System Model

We consider a mmWave massive MIMO wireless charging system. The power trans- mitter (PT) is equipped with N antennas, and serves K single-antenna 1 _{power re-}

ceivers (PR). We assume that the PT has MRF(≤ N ) RF chains due to the practical

cost-efficient considerations, and MRF antennas are selected to pair with these RF

chains. The system therefore can transmit MS(≤ MRF) streams for power transfer. In

the spatial domain, for each stream, a beamforming vector forms a beam which brings in the power gain to those PRs in the beam direction. Multiple streams can simulta- neously serve PRs in different locations, but also result in high baseband processing 1_{We consider single-antenna receiver in this work, and the results in this chapter can be general-}

ized to multiple-antenna receivers by alternatively optimizing for the receive beamformers and the transmit beamformer.

complexity. Hence the number of streams MS is an important system parameter and

we will consider the multi-stream (i.e., MS > 1) and uni-stream (i.e., MS = 1) cases

separately.

3.2.1

MmWave MIMO Channel Model in Spatial Domain

The first part of this thesis (Chapter 2 and Chapter 3) aims to optimize the perfor- mance of the multi-agent sequential test under these two constraints.

First we consider the classic fully digital MIMO architecture with N = MRF as

shown in Fig. 3.1(a). Let ˜hk ∈ CN denote the channel vector of the k-th PR. It

is known that the mmWave channel exhibits sparse scattering [60]. We employ the classic Saleh-Valenzula channel model to describe ˜hk with a line-of-sight (LoS) path

and L scattering paths as [58, 60] ˜ hk = s N %_k  αk,0a(θk,0) + L X l=1 αk,la(θk,l)  , (3.1)

where %k is the path-loss, αk,0 and αk,l are the complex channel gains of the LoS

and the l-th scattering path of PR k with the signal physical directions θk,0 and

θk,l respectively, and a(θ) is the steering vector. For uniform linear arrays (ULA)

the steering vector can be expressed as a(θ) = √1 N[1, e

j2π

λd sin(θ), · · · , ej(N −1) 2π

λ sin(θ)]H, where λ is the wavelength and d is the distance between adjacent antenna elements. Typically d = λ/2, which yields a(θ) = √1

N[1, e

jπ sin(θ)_{, · · · , e}j(N −1)π sin(θ)_]H_{. Denote}

˜

H = [˜h1, · · · , ˜hK] ∈ CN ×K as the channel matrix of all PRs.

Let vm ∈ CN be the beamforming vector of the power stream m, and ˆV =

[v1, · · · , vMS] ∈ C

N ×MS _{be the beamforming matrix. To satisfy the transmit power}

constraint, we have Tr( ˜VHV ) ≤ P where P is the total power supply at the PT. Let˜ s ∈ CMS _{be the power signal vector with E(ss}H_{) = I}

MS. Hence, the received signal vector for all PRs is

˜

Baseband Processing RF Chain RF Chain RF Chain Channel PR 1 PR 2 PR K

(a) Traditional MIMO system

Baseband Processing RF Chain RF Chain RF Chain Channel PR 1 PR 2 PR K Selection Network DLA

(b) Beamspace MIMO with lens antenna array

3.2.2

Beamspace MIMO with Lens Antenna Array

The channel model of beamspace MIMO can be transformed from the conventional MIMO in the spatial domain with carefully designed lens antenna array [58, 62]. As shown in Fig. 3.1(b), N antenna elements are attached to the lens to form a discrete lens array (DLA), and such DLA acts as an N × N discrete Fourier matrix U which transforms between the spatial domain and beamspace domain. In other words, the DLA turns an omni-directional signal into a beam. Specifically, the DLA matrix is U = [˜a(1), · · · , ˜a(N )], where column ˜a(m) = √1

N[1, e

jmπ_{, e}jm2π_{, · · · , e}jm(N −1)π_]H

represents the orthogonal array steering vector of the m-th antenna elements in the DLA. Hence, for any PR k, the channel vector for the beamspace MIMO becomes hk = UHh˜k. The received signal vector for the beamspace MIMO is

y = HHV s = ˜ˆ HHU ˆ_{V s ∈ C}K, (3.3)

where H = UH_{H is the beamspace MIMO channel matrix. We assume that the˜}

channel matrix H is known to the transmitter 2.

So far we have considered the beamspace MIMO system with full deployment of the N antennas, which requires N individual RF chains. In this chapter, we consider the hybrid beamforming architecture with MRF(≤ N ) RF chains using an-

tenna selection. There are two main reasons. The first is for cost efficiency, since reducing the number of RF chains will lower the total cost of the RF circuits given that the RF chain module is expensive. Secondly, the mmWave channel has lim- ited and sparse scattering, especially for the beamspace MIMO since the channel is sparse under the Fourier bases. This means that it is not necessary to use ev- ery antenna for power transmission, and it suffices to use the dominant beams. As a result, MRF antennas are selected, which form the effective channel matrix

H(A) = H[n, :]{n∈A} = U [n, :]H_{n∈A}H ∈ C˜ MRF×K where A is the set of indices of se-

lected antennas, and the corresponding beamforming matrix becomes V ∈ CMRF×MS_. 2_{Channel estimation can be efficiently performed for beamspace MIMO[62].}

The final signal model is then

y = H(A)H_{V s ∈ C}K. (3.4)

The received power vector of all PRs is

p = diag(VHH(A)H(A)HV ), (3.5)

where diag(·) returns a vector consisting of the diagonal elements of the input ma- trix. Note that in (3.5) we employ the linear RF energy harvesting model[45, 50]. Recent experiments in[64] show that the linear model is justified and can accurately characterize the relationship between the transmitted and received power.

Note that in wireless power transfer systems all PRs can harvest power from a common RF power stream, hence it is not necessary to restrict that K ≤ MS. In

fact, it is preferable to design a system with MS as low as possible since this would

make the beamformer design and baseband processing tasks much more efficient.