Adaptive Beamforming for High-Data-Rate Wireless Systems

Brian Lawrence

Department of Electrical & Computer Engineering McGill University Montr´eal, Canada

May 2010

A thesis submitted to McGill University in partial fulﬁllment of the requirements for the degree of M.Eng.

c 2010 Brian Lawrence ii

Abstract

Single-port beamforming, otherwise known as microwave or radio-frequency (RF) beamforming, is advantageous over digital beamforming in terms of cost and power consumption, as the former only requires one downconversion receiver and analog-to-digital converter (ADC) in total, while the latter requires one downconversion receiver and ADC per antenna. We develop a novel, single-port beamforming algorithm based on the minimum mean-square error (MMSE) criterion. This new method is designed to minimize the number of weight changes required to estimate the optimal weights, thereby making it easier to implement in practice, and more suitable for high-data-rate applications than the popular perturbation algorithms, which require their weights be continuously switched. Numerical simulations reveal that the proposed algorithm demonstrates promising performance in the presence of multiple interfering signals, and is less susceptible to performance degradation than the perturbation algorithm when subjected to ﬁnite-resolution phase shifters and amplitude-control devices. iii

Sommaire

La formation de faisceauxa ` accès simple, également appelé formation de faisceaux micro- ondes ou radiofréquences (RF), a l’avantage sur la formation de faisceaux numérique au niveau du coût et de la consommation d’énergie puisqu’elle ne requiert qu’un récepteur in- fradyne et convertisseur analogique-numérique (ADC) plutôt qu’un par antenne. Nous développons un nouvel algorithme de formation de faisceauxa ` accès simple basésur le critère d’erreur quadratique moyenne minimale (MMSE). Cette nouvelle méthode est con¸cue afin de minimiser le nombre de réglages servanta ` estimer les pondérations opti- males. Ceci facilite la mise en œuvre pratique et rend plus convenable les applications a` haut débit, làoù les algorithmes de perturbation populaires nécessite un ajustement continu. Les simulations numériques révèlent que l’algorithme proposédémontre une performance prometteuse en présence de multiples signaux d’interférences et est moins sujet que les algorithmes de perturbation aux pertes de performance causées par la résolution finie des déphaseurs et contrôles d’amplitude. iv

Acknowledgments

Firstly, I would like to thank my supervisor, Professor Ioannis Psaromiligkos, for his in- valuable guidance, assistance, and encouragement. Moreover, I would also like to express my gratitude for the financial support I received from both the Natural Sciences and En- gineering Research Council of Canada, and McGill University. I am greatly indebted to my colleagues and friends, Saeed Abdallah, Fran¸cois Côté , and Mohammad Chakroun, for providing helpful advice, insight, and stimulating discussions. Moreover, I am particularly grateful to Fran¸cois Côté for translating the thesis abstract. Finally, thank you to my parents, my sister, and to all my family and friends for their unconditional moral support and encouragement. v

Contents

1 Introduction 1

2 Background 3 2.1 Array Processing Fundamentals ...... 3 2.1.1 Narrowband Beamforming ...... 6 2.1.2 Broadband Beamforming ...... 11 2.2 Adaptive Beamforming Architectures ...... 12 2.2.1 Digital Beamforming ...... 13 2.2.2 Single-Port Beamforming ...... 14 2.3 System Model ...... 15 2.3.1 Eigenvalue Decomposition of R and R−1 ...... 18 2.4 Optimal Beamformers ...... 19 2.4.1 Conventional (Delay-and-Sum) Beamformer ...... 19 2.4.2 MVDR Beamformer ...... 21 2.4.3 MMSE Beamformer ...... 23 2.5 Adaptive Digital Beamforming ...... 24 2.5.1 Sample Matrix Inversion (SMI) ...... 25 2.5.2 Least-Mean-Squares (LMS) ...... 26 2.6 Adaptive Single-Port Beamforming ...... 27 2.6.1 Null Steering ...... 27 2.6.2 Aerial Beamforming ...... 28 2.6.3 Phased-Array ...... 28 2.6.4 Perturbation Algorithms ...... 28 2.6.5 Proposed Single-Port Algorithm ...... 30 vi Contents

3 Adaptive Beamforming for 60-GHz Receivers 31 3.1 Design Considerations of 60-GHz Receiver ...... 32 3.1.1 Narrowband Beamforming for 60-GHz Receivers ...... 32 3.1.2 Single-Port Architecture ...... 33 3.1.3 Interference Mitigation ...... 34 3.1.4 Exploiting the High Data Rates of 60-GHz Communications .... 34 3.2 60-GHz Channel Model ...... 35

4 Proposed Beamforming Algorithm 37 4.1 Proposed Algorithm ...... 37 4.2 Estimation of R ...... 38 4.2.1 Expected Value of ˜ˆr and R˜ˆ ...... 41 4.2.2 Covariance Matrix of ˜ˆr ...... 41 4.2.3 Hermitian Structure of R˜ˆ ...... 44 4.3 Estimation of z ...... 46 4.3.1 Expected Value of z˜ˆ ...... 47 4.3.2 Covariance Matrix of z˜ˆ ...... 48 4.4 Selection of W and W˜ ...... 49 4.4.1 Effect of Weights on Error of ˜ˆr ...... 50 4.4.2 Effect of Weights on Error of z˜ˆ ...... 52 4.5 Computationally-Efficient Algorithm to Estimate r and z ...... 53

4.6 Estimation of wmmse ...... 56

4.6.1 Estimation of wmmse By Means Of Diagonal Loading ...... 60

4.6.2 Estimation of wmmse Using Truncated Singular Value Decomposition (TSVD) ...... 61

4.6.3 Estimation of wmmse Based on Dominant-Mode Rejection (DMR) . 63 4.7 Discussion of Proposed Algorithm ...... 64

5 Numerical Simulations 67 5.1 Simulation Models ...... 67 5.1.1 AWGN Model ...... 68 5.1.2 Multipath Model ...... 69 5.1.3 MSE vs. Parameter Settings ...... 70 5.1.4 MSE vs. Sample Size ...... 73 Contents vii

5.1.5 Power Patterns ...... 77 5.1.6 Eﬀects of Phase and Amplitude Quantization ...... 77

6 Conclusions and Future Research 83

A Proof of Proposition 2 87

−1 B Weight Matrices ΨH and Ψ˜ H 93

C Details Concerning Implementation of Multipath Model 95

D Perturbation Algorithm Implementation 97

References 99 viii ix

List of Figures

2.1 Antenna array receiving plane-wave front...... 4 2.2 Uniform linear array (ULA)...... 5 2.3 Narrowband beamformer...... 9 2.4 Beam pattern of uniformly weighted, 10-element array...... 10 2.5 TDL broadband beamformer...... 12 2.6 Digital beamforming architecture...... 13 2.7 Single-port beamforming architecture...... 14

4.1 Eigenvalues of R˜ˆ calculated using (4.16) by single-port beamformer with (a) No interference; (b) One interfering signal...... 58 4.2 Eigenvalues of Rˆ calculated using (2.73) by digital beamformer with (a) No interference; (b) One interfering signal...... 59

4.3 MSE of beamformer implementing w˜ˆ mmse...... 59

4.4 MSE of beamformer implementing wˆ mmse...... 60

5.1 MSE of w˜ˆ dl as a function of υ in (a) AWGN model; (b) Multipath model. . 72

5.2 MSE of w˜ˆ tsvd as a function of σ˜ˆth in (a) AWGN model; (b) Multipath model. 72 ˆ 5.3 MSE of w˜ˆ dmr as a function of λ˜th in (a) AWGN model; (b) Multipath model. 73 5.4 MSE as a function of sample size in Scenario A in (a) AWGN model; (b) Multipath model...... 74 5.5 MSE as a function of sample size in Scenario B in (a) AWGN model; (b) Multipath model...... 75 5.6 MSE as a function of sample size in Scenario C in (a) AWGN model; (b) Multipath model...... 75 x List of Figures

5.7 MSE as a function of sample size in Scenario D in (a) AWGN model; (b) Multipath model...... 76 5.8 Power patterns pertaining to Scenario A in (a) AWGN model; (b) Multipath model...... 78 5.9 Power patterns pertaining to Scenario B in (a) AWGN model; (b) Multipath model...... 78 5.10 Power patterns pertaining to Scenario C in (a) AWGN model; (b) Multipath model...... 79 5.11 Power patterns pertaining to Scenario D in (a) AWGN model; (b) Multipath model...... 79

5.12 Eﬀects of quantization on w˜ˆ tsvd in (a) AWGN model; (b) Multipath model. 81

5.13 Eﬀects of quantization on w˜ˆ dmr in (a) AWGN model; (b) Multipath model. 81

5.14 Eﬀects of quantization on wpert in (a) AWGN model; (b) Multipath model. 82

5.15 Comparison of w˜ˆ tsvd, w˜ˆ dmr, and wpert when quantized to 3 bits in (a) AWGN model; (b) Multipath model...... 82

C.1 Block diagram of the TDLs used to model the channel between the qth transmitter and each of the M antennas at the receiver...... 96 xi

List of Acronyms

ADC Analog-to-Digital Converter

AIC Akaike Information Criterion

AOA Angle-Of-Arrival

AWGN Additive White Gaussian Noise

CIR Channel Impulse Response

CLMS Constrained Least-Mean-Squares

DMR Dominant-Mode Rejection

ESPAR Electronically Steerable Passive Radiator

ESPRIT Estimation of Signal Parameters via Rotation Invariance Techniques

FB Fractional Bandwidth

FIR Finite Impulse Response

FLOPS Floating-Point Operations

HDMI High-Deﬁnition Multimedia Interface

IF Intermediate Frequency

LOS Line Of Sight

LMS Least-Mean-Squares

MDL Minimum Description Length xii List of Figures

MSE Mean-Square Error

MMSE Minimum Mean-Square Error

MPDR Minimum Power Distortionless Response

MUSIC Multiple Signal Classiﬁcation

MVDR Minimum Variance Distortionless Response

NAMI Noise-Alone Matrix Inverse

PC Principal Component

RF Radio Frequency

SINR Signal-to-Interference-plus-Noise Ratio

SMI Sample Matrix Inversion

SNR Signal-to-Noise Ratio

SPNMI Signal-Plus-Noise Matrix Inverse

SVD Singular Value Decomposition

TDL Tapped Delay Line

TSVD Truncated Singular Value Decomposition

ULA Uniform Linear Array

WPAN Wireless Personal Area Network

WLAN Wireless Local Area Network 1

Chapter 1

Introduction

Antenna arrays can provide significant enhancements over the use of a single antenna in terms of power gain, electronic steering capabilities, and co-channel interference suppres- sion [1]. By combining the signals received at each antenna, a process known as beamforming, antenna arrays perform spatial filtering on signals in a space-time field. The filtering characteristics can be adjusted via electronic manipulation of the received signals, and as will be explained, this can be equivalently interpreted as applying a complex weighting to the signal on each antenna; an antenna array possessing the capability to perform such filtering is termed a beamformer. It is often desirable to design a beamformer that optimizes a performance criterion, such as the signal-to-noise ratio (SNR), signal-to-interference-plus-noise ratio (SINR), or the mean-square error (MSE). This is especially useful in situations involving a desired signal embedded in interfering signals and background noise. Such filters are known as optimal beamformers in the corresponding (e.g., SNR) sense, and are capable of enhancing a desired signal by canceling interference and minimizing noise. In practice, however, a designer will not have knowledge of the exact environment in which the antenna array operates. As a result, adaptive beamforming algorithms1 must be employed to estimate the optimal beamforming settings based on received information. Although there seems to be some ambiguity in literature concerning exact terminology, generally speaking, a system employing beamforming algorithms is referred to as a smart antenna,anadaptive array,or simply as a beamformer. The majority of adaptive beamforming research focuses on digital beamforming, where

1We use the terms adaptive beamforming algorithms and beamforming algorithms interchangeably. 2 Introduction the signal that is induced on each antenna is down-converted and sampled by a digital processor. Although digital beamforming, otherwise known as multi-port beamforming,is ideal for implementing beamforming algorithms, it requires a full down-converting receiver and analog-to-digital converter (ADC) be used per antenna, thus making it very expen- sive and energy demanding, and thereby limiting its application in commercial wireless devices. Conversely, single-port beamforming, otherwise known as microwave or radio- frequency (RF) beamforming, carries out the beamforming at RF using analog devices and then down-converts and samples the spatially-filtered signal. As a result, it requires only one down-converting receiver and ADC, thereby dramatically reducing cost and power consumption. However, single-port beamforming algorithms are more difficult to design than their digital counterparts due to the decrease in information available for processing. A popular method of single-port beamforming is the perturbation algorithm in which the weightings applied to the signals on each element are continuously varied, and the output of the beamformer observed as a means of estimating parameters required in beamforming algorithms. Not only do these algorithms require complex synchronization, but the speed at which perturbation algorithms can be accomplished is limited by the switching speed of the weights. In this thesis we propose a novel single-port beamforming algorithm that utilizes a minimal number of weight changes to perform adaptive beamforming. In Chapter 2, the fundamental principles of array processing are presented, and a system model is established. We introduce the concept of optimal beamforming, and provide descriptions of several digital and single-port beamforming algorithms. A potential application of our algorithm - 60-GHz wireless receivers - is discussed in Chapter 3, and we put forth several arguments as to why our algorithm may prove beneficial to such high-data-rate systems. Chapter 4 describes our novel, single-port algorithm. The algorithm requires a correlation matrix be estimated and then inverted. It will be shown that the correlation matrix estimate can be ill-conditioned and, consequently, we provide three alternative methods of overcoming this impediment, which ultimately results in three distinct, albeit closely related, algorithms being proposed. Finally, numerical simulations of our proposed beamforming algorithms are carried out in Chapter 5, and, for comparative purposes, simulations of a perturbation algorithm are also presented. 3

Chapter 2

Background

The primary objective of this chapter is to provide the reader with a solid understanding of array processing principles that are pertinent to understanding the proposed beamforming algorithm presented in the latter chapters of this work. More specifically, we introduce fundamental array processing concepts and differentiate between narrowband and broadband beamforming; we illustrate the idea of single-port versus digital beamformers and develop a system model; the topic of optimal beamforming is discussed and several adaptive beamforming algorithms are presented; finally, we briefly introduce our proposed single-port algorithm that will be described in detail in Chapter 4.

2.1 Array Processing Fundamentals

Let us consider the general case of an M-element1 array consisting of M arbitrarily arranged, omnidirectional antennas, situated within an external signal ﬁeld. Denoting the 3 position of the mth element in three-dimensional Euclidean space as pm ∈ R , and the signal induced on the mth element asx ˜(t, pm), we can represent the signals present on each element of the array in vector form as follows [2]: ⎡ ⎤ ⎡ ⎤ x˜(t, p1) x˜1(t) ⎢ ⎥ ⎢ ⎥ ⎢ . ⎥ ⎢ . ⎥ x˜(t) ⎣ . ⎦ = ⎣ . ⎦ , (2.1)

x˜(t, pM ) x˜M (t)

1Antenna and element are used interchangeably throughout this work. 4 Background

wherex ˜m(t) x˜(t, pm).

Fig. 2.1 Antenna array receiving plane-wave front [2].

The expression for x˜(t) in (2.1) is not very useful as it does not reveal any spatial characteristics of the signals being received. Therefore, we consider a more specific model in which the signals impinging upon the array emanate from a source located in the far field of the array and propagate in a homogeneous medium. Under such circumstances, the signal received at the array can be considered a plane-wave front [2, 3], as illustrated in Figure 2.1. If we assign p1 as the point of reference and position it at the origin of the coordinate system, the time delay experienced by the plane wave traveling from pm to p1 is T Θ pm τm(Θ)=− , (2.2) c where Θ ∈ R3 and c represent the plane wave’s normalized direction of propagation and speed, respectively [3]. Consequently, ifx ˜(t) x˜1(t) is redefined as the signal induced at the reference element by the plane wave, then (2.1) becomes ⎡ ⎤ x˜(t) ⎢ ⎥ ⎢ ⎥ ⎢ x˜(t + τ2(Θ)) ⎥ x˜(t)=⎢ . ⎥ , (2.3) ⎣ . ⎦

x˜(t + τM (Θ)) 2.1 Array Processing Fundamentals 5 and it is seen that the signals across the array are time-delayed versions of one another. The delays are proportional to the time required for the plane wave to reach successive elements, and are therefore directly dependent on Θ. Knowledge concerning the arrangement of the elements within the array will allow us to extract further spatial information about the incoming signal. This elemental arrangement is selected by the designer, and while numerous conﬁgurations have been studied in literature [2], the most analyzed [4] is referred to as the uniform linear array (ULA). The ULA is characterized by a linear arrangement of equispaced elements, as depicted in Figure 2.2. Assuming the system under analysis

Fig. 2.2 ULA receiving plane wave [3]. implements a ULA, the dimensions of Θ reduce to two. It follows that the plane wave can now be characterized in terms of the angle, θ ∈ [−π, π), that measures the deviation of the propagating direction vector from broadside of the array, as shown in Figure 2.2. We refer to θ as the angle-of-arrival (AOA). By, once again, designating the ﬁrst antenna as the reference, the delay in (2.2) simpliﬁes to

D(m − 1)sin(θ) τm(θ)= , (2.4) c 6 Background where D represents the spacing between elements. Consequently, (2.3) becomes ⎡ ⎤ ⎡ ⎤ x˜1(t) x˜(t) ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ x˜2(t) ⎥ ⎢ x˜(t + τ2(θ)) ⎥ x˜(t)=⎢ . ⎥ = ⎢ . ⎥ . (2.5) ⎣ . ⎦ ⎣ . ⎦

x˜M (t) x˜(t + τM (θ))

Throughout the remainder of this work we will assume a ULA implementation. The jus- tiﬁcation for this is two-fold: it facilitates the reader’s understanding of adaptive array concepts, and it is widely employed in research and practice.

2.1.1 Narrowband Beamforming

In contrast to systems such as microphone arrays (e.g., [5]), that receive signals at, or very close to baseband, this work focuses on wireless communication systems, and as a result x˜(t) can be considered a bandpass signal. It follows thatx ˜(t) can be represented in complex notation2 as [3] x˜(t)=x(t)ej2πfct, (2.6) where x(t) is the complex modulating function and fc is the carrier frequency. Substituting (2.6) in (2.5) produces ⎡ ⎤ x(t)ej2πfct ⎢ ⎥ ⎢ j2πfc(t+τ2(θ)) ⎥ ⎢ x(t + τ2(θ))e ⎥ x˜(t)=⎢ . ⎥ . (2.7) ⎣ . ⎦

j2πfc(t+τ (θ)) x(t + τM (θ))e M

If the bandwidth of x(t) is narrow such that

x(t) ≈ x(t + τm), for m =2, ..., M, (2.8) then (2.8) is referred to as the narrowband assumption/approximation for array signal processing [2, 3]. By incorporating the narrowband assumption of (2.8) into (2.7), the

2The physical signal received by the reference antenna is Re x(t)ej2πfct . However, at this point we only consider the analytic signal for ease of explanation. 2.1 Array Processing Fundamentals 7 received signal vector can be re-expressed as ⎡ ⎤ ⎡ ⎤ x(t)ej2πfct 1 ⎢ ⎥ ⎢ ⎥ ⎢ x(t)ej2πfc(t+τ2(θ)) ⎥ ⎢ ej2πfcτ2(θ) ⎥ ⎢ ⎥ j2πfct ⎢ ⎥ x˜(t)=⎢ . ⎥ = x(t)e ⎢ . ⎥ . (2.9) ⎣ . ⎦ ⎣ . ⎦ x(t)ej2πfc(t+τM (θ)) ej2πfcτM (θ)

We now deﬁne T s(θ) 1,ej2πfcτ2(θ),...,ej2πfcτM (θ) (2.10) which is referred to, among other names, as the steering vector [3], the array manifold vector [2], or the array vector [6]. Throughout the remainder of this work the term steering vector will be used when referencing s(θ). Making use of (2.4), as well as the relation fc = c/λc, where λc denotes the wavelength of the carrier signal, s(θ) can be further reduced to j2πD sin(θ) j4πD sin(θ) j2(M−1)πD sin(θ) T s(θ)= 1,e λc ,e λc ,...,e λc . (2.11)

By noting that the entries of s(θ) are dependent on both D and θ, we see that s(θ) encap- sulates the spatial properties of both the array and the received plane wave. Thus, signals received by the array will be associated with a steering vector based on their AOA. The uniqueness of this association is dependent upon the geometry of the array [3, 7]. In the proposition that follows it is shown that by setting D<λc/2 in a ULA conﬁguration, a unique association between s(θ) and θ results. λc ∈ − π π Proposition 1 A ULA with element spacing D< 2 ensures that each θ 2 , 2 is associated with a unique s(θ) [7].

Proof 2πD sin(θ) Let φ = λc , it follows that ⎡ ⎤ ⎡ ⎤ 1 1 ⎢ ⎥ ⎢ ⎥ ⎢ jφ ⎥ ⎢ jφ ⎥ ⎢ e ⎥ ⎢ e ⎥ ⎢ ⎥ ⎢ 2 ⎥ ⎢ ej2φ ⎥ ⎢ ejφ ⎥ s(φ)=⎢ ⎥ = ⎢ ⎥ . (2.12) ⎢ . ⎥ ⎢ . ⎥ ⎣ . ⎦ ⎣ . ⎦ (M−1) ej(M−1)φ ejφ

Since, ejφ = ej(φ±2zπ), z ∈ Z, therefore a one-to-one correspondence between φ and s(φ) 8 Background requires

−π ≤ φ<π,or, − ≤ 2πD sin(θ) π λc <π, or, − ≤ 2D sin(θ) 1 λc < 1. (2.13)

The range of θ bounds the value of sin(θ) such that −1 ≤ sin(θ) < 1. Substituting the maximum and minimum values of sin(θ) into (2.13), we see the inequality is satisﬁed if

2D < 1, or, λc λc D< . (2.14) 2

It is common practice to set D = λc/2 [2–4], and unless otherwise stated, this element spacing will be assumed from this point forward. Applying D = λc/2 to (2.11) results in an even further simpliﬁed version of s(θ), as shown below:

T s(θ)= 1,ejπ sin(θ),ej2π sin(θ),...,ej(M−1)π sin(θ) . (2.15)

Let us now consider the situation of receiving a desired signal with an AOA of θ0.So long as θ0 = 0, there will be delays amongst the signals on various antennas. As evident in (2.9), the narrowband assumption allows us to approximate these delays by phase shifts of the carrier frequency as opposed to time delays. Therefore, the task of coherently combining the signals reduces to the task of applying the appropriate phase shifts to each signal before the collection of signals is summed. A phase shift can be applied to the

jφw mth element through a complex weight, wm = |wm|e m , and as a result, the narrowband T beamformer uses complex weights, w =[w1,w2,...,wM ] , to ﬁlter x˜(t). A block diagram of a narrowband beamformer is illustrated in Figure 2.3. It follows that the output, y(t), of the narrowband beamformer is given by

y(t)=wH x˜(t) (2.16)

H j2πfct = w s(θ0)x(t)e (2.17) H = w s(θ0)˜x(t). (2.18) 2.1 Array Processing Fundamentals 9

Fig. 2.3 Narrowband beamformer [2].

From (2.18), it can be seen that y(t) consists of the received signal,x ˜(t), scaled by an inner H product, w s(θ0). This indicates thatx ˜(t) can be filtered through the selection of w.For instance, setting w = s(θ0), electrically steers the array towards the direction of θ0, and, as a result,x ˜(t) is amplified by a factor of M. Conversely, setting w⊥s(θ0), cancelsx ˜(t). It should be noted that when the weights are selected to steer the antenna towards a specific direction, θ , i.e., w = s(θ ), then θ is referred to as the look direction . | | | | H − π ≤ ≤ π We define the beam pattern, B(θ) ,as B(θ) w s(θ) , 2 θ 2 , for a given w3. Figure 2.4 presents a plot of |B(θ)| for a 10-element array implementing w = [1/10,...,1/10]T . The largest lobe is referred to as the main lobe, or equivalently the main beam, the smaller lobes are called the side lobes, and the nulls are the points at which |B(θ)| is minute, or equal to zero. Generally speaking, adjustments made to the phases of wm,m=1,...,M, will result in change of position for the main lobe, while the magni- tudes of the elements of wm,m=1,...,M, can be altered to steer the positions of the nulls. Thus, in theory, weights can be selected to steer the main lobe towards the desired direction, θ0, and position the nulls to reject interfering signals. The development in this section has been based, thus far, upon the narrowband assumption in (2.8). Therefore, it is of the utmost importance to understand when (2.8) is applicable. In [2], a heuristic definition of the narrowband assumption states that for (2.8) to be valid when receiving a signal with an AOA of θ0, we require

Bx˜ |τM (θ0)| << 1, (2.19)

3 H 2 π π The power pattern, P (θ), is closely related and we deﬁned it as P (θ) |w s(θ)| for − 2 ≤ θ ≤ 2 . 10 Background

Fig. 2.4 Beam pattern of uniformly weighted, 10-element array.

where Bx˜ denotes the bandwidth ofx ˜(t), and τM (θ0) is defined in (2.4). In [4] and [8], a metric known as the fractional bandwidth (FB), denoted by Fx˜ for signalx ˜(t), is defined as Bx˜ Fx˜ = , (2.20) fc and is used to determine the applicability of (2.8). More specifically, [4] states that the narrowband assumption should only be applied if Fx˜ < 0.01. Though useful, setting an upper limit of the FB is slightly suspect, as it doesn’t account for the number of antennas, AOA, etc., that impact (2.8). Consequently, this upper limit can be used as a general guideline but need not be followed imperatively. The use of fc in (2.20) arises from the dependence of D on λc, as explained in the previous section. Thus, as fc increases, the spacing D decreases, and the plane wave requires less time to propagate across the array. The key observation to note from (2.20) is that the amount of bandwidth allowable under the narrowband assumption is proportional to fc. A very involved examination of the narrowband assumption is presented in [9] and a narrowband test is hypothesized based on several parameters including the FB and the AOA of the received signal, as well as on the dimensions of the array. This test will be discussed in more detail in Chapter 3. In the following section we discuss the consequences of processing broadband4 signals with a narrowband beamformer. With the exception of Section 2.1.2, narrowband beam-

4The term broadband signal implies a signal that does not satisfy the narrowband assumption in (2.8). 2.1 Array Processing Fundamentals 11 forming systems will be the focus of this work and are assumed throughout the remainder of this thesis.

2.1.2 Broadband Beamforming

If Fx˜ is large such that the approximation in (2.8) no longer holds, then the relation presupposed in (2.9) fails, as x(t) can no longer be assumed coherent across the array. In such circumstances, the performance of the narrowband beamformer deteriorates [10]. One of the causes for this deterioration, which we will describe in this section, is referred to as array-induced inter-symbol interference [8], and occurs as a result of the narrowband beamformer failing to compensate for the time delays between xm(t) and x(t) for m = . If (2.8) is not satisﬁed then x˜(t) must be expressed in the same form as (2.7). Now, if the narrowband beamformer sets the weights to compensate for the phase shifts, i.e.,

1 j2πfcτm(θ) wm = M e for m =1,...,M, then the output of the narrowband beamformer becomes

H ˜ y(t)=w x(t) M j2πfct = x(t + τm(θ)) e , (2.21) m=1 and the modulating functions are no longer being added coherently. Consequently, broadband beamformers must be used to overcome the shortcomings of narrowband beamformers when processing signals possessing large FBs. One popular type of broadband beamformer [10–12] is shown in Figure 2.5, where T1(θ),...,TM (θ), are true time-delay lines (as opposed to the phase shifters employed by narrowband beamformers) that allow the array to be steered towards angle θ0. The variables wm,l and T are the tap weights and inter-tap delay spacing of the tapped-delay-line (TDL) filters, respectively. The TDL filters can either be implemented using analog components at passband, or digitally at baseband using a finite impulse response (FIR) filter, where the latter performs a quadrature demodulation after the steering time delays [2]. The TDLs are typically used for interference cancellation and optimization of the SINR or MSE [3]. They can be designed to have phase responses that vary with frequency and compensate for the phase discrepancies of different spectral components at each antenna [4]. However, disadvantages of broadband beamformers include higher costs and larger chip area due to the true 12 Background

Fig. 2.5 TDL broadband beamformer [3]. time-delay lines and TDLs. For these reasons, narrowband beamforming is desirable over broadband beamforming so long as the narrowband assumption holds. A practical example of a system which can assume narrowband beamforming despite processing signals of considerable bandwidth is one that adheres to the IEEE 802.11n wireless standard. The 802.11n standard speciﬁes that systems can operate at the 5-GHz frequency band while using a 40-MHz channel [13], which produces a minute FB of Fx˜ =0.008. Similarly, the proposed wireless systems that will operate around the 60-GHz band can also make the narrowband assumption despite receiving large-bandwidth signals (i.e., > 1 GHz). A detailed discussion concerning 60-GHz communication systems along with a justiﬁcation for such systems employing the narrowband assumption will be provided in Chapter 3.

2.2 Adaptive Beamforming Architectures

The manner in which the entries of w are physically applied to the received signals is an important design consideration that has substantial impact on the cost, size, and power consumption of an adaptive array. In general, they can be implemented digitally, by ﬁrst converting all received information to the digital domain, or in analog using devices such as analog phase shifters, ampliﬁers, and attenuators5. The former method is referred to as

5Throughout this work we will often refer to the analog device that adjusts the amplitude of a signal using the general term amplitude-control element. 2.2 Adaptive Beamforming Architectures 13 digital beamforming [14, 15] or multiport beamforming [15, 16], and the latter as single-port beamforming [17, 18], microwave beamforming [14, 15], RF beamforming [19], or single- receiver beamforming [3, 20, 21]. In this work, we will use the terms digital beamforming and single-port beamforming to differentiate between the two methods. General block diagrams of the digital beamforming architecture and single-port beamforming architecture are presented in Figures 2.6 and 2.7. As is common in array processing literature [2, 3], they do not show components such as preamplifiers, filters, etc.

2.2.1 Digital Beamforming

The architecture of a digital beamformer is displayed in Figure 2.6. The received signals, x˜m(t),m=1,...,M, are each downconverted to an intermediate frequency (IF), or to baseband before being sampled by an ADC. If downconverting to an IF, the signals must ﬁrst be sampled at the Nyquist frequency, and then digitally downconverted to obtain the corresponding complex-baseband representation [22]. Conversely, if each signal is downconverted to baseband by an I/Q demodulator, then two ADCs are required to sample both the in-phase and quadrature components of the baseband signal at a relaxed sampling rate [22]. Both procedures result in the digital signal processor (DSP) having a digital representation of the complex-baseband signals from each element, and as a result, the complex weight vector, w, can be easily applied in the digital domain.

Fig. 2.6 Digital beamforming architecture.

Digital beamformers are advantageous from a signal processing perspective, as they have access to the received signals at each element, and as a result are capable of utilizing advanced beamforming algorithms. However, implementation of these systems is expen- 14 Background sive and energy-demanding, as they require a full downconversion receiver and one, or two, ADC(s) per antenna. The high cost of the digital beamformer has limited its use in commercial wireless applications [15].

2.2.2 Single-Port Beamforming

The single-port beamforming architecture is shown in Figure 2.7. Beamforming is accomplished using analog devices such as phase shifters, ampliﬁers and/or attenuators at RF6 before the signal is downconverted and sampled. The actual beamforming algorithm (within the beamforming control block in Figure 2.7) is executed using a DSP.

Fig. 2.7 Single-port beamforming architecture.

The primary advantages of this structure are lower cost, reduced power consumption, and smaller size, in comparison to the digital beamformer [8, 15]. While the digital beamformer requires M downconversion receivers and M,or2M, ADC(s), the analog beamformer requires a total of one receiver and one, or two, ADC(s)7. However, beamforming algorithms become more diﬃcult to execute, as the system only has access to the output signal y(t). Therefore, many of the advanced beamforming techniques that have been proposed for digital beamforming systems cannot be implemented directly in single-port architectures. 6Analog beamforming can also be performed at an IF or at baseband [22, 23], however these implemen- tations are more costly as they require one downconversion receiver per antenna. 7Throughout this thesis we assume the signal is downconverted to baseband and the in-phase and quadrature-phase signals are sampled using two ADCs, as shown in Figure 2.7. 2.3 System Model 15

2.3 System Model

We consider an M-element array receiving a desired signal in the presence of Q unwanted directional signals and background noise. The signal incident on the array due to the qth, q =0,...,Q, far-ﬁeld point source can be expressed in vector form as j2πfct ˜iq(t)=Re iq(t)s(θq)e , (2.22) where iq(t)andθq are the complex modulating function and AOA, respectively, associated T jπsin(θq) j(M−1)πsin(θq) with the qth source, and s(θq)= 1,e ,...,e . It is assumed that 2 2 iq(t), q =0,...,Q, is a zero-mean process with a variance σq =E[|iq(t)| ], and iq1(t)is independent of iq2(t) for q1 = q2. If we now consider the single-port structure, as shown in Figure 2.7, the complex- baseband signal, y(t), at the input to the ADC can be represented by

y(t)=wH x(t), (2.23) where the signal vector, x(t), is modeled as

Q x(t)= iq(t)s(θq)+n(t), (2.24) q=0 and n(t) ∈ CM denotes a zero-mean, additive white Gaussian noise (AWGN) vector whose entries represent the background/electronic noise present on each element. It is assumed that the entries of n(t) are independent of iq(t) for q =0,...,Q. Moreover, in addition to being a zero-mean process, it is assumed that x(t) is also a stationary, ergodic process.

The signal in (2.23) is sampled at time instances kTs, where k is the discrete-time sample index and Ts is the sampling period, to produce the following discrete-time sequence,

y(k)=wH x(k), (2.25) where Q x(k)= iq(k)s(θq)+n(k), (2.26) q=0 and Ts has been omitted for ease of notation. By combining (2.25) with (2.26), we can 16 Background obtain an equivalent expression for y(k), Q H y(k)=w iq(k)s(θq)+n(k) , (2.27) q=0 and the discrete sequence given by (2.27) is the only information available to the single-port beamformer to carry out its beamforming algorithm. Similarly, the digital beamformer in Figure 2.6 samples the signals given by each entry of (2.24) to obtain (2.26). After x(k) has been obtained, the weights are applied digitally, as explained in Section 2.2.1, to obtain y(k) in (2.27). Thus, we see that the digital beamformer has access to x(k), while the single-port structure has knowledge of y(k) only. It should be noted that in adaptive beamforming algorithms, which will be discussed towards the end of this chapter, the weights change in time and this time dependency will be represented using w(t)orw(k). The array correlation matrix, R, which will be encountered frequently throughout the upcoming discussions, is deﬁned as

R =E x(t)xH (t) . (2.28)

Similarly, correlation matrices associated with the desired signal, interfering signals, and noise are given, respectively, as

∗ H 2 H R0 =E i0(t)s(θ0)i0(t)s (θ0) = σ0s(θ0)s (θ0), (2.29)

Q Q ∗ H 2 H RI = E iq(t)s(θq)iq(t)s (θq) = σq s(θq)s (θq), (2.30) q=1 q=1 and,

H 2 RN =E n(t)n (t) = σN I, (2.31)

2 ∗ where σN =E[nm(t)nm(t)] for m =1,...,M, denotes the variance of the noise on each element. Since we have assumed the sources are uncorrelated, we can express R equivalently 2.3 System Model 17 using

R = R0 + RI + RN , (2.32) Q 2 H 2 = σq s(θq)s (θq)+σN I. (2.33) q=0

It is also helpful to deﬁne a noise-plus-interference matrix,

RI+N = RI + RN . (2.34)

For a given w, the instantaneous output power of the beamformer at time t (which can be replaced by the sampling instance k if considering the digital beamformer) is

P (w,t)=|y(t)|2 = wH x(t)xH (t)w, (2.35) and the corresponding mean output power is

P (w)=EwH x(t)xH (t)w = wH Rw. (2.36)

Following a similar procedure, the output power due to the desired signal, noise, and interference-plus-noise are H P0(w)=w R0w, (2.37)

H PN (w)=w RN w, (2.38) and H PI+N (w)=w RI+N w, (2.39) respectively. Finally, we deﬁne the SNR and SINR, for a given w,as

H w R0w SNR = H , (2.40) w RN w 18 Background

and H w R0w SINR = H . (2.41) w RI+N w

2.3.1 Eigenvalue Decomposition of R and R−1

We will brieﬂy introduce the notion of expressing R in terms of its eigenvalues and eigenvectors. The usefulness of this representation will become more apparent during the discussion of our proposed beamformer in Chapter 4. The eigenvalue decomposition of R can be formulated as

R = QΛQH (2.42) ⎡ ⎤ ⎡ ⎤ H λ1 0 q ⎢ ⎥ ⎢ 1 ⎥ ⎢ ⎥ ⎢ H ⎥ ⎢ λ2 ⎥ ⎢q2 ⎥ = q1, q2,...,qM ⎢ . ⎥ ⎢ . ⎥ , (2.43) ⎣ .. ⎦ ⎣ . ⎦ H 0 λM qM where λm denotes the mth eigenvalue of R, whose corresponding eigenvector, qm, is the mth column vector of Q. Throughout the remainder of this thesis, it will be assumed that the eigenvalues, and their corresponding eigenvectors, have been arranged such that

λ1 ≥ λ2 ≥ ... ≥ λM . The hermitian structure of R enables us to conclude that Q is a unitary matrix, and λm,m=1,...,M, are real-valued. Additionally, the eigenvalues of R are known to be positive as a result of the matrix being positive deﬁnite. An equivalent formulation of (2.42) can be stated in terms of the summation,

M H R = λmqmqm. (2.44) m=1

The inverse of R can be expressed in a like manner as follows:

M −1 H −1 −1 −1 −1 H 1 H R =(Q ) Λ (Q) = QΛ Q = qmqm. (2.45) m=1 λm

If a desired plane wave as well as Q interfering plane waves impinge upon the array, 2.4 Optimal Beamformers 19 and (Q +1)

Q+1 M H H R = λmqmqm + λmqmqm, (2.46) m=1 m=Q+2 where the ﬁrst Q + 1 eigenvalues are referred to as principle eigenvalues [2], and correspond to the desired and interfering plane waves in addition to the background noise. The remaining M − (Q + 1) eigenvalues are associated with the background noise only, and under the assumption that the noise variance is equal at each antenna, it follows that

λQ+2 = λQ+3 = ...= λM .

2.4 Optimal Beamformers

The conventional, minimum variance distortionless response (MVDR), and minimum mean- square error (MMSE) beamformers are presented in this section. As will be explained, the conventional beamformer is suitable for environments that do not contain any directional interference, whereas the MVDR and MMSE beamformers are both capable of minimizing the eﬀects of interfering signals by maximizing the SINR and MSE, respectively, where the MSE will be deﬁned in Section 2.4.3. Without loss of generality, we use the continuous-time input-output relation y(t)=wH x(t) given by (2.23) to analyze the optimal weights in this section.

2.4.1 Conventional (Delay-and-Sum) Beamformer

The conventional beamformer implements a weight vector, wcnv, whose entries are of equal magnitude, but of varying phase. The phases are selected such that the desired signal, arriving from direction θ0, appears coherent across the array. This is accomplished by setting the weights as follows: 1 wcnv = s(θ0). (2.47) M Theorem 1 The weights, wcnv, deﬁned in (2.47) optimize the SNR when no directional interference is present [3].

Proof When no interference is present the signal in (2.24) can be represented as

x(t)=i0(t)s(θ0)+n(t), (2.48) 20 Background and the resulting output signal becomes

H y(t)=w (i0(t)s(θ0)+n(t)) . (2.49)

H The desired signal component in (2.49) is given by i0(t)w s(θ0), and the noise component by wH n(t). As stated in [3], maximizing the SNR of y(t) is equivalent to minimizing H 2 H H PN (w)=w RN w, while maintaining a constant signal power, P0(w)=σ0w s(θ0)s (θ0)w. As a result, the maximization of the output SNR can be obtained by solving the following optimization problem: H Minimize w RN w H (2.50) Subject to w s(θ0)=1. Following a similar procedure as given in [6], this type of constrained optimization problem can be solved through the use of Lagrange multipliers. The Lagrangian, J(w,β), of (2.50) can be deﬁned as H w RN w H J(w,β)= + β 1 − w s(θ0) , (2.51) 2 where β represents the Lagrange multiplier. Setting the gradient, ∇wJ(w,β), to zero produces

∇wJ(w,β)=RN w − βs(θ0)=0. (2.52)

Solving for w of (2.52) gives −1 w = RN βs(θ0). (2.53)

By substituting (2.53) into the constraint listed in (2.50), we obtain

1 β = H −1 , (2.54) s (θ0)RN s(θ0) which can be used in (2.53) to solve for w:

−1 RN s(θ0) w = H −1 . (2.55) s (θ0)RN s(θ0)

Substituting the expression for RN given by (2.31) into (2.55) produces

s(θ0) w = = wcnv. (2.56) M 2.4 Optimal Beamformers 21

Thus, the conventional beamformer is optimal in the SNR sense under the assumption of uncorrelated noise when no directional interference exists. However, in the presence of interference, the performance reduces substantially [3], as the beamformer makes no eﬀort to reject the unwanted signals. Furthermore, the AOA of the desired signal, θ0, must be known in order to calculate s(θ0). If the angle is not known, it must be estimated by means of an AOA algorithm such as multiple signal classiﬁcation (MUSIC) [24], minimum norm method [25], estimation of signal parameters via rotational invariance techniques (ESPRIT) [26], etc.

2.4.2 MVDR Beamformer

The MVDR beamformer strives to reject directional interference by minimizing the mean output power, P (w), while ensuring a unity response in the look direction. The weights of the MVDR beamformer are given by

−1 R s(θ0) wmvdr = H −1 , (2.57) s (θ0)R s(θ0) where θ0 is the AOA of the desired signal. As stated in [3], the weights of (2.57) are found to be the solution to the following optimization problem:

Minimize wH Rw H (2.58) Subject to w s(θ0)=1.

The claim made above can easily be veriﬁed by solving (2.58) in an analogous manner as was followed during the proof of Theorem (1). It can be further shown that wmvdr optimizes the SINR.

Theorem 2 The MVDR weights, wmvdr in (2.57), maximize the SINR at the output of the beamformer [3].

Proof Maximizing the SINR, is equivalent to minimizing the total interference-plus-noise H 2 H H power, PI+N (w)=w RI+N w, while maintaining a constant P0(w)=σ0w s(θ0)s (θ0)w [3]. Thus, maximizing the SINR reduces to solving the following optimization problem

H Minimize w RI+N w H (2.59) Subject to w s(θ0)=1, 22 Background

In the same manner as was followed in the proof of Theorem (1), the solution to this problem is found to be −1 RI+N s(θ0) w = H −1 . (2.60) s (θ0)RI+N s(θ0)

Now, in a similar method as described in [3], we can show that wmvdr in (2.57) is 2 H identical to (2.60). By making use of the relation R = R0 +RI+N = σ0s(θ0)s (θ0)+RI+N , and the Matrix Inversion Lemma 8, R−1 can be represented in the following form:

2 −1 H −1 σ R s(θ0)s (θ0)R −1 −1 − 0 I+N I+N R = RI+N H −1 2 . (2.61) 1+s (θ0)RI+N s(θ0)σ0

The numerator of wmvdr can therefore be rewritten as

2 −1 H −1 σ R s(θ0)s (θ0)R s(θ0) −1 −1 − 0 I+N I+N R s(θ0)=RI+N s(θ0) H −1 2 , (2.62) 1+s (θ0)RI+N s(θ0)σ0 and after algebraic manipulation, (2.62) can be reduced to

−1 −1 RI+N s(θ0) R s(θ0)= H −1 2 . (2.63) 1+s (θ0)RI+N s(θ0)σ0

Using (2.63), the denominator of (2.57) can be easily re-expressed as

H −1 H −1 s (θ0)RI+N s(θ0) s (θ0)R s(θ0)= H −1 2 , (2.64) 1+s (θ0)RI+N s(θ0)σ0 and combining (2.57), (2.63), and (2.64) we obtain

−1 RI+N s(θ0) wmvdr = H −1 . (2.65) s (θ0)RI+N s(θ0)

Therefore, we see from (2.65) that wmvdr in (2.57) is equivalent to the weights given by (2.60) that maximize the output SINR.

In practice, it is typically much easier to obtain an estimate of R than RI+N [3], which −1 is why we express wmvdr in terms of R . However, it should be noted that this equivalence

−1 8For some invertible matrix A ∈ CM×M , and a vector x ∈ CM , one can express [3]: A + xxH = −1 − A−1xxH A−1 A 1+xH A−1x 2.4 Optimal Beamformers 23

only holds in the absence of errors; if s(θ0) contains errors (e.g., the estimate of θ0 is inaccurate), the performance of a beamformer incorporating RI+N in (2.60) will perform better than a beamformer incorporating R in (2.57) [27]. The two are sometimes distinguished with diﬀerent designations. For instance, in [2], the beamformer incorporating the weights in (2.57) is called a minimum power distortionless response (MPDR) beamformer while MVDR is reserved for a system that utilizes (2.60); in [3], the names signal-plus-noise matrix inverse (SPNMI) processor and noise-alone matrix inverse (NAMI) processor are used to describe systems employing (2.57) and (2.60), respectively. As was the case in the conventional beamformer, the AOA of the desired signal must be known to the processor in order to implement wmvdr.

2.4.3 MMSE Beamformer

The MMSE beamformer, like the MVDR beamformer, is capable of canceling interference and minimizing noise. However, the MMSE is designed to minimize the MSE between a (known) reference signal, d(t), and the beamformer output y(t). Unlike the MVDR, the

MMSE does not require knowledge of θ0, but it must have knowledge of d(t), which can be accomplished by means of a training sequence sent by the transmitter and known to the receiver. The weights used by the MMSE are given as

−1 wmmse = R z, (2.66) where the vector z represents the correlation between d(t) and x(t),

z =E[x(t)d∗(t)] . (2.67)

Theorem 3 The MMSE weights, wmmse, minimize the MSE between d(t) and y(t) [3].

Proof In a similar manner to the procedure described in [3], we deﬁne the error (t)= d(t) − y(t), and solve the problem by representing the MSE, ξ(w), as a quadratic function 24 Background of w:

ξ(w)=E| (t)|2 ∗ =E[(d(t) − y(t)) (d(t) − y(t)) ]

=E|d(t)|2 + wH Rw − wH z − zH w. (2.68)

Taking the gradient of ξ(w) with respect to w produces

∇wξ(w)=2Rw − 2z, (2.69) and setting it to zero, results in

−1 w = R z = wmmse. (2.70)

A ﬁlter implementing these weights is commonly referred to as a Wiener ﬁlter [3]. The value of the MMSE is determined by substituting (2.66) into (2.68), and is found to be

MMSE = E |d(t)|2 − zH R−1z. (2.71)

For a given weight, w, the MSE can be calculated using

H ξ(w) = MMSE + (w − wmmse) R (w − wmmse) , (2.72) which is easily obtained [28] using (2.66), (2.68), and (2.71).

2.5 Adaptive Digital Beamforming

In the previous section, wmvdr was derived assuming θ0 and R were known. Even if θ0 is known a priori, by means of an AOA algorithm or otherwise, the issue still remains that an estimate of R is required to implement the beamformer. Likewise, calculating wmmse requires knowledge of both R and z, neither of which are known, and both of which must therefore be estimated. Two prominent, adaptive techniques designed to estimate the weights of the MVDR and MMSE beamformers will be discussed; these algorithms assume that x(k) is known to the 2.5 Adaptive Digital Beamforming 25 receiver (i.e., a digital beamforming architecture is employed). It must be pointed out that this is not a comprehensive review of adaptive algorithms for digital beamformers (a more complete listing can be found in [2]). The two algorithms presented in this section have been selected according to their pertinence to material discussed throughout the remainder of this work.

2.5.1 Sample Matrix Inversion (SMI)

An estimate of R, called the sample matrix, can be formulated using K samples of x(k) and calculating a time average as follows [2, 3, 6]:

K 1 Rˆ = x(k)xH (k). (2.73) K k=1

Estimating R in this fashion is referred to as a block-adaptive approach, as it requires a block of data to carry out the estimation [6]. It can easily be shown that Rˆ is an unbiased estimate of R [3]. It follows that the estimate of R−1, which we denote as Rˆ −1, can be calculated by taking the inverse of the sample matrix in (2.73), hence the name sample matrix inversion. −1 Therefore, Rˆ can be inserted directly into (2.57) to estimate wmvdr. The estimate of z in (2.67) can be found in a parallel manner to (2.73) using [6]

K 1 zˆ = x(k)d∗(k). (2.74) K k=1

Substituting Rˆ and zˆ into (2.66) allows us to estimate wmmse directly. Moreover, wmmse can be found by solving the following system:

Rwˆ mmse = zˆ. (2.75)

A simple, yet effective, adjustment that can be applied to (2.73) is the addition of a fixed diagonal matrix: K 1 H Rˆ dl = x(k)x (k)+νI, (2.76) K k=1 where ν is a constant. This procedure, known as diagonal loading, can result in significant 26 Background benefits [2], such as ensuring Rˆ is non-singular in circumstances where K is small, improving the SINR performance of the MVDR beamformer, and decreasing any negative side-effects caused by small fluctuating noise eigenvalues in Rˆ [2].

2.5.2 Least-Mean-Squares (LMS)

The LMS algorithm, sometimes speciﬁed as unconstrained LMS [3] to distinguish it from a related algorithm known as constrained LMS (CLMS)9, can be used to estimate the MMSE weights in (2.66) without requiring Rˆ −1 be calculated. As will be discussed, the weights are updated as new samples are acquired, and the technique is accordingly referred to as iterative. In accordance with the system model presented in Section 2.3, the index k is used to enumerate the iterative step. In Section 2.4.3 we deﬁned the performance surface, or cost function, as ξ(w), which is a quadratic function of w. This implies that ξ(w) has the shape of an elliptic paraboloid with one global minimum [6]. As previously explained, the weights corresponding to this minimum can easily be found by calculating the gradient of ξ(w), equating it to zero, and solving for wmmse.

Rather than calculating wmmse directly, the LMS algorithm [11] applies the method of steepest descent using an estimate of the gradient at each step, by updating the weights as follows: w(k +1)=w(k) − μ ∇ˆ wξ(w(k)) w=w(k) , (2.77) where μ is a positive scalar that represents the step size and H ∗ ∇ˆ wξ(w(k)) w=w(k) =2x(k)x (k)w(k) − 2x(k)d (k) (2.78) is the gradient estimate of the MSE at the kth iteration. A parameter of particular importance in (2.77) is μ, as it controls the speed of convergence and the steady-state error of the algorithm. Stability is ensured if [29]

1 0 <μ< , (2.79) λmax where λmax is the largest eigenvalue of R.

9The constrained LMS is discussed in Section 2.6.4. 2.6 Adaptive Single-Port Beamforming 27

2.6 Adaptive Single-Port Beamforming

The SMI and LMS algorithms that have been presented in the preceding sections require knowledge of x(k), which is not available to a single-port beamformer. In this section, we review several techniques that circumvent this obstacle and can accomplish beamforming using a single-port structure. We place particular emphasis on the well-known perturbation algorithm, and before concluding the chapter, we provide a brief introduction to our proposed single-port algorithm that will be explained in detail in Chapter 4.

2.6.1 Null Steering

The null-steering algorithm [30] is used to cancel interfering plane waves arriving from known directions by producing nulls in the direction of their AOAs. Following the notation of Section 2.3, the desired AOA is denoted by θ0, and the interfering AOAs by θ1,...,θQ.

Furthermore, a matrix A and a vector e1 are deﬁned as

A [s(θ0), s(θ1),...,s(θQ)] , (2.80) and T e1 =[1, 0,...,0] . (2.81)

A weight vector, wns, that enforces a unity response in the direction of θ0, while canceling the signals arriving from angles θ1,...,θQ, can be given by

H T wnsA = e1 . (2.82)

Assuming the number of columns in A is less than or equal to Q +1,wns can be solved as H T H H −1 wns = e1 A AA . (2.83)

A performance analysis of this null-steering beamformer is provided in [31]. Although the null-steering algorithm can be used by a single-port architecture, much literature related to this algorithm assumes a digital beamforming architecture is used, and therefore the received signal x(k) is known [31, 32]. The knowledge of x(k) is quite advantageous in estimating θ0,...,θQ, as it allows for the use of well-studied AOA algorithms, e.g., MUSIC and ESPRIT, which require an estimate of R as well as its eigenvalues and 28 Background eigenvectors; as explained in Section 2.5.1, the estimate of R can be calculated quite easily when x(k) is known, however such an estimate is not readily available for a single-port beamformer. Therefore, a more primitive algorithm, such as the Bartlett method [30], must be employed. In addition, due to the large number of AOA estimates that are required, poor estimation of the AOAs have a greater impact on the performance degradation of the beamformer.

2.6.2 Aerial Beamforming

Aerial beamforming is a method of further reducing the cost, size, and power dissipation of a single-port beamforming architecture, and is based upon the electromagnetic coupling between elements of an array [19]. The signal from only one active antenna element is downconverted, sampled, and used by an optimizing algorithm to adjust the loaded re- actance of the remaining passive elements. A well-studied aerial beamforming structure is known as the electronically steerable passive radiator (ESPAR) [33–35], which arranges the passive antennas at equal intervals around the active antenna. However, as stated in [14, 15], the inherent structure of ESPAR and aerial beamformers makes them slow and limits their performance.

2.6.3 Phased-Array

Numerous systems have been proposed that perform spatial ﬁltering by employing phase shifters without amplitude-control elements. Some proposals (e.g. [36]), assume a model in which only the desired signal is present, and consequently, strive to maximize the output power of the beamformer. Other phased-array antennas use a reference signal and beamform by attempting to minimize the error (e.g. [37, 38]). The phased-array antennas are advantageous from a cost perspective, however without amplitude control they are unable to implement optimal beamforming weights in the MMSE or SINR sense, and have little control over nulling interference.

2.6.4 Perturbation Algorithms

The gradient-based perturbation algorithm proposed in [20, 21] applies perturbations to the weights, observes the corresponding change in the output, and estimates the power gradient used to update the CLMS algorithm. For completeness, we will provide a brief description 2.6 Adaptive Single-Port Beamforming 29 of the CLMS algorithm. In contrast to the LMS algorithm, the CLMS algorithm does not require a reference signal, but does require knowledge of θ0. The CLMS algorithm strives to estimate wmvdr by iteratively ﬁnding the weights that minimize the output power, while H applying the constraint w (k)s(θ0) = 1 at each iteration. It follows that the weight vector at the kth iteration is updated using [39] s(θ0) w(k +1)=P` w(k) − μ ∇ˆ wP (w(k)) w=w(k) + , (2.84) M M where μ ∈ R denotes the step size, ∇ˆ wP (w(k)) w=w(k) ∈ R is the estimate of the power gradient at the kth iteration, and P` ∈ CM×M is the projection operator matrix deﬁned as H s(θ0)s (θ0) P` = I − . (2.85) M The estimate, ∇ˆ wP (w(k)) w=w(k) , at the kth iteration is calculated using a sequence, S, of perturbation vectors, S = {δ(1), δ(2),...,δ(L)} , (2.86) where δ( ) ∈ CM , =1,...,L. As a result, at each iteration the nominal weight, w(k), is perturbed L times; this is called a perturbation cycle [20]. Consequently, the instantaneous output power at the th perturbation of the kth iteration is

H H P (w(k), )=(w(k)+γpδ( )) x(ks + − 1)x (ks + − 1) (w(k)+γpδ( )) , (2.87) where γp ∈ R is the magnitude of the perturbation, and ks denotes the sample index at which the perturbation cycle begins. Using the values obtained from (2.87), an estimate of the power gradient can be made as follows:

L 1 ∇ˆ wP (w(k)) w=w(k) = P (w(k), ) δ( ). (2.88) p Lγ =1

If {δ(1),...,δ(L)} are chosen to satisfy several desirable properties including orthogonality, zero mean, and odd symmetry10, then it can be shown that (2.88) is unbiased for a given w(k) [20]. The perturbation sequences presented in [20] are of length L =4M.

10These properties are deﬁned in [20]. 30 Background

There have been many variations of the perturbation algorithm suggested in literature. A perturbation sequence is proposed in [17] that makes use of linear constraints, known a priori, to reduce the length of the perturbation sequence required to estimate the power gradient. In [15, 18, 40], perturbation-based systems are proposed that increase the algorithm’s speed of convergence by sampling the beamformer’s output fast enough as to allow successive samples to be highly correlated. A phase-only perturbation algorithm is presented in [16]. Although the algorithm presented in this section estimates the optimal MVDR weights via the CLMS algorithm, the use of perturbation sequences can be applied to other processors [3], and a similar procedure could be used to estimate the MMSE weights. The ability to estimate optimal beamforming weights in the SINR sense suggests that the perturbation method is a superior single-port algorithm than the previously discussed algorithms in this section. The downside of the perturbation algorithm, however, stems from its necessity to continuously switch the weights. This not only consumes more power, but is also more diﬃcult to implement in practice as it requires complex synchronization between the ADC, the analog weights, and the DSP; further complications arise due to the settling time of the analog weights. In addition, the weighting rate limits the speed of beamforming, and as a result, slow-switching devices that are cheap and power-eﬃcient, such as micro-electro mechanical systems (MEMS) phase shifters, are not well-suited for the perturbation algorithm.

2.6.5 Proposed Single-Port Algorithm

In Chapter 4 we propose a single-port beamforming algorithm that estimates the MMSE weights, wmmse. However, unlike the perturbation algorithms, which may require hundreds or thousands of weight changes, our algorithm requires the weights to be switched M 2 times. Each weighting will process a block of K samples before switching to the next weight. In this way, we alleviate many of the aforementioned diﬃculties associated with the perturbation algorithms. In the following chapter we discuss a prospective application of our algorithm - 60-GHz receivers. We stress that our algorithm is not designed speciﬁcally for a 60-GHz system, but include this discussion simply as an illustration of a relevant use for the algorithm. 31

Chapter 3

Adaptive Beamforming for 60-GHz Receivers

Wireless communication systems operating in the frequency band around 60 GHz have recently garnered considerable attention as a result of the vast quantity of unlicensed spectrum allocated to this frequency range, as listed in Table 3.1. The abundance of devices saturating the widely-popular unlicensed spectrum around 2.4 GHz and 5 GHz, coupled with the ever-increasing demand to maximize wireless data transmission rates, have led to design eﬀorts focused on 60-GHz systems capable of exploiting the available bandwidth. Al- though 60-GHz communication remains in its infancy, several standards have been recently released by ECMA International [41], IEEE 802.15.3c [42], and WirelessHD [43].

Table 3.1 Allocation of unlicensed bandwidth around 60 GHz [44, 45] Geographic Location Unlicensed Spectrum (GHz) USA 57 - 64 CANADA 57 - 64 JAPAN 59 - 66 AUSTRALIA 59.4 - 62.9 KOREA 57 - 64 EUROPE 57 - 66

Significant attenuation due to free-space path loss, oxygen absorption [46], and increased penetration losses [44, 47], proves to be the prominent challenge facing 60-GHz technology. As a result, these systems are inherently suited for short-distance applications such as 32 Adaptive Beamforming for 60-GHz Receivers wireless personal area networks (WPANs), wireless local area networks (WLANs), wireless high-definition multimedia interface (HDMI) streaming for high-definition televisions, and wireless docking stations [45, 48]. Due to the increased gain offered by an array over a single antenna, the use of antenna arrays in 60-GHz transmitters and receivers has been suggested as a means of overcoming the aforementioned free-space path loss and oxygen absorption problems. Furthermore, incorporating beamforming capabilities into the 60- GHz receiver provides a means of steering the beam of the array towards the transmitter as well as rejecting co-channel interference; these latter two properties allow multiple, co- channel WPANs and WLANs to operate in close proximity to one another. This chapter presents an analysis of several important design criteria that must be considered when designing a 60-GHz beamforming receiver, and briefly clarifies the ways in which our proposed beamforming algorithm, that will be presented in detail in Chapter 4, satisfies these criteria. Finally, we introduce the 60-GHz channel model that will be used in future simulations.

3.1 Design Considerations of 60-GHz Receiver

Minimization of cost, power consumption, and size, in addition to interference mitigation and fast-beamforming capabilities, are all essential design criteria for the commercial success of a 60-GHz beamforming receiver. In the discussions that follow, we explain the beneﬁts of our algorithm in the context of these criteria. However, given that our algorithm has been designed for a system in which the narrowband beamforming assumption holds, we ﬁrst discuss the validity of this assumption for 60-GHz communications.

3.1.1 Narrowband Beamforming for 60-GHz Receivers

A discussion explaining the validity of the narrowband beamforming approximation for a 60-GHz antenna array is given in [8], and there have been several proposed 60-GHz systems that incorporate phase shifters as opposed to time delays to perform beamforming [49–52]. There are several contributing factors that explain why this assumption holds for such a large bandwidth. First and foremost, as explained in Section 2.1.1, the narrowband approximation for array signal processing is not dependent solely upon the bandwidth, but rather on the fractional bandwidth, of the received signal. Furthermore, as explained in [9], small input SNRs, and a low number of antennas are also contributing factors to the 3.1 Design Considerations of 60-GHz Receiver 33 narrowband approximation. The projected signal bandwidths of 60-GHz systems do not create excessive fractional bandwidths, as they are offset by the high carrier frequency, and as was elucidated earlier in this chapter, the signals received are highly attenuated. Unlike military radar applications, which can require tens of thousands of antennas, the intended usage of the 60-GHz spectrum targets commercial WPAN-type systems, and consequently a small number of antennas, typically ranging from four [52] to 16 [53], is required [8]. According to [9], the narrowband approximation for an M-element ULA receiving an incoming signal, ˜iq(t),q=0,...,Q, is valid if Mσ2 sin( Mπς) q − 2 2 1 πς < 1, (3.1) 2σN M sin( 2 ) where sin(θq)F˜i ς = √ q , (3.2) 3 ˜ and F˜iq is the FB of iq(t). The definition in (3.1) cannot be used to make a general statement regarding the validity of the narrowband approximation for all 60-GHz arrays, as it relies 2 2 on system-specific parameters such as M, F˜iq , σq , and σN . Let us consider a 60-GHz system employing a 6-element ULA, and receiving a signal possessing a fractional bandwidth of ≈ F˜iq =1.705 GHz/61.380 GHz 0.028 (these are similar parameters to the 60-GHz system proposed in [49]). Furthermore, we test for the worst-case scenario, which occurs when

θq = ±π/2, as the plane wave will experience the longest delay between reaching the ﬁrst and Mth antenna. Substituting the aforementioned values into (3.1) and (3.2), we obtain the following inequalities:

2 2 σq σq 2 < 88.7, or, 2 < 19.5dB. (3.3) σN σN

Therefore, so long as the received signals satisfy (3.3), the narrowband beamforming assumption is valid for the 60-GHz signal under the worst-case scenario of θ0 = ±π/2.

3.1.2 Single-Port Architecture

As discussed in Section 2.2, the disadvantages of using a digital beamforming structure are the high cost and large power consumption that characterize the architecture as a 34 Adaptive Beamforming for 60-GHz Receivers result of the multiple downconversion receivers and ADCs required. This is particularly detrimental in a 60-GHz system, as it is required that each ADC be capable of sampling at gigabit rates. As emphasized in [54], the ADC is typically the highest power-consuming component of a receiver, and the large number of ADCs contributes signiﬁcantly to the cost. Accordingly, digital beamforming is not a viable option and many 60-GHz system designs perform analog beamforming [49, 51, 53]. Thus we see the immediate applicability of our single-port beamforming algorithm to a 60-GHz receiver.

3.1.3 Interference Mitigation

A study of the importance of interference mitigation in 60-GHz networks is provided in [53]. In particular, [53] emphasizes scenarios involving interference between adjacent 60- GHz networks that occur in dense environments, such as an oﬃce consisting of multiple cubicles, each equipped with its own WPAN. The article suggests that interference mitigation via beamforming techniques can oﬀer comparable performance to highly-complex MAC-layer techniques. Therefore, it is essential that the beamforming algorithm be capable of not only steering the array towards a desired direction, but also of rejecting interfering signals. Seeing as our proposed algorithm estimates the MMSE weights, wmmse, it inherently possesses interference-mitigation capabilities, as the MSE is minimized by canceling any interfering signals.

3.1.4 Exploiting the High Data Rates of 60-GHz Communications

Although the perturbation-based algorithm is designed for single-port structures and is capable of performing interference mitigation (since it estimates the wmvdr weights), it is not capable of taking advantage of the high data rates oﬀered by 60-GHz systems. Rather its beamforming speed is limited by the switching rate of the phase shifters and amplitude- control elements. In contrast, our algorithm collects a block of K samples using a single weight vector before switching to the next weight. In this way, the algorithm can exploit the high-data-rates of 60-GHz communications by taking a large number of samples in a very short period of time. 3.2 60-GHz Channel Model 35

3.2 60-GHz Channel Model

The theoretical development and analysis of the beamforming algorithm that will be developed in the following chapter will assume the system model developed in Section 2.3. However, some simulations of the proposed algorithm that are presented in Chapter 5 are performed assuming a model in which the signals sent by the desired and interfering transmitters experience a 60-GHz channel before arriving at the receiver. In this section, we will provide a general description of the channel model that will be used in these simulations. MATLAB code for a 60-GHz channel model was developed by the IEEE 802.15 Working Group for WPANs and a description of it is given in [55]. It is based on the extension [56] of the Saleh-Valenzuela model [57], which incorporates the clustering phenomenon of multipath components in both the temporal and spatial domains. The code creates channel realizations based off of experimental data for several different environments, such as residential, office, etc. In this work, we focus on the line of sight (LOS) model, whose baseband channel impulse response (CIR) is given by

L Pl jφp,l h(t, θ)=βLOSδ(t)δ(θ − θLOS)+ αp,le δ(t − Tl − τp,l)δ(θ − Θl − ωp,l), (3.4) l=1 p=1 which is a function of time, t, as well as AOA, θ. It should be noted that we have made an adjustment to the code in [55] to allow the LOS component to arrive from a non-zero angle, θLOS. The scalars βLOS and θLOS represent the magnitude and AOA of the LOS component, respectively. The total number of clusters is denoted by L ∈ Z+, while the total + number of rays within the lth cluster is given by Pl ∈ Z . The delay and AOA of the lth + + cluster are Tl ∈ R and Θl ∈ [−π, π), respectively. The variables τp,l ∈ R , ωp,l ∈ [−π, π), + αp,l ∈ R , and φp,l ∈ [−π, π) signify the delay, AOA, amplitude, and phase, respectively, of the pth ray within the lth cluster.

The LOS component is assumed to arrive at t = 0, and its magnitude is set to βLOS =1.

The amplitudes of the remaining rays, αp,l, as well as the Gaussian noise components, n(t), that will be included in the simulations are set in relation to βLOS = 1. The value of θLOS will be speciﬁed for each simulation.

Details concerning the generation of L , Pl, αp,l, φp,l, Tl, τp,l,Θl, and ωp,l are provided in [55]. Under the quasi-static assumption, we generate a random realization of the channel, and 36 Adaptive Beamforming for 60-GHz Receivers perform the simulation assuming the channel remains stationary. New channel realizations are generated for each Monte-Carlo simulation. We feel the quasi-static assumption is justiﬁed for several reasons. The transmitter and the receiver are stationary in most of the projected 60-GHz applications. Moreover, in situations where there is no movement in the room, the coherence time of the 60-GHz channel is much longer than the transmitted signal [58, 59]. Clearly, as movement within the channel increases, the channel becomes less stationary, however to the best of the authors’ knowledge an accurate time-varying channel model was not available at the time the simulations of this work were performed. This model provided by the IEEE does not consider wideband fading. 37

Chapter 4

Proposed Beamforming Algorithm

In this chapter, a beamforming algorithm is presented that is comprised of two key features:

1. The ability to beamform using a single-port architecture. This becomes particularly advantageous as the data rate being received increases, since the cost and power consumption of a digital beamformer increases drastically due to the numerous high- rate ADCs.

2. The capability to estimate the beamforming weights using a minimal number of weight changes. Consequently, the speed of beamforming is not limited by the switching speed of the phase shifters and amplitude-control elements.

The development and analysis of the proposed algorithm that will be presented in this chapter assumes the system model described in Section 2.3.

4.1 Proposed Algorithm

The proposed algorithm attempts to minimize the MSE between y(k) and a (known) training sequence, d(k)=i0(k). As explained in Section 2.4.3, the optimal MSE weights, wmmse, satisfy the following relation:

Rwmmse = z. (4.1)

Since R and z are unknown they are estimated from a ﬁnite set of observations, y(k), k =1,...,KM2. The estimates of R and z that are calculated by our algorithm will be denoted by R˜ˆ and z˜ˆ. The tilde notation is used to diﬀerentiate our proposed estimates from 38 Proposed Beamforming Algorithm the estimates Rˆ and zˆ presented in Section 2.5.1 for digital beamformers implementing the ˆ sample matrix inversion algorithm. Once R˜ and z˜ˆ are established, an estimate, w˜ˆ mmse,of wmmse can be computed by solving the linear system in (4.1). ˆ Sections 4.2 through 4.6 will present algorithms for calculating R˜ , z˜ˆ, and w˜ˆ mmse.

4.2 Estimation of R

As discussed in Section 2.3, the mean output power of an array, for a given w,is

P (w)=wH Rw. (4.2)

Equation (4.2) can also be expressed as the inner product of two vectors,

P (w˜ )=w˜ H r, (4.3)

2 where r ∈ CM is deﬁned as r vec(R), (4.4) and the vec operator creates a column vector from a matrix by stacking the columns of the 2 matrix below one another. The vector w˜ ∈ CM is deﬁned as

w˜ w∗ ⊗ w, (4.5) where ⊗ is the Kronecker product, and w∗ denotes the vector whose entries are the complex conjugates of w. It follows directly from the properties of the Kronecker product that

w˜ H = wT ⊗ wH . (4.6)

It can be seen from (4.4) that estimating R has been reduced to estimating r, which can be accomplished through exploiting the linear relationship in (4.3). By computing 2 2 M diﬀerent mean-power estimates, Pˆ(w˜ m),m=1,...,M , each obtained using a distinct weight vector w˜ m, and evidently a distinct wm, a linear system of equations can be 4.2 Estimation of R 39 established as follows, ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ H Pˆ(w˜ 1) w˜ r1 Pe(w˜ 1) ⎢ ⎥ ⎢ 1 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ˆ ⎥ ⎢ H ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ P (w˜ 2) ⎥ ⎢ w˜ 2 ⎥ ⎢ r2 ⎥ ⎢ Pe(w˜ 2) ⎥ ⎢ . ⎥ = ⎢ . ⎥ ⎢ . ⎥ + ⎢ . ⎥ , (4.7) ⎣ . ⎦ ⎣ . ⎦ ⎣ . ⎦ ⎣ . ⎦ ˆ H P (w˜ M 2 ) w˜ M 2 rM 2 Pe(w˜ M 2 ) where rm denotes the mth element of r, Pˆ(w˜ m) is the ﬁnite-sample estimate of P (w˜ m), and Pe(w˜ m) Pˆ(w˜ m) − P (w˜ m). The weights are selected such that the set of vectors

{w˜ 1, w˜ 2,...,w˜ M 2 } are linearly independent, and the matrix W [w1, w2,...,wM 2 ] has full row rank; the necessity of the latter condition becomes more apparent in Section 4.3. A protracted discussion concerning the selection of the weights is presented in Section

4.4. The mth mean-power estimate, Pˆ(w˜ m), is calculated using a block of K samples of y(k),k=(m − 1)K +1,...,mK, as follows,

mK 1 ∗ Pˆ(w˜ m)=Pˆ(wm)= y(k)y (k) K k=(⎛m−1)K+1 ⎞ mK H ⎝ 1 H ⎠ = w x(k)x (k) wm m K k=(m−1)K+1 H = wmRˆ (m)wm (4.8) H = w˜ mˆr(m), (4.9) ˆ 1 mK H ˆ where R(m) K k=(m−1)K+1 x(k)x (k), and ˆr(m) vec(R(m)). Note that in upcoming discussions, it will be more convenient at times to use one of the relations given by (4.8) or

(4.9), over the other. To maintain notational diligence, we will refer to (4.8) using Pˆ(wm), and to (4.9) using Pˆ(w˜ m), however both signify the same mean-power estimate. The matrix Rˆ (m) can be expressed alternatively as

Rˆ (m)=R + Re(m), (4.10) where Re(m) represents the error contained within Rˆ (m). It follows that ˆr(m) can be expressed similarly as

ˆr(m)=r + re(m), (4.11) 40 Proposed Beamforming Algorithm

and re(m) vec (Re(m)). By substituting (4.11) into (4.9) we obtain the relation for

Pˆ(w˜ m) that was used in (4.7),

H Pˆ(w˜ m)=w˜ m (r + re(m)) H = w˜ mr + Pe(w˜ m), (4.12)

H H where Pe(w˜ m)=w˜ mre(m)=wmRe(m)wm. The system in (4.7) can be expressed in matrix form as H Pˆ W˜ = W˜ r + Pe W˜ (4.13) = P W˜ + Pe W˜ , (4.14)

T T where W˜ [w˜ 1,...,w˜ M 2 ], Pˆ (W˜ ) Pˆ(w˜ 1),...,Pˆ(w˜ M 2 ) , P(W˜ ) [P (w˜ 1),...,P(w˜ M 2 )] , T and Pe(W˜ ) [Pe(w˜ 1),...,Pe(w˜ M 2 )] . For ease of notation we omit (W˜ ) from Pˆ (W˜ ),

P(W˜ ), Pe(W˜ ) and denote these vectors as Pˆ , P, and Pe, respectively. We select our estimate, ˜ˆr, to be the least squares (LS) solution of (4.13):

−1 ˜ˆr = W˜ H Pˆ . (4.15)

The estimate R˜ˆ is then obtained using R˜ˆ =unvec ˜ˆr , (4.16) where the unvec operator maps a vector to matrix form. In the following sections we present an analysis concerning the properties of ˜ˆr and R˜ˆ . 4.2 Estimation of R 41

4.2.1 Expected Value of ˜ˆr and R˜ˆ

Taking the expected value of Pˆ(wm) in (4.8) yields H E Pˆ(wm) =EwmRˆ (m)wm H = wmE Rˆ (m) wm H = wmRwm

= P (wm), (4.17)

ˆ which indicates that P (wm) is an unbiased estimate of P (wm), and we can therefore gen- eralize that E Pˆ = P = W˜ H r. Now, by taking the expected value of ˜ˆr in (4.15), −1 E ˜ˆr =E W˜ H Pˆ

−1 = W˜ H E Pˆ −1 = W˜ H W˜ H r = r, (4.18) we see that our estimate ˜ˆr is unbiased. Clearly, since R˜ˆ and ˜ˆr consist of the same entries in diﬀerent arrangements, we can conclude that R˜ˆ is also an unbiased estimator.

4.2.2 Covariance Matrix of ˜ˆr

In order to calculate the covariance matrix of ˜ˆr, it is beneﬁcial to ﬁrst examine the variance of Pˆ(wm). The estimate of Pˆ(wm) given by (4.8) can be equivalently expressed as

1 H Pˆ(wm)= w A(m)wm, (4.19) K m where mK A(m)= x(k)xH (k), (4.20) k=(m−1)K+1 42 Proposed Beamforming Algorithm possesses a complex Wishart distribution1 with K degrees of freedom and mean R, denoted 2 as A(m) ∼ CWM (R,K). The variance of Pˆ(wm), denoted σ ˆ , can be calculated as Pm 2 1 H σ ˆ = var wmA(m)wm Pm K2 1 H 2 H 2 = E w A(m)wm − E w A(m)wm K2 m m 1 H 2 2 H 2 = E (w A(m)wm) − K w Rwm K2 m m 1 H H 2 H 2 = w E A(m)wmw A(m) wm − K w Rwm . (4.21) K2 m m m

In [60], the following expression for a complex Wishart matrix, A ∼ CWM (R,K), and a deterministic matrix B ∈ CM×M is provided:

2 E (ABA) = K2RBR + K tr (BR) R, (4.22) where tr(·) denotes the trace operator. By incorporating this formula into (4.21), the 2 expression for σ ˆ can be reformulated as Pm 2 1 H 2 H H 2 H 2 σ ˆ = wm K RwmwmR + K tr wmwmR R wm − K wmRwm Pm K2 H 2 1 H H H 2 = w Rw + w tr wmw R Rwm − w Rwm m K m m m 1 H H = w Rwm tr wmw R K m m 1 H 2 = w Rwm , (4.23) K m H H where the ﬁnal equality was made using the trace property, wmRwm =tr wmwmR . 2 Now that an expression for σ ˆ has been deduced, we focus our attention on establishing Pm ˆ the covariance matrix of ˜r, which will be represented by Σ˜ˆr. By deﬁnition of the covariance

1To use the Complex Wishart distribution we must make the additional assumption that x(k),k= (m − 1)K +1,...,mK, are independent, identically distributed, Gaussian random vectors, and we will continue to make this assumption for the remainder of the chapter. This assumption is commonly employed when analyzing adaptive algorithms, e.g., LMS. 4.2 Estimation of R 43 matrix of a random vector we ﬁnd H ˆ ˆ ˆ ˆ Σ˜ˆr =E ˜r − E ˜r ˜r − E ˜r H =E ˜ˆr − r ˜ˆr − r

(4.24)

An expression for ˜ˆr − r can be obtained by combining (4.13) and (4.15),

−1 H H ˜ˆr − r = W˜ W˜ r + Pe − r −1 H = r + W˜ Pe − r −1 H = W˜ Pe. (4.25)

Accordingly, (4.24) becomes H ˜ H −1 ˜ H −1 Σ˜ˆr =E (W ) Pe (W ) Pe

H −1 H −1 =(W˜ ) E PePe (W˜ ) ˜ H −1 ˜ −1 =(W ) ΣPe (W) ˜ H −1 ˜ −1 =(W ) ΣPˆ (W) , (4.26)

ˆ where ΣPe and ΣPˆ are the covariance matrices of Pe and P, respectively. The equality

ΣPˆ =ΣPe is obtained in a straightforward manner using the relation in (4.14). 2 Each estimate Pˆ(wm),m=1,...,M , is assumed independent since each was calculated using a diﬀerent block of data. Consequently, the covariance matrix ΣPˆ is a diagonal 2 matrix whose mth entry is equal to σ ˆ , i.e., Pm ⎡ ⎤ 2 σ ˆ 0 ⎢ P1 ⎥ ⎢ 2 ⎥ σ ˆ ⎢ P2 ⎥ Σ ˆ = ⎢ ⎥ . (4.27) P ⎢ .. ⎥ ⎣ . ⎦ 2 0 σ ˆ PM2 44 Proposed Beamforming Algorithm

4.2.3 Hermitian Structure of R˜ˆ

In this section we will show that the estimate R˜ˆ , as presented in (4.16), is a Hermitian matrix. It must be clarified that individual entries within matrices and vectors will be referenced by enclosing the matrix or vector in square brackets and using subscripts to ˆ specify the entry. Thus, the entry at the uth row, and vth column of R˜ will be denoted as ˜ˆ ˆ ˆ R ; the uth entry of the vector ˜r will be represented as ˜r ; and finally, [wm]u signifies u,v u the uth entry of the vector wm. Throughout the remainder of this section it is implied that u ∈{1,...,M}, and v ∈{1,...,M}. By definition of a Hermitian matrix, R˜ˆ can be deemed Hermitian if the following properties hold: 1. R˜ˆ ∈ R. u,u ∗ 2. R˜ˆ = R˜ˆ , for u = v. u,v v,u By re-expressing (4.4) as ˜ˆr = R˜ˆ , (4.28) M(v−1)+u u,v the two properties of a Hermitian matrix listed above can be equivalently expressed in terms of ˜ˆr. As a result, it is implied that R˜ˆ is Hermitian if: 1. ˜ˆr ∈ R. M(u−1)+u ∗ 2. ˜ˆr = ˜ˆr , for u = v. M(v−1)+u M(u−1)+v

To prove that R˜ˆ is Hermitian we will show the latter two properties hold. As stated in (4.15), and restated below for convenience, the estimate of ˜ˆr, is given by

−1 ˜ˆr = W˜ H Pˆ . (4.29)

−1 −1 By denoting the mth column of W˜ as w˘ m, i.e., W˜ =[w˘ 1, w˘ 2,...,w˘ M 2 ], (4.29) becomes ⎡ ⎤ w˘ H ⎢ 1 ⎥ ⎢ H ⎥ ⎢ w˘ 2 ⎥ ˜ˆr = ⎢ . ⎥ Pˆ . (4.30) ⎣ . ⎦ H w˘ M 2 4.2 Estimation of R 45

The knowledge that Pˆ is a vector of real-valued entries leads us to believe that relations amongst vectors w˘ 1,...,w˘ M 2 , may reveal information concerning the relations between the elements of ˜ˆr. Proposition 2 states two useful properties about the vectors w˘ 1,...,w˘ M 2 .

2 Proposition 2 The vectors w˘ 1,...,w˘ M 2 , possess the following properties :

M 2 1. w˘ M(u−1)+u ∈ R .

∗ 2. w˘ M(v−1)+u = w˘ M(u−1)+v , u = v.

The proof of Proposition 2 is provided in Appendix A. Using (4.30) we can express the entries of ˜ˆr as H ˜ˆr = w˘ M(v−1)+uPˆ . (4.31) M(v−1)+u

Combining (4.31) with the ﬁrst property listed in Proposition 2 and the knowledge that Pˆ is composed of real-valued entries implies ˜ˆr ∈ R. (4.32) M(u−1)+u

Substituting the second property listed in Proposition 2 into (4.31), yields H ∗ ˜ˆr = w˘ M(u−1)+v Pˆ M(v−1)+u ∗ H = w˘ M(u−1)+vPˆ ∗ = ˜ˆr , (4.33) M(u−1)+v for u = v. Hence, by comparing the results presented in (4.32) and (4.33) with the properties of a Hermitian matrix that were listed earlier, we can conclude that R˜ˆ is a Hermitian matrix. 2 ∗ The notation w˘ M(u−1)+v denotes that the complex conjugate operation is applied to each entry of the vector w˘ M(u−1)+v. 46 Proposed Beamforming Algorithm

4.3 Estimation of z

The estimate z˜ˆ can be found in a manner analogous to ˜ˆr. First, we deﬁne the correlation between the output and the reference signal, for a given w,as

α(w) E[y(t)d∗(t)] = wH z. (4.34)

2 2 Making use of the samples, y(k),k=1,...,KM , and weights, wm,m=1,...,M , that were used to calculate ˜ˆr, we can obtain M 2 estimates of α(w), where the mth estimate is 2 represented asα ˆ(wm),m=1,...,M , and form the following system of equations: ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ H αˆ(w1) w z1 αe(w1) ⎢ ⎥ ⎢ 1 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ H ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ αˆ(w2) ⎥ ⎢ w2 ⎥ ⎢ z2 ⎥ ⎢ αe(w2) ⎥ ⎢ . ⎥ = ⎢ . ⎥ ⎢ . ⎥ + ⎢ . ⎥ , (4.35) ⎣ . ⎦ ⎣ . ⎦ ⎣ . ⎦ ⎣ . ⎦ H αˆ(wM 2 ) wM 2 zM αe(wM 2 ) where zm is the mth element of z, and αe(wm) αˆ(wm) − α(wm). The calculation of the estimateα ˆ(wm) is carried out using K samples of y(k),k=(m − 1)K +1,...,mK,inthe following averaging scheme,

mK 1 ∗ αˆ(wm)= y(k)d (k) K k=(⎛m−1)K+1 ⎞ mK 1 = wH ⎝ x(k)d∗(k)⎠ m K k=(m−1)K+1 H = wmzˆ(m), (4.36) 1 mK ∗ − where zˆ(m) K k=(m−1)K+1 x(k)d (k). Deﬁning the vector ze(m)=zˆ(m) z, and using (4.36), we can expressα ˆ(wm) as the same form as employed in (4.35),

H αˆ(wm)=wm (z + ze(m)) H = wmz + αe(wm), (4.37)

H T and we see that αe(wm)=wmze(m). Deﬁning αˆ (W) [ˆα(w1),...,αˆ(wM 2 )] , α (W) T T [α(w1),...,α(wM 2 )] , and αe (W) [αe(w1),...,αe(wM 2 )] , allows for (4.35) to be ex- 4.3 Estimation of z 47 pressed in matrix form as

αˆ (W)=α (W)+αe (W) H = W z + αe (W) , (4.38) and for notational convenience we will express the vectors αˆ (W), α (W), and αe (W), as

αˆ , α, and αe, from this point forward. Making use of (4.38), the estimate z˜ˆ is taken to be

−1 z˜ˆ = WWH Wαˆ , (4.39) which is the LS estimate of z.

4.3.1 Expected Value of z˜ˆ

Taking the expected value of (4.36) yields

H E[ˆα(wm)] = E w zˆm m⎡ ⎤ mK 1 = wH E ⎣ x(k)d∗(k)⎦ m K k=(m−1)K+1 H = wmz

= α(wm), (4.40) which implies thatα ˆ(wm) is an unbiased estimate of α(wm), and it follows directly that E[αˆ ]=α = WH z. Taking the expected value of our estimate z˜ˆ, in (4.39), leads to −1 E z˜ˆ =E WWH Wαˆ −1 = WWH W E[αˆ ] −1 = WWH WWH z = z, (4.41) and hence z˜ˆ is also an unbiased estimate of z. 48 Proposed Beamforming Algorithm

4.3.2 Covariance Matrix of z˜ˆ ˆ 2 To find an expression for the covariance matrix of z˜, it is helpful to first find σαˆm , the variance of the estimateα ˆ(wm):

2 | |2 −| |2 σαˆm =Eαˆ(wm) E[ˆα(wm)] ⎡⎛ ⎞ ⎛ ⎞∗⎤ mK mK 1 1 =E⎣⎝ y(k)d∗(k)⎠ ⎝ y(k)d∗(k)⎠ ⎦ K K k=(m−1)K+1 k=(m−1)K+1 ⎡ ⎤ ⎡ ⎤∗ mK mK 1 1 −E ⎣ y(k)d∗(k)⎦ E ⎣ y(k)d∗(k)⎦ K K k=(m−1)K+1 k=(m−1)K+1 ⎡ ⎤ mK mK 1 ⎣ H ∗ H ⎦ H H = E w x(k)d (k) x (n)wmd(n) − w zz wm K2 m m k=(m−1)K+1 n=(m−1)K+1 ⎛ ⎡ ⎤ ⎞ mK 1 ⎝ H ⎣ H 2⎦ ⎠ = w E x(k)x (k) |d(k)| wm K2 m k=(m−1)K+1 ⎛ ⎛ ⎞ ⎞ ⎜ ⎜ mK mK ⎟ ⎟ 1 ⎜ H ⎜ ∗ H ⎟ ⎟ + w E[x(k)d (k)] E x (n)d(n) wm K2 ⎝ m ⎝ ⎠ ⎠ k=(m−1)K+1 n=(m−1)K+1 n=k H H −wmzz wm 1 H H 2 2 H H H H = Kw E x(k)x (k)|d(k)| wm + K − K w zz wm − w zz wm. K2 m m m 1 H H 2 H = w E x(k)x (k)|d(k)| − zz wm. (4.42) K m

Let us deﬁne

Z E x(k)xH (k)|d(k)|2 − zzH , (4.43) which is the covariance matrix of x(k)d∗(k). Substituting Z into (4.42), yields

2 1 H σ = w Zwm. (4.44) αˆm K m 4.4 Selection of W and W49˜

ˆ Making use of the relation presented in (4.41), the covariance matrix of z˜, denoted as Σz˜ˆ, is found to be H ˆ ˆ ˆ ˆ Σz˜ˆ =E z˜ − E z˜ z˜ − E z˜ H =E z˜ˆ − z z˜ˆ − z (4.45)

An expression for z˜ˆ − z can be constructed by combining (4.38) and (4.39), H −1 H z˜ˆ − z = WW W W z + αe − z H −1 = WW Wαe. (4.46)

The expression in (4.45) can now be rewritten as H −1 H −1 H Σz˜ˆ =E (WW ) Wαe (WW ) Wαe

H −1 H H H −1 =(WW ) W E αeαe W (WW ) H −1 H H −1 =(WW ) W Σαe W (WW ) H −1 H H −1 =(WW ) W Σαˆ W (WW ) , (4.47)

where Σαe and Σαˆ are the covariance matrices of αe and αˆ , respectively. The entries of 2 Σαˆ are uncorrelated due to each estimateα ˆ(wm),m=1,...,M , being calculated using a separate block of data and therefore assumed independent. The diagonal entries of Σαˆ are given by (4.44), as shown below: ⎡ ⎤ σ2 0 ⎢ αˆ1 ⎥ ⎢ σ2 ⎥ ⎢ αˆ2 ⎥ Σαˆ = ⎢ . ⎥ . (4.48) ⎣ .. ⎦ 0 σ2 αˆM2

4.4 Selection of W and W˜

As previously stated, the matrix W, and consequently W˜ , can be chosen arbitrarily so long as 50 Proposed Beamforming Algorithm

1. W has full row rank.

2. W˜ has full rank.

However, we see from (4.26) and (4.47) that the covariance matrices of ˜ˆr and z˜ˆ are dependent on W and W˜ . This implies that the selection of the weights aﬀects the accuracy of ˜ˆr and z˜ˆ. Reducing the error contained within the vectors ˜ˆr and z˜ˆ is equivalent to minimizing the trace of their covariance matrices. The ensuing analysis will therefore focus on developing equations for tr (Σ˜ˆr), and tr (Σz˜ˆ).

4.4.1 Eﬀect of Weights on Error of ˜ˆr

The trace of ˜ˆr is given by ⎛ ⎡ ⎤ ⎞ 2 σ ˆ 0 ⎜ ⎢ P1 ⎥ ⎟ ⎜ ⎢ 2 ⎥ ⎟ −1 σ ˆ −1 ⎜ H ⎢ P2 ⎥ ⎟ tr (Σˆ)=tr⎜ W˜ ⎢ ⎥ W˜ ⎟ ˜r ⎜ ⎢ .. ⎥ ⎟ ⎝ ⎣ . ⎦ ⎠ 2 0 σPˆ ⎛ ⎡ M2 ⎤⎞ 2 σ ˆ 0 ⎜ ⎢ P1 ⎥⎟ 2 ⎜ −1 −1 ⎢ ⎥⎟ ⎜ ⎢ σPˆ ⎥⎟ =tr⎜ W˜ W˜ H ⎢ 2 ⎥⎟ ⎜ ⎢ .. ⎥⎟ ⎝ ⎣ . ⎦⎠ 2 0 σPˆ ⎛ ⎡ M2 ⎤⎞ H 2 w1 Rw1 0 ⎜ ⎢ 2 ⎥⎟ ⎜ −1 −1 ⎢ H ⎥⎟ 1 ⎜ H ⎢ w2 Rw2 ⎥⎟ = tr ⎜ W˜ W˜ ⎢ ⎥⎟ K ⎝ ⎣ ... ⎦⎠ H H 2 0 wM 2 RwM 2 ⎛ ⎡ ⎤⎞ H 2 w˜ 1 r 0 ⎜ ⎢ 2 ⎥⎟ ⎜ −1 −1 ⎢ H ⎥⎟ 1 ⎜ H ⎢ w˜ 2 r ⎥⎟ = tr ⎜ W˜ W˜ ⎢ ⎥⎟ . (4.49) K ⎝ ⎣ ... ⎦⎠ H 2 0 w˜ M 2 r

Finding the matrix W˜ that minimizes (4.49) proves to be an arduous, if at all possible, task. So instead, we focus on minimizing an upper bound of (4.49). Deﬁning w˜ max = 4.4 Selection of W and W51˜

max { w˜ 1 , w˜ 2 ,..., w˜ M 2 }, and making use of the Cauchy-Schwarz inequality, results in ! ! H 2 H 2 ! H !2 2 w˜ mr = w˜ mr ≤ w˜ m r 2 2 ≤ w˜ max r , (4.50)

which provides us with an upper bound of the entries of ΣPˆ . By combining (4.49) and (4.50) it can be found that −1 −1 1 ˜ ˜ H 2 2 tr (Σˆr) ≤ tr W W w˜ max r K H 1 2 2 ≤ tr W˜ −1 W˜ −1 w˜ R K max F ! !2 1 ! ˜ −1! 2 2 ≤ !W ! w˜ max R F , (4.51) K F where · F denotes the Frobenius norm of the matrix. The result presented in (4.51) indicates that the upper bound of tr (Σˆr) can be reduced by selecting weights that minimize ! !2 ! ˜ −1! 2 !W ! w˜ max. By observing that F

! !2 2 ! ˜ ! w˜ max ≤ !W! , (4.52) F and adjusting (4.51) to

! !2 ! !2 1 ! ˜ −1! ! ˜ ! 2 tr (Σˆr) ≤ !W ! !W! R F , (4.53) K F F we can express tr (Σˆr) in terms of the Frobenius norm condition number, κF W˜ ,as

1 2 tr (Σˆr) ≤ κF W˜ R , (4.54) K F ! ! ! ! ! ! ! −1! where κF W˜ = !W˜ ! !W˜ ! . F F The result provided in (4.54) signiﬁes that in order to minimize the error of ˜ˆr, the matrices W and W˜ should be designed based on minimizing κF (W˜ ). An obvious solution for ensuring a small κF (W˜ ) would be to select weights such that W˜ is a unitary matrix. 52 Proposed Beamforming Algorithm

However, as will be proven in Proposition 3, the inherent structure of W˜ prevents it from being unitary.

2 Proposition 3 The weights, wm,m=1,...,M , cannot be chosen such that W˜ is unitary, i.e. W˜ W˜ H = I.

Proof Using the relation given by (4.5) in conjunction with the mixed-product property of the Kronecker product [61], we can express the inner product of two vectors, w˜ m and w˜ z, m = z,as H T H ∗ w˜ mw˜ z = wm ⊗ wm (wz ⊗ wz) T ∗ H = wm(wz) ⊗ wmwz H H = wz wm ⊗ wmwz H 2 = |wz wm| . (4.55)

Assuming wm and wz are non-zero vectors, it can be observed that

w˜ m⊥w˜ z iﬀ wm⊥wz. (4.56)

2 Since, for a given m, m =1,...,M , the vector wm contains M entries, it follows that no set of vectors {w1,...,wM 2 } can be mutually orthogonal, and consequently, the set of vectors {w˜ 1,...,w˜ M 2 } cannot be mutually orthogonal. Therefore W˜ cannot be unitary.

Although we cannot provide a theoretically-justiﬁed, optimal matrix of weights to minimize the error of ˜ˆr, we can conclude that the selection of W, and its corresponding matrix

W˜ , should be based on minimizing the Frobenius condition number, κF (W˜ ). In practice this could be achieved by implementing a type of brute-force algorithm in which random weights are generated to construct W, the associated κF (W˜ ) is calculated, and this procedure is repeated numerous times after which the matrix W producing the lowest-valued

κF (W˜ ) is employed.

4.4.2 Eﬀect of Weights on Error of z˜ˆ

Following a similar procedure as was described in Section 4.4.1, we can minimize the error ˆ of the estimate z˜ by minimizing the trace of its covariance matrix, Σz˜ˆ. The trace of Σz˜ˆ is 4.5 Computationally-Eﬃcient Algorithm to Estimate r and z 53 found to be ⎛ ⎡ ⎤⎞ σ2 ⎜ ⎢ αˆ1 ⎥⎟ ⎜ ⎢ σ2 ⎥⎟ ⎜ H H −1 H −1 ⎢ αˆ2 ⎥⎟ tr (Σz˜ˆ)=tr⎜W WW WW W ⎢ . ⎥⎟ , (4.57) ⎝ ⎣ .. ⎦⎠ σ2 αˆM2

2 where σαˆm was deﬁned in (4.44), and can be reformulated as

2 1 H σ = w Zwm αˆm K m 1 = w˜ H (vec (Z)) . (4.58) K m

Applying the Cauchy-Schwarz inequality to (4.58) yields 2 2 1 σ = σ ≤ w˜ m vec(Z) . (4.59) αˆm αˆm K

As in the previous section, we deﬁne w˜ max = max { w˜ 1 , w˜ 2 ,..., w˜ M 2 } and obtain an upper bound to (4.57), 1 H H −1 H −1 tr (Σˆ) ≤ tr W WW WW W w˜ vec(Z) z˜ K max ! !2 1 ! H H −1! ≤ !W WW ! w˜ max vec(Z) . (4.60) K F

! !2 ! H H −1! Thus, by minimizing the term !W WW ! w˜ max, the upper bound of the F error contained within z˜ˆ can be minimized.

4.5 Computationally-Eﬃcient Algorithm to Estimate r and z

Throughout Sections 4.2 - 4.4 the development and analysis of ˜ˆr and z˜ˆ were performed in terms of arbitrary weight matrices W and W˜ . In this section, rather than assume W is arbitrary, we present a specific weight matrix, i.e., W = Ψ, and provide an algorithm that incorporates Ψ to estimate R and z efficiently. As will be seen, the estimate of R computed by this algorithm is equivalent to (4.16), and thus will be denoted as R˜ˆ throughout this section. However, the estimate of z computed by this algorithm will not be equivalent 54 Proposed Beamforming Algorithm to (4.39), and to differentiate the two we denote the estimate produced by the upcoming algorithm as z¯ˆ. Despite being computationally efficient, this algorithm does not guarantee the error of the estimates are minimized. Furthermore, it should be clarified that this section is being presented only as a demonstration to the reader that through the proper selection of W, the estimation of R and z can be carried out in a computationally efficient manner, albeit with a modification to the estimate of z. In Algorithm 1, a method of generating Ψ for an M-element array is presented, where H the mth column of Ψ is denoted as ψm. The resulting matrix of Ψ obtained for M =4 −1 is provided in Appendix B. Also provided in Appendix B is the matrix Ψ˜ H that −1 corresponds to Ψ, for M = 4. The sparse nature of Ψ˜ H can be exploited to circumvent the matrix multiplication in (4.15) required to compute ˜ˆr, and thereby reduce the number of floating point operations (FLOPS).

Algorithm 1 Generate Ψ

//All weight vectors, ψm,m=1,...,M, are initialized to zero.

//Set values of ﬁrst M weight vectors for m =1toM do [ψm]m =1 end for

//Set values of ﬁnal (M 2 − M) weight vectors for u =1toM do for v =(u +1)toM do m = m +1 [ψm]u =1 [ψm]v =1 m = m +1 [ψm]u = j [ψm]v =1 end for end for

The pseudo-code for computing R˜ˆ , as well as z¯ˆ, is presented in Algorithm 2. It is as- 2 sumed that both Pˆ(ψm),m=1,...,M , andα ˆ(ψm),m=1,...,M, have been obtained. The construction of R˜ˆ commences by ﬁnding the diagonal entries. Noting that the ﬁrst

M vectors of Ψ, i.e., ψm,m=1,...,M, are zero vectors whose mth entry is set to one, it 4.5 Computationally-Eﬃcient Algorithm to Estimate r and z 55

Algorithm 2 Generate R˜ˆ and z¯ˆ ˆ 1: //Calculate diagonal entries of R˜ ; calculate z¯ˆ. 2: for m=1toM do ˆ 3: R˜ = Pˆ(ψm) m,m ˆ ψ 4: z¯ m =ˆα( m) 5: end for ˆ 6: //Calculate oﬀ-diagonal entries of R˜ 7: for u =1toM do 8: for v = u +1toM do 9: m = m +1 " # " # ˆ ˜ˆ ˜ˆ P (ψm)− R − R ˜ˆ ˜ˆ u,u v,v 10: Re R =Re R = 2 u,v v,u 11: m = m +1 " # " # ˆ ˜ˆ ˜ˆ P (ψm)− R − R ˜ˆ − ˜ˆ u,u v,v 12: Im R = Im R = 2 u,v v,u 13: end for 14: end for follows that mK ˆ ∗ P (ψm)= [x(k)]m [x(k)]m , for m =1,...,M, (4.61) k=(m−1)K+1 ˆ and we accordingly assign R˜ = Pˆ (ψm). This operation is carried out in Line 3 of m,m Algorithm 2. The oﬀ-diagonal entries of R˜ˆ are computed in pairs by taking advantage of the complex-conjugate symmetry of R. The real part of each pairing, being the same, can be found using one calculation, given by Line 10. Similarly, the imaginary parts can both 2 be calculated using Line 12. Since we have assumed Pˆ(ψm),m=1,...,M , have already been computed, no FLOPS are required to calculate the diagonal entries of R˜ˆ and as a result the total number of FLOPS required to estimate R˜ˆ arises from the computation of M 2−M the non-diagonal entries. There are 2 pairs of complex conjugate entries and each pair requires:

• 1 FLOP to calculate the sum: [R]u,u +[R]v,v. 56 Proposed Beamforming Algorithm

• 2 additional FLOPS to calculate Line 10.

• 2 additional FLOPS to calculate Line 12 (the sign change isn’t counted as a FLOP). ˜ˆ M 2−M Thus, the total number of FLOPS required to calculate R is equal to 5 2 .In −1 comparison, calculating ˜ˆr by means of (4.15) when W˜ H is dense requires 2M 4 − M 2 FLOPS. The calculation of z¯ˆ is carried out in Line 4 in a similar manner to the calculations of ˆ the diagonal entries of R˜ . Due to the structure of the vectors ψm for m =1,...,M, it can be seen that mK ∗ αˆ(ψm)= [x(k)]m d (k),m=1,...,M, (4.62) k=(m−1)K+1

ˆ ψ and it follows that the estimate of z is found by equating z¯ m =ˆα( m), for m =1,...,M.

Consequently, since it has been assumed thatα ˆ(ψm),m=1,...,M, have already been computed, no FLOPS are required to calculate z˜ˆ. In comparison, the matrix multiplication used in (4.39) to calculate z˜ˆ requires 2M 3 − M FLOPS. One of the principal differences between the calculation of z¯ˆ and z˜ˆ, is that the former is carried out using M estimates of 2 2 α(ψm),m=1,...,M, whereas the latter uses M estimates of α(ψm),m=1,...,M . We see that the new estimate does not make use of the same quantity of information as z˜ˆ, but it is more efficient. The algorithm described in this section has been presented only to demonstrate that proper choice of W and a modification to the estimate of z can be implemented to increase computational efficiency. Still, as explained in Chapter 3, the number of elements, M, will typically be kept small, which indicates that the improvement in speed that will arise from this algorithm may not prove to be fruitful. For the remainder of this work, we will continue to consider z˜ˆ, and not z¯ˆ, as being the estimate of z.

4.6 Estimation of wmmse

A linear system, analogous in form to that of (4.1), can be formed using the estimates R˜ˆ and z˜ˆ as follows, ˆ Rw˜ mmse = z˜ˆ. (4.63) 4.6 Estimation of wmmse 57

A seemingly-straightforward approach to ﬁnding an estimate of wmmse would entail calculating ˆ −1 w˜ˆ mmse = R˜ z˜ˆ. (4.64)

However, calculating w˜ˆ in such a manner can result in a very erroneous estimate. As explained in [62], solving a linear system in which the design matrix, in this case R˜ˆ ,is ill-conditioned produces a very poor LS estimate. Through simulations, it will be shown that R˜ˆ suﬀers from near-zero eigenvalues, a deﬁning characteristic of an ill-conditioned matrix. Overcoming this problem involves applying some type of regularization to R˜ˆ . The eigenvalue decompositions of R˜ˆ and its inverse are

M ˆ ˆ ˆ ˆ H ˆ H R˜ = Q˜ Λ˜ Q˜ = λ˜mq˜ˆmq˜ˆm, (4.65) m=1 and, M ˆ −1 ˆ ˆ −1 ˆ H 1 H R˜ = Q˜ Λ˜ Q˜ = q˜ˆmq˜ˆ , (4.66) ˆ m m=1 λ˜ m ˆ ˆ where λ˜m and q˜ˆm are the mth eigenvalue and eigenvector of R˜ , and we continue to assume ˆ ˆ ˆ ˆ that λ˜1 ≥ λ˜2 ≥ ...≥ λ˜M . Having proven in Section 4.2.3 that R˜ is a hermitian matrix, it ˆ follows that ∀m, λ˜m ∈ R. To gain a better understanding of the eigenvalue characteristics of R˜ˆ , computer simulations of a 6-element, single-port beamformer are performed using the sampled beamformer output (2.27) provided in the system model of Section 2.3. Figure 4.1(a) displays a plot of ˆ the eigenvalues of R˜ as a function of K, the number of samples used to calculate Pˆ(wm) for each m =1,...,M, and assumes no interfering signals are present3. Figure 4.1(b) 2 repeats the simulations and includes an interfering plane wave of power σ1 = 1, and AOA ◦ θ1 = −π/18 (−10 ). In Figure 4.1(a), we observe the presence of one large eigenvalue which corresponds to the desired plane wave, and ﬁve smaller eigenvalues that are associated with the background noise. Likewise, in Figure 4.1(b), we observe two large eigenvalues that are caused by the plane waves, and four noise-related eigenvalues. It is also clear from Figure

3To avoid deﬂecting the focus of our discussion we will not provide a detailed description of the simulation model at this point; however, this section follows the same model as described in Section 5.1.1 (with the exception of the interference settings) and the reader may therefore refer to Section 5.1.1 to obtain a complete explanation of system settings. 58 Proposed Beamforming Algorithm

4.1(a) and 4.1(b) that R˜ˆ possesses negative eigenvalues, i.e., R˜ˆ is not positive definite. As a result some eigenvalues are equal, or in very close proximity, to zero, which signify that the matrix is ill-conditioned. For comparative purposes we simulate a digital beamformer that calculates Rˆ using (2.73), under the same simulation settings. The results are provided in Figures 4.2(a) and 4.2(b). In both plots we observe the presence of strong eigenvalues caused by the desired and interfering signals, and smaller eigenvalues associated with the background noise. However, the behavior of the noise eigenvalues in Figures 4.2(a) and 4.2(b) is much different than in Figures 4.1(a) and 4.1(b), as all eigenvalues display very little fluctuation and are positive, which is expected since the estimate Rˆ is a positive-definite matrix. The key observation

Fig. 4.1 Eigenvalues of R˜ˆ calculated using (4.16) by single-port beamformer with (a) No interference; (b) One interfering signal. to be made from these results is the near-zero values of the noise eigenvalues of R˜ˆ .To demonstrate the catastrophic eﬀect of the ill-conditioned R˜ˆ on the estimate in (4.64), we simulate the MSE performance of a beamformer, in the presence of an interfering signal, employing the weights, ˆ −1 w˜ˆ mmse = R˜ z, (4.67) as well as a second simulation using the weights,

−1 wˆ mmse = Rˆ z, (4.68) 4.6 Estimation of wmmse 59

Fig. 4.2 Eigenvalues of Rˆ calculated using (2.73) by digital beamformer with (a) No interference; (b) One interfering signal. where R˜ˆ and Rˆ are the same estimates whose eigenvalues are shown in Figures 4.1(b) and 4.2(b). The exact value of z is calculated as described in Section 5.1.1, and is employed in both (4.67) and (4.68) to ensure the simulation results reflect the effects of R˜ˆ and Rˆ on the MSE performance of the weights, and to prevent them from being affected by a faulty estimate of z. The simulation results are provided in Figures 4.3 and 4.4. Clearly, R˜ˆ will require some type of regularization to ensure the estimate, w˜ˆ mmse, is meaningful. In the following sections we suggest several techniques to improve our estimate, w˜ˆ mmse.

Fig. 4.3 MSE of beamformer implementing w˜ˆ mmse. 60 Proposed Beamforming Algorithm

Fig. 4.4 MSE of beamformer implementing wˆ mmse.

4.6.1 Estimation of wmmse By Means Of Diagonal Loading

The idea of diagonal loading was briefly introduced in Section 2.5.1, and is widely used in practice [2] as a means to improve the robustness of the MVDR beamformer against errors −1 contained within s(θ0), Rˆ , and Rˆ , as well as improve the beam pattern of the beamformer. These beneficial effects arise, in part, as a result of a decrease in the eigenvalue spread,

λmax/λmin,ofRˆ [2, 63, 64], where λmax and λmin denote the maximum and minimum eigenvalues. In theory, this technique may prove beneﬁcial to our proposed estimate, R˜ˆ , by pulling the eigenvalues away from zero; moreover, it is simple to implement. For these reasons, we feel that it is worthwhile to investigate its eﬀectiveness as a solution to our problem. As was stated in Section 2.5.1, diagonal loading adds a diagonal matrix ,υI, to the correlation matrix. It follows that the diagonally-loaded correlation matrix of the estimate R˜ˆ can be described in terms of its eigenvalue decomposition by ˆ ˆ ˆ ˆ H ˆ ˆ ˆ H R˜ dl = Q˜ Λ˜Q˜ + υI = Q˜ Λ+˜ υI Q˜ , (4.69) 4.6 Estimation of wmmse 61 and the corresponding inverse by ⎡ ⎤ −1 ˜ˆ ⎢ λ1 + υ 0 ⎥ ⎢ ⎥ ⎢ ˆ −1 ⎥ ⎢ λ˜2 + υ ⎥ ˜ˆ −1 ˜ˆ ⎢ ⎥ ˜ˆ H Rdl = Q ⎢ ⎥ Q . (4.70) ⎢ ... ⎥ ⎣ ⎦ ˆ −1 0 λ˜M + υ

Making use of (4.70), an estimate of wmmse using diagonal loading is given as

ˆ ˜ˆ −1ˆ w˜ dl = Rdl z˜. (4.71)

The simulations presented in the following chapter will test the eﬀectiveness of w˜ˆ dl for various values of υ.

4.6.2 Estimation of wmmse Using Truncated Singular Value Decomposition (TSVD)

The singular value decomposition (SVD) of R˜ˆ can be expressed as

M ˆ H R˜ = σ˜ˆmu˜ˆmv˜ˆm, (4.72) m=1 where σ˜ˆm denotes the mth singular value such that σ˜ˆ1 ≥ σ˜ˆ2 ≥ ... ≥ σ˜ˆM > 0, and u˜ˆm ˆ and v˜ˆm represent the mth left and right singular vectors of R˜ . A common method of regularizing an ill-conditioned matrix is called the TSVD [62], which modiﬁes the matrix by equating the smallest singular values to zero. This algorithm can be applied to solve (4.63) by deﬁning J ˆ H R˜ tsvd σ˜ˆmu˜ˆmv˜ˆm,J≤ M, (4.73) m=1 and solving the minimum 2-norm LS problem [65]: $ ! ! % !ˆ ˜ˆ ! min wmmse 2 ,S= wmmse !z˜ − Rtsvdwmmse! = min . (4.74) wmmse∈S 2 62 Proposed Beamforming Algorithm ! ! ! ˆ ! In other words, the TSVD solution is the solution of min !z˜ˆ − R˜ tsvdwmmse! that possesses the smallest two-norm value [65], and is given by

ˆ ˜ˆ −1 ˆ w˜ tsvd = Rtsvdz˜, (4.75) where J ˆ −1 1 H R˜ = v˜ˆmu˜ˆ . (4.76) tsvd ˆ m m=1 σ˜m

To select J, we choose to set a threshold, σ˜ˆth, such that 0 ≤ σ˜ˆth ≤ 1, and determine the smallest J ∈ [1,...,M − 1] for which

σ˜ˆJ+1 < σ˜ˆth. (4.77) M ˆ m=1 σ˜m

If ∀J (4.77) is not satisﬁed, then J = M. The Hermitian property of R˜ˆ indicates that its SVD is closely related to its eigenvalue ˆ M H decomposition. It is well-known [66] that the SVD, R˜ = σ˜ˆmu˜ˆmv˜ˆ , and the eigenvalue m=1 m ˜ˆ M ˜ˆ ˆ ˆH decomposition, R = m=1λmq˜mq˜m, of a Hermitian matrix are related by the following ˆ ˆ equalities: u˜ˆm = q˜ˆm; σ˜ˆm = λ˜m; and, v˜ˆm = sign λ˜m u˜ˆm. Consequently, we know that

J J 1 H 1 H q˜ˆmq˜ˆ = v˜ˆmu˜ˆ , (4.78) ˆ m ˆ m m=1 λ˜m m=1 σ˜m

ˆ so long as λ˜m ≥ 0,m=1,...,J. In other words, truncating the eigenvalue decomposition of R˜ˆ is equivalent to truncating the SVD of R˜ˆ , so long as the J largest eigenvalues that are not truncated are positive. Therefore, we’d like to point out that our proposed method of using the TSVD of R˜ˆ shares similarities with the Principal Component (PC) beamformer, which is an MVDR-based beamformer that truncates the eigenvalue decomposition of Rˆ [2]. The PC beamformer is shown to have an improved SINR performance [2] over the typical MVDR beamformer of (2.57) in situations where the number of samples used to estimate Rˆ is low, or the presumed AOA is erroneous. 4.6 Estimation of wmmse 63

4.6.3 Estimation of wmmse Based on Dominant-Mode Rejection (DMR)

Another MVDR-based beamformer, called the DMR beamformer [2], divides the eigenvalue decomposition of Rˆ into two terms:

J M H H Rˆ = λˆmqˆmqˆm + λˆmqˆmqˆm, (4.79) m=1 m=J+1 where the largest J eigenvectors are referred to as the dominant modes. The remaining M − J eigenvalues (and eigenvectors) are associated with the background noise. The DMR beamformer modiﬁes (4.79) by setting λˆJ+1 = λˆJ+2 = ... = λˆM = αdmr, where αdmr is a constant. Typically [2], αdmr is assigned to equal the average value of the noise eigenvalues,

M 1 αdmr λˆm. (4.80) − M J m=J+1

It follows that (4.79) can be rewritten as,

J M H H Rˆ dmr λˆmqˆmqˆm + αdmr qˆmqˆm, (4.81) m=1 m=J+1 whose inverse is J M ˆ −1 1 H 1 H Rdmr = qˆmqˆm + qˆmqˆm, (4.82) ˆ dmr m=1 λm α m=J+1 and (4.82) can then be used as a substitute for R−1 in (2.57). Much like the PC beamformer, the DMR beamformer is used to improve the performance of an MVDR beamformer in situations where a low number of samples are used to estimate Rˆ , or the estimated AOA is inaccurate [2]. In our system, the idea of setting the noise-related eigenvalues of R˜ˆ to a single value is promising, as it will allow for the near-zero eigenvalues to be increased. However, since noise eigenvalues of R˜ˆ can be negative, the averaging scheme in (4.80) may result in a very small value of αdmr. Consequently, we propose a DMR-based estimate of wmmse that will ˆ ˆ select J by setting a threshold, λ˜th, where 0 ≤ λ˜th ≤ 1, and in a manner parallel to the 64 Proposed Beamforming Algorithm algorithm proposed in Section 4.6.2, will ﬁnd the smallest J ∈ [1,...,M − 1], such that ˆ λ˜J+1 ˆ < λ˜th. (4.83) M ˜ˆ m=1 λm

The value of αdmr is then set to $ % ˆ ˆ αdmr = max λ˜J+1 ,...,λ˜M , (4.84)

˜ˆ −1 and Rdmr is calculated as

J M ˜ˆ −1 1 ˆ ˆH 1 ˆ ˆH Rdmr = q˜mq˜m + q˜mq˜m, (4.85) ˆ αdmr m=1 λ˜m m=J+1 where αdmr in (4.85) is given by (4.84). The DMR-based estimate, w˜ˆ dmr, is calculated using ˆ ˜ˆ −1 ˆ w˜ dmr = Rdmrz˜. (4.86)

We use the subscript dmr to indicate that it is based on the DMR beamformer, however we would like to emphasize that our estimate is a modified version of the DMR beamformer. As shown in [67], the number of signals impinging upon an array can be estimated by means of the Akaike Information Criterion (AIC), introduced in [68], or the minimum description length (MDL), introduced in [69, 70]. This presents an alternative method for calculating J,asJ can be assigned to equal the estimated number of directional signals being received by the array. We do not incorporate the AIC or MDL algorithms to estimate J in our proposed beamformers due to the fact that our correlation matrix estimate, R˜ˆ ,is different than Rˆ , where the latter was assumed in [67]. Thus, it is uncertain how accurately the AIC or MDL algorithms will perform when the correlation matrix is not guaranteed to be positive definite.

4.7 Discussion of Proposed Algorithm

In this chapter, we have proposed a single-port beamforming algorithm that calculates a correlation matrix estimate, R˜ˆ , as well as a cross-correlation matrix estimate, z˜ˆ. We have 4.7 Discussion of Proposed Algorithm 65 proposed three diﬀerent methods of conditioning R˜ˆ , and consequently we have proposed three unique beamforming weights that are given by w˜ˆ dl in (4.71), w˜ˆ tsvd in (4.75), and w˜ˆ dmr in (4.86). We will examine the performance of each weight in simulations that are presented in the following chapter. Similar to SMI-based algorithms, the proposed algorithm requires the environment to remain stationary while the weights are being estimated and while the receiver employs these weights. In practice, this limits the number of samples that can be used to form the estimates of R˜ˆ and z˜ˆ, and dictates how often the weights must be updated. 66 67

Chapter 5

Numerical Simulations

In this chapter, we simulate the performance of a single-port beamformer that implements our proposed algorithm to calculate the weights w˜ˆ dl, w˜ˆ tsvd, and w˜ˆ dmr, via computer simulations. The resulting MSE performance of the beamformer as a function of the number of samples used to estimate the weights will be presented. Moreover, the power patterns produced by each weight vector will be provided as a means of illustrating its steering and nulling capabilities. Finally, the quantization eﬀects caused by ﬁnite-resolution phase shifters and amplitude-control elements will be explored.

5.1 Simulation Models

All simulations of the proposed beamformers will be carried out under two different system models. The first, described in Section 2.3, assumes the sampled signal can be expressed in terms of a sum of uncorrelated directional signals and an AWGN component, as given by (2.27); it will be referred to as the AWGN model in this chapter. The second implements a model of a 60-GHz channel in a residential environment, and is designated the multipath model. We have chosen to study the performance under both models for two principal reasons. Firstly, the AWGN model is widely adopted in array processing literature, making it useful for assessing the potential capabilities of novel algorithms. On the other hand, as explained in Chapter 3, 60-GHz receivers employing antenna arrays represent a potential application of the proposed algorithm. We therefore find it fitting to simulate our system in a more realistic 60-GHz, multipath channel as well. In both models, the receiver employs a single-port beamforming architecture with a 6-element ULA of element spacing λc/2. 68 Numerical Simulations

5.1.1 AWGN Model

Following the system model we presented in Section 2.3, the AWGN model assumes the sampled output of the beamformer, y(k), is of the form given by (2.27). The modulating function of the desired signal, i0(k), is modeled as a QPSK signal, and those of the interfering signals, iq(k), q ≥ 1, are represented by zero-mean, complex, Gaussian random variables. Moreover, we sample i0(k) at its symbol rate, and assume that ∀q, iq(k1)is independent of iq(k2) for k1 = k2. Each entry of the background noise, n(k), is represented by a zero-mean, independent, complex, Gaussian random variable, and is independent of iq(k), ∀q. ◦ The AOA of the desired signal is θ0 = π/9 (20 ); the variance of the desired signal and 2 2 background noise are σ0 = 1, and σN =0.1, respectively; and, the interference is modeled under one of the following four scenarios:

• Scenario A: No interfering signals are present.

◦ • Scenario B: Three interfering signals are present, possessing AOAs of θ1 = −π/18 (−10 ), ◦ ◦ 2 2 2 θ2 =5π/18 (50 ), θ3 = −2π/9(−40 ), and variances of σ1 = σ2 = σ3 =0.5.

• Scenario C: Three interfering signals are present whose AOAs are identical to those 2 2 2 of Scenario B, and whose variances are increased to σ1 = σ2 = σ3 =1.

◦ • Scenario D: One interfering signal is present, possessing an AOA of θ1 = −π/18 (−10 ), 2 and a variance of σ1 = 10. ˆ The weights used to estimate R˜ and z˜ˆ are given by W = Ψ, where W =[w1,...,wM 2 ] was deﬁned in Section 4.2, and Ψ is constructed as outlined in Algorithm 1. We employ the MSE as a performance metric, and calculate it, for a given w, using

(2.72). Computing (2.72) requires the values of wmmse, MMSE, R, and z. These values can be calculated using (2.66), (2.71), (2.33), and

2 z = s(θ0)σ0, (5.1) respectively. 5.1 Simulation Models 69

5.1.2 Multipath Model

In this section we provide an overview of the multipath model conﬁguration. The multipath model considers the situation in which a transmitter is transmitting through a 60-GHz channel to an intended receiver at 1 gigasymbol per second (Gs/s). In addition to the desired transmitter, there are Q interfering transmitters introducing co-channel interference, and all Q + 1 transmitters possess an LOS with the receiver. The signal sent by the qth, q =0,...,Q, transmitter is j2πfct ˜iq(t)=Re iq(k)g(t − kTs)e , (5.2) k whose equivalent baseband signal1 is given by iq(t)= iq(k)g(t − kTs). (5.3) k

Similar to the AWGN model, the modulating function of the desired signal, i0(k), is QPSK, the modulating functions of the interfering signals, iq(k),q≥ 1, are zero-mean, complex, 2 2 Gaussian random variables. For q =0,...,Q, we define the variance as σq =E[|iq(k)| ], and iq(k1) is independent of iq(k2) for k1 = k2. The function g(t) is a root-raised-cosine, pulse-shaping filter. It is assumed that the pulse duration, Ts, is the same for the desired signal as well as the interferers, and the signals from all transmitters are synchronous. The baseband CIR of the 60-GHz channel, h(t, θ), used in the multipath simulations was described in Chapter 3 and the MATLAB code for generating it is provided by the IEEE; more specifically, we use Channel Model 1.2, which corresponds to an LOS channel in a residential environment. Each transmitter will experience a unique channel realization, and we denote the channel realization experienced by the qth transmitter as hq(t, θ). For simulation purposes, we find it helpful to re-express hq(t, θ) as the following vector:

L q Pq;l jφq;p,l hq(t)=βLOSδ(t)s(θq)+ αq;p,le s (Θq;l + ωq;p,l) δ(t − Tq;l − τq;p,l), (5.4) l=1 p=1 whereL q, Pq;l, αq;p,l, φq;p,l, Tq;l, τq;p,l,Θq;l, and ωq;p,l, represent the same parameters as were

1We assume perfect demodulation at the receiver. 70 Numerical Simulations described in Section 3.2, and the additional q subscript indicates that they are speciﬁc to the channel experienced by the qth transmitter. The mth row of hq(t) is denoted as hq,m(t) and represents the CIR between the qth transmitter and the mth element of the receiver, where the only discrepancy between hq,m(t) and hq,m (t) for m = m is caused by the diﬀerent phase shifts of the carrier signal. Convolving iq(t) with hq,m(t) for m =1,...,M, produces the received signal from the qth transmitter at each of the M antennas. The signal at the output of the beamformer, before sampling, is given by Q H y(t)=w (t) (iq(t) ∗ hq(t)) + n(t) , (5.5) q=0 where ⎡ ⎤ iq(t) ∗ hq,1(t) ⎢ ⎥ ⎢ ∗ ⎥ ⎢ iq(t) hq,2(t) ⎥ iq(t) ∗ hq(t) ⎢ . ⎥ . (5.6) ⎣ . ⎦

iq(t) ∗ hq,M (t) The signal y(t) is filtered using a matched, root raised-cosine filter, g(t), and the output of the filter is sampled at the symbol rate to obtain y(k). The AOAs and variances of the desired signal, interfering signal, and noise are the same values as those stated in Section 5.1.1, and the weights used to calculate R˜ˆ and z˜ˆ continue to be set as W = Ψ. Furthermore, we continue to simulate the system under the same four scenarios as described in the previous section. Due to the multipath environment, exact values of R and z are no longer known. The MSE, for a given w, is therefore determined by sampling an additional 100 symbols and calculating 100 1 2 ξ(w)= |y(k) − i0(k)| . (5.7) 100 k=1 More specific details concerning the implementation of the multipath model in MAT- LAB are provided in Appendix C.

5.1.3 MSE vs. Parameter Settings

Based on the material presented in Chapter 4, we know that w˜ˆ dl, w˜ˆ tsvd, and w˜ˆ dmr, are ˆ dependent upon the design parameters υ, σ˜ˆth, and λ˜th. Our ﬁrst set of simulations aim 5.1 Simulation Models 71 to reveal how these weights perform as the design parameters are varied. Figures 5.1(a),

5.2(a), and 5.3(a), illustrate the MSEs of w˜ˆ dl, w˜ˆ tsvd, and w˜ˆ dmr as functions of υ, σ˜ˆth, ˆ and λ˜th, respectively, in the AWGN model. In each of the aforementioned ﬁgures, four separate plots are provided that correspond to the MSE performance of w˜ˆ dl, w˜ˆ tsvd, and w˜ˆ dmr in each of the four scenarios; these four plots are labeled as Scenario A through Scenario D. To provide a frame of reference, the MMSEs corresponding to each scenario, calculated using (2.71), (2.33), and (5.1), are also plotted; these are labeled as MMSE Scenario A through MMSE Scenario D. The MMSEs of each scenario have very similar values, which makes it diﬃcult to distinguish their plots; nevertheless, all four MMSE plots are included for completeness. Figures 5.1(b), 5.2(b), and 5.3(b) illustrate the results of the same simulations carried out using the multipath model2. Simulations are carried out using 3600 samples to calculate R˜ˆ and z˜ˆ, and all results in Figures 5.1, 5.2, and 5.3, are averaged over 1000 independent simulations.

Figures 5.1(a) and 5.1(b) illustrate the plots corresponding to w˜ˆ dl as a function of υ, over the domain 0 ≤ υ ≤ 5. The ‘spikes’ that are prevalent in many of the plots are indications of a sudden increase in the MSE during one, or more, of the Monte Carlo simulations, ˆ and were likely caused by small eigenvalues of R˜ dl. In Figure 5.1(a), we see that w˜ˆ dl is stable in Scenario A and nearly reaches the MMSE at υ ≈ 0.4. In Scenarios B and C, the beamformer is unstable when υ is less than 1.3 and 3.1, respectively; we postulate that for small υ, the instability results from the diagonal loading increasing the negative eigenvalues ˆ closer to zero and thereby increasing the condition number of R˜ dl. However, w˜ˆ dl shows much more stable results when υ>1.3 in Scenario B, and when υ>3.1 in Scenario C. No amount of diagonal loading seems to improve the MSE performance of w˜ˆ dl in the presence of the strong interferer of Scenario D. In Figure 5.1(b), we see that the diagonal loading shows stability in Scenario A for υ ≥ 1, but reveals instability across the domain of υ in ˆ Scenarios B, C, and D. Due to the unstable nature of w˜ˆ dl, the diagonal loading of R˜ does not appear to be a viable solution, and consequently we no longer consider w˜ˆ dl throughout the remainder of this work.

The MSE of w˜ˆ tsvd as a function of σ˜ˆth is plotted in Figures 5.2(a) and 5.2(b). We observe that σ˜ˆth must be selected carefully to minimize the MSE, as choosing a value too high will result in the desired or interfering plane waves being attenuated and selecting a value too small will fail to remove the problematic singular values. Moreover, it is desirable to set a

2We plot the same MMSEs as were calculated for the AWGN model in multipath model ﬁgures. 72 Numerical Simulations

Fig. 5.1 MSE of w˜ˆ dl as a function of υ in (a) AWGN model; (b) Multipath model. threshold that works well across all scenarios, as an adaptable threshold would increase the receiver’s level of complexity. Consequently, we will set σ˜ˆth =0.07 for the remainder of the simulations. From Figures 5.2(a) and 5.2(b), it can be seen that this value of σ˜ˆth generates the lowest (or close to the lowest) MSE in all four scenarios under both the AWGN and multipath models.

Fig. 5.2 MSE of w˜ˆ tsvd as a function of σ˜ˆth in (a) AWGN model; (b) Multi- path model. 5.1 Simulation Models 73

ˆ The MSE performance of w˜ˆ dmr over 0 ≤ λ˜th ≤ 1 is shown in Figures 5.3(a) and 5.3(b).

We see in Scenario A, when no interference is present, w˜ˆ tsvd is capable of achieving a lower

MSE than w˜ˆ dmr in both the AWGN and multipath channels. However, when interference is introduced in Scenarios B, C, and D, w˜ˆ dmr is capable of producing a lower MSE than w˜ˆ tsvd. As was done with w˜ˆ tsvd, we select a threshold value that produces superior results in all four scenarios. However, this task is not straightforward since the minimum values of the plots of w˜ˆ dmr in all four scenarios do not align. Based on the empirical results shown ˆ ˆ in Figures 5.3(a) and 5.3(b), we select λ˜th =0.24. This value of λ˜th will be used in all upcoming simulations of w˜ˆ dmr in both the AWGN and multipath models.

ˆ Fig. 5.3 MSE of w˜ˆ dmr as a function of λ˜th in (a) AWGN model; (b) Multi- path model.

5.1.4 MSE vs. Sample Size

The MSE of w˜ˆ tsvd and w˜ˆ dmr versus the number of samples, ranging from 1 to 3600, used to calculate these weights in the four diﬀerent scenarios are plotted in Figures 5.4 through

5.7. For comparative purposes, we also plot the MSE performance of the weights, wpert, which are obtained from a perturbation-based algorithm that has been slightly modiﬁed from the one presented in Section 2.6.4 (a description of the calculation of wpert is provided in Appendix D). It must be emphasized that wpert has been plotted purely as a frame of reference, not as a performance benchmark upon which we are trying to improve. 74 Numerical Simulations

Comparing the relative performance between weights of the proposed algorithms, w˜ˆ tsvd and w˜ˆ dmr, with wpert is difficult because the perturbation algorithm is LMS-based and requires a step size, μ, be set, whose value directly affects the algorithm’s steady-state error and speed of convergence. For comparative purposes, we set μ such that the MSE of 3 ˆ wpert matches that of w˜ˆ tsvd in Scenario A at 3600 samples . Like the parameters λ˜ and σ˜ˆ, μ is not changed between different scenarios as, ideally, the beamformer should function in various scenarios without requiring its parameters be readjusted. All simulations involving wpert in the AWGN model incorporate μ =0.0025, while simulations in the multipath model use μ =0.0018. The plots in Figures 5.4 to 5.7 appear discrete-like due to the inherent structure of the algorithms. The MSE output of the perturbation algorithm is constant over a period of 4M = 24 samples, as that is the number of perturbations required to estimate the MSE gradient; when the number of samples is not a multiple of 24, the weights cannot be 2 updated. Likewise, multiples of M = 36 samples are required to estimate w˜ˆ tsvd and w˜ˆ dmr.

As a result, for an arbitrary number of samples, Nsamp, the estimate must be made using

Nsamp/36 samples, where · is the ﬂoor operator.

Fig. 5.4 MSE as a function of sample size in Scenario A in (a) AWGN model; (b) Multipath model.

As expected, we observe in Figures 5.4(a) and 5.4(b) that w˜ˆ tsvd outperforms w˜ˆ dmr

3 We set wpert to match the MSE of w˜ˆ tsvd rather than w˜ˆ dmr because w˜ˆ tsvd outperforms w˜ˆ dmr in Scenario A, as shown in Figures 5.4 a) and 5.4 b). 5.1 Simulation Models 75

Fig. 5.5 MSE as a function of sample size in Scenario B in (a) AWGN model; (b) Multipath model.

Fig. 5.6 MSE as a function of sample size in Scenario C in (a) AWGN model; (b) Multipath model. when there is no interfering signals present; this is especially true in the AWGN model.

In the presence of interfering plane waves w˜ˆ dmr displays superior MSE performance than w˜ˆ tsvd, as shown in Figures 5.5 to 5.7. Clearly, the perturbation algorithm responds better to interference than either w˜ˆ tsvd or w˜ˆ dmr, however its performance degrades considerably under the multipath model. As a result, w˜ˆ dmr shows a somewhat comparable performance to wpert in Scenarios B and C under the multipath model, which can be seen in Figures 76 Numerical Simulations

Fig. 5.7 MSE as a function of sample size in Scenario D in (a) AWGN model; (b) Multipath model.

5.5(b) and 5.6(b). The results shown in Figures 5.7(a) and 5.7(b) indicate that neither w˜ˆ tsvd nor w˜ˆ dmr perform well in the presence of very strong interference. Note the plot of wpert is missing in Figure 5.7(b) because it diverged in the simulations, indicating the step size is too large. This is not to say that wpert cannot be made stable for this scenario, but doing so requires μ to be decreased.

Overall, the results of w˜ˆ tsvd and w˜ˆ dmr illustrated in Figures 5.4, 5.5, and 5.6, are promising, although it is clear from these ﬁgures that neither w˜ˆ tsvd nor w˜ˆ dmr achieve the same

MSE performance in the presence of interference as wpert, particularly in the AWGN model simulations. However, we would like to remind the reader that these numerical simulations have been performed as functions of the number of samples used to calculate w˜ˆ tsvd, w˜ˆ dmr, and wpert. In a real system, the speed at which the analog weights can switch will have a dramatic impact on the perturbation algorithm, as it will aﬀect the speed at which the samples can be taken, and the perturbation algorithm requires 3600 weight changes for 3600 samples. Whereas, the speed at which the weights can switch will have a relatively negligible eﬀect on the proposed algorithm since the system requires 36 weight changes to obtain 3600 samples. Clearly, as the number of samples increases, the proposed algorithm becomes more attractive. 5.1 Simulation Models 77

5.1.5 Power Patterns

In this section, we examine the power patterns of the weights to obtain a more thorough understanding of their beam steering and nulling capabilities.

The power patterns of w˜ˆ tsvd, w˜ˆ dmr, and wpert, all calculated using 3600 samples and averaged over 100 simulations, are plotted in Figures 5.8 to 5.11. For comparative purposes, the power patterns produced by the optimal weights, wmmse, are also shown, where wmmse is calculated as described in Section 5.1.1. In all ﬁgures we mark the AOA of the desired signal, and any interfering signals that may be present, with a vertical, dotted line. The results pertaining to Scenario A are plotted in Figures 5.8(a) and 5.8(b), and reveal that w˜ˆ tsvd, w˜ˆ dmr, and wpert, all receive approximately the same power from angle θ0 in both the AWGN and multipath models. It can also be seen that w˜ˆ tsvd has lower side-lobe levels than w˜ˆ dmr, which explains the superior performance of w˜ˆ tsvd displayed in Figures 5.4(a) and 5.4(b) of the previous section.

Figures 5.9 and 5.10 reveal that w˜ˆ tsvd is generally the least capable of the three beamformers at canceling interference, as the nulls it produces at angles θ1, θ2, and θ3 are more shallow than those of w˜ˆ dmr or wpert. Moreover, we see that the superior performance of wpert in Figures 5.5 and 5.6 of the previous section can be attributed to the deep nulls that it forms at the AOAs of the interfering signals.

Figures 5.11(a) and 5.11(b) show that the performance degradation of w˜ˆ tsvd and w˜ˆ dmr in Scenario D, as observed in Figures 5.7(a) and 5.7(b), is not caused by an inability of these beamforming weights to form nulls in the direction of interference, but rather by the decrease in received power from angle θ0.

5.1.6 Eﬀects of Phase and Amplitude Quantization

We now consider the effects of finite-resolution phase shifters and amplitude-control elements. The finite resolution of these devices will create discrepancies between the weights calculated by the processor, and the actual weights applied to the incoming signals. A B-bit phase shifter can adjust the phase in increments of 2π/(2B). Thus, the set of all

B B B B possible phase shifts is φps = 0, 2π/(2 ), 4π/(2 ),...,(2 − 1)2π/(2 ) . Our program simulates the quantization eﬀect of the phase shifter on a weight vector, w, by rounding the phase of each entry of w to the nearest element of φps. We model a B-bit amplitude-control element as being capable of applying an amplitude from the discrete 78 Numerical Simulations

Fig. 5.8 Power patterns pertaining to Scenario A in (a) AWGN model; (b) Multipath model.

Fig. 5.9 Power patterns pertaining to Scenario B in (a) AWGN model; (b) Multipath model. set a = 0, 1/(2B − 1), 2/(2B − 1),...,1 . The amplitudes of w, denoted by |w|, are quantized by ﬁrst calculating |w|norm = |w| / |w|max, where |w|max is the largest entry of |w|, and then rounding the entries of |w|norm to the closest elements within the set of a. The scaling factor, 1/|w|max, is removed by multiplying the output of the beamformer, y(k), by |w|max. Under this model, the amplitudes of w are elements of the set

B B 0, |w|max/(2 − 1), 2|w|max/(2 − 1),...,|w|max . 5.1 Simulation Models 79

Fig. 5.10 Power patterns pertaining to Scenario C in (a) AWGN model; (b) Multipath model.

Fig. 5.11 Power patterns pertaining to Scenario D in (a) AWGN model; (b) Multipath model.

Our proposed algorithm can be designed to minimize the effects of quantization by ˆ ensuring that the weights w1,...,wM 2 , which are utilized to calculate R˜ and z˜ˆ, are constructed using only the discrete set of amplitudes and phases that can be produced by the finite-resolution phase shifters and amplitude-control elements. In this way, it can be made certain that there are no discrepancies between the weights w1,...,wM 2 , used by the digital processor and the actual weights applied by the analog devices during the calculation 80 Numerical Simulations of R˜ˆ and z˜ˆ. As a result, the only quantization occurs when the optimal weight estimates, w˜ˆ tsvd and w˜ˆ dmr, are implemented by the analog devices. Conversely, the weights used in the iterative-based perturbation algorithm are continuously subject to quantization effects during the weight perturbations that are carried out at each iteration.

Simulations illustrating the performance of w˜ˆ tsvd, w˜ˆ dmr, and wpert, using 3-, 4-, 5-, and 6-bit quantization (where both the phase shifter and the amplitude-control element possess the same resolution), are illustrated in Figures 5.12, 5.13, and 5.14, respectively. All simulations are performed using Scenario B, the number of samples used to calculate w˜ˆ tsvd, w˜ˆ dmr, and wpert range from 1 to 3600, and all results are averaged over 100 independent simulations.

Figures 5.12(a) and 5.12(b) reveal that quantizing w˜ˆ tsvd using a 3-bit resolution results in an MSE deterioration of approximately −2.5dB and −1.6dB, in the AWGN and multipath 4 models, respectively, at 3600 samples . Subjecting w˜ˆ tsvd to a 4-bit quantization results in losses of approximately −0.6dB and −0.5dB. In Figures 5.13(a) and 5.13(b), we see that w˜ˆ dmr exhibits a degradation in MSE of −2.5dB and −1.5dB, in the AWGN and multipath models, respectively, when quantized to a 3-bit resolution, and decreases of −0.9dB and −0.5dB under 4-bit quantization. The eﬀects of using 5-, and 6-bit resolution devices is negligible for both w˜ˆ tsvd and w˜ˆ dmr. Based on the preceding discussion it can be concluded that quantization has approximately the same relative eﬀect on w˜ˆ tsvd and w˜ˆ dmr.

Clearly, wpert is much more susceptible to performance degradation caused by quantization of weights, as is apparent in Figures 5.14(a) and 5.14(b). A noticeable decrease in MSE is displayed at all resolutions and over both models, however the decrease in MSE is less pronounced in the multipath model. The performance loss of wpert when quantized to a 3-bit resolution is −10dB and −6.5dB in the AWGN and multipath models, respectively. In Figures 5.15(a) and 5.15(b), we combine the plots from Figures 5.12 to 5.14 that correspond to 3-bit resolutions. In both the AWGN and multipath models, we see that w˜ˆ dmr displays the best performance. These results are of interest for practical implementation, as low resolution devices are generally cheaper and consume less power.

4All remaining measurements made in this section correspond to 3600 samples. 5.1 Simulation Models 81

Fig. 5.12 Eﬀects of quantization on w˜ˆ tsvd in (a) AWGN model; (b) Multi- path model.

Fig. 5.13 Eﬀects of quantization on w˜ˆ dmr in (a) AWGN model; (b) Multi- path model. 82 Numerical Simulations

Fig. 5.14 Eﬀects of quantization on wpert in (a) AWGN model; (b) Multi- path model.

Fig. 5.15 Comparison of w˜ˆ tsvd, w˜ˆ dmr, and wpert when quantized to 3 bits in (a) AWGN model; (b) Multipath model. 83

Chapter 6

Conclusions and Future Research

In this work, we have provided a comprehensive review concerning the fundamentals of array processing, and we have differentiated between digital beamforming structures and single-port beamforming structures. The single-port beamformer’s potential for commercial success in wireless applications is greater than that of the digital beamformer due to its lower cost, size, and power consumption. Conversely, it is more difficult to beamform with a single-port structure, as the signals present at each antenna are not available to the processor, and consequently the structure is not accommodating for well-studied digital beamforming algorithms. One popular method of single-port beamforming that was discussed was the iterative-based perturbation algorithm. It was explained that the dis- advantage of this method is the required constant switching of weights; this is not only complicated to design from a practical/hardware standpoint, but also limits the speed of beamforming to the switching speed of the weights. Accordingly, this algorithm is ill-suited for slow-switching, cheap, power efficient, analog components such as MEMs phase shifters. In response to these disadvantages, a single-port beamforming algorithm was proposed that uses a low number of weight changes to estimate the optimal MMSE weights, wmmse. A block of samples are collected for each weight vector, thus allowing the algorithm to take advantage of high-data-rate systems. Accordingly, it was argued in Chapter 3 that this ability to exploit high data rates along with the capability of mitigating interference makes the proposed algorithm well-suited for 60-GHz wireless receivers. Our algorithm calculates unbiased estimates of R and z using a total of M 2 unique weights. These estimates, R˜ˆ and z˜ˆ, can be used to obtain an estimate of the MMSE −1 ˆ −1 weights, wmmse = R z, by direct substitution, i.e., w˜ˆ = R˜ z˜ˆ. However, it was shown 84 Conclusions and Future Research via simulations that R˜ˆ can be ill-conditioned, thereby making w˜ˆ a very poor estimate. We suggested three different ways of ameliorating this problem based on diagonal loading,

TSVD, and DMR, and consequently proposed three different beamforming weights: w˜ˆ dl, w˜ˆ tsvd, and w˜ˆ dmr, respectively. The simulation results illustrating the MSE performance (versus the number of samples used to estimate the beamforming weights) revealed that w˜ˆ dl is not a viable option for beamforming. On the contrary, both w˜ˆ tsvd, and w˜ˆ dmr showed promising results, although neither seemed to perform particularly well in the presence of a very strong interfering signal. The weights w˜ˆ dmr displayed a superior performance over w˜ˆ tsvd in all scenarios involving interference. The MSE performances of both w˜ˆ tsvd and w˜ˆ dmr were inferior to that of the perturbation algorithm when interfering signals were present. Although it should be emphasized that the numerical simulations were carried out as a function of the number of samples employed to calculate the weights; in practice, the actual time required to beamform would be limited by the switching speed of the weights, which greatly impacts the perturbation algorithm, but has relatively little effect on the proposed algorithm. Finally, we saw that w˜ˆ tsvd and w˜ˆ dmr were less susceptible than wpert to performance degradation caused by quantization effects of finite-resolution, analog weights; in fact, both w˜ˆ tsvd and w˜ˆ dmr showed superior performance results when 3-bit amplitude-control elements and 3-bit phase shifters were employed. The research presented in this thesis can be extended in several areas. For instance, it would be interesting to perform a study, over a wide range of scenarios, revealing how many ˆ −1 samples would be required to obtain an acceptable estimate of wmmse using R˜ z˜ˆ without requiring any conditioning of R˜ˆ −1. For a high-data-rate application such as a 60-GHz wireless system, it may be found that an adequate number of samples can be accumulated over a small time interval to allow w˜ˆ = R˜ˆ −1z˜ˆ to be employed. It would also be very helpful to design alternative methods of conditioning R˜ˆ , such that the beamformer performs better in the presence of strong interferers. In Section 4.4.1, it was proposed that the selection of W should be based upon minimizing the Frobenius condition number, κF (W˜ ), and it was further suggested that this can be accomplished using a brute-force approach. Along the same lines as the work in [71], that however is not directly applicable to our case, it would be interesting to investigate the possibility of constructing a constrained optimization problem to solve for the minimum Frobenius condition number while ensuring the column vectors of W˜ satisfy (4.5), while 85

W has full row rank and W˜ has full rank. The system model and simulations presented in this work have assumed the components of the receiver are ideal. Ultimately, it would be beneﬁcial to simulate a realistic receiver that models many of the imperfections associated with practical antenna array receivers, e.g., noisy components, electromagnetic coupling between antennas, etc., and compare the proposed algorithm against the perturbation algorithm. 86 87

Appendix A

Proof of Proposition 2

As stated in Section 4.2.3, it is implied that u ∈{1,...,M} and v ∈{1,...,M} throughout the proof that follows.

Proof By deﬁnition of the inverse, ⎡ ⎤ 10 ⎢ ⎥ −1 ⎢ 1 ⎥ ˜ H ˜ H ⎢ ⎥ W W = ⎢ . ⎥ , or, (A.1) ⎣ .. ⎦ 01 ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ w˘ H w˜ H 10 ⎢ 1 ⎥ ⎢ 1 ⎥ ⎢ ⎥ ⎢ H ⎥ ⎢ H ⎥ ⎢ ⎥ ⎢ w˘ 2 ⎥ ⎢ w˜ 2 ⎥ ⎢ 1 ⎥ ⎢ . ⎥ ⎢ . ⎥ = ⎢ . ⎥ . (A.2) ⎣ . ⎦ ⎣ . ⎦ ⎣ .. ⎦ H H w˘ M 2 w˜ M 2 01

H H From (A.2), we will extract information about the vectors w˘ 1 ,...,w˘ M 2 , using known H H properties of w˜ 1 ,...,w˜ M 2 . Based on (4.6), a formula stating the relationship between the H individual elements of w˜ m and w can be established as

H ∗ w˜ m M(v−1)+u =[wm]u [wm]v . (A.3) 88 Proof of Proposition 2

From (A.3), it can be seen that when u = v,

H ∗ w˜ m M(u−1)+u =[wm]u [wm]u 2 = | [wm]u | . (A.4)

Alternatively, when u = v,

H H ∗ w˜ m M(v−1)+u = w˜ m M(u−1)+v . (A.5)

H H If we denote the mth column vector of W˜ as am, i.e., W˜ =[a1, a2,...,aM 2 ], then we can conclude from (A.4) and (A.5) that

M 2 aM(u−1)+u ∈ R , (A.6) and ∗ aM(v−1)+u = aM(u−1)+v , for u = v. (A.7)

Furthermore, (A.2) can be expressed as ⎡ ⎤ ⎡ ⎤ w˘ H 10 ⎢ 1 ⎥ ⎢ ⎥ ⎢ H ⎥ ⎢ ⎥ ⎢ w˘ 2 ⎥ ⎢ 1 ⎥ ⎢ . ⎥ [a1, a2,...,aM 2 ]=⎢ . ⎥ (A.8) ⎣ . ⎦ ⎣ .. ⎦ H w˘ M 2 01

We are now ready to prove the validity of the ﬁrst property stated in Proposition 2. H H We begin by examining the vector w˘ 1 in (A.8). Combining w˘ 1 a1 = 1 with the property M 2 in (A.6) stating that a1 ∈ R , yields

H Re w˘ 1 a1 =1, (A.9)

H Im w˘ 1 a1 =0. (A.10)

For u =2,...,M, we can use the relation in (A.8) to obtain

H Re w˘ 1 aM(u−1)+u =0, (A.11)

H Im w˘ 1 aM(u−1)+u =0. (A.12) 89

Thus, equations (A.9) to (A.12) establish the relationship between the real/imaginary com- H ponents of w˘ 1 , and the real-valued column vectors, aM(u−1)+u,u=1,...,M. The remaining M 2 − M column vectors of W˜ H possess the complex-conjugate symmetry described in (A.7). From (A.8), we see that for u = v,

H w˘ 1 aM(v−1)+u =0, (A.13) H w˘ 1 aM(u−1)+v =0, (A.14) and using (A.7), (A.13), and (A.14) we obtain (for u = v),

H Re w˘ 1 aM(u−1)+v =0, (A.15)

H Re w˘ 1 aM(v−1)+u =0, (A.16)

H Im w˘ 1 aM(u−1)+v =0, (A.17)

H Im w˘ 1 aM(v−1)+u =0. (A.18)

The following linear system can be constructed by combining (A.10),(A.12), (A.17), and (A.18),

H Im w˘ 1 [a1,...,aM 2 ]=[0,...,0]. (A.19)

H Since W˜ =[a1,...,aM 2 ] is full-rank, it follows that

H Im w˘ 1 =[0,...,0]. (A.20)

H Moreover, the expression in (A.9) indicates that Re w˘ 1 is a non-zero vector. Combining this observation with (A.20) requires that

M 2 w˘ 1 ∈ R . (A.21)

M 2 In a parallel manner it can be shown that w˘ M(u−1)+u ∈ R for u =2,...,M. Thus, the ﬁrst criteria listed in Proposition 2 holds. To prove the second statement of Proposition 2, we set u =2,v= 1, and from (A.7) we observe the relation between a2 and aM+1 as

∗ a2 =(aM+1) . (A.22) 90 Proof of Proposition 2

Referring to the system in (A.8), we can obtain the four following equations:

H w˘ 2 a2 =1, (A.23) H w˘ 2 aM+1 =0, (A.24) H w˘ M+1aM+1 =1, (A.25) H w˘ M+1a2 =0. (A.26)

Combining (A.22) through (A.26), we obtain the following eight inner products:

H Re w˘ 2 a2 =1/2, (A.27)

H Im w˘ 2 a2 = −j/2, (A.28)

H Re w˘ 2 aM+1 =1/2, (A.29)

H Im w˘ 2 aM+1 = j/2, (A.30) and,

H Re w˘ M+1 a2 =1/2, (A.31)

H Im w˘ M+1 a2 = j/2, (A.32)

H Re w˘ M+1 aM+1 =1/2, (A.33)

H Im w˘ M+1 aM+1 = −j/2. (A.34)

H H The relation between w˘ 2 , w˘ M+1, and the remaining vectors in [a1,...,aM 2 ] can easily be found using (A.6), (A.7), and (A.8). It follows that for {u, v} = {1, 2}, and {u, v} = {2, 1}, we obtain,

H Re w˘ 2 aM(v−1)+u =0, (A.35)

H Im w˘ 2 aM(v−1)+u =0, (A.36)

H Re w˘ M+1 aM(v−1)+u =0, (A.37)

H Im w˘ M+1 aM(v−1)+u =0. (A.38) 91

Combining (A.27) through (A.38), the following systems of equations can be constructed:

H 1 1 Re w˘ 2 [a1, a2, a3,...,aM , aM+1, aM+2,...]= 0, , 0,...,0, , 0,... , (A.39) 2 2

H 1 1 Re w˘ [a1, a2, a3,...,aM , aM+1, aM+2,...]= 0, , 0,...,0, , 0,... , (A.40) M+1 2 2 and,

H −j j Im w˘ 2 [a1, a2, a3,...,aM , aM+1, aM+2,...]= 0, , 0,...,0, , 0,... ,(A.41) 2 2

H j −j Im w˘ [a1, a2, a3,...,aM , aM+1, aM+2,...]= 0, , 0,...,0, , 0,... .(A.42) M+1 2 2

Since [a1,...,aM 2 ] is a full-rank matrix, it follows from (A.39) and (A.40) that

H H Re w˘ 2 =Re w˘ M+1 , (A.43) and similarly, (A.41) and (A.42) indicate that

H H Im w˘ 2 = − Im w˘ M+1 . (A.44)

Therefore, H H ∗ w˘ 2 = w˘ M+1 , (A.45) or equivalently ∗ w˘ 2 =(w˘ M+1) . (A.46)

This same logic that lead to the development of (A.46) can be applied to any other pair of vectors, w˘ M(v−1)+u, w˘ M(u−1)+v, u = v, to show

∗ w˘ M(v−1)+u = w˘ M(u−1)+v , for u = v, (A.47) which proves the second statement of Proposition 2 is valid. 92 93

Appendix B −1 Weight Matrices ΨH and Ψ˜ H

The matrix ΨH generated by Algorithm 1, for M = 4, is: ⎡ ⎤ 1000 ⎢ ⎥ ⎢ ⎥ ⎢ 0100⎥ ⎢ ⎥ ⎢ 0010⎥ ⎢ ⎥ ⎢ ⎥ ⎢ 0001⎥ ⎢ ⎥ ⎢ 1100⎥ ⎢ ⎥ ⎢ ⎥ ⎢ −j 100⎥ ⎢ ⎥ ⎢ 1010⎥ ⎢ ⎥ ⎢ − ⎥ H ⎢ j 010⎥ Ψ = ⎢ ⎥ (B.1) ⎢ 1001⎥ ⎢ ⎥ ⎢ ⎥ ⎢ −j 001⎥ ⎢ ⎥ ⎢ 0110⎥ ⎢ ⎥ ⎢ ⎥ ⎢ 0 −j 10⎥ ⎢ ⎥ ⎢ 0101⎥ ⎢ ⎥ ⎢ ⎥ ⎢ 0 −j 01⎥ ⎢ ⎥ ⎣ 0011⎦ 00−j 1

−1 The corresponding matrix Ψ˜ H is: −1 94 Weight Matrices ΨH and Ψ˜ H (B.2) ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ j 2 j 2 − 1 2 1 2 00 00 j 2 j 2 − 1 2 1 2 0000 0000 j 2 j 2 − 1 2 1 2 000000 000000 j 2 j 2 − 1 2 1 2 00000000 00000000 j 2 j 2 − 1 2 1 2 0000000000 0000000000 j 2 j 2 − 1 2 1 2 0000 00000000 0000000000 0000 00000000 0000000000 000000000000 000000000000 000000000000 000000000000 j j j j j j 2 2 2 2 2 2 + + + − − − 000 0 000000 000 0 000000 1 2 1 2 1 2 1 2 1 2 1 2 − − − − − − j j j j j j 2 2 2 2 2 2 + + + − − − 00 00 0 0 1 2 1 2 1 2 1 2 1 2 1 2 − − − − − − j j j j j j 2 2 2 2 2 2 + + + − − − 0 00 0 00 1 2 1 2 1 2 1 2 1 2 1 2 − − − − − − j j j j j j 2 2 2 2 2 2 + + + − − − 00 0 0010 0 00 1000 0 0001 0100 0 1 2 1 2 1 2 1 2 1 2 1 2 − − − − − − ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ = 1 − H ˜ Ψ 95

Appendix C

Details Concerning Implementation of Multipath Model

The root-raised-cosine, pulse-shaping filter, g(t), is designed with a roll-off factor of 0.5, a group delay of three symbols, and an upsampled rate of 20. Moreover, these settings are identical for each of the Q + 1 transmitters. The modulating signal, iq(k), of the qth transmitter in (5.3) is simulated by generating a random sequence of symbols, assumed to have a symbol rate of 1 Gs/s, and filtering this sequence using the aforementioned filter g(t). Thus, the sequence at the output of the transmit filter has a sampling frequency of 20 Gs/s, and represents the complex, equivalent baseband signal in continuous time being transmitted over the channel.

A unique realization of the CIR, hq(t, θ), is generated for each transmitter. The MAT- LAB code provided by IEEE generates a channel realization by producing three vectors.

The ﬁrst vector contains the amplitudes of the multipath components, αq;p,l, the second contains the time-of-arrivals (TOAs), Tq;l +τq;p,l, and the third contains the AOAs, Θq;l +ωq;p,l, where in all three cases l =1, ..., L q,p=1, ..., Pq;l. The TOAs are provided in continuous time, so we discretize them into time bins of duration 1/20 ns (chosen to match the sampling frequency of the transmitted signals). Two or more multipath components falling within the same bin are summed. The channel between the qth transmitter and the receiver is modeled using M = 6 separate TDL ﬁlters, where each TDL simulates hq,m(t), as described in Section 5.1.2. These TDL models are shown in Figure C.11. It should be noted that the number of CIRs shown in Figure C.1 is given in terms of M, however,

1The weights of the taps shown in Figure C.1 are an example of one possible realization. 96 Details Concerning Implementation of Multipath Model

as previously stated, M is set to 6 in our simulations. By convolving iq(t) with hq,m(t),

Fig. C.1 Block diagram of the TDLs used to model the channel between the qth transmitter and each of the M antennas at the receiver (in our simulations M = 6). a vector is produced that represents the received signal at the mth element of the array due to the qth transmitter. This procedure is repeated for each of the six elements to produce six vectors, each representing the received signal at that element. The aggregate signal received from all directional sources at the mth antenna is created by summing i0(t)∗h0,m(t)+...+iQ(t)∗hQ,m(t). AWGN components are then added and the signals are multiplied by the beamforming weights and summed to produce the output of the beamformer, y(t), as given by (5.5). Finally, this output sequence of the beamformer is ﬁltered with a root-raised-cosine, pulse-shaping ﬁlter, identical to g(t), and sampled at the symbol rate to obtain y(k). 97

Appendix D

Perturbation Algorithm Implementation

The majority of perturbation-based algorithms (e.g., [20, 21]) are designed to estimate the gradient of the output power, P (w(k)), for a given weight w(k). These algorithms assume that θ0 is known to the receiver and estimate the optimal MVDR weights using the CLMS algorithm, which was described in Section 2.6.4. Therefore, a comparison between these perturbation algorithms and our proposed algorithms is diﬃcult, as our proposed beamformers have no knowledge of θ0, but rather use a training sequence that is known at the receiver. However, as asserted in [3], the perturbation method can be easily applied to other processors. It follows that in an attempt to make the comparison fair, we modify (2.88) such that the perturbation algorithm calculates an estimate of the MSE gradient, ∇ˆ wξ(w(k)) w=w(k) , as opposed to the power gradient, using

L 1 2 ∇ˆ wξ(w(k)) w=w(k) = | (w(k), )| δ( ), (D.1) p Lγ =1

H where (w(k), )=d(ks + − 1) − (w(k)+γpδ( )) x(ks + − 1) (the notation follows that of Section 2.6.4). The estimate in (D.1) is then used in conjunction with (2.77) of the LMS algorithm to update w(k). Following the same procedure as described in [20, 21] to ﬁnd γp that minimizes the covariance of (2.88), we set γp to minimize the covariance of the MSE 98 Perturbation Algorithm Implementation gradient estimate as follows: 2 H H H 1/2 E[|d(k)| ]+wmmseRwmmse − wmmsez − z wmmse γp = , (D.2) 2tr (R) where the calculations of R and wmmse are given by (2.33) and (2.66), respectively. The sequence of perturbation vectors, S = {δ(1), δ(2),...,δ(L)}, that are used in the simulations are the same as those originally proposed in [20, 21]. There are a total of L =4M vectors that are given by

[δ( )]m =[f( )]m + j [f( )]m+M ,m=1, 2,...,M, =1, 2,...,4M, (D.3)

where [δ( )]m is the mth entry of the vector δ( ), and ⎧ √ ⎪ − ⎨ √2M, =2m 1, [f( )] = − 2M, =2m , (D.4) m ⎩⎪ 0, elsewhere in the range 1 ≤ ≤ 4M,

where m =1,...,2M, and [f( )]m is the (m )th entry of f( ). The weights switch at the same speed as the sampling rate of the ADC, which results in consecutive symbols being ﬁltered by diﬀerent weights. As stated in Chapter 3, this may not be physically realizable in a 60-GHz system. Nevertheless, for simulation purposes, we assume in the multipath model that the weights are capable of changing at this rate to allow the perturbation algorithm to be compared with the proposed beamformers. 99

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