# A Unified Beamforming and A/D Self-Interference Cancellation Design for Full
Duplex MIMO Radios

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

In this paper, we focus on reduced complexity full duplex Multiple-Input
Multiple-Output (MIMO) systems and present a joint design of digital transmit
and receive beamforming with Analog and Digital (A/D) self-interference
cancellation. We capitalize on a recently proposed multi-tap analog canceller
architecture, whose number of taps does not scale with the number of
transceiver antennas, and consider practical transmitter impairments for the
full duplex operation. Particularly, transmitter IQ imbalance and nonlinear
power amplification are assumed via relevant realistic models. Aiming at
suppressing the residual linear and nonlinear self-interference signal below
the noise floor, we propose a novel digital self-interference cancellation
technique that is jointly designed with the configuration of the analog taps
and digital beamformers. Differently from the state of the art, we design
pilot-assisted estimation of all involved wireless channels. Our
representative Monte Carlo simulation results demonstrate that our unified
full duplex MIMO design exhibits higher self-interference cancellation
capability with less analog taps compared to available techniques, which
results in improved achievable rate and bit error performance.

###### Index Terms:

Analog and digital cancellation, beamforming, full duplex, impairments, MIMO,
optimization, self-interference.

##  I Introduction

Recent advances in Full Duplex (FD) communication technology prove the
potential of a twofold increase in spectral efficiency over the conventional
frequency- and time-division duplex systems through simultaneous uplink and
downlink communication in the same frequency resources [1, 2, 3, 4].
Exploitation of Multiple-Input Multiple-Output (MIMO) communication provides
an efficient system performance boost due to increasing the system’s spatial
degrees of freedom (DoF) offered by the plurality of Transmitter (TX) and
Receiver (RX) antennas [5, 6, 7, 8, 9, 10]. Thus, enabling FD in conjunction
with MIMO operation can achieve the demanding throughput requirements of fifth
Generation (5G), and beyond, wireless communication systems with limited
spectrum resources.

The simultaneous transmission and reception in FD systems results in practice
in an in-band Self-Interference (SI) signal at the RX side due to the limited
isolation between the TX and RX blocks. Since the propagation path for this
signal is shorter than any other signal-of-interest from a far-end
communicating node, the SI signal at RX is many times stronger [1]. Therefore,
SI cancellation techniques are required to suppress SI below the RX noise
floor. In FD MIMO systems, the suppression techniques are particularly
challenging due to higher SI components as a consequence of the increased
number of antennas. Conventional SI suppression techniques in Single-Input
Single-Output (SISO) systems combine propagation domain isolation, analog
domain suppression, and digital cancellation[2, 11]. Propagation domain
isolation is accomplished by pathloss or antenna directionality[12], whereas
analog domain suppression is achieved by subtracting a processed copy of the
TX signal from the RX inputs to avoid saturation of the RX Radio Frequency
(RF) chain[1, 13]. Analog SI cancellation in FD MIMO systems can be
implemented through SISO replication. However, the hardware requirements for
such an approach scale with the number of TX/RX antennas, rendering the
implementation of analog SI a core design bottleneck[14]. The authors in [5,
6] presented spatial suppression techniques that alleviate the need for analog
SI cancellation, which was replaced solely by digital TX/RX beamforming. In
[7], a joint design of multi-tap analog cancellation and TX/RX beamforming,
where the number of taps does not scale with the product of TX and RX antenna
elements, was proposed. This approach reduces the hardware complexity of
conventional FD MIMO designs [8, 9, 10] based on analog SI cancellation
through SISO replication.

In practical FD MIMO systems, there exist impairments contaminating the
signal-of-interest when it is traversing through the TX block. These include
IQ imbalances and Power Amplifier (PA) nonlinearities inserted in the
transceiver RF chain. Such impairments need to be accurately estimated in
order to be extracted from the residual signal after analog SI cancellation.
Conventionally in FD SISO systems, the latter nonlinear components are
suppressed in the digital domain. The effect of PA nonlinearities is analyzed
to formulate digital cancellation techniques in [2, 15], whereas the impact of
IQ imbalances and resulting image components is investigated in [11]. A
comprehensive digital cancellation approach for single-antenna FD SISO systems
considering both transmitter and receiver nonlinearities and IQ imbalances is
presented in[16]. However, in FD MIMO, their impact will be pronounced
especially when the TX power is high [8, 10].

In this paper, we present a joint design of digital TX/RX beamforming with
Analog and Digital (A/D) SI cancellation for practical FD MIMO systems with TX
IQ imbalances and PA nonlinearities. Building upon the reduced complexity
multi-tap analog canceller architecture of [7], we design a novel digital SI
cancellation technique that is jointly designed with the configuration of the
analog taps and digital TX/RX beamformers. Selected simulation results
showcase the interplay of the individual components of our proposed FD MIMO
approach when TX impairments are considered. It is also shown that our design
exhibits improved performance with less analog canceller taps compared to
available approaches.

Notation: Vectors and matrices are denoted by boldface lowercase and boldface
capital letters, respectively. The transpose, Hermitian transpose, and
conjugate of $\mathbf{A}$ are denoted by $\mathbf{A}^{\rm T}$,
$\mathbf{A}^{\rm H}$, and $\mathbf{A}^{*}$, respectively, and
$\det(\mathbf{A})$ is $\mathbf{A}$’s determinant, while $\mathbf{I}_{n}$
($n\geq 2$) is the $n\times n$ identity matrix and $\mathbf{0}_{m\times n}$
($m\geq 2$ and $n\geq 1$) represents the $m\times n$ matrix with all zeros.
$\|\mathbf{a}\|$ stands for the Euclidean norm of $\mathbf{a}$,
$\mathbf{a}^{\circ n}$ denotes the Hadamard power operation to a factor of n,
operand $\odot$ represents the Hadamard entry-wise product, ${\rm
col}\\{\mathbf{a}_{1},\mathbf{a}_{2},\ldots,\mathbf{a}_{n}\\}$ denotes a
column vector after vertically concatenating vectors
$\mathbf{a}_{1},\mathbf{a}_{2},\ldots,\mathbf{a}_{n}$, and ${\rm
diag}\\{\mathbf{a}\\}$ denotes a square diagonal matrix with $\mathbf{a}$’s
elements in its main diagonal. $[\mathbf{A}]_{i,j}$, $[\mathbf{A}]_{(i,:)}$,
and $[\mathbf{A}]_{(:,j)}$ represent $\mathbf{A}$’s $(i,j)$-th element, $i$-th
row, and $j$-th column, respectively, while $[\mathbf{a}]_{i}$ denotes the
$i$-th element of $\mathbf{a}$. $\mathbb{C}$ represents the complex number
set, $\mathbb{E}\\{\cdot\\}$ is the expectation operator, and $|\cdot|$
denotes the amplitude of a complex number.

##  II System and Signal Models

Figure 1: The considered three-node communication system and the proposed FD
MIMO architecture. The FD MIMO node $k$ incorporates processing blocks for
analog SI cancellation and digital TX/RX beamforming similar to [7], as well
as for digital cancellation of residual SI. All these blocks are jointly
designed for efficient FD operation. The half-duplex multi-antenna nodes $q$
and $m$ communicate with node $k$ in the downlink and uplink directions,
respectively.

The considered wireless communication system of Fig. 1 is comprised of a FD
MIMO node $k$ wishing to communicate concurrently with two half duplex multi-
antenna nodes: node $q$ in the DownLink (DL) and node $m$ in the UpLink (UL)
communication. The FD MIMO node $k$ is assumed to be equipped with $N_{k}$ TX
and $M_{k}$ RX antenna elements. Each TX antenna is attached to a dedicated RF
chain; similarly holds for the RX RF chain and their attached antennas. The
half duplex multi-antenna nodes $q$ and $m$ are assumed to have $M_{q}$ and
$N_{m}$ antennas, respectively, with each of their antennas connected to a
respective RF chain. All three nodes are considered capable of performing
digital BF; for simplicity, we assume hereinafter that digital TX/RX
beamforming at the focused FD MIMO node $k$ is realized with linear filters.
In particular, we assume that node $k$ makes use of the precoding matrix
$\mathbf{V}_{k}\in\mathbb{C}^{N_{k}\times d_{k}}$ for processing its symbol
vector $\mathbf{s}_{k}\in\mathbb{C}^{d_{k}\times 1}$ at baseband. The
dimension of $\mathbf{s}_{k}$ satisfies $d_{k}\leq\min\\{M_{q},N_{k}\\}$,
which complies with the available spatial degrees of freedom for the downlink
$M_{q}\times N_{k}$ MIMO channel. Similarly, node $m$ processes its symbol
vector $\mathbf{s}_{m}\in\mathbb{C}^{d_{m}\times 1}$ with a baseband precoding
matrix $\mathbf{V}_{m}\in\mathbb{C}^{N_{m}\times d_{m}}$, where
$d_{m}\leq\min\\{M_{k},N_{m}\\}$. Without loss of generality, we assume that
$\mathbf{V}_{k}$ and $\mathbf{V}_{m}$ have unit norm columns, and we use
notations
$\mathbf{x}_{k}\triangleq\mathbf{V}_{k}\mathbf{s}_{k}\in\mathbb{C}^{N_{k}\times
1}$ and
$\mathbf{x}_{m}\triangleq\mathbf{V}_{m}\mathbf{s}_{m}\in\mathbb{C}^{N_{m}\times
1}$ for the output signals of the TX baseband blocks at nodes $k$ and $m$,
respectively.

###  II-A TX Signal Model Incorporating Practical Impairments

As shown in node $k$ of Fig. 1, the $N_{k}$ baseband signals in
$\mathbf{x}_{k}$ are fed to the TX RF chains for upconversion and
amplification. In this section, we introduce a baseband equivalent model for
each of these chains incorporating IQ imbalances and PA nonlinearities. We
assume that all TX RF chains are identical. For simplification, we do not
consider such impairments for the TX of node $m$, which is instead assumed to
be linear111In general practical TXs, nonlinear distortion components are $30$
dB lower than the linear signal[2]. For typical UL pathloss of $110$ dB, these
components fall well below the RX noise floor. Thus, ignoring these nonlinear
components in the UL signal model will have a small impact on link
performance..

After entering the TX RF chains of node $k$, the baseband signal goes through
the IQ mixer for upconversion to the carrier frequency. As stated in [11], IQ
imbalances in practical mixers induce a mirror image of the original signal
with certain image attenuation. By denoting $[\mathbf{x}_{k}]_{i}$ as one
sample of the baseband signal at the input of the $i$-th TX RF chain of node
$k$, the IQ mixer output can be written as [11]

$\begin{split}[\mathbf{x}_{k}]_{i}^{\rm
IQ}\triangleq\mu_{1}[\mathbf{x}_{k}]_{i}+\mu_{2}[\mathbf{x}_{k}]_{i}^{*},\end{split}$
(1)

where $\mu_{1}\triangleq(1+ge^{-j\theta})/2$ and
$\mu_{2}\triangleq(1-ge^{j\theta})/2$ with $g$ and $\theta$ representing the
gain and phase imbalances, respectively. The Image Rejection Ratio, defined as
${\rm IRR}\triangleq\left|\mu_{1}/\mu_{2}\right|^{2}$, represents the strength
of the IQ induced conjugate term[17].

Before transmission, the upconverted signal is fed into the PA for
amplification, while satisfying the TX power constraint. Note that practical
PAs exhibit varying degrees of nonlinearity. For brevity, we consider a quasi
memoryless PA model inducing only an odd-order nonlinearity, as the even-power
harmonics lie out of band and will be cutoff by the RF low pass filter at RX
[11, 18]. For this PA model, the baseband equivalent of each of its output
samples is given using (1) as

$\begin{split}[\mathbf{x}_{k}]_{i}^{\rm
PA}\\!&\triangleq\\!g_{1,i}[\mathbf{x}_{k}]_{i}\\!+g_{2,i}[\mathbf{x}_{k}]_{i}^{*}\\!+g_{3,i}[\mathbf{x}_{k}]_{i}^{3}\\!+\\!g_{4,i}[\mathbf{x}_{k}]_{i}^{2}[\mathbf{x}_{k}]_{i}^{*}\\\
&\quad+g_{5,i}[\mathbf{x}_{k}]_{i}([\mathbf{x}_{k}]_{i}^{*})^{2}+g_{6,i}([\mathbf{x}_{k}]_{i}^{*})^{3},\end{split}$
(2)

where the six gain components are derived as

$\begin{split}g_{1,i}&\triangleq\mu_{1}\nu_{1},\;g_{2,i}\triangleq\mu_{2}\nu_{1},\;g_{3,i}\triangleq\mu^{2}_{1}\mu^{*}_{2}\nu_{3},\\\
g_{4,i}&\triangleq(2|\mu_{1}|^{2}\mu_{1}\\!+\\!|\mu_{2}|^{2}\mu_{1})\nu_{3},\\\
g_{5,i}&\triangleq(2|\mu_{1}|^{2}\mu_{2}\\!+\\!|\mu_{2}|^{2}\mu_{2})\nu_{3},\,g_{6,i}\triangleq\mu^{*}_{1}\mu^{2}_{2}\nu_{3},\end{split}$
(3)

where $\nu_{1}$ denotes the PA linear gain,
$\nu_{3}\triangleq{\nu_{1}}{/}{{(\rm IIP3)}^{2}}$ is the gain of the third
order nonlinear distortions, and ${\rm IIP3}$ is the third order Input-
referred Intercept Point of the PA[19]. We have considered up to third order
distortions, which represents the strongest nonlinearity at the PA output in
practice[11].

Based on (2) and after some algebraic manipulations, the baseband
representation of the impaired transmitted signal from the $N_{k}$ TX antennas
of FD MIMO node $k$ at the DL direction can be expressed as

$\begin{split}\mathbf{\widetilde{x}}_{k}&\triangleq\mathbf{G}_{1,k}\mathbf{x}_{k}+\mathbf{G}_{2,k}\mathbf{x}_{k}^{*}+\mathbf{G}_{3,k}\mathbf{x}_{k}^{\circ
3}+\mathbf{G}_{4,k}\mathbf{x}_{k}^{\circ 2}\odot\mathbf{x}_{k}^{*}\\\
&\quad+\mathbf{G}_{5,k}\mathbf{x}_{k}\odot(\mathbf{x}^{*}_{k})^{\circ
2}+\mathbf{G}_{6,k}(\mathbf{x}_{k}^{*})^{\circ 3}\\\
&=\mathbf{G}_{1,k}\mathbf{V}_{k}\mathbf{s}_{k}+\boldsymbol{\delta}_{k}=\mathbf{G}_{k}\boldsymbol{\psi}_{k},\end{split}$
(4)

where
$\mathbf{G}_{1,k}\triangleq\text{diag}\\{g_{1,1},g_{1,2},\dots,g_{1,N_{k}}\\}$
is the power allocation matrix of the linear components and
$\mathbf{G}_{\ell,k}\triangleq\text{diag}\\{g_{\ell,1},g_{\ell,2},\dots,g_{\ell,N_{k}}\\}$
for $\ell=2,3,\ldots,6$ are the coefficient matrices for the remaining
nonlinear components defined in similar way. In (4), we also introduce the
notation $\boldsymbol{\delta}_{k}\in\mathbb{C}^{N_{k}\times 1}$ for the
nonlinear and conjugate signal components, as well as the augmented power
allocation matrix $\mathbf{G}_{k}\in\mathbb{C}^{N_{k}\times 6N_{k}}$, and the
vertically arranged signal vector
$\boldsymbol{\psi}_{k}\in\mathbb{C}^{6N_{k}\times 1}$ including the image and
nonlinear components. The latter matrices are given by

$\begin{split}\mathbf{G}_{k}&\triangleq\left[\mathbf{G}_{1,k}\,\,\mathbf{G}_{2,k}\,\,\mathbf{G}_{3,k}\,\,\mathbf{G}_{4,k}\,\mathbf{G}_{5,k}\,\,\mathbf{G}_{6,k}\right],\\\
\boldsymbol{\psi}_{k}&\triangleq\text{col}\\{\mathbf{x}_{k},\mathbf{x}_{k}^{*},\mathbf{x}_{k}^{\circ
3},\mathbf{x}_{k}^{\circ
2}\odot\mathbf{x}_{k}^{*},\mathbf{x}_{k}\odot(\mathbf{{x}}^{*}_{k})^{\circ
2},(\mathbf{{x}}_{k}^{*})^{\circ 3}\\}.\end{split}$ (5)

The signal transmissions from nodes $k$ and $m$ are power limited to ${\rm
P}_{k}$ and ${\rm P}_{m}$, respectively. Specifically, the DL signal is such
that
$\mathbb{E}\\{\|\mathbf{G}_{1,k}\mathbf{V}_{k}\mathbf{s}_{k}+\boldsymbol{\delta}_{k}\|^{2}\\}\leq{\rm
P}_{k}$, whereas the UL signal is constrained as
$\mathbb{E}\\{\|\mathbf{G}_{1,m}\mathbf{V}_{m}\mathbf{s}_{m}\|^{2}\\}\leq{\rm
P}_{m}$ with $\mathbf{G}_{1,m}\in\mathbb{C}^{N_{m}\times N_{m}}$ representing
the diagonal (linear) power allocation matrix of the precoded signal
components.

###  II-B Baseband Model for the RX Signals

Following the definitions in Sec. II-A, the baseband received signal
$\mathbf{y}_{q}\in\mathbb{C}^{M_{q}\times 1}$ at node $q$ is mathematically
expressed as

$\mathbf{y}_{q}\triangleq\mathbf{H}_{q,k}\mathbf{G}_{1,k}\mathbf{V}_{k}\mathbf{s}_{k}+\mathbf{H}_{q,k}\boldsymbol{\delta}_{k}+\mathbf{H}_{q,m}\mathbf{G}_{1,m}\mathbf{V}_{m}\mathbf{s}_{m}+\mathbf{n}_{q},$
(6)

where $\mathbf{H}_{q,k}\in\mathbb{C}^{M_{q}\times N_{k}}$ is the DL channel
matrix, $\mathbf{H}_{q,m}\in\mathbb{C}^{M_{q}\times N_{m}}$ denotes the
channel matrix for inter-node interference (i.e., between nodes $q$ and $m$),
and $\mathbf{n}_{q}\in\mathbb{c}^{M_{q}\times 1}$ represents the additive
white Gaussian noise (AWGN) vector at node $q$ with covariance matrix
$\sigma_{q}^{2}\mathbf{I}_{M_{q}}$. Assuming no inter-node interference
between node $q$ and $m$ (i.e., $\mathbf{H}_{q,m}=\mathbf{0}_{q,m}$ in (6))
due to appropriate node scheduling[20, 21], the linearly processed estimated
symbol vector $\hat{\mathbf{s}}_{k}\in\mathbb{C}^{d_{k}\times 1}$ for
$\mathbf{s}_{k}$ at the half duplex multi-antenna node $q$ is expressed as

$\begin{split}\hat{\mathbf{s}}_{k}\triangleq\mathbf{U}_{q}\left(\mathbf{H}_{q,k}\mathbf{G}_{1,k}\mathbf{V}_{k}\mathbf{s}_{k}+\mathbf{H}_{q,k}\boldsymbol{\delta}_{k}+\mathbf{n}_{q}\right),\end{split}$
(7)

where $\mathbf{U}_{q}\in\mathbb{C}^{M_{q}\times d_{k}}$ denotes the optimum
combining matrix.

Upon signal reception at the FD MIMO node $k$, analog SI cancellation is first
applied to the signals received at its RX antenna elements before these
signals enter to the RX RF chains, as shown in Fig. 1. Similar to [7] we use
notation $\mathbf{C}_{k}\in\mathbb{C}^{M_{k}\times N_{k}}$ to denote the
signal processing realized by the multi-tap analog canceller. The baseband
representation for the $M_{k}\times 1$ complex-valued output signal of the
analog canceller at node $k$ can be expressed as

$\widetilde{\mathbf{y}}_{k}\triangleq\mathbf{C}_{k}\left(\mathbf{G}_{1,k}\mathbf{V}_{k}\mathbf{s}_{k}+\boldsymbol{\delta}_{k}\right).$
(8)

In order to suppress the residual SI after applying analog cancellation, we
further consider digital SI cancellation from the received signals at the
outputs of the RX RF chains. We note that for the practical case of channel
estimation and TX impairments considered in this paper, digital SI
cancellation is necessary, as will be shown in the simulation results. After
A/D SI cancellation, the digitally converted and downsampled received signals
are linearly processed in baseband by the combining matrix
$\mathbf{U}_{k}\in\mathbb{C}^{d_{m}\times M_{k}}$, which we assume having unit
norm columns. Then, the estimated symbol vector
$\hat{\mathbf{s}}_{m}\in\mathbb{C}^{d_{m}\times 1}$ for $\mathbf{s}_{m}$ at
the FD MIMO node $k$ is derived as

$\hat{\mathbf{s}}_{m}\triangleq\mathbf{U}_{k}\left(\widetilde{{\mathbf{y}}}_{k}+\mathbf{y}_{k}+\overline{\mathbf{y}}_{k}+\boldsymbol{\xi}_{k}+\mathbf{n}_{k}\right),$
(9)

where the complex-valued $M_{k}$-element vectors $\mathbf{y}_{k}$,
$\overline{\mathbf{y}}_{k}$, and $\boldsymbol{\xi}_{k}$ are the baseband
representations of the received signal of interest, received SI signal, and
digital SI cancellation signal, respectively, at node $k$. In addition,
$\mathbf{n}_{k}\in\mathbb{C}^{M_{k}\times 1}$ denotes the received AWGN vector
at node $k$ with covariance matrix $\sigma_{k}^{2}\mathbf{I}_{M_{k}}$. Vector
$\mathbf{y}_{k}$ including the signal of interest is given by

$\mathbf{y}_{k}\triangleq\mathbf{H}_{k,m}\mathbf{G}_{1,m}\mathbf{V}_{m}\mathbf{s}_{m},$
(10)

where $\mathbf{H}_{k,m}\in\mathbb{C}^{M_{k}\times N_{m}}$ is the UL channel
matrix (i.e., between nodes $k$ and $m$), while $\overline{\mathbf{y}}_{k}$
representing the received SI signal with the TX impairments is expressed as

$\overline{\mathbf{y}}_{k}\triangleq\mathbf{H}_{k,k}\left(\mathbf{G}_{1,k}\mathbf{V}_{k}\mathbf{s}_{k}+\boldsymbol{\delta}_{k}\right)$
(11)

with $\mathbf{H}_{k,k}\in\mathbb{C}^{M_{k}\times N_{k}}$ being the SI wireless
channel seen at the RX antennas of node $k$ due to its own DL transmission.
Signal $\boldsymbol{\xi}_{k}\in\mathbb{C}^{M_{k}\times 1}$ for cancelling the
residual SI signal in the digital domain will be described in the sequel.

##  III Proposed Unified FD MIMO Design

In this section, we describe the joint design of digital TX/RX beamforming
with A/D SI cancellation. We first extend the approach of [7] to the case of
pilot-estimated channels, which we then co-design with a novel digital
cancellation technique to further suppress the residual SI signal.

###  III-A Digital TX/RX Beamforming and Analog SI Cancellation

Suppose that the UL, DL, and SI wireless channels in the considered system of
Fig. 1 are estimated using pilot signals as $\widehat{\mathbf{H}}_{k,m}$,
$\widehat{\mathbf{H}}_{q,k}$, and $\widehat{\mathbf{H}}_{k,k}$, respectively.
Using these estimations and the representations for the analog canceller and
digital TX/RX beamformers, estimates for the achievable UL and DL rates can be
respectively calculated as

$\displaystyle\widehat{\mathcal{R}}_{\text{UL}}\\!$
$\displaystyle\triangleq\log_{2}\\!\left(\\!\det\\!\left(\mathbf{I}_{M_{k}}\\!+\\!\|\mathbf{U}_{k}\widehat{\mathbf{H}}_{k,m}\mathbf{G}_{1,m}\mathbf{V}_{m}\|^{2}{\widehat{\mathbf{W}}_{k}}^{-1}\right)\\!\right)\\!,$
$\displaystyle\widehat{\mathcal{R}}_{\text{DL}}\\!$
$\displaystyle\triangleq\log_{2}\left(\det\left(\mathbf{I}_{M_{q}}\\!+\\!\|\mathbf{U}_{q}\widehat{\mathbf{H}}_{q,k}\mathbf{G}_{1,k}\mathbf{V}_{k}\|^{2}{\widehat{\mathbf{W}}_{q}}^{-1}\right)\right)\\!,$
(12)

where $\widehat{\mathbf{W}}_{k}$ and $\widehat{\mathbf{W}}_{q}$ denote the
estimated covariances matrices of interference plus noise at multi-antenna
nodes $k$ and $q$, respectively, which can be computed as

$\begin{split}\widehat{\mathbf{W}}_{k}\\!&\triangleq\\!\|\mathbf{U}_{k}(\widehat{\mathbf{H}}_{k,k}\\!+\\!\mathbf{C}_{k})\\!\left(\mathbf{G}_{1,k}\mathbf{V}_{k}+\boldsymbol{\delta}_{k}\right)\|^{2}\\!+\\!\sigma_{k}^{2}\|\mathbf{U}_{k}\|^{2},\\\
\widehat{\mathbf{W}}_{q}\\!&\triangleq\\!\|\mathbf{U}_{q}\widehat{\mathbf{H}}_{q,k}\boldsymbol{\delta}_{k}\|^{2}\\!+\\!\sigma_{q}^{2}\|\mathbf{U}_{q}\|^{2}.\end{split}$
(13)

The latter expressions have been obtained from (7) and (9) assuming estimation
of the TX impairments at RXs and that
$\breve{\mathbf{y}}_{k}=\mathbf{0}_{M_{k}\times 1}$ (i.e., yet no digital SI
cancellation).

Algorithm 1 Joint Digital Beamforming and SI Cancellation

0: $\mathbf{C}_{k}$, ${\rm P}_{k}$, ${\rm P}_{m}$, $\mu_{1}$, $\nu_{1}$,
$\mu_{2}$, $\nu_{3}$, $\sigma_{q}^{2}$, and $\sigma_{k}^{2}$.

0: $\mathbf{V}_{k}$, $\mathbf{G}_{1,k}$, $\mathbf{U}_{k}$, $\mathbf{V}_{m}$,
$\mathbf{G}_{1,m}$, and $\mathbf{U}_{q}$.

1: Obtain wireless channel estimates
$\widehat{\mathbf{H}}_{k,k},\widehat{\mathbf{H}}_{k,m}$, and
$\widehat{\mathbf{H}}_{q,k}$ using pilot signals.

2: Run [7, Algorithm 1] using $\widehat{\mathbf{H}}_{k,k}$ and
$\widehat{\mathbf{H}}_{q,k}$, and perform column normalization to the result
to obtain $\mathbf{V}_{k}$. Store
$\mathbf{F}_{k}\in\mathbb{C}^{N_{k}\times\alpha}$ with
$1\leq\alpha\leq\min\\{M_{q},N_{k}\\}$ during this computation for step $5$.

3: Set $[\mathbf{G}_{1,k}]_{(i,i)}=\sqrt{P_{k}/N_{k}},\forall
i=1,2,\dots,N_{k}$ and obtain $\mathbf{G}_{j,k}$ for $j=2,3,\ldots,6$ using
(3).

4: Set $\mathbf{V}_{m}$ as the right-singular vectors of
$\widehat{\mathbf{H}}_{k,m}$ and set
$[\mathbf{G}_{1,m}]_{(i,i)}=\sqrt{P_{m}/N_{m}},\forall i=1,2,\dots,N_{m}$.

5: Set $\mathbf{U}_{q}$ as the left-singular vectors of the effective DL
channel estimation matrix $\widehat{\mathbf{H}}_{q,k}\mathbf{F}_{k}$.

6: Transmit the $L$ precoded pilot symbols $\mathbf{X}_{k}$ and
$\mathbf{X}_{m}$.

7: Measure the residual SI signal $\mathbf{J}_{k}$ in (15) at the outputs of
the RX RF chains of node $k$.

8: Obtain estimate $\widehat{\breve{\mathbf{H}}}_{k,k}$ using (18) and
$\boldsymbol{\widehat{\Omega}}=\widehat{\breve{\mathbf{H}}}_{k,k}\mathbf{R}$,
and set the digital SI cancellation signal as $\boldsymbol{\xi}_{k}$ in (19).

Extending the design approach of [7], we focus on the estimated achievable FD
rate defined as the sum of $\widehat{\mathcal{R}}_{\text{UL}}$ and
$\widehat{\mathcal{R}}_{\text{DL}}$, and formulate the following general
optimization problem for the joint design of the $N$-tap ($0\leq
N<N_{k}M_{k}$) analog SI canceller and the digital TX/RX beamformers:

$\displaystyle\underset{\begin{subarray}{c}\mathbf{C}_{k},\mathbf{V}_{k},\mathbf{U}_{k},\\\
\mathbf{G}_{1,k},\mathbf{G}_{1,m},\\\
\mathbf{V}_{m},\mathbf{U}_{q}\end{subarray}}{\text{max}}$
$\displaystyle\widehat{\mathcal{R}}_{\text{DL}}+\widehat{\mathcal{R}}_{\text{UL}}$
s.t $\displaystyle\mathbf{C}_{k}\,\,\text{as in}\,\,[7,{\rm eq.(C2)}],$
$\displaystyle\mathbb{E}\\{\|\mathbf{G}_{1,k}\mathbf{V}_{k}\mathbf{s}_{k}+\boldsymbol{\delta}_{k}\|^{2}\\}\leq{\rm
P}_{k},$
$\displaystyle\mathbb{E}\\{\|\mathbf{G}_{1,m}\mathbf{V}_{m}\mathbf{s}_{m}\|^{2}\\}\leq{\rm
P}_{m},$ (14)
$\displaystyle\mathbf{G}_{1,k},\mathbf{G}_{1,m}:\,\text{real}\,\,\text{diagonal}\,\,\text{matrices},$
$\displaystyle\mathbf{V}_{k},\mathbf{U}_{k},\mathbf{V}_{m},\mathbf{U}_{q}:\,\text{unit}\,\,\text{norm}\,\,\text{columns},$
$\displaystyle\|(\widehat{\mathbf{H}}_{k,k}+\mathbf{C}_{k})(\mathbf{G}_{1,k}\mathbf{V}_{k}\mathbf{s}_{k}+\boldsymbol{\delta}_{k})\|^{2}<\lambda_{\rm
A},$

where $\lambda_{\rm A}$ denotes the real-valued threshold residual power level
after analog SI cancellation. This threshold depends on the dynamic range of
the Analog-to-Digital Converter (ADC) and determines the saturation of the RF
chains. A solution of (III-A) combined with a novel digital SI cancellation
technique will be presented in the sequel.

###  III-B Proposed Digital SI Cancellation

The solution of (III-A) determines the power of the SI signal after analog
cancellation. As shown in Fig. 1, we propose an additional digital
cancellation block to boost SI cancellation in the digital domain. This block
performs complementary to analog canceller ensuring SI suppression below RX
noise floor.

To design digital SI cancellation, we use the baseband precoded samples at the
TX of node $k$ to reconstruct the SI components, and then cancel them out at
the output of each RX RF chain of the node. Supposing the $L$ samples
$\mathbf{X}_{k}\triangleq\mathbf{V}_{k}\mathbf{S}_{k}\in\mathbb{C}^{N_{k}\times
L}$ (and
$\mathbf{X}_{m}\triangleq\mathbf{V}_{m}\mathbf{X}_{m}\in\mathbb{C}^{N_{m}\times
L}$ with perfect synchronization), the residual SI signal
$\mathbf{J}_{k}\in\mathbb{C}^{M_{k}\times L}$ after analog cancellation is
derived using (4) and (8)-(11) as

$\mathbf{J}_{k}\triangleq\breve{\mathbf{H}}_{k,k}\boldsymbol{\Psi}_{k}+\mathbf{H}_{k,m}\mathbf{G}_{1,m}\mathbf{X}_{m}+\mathbf{N}_{k},$
(15)

where $\breve{\mathbf{H}}_{k,k}\in\mathbb{C}^{M_{k}\times 6N_{k}}$ is the
residual SI matrix defined as
$\breve{\mathbf{H}}_{k,k}\triangleq\left(\mathbf{H}_{k,k}+\mathbf{C}_{k}\right)\mathbf{G}_{k}$
and $\boldsymbol{\Psi}_{k}\in\mathbb{C}^{6N_{k}\times L}$ contains the linear,
image, and nonlinear SI signal components. The latter definitions have been
obtained from (5) for $L$ samples.

The objective of the digital canceller is to estimate
$\breve{\mathbf{H}}_{k,k}$ and then subtract it at the ADC outputs of each RX
RF chain. Making use of the SI samples in (15) and the notation
$\boldsymbol{\Theta}\in\mathbb{C}^{M_{k}\times 6N_{k}}$, the Least Squares
(LS) estimation for $\breve{\mathbf{H}}_{k,k}$ minimizing the power of the
error matrix can be expressed as

$\widehat{\breve{\mathbf{H}}}_{k,k}\triangleq\operatorname*{arg\,min}_{\boldsymbol{\Theta}}\|\mathbf{J}_{k}-\boldsymbol{\Theta}\mathbf{\Psi}_{k}\|^{2}.$
(16)

This LS problem would have closed form solution if $\boldsymbol{\Psi}_{k}$ had
independent rows. However, $\boldsymbol{\Psi}_{k}$’s rows are high order
polynomials of linear and conjugate SI samples, as well as their interaction
products. Therefore, its rows are correlated which results in high level of
multi-collinearity. To tackle this problem, we propose orthogonalization of
$\boldsymbol{\Psi}_{k}$ using RQ decomposition. We use traditional Gram-
Schmidt orthogonalization to obtain $\boldsymbol{\Psi}_{k}=\mathbf{RQ}$, where
$\mathbf{Q}\in\mathbb{C}^{L\times L}$ has orthogonal rows and
$\mathbf{R}\in\mathbb{C}^{6N_{k}\times L}$ is a lower triangular matrix with
ones on the main diagonal. Using the latter expression decomposition, (16) can
be re-expressed as

$\begin{split}\widehat{\breve{\mathbf{H}}}_{k,k}=\operatorname*{arg\,min}_{\boldsymbol{\Theta}}\|\mathbf{J}_{k}-\boldsymbol{\Theta}\mathbf{RQ}\|^{2}.\end{split}$
(17)

Setting $\boldsymbol{\Omega}\triangleq\boldsymbol{\Theta}\mathbf{R}$ in (17)
results in a typical LS problem possessing the following closed form solution:

$\begin{split}\boldsymbol{\widehat{\Omega}}\triangleq&\operatorname*{arg\,min}_{\boldsymbol{\Omega}}\|{\mathbf{J}_{k}-\boldsymbol{\Omega}\mathbf{Q}}\|^{2}=\mathbf{J}_{k}\mathbf{Q}^{\rm
H}(\mathbf{QQ}^{\rm H})^{-1}.\end{split}$ (18)

Since
$\boldsymbol{\widehat{\Omega}}=\widehat{\breve{\mathbf{H}}}_{k,k}\mathbf{R}$,
using back substitution yields $\widehat{\breve{\mathbf{H}}}_{k,k}$.

At the FD MIMO node $k$, the digital SI cancellation block operates
complementary to analog SI cancellation and digital TX/RX beamforming. It
performs the estimation of $\breve{\mathbf{H}}_{k,k}$ as described in
(15)–(18) to derive the digital signal $\boldsymbol{\xi}_{k}$ in (9) as

$\begin{split}\boldsymbol{\xi}_{k}=-\widehat{\breve{\mathbf{H}}}_{k,k}\boldsymbol{\psi}_{k},\end{split}$
(19)

which is applied to the received signals after the ADCs. Our unified design of
digital TX/RX beamforming with A/D SI cancellation includes the following two
steps. We first find the $N$-tap analog SI canceller matrix $\mathbf{C}_{k}$
as described in [7, Sec. III], and then obtain the digital TX/RX beamformers
and digital SI cancellation signal as summarized in Algorithm 1.

##  IV Simulation Results and Discussion

In this section, we present performance evaluation comparisons among our
proposed FD MIMO design and relevant ones intended for the communication
scenario depicted in Fig. 1.

TABLE I: Simulation Parameters

Parameter |   Value
---|---
Signal Bandwidth |  1.4 MHz
Constellation  |  16-QAM
No. of Subcarriers  |  256
No. of Data Subcarriers  |  234
Cyclic Prefix Length  |  64
TX Power  |  20-40 dBm

  Parameter   Value  Node $k$ Noise Floor   $-$110.0
dBm Node $q$ Noise Floor   $-$90.0 dBm PA IIP3   15
dBm IRR   30 dB No. of ADC Bits   14 PAPR
 10 dB

Figure 2: Probability of saturation of any of the RX RF chains at the FD MIMO
node $k$ after analog SI cancellation versus the TX power in dBm.

###  IV-A Simulation Parameters

We perform an extensive waveform simulation following the FD MIMO architecture
illustrated in Fig. 1 and using baseband equivalent modeling of practical
transceiver components including IQ imbalance and third-order PA
nonlinearities [22]. We have considered $N_{k}=M_{k}=M_{q}=4$ for the TX and
RX antennas at the FD node $k$ and the RX antennas of the half duplex node
$q$. For the half duplex node $m$, we have assumed single symbol stream with
$\mathbf{V}_{m}=N_{m}=d_{m}=1$. Both DL and UL channels are assumed as block
Rayleigh fading channel with a pathloss of $110$ dB. The SI channel at the FD
MIMO node $k$ is simulated as a Rician fading one with a $K$-factor of $35$ dB
and pathloss of $40$ dB [4]. All involved wireless channels are estimated
using pilot symbols before the symbol data communication[23]. The generated
waveforms were considered to be Orthogonal Frequency Division Multiplexing
(OFDM) signals with a bandwidth of $1.4$ MHz, which is a supported bandwidth
for LTE. The additional parameters of the waveforms along with other system
level parameters are shown in Table I. In order to incorporate TX impairments
at the FD MIMO node $k$, the TX IRR was considered to be $30$ dB, which
implies that the image signal power is $30$ dB lower than the linear SI signal
[11]. The quasi memoryless nonlinear PAs are assumed to have an IIP3 value of
$15$ dBm [22]. Each ADC at the RX RF chains of node $k$ is considered to have
$14$-bit resolution with an effective dynamic range of $62.24$ dB for a Peak-
to-Average-Power-Ratio (PAPR) of $10$ dB [24]. Therefore, the residual SI
power after analog cancellation has to be below $-47.76$ dBm to avoid
saturation of each RX RF chain. Furthermore, non-ideal multi-tap analog
canceller is considered with steps of $0.02$ dB for attenuation and
$0.13^{\circ}$ for phase as in [7, 14]. We have used $1000$ Monte Carlo
simulation runs ($1000$ independent set of channels) to calculate the
performance of all considered designs. In each run, we have performed pilot-
assisted estimation of all involved wireless channels and considered $200$
OFDM symbols transmitted from both nodes $k$ and $m$ to emulate a packet-based
FD MIMO communication system.

Figure 3: Average UL and FD rates with respect to the TX power in dBm for the
proposed and the state-of-the-art FD MIMO designs.

###  IV-B Compared FD MIMO Designs

We compare the performance of our unified design of digital TX/RX beamforming
and A/D SI cancellation with that of spatial suppression design presented in
[5]. For the spatial suppression approach we have used a $16$-tap time domain
analog SI canceller, which corresponds to the highest hardware complexity
canceller for the considered number of transceiver antennas. In our unified
approach we have incorporated both digital SI cancellation and an analog
canceller with $12$ taps (i.e., $25$% less hardware for the canceller compared
with the spatial suppression design), where the taps were placed in orderly
column-by-column in $\mathbf{C}_{k}$. For the latter we use the term “Proposed
Unified Scheme with $12$ taps” and for the former the term “Spatial Scheme [5]
with $16$ taps” in the figures that follow. We have also considered the
reduced complexity design of [7] with only $8$ taps for the analog canceller
(i.e., $50$% less hardware for the canceller compared with the spatial
suppression design). This design performs joint analog SI cancellation with
digital TX/RX beamforming, but does not include digital SI cancellation and
transceiver RF chain impairments. We term this scheme as “Joint Scheme [7]
with $8$ taps” in the figures.

We also present performance comparisons among our proposed digital SI
cancellation technique and the widely linear digital cancellation proposed in
[11], as well as the nonlinear one designed in [2]. Both latter methods were
extended to the MIMO case since they were initially proposed for SISO systems.
In the results that follow we have computed the Bit Error Rate (BER) with
$16$-ary Quadrature Amplitude Modulation (QAM) as well as the achievable UL
and DL rates. These rates weres obtained from (III-A) and (13) after
substituting the estimated channels with the actual ones.

Figure 4: Average BER with $16$-QAM at the FD MIMO node $k$ versus the TX
power in dBm after cascading A/D SI cancellation designs.

###  IV-C SI Mitigation Capability, BER, and Achievable Rates

We start in Fig. 2 by comparing the SI mitigation capability of the three
considered designs as a function of the TX power ${\rm P}_{k}$ in dBm. Recall
that the residual SI after analog cancellation at the FD MIMO node $k$ should
be below $-47.76$ dBm to avoid saturation of each RX RF chain. As shown, both
the proposed scheme with $12$ taps and the spatial suppression approach with
$16$ taps provide residual SI signal power below the threshold for the whole
TX power range. However, this does not happen for the scheme of [7] with $8$
taps where digital SI cancellation is not considered. For TX power larger than
$30$ dBm with this scheme, there exists non negligible probability that the
residual SI power level is above the critical threshold $-47.76$ dBm.

The ergodic UL and FD rates as functions of the TX powers ${\rm P}_{k}$ and
${\rm P}_{m}$ in dBm are depicted in Fig. 3. It is evident that both schemes
[5] and [7] exhibit poor UL rates due to their inability to cancel the
residual nonlinear and image SI signal components. However, our proposed
unified scheme adequately suppresses all nonlinear SI components yielding
increased UL performance. Additionally, the proposed scheme significantly
outperforms both other in terms of achievable FD rate. In Fig. 4, we plot the
BER with $16$-QAM versus ${\rm P}_{k}$ and ${\rm P}_{m}$ in dBm for both the
proposed scheme and the scheme of [7] with $12$\- and $8$-tap analog SI
cancellation, respectively. For both schemes we have considered three
different digital SI cancellers: the proposed one, the widely linear [11], and
the nonlinear [2] cancellers. Recall that [7] was proposed without digital SI
cancellation which we herein incorporate. It is shown that our proposed
cancellation provides the best performance especially in high TX powers. More
importantly, it is demonstrated that, even when the proposed digital SI
cancellation is adopted at both schemes, the joint design of all blocks
(digital TX/RX beamformers and A/D SI cancellation) is necessary for achieving
the best performance.

##  V Conclusion and Future Work

In this paper, we have presented a unified design of digital TX/RX beamforming
with A/D SI cancellation for practical FD MIMO systems with TX IQ imbalances
and PA nonlinearities. To suppress the residual linear SI signal along with
its conjugate and nonlinear components, we proposed a novel digital SI
cancellation technique that is jointly designed with the configuration of the
multiple taps of the analog canceller and the digital TX/RX beamformers. Our
selected performance evaluation comparisons with state-of-the-art designs
demonstrated that our proposed approach is capable of achieving higher
achievable rates and BER performance with reduced hardware complexity for
analog SI cancellation. For future work, we intend to extend our unified
design to wideband channels incorporating also practical RX impairments.

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