text:
\section{Introduction}
\label{sec1}
Over the last few years, there has been a growing interest in leveraging the opening up
of the spectrum in the millimeter wave band ($\sim {\hspace{-0.03in}} 30$-$100$ GHz) in
realizing the emerging higher data rate demands of cellular
systems~\cite{khan,qualcomm,rappaport,boccardi1}. Communications in the millimeter wave
band suffers from increased path loss exponents, higher shadow fading, blockage and
penetration losses, etc., than sub-$6$ GHz systems leading to a poorer link margin than
legacy
systems~\cite{5G_whitepaper,3gpp_CM_rel14_38901,azar,shu_sun2016,vasanth_tap2018,vasanth_tap_blockage}.
However, by restricting attention to small cell coverage and by reaping the increased
array gains from the use of large antenna arrays at both the base-station and user ends,
significant rate improvements can be realized in practice.

Millimeter wave propagation is spatially {\em sparse} with few dominant clusters in
the channel relative to the number of
antennas~\cite{5G_whitepaper,3gpp_CM_rel14_38901,vasanth_it2,qualcomm2}.
Spatial sparsity of the channel along with the use of large antenna arrays motivates
a subset of physical layer beamforming schemes based on
{\em directional} transmissions for signaling. In this context, there have been a
number of studies on the design and performance analysis of directional
beamforming/precoding structures for single-user multi-input multi-output (MIMO) systems~\cite{rusek,hur,roh,brady_tcom,oelayach,vasanth_gcom15,raghavan_jstsp,rangan,ghosh,sun}.
These works~\cite{brady_tcom,oelayach,vasanth_gcom15,raghavan_jstsp} show that
directional schemes are not only good from an implementation standpoint, but are
also robust to phase changes across clusters and allow a smooth tradeoff between
peak beamforming gain and initial user discovery latency. There has also been
progress in generalizing such directional constructions for multi-user MIMO transmissions~\cite{sun,vasanth_gcom16,vasanth_jsac2017,ang_li}.

In this context, while legacy systems use as many radio frequency (RF)
chains\footnote{An RF chain includes (but is not limited to) analog-to-digital
converters (ADCs), digital-to-analog converters (DACs), mixers, low-noise and
power amplifiers (PAs), etc.} as the number of antennas, their higher cost,
energy consumption, area and weight at millimeter wave carrier frequencies has
resulted in the popularity of {\em hybrid beamforming}
systems~\cite{molisch1,molisch2,venkateswaran,adhikary}. A hybrid beamforming
system uses a smaller number of RF chains than the number of antennas, with the
one extreme case of a single RF chain being called the analog/RF beamforming
system and the other extreme of as many RF chains as the number of antennas
being called the digital beamforming system. Spatial sparsity of millimeter
wave channels ensures that having as many RF chains as the number of dominant
clusters in the channel is sufficient to reap the full array gain possible over
these channels.

A number of recent works have addressed hybrid beamforming for millimeter wave systems.
The problem of finding the optimal precoder and combiner with a hybrid architecture
is posed as a sparse reconstruction problem in~\cite{oelayach}, leading to algorithms
and solutions based on basis pursuit methods. While the solutions achieve good performance
in certain cases, to address the performance gap between the solution proposed
in~\cite{oelayach} and the unconstrained beamformer structure, an iterative scheme is
proposed in~\cite{alkhateeb,alkhateeb2} relying on a hierarchical training codebook for
adaptive estimation of millimeter wave channels. The authors in~\cite{alkhateeb,alkhateeb2}
show that a few iterations of the scheme are sufficient to achieve near-optimal performance.
In~\cite{sohrabi}, it is established that a hybrid architecture can approach the
performance of a digital architecture as long as the number of RF chains is twice that
of the data-streams. A heuristic algorithm with good performance is developed when this
condition is not satisfied. A number of other works such
as~\cite{noh16fdbk,bogale,gao_hybrid,magueta_hybrid} have also explored iterative/algorithmic
solutions for hybrid beamforming.

A common theme that underlies most of these works is the assumption of {\em phase-only}
control in the RF/analog domain for the hybrid beamforming architecture. This
assumption makes sense at the user end with a smaller number of antennas (relative to the
base-station end), where operating the PAs below their peak rating across RF chains can lead to
a substantially poor uplink performance. On the other hand, amplitude control (denoted as
{\em amplitude tapering} in the antenna theory literature) is necessary at the base-station end
with a large number of antennas for side-lobe management and mitigating out-of-band emissions.
Further, given that the base-station is a network resource, simultaneous amplitude and phase
control of the individual antennas across RF chains is feasible at millimeter wave base-stations
at a low-complexity\footnote{Any calibration complexity can be seen as a one-time effort
at the unit level for a large array and defrayed as a low network cost.} and
cost~\cite[pp.\ 285-289]{harish},~\cite{huang2015low,briqech2017low}.
In particular, the millimeter
wave experimental prototype demonstrated in~\cite{vasanth_comm_mag_16} allows simultaneous
amplitude and phase control. Thus, it is important to consider a hybrid architecture with
these constraints. Further, given the directional nature of the channel, a solution
should both inherit a directional structure and provide an intuitive description of
the beam weights. For example, a {\em black box}-type algorithmic solution that does
not provide an intuitive description of the beam weights is less preferable over a
solution that is constructed out of measurement reports obtained over an initial
{\em beam alignment} phase with a directional structure for the sounding beams.

\noindent {\bf \em \underline{Main Contributions:}}
With this backdrop, this work addresses these two fundamental issues in hybrid beamformer design.
It is assumed that the base-station trains all the users in the cell with a cell-specific
codebook of beamforming vectors over an initial beam alignment phase. Each user makes an
estimate\footnote{In a practical implementation such as the Third Generation Partnership
Project New Radio (3GPP 5G-NR) design, $P = 4$ is typically assumed both in terms of
measurements and reporting~\cite{3gpp_CM_rel14_38912}. The received ${\sf SNR}$ is estimated
as the received power of a beamformed link (corresponding to the beam pair under consideration)
using a certain reference symbol resource. This metric is typically known as the reference
symbol received power (RSRP) of the link.} of the top-$P$ (where $P \geq 1$) beams over this
phase and reports the beam indices to be used by the base-station as well as the measured/received
signal-to-noise ratios (${\sf SNR}$s). The simplest implementation at the base-station uses
only the best beam information for beam steering or zeroforcing as
in~\cite{vasanth_gcom16,vasanth_jsac2017}, with other beams serving as fall back options.

In contrast to this approach, we propose to {\em reconstruct} or {\em estimate} a rank-$P$
approximation of the channel matrix between the base-station and the user (at the base-station
end). To realize
this reconstruction, we envision the additional feedback of the phase of the received signal
estimate of the top-$P$ beams over the beam alignment phase and the cross-correlation information
of the top-$P$ beams at the user end with the beam used for multi-user reception. With
this novel construction, the base-station can remain agnostic of the user's top-$P$
beams in precoder design. In terms of overhead,
in 3GPP 5G-NR, these quantities can be fed back over the physical uplink control channel (PUCCH)
with a Type-II feedback scheme~\cite[Sec.\ 8.2.1.6.3, pp.\ 24-26]{3gpp_CM_rel14_38912}; see
Sec.~\ref{sec5b} for a detailed study that demonstrates this feedback overhead to be marginal.
Leveraging the rank-$P$ channel approximation, we propose the use of a zeroforcing structure
that is then quantized to meet the RF precoding constraints (amplitude and phase control) at
the base-station end for simultaneous transmissions.

To benchmark and compare the performance of the proposed scheme, we establish two upper bounds
for the sum rate. This is a fundamentally difficult problem given the non-convex dependence of
the sum rate on the beamforming vectors~\cite{cioffi_weighted,kobayashi,vasanth_arxiv2011}.
The first bound is based on an intuitive parsing and understanding of the zeroforcing
structure. The second bound is based on an alternating optimization of the
beamformer-combiner pair with signal-to-leakage and noise ratio
(${\sf SLNR}$)~\cite{tarighat} and signal-to-interference and noise ratio
(${\sf SINR}$) as optimization metrics. Numerical studies show that the proposed
scheme performs significantly better than a na{\"i}ve beam steering solution even for an initial
beam alignment codebook of poor resolution. Further, the proposed scheme is comparable with the
established upper bounds provided the beam alignment codebook resolution is moderate-to-good.
Thus, our work establishes the utility and efficacy of the proposed feedback techniques
as well as opens up avenues for further investigation of such approaches in
hybrid beamforming with millimeter wave systems.

\noindent {\bf \em \underline{Organization:}} This paper is organized as follows.
Sec.~\ref{sec2} develops the system setup and explains the RF precoder architectural
constraints adopted in this work. In Sec.~\ref{sec3}, we provide a background of the
initial beam alignment phase and the feedback mechanism necessary for the multi-user
beamforming envisioned in this work. Sec.~\ref{sec4} generates two upper bounds on
the sum rate to benchmark the performance of the proposed scheme. Sec.~\ref{sec5}
performs a number of numerical studies to understand the performance of the proposed
scheme relative to a na{\"i}ve beam steering solution as well as to the upper bounds
developed in Sec.~\ref{sec4}. Concluding remarks are provided in Sec.~\ref{sec6}.

\ignore{
\noindent {\bf \em \underline{Notations:}} Lower- and upper-case bold symbols are used to
denote vectors and matrices, respectively. The $i$-th entry of a vector ${\bf x}$ and the
$(i,j)$-th entry of a matrix ${\bf X}$ are denoted by ${\bf x}(i)$ and ${\bf X}(i,j)$,
respectively. The regular matrix transpose and complex conjugate Hermitian transpose
operations of a matrix are denoted by $(\cdot)^{\sf T}$ and $( \cdot)^{\dagger}$, respectively.
The inverse and principal square root of a square and a positive semi-definite matrix are denoted
as $(\cdot)^{-1}$ and $(\cdot)^{1/2}$, respectively. The trace and expectation operators
are denoted by ${\sf Tr}(\cdot)$ and $\expec[\cdot]$, respectively. The two-norm of a vector is
denoted as $\| \cdot \|$ with ${\mathbb{C}}$, ${\mathbb{R}}$ and ${\cal CN}$ standing for the set
of reals, complex and the complex normal random variable, respectively. Quantization operation of
an underlying variable with $B$ bits will commonly feature in this work and this operation is
denoted as ${\cal Q}_B(\cdot)$.
}

\noindent {\bf \em \underline{Notations:}} Lower- and upper-case bold symbols are used to
denote vectors and matrices, respectively. The $i$-th entry of a vector ${\bf x}$ and the
$(i,j)$-th entry of a matrix ${\bf X}$ are denoted by ${\bf x}(i)$ and ${\bf X}(i,j)$,
respectively. The regular matrix transpose and complex conjugate Hermitian transpose
operations of a matrix are denoted by $(\cdot)^{\sf T}$ and $( \cdot)^{\dagger}$, respectively.
The two-norm of a vector is denoted as $\| \cdot \|$ with ${\mathbb{C}}$, ${\mathbb{R}}$ and
${\cal CN}$ standing for the set of reals, complex numbers and the complex normal
random variable, respectively.

\section{System Setup}
\label{sec2}
We consider a cellular downlink scenario with a single base-station serving
$K_{\sf cell}$ potential users. The base-station and each user are assumed to
be equipped with planar arrays of dimensions $N_{\sf tx} \times N_{\sf tz}$
antennas and $N_{\sf rx} \times N_{\sf rz}$ antennas, respectively. At both
ends, the inter-antenna element spacing is $\lambda/2$ where $\lambda$ is the
wavelength of propagation. With $N_{\sf t} = N_{\sf tx} \cdot N_{\sf tz}$ and
$N_{\sf r} = N_{\sf rx} \cdot N_{\sf rz}$, the base-station and each user are
assumed to have $M_{\sf t} \leq N_{\sf t}$ and $M_{\sf r} \leq N_{\sf r}$ RF
chains, respectively.

\ignore{
\begin{figure*}[htb!]
\begin{center}
\includegraphics[height=3.2in,width=6.3in]  {figures/systemFigs.eps}
\caption{\label{fig_system_model}
System model capturing a cellular downlink scenario where a base-station
communicates with $K_{\sf cell} = 2$ users over an initial beam alignment
phase (beam patterns in dashed green), followed by multi-user beam design
(illustrated in red) based on the feedback from each user.}
\end{center}
\end{figure*}
}

For the channel ${\bf H}_k \in \complexs^{N_{\sf r} \times N_{\sf t}}$ between
the base-station and the $k$-th user (where $k = 1, \cdots, K_{\sf cell}$), we
assume an extended geometric propagation model over $L_k$
clusters/paths~\cite{saleh,3gpp_CM_rel14_38901}
\begin{eqnarray}
\label{eq:channel}
\bH_k = \sqrt{\frac{N_{\sf r} N_{\sf t} }{L_k}} \sum_{\ell = 1}^{L_k}
\alpha_{k,\ell} \bu_{k,\ell} \bv_{k,\ell}^{\dagger}.
\end{eqnarray}
In~(\ref{eq:channel}), $\alpha_{k,\ell}$, $\bu_{k,\ell}$ and $\bv_{k,\ell}$ denote
the complex gain, the array steering vector at the user end corresponding to
the angle of arrival (AoA) in azimuth/zenith, and the array steering vector at the
base-station corresponding to the angle of departure (AoD) in azimuth/zenith,
respectively. The
cluster gains are assumed to be independent and identically distributed (i.i.d.)
standard complex Gaussian random variables: $\alpha_{k,\ell} \sim \mathcal{CN}(0,1)$.
The normalization of the channel ensures that $\expec \big[
\trace(\bH_k \bH_k^\dagger ) \big] = N_{\sf r} N_{\sf t}$.

In terms of the system model, we focus on the narrowband aspects and assume that the
base-station serves $K \leq K_{\sf cell}$ users simultaneously with data along $M_{\sf t}$ RF
chains. The base-station precodes $r_m$ data-streams for the $m$-th user with
the $r_m \times 1$ symbol vector
${\bf s}_m$ using the $M_{\sf t} \times r_m$ digital/baseband precoder
${\bf F}_{ {\sf Dig},\hsppp m}$ which is then up-converted to the carrier
frequency by the use of the $N_{\sf t} \times M_{\sf t}$ RF precoder
${\bf F}_{\sf RF}$. This results in the following system equation at the $k$-th user
\begin{eqnarray}
{\bf y}_k = \sqrt{\frac{\rho}{K}} \hsppp {\bf H}_k {\bf F}_{\sf RF} \cdot
\left[ \sum_{m = 1}^K {\bf F}_{ {\sf Dig},\hsppp m} {\bf s}_m \right] + {\bf n}_k
\label{eq_system_model}
\end{eqnarray}
where $\rho$ is the pre-precoding ${\sf SNR}$ and ${\bf n}_k \sim {\cal CN}( {\bf 0},
\bI_{ N_{\sf r}})$ is the $N_{\sf r} \times 1$ white Gaussian noise vector
added at the $k$-th user. We assume that ${\bf s}_m$ are i.i.d.\ complex Gaussian
random vectors
with $\expec[\bs_m] = \mathbf{0}$ and $\expec[{\bf s}_m {\bf s}_m^\dagger] = \bI_{r_m}$.

At the $k$-th user, we assume that ${\bf y}_k$ is processed (down-converted) with an
$N_{\sf r} \times M_{\sf r}$ user-specific RF combiner ${\bf G}_{ {\sf RF}, \hsppp k}$
followed by a user-specific $M_{\sf r} \times r_k$ digital combiner ${\bf G}_{ {\sf Dig},
\hsppp k}$ to produce an estimate of ${\bf s}_k$ as follows
\begin{align}
& \widehat{\bf s}_k =
{\bf G}_{ {\sf Dig}, \hsppp k} ^{\dagger} {\bf G}_{ {\sf RF}, \hsppp k}^{\dagger}
{\bf y}_k
\\
& {\hspace{0.15in}}
=
\sqrt{\frac{\rho}{K}} \hsppp
{\bf G}_{ {\sf Dig}, \hsppp k} ^{\dagger} {\bf G}_{ {\sf RF}, \hsppp k}^{\dagger}
{\bf H}_k {\bf F}_{\sf RF} {\bf F}_{ {\sf Dig},\hsppp k} {\bf s}_k
+ \sqrt{\frac{\rho}{K}} \hsppp
{\bf G}_{ {\sf Dig}, \hsppp k} ^{\dagger} {\bf G}_{ {\sf RF}, \hsppp k}^{\dagger}
{\bf H}_k {\bf F}_{\sf RF} \sum_{m = 1, m \neq k}^K
{\bf F}_{ {\sf Dig},\hsppp m} {\bf s}_m + {\bf n}_k.
\label{eq_decoding}
\end{align}
The achievable rate ${\cal R}_k$ (in nats/s/Hz) at the $k$-th user when treating
multi-user interference as noise
is given as
\begin{eqnarray}
{\cal R}_k & = & \log \det \left( {\bf I}_{r_k} + \frac{\rho}{K}
{\bf G}_{ {\sf Dig}, \hsppp k} ^{\dagger} {\bf G}_{ {\sf RF}, \hsppp k}^{\dagger}
{\bf H}_k {\bf F}_{\sf RF} {\bf F}_{ {\sf Dig},\hsppp k}
{\bf F}_{ {\sf Dig},\hsppp k}^{\dagger} {\bf F}_{\sf RF}^{\dagger}
{\bf H}_k^{\dagger} {\bf G}_{ {\sf RF}, \hsppp k} {\bf G}_{ {\sf Dig}, \hsppp k}
\cdot {\bf \Sigma}_{\sf intf}^{-1} \right)
\end{eqnarray}
where ${\bf \Sigma}_{\sf intf}$ denotes the interference and noise covariance matrix
\begin{eqnarray}
{\bf \Sigma}_{\sf intf} & = & {\bf I}_{r_k} +
\frac{\rho}{K}
{\bf G}_{ {\sf Dig}, \hsppp k} ^{\dagger} {\bf G}_{ {\sf RF}, \hsppp k}^{\dagger}
{\bf H}_k {\bf F}_{\sf RF} \left( \sum_{m \neq k} {\bf F}_{ {\sf Dig},\hsppp m}
{\bf F}_{ {\sf Dig},\hsppp m}^{\dagger} \right) {\bf F}_{\sf RF}^{\dagger}
{\bf H}_k^{\dagger} {\bf G}_{ {\sf RF}, \hsppp k} {\bf G}_{ {\sf Dig}, \hsppp k}.
\end{eqnarray}

The traditional use of {\em finite-rate feedback} has been to convey the index of
a precoder matrix from an appropriately-designed codebook of precoders to assist
with adaptive transmissions to improve ${\cal R}_k$~\cite{david_review,vasanth_design}.
More generally, feedback from
users can also be used to aid in scheduling, channel estimation and
advanced/non-codebook based precoder design. In this work, as we will see later
in Sec.~\ref{sec3}, we assume that each user feeds back its top beam indices,
an estimate of the received ${\sf SNR}$ and signal phase, and cross-correlation
of the top receive beams to assist with the design of
a non-codebook based multi-user precoder structure.
In terms of precoder constraints, we make the assumption that
${\bf F}_{ {\sf Dig}, \hsppp m} \in {\mathbb{C}}^{ M_{\sf t} \times r_m}$.

For the RF precoder, we assume that the amplitude and phase of each entry in ${\bf F}_{\sf RF}$
are controlled by a finite precision gain controller and phase shifter, respectively.
In other words, the amplitude and phase come from a set of $2^{B_{\sf amp}}$ and
$2^{B_{\sf phase}}$ quantization levels
\begin{eqnarray}
|{\bf F}_{\sf RF}(i,j)| \in \left\{ A_1, \cdots, A_{2^{B_{\sf amp}}} \right\},
\hspp \hspp
\angle{ {\bf F}_{\sf RF}(i,j) } \in \left\{ \phi_1, \cdots, \phi_{2^{B_{\sf phase}}} \right\},
\label{eq_hybrid_precoder_constraints}
\end{eqnarray}
where $0 \leq A_1 < A_2 < \dots < A_{2^{B_{\sf amp}}}$. Prior work on hybrid
beamforming such as~\cite{oelayach,alkhateeb,alkhateeb2,sohrabi} etc., assume
that the RF precoder can only be controlled by a phase shifter. However, such
constraining assumptions are not reflective of practical
implementations~\cite{huang2015low,briqech2017low,vasanth_comm_mag_16}, where an
independent gain controller can be used in every RF chain for every antenna.
With these structural constraints on the precoder, the transmit power constraint
is captured by
\begin{eqnarray}
\sum_{m = 1}^K {\sf Tr} \left( {\bf F}_{ {\sf Dig}, \hsppp m}^{\dagger}
{\bf F}_{\sf RF}^{\dagger} {\bf F}_{\sf RF} {\bf F}_{ {\sf Dig}, \hsppp m} \right) \leq K.
\end{eqnarray}

We are interested in the design of RF and digital precoders with the sum rate,
${\cal R}_{\sf sum} \triangleq \sum_{k = 1}^K {\cal R}_k$, being the metric to maximize.
In general, we only need the constraints $\sum_{k = 1}^{K} r_k \leq M_{\sf t} \leq N_{\sf t}$
and $\max _{k} r_k \leq M_{\sf r} \leq N_{\sf r}$.
However, the considered sum rate optimization with such an assumption is quite
complicated. To overcome this complexity, we
consider a simple use-case
in this work.

\section{Multi-User Beamformer Design}
\label{sec3}
We are interested in the practically-motivated setting where each user is equipped
with only one RF chain and the base-station transmits one data-stream to each user that is simultaneously
scheduled. In this scenario, $M_{\sf r} = r_k = 1$ (for all $k = 1, \cdots, K$)
and $M_{\sf t} = K \leq N_{\sf t}$. The system decoding model in~(\ref{eq_system_model})
and~(\ref{eq_decoding}) reduce to
\begin{eqnarray}
\widehat{\bf s}_k =
{\bf G}_{ {\sf Dig}, \hsppp k} ^{\dagger} {\bf G}_{ {\sf RF}, \hsppp k}^{\dagger} {\bf y}_k
& = &
\underbrace{ {\bf G}_{ {\sf Dig}, \hsppp k} ^{\dagger} }_{1 \times 1}
\underbrace{ {\bf G}_{ {\sf RF}, \hsppp k}^{\dagger} }_{1 \times N_{\sf r}}
\cdot \left(
\sqrt{ \frac{\rho}{K}} \hsppp {\bf H}_k \hsppp
\underbrace{ {\bf F}_{\sf RF} }_{ N_{\sf t} \times K}
\cdot \underbrace{ {\bf F}_{\sf Dig} }_{K \times K}
\cdot \underbrace{ {\bf s} }_{K \times 1}  + {\bf n}_k \right) \\
& = & \sqrt{\frac{\rho}{K}} \cdot
{\bf g}_k^{\dagger} \hsppp {\bf H}_k \hsppp \left[ {\bf f}_1 {\bf s}_1 ,
\cdots, {\bf f}_K {\bf s}_K \right] + {\bf g}_k^{\dagger} \hsppp {\bf n}_k
\end{eqnarray}
where ${\bf F}_{\sf Dig} = \left[ {\bf F}_{{\sf Dig}, 1}, \cdots,
{\bf F}_{{\sf Dig},K} \right]$ and ${\bf s} = \left[ {\bf s}_1 , \cdots,
{\bf s}_K \right]^{\sf T}$,
and the second equation follows assuming\footnote{A simple realization of the
hybrid precoding architecture is achieved by setting ${\bf F}_{\sf Dig} = {\bf I}_K$
and the desired ${\bf f}_k$ for the $k$-th user is set as the $k$-th column of
${\bf F}_{\sf RF}$. The desired ${\bf f}_k$ is such that ${\bf f}_k^{\dagger}
{\bf f}_k \leq 1$ and meets the quantization constraints
in~(\ref{eq_hybrid_precoder_constraints}). In a practical implementation,
${\bf F}_{\sf Dig}$ could be primarily used for sub-band precoding and in the
narrowband context of this work, ${\bf F}_{\sf Dig} = {\bf I}_K$ would reflect
such an implementation-driven model.}
${\bf f}_k = {\bf F}_{\sf RF}
{\bf F}_{ {\sf Dig}, k}$ and ${\bf G}_{ {\sf RF}, \hsppp k} = {\bf g}_k$.
The power constraint
is equivalent to
$\sum_{m = 1}^K  {\bf f}_k^{\dagger} {\bf f}_k \leq K$
and ${\cal R}_k$ reduces to
\begin{eqnarray}
{\cal R}_k = \log \left( 1 + \frac{ \frac{\rho}{K} \cdot |{\bf g}_k^{\dagger}
{\bf H}_k {\bf f}_k|^2 } {1 + \frac{\rho}{K} \cdot \sum_{m \neq k}
|{\bf g}_k^{\dagger} {\bf H}_k {\bf f}_m|^2 } \right).
\end{eqnarray}

The focus of this section is to first develop an advanced feedback
mechanism and a systematic design of the multi-user beamforming
structure based on a directional representation of the channel. This
structure allows the base-station to combat multi-user interference in simultaneous
transmissions.

\subsection{Initial Beam Alignment}
\label{sec3a}
Enabling multi-user transmissions in practice is critically dependent on an
initial beam acquisition process (commonly known as the {\em beam alignment}
phase). In a practical implementation such as 3GPP 5G-NR, beam alignment
corresponds to a beam sweep over a block of secondary synchronization (SS)
signals transmitted over multiple ports/RF chains. The use of multiple
directional beams over multiple ports results in a composite beam pattern
at the base-station end (as seen from the user side). The composite pattern
can lead to uncertainty in the direction of the strongest path between the
base-station and the user. This directional ambiguity is subsequently resolved
with a beam refinement over the individual constituent beams that make the
composite beam on separate resource elements. Beam refinement allows identification
and ambiguity resolution of the constituent beams.

Such a ``post directional ambiguity resolved'' beam alignment process is
modeled by assuming that the base-station is equipped with an $N$ element
codebook ${\cal F}_{\sf tr}$
\begin{eqnarray}
\label{eq:bs_codebook_tx}
{\cal F}_{\sf tr} =
\Big\{ \bff_{{\sf tr},1}, \, \dots, \, \bff_{{\sf tr},N} \Big\},
\end{eqnarray}
and the $k$-th user is equipped with an $M$ element user-specific codebook
${\cal G}_{\sf tr}^k$
\begin{eqnarray}
\label{eq:bs_codebook_rx}
{\cal G}_{\sf tr}^k =
\left\{ \bg^\ind{k}_{{\sf tr},1}, \, \dots, \, \bg^\ind{k}_{{\sf tr},M} \right \}.
\end{eqnarray}
A typical design methodology for ${\cal F}_{\sf tr}$ is a hierarchical design with
different sets of beams that trade-off peak array gain at the cost of initial beam
acquisition latency. For example, at least from the 3GPP 5G-NR perspective, the
designs of ${\cal F}_{\sf tr}$ and ${\cal G}_{\sf tr}^k$ are intended to be
implementation-specific at the base-station and user ends, respectively.
Nevertheless, overarching design guidelines for beam broadening are provided
in~\cite{hur,raghavan_jstsp,song17codebook,noh17adaptive}. In particular, a broadened
beam can be generated by an optimal co-phasing of a number of array steering vectors
in appropriately chosen directions. Both the number of such vectors as well as their
steering directions can be optimized to produce a broadened beam. It must also be
pointed out that most of the beam broadening works have some variations in terms
of design principles and these variations themselves do not affect the flavor of
results reported in this paper.

In the beam alignment phase, the top-$P$ beam indices at the base-station and each
user that maximize an estimate of the received ${\sf SNR}$ are learned. In particular,
the received ${\sf SNR}$
corresponding to the $(m, n)$-th beam index pair at the $k$-th user is given as
\begin{eqnarray}
{\sf SNR}^\ind{k}_{\sf rx}(m,n) =
\abs{\left(\bg^\ind{k}_{{\sf tr},m}\right)^\dagger \bH_k \bff_{{\sf tr},n}}^2.
\end{eqnarray}
Let the beam pair indices at the $k$-th user be arranged in non-increasing
order of the received ${\sf SNR}$ and let the top-$P$ beam pair indices be
denoted as
\begin{eqnarray}
{\cal M} =
\Big\{ \left(m_1^k, \, n_1^k\right), \hspp \cdots, \hspp
\left(m_P^k, \, n_P^k \right) \Big\}.
\end{eqnarray}
With the simplified notation of
\begin{eqnarray}
{\sf SNR}^\ind{k}_{ {\sf rx}, \hsppp \ell} \triangleq
{\sf SNR}^\ind{k}_{\sf rx}(m_{\ell}^{k},n_{\ell}^{k}), \hspp \ell = 1, \cdots, P,
\end{eqnarray}
we have ${\sf SNR}^\ind{k}_{ {\sf rx}, \hsppp 1} \geq \cdots \geq
{\sf SNR}^\ind{k}_{ {\sf rx}, \hsppp P}$.
With the initial beam alignment methodology as described above,
we now leverage the top-$P$ beam information learned at the $k$-th user
to estimate the channel matrix ${\bf H}_k$ and to design ${\bf F}_{\sf RF}$ at
the base-station end.

\subsection{Channel Reconstruction and Beamformer Design}
\label{sec3b}
A typical use of the feedback information at the base-station is to select the
top/best beam indices for all the users and to leverage this information to construct
a multi-user transmission scheme. Such an approach is adopted in~\cite{vasanth_jsac2017},
where multi-user beam designs leveraging only the top beam pair index,
$\left(m_1^k, \, n_1^k\right)$, and intended to serve different objectives are proposed:
i) greedily (from each user's perspective) steering a beam to the best direction for
that user (called the {\em beam steering} scheme), ii) using the information collated
from different users to combat interference to other simultaneously scheduled users via
a zeroforcing solution (called the {\em zeroforcing} scheme), and iii) for leveraging
both the beam steering and interference management objectives via a generalized
eigenvector optimization (called the {\em generalized eigenvector} scheme). If the beam
pair $\left(m_1^k, \, n_1^k\right)$ is blocked or fades, the $k$-th user requests the
base-station to switch to the beam index $n_2^k$ and it switches to the beam with index
$m_2^k$ (and so on)~\cite{vasanth_tap_blockage}.

In this work, we propose to generalize the structures in~\cite{vasanth_jsac2017} by
leveraging {\em all} the top-$P$ beam pair indices fed back from each user.
In this direction, the base-station intends to {\em reconstruct} or {\em estimate} a rank-$P$
approximation of (a scaled version of) the channel matrix ${\bf H}_k$
corresponding to the $k$-th user as follows
\begin{eqnarray}
\widehat{\bf H}_k = \sum_{\ell = 1}^P \widehat{\alpha}_{k, \ell} \hsppp
\widehat{ \bu}_{k,\ell} \hsppp \widehat{\bv}_{k,\ell}^{\dagger},
\label{eq_channel_reconstruction}
\end{eqnarray}
where $\widehat{ \bu}_{k,\ell}$ and $\widehat{ \bv}_{k,\ell}$ are defined as
estimates of the array steering vectors $\bu_{k, \ell}$ and $\bv_{k,\ell}$, respectively.
Given the channel model structure in~(\ref{eq:channel}),~(\ref{eq_channel_reconstruction})
is simplified by estimating ${\bf v}_{k, \hsppp \ell}$ and $|\alpha_{k, \hsppp \ell}|$ by
${\bf f}_{ {\sf tr}, n_{\ell}^k}$ and $\gamma_{k, \ell}$, respectively, where
\begin{eqnarray}
\gamma_{k, \ell} \triangleq
\sqrt{ {\cal Q}_{ B_{\sf SNR} } \left(  {\sf SNR}^\ind{k}_{ {\sf rx}, \hsppp \ell}  \right) }
\end{eqnarray}
for some choice of $B_{\sf SNR}$. In the above description, ${\cal Q}_B(\cdot)$ denotes
an appropriately-defined $B$-bit quantization operation\footnote{A $B$-bit quantization
operation is precisely specified if $2^B$ disjoint intervals that exactly and entirely
span the range
of the quantity and a representative/quantized value from each interval are specified.}
of the quantity under consideration.
However, estimating $\widehat{\bf H}_k$ as in~(\ref{eq_channel_reconstruction}) is not
complete until we have an estimate for $\angle{ \alpha_{k,\ell} }$ and
${\bf u}_{k, \ell}$. The quantity $\angle{ \alpha_{k, \ell}} $ can be estimated
by the user with the same reference symbol resource (or pilot symbol) transmitted
during the beam training phase with no additional training overhead. Therefore,
we define $\varphi_{k,\ell}$ as the
$B_{\sf est, \hsppp phase}$-bit quantization of the phase of an estimate
$\widehat{\bf s}_{{\sf tr}, k,\ell}$ of the pilot symbol
${\bf s}_{{\sf tr}, k,\ell}$
\begin{eqnarray}
\varphi_{k,\ell} \triangleq {\cal Q}_{B_{\sf est, \hsppp phase}}
\left( \angle{ \widehat{\bf s}_{{\sf tr}, k,\ell} } \right), \hspp \hspp
{\sf where} \hspp \hspp
\widehat{\bf s}_{{\sf tr}, k,\ell} = \left( {\bf g}^{(k)}_{{\sf tr}, \hsppp m_{\ell}^k } \right)^{\dagger}
\left[ \sqrt{\rho} \hsppp {\bf H}_k {\bf f}_{{\sf tr}, \hsppp n_{\ell}^k }
{\bf s}_{{\sf tr}, k,\ell}
+
{\bf n}_{k,\ell} \right]
\end{eqnarray}
for some choice of $B_{\sf est, \hsppp phase}$. The noise term
${\bf n}_{k,\ell}$ captures the additive noise in the initial beam alignment
process corresponding to the top-$P$ beam pairs.

For ${\bf u}_{k, \ell}$,
we note that the base-station not only needs the beam indices
$\{ m_{\ell}^k \}$ that are useful for the user side, but also the useful part of the
user's codebook (${\cal G}_{\sf tr}^k$) since the base-station is typically unaware
of it.
To avoid this unnecessary complexity and feedback given the proprietary structure of
${\cal G}_{\sf tr}^k$, we assume that the $k$-th user uses a multi-user reception beam
${\bf g}_k$. In the simplest manifestation, ${\bf g}_k$ could be the best training
beam learned in the beam alignment phase, $\bg_{{\sf tr}, m_1^{k}}^{(k)}$.
However, a more sophisticated choice for ${\bf g}_k$ is not precluded. For example, an
iterative choice that maximizes the ${\sf SINR}$ (instead of the ${\sf SNR}$) could
be considered for ${\bf g}_k$.

We then note that the estimated ${\sf SINR}$, defined as,
\begin{eqnarray}
\widehat{\sf SINR}_k \triangleq
\frac{ \frac{\rho}{K} \cdot | {\bf g}_k^{\dagger} \widehat{\bf H}_k {\bf f}_k |^2 }
{ 1 + \frac{ \rho}{K} \cdot \sum_{m \neq k} | {\bf g}_k^{\dagger} \widehat{\bf H}_k {\bf f}_m |^2 }
\label{eq_estimated_SINR}
\end{eqnarray}
is only dependent on $\widehat{\bf H}_k$ in the form of ${\bf g}_k^{\dagger} \widehat{\bf H}_k$.
Building on this fact, each user
generates $\{ \beta_{k, \hsppp \ell} \}$, defined as,
\begin{eqnarray}
\beta_{k, \hsppp \ell} \triangleq
{\bf g}_k^{\dagger} \widehat{\bu}_{k,\ell} \hsppp \hspp
{\sf where} \hspp \hsppp
\widehat{\bu}_{k,\ell} = {\bf g}_{ {\sf tr}, \hsppp
m_{\ell}^k}^{(k)}.
\end{eqnarray}
It then quantizes the amplitude and phase of $\beta_{k, \ell}$ for some choice
of $B_{\sf corr, \hsppp amp}$ and $B_{\sf corr, \hsppp phase}$ and feeds them back
\begin{eqnarray}
\mu_{k, \ell} \triangleq {\cal Q}_{ B_{\sf corr,\hsppp amp}} \left( |\beta_{k,\ell}| \right),
\hspp \hspp
\nu_{k,\ell} \triangleq {\cal Q}_{ B_{\sf corr,\hsppp phase}} \left( \angle{ \beta_{k,\ell} }
\right).
\end{eqnarray}
For both $\varphi_{k,\ell}$ and $\nu_{k,\ell}$, without loss in generality, relative
phases with respect to $\varphi_{k,1}$ and $\nu_{k,1}$ (that is,
$\varphi_{k,\ell} - \varphi_{k,1}$ and $\nu_{k,\ell} - \nu_{k,1}$) can be reported.

The mappings between the quantities of interest and the approximated quantities as
well as the feedback overhead needed from each user to implement the proposed scheme
are described in Table~\ref{table_feedback}.
While the feedback overhead increases
linearly with $P$ (the rank of the channel approximation), there are diminishing
returns in terms of channel representation accuracy since the clusters captured
in $\widehat{\bf H}_k$ are sub-dominant as $P$ increases (and are eventually limited by
$L_k$). Thus, it is useful to select $P$ to trade-off these two conflicting objectives.

\begin{table*}[htb!]
\caption{Mappings between quantities describing ${\bf H}_k$ and
the approximated quantities, and their feedback overhead.}
\label{table_feedback}
\begin{center}
\begin{tabular}{|c||c||c|}
\hline
Quantity of Interest & Approximated Quantity & Feedback Overhead \\
\hline
\hline
Array steering vector at base-station end (${\bf v}_{k, \ell}$) &
Base-station beam indices ($n_{\ell}^k$) &
$P \cdot \log_2(N)$ \\ \hline
Gain of cluster coefficient ($|\alpha_{k, \ell}|$) &
Received ${\sf SNR}$ in beam alignment (${\sf SNR}_{ {\sf rx}, \hsppp \ell }^{(k)}$) &
$P \cdot B_{\sf SNR}$ \\ \hline
Phase of cluster coefficient ($\angle{\alpha_{k,\ell}}$) &
Estimated phase in beam alignment ($\angle{{\widehat{\bf s}}_{{\sf tr} , k, \ell} }$)
&
$(P-1) \cdot B_{\sf est, \hsppp phase}$ \\ \hline
Array steering vector at user end (${\bf u}_{k, \ell}$) &
Amplitude of codebook correlation
($|\beta_{k, \ell}|$) &
$P \cdot B_{\sf corr, \hsppp amp}$
\\
& Phase of codebook correlation
($\angle{\beta_{k,\ell}}$) &
$(P-1) \cdot B_{\sf corr, \hsppp phase}$
\\ \hline
\end{tabular}
\end{center}
\end{table*}

Following the above discussion, the $k$-th user feeds back the $P \times 5$ matrix
${\bf P}_k$, defined as
\begin{eqnarray}
{\bf P}_k \triangleq \left[
\begin{array}{ccccc}
n_{1}^k & \gamma_{k, 1} & 0 & \mu_{k,1} & 0 \\
n_{2}^k & \gamma_{k, 2} & \varphi_{k, 2} - \varphi_{k, 1} & \mu_{k,2}
& \nu_{k, 2} - \nu_{k, 1} \\
\vdots & \vdots & \vdots & \vdots & \vdots \\
n_{P}^k & \gamma_{k, P} & \varphi_{k, P} - \varphi_{k,1} & \mu_{k, P} & \nu_{k, P} - \nu_{k,1}
\end{array}
\right],
\end{eqnarray}
and the base-station approximates ${\bf g}_k^{\dagger} \widehat{\bf H}_k$
as follows
\begin{eqnarray}
{\bf g}_k^{\dagger} \widehat{\bf H}_k = \sum_{\ell = 1}^P
\mu_{k,\ell} \hsppp \gamma_{k,\ell} \cdot
e^{j (\varphi_{k, \ell} + \nu_{k,\ell} )}
\cdot \left( {\bf f}_{ {\sf tr}, n_{\ell}^k} \right)^{\dagger}.
\label{eq_eigenspace_reconst}
\end{eqnarray}
In other words, ${\bf g}_k^{\dagger} \widehat{\bf H}_k$ is represented as a linear
combination of the top-$P$ beams as estimated from ${\cal F}_{\sf tr}$ in the initial
beam alignment phase. The weights in this linear combination correspond to the relative
strengths of the clusters as distinguished by the codebook resolution (at both ends).

The base-station uses the channel matrix constructed for each user based on its feedback
information (${\bf g}_k^{\dagger} \widehat{\bf H}_k$) and generates a good
beamformer structure, illustrated in the next result, for use in multi-user
transmissions.
\begin{prop}
\label{prop_zf}
The zeroforcing beamformer structure is one where for every user that is simultaneously
scheduled, the beam ${\bf f}_k$ nulls the multi-user interference in
$\widehat{\sf SINR}_m, \hspp m \neq k$ with $\widehat{\sf SINR}_m$ as given
in~(\ref{eq_estimated_SINR}). The beams $\{ {\bf f}_m \}$ in the zeroforcing
structure are the unit-norm column vectors of the $N_{\sf t} \times K$ matrix
${\cal H}^{\dagger} \left( {\cal H} {\cal H}^{\dagger} \right)^{-1}$, where
${\cal H}$ is the $K \times N_{\sf t}$ matrix given as
\begin{eqnarray}
{\cal H} = \left[
\begin{array}{c}
{\bf g}_1^{\dagger} \widehat{\bf H}_1 \\
{\bf g}_2^{\dagger} \widehat{\bf H}_2 \\
\vdots \\
{\bf g}_K^{\dagger} \widehat{\bf H}_K
\end{array}
\right] =
\left[
\begin{array}{c}
\sum_{\ell = 1}^P \mu_{1,\ell} \hsppp \gamma_{1,\ell} \cdot
e^{j (\varphi_{1, \ell} + \nu_{1,\ell} )}
\cdot \left( {\bf f}_{ {\sf tr}, n_{\ell}^1} \right)^{\dagger}
\\
\sum_{\ell = 1}^P \mu_{2,\ell} \hsppp \gamma_{2,\ell} \cdot
e^{j (\varphi_{2, \ell} + \nu_{2,\ell} )}
\cdot \left( {\bf f}_{ {\sf tr}, n_{\ell}^2} \right)^{\dagger}
\\
\vdots \\
\sum_{\ell = 1}^P \mu_{K,\ell} \hsppp \gamma_{K,\ell} \cdot
e^{j (\varphi_{K, \ell} + \nu_{K,\ell} )}
\cdot \left( {\bf f}_{ {\sf tr}, n_{\ell}^K} \right)^{\dagger}
\end{array}
\right].
\end{eqnarray}
\end{prop}
\begin{proof}
See Appendices~\ref{app_ge} and~\ref{proof_prop_zf}.
\end{proof}

\section{Upper Bounds for ${\cal R}_{\sf sum}$}
\label{sec4}
We are interested in benchmarking the performance of the zeroforcing structure
against an upper bound on ${\cal R}_{\sf sum}$. The goal of optimizing
${\cal R}_{\sf sum}$ over $\{ {\bf f}_k, \hsppp {\bf g}_k \}$ with perfect channel
state information $\{ {\bf H}_k \}$ is a non-convex optimization
problem~\cite{cioffi_weighted,kobayashi,vasanth_arxiv2011}
that appears to be complicated. In this context, an alternate formulation based on the
signal-to-leakage and noise ratio metric~\cite{tarighat} that {\em simultaneously} maximizes the
array gain seen by the $k$-th user, $| {\bf g}_k^{\dagger} {\bf H}_k {\bf f}_k |^2$,
and minimizes the interfering array gain seen by the other users,
$| {\bf g}_m^{\dagger} {\bf H}_m {\bf f}_k |^2, \hspp m \neq k$ is relevant. Since these objectives
are in some sense conflicting and can be weighed differently, we consider the composite metric
\begin{eqnarray}
{\sf SLNR} _k \triangleq
\frac{ \eta_{k,k} \hsppp | {\bf g}_k^{\dagger} {\bf H}_k {\bf f}_k |^2 }
{ 1 + \sum_{m \neq k} \eta_{m,k} \hsppp
| {\bf g}_m^{\dagger} {\bf H}_m {\bf f}_k |^2 }
\label{eq_SLNR}
\end{eqnarray}
for an appropriate set of weighting factors $\eta_{m,k} \geq 0$ with
$m ,k \in \{ 1, \cdots, K \}$.

\subsection{Upper Bound Motivated by the Zeroforcing Structure}
\label{sec4a}
Building on Prop.~\ref{prop_zf}, we now develop an upper bound for ${\cal R}_{\sf sum}$
motivated by the zeroforcing structure. In this direction, we consider a
signal-to-leakage-type metric equivalent of~(\ref{eq_SLNR}) based on the estimated
channel matrix $\widehat{\bf H}_k$
\begin{eqnarray}
\widehat{ {\sf SLNR} }_k \triangleq
\frac{ \eta_{k,k} \hsppp | {\bf g}_k^{\dagger} \widehat{\bf H}_k {\bf f}_k |^2 }
{ 1 + \sum_{m \neq k} \eta_{m,k} \hsppp
| {\bf g}_m^{\dagger} \widehat{\bf H}_m {\bf f}_k |^2 }
\end{eqnarray}
for an appropriate set of weighting factors $\eta_{m,k} \geq 0$ with
$m ,k \in \{ 1, \cdots, K \}$.
\begin{prop}
\label{prop_ge}
Assuming that $\{ \widehat{\bf H}_m^{\dagger} {\bf g}_m \}$ and
$\{ \eta_{m,k} \}$ are known at the base-station, the choice of ${\bf f}_k$
that maximizes $\widehat{ {\sf SLNR}} _k$ is given
by the generalized eigenvector structure
\begin{eqnarray}
{\bf f}_k = \frac{ \left( {\bf I}_{ N_{\sf t}} + \sum_{m \neq k} \eta_{m,k}
\hsppp
\widehat{\bf H}_m^{\dagger} {\bf g}_m {\bf g}_m^{\dagger} \widehat{\bf H}_m
\right)^{-1} \widehat{\bf H}_k^{\dagger} {\bf g}_k
}
{ \Big\| \left( {\bf I}_{ N_{\sf t}} + \sum_{m \neq k} \eta_{m,k} \hsppp
\widehat{\bf H}_m^{\dagger} {\bf g}_m {\bf g}_m^{\dagger}
\widehat{\bf H}_m \right)^{-1} \widehat{\bf H}_k^{\dagger} {\bf g}_k \Big\| }.
\label{eq_ge}
\end{eqnarray}
\end{prop}
\begin{proof}
See Appendix~\ref{proof_prop_ge}.
\end{proof}

Several remarks are in order at this stage.

\begin{itemize}
\item
In the case where $\eta_{m,k}$ are set to zero for all $m \neq k$ (that is,
the focus is {\em not} on interference management), the solution in~(\ref{eq_ge}) reduces
to
\begin{eqnarray}
{\bf f}_k = \frac{ \widehat{\bf H}_k^{\dagger} {\bf g}_k}
{ \| \widehat{\bf H}_k^{\dagger} {\bf g}_k \| }
= \frac{ \sum_{\ell = 1}^P \mu_{k,\ell} \hsppp \gamma_{k,\ell} \cdot
e^{-j (\varphi_{k, \ell} + \nu_{k,\ell} )}
\cdot {\bf f}_{ {\sf tr}, n_{\ell}^k} }
{ \big\| \sum_{\ell = 1}^P \mu_{k,\ell} \hsppp \gamma_{k,\ell} \cdot
e^{-j (\varphi_{k, \ell} + \nu_{k,\ell} )}
\cdot {\bf f}_{ {\sf tr}, n_{\ell}^k} \big\| }.
\end{eqnarray}
This is not surprising, and the base-station {\em greedily} steers a beam along the
weighted set of top-$P$ beams from ${\cal F}_{\sf tr}$ for the $k$-th user. In other
words, the base-station generates a set of transmit weights that are matched to the
transmit angular spread of the channel as identified by the resolution of ${\cal F}_{\sf tr}$.

\item
In the case where $\eta_{m,k} = 0$ except if $m = k$ or $m = m'$ (for a specific
$m' \neq k$), it can be seen that ${\bf f}_k$ reduces to
\begin{eqnarray}
{\bf f}_k = \frac{ \widehat{\bf H}_k^{\dagger} {\bf g}_k -
\eta_{m',k} \cdot \left( {\bf g}_{m'}^{\dagger} \widehat{\bf H}_{m'}
\widehat{\bf H}_k^{\dagger} {\bf g}_k \right) \cdot
\widehat{\bf H}_{m'}^{\dagger} {\bf g}_{m'}  }
{ \big\| \widehat{\bf H}_k^{\dagger} {\bf g}_k -
\eta_{m',k} \cdot \left( {\bf g}_{m'}^{\dagger} \widehat{\bf H}_{m'}
\widehat{\bf H}_k^{\dagger} {\bf g}_k \right) \cdot
\widehat{\bf H}_{m'}^{\dagger} {\bf g}_{m'}
\big\| }.
\label{eq_fk_case2}
\end{eqnarray}
In other words, the specific design of ${\bf f}_k$ in~(\ref{eq_fk_case2})
removes a certain component of the beam corresponding to the $m'$-th user
from the beam corresponding to the $k$-th user.

\item
In the general case, while it gets much harder to simplify ${\bf f}_k$ in~(\ref{eq_ge}),
it can be seen that ${\bf f}_k$ has the structure
\begin{eqnarray}
{\bf f}_k = \frac{  \sum_{m = 1}^K \widehat{ \delta}_{m,k} \widehat{\bf H}_m^{\dagger} {\bf g}_m
} { \big\| \sum_{m = 1}^K \widehat{\delta}_{m,k} \widehat{\bf H}_m^{\dagger} {\bf g}_m
\big\| }
\label{eq_ge_structure}
\end{eqnarray}
for some complex scalars $\widehat{\delta}_{m,k}$. In other words, the optimal ${\bf f}_k$ is in the
span of $\{ \widehat{\bf H}_m^{\dagger} {\bf g}_m \}$ with the weights $\{ \widehat{\delta}_{m,k} \}$
that make the linear combination being a complicated function of $\{ \eta_{m,k} \}$ as well as
$\{ \widehat{\bf H}_m^{\dagger} {\bf g}_m \}$.

\item
The above observations are not entirely surprising given the Karhunen-Lo{\`e}ve
interpretation of the eigen-space of the channel(s)~\cite{akbar,tulino_ind,vasanth_it2}
and utilizing an expansion of ${\bf f}_k$ on this basis. Such an expansion is also consistent
with Prop.~\ref{prop_zf} which shows that in the pure interference management case
($\eta_{m,k} \rightarrow \infty$ for all $m \neq k$), ${\bf f}_k$ is given as
\begin{eqnarray}
{\bf f}_k = \frac{ \sum_{m = 1}^K {\cal G}_{m,k} \widehat{\bf H}_m^{\dagger} {\bf g}_m }
{ \big\| \sum_{m = 1}^K {\cal G}_{m,k} \widehat{\bf H}_m^{\dagger} {\bf g}_m
\big\|}
\label{eq_purezf}
\end{eqnarray}
where the $K \times K$ matrix ${\cal G} = \left( {\cal H} {\cal H}^{\dagger} \right)^{-1}$.

\item
On the other hand, from~(\ref{eq_eigenspace_reconst}), we note that
$\widehat{\bf H}_m^{\dagger} {\bf g}_m$ is itself a linear combination of the beams
from ${\cal F}_{\sf tr}$. Thus, ${\bf f}_k$ in~(\ref{eq_ge})
is a linear combination of beams from ${\cal F}_{\sf tr}$. In other words, the
design of ${\bf f}_k$ is equivalent to a search over $N$
scalar (complex) weights, where $N$ denotes the size of the initial beam
alignment codebook at the base-station end.
\end{itemize}

With this interpretation, while Prop.~\ref{prop_ge} considers only the
maximization of $\widehat{ {\sf SLNR}} _k$ (not even the sum rate with
$\widehat{\bf H}_k$), we can consider the optimization of ${\cal R}_{\sf sum}$
over ${\bf f}_k$ from a class ${\cal F}_k$, defined as
\begin{eqnarray}
{\cal F}_k \triangleq \left\{ {\bf f}_k \hspp : \hspp
{\bf f}_k =
\frac{  \sum_{n = 1}^{N}
\delta_{n,k} {\bf f}_{ {\sf tr}, \hsppp n}
} { \big\| \sum_{n = 1}^{N}
\delta_{n,k} {\bf f}_{ {\sf tr}, \hsppp n}
\big\| } \hspp {\sf such} \hspp {\sf that} \hspp \delta_{n,k} \in {\mathbb{C}}, \hspp
k = 1, \cdots, K  \right\}.
\label{eq_class_Fk}
\end{eqnarray}
\begin{thm}
Assume that the same multi-user beams ${\bf g}_k$
as in the zeroforcing scheme are used for reception at the $k$-th user. Let
$\{ \delta^{\star}_{n,k} \}$ be defined as the solution to the search
over the complex scalars $\{ \delta_{n,k} \}$
\begin{eqnarray}
\{ \delta^{\star}_{n,k} \} = \arg\max \limits_{ \{ \delta_{n,k} \hsppp : \hsppp
{\bf f}_k \hsppp \in \hsppp {\cal F}_k \} } {\cal R}_{\sf sum}.
\end{eqnarray}
With ${\bf g}_k$ as above and
\begin{eqnarray}
{\bf f}_k = \frac{  \sum_{n = 1}^{N}
\delta^{\star}_{n,k} {\bf f}_{ {\sf tr}, \hsppp n}
} { \big\| \sum_{n = 1}^{N}
\delta^{\star}_{n,k} {\bf f}_{ {\sf tr}, \hsppp n} \big\| },
\label{eq_fk_scalar}
\end{eqnarray}
we obtain an upper bound to
the sum rate with the zeroforcing scheme.
\qed
\end{thm}
The proof is trivial following the structure of ${\bf f}_k$ in the zeroforcing
scheme in~(\ref{eq_purezf}) and the definition of the class ${\cal F}_k$
in~(\ref{eq_class_Fk}).
Since the structure in~(\ref{eq_fk_scalar}) is obtained as a search over scalar
parameters, we call this upper bound a scalar optimization-based upper bound.
Further, while~(\ref{eq_fk_scalar}) is difficult to practically implement,
it provides a benchmark to compare the realizable zeroforcing scheme of
Prop.~\ref{prop_zf}.

Another important consequence of~(\ref{eq_fk_scalar})
is that the coefficients of ${\bf f}_k$ for either the zeroforcing or the upper bound
are (in general) not of equal amplitude. Thus, ${\bf f}_k$ has to be quantized for
implementation to ensure that the RF beamforming constraints are satisfied. In
particular, we compute $\widehat{\bf f}_k$ with an appropriate quantization scheme
as below
\begin{eqnarray}
|\widehat{\bf f}_k(i)| = \widetilde{\cal Q}_{ B_{\sf amp}} \left( |{\bf f}_k(i)| \right),
\hspp  \hspp \angle{ \widehat{\bf f}_k(i) } = \widetilde{\cal Q}_{ B_{\sf phase} } \left(
\angle{ {\bf f}_k(i) } \right),
\end{eqnarray}
and use them in transmissions for the $k$-th user. Good choices
for $\widetilde{\cal Q}(\cdot)$ will be discussed in Sec.~\ref{sec5b}.

\subsection{Bounding ${\cal R}_{\sf sum}$ with an
Alternating/Iterative Optimization}
\label{sec3b}
We now propose an iterative
maximization algorithm to optimize ${\cal R}_{\sf sum}$ over $\{ {\bf f}_k,
{\bf g}_k \}$. In this approach, we first optimize the ${\sf SLNR}$ metric
over ${\bf f}_k$ (assuming ${\bf g}_k$ is fixed), and then optimize the
${\sf SINR}$ metric over ${\bf g}_k$ (assuming ${\bf f}_k$ is fixed). The
algorithm is as follows:
\begin{enumerate}
\item Initialize $\{ {\bf g}_k^{(1)}, \hspp k = 1, \cdots, K \}$ randomly.

\item For $i = 1, \cdots, N_{\sf stop}$, where $N_{\sf stop}$ is chosen
according to a stopping criterion to determine convergence:
\begin{itemize}
\item
With $\{ {\bf g}_k = {\bf g}_k^{(i)}, k = 1, \cdots, K \}$ fixed, compute
${\bf f}_k^{(i)}$
as the solution to the following optimization
\begin{eqnarray}
{\bf f}_k^{(i)} = \arg \max \limits_{ {\bf f}_k }
\max_{ \{ \eta_{m,k} \} } {\sf SLNR}_k.
\end{eqnarray}
From Lemma~\ref{lem_ge_soln} in Appendix~\ref{app_ge}, the solution to the
above problem with $\{ \eta_{m,k} \}$ fixed can be seen to be
\begin{eqnarray}
{\bf f}_k =
\frac{ \Big( {\bf I}_{N_{\sf t}} + \sum_{m \neq k} \eta_{m,k} \hsppp
{\bf H}_m^{\dagger} {\bf g}_m^{(i)} {\bf g}_m^{(i) \hsppp \dagger}
{\bf H}_m  \Big)^{-1} {\bf H}_k^{\dagger} {\bf g}_{k}^{(i)} }
{ \Big\|
\Big( {\bf I}_{N_{\sf t}} + \sum_{m \neq k} \eta_{m,k} \hsppp
{\bf H}_m^{\dagger} {\bf g}_m^{(i)} {\bf g}_m^{(i) \hsppp \dagger}
{\bf H}_m  \Big)^{-1} {\bf H}_k^{\dagger} {\bf g}_{k}^{(i)} \Big\| }.
\end{eqnarray}
This candidate ${\bf f}_k$ has to be used to compute ${\sf SLNR}_k$ for
all possible weights $\{ \eta_{m,k} \}$ and optimized to produce $\{ {\bf f}_k^{(i)} \}$.

\item
With $\{ {\bf f}_k = {\bf f}_k^{(i)}, k = 1, \cdots, K \}$ fixed,
compute ${\bf g}_k^{(i+1)}$ as the solution to the following optimization
\begin{eqnarray}
{\bf g}_k^{(i+1)} = \arg \max \limits_{ {\bf g}_k } {\sf SINR}_k.
\end{eqnarray}
Again, from Lemma~\ref{lem_ge_soln} in Appendix~\ref{app_ge}, we have
\begin{eqnarray}
{\bf g}_k^{(i+1)} = \left( {\bf I}_{N_{\sf r}} +
\frac{\rho}{K} \sum_{m \neq k} {\bf H}_k {\bf f}_m^{(i)} {\bf f}_m^{(i), \dagger}
{\bf H}_k^{\dagger} \right)^{-1} {\bf H}_k {\bf f}_k^{(i)}.
\end{eqnarray}
\end{itemize}

\item Compute ${\cal R}_{\sf sum}$ with $\{ {\bf f}_k^{(N_{\sf stop})} \}$
and $\{  {\bf g}_k^{ (N_{\sf stop} + 1)} \}$ for a (potential) upper bound.
\end{enumerate}
Numerical studies show that for almost all channel realizations, the proposed
algorithm converges in a small number of steps ($N_{\sf stop} \approx 10$) to
lead to a tolerable level of difference between successive iterates of
${\cal R}_{\sf sum}$. Further, while we are unable to theoretically establish
that the proposed algorithm results in an upper bound to ${\cal R}_{\sf sum}$,
numerical studies (see Sec.~\ref{sec5c}) suggest that it leads to an upper bound
for almost all channel realizations.

\section{Numerical Studies}
\label{sec5}
We now present numerical studies in a single-cell downlink
framework to illustrate the advantages of the proposed beamforming solutions.
The channel model from~(\ref{eq:channel}) is used to generate a channel matrix
with $L_k = 6$ clusters, AoDs uniformly distributed in a $120^{\sf o} \times
30^{\sf o}$ coverage area, and AoAs uniformly distributed in a
$120^{\sf o} \times 120^{\sf o}$ coverage area for each of the
$k = 1, \cdots, K_{\sf cell}$ users in the cell. The AoD spread captures a
traditional three-sector approach with a $30^{\sf o}$ zenith coverage and the
AoA spread corresponds to the assumption of the use of multiple
subarrays~\cite{vasanth_tap2018} with the best subarray limited to a
$120^{\sf o} \times 120^{\sf o}$ coverage. $L_k = 6$ is justified from
millimeter wave channel measurements reported in~\cite{vasanth_tap2018,qualcomm2}.
The antenna
dimensions assumed in these studies are $N_{\sf tx} = 16$ and $N_{\sf tz} = 4$ at
the base-station end, and $N_{\sf rx} = 2$ and $N_{\sf rz} = 2$ at each user. We
consider simultaneous transmissions from the base-station to $K = 2$ out of
the $K_{\sf cell}$ users in the cell.

In terms of user scheduling, commonly used
criteria include a round robin or a proportionate fair scheduler. On the other hand,
a recently proposed directional scheduler~\cite{vasanth_jsac2017} leverages the
smaller beamwidths afforded by large antenna dimensions to schedule users with
dominant clusters that are spatially well-separated. In this work, the first of the
$K = 2$ users is scheduled randomly and the second user is chosen to ensure that
${\bf f}_{ {\sf tr}, n_1^2} \neq {\bf f}_{ {\sf tr}, n_1^1}$. In other words, the
considered scheduler implements a {\em directional avoidance} protocol with the
dominant cluster in the channel of the first user separated spatially from the
dominant cluster in the channel of the second user, as parsed by ${\cal F}_{\sf tr}$.
With this scheduler, we now primarily focus on the beamforming aspects.

For the initial beam alignment codebooks, based on the beam broadening principles
proposed in~\cite{raghavan_jstsp}, Figs.~\ref{fig_beampatterns_codebooks}(a)-(d)
illustrate the beam patterns in the azimuth plane for codebooks of sizes
$N = 32$, $N = 16$, $N = 8$ and $N = 4$, respectively,
to cover the $120^{\sf o} \times
30^{\sf o}$ AoD space with a $16 \times 4$ planar array at the base-station side.
The optimization proposed in~\cite{raghavan_jstsp} results in a discrete
Fourier transform (DFT) codebook solution for $N = 32$ and $N = 16$.
From
Fig.~\ref{fig_beampatterns_codebooks}, we observe that a beam codebook
of small size (e.g., $N = 4$) where each beam offers a broad directional
coverage can reduce the acquisition latency at the cost of peak and/or worst-case
array gain. On the other hand, a beam codebook of large size (e.g., $N = 32$)
where each beam can offer precision in terms of beamspace (and array gain)
comes at the cost of acquisition latency. For the codebooks at the user end,
two codebook sizes ($M = 4$ for a reduced acquisition latency and $M = 16$ for
performance improvement at the cost of acquisition latency) are considered with
similar beam design principles as for the base-station side.

\begin{figure*}[htb!]
\begin{center}
\begin{tabular}{cc}
\includegraphics[height=2.5in,width=3.1in] {figures/fig_CPE_32beams_onelayer_azimuth.eps}
&
\includegraphics[height=2.5in,width=3.1in] {figures/fig_CPE_16beams_onelayer_azimuth.eps}
\\
(a) & (b)
\\
\includegraphics[height=2.5in,width=3.1in] {figures/fig_CPE_8beams_onelayer_azimuth.eps}
&
\includegraphics[height=2.5in,width=3.1in] {figures/fig_CPE_4beams_onelayer_azimuth.eps}
\\
(a) & (b)
\end{tabular}
\caption{\label{fig_beampatterns_codebooks}
Beam patterns in the azimuth plane of four different base-station codebooks, all
covering a $120^{\sf o} \times 30^{\sf o}$ coverage area, with (a) $N = 32$,
(b) $N = 16$, (c) $N = 8$, and (d) $N = 4$ elements in ${\cal F}_{\sf tr}$.}
\end{center}
\end{figure*}

At this stage, it is worth noting that a number of system parameters impact the
performance of the proposed multi-user schemes such as: i) Granularity of
${\cal F}_{\sf tr}$ and ${\cal G}_{\sf tr}^k$ (initial beam alignment codebook
sizes), ii) Coarseness of channel approximation (rank-$P$), iii) Finite-rate
feedback of channel reconstruction parameters, and iv) Quantization of the resulting
multi-user beams.

\begin{figure*}[htb!]
\begin{center}
\begin{tabular}{cc}
\includegraphics[height=2.5in,width=3.0in]{figures/sumrate_txCodebook_4_16_rxCodebook_4_zf_v4.eps}
& {\hspace{0.1in}}
\includegraphics[height=2.5in,width=3.0in]{figures/sumrate_txCodebook_8_32_rxCodebook_4_zf_v4.eps}
\\
{\vspace{0.15in}}
(a) & (b) \\
\includegraphics[height=2.5in,width=3.0in]{figures/sumrate_txCodebook_4_16_rxCodebook_16_zf_v4.eps}
& {\hspace{0.1in}}
\includegraphics[height=2.5in,width=3.0in]{figures/sumrate_txCodebook_8_32_rxCodebook_16_zf_v4.eps}
\\
(c) & (d)
\end{tabular}
\caption{\label{fig:txCodebook}
CDF of sum rates for a beam steering scheme and the proposed zeroforcing scheme for
different choices of $N$ with $M = 4$ in (a) and (b), and $M = 16$ in (c) and (d).}
\end{center}
\end{figure*}

\subsection{Impact of Initial Beam Alignment Codebook}
\label{sec5a}
In the first study, we consider the relative performance of the zeroforcing
scheme (proposed in Prop.~\ref{prop_zf}) relative to a baseline beam steering
scheme with different initial beam alignment codebooks. We assume that the
system has infinite-precision feedback of channel reconstruction parameters
and infinite-precision resolution in the quantization of multi-user beams. We
also compare the performance of the proposed schemes with the zeroforcing scheme
presented in~\cite{vasanth_gcom16,vasanth_jsac2017}, where the system is assumed
to be able to find perfectly aligned directional beams in the training phase.
Fig.~\ref{fig:txCodebook} illustrates this comparative performance with different
choices of $P$ in approximating ${\bf g}_k^{\dagger} \widehat{\bf H}_k$ and different
codebook sizes ($N$ and $M$).

While it is intuitive that there should be diminishing performance as $P$ increases
(since increasing $P$ beyond the channel rank $L_k$ is not expected to improve
performance), whether this saturation in
performance is observed with a low-rank channel approximation is dependent on the
resolution of the codebooks. In particular, increasing $P$
when the codebook granularity is already poor (small $M$ and $N$) does not lead to
any performance improvement than observed with $P = 1$ (beam steering). On the other hand,
with a high resolution for ${\cal F}_{\sf tr}$ (large $N$), even a rank-$2$
approximation appears to be sufficient to reap most of the performance improvement
gains. This is because the performance of the baseline (beam steering) scheme is
already quite good and significant relative improvement over it with increasing $P$
has a lower likelihood unless the channel has a large number of similar gain clusters (a
low-probability event). When $M$ is large and $N$ is small, the beam steering performance
is poor and the channel can be better approximated with the higher codebook resolution
of ${\cal G}_{\sf tr}^k$ leading to a sustained performance improvement for even
up to $P = 4$. For example, with $N = 4$ or $8$ and $M = 16$, zeroforcing based on a
rank-$4$ channel approximation leads to around $2$ bps/Hz improvement at the median level.

In terms of performance comparison, note that the scheme
from~\cite{vasanth_gcom16,vasanth_jsac2017} assumes $P = 1$ but infinite-precision
in terms of beam alignment ($N = M \rightarrow \infty$). Thus, it is not surprising
that as $N$ and $M$ increase, the performance of the proposed schemes compare well
with that of~\cite{vasanth_gcom16,vasanth_jsac2017}. For lower codebook
resolutions, the proposed schemes overcome the codebook disadvantage by leveraging
a better channel approximation as $P$ increases.
These observations suggest that the optimal choice of the rank in approximating
${\bf g}_k^{\dagger} \widehat{\bf H}_k$ (which in turn determines the feedback overhead)
depends not only on the rank of the true channel ${\bf H}_k$, but also on the codebook
granularities. In general, a higher $P$ (and feedback overhead) is necessary if the
codebook resolution is rich enough at the user end to allow the parsing of the channel
better, but poor enough at the base-station end to allow a sustained performance
improvement with increasing $P$. In particular, we provide the following heuristic
design guidelines based on our studies
\begin{eqnarray}
P = \left\{ \begin{array}{cl}
1 & {\sf if} \hspp M \hspp {\sf and} \hspp N \hspp {\sf are} \hspp {\sf small} \\
2 & {\sf if} \hspp M \hspp {\sf is} \hspp {\sf small} \hspp {\sf and} \hspp
N \hspp {\sf is} \hspp {\sf large}
\\
4 & {\sf if} \hspp M \hspp {\sf is} \hspp {\sf large}.
\end{array}
\right.
\end{eqnarray}

\begin{figure*}[htb!]
\begin{center}
\begin{tabular}{cc}
\includegraphics[height=2.5in,width=3.0in] {figures/sumrate_snrQuant.eps}
& {\hspace{0.1in}}
  \includegraphics[height=2.5in,width=3.0in] {figures/sumrate_phaseQuant.eps}
\\
(a) & (b)
\\
  \includegraphics[height=2.5in,width=3.0in]{figures/sumrate_corrQuant.eps}
& {\hspace{0.1in}}
  \includegraphics[height=2.5in,width=3.0in]{figures/sumrate_fdbkQuant_v2.eps}
\\
(c) & (d)
\end{tabular}
\caption{\label{fig:fdbkQuant}
CDF of sum rates of the different multi-user schemes with finite-rate feedback
of (a) only received ${\sf SNR}$s, (b) only received signal phases,
(c) only user side cross-correlation information, and (d) all the parameters
quantized simultaneously.
}
\end{center}
\end{figure*}

\subsection{Quantizer Design}
\label{sec3c}
Towards the second study, we utilize different quantization functions to
quantize the different parameters needed in channel reconstruction. For a
phase term $\theta$ with a dynamic range of $[0, 2\pi)$
(e.g., $\angle{ \widehat{\bf s}_{ {\sf tr}, k,\ell} }$ and $\angle{ \beta_{k,\ell} }$), we use a
uniform quantizer of the form
\begin{eqnarray}
{\cal Q}_{B}(\theta) = \frac{2 \pi}{2^B} \cdot {\sf round}\left( \frac{2^B}{2 \pi}
\cdot \theta
\right),
\label{eq_quant_phase}
\end{eqnarray}
where ${\sf round}(\cdot)$ stands for a function that rounds off the underlying quantity to
the nearest integer. For an amplitude term $\alpha$ with a dynamic range of $[0, 1]$
(e.g., $|\beta_{k,\ell}|$), we use a non-uniform quantizer of the form
\begin{eqnarray}
{\cal Q}_B(\alpha) = \frac{ {\sf round}\left( (2^B - 1) \cdot \alpha \right) }{2^B - 1}.
\label{eq_ampl_quantize}
\end{eqnarray}
The reason for scaling with respect to $2^B - 1$ in~(\ref{eq_ampl_quantize}) instead of by
$2^B$ is because we want the quantized set to include both $0$ and $1$ for proper
cross-correlation quantization. For example, in the typical case where the multi-user
reception beam ${\bf g}_k = {\bf g}_{{\sf tr}, m_1^{k}}^{(k)}$, we have $|\beta_{k,1}| = 1$
and the use of a uniform amplitude quantizer will not allow the correct reproduction of
this important quantity at the base-station end.

Quantization of the ${\sf SNR}$ is performed on a dB scale rather than on a linear
scale. This is intuitive since ${\sf SNR}$ measurements have a wide dynamic range.
The proposed ${\sf SNR}$ quantizer is similar to quantizations considered in Fourth
Generation (4G) systems. In particular, for a received ${\sf SNR}$ term
$\varrho$ (in dB) with a
theoretically unbounded range (e.g., $10 \log_{10} \big( {\sf SNR}_{ {\sf rx},
\hsppp \ell }^{(k)} \big)$), we first cap $\varrho$ to a maximum value of
$\varrho_{\sf max}$ and quantize a spread of $\Delta$ (in dB) with $2^B$ quantization
levels (denoted as $\varrho_i$) as follows:
\begin{eqnarray}
\varrho_i = \varrho_{\sf max} - \frac{\Delta}{2^B - 1} \cdot i , \hspp i = 0, \cdots,
2^B - 1.
\end{eqnarray}
The quantization of $\varrho$ is given as
\begin{eqnarray}
{\cal Q}_B(\varrho) = \varrho_{i^{\star}} \hspp {\sf where} \hspp
i^{\star} = \arg \min \limits_{i = 0, \cdots, 2^B - 1} |\varrho - \varrho_i | .
\end{eqnarray}
The parameters $\varrho_{\sf max}$ and $\Delta$ correspond to the maximum quantizer
level value and the distance between adjacent quantizer levels, respectively. In
our numerical
studies, we use $\varrho_{\sf max} = 30$ dB with $\Delta = 24$ dB for $B = 2$ bits,
and $\Delta = 30$ dB for $B = 4$ bits.

A similar approach is pursued in quantizing the amplitudes of the multi-user beam.
While these amplitudes do not span a wide range, the relative variation across the
antenna array can show wide variations. Specifically,
the infinite-precision zeroforcing beams generated in Prop.~\ref{prop_zf} are
quantized to meet the RF constraints in~(\ref{eq_hybrid_precoder_constraints})
as described next. Since $\| {\bf f}_k \| = 1$, we assume that on average
${\bf f}_k(i) \approx \frac{1}{ \sqrt{ N_{\sf t} } }$. By scaling $|{\bf f}_k(i)|^2$
by $N_{\sf t}$, we can ensure that $10 \log_{10} \left(N_{\sf t} \cdot
|{\bf f}_k(i)|^2 \right)$ is centered around $0$ dB and for this quantity, we
generate $2^B$ quantization levels in dB scale (denoted as $f_i$) corresponding to
a step size of $\Delta f$ (in dB) as follows:
\begin{eqnarray}
f_i = \Delta f \cdot \left[ i + 1 - 2^{B-1} \right], \hspp i = 0, \cdots, 2^B -1.
\end{eqnarray}
With these levels that are spaced $\Delta f$ apart, we obtain the quantized beam weights as
\begin{eqnarray}
| {\widehat{\bf f}}_k(i) | = \widetilde{\cal Q}_B( | {\bf f}_k(i) | )
= \frac{1}{ \sqrt{ N_{\sf t}} } \cdot \left\{
\begin{array}{cl}
0 & {\sf if} \hspp 10 \log_{10} \left(N_{\sf t} \cdot |{\bf f}_k(i)|^2 \right)
< - \Delta f \cdot (2^{B-1} - 1) \\
10^{ \frac{ f_{j^{\star}} }{20} } & {\sf otherwise},
\end{array}
\right.
\end{eqnarray}
where
\begin{eqnarray}
j^{\star} = \arg \min_{j }
10 \log_{10} \left(N_{\sf t} \cdot |{\bf f}_k(i)|^2 \right) - f_j
\hspp {\sf provided} \hspp 10 \log_{10} \left(N_{\sf t} \cdot |{\bf f}_k(i)|^2 \right)
> f_j.
\label{eq_constraint}
\end{eqnarray}
The constraint in~(\ref{eq_constraint}) ensures that $\sum_i | {\widehat{\bf f}}_k(i) |^2
\leq 1$. In our numerical studies, we use $\Delta f = 1$ dB for $B = 4$ bits leading
to a range of $-7$ to $8$ dB for $f_i$. We also use $\Delta f = 0.25$ dB for $B = 6$
bits leading to a range of $-7.75$ to $8$ dB for $f_i$. For the phase quantities (that
is, $\widetilde{\cal Q}_B( \angle{ {\bf f}_k(i) } )$), we reuse
${\cal Q}_B( \angle{ {\bf f}_k(i) } )$ as in~(\ref{eq_quant_phase}).

\subsection{Finite-Rate Feedback}
\label{sec5b}
With the quantizer design as described in~Sec.~\ref{sec3c},
we now consider the impact of finite-rate feedback of the
quantities of interest necessary for the channel reconstruction step. As noted
from Table~\ref{table_feedback}, each user quantizes and feeds back to the
base-station: i) the base-station beam indices, ii) the received
${\sf SNR}$s, iii) the received signal's phases, and iv) user side codebook
correlation information (amplitude and phases). To reduce clutter in
presentation, in our studies illustrated in Fig.~\ref{fig:fdbkQuant}, we
only focus on the $N = 8$ and $N = 32$ codebooks for beam alignment with
$M = 16$ at the user side. Fig.~\ref{fig:fdbkQuant}(a) considers the impact of
$B_{\sf SNR}$ (the number of bits used in received ${\sf SNR}$ quantization) while
infinite-precision is used for the signal phases and codebook correlation. This
figure shows that the proposed scheme is robust to $B_{\sf SNR}$ in the sense
that for both the $P = 2$ and $P = 4$ cases, the performance improvement is minimal
as $B_{\sf SNR}$ is increased from $2$ bits to $4$ bits.

On the other hand, Fig.~\ref{fig:fdbkQuant}(b) considers the impact of
$B_{ {\sf est}, \hsppp {\sf phase}}$ (the number of bits used in received
signal phase quantization) while infinite-precision is used for received ${\sf SNR}$
and codebook correlation. In the third experiment, we study the impact
of codebook correlation quantization with infinite-precision for the other two quantities.
To simplify this investigation, we assume that $B_{\sf corr, \hsppp phase} =
B_{\sf corr, \hsppp amp} = B_{\sf corr}$ and Fig.~\ref{fig:fdbkQuant}(c) considers
the impact of $B_{\sf corr}$ on performance. Both Figs.~\ref{fig:fdbkQuant}(b)
and~(c) show that increasing $B_{ {\sf est}, \hsppp {\sf phase}}$ or $B_{\sf corr}$
has maximal impact on performance for
large $P$. In other words, if
the channel approximation gets better, it becomes pertinent to quantize the
phase terms and codebook correlation information in the channel reconstruction
with a finer resolution.

While Figs.~\ref{fig:fdbkQuant}(a)-(c) study the quantization of each parameter
of interest separately, we now consider the impact of finite-rate quantization of
all the parameters necessary for channel reconstruction (relative to infinite-precision
quantization). For this, we consider the case where $B_{\sf SNR} =
B_{ {\sf est}, \hsppp {\sf phase}} = B_{ {\sf corr}, \hsppp {\sf amp}} =
B_{ {\sf corr}, \hsppp {\sf phase}} = 3$ bits with $M = 16$. From Fig.~\ref{fig:fdbkQuant}(d),
we observe that the proposed joint quantization scheme performs comparable with a scheme that
uses infinite-precision for all the parameters of interest.

\begin{table*}[htb!]
\caption{Feedback overhead ($B_{\sf feedback}$) for different choices of $P$ and $N$}
\label{table_fb_overhead}
\begin{center}
\begin{tabular}{|c||c|c|c|c|}
\hline
& $N = 4$ & $N = 8$ & $N = 16$ & $N = 32$ \\
\hline
$P = 2$ &
14$ &
16$
&
18$ &
20$ \\
\hline
$P = 4$ &
44$ &
48$
&
52$ &
56$
\\
\hline
\end{tabular}
\end{center}
\end{table*}

At this stage, it is important to note that the feedback overhead of $\varphi_{k,\ell}$ and
$\nu_{k,\ell}$ can be combined\footnote{Similarly, it might be envisioned that the feedback
of $\gamma_{k,\ell}$ and $\mu_{k,\ell}$ can be combined, but their dynamic ranges are different.
Feedback overhead reduction could be a useful topic of study in future research.} since
they are always used in the form $\varphi_{k,\ell} + \nu_{k,\ell}$
(see~(\ref{eq_eigenspace_reconst})). Thus, based on the above studies, we make the following
heuristic design guidelines on the feedback overhead
\begin{eqnarray}
B_{\sf SNR}
= 2 \hspp {\sf bits}, \hspp \hspp
B_{ {\sf est}, \hsppp {\sf phase}} + B_{ {\sf corr}, \hsppp {\sf phase}} =
B_{{\sf corr}, \hsppp {\sf amp}} =
P \hspp {\sf bits}.
\end{eqnarray}
Combining this information with Table~\ref{table_feedback}, the total feedback
overhead from each user is given as
\begin{align}
B_{\sf feedback} & = P \cdot \left[ \log_2(N) + B_{\sf SNR} + B_{ {\sf corr}, \hsppp {\sf amp}}
\right] + (P-1) \cdot \left[ B_{ {\sf est}, \hsppp {\sf phase}} +
B_{ {\sf corr}, \hsppp {\sf phase}}  \right] \hspp \hspp ({\sf in} \hspp {\sf bits})
\\
& = P \cdot \left[ \log_2(N) + 2 + P \right] + (P-1) \cdot P
\\
& = P \cdot \left[ \log_2(N) + 2P + 1 \right] \hspp {\sf bits}.
\end{align}
$B_{\sf feedback}$ is presented in Table~\ref{table_fb_overhead} for the choices
$P \in \{2, 4\}$ and $N \in \{ 4, 8, 16 , 32 \}$. From Table~\ref{table_fb_overhead},
a $56$ bit control payload appears to be sufficient to convey the information
necessary for multi-user beamforming across different choices of $M$, $N$ and $P$.
On a first glance, while this may appear to be onerous, similar feedback overheads
are currently considered viable in 3GPP 5G-NR design. In particular, two types
of feedback methods are being
studied~\cite[Sec.\ 8.2.1.6.3, pp.\ 24-26]{3gpp_CM_rel14_38912}: i) Type-I feedback
of both the beam indices and RSRPs of the top-$4$ beams, and ii) a more general
Type-II feedback that can include feedback of covariance matrices, co-phasing
factors with different codebook structures, etc. Further, the time-scales at
which this information has to be reported is on the order of the coherence time
of the channel (which varies from a few milliseconds at high speeds to a few
hundreds of milliseconds in an indoor or low speed
scenario~\cite{vasanth_comm_mag_16,vasanth_tap_blockage})
allowing multiple long PUCCH
instances for beam reporting. Also, this control information can be fed back
on legacy carriers such as 4G links in a non-standalone deployment. Thus,
the feedback
overhead necessary for realizing the proposed schemes are practically viable.

\subsection{Quantization of Multi-User Beams and Comparison with Upper Bounds}
\label{sec5c}
In the third study, the effect of quantizing the multi-user beams to ensure that
it fits the RF precoder constraints as in~(\ref{eq_hybrid_precoder_constraints})
is considered. In general, if a low rate quantization is used ($B_{\sf amp}$ or
$B_{\sf phase}$) as $P$ increases, the resultant multi-user beam's sum rate
performance could be worse than that with beam steering. In particular,
from Fig.~\ref{fig:bfQuant}(a), we observe that a higher phase resolution
($B_{\sf phase}$) is necessary for improved performance as the codebook resolution
improves (large $N$) or when $P$ increases. On the other hand, from
Fig.~\ref{fig:bfQuant}(b), we observe that an amplitude resolution $B_{\sf amp}$)
on the order of $4$-$6$ bits can produce a performance comparable with the unquantized
scheme.

In Fig.~\ref{fig:upperBounds}, we finally compare the performance of the proposed
zeroforcing scheme with the beam steering scheme and the bounds established in
Sec.~\ref{sec4}. We also benchmark the performance with a fully-digital system
employing: i) maximal ratio transmission/maximal ratio combining (MRT/MRC) beams
in the initial alignment phase, and ii) a zeroforcing scheme performed using the
MRT/MRC beams as in~\cite{vasanth_gcom16,vasanth_jsac2017}. Note that the MRT/MRC
scheme is different from that employed in~\cite{vasanth_gcom16,vasanth_jsac2017}
where perfect beam steering vectors are used in deriving the zeroforcing
structure. In terms of differences between these structures, the readers are
referred to~\cite{vasanth_gcom15}. For the proposed scheme, an $M = 16$ codebook
is used at the user end. Figs.~\ref{fig:upperBounds}(a)-(b) and Fig.~\ref{fig:upperBounds1}
illustrate the trends with $N = 8$, $N = 32$, and $N = 256$ codebooks, respectively.
For $N = 256$, we employ a DFT codebook at the base-station covering the $120^\circ
\times 30^\circ$ AoD space.

With low-resolution quantization,
we note that there
is a considerable performance gap between the zeroforcing scheme and the scalar
optimization-based upper bound (up to $2$ bps/Hz). On the other hand, this gap
reduces as $N$ increases suggesting the good
performance of the zeroforcing scheme. Nevertheless, the performance gap between the
proposed zeroforcing scheme and the upper bounds suggests the possible utility of
more advanced feedback mechanisms, a topic for future research. In all the plots, there
is a considerable gap between the performance of the upper bounds with the
fully-digital system. Plausible explanations for this observation include the use
of small arrays at the user end ($2 \times 2$) and $L_k = 6$ clusters in the channel.
A more complex hybrid precoding architecture achieved by
optimally choosing $\bF_{\sf Dig}$ with respect to some performance metric may assist
in bridging this gap. It is also to be pointed out
that while the alternate
optimization-based sum rate serves as an upper bound for most channel realizations,
for some realizations (especially at low ${\sf SINR}$ values where the ${\sf SLNR}$
optimization has a different behavior than the sum rate maximization), this connection
breaks down.

\begin{figure*}[htb!]
\begin{center}
\begin{tabular}{cc}
\includegraphics[height=2.4in,width=3.0in]{figures/sumrate_beamPhaseQuant.eps}
& {\hspace{0.1in}}
\includegraphics[height=2.4in,width=3.0in]{figures/sumrate_beamAmpQuant.eps}
\\
(a) & (b)
\end{tabular}
\caption{\label{fig:bfQuant}CDF of sum rates of the different schemes with
quantization constraints on the multi-user beam's (a) phases and (b) amplitudes.}
\end{center}
\end{figure*}

\begin{figure*}[htb!]
\begin{center}
\begin{tabular}{cc}
\includegraphics[height=2.4in,width=3.0in]{figures/sumrate_txCodebook_8_rxCodebook_16_upperBounds_v2.eps}
& {\hspace{0.1in}}
\includegraphics[height=2.4in,width=3.0in]{figures/sumrate_txCodebook_32_rxCodebook_16_upperBounds_v2.eps}
\\
(a) & (b)
\end{tabular}
\caption{\label{fig:upperBounds}CDF of sum rates of the multi-user schemes
compared with the two upper bounds using a $M = 16$ codebook with (a) $N = 8$, (b) $N = 32$.
}
\end{center}
\end{figure*}

\begin{figure*}[htb!]
\begin{center}
\includegraphics[height=2.4in,width=3.0in]{figures/sumrate_txCodebook_256_rxCodebook_16_upperBounds_v2.eps} \caption{\label{fig:upperBounds1}
CDF of sum rates of the multi-user schemes
compared with the two upper bounds using a $M = 16$ codebook with $N = 256$.}
\end{center}
\end{figure*}

\ignore{
\begin{figure*}[htb!]
\begin{center}
\begin{tabular}{cc}
\includegraphics[height=2.5in,width=3.3in] {figures/sumrate_txCodebook_16_rxCodebook_16.eps}
&
\includegraphics[height=2.5in,width=3.3in] {figures/sumrate_txCodebook_32_rxCodebook_16.eps}
\\
(a) & (b)
\end{tabular}
\caption{\label{fig:sumrates}
CDF of sum rates with MU-MIMO schemes using $M = 16$ for (a) $N = 16$ and
(b) $N = 32$.}
\end{center}
\end{figure*}
}

\section{Concluding Remarks}
\label{sec6}
The focus of this work has been the development of a feedback mechanism to convey estimates
of certain quantities of interest from an initial beam alignment phase to enable the
base-station to construct an advanced RF precoding structure for multi-user transmissions.
These quantities of interest include the top-$P$ (where $P \geq 1$) base-station side beam
indices, phases and amplitudes of an appropriate received signal estimate, as well as
the cross-correlation
information of the beams at the user end. This feedback is leveraged to reconstruct/estimate
a rank-$P$ approximation of the channel matrix of interest at the base-station end and generate
a zeroforcing structure for multi-user interference management. Numerical studies show that the
additional feedback overhead is marginal, but the relative performance improvement over a simplistic
beam steering scheme is significant even with a very coarse initial beam alignment codebook.

This study reinforces the importance of the development of low-complexity (in terms of feedback
overhead as well as implementation) yet good (in terms of performance and structure)
feedback techniques for
large-MIMO systems~\cite{david_review,vasanth_design}. While this work has only scratched the
surface of such techniques, a number of possible future research directions are worth considering.
Benchmarking the performance of any proposed feedback technique with a tight upper bound (for
the sum rate) is an area of fundamental difficulties due to the non-convex nature of the
problem~\cite{cioffi_weighted,kobayashi,vasanth_arxiv2011} and is richly rewarding. Understanding
the fundamental limits of hybrid precoders beyond the phase-only control architecture that is
common in the literature, as well as providing a directional intuition into the structure of the
precoder construction (in contrast to a {\em black box} optimization solution) are of importance
in practical implementations. While the solutions proposed in this work can be
readily extended to polarization-diversity transmissions, extending it to the case where the
users possess two (or more) RF chains with the base-station communicating over two {\em spatial}
layers is of
importance from a 3GPP 5G-NR deployment perspective. Study of different hybrid beamforming
architectures such as the sub-connected structure in~\cite{gao_hybrid} and comparison
with the proposed scheme(s) would be of interest. Sensitivity of such advanced schemes to
impairments such as Doppler and
phase noise are also worth exploring more carefully.


This evening Westbourne Studios played host to a few readings of plays written by young Muslims aged between 16-25 as part of a initiative run by the Royal Court Theatre.
As well as being written by Muslims, the plays seemed to be for Muslims too. For God's Sake (by Siama Shah, 16!) discussed the identity issues between two young British born Pakistanis, The Next Step (by Hassan Minas) touched on the different Muslims communities, Arab in the West (by Osamah Al-Tamimy) was focussing on the life of an new Arab immigrant to Britain, while Shades (by Alia Bano) tackled the topic of marriage and relationships between British Muslims. Of course I had just the extract to go on, so the full plays may really be about other things.
Although they were just readings, they were as enjoyable, engaging and meaningful as any full production. The lack of props didn't seem to hinder the acting, some of which was really good. And even though they were very short, they provoked enough discussion and debate afterwards. Unfortunately the Q&A session with the actors and writers wasn't as fulfilling as I hoped it would be, mainly due to a meek audience. Still I had my question about crossing lines of offence answered adequately by Shah.
In fact the most disappointing parts of the extracts were their lengths - it was all over so quickly, leaving most of the audience gagging for more. When I asked about the future of the performances, we were told that that would only happen with demand (and as a consequence, funding), but from what I saw tonight any full production at the Royal Court would definitely be something I'd be interested in.
Hey, have you seen the Royal Court Website, one of the writer's you mention, Alia Bano's play is running next year and another called Salaam Mr Bush is having a reading.
Thanks for the tip; we were pointed to the Royal Court when we were at the readings, so it's good to see that at least one has made it!


Grit Real Estate Income Group ('Grit' or 'the Company'), is a multi-listed real estate income group operating in carefully selected African countries and is currently listed on the Main Market of the London Stock Exchange ('LSE'), the Main Board of the Johannesburg Stock Exchange Limited ('JSE') and the Official Market of the Stock Exchange of Mauritius Limited ('SEM'). Grit has a robust property portfolio with diversification across sectors and geography, backed by comprehensive risk mitigation policies and procedures. The Company optimises its structural investments, underpinned by solid property fundamentals, to achieve compelling US Dollar returns. Its ability to sustain distribution is supported by predominantly US Dollar and EURO-denominated long-term leases with blue chip multinational tenants, eliminating currency risk and delivering strong sustainable income. Since listing on the JSE in 2014, Grit has grown its property portfolio from USD140 million to over USD796 million and has successfully delivered ten consecutive dividend distributions in line with Company guidance. Grit is a constituent of the SEM-10 index, the S&P Africa Frontier (Global) index as well as an underlying constituent of the JSE-listed Cloud Atlas AMI series BIG 50 ex-SA ETF.


National Research Council. 1987. Guidelines for Noise and Vibration Levels for the Space Station. Washington, DC: The National Academies Press. https://doi.org/10.17226/19173.
Click here to obtain permission for Guidelines for Noise and Vibration Levels for the Space Station.


text:
Labadia de Saint-Wandrille, antigament abadia de Fontenelle, és un establiment monàstic benedictí situat al municipi de Saint-Wandrille-Rançon, en el departament de Seine-Maritime, a la regió de Normandia (França).

Història
Sant Wandrille va fundar l'any 649 l'abadia de Fontenelle, sobre una propietat concedida per Erquinoald, majordom de palau de Nèustria. El 787 per ordre de Carlemany, Landry, abat de Jumièges, i Ricard, comte de Rouen proveïren un políptic, perdut en l'actualitat. Va ser la tercera abadia de la província de Rouen després de les de Saint-Ouen de Rouen i de Saint-Evrault.

En una carta de Carles el Calb datada el 21 de març de 854, s'indica que els religiosos de Fontenelle posseïen béns a Li Pecq (Yvelines), a Chaussy-en-Vexin (Val-d'Oise), a Pierrepont, a la comuna de Grandcourt (Somme), Bution (lloc encara sense localitzar, però probablement prop d'Arpajon) i a Marcoussis a l'Essonne. L'abadia va ser saquejada pels normands cap al 858 i abandonada.

Cap a 960, Ricard I de Normandia propicià el restabliment dels monjos portats per Gérard de Brogne. Robert I de Normandia ordenà que se'ls restituïssin els béns usurpats.

Durant les guerres de Religió, al maig de 1562 els protestants i els seus seguidors saquejaren l'abadia. La Revolució francesa va suprimir la institució, i els edificis gòtics es transformaren en pedrera.

Edificis
 Únic claustre gòtic complet en Alta Normandia
 Gran refetor
 Pati d'honor
 Ruïnes de l'abacial gòtica
 Granja medieval

Referències

Enllaços externs

 Plana de l'abadia

Sena Marítim
Monuments històrics de Normandia
Monestirs de França
Monestirs de l'orde de Sant Benet

redpajama_set_name: RedPajamaWikipedia


One of the greatest challenges about the human condition is not knowing what's around the corner. When so many of us, it seems to me, yearn to know our future before it has arrived.
I found myself like this today, trying to work on stuff for a market on the weekend, but restless, I couldn't settle and enjoy the task at hand. I gave up and I went in to my daughter's Kindy and hour early and helped out.
It was so refreshing to be among young minds whose only concern is the present "Will you be my friend"; "Will you push my on the swing"; "Come play 'cats' with me".
For them the future is Birthdays and Santa and Grandad coming for dinner on Thursday.
By the end of tomorrow I may or may not have taken a huge step along a path that changes our path so completely. I can not know if we'll be successful and our chances are slim, but still I find myself playing "What if...."
I know what you're saying ... the what ifs - they are so much to juggle sometimes aren't they.


text:
Home News Defending champions Western Sydney make ideal start
Defending champions Western Sydney make ideal start
Defending champions Western Sydney Wanderers underlined their determination to retain the AFC Champions League title this season with a hard-fought 3-1 win over Japan's Kashima Antlers on Wednesday, while 2014 runners-up Al Hilal also began their campaign with victory by the same margin after defeating Uzbekistan side Lokomotiv 3-1 in Riyadh.
Kashima defender Gen Shoji had put the holders in front with an own goal only for Shoma Doi to level for the J.League side just after the midway point of the second half.
But goals from former Sanfrecce Hiroshima midfielder Yojiro Takahagi on 86 minutes and Mark Bridge in stoppage time saw the Australian side claim all three points.
And Western Sydney were joined at the top of Group H by China's Guangzhou Evergrande after new signing Ricardo Goulart scored the game's only goal to give the 2013 champions a 1-0 win over their opponents from the final two years ago, FC Seoul of Korea Republic.
"It was a very tough game and Kashima are an exceptionally good team with fabulous history, and for us to come here to start our AFC Champions League defense with a positive a result is fantastic achievement for us," said Western Sydney assistant coach Ian Crook.
"Our performance in the A-League at times is very good, but maybe we haven't got the results that we think we deserve, but certainly winning games of football helps confidence, and all players and all teams need confidence to win games."
Two-time Asian champions Al Hilal were also able to make a winning start to their campaign having reached the final last year as goals from Saud Kariri, Yousef Al Salem and Thiago Neves secured the Saudi Arabian club a 3-1 win against Lokomotiv at the King Fahd International Stadium.
And with Iranian champions Foolad Khouzestan drawing with 2011 champions Al Sadd in Ahwaz, Al Hilal are the early pacesetters in Group C ahead of their meeting with the Qatari side next week.
Former winners Gamba Osaka, meanwhile, suffered a setback in their Group F opener as debutants Guangzhou R&F of China caused the tournament's first upset with a 2-0 victory on Tuesday in Japan.
Moroccan striker Abderrazzaq Hamedallah set Cosmin Contra's side on their way after 10 minutes at Expo '70 Stadium and then Wang Song's deflected free-kick wrapped up the points on the 80th minute.
"I'm very happy, not for me but for the club and the players, because they worked very hard on the pitch for these three points, and these three points are very, very good for us," said Romanian coach Contra.
And the Chinese Super League club were joined at the top of the Group F table by Thailand champions Buriram United who defeated Korean FA Cup winners Seongnam FC 2-1 in the group's other fixture at Buriram Stadium on Tuesday evening.
Elsewhere, two-time Asian champions Suwon Bluewings edged 2007 AFC Champions League winners Urawa Reds 2-1 at Suwon World Cup Stadium in Group G on Wednesday, while midfielder Shao Jiayi scored the game's only goal in stoppage time as China's Beijing Guoan beat Australian champions Brisbane Roar.
Also in the East, Chinese FA Cup winners Shandong Luneng saw off Vietnamese champions Binh Duong in a rollercoaster 3-2 away victory to take an early lead in Group E, while the meeting between
2006 champions Jeonbuk Motors of Korea and Kashiwa Reysol of Japan ended in a goalless stalemate.
Inaugural AFC Champions League winners Al Ain were unable to break the deadlock against Al Shabab of Saudi Arabia in their AFC Champions League Group B opener and the two sides share second spot behind early pacesetters Pakhtakor, after the Uzbekistan champions defeated tournament newcomers Naft Tehran of Iran 2-1 in Tashkent.
In Group A, Iran's Persepolis celebrated their return to the AFC Champions League after a three year absence with a deserved 3-0 win > over Qatar champions Lekhwiya , while Saudi Arabia champions Al Nassr shared a 1-1 draw with visitors Bunyodkor of Uzbekistan.
Finally in Group D, 2011 AFC Cup winners Nasaf beat Iran's Tractorsazi Tabriz 2-1, while Emirati champions Al Ahli shared a 3-3 draw at home to Saudi Arabia's Al Ahli.
CAPTION – Australia's Western Sydney Wanderers began the defence of their AFC Champions League title with a 3-1 victory in Japan over Kashima Antlers on Wednesday.
Previous articleOne-millionth BMW 3 Series Sedan produced at BMW South Africa's Rosslyn Plant
Next articleLiverpool focus on home after Istanbul shock exit
Dr Nathan hails Shalin
For 23 read 24 in WTCR
Sid's take on Shalin
Macarthur 'looked like a basketball team' against Jets, says Milicic

redpajama_set_name: RedPajamaCommonCrawl


text:
Home » Lifestyle » 10 Skater Girl Outfits That are Cool and Carefree
10 Skater Girl Outfits That are Cool and Carefree
July 13, 2021 Sophia Vergara
In 2021, we can safely say that the trends of the past decade are making a comeback, and so are the aesthetics of the female skater. A subculture that has been around for over 70 years, the skateboarder girl became popular in the late 50s and early 60s. The 80s and 90s were a big era for the skate scene, skyrocketing the sport in mainstream popularity. Today, aesthetic influences from the 90s and 2000s have inspired some of the latest skater styles, even if you don't know how to kickflip. If you're hoping to shake up your look in a casual, carefree way, stay tuned. Here's how you can mimic that niche, history-rich aesthetic while looking super cute at the same time.
RELATED: 25 best streetwear brands you need to know
What is a Skater Girl?
A skateboarder is a subculture of people who love skateboarding or the aesthetic that surrounds it. While skateboarding itself dates back to the 1940s, it wasn't until the 1960s that women began to popularize the trend, including Patti McGee. The scene's heyday was the 70s, then morphed again in the late 80s and early 90s, with influences like Cara-Beth Burnside and Elissa Streamer. It doesn't matter if you're a pro or just starting out, anyone can easily rock the look and feel authentic.
How do skaters dress?
While the specific style has undergone transformations in each era, the fundamentals have remained the same. Skaters generally wear comfortable, loose clothing, which allows for more movement as you move around the board. Graphic tees and high waisted mom jeans with sneakers are a classic ensemble. Replace the t-shirt with a cropped top when you're in hot weather. Sweatshirts, denim shorts, Converse tops or Vans and bucket hats are also popular choices for those who love aesthetics and sport.
Skater Girl Outfits
As well as being an essential part of your outfit, wearing the right shoes is crucial for your comfort and riding style. The original skate shoes include the Vans, which were launched in the 60s and are an integral part of the subculture and aesthetic. Popular choices that have a retro feel include Vans Old Skool or Vans Era. Alternatively, you can wear Converse All Stars, either in High Tops or Low Tops – the authentic style usually has a thick, flat sole, giving you the best grip on the board.
SHOP: SKATE SHOES
Much like skate shoes, the pants you wear while on the board are essential to your aesthetics and movement. While you're skating you'll be ducking a lot, so you need to find something appropriate for the actions and cute enough for the Insta grid. Unleash your inner skater girl by wearing oversized chinos or drawstrings in black, dark hues or a bright hue like cherry. Pair it with a cropped t-shirt, oversized sweater, or even a hoodie. You can add skate shoes with high socks in white, black or a funky pattern for an authentic edge. Remember, you need to be comfortable in whatever outfit you're wearing – this aesthetic is cool, laid-back, and all about being able to move.
SHOP: SKATE PANTS
Skate Girl Jeans
As one of the simplest clothing styles, humble jeans can totally elevate your skate outfit. A major element of the skater girl aesthetic is loose, high-waisted denim, which you can pair with a cute crop, oversized graphic tee, or hoodie. For the colder months, experiment with layering clothes, like a plain t-shirt with a chunky argyle knit cardigan or a long-sleeved shirt under a cute t-shirt. Add a pair of chunky white flat bottom sneakers for an authentic skater feel to finish the look.
SHOP: SKATE GIRL JEANS
Showcasing your favorite grunge or punk band on your clothing is not only a staple for a skate aesthetic, but it's also a great way to express your personality. A major element of this style, wide and loose t-shirts with prints can completely elevate an outfit. Wear them as they are or layer them with long sleeves underneath – it all depends on the aesthetic you're emulating. For a soft skater aesthetic, crop tops and tank tops are a big trend paired with shorts, tennis skirts or baggy jeans. Skatepunk girls and traditional skaters will usually opt for oversized tees with a large print on the front, either of a nostalgic TV show like Beavis and Butt-Head, a punk band like The Clash, or a grunge band like Nirvana.
SHOP: SKATE T-SHIRTS
During the warmer months and in warmer climates, wearing baggy jeans can become uncomfortable. That's where denim shorts come in – they're a great alternative to long pants that still have an authentic skater edge to them. This set was made popular in the 70s, which were considered the "golden age" of skateboarding. Iconic faces from the era include skateboarders like Laura Thornhill, Ellen O Neal and models like Farrah Fawcett-Majors. All played a huge role in wearing shorts on set. Today, the sweet girl aesthetic includes denim cuts with crop tops and tank tops, usually in brighter colors or white.
SHOP: SKATE SHORTS
Taking inspiration from Southern California grunge scenes, oversized clothing plays a huge role in the overall aesthetic. When freezing temperatures set in, swap out a long-sleeved shirt for a loose-fitting hoodie. They're ultra-comfortable and versatile, meaning you can skate, walk, and kick-flip without feeling cramped. They also act as a great padding blanket if you slip or fall off the board. Go for shades like charcoal, blue, bottle green or even sand. Just like other tops in this aesthetic, wear them with baggy pants or jeans.
SHOP: HOODIES
While most skaters won't wear a lot of accessories when performing an awkward step or grind, there are some cute ways to tweak an outfit for an authentic skater feel. Head coverings like a beanie, cap, or bucket hat are great ways to keep hair out of your face and protect your head. You should have a bag with you that you can easily carry when you are on the go. In this case, opt for a backpack or even a fanny pack to keep your belongings safe.
SHOP: SKATE ACCESSORIES
Best skater brands
With the history of female skaters dating back to the 60s, some brands have stayed true to the authentic feel of the sport that still exists today. Some of the top brands include Vans, which debuted in 1966 and continues to produce some of the most popular sneakers to this day. Others include Santa Cruz, HUF, Element and DC. These all hail from the California skate scene over the last century and make for great skate shoes, boards and apparel to try on when you're on the go. Other labels that have made the rounds include Stüssy and Supreme, which have started to enter the luxury space in recent years.
How to Get the Skater Girl Style
Wear loose t-shirts and sweatshirts for comfort and a laid-back style
Opt for sneakers with flat and thick soles
Try graphic tees and layer long sleeve shirts
Stay comfortable with mom jeans or loose chinos
During warmer months, replace pants with shorts.
Accessories like beanies, backpacks and bucket hats are essential to style.
Car FAQs
What should a skateboarder wear?
The beauty of being a skater outfit is that you can wear almost anything that makes you feel comfortable. Popular choices include oversized graphic tees and vintage styles or iconic skate brands like Vans, Stüssy or Supreme. You can wear loose mom jeans or high waisted drawstrings for optimal movement and style. You can create a silhouette by pairing a crop or tank top if you choose looser pants.
How do you become a skateboarder?
To become a skateboarder, you have to learn how to skateboard and look the part. Start with an oversized t-shirt with a stripe down the front, like Slasher or Nirvana. Then add a long sleeve shirt underneath for layering, baggy jeans and a pair of chunky sneakers. Of course, a skateboarder wouldn't be the same without a real skateboard. You can find great affordable boards online that look fantastic to ride and look great on TikTok or Instagram.
What is a skater aesthetic?
The skater girl aesthetic is a subculture that grew out of the grunge aesthetic of the late 80s and early 90s. While original elements of this style will generally do away with hyper-feminine silhouettes and typical color palettes. of the time. Now it combines the aesthetic that has recently emerged due to TikTok trends, which includes Y2K influence and trends in different subcultures.
Subscribe to our mailing list and get interesting news and updates delivered to your inbox.
We respect your privacy and take its protection seriously.
Disclaimer: Curated and re-published here. We do not claim anything as we translated and re-published using Google translator. All images and Tattoo Design ideas and images shared only for information purpose. Ideas and information collected through Google re-written in accordance with guidelines. We strictly follow Google Webmaster guidelines. You can reach us @ chiefadmin@tipsclear.com. We resolve the issues within hour to keep the work on top priority.
Previous: How Listening To Podcasts Can Improve Your life
Next: Memorial Tattoos – Design Ideas In The Memory Of A Loved One


Brown meat and onion. Drain grease. Add sauce. Simmer. Boil noodles until tender; drain. In crock pot, put a layer of sauce, noodles, meat/sauce mixture, cheses, pepperoni and as much green peppers and mushrooms as you want. Original recipe I had said to repeat layers one more time, but I did the layers over a few times more. Cook on HIGH for 30 minutes and then on LOW for 1 hour (I cooked on HIGH for 45 minutes and LOW for 2 hours), until cheeses melts. This can be made in a deep casserole dish in the oven at 350 degrees F. until cheese melts.
Your Crockpot Pizza is ready. Buon appetito!


As a high school runner, Lukas Verzbicas was about as special as they come.
The two-time Class 3A IHSA state cross country champion from Orland Park Sandburg could have won a third state title last year had he not opted to graduate early to being his collegiate career at Oregon.
When he left college to pursue his great passion for triathlons and turned pro, you had to figure it wouldn't be long before we would see him in the Olympics.
But on July 31, Verzbicas' career and life took a dramatic turn. He was involved in a serious bicycle crash while on a training run.
The two-time Class 3A IHSA state cross country champion from Orland Park Sandburg could have won a third state title last year had he not opted to graduate early to being his collegiate career at Oregon. He just missed setting the course record in 2010 with his 13:54.
But on July 31, Verzbicas' career and life took a dramatic turn. He was involved in a serious bicycle crash while on a training run and suffered severe injuries — broken vertebrae, claivicle and thorax and a collapsed lung. He has had several surgeries and has just begun rehab, which on Wednesday included just being able to stand and walk a few steps.
But clearly, he has a long recovery ahead.
Here is a video of Lukas taking his first steps since the accident. If you remember that young man who ran at Detweiller Park two years ago, this is a heartbreaking sight.
But he's also a strong and determined athlete and if there is a way to come back from this and compete again at the highest level, he will do it.