# Channel Estimation Techniques for Millimeter-Wave Communication Systems: Achievements and Challenges

Kais Hassan, Mohammad Masarra, Marie Zwingelstein, Iyad Dayoub

► To cite this version:

Kais Hassan, Mohammad Masarra, Marie Zwingelstein, Iyad Dayoub. Channel Estimation Techniques for Millimeter-Wave Communication Systems: Achievements and Challenges. IEEE Open Journal of the Communications Society, 2020, 1, pp.1336-1363. 10.1109/OJCOMS.2020.3015394. hal-03455706

HAL Id: hal-03455706

https://hal.science/hal-03455706

Submitted on 7 May 2024

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_1.png" /></figure>
# Channel Estimation Techniques for Millimeter-Wave Communication Systems: Achievements and Challenges

Kais Hassan, Mohammad Masarra, Marie Zwingelstein, Iyad Dayoub

► To cite this version:

Kais Hassan, Mohammad Masarra, Marie Zwingelstein, Iyad Dayoub. Channel Estimation Techniques for Millimeter-Wave Communication Systems: Achievements and Challenges. IEEE Open Journal of the Communications Society, 2020, 1, pp.1336-1363. 10.1109/OJCOMS.2020.3015394. hal-03455706

HAL Id: hal-03455706

https://hal.science/hal-03455706v1

Submitted on 7 May 2024

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
Manuscript received 26 June 2020; revised 29 July 2020; accepted 30 July 2020. Date of publication 12 August 2020; date of current version 25 September 2020.

Digital Object Identifier 10.1109/OJCOMS.2020.3015394

# Channel Estimation Techniques for Millimeter-Wave Communication Systems: Achievements and Challenges

KAIS HASSAN<sup>D 1</sup>, MOHAMMAD MASARRA<sup>2</sup>, MARIE ZWINGELSTEIN<sup>2</sup>, AND IYAD DAYOUB<sup>D 2</sup> (Senior Member, IEEE)

CORRESPONDING AUTHOR: I. DAYOUB (e-mail: iyad.dayoub@uphf.fr)

This work was supported by the French Ministry of Higher Education, Research and Innovation (MESRI) through the funding of a doctoral contract.

**ABSTRACT** The fifth-generation (5G) of cellular networks and beyond requires massive connectivity, high data rates, and low latency. Millimeter-wave (mmWave) communications is a key 5G enabling technology to meet these requirements thanks to its technical potentials that can be integrated with other 5G enablers such as ultra-dense networks (UDNs) and massive multiple-input-multiple-output (massive MIMO) systems. However, some technical challenges, which are mainly related to specific characteristics of mmWave propagation, must be addressed. All the aforementioned points will be discussed in this article before presenting the different existing architectures of massive MIMO mmWave systems. This survey mainly aims at presenting a comprehensive state-of-the-art review of the channel estimation techniques associated with the different mmWave system architectures. Subsequently, we will provide a comparison among existing solutions in terms of their respective benefits and shortcomings. Finally, some open directions of research are discussed, and challenges that wait to be met are pointed out.

**INDEX TERMS** Millimeter-wave communications, massive MIMO, channel estimation, 5G, cellular systems, spatial channel model, lens antenna array, hybrid architecture, few-bit ADCs, compressive sensing, beamforming.

## NOMENCLATURE

<table><tr><td>2D</td><td>Two Dimensional</td><td>BS</td><td>Base Station</td></tr><tr><td>3D</td><td>Three Dimensional</td><td>CMOS</td><td>Complementary Metal Oxide Semiconductor</td></tr><tr><td>4G</td><td>Fourth Generation</td><td>C-OMP</td><td>Covariance Orthogonal Matching Pursuit</td></tr><tr><td>5G</td><td>Fifth Generation</td><td>CoSaMP</td><td>Compressive Sampling Match Pursuit</td></tr><tr><td>AAVE</td><td>Antenna Array with Virtual Elements</td><td>CP</td><td>CANDECOMP/PARAFAC</td></tr><tr><td>ADC</td><td>Analog-to-Digital Converter</td><td>CS</td><td>Compressive Sensing</td></tr><tr><td>AMP</td><td>Approximate Message Passing</td><td>CSI</td><td>Channel State Information</td></tr><tr><td>AoA</td><td>Angle-of-Arrival</td><td>DC</td><td>Dual Crossing</td></tr><tr><td>AoD</td><td>Angle-of-Departure</td><td>DC-OMP</td><td>Dynamic Covariance Orthogonal Matching Pursuit</td></tr><tr><td>AWGN</td><td>Additive White Gaussian Noise</td><td>DCS</td><td>Distributed Compressed Sensing</td></tr><tr><td>BiG-AMP</td><td>Bilinear Generalized Approximate Message Passing</td><td>DFT</td><td>Discrete Fourier Transform</td></tr><tr><td>BIHT</td><td>Binary Iterative Hard Thresholding</td><td></td><td></td></tr></table>

<footnote n="1">¹Laboratoire d'Acoustique de l'Université du Mans-UMR CNRS 6613, Le Mans University, 72085 Le Mans, France</footnote>

<footnote n="2">²University Polytechnic Hauts-de-France, CNRS, L'Université de Lille, ISEN, Centrale Lille, UMR 8520-IEMN-Institut D'Electronique de Microélectronique Et de Nanotechnologie-DOAE-Département Opto-Acousto-Electronique, 59313 Valenciennes, France</footnote>
Manuscript received 26 June 2020; revised 29 July 2020; accepted 30 July 2020. Date of publication 12 August 2020; date of current version 25 September 2020.

Digital Object Identifier 10.1109/OJCOMS.2020.3015394

# Channel Estimation Techniques for Millimeter-Wave Communication Systems: Achievements and Challenges

KAIS HASSAN<sup>D 1</sup>, MOHAMMAD MASARRA<sup>2</sup>, MARIE ZWINGELSTEIN<sup>2</sup>, AND IYAD DAYOUB<sup>D 2</sup> (Senior Member, IEEE)

¹Laboratoire d'Acoustique de l'Université du Mans-UMR CNRS 6613, Le Mans University, 72085 Le Mans, France

²University Polytechnic Hauts-de-France, CNRS, L'Université de Lille, ISEN, Centrale Lille, UMR 8520-IEMN-Institut D'Electronique de Microélectronique Et de Nanotechnologie-DOAE-Département Opto-Acousto-Electronique, 59313 Valenciennes, France

CORRESPONDING AUTHOR: I. DAYOUB (e-mail: iyad.dayoub@uphf.fr)

This work was supported by the French Ministry of Higher Education, Research and Innovation (MESRI) through the funding of a doctoral contract.

**ABSTRACT** The fifth-generation (5G) of cellular networks and beyond requires massive connectivity, high data rates, and low latency. Millimeter-wave (mmWave) communications is a key 5G enabling technology to meet these requirements thanks to its technical potentials that can be integrated with other 5G enablers such as ultra-dense networks (UDNs) and massive multiple-input-multiple-output (massive MIMO) systems. However, some technical challenges, which are mainly related to specific characteristics of mmWave propagation, must be addressed. All the aforementioned points will be discussed in this article before presenting the different existing architectures of massive MIMO mmWave systems. This survey mainly aims at presenting a comprehensive state-of-the-art review of the channel estimation techniques associated with the different mmWave system architectures. Subsequently, we will provide a comparison among existing solutions in terms of their respective benefits and shortcomings. Finally, some open directions of research are discussed, and challenges that wait to be met are pointed out.

**INDEX TERMS** Millimeter-wave communications, massive MIMO, channel estimation, 5G, cellular systems, spatial channel model, lens antenna array, hybrid architecture, few-bit ADCs, compressive sensing, beamforming.

## NOMENCLATURE

<table><tr><td>2D</td><td>Two Dimensional</td><td>BS</td><td>Base Station</td></tr><tr><td>3D</td><td>Three Dimensional</td><td>CMOS</td><td>Complementary Metal Oxide Semiconductor</td></tr><tr><td>4G</td><td>Fourth Generation</td><td>C-OMP</td><td>Covariance Orthogonal Matching Pursuit</td></tr><tr><td>5G</td><td>Fifth Generation</td><td>CoSaMP</td><td>Compressive Sampling Match Pursuit</td></tr><tr><td>AAVE</td><td>Antenna Array with Virtual Elements</td><td>CP</td><td>CANDECOMP/PARAFAC</td></tr><tr><td>ADC</td><td>Analog-to-Digital Converter</td><td>CS</td><td>Compressive Sensing</td></tr><tr><td>AMP</td><td>Approximate Message Passing</td><td>CSI</td><td>Channel State Information</td></tr><tr><td>AoA</td><td>Angle-of-Arrival</td><td>DC</td><td>Dual Crossing</td></tr><tr><td>AoD</td><td>Angle-of-Departure</td><td>DC-OMP</td><td>Dynamic Covariance Orthogonal Matching Pursuit</td></tr><tr><td>AWGN</td><td>Additive White Gaussian Noise</td><td>DCS</td><td>Distributed Compressed Sensing</td></tr><tr><td>BiG-AMP</td><td>Bilinear Generalized Approximate Message Passing</td><td>DFT</td><td>Discrete Fourier Transform</td></tr><tr><td>BIHT</td><td>Binary Iterative Hard Thresholding</td><td></td><td></td></tr></table>

This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/
<table><tr><td>DGMP</td><td>Distributed Grid Matching Pursuit</td></tr><tr><td>DL</td><td>DownLink</td></tr><tr><td>DS-OMP</td><td>Dynamic Simultaneous Orthogonal Matching Pursuit</td></tr><tr><td>EM</td><td>Expectation-Maximization</td></tr><tr><td>FBMC</td><td>Filter Bank based Multi-Carrier</td></tr><tr><td>FDD</td><td>Frequency-Division Duplexing</td></tr><tr><td>FFT</td><td>Fast Fourier Transform</td></tr><tr><td>GAMP</td><td>Generalized Approximate Message Passing</td></tr><tr><td>GOMP</td><td>Generalized Orthogonal Matching Pursuit</td></tr><tr><td>IHT</td><td>Iterative Hard Thresholding</td></tr><tr><td>JCE+LS</td><td>Joint Channel Estimation + Local Search</td></tr><tr><td>LASSO</td><td>Least Absolute Shrinkage and Selection Operator</td></tr><tr><td>LMMSE</td><td>Linear Minimum Mean Squared Estimator</td></tr><tr><td>LOS</td><td>Line-of-Sight</td></tr><tr><td>LSE</td><td>Least-Squares Estimation</td></tr><tr><td>MG-OMP</td><td>Multi-Grid Orthogonal Matching Pursuit</td></tr><tr><td>MIMO</td><td>Multiple-Input Multiple-Output</td></tr><tr><td>MMP</td><td>Multipath Matching Pursuit</td></tr><tr><td>MMSE</td><td>Minimum Mean Square Error</td></tr><tr><td>mmWave</td><td>Millimeter-Wave</td></tr><tr><td>M-OMP</td><td>Multiple Measurement Vector Orthogonal Matching Pursuit</td></tr><tr><td>MRC</td><td>Maximum Ratio Combining</td></tr><tr><td>MT</td><td>Mobile Terminal</td></tr><tr><td>MU</td><td>Multi-User</td></tr><tr><td>MUSIC</td><td>MULTiple SIgnal Classification</td></tr><tr><td>NLOS</td><td>Non-Line-of-Sight</td></tr><tr><td>OFDM</td><td>Orthogonal Frequency-Division Multiplexing</td></tr><tr><td>OMP</td><td>Orthogonal Matching Pursuit</td></tr><tr><td>RF</td><td>Radio-Frequency</td></tr><tr><td>RX</td><td>Receiver</td></tr><tr><td>SC-FDE</td><td>Single Carrier-Frequency Domain Equalization</td></tr><tr><td>SdMP</td><td>Stage-Determined Matching Pursuit</td></tr><tr><td>SINR</td><td>Signal-to-Interference-plus-Noise Ratio</td></tr><tr><td>SISO</td><td>Single-Input Single-Output</td></tr><tr><td>SNR</td><td>Signal-to-Noise Ratio</td></tr><tr><td>S-OMP</td><td>Simultaneous Orthogonal Matching Pursuit</td></tr><tr><td>SSAMP</td><td>Structured Sparsity Adaptive Matching Pursuit</td></tr><tr><td>SS-SW-OMP+Th</td><td>Subcarrier Selection SW-OMP + Thresholding</td></tr><tr><td>SU</td><td>Single-User</td></tr><tr><td>SURE</td><td>Stein's Unbiased Risk Estimate</td></tr><tr><td>SVD</td><td>Singular Value Decomposition</td></tr><tr><td>SW-OMP</td><td>Simultaneous Weighted Orthogonal Matching Pursuit</td></tr></table>

<table><tr><td>TDD</td><td>Time-Division Duplexing</td></tr><tr><td>TX</td><td>Transmitter</td></tr><tr><td>UL</td><td>UpLink</td></tr><tr><td>ULA</td><td>Uniform Linear Array</td></tr><tr><td>ULPA</td><td>Uniform Linear Planar Array</td></tr><tr><td>VAMP</td><td>Vector Approximate Message Passing</td></tr><tr><td>ZF</td><td>Zero Forcing.</td></tr></table>

I. INTRODUCTION AND BACKGROUND

THE NEXT decade will encounter many emerging applications such as connected cars, augmented reality, virtual reality, mixed reality, 3D video, ultra-high definition video, industrial IoT, smart cities, connected healthcare, etc. These applications require to massively connect new devices and to exchange more data. For instance, the global mobile traffic is expected to grow annually by 30 percent between 2018 and 2024, and the capacity demand in the next decade is expected to witness a 1000-fold increase [1]. Among the fifth generation (5G) promises, when compared to fourth generation (4G), is to increase user data rate by 10 to 100 (up to 10 Gbit/s), to reduce latency by 10, to increase connectivity density by 10, and also to reduce the cost and power consumption [2]. Due to this hugely increasing demand for data traffic and massive connectivity as well as to the scarcity in the sub-6 GHz radio spectrum, researchers are trying to propose new solutions. These are mainly based either on the new signal processing techniques, or on densifying the network, or on exploiting additional frequency bands.

In regards of exploiting new bands, the millimeter-wave (mmWave) spectrum is between 30 GHz and 300 GHz (i.e., wavelengths between 1 mm to 10 mm) where the large unused bandwidth in these bands can allow the wireless systems to support the enormously increase in capacity demand since capacity of wireless systems increases when the exploited bandwidth increases. Hence, mmWave communications will play an essential role in 5G and upcoming future generations of cellular networks [3]–[6].

Despite their bandwidth attractivity, mmWave communications systems suffer from high free-space path-loss when compared to the sub-6 GHz ones. In addition, a substantial attenuation is observed in some mmWave bands due to atmospheric absorption, and rain and snow effects. Therefore, the mmWave signal risks to propagate over a few meters in some scenarios, and hence mmWave communications may be only suitable for very close-range communications such as in indoor applications. This difficulty can be overcome for outdoor mobile communications, where a greater transmission range is expected, by either increasing the transmission power or by using high-gain, high-directional antennas. Since the transmission power is always limited by regulations, mmWave systems must enable narrow steering beams such as the transmitter and the receiver steer towards each other which results in high directional gain in the wanted directions and low gain in the unwanted ones. The desirable high directivity is based on signal processing techniques such as
<table><tr><td>DGMP</td><td>Distributed Grid Matching Pursuit</td></tr><tr><td>DL</td><td>DownLink</td></tr><tr><td>DS-OMP</td><td>Dynamic Simultaneous Orthogonal Matching Pursuit</td></tr><tr><td>EM</td><td>Expectation-Maximization</td></tr><tr><td>FBMC</td><td>Filter Bank based Multi-Carrier</td></tr><tr><td>FDD</td><td>Frequency-Division Duplexing</td></tr><tr><td>FFT</td><td>Fast Fourier Transform</td></tr><tr><td>GAMP</td><td>Generalized Approximate Message Passing</td></tr><tr><td>GOMP</td><td>Generalized Orthogonal Matching Pursuit</td></tr><tr><td>IHT</td><td>Iterative Hard Thresholding</td></tr><tr><td>JCE+LS</td><td>Joint Channel Estimation + Local Search</td></tr><tr><td>LASSO</td><td>Least Absolute Shrinkage and Selection Operator</td></tr><tr><td>LMMSE</td><td>Linear Minimum Mean Squared Estimator</td></tr><tr><td>LOS</td><td>Line-of-Sight</td></tr><tr><td>LSE</td><td>Least-Squares Estimation</td></tr><tr><td>MG-OMP</td><td>Multi-Grid Orthogonal Matching Pursuit</td></tr><tr><td>MIMO</td><td>Multiple-Input Multiple-Output</td></tr><tr><td>MMP</td><td>Multipath Matching Pursuit</td></tr><tr><td>MMSE</td><td>Minimum Mean Square Error</td></tr><tr><td>mmWave</td><td>Millimeter-Wave</td></tr><tr><td>M-OMP</td><td>Multiple Measurement Vector Orthogonal Matching Pursuit</td></tr><tr><td>MRC</td><td>Maximum Ratio Combining</td></tr><tr><td>MT</td><td>Mobile Terminal</td></tr><tr><td>MU</td><td>Multi-User</td></tr><tr><td>MUSIC</td><td>MULTiple SIgnal Classification</td></tr><tr><td>NLOS</td><td>Non-Line-of-Sight</td></tr><tr><td>OFDM</td><td>Orthogonal Frequency-Division Multiplexing</td></tr><tr><td>OMP</td><td>Orthogonal Matching Pursuit</td></tr><tr><td>RF</td><td>Radio-Frequency</td></tr><tr><td>RX</td><td>Receiver</td></tr><tr><td>SC-FDE</td><td>Single Carrier-Frequency Domain Equalization</td></tr><tr><td>SdMP</td><td>Stage-Determined Matching Pursuit</td></tr><tr><td>SINR</td><td>Signal-to-Interference-plus-Noise Ratio</td></tr><tr><td>SISO</td><td>Single-Input Single-Output</td></tr><tr><td>SNR</td><td>Signal-to-Noise Ratio</td></tr><tr><td>S-OMP</td><td>Simultaneous Orthogonal Matching Pursuit</td></tr><tr><td>SSAMP</td><td>Structured Sparsity Adaptive Matching Pursuit</td></tr><tr><td>SS-SW-OMP+Th</td><td>Subcarrier Selection SW-OMP + Thresholding</td></tr><tr><td>SU</td><td>Single-User</td></tr><tr><td>SURE</td><td>Stein's Unbiased Risk Estimate</td></tr><tr><td>SVD</td><td>Singular Value Decomposition</td></tr><tr><td>SW-OMP</td><td>Simultaneous Weighted Orthogonal Matching Pursuit</td></tr></table>

<table><tr><td>TDD</td><td>Time-Division Duplexing</td></tr><tr><td>TX</td><td>Transmitter</td></tr><tr><td>UL</td><td>UpLink</td></tr><tr><td>ULA</td><td>Uniform Linear Array</td></tr><tr><td>ULPA</td><td>Uniform Linear Planar Array</td></tr><tr><td>VAMP</td><td>Vector Approximate Message Passing</td></tr><tr><td>ZF</td><td>Zero Forcing.</td></tr></table>

I. INTRODUCTION AND BACKGROUND

THE NEXT decade will encounter many emerging applications such as connected cars, augmented reality, virtual reality, mixed reality, 3D video, ultra-high definition video, industrial IoT, smart cities, connected healthcare, etc. These applications require to massively connect new devices and to exchange more data. For instance, the global mobile traffic is expected to grow annually by 30 percent between 2018 and 2024, and the capacity demand in the next decade is expected to witness a 1000-fold increase [1]. Among the fifth generation (5G) promises, when compared to fourth generation (4G), is to increase user data rate by 10 to 100 (up to 10 Gbit/s), to reduce latency by 10, to increase connectivity density by 10, and also to reduce the cost and power consumption [2]. Due to this hugely increasing demand for data traffic and massive connectivity as well as to the scarcity in the sub-6 GHz radio spectrum, researchers are trying to propose new solutions. These are mainly based either on the new signal processing techniques, or on densifying the network, or on exploiting additional frequency bands.

In regards of exploiting new bands, the millimeter-wave (mmWave) spectrum is between 30 GHz and 300 GHz (i.e., wavelengths between 1 mm to 10 mm) where the large unused bandwidth in these bands can allow the wireless systems to support the enormously increase in capacity demand since capacity of wireless systems increases when the exploited bandwidth increases. Hence, mmWave communications will play an essential role in 5G and upcoming future generations of cellular networks [3]–[6].

Despite their bandwidth attractivity, mmWave communications systems suffer from high free-space path-loss when compared to the sub-6 GHz ones. In addition, a substantial attenuation is observed in some mmWave bands due to atmospheric absorption, and rain and snow effects. Therefore, the mmWave signal risks to propagate over a few meters in some scenarios, and hence mmWave communications may be only suitable for very close-range communications such as in indoor applications. This difficulty can be overcome for outdoor mobile communications, where a greater transmission range is expected, by either increasing the transmission power or by using high-gain, high-directional antennas. Since the transmission power is always limited by regulations, mmWave systems must enable narrow steering beams such as the transmitter and the receiver steer towards each other which results in high directional gain in the wanted directions and low gain in the unwanted ones. The desirable high directivity is based on signal processing techniques such as
beamforming which requires to increase the number of antennas of the antenna arrays at the transmitter and/or receiver, i.e., massive antenna arrays are needed. This approach is facilitated by the short wavelength of mmWave signals which makes possible to compact more antennas while keeping the array size small. Another aspect of mmWave system design comes from the impossibility to directly apply the traditional digital transceiver architectures, which are employed in sub-6 GHz, directly to mmWave systems because of the high power consumption of mmWave radio-frequency (RF) chains. Recently, new tailored multiple-input-multiple-output (MIMO) architectures were proposed to solve this problem namely fully-analog, hybrid and few-bit ADCs architectures. The objective is to reduce the total consumed power by either reducing the number of RF chains or the power consumption per each one.

Hence, mmWave communications are a key technology enabler for the soon expected 5G and beyond, not only thanks to the huge available bandwidth, but also by means of its complementary with other 5G enablers such as ultra-dense networks (UDNs) and massive MIMO. MmWave links are suitable for wireless backhauling and short-range communications in the small cells of UDNs while still capable of mitigating interferences by using the high-directional antenna arrays. This high directivity is enabled by the beamforming ability of massive MIMO. However, the evolution of mmWave communications is hindered by challenges that require more insight to propose the needed solutions to reach their full potential. Some of these challenges come from the propagation characteristics in the mmWave bands (the path-loss, penetration loss, and the atmospheric attenuation), others come from the high power consumption and the hardware impairments of mmWave circuits.

## THE CONTRIBUTIONS OF THE SURVEY

An important part of any telecommunications system is channel estimation which is essential to optimize link performance. Acquiring channel state information (CSI) is particularly challenging for mmWave systems because of the massive number of antennas, the complex architecture of the transceiver and the large exploited bandwidth.

The aim of this survey is to propose a comprehensive overview of the existing channel estimation techniques for mmWave systems. We attempt to present each estimation method in a clear and concise manner while providing sufficient level of details to offer a fair comparison among them in terms of performance, studied scenario, and MIMO architecture design. The studied scenario concerns the different assumptions that were considered in existing research works, such as i) is it a single-user or a multi-user system? ii) is the channel a frequency-selective channel? iii) which duplexing mode is used? iv) is it a 2D or 3D channel estimation? v) which channel model is adopted? etc. One important thing to note is that the channel estimation techniques are generally tailored to the MIMO architecture of the transmitter and receiver, as well as to the employed antenna array. As

<table><caption>TABLE 1. Review of the main topics studied in the existing surveys on mmWave communications systems.</caption><tr><td>Access and Coverage</td><td>[5]</td></tr><tr><td>Channel Estimation</td><td>[5], [8], [12]</td></tr><tr><td>Channel Measurement</td><td>[5], [6], [12]</td></tr><tr><td>Channel Modelling</td><td>[5]–[7], [12]</td></tr><tr><td>Cross-Layer Design</td><td>[10], [12]</td></tr><tr><td>Key Challenges and Technical Potentials</td><td>[3], [5]</td></tr><tr><td>MIMO Architecture</td><td>[4], [5], [9]</td></tr><tr><td>Performance Analysis</td><td>[4]</td></tr><tr><td>Propagation Characteristics</td><td>[3], [4], [6], [7]</td></tr><tr><td>Standardization</td><td>[10], [12]</td></tr></table>

a result, some methods are powerful when used with planar array antenna and hybrid architecture but their performance will degrade when it comes to lens antenna array systems or to few-bit analog-to-digital converters (ADCs) ones.

To the best of our knowledge, there has been no survey that discussed the channel estimation methods for mmWave systems, although there are few surveys available such as [3]–[15] which typically overview mmWave systems without specifically dealing with the channel estimation aspect. Table 1 shows a review of some topics related to mmWave systems and the surveys in which each topic was largely reviewed. For instance, a chapter in [8] was dedicated to present six channel estimation papers which were published between 2014 and 2016. Some channel estimation papers with hybrid architecture were reviewed in [5], [12]. However, a further inclusive review of this aspect is needed. In particular, channel estimation techniques with few-bit ADCs architecture, and with lens antenna were not considered in the above mentioned surveys.

## SCOPE AND ORGANIZATION

This inclusive survey focuses on the channel estimation methods for mmWave communication systems and provides a summary of the latest research progress in this domain. The outline of this article can be summarised as:

* First, in Section II the characteristics of mmWave channels such as the differences between the propagation behavior in the mmWave bands and in the sub-6 GHz ones are illustrated. This will help to understand the technical potential of mmWave communications, the challenges which have to be addressed to reach this potential, and the enabler-technologies to reply to these challenges.

* Then in Section III, we discuss the different existing system architectures which are proposed for massive MIMO mmWave systems. The spatial MIMO channel models which present the special nature of highly directive mmWave communications are also introduced before presenting the massive MIMO system model.

* Afterwards, compressive sensing (CS) as a mathematical tool for channel estimation is presented in Section IV.

* In the following sections, the channel estimation techniques for mmWave systems are comprehensively
beamforming which requires to increase the number of antennas of the antenna arrays at the transmitter and/or receiver, i.e., massive antenna arrays are needed. This approach is facilitated by the short wavelength of mmWave signals which makes possible to compact more antennas while keeping the array size small. Another aspect of mmWave system design comes from the impossibility to directly apply the traditional digital transceiver architectures, which are employed in sub-6 GHz, directly to mmWave systems because of the high power consumption of mmWave radio-frequency (RF) chains. Recently, new tailored multiple-input-multiple-output (MIMO) architectures were proposed to solve this problem namely fully-analog, hybrid and few-bit ADCs architectures. The objective is to reduce the total consumed power by either reducing the number of RF chains or the power consumption per each one.

Hence, mmWave communications are a key technology enabler for the soon expected 5G and beyond, not only thanks to the huge available bandwidth, but also by means of its complementary with other 5G enablers such as ultra-dense networks (UDNs) and massive MIMO. MmWave links are suitable for wireless backhauling and short-range communications in the small cells of UDNs while still capable of mitigating interferences by using the high-directional antenna arrays. This high directivity is enabled by the beamforming ability of massive MIMO. However, the evolution of mmWave communications is hindered by challenges that require more insight to propose the needed solutions to reach their full potential. Some of these challenges come from the propagation characteristics in the mmWave bands (the path-loss, penetration loss, and the atmospheric attenuation), others come from the high power consumption and the hardware impairments of mmWave circuits.

## THE CONTRIBUTIONS OF THE SURVEY

An important part of any telecommunications system is channel estimation which is essential to optimize link performance. Acquiring channel state information (CSI) is particularly challenging for mmWave systems because of the massive number of antennas, the complex architecture of the transceiver and the large exploited bandwidth.

The aim of this survey is to propose a comprehensive overview of the existing channel estimation techniques for mmWave systems. We attempt to present each estimation method in a clear and concise manner while providing sufficient level of details to offer a fair comparison among them in terms of performance, studied scenario, and MIMO architecture design. The studied scenario concerns the different assumptions that were considered in existing research works, such as i) is it a single-user or a multi-user system? ii) is the channel a frequency-selective channel? iii) which duplexing mode is used? iv) is it a 2D or 3D channel estimation? v) which channel model is adopted? etc. One important thing to note is that the channel estimation techniques are generally tailored to the MIMO architecture of the transmitter and receiver, as well as to the employed antenna array. As

<table><caption>TABLE 1. Review of the main topics studied in the existing surveys on mmWave communications systems.</caption><tr><td>Access and Coverage</td><td>[5]</td></tr><tr><td>Channel Estimation</td><td>[5], [8], [12]</td></tr><tr><td>Channel Measurement</td><td>[5], [6], [12]</td></tr><tr><td>Channel Modelling</td><td>[5]–[7], [12]</td></tr><tr><td>Cross-Layer Design</td><td>[10], [12]</td></tr><tr><td>Key Challenges and Technical Potentials</td><td>[3], [5]</td></tr><tr><td>MIMO Architecture</td><td>[4], [5], [9]</td></tr><tr><td>Performance Analysis</td><td>[4]</td></tr><tr><td>Propagation Characteristics</td><td>[3], [4], [6], [7]</td></tr><tr><td>Standardization</td><td>[10], [12]</td></tr></table>

a result, some methods are powerful when used with planar array antenna and hybrid architecture but their performance will degrade when it comes to lens antenna array systems or to few-bit analog-to-digital converters (ADCs) ones.

To the best of our knowledge, there has been no survey that discussed the channel estimation methods for mmWave systems, although there are few surveys available such as [3]–[15] which typically overview mmWave systems without specifically dealing with the channel estimation aspect. Table 1 shows a review of some topics related to mmWave systems and the surveys in which each topic was largely reviewed. For instance, a chapter in [8] was dedicated to present six channel estimation papers which were published between 2014 and 2016. Some channel estimation papers with hybrid architecture were reviewed in [5], [12]. However, a further inclusive review of this aspect is needed. In particular, channel estimation techniques with few-bit ADCs architecture, and with lens antenna were not considered in the above mentioned surveys.

## SCOPE AND ORGANIZATION

This inclusive survey focuses on the channel estimation methods for mmWave communication systems and provides a summary of the latest research progress in this domain. The outline of this article can be summarised as:

* First, in Section II the characteristics of mmWave channels such as the differences between the propagation behavior in the mmWave bands and in the sub-6 GHz ones are illustrated. This will help to understand the technical potential of mmWave communications, the challenges which have to be addressed to reach this potential, and the enabler-technologies to reply to these challenges.

* Then in Section III, we discuss the different existing system architectures which are proposed for massive MIMO mmWave systems. The spatial MIMO channel models which present the special nature of highly directive mmWave communications are also introduced before presenting the massive MIMO system model.

* Afterwards, compressive sensing (CS) as a mathematical tool for channel estimation is presented in Section IV.

* In the following sections, the channel estimation techniques for mmWave systems are comprehensively
<figure><img src="image_2.png" /><figcaption>FIGURE 1. The outline of this survey.</figcaption></figure>

reviewed. It is organised as follows: Existing chan-
nel estimation techniques for hybrid architecture, for
lens antenna array architecture, and for few-bit ADCs
architectures are respectively introduced in Section V,
Section VI and Section VII. Conclusions are offered at
the end of each section.

* The learned lessons are presented in Section VIII before identifying some open research directions. Finally, Section IX completes the paper with concluding remarks.

The skeleton structure of our survey is illustrated in
Figure 1.

NOTATION

Lightface letters denote scalars. Boldface lower- and upper-
case letters denote column vectors and matrices, respectively.
[A]$_{n,m}$ and xᵢ stand for the entries of matrix A and vector x,
respectively. [A]$_{n,·}$ is the n-th row of matrix A. The M × M
identity matrix is I_M. 1_M is an M × 1 unity vector. The
Kronecker product of matrix A and matrix B is expressed
as A ⊗ B. A^H is the Hermitian (conjugate transpose) of matrix
A. ||x||_0 is the number of non-zero entries of the vector x. E[·]
is the statistical expectation. Letter j is the imaginary unit,
i.e., j² = -1. [x] denotes the ceiling function. An ellipsis
“...” refers to the continuation of an equation. Finally, a ∝ b
means a is proportional to b.

II. CHARACTERISTICS OF MMWAVE COMMUNICATIONS

An increased attention has recently been drawn to the huge
spectrum in the mmWave frequency bands to meet the
increase in the global mobile data traffic and to achieve
up of hundreds of times more capacity compared to current
4G cellular networks [16]–[18].

MmWave communications have been employed for appli-
cations such as radar and point-to-point communication for

100 years ago [19]. In the last decade, some mmWave stan-
dards have emerged including IEEE 802.15.3c [20], IEEE
802.11ad [21] and WirelessHD [22] which are intended
to wireless personal area networks, wireless local area
networks and wireless HDMI, respectively. Recently, the
mobile network research community dedicated a lot of atten-
tion to the sub-100 GHz systems operating in the 28 GHz,
38 GHz, 60 GHz, 71 GHz and 81 GHz bands, while the
band above 100 GHz has been studied by only a few very
recent papers [23]. Some proof of concept studies had been
conducted in the last years. In May 2013, Samsung real-
ized a 1.056 Gb/s transmission in the 28 GHz band to a
distance of up to 2 km [24]. In April 2015, a peak rate
of 15 Gbps at 73 GHz was achieved by Nokia in collab-
oration with National Instruments [5]. In February 2018,
Deutsche Telekom and Huawei have successfully completed
the world's first multi-cell field tests at 73 GHz [25].

However, the propagation characteristics at mmWave
frequencies are different from that at the traditional sub-
6 GHz ones, that is why we will show in the following
subsections the main technical advantages and challenges of
the mmWave technology.

A. PROPAGATION CHARACTERISTICS

A major difference between sub-6 GHz and mmWave
systems in terms of propagation characteristics is in their
free-space path-loss (PL<sub>FS</sub>) [26] which is described by,

$$
\mathrm{PL}_{\mathrm{FS}} \propto \frac{d^n}{\lambda^2}, \qquad (1)
$$

where *d* is the distance between the transmitter and receiver, *n* is the path-loss exponent which typically equals 2, and *λ* is the wavelength of the signal. However, *n* is less than 2 in some scenarios of cellular networks and indoor applications [27]. On the other hand, path-loss exponent can reach the value of 6 in some severe environments of propagation [28]. A comparison between the microwave propagation in the 1.8 GHz GSM band and the mmWave one in the 73 GHz bands, under the same configuration (transmission distance, propagation environment, antenna array), shows an additional loss of 32 dB in the mmWave band [29]. Recent urban model experiments also show a degradation of 40 dB in path-loss at 28 GHz compared to 2.8 GHz [28], [30]. Thus, the severe path-loss of mmWave propagation required to be compensated by a strong directionality.

The atmospheric attenuation of mmWave signals, which
is caused particularly by the absorption of oxygen and water
vapor, as well as the scattering of rain, must be added to
the path-loss. The atmospheric attenuation depends on the
operating frequency. The atmospheric oxygen absorption is
especially severe at 60 GHz and 120 GHz, and the water
vapor absorption is particularly very high at 180 GHz [31].
Also, rain attenuation at mmWave frequencies is much
greater than that of sub-6 GHz frequencies [32].

Another factor which affects the propagation of mmWave
signals is their severe penetration loss which results in sus-
ceptibility to the static and dynamic blockage effect. In fact,
<figure><img src="image_2.png" /><figcaption>FIGURE 1. The outline of this survey.</figcaption></figure>

reviewed. It is organised as follows: Existing chan-
nel estimation techniques for hybrid architecture, for
lens antenna array architecture, and for few-bit ADCs
architectures are respectively introduced in Section V,
Section VI and Section VII. Conclusions are offered at
the end of each section.

* The learned lessons are presented in Section VIII before identifying some open research directions. Finally, Section IX completes the paper with concluding remarks.

The skeleton structure of our survey is illustrated in
Figure 1.

NOTATION

Lightface letters denote scalars. Boldface lower- and upper-
case letters denote column vectors and matrices, respectively.
[A]$_{n,m}$ and xᵢ stand for the entries of matrix A and vector x,
respectively. [A]$_{n,·}$ is the nᵗʰ row of matrix A. The M × M
identity matrix is I_M. 1_M is an M × 1 unity vector. The
Kronecker product of matrix A and matrix B is expressed
as A⊗B. A^H is the Hermitian (conjugate transpose) of matrix
A. ||x||₀ is the number of non-zero entries of the vector x. E[·]
is the statistical expectation. Letter j is the imaginary unit,
i.e., j² = -1. [x] denotes the ceiling function. An ellipsis
“...” refers to the continuation of an equation. Finally, a ∝ b
means a is proportional to b.

II. CHARACTERISTICS OF MMWAVE COMMUNICATIONS

An increased attention has recently been drawn to the huge
spectrum in the mmWave frequency bands to meet the
increase in the global mobile data traffic and to achieve
up of hundreds of times more capacity compared to current
4G cellular networks [16]–[18].

MmWave communications have been employed for appli-
cations such as radar and point-to-point communication for

100 years ago [19]. In the last decade, some mmWave stan-
dards have emerged including IEEE 802.15.3c [20], IEEE
802.11ad [21] and WirelessHD [22] which are intended
to wireless personal area networks, wireless local area
networks and wireless HDMI, respectively. Recently, the
mobile network research community dedicated a lot of atten-
tion to the sub-100 GHz systems operating in the 28 GHz,
38 GHz, 60 GHz, 71 GHz and 81 GHz bands, while the
band above 100 GHz has been studied by only a few very
recent papers [23]. Some proof of concept studies had been
conducted in the last years. In May 2013, Samsung real-
ized a 1.056 Gb/s transmission in the 28 GHz band to a
distance of up to 2 km [24]. In April 2015, a peak rate
of 15 Gbps at 73 GHz was achieved by Nokia in collab-
oration with National Instruments [5]. In February 2018,
Deutsche Telekom and Huawei have successfully completed
the world's first multi-cell field tests at 73 GHz [25].

However, the propagation characteristics at mmWave
frequencies are different from that at the traditional sub-
6 GHz ones, that is why we will show in the following
subsections the main technical advantages and challenges of
the mmWave technology.

A. PROPAGATION CHARACTERISTICS

A major difference between sub-6 GHz and mmWave
systems in terms of propagation characteristics is in their
free-space path-loss (PL<sub>FS</sub>) [26] which is described by,

$$
\mathrm{PL}_{\mathrm{FS}} \propto \frac{d^n}{\lambda^2}, \qquad (1)
$$

where *d* is the distance between the transmitter and receiver, *n* is the path-loss exponent which typically equals 2, and *λ* is the wavelength of the signal. However, *n* is less than 2 in some scenarios of cellular networks and indoor applications [27]. On the other hand, path-loss exponent can reach the value of 6 in some severe environments of propagation [28]. A comparison between the microwave propagation in the 1.8 GHz GSM band and the mmWave one in the 73 GHz bands, under the same configuration (transmission distance, propagation environment, antenna array), shows an additional loss of 32 dB in the mmWave band [29]. Recent urban model experiments also show a degradation of 40 dB in path-loss at 28 GHz compared to 2.8 GHz [28], [30]. Thus, the severe path-loss of mmWave propagation required to be compensated by a strong directionality.

The atmospheric attenuation of mmWave signals, which
is caused particularly by the absorption of oxygen and water
vapor, as well as the scattering of rain, must be added to
the path-loss. The atmospheric attenuation depends on the
operating frequency. The atmospheric oxygen absorption is
especially severe at 60 GHz and 120 GHz, and the water
vapor absorption is particularly very high at 180 GHz [31].
Also, rain attenuation at mmWave frequencies is much
greater than that of sub-6 GHz frequencies [32].

Another factor which affects the propagation of mmWave
signals is their severe penetration loss which results in sus-
ceptibility to the static and dynamic blockage effect. In fact,
some obstacles, such as human bodies, doors, glass and walls, attenuate and even can block mmWaves signals. For instance, attenuation of a 28 GHz signal can be as high as 24 dB and 45 dB if penetrating respectively through two walls and four doors [33]. Hence, it is not realistic to employ outdoor base stations (BSs) to serve indoor users for example.

The severe path-loss, the vulnerability of mmWave trans-
missions to blockages, and the atmospheric attenuation affect
the choice of the operating frequencies, and require important
changes to the system design and architecture.

B. TECHNICAL POTENTIAL

Despite the challenging characteristics of the mmWave chan-
nel, mmWave communications will be an important part of
the 5G cellular system and beyond. This is due to the poten-
tial of mmWave communications in terms of large bandwidth
availability and short wavelength.

1) LARGE BANDWIDTH

The 5G requirements for very high data rate communi-
cations may not be achievable by focusing only on the
heavily-occupied conventional sub-6 GHz frequency bands.
On the contrary, an abandon of the mmWave bands is
available to be exploited opening the door to multi-Gigabit
data exchange in spite of the low spectral efficiency since
a very large bandwidth is hopefully sufficient to support
very high data rates [34]. For instance, more than 12 GHz
of bandwidth is available between 60 GHz and 90 GHz
which is also called the E-band. Furthermore, the low spec-
tral efficiency means less complexity and more robustness
making mmWave systems more feasible. Actually, the 3rd
Generation Partnership Project (3GPP) Release 15 selected
the 24~29 GHz and 37~43 GHz frequency bands for the
deployment of 5G mmWave systems [35]. In the future,
the International Telecommunication Union (ITU) and the
3GPP will allocate two frequency bands around 40 GHz
and 100 GHz for commercial use [13]. In July 2016, the
Federal Communications Commission (FCC) dedicated sev-
eral mmWave bands for wireless services including around
28 GHz and 39 GHz for licensed allocation, and 64~71 GHz
for unlicensed usage [36].

2) SHORT WAVELENGTH

The short wavelength is the main reason behind the strong
path-loss of mmWave signals and hence the need for high
directivity antennas which can be realised with antenna
arrays and beamforming techniques. Narrower beams require
higher number of antennas in the array which is compatible
with mmWave communications thanks to the short wave-
length. Indeed, the shorter the wavelength is, the smaller the
antenna arrays' sizes are (since the distance between any two
antennas in the array is basically half of the wavelength) and
thus the higher the number of antennas to be packed into the
same physical array size is, and also the higher the beam-
forming gain is [5], [13], [16]. It is worth noting that a

typical antenna length at 60 GHz is less than 2.5 mm for
example [13]. Another benefit of highly directional commu-
nications, based on narrow beams at the transmitter and at
receiver, is that it helps to secure the mmWave communica-
tion link against eavesdropping and jamming as well as to
increase the interference immunity [5], [37].

C. ENABLING TECHNOLOGIES

The mmWave communications technology is, at the same
time, enabled by and complementary to some other technolo-
gies such as massive MIMO, advances in signal processing,
network densification and finally advances in circuit design
and integration.

MASSIVE MIMO

The use of a large number of antennas at BSs and for
mobile terminals (MTs) is called Massive MIMO and is
an essential technology to increase the capacity of cellular
networks. Massive MIMO has been studied for conventional
sub-6 GHz systems, and is also very crucial for mmWave
systems where high directivity is mandatory [38]. Higher
frequency bands offered by mmWave systems grant the
ability to design antenna arrays with a huge number of
antennas [39]. However, this will not be possible without
the recent advances in complementary metal oxide semi-
conductor (CMOS) circuits which increase the capacity of
circuit integration [40].

ENHANCED SIGNAL PROCESSING TECHNIQUES

It is clear that the conventional *fully-digital* MIMO system,
in which one RF chain is dedicated to each antenna, is
infeasible for mmWave systems because of the high imple-
mentation cost and high energy consumption [18], [41], [42].
Indeed, it has been shown that RF components can consume
up to 70% of the total transceiver power consumption [43].
For these reasons, several mmWave *hybrid* architectures
were proposed as alternatives to the fully-digital one, see
Section III-A. These solutions offer a good balance between
system performance and hardware complexity. However, they
require to develop new signal processing techniques for
beamforming, channel estimation, and more generally for
improving system performance.

NETWORK DENSIFICATION

Network densification means increasing the density of BSs deployment in the network via several tiers and an hierarchy of macro, micro, pico and femto cells, and leads to a multi-tier heterogeneous network [44]. Increasing the small cells density increases capacity and spectral efficiency of served users at the expense of increasing the cost of interference management [45]. MmWave communications are a very good candidate for small cell deployment mainly for three reasons i) a small cell means a short-range communication which makes the high path-loss of mmWave communications less unfavorable, ii) the larger bandwidth of mmWave frequencies means higher capacity which is the main goal of small cells
some obstacles, such as human bodies, doors, glass and walls, attenuate and even can block mmWaves signals. For instance, attenuation of a 28 GHz signal can be as high as 24 dB and 45 dB if penetrating respectively through two walls and four doors [33]. Hence, it is not realistic to employ outdoor base stations (BSs) to serve indoor users for example.

The severe path-loss, the vulnerability of mmWave trans-
missions to blockages, and the atmospheric attenuation affect
the choice of the operating frequencies, and require important
changes to the system design and architecture.

B. TECHNICAL POTENTIAL

Despite the challenging characteristics of the mmWave chan-
nel, mmWave communications will be an important part of
the 5G cellular system and beyond. This is due to the poten-
tial of mmWave communications in terms of large bandwidth
availability and short wavelength.

1) LARGE BANDWIDTH

The 5G requirements for very high data rate communi-
cations may not be achievable by focusing only on the
heavily-occupied conventional sub-6 GHz frequency bands.
On the contrary, an abandon of the mmWave bands is
available to be exploited opening the door to multi-Gigabit
data exchange in spite of the low spectral efficiency since
a very large bandwidth is hopefully sufficient to support
very high data rates [34]. For instance, more than 12 GHz
of bandwidth is available between 60 GHz and 90 GHz
which is also called the E-band. Furthermore, the low spec-
tral efficiency means less complexity and more robustness
making mmWave systems more feasible. Actually, the 3rd
Generation Partnership Project (3GPP) Release 15 selected
the 24~29 GHz and 37~43 GHz frequency bands for the
deployment of 5G mmWave systems [35]. In the future,
the International Telecommunication Union (ITU) and the
3GPP will allocate two frequency bands around 40 GHz
and 100 GHz for commercial use [13]. In July 2016, the
Federal Communications Commission (FCC) dedicated sev-
eral mmWave bands for wireless services including around
28 GHz and 39 GHz for licensed allocation, and 64~71 GHz
for unlicensed usage [36].

2) SHORT WAVELENGTH

The short wavelength is the main reason behind the strong
path-loss of mmWave signals and hence the need for high
directivity antennas which can be realised with antenna
arrays and beamforming techniques. Narrower beams require
higher number of antennas in the array which is compatible
with mmWave communications thanks to the short wave-
length. Indeed, the shorter the wavelength is, the smaller the
antenna arrays' sizes are (since the distance between any two
antennas in the array is basically half of the wavelength) and
thus the higher the number of antennas to be packed into the
same physical array size is, and also the higher the beam-
forming gain is [5], [13], [16]. It is worth noting that a

typical antenna length at 60 GHz is less than 2.5 mm for
example [13]. Another benefit of highly directional commu-
nications, based on narrow beams at the transmitter and at
receiver, is that it helps to secure the mmWave communica-
tion link against eavesdropping and jamming as well as to
increase the interference immunity [5], [37].

C. ENABLING TECHNOLOGIES

The mmWave communications technology is, at the same
time, enabled by and complementary to some other technolo-
gies such as massive MIMO, advances in signal processing,
network densification and finally advances in circuit design
and integration.

MASSIVE MIMO

The use of a large number of antennas at BSs and for
mobile terminals (MTs) is called Massive MIMO and is
an essential technology to increase the capacity of cellular
networks. Massive MIMO has been studied for conventional
sub-6 GHz systems, and is also very crucial for mmWave
systems where high directivity is mandatory [38]. Higher
frequency bands offered by mmWave systems grant the
ability to design antenna arrays with a huge number of
antennas [39]. However, this will not be possible without
the recent advances in complementary metal oxide semi-
conductor (CMOS) circuits which increase the capacity of
circuit integration [40].

ENHANCED SIGNAL PROCESSING TECHNIQUES

It is clear that the conventional *fully-digital* MIMO system,
in which one RF chain is dedicated to each antenna, is
infeasible for mmWave systems because of the high imple-
mentation cost and high energy consumption [18], [41], [42].
Indeed, it has been shown that RF components can consume
up to 70% of the total transceiver power consumption [43].
For these reasons, several mmWave *hybrid* architectures
were proposed as alternatives to the fully-digital one, see
Section III-A. These solutions offer a good balance between
system performance and hardware complexity. However, they
require to develop new signal processing techniques for
beamforming, channel estimation, and more generally for
improving system performance.

NETWORK DENSIFICATION

Network densification means increasing the density of BSs deployment in the network via several tiers and an hierarchy of macro, micro, pico and femto cells, and leads to a multi-tier heterogeneous network [44]. Increasing the small cells density increases capacity and spectral efficiency of served users at the expense of increasing the cost of interference management [45]. MmWave communications are a very good candidate for small cell deployment mainly for three reasons i) a small cell means a short-range communication which makes the high path-loss of mmWave communications less unfavorable, ii) the larger bandwidth of mmWave frequencies means higher capacity which is the main goal of small cells
<figure><img src="image_3.png" /><figcaption>FIGURE 2. General system architecture of mmWave transceiver.</figcaption></figure>

deployment, iii) and the high directivity of mmWave com-
munications, at the transmitter and receiver, results in better
interference control.

III. MASSIVE MIMO FOR MMWAVE SYSTEMS

Millimeter-wave communication systems need to employ a
large number of antennas at the transmitter and receiver
sides. Some early research works suggested arrays of 32
to 256 antennas at the BS and 4 to 16 ones at the mobile
terminal [17], [46]. Fortunately, it is now possible to pack
such number of antennas into small packages [47], [48].
However, other aspects such as the power consumption and
cost influence the maximum number of antennas that can be
integrated in the mmWave system. If digital signal process-
ing techniques are employed for baseband precoding at the
transmitter (TX) and combining at the receiver (RX) then one
dedicated RF chain per antenna is needed, where each RF
chain includes an ADC. These techniques are usually used
for sub-6 GHz systems. However, they are not affordable
actually at mmWave frequencies as the bandwidths become
wider and the antenna arrays become larger. This results in
high-resolution ADCs and high energy consumption of each
RF chain. For instance, 30 mW per RF chain is consumed
at sub-6 GHz frequencies, while 250 mW per RF chain is
needed at mmWave frequency bands [18].

Figure 2 is a block diagram that illustrates the general
architecture of a hybrid mmWave transceiver. Without loss
of generality, a BS equipped with M<sub>B</sub> RF chains and N<sub>B</sub>
antennas (M<sub>B</sub> < N<sub>B</sub>) is assumed to communicate with an MT
equipped with M<sub>T</sub> RF chains and N<sub>T</sub> antennas (M<sub>T</sub> < N<sub>T</sub>)
in order to exchange some data streams [4]. In practice, the
number of RF chains at the MTs is usually less than that
at the BSs. A point to point communication is considered
here, however, this architecture could be easily extended to
the case of multi-users. The combined role of the base-
band digital processing and the analog circuit of the hybrid
architecture is to direct the beams at the transmitter or/and
receiver. In contrast with the traditional fully-digital architec-
ture that does not consider an analog circuit behind (before)
the RF chains at the transmitter (receiver) and considers one
RF chain per antenna, i.e., M<sub>B</sub> = N<sub>B</sub>, M<sub>T</sub> = N<sub>T</sub> [5], [49]. For

this reason, fully-digital architecture must be avoided nowa-
days when it comes to mmWave communication systems
although it could be possible at some point in the future.
This leads to introducing some new hybrid architectures in
order to make the massive MIMO mmWave systems fea-
sible. The idea is to figure out how to reduce the number
of RF chains and the resolution of the ADCs which con-
sequently will reduce the total power consumption and the
total cost. Different transceiver architectures for mmWave
communication will be introduced in the following section.

A. MIMO ARCHITECTURES FOR MMWAVE
COMMUNICATIONS

1) FULLY-ANALOG ARCHITECTURE

One solution is to reduce the number of RF chains to
one and to perform beamforming entirely using the ana-
log processing. As illustrated in Figure 3(a) [5], [49], this
is achieved by analog shifters employed in the RF domain
to adjust the phase of the RF signals at each antenna. This
solution drastically reduces the hardware cost and power
consumption.

Many studies have been carried out on the fully-analog
architecture. A codebook-based analog beamformer for
mmWave wireless personal area networks via beam switch-
ing is discussed in [50]. References [51], [52] which
propose an alternative codebook-based beamforming for out-
door backhaul based on a joint transmitter-receiver beam
scanning employing a tree-structured codebook. Moreover,
an iterative channel estimation and analog beamforming
algorithm has been investigated in [53]–[56], where the
coefficients at the transmitter and receiver sides are calcu-
lated to reach asymptotically singular value decomposition
(SVD) elements.

Unfortunately, these solutions suffer from hardware lim-
itations due to the constant modulus constraint imposed by
the implementation of the phase shifters, hence the signals
can only be partially adjusted [5], [49]. Another draw-
back of the fully-analog architecture is that it supports
only one data stream, which makes its usage for multi-
user communications tricky. However, this architecture is
still a good fit for some standards such as WirelessHD for
instance.
<figure><img src="image_3.png" /><figcaption>FIGURE 2. General system architecture of mmWave transceiver.</figcaption></figure>

deployment, iii) and the high directivity of mmWave com-
munications, at the transmitter and receiver, results in better
interference control.

III. MASSIVE MIMO FOR MMWAVE SYSTEMS

Millimeter-wave communication systems need to employ a
large number of antennas at the transmitter and receiver
sides. Some early research works suggested arrays of 32
to 256 antennas at the BS and 4 to 16 ones at the mobile
terminal [17], [46]. Fortunately, it is now possible to pack
such number of antennas into small packages [47], [48].
However, other aspects such as the power consumption and
cost influence the maximum number of antennas that can be
integrated in the mmWave system. If digital signal process-
ing techniques are employed for baseband precoding at the
transmitter (TX) and combining at the receiver (RX) then one
dedicated RF chain per antenna is needed, where each RF
chain includes an ADC. These techniques are usually used
for sub-6 GHz systems. However, they are not affordable
actually at mmWave frequencies as the bandwidths become
wider and the antenna arrays become larger. This results in
high-resolution ADCs and high energy consumption of each
RF chain. For instance, 30 mW per RF chain is consumed
at sub-6 GHz frequencies, while 250 mW per RF chain is
needed at mmWave frequency bands [18].

Figure 2 is a block diagram that illustrates the general
architecture of a hybrid mmWave transceiver. Without loss
of generality, a BS equipped with M<sub>B</sub> RF chains and N<sub>B</sub>
antennas (M<sub>B</sub> < N<sub>B</sub>) is assumed to communicate with an MT
equipped with M<sub>T</sub> RF chains and N<sub>T</sub> antennas (M<sub>T</sub> < N<sub>T</sub>)
in order to exchange some data streams [4]. In practice, the
number of RF chains at the MTs is usually less than that
at the BSs. A point to point communication is considered
here, however, this architecture could be easily extended to
the case of multi-users. The combined role of the base-
band digital processing and the analog circuit of the hybrid
architecture is to direct the beams at the transmitter or/and
receiver. In contrast with the traditional fully-digital architec-
ture that does not consider an analog circuit behind (before)
the RF chains at the transmitter (receiver) and considers one
RF chain per antenna, i.e., M<sub>B</sub> = N<sub>B</sub>, M<sub>T</sub> = N<sub>T</sub> [5], [49]. For

this reason, fully-digital architecture must be avoided nowa-
days when it comes to mmWave communication systems
although it could be possible at some point in the future.
This leads to introducing some new hybrid architectures in
order to make the massive MIMO mmWave systems fea-
sible. The idea is to figure out how to reduce the number
of RF chains and the resolution of the ADCs which con-
sequently will reduce the total power consumption and the
total cost. Different transceiver architectures for mmWave
communication will be introduced in the following section.

A. MIMO ARCHITECTURES FOR MMWAVE
COMMUNICATIONS

1) FULLY-ANALOG ARCHITECTURE

One solution is to reduce the number of RF chains to
one and to perform beamforming entirely using the ana-
log processing. As illustrated in Figure 3(a) [5], [49], this
is achieved by analog shifters employed in the RF domain
to adjust the phase of the RF signals at each antenna. This
solution drastically reduces the hardware cost and power
consumption.

Many studies have been carried out on the fully-analog
architecture. A codebook-based analog beamformer for
mmWave wireless personal area networks via beam switch-
ing is discussed in [50]. References [51], [52] which
propose an alternative codebook-based beamforming for out-
door backhaul based on a joint transmitter-receiver beam
scanning employing a tree-structured codebook. Moreover,
an iterative channel estimation and analog beamforming
algorithm has been investigated in [53]–[56], where the
coefficients at the transmitter and receiver sides are calcu-
lated to reach asymptotically singular value decomposition
(SVD) elements.

Unfortunately, these solutions suffer from hardware lim-
itations due to the constant modulus constraint imposed by
the implementation of the phase shifters, hence the signals
can only be partially adjusted [5], [49]. Another draw-
back of the fully-analog architecture is that it supports
only one data stream, which makes its usage for multi-
user communications tricky. However, this architecture is
still a good fit for some standards such as WirelessHD for
instance.
<figure><img src="image_4.png" /><figcaption>FIGURE 3. Different MIMO architectures (a) fully-analog architecture; (b) hybrid architecture; (c) lens array architecture (d) few-bit ADCs architecture.</figcaption></figure>

## 2) HYBRID ARCHITECTURE

The hybrid architecture represents a compromise between the fully-digital architecture (where the number of RF chains equals that of the antennas) and the fully-analog one (where only one RF chain is employed). Precoding (at the transmitter) and combining (at the receiver) are performed in both the analog and the digital domains as shown in Figure 3(b). The idea is to employ a dimension-reduced precoder/combiner with a small number of RF chains ($M_B \ll N_B$, $M_T \ll N_T$) while still relying on full-size analog precoder/combiner. Despite the reduced number of RF chains, the performance of these architectures has been shown to be not far from the fully-digitally ones [18], [46], [57]. This can be explained by the sparse nature of the mmWave channel (since the number of scatters is small), which makes the channel matrix low-rank [6], [58], see Section IV for further illustration about sparsity. In addition, this solution supports easily several users/data streams, their number being equal or less than the number of RF chains. The 3GPP included the hybrid architecture in its recommendation for 5G systems in 2016 [59].

Furthermore, the analog processing (precoding/combining) can be realized through different analog networks with phase shifters, namely the fully-connected networks [60] and the sub-connected ones [46], [61]. The idea is either to connect each RF chain to all the antennas, or to connect it to a subset of them. It seems that the sub-connected choice could be practically more interesting since it reduces the cost and complexity while still achieving a good performance when

compared to the fully-connected network [46], [61], [62]. Another alternative is to replace the phase shifters (and all the required complementary components such as power combiners/splitters, control lines) with switches [62], [63]. Using switching networks could reduce the power consumption and complexity at the expense of lesser spectral-efficiency. One perspective is to combine switches and phase shifters to obtain an advantageous trade-off between them [64].

## 3) LENS ARRAY ARCHITECTURE

All the above mentioned architectures consider 1D or 2D planar antenna arrays. An attractive emerging approach is to combine lens antenna arrays with switching networks [65], [66]. A lens array is an electromagnetic lens with feed antennas located on the focal surface of the lens, which allows to concentrate the signal arriving from different directions on different antennas as depicted in Figure 3(c). Hence, the spatial model of the channel can be seen as a beamspace model. Furthermore, the beamspace of mmWave channels is sparse since the scattering is not rich and the power is propagated over a small number of paths [6], [58]. Due to the sparsity of the mmWave channel, a reduced-size switching network is used to select the dominant beams in the beamspace, which greatly reduces the number of RF chains [65], [66]. The authors in [66] had shown that the number of required switches (or RF chains) to achieve near-optimal capacity in lens array systems depends on the number of signal beams (and not on the number of antennas on the lens), and they proposed a spatial
<figure><img src="image_4.png" /><figcaption>FIGURE 3. Different MIMO architectures (a) fully-analog architecture; (b) hybrid architecture; (c) lens array architecture (d) few-bit ADCs architecture.</figcaption></figure>

## 2) HYBRID ARCHITECTURE

The hybrid architecture represents a compromise between the fully-digital architecture (where the number of RF chains equals that of the antennas) and the fully-analog one (where only one RF chain is employed). Precoding (at the transmitter) and combining (at the receiver) are performed in both the analog and the digital domains as shown in Figure 3(b). The idea is to employ a dimension-reduced precoder/combiner with a small number of RF chains ($M_B \ll N_B$, $M_T \ll N_T$) while still relying on full-size analog precoder/combiner. Despite the reduced number of RF chains, the performance of these architectures has been shown to be not far from the fully-digitally ones [18], [46], [57]. This can be explained by the sparse nature of the mmWave channel (since the number of scatters is small), which makes the channel matrix low-rank [6], [58], see Section IV for further illustration about sparsity. In addition, this solution supports easily several users/data streams, their number being equal or less than the number of RF chains. The 3GPP included the hybrid architecture in its recommendation for 5G systems in 2016 [59].

Furthermore, the analog processing (precoding/combining) can be realized through different analog networks with phase shifters, namely the fully-connected networks [60] and the sub-connected ones [46], [61]. The idea is either to connect each RF chain to all the antennas, or to connect it to a subset of them. It seems that the sub-connected choice could be practically more interesting since it reduces the cost and complexity while still achieving a good performance when

compared to the fully-connected network [46], [61], [62]. Another alternative is to replace the phase shifters (and all the required complementary components such as power combiners/splitters, control lines) with switches [62], [63]. Using switching networks could reduce the power consumption and complexity at the expense of lesser spectral-efficiency. One perspective is to combine switches and phase shifters to obtain an advantageous trade-off between them [64].

## 3) LENS ARRAY ARCHITECTURE

All the above mentioned architectures consider 1D or 2D planar antenna arrays. An attractive emerging approach is to combine lens antenna arrays with switching networks [65], [66]. A lens array is an electromagnetic lens with feed antennas located on the focal surface of the lens, which allows to concentrate the signal arriving from different directions on different antennas as depicted in Figure 3(c). Hence, the spatial model of the channel can be seen as a beamspace model. Furthermore, the beamspace of mmWave channels is sparse since the scattering is not rich and the power is propagated over a small number of paths [6], [58]. Due to the sparsity of the mmWave channel, a reduced-size switching network is used to select the dominant beams in the beamspace, which greatly reduces the number of RF chains [65], [66]. The authors in [66] had shown that the number of required switches (or RF chains) to achieve near-optimal capacity in lens array systems depends on the number of signal beams (and not on the number of antennas on the lens), and they proposed a spatial
multiplexing scheme for mmWave communications, namely the path-division multiplexing.

## 4) FEW-BIT ADCS ARCHITECTURE

Reducing the power consumption can be done either by reducing the number of RF chains (as proposed in the above presented architectures) and/or by reducing the energy consumed per RF chain. The latest is the key motivation behind proposing to replace the high-resolution ADCs with few-bit (e.g., 1 to 4 bit) ADCs, since high-frequency high-resolution ADCs are well known to consume a lot of power [18], [67], [68]. The mmWave system with few-bit ADCs is shown in Figure 3(d). Using few-bit ADCs could even open the door to the feasibility of fully-digital mmWave systems [18]. However, the high non-linearity errors of quantization and the wide bands of mmWave channels pose a lot of challenges to the design of signal processing solutions for mmWave communications despite some recent research works [69], [70].

## B. SPATIAL MMWAVE MIMO CHANNEL MODELING

MmWave channel models are classified into two categories [6], physical models and analytical models. The physical models are based on the electromagnetic characteristics of the signal propagation between the transmit and receive antenna arrays. They can efficiently reflect the measured parameters and they are popular for MIMO channels, hence they are a good choice for mmWave MIMO channels. On the other hand, the analytical models are based on the mathematical analysis of the channel, and they are convenient for algorithm development and system analysis. The physical channel models are divided into two categories. Deterministic models, which characterize the real effects of the environment on the system, but need high computational complexity, and stochastic channel models, which require low computational complexity, hence they are the popular choice for mmWave system design and simulation. Thus the considered channel model for this article is the Saleh-Valenzuela stochastic channel model.

Since the number of scatters is limited [6], [58], most research works had adopted the geometric channel model to describe mmWave channels [4], [60], [61]. Let us consider again the system presented in Figure 2. In the geometric channel model, the $N_T \times N_B$ complex matrix $H_{DL}$ of the narrowband downlink (DL) channel is expressed as,

$$H_{DL} = \sqrt{\frac{N_B N_T}{L \alpha_{PL,DL}}} \sum_{l=1}^{L} \alpha_{l,DL} a_T(\theta_{l,T}, \phi_{l,T}) a_B^H(\theta_{l,B}, \phi_{l,B}), (2)$$

where L is number of paths between the BS and MT (L is small), $\alpha_{PL,DL}$ is the average path-loss, $\alpha_{l,DL}$ is the complex gain of the lth path in the DL channel, and $(\theta_{l,T}, \phi_{l,T})$ and $(\theta_{l,B}, \phi_{l,B})$ are the azimuth and elevation angles of arrival (AoAs) and the azimuth and elevation angles of departure (AoDs) respectively. Finally, $a_B(\alpha_T)$ is the $N_B \times 1$ ($N_T \times 1$) complex antenna array response (steering) vector at the BS

(at the MT). Spatially quantized AoDs and AoAs can be employed to represent the channel over the whole space, i.e., $\theta_{l,T}, \phi_{l,T}, \theta_{l,B}, \phi_{l,B}$ can be obtained from a uniform grid of $N_G$ points on [0, $2\pi$] with $N_G \gg L$ [71], [72]. This approach is also called the virtual MIMO channel representation [73]. In this approach, the channel in (2) can be reformulated as,

$$H_{DL} = A_T H_\alpha A_B^H, \quad (3)$$

where $A_B \in \mathbb{C}^{N_B \times L}$ and $A_T \in \mathbb{C}^{N_T \times L}$ are, respectively, the aggregation of the L steering vectors $a_B(\theta_{l,T}, \phi_{l,T})$ and $a_T(\theta_{l,T}, \phi_{l,T})$ for $l = 1, 2, \dots, L$, and $H_\alpha = \sqrt{\frac{N_B N_T}{L \alpha_{PL,DL}}} \text{diag}(\alpha_{1,DL}, \alpha_{2,DL}, \dots, \alpha_{L,DL})$.

The channel is frequency-selective when the system is wideband, hence equation (2) must be rewritten to represent the D-delay DL channel model. The d-th delay tap is given by [74],

$$H_{DL}(d) = \sqrt{\frac{N_B N_T}{L \alpha_{PL,DL}}} \sum_{l=1}^{L} \alpha_{l,DL} p_B(dT_s - \tau_l) \dots a_T(\theta_{l,T}, \phi_{l,T}) a_B^H(\theta_{l,B}, \phi_{l,B}), \quad (4)$$

where $p_B(\cdot)$ is the combination of pulse shaping filter and other filters at the BS, $T_s, \tau_l$ are the sampling period and the delay. For orthogonal frequency-division multiplexing (OFDM) modulation, the MIMO channel frequency response matrix at each subcarrier k follows the expression [74],

$$H_{DL}(k) = \sqrt{\frac{N_B N_T}{L \alpha_{PL,DL}}} \sum_{l=1}^{L} \alpha_{l,DL} q_l(k) \dots a_T(\theta_{l,T}, \phi_{l,T}) a_B^H(\theta_{l,B}, \phi_{l,B}), \quad (5)$$

where $q_l(k)$ is given by,

$$q_l(k) = \sum_{d=1}^{D} p_B(dT_s - \tau_l) e^{-\frac{2\pi j k d}{k_s}}, \quad (6)$$

and $K_s$ is the number of subcarriers.

The response vectors for two antenna array configurations, namely the uniform linear planar array (ULPA) and the lens array, are given as follows.

### 1) ULPA RESPONSE VECTOR

A ULPA is a rectangular array that consists of $N_h$ antennas in each row and $N_v$ antennas in each column, in the horizontal and vertical directions, uniformly separated by a distance d. Typically, $d = \lambda/2$ where $\lambda$ is the signal wavelength. The total number of antennas within the array is $N = N_h \times N_v$, and each row or column is a uniform linear array (ULA) which is considered as a special case of ULPA. The response vector of an N-element ULA is given by,

$$a_{\text{ULA}}(\theta) = \frac{1}{\sqrt{N}} \left[ 1 e^{j2\pi \frac{d}{\lambda} \sin(\theta)} \dots e^{j(N-1)2\pi \frac{d}{\lambda} \sin(\theta)} \right]^T \quad (7)$$

The array response for the ULPA configuration is given by [75], [76],

$$a_{\text{ULPA}}(\theta, \phi) = a_h(\theta) \otimes a_v(\phi) \quad (8)$$
multiplexing scheme for mmWave communications, namely the path-division multiplexing.

## 4) FEW-BIT ADCS ARCHITECTURE

Reducing the power consumption can be done either by reducing the number of RF chains (as proposed in the above presented architectures) and/or by reducing the energy consumed per RF chain. The latest is the key motivation behind proposing to replace the high-resolution ADCs with few-bit (e.g., 1 to 4 bit) ADCs, since high-frequency high-resolution ADCs are well known to consume a lot of power [18], [67], [68]. The mmWave system with few-bit ADCs is shown in Figure 3(d). Using few-bit ADCs could even open the door to the feasibility of fully-digital mmWave systems [18]. However, the high non-linearity errors of quantization and the wide bands of mmWave channels pose a lot of challenges to the design of signal processing solutions for mmWave communications despite some recent research works [69], [70].

## B. SPATIAL MMWAVE MIMO CHANNEL MODELING

MmWave channel models are classified into two categories [6], physical models and analytical models. The physical models are based on the electromagnetic characteristics of the signal propagation between the transmit and receive antenna arrays. They can efficiently reflect the measured parameters and they are popular for MIMO channels, hence they are a good choice for mmWave MIMO channels. On the other hand, the analytical models are based on the mathematical analysis of the channel, and they are convenient for algorithm development and system analysis. The physical channel models are divided into two categories. Deterministic models, which characterize the real effects of the environment on the system, but need high computational complexity, and stochastic channel models, which require low computational complexity, hence they are the popular choice for mmWave system design and simulation. Thus the considered channel model for this article is the Saleh-Valenzuela stochastic channel model.

Since the number of scatters is limited [6], [58], most research works had adopted the geometric channel model to describe mmWave channels [4], [60], [61]. Let us consider again the system presented in Figure 2. In the geometric channel model, the $N_T \times N_B$ complex matrix $H_{DL}$ of the narrowband downlink (DL) channel is expressed as,

$$H_{DL} = \sqrt{\frac{N_B N_T}{L \alpha_{PL,DL}}} \sum_{l=1}^{L} \alpha_{l,DL} a_T(\theta_{l,T}, \phi_{l,T}) a_B^H(\theta_{l,B}, \phi_{l,B}), (2)$$

where L is number of paths between the BS and MT (L is small), $\alpha_{PL,DL}$ is the average path-loss, $\alpha_{l,DL}$ is the complex gain of the lth path in the DL channel, and $(\theta_{l,T}, \phi_{l,T})$ and $(\theta_{l,B}, \phi_{l,B})$ are the azimuth and elevation angles of arrival (AoAs) and the azimuth and elevation angles of departure (AoDs) respectively. Finally, $a_B(\alpha_T)$ is the $N_B \times 1$ ($N_T \times 1$) complex antenna array response (steering) vector at the BS

(at the MT). Spatially quantized AoDs and AoAs can be employed to represent the channel over the whole space, i.e., $\theta_{l,T}, \phi_{l,T}, \theta_{l,B}, \phi_{l,B}$ can be obtained from a uniform grid of $N_G$ points on [0, $2\pi$] with $N_G \gg L$ [71], [72]. This approach is also called the virtual MIMO channel representation [73]. In this approach, the channel in (2) can be reformulated as,

$$H_{DL} = A_T H_{\alpha} A_B^H, \quad (3)$$

where $A_B \in \mathbb{C}^{N_B \times L}$ and $A_T \in \mathbb{C}^{N_T \times L}$ are, respectively, the aggregation of the L steering vectors $a_B(\theta_{l,T}, \phi_{l,T})$ and $a_T(\theta_{l,T}, \phi_{l,T})$ for $l = 1, 2, \dots, L$, and $H_{\alpha} = \sqrt{\frac{N_B N_T}{L \alpha_{PL,DL}}} \text{diag}(\alpha_{1,DL}, \alpha_{2,DL}, \dots, \alpha_{L,DL})$.

The channel is frequency-selective when the system is wideband, hence equation (2) must be rewritten to represent the D-delay DL channel model. The d-th delay tap is given by [74],

$$H_{DL}(d) = \sqrt{\frac{N_B N_T}{L \alpha_{PL,DL}}} \sum_{l=1}^{L} \alpha_{l,DL} p_B(dT_s - \tau_l) \dots a_T(\theta_{l,T}, \phi_{l,T}) a_B^H(\theta_{l,B}, \phi_{l,B}), \quad (4)$$

where $p_B(\cdot)$ is the combination of pulse shaping filter and other filters at the BS, $T_s, \tau_l$ are the sampling period and the delay. For orthogonal frequency-division multiplexing (OFDM) modulation, the MIMO channel frequency response matrix at each subcarrier k follows the expression [74],

$$H_{DL}(k) = \sqrt{\frac{N_B N_T}{L \alpha_{PL,DL}}} \sum_{l=1}^{L} \alpha_{l,DL} q_l(k) \dots a_T(\theta_{l,T}, \phi_{l,T}) a_B^H(\theta_{l,B}, \phi_{l,B}), \quad (5)$$

where $q_l(k)$ is given by,

$$q_l(k) = \sum_{d=1}^{D} p_B(dT_s - \tau_l) e^{-\frac{2\pi j k d}{k_s}}, \quad (6)$$

and $K_s$ is the number of subcarriers.

The response vectors for two antenna array configurations, namely the uniform linear planar array (ULPA) and the lens array, are given as follows.

### 1) ULPA RESPONSE VECTOR

A ULPA is a rectangular array that consists of $N_h$ antennas in each row and $N_v$ antennas in each column, in the horizontal and vertical directions, uniformly separated by a distance d. Typically, $d = \lambda/2$ where $\lambda$ is the signal wavelength. The total number of antennas within the array is $N = N_h \times N_v$, and each row or column is a uniform linear array (ULA) which is considered as a special case of ULPA. The response vector of an N-element ULA is given by,

$$a_{\text{ULA}}(\theta) = \frac{1}{\sqrt{N}} \left[ 1 \ e^{j2\pi \frac{d}{\lambda} \sin(\theta)} \ \dots \ e^{j(N-1)2\pi \frac{d}{\lambda} \sin(\theta)} \right]^T \quad (7)$$

The array response for the ULPA configuration is given by [75], [76],

$$a_{\text{ULPA}}(\theta, \phi) = a_h(\theta) \otimes a_v(\phi) \quad (8)$$
where $\theta$ and $\phi$ are, respectively, the azimuth and elevation angles, and $a_h$ and $a_v$ are the horizontal and vertical steering vectors which are derived from (7) with the corresponding angle and number of antennas.

Recently, 3D beamforming techniques for mmWave communications had been introduced [77]–[79]. However, few papers studied the estimation of the 3D mmWave channel [80]–[82]. On the other hand, most existing research works had considered only the 2D beamforming case, i.e., the elevation is neglected and only the horizontal scattering is taken into consideration. Under these assumptions, the ULPA response is rewritten as,

$$a_{\text{ULPA}}(\theta) = a_h(\theta) \otimes 1_{N_v} \quad (9)$$

where $1_{N_v}$ is an $N_v \times 1$ unity vector.

## 2) LENS ARRAY RESPONSE VECTOR

Modeling the lens antenna array is typically based on separating the energy-focusing electromagnetic lens from the feeding antenna array where the lens effect is approximated by a spatial discrete Fourier transform (DFT) matrix, U, of size $N \times N$ [65], [81]–[84]. The DFT matrix is composed of $N$ orthogonal array response vectors steering towards $N$ beams scanning the entire space, and is defined as [65], [82],

$$[\mathbf{U}]_{i \times j}, : = [a(\xi_i, \xi_j)]^H, \quad (10)$$

where $\xi_i = \frac{1}{N_h}(i - \frac{N_h+1}{2})$ for $i = 1, 2, \dots, N_h$, $\xi_j = \frac{1}{N_v}(j - \frac{N_v+1}{2})$ for $j = 1, 2, \dots, N_v$, and $N_h, N_v$ are, respectively, the number of horizontal and vertical antennas of the lens array ($N = N_h \times N_v$). The idea is that this special antenna configuration transforms the above presented spatial channel into a beamspace model. This allows us to apply a beam selection technique in the beamspace domain directly without affecting the width of beams nor the gain of the antenna pattern. This model is referred to as the *DFT model*.

Another approach for modeling lens antenna arrays is to study the lens and the matching antenna array as one integrated unit and to derive an approximated closed-form of the array response vector [66], [85], [86]. The 3D lens antenna array can steer towards given azimuth and elevation angles, $\theta$ and $\phi$, based on the following response vector [85],

$$a_m(\theta, \phi) = e^{-j\phi_0} \sqrt{\tilde{D}_h \tilde{D}_v \text{sinc}(m_v - \tilde{D}_v \sin\phi)} \dots \text{sinc} \\ \times (m_h - \tilde{D}_h \sin\theta \cos\phi), \quad (11)$$

where $m_h = 1, \pm 1, \pm 2, \dots, \pm \frac{N_h-1}{2}$ and $m_v = 1, \pm 1, \pm 2, \dots, \pm \frac{N_v-1}{2}$ are the horizontal and vertical indices of each antenna. $\tilde{D}_h \triangleq \frac{D_h}{\lambda}$ and $\tilde{D}_v \triangleq \frac{D_v}{\lambda}$ denote the electric dimensions of the lens which are the physical dimensions, $D_h$ and $D_v$, normalized by the wavelength, and $\phi_0$ is a common phase shift from the lens aperture to the array. This model is referred to as the *approximated array response model*.

## C. MMWAVE MIMO SYSTEM MODEL

Let us first consider a narrowband system model for a hybrid architecture. The downlink is described by the $N_T \times 1$ received signal vector at the MT, which is given by

$$r_T = H_{\text{DL}}F_B S_B + n, \quad (12)$$

where $H_{\text{DL}}$ is the DL channel matrix as defined in equation (2), $S_B$ is the $D_B \times 1$ normalized transmitted symbols vector with $E[S_B S_B^H] = (P_B/D_B)I_{D_B}$, and $P_B$ is the average transmission power of the BS. Also, $F_B = F_{\text{RF,B}}F_{\text{BB,B}}$ is the $N_B \times D_B$ precoding matrix of the BS which combines the baseband precoder $F_{\text{BB,B}} \in \mathbb{C}^{M_B \times D_B}$ and the RF precoder $F_{\text{RF,B}} \in \mathbb{C}^{N_B \times M_B}$. Finally, the vector n of size $N_T \times 1$ is the additive white Gaussian noise (AWGN). The received vector r is processed at the MT such as,

$$y_T = W_T^H H_{\text{DL}} F_B S_B + W_T^H n, \quad (13)$$

where $W_T = W_{\text{RF,T}} W_{\text{BB,T}}$ is the $N_T \times D_T$ combiner matrix of the MT which is assumed to consecutively apply the RF combiner $W_{\text{RF,T}} \in \mathbb{C}^{N_T \times M_T}$ and the baseband combiner $W_{\text{BB,T}} \in \mathbb{C}^{M_T \times D_T}$. It is worth noting that the constant modulus constraint must be applied to the RF precoder and to the RF combiner when analog shifters are used, that is all the entries of the matrix must have the same magnitude. Finally, the channel matrix $H_{\text{DL}}$ in (13) must be replaced by the beamspace channel matrix, $\tilde{H}_{\text{DL}}$, in the case of lens antenna array, when the lens effect is separately modeled as in equation (10), where $\tilde{H}_{\text{DL}} = UH_{\text{DL}}$.

The same logic can be employed for the uplink (UL) by replacing $H_{\text{DL}}$ by the UL channel matrix $H_{\text{UL}}$ and by reversing the combiners and precoders roles. Under the widely-adopted assumption of *channel reciprocity* [87], the UL channel is a Hermitian transposition of the DL channel $H_{\text{UL}} = H_{\text{DL}}^H$, and one between them must be estimated to have CSI [8]. In practice, it is not obvious how to guarantee the channel reciprocity in mmWave communications, even for time-division duplexing (TDD) systems (where the same carrier frequency is used for both DL and UL), due to different issues, for instance the synchronization and calibration errors of RF chains [8]. However, it should be pointed out that the number of paths and the AoAs/AoDs for each path are strongly correlated for both DL and UL channels even for frequency-division duplexing (FDD) communication systems where separated DL and UL frequencies are employed. This property is called the *path reciprocity* [86]. Assuming path reciprocity, the $N_B \times N_T$ complex matrix $H_{\text{UL}}$ of the UL channel is given by,

$$H_{\text{UL}} = \sqrt{\frac{N_B N_T}{L \alpha_{\text{PL,UL}}}} \sum_{l=1}^{L} \alpha_{l, \text{UL}} a_T(\theta_{l, \text{T}}, \phi_{l, \text{T}}) a_B^H(\theta_{l, \text{B}}, \phi_{l, \text{B}}), \quad (14)$$

where $\alpha_{l, \text{UL}}$ is the complex gain of the $l^{th}$ path in the UL channel, and is in general different from $\alpha_{l, \text{DL}}$ as introduced in Equation (2).
where $\theta$ and $\phi$ are, respectively, the azimuth and elevation angles, and $a_h$ and $a_v$ are the horizontal and vertical steering vectors which are derived from (7) with the corresponding angle and number of antennas.

Recently, 3D beamforming techniques for mmWave communications had been introduced [77]–[79]. However, few papers studied the estimation of the 3D mmWave channel [80]–[82]. On the other hand, most existing research works had considered only the 2D beamforming case, i.e., the elevation is neglected and only the horizontal scattering is taken into consideration. Under these assumptions, the ULPA response is rewritten as,

$$a_{\text{ULPA}}(\theta) = a_h(\theta) \otimes 1_{N_v} \quad (9)$$

where $1_{N_v}$ is an $N_v \times 1$ unity vector.

## 2) LENS ARRAY RESPONSE VECTOR

Modeling the lens antenna array is typically based on separating the energy-focusing electromagnetic lens from the feeding antenna array where the lens effect is approximated by a spatial discrete Fourier transform (DFT) matrix, U, of size $N \times N$ [65], [81]–[84]. The DFT matrix is composed of $N$ orthogonal array response vectors steering towards $N$ beams scanning the entire space, and is defined as [65], [82],

$$[\mathbf{U}]_{i \times j}, : = [a(\xi_i, \xi_j)]^H, \quad (10)$$

where $\xi_i = \frac{1}{N_h}(i - \frac{N_h+1}{2})$ for $i = 1, 2, \dots, N_h$, $\xi_j = \frac{1}{N_v}(j - \frac{N_v+1}{2})$ for $j = 1, 2, \dots, N_v$, and $N_h, N_v$ are, respectively, the number of horizontal and vertical antennas of the lens array ($N = N_h \times N_v$). The idea is that this special antenna configuration transforms the above presented spatial channel into a beamspace model. This allows us to apply a beam selection technique in the beamspace domain directly without affecting the width of beams nor the gain of the antenna pattern. This model is referred to as the *DFT model*.

Another approach for modeling lens antenna arrays is to study the lens and the matching antenna array as one integrated unit and to derive an approximated closed-form of the array response vector [66], [85], [86]. The 3D lens antenna array can steer towards given azimuth and elevation angles, $\theta$ and $\phi$, based on the following response vector [85],

$$a_m(\theta, \phi) = e^{-j\phi_0} \sqrt{\tilde{D}_h \tilde{D}_v \text{sinc}(m_v - \tilde{D}_v \sin\phi)} \dots \text{sinc} \\ \times (m_h - \tilde{D}_h \sin\theta \cos\phi), \quad (11)$$

where $m_h = 1, \pm 1, \pm 2, \dots, \pm \frac{N_h-1}{2}$ and $m_v = 1, \pm 1, \pm 2, \dots, \pm \frac{N_v-1}{2}$ are the horizontal and vertical indices of each antenna. $\tilde{D}_h \triangleq \frac{D_h}{\lambda}$ and $\tilde{D}_v \triangleq \frac{D_v}{\lambda}$ denote the electric dimensions of the lens which are the physical dimensions, $D_h$ and $D_v$, normalized by the wavelength, and $\phi_0$ is a common phase shift from the lens aperture to the array. This model is referred to as the *approximated array response model*.

## C. MMWAVE MIMO SYSTEM MODEL

Let us first consider a narrowband system model for a hybrid architecture. The downlink is described by the $N_T \times 1$ received signal vector at the MT, which is given by

$$r_T = H_{\text{DL}}F_B S_B + n, \quad (12)$$

where $H_{\text{DL}}$ is the DL channel matrix as defined in equation (2), $S_B$ is the $D_B \times 1$ normalized transmitted symbols vector with $E[S_B S_B^H] = (P_B/D_B)I_{D_B}$, and $P_B$ is the average transmission power of the BS. Also, $F_B = F_{\text{RF,B}}F_{\text{BB,B}}$ is the $N_B \times D_B$ precoding matrix of the BS which combines the baseband precoder $F_{\text{BB,B}} \in \mathbb{C}^{M_B \times D_B}$ and the RF precoder $F_{\text{RF,B}} \in \mathbb{C}^{N_B \times M_B}$. Finally, the vector n of size $N_T \times 1$ is the additive white Gaussian noise (AWGN). The received vector r is processed at the MT such as,

$$y_T = W_T^H H_{\text{DL}} F_B S_B + W_T^H n, \quad (13)$$

where $W_T = W_{\text{RF,T}} W_{\text{BB,T}}$ is the $N_T \times D_T$ combiner matrix of the MT which is assumed to consecutively apply the RF combiner $W_{\text{RF,T}} \in \mathbb{C}^{N_T \times M_T}$ and the baseband combiner $W_{\text{BB,T}} \in \mathbb{C}^{M_T \times D_T}$. It is worth noting that the constant modulus constraint must be applied to the RF precoder and to the RF combiner when analog shifters are used, that is all the entries of the matrix must have the same magnitude. Finally, the channel matrix $H_{\text{DL}}$ in (13) must be replaced by the beamspace channel matrix, $\tilde{H}_{\text{DL}}$, in the case of lens antenna array, when the lens effect is separately modeled as in equation (10), where $\tilde{H}_{\text{DL}} = UH_{\text{DL}}$.

The same logic can be employed for the uplink (UL) by replacing $H_{\text{DL}}$ by the UL channel matrix $H_{\text{UL}}$ and by reversing the combiners and precoders roles. Under the widely-adopted assumption of *channel reciprocity* [87], the UL channel is a Hermitian transposition of the DL channel $H_{\text{UL}} = H_{\text{DL}}^H$, and one between them must be estimated to have CSI [8]. In practice, it is not obvious how to guarantee the channel reciprocity in mmWave communications, even for time-division duplexing (TDD) systems (where the same carrier frequency is used for both DL and UL), due to different issues, for instance the synchronization and calibration errors of RF chains [8]. However, it should be pointed out that the number of paths and the AoAs/AoDs for each path are strongly correlated for both DL and UL channels even for frequency-division duplexing (FDD) communication systems where separated DL and UL frequencies are employed. This property is called the *path reciprocity* [86]. Assuming path reciprocity, the $N_B \times N_T$ complex matrix $H_{\text{UL}}$ of the UL channel is given by,

$$H_{\text{UL}} = \sqrt{\frac{N_B N_T}{L \alpha_{\text{PL,UL}}}} \sum_{l=1}^{L} \alpha_{l, \text{UL}} a_{\text{T}}(\theta_{l, \text{T}}, \phi_{l, \text{T}}) a_{\text{B}}^{H}(\theta_{l, \text{B}}, \phi_{l, \text{B}}), \quad (14)$$

where $\alpha_{l, \text{UL}}$ is the complex gain of the $l^{th}$ path in the UL channel, and is in general different from $\alpha_{l, \text{DL}}$ as introduced in Equation (2).
For wideband systems, equation (13) is reformulated to express the received signal at each subcarrier $k$. It is rewritten as

$$y_T(k) = W_T^H(k)H_{DL}(k)F_B(k)S_B(k) + W_T^H(k)n(k), \quad (15)$$

where $H_{DL}(k)$ is given by (5).

In the following sections, we will explain why the CS techniques are good candidates to estimate the sparse mmWave channels. Afterword, an inclusive review of existing channel estimation methods for mmWave systems with hybrid architecture, with lens antenna array, and with few-bit ADCs, is respectively presented in Section V, Section VI and Section VII.

## IV. COMPRESSIVE SENSING: MATHEMATICAL TOOL FOR CHANNEL ESTIMATION

The multi-path signal components of mmWave systems tend to be distributed into few clusters such as mmWave channels look sparse [60], [75], [88]. This sparsity characteristic is also verified by measurements, for instance, [28], [30], [89], [90] showed that mmWave channels typically exhibit only 3-4 scattering clusters in dense-urban non-line-of-sight (NLOS) environments. Thus a convenient representation of such channels needs a comprehensive study of sparsity. The traditional training-based channel estimation methods seem to be not optimal under these sparse conditions due to the huge channel size (big number of antennas at the TX and RX) and the fact that most of what we get will be thrown away. Hence, the aim is to estimate the non-zero elements of the channel and one approach to solve this type of problems is to use CS techniques [91]–[93]. CS is widely employed in wireless communication applications like channel estimation, spectrum sensing for cognitive radio, and localization among others [94]. CS tools handle the problem of estimating any sparse signal by directly acquiring a compressive signal representation with a lot fewer number of samples than that required by the Shannon-Nyquist theorem, and from which the sparse signal can be recovered through an optimization process [92], [95].

A signal represented by an $n \times 1$ vector $s \in \mathbb{C}^n$ is said to be exactly $k$-sparse signal if all but just $k \ll n$ values in the vector are zeros, in other words, there is a very small number of non-zero values in the vector and the rest are zero value elements. Mathematically, we can represent it as $\|s\|_0 \le k$. The same idea can be generalized to 2D and 3D signals. Consider a discrete-time signal $x$, which can be represented by an $n \times 1$ vector in $\mathbb{C}^n$. If $x$ is not sparse, it can be transformed into another domain via a transformation matrix $\Psi \in \mathbb{C}^{n \times n}$ as follows

$$x = \Psi s \quad (16)$$

such as $s$ represents an exactly $k$-sparse signal. It is very essential to employ a careful transformation matrix which can further expose the sparse nature of the original signal.

The objective of CS tools is to compress the dimension of measurements by projecting the high-dimensional sparse

signals of dimension $n$ into a reduced-dimension spaces of dimension $m \ll n$ via a measurement or sensing matrix $\Phi$ such as

$$y = \Phi x = \Phi \Psi s = \Theta s \quad (17)$$

where $\Theta = \Phi \Psi$ is an $m \times n$ matrix. This requires first to design a stable sensing (measurement) matrix $\Phi$, and consequently a matrix $\Theta$, before proposing a reconstruction algorithm to recover the signal $x$ from only $m \approx k$ measurement vector $y$ with the best reconstruction reliability and hence the minimum information loss. This problem is ill-posed in general, but can be resolved for the class of signals that have a sparse expansion. This requires that the design of the compression matrix $\Theta$ respects some properties such as the restricted isometry property and the small coherence [96]–[98]. The first represents a necessary and sufficient condition for the CS problem to be well conditioned, while the latter makes the CS technique more effective.

Many algorithms have been proposed to recover $k$-sparse signals with high probability [99], where some of them used tractable mixed-norm optimization methods [100]–[102], efficient greedy algorithms [98], [103]–[105], fast iterative thresholding methods [106], statistical sparse recovery [107], and many more [94], [95], [98], [108]. We give a glance and point out some of these algorithms. The non-convex $\ell_0$-norm problem is transformed into a convex $\ell_1$-norm one, and basis pursuit (BP) solutions [100], such as least absolute shrinkage and selection operator (LASSO) [102], are used. However, the BP approach is rarely implemented in real-time wireless applications because of its high computational cost. Orthogonal matching pursuit (OMP) is a greedy algorithm [98] which was proposed as an effective alternative to the BP ones [100]. In [105], compressive sampling matching pursuit (CoSaMP), which is a parallel greedy algorithm, was introduced. Moreover, the authors in [104] proposed a modified version of the OMP, called multi-grid OMP (MG-OMP), in order to reduce the complexity, and to make the reconstruction more adaptive. Another low-complexity approach, which tries to solve the problem by iteratively refining the sparse estimate, is a thresholding approach illustrated in [106]. Three algorithms were proposed in [109]–[111] that select more than one candidate per iteration to decrease the running time, while for OMP one candidate is selected per each iteration. An algorithm termed as multipath matching pursuit (MMP) was proposed in [109], where it performs the tree search, in which all combinations of $k$-sparse indices are the candidates in the tree, and the algorithm tries to find the best candidate from this tree that minimizes the residual. The aforementioned algorithm imposes reasonable computational overhead while achieving better performance over existing greedy algorithms. A generalization algorithm of the OMP was proposed in [110] termed as generalized OMP (GOMP) that selects more than one index per iteration corresponding to largest correlation in magnitude with the residual. Similar to the MMP and GOMP, a stage-determined matching pursuit (SdMP) was proposed in [111] that aims
For wideband systems, equation (13) is reformulated to express the received signal at each subcarrier $k$. It is rewritten as

$$y_T(k) = W_T^H(k)H_{DL}(k)F_B(k)S_B(k) + W_T^H(k)n(k), \quad (15)$$

where $H_{DL}(k)$ is given by (5).

In the following sections, we will explain why the CS techniques are good candidates to estimate the sparse mmWave channels. Afterword, an inclusive review of existing channel estimation methods for mmWave systems with hybrid architecture, with lens antenna array, and with few-bit ADCs, is respectively presented in Section V, Section VI and Section VII.

## IV. COMPRESSIVE SENSING: MATHEMATICAL TOOL FOR CHANNEL ESTIMATION

The multi-path signal components of mmWave systems tend to be distributed into few clusters such as mmWave channels look sparse [60], [75], [88]. This sparsity characteristic is also verified by measurements, for instance, [28], [30], [89], [90] showed that mmWave channels typically exhibit only 3-4 scattering clusters in dense-urban non-line-of-sight (NLOS) environments. Thus a convenient representation of such channels needs a comprehensive study of sparsity. The traditional training-based channel estimation methods seem to be not optimal under these sparse conditions due to the huge channel size (big number of antennas at the TX and RX) and the fact that most of what we get will be thrown away. Hence, the aim is to estimate the non-zero elements of the channel and one approach to solve this type of problems is to use CS techniques [91]–[93]. CS is widely employed in wireless communication applications like channel estimation, spectrum sensing for cognitive radio, and localization among others [94]. CS tools handle the problem of estimating any sparse signal by directly acquiring a compressive signal representation with a lot fewer number of samples than that required by the Shannon-Nyquist theorem, and from which the sparse signal can be recovered through an optimization process [92], [95].

A signal represented by an $n \times 1$ vector $s \in \mathbb{C}^n$ is said to be exactly $k$-sparse signal if all but just $k \ll n$ values in the vector are zeros, in other words, there is a very small number of non-zero values in the vector and the rest are zero value elements. Mathematically, we can represent it as $\|s\|_0 \le k$. The same idea can be generalized to 2D and 3D signals. Consider a discrete-time signal $x$, which can be represented by an $n \times 1$ vector in $\mathbb{C}^n$. If $x$ is not sparse, it can be transformed into another domain via a transformation matrix $\Psi \in \mathbb{C}^{n \times n}$ as follows

$$x = \Psi s \quad (16)$$

such as $s$ represents an exactly $k$-sparse signal. It is very essential to employ a careful transformation matrix which can further expose the sparse nature of the original signal.

The objective of CS tools is to compress the dimension of measurements by projecting the high-dimensional sparse

signals of dimension $n$ into a reduced-dimension spaces of dimension $m \ll n$ via a measurement or sensing matrix $\Phi$ such as

$$y = \Phi x = \Phi \Psi s = \Theta s \quad (17)$$

where $\Theta = \Phi \Psi$ is an $m \times n$ matrix. This requires first to design a stable sensing (measurement) matrix $\Phi$, and consequently a matrix $\Theta$, before proposing a reconstruction algorithm to recover the signal $x$ from only $m \approx k$ measurement vector $y$ with the best reconstruction reliability and hence the minimum information loss. This problem is ill-posed in general, but can be resolved for the class of signals that have a sparse expansion. This requires that the design of the compression matrix $\Theta$ respects some properties such as the restricted isometry property and the small coherence [96]–[98]. The first represents a necessary and sufficient condition for the CS problem to be well conditioned, while the latter makes the CS technique more effective.

Many algorithms have been proposed to recover $k$-sparse signals with high probability [99], where some of them used tractable mixed-norm optimization methods [100]–[102], efficient greedy algorithms [98], [103]–[105], fast iterative thresholding methods [106], statistical sparse recovery [107], and many more [94], [95], [98], [108]. We give a glance and point out some of these algorithms. The non-convex $\ell_0$-norm problem is transformed into a convex $\ell_1$-norm one, and basis pursuit (BP) solutions [100], such as least absolute shrinkage and selection operator (LASSO) [102], are used. However, the BP approach is rarely implemented in real-time wireless applications because of its high computational cost. Orthogonal matching pursuit (OMP) is a greedy algorithm [98] which was proposed as an effective alternative to the BP ones [100]. In [105], compressive sampling matching pursuit (CoSaMP), which is a parallel greedy algorithm, was introduced. Moreover, the authors in [104] proposed a modified version of the OMP, called multi-grid OMP (MG-OMP), in order to reduce the complexity, and to make the reconstruction more adaptive. Another low-complexity approach, which tries to solve the problem by iteratively refining the sparse estimate, is a thresholding approach illustrated in [106]. Three algorithms were proposed in [109]–[111] that select more than one candidate per iteration to decrease the running time, while for OMP one candidate is selected per each iteration. An algorithm termed as multipath matching pursuit (MMP) was proposed in [109], where it performs the tree search, in which all combinations of $k$-sparse indices are the candidates in the tree, and the algorithm tries to find the best candidate from this tree that minimizes the residual. The aforementioned algorithm imposes reasonable computational overhead while achieving better performance over existing greedy algorithms. A generalization algorithm of the OMP was proposed in [110] termed as generalized OMP (GOMP) that selects more than one index per iteration corresponding to largest correlation in magnitude with the residual. Similar to the MMP and GOMP, a stage-determined matching pursuit (SdMP) was proposed in [111] that aims
at selecting more than one index per iteration that surpass a carefully designed threshold. A small difference compared to MMP and GOMP is that the StMP adds a pruning step after the end of some latter iterations (after satisfying a certain sparsity level condition) in order to refine the selection. Finally, approximate message passing algorithm (AMP), which combines the thresholding methods with the message passing ones, performs well for highly structured measurement matrices [93].

## A. A SPARSE FORMULATION OF THE MMWAVE CHANNEL ESTIMATION PROBLEM

The mmWave system model is described in equation (13). In what follows, some indices will be omitted for simplicity. Let us consider the received signal at $Q$ successive instants when the same precoder, $F$, and the same combiner, $W$, are employed. To exploit the sparse nature of the channel, the concatenated matrix is vectorized, the $Q \times D_T$ resultant vector, $y_v$, can then be written as [71], [90],

$$y_v = \sqrt{P}F^T A_{B,D}^* Z_B \otimes W^H A_{T,D}^* Z_T + n_v, \quad (18)$$

where the $N_G \times 1$ sparse vectors, $Z_B$ and $Z_T$, have non-zero elements that correspond to the actual AoDs and AoAs. $A_{B,D}$ and $A_{T,D}$ are, respectively, the $N_B \times N_G$ and $N_T \times N_G$ beamforming dictionary matrices at the BS and MT. Each dictionary consists of column vectors which represent the complex antenna steering vectors corresponding to the $N_G$ spatially quantized directions, i.e., it is assumed that each direction is defined by an angle which is taken from $\{0, \frac{2\pi}{N_G}, \dots, \frac{2\pi(N_G-1)}{N_G}\}$. Equation (18) introduces a sparse formulation of this problem by employing a compression matrix which can be expressed as a function of beamforming dictionaries, precoder and combiner matrices [71], [90]. This allows to estimate the channel by detecting and estimating the non-zero elements of $Z_B$ and $Z_T$ with a small number of measurements. The CS tools will be able to guarantee that if only the compression matrix is well-designed which in turn requires an efficient design of the precoder and the combiner.

Hence, the estimation of mmWave channels is enabled by reformulating the matrix system model into a sparse problem, before compressing the sparse matrices into reduced-size ones, and finally employing the appropriate CS reconstruction method. Basically, CS-based methods have a significant improvement over most traditional ones, a conclusion made in almost all recent studies on mmWave channel estimation [18]. Such conclusion motivated the practical usage of CS solutions for estimating mmWave channels as will be shown in the upcoming sections [69], [71], [74], [88], [104], [112]–[126].

## V. CHANNEL ESTIMATION TECHNIQUES FOR HYBRID ARCHITECTURE SYSTEMS

For the sake of attaining more precoding gains as well as to be able to precode simultaneously multiple data streams, numerous studies have been proposed to split the precoding process into analog and digital domains [60], [71].

<figure><img src="image_5.png" /><figcaption>FIGURE 4. An example of the beam patterns adopted in the first (a) and second (b) stages in [71] when $K=3$, where $S_k, k \in \{1, \dots, K\}$ denotes the sub-ranges in each stage.</figcaption></figure>

Moreover, to overcome the limitations of analog-only beamforming mmWave systems, constrained by the analog phase shifters amplitude which has to be constant, and by the potentially low-resolution signal phase control, several works have adopted the joint analog-digital hybrid architecture. Some channel estimation methods with this hybrid architecture have been proposed taking into account the aforementioned constraints. Most of these works assume a frequency-flat narrowband mmWave channel model, [62], [71], [104], [127]–[131]. However, several papers have been proposed focusing on the frequency-selective channel case [118]–[120]. In the following paragraphs, we will first present the different approaches for mmWave channel estimation with hybrid architecture when the system is narrowband, before presenting the research works under the frequency-selective channel assumption.

### A. TECHNIQUES FOR NARROWBAND SYSTEMS

Different approaches are employed to estimate the frequency-flat mmWave channel including divide-and-conquer, ping-pong, mode-by-mode approaches and many others. These approaches are explained below in addition to other methods, and they are listed in Table 2 with their respective references. Furthermore, Table 3 refers to the definition of the parameters used in the upcoming sections, note that the indices B, T stand for the BS and MT, respectively.

#### 1) DIVIDE-AND-CONQUER APPROACH

In [71] a low-complexity adaptive channel estimation algorithm has been proposed for narrowband mmWave channel with large antenna arrays and a few number of RF units at both the BS and MT sides. The authors assumed that the amplitude of phase shifters is constant, and that the phases are quantized. The algorithm divides the estimation process into several stages as shown in Figure 4. At each stage, the AoAs/AoDs angular ranges are divided into $K$ non-overlapped angular sub-ranges, $K$ beam patterns are used to send the pilot signal and $K$ beam patterns are used to combine the signal at the receiver. So each beam pattern at the
at selecting more than one index per iteration that surpass a carefully designed threshold. A small difference compared to MMP and GOMP is that the StMP adds a pruning step after the end of some latter iterations (after satisfying a certain sparsity level condition) in order to refine the selection. Finally, approximate message passing algorithm (AMP), which combines the thresholding methods with the message passing ones, performs well for highly structured measurement matrices [93].

## A. A SPARSE FORMULATION OF THE MMWAVE CHANNEL ESTIMATION PROBLEM

The mmWave system model is described in equation (13). In what follows, some indices will be omitted for simplicity. Let us consider the received signal at $Q$ successive instants when the same precoder, $F$, and the same combiner, $W$, are employed. To exploit the sparse nature of the channel, the concatenated matrix is vectorized, the $Q \times D_T$ resultant vector, $y_v$, can then be written as [71], [90],

$$y_v = \sqrt{P}F^T A_{B,D}^* Z_B \otimes W^H A_{T,D}^* Z_T + n_v, \quad (18)$$

where the $N_G \times 1$ sparse vectors, $Z_B$ and $Z_T$, have non-zero elements that correspond to the actual AoDs and AoAs. $A_{B,D}$ and $A_{T,D}$ are, respectively, the $N_B \times N_G$ and $N_T \times N_G$ beamforming dictionary matrices at the BS and MT. Each dictionary consists of column vectors which represent the complex antenna steering vectors corresponding to the $N_G$ spatially quantized directions, i.e., it is assumed that each direction is defined by an angle which is taken from $\{0, \frac{2\pi}{N_G}, \dots, \frac{2\pi(N_G-1)}{N_G}\}$. Equation (18) introduces a sparse formulation of this problem by employing a compression matrix which can be expressed as a function of beamforming dictionaries, precoder and combiner matrices [71], [90]. This allows to estimate the channel by detecting and estimating the non-zero elements of $Z_B$ and $Z_T$ with a small number of measurements. The CS tools will be able to guarantee that if only the compression matrix is well-designed which in turn requires an efficient design of the precoder and the combiner.

Hence, the estimation of mmWave channels is enabled by reformulating the matrix system model into a sparse problem, before compressing the sparse matrices into reduced-size ones, and finally employing the appropriate CS reconstruction method. Basically, CS-based methods have a significant improvement over most traditional ones, a conclusion made in almost all recent studies on mmWave channel estimation [18]. Such conclusion motivated the practical usage of CS solutions for estimating mmWave channels as will be shown in the upcoming sections [69], [71], [74], [88], [104], [112]–[126].

## V. CHANNEL ESTIMATION TECHNIQUES FOR HYBRID ARCHITECTURE SYSTEMS

For the sake of attaining more precoding gains as well as to be able to precode simultaneously multiple data streams, numerous studies have been proposed to split the precoding process into analog and digital domains [60], [71]. Moreover, to overcome the limitations of analog-only beamforming mmWave systems, constrained by the analog phase shifters amplitude which has to be constant, and by the potentially low-resolution signal phase control, several works have adopted the joint analog-digital hybrid architecture. Some channel estimation methods with this hybrid architecture have been proposed taking into account the aforementioned constraints. Most of these works assume a frequency-flat narrowband mmWave channel model, [62], [71], [104], [127]–[131]. However, several papers have been proposed focusing on the frequency-selective channel case [118]–[120]. In the following paragraphs, we will first present the different approaches for mmWave channel estimation with hybrid architecture when the system is narrowband, before presenting the research works under the frequency-selective channel assumption.

<figure><img src="image_5.png" /><figcaption>FIGURE 4. An example of the beam patterns adopted in the first (a) and second (b) stages in [71] when $K = 3$, where $S_k, k \in \{1, \dots, K\}$ denotes the sub-ranges in each stage.</figcaption></figure>

### A. TECHNIQUES FOR NARROWBAND SYSTEMS

Different approaches are employed to estimate the frequency-flat mmWave channel including divide-and-conquer, ping-pong, mode-by-mode approaches and many others. These approaches are explained below in addition to other methods, and they are listed in Table 2 with their respective references. Furthermore, Table 3 refers to the definition of the parameters used in the upcoming sections, note that the indices B, T stand for the BS and MT, respectively.

#### 1) DIVIDE-AND-CONQUER APPROACH

In [71] a low-complexity adaptive channel estimation algorithm has been proposed for narrowband mmWave channel with large antenna arrays and a few number of RF units at both the BS and MT sides. The authors assumed that the amplitude of phase shifters is constant, and that the phases are quantized. The algorithm divides the estimation process into several stages as shown in Figure 4. At each stage, the AoAs/AoDs angular ranges are divided into $K$ non-overlapped angular sub-ranges, $K$ beam patterns are used to send the pilot signal and $K$ beam patterns are used to combine the signal at the receiver. So each beam pattern at the
<table><caption>TABLE 2. Channel estimation algorithms for narrowband hybrid mmWave systems.</caption><thead><tr><th>Reference</th><th>Year</th><th>Approach</th><th>Scenario</th><th>UL/DL</th><th>2D/3D</th><th>Complexity</th><th>Description</th></tr></thead><tbody><tr><td>Alkhateeb et al. [71]</td><td>2014</td><td>Divide-and-conquer</td><td>Single-user</td><td>DL</td><td>2D</td><td>O(KL<sup>2</sup>[KL/M] log<sub>K</sub>(N<sub>G</sub>/L))</td><td>OMP, LSE</td></tr><tr><td>Lee et al. [104]</td><td>2014</td><td>Open-loop</td><td>Single-user</td><td>Not specified</td><td>2D</td><td>O(LK<sub>B</sub>K<sub>T</sub>N<sub>G</sub><sup>2</sup>)</td><td>OMP, MG-OMP, LSE</td></tr><tr><td>He et al. [132]</td><td>2014</td><td>Mode-by-mode</td><td>Single-user</td><td>UL/DL</td><td>2D</td><td>Not specified</td><td>A temporally correlated NLOS channel, based on the TDD correlation statistics</td></tr><tr><td>Schniter et al. [88]</td><td>2014</td><td>Aperture shaping</td><td>Not specified</td><td>Not specified</td><td>2D</td><td>Not specified</td><td>LASSO, LMMSE</td></tr><tr><td>Alkhateeb et al. [112]</td><td>2014</td><td>Ping-ping</td><td>Single-user</td><td>UL/DL</td><td>2D</td><td>O(KL<sup>2</sup>N<sub>B</sub> log<sub>K</sub>(N<sub>G</sub>/L))</td><td>OMP, no feedback is needed</td></tr><tr><td>Payami et al. [133]</td><td>2015</td><td>Ping-pong</td><td>Single-user</td><td>Not specified</td><td>2D</td><td>|N/M| measurements to scan all N directions</td><td>The training time doesn't scale with the number of multi-path components</td></tr><tr><td>Kokshoorn et al. [134]</td><td>2015</td><td>Overlapped beam patterns</td><td>Single-user</td><td>Not specified</td><td>2D</td><td>K<sup>2</sup>/log<sub>2</sub>(k+1) reduction compared to [71]</td><td>MRC, used to track fast changing channels</td></tr><tr><td>Peng et al. [113]</td><td>2015</td><td>AAVE</td><td>Single-user</td><td>UL</td><td>2D</td><td>2M time slots</td><td>CS-based technique, enhances the angular estimation resolution</td></tr><tr><td>Montagner et al. [135]</td><td>2015</td><td>2D DFT</td><td>Single-user</td><td>Not specified</td><td>2D</td><td>[N<sub>D<sub>T</sub></sub>/M<sub>T</sub>][N<sub>D<sub>B</sub></sub>/M<sub>B</sub>] M<sub>B</sub> time slots</td><td>DFT, iterative cancellation method</td></tr><tr><td>Mendez-rial et al. [114]</td><td>2015</td><td>Switches</td><td>Single-user</td><td>DL</td><td>2D</td><td>Not specified</td><td>OMP, M-OMP</td></tr><tr><td>Chiang et al. [115]</td><td>2016</td><td>SVD avoidance</td><td>Single-user</td><td>Not specified</td><td>Not specified</td><td>Not specified</td><td>OMP, it exploits the orthogonality between the array propagation vectors</td></tr><tr><td>Lu et al. [136]</td><td>2016</td><td>Adaptive DFT</td><td>Single-user</td><td>DL</td><td>2D</td><td>[N<sub>D<sub>T</sub></sub>/M<sub>T</sub>][N<sub>D<sub>B</sub></sub>/M<sub>B</sub>] M<sub>B</sub> time slots</td><td>DFT, a feedback is adopted to improve the accuracy</td></tr><tr><td>Han et al. [116]</td><td>2016</td><td>Two-stage asymmetric</td><td>Multi-user</td><td>DL</td><td>2D</td><td>Not specified</td><td>Exhaustive search, CS</td></tr><tr><td>Park et al. [117]</td><td>2016</td><td>Spatial covariance</td><td>Single-user</td><td>UL</td><td>Not specified</td><td>Not specified</td><td>OMP, S-OMP, C-OMP, DS-OMP, DC-OMP</td></tr><tr><td>Zhou et al. [137]</td><td>2016</td><td>CANDE-COMP/PARAFAC (CP)</td><td>Multi-user</td><td>UL</td><td>2D</td><td>O(N<sub>G1</sub><sup>2</sup>N<sub>G2</sub><sup>2</sup> + T'TM<sub>B</sub>)</td><td>CP, referred to as tensor rank decomposition</td></tr><tr><td>Guo et al. [138]</td><td>2017</td><td>2D beamspace MUSIC</td><td>Single-user</td><td>Not specified</td><td>2D</td><td>The main complexity comes from (i) the eigenvalue decomposition: O(M<sub>B</sub><sup>2</sup>M<sub>T</sub><sup>2</sup>) (ii) the grid search: O(N<sub>G2</sub>N<sub>G1</sub>M<sub>B</sub><sup>2</sup>M<sub>T</sub><sup>2</sup>)</td><td>MUSIC, LSE, it exploits the large-scale fading property of path directions</td></tr><tr><td>Guo et al. [139]</td><td>2019</td><td>Dimension-deficient</td><td>Single-user</td><td>UL</td><td>2D</td><td>O(MN<sub>B</sub>(N<sub>T</sub>L)<sup>3</sup>)</td><td>CoSaMP, it reduces the influence of accidental errors</td></tr></tbody></table>

transmitter is combined by *K* beam patterns at the receiver, as a result, each stage needs *K*² time slots to span all the combinations of transmit-receive beam patterns. The beam patterns are taken from a predefined codebook designed and proposed by the authors. The process is then continued by calculating the magnitudes of the *K*² received signals to determine or conquer the next AoA/AoD angular sub-range for the next stage. At each stage, the process pursues the same way as in the previous stage, in which it divides the chosen AoD sub-range at the transmitter and AoA sub-range at the receiver into *K* sub-ranges. It proceeds this way until it achieves the desired AoA/AoD resolution. This algorithm, initially proposed for single-path and multi-path cases, needs $K^2[\log_K(\max(N_B, N_T))]$ time slots for each channel path. Such algorithm is not fast enough to track the rapid variations of mmWave channels which could be a major drawback. Moreover, this algorithm requires an exclusive feedback channel.

## 2) PING-PONG APPROACHES

Based on the same codebook design, the authors appended their work in [71] by proposing another estimation algorithm that doesn't need feedback in [112]. The developed algorithm uses ping-pong iterations, which requires a complexity of $O(KL^2N_B \log_K(N_G/L))$, to acquire the channel parameters, where the AoAs, and AoDs are taken from a uniform grid of $N_G$ points. In [133] a two-stage algorithm for single-user (SU) channel estimation is presented as well as a codebook design which is similar to that in [112]. The algorithm is characterized by a two-stage handshaking

between the transmitter and the receiver. During the first stage, the transmitter uses one transmit antenna to send an omni-directional signal and correspondingly the receiver senses and scans its multiple directions to detect the AoA angles, subsequently the roles are exchanged at the second stage such that the receiver sends a pilot signal at the detected angles only. The two-stage estimation algorithm is illustrated in Figure 5. It needs $[N/M]$ measurements to scan all $N$ required directions, where $M$ is the number of the RF chains in the system. Moreover, its training time doesn't scale with the number of multi-path components, unlike the channel estimation method in [112].

## 3) OVERLAPPED BEAM PATTERNS APPROACH

Reference [134] outlined a fast channel estimation algorithm by proposing the overlapped beam patterns estimation

<table><caption>TABLE 3. Parameters guide for methods presented in the article.</caption><thead><tr><th></th><th>Parameters sheet</th></tr></thead><tbody><tr><td>N</td><td>The number of antennas</td></tr><tr><td>M</td><td>The number of RF chains</td></tr><tr><td>L</td><td>The number of channel paths</td></tr><tr><td>K</td><td>The number of beam patterns or beamforming vectors</td></tr><tr><td>n, k</td><td>n is the size of a sparse vector and k is its sparsity</td></tr><tr><td>N<sub>G</sub>, N<sub>G,1</sub>, N<sub>G,2</sub></td><td>The numbers of points of the uniform grids</td></tr><tr><td>K<sub>s</sub></td><td>The number of subcarriers</td></tr><tr><td>K<sub>p</sub></td><td>A reduced number of subcarriers</td></tr><tr><td>D</td><td>The delay spread of the channel</td></tr><tr><td>T, T'</td><td>The number of frames and sub-frames, respectively.</td></tr><tr><td>A<sub>v</sub></td><td>The number of virtual antennas</td></tr><tr><td>j</td><td>The index of a current iteration in an algorithm</td></tr><tr><td>N<sub>TS</sub></td><td>The training sequence length</td></tr><tr><td>N<sub>DT</sub>, M<sub>D<sub>B</sub></sub></td><td>Suitable integers depend on N<sub>TS</sub></td></tr><tr><td>Q</td><td>The number of snapshots or time instants</td></tr><tr><td>N<sub>f</sub></td><td>The number of non-overlapping antenna subsets</td></tr></tbody></table>
<table><caption>TABLE 2. Channel estimation algorithms for narrowband hybrid mmWave systems.</caption><thead><tr><th>Reference</th><th>Year</th><th>Approach</th><th>Scenario</th><th>UL/DL</th><th>2D/3D</th><th>Complexity</th><th>Description</th></tr></thead><tbody><tr><td>Alkhateeb et al. [71]</td><td>2014</td><td>Divide-and-conquer</td><td>Single-user</td><td>DL</td><td>2D</td><td>O(KL<sup>2</sup>[KL/M] log<sub>K</sub>(N<sub>G</sub>/L))</td><td>OMP, LSE</td></tr><tr><td>Lee et al. [104]</td><td>2014</td><td>Open-loop</td><td>Single-user</td><td>Not specified</td><td>2D</td><td>O(LK<sub>B</sub>K<sub>T</sub>N<sub>G</sub><sup>2</sup>)</td><td>OMP, MG-OMP, LSE</td></tr><tr><td>He et al. [132]</td><td>2014</td><td>Mode-by-mode</td><td>Single-user</td><td>UL/DL</td><td>2D</td><td>Not specified</td><td>A temporally correlated NLOS channel, based on the TDD correlation statistics</td></tr><tr><td>Schniter et al. [88]</td><td>2014</td><td>Aperture shaping</td><td>Not specified</td><td>Not specified</td><td>2D</td><td>Not specified</td><td>LASSO, LMMSE</td></tr><tr><td>Alkhateeb et al. [112]</td><td>2014</td><td>Ping-ping</td><td>Single-user</td><td>UL/DL</td><td>2D</td><td>O(KL<sup>2</sup>N<sub>B</sub> log<sub>K</sub>(N<sub>G</sub>/L))</td><td>OMP, no feedback is needed</td></tr><tr><td>Payami et al. [133]</td><td>2015</td><td>Ping-pong</td><td>Single-user</td><td>Not specified</td><td>2D</td><td>|N/M| measurements to scan all N directions</td><td>The training time doesn't scale with the number of multi-path components</td></tr><tr><td>Kokshoorn et al. [134]</td><td>2015</td><td>Overlapped beam patterns</td><td>Single-user</td><td>Not specified</td><td>2D</td><td>K<sup>2</sup>/log<sub>2</sub>(k+1) reduction compared to [71]</td><td>MRC, used to track fast changing channels</td></tr><tr><td>Peng et al. [113]</td><td>2015</td><td>AAVE</td><td>Single-user</td><td>UL</td><td>2D</td><td>2M time slots</td><td>CS-based technique, enhances the angular estimation resolution</td></tr><tr><td>Montagner et al. [135]</td><td>2015</td><td>2D DFT</td><td>Single-user</td><td>Not specified</td><td>2D</td><td>[N<sub>D<sub>T</sub></sub>/M<sub>T</sub>][N<sub>D<sub>B</sub></sub>/M<sub>B</sub>] M<sub>B</sub> time slots</td><td>DFT, iterative cancellation method</td></tr><tr><td>Mendez-rial et al. [114]</td><td>2015</td><td>Switches</td><td>Single-user</td><td>DL</td><td>2D</td><td>Not specified</td><td>OMP, M-OMP</td></tr><tr><td>Chiang et al. [115]</td><td>2016</td><td>SVD avoidance</td><td>Single-user</td><td>Not specified</td><td>Not specified</td><td>Not specified</td><td>OMP, it exploits the orthogonality between the array propagation vectors</td></tr><tr><td>Lu et al. [136]</td><td>2016</td><td>Adaptive DFT</td><td>Single-user</td><td>DL</td><td>2D</td><td>[N<sub>D<sub>T</sub></sub>/M<sub>T</sub>][N<sub>D<sub>B</sub></sub>/M<sub>B</sub>] M<sub>B</sub> time slots</td><td>DFT, a feedback is adopted to improve the accuracy</td></tr><tr><td>Han et al. [116]</td><td>2016</td><td>Two-stage asymmetric</td><td>Multi-user</td><td>DL</td><td>2D</td><td>Not specified</td><td>Exhaustive search, CS</td></tr><tr><td>Park et al. [117]</td><td>2016</td><td>Spatial covariance</td><td>Single-user</td><td>UL</td><td>Not specified</td><td>Not specified</td><td>OMP, S-OMP, C-OMP, DS-OMP, DC-OMP</td></tr><tr><td>Zhou et al. [137]</td><td>2016</td><td>CANDE-COMP/PARAFAC (CP)</td><td>Multi-user</td><td>UL</td><td>2D</td><td>O(N<sub>G1</sub><sup>2</sup>N<sub>G2</sub><sup>2</sup> + T'TM<sub>B</sub>)</td><td>CP, referred to as tensor rank decomposition</td></tr><tr><td>Guo et al. [138]</td><td>2017</td><td>2D beamspace MUSIC</td><td>Single-user</td><td>Not specified</td><td>2D</td><td>The main complexity comes from (i) the eigenvalue decomposition: O(M<sub>B</sub><sup>2</sup>M<sub>T</sub><sup>2</sup>) (ii) the grid search: O(N<sub>G2</sub>N<sub>G1</sub>M<sub>B</sub><sup>2</sup>M<sub>T</sub><sup>2</sup>)</td><td>MUSIC, LSE, it exploits the large-scale fading property of path directions</td></tr><tr><td>Guo et al. [139]</td><td>2019</td><td>Dimension-deficient</td><td>Single-user</td><td>UL</td><td>2D</td><td>O(MN<sub>B</sub>(N<sub>T</sub>L)<sup>3</sup>)</td><td>CoSaMP, it reduces the influence of accidental errors</td></tr></tbody></table>

transmitter is combined by *K* beam patterns at the receiver, as a result, each stage needs *K*² time slots to span all the combinations of transmit-receive beam patterns. The beam patterns are taken from a predefined codebook designed and proposed by the authors. The process is then continued by calculating the magnitudes of the *K*² received signals to determine or conquer the next AoA/AoD angular sub-range for the next stage. At each stage, the process pursues the same way as in the previous stage, in which it divides the chosen AoD sub-range at the transmitter and AoA sub-range at the receiver into *K* sub-ranges. It proceeds this way until it achieves the desired AoA/AoD resolution. This algorithm, initially proposed for single-path and multi-path cases, needs $K^2[\log_K(max(N_B, N_T))]$ time slots for each channel path. Such algorithm is not fast enough to track the rapid variations of mmWave channels which could be a major drawback. Moreover, this algorithm requires an exclusive feedback channel.

## 2) PING-PONG APPROACHES

Based on the same codebook design, the authors appended their work in [71] by proposing another estimation algorithm that doesn't need feedback in [112]. The developed algorithm uses ping-pong iterations, which requires a complexity of $O(KL^2N_B \log_K(N_G/L))$, to acquire the channel parameters, where the AoAs, and AoDs are taken from a uniform grid of $N_G$ points. In [133] a two-stage algorithm for single-user (SU) channel estimation is presented as well as a codebook design which is similar to that in [112]. The algorithm is characterized by a two-stage handshaking

between the transmitter and the receiver. During the first stage, the transmitter uses one transmit antenna to send an omni-directional signal and correspondingly the receiver senses and scans its multiple directions to detect the AoA angles, subsequently the roles are exchanged at the second stage such that the receiver sends a pilot signal at the detected angles only. The two-stage estimation algorithm is illustrated in Figure 5. It needs $[N/M]$ measurements to scan all $N$ required directions, where $M$ is the number of the RF chains in the system. Moreover, its training time doesn't scale with the number of multi-path components, unlike the channel estimation method in [112].

## 3) OVERLAPPED BEAM PATTERNS APPROACH

Reference [134] outlined a fast channel estimation algorithm by proposing the overlapped beam patterns estimation

<table><caption>TABLE 3. Parameters guide for methods presented in the article.</caption><thead><tr><th></th><th>Parameters sheet</th></tr></thead><tbody><tr><td>N</td><td>The number of antennas</td></tr><tr><td>M</td><td>The number of RF chains</td></tr><tr><td>L</td><td>The number of channel paths</td></tr><tr><td>K</td><td>The number of beam patterns or beamforming vectors</td></tr><tr><td>n, k</td><td>n is the size of a sparse vector and k is its sparsity</td></tr><tr><td>N<sub>G</sub>, N<sub>G,1</sub>, N<sub>G,2</sub></td><td>The numbers of points of the uniform grids</td></tr><tr><td>K<sub>s</sub></td><td>The number of subcarriers</td></tr><tr><td>K<sub>p</sub></td><td>A reduced number of subcarriers</td></tr><tr><td>D</td><td>The delay spread of the channel</td></tr><tr><td>T, T'</td><td>The number of frames and sub-frames, respectively.</td></tr><tr><td>A<sub>v</sub></td><td>The number of virtual antennas</td></tr><tr><td>j</td><td>The index of a current iteration in an algorithm</td></tr><tr><td>N<sub>TS</sub></td><td>The training sequence length</td></tr><tr><td>N<sub>DT</sub>, M<sub>D<sub>B</sub></sub></td><td>Suitable integers depend on N<sub>TS</sub></td></tr><tr><td>Q</td><td>The number of snapshots or time instants</td></tr><tr><td>N<sub>f</sub></td><td>The number of non-overlapping antenna subsets</td></tr></tbody></table>
<figure><img src="image_6.png" /><figcaption>FIGURE 5. Illustration of the two-stage estimation algorithm in [133], the blue colored arrows represent the AoA/s and AoD/s at the BS and MT; (a) the BS sends an omni-directional signal, (b) the MT scans the directions, (c) the MT sends through its AoDs, (d) the BS scans the directions.</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>FIGURE 6. The overlapped beam patterns adopted in the first (a) and second (b) stages when $K = 3$.</figcaption></figure>

concept. This concept reduces the time slots needed to estimate the channel by $\frac{K^2}{\log_2(K+1)}$ compared to [71], with a slight degradation in performance. This small sacrifice in degradation can be accepted when tracking fast changing channels is required. Indeed, the concept is to use a smaller number of overlapped beam patterns for the same number of sub-ranges to estimate the channel, instead of assigning one beam pattern for each sub-range as in [71] in which the number of beam patterns should be equal to the number of sub-ranges. The algorithm is illustrated in Figure 6.

## 4) OPEN-LOOP APPROACH

An *open-loop* channel estimation technique that doesn't need a feedback loop was proposed in [104]. The algorithm is provided for an SU mmWave system, and is based on CS techniques. As for many channel estimation algorithms, the authors used the OMP and least-squares estimation (LSE) techniques to perform the estimation of the AoAs/AoDs and gains, respectively. However, they also proposed the adaptive MG-OMP, this new version of OMP enhances the estimates of AoAs/AoDs by refining these estimates just around the regions where the AoAs/AoDs are present which reduces MG-OMP complexity over OMP. Through their computer simulations, the authors showed that CS techniques, OMP and MG-OMP, outperforms the LSE ones. In addition, the complexity reduction gained by the MG-OMP was estimated.

## 5) MODE-BY-MODE APPROACH

Another scheme was proposed in [132] to estimate a temporally correlated NLOS mmWave MIMO channel. The

authors, first modified the parametric channel model to an evolution temporally correlated MIMO channel model to successfully track the channel variations. The system is based on the TDD correlation statistics and it exploits the reciprocity of the channel. The proposed algorithm updates each column of the analog precoder and combiner, this approach is called the *mode-by-mode* approach, in which each mode represents one column of the analog precoder and combiner. For each mode, a codebook is built using a group of rotation matrices that rotate the previous mode to reconstruct the new mode, then the algorithm chooses the codeword that maximizes the received power. The digital precoder and combiner are then constructed using conventional pilot-aided estimation of the effective channel.

## 6) APERTURE SHAPING APPROACH

In [88], Schniter and Sayeed proposed a technique termed as *aperture shaping* to enhance the sparsity of the virtual MIMO channel. Briefly, aperture shaping is performed by applying a fixed gain at each antenna at both the TX and RX. The shaping coefficients are optimised to maximize the signal-to-interference ratio. In addition, they implemented a mmWave system that uses modulation and demodulation techniques based on fast Fourier transform (FFT) to further expose the channel sparsity. Finally, waterfilling technique and Lanczos algorithm were employed to design spectrally efficient precoding and decoding. They solved the sparsity problem using LASSO [102], where it was shown that their procedure approaches the perfect-CSI capacity for a mmWave system.

## 7) AAVE APPROACH

Based on the same assumptions as in [71], the authors in [113] proposed a new concept called antenna array with virtual elements (AAVE), that extends the real antenna arrays at both the transmitter and receiver to a new one by appending some $A_v$ virtual antennas without affecting the physical array. The idea behind AAVE is to add some virtual antennas to the real physical antennas at both the BS and MT, in order to enhance the angular estimation resolution. However, the scheme assures that no data is sent over the virtual antennas to guarantee that no physical change has been introduced to the antenna arrays. Based on AAVE, [113]
<figure><img src="image_6.png" /><figcaption>FIGURE 5. Illustration of the two-stage estimation algorithm in [133], the blue colored arrows represent the AoA/s and AoD/s at the BS and MT; (a) the BS sends an omni-directional signal, (b) the MT scans the directions, (c) the MT sends through its AoDs, (d) the BS scans the directions.</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>FIGURE 6. The overlapped beam patterns adopted in the first (a) and second (b) stages when $K = 3$.</figcaption></figure>

concept. This concept reduces the time slots needed to estimate the channel by $\frac{K^2}{\log_2(K+1)}$ compared to [71], with a slight degradation in performance. This small sacrifice in degradation can be accepted when tracking fast changing channels is required. Indeed, the concept is to use a smaller number of overlapped beam patterns for the same number of sub-ranges to estimate the channel, instead of assigning one beam pattern for each sub-range as in [71] in which the number of beam patterns should be equal to the number of sub-ranges. The algorithm is illustrated in Figure 6.

## 4) OPEN-LOOP APPROACH

An *open-loop* channel estimation technique that doesn't need a feedback loop was proposed in [104]. The algorithm is provided for an SU mmWave system, and is based on CS techniques. As for many channel estimation algorithms, the authors used the OMP and least-squares estimation (LSE) techniques to perform the estimation of the AoAs/AoDs and gains, respectively. However, they also proposed the adaptive MG-OMP, this new version of OMP enhances the estimates of AoAs/AoDs by refining these estimates just around the regions where the AoAs/AoDs are present which reduces MG-OMP complexity over OMP. Through their computer simulations, the authors showed that CS techniques, OMP and MG-OMP, outperforms the LSE ones. In addition, the complexity reduction gained by the MG-OMP was estimated.

## 5) MODE-BY-MODE APPROACH

Another scheme was proposed in [132] to estimate a temporally correlated NLOS mmWave MIMO channel. The

authors, first modified the parametric channel model to an evolution temporally correlated MIMO channel model to successfully track the channel variations. The system is based on the TDD correlation statistics and it exploits the reciprocity of the channel. The proposed algorithm updates each column of the analog precoder and combiner, this approach is called the *mode-by-mode* approach, in which each mode represents one column of the analog precoder and combiner. For each mode, a codebook is built using a group of rotation matrices that rotate the previous mode to reconstruct the new mode, then the algorithm chooses the codeword that maximizes the received power. The digital precoder and combiner are then constructed using conventional pilot-aided estimation of the effective channel.

## 6) APERTURE SHAPING APPROACH

In [88], Schniter and Sayeed proposed a technique termed as *aperture shaping* to enhance the sparsity of the virtual MIMO channel. Briefly, aperture shaping is performed by applying a fixed gain at each antenna at both the TX and RX. The shaping coefficients are optimised to maximize the signal-to-interference ratio. In addition, they implemented a mmWave system that uses modulation and demodulation techniques based on fast Fourier transform (FFT) to further expose the channel sparsity. Finally, waterfilling technique and Lanczos algorithm were employed to design spectrally efficient precoding and decoding. They solved the sparsity problem using LASSO [102], where it was shown that their procedure approaches the perfect-CSI capacity for a mmWave system.

## 7) AAVE APPROACH

Based on the same assumptions as in [71], the authors in [113] proposed a new concept called antenna array with virtual elements (AAVE), that extends the real antenna arrays at both the transmitter and receiver to a new one by appending some $A_v$ virtual antennas without affecting the physical array. The idea behind AAVE is to add some virtual antennas to the real physical antennas at both the BS and MT, in order to enhance the angular estimation resolution. However, the scheme assures that no data is sent over the virtual antennas to guarantee that no physical change has been introduced to the antenna arrays. Based on AAVE, [113]
develops a CS angle estimation method with less overhead and delay than [71]. The proposed method with AAVE can obtain a resolution of $O(1/A_v)$ which is better than $O(1/N_G)$ in [71], [134], [140].

## 8) 2D DFT APPROACH

In [135], a mmWave channel estimation method was proposed based on the 2D DFT. The estimation of the channel parameters is done using the iterative cancellation method. In detail, the approach estimates the channel parameters for each path using the DFT samples, after canceling the previous estimated parameters in each iteration. This technique requires a training sequence of $N_{TS} = \lceil \frac{N_D}{M_T} \rceil \lfloor \frac{N_D}{M_B} \rfloor M_B$ time slots, where $N_D$ and $M_D$ are two suitable integer parameters to be chosen depending on the length $N_{TS}$ of the training sequence. The method showed low complexity and performed close to a known channel system. Another channel estimation algorithm for a DL SU mmWave system was proposed in [136] based on the DFT algorithm in [135]. The algorithm uses exactly the same DFT technique provided in [135], while adopting a feedback to the BS to improve the estimation accuracy. The algorithm is illustrated as follows: First, the BS sends its training sequence to the MT, the MT estimates the channel using the DFT technique, and determines if the estimation is accurate based on an adjustable threshold. Second, the MT returns YES or NO to the BS depending on the accuracy of its estimation. Third, the BS readjusts the length of the training sequence adaptively according to the feedback.

## 9) SWITCHES APPROACH; SVD AVOIDANCE APPROACH

In [114], the authors proposed a new hybrid architecture for a DL SU mmWave system using switches as an alternative for phase shifters, to reduce cost, complexity and power consumption, especially at MTs where these parameters are crucial. Figure 7 depicts the proposed hybrid architecture which is slightly different from the aforementioned one as it was presented in Figure 3(b). A new CS-based channel estimation algorithm was also developed. A multiple measurement vector OMP (M-OMP) in [141] was considered instead of the single measurement vector OMP. Phase shifters architecture versus switches architecture were compared, the results showed a slight better performance of the architecture implementing switches over phase shifters one. A modified version of the aforementioned algorithm was proposed in [115] for an SU mmWave system. The authors intended to reduce the computational complexity and feedback overhead to the TX. The algorithm is also based on the OMP, but unlike to [114] it avoids the computation of the channel SVD at the RX by exploiting the orthogonality between the array propagation vectors. In contrast to [114], where the whole reconstructed precoder is sent back to the TX through the feedback link, the algorithm in [115] diminishes the feedback overhead, since it aims to reconstruct the precoder at the TX, after obtaining the codebook indices through the feedback link.

<figure><img src="image_8.png" /><figcaption>FIGURE 7. Hybrid mmWave MT architecture implementing switches instead of phase shifters.</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>FIGURE 8. The two-stage asymmetric estimation approach in [116].</figcaption></figure>

## 10) TWO-STAGE ASYMMETRIC APPROACH

An asymmetric channel estimation approach was proposed in [116] for a DL multi-user (MU) mmWave system. The MU system is characterized by a hybrid BS and analog-only beamforming at the MTs with one RF chain for each MT. The proposed algorithm is a two-stage asymmetric approach, an exhaustive search stage, followed by a CS estimation stage. At the first phase, the BS sends the omni-directional training signals to the MTs, the MTs search in exhaustive manner for the best combining vectors to find the AoA. Following the first phase, a second phase of CS estimation is performed, where the MTs fix their receiving beams to the best ones found at the first phase, while utilizing CS tools to estimate their channel parameters. The two-stage algorithm is depicted in Figure 8.

## 11) SPATIAL COVARIANCE APPROACH

In [117], a channel estimation algorithms based on estimating the spatial covariance for a TDD UL SU mmWave channel were proposed. Unlike the two-step approach, where first it is needed to estimate the channel, followed by a second step of channel covariance calculation. The authors proposed to estimate the channel covariance directly, and without estimating the channel explicitly. To overcome the need of estimating the channel explicitly, the authors exploited the Hermitian property of the spatial covariance channel matrix. A sparse Hermitian matrix is shown in Figure 9(c). Furthermore, the authors designed the covariance OMP (C-OMP) algorithm, which is based on the OMP and simultaneous OMP
develops a CS angle estimation method with less overhead and delay than [71]. The proposed method with AAVE can obtain a resolution of $O(1/A_v)$ which is better than $O(1/N_G)$ in [71], [134], [140].

## 8) 2D DFT APPROACH

In [135], a mmWave channel estimation method was proposed based on the 2D DFT. The estimation of the channel parameters is done using the iterative cancellation method. In detail, the approach estimates the channel parameters for each path using the DFT samples, after canceling the previous estimated parameters in each iteration. This technique requires a training sequence of $N_{TS} = \lceil \frac{N_D}{M_T} \rceil \lfloor \frac{N_D}{M_B} \rfloor M_B$ time slots, where $N_D$ and $M_D$ are two suitable integer parameters to be chosen depending on the length $N_{TS}$ of the training sequence. The method showed low complexity and performed close to a known channel system. Another channel estimation algorithm for a DL SU mmWave system was proposed in [136] based on the DFT algorithm in [135]. The algorithm uses exactly the same DFT technique provided in [135], while adopting a feedback to the BS to improve the estimation accuracy. The algorithm is illustrated as follows: First, the BS sends its training sequence to the MT, the MT estimates the channel using the DFT technique, and determines if the estimation is accurate based on an adjustable threshold. Second, the MT returns YES or NO to the BS depending on the accuracy of its estimation. Third, the BS readjusts the length of the training sequence adaptively according to the feedback.

## 9) SWITCHES APPROACH; SVD AVOIDANCE APPROACH

In [114], the authors proposed a new hybrid architecture for a DL SU mmWave system using switches as an alternative for phase shifters, to reduce cost, complexity and power consumption, especially at MTs where these parameters are crucial. Figure 7 depicts the proposed hybrid architecture which is slightly different from the aforementioned one as it was presented in Figure 3(b). A new CS-based channel estimation algorithm was also developed. A multiple measurement vector OMP (M-OMP) in [141] was considered instead of the single measurement vector OMP. Phase shifters architecture versus switches architecture were compared, the results showed a slight better performance of the architecture implementing switches over phase shifters one. A modified version of the aforementioned algorithm was proposed in [115] for an SU mmWave system. The authors intended to reduce the computational complexity and feedback overhead to the TX. The algorithm is also based on the OMP, but unlike to [114] it avoids the computation of the channel SVD at the RX by exploiting the orthogonality between the array propagation vectors. In contrast to [114], where the whole reconstructed precoder is sent back to the TX through the feedback link, the algorithm in [115] diminishes the feedback overhead, since it aims to reconstruct the precoder at the TX, after obtaining the codebook indices through the feedback link.

<figure><img src="image_8.png" /><figcaption>FIGURE 7. Hybrid mmWave MT architecture implementing switches instead of phase shifters.</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>FIGURE 8. The two-stage asymmetric estimation approach in [116].</figcaption></figure>

## 10) TWO-STAGE ASYMMETRIC APPROACH

An asymmetric channel estimation approach was proposed in [116] for a DL multi-user (MU) mmWave system. The MU system is characterized by a hybrid BS and analog-only beamforming at the MTs with one RF chain for each MT. The proposed algorithm is a two-stage asymmetric approach, an exhaustive search stage, followed by a CS estimation stage. At the first phase, the BS sends the omni-directional training signals to the MTs, the MTs search in exhaustive manner for the best combining vectors to find the AoA. Following the first phase, a second phase of CS estimation is performed, where the MTs fix their receiving beams to the best ones found at the first phase, while utilizing CS tools to estimate their channel parameters. The two-stage algorithm is depicted in Figure 8.

## 11) SPATIAL COVARIANCE APPROACH

In [117], a channel estimation algorithms based on estimating the spatial covariance for a TDD UL SU mmWave channel were proposed. Unlike the two-step approach, where first it is needed to estimate the channel, followed by a second step of channel covariance calculation. The authors proposed to estimate the channel covariance directly, and without estimating the channel explicitly. To overcome the need of estimating the channel explicitly, the authors exploited the Hermitian property of the spatial covariance channel matrix. A sparse Hermitian matrix is shown in Figure 9(c). Furthermore, the authors designed the covariance OMP (C-OMP) algorithm, which is based on the OMP and simultaneous OMP
<table><caption>TABLE 4. Channel estimation algorithms for wideband hybrid mmWave systems.</caption><thead><tr><th>Reference</th><th>Year</th><th>Scenario</th><th>UL/DL</th><th>2D/3D</th><th>Complexity</th><th>Description</th></tr></thead><tbody><tr><td>Venugopal et al. [118]</td><td>2017</td><td>Single-user or multi-user</td><td>DL</td><td>2D</td><td>Not specified</td><td>CS, LSE, high complexity</td></tr><tr><td>Venugopal et al. [119]</td><td>2017</td><td>Single-user or multi-user</td><td>Not specified</td><td>2D or 3D</td><td>Not specified</td><td>OMP, DGMP, LSE, MMSE, SC-FDE or OFDM are considered</td></tr><tr><td>Rodriguez-fernandez et al. [120]</td><td>2017</td><td>Single-user</td><td>Not specified</td><td>2D</td><td>Not specified</td><td>OMP, MSE, LSE, exploits the common support between the subcarriers to reduce complexity</td></tr><tr><td>Rodriguez-fernandez et al. [121]</td><td>2018</td><td>Not specified</td><td>Not specified</td><td>2D</td><td>SW-OMP:<br>O(M<sub>T</sub>T(K<sub>s</sub>(N<sub>G1</sub>N<sub>G2</sub> - (j - 1))));<br>SS-SW-OMP+Th:<br>O(M<sub>T</sub>T(K<sub>p</sub>(N<sub>G1</sub>N<sub>G2</sub> - (j - 1)))))</td><td>SW-OMP, SS-SW-OMP+Th, provides a trade-off between complexity and achievable rate</td></tr><tr><td>Gao et al. [122]</td><td>2016</td><td>Multi-user</td><td>UL</td><td>2D</td><td>Not specified</td><td>DGMP, exploits the angle-domain sparsity of frequency-selective fading channels</td></tr><tr><td>Zhou et al. [143]</td><td>2017</td><td>Multi-user</td><td>DL</td><td>2D</td><td>O(T'TK<sub>s</sub>)</td><td>CP, higher estimation accuracy compared to OMP</td></tr><tr><td>Araujo et al. [124]</td><td>2014</td><td>Single-user</td><td>DL</td><td>3D</td><td>O(log(N<sub>B</sub>))</td><td>OMP, search region, a coarse stage followed by a refinement stage</td></tr><tr><td>Gonzalez-coma et al. [74]</td><td>2018</td><td>Multi-user</td><td>UL</td><td>2D</td><td>O(M<sub>T</sub>T(K<sub>s</sub>(N<sub>G1</sub>N<sub>G2</sub> - (j - 1)))))</td><td>SW-OMP</td></tr></tbody></table>

<figure><img src="image_10.png" /><figcaption>FIGURE 9. Different sparse matrix types (a) Unstructured sparse matrix (b) Sparse rows matrix (c) Sparse Hermitian matrix.</figcaption></figure>

(S-OMP), to exploit this property and to estimate the covariance of the channel. C-OMP employs a quadratic form in its covariance calculations unlike OMP and S-OMP where a linear form is used. As known, for a perfect recovery using CS techniques, many measurements are required especially for time-varying channels. Accordingly, the authors also proposed dynamic S-OMP (DS-OMP), which was inspired by S-OMP, to mitigate the number of measurements, similarly, dynamic C-OMP (DC-OMP) was introduced. Both DS-OMP and DC-OMP can be applied to a time-varying analog combining matrix in a time-varying channel.

## 12) OVER-COMPLETE DICTIONARY APPROACH

Another channel estimation algorithm for a fully-connected UL SU mmWave system has been developed in [139]. The authors described the channel estimation problem by *dimension-deficient* which results from the fact that the number of RF chains is a lot smaller than the number of antennas, hence the received signal does not contain full CSI. The authors treated this issue by adopting an adaptive over-complete dictionary, then by estimating the channel parameters using CoSaMP which improves the OMP technique by reducing the influence of accidental errors. The proposed algorithm showed more robustness against noise, better performance compared with non-adaptive CS techniques, with spectral efficiency close to perfect CSI.

## 13) OTHER APPROACHES

Some works tried to compete with CS algorithms in estimating the mmWave channel. Based on a technique

termed CANDECOMP/PARAFAC (CP) decomposition, an interesting method was carried out in [137] and was also compared to some CS techniques. In this regard, the UL MU mmWave channel can be estimated by means of the CP decomposition method. This procedure also referred to as tensor rank decomposition, can be viewed as a generalisation of the matrix singular value decomposition to tensors. Please refer to [142] for more details. The authors stated that the CP method can be advantageous when compared to CS techniques in terms of computational complexity due to the utilization of tensors. In addition, while it is somewhat troublesome to examine the right recovery condition for generic dictionaries, this is not the case for the CP method, hence it is more easier to analyze and find the exact size of training overhead. Moreover, unlike CS methods, the CP doesn't require the quantization of the continuous parameter space, consequently, no grid quantization errors. The overall computational complexity of the proposed CP method was shown to be $O(N_{G1}^2N_{G2}^2+T'TM_B)$, where T and T' are respectively the number of frames and sub-frames (each frame is divided into a number of sub-frames), and $N_{G2}$ and $N_{G1}$ are respectively the search grid sizes within the considered beamforming sectors. Another contribution to estimate an SU mmWave channel was achieved in [138] based on a 2D beamspace multiple signal classification (MUSIC) method. The MUSIC method is used to estimate the path directions while the LSE one estimates the path gains. The suggested algorithm reduces the computational overhead by exploiting the large-scale fading property of the path directions which are believed to remain unchanged for each frame according to the measurement results obtained in [29]. Hence the costly computation of path directions is executed only once per each frame. The main computational complexity analysis of the algorithm brings out two major tasks (i) the eigenvalue decomposition whose complexity is $O(M_B^3M_T^3)$ (ii) the grid search whose complexity is $O(N_{G2}N_{G1}M_B^2M_T^2)$.

## B. TECHNIQUES FOR WIDEBAND SYSTEMS

Most of the work done on mmWave channels estimation assumed the narrowband case. Such assumption was found
<table><caption>TABLE 4. Channel estimation algorithms for wideband hybrid mmWave systems.</caption><thead><tr><th>Reference</th><th>Year</th><th>Scenario</th><th>UL/DL</th><th>2D/3D</th><th>Complexity</th><th>Description</th></tr></thead><tbody><tr><td>Venugopal et al. [118]</td><td>2017</td><td>Single-user or multi-user</td><td>DL</td><td>2D</td><td>Not specified</td><td>CS, LSE, high complexity</td></tr><tr><td>Venugopal et al. [119]</td><td>2017</td><td>Single-user or multi-user</td><td>Not specified</td><td>2D or 3D</td><td>Not specified</td><td>OMP, DGMP, LSE, MMSE, SC-FDE or OFDM are considered</td></tr><tr><td>Rodriguez-fernandez et al. [120]</td><td>2017</td><td>Single-user</td><td>Not specified</td><td>2D</td><td>Not specified</td><td>OMP, MSE, LSE, exploits the common support between the subcarriers to reduce complexity</td></tr><tr><td>Rodriguez-fernandez et al. [121]</td><td>2018</td><td>Not specified</td><td>Not specified</td><td>2D</td><td>SW-OMP:<br>O(M<sub>T</sub>T(K<sub>s</sub>(N<sub>G1</sub>N<sub>G2</sub> - (j - 1))));<br>SS-SW-OMP+Th:<br>O(M<sub>T</sub>T(K<sub>p</sub>(N<sub>G1</sub>N<sub>G2</sub> - (j - 1)))))</td><td>SW-OMP, SS-SW-OMP+Th, provides a trade-off between complexity and achievable rate</td></tr><tr><td>Gao et al. [122]</td><td>2016</td><td>Multi-user</td><td>UL</td><td>2D</td><td>Not specified</td><td>DGMP, exploits the angle-domain sparsity of frequency-selective fading channels</td></tr><tr><td>Zhou et al. [143]</td><td>2017</td><td>Multi-user</td><td>DL</td><td>2D</td><td>O(T'TK<sub>s</sub>)</td><td>CP, higher estimation accuracy compared to OMP</td></tr><tr><td>Araujo et al. [124]</td><td>2014</td><td>Single-user</td><td>DL</td><td>3D</td><td>O(log(N<sub>B</sub>))</td><td>OMP, search region, a coarse stage followed by a refinement stage</td></tr><tr><td>Gonzalez-coma et al. [74]</td><td>2018</td><td>Multi-user</td><td>UL</td><td>2D</td><td>O(M<sub>T</sub>T(K<sub>s</sub>(N<sub>G1</sub>N<sub>G2</sub> - (j - 1)))))</td><td>SW-OMP</td></tr></tbody></table>

<figure><img src="image_10.png" /><figcaption>FIGURE 9. Different sparse matrix types (a) Unstructured sparse matrix (b) Sparse rows matrix (c) Sparse Hermitian matrix.</figcaption></figure>

(S-OMP), to exploit this property and to estimate the covariance of the channel. C-OMP employs a quadratic form in its covariance calculations unlike OMP and S-OMP where a linear form is used. As known, for a perfect recovery using CS techniques, many measurements are required especially for time-varying channels. Accordingly, the authors also proposed dynamic S-OMP (DS-OMP), which was inspired by S-OMP, to mitigate the number of measurements, similarly, dynamic C-OMP (DC-OMP) was introduced. Both DS-OMP and DC-OMP can be applied to a time-varying analog combining matrix in a time-varying channel.

## 12) OVER-COMPLETE DICTIONARY APPROACH

Another channel estimation algorithm for a fully-connected UL SU mmWave system has been developed in [139]. The authors described the channel estimation problem by dimension-deficient which results from the fact that the number of RF chains is a lot smaller than the number of antennas, hence the received signal does not contain full CSI. The authors treated this issue by adopting an adaptive over-complete dictionary, then by estimating the channel parameters using CoSaMP which improves the OMP technique by reducing the influence of accidental errors. The proposed algorithm showed more robustness against noise, better performance compared with non-adaptive CS techniques, with spectral efficiency close to perfect CSI.

## 13) OTHER APPROACHES

Some works tried to compete with CS algorithms in estimating the mmWave channel. Based on a technique

termed CANDECOMP/PARAFAC (CP) decomposition, an interesting method was carried out in [137] and was also compared to some CS techniques. In this regard, the UL MU mmWave channel can be estimated by means of the CP decomposition method. This procedure also referred to as tensor rank decomposition, can be viewed as a generalisation of the matrix singular value decomposition to tensors. Please refer to [142] for more details. The authors stated that the CP method can be advantageous when compared to CS techniques in terms of computational complexity due to the utilization of tensors. In addition, while it is somewhat troublesome to examine the right recovery condition for generic dictionaries, this is not the case for the CP method, hence it is more easier to analyze and find the exact size of training overhead. Moreover, unlike CS methods, the CP doesn't require the quantization of the continuous parameter space, consequently, no grid quantization errors. The overall computational complexity of the proposed CP method was shown to be $O(N_{G1}^2N_{G2}^2+T'TM_B)$, where T and T' are respectively the number of frames and sub-frames (each frame is divided into a number of sub-frames), and $N_{G2}$ and $N_{G1}$ are respectively the search grid sizes within the considered beamforming sectors. Another contribution to estimate an SU mmWave channel was achieved in [138] based on a 2D beamspace multiple signal classification (MUSIC) method. The MUSIC method is used to estimate the path directions while the LSE one estimates the path gains. The suggested algorithm reduces the computational overhead by exploiting the large-scale fading property of the path directions which are believed to remain unchanged for each frame according to the measurement results obtained in [29]. Hence the costly computation of path directions is executed only once per each frame. The main computational complexity analysis of the algorithm brings out two major tasks (i) the eigenvalue decomposition whose complexity is $O(M_B^3M_T^3)$ (ii) the grid search whose complexity is $O(N_{G2}N_{G1}M_B^2M_T^2)$.

## B. TECHNIQUES FOR WIDEBAND SYSTEMS

Most of the work done on mmWave channels estimation assumed the narrowband case. Such assumption was found
to be not realistic for aforementioned channels, but was considered as a first step in the development of mmWave channel estimation. MmWave channels are wideband channels in reality, and to be useful, channel estimation algorithms have to take this into consideration. Recently, some works have been proposed to estimate the frequency-selective channels for mmWave hybrid systems, we review these works in the following. Table 4 provides a list of these research works.

## 1) TIME-DOMAIN APPROACH

In [118] a time-domain channel estimation algorithm was proposed for both single- and multi-user fully-connected hybrid single-carrier mmWave systems. The authors had taken into account the bandlimiting filter and the time required to reconfigure RF circuits. The proposed algorithm is based on CS to estimate the AoAs/AoDs, and on the LSE or the minimum mean square error (MMSE) to estimate the path gains. Simulation results showed that the proposed algorithm provides low estimation error using small training overhead, but the main drawback of this algorithm lies into its complexity.

## 2) FREQUENCY-DOMAIN APPROACHES

In [121], two frequency-domain channel estimation algorithms for a fully-connected OFDM mmWave system were proposed based on the S-OMP method [144] and tested on real frequency-selective channel models. The proposed algorithms provide a trade-off between complexity and achievable rate, and consider the effects of the bandlimiting filter and the time required to reconfigure RF circuits. The first algorithm termed as the simultaneous weighted OMP (SW-OMP) provides the best performance compared to the second one since it exploits the information on the support coming from every subcarrier of the OFDM system. While the second algorithm termed as subcarrier selection SW-OMP + thresholding (SS-SW-OMP+Th) aims to exploit information from a reduced number of subcarriers, hence it provides a lower complexity compared to SW-OMP. Both algorithms achieve the Cramer-Rao lower bound. Comparisons were done for the proposed algorithms and other frequency-selective mmWave channel estimation algorithms including structured sparsity adaptive matching pursuit (SSAMP) [145] and distributed grid matching pursuit (DGMP) [122]. The simulation results show a very good performance of the proposed algorithms.

Another frequency-domain algorithm was developed in [120], for a fully-connected OFDM mmWave system. The algorithm was compared with the time-domain algorithm proposed in [118], where it provides the same estimation error as the time-domain one, however with a lesser complexity. The authors opted to use OMP and LSE techniques to estimate the sparse channel, and exploited the common support between the subcarriers to gain more reduction in estimation complexity. In addition, they proposed to refine more the estimates using an algorithm they termed as the joint channel estimation + local search (JCE+LS), which is

based on minimum square error (MSE) to know whether an improvement of the estimates is needed or not.

In [122], a frequency-domain CS-based UL MU channel estimation algorithm was proposed for OFDM mmWave system. The proposed DGMP algorithm aims to exploit the angle-domain sparsity of frequency-selective fading channels, and solves the problem of the leakage power caused by the continuous AoA/AoD. The simulation results showed a good performance of the proposed algorithm.

Based on the SW-OMP algorithm developed in [121], the authors in [74] proposed a joint UL MU channel estimation method for an OFDM mmWave system. Afterwards, the UL channel estimates and the reciprocity of TDD scheme were exploited to jointly design the precoders and combiners in the DL.

## 3) TWO-STAGE APPROACH

A two-stage channel estimation algorithm has been proposed in [123] for an indoor SU DL mmWave system, implementing FDD and OFDM, and operating at 60 GHz. The estimation algorithm is based on two stages, a coarse stage followed by an amelioration stage. The first stage of the algorithm is performed based on the OMP CS technique, which estimates the parameters of the channel coarsely. Subsequently, the second stage is carried out to refine the estimates based on the maximization of the energy of the received signal and using the so-called *search region* algorithm as described in the paper. The proposed method can achieve a low pilot overhead of $O(\log N_B)$ compared to the traditional LSE that requires a length of $O(N_B)$, and shows a quite well performance.

## 4) OTHER APPROACHES

The authors in [119] developed channel estimation algorithms in the frequency-domain, in the time-domain, as well as in the combined time/frequency domain. Both single- and multi-user fully-connected hybrid mmWave systems implementing either single carrier-frequency domain equalization (SC-FDE) or OFDM are considered. The estimators are based on CS, where OMP is the dominant technique used to estimate AoAs/AoDs, while LSE is used to estimate channel gains, whether for the time-domain or the frequency-domain. However, for the combined time/frequency domain, OMP or DGMP [122] are used to estimate AoAs/AoDs, and LSE or MMSE to recover the gains. The results showed a good error performance with low overhead, and a further reduction in complexity offered by the combined time/frequency algorithm.

Similarly to the CANDECOMP/PARAFAC narrowband mmWave channel estimation algorithm proposed in [137], an MU DL channel estimation technique for OFDM mmWave systems was proposed in [143]. The authors developed the Cramer-Rao bound results to describe the best asymptotically achievable performance of the algorithm, then they compared the proposed algorithm to the OMP method, where it was
to be not realistic for aforementioned channels, but was considered as a first step in the development of mmWave channel estimation. MmWave channels are wideband channels in reality, and to be useful, channel estimation algorithms have to take this into consideration. Recently, some works have been proposed to estimate the frequency-selective channels for mmWave hybrid systems, we review these works in the following. Table 4 provides a list of these research works.

## 1) TIME-DOMAIN APPROACH

In [118] a time-domain channel estimation algorithm was proposed for both single- and multi-user fully-connected hybrid single-carrier mmWave systems. The authors had taken into account the bandlimiting filter and the time required to reconfigure RF circuits. The proposed algorithm is based on CS to estimate the AoAs/AoDs, and on the LSE or the minimum mean square error (MMSE) to estimate the path gains. Simulation results showed that the proposed algorithm provides low estimation error using small training overhead, but the main drawback of this algorithm lies into its complexity.

## 2) FREQUENCY-DOMAIN APPROACHES

In [121], two frequency-domain channel estimation algorithms for a fully-connected OFDM mmWave system were proposed based on the S-OMP method [144] and tested on real frequency-selective channel models. The proposed algorithms provide a trade-off between complexity and achievable rate, and consider the effects of the bandlimiting filter and the time required to reconfigure RF circuits. The first algorithm termed as the simultaneous weighted OMP (SW-OMP) provides the best performance compared to the second one since it exploits the information on the support coming from every subcarrier of the OFDM system. While the second algorithm termed as subcarrier selection SW-OMP + thresholding (SS-SW-OMP+Th) aims to exploit information from a reduced number of subcarriers, hence it provides a lower complexity compared to SW-OMP. Both algorithms achieve the Cramer-Rao lower bound. Comparisons were done for the proposed algorithms and other frequency-selective mmWave channel estimation algorithms including structured sparsity adaptive matching pursuit (SSAMP) [145] and distributed grid matching pursuit (DGMP) [122]. The simulation results show a very good performance of the proposed algorithms.

Another frequency-domain algorithm was developed in [120], for a fully-connected OFDM mmWave system. The algorithm was compared with the time-domain algorithm proposed in [118], where it provides the same estimation error as the time-domain one, however with a lesser complexity. The authors opted to use OMP and LSE techniques to estimate the sparse channel, and exploited the common support between the subcarriers to gain more reduction in estimation complexity. In addition, they proposed to refine more the estimates using an algorithm they termed as the joint channel estimation + local search (JCE+LS), which is

based on minimum square error (MSE) to know whether an improvement of the estimates is needed or not.

In [122], a frequency-domain CS-based UL MU channel estimation algorithm was proposed for OFDM mmWave system. The proposed DGMP algorithm aims to exploit the angle-domain sparsity of frequency-selective fading channels, and solves the problem of the leakage power caused by the continuous AoA/AoD. The simulation results showed a good performance of the proposed algorithm.

Based on the SW-OMP algorithm developed in [121], the authors in [74] proposed a joint UL MU channel estimation method for an OFDM mmWave system. Afterwards, the UL channel estimates and the reciprocity of TDD scheme were exploited to jointly design the precoders and combiners in the DL.

## 3) TWO-STAGE APPROACH

A two-stage channel estimation algorithm has been proposed in [123] for an indoor SU DL mmWave system, implementing FDD and OFDM, and operating at 60 GHz. The estimation algorithm is based on two stages, a coarse stage followed by an amelioration stage. The first stage of the algorithm is performed based on the OMP CS technique, which estimates the parameters of the channel coarsely. Subsequently, the second stage is carried out to refine the estimates based on the maximization of the energy of the received signal and using the so-called *search region* algorithm as described in the paper. The proposed method can achieve a low pilot overhead of $O(\log N_B)$ compared to the traditional LSE that requires a length of $O(N_B)$, and shows a quite well performance.

## 4) OTHER APPROACHES

The authors in [119] developed channel estimation algorithms in the frequency-domain, in the time-domain, as well as in the combined time/frequency domain. Both single- and multi-user fully-connected hybrid mmWave systems implementing either single carrier-frequency domain equalization (SC-FDE) or OFDM are considered. The estimators are based on CS, where OMP is the dominant technique used to estimate AoAs/AoDs, while LSE is used to estimate channel gains, whether for the time-domain or the frequency-domain. However, for the combined time/frequency domain, OMP or DGMP [122] are used to estimate AoAs/AoDs, and LSE or MMSE to recover the gains. The results showed a good error performance with low overhead, and a further reduction in complexity offered by the combined time/frequency algorithm.

Similarly to the CANDECOMP/PARAFAC narrowband mmWave channel estimation algorithm proposed in [137], an MU DL channel estimation technique for OFDM mmWave systems was proposed in [143]. The authors developed the Cramer-Rao bound results to describe the best asymptotically achievable performance of the algorithm, then they compared the proposed algorithm to the OMP method, where it was
shown to provide a complexity similar to the OMP method, but with higher estimation accuracy.

**Conclusion:** The usage of analog-only beamforming for channel estimation in mmWave systems is constrained by many limitations, which can be avoided using the hybrid architecture. In this section we have seen different channel estimation techniques for both narrowband and wideband hybrid mmWave systems. It can be said that there exist plenty of research works on channel estimation for the narrowband case, while only few papers have been proposed for the wideband one. It is known that mmWave channels are wideband in nature, hence some further future steps have to be taken in this regard. Clearly the aim of most of the work that has been delivered so far for hybrid architecture was for 2D channel estimation, this work should be extended to the 3D case also.

# VI. CHANNEL ESTIMATION TECHNIQUES FOR SYSTEMS WITH LENS ANTENNA ARRAY

In this section, we focus on the channel estimation problem for the mmWave massive MIMO systems with lens antenna array. Under this assumption, the conventional channel estimation techniques for fully-digital architectures are obviously not applicable since the number of RF chains is limited. In addition, the above presented channel estimation techniques for hybrid architectures are also unsuitable for lens antenna arrays mmWave systems for mainly two reasons. The first one is related to the simple switching network employed at the BS and/or MT to simplify the analog beamformers of the hybrid schemes where the RF chains are connected with all the antennas with phase shifters and power splitters. The simplicity of the analog precoding/combining circuits in mmWave systems with lens antenna array comes on the price of stronger constraint when compared to the constant modulus one of the phase shifters network [83], [86]. Secondly, lens antenna arrays are able to focus the signal with distinct AoAs onto different antennas, contrary to the ULPA where the energy is distributed uniformly on all the antennas of the array. That is why several research works tries to develop tailored channel estimation techniques for mmWave systems with lens antenna array in order to exploit its focusing-energy feature [83], [86], [146]–[150].

The authors in [148] considered a narrowband model for the UL MU mmWave channel consisting of a massive-MIMO BS and *U* single-antenna MTs. The beamspace channel is estimated via a sparsity mask detection method where a beam training procedure between the BS and the *U* users is first employed to determine which beams with large power are simultaneously used by the BS; this defines the sparsity mask [151], [152]. Then, the dimension of the beamspace channel is reduced, and conventional algorithms are used to estimate the dimension-reduced channel. However, the BS overhead is still proportional to the dimension of its antenna array, which is large.

In [149], a BS equipped with a 2D lens array and a zero forcing (ZF) precoder, communicating with *U*

single-antenna MTs is considered. The beams that will be employed by the BS during the data transmission are identified via a beam selection technique. This research work applied three different criteria to select beams, namely the magnitude of the path [151], the signal-to-interference-plus-noise ratio (SINR) at the MT side [153], the capacity of the full system [154]. The authors demonstrated that the maximum magnitude beam selection scheme can achieve near-optimal performances in both line-of-sight (LOS) and multi-path scenarios. The complexity and the overhead of the above presented methods can be reduced by utilizing CS tools [94], [155], [156] which can further exploit the sparsity of beamspace channel.

# 5) SUPPORT-DETECTION-BASED CHANNEL ESTIMATION

In [83], the beamspace channel estimation problem in a TDD system with lens antenna array was investigated. A mmWave massive MIMO BS equipped with $N_B$ antennas and $M_B$ RF chains communicates simultaneously with *U* single-antenna users. Under the channel reciprocity assumption, the authors proposed to estimate the uplink channel which is just a transposition of the downlink one. They adopted the DFT model to describe the channel, i.e., the lens antenna array plays the role of a spatial DFT matrix which contains the array steering vectors of different orthogonal directions. CS tools were used to propose a support detection (SD)-based beamspace channel estimation with low pilot overhead. Before that a strategy for pilot transmission and adaptive selection network was designed. Thereafter, we will briefly present the three phases of this channel estimation method.

*   **Pilot Transmission:** Known mutually orthogonal pilot sequences are transmitted by the *U* users to the BS over a certain duration lesser than the channel coherence time such as the beamspace channel remains unchanged during its estimation time.

*   **Adaptive selecting network:** During the pilot transmission, the BS combines the received UL signal via an adaptive selecting network which consists of a small number of 1-bit phase shifters (equivalent to switches in terms of simplicity and energy consumption) to efficiently measure the beamspace channel. CS tools were employed to design a Bernoulli random matrix adaptive combiner such that the required recovery accuracy is guaranteed. The application of the proposed adaptive selecting network makes it possible to formulate the beamspace channel estimation problem as a sparse recovery problem, which significantly reduces the number of required pilot symbols.

*   **SD-based channel estimation scheme:** The channel estimation problem is decomposed into a series of sub-problems where only one sparse channel component is considered for each sub-problem. The idea is to iteratively detect the support (i.e., the index set of non-zero elements) in a sparse vector containing the information of a specific propagation direction at each
shown to provide a complexity similar to the OMP method, but with higher estimation accuracy.

**Conclusion:** The usage of analog-only beamforming for channel estimation in mmWave systems is constrained by many limitations, which can be avoided using the hybrid architecture. In this section we have seen different channel estimation techniques for both narrowband and wideband hybrid mmWave systems. It can be said that there exist plenty of research works on channel estimation for the narrowband case, while only few papers have been proposed for the wideband one. It is known that mmWave channels are wideband in nature, hence some further future steps have to be taken in this regard. Clearly the aim of most of the work that has been delivered so far for hybrid architecture was for 2D channel estimation, this work should be extended to the 3D case also.

## VI. CHANNEL ESTIMATION TECHNIQUES FOR SYSTEMS WITH LENS ANTENNA ARRAY

In this section, we focus on the channel estimation problem for the mmWave massive MIMO systems with lens antenna array. Under this assumption, the conventional channel estimation techniques for fully-digital architectures are obviously not applicable since the number of RF chains is limited. In addition, the above presented channel estimation techniques for hybrid architectures are also unsuitable for lens antenna arrays mmWave systems for mainly two reasons. The first one is related to the simple switching network employed at the BS and/or MT to simplify the analog beamformers of the hybrid schemes where the RF chains are connected with all the antennas with phase shifters and power splitters. The simplicity of the analog precoding/combining circuits in mmWave systems with lens antenna array comes on the price of stronger constraint when compared to the constant modulus one of the phase shifters network [83], [86]. Secondly, lens antenna arrays are able to focus the signal with distinct AoAs onto different antennas, contrary to the ULPA where the energy is distributed uniformly on all the antennas of the array. That is why several research works tries to develop tailored channel estimation techniques for mmWave systems with lens antenna array in order to exploit its focusing-energy feature [83], [86], [146]–[150].

The authors in [148] considered a narrowband model for the UL MU mmWave channel consisting of a massive-MIMO BS and *U* single-antenna MTs. The beamspace channel is estimated via a sparsity mask detection method where a beam training procedure between the BS and the *U* users is first employed to determine which beams with large power are simultaneously used by the BS; this defines the sparsity mask [151], [152]. Then, the dimension of the beamspace channel is reduced, and conventional algorithms are used to estimate the dimension-reduced channel. However, the BS overhead is still proportional to the dimension of its antenna array, which is large.

In [149], a BS equipped with a 2D lens array and a zero forcing (ZF) precoder, communicating with *U*

single-antenna MTs is considered. The beams that will be employed by the BS during the data transmission are identified via a beam selection technique. This research work applied three different criteria to select beams, namely the magnitude of the path [151], the signal-to-interference-plus-noise ratio (SINR) at the MT side [153], the capacity of the full system [154]. The authors demonstrated that the maximum magnitude beam selection scheme can achieve near-optimal performances in both line-of-sight (LOS) and multi-path scenarios. The complexity and the overhead of the above presented methods can be reduced by utilizing CS tools [94], [155], [156] which can further exploit the sparsity of beamspace channel.

## 5) SUPPORT-DETECTION-BASED CHANNEL ESTIMATION

In [83], the beamspace channel estimation problem in a TDD system with lens antenna array was investigated. A mmWave massive MIMO BS equipped with $N_B$ antennas and $M_B$ RF chains communicates simultaneously with *U* single-antenna users. Under the channel reciprocity assumption, the authors proposed to estimate the uplink channel which is just a transposition of the downlink one. They adopted the DFT model to describe the channel, i.e., the lens antenna array plays the role of a spatial DFT matrix which contains the array steering vectors of different orthogonal directions. CS tools were used to propose a support detection (SD)-based beamspace channel estimation with low pilot overhead. Before that a strategy for pilot transmission and adaptive selection network was designed. Thereafter, we will briefly present the three phases of this channel estimation method.

*   **Pilot Transmission:** Known mutually orthogonal pilot sequences are transmitted by the *U* users to the BS over a certain duration lesser than the channel coherence time such as the beamspace channel remains unchanged during its estimation time.

*   **Adaptive selecting network:** During the pilot transmission, the BS combines the received UL signal via an adaptive selecting network which consists of a small number of 1-bit phase shifters (equivalent to switches in terms of simplicity and energy consumption) to efficiently measure the beamspace channel. CS tools were employed to design a Bernoulli random matrix adaptive combiner such that the required recovery accuracy is guaranteed. The application of the proposed adaptive selecting network makes it possible to formulate the beamspace channel estimation problem as a sparse recovery problem, which significantly reduces the number of required pilot symbols.

*   **SD-based channel estimation scheme:** The channel estimation problem is decomposed into a series of sub-problems where only one sparse channel component is considered for each sub-problem. The idea is to iteratively detect the support (i.e., the index set of non-zero elements) in a sparse vector containing the information of a specific propagation direction at each
<figure><img src="image_11.png" /><figcaption>FIGURE 10. Existing duplexing schemes for mmWave MIMO systems (a) TDD scheme (b) FDD scheme.</figcaption></figure>

time. Thereafter, the influence of each channel compo-
nent is removed from the mmWave beamspace channel,
and the support of the next one is detected in a similar
way. After completing the detection of the supports of
all channel components, the beamspace channel can be
estimated with low pilot overhead.

The SD-based channel method can outperform the clas-
sical CS algorithms (such as OMP and CoSaMP) which
also iteratively estimate all the positions of non-zero ele-
ments one by one. The main reason is that the SD-based
method only estimates the position of the strongest element,
by contrast to the CS algorithms that can try to estimate an
element whose power is not strong enough. This becomes
more crucial when the transmit power of users is limited.
Simulation results confirm that this method provides satis-
fying performance and low pilot overhead even in the low
signal-to-noise ratio (SNR) region.

ESTIMATION OF APPROXIMATED LENS ARRAY
RESPONSE

The research work in [86] studies the mmWave channel estimation problem for a general lens antenna system, i.e., a narrowband or wideband system using a TDD or FDD mode where BS and MT are equipped with lens antenna arrays. The specificity of this article comes from the fact that it is one of the pioneer works that have adopted the approxi- mated lens array response model as defined in equation (11). The authors assumed that the channel is a quasi-static block- fading one such that the UL/DL channels can be considered constant over some period and change independently from one period to the next. Each period is divided into three main phases: energy-based antenna selection, reduced MIMO channel estimation and finally data transmission. It is pos- sible to apply this sequence of phases for TDD and FDD schemes as shown in Figure 10. Note that the assumption of channel reciprocity was not needed in this article, and even the authors explained that the path reciprocity assumption is more realist and sufficient to propose an efficient tailored channel estimation method.

In the first phase, the angle-dependent energy focusing of
the lens array enables the antenna selection at BS and MT via
a simple energy detection and comparison since only a small

subset of antennas would receive (steer) significant power
from (to) the transmitter (receiver). The antenna selection
at the MT is assisted by the BS which first isotropically
sends identical training symbols such as those along the
few channel paths would be received at the MT. A similar
technique is applied to select the BS antennas but this time
the MT only sends his signal via its selected antennas which
are associated with the most powerful paths of the channel.
The employed procedure is illustrated in Figure 11.

After antenna selection, the matrices of the UL and
DL channel impulse responses, as defined in (2) and (14)
respectively, are reduced to the corresponding submatrices
associated with the selected antennas. The following phase is
to estimate the dimension-reduced matrices which exhibit a
highly sparse structure. The UL channel matrix is estimated
by sending orthogonal training sequences from each of the
selected antennas at the MT. The authors assumed that the
received signals of the *L* paths at the BS are focused on *N*<sub>*f*</sub>
non-overlapping antenna subsets when the angle resolution
of the lens antenna array is sufficiently large. Afterwards, a
correlation between the received sequences and the training
sequence with simple peak searching is employed to estimate
the channel gains. This method is efficient since the path-
division property of lens antenna array allows to drastically
reduce both inter-stream interference within each path and
inter-stream interference in other paths. However, the inter-
stream interference among paths is much more significant
in conventional antenna arrays and that is why the channel
impulse response can't be directly estimated.

MESSAGE PASSING METHODS

In [157], the authors investigated the channel estimation
problem for a mmWave system with BS and MT equipped
with lens antenna arrays. The DFT channel model was
adopted such as the channel coefficients are approximately
modeled as a Bernoulli-Gaussian distribution. Thanks to the
sparse nature of the channel, this research work proposed
an iterative estimation method based on two main steps :
the LSE and sparse message passing (SMP). The block dia-
gram of this method is presented in Figure 12. The SMP
is similar to the belief propagation decoding process and its
aim is to detect the exact locations of the non-zero entries
of the sparse channel matrix or its vectorized version. This
step required first to introduce a factor graph representa-
tion of the channel such as it is possible to identify two
types of nodes : the entries of the received signal vector,
and that of a position vector which represents the positions
of non-zero coefficients in the channel vector. The extrin-
sic information on each edge is calculated by the messages
on the other edges that are connected with the same node.
Then, LSE is applied to estimate the values of the non-zero
elements whose positions were previously detected by the
SMP. The new channel vector will replace the old input of
the SMP step in the next iteration. Of course, this iterative
process requires to be initialized via a coarse LSE estima-
tion of the channel vector without taking into consideration
<figure><img src="image_11.png" /><figcaption>FIGURE 10. Existing duplexing schemes for mmWave MIMO systems (a) TDD scheme (b) FDD scheme.</figcaption></figure>

time. Thereafter, the influence of each channel compo-
nent is removed from the mmWave beamspace channel,
and the support of the next one is detected in a similar
way. After completing the detection of the supports of
all channel components, the beamspace channel can be
estimated with low pilot overhead.

The SD-based channel method can outperform the clas-
sical CS algorithms (such as OMP and CoSaMP) which
also iteratively estimate all the positions of non-zero ele-
ments one by one. The main reason is that the SD-based
method only estimates the position of the strongest element,
by contrast to the CS algorithms that can try to estimate an
element whose power is not strong enough. This becomes
more crucial when the transmit power of users is limited.
Simulation results confirm that this method provides satis-
fying performance and low pilot overhead even in the low
signal-to-noise ratio (SNR) region.

ESTIMATION OF APPROXIMATED LENS ARRAY RESPONSE

The research work in [86] studies the mmWave channel estimation problem for a general lens antenna system, i.e., a narrowband or wideband system using a TDD or FDD mode where BS and MT are equipped with lens antenna arrays. The specificity of this article comes from the fact that it is one of the pioneer works that have adopted the approximated lens array response model as defined in equation (11). The authors assumed that the channel is a quasi-static blockfading one such that the UL/DL channels can be considered constant over some period and change independently from one period to the next. Each period is divided into three main phases: energy-based antenna selection, reduced MIMO channel estimation and finally data transmission. It is possible to apply this sequence of phases for TDD and FDD schemes as shown in Figure 10. Note that the assumption of channel reciprocity was not needed in this article, and even the authors explained that the path reciprocity assumption is more realist and sufficient to propose an efficient tailored channel estimation method.

In the first phase, the angle-dependent energy focusing of
the lens array enables the antenna selection at BS and MT via
a simple energy detection and comparison since only a small

subset of antennas would receive (steer) significant power
from (to) the transmitter (receiver). The antenna selection
at the MT is assisted by the BS which first isotropically
sends identical training symbols such as those along the
few channel paths would be received at the MT. A similar
technique is applied to select the BS antennas but this time
the MT only sends his signal via its selected antennas which
are associated with the most powerful paths of the channel.
The employed procedure is illustrated in Figure 11.

After antenna selection, the matrices of the UL and
DL channel impulse responses, as defined in (2) and (14)
respectively, are reduced to the corresponding submatrices
associated with the selected antennas. The following phase is
to estimate the dimension-reduced matrices which exhibit a
highly sparse structure. The UL channel matrix is estimated
by sending orthogonal training sequences from each of the
selected antennas at the MT. The authors assumed that the
received signals of the *L* paths at the BS are focused on *N*<sub>*f*</sub> non-overlapping antenna subsets when the angle resolution
of the lens antenna array is sufficiently large. Afterwards, a
correlation between the received sequences and the training
sequence with simple peak searching is employed to estimate
the channel gains. This method is efficient since the path-
division property of lens antenna array allows to drastically
reduce both inter-stream interference within each path and
inter-stream interference in other paths. However, the inter-
stream interference among paths is much more significant
in conventional antenna arrays and that is why the channel
impulse response can't be directly estimated.

MESSAGE PASSING METHODS

In [157], the authors investigated the channel estimation
problem for a mmWave system with BS and MT equipped
with lens antenna arrays. The DFT channel model was
adopted such as the channel coefficients are approximately
modeled as a Bernoulli-Gaussian distribution. Thanks to the
sparse nature of the channel, this research work proposed
an iterative estimation method based on two main steps :
the LSE and sparse message passing (SMP). The block dia-
gram of this method is presented in Figure 12. The SMP
is similar to the belief propagation decoding process and its
aim is to detect the exact locations of the non-zero entries
of the sparse channel matrix or its vectorized version. This
step required first to introduce a factor graph representa-
tion of the channel such as it is possible to identify two
types of nodes : the entries of the received signal vector,
and that of a position vector which represents the positions
of non-zero coefficients in the channel vector. The extrin-
sic information on each edge is calculated by the messages
on the other edges that are connected with the same node.
Then, LSE is applied to estimate the values of the non-zero
elements whose positions were previously detected by the
SMP. The new channel vector will replace the old input of
the SMP step in the next iteration. Of course, this iterative
process requires to be initialized via a coarse LSE estima-
tion of the channel vector without taking into consideration