convergence is slower whenlarge arrays are used because, in such cases, the codebook hasmore levels and its few last levels contain a high number ofbeamformers with very narrow beams. The selection of suchdirective beams is more prone to error than those in codebookswith lower resolution. Although the backtracking mechanismcan correct for the errors made, this comes at the expense of alarger number of iterations needed for convergence. Note thatthe number of PP iterations needed for convergence for alltopologies is still much below what is needed for exhaustivesearch: this latter requires as an example, for NA=NB=1024with NRF

A =NRFB =8, NA/NRF

A =NB/NRFB =128 PP iterations

to find the best beams, compared with only 16 for ouralgorithm.

Fig. 3 shows the algorithm’s performance over the samechannel as the one used in Fig. 2, but with the number ofMIMO streams, NS , taking different values (4, 6 and 8) andthe two devices being equipped with identical arrays made ofNA=NB=128 elements and use 8 RF chains each. The purposeof these simulations is to investigate how does the ratio ofthe number of MIMO streams to the number of RF chainsNS/NRF affect the algorithm’s performance. We can clearlysee that the algorithm behaves in three different ways depend-ing on the aforementioned ratio: 1) When NS NRF /2, theconvergence is very quick and no backtracking is performed.2) When NRF /2 < NS < NRF , the algorithm’s convergenceis slowed down and one observes some irregularity of theconvergence behavior over iterations which is due to the

0 5 10 15 20 25 30 35 40


0

10

20

30

40

50

60

70

80

90

100

110

Sp

ectr

al E

ffic

ien

cy


Digital PPMBT

Hybrid PPMBT

NS=4

NS=6

NS=8

Fig. 3: Spectral Efficiency (bits/s/Hz) attained by the al-gorithm over PP Iterations for different NS values. De-vices A and B are equipped with a hybrid array.NA=NB=128, NRF

A =NRFB =8, L=8, =30dB.

backtracking mechanism. 3) When NRF = NS , the algorithmfails to provide acceptable performance.

Next, in Fig. 4, we fix the size of antenna arrays andnumber of RF chains used at both devices to NA=NB=64and NRF

A =NRFB =8, and evaluate the algorithm’s performance

over a channel with L = 7 multipath components at differentSNR values. In Fig. 4a, we show the algorithm’s performanceagainst SNR after training convergence for different NS val-ues. The results show that the number of streams that thetraining algorithm can handle efficiently grows with SNR,as is to be expected. At low SNR regimes (15 to 0 dB)the algorithm works better when it attempts to estimate onlythe dominant singular vector of the channel; if more singularvectors are estimated, the large estimation error degrades theoverall spectral efficiency of the system. As the SNR grows,an increasing number of singular vectors can be accuratelyestimated by the algorithm and, hence, increasing the numberof streams provides significant spectral efficiency gains. Forall cases, we observe that the performance of the HybridPPMBT is very close to that of the fully-digital counterpart,and approaches the SVD precoding performance as the SNRgrows. In Fig. 4b the convergence behavior of the trainingalgorithm with different NS settings is evaluated at their re-spective SNR values of interest. We observe that the method’sspectral efficiency tends to saturate at around 10 ping pongiterations for moderate and high SNR. Convergence for lowerSNR values is, however, slower. We attribute this effect tothe fact that the large noise power induces numerous incorrectbeam selection errors in the updates of the analog precoder;although the backtracking feature of the algorithm can helpcorrect some of them, the price to pay is a longer trainingtime. In any case, we remark that the algorithm reaches about70% of the spectral efficiency obtained at convergence withinthe first 6 iterations, regardless of the SNR value and numberof spatial streams.

-20 -15 -10 -5 0 5 10 15

SNR(dB)

0

5

10

15

20

25

30

35

40

45

50

Sp

ectr

al E

ffic

ien

cy


Digital PPMBT

Hybrid PPMBT

NS=2

NS=1

NS=4

(a)

0 5 10 15 20 25


0

5

10

15

20

25

30

35

40

45

50

Sp

etc

ral E

ffic

ien

cy


Digital PPMBT

Hybrid PPMBT

!=0dB, NS=2

!=15dB, NS=4

!=-15dB, NS=1

(b)Fig. 4: Spectral Efficiency (bits/s/Hz) attained by the al-gorithm over PP Iterations for different NS values. De-vices A and B are equipped with a hybrid array.NA=NB=64, NRF

A =NRFB =8, L=7. (a):Spectral Efficiency over

SNR, (b): Spectral Efficiency over PP Iterations

To conclude, we evaluate our proposed training procedure ina system in which only one of the devices is equipped with alarge, hybrid array (NA=256, NRF

A =16), while the other hasa digitally-controlled array of moderate size (NB=4). Suchtopologies can be seen as massive MIMO systems operatingat microwave frequencies. In addition, to reflect the richerscattering experienced in such frequency bands [18], we adoptthe following channel model:

H=p

NANB

LABAH

A (5)where AA contains in its columns the steering vectorsaA(A,p), p=1, 2, . . . , P , AB is defined analogously, L isagain the number of multipath components, and is a LLmatrix with i.i.d. standard complex Gaussian entries. Thissetting will allow to test the validity of our approach inchannels with richer scattering and its robustness against thesparse assumption of the channel.

0 2 4 6 8 10 12 14 16


0

10

20

30

40

50

60

70

80

Sp

ectr

al E

ffic

ien

cy


Digital PPMBT

Hybrid PPMBT

!=-15dB

!=30dB

!=0dB

Fig. 5: Spectral Efficiency (bits/s/Hz) attained by the al-gorithm over PP Iterations. Device A is equipped with ahybrid array and device B has a full digital architecture.NA=128, NB=4, NRF

A =16, NS=4, L=40.

Fig. 5 shows the algorithm’s performance over a rich scat-tering channel (L=40) when NA=128, NRF

A =16, NB=NS=4.It can be seen that the proposed scheme performs remarkablywell and very close to the full digital and optimal SVD basedprecoder solutions at low, mid and high SNRs. These resultsshow that, although the algorithm was originally designed toexploit the sparse nature of mmWave channels, it is robust tochannels with richer scattering.

V. CONCLUSION

We proposed a method to derive precoders and combin-ers for multi-stream MIMO transmission between two de-vices equipped with hybrid digital-analog antenna arrays.The method relies on a low-complexity “multi-beam splitand drop with backtracking” procedure to update the analogprecoders, while digital precoders are computed with the QR-decomposition based method in [10]. For sufficiently largeSNR, the resulting precoders approximate well the uncon-strained SVD-based precoders, as our numerical assessmentshows. We envision that the proposed algorithm can be espe-cially useful in mmWave communication systems.

Compared to the state-of-art methods, our approach offersthe advantage of computational simplicity while achievinghigh-spectral efficiency with moderate training overhead. Thenumerical results show that the method achieves conver-gence within NRF (log2(N/NRF )+1) ping pong iterationsin the low SNR regime and log2(N/NRF )+1 iterations inthe mid and high SNR regime, assuming both transceiversare equipped with arrays made of N elements and NRF

RF chains. Although the method was developed with sparsechannels in mind, the performance assessment shows that it isrobust against this assumption and also performs well in richscattering channels.

Also, in order to further reduce the training overhead, theproposed scheme can be interleaved with transmission of

payload with increasing data-rate. This, the extension to multi-user environments and to time varying channels will be thesubject of our future work.

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[11] E. de Carvalho and J. B. Andersen, “Ping-pong beam training forreciprocal channels with delay spread,” in 49th Asilomar Conference onSignals, Systems and Computers, Pacific Grove, CA, USA, Nov. 2015,pp. 1752–1756.

[12] D. Ogbe, D. J. Love, and V. Raghavan, “Noisy beam alignment tech-niques for reciprocal MIMO channels,” IEEE Transactions on SignalProcessing, vol. 65, no. 19, pp. 5092–5107, Oct. 2017.

[13] C. N. Manchon, E. de Carvalho, and J. B. Andersen, “Ping-pongbeam training with hybrid digital-analog antenna arrays,” in IEEEInternational Conference on Communications (ICC), Paris, France, May2017, pp. 1–7.

[14] H. Ghauch, T. Kim, M. Bengtsson, and M. Skoglund, “Subspace esti-mation and decomposition for large millimeter-wave MIMO systems,”IEEE Journal of Selected Topics in Signal Processing, vol. 10, no. 3,pp. 528–542, Apr. 2016.

[15] X. Cheng, N. Lou, and S. Li, “Spatially sparse beamforming trainingfor millimeter wave MIMO systems,” IEEE Transactions on WirelessCommunications, vol. 16, no. 5, pp. 3385–3400, May 2017.

[16] M. F. Duarte and Y. C. Eldar, “Structured compressed sensing: Fromtheory to applications,” IEEE Transactions on Signal Processing, vol. 59,no. 9, pp. 4053–4085, Sept 2011.

[17] A. G. Burr, “Capacity bounds and estimates for the finite scatterersMIMO wireless channel,” IEEE Journal on Selected Areas in Commu-nications, vol. 21, no. 5, pp. 812–818, Jun. 2003.

[18] J. Andersen, “Propagation aspects of MIMO channel modeling,” inSpace-Time Wireless Systems, H. Bolcskei, C. Papadias, D. Gesbert, andA.-J. van der Veen, Eds. Cambridge University Press, 2006, ch. 1.

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