# Performance Analysis of Self-Interference Cancellation in Full-Duplex
Massive MIMO Systems: Subtraction versus Spatial Suppression

Soo-Min Kim, Yeon-Geun Lim,
Linglong Dai, and Chan-Byoung Chae This research was supported by IITP grant
funded by the Korea government (2018-0-00923, 2021-0-00486). An earlier
version of this paper was presented, in part, at the IEEE Global
Telecommunications Conference (GLOBECOM) in 2016 [1].S.-M. Kim and C.-B. Chae
are with Yonsei University, Korea. Email: {sm.kim<EMAIL_ADDRESS>Y.-G.
Lim is with Samsung Electronics, Korea. Email<EMAIL_ADDRESS>L. Dai is
with Tsinghua University, Beijing 100084, China. Email<EMAIL_ADDRESS>S.-M. Kim and Y.-G. Lim are co-1st authors.

###### Abstract

Massive multiple-input multiple-output (MIMO) and full-duplex (FD) are
promising candidates for achieving the spectral efficiency to meet the needs
of 5G communications. One essential key to realizing practical FD massive MIMO
systems is how to effectively mitigate the self-interference (SI).
Conventionally, however, the performance comparison of different SI methods by
reflecting the actual channel characteristics was insufficient in the
literature. Accordingly, this paper presents a performance analysis of SI
cancellation (SIC) methods in FD massive MIMO systems. Analytical and
numerical results confirm that, in an imperfect channel-estimation case, the
ergodic rates performance of the spatial suppression in the uplink outperforms
those of the SI subtraction, due to the correlation between the precoder and
the estimation error of the SI channel. In addition, we discuss which method
performs better under different given system constraints such as uplink and
downlink sum rates, the total transmit power, and the power scaling law.

###### Index Terms:

Massive MIMO, full-duplex, self-interference cancellation, and spatial
suppression.

##  I Introduction

The fifth-generation (5G) communication systems can support various types of
services, including enhanced mobile broadband (eMBB), massive machine-type
communication, and ultra-reliable low-latency communication [2, 3]. As mobile
data traffic is expected to increase, eMBB has become a crucial part of 5G
communications. To support the tremendous data traffic of 5G, researchers
should jointly consider key issues such as spectral efficiency (bps/Hz) and
spectrum extension (Hz/cell). As researchers developed key issues over the
past few decades, massive multiple-input multiple-output (MIMO) technology has
been suggested as a candidate to achieve high spectral efficiency. At the same
time, researchers [4, 5] intending to double the spectrum extension without
shifting the system operating frequency band have studied full-duplex (FD)
transmission.

To maximize spectral efficiency up to tens of bps/Hz and conserve energy, the
researchers in [6, 7, 8, 9, 10] have proposed massive MIMO with tens of
antennas as a core technology of 5G. They considered using simple linear
precoders and filters to mitigate the interferences in massive MIMO systems.
One key challenge in commercializing massive MIMO systems is the estimation of
the high-dimensional MIMO channel matrix. This paper focuses on the channel
estimation based on practical channel characteristics [7, 8, 9]. Specifically,
the authors in [7, 8] studied the downlink performance of the maximum ratio
transmission (MRT) and zero-forcing (ZF) precoders while assuming either
perfect or imperfect channel estimation at the base station (BS). The authors
in [9] investigated the uplink performance of the maximum ratio combining
(MRC), ZF, and minimum mean square error (MMSE) filters with perfect or
imperfect channel estimation.

Another relevant technology is FD transmission, which, compared to half-duplex
transmission, can double the spectrum efficiency [5, 11, 12] and reduce
transmission latency [13]. In contrast to half-duplex transmission, self-
interference (SI), which is a transmitted downlink signal but is directly
received at the BS when receiving an uplink signal, should be mitigated. The
level of SI is usually higher than that of the receiver noise floor, and thus,
any FD system requires SI cancellation (SIC) in both the analog and digital
domains [14]. There are typically two types of classical SIC methods in a
digital-domain, that is, SI subtraction and spatial suppression. Specifically,
SI subtraction subtracts replicated interference signal from the input signal,
while spatial suppression applies multi-antenna techniques that utilize extra
degrees of freedom offered by spatial dimensions for linear receive and
transmit filtering (It is now considered in the form of null-space projection
(NSP) [15]). From the digital-domain SIC perspective, the authors in [16]
compared SI subtraction and spatial suppression in a MIMO relay system with a
small number of antennas (up to four antennas) at the BS and user. They showed
that the performance of the spatial suppression was depended on the rank of
the SI channel. The authors paid little attention to the performance analysis
of practical massive MIMO systems, such as user rate and uplink/downlink
performance, because they focused on introducing advantages and disadvantages
for each scheme according to the number of antennas and the rank of the
channel. In a practical small MIMO system, SI subtraction is a widely
considered SIC technique owing to the limitation of the requirement of
additional antennas. The researchers in [17, 18, 19, 20] prototyped SI
subtraction on a software-defined radio platform in real time. They combined a
dual-polarization antenna-based analog cancellation with SI subtraction and
realized a spectral efficiency 1.9 times higher than that of the half-duplex
transmission. Nevertheless, spatial suppression has received limited attention
in small MIMO systems.

As aforementioned, in 5G and beyond 5G systems, integrated techniques are
required to efficiently fulfill the requirements of eMBB. In particular, we
should integrate spectrum extension technologies with massive MIMO to enhance
both the data rate of the network and each user. Conventional massive MIMO
systems assume a small number of antennas at the users (for example, a single
antenna) with a per-user rate constraint, which is limited by the used
modulation technologies and allocated bandwidth. In addition, massive MIMO
systems are typically launched in the sub-6-GHz range, whereas hybrid
beamforming technologies are considered in the millimeter-wave range [21].
Therefore, FD is a potential solution to the spectrum extension of massive
MIMO systems. In both [22] and [23], the researchers investigated asymptotic
ergodic sum rates and channel estimation methods in FD-assisted massive MIMO
systems. They considered the condition of specific system constraints wherein
the performance of an FD system outperforms that of a half-duplex system. The
authors in [24] proposed spatial suppression-based algorithms such as extended
ZF and extended MMSE precoders. As the conventional massive MIMO system serves
a relatively smaller number of users than the number of antennas at the BS,
the extended precoders can effectively mitigate SI by utilizing the
dimensional feasibility in the spatial domain. As an interference-free channel
was assumed in [24], the analysis is not general. The researchers in [25]
analyzed the ratio of transmit antennas and receive antennas in terms of the
numbers of both the downlink and uplink users. The authors in [26] utilized a
spatial suppression precoder based on singular-value decomposition and
proposed an uplink power control mechanism for cellular systems using a
Poisson point process model. Furthermore, the authors in [27] showed the
performance trade-off between the degree of freedom and the SI mitigation
level, while the authors in [28] presented a combination of spatial
suppression and an analog domain cancellation solution that can relax the
requirement of the number of antennas and reduce the complexity of SI
cancellation. They demonstrated the performance analysis and feasibility of
the spatial suppression method under imperfect channel estimation. However,
there is insufficient explanation of whether SI suppression is optimal in the
considered environments.

In this paper, motivated by the potential of utilizing spatial suppression in
massive MIMO systems but the lack of prior studies on its advantages compared
to SI subtraction, we compared these two SIC methods in the digital domain.
Prior works typically model SI channel coefficients as simple independent
identically distributed (i.i.d.) complex Gaussian random variables. In this
prior channel model, the researchers assumed that the uplink performances of
both SIC methods were the same [24, 25]. Thus, SI subtraction was always a
better choice, because spatial suppression requires additional spatial
resources at the downlink, which causes a difference in the number of antennas
required to obtain a diversity gain for the downlink. In a practical channel,
however, an SI channel would be highly correlated because of the higher power
in the direct path. Therefore, in the imperfect channel, spatial suppression
can be better for the uplink, depending on the magnitude of the channel
estimation error. From this point of view, we analyzed the performance
according to the diversity gain, number of antennas, and channel estimation
error by performing the perfect channel analysis together. Motivated by [16],
in small FD MIMO systems, SI subtraction is better because the diversity gain
for the downlink is limited because there are fewer antennas. However, in FD
massive MIMO systems, because this limit is weaker, spatial suppression may be
better. We mathematically prove this content accurately. In addition, the
considered small FD MIMO system is not sufficient to meet the requirements of
massive MIMO systems in beyond 5G. This paper presents a comparison of the two
SIC methods with the following considerations in a system: i) channel
estimation error, ii) spatial correlation, and iii) asymmetric data traffic
with a non-full buffer, and power scaling law at the downlink. Based on the
analysis and numerical results, we determined which method is better in a
certain scenario. In our prior work [1], two methods were compared in perfect
and imperfect SI channel conditions, particularly in uncorrelated conditions.
Furthermore, the two methods were compared through mathematical analysis and
proofs in correlated conditions. The performance was also evaluated by
considering the practical system constraints mentioned above. Through these
analyses, results were obtained using a system-level simulator based on a 3D
ray-tracing tool to obtain the real SI channel. The additive quantization
noise model (AQNM) was also considered as a channel estimation error model to
design the system more practically. Specifically, the main contributions of
this study are summarized as follows.

  * •

Performance analysis with imperfect channel estimation: First, we provide the
performance analysis to show that the uplink performance of spatial
suppression is better than that of SI subtraction with imperfect channel
estimation. The channel estimation error would be high, particularly in
practical full-digital MIMO systems that usually use analog-to-digital
converters (ADCs) with few bits (for example, 4-5 bits) for a cost-effective
design [29, 30, 31]. The SIC algorithms having improved performance with
imperfect channel estimation are yet to be determined. We also derive the
achievable ergodic sum rates of the two SIC methods, then we show that the
uplink rate of spatial suppression is better than that of SI subtraction with
an imperfect channel estimation error.

  * •

Performance analysis with spatial correlation: As noted above, SI channels
would be highly correlated. We prove this baseline channel condition by
measuring the channel parameters with a 3D ray-tracing tool. Next, we
investigate the impact of spatial correlation on the expected SI power. Based
on this analysis, we conclude that the uplink rate of spatial suppression
improves beyond that of SI subtraction as the spatial correlation increases.
Notably, in the conventional analysis, these correlation terms of the SI
channel are assumed as independent conditions [24, 25].

  * •

Performance comparison with system constraints: Based on our analysis, we
compare the two SIC methods when the system constraints are considered. These
constraints include the ratio of downlink and uplink traffic, ratio of
transmit and receive RF chains, transmit power at the BS, and power scaling
law at the BS. Because of the performance trade-off between these two SIC
schemes in a certain link direction, our analysis shows that each method
exhibits different performances for different system constraints. These
comparisons provide insights into the SIC selection criterion for certain
system requirements.

The remainder of this paper is organized as follows. In Section II, we
introduce the system model and problem statement with respect to
precoding/receive combining and SIC methods. In Section III, we present
certain mathematical motivations and preliminaries that are useful for
analysis. Then we compare the SIC algorithms under different channel
conditions. In Section IV, we compare SIC algorithms with respect to the
system constraints, such as the transmit power and traffic load. The numerical
results are presented in Section V. Finally, Section VI presents our
conclusions.

Figure 1: A block diagram of an FD multi-user massive MIMO system.

##  II System Model

In this section, we firstly introduce the system model of an asymmetric FD
system with massive MIMO and the basic notation used in this
paper.111Throughout this paper, we use upper- and lower-case boldfaces to
denote matrix $\boldsymbol{A}$ and vector $\boldsymbol{a}$, respectively. We
denote the inverse, transpose, and Hermitian of matrix $\boldsymbol{A}$ as
$\boldsymbol{A}^{-1}$, $\boldsymbol{A}^{T}$, and $\boldsymbol{A}^{*}$,
respectively. In addition, we denote the conjugate of a scalar $a$ as $a^{*}$.
$\text{Var}[X]$ and $\mathbb{E}[X]$ respectively represent the variance and
average of the random variable $X$, while $\text{Cov}(X,Y)$ represents the
covariance between two random variables $X$ and $Y$. $\text{Sort}(X_{k})$ is
referred to as a sort function of sequence $X_{k}$ in the descending order.
Then SINR, precoding/receive combining matrix design, and SIC methods are
presented.

###  II-A FD BS with Massive Arrays

We consider FD multi-user MIMO channels, as illustrated in Fig. 1, with a
large number of antennas at the BS that serves $K_{\text{DL}}$ downlink users
and $K_{\text{UL}}$ uplink users equipped with one antenna for each. It is
assumed that the BS is equipped with $M_{\text{total}}$ ($M_{\text{total}}>>K$
[10]) RF chains that are divided into $M$ transmit and $N$ receive RF chains
by considering their physical size, whereas each user has a single transmit
and single receive RF chain. All users are assumed to operate in a half-duplex
mode, so the hardware complexity for the SIC at the user is zero, which is
beneficial for their battery uses. This concept has been widely adopted in
massive MIMO systems in cellular scenarios [24, 26, 23]. In addition, we
assume an asymmetric data traffic load (i.e., the downlink has more data
traffic than the uplink). In cellular systems, there hardly exists any user
equipment (UE) requiring contents with high data rates for both links within
the same coherence time. This stands in contrast to device-to-device
scenarios. Thus, supporting the best-performing UE in a half-duplex mode can
achieve greater scheduling gain than that in an FD mode. We consider that both
the RF chain usage and data traffic load are asymmetric. Therefore, we assume
$M\geq N+K_{\text{DL}}$ and $N\geq K_{\text{UL}}$. This asymmetric antenna
usage ensures a small and low-cost BS by reducing the receive RF chains that
include low-noise amplifiers (LNA) within the fixed size of
$M_{\text{total}}$. Without loss of generality, let us consider that each
propagation gain of the SI channels between BS antennas, denoted by
$\beta^{\text{SI}}$, is the same as their expectation. In addition, the
notations defined in this paper are shown in Table I.

Table I:  Symbol |  Description
---|---
$M_{\text{total}}$(=$M+N$) |  The number of RF chains (transmit RF chains $+$ receive RF chains)
$K_{\text{DL}}$/ $K_{\text{UL}}$ |  The number of downlink/ uplink users
$\rho_{\text{d}}(=\text{P}_{\text{d}}/\text{P}_{\text{n}})$ |  The total transmit signal-to-noise ratio (SNR) at the BS
$\rho_{\text{u}}(=\text{P}_{\text{u}}/\text{P}_{\text{n}})$ |  The transmit SNR at the user
$\rho^{\text{DL}}_{k_{\text{d}}}(=\rho_{\text{d}}\beta^{\text{DL}}_{k_{\text{d}}})$ |  The received downlink SNR at the users
$\rho^{\text{UU}}_{k_{\text{d}},k_{\text{u}}}(=\rho_{\text{u}}\beta^{\text{UU}}_{k_{\text{d}},k_{\text{u}}})$ |  The received SI SNR at the users
$\rho^{\text{UL}}_{k_{\text{u}}}(=\rho_{\text{u}}\beta^{\text{UL}}_{k_{\text{u}}})$ |  The received uplink SNR at the BS
$\rho^{\text{SI}}(=\rho_{\text{d}}\beta^{\text{SI}})$ |  The received SI SNR at the BS
$\text{P}_{\text{d}}$, $\text{P}_{\text{u}}$, $\text{P}_{\text{n}}$ |  The total transmit power, the transmit power of the user, the noise power of the additive white Gaussian noise (AWGN) of the channel
$\beta^{\text{DL}}_{k_{\text{d}}}$/ $\beta^{\text{UL}}_{k_{\text{u}}}$ |  The propagation gain between the BS and the $k_{\text{d}}$-th DL user / the $k_{\text{u}}$-th UL user
$\beta^{\text{UU}}_{k_{\text{d}},k_{\text{u}}}$ |  The propagation gain from the $k_{\text{u}}$-th user to the $k_{\text{d}}$-th user
$\boldsymbol{H}$/ $\boldsymbol{R}$/ $\boldsymbol{E}$ |  The channel/ correlation/ estimated error matrix
$\tau$ |  The number of pilot symbols for the channel
$\epsilon$ |  The variance of the estimation error for the channel
$\alpha_{\text{anc}}$ |  The level of passive analog SIC
$\alpha_{\text{tx}}$, $\boldsymbol{n}_{\text{tx}}$ |  The transmitter noise variance, the transmitter noise vector
$\boldsymbol{n}_{\text{q}}$, $\boldsymbol{n}_{\text{p}}$ |  The additive quantization noise vector, the AWGN vector at SI
$\boldsymbol{n}_{{}_{\text{DL}}}$/ $\boldsymbol{n}_{{}_{\text{UL}}}$ |  The AWGN vector at downlink/ uplink
$s$/ $x$ |  The transmit symbol at downlink/ uplink
$\boldsymbol{G}$, $\boldsymbol{W}$ |  The transmit precoding matrix, the receive combining matrix
$\gamma$ |  The SINR
$\Omega$ |  The SI power term

###  II-B Channel Model

####  II-B1 Spatially Correlated Channels

The downlink, uplink, and SI channels are modeled by the $K_{\text{DL}}\times
M$ matrix
$\boldsymbol{H}_{\text{DL}}=\boldsymbol{R}_{\text{RX(DL)}}^{1/2}\boldsymbol{H}_{\text{DL,iid}}\boldsymbol{R}_{\text{TX(DL)}}^{1/2}$,
$N\times K_{\text{UL}}$ matrix
$\boldsymbol{H}_{\text{UL}}=\boldsymbol{R}_{\text{RX(UL)}}^{1/2}\boldsymbol{H}_{\text{UL,iid}}\boldsymbol{R}_{\text{TX(UL)}}^{1/2}$,
and $N\times M$ matrix
$\boldsymbol{H}_{\text{SI}}=\boldsymbol{R}_{\text{RX(SI)}}^{1/2}\boldsymbol{H}_{\text{SI,iid}}\boldsymbol{R}_{\text{TX(SI)}}^{1/2}$,
which are based on the Kronecker model. The element of
$\boldsymbol{R}_{\text{TX}}$ and $\boldsymbol{R}_{\text{RX}}$ is
$r_{\text{TX/RX}}^{ij}=J_{0}(2\pi{d_{\text{TX/RX}}^{ij}}/\lambda)$, where
$d_{\text{TX/RX}}^{ij}$ is the distance between the $i$-th array and the
$j$-th array of the transmit/receive antenna at the BS, $J_{0}$ and $\lambda$
are a zero-order Bessel function of the first kind and a carrier wavelength,
respectively. The specific parameters used in the simulation are presented in
Section V. The elements of $\boldsymbol{H}_{\text{DL,iid}}$,
$\boldsymbol{H}_{\text{UL,iid}}$, and $\boldsymbol{H}_{\text{SI,iid}}$ are
i.i.d. complex Gaussian random variables with zero mean and unit variance. The
transmit correlation matrices in the downlink, uplink and SI channels are
denoted as $\boldsymbol{R}_{\text{TX(DL)}}$, $\boldsymbol{R}_{\text{TX(UL)}}$,
and $\boldsymbol{R}_{\text{TX(SI)}}$, respectively, while the receive
correlation matrices in these channels are denoted as
$\boldsymbol{R}_{\text{RX(DL)}}$, $\boldsymbol{R}_{\text{RX(UL)}}$, and
$\boldsymbol{R}_{\text{RX(SI)}}$, respectively. It should be noted that, in
contrast to the uplink and downlink channels, the SI channel is highly
correlated in the spatial domain. Despite the high level of analog SIC, the
power from direct paths (which are line-of-sight (LOS) paths, reflected paths
around the antenna, and circuit leakage) is still greater than the power from
non-line-of-sight (NLOS) paths. In addition, because the angular spread of the
LOS path is very small, the spatial correlation of the SI channel is high.
Thus, it is assumed that the SI channel is more highly correlated than the
downlink and uplink channels. Furthermore, because the SI channel has a strong
LOS component (a Rician channel model), a highly correlated channel and a real
SI channel (which are equivalent channel models with a strong LOS component)
are considered to supplement the SI channel modeling.

####  II-B2 MMSE-Based Channel Estimation

Let $\boldsymbol{\hat{H}}_{\text{DL}}$, $\boldsymbol{\hat{H}}_{\text{UL}}$,
and $\boldsymbol{\hat{H}}_{\text{SI}}$ be the estimated channel matrices of
$\boldsymbol{H}_{\text{DL}}$, $\boldsymbol{H}_{\text{UL}}$, and
$\boldsymbol{H}_{\text{SI}}$, respectively. Because estimation errors are
independent of the original channels, the estimated channels can be modeled
using estimated error matrices ($\boldsymbol{E}$) that have i.i.d. elements
with the distribution $\mathcal{CN}(0,\epsilon^{2})$ as follows:
$\boldsymbol{\hat{H}}_{\text{DL/UL/SI}}=\boldsymbol{H}_{\text{DL/UL/SI}}+\boldsymbol{E}_{\text{DL/UL/SI}}.$
It is assumed that an MMSE-based channel estimation is used at the BS, which
is also used in the 3rd Generation Partnership Project (3GPP) [32]. Other
estimation schemes, such as least square-based estimation, can also be adapted
to this scheme [9]. In practice, the estimation error of the SI channel is
affected by transmitter noise caused by hardware impairments222We consider the
transmitter noise only in the SI channel, because its level is higher than
that of the receiver noise [33]. In the downlink and uplink channels, this
noise can be negligible owing to the large path loss. In addition, we
disregard the receiver distortion resulted from the hardware impairment at the
receiver because it is much lower than the receiver noise., such as the
nonlinearity of the power amplifier, a quantization error, and phase noise
[33]. Thus, it is difficult to model the estimated SI channel while taking
into consideration hardware impairments. The variance of the estimation error
is simply modeled as the normalized-MSE (NMSE) value in (II-B2) between the
estimated channel and the original channel. This error depends on the
performance of the estimation algorithm and the specifications of the
hardware. Because we consider a scenario of ADCs with a few bits (e.g., 4-5
bits) for realizing a cost-effective design [29, 30, 31], the variance of the
estimation error of the SI channel is given by
$\epsilon_{{}_{\text{SI}}}^{2}=\text{NMSE}$, which is derived from the AQNM as
shown in II-B3.

###### Remark 1

It should be noted that the variance of the estimation error of the SI channel
can also be modeled by the number of pilot symbols. Let the pilot symbols be
transmitted orthogonally under TDD-based transmission. The numbers of pilot
symbols for the SI channels are given by $\tau_{{}_{\text{SI}}}(\geq N)$ ($N$
transmit RF chains are connected to $N$ receive antennas through the
switches). From [23], the variance of the estimation error of the SI channel
is given by

$\displaystyle\epsilon_{{}_{\text{SI}}}^{2}=\frac{\alpha_{\text{tx}}\rho_{\text{d}}N\frac{\beta^{\text{SI}}}{\alpha_{\text{anc}}}+1}{\tau_{{}_{\text{SI}}}\rho_{\text{d}}\frac{\beta^{\text{SI}}}{\alpha_{\text{anc}}}+\alpha_{\text{tx}}\rho_{\text{d}}N\frac{\beta^{\text{SI}}}{\alpha_{\text{anc}}}+1}=\frac{\alpha_{\text{tx}}N\frac{\rho^{\text{SI}}}{\alpha_{\text{anc}}}+1}{\tau_{{}_{\text{SI}}}\frac{\rho^{\text{SI}}}{\alpha_{\text{anc}}}+\alpha_{\text{tx}}N\frac{\rho^{\text{SI}}}{\alpha_{\text{anc}}}+1},$
(1)

where $\alpha_{\text{anc}}$ is the level of the passive analog SIC, and the
transmitter noise is a complex Gaussian random variable with zero mean and
smaller variance than the receiver noise (denoted by $\alpha_{\text{tx}}$
($\alpha_{\text{tx}}\ll 1$)).

In the case of the downlink/uplink channel estimation, we assume there are no
error terms arising from the hardware impairments. Thus, the variances of the
estimation error for the downlink and uplink channels are given by [9]

$\displaystyle\epsilon_{{}_{\text{DL}},k_{\text{d}}}^{2}=\frac{1}{K_{\text{DL}}\rho_{\text{u}^{\text{(DL)}}}\beta^{\text{DL}}_{k_{\text{d}}}+1},~{}~{}\epsilon_{{}_{\text{UL}},k_{\text{u}}}^{2}=\frac{1}{K_{\text{UL}}\rho_{\text{u}}\beta^{\text{UL}}_{k_{\text{u}}}+1}.$
(2)

Here, the received pilot SNRs of the downlink and uplink users are
$K_{\text{DL}}\rho_{\text{u}}^{\text{(DL)}}$ and
$K_{\text{UL}}\rho_{\text{u}}$, respectively. In addition, the variances of
the estimated channel for the $k_{\text{d}}$-th and $k_{\text{u}}$-th users
are given by

$\displaystyle\text{Var}[\hat{h}^{\text{DL}}_{k_{\text{d}}m}]$
$\displaystyle=\frac{K_{\text{DL}}\rho_{\text{u}^{\text{(DL)}}}\beta^{\text{DL}}_{k_{\text{d}}}}{K_{\text{DL}}\rho_{\text{u}^{\text{(DL)}}}\beta^{\text{DL}}_{k_{\text{d}}}+1},$
$\displaystyle\text{Var}[\hat{h}^{\text{UL}}_{nk_{\text{u}}}]$
$\displaystyle=\frac{K_{\text{UL}}\rho^{\text{UL}}_{k_{\text{u}}}}{K_{\text{UL}}\rho^{\text{UL}}_{k_{\text{u}}}+1}.$
(3)

If the received pilot SNRs are high (i.e., both SNRs approach infinity), one
may consider perfect downlink and uplink channel estimations, which means
$\text{Var}[\hat{h}^{\text{DL}}_{k_{\text{d}}m}]\rightarrow 1$ and
$\text{Var}[\hat{h}^{\text{UL}}_{nk_{\text{u}}}]\rightarrow 1$, and these are
the same as $\text{Var}[{h}^{\text{DL}}_{k_{\text{d}}m}]$ and
$\text{Var}[{h}^{\text{UL}}_{nk_{\text{u}}}]$, respectively. It should be
noted that any estimation algorithm dependent on the transmit pilot power can
be modeled similarly.

####  II-B3 Channel Estimation Error Model

We assume that each RF chain is connected to a $p$ bits ADC at the BS. To
represent a quantization error in a channel estimation error, we use the AQNM
[30, 31] as follows:
$\boldsymbol{U}_{\text{SIQ}}=\alpha\boldsymbol{U}_{\text{SI}}+\boldsymbol{N}_{\text{q}},$
where $\boldsymbol{U}_{\text{SIQ}}$ is the $M\times N$ output matrix of the
ADC corresponding to the $M\times N$ input matrix
$\boldsymbol{U}_{\text{SI}}$, which is the received SI signal at the BS.
Accordingly, $\alpha$ can be computed as $1-\frac{p^{-2}(\pi\sqrt{3})}{2}$.
Here, $\boldsymbol{U}_{\text{SI}}$ can be modeled as [9]

$\displaystyle\boldsymbol{U}_{\text{SI}}=\sqrt{\frac{\rho_{\text{SI}}}{\alpha_{\text{anc}}}}\boldsymbol{\text{H}}_{\text{SI}}^{T}(\boldsymbol{P}+\boldsymbol{N}_{\text{tx}})+\boldsymbol{N}_{\text{p}},$
(4)

where $\boldsymbol{N}_{\text{tx}}$, $\boldsymbol{P}$, and
$\boldsymbol{N}_{\text{p}}$ are the transmitter noise, the $M\times N$ matrix
that satisfies $\boldsymbol{P}\boldsymbol{P}^{H}=\boldsymbol{I}$, and the
$M\times N$ complex Gaussian random variables with i.i.d. elements following
the distribution $\mathcal{CN}(0,\epsilon^{2})$, respectively.
$\boldsymbol{N}_{\text{q}}$ is the $M\times N$ additive quantization noise
matrix such that $\boldsymbol{U}_{\text{SI}}$ and $\boldsymbol{N}_{\text{q}}$
are uncorrelated. It should be noted that the variance of
$\boldsymbol{N}_{\text{q}}$ is given by

$\displaystyle\text{Var}[\boldsymbol{N}_{\text{q}}]$
$\displaystyle=\alpha(1-\alpha)\text{Var}[{\boldsymbol{U}_{\text{SI}}}].$
$\displaystyle\overset{(2)}{=}\alpha(1-\alpha)(\left(\frac{\rho_{\text{SI}}}{\alpha_{\text{anc}}}\right)(1+\alpha_{\text{tx}})+1).$
(5)

Thus, the MMSE estimate of $\boldsymbol{\hat{H}}_{\text{SI}}$ given
$\boldsymbol{U}$ is
$\widehat{\boldsymbol{H}_{\text{SI}}}=(\sqrt{\frac{\alpha_{\text{anc}}}{\rho_{\text{d}}\beta_{\text{SI}}}}\boldsymbol{\text{U}}_{\text{SIQ}}^{T}\boldsymbol{P}\ast\widetilde{\boldsymbol{D}})^{T},$
where
$\widetilde{\boldsymbol{D}}=(\frac{\alpha_{\text{anc}}}{\rho_{\text{d}}}\beta_{\text{SI}}+I)^{-1}$.

###  II-C SINR at Uplink and Downlink

Let $\boldsymbol{g}_{k_{\text{d}}}$ and $s_{k_{\text{d}}}$ denote the
$k_{\text{d}}$-th column vector of the transmit precoding matrix
$\boldsymbol{G}$ and the transmit symbol for the $k_{\text{d}}$-th user at the
downlink, respectively. Similarly, let $\boldsymbol{w}_{k_{\text{u}}}$ denote
the $k_{\text{u}}$-th column vector of the receive combining matrix
$\boldsymbol{W}$ and $x_{k_{\text{u}}}$ represent the transmit symbol for the
$k_{\text{u}}$-th user at the uplink. In addition, let
$\boldsymbol{n}_{{}_{\text{DL}}}$ and $\boldsymbol{n}_{{}_{\text{UL}}}$ be the
additive white Gaussian receiver noise vector at the downlink and uplink,
respectively. The received signal at the $k_{\text{d}}$-th user is then
expressed as

$\displaystyle y_{k_{\text{d}}}$
$\displaystyle=\underbrace{\sqrt{\rho_{{}_{\text{DL}}}}\boldsymbol{\hat{h}}_{\text{DL},k_{\text{d}}}^{T}\boldsymbol{g}_{k_{\text{d}}}s_{k_{\text{d}}}}_{\text{desired
signal}}+\underbrace{\sqrt{\rho_{{}_{\text{DL}}}}\sum_{\ell=1,\ell\neq
k_{\text{d}}}^{K_{\text{DL}}}\boldsymbol{\hat{h}}_{\text{DL},k_{\text{d}}}^{T}\boldsymbol{g}_{\ell}s_{\ell}}_{\text{user-
interference}}$
$\displaystyle~{}~{}~{}+\underbrace{\sqrt{\rho_{{}_{\text{DL}}}}\sum_{\ell=1}^{K_{\text{DL}}}\boldsymbol{e}_{\text{DL},k_{\text{d}}}^{T}\boldsymbol{g}_{\ell}s_{\ell}}_{\text{estimation
error}}+\underbrace{\sum_{k_{\text{u}}=1}^{K_{\text{UL}}}\sqrt{\rho^{\text{UU}}_{k_{\text{d}},k_{\text{u}}}}h_{k_{\text{d}}k_{\text{u}}}^{\text{UU}}x_{k_{\text{u}}}}_{\text{user-
to-user-interference}}+n_{{}_{\text{DL}},k_{\text{d}}},$ (6)

where $\boldsymbol{\hat{h}}_{\text{DL},k_{\text{d}}}$ represents the
$k_{\text{d}}$-th column vector of $\boldsymbol{\hat{H}}_{\text{DL}}^{T}$ and
$h_{k_{\text{d}}k_{\text{u}}}$ is the interference channel from the
$k_{\text{u}}$-th user to the $k_{\text{d}}$-th user. The received signal for
the $k_{\text{u}}$-th user at the BS is expressed as

$\displaystyle u_{k_{\text{u}}}$
$\displaystyle=\underbrace{\sqrt{\rho^{\text{UL}}_{k_{\text{u}}}}\boldsymbol{w}_{k_{\text{u}}}^{T}\boldsymbol{\hat{h}}_{\text{UL},k_{\text{u}}}x_{k_{\text{u}}}}_{\text{desired
signal}}+\underbrace{\sum_{\ell=1,\ell\neq{k_{\text{u}}}}^{K_{\text{UL}}}\sqrt{\rho^{\text{UL}}_{\ell}}\boldsymbol{w}_{k_{\text{u}}}^{T}\boldsymbol{\hat{h}}^{\text{UL}}_{\ell}x_{\ell}}_{\text{user-
interference}}$
$\displaystyle~{}~{}~{}+\underbrace{\sum_{\ell=1}^{K_{\text{UL}}}\sqrt{\rho^{\text{UL}}_{\ell}}\boldsymbol{w}_{k_{\text{u}}}^{T}\boldsymbol{e}^{\text{UL}}_{\ell}x_{\ell}}_{\text{estimation
error}}+\underbrace{\sqrt{\rho^{\text{SI}}/\alpha_{\text{anc}}}\boldsymbol{w}_{k_{\text{u}}}^{T}\boldsymbol{H}_{\text{SI}}\boldsymbol{G}\boldsymbol{\bar{s}}}_{\text{self-
interference}}+\boldsymbol{w}_{k_{\text{u}}}^{T}\boldsymbol{n}_{{}_{\text{UL}}},$
(7)

where $\boldsymbol{\hat{h}}_{\text{UL},k_{\text{u}}}$ is the $k_{\text{u}}$-th
column vector of $\boldsymbol{\hat{H}}_{\text{UL}}$. In addition,
$\boldsymbol{\bar{s}}$ represents
$\boldsymbol{s}+\boldsymbol{n}_{{}_{\text{tx}}}$. To satisfy a power
constraint, the normalized transmit beamforming vectors (the columns of the
precoding matrix) with vector normalization are given as
$\boldsymbol{g}_{k_{\text{d}}}=\boldsymbol{f}_{k_{\text{d}}}/(\sqrt{K_{\text{DL}}}||\boldsymbol{f}_{k_{\text{d}}}||)$,
where $1\leq k_{\text{d}}\leq K_{\text{DL}}$ [10].

From (6) and (7), we can derive the SINRs of the $k_{\text{d}}$-th user at the
downlink and the $k_{\text{u}}$-th user at the uplink, respectively, as
follows:

$\displaystyle{\gamma}_{k_{\text{d}}}^{\text{DL}}$
$\displaystyle=\frac{\rho^{\text{DL}}_{k_{\text{d}}}\left|\boldsymbol{\hat{h}}_{\text{DL},k_{\text{d}}}^{T}\frac{\boldsymbol{f}_{k_{\text{d}}}}{\sqrt{K_{\text{DL}}}||\boldsymbol{f}_{k_{\text{d}}}||}\right|^{2}}{\mathcal{I}_{\text{error}}^{\text{DL}}+\mathcal{I}_{\text{user}}^{\text{DL}}+\sum_{k_{\text{u}}=1}^{K_{\text{UL}}}\rho^{\text{UU}}_{k_{\text{d}},k_{\text{u}}}|h_{k_{\text{d}}k_{\text{u}}}^{\text{UU}}|^{2}+1},$
$\displaystyle{\gamma}_{k_{\text{u}}}^{\text{UL}}$
$\displaystyle=\frac{\rho^{\text{UL}}_{k_{\text{u}}}|\boldsymbol{w}_{k_{\text{u}}}^{T}\boldsymbol{\hat{h}}_{\text{UL},k_{\text{u}}}|^{2}}{\mathcal{I}_{\text{error}}^{\text{UL}}+\mathcal{I}_{\text{user}}^{\text{UL}}+\frac{\rho^{\text{SI}}}{\alpha_{\text{anc}}}\Omega(1+\alpha_{\text{tx}})+||\boldsymbol{w}_{k_{\text{u}}}||^{2}},$
(8)

where
$\mathcal{I}_{\text{error}}^{\text{DL}}=\rho^{\text{DL}}_{k_{\text{d}}}\sum_{\ell=1}^{K_{\text{DL}}}\left|\boldsymbol{e}_{\text{DL},k_{\text{d}}}^{T}\frac{\boldsymbol{f}_{\ell}}{\sqrt{K_{\text{DL}}}||\boldsymbol{f}_{\ell}||}\right|^{2}$,
$\mathcal{I}_{\text{user}}^{\text{DL}}=\rho^{\text{DL}}_{k_{\text{d}}}\sum_{\ell=1,\ell\neq
k_{\text{d}}}^{K_{\text{DL}}}\left|\boldsymbol{\hat{h}}_{\text{DL},k_{\text{d}}}^{T}\frac{\boldsymbol{f}_{\ell}}{\sqrt{K_{\text{DL}}}||\boldsymbol{f}_{\ell}||}\right|^{2}$,
$\mathcal{I}_{\text{error}}^{\text{UL}}=\sum_{\ell=1}^{K_{\text{UL}}}\rho^{\text{UL}}_{\ell}|\boldsymbol{w}_{k_{\text{u}}}^{T}\boldsymbol{e}_{\text{UL},\ell}|^{2}$,
$\mathcal{I}_{\text{user}}^{\text{UL}}=\sum_{\ell=1,\ell\neq
k_{\text{u}}}^{K_{\text{UL}}}\rho^{\text{UL}}_{\ell}|\boldsymbol{w}_{k_{\text{u}}}^{T}\boldsymbol{\hat{h}}_{\text{UL},\ell}|^{2}$,
and
$\Omega=||\boldsymbol{w}_{k_{\text{u}}}^{T}\boldsymbol{H}_{\text{SI}}\boldsymbol{G}||^{2}$.

###  II-D Precoding/Receive Combining Matrix Design and SIC Methods

Because the FD BS always perfectly knows its precoding/receive matrix and
transmitted downlink symbols, it can mitigate SI by directly subtracting
$\sqrt{\rho^{\text{SI}}/\alpha_{\text{anc}}}\boldsymbol{w}_{k_{\text{u}}}^{T}\hat{\boldsymbol{H}}_{\text{SI}}\boldsymbol{G}\boldsymbol{s}$,
of which the SIC is referred to as the SI subtraction. The received signal
after the SI subtraction is

$\displaystyle
u_{\text{stt},k_{\text{u}}}=u_{k_{\text{u}}}-\sqrt{\rho^{\text{SI}}/\alpha_{\text{anc}}}\boldsymbol{w}_{k_{\text{u}}}^{T}\hat{\boldsymbol{H}}_{\text{SI}}\boldsymbol{G}\boldsymbol{\bar{s}}.$
(9)

Here, the subscript stt indicates the SI subtraction. The ZF precoder for the
downlink is assumed at the BS in the cases of the FD BS with the SI
subtraction and without SIC:

$\displaystyle\boldsymbol{G}_{\text{ZF}}=\boldsymbol{\hat{H}}_{\text{DL}}^{*}(\boldsymbol{\hat{H}}_{\text{DL}}\boldsymbol{\hat{H}}_{\text{DL}}^{*})^{-1}.$
(10)

In addition, because the precoding/combining matrices are computed only once
per coherence block, the complexity difference between MF and ZF/MMSE is
relatively small (the bulk of the complexity comes from FFTs and matrix-vector
multiplications performed on a per-symbol basis [34]). In addition, because
the FD massive MIMO considered has many antennas ($>>the\ number\ of\ users$)
and a high SNR environment, both ZF and MMSE have similar performance.
Furthermore, ZF can provide a clear performance comparison with SI subtraction
to satisfy our main contribution. For these reasons, the ZF is considered for
downlink transmission. Future work will consider various precoding/combining
matrices, including MRT/MRC, generalized MMSE and so on. From the antenna
configuration of $M\geq N+K$, additional antennas can be utilized for spatial
suppression [16, 24, 26]. Spatial suppression-based algorithms for massive
MIMO systems were proposed in [24, 26]. We adopt the extended ZF in [24] for
spatial suppression, which requires minimum computational complexity for
nulling.

$\displaystyle\boldsymbol{G}_{\text{sps}}=\boldsymbol{\hat{H}}_{\text{ext}}^{*}(\boldsymbol{\hat{H}}_{\text{ext}}\boldsymbol{\hat{H}}_{\text{ext}}^{*})^{-1},~{}~{}~{}~{}\boldsymbol{\hat{H}}_{\text{ext}}=\begin{bmatrix}\boldsymbol{\hat{H}}_{\text{DL}}\\\
\boldsymbol{\hat{H}}_{\text{SI}}\end{bmatrix}.$ (11)

Here, the subscript sps indicates the spatial suppression. It should be noted
that the vector set, from the beginning of the column vector of
$\boldsymbol{G}_{\text{sps}}$ to the $K_{\text{DL}}$-th column vector, is the
same as the null-space projection precoder in [16], where the rank of
$\boldsymbol{H}_{\text{SI}}$ is $N$, and the nullity of that is
$K_{\text{DL}}$.

To eliminate user-interference signals at the uplink, we use the following ZF
filter in both the transceiver with the SI subtraction/spatial suppression and
without the SIC:

$\displaystyle\boldsymbol{W}=({\boldsymbol{\hat{H}}}_{\text{UL}}^{*}{\boldsymbol{\hat{H}}}_{\text{UL}})^{-1}{\boldsymbol{\hat{H}}^{*}_{\text{UL}}}.$
(12)

##  III Performance Comparison in Different Channel Conditions

In this section, we compare the performance of two SIC methods with channel
conditions which are correlated/uncorrelated and perfect/imperfect.
Specifically, we provide mathematical motivations and preliminaries at first.
Then, based on these preliminaries, we compare the two methods by SINR or the
expected SI power term in uncorrelated channels. Finally, in subsection III-C,
we compare them by the expected SI power term in correlated channels.

###  III-A Mathematical Motivations and Preliminaries

In this subsection, we provide mathematical motivations and preliminaries to
investigate the ergodic sum rate and SI power for FD massive MIMO systems. We
also present an asymptotic analysis of FD massive MIMO systems under the
perfect channel estimation conditions.

####  III-A1 SI Power Term in Massive MIMO Systems

In the FD system, the SI level degrades the uplink performance as a
bottleneck, which is represented as the SI power term $\Omega$ in (8). Let
$\boldsymbol{A}$ be an $N\times M$ random matrix, the elements of which are
complex random variables with zero mean and $\sigma_{A}^{2}$. In addition,
$w_{k_{\text{u}}n}$, $a_{nm}$, and $g_{m\ell}$ are the $n$-th elements of
$\boldsymbol{w}_{k_{\text{u}}}$, the $(n,m)$-th element of $\boldsymbol{A}$,
and the $(m,\ell)$-th element of the $M\times L$ matrix $\boldsymbol{G}$ (the
precoder matrix), respectively. Then, the quadratic form of
$||\boldsymbol{w}_{k_{\text{u}}}^{T}\boldsymbol{A}\boldsymbol{G}||^{2}$ can be
expressed as

$\displaystyle||\boldsymbol{w}_{k_{\text{u}}}^{T}\boldsymbol{A}\boldsymbol{G}||^{2}=\sum_{m=1}^{L}\sum_{i=1}^{N}\sum_{j=1}^{N}\sum_{k=1}^{M}\sum_{\ell=1}^{M}w_{k_{\text{u}}i}w_{k_{\text{u}}j}^{*}a_{ik}a_{j\ell}^{*}g_{km}g_{\ell
m}^{*}.$ (13)

It should be noted that the SI power term can be derived by replacing
$\boldsymbol{A}$ in (13) with $\boldsymbol{H}_{\text{SI}}$ or
$\boldsymbol{E}_{\text{SI}}$ according to the SIC method.

###### Lemma 1

_(decomposition of the expected SI power term)_ The expectation of (13) can be
decomposed into

$\displaystyle\mathbb{E}\left\\{||\boldsymbol{w}_{k_{\text{u}}}^{T}\boldsymbol{A}\boldsymbol{G}||^{2}\right\\}$
$\displaystyle=\sum_{m=1}^{L}\sum_{i=1}^{N}\sum_{j=1}^{N}\sum_{k=1}^{M}\sum_{\ell=1}^{M}\mathbb{E}\left\\{w_{k_{\text{u}}i}w_{k_{\text{u}}j}^{*}a_{ik}a_{j\ell}^{*}g_{km}g_{\ell
m}^{*}\right\\}$
$\displaystyle\overset{(a)}{=}\sum_{m=1}^{L}\sum_{i=1}^{N}\sum_{j=1}^{N}\sum_{k=1}^{M}\sum_{\ell=1}^{M}\mathbb{E}\left\\{w_{k_{\text{u}}i}w_{k_{\text{u}}j}^{*}\right\\}\mathbb{E}\left\\{a_{ik}a_{j\ell}^{*}g_{km}g_{\ell
m}^{*}\right\\}$
$\displaystyle=\underbrace{\sum_{m=1}^{L}\sum_{i=1}^{N}\sum_{k=1}^{M}\sum_{\ell=1}^{M}\mathbb{E}\left\\{|w_{k_{\text{u}}i}|^{2}\right\\}\mathbb{E}\left\\{a_{ik}a_{i\ell}^{*}g_{km}g_{\ell
m}^{*}\right\\}}_{\text{expectation term related to i.i.d.
$\boldsymbol{H}_{UL}$}}+\widetilde{\Omega}_{\boldsymbol{A}(i\neq j)}$
$\displaystyle=\sum_{i=1}^{N}\mathbb{E}\left\\{|w_{k_{\text{u}}i}|^{2}\right\\}$
$\displaystyle~{}~{}\times\sum_{m=1}^{L}\sum_{k=1}^{M}\left[\underbrace{\mathbb{E}\left\\{|a_{ik}|^{2}\right\\}\mathbb{E}\left\\{|g_{km}|^{2}\right\\}}_{\text{expectation
term where $\boldsymbol{A}$ and $\boldsymbol{G}$ are
independent}}+\zeta\right]$
$\displaystyle~{}~{}+\widetilde{\Omega}_{\boldsymbol{A}(i\neq j)},$ (14)

where

$\displaystyle\widetilde{\Omega}_{\boldsymbol{A}(i\neq j)}$
$\displaystyle=\underbrace{\sum_{m=1}^{L}\sum_{i=1}^{N}\sum_{j=1,j\neq
i}^{N}\sum_{k=1}^{M}\sum_{\ell=1}^{M}\mathbb{E}\left\\{w_{k_{\text{u}}i}w_{k_{\text{u}}j}^{*}\right\\}\mathbb{E}\left\\{a_{ik}a_{j\ell}^{*}g_{km}g_{\ell
m}^{*}\right\\}}_{\text{residual expectation term due to correlated
$\boldsymbol{H}_{UL}$}},$
$\displaystyle\zeta=\underbrace{\text{Cov}(|a_{ik}|^{2},|g_{km}|^{2})+\sum_{\ell=1,\ell\neq
k}^{M}\mathbb{E}\left\\{a_{ik}a_{i\ell}^{*}g_{km}g_{\ell
m}^{*}\right\\}}_{\text{residual expectation term due to correlation between
$\boldsymbol{A}$ and $\boldsymbol{G}$}}.$

Equation (_a_) is derived based on the fact that $w_{k_{\text{u}}n}$ is
independent of both $a_{nm}$ and $g_{m\ell}$.

This lemma implies that the expected SI power varies according to the
correlation between $\boldsymbol{A}$ and $\boldsymbol{G}$ as well as the
correlation among their own elements.333The right side of $\zeta$ includes the
terms that occurred due to the correlated $\boldsymbol{G}$. We discuss this in
_Lemma_ 5. Using this lemma, we compare the two SIC methods in correlated
channel models. It should be noted that, the conventional analysis does not
focus on these correlation terms by assuming independent conditions [24, 25].

###### Lemma 2

_(expected SI power term with independent conditions)_ The following equation
holds when $\boldsymbol{w}_{k_{\text{u}}}$, $\boldsymbol{A}$, and
$\boldsymbol{G}$ are independent of each other and, consist of i.i.d. random
variables [24, 25]:

$\displaystyle\mathbb{E}\left\\{||\boldsymbol{w}_{k_{\text{u}}}^{T}\boldsymbol{A}\boldsymbol{G}||^{2}\right\\}=\sigma_{A}^{2}\mathbb{E}\left\\{||\boldsymbol{w}_{k_{\text{u}}}||^{2}\right\\}.$

###### Proof:

See Appendix A. ∎

####  III-A2 Approximation of Ergodic Sum Rate of Massive MIMO Systems

###### Remark 2

Let $S$, $I$, and $N$ be the norms of the random vectors of the desired signal
power, interference power, and noise power terms, respectively. From [10,
Lemma 4], the ergodic achievable sum rate of massive MIMO systems in the
low/high SNR regime can be approximated as follows:

$\displaystyle\mathbb{E}\left\\{\log_{2}\left\\{1+\frac{S}{I+N}\right\\}\right\\}\approx\mathcal{R}=\log_{2}\left\\{1+\frac{\mathbb{E}\\{S\\}}{\mathbb{E}\\{I+N\\}}\right\\},$
(15)

where
$\mathbb{E}\\{S\\}\mathbb{E}\left\\{\frac{1}{I+N}\right\\}\approx\frac{\mathbb{E}\\{S\\}}{\mathbb{E}\\{I+N\\}}$
or
$\mathbb{E}\left\\{\frac{S}{I+N}\right\\}\approx\frac{\mathbb{E}\\{S\\}}{\mathbb{E}\\{I+N\\}}$.
From (15), we let $\bar{\gamma}$
$(=\frac{\mathbb{E}\\{S\\}}{\mathbb{E}\\{I+N\\}})$ be an approximation of
$\gamma$.

####  III-A3 Performance Analysis with Perfect Channel Estimation

First, we begin with a brief overview of the ergodic achievable sum rate of
the two SIC schemes with perfect channel state information
($\epsilon_{{}_{\text{DL/UL/SI}}}$= 0) in spatially uncorrelated channels. In
summary, from _Remark_ 2, the approximations of the ergodic rate of the
$k_{\text{d}}$-th user with the SI subtraction and the spatial suppression at
downlink are expressed, respectively, as follows:

$\displaystyle\mathcal{R}_{k_{\text{d}},\text{stt}}^{\text{DL}}$
$\displaystyle=\log_{2}\left\\{1+\frac{\frac{\rho^{\text{DL}}_{k_{\text{d}}}}{K_{\text{DL}}}\mathbb{E}\left\\{\frac{1}{||\boldsymbol{f}_{k_{\text{d}},\text{ZF}}||^{2}}\right\\}}{\sum_{k_{\text{u}}=1}^{K_{\text{UL}}}\rho^{\text{UU}}_{k_{\text{d}},k_{\text{u}}}\mathbb{E}\\{|h_{k_{\text{d}}k_{\text{u}}}^{\text{UU}}|^{2}\\}+1}\right\\}$
$\displaystyle=\log_{2}\left\\{1+\frac{\rho^{\text{DL}}_{k_{\text{d}}}(M-K_{\text{DL}}+1)}{K_{\text{DL}}(\sum_{k_{\text{u}}=1}^{K_{\text{UL}}}\rho^{\text{UU}}_{k_{\text{d}},k_{\text{u}}}+1)}\right\\},$
(16) $\displaystyle\mathcal{R}_{k_{\text{d}},\text{sps}}^{\text{DL}}$
$\displaystyle=\log_{2}\left\\{1+\frac{\frac{\rho^{\text{DL}}_{k_{\text{d}}}}{K_{\text{DL}}}\mathbb{E}\left\\{\frac{1}{||\boldsymbol{f}_{k_{\text{d}},\text{sps}}||^{2}}\right\\}}{\sum_{k_{\text{u}}=1}^{K_{\text{UL}}}\rho^{\text{UU}}_{k_{\text{d}},k_{\text{u}}}\mathbb{E}\\{|h_{k_{\text{d}}k_{\text{u}}}^{\text{UU}}|^{2}\\}+1}\right\\}$
$\displaystyle=\log_{2}\left\\{1+\frac{\rho^{\text{DL}}_{k_{\text{d}}}(M-N-
K_{\text{DL}}+1)}{K_{\text{DL}}(\sum_{k_{\text{u}}=1}^{K_{\text{UL}}}\rho^{\text{UU}}_{k_{\text{d}},k_{\text{u}}}+1)}\right\\}.$
(17)

In addition, the approximations of the ergodic rate of the $k_{\text{u}}$-th
user with both SIC methods at the uplink are the same and are given by

$\displaystyle\mathcal{R}_{k_{\text{u}},\text{stt/sps}}^{\text{UL}}$
$\displaystyle=\log_{2}\left\\{1+\mathbb{E}\left\\{\frac{\rho^{\text{UL}}_{k_{\text{u}}}}{||\boldsymbol{w}_{k_{\text{u}}}||^{2}}\right\\}\right\\}$
$\displaystyle=\log_{2}\left\\{1+\rho^{\text{UL}}_{k_{\text{u}}}(N-K_{\text{UL}}+1)\right\\},$
(18)

where $\mathcal{I}_{\text{error}}$, $\mathcal{I}_{\text{user}}=0$,
$\mathbb{E}\left\\{\frac{1}{||\boldsymbol{f}_{k_{\text{d}},\text{ZF}}||^{2}}\right\\}=M-K_{\text{DL}}+1$,
$\mathbb{E}\left\\{\frac{1}{||\boldsymbol{f}_{k_{\text{d}},\text{sps}}||^{2}}\right\\}=M-(N+K_{\text{DL}})+1$,
and
$\mathbb{E}\left\\{\frac{1}{||\boldsymbol{w}_{k_{\text{u}}}||^{2}}\right\\}=N-K_{\text{UL}}+1$
(results are derived from the diversity order of ZF [10, 35]). From (16)-(18),
we obtain

$\displaystyle\mathcal{R}_{\text{stt}}$
$\displaystyle=\sum_{k_{\text{d}}=1}^{K_{\text{DL}}}\log_{2}\left\\{1+\frac{\rho^{\text{DL}}_{k_{\text{d}}}(M-K_{\text{DL}}+1)}{K_{\text{DL}}(\sum_{k_{\text{u}}=1}^{K_{UL}}\rho^{\text{UU}}_{k_{\text{d}},k_{\text{u}}}+1)}\right\\}$
$\displaystyle~{}~{}+\sum_{k_{\text{u}}=1}^{K_{\text{UL}}}\log_{2}\left\\{1+\rho^{\text{UL}}_{k_{\text{u}}}(N-K_{\text{UL}}+1)\right\\},$
(19) $\displaystyle\mathcal{R}_{\text{sps}}$
$\displaystyle=\sum_{k_{\text{d}}=1}^{K_{\text{DL}}}\log_{2}\left\\{1+\frac{\rho^{\text{DL}}_{k_{\text{d}}}(M-N-
K_{\text{DL}}+1)}{K_{\text{DL}}(\sum_{k_{\text{u}}=1}^{K_{UL}}\rho^{\text{UU}}_{k_{\text{d}},k_{\text{u}}}+1)}\right\\}$
$\displaystyle~{}~{}+\sum_{k_{\text{u}}=1}^{K_{\text{UL}}}\log_{2}\left\\{1+\rho^{\text{UL}}_{k_{\text{u}}}(N-K_{\text{UL}}+1)\right\\}.$
(20)

It should be noted that the approximations of the ergodic sum rate of the
half-duplex with ZF precoding and ZF receive combining is the same as
$\frac{1}{2}\mathcal{R}_{\text{stt}}$ with
$\rho^{\text{UU}}_{k_{\text{d}},k_{\text{u}}}=0$ $(\forall
k_{\text{d}},\forall k_{\text{u}})$ [10].

From (19) and (20), the sum rate of the SI subtraction is, in the Genie-aided
system, always better than that of the spatial suppression owing to the same
performance at the uplink and the performance degradation at the downlink.
This is because the SI subtraction and spatial suppression transceivers can
perfectly cancel out the SI at the uplink, and the spatial suppression
requires an additional dimension for the nulling of the ZF precoder at the
downlink.

###  III-B Asymptotic Rate in Uncorrelated Channels

In this subsection, we present the asymptotic analysis of FD massive MIMO
systems according to the two SIC methods: SI subtraction and spatial
suppression. First, we assume uncorrelated channels for simplifying
mathematical expressions to obtain insights into how the imperfect channel
estimation affects the performance of the two SIC schemes.

In the FD system, the SI level degrades the uplink performance as a
bottleneck, which is represented as the SI power term $\Omega$ in (8). Let
${\bar{\Omega}}=\mathbb{E}\\{\Omega\\}$ be the expected SI power. If there is
no SIC at the BS ($\alpha_{\text{anc}}=0$ dB), from _Lemma_ 2, the expected SI
power is given by

$\displaystyle{\bar{\Omega}}_{\text{woSIC}}$
$\displaystyle=\mathbb{E}\left\\{||\boldsymbol{w}_{ku}^{T}\boldsymbol{H}_{\text{SI}}\boldsymbol{G}||^{2}\right\\}=\mathbb{E}\left\\{||\boldsymbol{w}_{k_{\text{u}}}||^{2}\right\\}\mathbb{E}\left\\{|h^{\text{SI}}_{n,m}|^{2}\right\\}$
$\displaystyle=\mathbb{E}\left\\{||\boldsymbol{w}_{k_{\text{u}}}||^{2}\right\\}.$
(21)

Similarly, because $\boldsymbol{E}_{\text{SI}}$ and
$\boldsymbol{G}_{\text{ZF}}$ are independent, from (14), the quadratic form of
the expected SI power of the SI subtraction is given by

$\displaystyle{\bar{\Omega}}_{\text{stt}}$
$\displaystyle=\mathbb{E}\left\\{||\boldsymbol{w}_{k_{\text{u}}}^{T}(\boldsymbol{H}_{\text{SI}}-\boldsymbol{\hat{H}}_{\text{SI}})\boldsymbol{G}_{\text{ZF}}||^{2}\right\\}=\mathbb{E}\left\\{||\boldsymbol{w}_{k_{\text{u}}}^{T}\boldsymbol{E}_{\text{SI}}\boldsymbol{G}_{\text{ZF}}||^{2}\right\\}$
$\displaystyle=\epsilon_{{}_{\text{SI}}}^{2}\mathbb{E}\left\\{||\boldsymbol{w}_{k_{\text{u}}}||^{2}\right\\}.$
(22)

In contrast to the SI subtraction, $\boldsymbol{E}_{\text{SI}}$ and
$\boldsymbol{G}_{\text{sps}}$ are dependent when spatial suppression is used,
because $\boldsymbol{G}_{\text{sps}}$ is a function of
$\boldsymbol{\hat{H}}_{\text{SI}}=\boldsymbol{H}_{\text{SI}}+\boldsymbol{E}_{\text{SI}}$.
Thus, from _Lemma_ 1, we can derive the quadratic form of the expected SI
power of the spatial suppression as follows:

$\displaystyle{\bar{\Omega}}_{\text{sps}}$
$\displaystyle=\mathbb{E}\left\\{||\boldsymbol{w}_{k_{\text{u}}}^{T}\boldsymbol{H}_{\text{SI}}\boldsymbol{\bar{G}}_{\text{sps}}||^{2}\right\\}=\mathbb{E}\left\\{||\boldsymbol{w}_{k_{\text{u}}}^{T}(\boldsymbol{\hat{H}}_{\text{SI}}-\boldsymbol{E}_{\text{SI}})\boldsymbol{\bar{G}}_{\text{sps}}||^{2}\right\\}$
$\displaystyle=\mathbb{E}\left\\{||\boldsymbol{w}_{k_{\text{u}}}^{T}\boldsymbol{E}_{\text{SI}}{\boldsymbol{\bar{G}}}_{\text{sps}}||^{2}\right\\}$
$\displaystyle=\sum_{i=1}^{N}\mathbb{E}\left\\{|w_{k_{\text{u}}i}|^{2}\right\\}\sum_{m=1}^{K_{\text{DL}}}\sum_{k=1}^{M}\mathbb{E}\left\\{|E_{ik}|^{2}\right\\}\mathbb{E}\left\\{|g_{km,\text{sps}}|^{2}\right\\}$
$\displaystyle~{}~{}+\sum_{i=1}^{N}\mathbb{E}\left\\{|w_{k_{\text{u}}i}|^{2}\right\\}\sum_{m=1}^{K_{\text{DL}}}\sum_{k=1}^{M}\zeta_{\text{sps}}$
$\displaystyle=\epsilon_{{}_{\text{SI}}}^{2}\mathbb{E}\left\\{||\boldsymbol{w}_{k_{\text{u}}}||^{2}\right\\}+\sum_{i=1}^{N}\mathbb{E}\left\\{|w_{k_{\text{u}}i}|^{2}\right\\}\sum_{m=1}^{K_{\text{DL}}}\sum_{k=1}^{M}\zeta_{\text{sps}},$
(23)

where $\widetilde{\Omega}_{\boldsymbol{A}(i\neq j)}=0$,
$\zeta_{\text{sps}}=\text{Cov}(|E_{ik}|^{2},|g_{km,\text{sps}}|^{2})+\sum_{\ell=1,\ell\neq
k}^{M}\mathbb{E}\left\\{E_{ik}E_{i\ell}^{*}g_{km,\text{sps}}g_{\ell
m,\text{sps}}^{*}\right\\}\\!$, and $E_{nm}$ is the $(n,m)$-th element of
$\boldsymbol{E}_{\text{SI}}$. In addition,
${\boldsymbol{\bar{G}}}_{\text{sps}}$ is the subset from the first to the
$K_{\text{DL}}$-th columns of $\boldsymbol{G}_{\text{sps}}$ and
$\boldsymbol{\hat{H}}_{\text{SI}}{\boldsymbol{\bar{G}}}_{\text{sps}}=\boldsymbol{0}$.

###### Lemma 3

The residual expectation term in $\bar{\Omega}_{\text{sps}}$ owing to the
correlation between $\boldsymbol{E}_{\text{SI}}$ and
$\boldsymbol{\bar{G}}_{\text{sps}}$ is negative:

$\displaystyle\zeta_{\text{sps}}$
$\displaystyle=\text{Cov}(|E_{ik}|^{2},|g_{km,\text{sps}}|^{2})$
$\displaystyle~{}~{}+\sum_{\ell=1,\ell\neq
k}^{M}\mathbb{E}\left\\{E_{ik}E_{i\ell}^{*}g_{km,\text{sps}}g_{\ell
m,\text{sps}}^{*}\right\\}$ $\displaystyle~{}~{}\leq 0.$

###### Proof:

See Appendix B. ∎

Figure 2: Concepts of digital SIC schemes in the vector space.

By applying _Lemma_ 3 to the quadratic forms of $\bar{\Omega}_{\text{stt}}$
and $\bar{\Omega}_{\text{sps}}$ ((22), (23)), we can confirm the inequality

$\displaystyle\bar{\Omega}_{\text{stt}}\geq\bar{\Omega}_{\text{sps}}.$ (24)

To illustrate the implicit meaning of _Lemma_ 3, we present Fig. 2, which
illustrates an example of the effect of spatial suppression on the residual SI
by representing a manifold as a Euclidean space. This shows that, with
imperfect channel estimation, the difference vector between the SI channel
vector and their estimated vector (i.e., the estimation error vector) is
additionally projected onto the null vector of the estimated SI channel
vector. Thus, the length of the residual SI vector of the spatial suppression
can be shorter than that of SI subtraction. Therefore, the expected SI power
of the spatial suppression can be mitigated to a greater extent than that of
SI subtraction.

It is trivial from (19) and (20) that the SI subtraction outperforms the
spatial suppression in the perfect channel estimation condition, i.e.,
$\mathcal{R}_{\text{stt}}\geq\mathcal{R}_{\text{sps}}$. However, as the
ergodic uplink performance of the spatial suppression is the same as that of
the SI subtraction in the perfect channel estimation condition, it is better
than that of the SI subtraction as determined from (24), i.e.,
$\bar{\Omega}_{\text{stt}}\geq\bar{\Omega}_{\text{sps}}$.

Next, we present some approximations of the uplink SINR for each SIC method to
support the obtained results. Using the approximation methods of [9] and
_Remark_ 2, the approximation of the uplink SINR without the SIC can be
derived from (2), (8), and
${\bar{\Omega}_{\text{woSIC}}}=\mathbb{E}\left\\{||\boldsymbol{w}_{k_{\text{u}}}||^{2}\right\\}$
as follows:

$\displaystyle\bar{\gamma}_{\text{woSIC},k_{\text{u}}}^{\text{UL}}$
$\displaystyle=\frac{\rho^{\text{UL}}_{k_{\text{u}}}}{\mathbb{E}\\{||\boldsymbol{w}_{k_{\text{u}}}||^{2}\\}\sum_{\ell=1}^{K_{\text{UL}}}\frac{\rho^{\text{UL}}_{\ell}}{K_{\text{UL}}\rho^{\text{UL}}_{\ell}+1}+AA+\mathbb{E}\\{||\boldsymbol{w}_{k_{\text{u}}}||^{2}\\}}$
$\displaystyle=\frac{(K_{\text{UL}}\rho^{\text{UL}}_{k_{\text{u}}}+1)(N-K_{\text{UL}}+1)}{K_{\text{UL}}(\sum_{\ell=1}^{K_{\text{UL}}}\frac{\rho^{\text{UL}}_{\ell}}{K_{\text{UL}}\rho^{\text{UL}}_{\ell}+1}+1)+\frac{\rho^{\text{SI}}}{\alpha_{\text{anc}}}K_{\text{UL}}(1+\alpha_{\text{tx}})},$
(25)

where
$AA=\frac{\rho^{\text{SI}}}{\alpha_{\text{anc}}}\mathbb{E}\\{||\boldsymbol{w}_{k_{\text{u}}}||^{2}\\}(1+\alpha_{\text{tx}})$,
$|\boldsymbol{w}_{k_{\text{u}}}^{T}\boldsymbol{\hat{h}}_{\text{UL},k_{\text{u}}}|^{2}=1$,
$|\boldsymbol{w}_{k_{\text{u}}}^{T}\boldsymbol{\hat{h}}_{\text{UL},\ell}|^{2}=0$,
and
$\mathbb{E}\left\\{\frac{1}{||\boldsymbol{w}_{k_{\text{u}}}||^{2}}\right\\}=\frac{K_{\text{UL}}\rho^{\text{UL}}_{k_{\text{u}}}+1}{K_{\text{UL}}\rho^{\text{UL}}_{k_{\text{u}}}}(N-K_{\text{UL}}+1)$,
which is calculated from (3).

Similarly, from (8) and (22), we can derive the approximation of the uplink
SINR with the SI subtraction as follows

$\displaystyle\bar{\gamma}_{\text{stt},k_{\text{u}}}^{\text{UL}}=\frac{(K_{\text{UL}}\rho^{\text{UL}}_{k_{\text{u}}}+1)(N-K_{\text{UL}}+1)}{K_{\text{UL}}(\sum_{\ell=1}^{K_{\text{UL}}}\frac{\rho^{\text{UL}}_{\ell}}{K_{\text{UL}}\rho^{\text{UL}}_{\ell}+1}+1)+\frac{\rho^{\text{SI}}}{\alpha_{\text{anc}}}\epsilon_{{}_{\text{SI}}}^{2}K_{\text{UL}}(1+\alpha_{\text{tx}})}.$
(26)

Finally, we present the uplink SINR for spatial suppression. To obtain the
independent condition, we should consider, given that
$\boldsymbol{H}_{\text{SI}}$ and $\boldsymbol{G}_{\text{sps}}$ are not
independent of each other, the best and worst estimation errors to obtain the
independent condition. In the case of the best channel estimation error
($\epsilon_{{}_{\text{SI}}}\rightarrow 0$), the estimated SI channel tends to
the original SI channel
($\boldsymbol{\hat{H}}_{\text{SI}}\rightarrow\boldsymbol{H}_{\text{SI}}$), and
thus $\boldsymbol{E}_{\text{SI}}$ and $\boldsymbol{G}_{\text{sps}}$ tend to be
independent. Accordingly, from _Lemma_ 2, the expected SI power of the spatial
suppression (the best case) is given by

$\displaystyle{\bar{\Omega}}_{\text{sps}(\epsilon_{{}_{\text{SI}}}\rightarrow
0)}$
$\displaystyle=\mathbb{E}\left\\{||\boldsymbol{w}_{k_{\text{u}}}^{T}\boldsymbol{H}_{\text{SI}}\bar{\boldsymbol{G}}_{\text{sps}}||^{2}\right\\}$
$\displaystyle=\mathbb{E}\left\\{||\boldsymbol{w}_{k_{\text{u}}}^{T}(\boldsymbol{\hat{H}}_{\text{SI}}-\boldsymbol{E}_{\text{SI}})\bar{\boldsymbol{G}}_{\text{sps}}||^{2}\right\\}$
$\displaystyle=\mathbb{E}\left\\{||\boldsymbol{w}_{k_{\text{u}}}^{T}\boldsymbol{E}_{\text{SI}}\bar{\boldsymbol{G}}_{\text{sps}}||^{2}\right\\}$
$\displaystyle=\epsilon_{{}_{\text{SI}}}^{2}\mathbb{E}\left\\{||\boldsymbol{w}_{k_{\text{u}}}||^{2}\right\\}.$
(27)

Similarly, in the case of the worst channel estimation error
($\epsilon_{{}_{\text{SI}}}\rightarrow\infty$), the estimated SI channel tends
to the channel estimation error matrix
($\boldsymbol{\hat{H}}_{\text{SI}}\rightarrow\boldsymbol{E}_{\text{SI}}$) and
$\boldsymbol{\hat{H}}_{\text{ext}}$ tends to
$\begin{bmatrix}\boldsymbol{\hat{H}}_{\text{DL}}\\\
\boldsymbol{E}_{\text{SI}}\end{bmatrix}$. Therefore,
$\boldsymbol{H}_{\text{SI}}$ and $\boldsymbol{G}_{\text{sps}}$ tend to be
independent. Thus, the expected SI power of the spatial suppression (the worst
case) is given by

$\displaystyle{\bar{\Omega}}_{\text{sps}(\epsilon_{{}_{\text{SI}}}\rightarrow\infty)}=\mathbb{E}\left\\{||\boldsymbol{w}_{k_{\text{u}}}^{T}\boldsymbol{H}_{\text{SI}}\bar{\boldsymbol{G}}_{\text{sps}}||^{2}\right\\}=\mathbb{E}\left\\{||\boldsymbol{w}_{k_{\text{u}}}||^{2}\right\\}.$
(28)

From (27) and (28), the approximation of the expected SI power of spatial
suppression is expressed as
${\bar{\Omega}}_{\text{sps}}\approx\frac{\mathbb{E}\left\\{||\boldsymbol{w}_{k_{\text{u}}}||^{2}\right\\}}{1/\epsilon_{{}_{\text{SI}}}^{2}+1}.$
This approximation approach is a simple application of harmonic mean. The
approximation of the uplink SINR with spatial suppression is then given by

$\displaystyle\bar{\gamma}_{\text{sps},k_{\text{u}}}^{\text{UL}}$
$\displaystyle=\frac{(K_{\text{UL}}\rho^{\text{UL}}_{k_{\text{u}}}+1)(N-K_{\text{UL}}+1)}{K_{\text{UL}}(\sum_{\ell=1}^{K_{\text{UL}}}\frac{\rho^{\text{UL}}_{\ell}}{K_{\text{UL}}\rho^{\text{UL}}_{\ell}+1}+1)+\frac{\rho^{\text{SI}}}{\alpha_{\text{anc}}}\left(\frac{1}{1/\epsilon_{{}_{\text{SI}}}^{2}+1}\right)K_{\text{UL}}(1+\alpha_{\text{tx}})}.$
(29)

###  III-C Expected SI Power in Correlated Channels

In this subsection, we compare the expected SI power of the two SIC methods by
considering the channel correlation. The estimated channel
$\hat{\boldsymbol{H}}$ is correlated when the channel is correlated while the
estimation errors, elements of $\boldsymbol{E}$, are uncorrelated [36]. We
first investigate how the correlated $\hat{\boldsymbol{H}}_{\text{SI}}$
affects the level of SI power when $\boldsymbol{G}_{\text{sps}}$ is used.

###### Lemma 4

When considering the correlated $\boldsymbol{H}_{\text{SI}}$,
$\zeta_{\text{sps}}$ is given by

$\displaystyle\zeta_{\text{sps}}=\text{Cov}(|E_{ik}|^{2},|g_{km,\text{sps}}|^{2})+\\!\\!\\!\\!\\!\sum_{\ell=1,\ell\neq
k}^{M}\\!\\!\\!\\!\\!\epsilon_{{}_{\text{SI}}}^{2}r_{im,\text{sps}}\frac{\sqrt{qr_{k\ell,\text{SI}}^{2}+1}}{MK_{\text{DL}}},$

where
$r_{im,\text{sps}}=\text{Corr}\left(E_{ik}E_{i\ell}^{*},g_{km,\text{sps}}g_{\ell
m,\text{sps}}^{*}\right)$ and
$r_{k\ell,\text{SI}}=\text{Corr}\left(g_{km,\text{sps}},g_{\ell
m,\text{sps}}^{*}\right)$. In addition, $q$ is a constant that depends on the
distribution of $|g_{km,\text{sps}}|^{2}$ (if the distribution is approximated
to an exponential distribution, then $q=2$).

###### Proof:

See Appendix C. ∎

Because $r_{k\ell,\text{SI}}^{2}$ is proportional to the spatial correlation
of the highly correlated SI channel, from _Lemma_ 3 and _Lemma_ 4, we can
conclude that as the correlation of the SI channel increases, the expected SI
power of the spatial suppression decreases. Meanwhile, the correlation of the
SI channel does not affect the expected SI power of the SI subtraction because
$r_{im,\text{sps}}=0$.

Next, we will show that the uplink performance of spatial suppression is
better than that of the SI subtraction in the correlated channels with the
practical angle distribution that is based on a Bessel function (e.g., Laplace
distribution and wrapped Gaussian distribution [37]).

###### Lemma 5

In the correlated channels of the $\boldsymbol{H}_{\text{SI}}$ and
$\boldsymbol{H}_{\text{DL/UL}}$,
$\bar{\Omega}_{\text{stt}}\geq\bar{\Omega}_{\text{sps}}$ holds, which can be
expressed as

$\displaystyle\bar{\Omega}_{\text{sps}}-\bar{\Omega}_{\text{stt}}$
$\displaystyle=\sum_{i=1}^{N}\mathbb{E}\left\\{|w_{k_{\text{u}}i}|^{2}\right\\}\sum_{m=1}^{L}\sum_{k=1}^{M}\zeta_{\text{sps}}$
$\displaystyle~{}~{}+c\sum_{i=1}^{N}\sum_{j=1,j\neq
i}^{N}\text{Cov}\left(w_{k_{\text{u}}i},w_{k_{\text{u}}j}^{*}\right)\leq 0,$

where
$c=\sum_{m=1}^{L}\sum_{k=1}^{M}\sum_{\ell=1}^{M}\mathbb{E}\left\\{E_{ik}E_{j\ell}^{*}g_{km,\text{sps}}g_{\ell
m,\text{sps}}^{*}\right\\}$.

###### Proof:

See Appendix D. ∎

Figure 3: The digital map for the 3D ray-tracing simulation and the absolute
values of the measured $\boldsymbol{R}_{\text{TX(SI)}}$ through the 3D ray-
tracing simulation.

##  IV Performance Comparison with System Constraints

In this section, we compare the performance of the two SIC methods while
considering the following system constraints: the downlink-to-uplink data
traffic ratio under a non-full buffer scenario, the number of transmit and
receive RF chains ratio with transmit power at the BS, and power scaling law
at the BS. We hope that this discussion elucidates FD massive MIMO performance
under various system constraints.

###  IV-A Optimal Antenna Ratio in a Non-Full Buffer Scenario

We first show that an asymmetric RF configuration is advantageous in a non-
full-buffer asymmetric data traffic scenario. The traffic load density at the
downlink and uplink are denoted by $\mu^{\text{DL}}$ and $\mu^{\text{UL}}$,
and $T^{\text{DL}}=\mu^{DL}/\mathcal{R}^{\text{DL}}$; the transmission time
for the given traffic loads is
$T^{\text{UL}}=\mu^{UL}/\mathcal{R}^{\text{UL}}$, represented in [38]. We
define the required downlink and uplink rates as the data rates for achieving
$T^{\text{DL}}=T^{\text{UL}}=T$ in the FD mode within a certain time ($T$) to
avoid half-duplex transmission, and their ratio is represented as $\eta$,
which is the ratio of downlink traffic load to the uplink traffic load,
$\eta=\mu^{DL}/\mu^{UL}$. Simple performance analysis for the optimal RF chain
ratio is given as follows.

###### Remark 3

Assuming that the per-user data traffic is identical under asymmetric data
traffic scenarios, the required downlink rate to uplink rate ratio can be
expressed as $\eta=K_{\text{DL}}/K_{\text{UL}}$. In this scenario, we can
obtain insight into the optimal ratio of the numbers of transmit RF chains and
receive RF chains, and the optimal ratio of the SI subtraction and the spatial
suppression given by $M/N=\eta$ and $(M-N)/N=\eta$, respectively, derived from
the result in [25].

_Remark_ 3 implies that an asymmetric RF configuration is better than a
symmetric RF configuration. In addition, this implication holds even if
different per-user data traffic is assumed. It should be noted that the
performance with the optimal RF chain ratio is not investigated in the
following subsections, because the traffic ratio changes while the RF chain
ratio at the BS is fixed. Instead, we present insights into the trade-off
between the two SIC methods by considering arbitrary data traffic.

###  IV-B Performance Gap between Downlink and Uplink Rate with the Same
Transmit Power

We assume that the numbers of downlink and uplink users are equal (i.e.,
$K=K_{\text{DL}}=K_{\text{UL}}$), while the downlink data traffic and uplink
data traffic differ. In addition, the propagation gains have the same values
for all $k_{\text{d}}$ and $k_{\text{u}}$. We analyze the performance from a
per-user asymmetric data traffic perspective and consider, for both links,
high SINR regimes. SINRs can be expressed in terms of the received downlink
SNR ($\varphi$, proportional to $\rho_{\text{d}}$), where the desired signal
power is relatively higher than others at the downlink, and the SI power is
relatively smaller than the desired power at the uplink. For example, from
(16)-(18), $\varphi$ can be denoted by a linear function of $\rho_{\text{d}}$
( $\varphi=\kappa\rho_{\text{d}}$) in the perfect channel estimation
condition, where
$\kappa=\frac{\beta^{\text{DL}}_{k_{\text{d}}}}{K_{\text{DL}}(\sum_{k_{\text{u}}=1}^{K_{\text{UL}}}\rho^{\text{UU}}_{k_{\text{d}},k_{\text{u}}}+1)}$.
In these regimes, we can approximate the SINRs as follows:
$\bar{\gamma}_{\text{stt}}^{\text{DL}}(\varphi)=(M-K)\varphi$,
$\bar{\gamma}_{\text{sps}}^{\text{DL}}(\varphi)=(M-N-K)\varphi$,
$\bar{\gamma}_{\text{stt}}^{\text{UL}}(\varphi)=\frac{\mathcal{S}}{\mathcal{I}+\omega_{\text{stt}}\varphi}$,
$\bar{\gamma}_{\text{sps}}^{\text{UL}}(\varphi)=\frac{S}{\mathcal{I}+(\omega_{\text{stt}}-\omega_{{}_{\varDelta}})\varphi}$,
where $\mathcal{S}$, $\mathcal{I}$, $\omega_{\text{stt}}$, and
$\omega_{\text{sps}}(=\omega_{\text{stt}}-\omega_{{}_{\varDelta}})$ are scaled
terms of the desired power, interference power, SI power of the SI
subtraction, and SI power of the spatial suppression, respectively, by
$\varphi$:

$\displaystyle\bar{\gamma}_{\text{sps}}^{\text{DL}}(\varphi)=\frac{M-K-N}{M-K}\bar{\gamma}_{\text{stt}}^{\text{DL}}(\varphi)=\bar{\gamma}_{\text{stt}}^{\text{DL}}(\varphi)-\delta^{\text{DL}}\bar{\gamma}_{\text{stt}}^{\text{DL}}(\varphi),$
$\displaystyle\bar{\gamma}_{\text{stt}}^{\text{UL}}(\varphi)=\frac{\mathcal{I}+(\omega_{\text{stt}}-\omega_{\varDelta})\varphi}{\mathcal{I}+\omega_{\text{stt}}\varphi}\bar{\gamma}_{\text{sps}}^{\text{UL}}(\varphi)=\bar{\gamma}_{\text{sps}}^{\text{UL}}(\varphi)-\delta^{\text{UL}}\bar{\gamma}_{\text{sps}}^{\text{UL}}(\varphi),$
(30)

where $\delta^{\text{DL}}=\frac{N}{M-K}>0$, and
$\delta^{\text{UL}}=\frac{\omega_{\varDelta}\varphi}{\mathcal{I}+\omega_{\text{stt}}\varphi}\geq
0$. Thus, the approximations of the per-user rates are given by

$\displaystyle\mathcal{R}_{\text{sps}}^{\text{DL}}(\varphi)=\mathcal{R}_{\text{stt}}^{\text{DL}}(\varphi)+\log_{2}\left(1-\delta^{\text{DL}}\frac{\bar{\gamma}_{\text{stt}}^{\text{DL}}(\varphi)}{1+\bar{\gamma}_{\text{stt}}^{\text{DL}}(\varphi)}\right),$
$\displaystyle\mathcal{R}_{\text{stt}}^{\text{UL}}(\varphi)=\mathcal{R}_{\text{sps}}^{\text{UL}}(\varphi)+\log_{2}\left(1-\delta^{\text{UL}}\frac{\bar{\gamma}_{\text{sps}}^{\text{UL}}(\varphi)}{1+\bar{\gamma}_{\text{sps}}^{\text{UL}}(\varphi)}\right).$
(31)

Thus, the gaps in the approximated rates between the two SIC methods at each
link are expressed as

$\displaystyle\varDelta^{\text{DL}}(\varphi)=-\log_{2}\left(1-\delta^{\text{DL}}\frac{\bar{\gamma}_{\text{stt}}^{\text{DL}}(\varphi)}{1+\bar{\gamma}_{\text{stt}}^{\text{DL}}(\varphi)}\right)\geq
0,$
$\displaystyle\varDelta^{\text{UL}}(\varphi)=-\log_{2}\left(1-\delta^{\text{UL}}\frac{\bar{\gamma}_{\text{sps}}^{\text{UL}}(\varphi)}{1+\bar{\gamma}_{\text{sps}}^{\text{UL}}(\varphi)}\right)\geq
0.$ (32)

By considering that
$\mathcal{R}_{\varDelta}(\varphi)=\mathcal{R}_{\text{stt}}^{\text{DL}}(\varphi)-\mathcal{R}_{\text{sps}}^{\text{UL}}(\varphi)$
is the cross point when the gaps between the downlink and uplink rates of each
SIC method are the same, we obtain

$\displaystyle\mathcal{R}_{\text{stt}}^{\text{DL}}(\varphi)-\mathcal{R}_{\text{stt}}^{\text{UL}}(\varphi)=\mathcal{R}_{\varDelta}(\varphi)+\varDelta^{\text{UL}}(\varphi),$
$\displaystyle\mathcal{R}_{\text{sps}}^{\text{DL}}(\varphi)-\mathcal{R}_{\text{sps}}^{\text{UL}}(\varphi)=\mathcal{R}_{\varDelta}(\varphi)-\varDelta^{\text{DL}}(\varphi),$
(33)

and we can obtain the inequality as
$\frac{\mathcal{R}_{\text{stt}}^{\text{DL}}}{\mathcal{R}_{\text{stt}}^{\text{UL}}}\geq\frac{\mathcal{R}_{\text{sps}}^{\text{DL}}}{\mathcal{R}_{\text{sps}}^{\text{UL}}}.$

Therefore, because the gap between the downlink and uplink rates of the SI
subtraction is higher than that of the spatial suppression for the given
transmit power, the SI subtraction is more desirable when the gap between the
downlink and uplink data traffic is higher (i.e.,
$\eta_{\text{stt}}\geq\eta_{\text{sps}}$), and vice versa.

###  IV-C Power Scaling Law in Downlink

Next, we consider a more practical constraint for asymmetric data traffic
scenarios. In practical FD systems, achieving the maximum downlink rate can
result in a bottleneck for supporting more downlink traffic and reducing the
total transmission time of data traffic in the buffer at both links. Here, the
maximum downlink rate is defined as the achievable rate by applying the best
modulation order and coding rate. We denote it as
$\tilde{\mathcal{R}}_{\text{max}}^{\text{DL}}$. In addition, we define the
minimum required SNRs to achieve the maximum downlink rates as follows:

$\displaystyle\tilde{\varphi}_{\text{stt}}=\underset{\varphi}{\arg\min}\left\\{\tilde{\mathcal{R}}_{\text{max}}^{\text{DL}}\leq\log_{2}(1+\bar{\gamma}^{\text{DL}}_{\text{stt}}(\varphi))\right\\},$
$\displaystyle\tilde{\varphi}_{\text{sps}}=\underset{\varphi}{\arg\min}\left\\{\tilde{\mathcal{R}}_{\text{max}}^{\text{DL}}\leq\log_{2}(1+\bar{\gamma}^{\text{DL}}_{\text{sps}}(\varphi))\right\\},$
(34)

where $\tilde{\varphi}_{\text{stt}}\leq\tilde{\varphi}_{\text{sps}}$. From the
power scaling law in the downlink given by
$\tilde{\mathcal{R}}_{\text{max}}^{\text{DL}}=\mathcal{R}_{\text{stt}}^{\text{DL}}(\tilde{\varphi}_{\text{stt}})=\mathcal{R}_{\text{sps}}^{\text{DL}}(\tilde{\varphi}_{\text{sps}})$,
we have
$\tilde{\varphi}_{\text{stt}}=\frac{M-K-N}{M-K}\tilde{\varphi}_{\text{sps}}.$
We compare the uplink SINR of the SI subtraction at
$\tilde{\varphi}_{\text{stt}}$ with that of the spatial suppression at
$\tilde{\varphi}_{\text{sps}}$ as follows:

$\displaystyle\frac{\bar{\gamma}_{\text{stt}}^{\text{UL}}(\tilde{\varphi}_{\text{stt}})}{\bar{\gamma}_{\text{sps}}^{\text{UL}}(\tilde{\varphi}_{\text{sps}})}$
$\displaystyle=\frac{\mathcal{I}+(\omega_{\text{stt}}-\omega_{\varDelta})\tilde{\varphi}_{\text{sps}}}{\mathcal{I}+\omega_{\text{stt}}\tilde{\varphi}_{\text{stt}}}=\frac{\mathcal{I}+(\omega_{\text{stt}}-\omega_{\varDelta})\tilde{\varphi}_{\text{sps}}}{\mathcal{I}+\omega_{\text{stt}}\frac{M-K-N}{M-K}\tilde{\varphi}_{\text{sps}}}$
$\displaystyle=\frac{\mathcal{I}+(\omega_{\text{stt}}-\omega_{\varDelta})\tilde{\varphi}_{\text{sps}}}{\mathcal{I}+(\omega_{\text{stt}}-\frac{N}{M-K}\omega_{\text{stt}})\tilde{\varphi}_{\text{sps}}}.$
(35)

In the correlated SI channel, the cross point of the uplink performance can be
expressed as

$\displaystyle\mathcal{R}^{\text{UL}}_{\text{cross}}$
$\displaystyle=\frac{\omega_{\varDelta}}{\frac{N}{M-K}\omega_{\text{stt}}}$
$\displaystyle=\underbrace{\frac{M-K}{N}}_{\begin{subarray}{c}\text{trade-off
from}\\\ \text{an RF
configuration}\end{subarray}}\cdotp\underbrace{\sum^{K}_{m}\sum^{M}_{k}\sum^{M}_{\ell}\frac{\sqrt{qr_{k\ell,\text{SI}}^{2}+1}}{MK}r_{im,\text{sps}}}_{\text{trade-
off from the channel estimation error}},$ (36)

where $\mathcal{R}^{\text{UL}}_{\text{cross}}\geq 1$ represents
${\bar{\gamma}_{\text{sps}}^{\text{UL}}(\tilde{\varphi}_{\text{sps}})\geq\bar{\gamma}_{\text{stt}}^{\text{UL}}(\tilde{\varphi}_{\text{stt}})}$.

According to the discussion in subsection IV-B, an increase in the number of
transmit RF chains and the use of SI subtraction is desirable to support
massive downlink traffic when the same transmit power is assumed. However, if
the transmit power is scaled down to conserve energy, SI subtraction with such
an RF configuration under performs the spatial suppression.

Figure 4: Downlink/uplink sum rate with perfect channel estimation.

##  V Performance Evaluation

###  V-A Simulation Parameters

For performing numerical comparisons, we assumed that the FD BS has 88 RF
chains ($M=64$, and $N=24$) and serves 12 users for each downlink and uplink.
The BS is equipped with dual-polarization antennas with a 15° tilt. We assume
that 24 transmit RF chains and 24 receive RF chains are connected to
horizontally polarized Port-1 and vertically polarized Port-2, respectively,
while 40 transmit RF chains are connected to both ports. The uniform linear
array with unit wavelength ($1\lambda_{\text{c}}$) spacing is assumed at 2.52
GHz of the center frequency. We also assume that $\alpha_{\text{anc}}=30$ dB,
which is reproduced from the cross-polarization ratio (XPR) of the dual-
polarization antenna. We assume that the variance of the transmitter noise is
large, and set it as $\alpha_{\text{tx}}=-10$ dB owing to the chip hardware
cost for installing several RF chains. Finally, the simulation parameters are
set as $\rho_{{}_{\text{UL}}}=10$ dB, $\beta^{\text{SI}}=-40$ dB,
$\beta^{\text{DL}}=-80$ dB, $\beta^{\text{UL}}=-80$ dB, and
$\beta^{\text{UU}}=-100$ dB, which are reproduced from the typical values of
the path loss at the reference meter (1 m) and those in small cell
environments.

###  V-B Channel Model for SI Channel

####  V-B1 Scenario I: SI Channel with Simple Correlation Matrices

The correlation matrices are simply modeled as

$\displaystyle\boldsymbol{R}_{\text{TX(SI)}}=\begin{bmatrix}1&r_{\text{TX,SI}}&\dots&r_{\text{TX,SI}}^{M-2}&r_{\text{TX,SI}}^{M-1}\\\
\vdots&&\ddots&&\vdots\\\
r_{\text{TX,SI}}^{M-1}&r_{\text{TX,SI}}^{M-2}&\dots&r_{\text{TX,SI}}&1\end{bmatrix},$
$\displaystyle\boldsymbol{R}_{\text{RX(SI)}}=\begin{bmatrix}1&r_{\text{RX,SI}}&\dots&r_{\text{RX,SI}}^{N-2}&r_{\text{RX,SI}}^{N-1}\\\
\vdots&&\ddots&&\vdots\\\
r_{\text{RX,SI}}^{N-1}&r_{\text{RX,SI}}^{N-2}&\dots&r_{\text{RX,SI}}&1\end{bmatrix}.$

According to [39], we set $r_{\text{TX/RX,SI}}$ as 0.2 at a low correlation
and 0.8 at a high correlation.

Figure 5: Ergodic sum rate of the SI subtraction and the spatial suppression
at high correlation and low correlation in the SI channel. The legends _H_ and
_L_ denote high correlation and low correlation, respectively.

####  V-B2 Scenario II: Practical SI Channel Measured from a 3D Ray-Tracing
Tool

Figure 6: Ergodic uplink sum rate of the SI subtraction and the spatial
suppression at high correlation with different RF configuration: a)
$M=64,N=27$; b) $M=64,N=40$; c) $M=128,N=27$.

We develop a system-level simulator based on a 3D ray-tracing tool (WiSE,
developed by Bell Labs [40]). We produce a customized digital map of a real
building for an indoor scenario. Fig. 3 presents the test site from a third-
floor perspective. The digital map includes concrete walls and floors,
metallic doors, glass windows, and sheetrock ceilings. The BS is located
immediately under a 3 m-high ceiling in a room facing the door to the south.
We use both the co-polarization and cross-polarization patterns of the
fabricated dual-polarization antenna, which are measured from an anechoic
chamber [18].444For modeling a more realistic dual-polarized channel, the XPR
values of each ray are applied to co- and cross-polarization antenna patterns
instead of using static XPR values in a polarization transfer matrix. To
realize an SI channel, we measure the power delay profiles of the $\ell$-th
channel tap ($P_{\ell}$) and angle parameters of the $\ell$-th channel tap,
such as the azimuth angle of arrival (AOA, $\phi_{\ell}^{\text{AOA}}$),
azimuth angle of departure (AOD, $\phi_{\ell}^{\text{AOD}}$), zenith angle of
arrival (ZOA, $\theta_{\ell}^{\text{ZOA}}$), and zenith angle of departure
(ZOD, $\theta_{\ell}^{\text{ZOD}}$). The BS and users are equipped with the
manufactured antenna and the isotropic antenna, respectively.

Next, we realized the SI channel in a realistic indoor scenario. We denote co-
polarization and cross-polarization patterns of Port 1 by
$G_{\text{rx}}^{\text{V}}$ and $G_{\text{rx}}^{\text{H}}$, respectively, and
co-polarization and cross-polarization patterns of Port 2 by
$G_{\text{tx}}^{\text{H}}$ and $G_{\text{tx}}^{\text{V}}$, respectively. Using
a 3D ray-tracing simulation, we measured the $L_{\text{tap}}$ of the channel
taps. Based on a clustered channel model [39], the $(n,m)$-th element of the
measured SI channel can be obtained as follows:

$\displaystyle[\boldsymbol{H}_{\text{SI}}]_{nm}$
$\displaystyle=\sum_{\ell=1}^{L_{\text{tap}}}P_{\ell}\begin{bmatrix}G_{\text{rx}}^{\text{H}}\left(\theta_{\ell}^{\text{ZOA}},\phi_{\ell}^{\text{AOA}}\right)\\\
G_{\text{rx}}^{\text{V}}\left(\theta_{\ell}^{\text{ZOA}},\phi_{\ell}^{\text{AOA}}\right)\end{bmatrix}^{T}$
$\displaystyle\times\begin{bmatrix}\exp\left(j\Phi^{\text{HH}}_{\ell}\right)&\exp\left(j\Phi^{\text{VH}}_{\ell}\right)\\\
\exp\left(j\Phi^{\text{HV}}_{\ell}\right)&\exp\left(j\Phi^{\text{VV}}_{\ell}\right)\end{bmatrix}\begin{bmatrix}G_{\text{tx}}^{\text{H}}\left(\theta_{\ell}^{\text{ZOD}},\phi_{\ell}^{\text{AOD}}\right)\\\
G_{\text{tx}}^{\text{V}}\left(\theta_{\ell}^{\text{ZOD}},\phi_{\ell}^{\text{AOD}}\right)\end{bmatrix}$
$\displaystyle\times\exp{\left(j\frac{2\pi}{\lambda_{\text{c}}}\boldsymbol{\hat{r}}_{\text{rx},\ell}\boldsymbol{\bar{d}}_{\text{rx},n}\right)}\exp{\left(j\frac{2\pi}{\lambda_{\text{c}}}\boldsymbol{\hat{r}}_{\text{tx},\ell}\boldsymbol{\bar{d}}_{\text{tx},m}\right)},$
(37)

where $\boldsymbol{\bar{d}}_{\text{tx},m}$ and
$\boldsymbol{\bar{d}}_{\text{rx},m}$ are the location vectors of the transmit
antenna $m$ and receive antenna $n$, respectively. In addition,
$\boldsymbol{\hat{r}}_{\text{tx},\ell}$ and
$\boldsymbol{\hat{r}}_{\text{rx},\ell}$ are spherical unit vectors with
$\phi_{\ell}^{\text{AOA}}$ and $\theta_{\ell}^{\text{ZOA}}$ and with
$\phi_{\ell}^{\text{AOD}}$ and $\theta_{\ell}^{\text{ZOD}}$, respectively.
From (37), the measured correlation matrices can be calculated from
$\boldsymbol{R}_{\text{TX(SI)}}=\mathbb{E}[\boldsymbol{H}_{\text{SI}}^{*}\boldsymbol{H}_{\text{SI}}]$
and
$\boldsymbol{R}_{\text{RX(SI)}}=\mathbb{E}[\boldsymbol{H}_{\text{SI}}\boldsymbol{H}_{\text{SI}}^{*}]$.
The absolute values of the measured $\boldsymbol{R}_{\text{TX(SI)}}$ are
presented in Fig. 3. We can confirm that the practical SI channel is highly
correlated owing to the high direct path power and small angular spread.

###  V-C Numerical Result

We first compare the proposed analysis and simulation results where perfect
channel estimation is assumed at the BS. Fig. 4 shows that the results from
III-A3 are approximately the same as the ergodic downlink/uplink sum rate of
the FD-BS without SIC and with SI subtraction/spatial suppression. Because the
SI is perfectly canceled out owing to perfect knowledge of the SI channel,
both the uplink performance of the spatial suppression and the SI subtraction
are identical, and they use the same ZF receiver. That is to say, there is a
performance gap only in the downlink case. An additional spatial domain $(N)$
is needed to cancel out the SI channel at the FD-BS with spatial suppression.
Thus, we confirm that the SI subtraction outperforms the spatial suppression
in the perfect channel estimation condition. For all the results below,
imperfect channel estimation is assumed and simulated by using MATLAB.

Figure 7: Ergodic uplink sum rate of the SI subtraction and the spatial
suppression in the practical SI channel.

####  V-C1 Scenario I

Fig. 5 shows the ergodic uplink sum rate of the SI subtraction and spatial
suppression at high and low correlations. The SI subtractions always exhibit
the same performance irrespective of the correlation of the SI channel.
Meanwhile, as the correlation of the SI channel increases, the performance of
the spatial suppression increases owing to the projection onto the estimated
SI channel manifold. This result verifies _Lemma_ 4.

Fig. 6 presents the ergodic uplink sum rate of the SI subtraction and the
spatial suppression at high correlation with different RF configurations when
the power scaling law at the downlink is assumed. A performance trade-off, as
shown in (36), occurs in the RF configuration. In the uplink, spatial
suppression outperforms the SI subtraction when $M/N$ is high, and vice versa.
Under these constraints, both methods have the same downlink sum rate under
these constraints. Furthermore, the opposite result is observed as compared to
_Remark_ 3.

Figure 8: The performance gap between the ergodic downlink sum rate and the
ergodic uplink sum rate of the SI subtraction and the spatial suppression in
the practical SI channel.

####  V-C2 Scenario II

Fig. 7 illustrates the ergodic uplink sum rate of the SI subtraction and
spatial suppression when the measured SI channel is considered. To investigate
the correlation impact, we additionally evaluated the sum rate with an i.i.d.
SI channel. When $M$ increases and the correlation of the SI channel
increases, no performance gain can be obtained by the SI subtraction. In
addition, there is a significant performance gain with spatial suppression in
the practical SI channel because of the high correlation.

Fig. 8 presents the performance gap between the ergodic downlink sum rate and
the ergodic uplink sum rate of the two SIC schemes. From the analysis in
Section IV, $\varDelta^{\text{DL}}$ increases as $\delta^{\text{DL}}$
increases, which means that an additional spatial domain ($N$) is needed to
cancel out the SI channel with spatial suppression. In addition,
$\varDelta^{\text{UL}}$ increases as $\delta^{\text{UL}}$ increases, where
$\delta^{\text{UL}}$ is proportional to $\epsilon_{{}_{\text{SI}}}$ and the
correlation of the SI channel. Thus, the gap between the downlink and uplink
rates of the SI subtraction is higher than that of the spatial suppression in
the high SINR regimes in both links.

From the analysis and numerical results, we have shown which SIC method has
better downlink and uplink sum rates in different channel conditions. In terms
of total sum rates, Table II summarizes the method most suitable under several
given system constraints.

Table II: The desired SIC technique under given system constraints with the corresponding formulas linked as references.666It is worth noting that power scaling law in the downlink is assumed under the constraints marked with (*). Also (**) works when the gap between the downlink and uplink data traffic is greater (i.e., $\eta_{\text{stt}}\geq\eta_{\text{sps}}$). Given System Constraints | Desired SIC Technique
---|---
Perfect Channel Estimation (19, 20) | SI Subtraction
Highly Correlated Channel (*) (36) | Spatial Suppression
High $M/N$ (*) (36) | Spatial Suppression
Total Transmit Power (**) (IV-B) | SI Subtraction

##  VI Conclusions

In this paper, two SIC methods in FD massive MIMO systems, i.e., SI
subtraction and spatial suppression, have been investigated by considering the
SI channel’s estimation error and spatial correlation. In addition, we
conducted an SI channel measurement campaign using a 3D ray-tracing
simulation. The analysis and numerical results reveal that, in the uplink with
imperfect channel estimation, the ergodic performance of the spatial
suppression is better than that of the SI subtraction. This performance gain
is caused by the correlation between the spatial suppression precoder and the
estimation error of the SI channel. Moreover, we provided insight into which
SIC algorithm is better under given system constraints, such as the
downlink/uplink traffic ratio, number of RF chains, total transmit power, and
power scaling law at the BS.

###  -A Proof of Lemma 2

From _Lemma_ 1, all residual expectation terms in (14) are zero because of the
independent conditions that result in
$\mathbb{E}\left\\{w_{k_{\text{u}}i}w_{k_{\text{u}}j}^{*}\right\\}=\mathbb{E}\left\\{w_{k_{\text{u}}i}\right\\}\mathbb{E}\left\\{w_{k_{\text{u}}j}^{*}\right\\}=0$,
$\text{Cov}(|a_{ik}|^{2},|g_{km}|^{2})=0$, and
$\mathbb{E}\left\\{a_{ik}a_{i\ell}^{*}g_{km}g_{\ell m}^{*}\right\\}=0$, where
$\ell\neq k$. Then, (14) is given by

$\displaystyle\mathbb{E}\left\\{||\boldsymbol{w}_{k_{\text{u}}}^{T}\boldsymbol{A}\boldsymbol{G}||^{2}\right\\}$
$\displaystyle=\sum_{i=1}^{N}\mathbb{E}\left\\{|w_{k_{\text{u}}i}|^{2}\right\\}\sum_{m=1}^{L}\sum_{k=1}^{M}\mathbb{E}\left\\{|a_{ik}|^{2}\right\\}\mathbb{E}\left\\{|g_{km}|^{2}\right\\}$
$\displaystyle=\sum_{i=1}^{N}\mathbb{E}\left\\{|w_{k_{\text{u}}i}|^{2}\right\\}\sum_{m=1}^{L}\sum_{k=1}^{M}\sigma_{A}^{2}\mathbb{E}\left\\{|g_{km}|^{2}\right\\}$
$\displaystyle\overset{(b)}{=}\sigma_{A}^{2}\mathbb{E}\left\\{||\boldsymbol{w}_{k_{\text{u}}}||^{2}\right\\}.$
(38)

Equation (_b_) is derived based on
$\sum_{i=1}^{N}\mathbb{E}\left\\{|w_{k_{\text{u}}i}|^{2}\right\\}=\mathbb{E}\left\\{||\boldsymbol{w}_{k_{\text{u}}}||^{2}\right\\}$
and $\sum_{m=1}^{L}\sum_{k=1}^{M}\mathbb{E}\left\\{|g_{km}|^{2}\right\\}=1$,
owing to the definitions of $L^{2}$ norm and vector normalization.

###  -B Proof of Lemma 3

From $\mathbb{E}\left\\{E_{ik}E_{i\ell}^{*}\right\\}=0$ and
$\mathbb{E}\left\\{g_{km,\text{sps}}g_{\ell m,\text{sps}}^{*}\right\\}=0$,
$\zeta_{\text{sps}}$ is given by

$\displaystyle\zeta_{\text{sps}}=\text{Cov}(|E_{ik}|^{2},|g_{km,\text{sps}}|^{2})$
$\displaystyle~{}~{}+\sum_{\ell=1,\ell\neq k}^{M}$
$\displaystyle\left[\text{Cov}(E_{ik}E_{i\ell}^{*},g_{km,\text{sps}}g_{\ell
m,\text{sps}})+\mathbb{E}\left\\{E_{ik}E_{i\ell}^{*}\right\\}\mathbb{E}\left\\{g_{km,\text{sps}}g_{\ell
m,\text{sps}}^{*}\right\\}\right]$
$\displaystyle=\text{Cov}(|E_{ik}|^{2},|g_{km,\text{sps}}|^{2})+\sum_{\ell=1,\ell\neq
k}^{M}\text{Cov}(E_{ik}E_{i\ell}^{*},g_{km,\text{sps}}g_{\ell
m,\text{sps}}^{*}).$ (39)

$\boldsymbol{\bar{G}}_{\text{sps}}$ is the Pseudo-inverse of
$\begin{bmatrix}\boldsymbol{\hat{H}}_{\text{DL}}\\\
\boldsymbol{H}_{\text{SI}}+\boldsymbol{E}_{\text{SI}}\end{bmatrix}$, the
random variables, $E_{ik}$ and $g_{km,\text{sps}}$, tend to show opposite
behaviors. Thus, each covariance term is negative as follows:

$\displaystyle\text{Cov}(|E_{ik}|^{2},|g_{km,\text{sps}}|^{2})\leq 0,$
$\displaystyle\text{Cov}(E_{ik}E_{i\ell}^{*},g_{km,\text{sps}}g_{\ell
m,\text{sps}}^{*})\leq 0.$ (40)

From (39)-(40), we obtain $\zeta_{\text{sps}}\leq 0$.

###  -C Proof of Lemma 4

Let $z_{k}$ and $z_{\ell}$ be uncorrelated random variables with the same
distribution of $g_{km,\text{sps}}$. From the Cholesky decomposition, two
correlated $g_{km,\text{sps}}$ and $g_{\ell m,\text{sps}}$ can be expressed as

$\displaystyle g_{km,\text{sps}}=z_{k},~{}g_{\ell
m,\text{sps}}^{*}=r_{k\ell}z_{k}+\sqrt{1-r_{k\ell}^{2}}z_{\ell}.$ (41)

Then, we obtain the variance of their multiplication as follows:

$\displaystyle\text{Var}\left\\{g_{km,\text{sps}}g_{\ell
m,\text{sps}}^{*}\right\\}$
$\displaystyle=\mathbb{E}\\{|g_{km,\text{sps}}|^{2}|g_{\ell
m,\text{sps}}|^{2}\\}+[\mathbb{E}\\{g_{km,\text{sps}}g_{\ell
m,\text{sps}}^{*}\\}]^{2}$
$\displaystyle=r_{k\ell}^{2}\mathbb{E}\\{z_{k}\\}+2r_{k\ell}\sqrt{1-r_{k\ell}^{2}}\mathbb{E}\\{|z_{k}|^{2}z_{k}z_{\ell}\\}+r_{k\ell}\mathbb{E}\\{|z_{k}|^{2}\\}$
$\displaystyle~{}~{}+(1-r_{k\ell}^{2})\mathbb{E}\\{|z_{k}|^{2}|z_{\ell}|^{2}\\}+\sqrt{1-r_{k\ell}^{2}}\mathbb{E}\\{z_{k}z_{\ell}\\}$
$\displaystyle=r_{k\ell}^{2}\mathbb{E}\\{|g_{km,\text{sps}}|^{4}\\}+\mathbb{E}\\{|g_{km,\text{sps}}|^{2}\\}^{2},$
(42)

where $\mathbb{E}\\{|z_{k}|^{2}z_{k}z_{\ell}\\}=0$, and
$\mathbb{E}\\{z_{k}z_{\ell}\\}=0$. Thus, when $g_{km,\text{sps}}$ and $g_{\ell
m,\text{sps}}$ are correlated, the variance of their multiplication is given
by

$\displaystyle\text{Var}\left\\{g_{km,\text{sps}}g_{\ell
m,\text{sps}}^{*}\right\\}=r_{k\ell}^{2}\mathbb{E}\\{|g_{km,\text{sps}}|^{4}\\}+\mathbb{E}\\{|g_{km,\text{sps}}|^{2}\\}^{2}.$
(43)

From (39) and (43), we obtain

$\displaystyle\zeta_{\text{sps}}$
$\displaystyle=\text{Cov}(|E_{ik}|^{2},|g_{km,\text{sps}}|^{2})~{}~{}$
$\displaystyle~{}~{}+\\!\\!\\!\\!\\!\sum_{\ell=1,\ell\neq
k}^{M}\\!\\!\\!\\!\\!\text{Cov}\left\\{E_{ik}E_{i\ell}^{*},g_{km,\text{sps}}g_{\ell
m,\text{sps}}^{*}\right\\}$
$\displaystyle=\text{Cov}(|E_{ik}|^{2},|g_{km,\text{sps}}|^{2})$
$\displaystyle~{}~{}+\\!\\!\\!\\!\\!\sum_{\ell=1,\ell\neq
k}^{M}\\!\\!\\!\\!\\!r_{im,\text{sps}}\sqrt{\text{Var}[E_{ik}E_{i\ell}^{*}]\text{Var}[g_{km,\text{sps}}g_{\ell
m,\text{sps}}]}$
$\displaystyle=\text{Cov}(|E_{ik}|^{2},|g_{km,\text{sps}}|^{2})$
$\displaystyle~{}~{}+\sum_{\ell=1,\ell\neq
k}^{M}\epsilon_{{}_{\text{SI}}}^{2}r_{im,\text{sps}}\sqrt{r_{k\ell}^{2}\mathbb{E}\\{|g_{km,\text{sps}}|^{4}\\}+\mathbb{E}\\{|g_{km,\text{sps}}|^{2}\\}^{2}}$
$\displaystyle=\text{Cov}(|E_{ik}|^{2},|g_{km,\text{sps}}|^{2})+\\!\\!\\!\\!\\!\sum_{\ell=1,\ell\neq
k}^{M}\epsilon_{{}_{\text{SI}}}^{2}r_{im,\text{sps}}\frac{\sqrt{qr_{k\ell,\text{SI}}^{2}+1}}{MK_{\text{DL}}},$
(44)

where $\mathbb{E}[|g_{km}|^{2}]=\frac{1}{MK}$ and
$\mathbb{E}[|g_{km}|^{4}]=\frac{q}{(MK)^{2}}$.

###  -D Proof of Lemma 5

We assume
$b^{(+)}_{k}=\text{Sort}(|\text{Cov}\left(w_{k_{\text{u}}i},w_{k_{\text{u}}j}^{*}\right)|)$
for $\text{Cov}\left(w_{k_{\text{u}}i},w_{k_{\text{u}}j}^{*}\right)\geq 0$ and
$b^{(-)}_{\ell}=\text{Sort}(|\text{Cov}\left(w_{k_{\text{u}}i},w_{k_{\text{u}}j}^{*}\right)|)$
for $\text{Cov}\left(w_{k_{\text{u}}i},w_{k_{\text{u}}j}^{*}\right)\leq 0$.
For practical angle distributions, the correlation function in terms of
antenna spacing can be expressed by a zero-order Bessel function [41]. The
main lobe of a Bessel function is larger than the side lobes, so
$b^{(+)}_{1}>b^{(-)}_{1}$ holds. Thus, in the correlated
$\boldsymbol{H}_{\text{UL}}$, we can obtain

$\displaystyle\sum_{i=1}^{N}\sum_{j=1,j\neq
i}^{N}\text{Cov}\left(w_{k_{\text{u}}i},w_{k_{\text{u}}j}^{*}\right)\geq 0.$
(45)

From  _Lemma_ 1, we check the quadratic forms of $\bar{\Omega}_{\text{stt}}$
and $\bar{\Omega}_{\text{sps}}$ as follows:

$\displaystyle{\bar{\Omega}}_{\text{stt}}=\sum_{i=1}^{N}\mathbb{E}\left\\{|w_{k_{\text{u}}i}|^{2}\right\\}\sum_{m=1}^{L}\sum_{k=1}^{M}\mathbb{E}\left\\{|E_{ik}|^{2}\right\\}\mathbb{E}\left\\{|g_{km,\text{ZF}}|^{2}\right\\},$
(46)

where $\bar{\Omega}_{\boldsymbol{E}_{\text{SI}},(i\neq j),\text{stt}}=0$,
since $\mathbb{E}\left\\{E_{ik}E_{j\ell}^{*}g_{km,\text{ZF}}g_{\ell
m,\text{ZF}}^{*}\right\\}=\mathbb{E}\left\\{E_{ik}E_{j\ell}^{*}\right\\}\mathbb{E}\left\\{g_{km,\text{ZF}}g_{\ell
m,\text{ZF}}^{*}\right\\}=0$ when $i\neq j$,

$\displaystyle{\bar{\Omega}}_{\text{sps}}$
$\displaystyle=\sum_{i=1}^{N}\sum_{j=1}^{N}\mathbb{E}\left\\{w_{k_{\text{u}}i}w_{k_{\text{u}}j}^{*}\right\\}\sum_{m=1}^{L}\sum_{k=1}^{M}\sum_{\ell=1}^{M}\mathbb{E}\left\\{E_{ik}E_{j\ell}^{*}g_{km,\text{sps}}g_{\ell
m,\text{sps}}^{*}\right\\}.$ (47)

From (46), the expected SI power term of the SI subtraction is not affected by
the correlated $\boldsymbol{H}_{\text{DL}}$. Similar to (39), we have
$\mathbb{E}\left\\{E_{ik}E_{j\ell}^{*}g_{km,\text{sps}}g_{\ell
m,\text{sps}}^{*}\right\\}=\text{Cov}\left(E_{ik}E_{j\ell}^{*},g_{km,\text{sps}}g_{\ell
m,\text{sps}}^{*}\right)$. Then, we can obtain

$\displaystyle
c=\sum_{m=1}^{L}\sum_{k=1}^{M}\sum_{\ell=1}^{M}\mathbb{E}\left\\{E_{ik}E_{j\ell}^{*}g_{km,\text{sps}}g_{\ell
m,\text{sps}}^{*}\right\\}\leq 0,$ (48)

where $i\neq j$. From (46) and (47), the gap between
$\bar{\Omega}_{\text{stt}}$ and $\bar{\Omega}_{\text{sps}}$ in the correlated
channels can be derived as

$\displaystyle\bar{\Omega}_{\text{sps}}-\bar{\Omega}_{\text{stt}}$
$\displaystyle=\sum_{i=1}^{N}\mathbb{E}\left\\{|w_{k_{\text{u}}i}|^{2}\right\\}\sum_{m=1}^{L}\sum_{k=1}^{M}\zeta_{\text{sps}}$
$\displaystyle~{}~{}+\sum_{i=1}^{N}\sum_{j=1,j\neq
i}^{N}\mathbb{E}\left\\{w_{k_{\text{u}}i}w_{k_{\text{u}}j}^{*}\right\\}$
$\displaystyle~{}~{}\times\sum_{m=1}^{L}\sum_{k=1}^{M}\sum_{\ell=1}^{M}\mathbb{E}\left\\{E_{ik}E_{j\ell}^{*}g_{km,\text{sps}}g_{\ell
m,\text{sps}}^{*}\right\\}$
$\displaystyle=\sum_{i=1}^{N}\mathbb{E}\left\\{|w_{k_{\text{u}}i}|^{2}\right\\}\sum_{m=1}^{L}\sum_{k=1}^{M}\zeta_{\text{sps}}$
$\displaystyle~{}~{}+c\sum_{i=1}^{N}\sum_{j=1,j\neq
i}^{N}\text{Cov}\left(w_{k_{\text{u}}i},w_{k_{\text{u}}j}^{*}\right).$ (49)

From (45), (48), and _Lemma_ 3, (49) can be expressed as
$\bar{\Omega}_{\text{sps}}-\bar{\Omega}_{\text{stt}}\leq 0$.

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