CENTRO DE TECNOLOGIA
DEPARTAMENTO DE ENGENHARIA DE TELEINFORM ´ATICA
PROGRAMA DE P ´OS-GRADUAC¸ ˜AO EM ENGENHARIA DE TELEINFORM ´ATICA
LUCAS NOGUEIRA RIBEIRO
SIGNAL PROCESSING METHODS FOR LARGE-SCALE MULTI-ANTENNA SYSTEMS
FORTALEZA 2019
SIGNAL PROCESSING METHODS FOR LARGE-SCALE MULTI-ANTENNA SYSTEMS
Tese apresentada ao Curso de Doutorado em Engenharia de Teleinform´atica do Pro-grama de P´os-Graduac¸˜ao em Engenharia de Teleinform´atica do Centro de Tecnologia da Universidade Federal do Cear´a, como requisito parcial `a obtenc¸˜ao do t´ıtulo de doutor em Engen-haria de Teleinform´atica. ´Area de Concentrac¸˜ao: Sinais e Sistemas
Orientador: Prof. Dr. Andr´e Lima F´errer de Almeida
Coorientador: Prof. Dr. Jo˜ao C´esar Moura Mota
FORTALEZA 2019
Biblioteca Universitária
Gerada automaticamente pelo módulo Catalog, mediante os dados fornecidos pelo(a) autor(a)
R369s Ribeiro, Lucas Nogueira.
Signal processing methods for large-scale multi-antenna systems / Lucas Nogueira Ribeiro. – 2019. 187 f. : il. color.
Tese (doutorado) – Universidade Federal do Ceará, Centro de Tecnologia, Programa de Pós-Graduação em Engenharia de Teleinformática, Fortaleza, 2019.
Orientação: Prof. Dr. André Lima Férrer de Almeida. Coorientação: Prof. Dr. João César Moura Mota.
1. Processamento de Sinais. 2. Antenas. 3. Comunicações Sem-Fio. I. Título.
SIGNAL PROCESSING METHODS FOR LARGE-SCALE MULTI-ANTENNA SYSTEMS
Thesis defended at the Teleinformatics Engineering Doctorate Program at the Teleinformatics Engineering Post-Graduate Program of the Technology Center at the Federal University of Cear´a, as a requirement to obtain the doctor degree in Teleinformatics Engineering. Concentration Area: Signals and Systems.
Approved on: October 10, 2019
EXAMINING COMMITTEE
Prof. Dr. Andr´e Lima F´errer de Almeida (Advisor)
Universidade Federal do Cear´a, Brazil
Prof. Dr. Jo˜ao C´esar Moura Mota (Co-Advisor)
Universidade Federal do Cear´a, Brazil
Prof. Dr. Walter da Cruz Freitas Junior Universidade Federal do Cear´a, Brazil
Univ.-Prof. Dr.-Ing. Martin Haardt Technische Universit¨at Ilmenau, Germany
Prof. Dr. Nuria Gonz´alez-Prelcic The University of Texas at Austin, USA
Asst. Prof. Dipl.-Ing. Dr.techn. Stefan Schwarz Technische Universit¨at Wien, Austria
This study was financed in part by the Coordenac¸˜ao de Aperfeic¸oamento de Pessoal de N´ıvel Superior - Brasil (CAPES) - Finance Code 001, CNPq and FUNCAP. I am also thankful to the Erasmus Mundus SMART2 program for supporting my research stay at TU Wien.
I would like to thank my academic advisors, Prof. Andr´e de Almeida, and Prof. Jo˜ao C´esar Mota. Our collaboration began almost10 years ago in the “Iniciac¸˜ao Cient´ıfica” program during my undergraduate studies. Ever since, I can count on their support, guidance, and friendship.
I am also very grateful to Prof. Stefan Schwarz and Prof. Markus Rupp for receiving me at the TU Wien Christian Doppler Laboratory in2017. I have learned a great deal from their scientific rigor and patience when working on the papers we have co-authored.
I would like to thank all the members of the Jury for accepting being examiners of this work.
Esta tese jamais teria sido escrita sem o apoio da minha fam´ılia, em especial, meus pais. Muito obrigado por tudo!
A demanda de tr´afego em sistemas de comunicac¸˜oes sem-fio tem crescido a largos passos devido ao uso generalizado de sistemas celulares e `a emergˆencia da Internet of Things. Para lidar com as demandas de alto tr´afego, a quinta gerac¸˜ao (5G) da tecnologia celular prevˆe sistemas transceptores operando no espectro de ondas milim´etricas com arranjos de antenas de larga escala. Por´em, esse projeto de sistema 5G enfrenta diversos desafios de engenharia. Os m´etodos e as arquiteturas de processamento de sinais utilizados em sistemas multi-antenas cl´assicos s˜ao inadequados nesse cen´ario de larga escala. T´ecnicas tradicionais de processamento de sinais tornam-se computa-cionalmente custosas, e as arquiteturas cl´asicas de front-end de r´adio-frequˆencia oferecem uma baixa eficiˆencia energ´etica. Esta tese apresenta soluc¸˜oes de baixa complexidade e energeticamente eficientes para o projeto de sistemas multi-antenna de larga escala. Primeiramente, prop˜oe-se filtros multilineares para reduzir a complexidade no processamento de recepc¸˜ao de larga escala. Mostra-se que os m´etodos de filtragem multilinear reduzem significantemente a complexidade computacional com uma pequena perda de desempenho em relac¸˜ao `a abordagem cl´assica linear. Quanto `a eficiˆencia energ´etica das arquiteturas de transceptores, investiga-se sistemas
multiple-input multiple-output (MIMO) massivos h´ıbridos anal´ogicos/digitais (A/D) com conversores
de dados de baixa resoluc¸˜ao. Prop˜oe-se esquemas eficientes de pr´e-codificac¸˜ao para sistemas h´ıbridos A/D com redes de deslocamento de fase completamente e parcialmente conectadas. Apresenta-se tamb´em esquemas de baixo custo para transceptores MIMO duplamente massivos, em que a estac¸˜ao base bem como os dispositivos-usu´arios possuem arranjos de antenas de larga escala. Prop˜oe-se a estrat´egia de filtragem em m´ultiplas camadas para reduzir a complexidade computacional e os requerimentos de channel state information do projeto de transceptor. Fi-nalmente, considera-se o problema de estimac¸˜ao de canal sob problemas de sincronismo. Mais especificamente, desenvolve-se algoritmos baseados em tensores para a estimac¸˜ao de canal na presenc¸a de deslocamento de frequˆencia da portadora (CFO) e ru´ıdo de fase (PN). Trata-se inicialmente do cen´ario com desvanecimento plano em frequˆencia assumindo medidas corromp-idas pelo CFO. Em seguida, foca-se no cen´ario com desvanecimento seletivo em frequˆencia considerando ambos CFO e PN.
Palavras-chave: Beamforming. 5G. MIMO massivo. Tensores. ´Algebra Multilinear. Ondas milim´etricas. Estimac¸˜ao de Canal.
The data traffic demand in wireless communication systems has been growing at a fast pace with the widespread use of cellular systems and the emergence of the Internet of Things. To meet the large traffic requirements, the fifth-generation (5G) of cellular technology envisions transceiver systems operating at the millimeter wave spectrum with large-scale antenna arrays. However, this 5G system design faces many engineering challenges. Signal processing methods and architectures employed in classic multi-antenna systems are inadequate in the large-scale scenario. Standard signal processing techniques become computationally expensive and classical radio-frequency front-end architectures exhibit low energy efficiency. This thesis presents low-complexity and energy-efficient solutions to the design of large-scale multi-antenna systems. First, we propose multilinear filters to tackle the complexity issue in large-scale receive processing. We show that the proposed multilinear filtering methods drastically reduces the computational complexity with a slight performance deterioration compared to the classical linear approach. Concerning the energy efficiency of transceiver architectures, we investigate hybrid analog/dig-ital (A/D) massive multiple-input multiple-output (MIMO) systems with low-resolution data converters. We present efficient precoding schemes for hybrid A/D systems with fully- and partially-connected phase-shifting networks. We also introduce low-complexity double-sided massive MIMO transceiver schemes, where both the base station and the user equipment employ large-scale antenna arrays. In particular, we leverage the multi-layer filtering strategy to reduce the computational complexity and channel state information requirements of the transceiver design. Finally, we consider the problem of channel estimation under synchronization impairments. More specifically, we develop tensor-based algorithms for channel estimation in the presence of carrier frequency offset (CFO) and phase noise (PN). We start with the frequency-flat case by assuming CFO-corrupted measurements. Then, we turn our attention to the frequency-selective case by including both CFO and PN.
Keywords: Beamforming. 5G. Massive MIMO. Tensors. Multilinear Algebra. Millimeter Wave. Channel Estimation.
Figure 1.1 – Mobile data traffic forecast. . . 21
Figure 1.2 – Thesis structure. . . 28
Figure 2.1 – UPA in they-z plane. . . 32
Figure 2.2 – Number of flops as function of array size forR = 4 wavefronts. . . 47
Figure 2.3 – KMMSE BER performance for different regularization parameterδ. . . 48
Figure 2.4 – Condition number of (2.45) for different regularization parameterδ. . . 48
Figure 2.5 – Separable beamformers BER performance. . . 49
Figure 2.6 – MMSE AF squared magnitude. . . 49
Figure 2.7 – TMMSE AF squared magnitude. . . 50
Figure 2.8 – KMMSE AF squared magnitude. . . 51
Figure 2.9 – Horizontal KMMSE AF squared magnitude. . . 52
Figure 2.10–Vertical KMMSE AF squared magnitude. . . 53
Figure 2.11–ATLMS learning curves for different iteration intervalsKhandKv. µ = 0.5. 54 Figure 2.12–ATLMS learning curves for different iteration intervalsKhandKv. µ = 0.1. 55 Figure 2.13–NLMS learning curves for different step sizeµ. . . 56
Figure 2.14–TLMS learning curves for different step sizeµ. . . 56
Figure 2.15–ATLMS learning curves for different step sizeµ. Kh = Kv = 10. . . 57
Figure 3.1 – Computational complexity of LCMV-type filters. . . 70
Figure 3.2 – Output SINR of LCMV-type filters as function of SNR . . . 70
Figure 3.3 – Output SINR of LCMV-type filters as function ofK. . . 71
Figure 3.4 – Output SINR of LCMV-type filters as function of array size. . . 71
Figure 3.5 – Computational complexity of Frost-type filters. . . 74
Figure 3.6 – Output SINR of Frost-type filters forNh = Nv = 8. . . 74
Figure 3.7 – Output SINR of Frost-type filters forNh = Nv = 14. . . 75
Figure 4.1 – Computational complexity as function ofK. N = 512, I = 2, R = 3. . . . 82
Figure 4.2 – Computational complexity as function ofN . K = 600, I = 2, D = 3. . . . 83
Figure 4.3 – LR-TMMSE SINR as function of SNR.N = 512, K = 600, D = 3. . . 84
Figure 4.4 – LR-TMMSE SINR as function of SNR.N = 512, K = 600, R = 3. . . 85
Figure 4.5 – LR-TMMSE SINR as function ofK. N = 512, SNR = 20 dB, D = 3. . . . 86
Figure 4.6 – LR-TMMSE SINR as function ofK. N = 512, SNR = 20 dB, R = 3. . . . 86
Figure 5.4 – Spectral efficiency of digital precoders. . . 109
Figure 5.5 – Spectral efficiency of hybrid precoders (ignoring RF hardware losses). . . . 109
Figure 5.6 – Spectral efficiency of hybrid precoders (considering RF hardware losses). . . 110
Figure 5.7 – Static power consumption. . . 111
Figure 5.8 – Computational power consumption. . . 112
Figure 5.9 – Energy-spectral efficiency curves for varying DAC resolution. SNR= 0 dB. 113 Figure 5.10–Energy-spectral efficiency curves for varying DAC resolution. SNR = −15 dB.114 Figure 5.11–Energy efficiency as function of transmit array sizeNt. SNR = 0 dB. . . 114
Figure 5.12–Energy efficiency as function of transmit array sizeNt. SNR = −15 dB. . . 115
Figure 6.1 – Illustration of the considered double-sided massive MIMO system model. . 117
Figure 6.2 – Outer layer methods at poor scattering. . . 130
Figure 6.3 – Outer layer methods at fair scattering. . . 130
Figure 6.4 – Outer layer methods at rich scattering. . . 131
Figure 6.5 – Inner layer methods at poor scattering,U = 4 UEs. . . 133
Figure 6.6 – Inner layer methods at poor scattering,U = 32 UEs. . . 134
Figure 6.7 – Inner layer methods at poor scattering,Mt = Mr = 4. . . 135
Figure 6.8 – Inner layer methods at fair scattering andMt= Mr = 16. . . 135
Figure 6.9 – Inner layer methods at rich scattering andMt = Mr = 32. . . 136
Figure 6.10–MET-MER benchmarking at poor scattering. . . 137
Figure 6.11–MET-BD benchmarking at poor scattering. . . 137
Figure 6.12–MET-MMSE benchmarking at poor scattering. . . 138
Figure 6.13–BD-MER benchmarking at poor scattering. . . 138
Figure 7.1 – Illustration of theNppilots for channel estimation. . . 141
Figure 7.2 – NMSE vs. SNR.Mt= 32 transmissions. . . 147
Figure 7.3 – NMSE vs. Mt. SNR= 0 dB. . . 147
Figure 7.4 – Achievable rate vs. SNR.Mt= 32 transmissions. . . 149
Figure 7.5 – Achievable rate vs. Mt. SNR= 0 dB. . . 149
Figure 8.1 – Illustration of the considered time-domain protocol. . . 151
Figure 8.2 – Angles of arrival average normalized mean square error. . . 160
Figure 8.3 – Angles of departure average normalized mean square error. . . 161
Figure 8.6 – Carrier frequency offset normalized mean square error. . . 162 Figure 8.7 – Fading gains normalized mean square error. . . 163 Figure A.1 – Matrix unfoldings of a third-order tensor. . . 177
Table 3.1 – Computational complexity of the TLCMV beamformer (Algorithm 3.1). . . . 62
Table 3.2 – Computational complexity of the KLCMV beamformer (Algorithm 3.2). . . 64
Table 3.3 – Computational complexity of Frost’s beamformer (Algorithm 3.3). . . 66
Table 3.4 – Computational complexity of the TFROST beamformer (Algorithm 3.4). . . 68
Table 3.5 – Computational complexity of KFROST beamformer (Algorithm 3.5). . . 69
Table 5.1 – Power consumption and loss of the RF front-end components. . . 100
Table 5.2 – FLOPS counting in Algorithm 5.1. . . 106
Table 5.3 – FLOPS counting in Algorithm 5.2. . . 106
Table B.1 – Computational complexity of the MMSE filter. . . 183
4G fourth-generation
5G fifth-generation
A/D analog/digital
ADC analog/digital converter
AF array factor
AGC automatic gain control
AQN additive quantization noise
ATLMS alternating tensor least mean square AWGN additive white Gaussian noise
BD block diagonalization
BER bit error ratio
BS base station
CFO carrier frequency offset CME covariance matrix eigenfilter
CP canonical polyadic
CP-OMP canonical polyadic orthogonal matching pursuit
CS compressive sensing
CSI channel state information DAC digital/analog converter FLOPS floating point operations GSC generalized sidelobe canceler HPF hybrid precoding fully connected HPP hybrid precoding partially connected i.i.d. independent and identically distributed
KFROST Kronecker Frost
KLCMV Kronecker linear constrained minimum variance KMMSE Kronecker minimum mean square error
LCMV linear constrained minimum variance
LMS least mean square
LR-TMMSE low-rank tensor minimum mean square error
MER maximum eigenmode reception
MET maximum eigenmode transmission
MIMO multiple-input multiple-output
MMSE minimum mean square error
mmWave millimeter wave
MSE mean square error
MU-MIMO multi-user MIMO
MUSIC multiple signal classification NLMS normalized least mean square NMSE normalized mean square error OMP orthogonal matching pursuit
PA power amplifier
PN phase noise
PPS power-dominant path selection
PSN phase-shifting network
PZF partial zero forcing
QPSK quadrature phase shift-keying
RF radio-frequency
RX receiver
SINR signal to interference and noise ratio SIR signal to interference ratio
SNR signal to noise ratio
SPS semi-orthogonal path selection SQNR signal to quantization noise ratio
SU-MIMO single-user MIMO
SVD singular value decomposition
TDD time-division duplex
TFROST tensor Frost
TLCMV tensor linear constrained minimum variance TLMS tensor least mean square
UE user equipment
ULA uniform linear array
UPA uniform planar array
FS-CP-OMP frequency-selective canonical polyadic orthogonal matching pursuit
imaginary unit√−1
x scalar
x vector
X matrix
X tensor
[X ](n) n-mode tensor unfolding matrix
R the real numbers field
C the complex numbers field
[·]i1,...,iN (i1, . . . , iN)-entry of the argument array (vector, matrix or tensor)
(·)∗
complex conjugate
(·)T transpose
(·)H conjugate transpose (Hermitian)
(·)−1 inverse operator (·)† Moore-Penrose pseudo-inverse k · k1 `1 norm k · k2 `2 norm k · kF Frobenius norm
h· , ·i tensor inner product E [·] statistical expectation
| · | absolute value (complex magnitude)
∠· phase (complex argument)
b·c floor operation
⊗ Kronecker product
◦ outer product
elementwise (Hadamard) product
Khatri-Rao product
#(·) argument set’s cardinality
!
= equality by construction
IN N -dimensional identity matrix
0M ×N (M × N )-dimensional null matrix
IR,I R-th order identity tensor with equal dimensions I
Tr(·) matrix trace
det(·) matrix determinant
Diag(·) transforms an input vector into a diagonal matrix
diag(·) forms a diagonal matrix out of the argument matrix main diagonal Blkdiag(·) forms a block-diagonal matrix from the matrix arguments
rank(·) argument matrix’s rank
kA k-rank of matrix A
span(·) space spanned by the argument vectors null(·) null space of the input matrix
vec(·) vectorizes the input array
unvec(·) reshapes a column vector into a matrix U (a, b) uniform distribution froma to b
CN (µ, Σ) complex Gaussian distribution with meanµ and covariance matrix Σ Qb(·) b-bits uniform scalar quantizer
1 INTRODUCTION . . . . 20
1.1 Motivation and Thesis Scope . . . 20
1.2 State of the Art . . . 22
1.3 Thesis Outline and Contributions . . . 26
1.3.1 Publications . . . . 28
2 SEPARABLE MMSE BEAMFORMING . . . . 31
2.1 Overview . . . 31 2.2 System Model . . . 32 2.3 Methods . . . 35 2.3.1 MMSE-type Beamformers . . . 36 2.3.2 LMS-type Algorithms . . . . 44 2.4 Simulation Results . . . 46 2.5 Summary . . . 55 3 SEPARABLE LCMV BEAMFORMING . . . . 58 3.1 Overview . . . 58 3.2 Methods . . . 58 3.2.1 LCMV-Type Beamformers . . . 59 3.2.2 Frost-Type Beamformers . . . . 64 3.3 Simulation Results . . . 68 3.4 Summary . . . 73
4 LOW-RANK MULTILINEAR EQUALIZER DESIGN . . . . 76
4.1 Overview . . . 76
4.2 System Model . . . 76
4.3 Methods . . . 78
4.3.1 Low-Rank Tensor MMSE Filter . . . . 79
4.4 Simulation Results . . . 82
4.5 Summary . . . 84
5 ENERGY EFFICIENCY OF MMWAVE MASSIVE MIMO SYSTEMS WITH LOW-RESOLUTION DACS . . . . 88
5.1 Overview . . . 88
5.2.2 Quantized Signal Model . . . . 92
5.2.3 Channel Model . . . . 96
5.2.4 Power Consumption and Loss Models . . . . 97
5.3 Precoding Strategies . . . 101
5.3.1 Analog Precoding Strategies . . . 102
5.3.2 Digital Precoding Strategy . . . 104
5.3.3 Computational Complexity Analysis . . . 105
5.3.4 Achievable Rate Bounds . . . 106
5.4 Simulation Results . . . 107
5.4.1 Spectral Efficiency . . . 108
5.4.2 Power Consumption . . . 110
5.4.3 Energy Efficiency . . . 111
5.5 Summary . . . 113
6 DOUBLE-SIDED MASSIVE MIMO TRANSCEIVER DESIGN . . . . 116
6.1 Overview . . . 116
6.2 System Model . . . 117
6.2.1 Channel Model . . . 118
6.2.2 Layered Transceiver Architecture . . . 119
6.2.3 Channel State Information Acquisition . . . 120
6.3 Transceiver Schemes . . . 121
6.3.1 Outer Layer Filtering . . . 122
6.3.2 Inner Layer Filtering . . . 124
6.3.3 Complexity Analysis . . . 127
6.4 Simulation Results . . . 129
6.4.1 Outer Layer Filters . . . 131
6.4.2 Inner Layer Filters . . . 132
6.4.3 Benchmarking . . . 134
6.5 Summary . . . 136
7 THE FREQUENCY-FLAT CHANNEL ESTIMATION SCENARIO . . 140
7.1 Overview . . . 140
7.2 System Model . . . 141
7.5 Summary . . . 148
8 THE FREQUENCY-SELECTIVE CHANNEL ESTIMATION SCENARIO150 8.1 Overview . . . 150
8.2 System Model . . . 150
8.3 Channel Estimation . . . 154
8.3.1 Uniqueness of the CP Factorization . . . 158
8.4 Simulation Results . . . 159
8.5 Summary . . . 160
9 CONCLUSION AND PERSPECTIVES . . . 164
REFERENCES . . . 166
APPENDICES . . . 175
APPENDIX A – Tensor Primer . . . 176
APPENDIX B – MMSE and LCMV Filters . . . 182
1 INTRODUCTION
1.1 Motivation and Thesis Scope
Wireless communication systems have allowed widespread access to the Internet throughout the world. Places that were expensive to reach using fiber cables are now being served by the telecoms industry with cellular systems such as the fourth-generation (4G) long-term evolution (LTE) technology. This global connectivity brought by cellular networks has tremendously shaped our society and economy. It is hard to imagine the daily routine without the easy Internet access provided by modern wireless technologies. Many markets and services have emerged and continue to grow thanks to the possibility of broadband mobile access to the Internet. As a consequence of the expansion of wireless communications over the years, the demand for data traffic keeps increasing, as illustrated in Figure 1.1. It is thus necessary to develop novel technologies capable of providing the expected data traffic in the coming years.
The fifth-generation (5G) of cellular systems has been designed to meet the require-ments and expectations of a connected society. It has been envisioned to operate in three use scenarios (DAHLMAN; PARKVALL; SKOLD, 2018):
1. Enhanced mobile broadband – the evolution of mobile broadband services cur-rently implemented. It will offer better user experience and data rates;
2. Massive machine-to-machine communications – It will allow to serve a massive number of devices, e.g., sensors, actuators, robots, in a very energy-efficient way; 3. Ultra-reliable and low-latency communications – Envisioned to serve devices with very low latency and high-reliability requirements, e.g., factory automation, autonomous vehicles, among others.
One can adopt two strategies to enable enhanced mobile broadband systems. The first strategy would be increasing system bandwidth. This is hard to realize in the congested and highly-regulated sub-6 GHz band. As an alternative, millimeter wave (mmWave) bands are less congested and loosely regulated, allowing 5G systems to operate on larger bandwidths. The second strategy would be deploying large-scale multi-antenna arrays to increase achieve large data rates by beamforming. This solution is also known as massive multiple-input multiple-output (MIMO) and it has been shown to yield great performance on sub-6 GHz systems (MARZETTA et al., 2016). Currently, there is a great interest in developing efficient large-scale multi-antenna array systems, especially at the mmWave range (HEATH et al., 2016).
Figure 1.1 – Mobile data traffic forecast.
Source – Ericsson’s mobile data traffic growth outlook. Available at hhttps://www.ericsson.com/en/mobility-report/ reports/november-2018/mobile-data-traffic-growth-outlooki.
The development of large-scale multi-antenna systems faces some engineering chal-lenges. Signal processing methods and system architectures employed in classical small-scale scenarios become inadequate. Standard signal processing techniques may exhibit prohibitive com-putational complexity in large-scale scenarios. For example, classical channel state information (CSI) acquisition and feedback methods as well as standard linear filtering techniques become too complex. Classical small-scale transceivers are typically implemented with fully-digital radio-frequency (RF) front-end and high-resolution analog/digital converter (ADC) and digital/analog converter (DAC) devices. This architecture is unfeasible in the large-scale scenario due to its large power requirements. Therefore, the development and investigation of alternative transceiver architectures are necessary to efficiently implement large-scale systems. The CSI acquisition in the mmWave massive MIMO scenario is especially challenging. This is mainly because of the high-dimensional CSI and the low-precision of mmWave RF hardware, which makes impairments such as phase noise (PN) and carrier frequency offset (CFO) more pronounced. Fortunately, the mmWave range is characterized by angle-domain sparsity, which allows the development of relatively low-complexity compressive sensing (CS)-based channel estimation techniques. However, PN and CFO may hinder the sparsity assumption and degrade the performance of CS-based channel estimation methods.
1. Computationally-efficient filtering methods design;
2. Energy-efficient and practical mmWave transceiver design; 3. MmWave channel estimation under imperfect hardware. The state of the art of these problems is discussed in the following section.
1.2 State of the Art
The first attempts to reduce the computational complexity of filtering methods can be traced back to some decades ago. One can cite the work of Treitel (TREITEL; SHANKS, 1971) which presents a low-dimensional multi-stage representation for bi-dimensional filters based on the eigendecomposition of the coefficient matrix, providing computational savings. However, the computation of large coefficient matrices is still expensive in general. In the context of massive MIMO systems (LARSSON et al., 2014), zero forcing (ZF)/minimum mean square error (MMSE) detection schemes typically involve matrix inversions, which are particularly expensive at large-scale systems. To avoid that, some works propose approximating the matrix inverse by the Neumann series expansion (TANG et al., 2016; JIANG et al., 2018). Unfortunately, this approximation may not always converge and its overall computational complexity might still be comparable to that of matrix inversion, owing to a large number of matrix multiplications. As an alternative, other researchers suggested calculating these linear detectors by iterative algorithms, such as the Jacobi method (QIN; YAN; HE, 2016; YIN et al., 2017). However, such methods may still suffer from slow convergence rate and high overall computational complexity. A popular solution consists of ZF/MMSE filters based on the QR decomposition implemented on systolic arrays (HARRIS et al., 2016).
Another strategy to reduce the number of calculations of filtering methods is to exploit any special structure the filter may present. For instance, the array factor of multi-dimensional arrays can be factorized into components corresponding to each array dimension (TREES, 2002). From this property, the steering vector of multi-dimensional arrays assumes a Kronecker fac-torization which can be exploited to enhance the performance of array processing techniques. Kronecker separable systems have been investigated in the context of system identification (RUPP; SCHWARZ, 2015), beamforming (MIRANDA et al., 2015), and echo cancellation (PALEO-LOGU; BENESTY; CIOCHINA, 2018; ELISEI-ILIESCU et al., 2018). In (RUPP; SCHWARZ, 2015), the authors introduce a tensor least mean square (TLMS) algorithm to identify bilinear separable systems. They observed that this representation reduces the number of parameters to
be identified, increasing the convergence rate of the stochastic gradient. The authors in (PALEO-LOGU; BENESTY; CIOCHINA, 2018) propose a Wiener-Hopf-based method to identify the response of low-rank systems, i.e., systems whose impulse response can be approximated as a sum of Kronecker-separable terms. The proposed methodology is extended in (ELISEI-ILIESCU
et al., 2018) for fast recursive least squares methods.
Energy-efficient design of mmWave massive MIMO systems is crucial for achieving their full potential. Hybrid analog/digital (A/D) transceiver architectures have been proposed to enable mmWave massive MIMO systems (HEATH, 2016). They employ digital filtering (precoding/decoding) at baseband, and perform beamforming in the RF domain by analog components. The most popular implementation of this RF beamformer consists of an active phase-shifting network (PSN) connecting the outputs of the baseband filter to the antennas, which is known as fully-connected PSN. This implementation, however, is associated with large power consumption as a considerable number of active phase-shift elements is required. As an alternative, one can employ sub-array beamforming, reducing the number of phase-shifters, and, consequently, the power consumption. In this case, the PSN is said to be partially-connected. It has been claimed that hybrid precoding provides a data throughput close to that of fully-digital systems (AYACH
et al., 2014). However, insertion losses of RF hardware are usually disregarded in the analysis of
such hybrid systems. If these losses are not properly compensated for, then their spectral efficiency might be much smaller in practice than what is expected. The work of (M ´ENDEZ-RIAL et
al., 2016) investigates the spectral and energy efficiencies of hybrid systems with switches and
phase-shifters to perform analog beamforming. To evaluate energy efficiency, they define a power consumption model for both types of RF beamforming considering different interconnection of components. They also propose a channel estimation technique based on compressive sensing. According to simulation results, all hybrid architectures yield similar spectral efficiency for given power consumption. In (TSINOS et al., 2017), the energetic performance of single/multi-carrier full-resolution hybrid transceivers is investigated. A transceiver optimization problem based on energy efficiency maximization is presented and solved by the alternating direction of multipliers method. The power consumption model proposed therein considers the computational power expenditure and RF hardware losses, which are usually ignored in most energy efficiency models. This model assumes the application of power amplifiers (PAs) and low-noise amplifiers that compensate for the analog beamforming losses. Based on this assumption, the paper claims that hybrid precoders with fully-connected PSN can be energy-efficient when the transmit power is high, even more than those with partially-connected PSN. This conclusion, however, contradicts
the results of (GARCIA-RODRIGUEZ et al., 2016), where the spectral efficiency of a hybrid precoder is examined under a realistic RF model. Therein, the fully-connected PSN is modeled as a bank of RF components described by their S-parameters. The obtained results show that SNR losses are significant, going up to25 dB for the given scenario. Unfortunately, with the current mmWave amplifier technology, one cannot assume that these losses are simply compensated for as in (TSINOS et al., 2017). Therefore, to make a more realistic comparison between hybrid and digital transmitters, the effect of RF losses on the spectral efficiency has to be considered.
Another energy-efficient approach to mmWave massive MIMO consists of using low-resolution DACs/ADCs (HEATH et al., 2016). At the receive side, the high-resolution ADC chains are the most power hungry part, motivating the application of low-resolution devices to reduce their power consumption. At the transmit side, however, power expenditure is dominated by PAs, which are usually required to operate within the high linearity regime to avoid distortion of the signal constellation. Employing low-resolution DACs relaxes the linearity requirement, allowing the amplifiers to operate closer to saturation, thus increasing their efficiency. A single-user MIMO (SU-MIMO) model with low-resolution quantization at the transmit side is introduced in (MEZGHANI; GHIAT; NOSSEK, 2009). A linear approximation for DAC quantization based on the Bussgang Theorem (BUSSGANG, 1952) is presented, allowing the derivation of a MMSE precoder optimized for tackling the quantization effects. Bit error ratio (BER) results reveal that this optimized MMSE filter outperforms the plain MMSE solution. In (JACOBSSON et al., 2017), a narrowband multi-user MIMO (MU-MIMO) system employing low-resolution DACs at the base station is considered. The authors investigate the performance of linear precoders with coarse quantization and propose a variety of non-linear precoders based on relaxations of the MMSE-optimal downlink precoding problem. Achievable rate expressions are obtained and simulation results suggest that performance achieved with infinite-resolution DAC can be attained by using 3 or 4 bits of resolution for the given scenario. Furthermore, it was shown that the presented non-linear precoding algorithms significantly outperform the linear filtering solutions for1-bit quantization.
Double-sided massive MIMO refers to the scenario wherein both base station (BS) and user equipment (UE) employ large antenna arrays. Therefore, this extension is even more suited than the standard massive MIMO implementation to operate at mmWave ranges, since it offers larger beamforming gain to offset the important signal propagation losses. Implementing this double-sided scenario in classical BS-smartphone links may not be realistic due to phys-ical constraints in the latter. However, we can mention many application scenarios that may
strongly benefit from this technology, including MIMO heterogeneous networks with wireless backhauling (NI et al., 2019), terahertz communication systems (AKYILDIZ; JORNET, 2016) and mmWave unmanned aerial vehicle communications (ZHANG et al., 2019). Low-complexity transceivers for double-sided massive MIMO systems were first investigated in (SCHWARZ; RUPP, 2016). The authors were interested in evaluating the effect of spatial antenna correlation on system performance. To this end, the Kronecker correlation model was adopted and the system performance was evaluated assuming linear transceiver schemes and perfect CSI. It was found that the impact of antenna correlation on performance strongly depends on the transceiver architecture. Specifically, ZF precoding and maximum eigenmode reception (MER) showed robustness against strong antenna correlation provided that the number of served users is not as large as the number of BS antennas. Hybrid A/D and fully-digital double-sided massive MIMO transceivers were investigated in (BUZZI; D’ANDREA, 2018). Partial zero forcing (PZF) and channel matching were proposed for both hybrid A/D and fully-digital strategies. However, it is not discussed whether the proposed transceiver architectures have practical CSI requirements. The transceiver strategies of (SCHWARZ; RUPP, 2016) and (BUZZI; D’ANDREA, 2018) rely on the perfect knowledge of the channel matrix of all users. As the size of these matrices is very large (due to the double-sided massive MIMO assumption), feedback and channel estimation techniques may become overwhelming.
Fundamental to massive MIMO performance is the CSI accuracy. Standard massive MIMO processing techniques, such as the ZF and MMSE filters, require accurate and updated CSI. However, large-scale multi-antenna arrays makes the channel estimation task harder than in classical MIMO systems, due to the extensive number of parameters to be estimated. MmWave channels are typically sparse in the angular domain, which allow the application of low-complexity CS channel estimation methods (HEATH et al., 2016). RF oscillators at the mmWave range are more sensitive to hardware impairments that may disable the use of CS-based channel estimation techniques. These synchronization impairments include carrier frequency offset (CFO) and phase noise (PN). Some proposed mmWave channel estimation methods take CFO into account. In (ZHANG et al., 2015), a CFO-aware iterative channel estimation method based on the least squares (LS) solution is presented for a single-input-single-output mmWave system based on single-carrier frequency-domain equalization. In (ABARI et al., 2016), a beam alignment technique which considers the effects of CFO is presented. This method relies on beam directions hashing, which allows us to easily track energy changes across directions and find the correct alignment. A channel estimation technique robust to both CFO and PN is introduced in (MYERS;
HEATH, 2017). Therein, the received signal is modeled as a tensor and the channel parameters are estimated by a customized orthogonal matching pursuit (OMP) algorithm. However, the methods presented in (ABARI et al., 2016; MYERS; HEATH, 2017) apply only to analog filtering-based systems. To fill this gap out, the work of (RODRIGUEZ-FERNANDEZ et al., 2018) presents a joint CFO and channel estimation method based on maximum likelihood and OMP for a hybrid A/D system.
1.3 Thesis Outline and Contributions
This thesis presents novel signal processing methods for large-scale multi-antenna arrays with practical requirements and low computational complexity. We organize this thesis in three parts:
1. Multilinear filtering;
2. MmWave massive MIMO transceiver design;
3. MmWave channel estimation with synchronization impairments.
Each part is focused on solving one of the three large-scale multi-antenna systems implementation challenges listed in Section 1.1.
The first part presents novel low-complexity multilinear filtering techniques for large-scale multi-antenna beamforming and equalization problems. It comprises three chapters:
• Chapter 2 introduces Kronecker-separable beamforming filters which take advan-tage of the bi-dimensional array geometry to reduce computational costs. The Kronecker factors are obtained using two strategies: alternating optimization and sub-array MMSE beamforming with Tikhonov regularization. We discuss their computational complexity and adaptive implementations;
• Chapter 3 extends the methodology introduced in Chapter 2 to filter design prob-lems with linear constraints. We formulate Kronecker-separable beamforming filters under the linear constrained minimum variance (LCMV) filtering frame-work and propose alternating optimization block methods and adaptive solutions. Computational complexity and simulation results are presented to discuss the performance of the proposed methods;
• Chapter 4 presents the development of the Kronecker-separable beamforming towards low-rank filters. The Kronecker separability assumption is too stringent in many practical scenarios, which limits the applicability of the methods proposed
in Chapters 2 and 3. This chapter introduces a low-complexity equalizer for large-scale multi-antenna arrays based on the MMSE filter. We conduct a computational complexity analysis and numerical simulations to investigate the performance of the proposed low-rank equalizer.
The second part focuses on the energy-efficient and practical implementation of mmWave massive MIMO systems and it contains two chapters:
• Chapter 5 investigates the energy efficiency of quantized hybrid A/D transmit-ters equipped with a fully/partially-connected PSN composed of active/passive phase-shifters and compare it to that of quantized digital precoders. It introduces a quantized SU-MIMO system model based on an additive quantization noise ap-proximation considering realistic power consumption and loss models to evaluate the spectral and energy efficiencies of the transmit precoding methods;
• Chapter 6 proposes practical transceiver structures for double-sided massive MIMO systems. In this chapter, we leverage the multi-layer filtering architecture and propose novel layered transceiver schemes with practical channel state in-formation requirements to simplify the complexity of our double-sided massive MIMO system. We conduct a comprehensive simulation campaign to investigate the performance of the proposed transceivers under different channel propagation conditions and to identify the most suitable strategy.
The third part refers to the mmWave channel estimation problem and consists of two chapters:
• Chapter 7 introduces a joint CFO and channel estimation method based on tensor modeling and CS techniques for the frequency-flat scenario. We establish pa-rameter identifiability conditions and evaluate the performance of the proposed method through numerical simulations;
• Chapter 8 extends the technique of Chapter 7 to the frequency-selective scenario. The extensions jointly estimate the angles of arrival and departure, path delays, channel fading and now consider CFO as well as PN distortions. We determine the identifiability conditions and benchmark the proposed method with numerical simulations.
The thesis parts are self-contained so that the reader can read them independently without loss of information. We refer the reader to Appendix A for a multilinear algebra (tensor) primer, we outline the classical MMSE and LCMV filters in Appendix B and we present the
Figure 1.2 – Thesis structure. Chapter 2 Tensor MMSE Chapter 3 Tensor LCMV Chapter 4 Low-Rank Filters Chapter 5 Energy Efficiency of Massive MIMO Chapter 6 Double-Sided Massive MIMO Design Chapter 7 Narrowband Chapter 8 Wideband
Part I: Multilinear Filtering Part II: MmWave Massive MIMO Transceiver Design
Part III: MmWave Channel Estimation with Syncronization Impairments
Tensor Algebra MmWaveSystems
Large-Scale Multi-Antenna Systems Beamforming Equalization CFO and PN Estimation Hybrid A/D and Multilayer Filtering Source – Author.
proofs of some results in Appendix C. The interconnection between the thesis parts and chapters is depicted in Figure 1.2.
1.3.1 Publications
The contributions in this thesis are also available in the following papers:
(J1) RIBEIRO, L. N; SCHWARZ, S.; DE ALMEIDA, A. L. F. Energy efficiency of mmWave massive MIMO precoding with low-resolution DACs. IEEE Journal
of Selected Topics in Signal Processing, v. 12, n. 2, p. 298–312, May, 2018; (J2) RIBEIRO, L. N; DE ALMEIDA, A. L. F.; MOTA, J. C. M. Separable linearly
constrained minimum variance beamformers. Signal Processing, v. 158, p. 15–25, May 2019;
(J3) RIBEIRO, L. N; DE ALMEIDA, A. L. F.; NOSSEK, J. A.; MOTA, J. C. M. Low-complexity separable beamformers for massive antenna array systems. IET
Signal Processing, v. 13, n. 4, p. 434–442, June 2019;
(J4) RIBEIRO, L. N; SCHWARZ, S.; DE ALMEIDA, A. L. F. Double-sided massive MIMO transceivers for mmWave communications. To appear at IEEE Access. (C1) RIBEIRO, L. N.; SCHWARZ, S. ; RUPP, M. ; de ALMEIDA, A. L. F. ; MOTA, J. C.M. A low-complexity equalizer for massive MIMO systems based on array
separability. In: Proc. 25th European Signal Processing Conference (EUSIPCO). Kos, Greece, 2017.
(C2) RIBEIRO, L. N.; SOKAL, B ; DE ALMEIDA, A. L. F. ; MOTA, J.C.M. Sepa-rable least-mean squares beamforming. In: Proc. XXXVI Simp´osio Brasileiro de Telecomunicac¸˜oes e Processamento de Sinais (SBRT). Campina Grande, PB, Brazil, 2018.
(C3) RIBEIRO, L. N.; DE ALMEIDA, A. L. F. ; MYERS, N. J. ; HEATH JR., R. W. Tensor-based estimation of mmWave MIMO channels with carrier frequency offset. In: Proc. IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP). Brighton, UK, 2019.
(C4) RIBEIRO, L. N.; DE ALMEIDA, A. L. F. ; MOTA, J. C. M. Low-rank tensor MMSE equalization. In: Proc. 16th International Symposium on Wireless Communication Systems (ISWCS). Oulu, Finland, 2019.
(C5) RIBEIRO, L. N; DE ALMEIDA, A. L. F. Joint phase noise and carrier frequency offset estimation in frequency-selective mmWave MIMO channels. Under
2 SEPARABLE MMSE BEAMFORMING
2.1 Overview
The MMSE filter is a classical solution to the beamforming problem. In the context of large-scale beamforming systems, it may be impractical due to the inversion of a large covariance matrix. Its stochastic gradient implementation, the classical least mean square (LMS) algorithm, and its popular variations, such as the normalized least mean square (NLMS) algorithm, may suffer from a slow convergence rate at the large-scale scenario.
In this chapter, we leverage the array separability property of planar antenna arrays to reduce the number of calculations of these classical beamforming methods. Instead of optimizing a full linear beamforming filter, we propose designing the low-dimensional factors of a Kronecker-separable filter. We present two strategies to design a Kronecker Kronecker-separable filter:
1. In the first strategy, the mean square error (MSE) function is minimized by means of alternating optimization;
2. The second strategy consists of a closed-form solution based on the Khatri-Rao factorization of the separable array manifold matrix. Each sub-beamformer is obtained by performing sub-array MMSE beamforming with Tikhonov regular-ization.
We also investigate the separable implementation of the NLMS algorithm. The simulation results presented in this chapter indicate that the proposed methods reduce the computational complexity without much beamforming performance losses. Moreover, adaptive implementations also achieve faster convergence when the array separability is exploited.
The remainder of this chapter is organized as follows. The system model is presented in Section 2.2, the proposed methods are introduced in Section 2.3 and the simulation results are finally presented in Section 2.4.
The content of this chapter has been published as
• RIBEIRO, L. N; DE ALMEIDA, A. L. F.; NOSSEK, J. A.; MOTA, J. C. M. Low-complexity separable beamformers for massive antenna array systems. IET
Signal Processing, v. 13, n. 4, p. 434–442, June 2019;
• RIBEIRO, L. N.; SOKAL, B ; DE ALMEIDA, A. L. F. ; MOTA, J.C.M. Sepa-rable least-mean squares beamforming. In: Proc. XXXVI Simp. Brasileiro de Telecom. e Proc. de Sinais (SBRT). Campina Grande, PB, Brazil, 2018.
Figure 2.1 – UPA in they-z plane.
unit ball wavefront
Source – Author
2.2 System Model
Let us consider a multi-antenna system equipped with a uniform planar array (UPA) of N = NhNv omni-directional antennas, withNh columns and Nv rows along theyz plane,
as depicted in Figure 2.1. The antenna array is designed to operate at wavelengthλ and the inter-element spacing in both horizontal and vertical directions isλ/2.
We assume thatR narrowband wavefronts with the same wavelength λ in the far-field propagation impinges onto the UPA from directions (φr, θr), r ∈ {1, . . . , R}. These
wave-fronts carry mutually independent digitally-modulated signals with zero mean and varianceσ2 s,r,
r ∈ {1, . . . , R}. The discrete-time model for the received signal at the n-th antenna at instant k is then given by:
xn[k] = R
X
r=1
an(φr, θr)sr[k] + bn[k], (2.1)
wherean(φr, θr) denotes the array response to the r-th wavefront at the n-th antenna, sr[k]
the complex-envelope of the digitally-modulated symbol, andbn[k] the complex additive white
Gaussian noise (AWGN) with zero mean and varianceσ2
spaced with half-wavelength spacing, the array response can be written as (TREES, 2002) an(φr, θr) = e−π[(nh−1) sin φrsin θr+(nv−1) cos θr]. (2.2)
with n = nh + (nv − 1)Nh, nh ∈ {1, . . . , Nh}, nv ∈ {1, . . . , Nv}. For notation
simplic-ity, we parameterize the array response by the horizontal and vertical direction cosines pr =
sin φrsin θr andqr = cos θr, respectively. Let us define the array steering vectora(pr, qr) =
[a1(pr, qr), . . . , aN(pr, qr)]Tand symbols vectors[k] = [s1[k], . . . , sR[k]]T. Using matrix
nota-tion, the received signals can be expressed as
x[k] = [x1[k], . . . , xN[k]]T= As[k] + b[k], (2.3)
whereA = [a(p1, q1), . . . , a(pR, qR)] ∈ CN ×Rdenotes the array manifold matrix, andb[k] =
[b1[k], . . . , bN[k]]Tthe AWGN vector. Note that the model (2.3) is valid only for a specific angular
range where the antenna response is considered to be omni-directional. Model (2.3) can be useful to describe MU-MIMO systems with line-of-sight propagation. The multipath propagation scenario is considered in Chapter 4.
The array response (2.2) can be separated into horizontal and vertical contributions due to the UPA bi-dimensionality (TREES, 2002). More specifically, (2.2) can be factorized as
an(pr, qr) = anh(pr)anv(qr), (2.4)
whereanh(pr) = e
−π(nh−1)pr and a
nv(qr) = e
−π(nv−1)qr. The sub-array steering vectors are
then defined as ah(pr) = [a1(pr), . . . , aNh(pr)] T ∈ CNh, (2.5) av(qr) = [a1(qr), . . . , aNv(qr)] T ∈ CNv. (2.6)
The separable representation in (2.4) leads to the Kronecker factorization of the array steering vectors:
a(pr, qr) = av(qr) ⊗ ah(pr) ∈ CN, (2.7)
and, consequently, the array manifold matrix can be rewritten as
A = [av(q1) ⊗ ah(p1), . . . , av(qR) ⊗ ah(pR)] = Av Ah, (2.8)
where
Ah = [ah(p1), . . . , ah(pR)] ∈ CNh×R, (2.9)
stand for the vertical and horizontal sub-array manifold matrices, respectively.
We can represent the signal model (2.3) in a useful manner with tensor algebra. Assuming array separability, the received signal at then-th antenna can be rewritten as
xnh,nv[k] = R X r=1 = a(v) nv(qr)a (h) nh(pr)sr[k] + bnh,nv[k], (2.11)
Define the received signal matrix[X[k]]nh,nv = xnh,nv[k], the array manifold tensor [A]nh,nv,r =
a(h)nh(pr)a
(v)
nv(qr), and the AWGN matrix [B[k]]nh,nv = bnh,nv[k]. Consistent index ordering in
(2.11) is important for model correctness. The indices n, nh and nv are connected as n =
nh+ (nv− 1)Nh,nh ∈ {1, . . . , Nh}, nv ∈ {1, . . . , Nv}. We assume that nh changes faster than
nv. This notation is used throughout the thesis, unless stated otherwise. The received signal
matrix can be expressed in terms ofn-mode products (see Appendix A) as
X[k] = A ×3s[k]T+ B[k] ∈ CNh×Nv. (2.12)
The array manifold tensor A is a three-dimensional array with dimensions Nh×
Nv × R whose elements are defined as
[A]nh,nv,r = a
(h)
nh(pr) · a
(v)
nv(qr), (2.13)
fornh ∈ {1, . . . , Nh}, nv ∈ {1, . . . , Nv} and r ∈ {1, . . . , R}. The two first array modes refer to
the physical array dimensions, whereas the third one represents the transmitted signal dimension, i.e., the number of wavefronts. This tensor can be unfolded into matrices in three different manners (see Appendix A):
AURA,h = [A](1) =ah(p1)av(q1)T, . . . , ah(pR)av(qR)T ∈ CNh×NvR, (2.14a)
AURA,v = [A](2) =av(q1)ah(p1)T, . . . , av(qR)ah(pR)T ∈ CNv×NhR, (2.14b)
A = [A](3) = (Av Ah)T= AT ∈ CR×NvNh. (2.14c)
From (2.14), it follows that the received signals vectorx[k] can be expressed as X[k] = AURA,h(s[k] ⊗ INv) + B[k] ∈ C Nh×Nv (2.15a) XT[k] = A URA,v(s[k] ⊗ INh) + B T [k] ∈ CNv×Nh, (2.15b)
where B[k] ∈ CNh×Nv is the matrix obtained by reshaping the AWGN vector b[k].
Equa-tions (2.11)–(2.15) emphasize the separable property of the UPA in our system model. These models will be leveraged to design the low-cost beamformers throughout the chapter. It is im-portant to mention thatX[k] and x[k] are related through a simple vectorization operation, i.e., x[k] = vec (X[k]).
The spatial covariance matrices of the AWGN term, symbols, and received signals vectors are respectively given by
Rbb = Eb[k]b[k]H = σb2IN, (2.16) Rss = Es[k]s[k]H = Diag σ2s,1, σ 2 s,2, . . . , σ 2 s,R , (2.17) Rxx = Ex[k]x[k]H = ARssAH+ Rbb. (2.18)
In practice, the covariance matrix of the observed signals is typically unknown and need thus to be estimated. In this case, we consider the following sample estimate overK snapshots:
ˆ Rxx = 1 K K−1 X k=0 x[k]xH[k] (2.19)
Hereafter, we assume without loss of generality that the first wavefront (r = 1) is the signal of interest while the otherR − 1 signals are regarded as interference. The desired symbols are now referred to assds[k] with variance σds2 . We can further express the received signals covariance
matrix as Rxx = σds2 a(pd, qd)a(pd, qd)H+ R X r>1 σ2
s,ra(pr, qr)a(pr, qr)H+ Rbb (2.20a)
= Rdd+ Rii+ Rbb, (2.20b)
where Rdd and Rii represent the covariance matrix of the desired and interference signals
signature, respectively. Now, we define the SNR as SNR= Tr(Rdd) Tr(Rbb) = σ 2 ds σ2 b
and the signal to interference ratio (SIR) as SIR= Tr(Rdd) Tr(Rii) = σ 2 ds PR r=2σs,r2 . 2.3 Methods
We are interested in spatially filtering the received signalsx[k] to extract sds[k], the
signal ofd-th (desired) wavefront, while attenuating the interfering signals. To this end, one can design the beamforming filterw ∈ CN so its outputy[k] = wHx[k] approximates the desired
signal. One can, for example, set this filter to minimize the MSE function JMSE(w) = E|sds[k] − wHx[k]|2 = σs2− p
H
xsw − w Hp
wherepxs = E [x[k]s∗ds[k]] = ARssed ∈ CN denotes the cross-covariance vector ander ∈ CR
ther-th canonical vector in the R-dimensional space. The MMSE beamformer yields the global minimum ofJMSE(w) and is given by the Wiener filter wopt= R−1xxpxs (see Appendix B). For
large array systems, the computation of this filter becomes impractical since it involves the inversion of a very large covariance matrix. Iterative algorithms, such as the gradient descent method, can be used to simplify the calculations, however each of their iterations can still be computationally expensive.
To simplify the calculations of the MMSE beamforming filter, we impose the fol-lowing Kronecker structure: w = wv⊗ wh,wm ∈ CNm,m ∈ {v, h}. Such a representation is
motivated by the computational reduction of the beamformer design, since only(Nv+ Nh)
param-eters need to be optimized instead ofNvNh when separability is not considered. In order to gain
more insight into the array separability, let us consider an example withN antennas and R = 1 impinging wavefront. The received signal in this case is given byx[k] = a(pd, qd)s[k] + b[k].
The output signal for the filterw is then written as
y[k] = wHx[k] = AF · s[k] + wHb[k], (2.22)
where AF= wHa(p
d, qd) denotes the array factor (AF). Note that it can be rewritten by applying
the mixed-product property (A.8c) as follows:
AF= (wv⊗ wh)H(av(qd) ⊗ ah(pd)) (2.23a)
=wH
vav(qd) · whHah(pd) . (2.23b)
Equation (2.23b) shows that the AF is given by the product of the sub-array factors. This property does not depend on the beam pattern of the antenna elements, since it operates on the AF factorization. The steering vectors of some array geometries, such as circular arrays, for example, do not allow a Kronecker factorization. In this case, we cannot directly apply the methods proposed in this chapter and the low-rank approach of Chapter 4 could be considered.
2.3.1 MMSE-type Beamformers
2.3.1.1 Tensor MMSE Beamformer
Let us now rewrite the beamformer outputy[k] = wHx[k] in terms of w
h andwv
received signals (2.11): y[k] = N X n=1 [w]∗nxn[k] = Nh X nh=1 Nv X nv=1 [wh]∗nh[wv] ∗ nvxnh,nv[k]. (2.24)
Using matrix notation, (2.24) can be rewritten as y[k] = wHhX[k]w ∗ v = w H vX[k] T wh∗. (2.25)
The MSE function (2.21) can now be reformulated as the following bi-linear function JMSE(wh, wv) = E h sds[k] − wHhX[k]w ∗ v 2i = Eh sds[k] − wHvX[k] T w∗h 2i . (2.26) Unfortunately, jointly minimizing (2.26) is not straightforward. The gradient of JMSE(wh, wv) with respect to any of its vector variables depends on the other variable. This
coupling hinders the direct application of methods such as gradient descent, calling for alternating minimization techniques. To this end, let us define the horizontal and vertical sub-array input signals uh[k] = X[k]w∗v ∈ C Nh, (2.27a) uv[k] = X[k]Twh∗ ∈ C Nv. (2.27b) and rewrite (2.26) as JMSE(wh, wv) = E h sds[k] − whHuh[k] 2i (2.28a) = Eh sds[k] − wvHuv[k] 2i . (2.28b)
It is easy to recognize (2.28a) and (2.28b) as linear functions ofwh andwv, respectively, when
the other vector variable is fixed. The proposed beamforming method, referred to as MMSE, consists of sequentially minimizing (2.28a) and (2.28b) using the MMSE solution for each sub-filter until a convergence criterion is satisfied. The sub-beamformers are calculated according to the following result:
Theorem 1. The minimizers of (2.28a) and (2.28b) conditioned onwv andwhare respectively
given by
wh = R−1hhphs, (2.29)
where Rhh = Euh[k]uh[k]H = AURA,h(Rss⊗ wv∗w T v)A H URA,h+ σ 2 bkwvk22INh ∈ C Nh×Nh, (2.31) Rvv = Euv[k]uv[k]H = AURA,v(Rss⊗ wh∗w T h)A H URA,v+ σ 2 bkwhk22INv ∈ C Nv×Nv (2.32)
denote the covariance matrices of the sub-array input signals, and
phs = E [uh[k]s∗ds[k]] = AURA,h(Rssed⊗ w∗v) ∈ C
Nh, (2.33)
pvs = E [uv[k]s∗ds[k]] = AURA,v(Rssed⊗ wh∗) ∈ C
Nv (2.34)
the cross-covariance vectors between the sub-array input signals and the signal of interest.
Proof. See Appendix C.
Theorem 1 can be applied when the signals’ statistics (RssandRbb) and the array
manifold matrix are known. However, such information might not be available in practice, and thus the sub-array covariance matrices and cross-covariance vectors need to be estimated. It can be done by using sample estimates overK time snapshots. In this sense, the covariance matrices RhhandRvvcan be estimated as
ˆ Rhh= 1 K K−1 X k=0 uh[k]uHh[k] = 1 K K−1 X k=0 X[k]wv∗wT vX[k] H, (2.35) ˆ Rvv = 1 K K−1 X k=0 uv[k]uHv[k] = 1 K K−1 X k=0 XT[k]w∗ hw T hX ∗ [k], (2.36)
and the cross-covariance vectorsphsandpvs as
ˆ phs = 1 K K−1 X k=0 uh[k]s∗ds[k] = 1 K K−1 X k=0 X[k]w∗ vs ∗ ds[k], (2.37) ˆ pvs = 1 K K−1 X k=0 uv[k]s∗ds[k] = 1 K K−1 X k=0 XT[k]wh∗s ∗ ds[k]. (2.38)
Note that uh[k] and uv[k] can be easily formed by observing x[k], reshaping into X[k], and
applying (2.27a) and (2.27b), respectively. The steps to compute the tensor minimum mean square error (TMMSE) beamformer are summarized in Algorithm 2.1.
The MMSE filter is known to be computationally complex. However, if the array manifold matrixA and the signal statistics RssandRbbare known, one can employ the matrix
inversion lemma (PETERSEN; PEDERSEN, 2012) to the MMSE filter and obtain the low-complexity MMSE expression (2.55), in which aR ×R matrix is inverted. When this information is not available, sample estimates are needed to compute the MMSE filter. In this case, the matrix
Algorithm 2.1:Tensor minimum mean square error (TMMSE) algorithm
1 Randomly initializewhandwv; 2 repeat
3 FormRhhandphs; 4 wh ← R−1hhphs; 5 FormRvvandpvs; 6 wv ← R−1vvpvs;
7 untilconvergence criterion triggers; 8 wTMMSE← wv⊗ wh;
inversion lemma cannot be applied, and, thus anN × N inverse covariance matrix has to be calculated. Such an operation has complexityO(N3), which can be overwhelming in practical
scenarios. In this case, the proposed methods can be used since they are much less expensive in computational terms, as we show in the following.
The TMMSE filter calculates its beamformer coefficients through an iterative process ofI iterations, in which Nh- andNv-dimensional matrices are inverted. Therefore, the TMMSE
filter requires O(I(N3 h + N
3
v)) operations. Therefore, this method is less complex than the
classical approach provided thatI, NhandNvare not too large. The authors in (YENER; YATES;
ULUKUS, 2001) discussed the convergence of alternating MMSE-based methods and concluded that they are monotonically convergent. Other numerical properties such as convergence rate and stability are not discussed and, to the best of our knowledge, the investigation of these aspects remains an open problem. The analytical convergence study of the proposed method is beyond the scope of this work.
2.3.1.2 Kronecker MMSE Beamformer
The mixed product property (A.11a) suggests that a Kronecker separable beamformer can be individually applied to the corresponding sub-array manifold matrix in (2.8). The filtering operationy[k] = wHx[k] can be rewritten as
y[k] = (wv ⊗ wh)H(Av Ah)s[k] + wHb[k] = (wHvAv) (whHAh) s[k] + wHb[k]. (2.39)
Therefore, instead of optimizing anN -dimensional beamformer for A, we propose designing two independent low-dimensional beamformers for Ah andAv. Each sub-beamformer is fed
horizontal and vertical observed signals:
xh[k] = Ahs[k] + bh[k] ∈ CNh, (2.40a)
xv[k] = Avs[k] + bv[k] ∈ CNv, (2.40b)
wherebh[k] ∈ CNhandbv[k] ∈ CNv represent the AWGN vectors observed at the horizontal and
vertical sub-arrays, respectively. These vectors are defined as [bh[k]]nh = bnh+(nv−1)Nh[k] nv=1, (2.41) [bv[k]]nv = bnh+(nv−1)Nh[k] nh=1. (2.42)
We propose to optimize each sub-beamformer according to the MMSE criterion. However, the direct application of the MMSE filter to each sub-beamformer may be subject to numerical problems. Often, in many practical scenarios, different signals are closely separated in an angular domain (azimuth or elevation). In this case, either the vertical or horizontal sub-array manifold matrices would become rank deficient, turning the MSE minimization problem ill-posed. To overcome this issue, we resort to Tikhonov regularization (PALOMAR; ELDAR, 2010), which avoids singular covariance matrices by penalizing large-norm solutions. The proposed beamforming method, hereafter referred to as Kronecker minimum mean square error (KMMSE), independently minimizes the following objective functions
JMSE(h) (wh, δh) = E|sds[k] − wHhxh[k]|2 + δhkwhk22, (2.43)
JMSE(v) (wv, δv) = E|sds[k] − wvHxv[k]|2 + δvkwvk22, (2.44)
whereδh, δv ≥ 0 are the regularization parameters. Define
Rm= AmRssAHm+ Rbb,m, (2.45)
pm = AmRssed, (2.46)
withRbb,m = σb2INm form ∈ {h, v}. The minimizers for (2.43) and (2.44) are thus given by
wm = (Rm+ δmINm)
−1
pmform ∈ {h, v}. Due to regularization, the KMMSE output signal is
not guaranteed to have the same power as the desired signal. Thus, we employ the following scaling to correct the KMMSE output power: yKMMSE[k] = (σs/σp)p[k], where p[k] = (wv⊗ wh)Hx[k]
and σp denotes the standard deviation of p[k]. In a hardware implementation, this scaling
correction can be performed by the automatic gain control (AGC) circuit. The computation of the KMMSE filter is summarized in Algorithm 2.2.
In practice, one might not have a priori knowledge of the sub-array manifold matrices (Ah andAv) and signals’ statistics. It is possible to estimate (2.45) and (2.46) using the received
signals from the horizontal and vertical sub-arrays, represented by ¯
xm[k] = Ams[k] + bm[k], m ∈ {h, v}. (2.47)
For the horizontal sub-array, we define
[ ¯xh[k]]nh = xnh+(nv−1)Nh[k]
nv=1 = xnh[k],
withnh ∈ {1, . . . , Nh} and r ∈ {1, . . . , R}. Similarly, for the vertical sub-array:
[ ¯xv[k]]nv = xnh+(nv−1)Nh[k]
nh=1 = x1+(nv−1)Nh[k],
withnv ∈ {1, . . . , Nv} and r ∈ {1, . . . , R}. The covariance matrices are then estimated as
ˆ Rh = 1 K K−1 X k=0 ¯ xh[k] ¯xh[k]H, (2.48) ˆ Rv = 1 K K−1 X k=0 ¯ xv[k] ¯xv[k]H, (2.49)
and the cross-covariance vectors as ˆ ph = 1 K K−1 X k=0 ¯ xh[k]s∗ds[k], (2.50) ˆ pv = 1 K K−1 X k=0 ¯ xv[k]s∗ds[k]. (2.51)
The proposed closed-form KMMSE beamformer can be seen as a sub-optimal solution which relies on a covariance matrix approximation. According to Kronecker product properties (A.8c) and (A.8f), the KMMSE beamformer can be expressed as
wKMMSE = wv ⊗ wh (2.52a) =(Rv + δINv) −1 pv ⊗ (Rh+ δINh) −1 ph (2.52b) = [(Rv+ δINv) ⊗ (Rh+ δINh)] −1 (pv⊗ ph). (2.52c)
The Kronecker product of covariance matrices in (2.52c) can be regarded as an approximation ofRxx. Also, it is straightforward to see in (2.52c) that the cross-covariance vectorpxs can be
2.3.1.2.1 KMMSE Asymptotic Analysis
We now conduct an asymptotic analysis of KMMSE to provide insights on its perfor-mance. First, consider the classical MMSE filter
wopt= R−1xxpxs= ARssAH+ Rbb
−1
ARssed. (2.53)
Applying the matrix inversion lemma (PETERSEN; PEDERSEN, 2012), its Hermitian vector can be written as wH opt = eTd R −1 ss + AHR −1 bb A −1 AHR−1 bb . (2.54)
From the signal statistics assumptions in Section 2.2, we have wH opt = e T d σ2 b σ2 s IR+ AHA −1 AH. (2.55) Now we rewrite the Kronecker factors of the KMMSE filter using (2.55) and forδ = 0 to obtain
wH KMMSE = " eT d σ2 b σ2 s IR+ AHvAv −1 AH v # ⊗ " eT d σ2 b σ2 s IR+ AHhAh −1 AH h # . (2.56) At high SNR, the noise power drops andσ2
b → 0. If the inverse matrix (AHmAm)−1
exists form ∈ {h, v}, then
wH KMMSE → e T dA † v ⊗ eT dA † h . (2.57)
As expected from the MMSE solution, each sub-array beamformer converges to a ZF filter. Using (A.11a) and (2.57), we see that the KMMSE output signal at high SNR converges to
y[k] →h eTdA † vAv eTdA † hAh i s[k] = sds[k]. (2.58) The inverse(AH
mAm)−1exists if and only ifAHmAmis not rank deficient, i.e., the wavefronts arrive
from sufficiently different directions. However, when the wavefronts are closely spaced in the angular domain,AH
mAm becomes ill-conditioned and the ZF filter performs poorly. Fortunately,
whenδ > 0, the inverse matrix is defined, allowing for the desired signal to be recovered. At low SNR, the term σb2
σ2
sIRdominates and (2.56) goes to
wH KMMSE = eT d σ2 s σ2 b AH v ⊗ eT d σ2 s σ2 b AH h = σ 4 s σ4 b av(φd, θd)H⊗ ah(φd, θd)H. (2.59)
Equation (2.59) shows that, as in the classical MMSE filter, the factors of wKMMSE converge
Algorithm 2.2:Kronecker minimum mean square error (KMMSE) algorithm 1 Selectδh, δv ≥ 0; 2 FormRhandph; 3 wh ← (Rh+ δhIN h) −1p h; 4 FormRv andpv; 5 wv ← (Rv + δvINv)−1pv; 6 wKMMSE← wv⊗ wh;
signal can be written asy[k] → σ4s
σ4
b[(av(φd, θd)
HA
v) (ah(φd, θd)HAh)]s[k] + wKMMSEH b[k]. If
the incoming wavefronts are sufficiently separated in the angular domain under the large-scale system assumption, the array steering vectors are almost orthogonal. In this case,
y[k] → σ 4 s σ4 b sds[k] + wHKMMSEb[k]. (2.60)
The analysis above shows that the proposed KMMSE filter is able to recover the desired signal from the received signals despite the covariance matrix approximations. Note that it is based on the assumption that the incoming signals are sufficiently separated in the angular domain.
The proposed beamforming methods work with low-dimensional sub-array manifold matrices to decrease their computational complexity. However, this also reduces their degrees of freedom, which are important for attenuating interfering signals. An MMSE filter designed forN antennas has N degrees of freedom, i.e., it is capable of recovering the desired signal and cancelingN − 1 undesired sources. Our separable beamforming framework, by contrast, offers NhandNvdegrees of freedom for the horizontal and vertical sub-arrays. Therefore, the separable
filter performance is limited by the least degree of freedom. Hence, the proposed methods are capable of recovering the desired signal and rejectingmin(Nh, Nv) − 1 undesired sources. The
proposed separable beamforming framework exchanges degrees of freedom for computational complexity reduction.
The KMMSE filter is much simpler than the previous methods since it performs sub-array beamforming using closed-form solutions. To obtain the beamformer coefficients, one needs to invertNh- andNv-dimensional matrices only once. Thus, this method carries out
O(N3
