# Direction-Assisted Beam Management in Full Duplex Millimeter Wave Massive
MIMO Systems

Md Atiqul Islam2, George C. Alexandropoulos4, and Besma Smida2  2Department of
Electrical and Computer Engineering, University of Illinois at Chicago, USA
4Department of Informatics and Telecommunications, National and Kapodistrian
University of Athens, Greece
emails<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Recent applications of the Full Duplex (FD) technology focus on enabling
simultaneous control communication and data transmission to reduce the control
information exchange overhead, which impacts end-to-end latency and spectral
efficiency. In this paper, we present a simultaneous direction estimation and
data transmission scheme for millimeter Wave (mmWave) massive Multiple-Input
Multiple-Output (MIMO) systems, enabled by a recent FD MIMO technology with
reduced hardware complexity Self-Interference (SI) cancellation. We apply the
proposed framework in the mmWave analog beam management problem, considering a
base station equipped with a large transmit antenna array realizing downlink
analog beamforming and few digitally controlled receive antenna elements used
for uplink Direction-of-Arrival (DoA) estimation. A joint optimization
framework for designing the DoA-assisted analog beamformer and the analog as
well as digital SI cancellation is presented with the objective to maximize
the achievable downlink rate. Our simulation results showcase that the
proposed scheme outperforms its conventional half-duplex counterpart, yielding
reduced DoA estimation error and superior downlink data rate.

###### Index Terms:

Full duplex, mmWave, massive MIMO, direction estimation, joint communication
and sensing, optimization.

##  I Introduction

In-band Full Duplex (FD) massive Multiple-Input Multiple-Output (MIMO)
communication is a promising technology for the increasingly demanding data
rate requirements of future wireless networks due to its inherent capability
to enable simultaneous UpLink (UL) and DownLink (DL) communications within the
entire frequency band [1, 2, 3]. In addition to concurrent UL/DL data
transmission in reciprocal communications, the FD technology has been recently
considered for simultaneous DL data transmission and UL Channel State
Information (CSI) estimation, where FD Base Stations (BSs) transmit data
symbols in the DL via CSI-dependent multi-user BeamForming (BF), while
receiving UL training symbols from the user terminals [4, 5, 6]. The principal
bottleneck of any FD system is the in-band Self-Interference (SI) signal at
the Receiver (RX) side, resulting from the simultaneous transmission and
reception. Recently in [7, 8, 9], FD MIMO systems operating at millimeter Wave
(mmWave) frequencies were introduced, where SI suppression was achieved
through a combination of propagation domain isolation, analog domain
suppression, and digital SI cancellation techniques.

Wireless communications at mmWave frequencies are mainly realized via highly
directional BF, which is enabled by massive MIMO transceivers [10] that are
capable of analog or hybrid Analog and Digital (A/D) beamforming [11]. To
alleviate the hardware cost in those transceiver architectures, the large-
scale antenna arrays are connected to small numbers of Radio Frequency (RF)
chains via analog preprocessing networks comprised of phase shifters [12].
However, the selection of the adequate analog beams from the predefined
codebooks and the design of the digital beamformers require CSI knowledge,
which is hard to acquire with realistically affordable latency, especially
under mobility conditions (see, e.g.,[13] and references therein). This
challenge will be naturally more pronounced in the envisioned FD massive MIMO
systems operating at mmWave frequencies, where multiple UL and DL channels
need to be estimated. In practice, the analog BF in mmWave massive MIMO
systems is designed via beam switching between the communicating nodes in
order to find a pair of beams from their available beam codebooks meeting a
link performance indicator threshold [14, 15]. However, such time-consuming
beam training procedures incur significant configuration overhead deprived of
data transmission. Hence, beam misalignment may yield poor performance under
mobility scenarios [16]. Alternatively, exploiting UL/DL reciprocity or
position information, the DL BF can be performed in the direction of the UL
dominant Direction-of-Arrival (DoA), thus, reducing the beam sweeping overhead
[10, 17].

Figure 1: The considered FD massive MIMO communication system operating at
mmWave frequencies: an FD massive MIMO Base Station (BS) capable of analog BF
communicates with a mobile single-antenna FD User (UE), where the latter node
transmits training signals in the UL for DoA estimation.

In this paper, we present a DoA-assisted beam management framework for FD
mmWave massive MIMO systems, where the BS is equipped with a large antenna
array realizing DL analog BF and few digitally controlled receive antenna
elements used for UL DoA estimation. Capitalizing on the recently presented FD
hardware architecture for hybrid A/D BF in [9], we propose a Simultaneous DoA
estimation and Data Transmission (SDDT) scheme for boosting beam management in
FD mmWave massive MIMO communications. Enabled by FD and leveraging channel
reciprocity, we simultaneously estimate the UL dominant DoA and transmit
analog beamformed data in the DL direction. We present a joint design of the
DoA-assisted analog BF as well as the A/D SI cancellation units, targeting the
maximization of the achievable DL rate. Our extensive simulation results
considering a mmWave channel model showcase the FD-enabled gains of DoA-
assisted beam management under various user mobility conditions. In
particular, the proposed FD-based SDDT scheme achieves approximately
$1.2\times$ the DL rate of conventional Half Duplex (HD) mmWave massive MIMO
systems.

##  II System and Signal Models

We consider an FD mmWave massive MIMO Base Station (BS) node $b$ equipped with
$N_{b}$ TX and $M_{b}$ RX antenna elements communicating with a mobile single-
antenna FD User Equipment (UE) node $u$. In particular, in the TX of the BS
node $b$, a single RF chain is connected via phase shifters with a large
antenna array of $N_{b}$ elements, whereas each of the few $M_{b}$ RX antennas
is connected to a dedicated RF chain. In this work, a small number for $M_{b}$
is sufficient for UL DoA estimation; extension for large $M_{b}$ and analog
combining is left for future work. Using the values of the $N_{b}$ phase
shifters, we can formulate the TX analog beamforming vector
$\mathbf{v}_{b}\in\mathbb{C}^{N_{b}\times 1}$, whose elements are assumed to
have constant magnitude, i.e., $|[\mathbf{v}_{b}]_{n}|^{2}=1/N_{b}$ $\forall
n=1,2,\ldots,N_{b}$. Besides, we assume that the angles of the analog phase
shifters are quantized and have a finite set of possible values. Therefore,
$\mathbf{v}_{b}\in\mathbb{F}_{\rm TX}$, where $\mathbb{F}_{\rm TX}$ represents
the predefined beam codebook including ${\rm card}(\mathbb{F}_{\rm TX})$
distinct vectors (or analog beams) [18, 13]. We assume that, for every channel
use, the BS node $b$ transmits the complex-valued information data symbol
$s_{b}$ (chosen from a discrete modulation set) after analog BF with
$\mathbf{v}_{b}$. Similarly, the single-antenna FD UE node $u$ sends the
training symbol ${s}_{u}$ through the UL channel. The signal transmissions at
both node $b$ and node $u$ are power limited according to
$\mathbb{E}\\{\|\mathbf{v}_{b}{s}_{b}\|^{2}\\}\leq{\rm P}_{b}$ and
$\mathbb{E}\\{|{s}_{u}|^{2}\\}\leq{\rm P}_{u}$, respectively.

###  II-A Channel Model

All the considered mmWave UL/DL channels consist of multiple propagation
paths. In the spatial domain, each propagation path can be characterized by
its Direction-of-Arrival (DoA)/ Direction-of-Departure (DoD) and the
corresponding power as well as phase components [19]. For a certain coherence
block, the UL channel $\mathbf{h}_{b,u}\in\mathbb{C}^{M_{b}\times 1}$
including a Line-of-Sight (LoS) component and $L_{p}\\!-\\!1$ non-Line-of-
Sight (nLoS) paths is mathematically expressed as

$\begin{split}\mathbf{h}_{b,u}\triangleq\beta_{\rm
LoS}\mathbf{a}_{M_{b}}(\theta_{\rm
LoS})+\sum\limits_{\ell=1}^{L_{p}-1}\beta_{{\rm
nLoS},\ell}\mathbf{a}_{M_{b}}(\theta_{{\rm nLoS},\ell}),\end{split}$ (1)

where $\beta_{\rm LoS}\in\mathbb{C}$ and $\theta_{\rm LoS}\in[0\,\,2\pi]$
represent the complex gain and the DoA of the LoS path, respectively. Here,
$\beta_{{\rm nLoS},\ell}\in\mathbb{C}$ and $\theta_{{\rm
nLoS},\ell}\in[0\,\,2\pi]$ $\forall\ell=\\{1,2,\ldots,L_{p}-1\\}$ are the
complex gains and the DoAs of the nLoS path, respectively. Considering a
Uniform Linear Array (ULA), the response vector $\mathbf{a}_{M_{b}}(\theta)$
for $M_{b}$ antenna elements and any DoA $\theta$ is formulated as [18]

$\begin{split}\mathbf{a}_{M_{b}}(\theta)\\!\triangleq\\!\frac{1}{\sqrt{M_{b}}}\big{[}1,e^{j\frac{2\pi}{\lambda}d\sin(\theta)},\ldots,e^{j\frac{2\pi}{\lambda}(M_{b}-1)d\sin(\theta)}]^{\rm
T},\end{split}$ (2)

where $\lambda$ is the propagation signal wavelength and $d$ denotes the
distance between adjacent antenna elements.

For the considered FD system, the UL and DL channels are reciprocal resulting
in similar complex gains and DoAs/DoDs [18, 10]. Therefore, similar to (1),
the DL channel $\mathbf{h}_{u,b}^{\rm T}\in\mathbb{C}^{1\times N_{b}}$ can be
expressed as follows:

$\begin{split}\mathbf{h}_{u,b}^{\rm T}\triangleq\beta_{\rm
LoS}\mathbf{a}_{N_{b}}^{\rm H}(\theta_{\rm
LoS})+\sum\limits_{\ell=1}^{L_{p}-1}\beta_{{\rm
nLoS},\ell}\mathbf{a}_{N_{b}}^{\rm H}(\theta_{{\rm nLoS},\ell}).\end{split}$
(3)

In this paper, our goal is to estimate the LoS DoA $\theta_{\rm LOS}$ using
the UL training symbols and find the DoA-assisted analog beamformer
$\mathbf{v}_{b}$ that maximizes the DL rate. However, due to the FD operation
at both nodes, the simultaneous DL data and UL training transmission induce SI
signal in the RXs of the BS and UE. Following [9, 20], we consider the mmWave
clustered model for the SI channels denoted by
$\mathbf{H}_{b,b}\in\mathbb{C}^{M_{b}\times N_{b}}$ for the BS node $b$ and
${h}_{u,u}\in\mathbb{C}$ for the UE node $u$, where $\kappa$ represent the
Rician factor.

###  II-B Received Signal Model

At the RX of the BS node $b$, the training symbols transmitted from the UE
node $u$ are received along with the SI signal induced by the simultaneous
data and training signal transmissions. Due to limited propagation
attenuation, the strong SI signal is capable of driving the RX RF chains into
saturation. Therefore, as shown in Fig. 1, an $N$-tap analog SI canceller is
utilized to suppress the SI at the inputs of the RX RF chains, which is based
on the TX RF chain output. Exploiting the low-complexity cancellation in [9],
we employ the canceller taps at the TX RF chain output at the BS node $b$
resulting in an analog SI canceller where the number of taps does not scale
with the number of antenna elements. To suppress the residual SI signal after
analog cancellation, a digital SI canceller is utilized at the BS baseband.
Denoting $\mathbf{c}_{b},\mathbf{d}_{b}\in\mathbb{C}^{M_{b}\times 1}$ as the
analog and digital SI cancellers, respectively, the baseband received signal
at the BS node $b$, $\mathbf{y}_{b}\in\mathbb{C}^{M_{b}\times 1}$, is given by

$\begin{split}\mathbf{y}_{b}&\triangleq\mathbf{h}_{b,u}s_{u}+(\mathbf{H}_{b,b}\mathbf{v}_{b}+\mathbf{c}_{b}+\mathbf{d}_{b})s_{b}+\mathbf{n}_{b},\end{split}$
(4)

where $\mathbf{n}_{b}\in\mathbb{C}^{M_{b}\times 1}$ is the zero-mean Additive
White Gaussian Noise (AWGN) with variance $\sigma^{2}_{b}\mathbf{I}_{M_{b}}$.
Similarly, the DL signal $y_{u}\in\mathbb{C}$ received at the UE node $u$
after analog and digital SI cancellation can be expressed as

$\begin{split}y_{u}&\triangleq\mathbf{h}_{u,b}^{\rm
T}\mathbf{v}_{b}s_{b}+({h}_{u,u}+c_{u}+d_{u})s_{u}+n_{u},\end{split}$ (5)

where $n_{u}\in\mathbb{C}$ represents the AWGN with variance $\sigma^{2}_{u}$.
As previously mentioned, for proper FD-based reception, the RXs’ RF chains
need to be unsaturated from any residual SI stemming out analog SI
cancellation at both nodes $b$ and $u$ [3, 21]. Denoting the residual SI power
thresholds as $\lambda_{b}$ and $\lambda_{u}$ at node $b$ and node $u$,
respectively, the residual SI power constraints ${\rm
P}_{b}|[({\mathbf{H}}_{b,b}\mathbf{v}_{b}+\mathbf{c}_{b})]_{(j,:)}|^{2}\\!\leq\\!\lambda_{b}\,\forall
j=1,2,\ldots,M_{b}$ and ${\rm P}_{u}|({h}_{u,u}+c_{u})|^{2}\leq\lambda_{u}$
are necessary to be satisfied for successful reception after analog SI
suppression.

##  III DoA Estimation and DL Data Transmission

In this section, we introduce the proposed DoA-assisted analog beam management
protocol along with the DoA tracking scheme for the considered UL/DL mmWave
channel with the proposed FD massive MIMO system.

###  III-A UL/DL Channel Evolution Properties

We assume wireless communications in Time Division Duplexing (TDD) manner,
where the considered channels remain constant for all the channel uses in a
time slot, and the channel properties (i.e., DoAs/DoDs, complex path gains) of
the successive time slots are temporally correlated. Each time slot of $T_{s}$
time units contains $L$ symbols (i.e., $L$ channel uses). For any consecutive
$(i\\!-1\\!)$ and $i$ time slots, the evolution of LoS DoA component is
expressed similar to [22] as

$\begin{split}\theta_{\rm LoS}[i]\triangleq\theta_{\rm
LoS}[i-1]+\Delta\theta,\end{split}$ (6)

where $\Delta\theta$ depends on the velocity of the UE node $u$ and the time
slot duration $T_{s}$. We also assume that the UE node is moving with the
constant velocity $v$, hence,
$\Delta\theta\triangleq\text{arctan}\left(\frac{vT_{s}}{d_{\rm BS}}\right)$
with $d_{\rm BS}$ being the distance between the UE and the BS nodes. The
evolution model for the LoS complex path gain $\beta_{\rm LOS}$ is given by
the first-order Gauss-Markov model as [22]

$\begin{split}\beta_{\rm LoS}[i]\triangleq\rho\beta_{\rm
LoS}[i-1]+\epsilon[i-1],\end{split}$ (7)

where $\rho$ is the correlation coefficient, and $\epsilon[i-1]$ is the zero-
mean complex Gaussian noise distributed as $\mathcal{CN}(0,1-\rho^{2})$. The
DoAs and complex path gains of the nLoS components are assumed to change
randomly between consecutive time slots. In this paper, we propose to track
only the LoS DoA component for each time slot, and then use it analog beam
selection. The estimation of all complex path gains and their correlation
coefficients is left for future investigation.

Figure 2: (a) Proposed FD SDDT Scheme, (b) Conventional HD DoA with update
after threshold, and (c) Conventional HD DoA estimation in each slot.

###  III-B Proposed FD-Based SDDT Protocol

The proposed protocol for simultaneous DoA estimation and data transmission is
illustrated in Fig. 2(a), where the DL channel is dedicated for data
transmission, while the UL is accessed to transmit training symbols. The
procedures of DoA-based tracking and transmission are described as follows:

  1. 1.

DL Data transmission at any $i$th time slot leverages the DOA estimation for
the previous $(i\\!-\\!1)$th time slot, which is realized via simultaneous UL
transmission of $L$ orthogonal training symbols sent from the UE node $u$.

  2. 2.

Capitalizing on the UL/DL reciprocity, any LoS component estimation
$\widehat{\theta}_{\rm LoS}[i-1]$ is used during the next $i$th time slot for
the DL data transmission precoding.

  3. 3.

The DL analog beamformer $\mathbf{v}_{b}[i]\in\mathbb{F}_{\rm TX}$ at each
$i$th time slot is chosen according to $\widehat{\theta}_{\rm LoS}[i-1]$
subject to the satisfaction of the residual SI power constraints.

In Fig. 2(b), a conventional HD DoA-assisted analog beamforming scheme is
illustrated, where, contrary to the FD case, a fraction of the available
symbols in a time slot is dedicated for DoA estimation. Once the DoA of a time
slot is estimated, its analog beamforming and that of the succeeding ones
utilizes the same DoA estimation, unless the received Signal-to-Noise Ratio
(SNR) falls below a certain threshold; in this case, a DoA estimation update
occurs. An HD data transmission protocol with DoA estimation at each time slot
is shown in Fig. 2(c).

###  III-C UL DoA Estimation

As previously discussed, the UL training symbols at any $(i\\!-\\!1)$th time
slot are used for this slot’s LoS DoA estimation, which is leveraged for
analog BF at the $i$th time slot. For this estimation, we deploy the MUltiple
SIgnal Classification (MUSIC) algorithm; other DoA estimation techniques can
be used as well [23]. Starting from (4), the received $L$ training symbols at
any $(i\\!-\\!1)$th time slot can be grouped in
$\mathbf{Y}_{b}[i\\!-\\!1]\in\mathbb{C}^{M_{b}\times L}$. For the MUSIC DoA
estimation, the received signal covariance matrix
$\mathbf{R}_{b}\in\mathbb{C}^{M_{b}\times M_{b}}$ can be estimated as

$\begin{split}{\mathbf{R}}_{b}\\!\triangleq\\!\mathbb{E}\\{\mathbf{Y}_{b}[i\\!-\\!1]\mathbf{Y}_{b}[i\\!-\\!1]^{\rm
H}\\}\\!,\,\widehat{\mathbf{R}}_{b}\\!\triangleq\\!\frac{1}{L}\mathbf{Y}_{b}[i\\!-\\!1]\mathbf{Y}_{b}^{\rm
H}[i\\!-\\!1].\end{split}$ (8)

By taking the eigenvalue decomposition of the estimated sample covariance
matrix $\widehat{\mathbf{R}}_{b}$, it is deduced that:

$\begin{split}\widehat{\mathbf{R}}_{b}\triangleq\mathbf{U}{\rm
diag}\\{\eta_{1},\eta_{2},\ldots,\eta_{M_{b}}\\}\mathbf{U}^{\rm
H},\end{split}$ (9)

where $\eta_{1}\geq\eta_{2}\geq\ldots\geq\eta_{M_{b}}$ are the eigenvalues of
$\widehat{\mathbf{R}}_{b}$ and $\mathbf{U}\in\mathbb{C}^{M_{b}\times M_{b}}$
contains their corresponding eigenvectors. Since we are interested in
estimating the DoA of the LoS component, the matrix $\mathbf{U}$ can be
partitioned as $\mathbf{U}=[\mathbf{u}_{s}|\mathbf{U}_{n}]$, where the columns
in $\mathbf{U}_{n}\in\mathbb{C}^{M_{b}\times M_{b}-1}$ are the eigenvectors
spanning the noise subspace and $\mathbf{u}_{s}$ is the signal space
eigenvector. We next project the search vector $\mathbf{a}_{M_{b}}(\theta)$
$\forall\theta\in[0\,\,2\pi]$ onto the noise subspace $\mathbf{U}_{n}$ and
calculate the spectral peak as

$\begin{split}S(\theta)\triangleq\frac{1}{\mathbf{a}_{M_{b}}^{\rm
H}(\theta)\mathbf{U}_{n}\mathbf{U}_{n}^{\rm
H}\mathbf{a}_{M_{b}}(\theta)}.\end{split}$ (10)

Finally, the position $\theta$ of the spectral peak is the estimated LoS DoA
$\widehat{\theta}_{\rm LoS}[i\\!-\\!1]$ corresponding to this $(i\\!-\\!1)$th
time slot.

###  III-D DoA-Assisted Analog Beamforming and DL Rate

Capitalizing on the UL/DL reciprocity and the estimated LoS DoA component, we
propose to approximate the DL channel at each $i$th time slot as follows:

$\begin{split}\widehat{\mathbf{h}}_{u,b}^{\rm
T}[i]\triangleq\mathbf{a}_{N_{b}}^{\rm H}(\widehat{\theta}_{\rm
LoS}[i-1]).\end{split}$ (11)

This approximation is further used during this $i$th time slot to find the
best analog beamformer $\mathbf{v}_{b}[i]$ at the BS node $b$, via searching
in the available beam codebook $\mathbb{F}_{\rm TX}$. Therefore, the
instantaneous achievable DL rate per channel use for the proposed FD-based
SDDT scheme at this time slot is given by

$\begin{split}\mathcal{R}_{\rm DL}[i]=\log_{2}\left(1+\frac{{\rm
P}_{b}|\widehat{\mathbf{h}}_{u,b}^{\rm
T}[i]\mathbf{v}_{b}[i]|^{2}}{\sigma_{u}^{2}+\sigma_{r,u}^{2}[i]}\right),\end{split}$
(12)

where $\sigma_{r,u}^{2}[i]\triangleq{\rm
P}_{u}|({h}_{u,u}[i]+c_{u}[i]+d_{u}[i])|^{2}$ is the residual signal power
after both A/D SI cancellation at the UE node $u$.

##  IV Proposed SDDT Optimization Framework

In this section, we focus on the joint design of the analog beamformer
$\mathbf{v}_{b}$, the analog SI cancellers $\mathbf{c}_{b}$ and $c_{u}$, as
well as the digital canceller $\mathbf{d}_{b}$ and $d_{u}$ at the BS node $b$
and the UE node $u$, which maximize the estimated achievable DL rate in (12)
at each $i$th time slot. Based on the availability of the DL channel
estimation $\widehat{\mathbf{h}}_{u,b}[i]$, as well as the SI estimations
$\widehat{\mathbf{H}}_{b,b}[i]$ and $\widehat{h}_{u,u}[i]$, we consider the
optimization problem:

$\displaystyle\underset{\begin{subarray}{c}\mathbf{v}_{b}[i],\mathbf{c}_{b}[i],\mathbf{d}_{b}[i]\\\
c_{u}[i],d_{u}[i]\end{subarray}}{\text{max}}$
$\displaystyle\log_{2}\left(1+\frac{{\rm
P}_{b}|\widehat{\mathbf{h}}_{u,b}^{\rm
T}[i]\mathbf{v}_{b}[i]|^{2}}{\sigma_{u}^{2}+\sigma_{r,u}^{2}[i]}\right)$ s.t.
$\displaystyle\\!\\!\\!{\rm
P}_{b}|[(\widehat{\mathbf{H}}_{b,b}[i]\mathbf{v}_{b}[i]\\!+\\!\mathbf{c}_{b}[i])]_{(j,:)}|^{2}\\!\leq\\!\lambda_{b}\\!\,\forall
j\\!=\\!1,\\!\ldots,\\!M_{b},$ $\displaystyle\\!\\!\\!{\rm
P}_{u}|(\widehat{h}_{u,u}[i]+c_{u}[i])|^{2}\leq\lambda_{u},$
$\displaystyle\\!\\!\\!\mathbb{E}\\{\|\mathbf{v}_{b}[i]s_{b}[i]\|^{2}\\}\leq{\rm
P}_{b},\,\,\text{and}\,\,\,\mathbb{E}\\{|s_{u}[i]|^{2}\\}\leq{\rm P}_{u},$
$\displaystyle\\!\\!\\!\mathbf{v}_{b}[i]\in\mathbb{F}_{\rm TX}.$ (13)

The first and second constraints impose the RX RF chain saturation thresholds
$\lambda_{b}$ and $\lambda_{u}$ after analog cancellation at nodes $b$ and
$u$, respectively. These saturation thresholds ensure successful reception of
the training symbols and decoding of BS’s data symbols. The next two
constraints in (IV) refer to the nodes’ average transmit powers. The final
constraint enforces the predefined analog codebook for the BS beamformer.

Algorithm 1 Proposed FD massive MIMO SDDT Design

0: $\widehat{\mathbf{H}}_{b,b}[i]$, $\widehat{h}_{u,u}[i]$, ${\rm P}_{b}$,
${\rm P}_{u}$, $\widehat{\theta}_{\rm LoS}[i-1]$, and $M_{b}$.

0: $\mathbf{v}_{b}[i],\mathbf{c}_{b}[i],c_{u}[i],\mathbf{d}_{b}[i]$, and
$d_{u}[i]$.

1: Obtain the DL channel estimate $\widehat{\mathbf{h}}_{u,b}[i]$ using
$\widehat{\theta}_{\rm LoS}[i-1]$ as described in (11).

2: Obtain the analog beamformer
$\mathbf{v}_{b}[i]=\underset{\mathbf{v}\in\mathbb{F}_{\rm TX}}{\text{arg
max}}\frac{|\widehat{\mathbf{h}}_{u,b}^{\rm
T}[i]\mathbf{v}|^{2}}{\|\widehat{\mathbf{H}}_{b,b}[i]\mathbf{v}\|^{2}}$ using
exhaustive search.

3: for $n=1,2,\ldots,M_{b}-1$ do

4: Set analog SI canceller
$\mathbf{c}_{b}[i]=\begin{bmatrix}-[\widehat{\mathbf{H}}_{b,b}[i]\mathbf{v}_{b}[i]]_{1:n}\\\
\mathbf{0}_{(M_{b}-n:M_{b})}\end{bmatrix}$.

5: Set $c_{u}[i]=-\widehat{h}_{u,u}[i]$.

6: if ${\rm
P}_{b}|[(\widehat{\mathbf{H}}_{b,b}[i]\mathbf{v}_{b}[i]\\!+\\!\mathbf{c}_{b}[i])]_{(j,:)}|^{2}\leq\lambda_{b}\,\forall
j=1,\ldots,M_{b}$, and ${\rm
P}_{u}|(\widehat{h}_{u,u}[i]+c_{u}[i])|^{2}\leq\lambda_{u}$ then

7: Output $\mathbf{v}_{b}[i]$, $\mathbf{c}_{b}[i]$, $c_{u}[i]$,
$\mathbf{d}_{b}[i]=-(\widehat{\mathbf{H}}_{b,b}[i]\mathbf{v}_{b}[i]+\mathbf{c}_{b}[i])$,
$d_{u}[i]=-(\widehat{h}_{u,u}[i]+c_{u}[i])$, and terminate the iterations.

8: end if

9: end for

10: Set $\mathbf{c}_{b}[i]=-\widehat{\mathbf{H}}_{b,b}[i]\mathbf{v}_{b}[i]$
and $c_{u}[i]=-\widehat{h}_{u,u}[i]$.

11: if ${\rm
P}_{b}|[(\widehat{\mathbf{H}}_{b,b}[i]\mathbf{v}_{b}[i]\\!+\\!\mathbf{c}_{b}[i])]_{(j,:)}|^{2}\leq\lambda_{b}\,\forall
j=1,2,\ldots,M_{b}$, and ${\rm
P}_{u}|(\widehat{h}_{u,u}[i]+c_{u}[i])|^{2}\leq\lambda_{u}$ then

12: Output $\mathbf{v}_{b}[i]$, $\mathbf{c}_{b}[i]$, $c_{u}[i]$,
$\mathbf{d}_{b}[i]=-(\widehat{\mathbf{H}}_{b,b}[i]\mathbf{v}_{b}[i]+\mathbf{c}_{b}[i])$,
$d_{u}[i]=-(\widehat{h}_{u,u}[i]+c_{u}[i])$, and stop the algorithm.

13: else

14: Output that the $\mathbf{c}_{b}[i]$ realizations or $c_{u}[i]$ do not meet
the RX RF saturation constraints.

15: end if

The optimization problem in (IV) is a non-convex problem with coupling
variables, hence, quite difficult to tackle. In this work, we solve it
suboptimally using alternating optimization, leaving other possibilities for
future work. First, we find the BS analog beamformer $\mathbf{v}_{b}[i]$ via
the following problem:

$\begin{split}\mathbf{v}_{b}[i]=\underset{\mathbf{v}\in\mathbb{F}_{\rm
TX}}{\text{arg max}}\frac{|\widehat{\mathbf{h}}_{u,b}^{\rm
T}[i]\mathbf{v}|^{2}}{\|\widehat{\mathbf{H}}_{b,b}[i]\mathbf{v}\|^{2}}.\end{split}$
(14)

The exhaustive search for this problem can be easily implemented by a simple
look-up table given the estimated SI channel $\widehat{\mathbf{H}}_{b,b}[i]$
and the estimated DL channel $\widehat{\mathbf{h}}_{u,b}[i]$. Following the
analog SI canceller structure in [9] and using $\mathbf{v}_{b}[i]$, we next
seek the $N$-tap analog canceller $\mathbf{c}_{b}[i]$ with $1\leq N\leq M_{b}$
that satisfies the first threshold constraint. The single-tap canceller at
node $u$ is obtained as $c_{u}[i]=-\widehat{h}_{u,u}[i]$. To maximize the
signal-to-interference-plus-noise ratio, the digital cancellers
$\mathbf{d}_{b}$ and $d_{u}$ are set as the respective complementary residual
SI channels after analog SI cancellation. Our solution for the optimization
problem (IV) is summarized in Algorithm 1.

Figure 3: Error of LoS DoA estimation w.r.t the time slot index for the
$64\times 2$ FD massive MIMO node $b$ communicating with the mobile single-
antenna FD UE $u$ transmitting UL training symbols with $10$dBm transmit
power.

##  V Numerical Results

In this section, we present simulation results for the performance of the
proposed SDDT scheme for FD mmWave massive systems in comparison with HD
counterparts.

###  V-A Simulation Parameters

We perform an extensive waveform simulation following the FD massive MIMO
architecture illustrated in Fig. 1 when operating at mmWave frequencies, where
a $64\times 2$ FD massive MIMO node $b$ is communicating with a single-antenna
FD UE node $u$. We have assumed that the single TX RF chain at the BS node $b$
is connected to a ULA consisting of $N_{b}=64$ antenna elements. In contrast,
on the RX side, $M_{b}=2$ RX antennas are connected to their dedicated RX RF
chains. The UE node $u$ employs a non-ideal single-tap SI canceller, where the
FD massive MIMO BS node $b$ deploys an $N=2$ taps analog SI canceller [9]. The
multi-path UL and DL channels are simulated as mmWave channels at $28$ GHz
each with one LoS and $4$ nLoS channel paths. Considering the distance $d_{\rm
BS}=100m$ of the BS node $b$ from the UE, the pathloss of the both UL and DL
channels is assumed to be $100$dB with a $25$ dB Rician factor between the LoS
and nLoS paths [19]. In addition, the SI channels are modeled as Rician fading
channels with a $\kappa$-factor of $35$dB and pathloss $40$dB [3]. The RX
noise floors at all nodes were assumed to be $-100$dBm. To this end, the RXs
have an effective dynamic range of $60$dB provided by $14$-bit Analog-to-
Digital Converters (ADC) for a Peak-to-Average-Power-Ratio (PAPR) of $10$ dB.
Therefore, the residual SI power after analog SI cancellation at the input of
each RX RF chain has to be below $-40$dBm to avoid signal saturation. For the
considered FD massive MIMO architecture, we have assumed that the UE $u$ is
moving at a constant velocity of $120$km/h for a duration of $100$ time slots,
where each time slot is considered to be $T_{s}=10$msec with $L=400$ symbols.
For the BS analog beamformer, we have used a $6$-bit beam codebook based on
the Discrete Fourier Transform (DFT) matrix. We have used $1000$ independent
Monte Carlo simulation runs to calculate the performance of all considered DoA
estimation and data transmission designs.

Figure 4: MSE of the DoA estimation with respect to UL transmit power in dBm
for the $64\times 2$ FD massive MIMO node $b$ communicating with the single-
antenna FD UE $u$ transmitting training symbols in the UL direction.

###  V-B Compared HD Massive MIMO Designs

We compare the performance of the proposed FD-based SDDT scheme with three HD
DoA-assisted DL analog beam management techniques. First, we consider the “HD
with Initial DoA” case, where DoA is estimated using the UL training symbols
only in the first time slot, and for the rest of the $99$ time slots, the same
DoA is utilized for finding the DL analog beam. Secondly, we simulate the “HD
DoA in Each Slot” scheme, where, for every time slot, a fraction of the UL
training symbols is utilized for DoA estimation, and the rest of the symbols
are dedicated for data transmission with analog BF. Finally, we consider the
“HD DoA with Update” technique, where DL analog BF in each time slot is
performed using the DoA estimation of the previous time slot, unless more than
$3$dB SNR loss occurs in the received DL signal, which triggers the
initialization of a DoA-based analog beamformer update. In addition to the
latter HD cases, we have simulated the achievable DL rate for the proposed FD-
based SDDT scheme with ideal DoA estimation.

###  V-C DoA Estimation Error and Average DL Rate

In Fig. 3, the LoS DoA estimation error in radians is illustrated with respect
to the time slot index. We have considered that the FD massive MIMO node $b$
communicates with the mobile single-antenna FD UE $u$, while the latter sends
training symbols in the UL for DoA estimation with power $10$dBm. It is shown
in the figure that the “HD with Initial DoA” scheme provides higher DoA
estimation error with increasing time slot index. This happens because it only
estimates the DoA in the first time slot. As the UE moves, the initial DoA
becomes outdated, and hence, a DoA estimation as well as an analog beam update
are required. As illustrated in the inset plot, the “HD DoA with Update”
scheme performs an LoS DoA update after reaching approximately $0.1$rad
estimation error, while the “HD DoA in Each Slot” results in a steady error as
the estimation is performed in each time slot. It is, however, evident from
Fig. 3 that the proposed FD-based SDDT scheme with a $2$-tap analog canceller
provides substantially lower DoA estimation error compared to the HD cases
across all time slots for the considered $10$dBm UL transmit power. In Fig. 4,
we plot the Mean Squared Error (MSE) of the DoA estimation for all considered
schemes as a function of the UL transmit power in the range $0-20$dBm. It is
clear from the figure that the MSE of the DoA estimation reduces with
increasing UL transmit power. This happens because the received SNR at the BS
increases. It is also shown that the proposed scheme outperforms all
considered HD-based counterparts.

Figure 5: Achievable DL rate with respect to DL transmit power in dBm for the
$64\times 2$ FD massive MIMO node $b$ communicating with the mobile single-
antenna FD UE $u$ sending training symbols with $10$dBm transmit power.

Figure 5 illustrates the achievable DL rate of the proposed SDDT scheme as a
function of the DL transmit power, considering that the $64\times 2$ BS node
$b$ is transmitting data in the DL via analog BF to the constant-velocity
mobile UE node $u$, while receiving UL training symbols with $10$dBm transmit
power. It is demonstrated that the “HD with Initial DoA” provides the lowest
DL rate for all DL transmit power levels, as it suffers from the highest DoA
estimation error. It is also shown that, although the “HD DoA with Update”
scheme achieves higher DoA estimation error compared to the “HD DoA in Each
Slot” approach, as depicted in Fig. 3, it achieves higher DL rate across all
DL transmit powers. This is due to the fact that the later HD DoA-assisted
beam management scheme utilizes $10\%$ symbols in each time slot for DoA
estimation, whereas the “HD DoA with Update” approach is capable of sending
data symbols for the whole time slot when it doesn’t have to update the DoA
estimation. It is finally evident that the proposed scheme with $2$-tap analog
cancellation achieves a higher DL rate compared to all the HD-based schemes fo
all DL transmit powers. In particular, for the large DL transmit power of
$40$dB, the proposed scheme for FD mmWave massive MIMO systems results in
$120\%$ of the achievable DL rate with the best HD-based counterpart.

##  VI Conclusion

In this paper, we presented a novel DoA-assisted analog beam management scheme
for FD mmWave massive MIMO systems. We considered an FD massive MIMO BS with
an analog beamformer serving a mobile single-antenna FD user moving at a
constant velocity. By adopting the MUSIC DoA estimation technique as an
example, we presented a joint design of the DoA-assisted analog beamformer and
A/D SI cancellation at the BS node maximizing the DL rate. Our performance
evaluation results considering a realistic mmWave channel model demonstrated
the superior achievable rates of the proposed FD-based SDDT scheme. In future
work, the simultaneous DoA estimation and data transmission protocol will be
considered for multi-user FD massive MIMO systems with both A/D precoders and
combiners at all involved nodes.

## Acknowledgments

This work was partially funded by the National Science Foundation CAREER award
#1620902.

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