# Multicarrier-Division Duplex for Solving the Channel Aging Problem in
Massive MIMO Systems

Bohan Li, Lie-Liang Yang, Fellow, IEEE, Robert G. Maunder, Senior Member,
IEEE, Songlin Sun, Senior Member, IEEE, Pei Xiao, Senior Member, IEEE B. Li,
L.-L. Yang and R. Maunder are with the School of Electronics and Computer
Science, University of Southampton, SO17 1BJ, UK. (E-mail: bl2n18, lly,
<EMAIL_ADDRESS>http://www-mobile.ecs.soton.ac.uk/lly). S. Sun is with the
School of Information and Communication Engineering, Beijing University of
Posts and Telecommunications (BUPT). P. Xiao is with 5GIC & 6GIC, Institute
for Communication Systems (ICS), University of Surrey, Guildford GU2 7XH, UK.
(Email<EMAIL_ADDRESS>The project was supported in part by the EPSRC,
UK, under Project EP/P034284/1 and EP/P03456X/1, and in part by the Innovate
UK project.

###### Abstract

The separation of training and data transmission as well as the frequent
uplink/downlink (UL/DL) switching make time-division duplex (TDD)-based
massive multiple-input multiple-output (mMIMO) systems less competent in fast
time-varying scenarios due to the resulted severe channel aging. To this end,
a multicarrier-division duplex (MDD) mMIMO scheme associated with two types of
well-designed frame structures are introduced for combating channel aging when
communicating over fast time-varying channels. To compare with TDD, the
corresponding frame structures related to 3GPP standards and their variant
forms are presented. The MDD-specific general Wiener predictor and decision-
directed Wiener predictor are introduced to predict the channel state
information, respectively, in the time domain based on UL pilots and in the
frequency domain based on the detected UL data, considering the impact of
residual self-interference (SI). Moreover, by applying the zero-forcing
precoding and maximum ratio combining, the closed-form approximations for the
lower bounded rate achieved by TDD and MDD systems over time-varying channels
are derived. Our main conclusion from this study is that the MDD, endowed with
the capability of full-duplex but less demand on SI cancellation than in-band
full-duplex (IBFD), outperforms both the conventional TDD and IBFD in
combating channel aging.

###### Index Terms:

Massive multiple-input multiple-output, channel aging, time-varying,
multicarrier-division duplex, time-division duplex, full-duplex, Wiener
predictor, performance analysis.

##  I Introduction

Massive multiple-input multiple-output (mMIMO) has been regarded as one of the
most promising technologies for significantly improving the spectral
efficiency (SE) in the fifth-generation (5G) wireless systems [1]. However,
due to the large number of antennas equipped at base station (BS), the
acquisition of channel state information (CSI) becomes highly challenging. As
the result, time-division duplex (TDD) is deemed to be the only feasible mode
in practical mMIMO systems to date, which exploits the channel reciprocity to
acquire downlink (DL) CSI from uplink (UL) pilots [2]. As its counterpart, in
frequency-division duplex (FDD) systems, the overhead for training/feedback is
proportional to the size of the antenna arrays at BS, resulting in less
resources for sending desired data in one coherence block [3]. On the other
hand, with the growing demand on the mobile services operated in ground
vehicles, high-speed trains, unmanned aerial vehicles (UAVs), etc., the
performance of TDD systems is no longer dominated by pilot contamination but
the channel aging problem. In other words, due to fast time-varying, the
channel at the time when it is estimated via UL training may be very different
from that at the time when it is applied for DL preprocessing and UL
detection. Moreover, the UL/DL switching intervals required by TDD systems
exacerbate the situation. This CSI mismatch may lead to significant
degradation of the SE of both UL and DL.

In order to support the robust communications over fast time-varying channels,
channel prediction based on the Wiener filter or Kalman filter has been
studied. In [4], the authors considered the channel aging problem in mMIMO
systems and applied the Wiener predictor (WP) to predict future CSI. The
results show that using channel prediction can partially relieve the channel
aging effect and render an increase of sum rate. The authors of [5] analyzed
the effect of WP for time-varying channels on the system performance. However,
in these two papers, the one-step WP relies on the latest channels estimated
using the UL pilots that are always available in the front of the symbol to be
predicted. Unfortunately, in practical TDD systems, UL pilots are unavailable
during DL transmission and hence, the accuracy of channel prediction degrades
with time, causing the performance degradation. For this sake, in [6], the
authors considered a more practical scenario, where no UL pilots are sent
during the Kalman filter assisted channel prediction for DL transmission. The
results show that the Kalman predictor is capable of improving performance
when compared with the case without prediction. However, due to the time-
varying effect, the achievable DL rate reduces quickly with time as the result
of the fact that the predicted CSI becomes less and less accurate. Recently in
[7], a learning-based approach was introduced, showing that it can outperform
the conventional non-linear Kalman predictor in dealing with the channel aging
problem. Again, the prediction accuracy of CSI deteriorates as the interval
between the channel being predicted and the pilot symbols becomes larger.

In contrast to the half-duplex (HD) mode of TDD, in-band full-duplex (IBFD)
has the inherent advantage for solving the channel aging problem. This is
because in IBFD systems, DL and UL occur concurrently, which enables to
acquire the latest CSI during data transmission without invoking DL/UL
switching. However, to the best of our knowledge, there are no open references
which have considered the IBFD-relied transmission design for the multicarrier
mMIMO systems communicating over fast time-varying channels. One conceivable
reason may be that the self-interference (SI) problem in IBFD-relied systems
is still intractable. This becomes even more challenging in the high-mobility
communication scenarios. Note that, although channel aging is not considered,
in [8], the channel acquisition relying on a hybrid IBFD and TDD mode was
demonstrated to be efficient. With this hybrid scheme, partial DL
transmissions are activated during UL training, leading to an increased SE.
Moreover, the effect of SI cancellation (SIC) on the performance of channel
estimation was studied with the IBFD systems, showing that if SIC is
insufficient, IBFD is unable to outperform the conventional TDD. Similar
observations were also obtained in [9].

To overcome the weakness of both IBFD and TDD over fast time-varying
communication channels, inspired by [10], we propose the multicarrier division
duplex (MDD) associated with the dedicated frame design so as to relieve the
channel aging problem, which constitutes the main motivation of this paper.
The rationale can be briefly explained as follows. On the one hand, in MDD
systems, both DL and UL transmissions can occur concurrently within the same
frequency band but on different subcarriers. Hence, when needed, UL pilots can
be continuously (or frequently) transmitted during DL transmissions.
Therefore, CSI can be updated in time and does not become outdated as in TDD
systems. On the other hand, MDD is capable of circumventing the stringent
requirement for SIC in IBFD systems. According to [11], in practical IBFD
systems, in addition to the SIC in the propagation- and analog-domain, the
digital-domain SIC has to cancel the main SI signal component by at least 30
dB. Achieving this is very power-consuming and technically demanding,
especially for the relatively small-sized mobile terminals (MTs), e.g., UAVs
and smartphones. By contrast, in the MDD-assisted systems, this amount of SIC
can be attained at nearly no extra cost of system resources, owing to the
embedded fast Fourier transform (FFT) operation [9], which allows to ideally
separate UL signals from DL signals in the digital-domain. For more detailed
comparison between MDD and the existing duplex modes, e.g., TDD, FDD and IBFD,
in terms of system architecture, initial access of physical layer and
resource-allocation, the readers are referred to our previous works [9, 12,
13, 14].

Owing to the aforementioned merits of MDD, it is important to investigate the
MDD’s implementation in fast time-varying scenarios, and to unlock its
advantages over the conventional TDD and IBFD systems. Therefore, our main
contributions of this paper are summarized as follows:

  * •

Firstly, a model for the MDD-based multiuser mMIMO systems communicating over
time-varying channels is presented. To alleviate the channel aging problem, we
propose two types of frame structures dedicated for MDD, which enable UL
pilots (or UL data transmissions) to occur concurrently with DL transmissions,
so that CSI can be promptly updated whenever needed. For comparison, the
corresponding TDD frame structures related to the 3GPP standards [15, 16] as
well as their modified forms for supporting different mobility scenarios are
introduced.

  * •

Secondly, to operate with the proposed frame structures, we introduce two
finite impulse response WPs that consider both channel estimation and residual
SI errors, namely the general WP and the decision-directed WP (DD-WP),
dedicated to MDD systems. To be more specific, the general WP directly
predicts the time-domain CSI based on the observations collected from the UL
channel estimation. By contrast, the DD-WP leverages both the UL pilots and
the detected UL data symbols to predict the frequency-domain CSI. Along with
these WPs, the impact of residual SI and the order of WPs on the performance
of MDD systems are studied and compared.

  * •

Thirdly, by assuming the zero-forcing (ZF) precoding for DL transmission and
the maximum ratio combining (MRC) for UL detection, the closed-form
expressions for approximating the lower bounded average sum rates of both the
TDD and MDD systems are derived, when the proposed WPs are operated with two
general types of frame structures.

Our studies and simulation results show that TDD systems suffer from the
channel aging problem, whose performance degrades significantly with time,
when channels vary fast. By contrast, the MDD systems endowed with the FD
capability can effectively mitigate the channel aging problem and hence, are
capable of significantly outperforming their TDD counterparts, when
communicating over fast time-varying channels. Moreover, the studies
demonstrate that the SIC in FD systems plays a paramount role in channel
prediction. In the case of imperfect SIC, MDD becomes more competent than IBFD
for operation in high-mobility communications scenarios.

The rest of the paper is organized as follows. In Section II, the model for
the MDD-based system is described. In Section III, two general frame
structures for TDD and MDD systems are introduced. Section IV presents the
principles of channel estimation and two approaches for channel prediction.
Section V analyzes the lower bounded average sum rates of MDD and TDD systems.
Section VI provides the performance results to compare TDD, MDD and IBFD
systems. Finally, conclusions are drawn in Section VII.

Throughout the paper, the following notations are used: $\boldsymbol{A}$,
$\boldsymbol{a}$ and $a$ stand for matrix, vector, and scalar, respectively;
$\mathcal{A}$ and $\left|\mathcal{A}\right|$ represent the set and the
cardinality of set, respectively; $(\boldsymbol{a})_{i}$ denotes the $i$-th
element of $\boldsymbol{a}$; $\boldsymbol{A}^{(i,:)}$,
$\boldsymbol{A}^{(:,j)}$ and $(\boldsymbol{A})_{i,j}$ denote the $i$-th row,
the $j$-th column and the $(i,j)$-th element of $\boldsymbol{A}$,
respectively; Furthermore, $\left|\boldsymbol{A}\right|$,
$\boldsymbol{A}^{*}$, $\boldsymbol{A}^{T}$, $\boldsymbol{A}^{-1}$ and
$\boldsymbol{A}^{H}$ represent, respectively, the determinant, complex
conjugate, transpose, inverse and Hermitian transpose of $\boldsymbol{A}$; The
Euclidean norm of a vector and the Frobenius norm of a matrix are denoted as
$\left\|\cdot\right\|_{2}$ and $\left\|\cdot\right\|_{F}$, respectively;
$\boldsymbol{I}_{N}$ denotes a $(N\times N)$ identity matrix;
$\mathcal{CN}(\boldsymbol{0},\boldsymbol{A})$ represents the zero-mean complex
Gaussian distribution with covariance matrix $\boldsymbol{A}$; Furthermore,
$\text{Tr}(\cdot)$, $\text{log}(\cdot)$ and $\mathbb{E}[\cdot]$ denote the
trace, logarithmic and expectation operators, respectively.

##  II System Model

Consider a single-cell mMIMO OFDM system having a base station (BS) equipped
with $N$ antennas and $D$ single-antenna MTs randomly distributed in the cell.
The system is operated in the MDD mode, allowing DL and UL to communicate
concurrently in the same frequency band but on different subcarriers.
Following the concept of mMIMO, the number of antennas at BS is assumed to be
much larger than the number of served MTs [3], i.e., $N\gg D$. Furthermore,
based on the principles of MDD [9], subcarriers are divided into two mutually
exclusive subsets, namely a DL subcarrier subset $\mathcal{M}$ with $M$
subcarriers, and a UL subcarrier subset $\bar{\mathcal{M}}$ with $\bar{M}$
subcarriers, i.e., $|\mathcal{M}|=M$ and $|\bar{\mathcal{M}}|=\bar{M}$. The
total number of subcarriers is expressed as $M_{\text{sum}}=M+\bar{M}$.
Similar to the other works on resource-allocation, e.g.,[17, 18, 19], the
DL/UL subcarrier allocation (SA) is carried out at BS based on the CSI and the
requirements of MTs’ quality of services. The allocation results are informed
to MTs through control channels during the initial access procedure. Below we
assume that the DL/UL SA results have been obtained, and that each MT knows
its assigned subcarriers. The readers interested in the SA in MDD systems are
referred to [13] for details.

###  II-A Channel Model

We assume that the channels are frequency-selective in terms of the
$M_{\text{sum}}$ subcarriers but each subcarrier experiences flat fading. To
consider the influence of channel aging, we assume that the channel
coefficients do not change within one OFDM symbol, but vary from one symbol to
the next. Therefore, the time-domain CSI of the $L$-tap channel between the
$d$-th MT and the $n$-th BS antenna over the $i$-th OFDM symbol duration can
be expressed as

$\boldsymbol{g}_{n,d}[i]=\left[g_{n,d}[i,1],...,g_{n,d}[i,l],...,g_{n,d}[i,L]\right]^{T}$
(1)

where $g_{n,d}[i,l]=\nu_{n,d}[i,l]\sqrt{\beta_{d}}$ and
$\nu_{n,d}[i,l]\sim\mathcal{CN}(0,1/L)$ is the small-scale fading, while
$\beta_{d}$ represents the large-scale fading, which only depends on the
distance between MT $d$ and BS and is assumed to remain constant over one
communication frame. For any user-antenna pair, the channels of different taps
are assumed to be independent. Hence we have
$\boldsymbol{R}_{g}^{d}=\mathbb{E}\left[\boldsymbol{g}_{n,d}[i]\boldsymbol{g}_{n,d}^{H}[i]\right]=\frac{\beta_{d}}{L}\boldsymbol{I}_{L}$.

According to the principles of OFDM [10], the frequency-domain CSI
$\boldsymbol{h}_{n,d}[i]$ over the $i$-th OFDM symbol duration can be obtained
as

$\boldsymbol{h}_{n,d}[i]=\boldsymbol{F}\boldsymbol{\varPsi}\boldsymbol{g}_{n,d}[i]$
(2)

where $\boldsymbol{F}\in\mathbb{C}^{M_{\text{sum}}\times M_{\text{sum}}}$ is
the FFT matrix with
$\left(\boldsymbol{F}\right)_{p,q}=\frac{1}{\sqrt{M_{\text{sum}}}}e^{-j2\pi(p-1)(q-1)/M_{\text{sum}}}$,
$\boldsymbol{\varPsi}\in\mathbb{C}^{{M_{\text{sum}}\times L}}$ is constructed
by the first $L$ columns of $\boldsymbol{I}_{M_{\text{sum}}}$. Furthermore,
the DL subchannels $\boldsymbol{h}_{n,d}^{\text{DL}}[i]$ and UL subchannels
$\boldsymbol{h}_{n,d}^{\text{UL}}[i]$ between the $n$-th antenna at BS and the
$d$-th MT over the $i$-th OFDM symbol can be obtained from (2), which can be
expressed as

$\displaystyle\boldsymbol{h}_{n,d}^{\text{DL}}[i]$
$\displaystyle={\boldsymbol{\Phi}_{\text{DL}}}\boldsymbol{h}_{n,d}[i]$
$\displaystyle=\left[h_{n,d}[i,1],...,h_{n,d}[i,m],...,h_{n,d}[i,M]\right]^{T}$
(3) $\displaystyle\boldsymbol{h}_{n,d}^{\text{UL}}[i]$
$\displaystyle={\boldsymbol{\Phi}_{\text{UL}}}\boldsymbol{h}_{n,d}[i]$
$\displaystyle=\left[h_{n,d}[i,1],...,h_{n,d}[i,\bar{m}],...,h_{n,d}[i,\bar{M}]\right]^{T}$
(4)

where
$\boldsymbol{\Phi}_{\text{DL}}=\boldsymbol{I}_{M_{\text{sum}}}^{(\mathcal{M},:)}$
and
$\boldsymbol{\Phi}_{\text{UL}}=\boldsymbol{I}_{M_{\text{sum}}}^{(\mathcal{\bar{M}},:)}$
are the mapping matrices, constructed from $\boldsymbol{I}_{M_{\text{sum}}}$
by choosing its rows corresponding to the particular subcarriers assigned to
DL and UL, respectively.

Since in the analog-domain the MDD systems are operated in the FD mode, there
is SI at both BS and MTs. In particular, although the MDD-based systems employ
the inherent advantage to suppress SI in the digital-domain by the FFT
operation111In IBFD-based systems, in order to mitigate the digital-domain SI,
the receiver has to estimate the channel between the DAC at transmitter and
the ADC at receiver so as to reconstruct the transmitted signal, which is then
subtracted from the received signal [20]. Explicitly, this process imposes a
heavy burden on system overhead, especially when a system with a large number
of subcarriers is considered., the DL and UL subcarrier signals are coupled in
the analog-domain before the ADC at the BS receiver. As in the traditional
IBFD systems [21], without sufficient SIC in the propagation- and analog-
domain prior to ADC, the high-power SI imposed by the DL subcarriers from the
BS transmitter may overwhelm the desired UL signals from MTs, which leads to
the inefficient operation of ADC and consequently significant quantization
error in digital-domain. Due to the short separation between transmitter and
receiver, the SI channels are assumed to be flat fading and identical to all
subcarriers, and also quasi-static within one frame. Specifically, the SI
channels at BS and MTs follow the distributions of
$(\boldsymbol{H}_{\text{SI}})_{i,j}\sim\mathcal{CN}(0,1)$, where
$\boldsymbol{H}_{\text{SI}}\in\mathbb{C}^{N\times N}$, and
$h_{\text{SI}}\sim\mathcal{CN}(0,1)$, respectively.

###  II-B Channel Aging

As we stated in Section I, the relative mobility between BS and MTs leads to
time-varying channels, causing that the channel varies between the time when
it is estimated and the time when the estimated CSI is applied for
transmissions. This is the well-known channel aging problem. To model the
channel aging, we introduce the Jakes model, which has the normalized
discrete-time autocorrelation function at the BS given by [7]

$R[k]=J_{0}(2\pi f_{D}T_{s}|k|),$ (5)

where $J_{0}(\cdot)$ is the zeroth-order Bessel function of the first kind,
$T_{s}$ is the OFDM symbol duration, $f_{D}$ is the maximum Doppler frequency
shift, and $|k|$ is the delay in terms of the number of symbols.

Furthermore, for the sake of analyzing MDD frame structures with WP and DD-WP
in the following sections, we adopt the autoregressive model of order 1,
denoted as AR(1), for approximating the temporal correlation between adjacent
symbols. In this case, the channel coefficient $\boldsymbol{g}_{n,d}[i]$ of
(1) can be formulated as [22]

$\boldsymbol{g}_{n,d}[i]=\alpha\boldsymbol{g}_{n,d}[i-1]+\boldsymbol{v}_{n,d}[i]$
(6)

where $\alpha=J_{0}(2\pi f_{D}T_{s})$, which is assumed to be perfectly known
at BS, $\boldsymbol{v}_{n,d}[i]$ is an uncorrelated channel error vector due
to channel aging, which can be modeled as a stationary Gaussian random process
with i.i.d. entries and the distribution of
$\mathcal{CN}\left(\boldsymbol{0},(1-\alpha^{2})\boldsymbol{R}_{g}^{d}\right)$.
According to the channel model of (6), $\boldsymbol{g}_{n,d}[i]$ is a
stationary Markov random process, and has the auto-correlation of
$\mathbb{E}\left[\boldsymbol{g}_{n,d}[i-q]\boldsymbol{g}_{n,d}^{H}[i-p]\right]=\alpha^{\left|p-q\right|}\boldsymbol{R}_{g}^{d}$.
Moreover, according to (II-A) and (II-A), we can obtain the DL subchannels
satisfying

$h_{n,d}[i,m]=\alpha
h_{n,d}[i-1,m]+\underbrace{\boldsymbol{\psi}_{m}\boldsymbol{v}_{n,d}[i]}_{\tilde{v}_{n,d}^{m}[i]}$
(7)

where
$\boldsymbol{\psi}_{m}=\boldsymbol{I}_{\text{sum}}^{(m,:)}\boldsymbol{F}\boldsymbol{\varPsi}$
and
$\tilde{v}_{n,d}^{m}[i]\in\mathcal{CN}\left(0,(1-\alpha^{2})R_{h}^{d}\right)$
with
$R_{h}^{d}=\mathbb{E}\left[h_{n,d}[i,m]h_{n,d}^{H}[i,m]\right]={\beta_{d}}/{M_{\text{sum}}}$.
Similarly, the UL subchannels $h_{n,d}[i,\bar{m}]$ can be generated.

###  II-C Transmission Model

####  II-C1 Downlink Transmission

For DL transmission, the signal
$\boldsymbol{s}_{\text{DL}}[i,m]\in\mathbb{C}^{N\times 1}$ transmitted by BS
on the $m$-th subcarrier in the $i$-th OFDM symbol duration is

$\boldsymbol{s}_{\text{DL}}[i,m]=\sqrt{p_{\text{DL}}}\boldsymbol{F}^{\text{ZF}}[i,m]\boldsymbol{x}[i,m]$
(8)

where $p_{\text{DL}}$ is the power budget per DL subcarrier and the total BS
transmit power is $P_{\text{DL}}=p_{\text{DL}}M$,
$\boldsymbol{x}[i,m]=\left[x_{1}[i,m],...,x_{D}[i,m]\right]^{T}$ contains the
information symbols normalized to satisfy
$\mathbb{E}\left[\boldsymbol{x}[i,m]\boldsymbol{x}[i,m]^{H}\right]=\boldsymbol{I}_{D}$.
We assume that the ZF precoding is applied at transmitter, giving
$\boldsymbol{F}^{\text{ZF}}[i,m]=\boldsymbol{H}_{\text{DL}}^{H}[i,m]\left(\boldsymbol{H}_{\text{DL}}[i,m]\boldsymbol{H}_{\text{DL}}^{H}[i,m]\right)^{-1}$,
where
$\boldsymbol{F}^{\text{ZF}}[i,m]=\left[\boldsymbol{f}_{1}[i,m],...,\boldsymbol{f}_{D}[i,m]\right]$
with the constraint of
$\left\|\boldsymbol{f}_{d}[i,m]\right\|_{2}=1/\sqrt{D}$, such that
$\mathbb{E}\left[\left\|\boldsymbol{s}_{\text{DL}}[i,m]\right\|_{2}^{2}\right]=p_{\text{DL}}$,
and
$\boldsymbol{H}_{\text{DL}}[i,m]=\left[\boldsymbol{h}_{1}[i,m],...,\boldsymbol{h}_{d}[i,m],...\boldsymbol{h}_{D}[i,m]\right]^{H}$,
where
$\boldsymbol{h}_{d}[i,m]=\left[h_{1,d}[i,m],h_{2,d}[i,m],...,h_{N,d}[i,m]\right]^{T}$.
It is noteworthy that the analysis in this paper is based on the equal power
allocation among MTs. Intuitively, an appropriate power allocation method,
e.g., that in our previous work [13], will increase the achievable sum rates,
owing to both user and subcarrier diversity being exploited. The received
signal at the $d$-th MT can be expressed as

$\begin{split}&y_{d}[i,m]=\sqrt{p_{\text{DL}}}\boldsymbol{h}_{d}^{H}[i,m]\boldsymbol{f}_{d}[i,m]x_{d}[i,m]\\\
&+\sqrt{p_{\text{DL}}}\sum_{k\neq
d}\boldsymbol{h}_{d}^{H}[i,m]\boldsymbol{f}_{k}[i,m]x_{k}[i,m]+z^{\text{SI}}_{d}[i]+z_{d}[i,m]\end{split}$
(9)

where
$z_{d}^{\text{SI}}[i]\sim\mathcal{CN}\left(0,\xi_{\text{MT}}p_{\text{UL}}\bar{M}\right)$
denotes the residual SI222The residual SI consists of the combined effect of
the additive noise introduced by automatic gain control (AGC), non-linearity
of ADC and the phase noise generated by oscillator due to RF imperfection
[23]. signal imposed by the UL transmission of MT $d$ with $p_{\text{UL}}$
being the power budget per UL subcarrier, such that the total MT transmit
power is $P_{\text{UL}}=p_{\text{UL}}\bar{M}$. Note that $\xi_{\text{MT}}$ in
(9) and $\xi_{\text{BS}}$ in (10) denote the SIC capability that is provided
by the existing SIC techniques, such as, the antenna circulator [11], spatial
beamforming [12], dual-port polarized antenna [24], multi-tap RF canceller
[25], etc., implemented in propagation- and analog-domain, as well as the FFT
operation operated in digital-domain. In (9), the covariance of
$z_{d}^{\text{SI}}[i]$ is calculated in detail in Appendix A. In addition,
$z_{d}[i,m]\sim\mathcal{CN}(0,\sigma^{2})$ denotes the additive white Gaussian
noise. Note also that, the interference caused by the other MTs’ UL
transmission on the DL reception is ignored. The reason is as follows. First,
in analogy to the conventional OFDMA systems [17], the inter-MT interference
(IMI) and the desired DL signal are located on the mutually orthogonal
subcarriers. Secondly, the large-scale fading of MT-MT link and the MT’s
relatively lower transmit power allow the received signal of an MT to
efficiently pass the ADC. Consequently, the IMI can be readily removed by the
FFT operation in the digital-domain, when all MTs and the BS are assumed to be
synchronized within an allowable time window [14].

####  II-C2 Uplink Transmission

For the UL transmission, the signal received at BS over the $\bar{m}$-th UL
subcarrier and $i$-th OFDM symbol duration can be expressed as

$\boldsymbol{s}_{\text{UL}}[i,\bar{m}]=\sum_{d=1}^{D}\boldsymbol{h}_{d}[i,\bar{m}]\sqrt{p_{\text{UL}}}x_{d}[i,\bar{m}]+\boldsymbol{z}^{\text{SI}}[i]+\boldsymbol{z}[i,\bar{m}]$
(10)

where
$\boldsymbol{h}_{d}[i,\bar{m}]=\left[h_{1,d}[i,\bar{m}],h_{2,d}[i,\bar{m}],...,h_{N,d}[i,\bar{m}]\right]^{T}$,
$\boldsymbol{z}[i,\bar{m}]\sim\mathcal{CN}(\boldsymbol{0},\sigma^{2}\boldsymbol{I}_{N})$.
$\boldsymbol{z}^{\text{SI}}[i]$ is the SI generated by BS transmissions, which
is modeled as
$\boldsymbol{z}^{\text{SI}}[i]\sim\mathcal{CN}\left(\boldsymbol{0},\xi_{\text{BS}}p_{\text{DL}}M\boldsymbol{I}_{N}\right)$.
The covariance of $\boldsymbol{z}^{\text{SI}}[i]$ is derived in detail in
Appendix A. In this paper, we assume that the MRC is used for signal
detection. Then we have
$\boldsymbol{W}^{\text{MRC}}[i,\bar{m}]=\boldsymbol{H}_{\text{UL}}[i,\bar{m}]$,
where
$\boldsymbol{H}_{\text{UL}}[i,\bar{m}]=\left[\boldsymbol{h}_{1}[i,\bar{m}],...,\boldsymbol{h}_{D}[i,\bar{m}]\right]$.
Let us write
$\boldsymbol{W}^{\text{MRC}}[i,\bar{m}]=\left[\boldsymbol{w}_{1}[i,\bar{m}],...,\boldsymbol{w}_{D}[i,\bar{m}]\right]$.
Then the decision variable for MT $d$ can be expressed as

$\begin{split}y_{d}[i,\bar{m}]&=\sqrt{p_{\text{UL}}}\boldsymbol{w}_{d}^{H}[i,\bar{m}]\boldsymbol{h}_{d}[i,\bar{m}]x_{d}[i,\bar{m}]\\\
&+\sqrt{p_{\text{UL}}}\sum_{k\neq
d}\boldsymbol{w}_{d}^{H}[i,\bar{m}]\boldsymbol{h}_{k}[i,\bar{m}]x_{k}[i,\bar{m}]\\\
&+\boldsymbol{w}_{d}^{H}[i,\bar{m}]\boldsymbol{z}^{\text{SI}}[i]+\boldsymbol{w}_{d}^{H}[i,m]\boldsymbol{z}[i,\bar{m}]\end{split}$
(11)

where the second term at the right-hand side is multiuser interference, while
the third term is due to SI.

##  III Frame Structure Design

Due to the effect of channel aging, channel coefficients may vary from symbol
to symbol, leading to poor system performance in high-mobility communication
scenarios [4]. To overcome the time-varying fading problem, in conventional
TDD mode, channel prediction is usually implemented after the training phase
to reduce CSI error. However, when the relative velocity between transmitter
and receiver increases, the channel becomes less correlated, increasing the
prediction error, which in turn renders the later detected symbols less
reliable. One way to mitigate this problem is to employ extra training symbols
for updating the CSI during data transmission. Following this observation, if
the channel aging problem becomes severe, more training symbols are inserted,
which results in more frequent switching between UL training and DL data
transmission, as well as the decrease of useful data rate. To this end, our
main motivation of this paper is to demonstrate that the MDD mode is capable
of efficiently solving the channel aging problem in fast fading communication
scenarios. For demonstration and comparison, in the following subsections, we
will present two types of frame structures for coping with the channel aging
problem and also for analyzing the performance of MDD systems in terms of sum-
rate.

Figure 1: Type I frame structure for the conventional TDD and proposed MDD
systems.

###  III-A Frame Structure: Type I

In this subsection, we consider a simple scenario, where the frame structure
only includes UL training and DL transmission, as shown in Fig. 1. The length
of one frame333Note that one frame defined in this paper may include several
subframes used in 4G LTE systems or slots in 5G NR systems, where each
subframe or slot contains 14 OFDM symbols. Specifically in 4G LTE, one frame
consists of 10 subframes and two of them are used for training [15]. The frame
structure in 5G NR is more flexible than that in 4G LTE[16]. is assumed to be
equal to $T$ OFDM symbol durations.

####  III-A1 TDD

As shown in Fig. 1, in the TDD mode, a frame consists of $\tau_{p}$ training
symbols transmitted by UL and $T-\tau_{p}-1$ symbols used for DL transmission.
The switching from UL training to DL data transmission or from DL data
transmission to the next training phase incurs a fixed cost, which is assumed
to be half of an OFDM symbol.444The length of the switching period in LTE can
be a different value, such as 1, 2, 3, 4, 9 or 10 OFDM symbols [15]. We choose
0.5 to minimize the influence of the switching interval on the evaluation of
TDD system’s performance. To cope with channel aging, the channel prediction
is implemented by a $\tau_{p}$-th order WP555To ease analysis and facilitate
comparison, only WP is considered in this paper. The Kalman predictor and the
state-of-the-art deep learning approaches will be studied in our future work.
after the training phase, as shown in Fig. 1, and all the precoding/detection
are operated at BS based on the predicted CSI. It is noteworthy that in
practice, the $\tau_{p}$ training symbols can be first disassembled into
several groups, which are then evenly distributed over one frame, so as to
relieve the effect of accumulated prediction errors. However, the accompany of
the increased switching intervals results in the reduced time for DL
transmission and hence the decreased system’s efficiency.

####  III-A2 MDD

In the context of MDD, as UL/DL transmissions are on different subcarriers,
CSI estimation and prediction can be implemented in the FD mode. That is to
say, UL pilots can be kept active to update CSI in real time within one OFDM
symbol. In detail, as shown in Fig. 1, UL training is always activated and
hence, the channels for DL transmission can be predicted by a WP based on the
newest $\tau_{p}$ pilots. However, in low-speed scenarios, the excessive
transmission of UL pilots inevitably increases energy consumption and system
complexity, which is unnecessary as the channels change slowly. Moreover, in
the case where only the DL transmission is needed, continuous UL training on
$\bar{M}$ subcarriers leads to only $M$ of the $M_{\text{sum}}$ subcarriers
being used for the DL transmission. Therefore, in practice, the pilots within
one MDD frame can be partially transmitted (as illustrated by the example
presented in Fig. 1) so that some UL subcarriers can be ‘borrowed’ for DL data
transmission to reduce the overhead of SI mitigation. In Section VI, the
TDD/MDD frame structures with different distributions of pilots will be
investigated.

Figure 2: Type II frame structure for conventional TDD and proposed MDD
systems.

###  III-B Frame Structure: Type II

A more general frame structure consisting of the UL training followed by both
UL transmission and DL transmission is considered for both TDD and MDD, as
shown in Fig. 2.

####  III-B1 TDD

In the TDD mode, as shown in Fig. 2, $\tau_{p}$ pilot symbols are transmitted
in Phase I for initializing the transmission. During Phase II, the detected UL
data symbols and possibly a part of pilots are used to predict channels by a
$\tau_{p}$-th order decision-directed (DD)-WP. With this setting, during the
UL transmission, the channel knowledge can be continuously updated with the DD
prediction principle. By contrast, during the DL transmission, channel
updating is unavailable as there is no UL transmission. Consequently, the
transmissions of the $T-\tau_{p}-\tau_{u}-1$ DL symbols are based on the
outdated CSI, which is similar to the situation with the Type I frame
structure.

####  III-B2 MDD

As shown in Fig. 2, the Type II frame structure in MDD systems can be designed
in a very efficient way. Firstly, for fair comparison with TDD, in Phase I,
$\tau_{p}$ pilot symbols are still transmitted for the implementation of a
$\tau_{p}$-th order DD-WP in Phase II. However, different from TDD, during the
transmission of pilot symbols, DL symbols can also be transmitted with the aid
of the CSI predicted by a $\kappa_{p}$-th order WP, where
$1\leq\kappa_{p}\leq\tau_{p}$. In the extreme case, when $\kappa_{p}=1$,
$\tau_{p}-1$ more DL symbols than TDD can be transmitted in Phase I, which may
significantly increase the SE. During Phase II, both DL and UL can transmit
$T-\tau_{p}$ symbols. Furthermore, a continuously updated $\tau_{p}$-th order
DD-WP can be implemented to predict the channels using the pilots and the
detected UL symbols.

##  IV Channel Estimation and Prediction In MDD systems

In this section, we study the channel estimation and prediction in MDD
systems, when the proposed frame structures in Section III are introduced.

###  IV-A Channel Estimation

Let us assume that all MTs synchronously transmit their frequency-domain pilot
sequences (FDPS) over $\bar{M}$ UL subcarriers. Specifically, the FDPS
transmitted by the $d$-th MT is expressed as
$\boldsymbol{x}_{p,d}=\left[x_{p,d}[1],...,x_{p,d}[\bar{m}],...,x_{p,d}[\bar{M}]\right]^{T}$.
Then, the received signal at the $n$-th antenna of BS corresponding to the
$i$-th symbol is given by

$\begin{split}\boldsymbol{y}_{p,n}[i]&=\sqrt{p_{\text{UL}}}\sum\limits_{d=1}^{D}\boldsymbol{X}_{p,d}\boldsymbol{h}_{n,d}^{\text{UL}}[i]+\boldsymbol{z}^{\text{SI}}_{n}[i]+\boldsymbol{z}_{p,n}[i]\\\
&=\sqrt{p_{\text{UL}}}\sum\limits_{d=1}^{D}\boldsymbol{X}_{p,d}{\boldsymbol{\Phi}_{\text{UL}}}\boldsymbol{F}\boldsymbol{\varPsi}\boldsymbol{g}_{n,d}[i]+\boldsymbol{z}^{\text{SI}}_{n}[i]+\boldsymbol{z}_{p,n}[i]\end{split}$
(12)

where $\boldsymbol{X}_{p,d}=\text{diag}\left\\{\boldsymbol{x}_{p,d}\right\\}$,
$\boldsymbol{z}_{p,n}[i]\sim\mathcal{CN}(\boldsymbol{0},\sigma^{2}\boldsymbol{I}_{\bar{M}})$.
Note that, in MDD systems, BS may experience SI during UL training, if DL
symbols are transmitted at the same time. For example, in the frame structure
as shown in Fig. 1, when $1\leq i\leq\tau_{p}$, we have
$\boldsymbol{z}^{\text{SI}}_{n}[i]=0$, as there is no DL transmissions. By
contrast, when $\tau_{p}<i\leq T$,
$\boldsymbol{z}^{\text{SI}}_{n}[i]\sim\mathcal{CN}\left(0,\xi_{\text{BS}}p_{\text{DL}}M\boldsymbol{I}_{\bar{M}}\right)$.

According to our previous study [12], the FDPS of MT $d$ can be designed as

$\boldsymbol{x}_{p,d}=\left[1,e^{j2\pi\frac{(d-1)\xi}{\bar{M}}},e^{j2\pi\frac{2(d-1)\xi}{\bar{M}}},\cdots,e^{j2\pi\frac{(\bar{M}-1)(d-1)\xi}{\bar{M}}}\right]^{T}$
(13)

where $\xi=\left\lfloor\frac{\bar{M}}{D}\right\rfloor$. With the assumption
that $\bar{M}\geq DL$ and the $\bar{M}$ number of UL subcarriers are evenly
distributed,
$\boldsymbol{P}_{d}=\boldsymbol{X}_{p,d}{\boldsymbol{\Phi}_{\text{UL}}}\boldsymbol{F}\boldsymbol{\varPsi}$
for $d=1,...,D$ are mutually orthogonal, i.e.,

$\displaystyle\begin{cases}\boldsymbol{P}_{d}^{H}\boldsymbol{P}_{d}=\frac{\bar{M}}{M_{\text{sum}}}\boldsymbol{I}_{L},\\\
\boldsymbol{P}_{d}^{H}\boldsymbol{P}_{k}=\boldsymbol{0}_{L},~{}\forall~{}{d}\neq{k}\end{cases}$
(14)

Then, the noisy observation of $\boldsymbol{g}_{n,d}[i]$ is given by

$\displaystyle\tilde{\boldsymbol{y}}_{p,n}^{d}[i]$
$\displaystyle=\boldsymbol{P}_{d}^{H}\boldsymbol{y}_{p,n}[i]$
$\displaystyle=\frac{\sqrt{p_{\text{UL}}}\bar{M}}{M_{\text{sum}}}\boldsymbol{g}_{n,d}[i]+\boldsymbol{P}_{d}^{H}(\boldsymbol{z}^{\text{SI}}[i]+\boldsymbol{z}_{p,n}[i])$
(15)

Correspondingly, the MMSE estimate to $\boldsymbol{g}_{n,d}[i]$ is given by

$\hat{\boldsymbol{g}}_{n,d}[i]=\frac{\frac{\sqrt{p_{\text{UL}}}\bar{M}}{M_{\text{sum}}}\frac{\beta_{d}}{L}}{\frac{p_{\text{UL}}\bar{M}^{2}}{M_{\text{sum}}^{2}}\frac{\beta_{d}}{L}+\frac{\xi_{\text{BS}}p_{\text{DL}}M\bar{M}}{M_{\text{sum}}}+\frac{\bar{M}}{M_{\text{sum}}}\sigma^{2}}\tilde{\boldsymbol{y}}_{p,n}^{d}[i]$
(16)

###  IV-B Channel Prediction

####  IV-B1 General Wiener Predictor

In this subsection, we focus on the prediction of $\boldsymbol{g}_{n,d}[i+1]$
based on the current and previous received training signals, which are
expressed as
$\bar{\boldsymbol{y}}_{p,n}^{d}[i]=\left[\tilde{\boldsymbol{y}}_{p,n}^{d,H}[i],\tilde{\boldsymbol{y}}_{p,n}^{d,H}[i-1],...,\tilde{\boldsymbol{y}}_{p,n}^{d,H}[i+1-\tau_{p}]\right]^{H}$,
where $\tilde{\boldsymbol{y}}_{p,n}^{d}[i]$ is given by (IV-A). We first
assume a $\tau_{p}$-th order WP, with the weights
$\boldsymbol{V}_{n,d}=\left[\boldsymbol{V}_{n,d}^{1},...,\boldsymbol{V}_{n,d}^{q},...,\boldsymbol{V}_{n,d}^{\tau_{p}}\right]$,
$\boldsymbol{V}_{n,d}^{q}\in\mathbb{C}^{L\times L}$. Then, the predictor can
be formulated as

$\check{\boldsymbol{g}}_{n,d}^{\text{WP}}[i+1]=\sum\limits_{q=1}^{\tau_{p}}\boldsymbol{V}_{n,d}^{q}\tilde{\boldsymbol{y}}_{p,n}^{d}[i+1-q]=\boldsymbol{V}_{n,d}\bar{\boldsymbol{y}}_{p,n}^{d}[i]$
(17)

According to the principles of Wiener filter [4, 26], the solution to
$\boldsymbol{V}_{n,d}$ is

$\boldsymbol{V}_{n,d}=\boldsymbol{R}_{g\bar{y}}^{d}[1]\left(\boldsymbol{R}_{\bar{y},i}^{d}[0]\right)^{-1}$
(18)

where the cross-correlation between the real channel of the next symbol and
the observations is

$\boldsymbol{R}_{g\bar{y}}^{d}[1]=\mathbb{E}\left[\boldsymbol{g}_{n,d}[i+1]\bar{\boldsymbol{y}}_{p,n}^{d,H}[i]\right]=\frac{\sqrt{p_{\text{UL}}}\bar{M}}{M_{\text{sum}}}[\boldsymbol{\delta}(\tau_{p},\alpha)\otimes\boldsymbol{R}_{g}^{d}]$
(19)

where
$\boldsymbol{\delta}(\tau_{p},\alpha)=[\alpha,\alpha^{2},...,\alpha^{\tau_{p}}]$.
In (18), the autocorrelation matrix of the training signals is

$\displaystyle\boldsymbol{R}_{\bar{y},i}^{d}[0]$
$\displaystyle=\bar{\boldsymbol{y}}_{p,n}^{d}[i]\bar{\boldsymbol{y}}_{p,n}^{d,H}[i]$
$\displaystyle=\boldsymbol{\xi}(\tau_{p},\alpha)\otimes(\frac{p_{\text{UL}}\bar{M}^{2}}{M_{\text{sum}}^{2}}\boldsymbol{R}_{g}^{d})+\frac{\bar{M}\sigma^{2}}{M_{\text{sum}}}\boldsymbol{I}_{L\tau_{p}}+\boldsymbol{\Xi}_{\text{SI}}[i]$
(20)

where

$\displaystyle\boldsymbol{\xi}(\tau_{p},\alpha)$
$\displaystyle=\left[\begin{array}[]{cccc}1&\alpha&\cdots&\alpha^{\tau_{p}-1}\\\
\alpha&1&\cdots&\alpha^{\tau_{p}-2}\\\ \vdots&\vdots&\ddots&\vdots\\\
\alpha^{\tau_{p}-1}&\alpha^{\tau_{p}-2}&\cdots&1\end{array}\right]$ (25)

It can be observed that in the Type I frame structure for fast-fading
channels, the amount of SI is dependent on the observation time, which leads
to

$\displaystyle\boldsymbol{\Xi}_{\text{SI}}[i]$
$\displaystyle=\begin{cases}\boldsymbol{0}_{L\tau_{p}\times L\tau_{p}},\ \
i=\tau_{p}\\\
\left[\begin{array}[]{cc}\frac{\xi_{\text{BS}}p_{\text{DL}}M\bar{M}}{M_{\text{sum}}}\boldsymbol{I}_{(i-\tau_{p})L}&\boldsymbol{0}\\\
\boldsymbol{0}&\boldsymbol{0}\\\ \end{array}\right],\ \ \tau_{p}<i\leq
2\tau_{p}\\\
\frac{\xi_{\text{BS}}p_{\text{DL}}M\bar{M}}{M_{\text{sum}}}\boldsymbol{I}_{\tau_{p}L},\
\ i>2\tau_{p}\end{cases}$ (26)

According to the properties of WP, $\boldsymbol{g}_{n,d}[i+1]$ can be
orthogonally decomposed into

$\boldsymbol{g}_{n,d}[i+1]=\check{\boldsymbol{g}}_{n,d}^{\text{WP}}[i+1]+\check{\boldsymbol{v}}_{n,d}[i+1]$
(27)

where
$\check{\boldsymbol{v}}_{n,d}[i+1]\sim\mathcal{CN}(\boldsymbol{0},\boldsymbol{R}_{g}^{d}-\boldsymbol{\Upsilon}_{d,i})$
is the uncorrelated channel prediction error vector with
$\boldsymbol{\Upsilon}_{d,i}=\boldsymbol{R}_{g\bar{y}}^{d}[1]{\boldsymbol{R}_{\bar{y},i}^{d}}^{-1}[0]\boldsymbol{R}_{g\bar{y}}^{d,H}[1]$.

Similar to (7), the predicted DL subchannels can be attained, which are
expressed as

$h_{n,d}[i+1,m]=\check{h}_{n,d}^{\text{WP}}[i+1,m]+\underbrace{\boldsymbol{\psi}_{m}\check{\boldsymbol{v}}_{n,d}[i+1]}_{\check{v}^{\text{WP}}_{n,d}[i+1,m]},\
m=1,...,M$ (28)

where
$\check{h}_{n,d}^{\text{WP}}[i+1,m]\sim\mathcal{CN}\left(0,\sigma^{2}_{\check{h}_{d,i,m}^{\text{WP}}}\right)$
with
$\sigma^{2}_{\check{h}_{d,i,m}^{\text{WP}}}=\boldsymbol{\psi}_{m}\boldsymbol{\Upsilon}_{d,i}\boldsymbol{\psi}_{m}^{H}$,
and
$\check{v}^{\text{WP}}_{n,d}[i+1,m]\sim\mathcal{CN}(0,\sigma^{2}_{\check{v}_{d,i,m}^{\text{WP}}})$
with
$\sigma^{2}_{\check{v}_{d,i,m}^{\text{WP}}}=R_{h}^{d}-\sigma^{2}_{\check{h}_{d,i,m}^{\text{WP}}}$.
Moreover, as the channels of different antennas are assumed to be independent,
the channels between BS antennas and the $d$-th MT can be collected to a
vector as

$\check{\boldsymbol{h}}^{\text{WP}}_{d}[i+1,m]=\boldsymbol{h}_{d}[i+1,m]+\check{\boldsymbol{v}}_{d}^{\text{WP}}[i+1,m]$
(29)

where
$\check{\boldsymbol{h}}^{\text{WP}}_{d}[i+1,m]\sim\mathcal{CN}(\boldsymbol{0},\sigma^{2}_{\check{h}_{d,i,m}^{\text{WP}}}\boldsymbol{I}_{N})$
and
$\check{\boldsymbol{v}}_{d}^{\text{WP}}[i+1,m]\sim\mathcal{CN}(\boldsymbol{0},\sigma^{2}_{\check{v}_{d,i,m}^{\text{WP}}}\boldsymbol{I}_{N}$).

####  IV-B2 Decision-Directed Wiener Predictor

In the proposed Type II frame structure, as UL transmission is activated,
channels can be predicted in the decision-directed (DD) principles [27].
Assume that a $\tau_{p}$-th order DD-WP is performed based on the $\tau_{p}$
latest detected symbols received by BS. According to (10), the received signal
by the $n$-th antenna of BS over the $\bar{m}$-th UL subcarrier and the $i$-th
symbol duration is

$s_{n}[i,\bar{m}]=\sqrt{p_{\text{UL}}}\boldsymbol{x}^{T}[i,\bar{m}]\boldsymbol{h}_{n}[i,\bar{m}]+z^{\text{SI}}[i]+z[i,\bar{m}]$
(30)

where
$\boldsymbol{x}[i,\bar{m}]=\left[{x}_{1}[i,\bar{m}],...,{x}_{D}[i,\bar{m}]\right]^{T}$,
$\boldsymbol{h}_{n}[i,\bar{m}]=\left[h_{n,1}[i,\bar{m}],...,h_{n,D}[i,\bar{m}]\right]^{T}$
and
$z^{\text{SI}}[i]\sim\mathcal{CN}\left(0,\xi_{\text{BS}}p_{\text{DL}}M\right)$.

Based on the DD signals of (30), a $\tau_{p}$-th order DD-WP is employed to
predict the channels $\boldsymbol{h}_{n}[i+1,\bar{m}]$ as
$\check{\boldsymbol{h}}_{n}^{\text{DD}}[i+1,\bar{m}]=\boldsymbol{V}_{\text{DD}}\bar{\boldsymbol{s}}_{n}[i,\bar{m}]$,
where
$\bar{\boldsymbol{s}}_{n}[i,\bar{m}]=[s_{n}[i,\bar{m}],...,s_{n}[i+1-\tau_{p},\bar{m}]]^{T}$,
and $\boldsymbol{V}_{\text{DD}}$ can be expressed as

$\boldsymbol{V}_{\text{DD}}=\boldsymbol{R}_{h\bar{s}}[1,\bar{m}]\boldsymbol{R}_{\bar{s},i}^{-1}[0,\bar{m}]$
(31)

with
$\boldsymbol{R}_{h\bar{s}}[1,\bar{m}]=\mathbb{E}\left[\boldsymbol{h}_{n}[i+1,\bar{m}]\bar{\boldsymbol{s}}_{n}^{H}[i,\bar{m}]\right]$
and
$\boldsymbol{R}_{\bar{s},i}[0,\bar{m}]=\mathbb{E}\left[\bar{\boldsymbol{s}}_{n}[i,\bar{m}]\bar{\boldsymbol{s}}_{n}^{H}[i,\bar{m}]\right]$,
which are detailed below.

First, the cross-correlation matrix in (31) can be obtained with the aid of
(7) and (30), which is

$\begin{split}&\boldsymbol{R}_{h\bar{s}}[1,\bar{m}]=\mathbb{E}\left[\boldsymbol{h}_{n}[i+1,\bar{m}]\bar{\boldsymbol{s}}_{n}^{H}[i,\bar{m}]\right]\\\
&=\sqrt{p_{\text{UL}}}\left[\alpha\boldsymbol{R}_{h}{\boldsymbol{x}}^{*}[i,\bar{m}],...,\alpha^{\tau_{p}}\boldsymbol{R}_{h}{\boldsymbol{x}}^{*}[i+1-\tau_{p},\bar{m}]\right]\\\
&=\sqrt{p_{\text{UL}}}\boldsymbol{A}(\tau_{p},\alpha)\boldsymbol{Q}(\tau_{p},D)\boldsymbol{B}^{H}(\tau_{p},D)\end{split}$
(32)

where

$\begin{split}&\boldsymbol{R}_{h}=\text{diag}(R_{h}^{1},...,R_{h}^{D}),\
\boldsymbol{Q}(\tau_{p},D)=\boldsymbol{R}_{h}\otimes\boldsymbol{I}_{\tau_{p}}\\\
&\boldsymbol{A}(\tau_{p},\alpha)=\boldsymbol{I}_{D}\otimes\boldsymbol{\delta}(\tau_{p},\alpha),\\\
&\boldsymbol{B}(\tau_{p},D)=\left[\text{diag}(\hat{\boldsymbol{x}}_{1}[\bar{m}]),...,\text{diag}(\hat{\boldsymbol{x}}_{d}[\bar{m}]),...,\text{diag}(\hat{\boldsymbol{x}}_{D}[\bar{m}])\right]\end{split}$
(33)

and in $\boldsymbol{B}(\tau_{p},D)$,
$\hat{\boldsymbol{x}}_{d}[\bar{m}]=\left[\hat{x}_{d}[i,\bar{m}],...,\hat{x}_{d}[i+1-\tau_{p},\bar{m}]\right]^{T}$
are UL detected symbols. To obtain (32), each element in
$\hat{\boldsymbol{x}}_{d}[\bar{m}]$ needs to be correctly detected. This can
be achieved by measuring the reliability of each detection in terms of the
likelihood value of a detected symbol. If the likelihood value is larger than
a preset threshold resulting in high reliability, the symbol is then included
for DD prediction. Otherwise, the symbol will not be used for DD channel
prediction.

The auto-correlation matrix $\boldsymbol{R}_{\bar{s},i}[0,\bar{m}]$ in (31) is
given by

$\begin{split}\boldsymbol{R}_{\bar{s},i}[0,\bar{m}]&=\mathbb{E}\left[\bar{\boldsymbol{s}}_{n}[i,\bar{m}]\bar{\boldsymbol{s}}_{n}^{H}[i,\bar{m}]\right]\\\
&=p_{\text{UL}}\boldsymbol{B}(\tau_{p},D)\boldsymbol{C}(\tau_{p},D)\boldsymbol{Q}(\tau_{p},D)\boldsymbol{B}^{H}(\tau_{p},D)\\\
&+\sigma^{2}\boldsymbol{I}_{\tau_{p}}+\boldsymbol{\Xi}_{\text{SI}}^{\text{DD}}[i]\end{split}$
(34)

where
$\boldsymbol{C}(\tau_{p},D)=\boldsymbol{I}_{D}\otimes\boldsymbol{\xi}(\tau_{p},\alpha)$.
When the proposed Type II frame structure is employed, SI is also present at
the observation time, which leads to

$\boldsymbol{\Xi}_{\text{SI}}^{\text{DD}}[i]=\begin{cases}\left[\begin{array}[]{cc}\xi_{\text{BS}}p_{\text{DL}}M\boldsymbol{I}_{\tau_{p}-1}&\boldsymbol{0}\\\
\boldsymbol{0}&0\\\ \end{array}\right],\ \ i=\tau_{p}\\\
\xi_{\text{BS}}p_{\text{DL}}M\boldsymbol{I}_{\tau_{p}},\ \
i>\tau_{p}\end{cases}$ (35)

With the aid of the orthogonal decomposition, we have the expression

$\boldsymbol{h}_{n}[i+1,\bar{m}]=\check{\boldsymbol{h}}_{n}^{\text{DD}}[i+1,\bar{m}]+\check{\boldsymbol{e}}^{\text{DD}}_{n}[i+1,\bar{m}]$
(36)

where
$\check{\boldsymbol{e}}^{\text{DD}}_{n}[i+1,\bar{m}]\sim\mathcal{CN}(\boldsymbol{0},\boldsymbol{R}_{h}-\boldsymbol{\Theta}_{i}[\bar{m}])$
is the uncorrelated channel prediction error vector, and
$\boldsymbol{\Theta}_{i}[\bar{m}]=\boldsymbol{R}_{h\bar{s}}[1,\bar{m}]\boldsymbol{R}_{\bar{s},i}^{-1}[0,\bar{m}]\boldsymbol{R}_{h\bar{s}}^{H}[1,\bar{m}]$,
which is the covariance matrix of
$\check{\boldsymbol{h}}_{n}^{\text{DD}}[i+1,\bar{m}]$. Furthermore, we can
write
$\check{\boldsymbol{h}}_{n,d}^{\text{UL}}[i+1]=\left[\check{h}_{n,d}^{\text{DD}}[i+1,1],...,\check{h}_{n,d}^{\text{DD}}[i+1,\bar{M}]\right]^{T}$,
which follows the distribution of
$\check{\boldsymbol{h}}_{n,d}^{\text{UL}}[i+1]\sim\mathcal{CN}(\boldsymbol{0},\boldsymbol{\Gamma}_{d,i})$,
with $\boldsymbol{\Gamma}_{d,i}$ derived in detail in Appendix B.

Having obtained the frequency-domain prediction of $\boldsymbol{h}_{n,d}[i+1]$
over UL subcarriers, the corresponding time-domain CSI can be found as

$\check{\boldsymbol{g}}_{n,d}^{\text{DD}}[i+1]=\boldsymbol{J}\check{\boldsymbol{h}}_{n,d}^{\text{UL}}[i+1]$
(37)

provided that $\bar{M}\geq L$. In (37),
$\boldsymbol{J}=\left(\left(\boldsymbol{\Phi}_{\text{UL}}\boldsymbol{F}\boldsymbol{\varPsi}\right)^{H}\left(\boldsymbol{\Phi}_{\text{UL}}\boldsymbol{F}\boldsymbol{\varPsi}\right)\right)^{-1}\left(\boldsymbol{\Phi}_{\text{UL}}\boldsymbol{F}\boldsymbol{\varPsi}\right)^{H}$.
Finally, upon following the same approach described in (27)-(28), the DL
channel vector between BS and the $d$-th MT with respect to the $m$-th DL
subcarrier and the $(i+1)$ symbol duration can be predicted as
$\check{\boldsymbol{h}}^{\text{DD}}_{d}[i+1,m]$, which follows

$\check{\boldsymbol{h}}^{\text{DD}}_{d}[i+1,m]=\boldsymbol{h}_{d}[i+1,m]+\check{\boldsymbol{v}}_{d}^{\text{DD}}[i+1,m],\
m=1,...,M$ (38)

where
$\check{\boldsymbol{h}}_{d}^{\text{DD}}[i+1,m]\sim\mathcal{CN}(\boldsymbol{0},\sigma^{2}_{\check{h}_{d,i,m}^{\text{DD}}}\boldsymbol{I}_{N})$,
$\sigma^{2}_{\check{h}_{d,i,m}^{\text{DD}}}=\boldsymbol{\psi}_{m}\boldsymbol{J}\boldsymbol{\Gamma}_{d,i}\boldsymbol{J}^{H}\boldsymbol{\psi}_{m}^{H}$,
and
$\check{\boldsymbol{v}}_{d}^{\text{DD}}[i+1,m]\sim\mathcal{CN}(\boldsymbol{0},\sigma^{2}_{\check{v}_{d,i,m}^{\text{DD}}}\boldsymbol{I}_{N}$),
$\sigma^{2}_{\check{v}_{d,i,m}^{\text{DD}}}=R_{h}^{d}-\sigma^{2}_{\check{h}_{d,i,m}^{\text{DD}}}$.

##  V Performance Analysis Over fast-fading channels

In this section, we derive the ergodic rates attainable by the MDD and TDD
systems with the proposed channel acquisition, where the two types of frame
structures introduced in Section III are respectively applied.

###  V-A Type I Frame Structure

The ergodic achievable rate of the MDD systems with the Type I frame structure
shown in Fig. 1 can be expressed as

$R^{\text{MDD-1}}(\tau_{p})=\frac{1}{TM_{\text{sum}}}\sum\limits_{d=1}^{D}\sum\limits_{i=\tau_{p}+1}^{T}\sum\limits_{m=1}^{M}R_{d,i,m}^{\text{MDD-1}}$
(39)

where $R_{d,i,m}^{\text{MDD-1}}$ denotes the rate obtained by the $d$-th MT
over the $m$-th DL subcarrier during the $i$-th symbol duration. Note that
since no DL pilots are employed, only the expected effective channel gain is
available at MTs. Hence, the mean effective channel gain is treated as the CSI
for signal detection at MTs’ sides. The effective channel gain is
$\omega_{d,d}^{\text{ZF}}[i,m]=\boldsymbol{h}_{d}^{H}[i,m]\check{\boldsymbol{f}}_{d}[i,m]$,
where $\check{\boldsymbol{f}}_{d}[i,m]$ is derived by the ZF principle with
the predicted channel $\check{\boldsymbol{h}}^{\text{WP}}_{d}[i+1,m]$ obtained
in (29). Then, we can derive the lower bounded achievable ergodic rate of
$R_{d,i,m}^{\text{MDD-1-LB}}$ as shown in (40).

$R_{d,i,m}^{\text{MDD-1}}\geq R_{d,i,m}^{\text{MDD-1--
LB}}=\text{log}_{2}\Big{(}1+\frac{p_{\text{DL}}\left|\mathbb{E}\left[\omega_{d,d}^{\text{ZF}}[i,m]\right]\right|^{2}}{p_{\text{DL}}\text{var}\left\\{\omega_{d,d}^{\text{ZF}}\left[i,m\right]\right\\}+p_{\text{DL}}\sum\limits_{k=1,k\neq
d}^{D}\mathbb{E}\left[\left|\omega_{d,k}^{\text{ZF}}\left[i,m\right]\right|^{2}\right]+\text{var}\left\\{z^{\text{SI}}_{d}\right\\}+\sigma^{2}}\Big{)}$
(40)

$\displaystyle
R_{d,i,m}^{\text{MDD-1-LB}}\approx\text{log}_{2}\Big{(}1+\frac{p_{\text{DL}}\left(N-D+1\right)\boldsymbol{\psi}_{m}\boldsymbol{\Upsilon}_{d,i}\boldsymbol{\psi}_{m}^{H}}{0.25p_{\text{DL}}\boldsymbol{\psi}_{m}\boldsymbol{\Upsilon}_{d,i}\boldsymbol{\psi}_{m}^{H}+p_{\text{DL}}D\left(\frac{\beta_{d}}{M_{\text{sum}}}-\boldsymbol{\psi}_{m}\boldsymbol{\Upsilon}_{d,i}\boldsymbol{\psi}_{m}^{H}\right)+\xi_{\text{MT}}p_{\text{UL}}D\bar{M}+D\sigma^{2}}\Big{)}$
(41)

According to Lemma 4 in [28], the expectation and variance of the effective
channel gain $\omega_{d,d}^{\text{ZF}}[i,m]$ can be approximated as

$\displaystyle\mathbb{E}\left[\omega_{d,d}^{\text{ZF}}\left[i,m\right]\right]$
$\displaystyle=\mathbb{E}\left[1/(\sqrt{D}\left\|\check{\boldsymbol{f}}_{d}[i,m]\right\|)\right]$
$\displaystyle\approx\sqrt{\frac{N-D+1}{D}}\sqrt{\sigma^{2}_{\check{h}_{d,i,m}^{\text{WP}}}}$
(42)

and

$\displaystyle\text{var}\left\\{\omega_{d,d}^{\text{ZF}}[i,m]\right\\}\approx\frac{1}{D}\Big{(}\frac{1}{4}\sigma^{2}_{\check{h}_{d,i,m}^{\text{WP}}}+\sigma^{2}_{\check{v}_{d,i,m}^{\text{WP}}}\Big{)},$
(43)

respectively, where $\check{\boldsymbol{f}}_{d}[i,m]$ is the unnormalized
column of the matrix $\check{\boldsymbol{F}}^{\text{ZF}}[i,m]$ seen in (8).
For the interference signal from the other MTs, as seen in the denominator of
(40), we have

$\displaystyle\mathbb{E}\left[\left|\omega_{d,k}^{\text{ZF}}\left[i,m\right]\right|^{2}\right]$
$\displaystyle=\mathbb{E}\left[\left|\check{\boldsymbol{h}}_{d}^{\text{WP},H}[i,m]\check{\boldsymbol{f}}_{k}[i,m]+\check{\boldsymbol{v}}_{d}^{\text{WP},H}[i+1,m]\check{\boldsymbol{f}}_{k}[i,m]\right|^{2}\right]$
$\displaystyle=\mathbb{E}\left[\left|\check{\boldsymbol{v}}_{d}^{\text{WP},H}[i+1,m]\check{\boldsymbol{f}}_{k}[i,m]\right|^{2}\right]$
$\displaystyle=\frac{1}{D}\sigma^{2}_{\check{v}_{d,i,m}^{\text{WP}}}$ (44)

Consequently, upon substituting (28), (V-A), (V-A) and (A-1) into (40), the
approximation to the lower bounded achievable ergodic rate of the $d$-th MT on
the $m$-th DL subcarrier within the $i$-th OFDM symbol duration can be
expressed as (41). Finally, $R^{\text{MDD-1-LB}}(\tau_{p})$ is obtained by
substituting (41) into (39).

On the other hand, the achievable ergodic rate of the TDD systems with the
Type I frame structure shown in Fig. 1 can be written as

$\displaystyle R^{\text{TDD-1}}(\tau_{p})$
$\displaystyle=\frac{1}{TM_{\text{sum}}}\sum\limits_{d=1}^{D}\Big{(}\sum\limits_{i=\tau_{p}+2}^{T-1}\sum\limits_{m=1}^{M_{\text{sum}}}R_{d,i,m}^{\text{TDD-1}}$
$\displaystyle+\frac{1}{2}\sum\limits_{m=1}^{M_{\text{sum}}}\left(R_{d,\tau_{p}+1,m}^{\text{TDD-1}}+R_{d,T,m}^{\text{TDD-1}}\right)\Big{)}$
(45)

Furthermore, the approximated lower bounded achievable ergodic rate of TDD
systems can be derived by following nearly the same steps as above for the MDD
systems, except only the following two points. The first one is that all the
channels within one frame are predicted relying on the first $\tau_{p}$
training symbols sent at the start of the frame, yielding
$\boldsymbol{\delta}(\tau_{p},\alpha)$ in (19) being changed to
$[\alpha^{i-\tau_{p}},\alpha^{i+1-\tau_{p}},...,\alpha^{i-1}]$. The second
point is that the TDD systems do not experience SI. Therefore, we have
$\boldsymbol{\Xi}_{\text{SI}}[i]=0,\forall i$ in (26).

###  V-B Type II Frame Structure

The ergodic achievable rate of the MDD systems with the Type II frame
structure shown in Fig. 2 can be expressed as

$\displaystyle
R^{\text{MDD-2}}(\tau_{p}.\kappa_{p})=\frac{1}{TM_{\text{sum}}}\sum\limits_{d=1}^{D}\Big{(}\sum\limits_{i=\kappa_{p}+1}^{\tau_{p}}\sum\limits_{m=1}^{M}R_{d,i,m}^{\text{MDD-1}}$
$\displaystyle+\sum\limits_{i=\tau_{p}+1}^{T}\sum\limits_{m=1}^{M}R_{d,i,m}^{\text{MDD-2-DL}}+\sum\limits_{i=\tau_{p}+1}^{T}\sum\limits_{\bar{m}=1}^{\bar{M}}R_{d,i,\bar{m}}^{\text{MDD-2-UL}}\Big{)}$
(46)

where $R_{d,i,m}^{\text{MDD-1}}$ denotes the rate attained during Phase I,
$R_{d,i,m}^{\text{MDD-2-DL}}$ and $R_{d,i,\bar{m}}^{\text{MDD-2-UL}}$ denote
respectively the DL and UL rates achievable during Phase II. To compute the
lower bounded rate $R^{\text{MDD-2-LB}}(\tau_{p}.\kappa_{p})$, in (V-B), the
lower bound for $R_{d,i,m}^{\text{MDD-1}}$ is given by (41), while the lower
bound for $R_{d,i,m}^{\text{MDD-2-DL}}$ can be obtained from the approach for
computing $R_{d,i,m}^{\text{MDD-1-LB}}$ by replacing
$\check{h}_{d,i,m}^{\text{WP}}$ using $\check{h}_{d,i,m}^{\text{DD}}$.
Therefore, we only need to derive the lower bound for the achievable UL
ergodic rate, i.e., $R_{d,i,\bar{m}}^{\text{MDD-2-UL}}$. Denote the effective
channel gain as
$\omega_{d,d}^{\text{MRC}}[i,\bar{m}]=\check{\boldsymbol{w}}_{d}^{H}[i,\bar{m}]\boldsymbol{h}_{d}[i,\bar{m}]$.
Then, we have (V-B) at the top of next page,

$\displaystyle R_{d,i,\bar{m}}^{\text{MDD-2-UL}}\geq
R_{d,i,\bar{m}}^{\text{MDD-2-UL-LB}}$
$\displaystyle=\text{log}_{2}\Big{(}1+\frac{p_{\text{UL}}\left|\mathbb{E}\left[\omega_{d,d}^{\text{MRC}}[i,\bar{m}]\right]\right|^{2}}{p_{\text{UL}}\sum\limits_{k=1,k\neq
d}^{D}\mathbb{E}\left[\left|\omega_{d,k}^{\text{MRC}}\left[i,\bar{m}\right]\right|^{2}\right]+\left(\text{var}\left\\{\boldsymbol{z}^{\text{SI}}[i]\right\\}+\sigma^{2}\right)\mathbb{E}\left[\left\|\check{\boldsymbol{w}}_{d}[i,\bar{m}]\right\|_{2}^{2}\right]}\Big{)}$
$\displaystyle\overset{(a)}{=}\text{log}_{2}\Big{(}1+\frac{p_{\text{UL}}\left|\mathbb{E}\left[\check{\boldsymbol{h}}_{d}^{\text{DD},H}[i,\bar{m}]\boldsymbol{h}_{d}[i,\bar{m}]\right]\right|^{2}}{p_{\text{UL}}\sum\limits_{k=1,k\neq
d}^{D}\mathbb{E}\left[\left|\check{\boldsymbol{h}}_{d}^{\text{DD},H}[i,\bar{m}]\boldsymbol{h}_{k}[i,\bar{m}]\right|^{2}\right]+\left(\xi_{\text{BS}}p_{\text{DL}}M+\sigma^{2}\right)\mathbb{E}\left[\left\|\check{\boldsymbol{h}}_{d}^{\text{DD}}[i,\bar{m}]\right\|_{2}^{2}\right]}\Big{)}$
(47)

where (a) is obtained by using
$\check{\boldsymbol{w}}_{d}[i,\bar{m}]=\check{\boldsymbol{h}}_{d}^{\text{DD}}[i,\bar{m}]$
to implement the MRC receiver. In particular, the expectation term in the
numerator of (V-B) can be derived as

$\displaystyle\left|\mathbb{E}\left[\check{\boldsymbol{h}}_{d}^{\text{DD},H}[i,\bar{m}]\boldsymbol{h}_{d}[i,\bar{m}]\right]\right|^{2}$
$\displaystyle=\left|\mathbb{E}\left[\check{\boldsymbol{h}}_{d}^{\text{DD},H}[i,\bar{m}]\left(\check{\boldsymbol{h}}_{d}^{\text{DD}}[i,\bar{m}]+\check{\boldsymbol{v}}_{d}^{\text{DD}}[i,\bar{m}]\right)\right]\right|^{2}$
$\displaystyle\overset{(b)}{=}\left|\mathbb{E}\left[\check{\boldsymbol{h}}_{d}^{\text{DD},H}[i,\bar{m}]\check{\boldsymbol{h}}_{d}^{\text{DD}}[i,\bar{m}]\right]\right|^{2}$
$\displaystyle=\mathbb{E}\left[\left\|\check{\boldsymbol{h}}_{d}^{\text{DD}}[i,\bar{m}]\right\|_{2}^{4}\right]$
(48)

where (b) is due to the fact that
$\check{\boldsymbol{v}}_{d}^{\text{DD}}[i,m]$ has zero mean and is independent
of $\check{\boldsymbol{h}}_{d}[i,\bar{m}]$. The first expectation in the
denominator of (V-B) can be derived as

$\displaystyle\mathbb{E}\left[\left|\check{\boldsymbol{h}}_{d}^{\text{DD},H}[i,\bar{m}]\boldsymbol{h}_{k}[i,\bar{m}]\right|^{2}\right]$
$\displaystyle=\mathbb{E}\left[\left|\check{\boldsymbol{h}}_{d}^{\text{DD},H}[i,\bar{m}]\check{\boldsymbol{h}}_{k}^{\text{DD}}[i,\bar{m}]+\check{\boldsymbol{h}}_{d}^{\text{DD},H}[i,\bar{m}]\check{\boldsymbol{v}}_{k}^{\text{DD}}[i,\bar{m}]\right|^{2}\right]$
$\displaystyle\overset{(c)}{\leq}\mathbb{E}\left[\left|\check{\boldsymbol{h}}_{d}^{\text{DD},H}[i,\bar{m}]\check{\boldsymbol{h}}_{k}^{\text{DD}}[i,\bar{m}]\right|^{2}\right]+\mathbb{E}\left[\left|\check{\boldsymbol{h}}_{d}^{\text{DD},H}[i,\bar{m}]\check{\boldsymbol{v}}_{k}^{\text{DD}}[i,\bar{m}]\right|^{2}\right]$
$\displaystyle\overset{(d)}{=}R_{h}^{k}\mathbb{E}\left[\left\|\check{\boldsymbol{h}}_{d}^{\text{DD}}[i,\bar{m}]\right\|_{2}^{2}\right]$
(49)

where (c) and (d) follow from the fact that
$\check{\boldsymbol{h}}_{d}^{\text{DD}}[i,\bar{m}]$,
$\check{\boldsymbol{h}}_{k}^{\text{DD}}[i,\bar{m}]$ and
$\check{\boldsymbol{v}}_{k}^{\text{DD}}[i,\bar{m}]$ are independent vectors
and $\check{\boldsymbol{v}}_{k}^{\text{DD}}[i,\bar{m}]$ has zero mean.
Consequently, upon substituting (36), (V-B) and (V-B) into (V-B), we can
obtain the approximated lower-bounded UL rate, which is given by

$R_{d,i,\bar{m}}^{\text{MDD-2-UL-
LB}}\approx\text{log}_{2}\Big{(}1+\frac{p_{\text{UL}}N\left(\boldsymbol{\Theta}_{i}[\bar{m}]\right)_{d,d}}{p_{\text{UL}}\sum\limits_{k=1,k\neq
d}^{D}\frac{\beta_{k}}{M_{\text{sum}}}+\xi_{\text{BS}}p_{\text{DL}}M+\sigma^{2}}\Big{)}$
(50)

In the context of TDD, the achievable ergodic rate in the case of the Type II
frame structure shown in Fig. 2 can be expressed as

$\displaystyle R^{\text{TDD-2}}(\tau_{p},\tau_{u})$
$\displaystyle=\frac{1}{TM_{\text{sum}}}\sum\limits_{d=1}^{D}\bigg{(}\sum\limits_{i=\tau_{p}+1}^{\tau_{p}+\tau_{u}}\sum\limits_{m=1}^{M_{\text{sum}}}R_{d,i,m}^{\text{TDD-2-UL}}$
$\displaystyle+\sum\limits_{i=\tau_{p}+\tau_{u}+2}^{T}\sum\limits_{m=1}^{M_{\text{sum}}}R_{d,i,m}^{\text{TDD-2-DL}}$
$\displaystyle+\frac{1}{2}\sum\limits_{m=1}^{M_{\text{sum}}}\left(R_{d,\tau_{p}+\tau_{u}+1,m}^{\text{TDD-2-UL}}+R_{d,T,m}^{\text{TDD-2-DL}}\right)\bigg{)}$
(51)

Based on (V-B), the approximated lower bound
$R^{\text{TDD-2-LB}}(\tau_{p},\tau_{u})$ can be obtained in the way similar to
that for MDD systems. Therefore, the derivation is omitted to avoid
redundancy.

TABLE I: Simulation parameters Default parameters | Value
---|---
Distance between BS and MTs ($D_{d}$) | $[50,100]\ m$
Number of BS antennas ($N$) | 32
Number of MTs ($D$) | 8
Number of DL/UL subcarriers for TDD ($M_{\text{sum}}$) | 96
Number of DL/UL subcarriers for MDD ($M/\bar{M}$) | 64 / 32
BS transmit power ($p_{\text{DL}}M$) | $30\ \text{dBm}$
MT transmit power ($p_{\text{UL}}\bar{M}$) | $20\ \text{dBm}$
Noise power | -94 dBm
Large-scale fading ($\beta_{d}$) | $(D_{d})^{3.8}$
Delay taps ($L$) | 4
Carrier center frequency | 5 GHz
Subcarrier bandwidth | 15 kHz
OFDM symbol duration ($T_{s}$) | 66.67 $\mu s$
Total symbols in one frame ($T$) | $28$
Switching interval in TDD | 0.5
Digital modulation | 16 QAM
SIC capability at BS ($\xi_{\text{BS}}$) | 130 dB
SIC capability at MT ($\xi_{\text{MT}}$) | 120 dB

##  VI Simulation Results and Discussion

In this section, we present the numerical results for the performance
comparison of the MDD and TDD systems with two types of frame structures, as
shown in Fig. 1 and Fig. 2, respectively. We also compare the performance of
IBFD and MDD systems with the proposed frame structures, when SIC capability
is limited. Furthermore, we validate the analytical results derived in Section
V by Monte Carlo simulations over 1000 channel realizations. Throughout this
section, we assume a single-cell network with one BS and multiple single-
antenna MTs. Some key parameters are listed in Table I. It is noteworthy that
to demonstrate the tendency of the rate change within one frame, the switching
intervals in the TDD mode are not considered, while the X-axis represents the
index of OFDM symbol. By contrast, the switching intervals are taken into
account in evaluating the average sum rate over one frame, as shown in Eqs.
(V-A) and (V-B).

###  VI-A Type I Frame Structure: TDD Vs MDD

In this subsection, TDD and MDD systems are compared under the Type I frame
structure. For TDD systems, a total of 7 pilots are transmitted within one
frame. By contrast, for MDD systems, UL pilots can always be transmitted or be
transmitted in part. Note that, in the following figures, ‘TDD-1’ denotes the
scheme that all pilots are consecutively transmitted at the beginning of the
frame and a $7$-th order WP is applied. The notations of ‘TDD-1, Ideal’ and
‘TDD-1, NoP’ indicate respectively that the channel is ideal without the
channel aging problem and that the CSI acquisition does not rely on channel
prediction. ‘MDD-1(z)’ denotes the scheme that UL pilots are continuously sent
by MTs and a $z$-th order WP is applied. Furthermore, the results of ‘TDD-1’
and ‘MDD-1(z)’ are plotted by using the lower-bounded rates given in (39),
(V-A), (V-B) and (V-B), while the results of ‘TDD-1, Apx’ and ‘MDD-1(z), Apx’
represent the correspondingly approximated rates derived in Section V.

Figure 3: Sum rate versus OFDM symbol index of the TDD systems with the
different assumptions about channel acquisition and relative velocity.

In Fig. 3, we evaluate the performance of the TDD systems operated in
different mobility scenarios with or without channel prediction. We observe
that when the relative velocity is 20 km/h, the TDD system with WP outperforms
the TDD system without channel prediction. However, as the symbol index
increases, meaning that CSI becomes more outdated, the performance gap between
the above two strategies becomes narrower. Therefore, in the TDD mode, the
latter symbols benefit less from channel prediction, as only 7 symbols are
used for prediction and the prediction error increases over time.
Additionally, as seen in Fig. 3, as the relative speed increases, the
advantage of employing channel prediction diminishes, and the performance
drops faster and becomes significantly worse than the system with precise CSI.
The results in Fig. 3 explicitly indicate that the traditional TDD mode is not
feasible for communication in high-mobility scenarios, even when channel
prediction is employed.

(a) Normalized MSE (b) Sum rate

Figure 4: Performance comparison of the MDD and TDD systems, when Type I frame
structure and 7-th order WP are employed.

Based on the Type I frame structure presented in Fig. 1, we demonstrate the
effect of the time-varying channels on the performance of TDD and MDD systems
in Fig. 4. It can be observed from Fig. 4(a) that the accuracy of channel
prediction in MDD systems is nearly constant after 7 pilot symbols, regardless
of the relative traveling velocity of 100 km/h or 200 km/h. This is because in
MDD systems, the UL pilots can continuously provide BS with the most updated
CSI for DL transmission, which hence guarantees a stable system sum rate, as
shown in Fig. 4(b). Moreover, the difference between the sum rates achieved by
the MDD systems at 100 km/h and 200 km/h is relatively small, meaning that MDD
systems are robust to time-varying fading channels. By contrast, although the
sum rate of TDD systems is higher than that of MDD systems within the first
several symbols, for both the velocities considered, it reduces quickly with
time in terms of the OFDM symbol indices, due to the reduced accuracy of
channel prediction with time. Furthermore, when the velocity is changed from
100 km/h to 200 km/h, the sum rate deteriorates significantly. Therefore, the
performance of TDD systems is sensitive to the mobility of wireless channels.
The rationale is that in TDD systems, UL training and DL have to be performed
in an alternative way. Once it is switched to DL transmission, the CSI has to
be predicted based on the old data, which may be outdated, if mobility is
high. Additionally, in Fig. 4, it is demonstrated that the approximations of
the lower bounded ergodic rates derived in Section V match closely to the
lower bounded sum rates attained via Monte Carlo simulations.

(a) Sum rate versus OFDM symbol index. (b) Average sum rate within one frame
versus relative velocity.

Figure 5: Performance comparison of the MDD systems with Type I frame
structure and different orders of WPs.

(a) Evenly spaced pilots in TDD Type I frame structure. (b) Two groups of
pilots in TDD Type I frame structure.

Figure 6: Alternative pilot distribution methods for the Type I frame
structure in TDD mode, when the total number of pilots is 7.

(a) $T=28$ (b) $T=56$

Figure 7: Average sum rate versus relative velocity, when the Type I frame
structure is used.

In Fig. 5, we investigate the performance of MDD systems, when Type I frame
structure and different orders of WPs are employed. According to Fig. 1, a
higher order WP uses more channel observations received in the past to predict
channels. It is observed from Fig. 5(a) that at the relative speed of 50 km/h,
the $7$-th order WP in general significantly outperforms the 1-st order WP.
However, when the speed is increased to 250 km/h, the 1-st order WP achieves
nearly the same performance as the $7$-th order WP. The reason behind is that
when channel varies fast, the prediction of the current channel is more
dependent on the nearest pilot symbols and the observations in the past become
less correlated to the prediction. Fig. 5(b) further shows that when pilots
are continuously transmitted under the Type I frame structure, the $1$-st
order WP attains a better performance averaged over one frame, owing to the
fact that in this case, 6 more DL symbols can be transmitted, even though the
slightly reduced prediction accuracy results in some performance loss.

Next, to comprehensively compare the performance of the TDD and MDD systems
employing the Type I frame, we consider two more pilot distribution methods
for TDD, namely the TDD-1-ES and TDD-1-TG, as shown in Fig. 6. Specifically,
with the TDD-1-ES, 7 training symbols are evenly distributed within one frame.
By contrast, with the TDD-1-TG, 7 training symbols are first divided into two
groups, which are then inserted into the frame. Note that the TDD-1-TG is
similar to the frame structure applied in 3GPP 36.211, where two pilot
subframes are evenly distributed within one frame [29]. For MDD systems, we
also introduce an alternative Type I frame structure denoted as MDD-1-PA, as
depicted in Fig. 1. With the MDD-1-PA, $\tau_{p}=1$, and a total of 7 pilots
are activated, which are evenly distributed within one frame.

In Fig. 7, we compare the average sum rate of the MDD and TDD systems with the
various Type I frame structures, and when two frame lengths are considered.
Note that, as the frame length increases, the number of pilots is set to
remain the same in MDD-1-PA and three TDD structures, while the number of DL
symbols increases. It can be observed from Fig. 7 that MDD-1-PA outperforms
all the other arrangements, when the relative speed is lower than 280 km/h in
Fig. 7(a) and 210 km/h in Fig. 7(b). This is because when only a fraction of
pilots are transmitted, the saved UL resources (subcarriers) can be exploited
to transmit DL data, provided that the CSI updating is enough for protecting
the DL transmission. By contrast, MDD-1(1) is superior to the MDD-1-PA in the
high-speed case. In this case, although there are only $M$ subcarriers used
for DL transmission, the continuously transmitted pilots on the $\bar{M}$ UL
subcarriers guarantees the reliable channel prediction, which the MDD-1-PA
structure is however unable to provide. With respect to TDD, the advantages of
the three frame structures are dependent on the relative velocity and the
length of frames. As shown in Fig. 7(a) and 7(b), in general, TDD-1 and TDD-TG
have relatively better performance, when the relative velocity is low, owing
to the use of less switching intervals. By contrast, TDD-1-ES benefits from
the frequent transmission of training symbols within one frame and hence
performs better in the high-speed long-frame case.

(a) Normalized MSE (b) Sum rate

Figure 8: Performance comparison of the MDD and TDD systems, when Type II
frame structure is applied. Figure 9: Distribution of two groups of pilots in
TDD Type II frame structure, when the total number of pilots is 7.

###  VI-B Type II Frame Structure: TDD Vs MDD

In this subsection, we compare the performance of MDD and TDD systems operated
with the Type II frame structure. In the following figures, ‘TDD-2’ and
‘MDD-2’ denote respectively the TDD and MDD systems employing the proposed
frame structure as shown in Fig. 2 with $\tau_{p}=7$, $\tau_{u}=7$ and
$\kappa_{p}=1$.

The performance comparison of MDD and TDD with respect to symbol index is
plotted in Fig. 8, when Type II frame structure is considered. In conjunction
with Fig. 2, we can see that, as the MDD system is able to operate DL and UL
simultaneously during Phase I, DL data symbols can be transmitted while
collecting the channel observations for prediction in Phase II.From Fig. 8(a)
we observe that the WP in Phase I yields less prediction errors when compared
with the DD-WP in Phase II. This is as expected, since the pilots used for WP
in Phase I are known to BS. In Phase II, although the performance of channel
prediction is better without the effect of SI, TDD system only activates the
UL transmission from the 8-th to the 14-th symbol, leading to lower sum rate
when compared with the MDD system, which is capable of transmitting data over
both DL and UL. At the 15-th symbol, both MDD and TDD systems apply the $7$-th
DD-WP and the TDD system attains a higher sum rate, as it uses all the
subcarriers for DL transmission. However, similar to the Type I frame
structure, new UL data symbols are not available for the TDD system to update
channel causing the degraded accuracy of channel prediction. Consequently, its
sum rate decreases, as symbol index increases.

(a) $T=28$ (b) $T=56$

Figure 10: Average sum rate versus the relative velocity, when the Type II
frame structure is used. Figure 11: Sum rate comparison of MDD, IBFD and TDD
with respect to OFDM symbol index, when Type I frame structure and the
relative velocity of 150 km/h are assumed.

In Fig. 10, we compare the average sum rate versus the relative speed of the
MDD and TDD systems, when the Type II frame structure is employed. Here, we
also consider an optional frame structure for TDD systems, which is shown in
Fig. 9, where two groups of pilots are separately inserted into one frame. We
can observe from Fig. 10 that owing to the DL transmission occurred in both
Phase I and the start of Phase II, MDD significantly outperforms TDD over the
whole velocity, ranging from 20 km/h to 300 km/h, for both the frame-length
considered. Specifically, for the TDD in low-mobility case, the rate achieved
by the TDD-2-TG is considerably lower than that achieved by the TDD-2.
However, benefiting from the extra training in the middle of the frame, the
TDD-2-TG finally surpasses the TDD-2 at 240 km/h when $T=28$, and at 140 km/h
when $T=56$. The results in Fig. 10 imply that in the general communication
scenarios where UL training, UL transmission and DL transmissions are all
required at the same time, MDD shows significant advantages over TDD, when the
channel aging problem is encountered.

###  VI-C Influence of SIC on IBFD and MDD

In this part, we demonstrate the impact of SIC on the performance of MDD and
IBFD systems. To this end, we consider the conventional IBFD with the Type I
frame structure denoted as ‘IBFD-1’, where UL pilots are unremittingly
transmitted over $\bar{M}$ subcarriers and a $7$-th order WP is applied for
channel prediction. Note that, unlike the MDD-1 that only uses $M$ subcarriers
for DL transmission, as stated in Section VI-A, IBFD-1 uses all the
$M_{\text{sum}}$ subcarriers for the DL transmission, which imposes an extra
SIC burden in digital-domain. According to [11], the digital-domain SIC in
IBFD systems is expected to cancel the main linear signal component by more
than 30 dB. By contrast, MDD systems are almost free from linear SI in the
digital-domain owing to the FFT operation, which separates the UL subcarriers
from the DL subcarriers. Based on this observation, it is sensible to assume
that with the same amount of system resources used for SI suppression, MDD can
always achieve 30 dB more SIC than IBFD. From Fig. 11, it is shown that when
both IBFD and MDD systems have the same SI mitigation capability666This means
IBFD costs more computation effort and system complexity than MDD for SIC, and
hence achieves lower energy efficient [14]., IBFD apparently outperforms both
TDD and MDD. However, if IBFD can only mitigate the SI by 100 dB777Note that
100 dB of SI is often assumed in IBFD systems, which is regarded as the
bottleneck value based on the existing SIC approaches [30, 31]., the
achievable sum rate of IBFD systems drops dramatically, and is much lower than
that of MDD systems.

##  VII Conclusions

A MDD-assisted mMIMO has been studied and comprehensively compared with the
TDD-relied mMIMO over fast time-varying channels. Two types of MDD frame
structures incorporated respectively with WP and DD-WP have been proposed to
combat channel aging. For comparison, the frames in TDD mMIMO have been
structured according to the 3GPP standards or improved for handling the
channel aging problem. MDD has also been compared with the IBFD by considering
their SIC capabilities. Moreover, the approximated lower bounds for the
achievable rates of the MDD and TDD mMIMO systems communicating based on the
proposed frame structures have been derived. According to our studies and
performance comparison, we have the following main observations. First, when
the Type I frame structure is employed, benefiting from the instantly
available CSI updating while without involving UL/DL switching, MDD
outperforms TDD over time-varying channels, which is more significant when
mobility becomes higher. Second, the above performance advantage of MDD over
TDD retains under the Type II frame structure, owing to the MDD’s merits that
it can proactively start DL transmission in Phase I, while simultaneously
updating the CSI with the aid of the UL transmission in Phase II. Third, it is
shown that SIC is critical to IBFD systems. By taking the advantage of near-
free SI in the digital-domain, MDD is capable of achieving better performance
than IBFD in the presence of channel aging, if SIC in IBFD systems is
imperfect. Additionally, the approximated lower bounded rates derived in this
paper have been validated by the Monte-Carlo simulations, showing that they
agree well with each other.

##  Appendix A Calculation of SI Power

####  A-1 SI in Equation (9)

According to the model for residual SI [23], the covariance of
$z_{d}^{\text{SI}}[i]$ is

$\displaystyle\text{cov}\left\\{z_{d}^{\text{SI}}[i]\right\\}$
$\displaystyle=\mathbb{E}\left[h_{\text{SI}}x_{d}[i]x_{d}^{H}[i]h_{\text{SI}}^{H}\right]$
$\displaystyle=\xi_{\text{MT}}p_{\text{UL}}\bar{M}$ (52)

when using $\mathbb{E}\left[h_{\text{SI}}h_{\text{SI}}^{H}\right]=1$. (A-1) is
obtained by substituting
$x_{d}[i]=\sqrt{p_{\text{UL}}}\sum\limits_{\bar{m}=1}^{\bar{M}}x_{d}[i,\bar{m}]$.

####  A-2 SI in Equation (10)

To compute the covariance of $z^{\text{SI}}[i]$ in (10), we again refer to the
residual model in [23] and have

$\displaystyle\text{cov}\left\\{\boldsymbol{z}^{\text{SI}}[i]\right\\}=\xi_{\text{BS}}\text{diag}\left(\mathbb{E}\left[\boldsymbol{H}_{\text{SI}}\boldsymbol{s}_{\text{DL}}[i]\boldsymbol{s}_{\text{DL}}^{H}[i]\boldsymbol{H}_{\text{SI}}^{H}\right]\right)$
(53)

Upon substituting
$\boldsymbol{s}_{\text{DL}}[i]=\sum\limits_{m=1}^{M}\sqrt{p_{\text{DL}}}\boldsymbol{F}^{\text{ZF}}[i,m]\boldsymbol{x}[i,m]$
into (53), we can obtain the SI power at the $n$-th receive element as

$\displaystyle\text{cov}\left\\{\boldsymbol{z}^{\text{SI}}[i]\right\\}_{n,n}=\xi_{\text{BS}}\mathbb{E}\left[\boldsymbol{H}_{\text{SI}}^{(n,:)}\boldsymbol{s}_{\text{DL}}[i]\boldsymbol{s}_{\text{DL}}^{H}[i]\boldsymbol{H}_{\text{SI}}^{(n,:),H}\right]$
$\displaystyle=\xi_{\text{BS}}\mathbb{E}\left[\text{Tr}\left(p_{\text{DL}}\sum\limits_{m=1}^{M}\boldsymbol{H}_{\text{SI}}^{(n,:)}\boldsymbol{F}^{\text{ZF}}[i,m]\boldsymbol{F}^{\text{ZF},H}[i,m]\boldsymbol{H}_{\text{SI}}^{(n,:),H}\right)\right]$
$\displaystyle\overset{(a)}{=}\xi_{\text{BS}}p_{\text{DL}}\sum\limits_{m=1}^{M}\left\|\boldsymbol{F}^{\text{ZF}}[i,m]\right\|_{F}^{2}$
$\displaystyle\overset{(b)}{=}\xi_{\text{BS}}p_{\text{DL}}M$ (54)

where $(a)$ is obtained using
$\mathbb{E}[\boldsymbol{H}_{\text{SI}}^{(n,:),H}\boldsymbol{H}_{\text{SI}}^{(n,:)}]=\boldsymbol{I}_{N}$
and $(b)$ is due to the power normalization
$\left\|\boldsymbol{F}^{\text{ZF}}[i,m]\right\|_{F}^{2}=1$. Consequently,
$\text{cov}\left\\{\boldsymbol{z}^{\text{SI}}[i]\right\\}=\xi_{\text{BS}}p_{\text{DL}}M\boldsymbol{I}_{N}$.

##  Appendix B Covariance matrix of predicted UL subcarrier channels in (37)

The covariance matrix $\boldsymbol{\Gamma}_{d,i}$ in (37) can be written as
(58).

$\displaystyle\boldsymbol{\Gamma}_{d,i}$
$\displaystyle=\mathbb{E}\left[\check{\boldsymbol{h}}_{n,d}^{\text{UL}}[i+1]\check{\boldsymbol{h}}_{n,d}^{\text{UL},H}[i+1]\right]=\left[\begin{array}[]{ccc}\mathbb{E}\left[\check{h}_{n,d}^{\text{DD}}[i+1,1]\check{h}_{n,d}^{\text{DD},H}[i+1,1]\right]&\cdots&\mathbb{E}\left[\check{h}_{n,d}^{\text{DD}}[i+1,1]\check{h}_{n,d}^{\text{DD},H}[i+1,\bar{M}]\right]\\\
\vdots&\ddots&\vdots\\\
\mathbb{E}\left[\check{h}_{n,d}^{\text{DD}}[i+1,\bar{M}]\check{h}_{n,d}^{\text{DD},H}[i+1,1]\right]&\cdots&\mathbb{E}\left[\check{h}_{n,d}^{\text{DD}}[i+1,\bar{M}]\check{h}_{n,d}^{\text{DD},H}[i+1,\bar{M}]\right]\end{array}\right]$
(58)

As for the diagonal elements in $\boldsymbol{\Gamma}_{d,i}$, according to
(36), we have

$h_{n,d}[i+1,\bar{m}]=\check{h}_{n,d}^{\text{DD}}[i+1,\bar{m}]+\check{e}^{\text{DD}}_{n,d}[i+1,\bar{m}]$
(59)

Hence, the variance of $\check{h}_{n,d}^{\text{DD}}[i+1,\bar{m}]$ is
$\left(\boldsymbol{\Theta}_{i}[\bar{m}]\right)_{d,d}$, given in (36). For the
off-diagonal elements of $\boldsymbol{\Gamma}_{d,i}$, we have

$\displaystyle\mathbb{E}\left[\check{h}_{n,d}^{\text{DD}}[i+1,\bar{m}_{1}]\check{h}_{n,d}^{\text{DD},H}[i+1,\bar{m}_{2}]\right]$
$\displaystyle=\mathbb{E}\Big{[}(h_{n,d}[i+1,\bar{m}_{1}]-\check{e}_{n,d}^{\text{DD}}[i+1,\bar{m}_{1}])(h_{n,d}^{H}[i+1,\bar{m}_{2}]$
$\displaystyle-\check{e}_{n,d}^{\text{DD},H}[i+1,\bar{m}_{2}])\Big{]}$
$\displaystyle\overset{(a)}{=}\mathbb{E}\left[h_{n,d}[i+1,\bar{m}_{1}]h_{n,d}^{H}[i+1,\bar{m}_{2}]\right]$
$\displaystyle\overset{(b)}{=}\mathbb{E}\left[\boldsymbol{\psi}_{\bar{m}_{1}}\boldsymbol{g}_{n,d}[i+1]\boldsymbol{g}^{H}_{n,d}[i+1]\boldsymbol{\psi}^{H}_{\bar{m}_{2}}\right]$
$\displaystyle=\frac{\beta_{d}}{L}\boldsymbol{\psi}_{\bar{m}_{1}}\boldsymbol{\psi}^{H}_{\bar{m}_{2}},\
\bar{m}_{1}\neq\bar{m}_{2}$ (60)

where (a) is obtained based on the fact that the prediction error of the
$\bar{m}_{1}$-th subcarrier channel is independent of the $\bar{m}_{2}$-th
subcarrier channel and vice versa, while (b) can be derived from the FFT
operation. In summary, the covariance matrix $\boldsymbol{\Gamma}_{d,i}$ can
be expressed as

$\displaystyle\boldsymbol{\Gamma}_{d,i}=\left[\begin{array}[]{ccc}\left(\boldsymbol{\Theta}_{i}[1]\right)_{d,d}&\cdots&\frac{\beta_{d}}{L}\boldsymbol{\psi}_{1}\boldsymbol{\psi}^{H}_{\bar{M}}\\\
\vdots&\ddots&\vdots\\\
\frac{\beta_{d}}{L}\boldsymbol{\psi}_{\bar{M}}\boldsymbol{\psi}^{H}_{1}&\cdots&\left(\boldsymbol{\Theta}_{i}[\bar{M}]\right)_{d,d}\end{array}\right]$
(64)

## References

  * [1] E. G. Larsson, O. Edfors, F. Tufvesson, and T. L. Marzetta, “Massive MIMO for next generation wireless systems,” _arXiv preprint arXiv:1304.6690_ , 2013.
  * [2] E. Björnson, E. G. Larsson, and T. L. Marzetta, “Massive MIMO: Ten myths and one critical question,” _IEEE Communications Magazine_ , vol. 54, no. 2, pp. 114–123, 2016.
  * [3] L. Sanguinetti, E. Björnson, and J. Hoydis, “Toward massive MIMO 2.0: Understanding spatial correlation, interference suppression, and pilot contamination,” _IEEE Transactions on Communications_ , vol. 68, no. 1, pp. 232–257, 2019.
  * [4] K. T. Truong and R. W. Heath, “Effects of channel aging in massive MIMO systems,” _Journal of Communications and Networks_ , vol. 15, no. 4, pp. 338–351, 2013.
  * [5] A. K. Papazafeiropoulos and T. Ratnarajah, “Deterministic equivalent performance analysis of time-varying massive MIMO systems,” _IEEE Transactions on Wireless Communications_ , vol. 14, no. 10, pp. 5795–5809, 2015\.
  * [6] S. Kashyap, C. Mollén, E. Björnson, and E. G. Larsson, “Performance analysis of (TDD) massive MIMO with kalman channel prediction,” in _2017 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)_. IEEE, 2017, pp. 3554–3558.
  * [7] J. Yuan, H. Q. Ngo, and M. Matthaiou, “Machine learning-based channel prediction in massive MIMO with channel aging,” _IEEE Transactions on Wireless Communications_ , vol. 19, no. 5, pp. 2960–2973, 2020.
  * [8] J. Mirza, G. Zheng, K.-K. Wong, S. Lambotharan, and L. Hanzo, “On the performance of multiuser MIMO systems relying on full-duplex CSI acquisition,” _IEEE Transactions on Communications_ , vol. 66, no. 10, pp. 4563–4577, 2018.
  * [9] R. Rajashekar, C. Xu, N. Ishikawa, L.-L. Yang, and L. Hanzo, “Multicarrier division duplex aided millimeter wave communications,” _IEEE Access_ , vol. 7, pp. 100 719–100 732, 2019.
  * [10] L.-L. Yang, _Multicarrier Communications_. John Wiley & Sons, 2009.
  * [11] D. Bharadia, E. McMilin, and S. Katti, “Full duplex radios,” in _ACM SIGCOMM Computer Communication Review_ , vol. 43, no. 4. ACM, 2013, pp. 375–386.
  * [12] B. Li, L.-L. Yang, R. G. Maunder, and S. Sun, “Self-interference cancellation and channel estimation in multicarrier-division duplex systems with hybrid beamforming,” _IEEE Access_ , vol. 8, pp. 160 653–160 669, 2020.
  * [13] B. Li, L.-L. Yang, R. G. Maunder, and S. Sun, “Resource allocation in millimeter-wave multicarrier-division duplex systems with hybrid beamforming,” _IEEE Transactions on Vehicular Technology_ , vol. 70, no. 8, pp. 7921–7935, 2021.
  * [14] B. Li, L.-L. Yang, R. G. Maunder, P. Xiao, and S. Sun, “Multicarrier-division duplex: A duplexing technique for the shift to 6G wireless communications,” _IEEE Vehicular Technology Magazine_ , vol. 16, no. 4, pp. 57–67, 2021.
  * [15] 3GPP, “Physical layer procedures (Release 15), TS 36.213.”
  * [16] 3GPP, “NR; physical channels and modulation (Release 15), TS 38.211.”
  * [17] C. Nam, C. Joo, and S. Bahk, “Joint subcarrier assignment and power allocation in full-duplex OFDMA networks,” _IEEE Transactions on Wireless Communications_ , vol. 14, no. 6, pp. 3108–3119, 2015.
  * [18] X. Li, X. Ge, X. Wang, J. Cheng, and V. C. Leung, “Energy efficiency optimization: Joint antenna-subcarrier-power allocation in OFDM-DASs,” _IEEE Transactions on Wireless Communications_ , vol. 15, no. 11, pp. 7470–7483, 2016.
  * [19] Y. J. Zhang and K. B. Letaief, “Multiuser adaptive subcarrier-and-bit allocation with adaptive cell selection for OFDM systems,” _IEEE Transactions on Wireless Communications_ , vol. 3, no. 5, pp. 1566–1575, 2004\.
  * [20] A. Sabharwal, P. Schniter, D. Guo, D. W. Bliss, S. Rangarajan, and R. Wichman, “In-band full-duplex wireless: Challenges and opportunities.” _IEEE Journal on Selected Areas in Communications_ , vol. 32, no. 9, pp. 1637–1652, 2014\.
  * [21] K. E. Kolodziej, B. T. Perry, and J. S. Herd, “In-band full-duplex technology: Techniques and systems survey,” _IEEE Transactions on Microwave Theory and Techniques_ , 2019.
  * [22] K. E. Baddour and N. C. Beaulieu, “Autoregressive modeling for fading channel simulation,” _IEEE Transactions on Wireless Communications_ , vol. 4, no. 4, pp. 1650–1662, 2005.
  * [23] B. P. Day, A. R. Margetts, D. W. Bliss, and P. Schniter, “Full-duplex MIMO relaying: Achievable rates under limited dynamic range,” _IEEE Journal on Selected Areas in Communications_ , vol. 30, no. 8, pp. 1541–1553, 2012.
  * [24] B. Debaillie, D.-J. van den Broek, C. Lavin, B. van Liempd, E. A. Klumperink, C. Palacios, J. Craninckx, B. Nauta, and A. Pärssinen, “Analog/RF solutions enabling compact full-duplex radios,” _IEEE Journal on Selected Areas in Communications_ , vol. 32, no. 9, pp. 1662–1673, 2014.
  * [25] K. E. Kolodziej, J. G. McMichael, and B. T. Perry, “Multitap RF canceller for in-band full-duplex wireless communications,” _IEEE Transactions on Wireless Communications_ , vol. 15, no. 6, pp. 4321–4334, 2016.
  * [26] S. S. Haykin, _Adaptive Filter Theory_. Pearson Education India, 2005.
  * [27] D. Schafhuber and G. Matz, “MMSE and adaptive prediction of time-varying channels for OFDM systems,” _IEEE Transactions on Wireless Communications_ , vol. 4, no. 2, pp. 593–602, 2005.
  * [28] A. Khansefid and H. Minn, “Achievable downlink rates of MRC and ZF precoders in massive MIMO with uplink and downlink pilot contamination,” _IEEE Transactions on Communications_ , vol. 63, no. 12, pp. 4849–4864, 2015\.
  * [29] 3GPP, “Evolved Universal Terrestrial Radio Access (E-UTRA); Physical channels and modulation (Release 8), TS 36.211.”
  * [30] B. Di, S. Bayat, L. Song, Y. Li, and Z. Han, “Joint user pairing, subchannel, and power allocation in full-duplex multi-user OFDMA networks,” _IEEE Transactions on Wireless Communications_ , vol. 15, no. 12, pp. 8260–8272, 2016\.
  * [31] D. W. K. Ng, Y. Wu, and R. Schober, “Power efficient resource allocation for full-duplex radio distributed antenna networks,” _IEEE Transactions on Wireless Communications_ , vol. 15, no. 4, pp. 2896–2911, 2016.


$\displaystyle\mathrm{Var}\left(\Big{(}\Gamma_{N}-\sum_{i<N}w_{v,i}^{\star}\Gamma_{i}\Big{)}\sum_{t\geq
T}v_{v,t}^{\star}\tilde{\Lambda}_{t}+\sum_{t\geq
T}v_{v,t}^{\star}\varepsilon_{n,t}\Big{|}T_{0}=T,N_{0}=N,(\Gamma_{i})_{i\geq
1}\right)\geq||v_{v}^{\star}||_{2}^{2}\sigma_{\varepsilon}^{2}\geq\sigma_{\varepsilon}^{2}/T_{1},$
(21)

since $\sigma_{\varepsilon}^{2}>0$ and $||v_{v}^{\star}||_{2}^{2}\geq 1/T_{1}$
(since $1^{\top}v_{v}=1$).

We can follow verbatim the proof of Theorem 4.1 (paragraph “CLT for fixed
$n$), with $(\Gamma_{n}-\sum_{i<N}w_{v,i}^{\star}\Gamma_{i})$ in lieu of
$\Gamma_{n}$ in the proof of Theorem 4.1, and obtain

$\displaystyle\frac{\mathbb{E}\Big{[}\Big{\\{}\Big{(}\Gamma_{n}-\sum_{i<N}w_{v,i}^{\star}\Gamma_{i}\Big{)}\sum_{t\geq
T}v_{v,t}^{\star}\tilde{\Lambda}_{t}+\sum_{t\geq
T}v_{v,t}^{\star}\varepsilon_{n,t}\Big{\\}}^{3}\Big{|}(\Gamma_{i})_{i\geq
1},N_{0}=N,T_{0}=T\Big{]}}{\mathrm{Var}\left(\Big{(}\Gamma_{n}-\sum_{i<N}w_{v,i}^{\star}\Gamma_{i}\Big{)}\sum_{t\geq
T}v_{v,t}^{\star}\tilde{\Lambda}_{t}+\sum_{t\geq
T}v_{v,t}^{\star}\varepsilon_{n,t}\Big{|}T_{0}=T,N_{0}=N,(\Gamma_{i})_{i\geq
1}\right)^{3/2}}=o(1).$ (22)

#### Collecting the terms

We can write, as $T_{1}\rightarrow\infty$, for any $N_{1}$,

$\displaystyle\hat{\tau}(w_{h}^{\star},w_{v}^{\star},v_{h}^{\star},v_{v}^{\star})-\tau=\sum_{n\geq
N}v_{h,n}\Big{\\{}\underbrace{\Big{(}\Gamma_{n}-\sum_{i<N}w_{v,i}^{\star}\Gamma_{i}\Big{)}\sum_{t\geq
T}v_{v,t}^{\star}\tilde{\Lambda}_{t}+\sum_{t\geq
T}v_{v,t}^{\star}\varepsilon_{n,t}}_{(A)}+\mathcal{O}_{p}(T_{0}^{-1/6})\Big{\\}}.$
(23)

Therefore, because $T_{1}N_{1}/T_{0}^{-1/3}=o(1)$, it follows from Equation
(21) that $(A)$ has a slower convergence rate than $T_{0}^{-1/6}$ and the
second component in the right-hand side of Equation (23) is asymptotically
negligible relative to the first one.

#### Asymptotic normality and conclusions

Here, $(A)$ in Equation (23) is asymptotically normal by Equation (22) and
Lyapunov’s central limit theorem. Because $\varepsilon_{i,t}$ are independent
across units and time, of $\Lambda_{t}$, and because $v_{h}^{\top}1=1$, the
central limit theorem for Equation (23) directly applies for the variance as
defined in Theorem 4.2.

#### Conclusions

The case under Assumption 5 follows similarly after inverting the role of the
loadings and factors.

###  C.5 Proof of Theorem A.1

Under Assumption 8, following verbatim the proof of Theorem 4.2 while
inverting the loadings and factors, we can write under the null that $\tau=0$
(letting $\Gamma_{i}=\gamma_{i}+\tilde{\Gamma}_{i}$)

$\hat{\tau}^{pl}(j;w_{v}^{\star},w_{h}^{\star},v_{h}^{\star})=\sum_{t\geq
T}v_{h,t}^{\star}\Big{\\{}\Gamma_{j}\tilde{\Lambda}_{t}+\varepsilon_{j,t}-\Gamma_{j}\sum_{s<T}w_{h,s}^{\star}\tilde{\Lambda}_{s}\Big{\\}}+\mathcal{O}_{p}(N_{0}^{-1/6}).$

We can then follow verbatim the proof of Theorem 4.2 (for fixed unit $n$),
after inverting the factors with the loadings to claim asymptotic normality.
Exchangeability of each estimator
$\hat{\tau}^{pl}(j;w_{v}^{\star},w_{h}^{\star},v_{h}^{\star})$ directly
follows from Assumption 3.

###  C.6 Proof of Corollary 2

We can write

$\displaystyle\frac{\hat{\tau}^{h}(\hat{w}_{h},\hat{v}_{h},\hat{\beta}_{h})-\tau}{\bar{\mathbb{V}}_{h}^{1/2}(N,T,q_{h})}=\underbrace{\frac{\hat{\tau}^{h}(w_{h}^{\star},v_{h}^{\star},\beta_{h}^{\star})-\tau}{\bar{\mathbb{V}}_{h}^{1/2}(N,T,q_{h})}}_{(A)}+\underbrace{\frac{\hat{\tau}^{h}(\hat{w}_{h},\hat{v}_{h},\hat{\beta}_{h})-\hat{\tau}^{h}(w_{h}^{\star},v_{h}^{\star},\beta_{h}^{\star})}{\bar{\mathbb{V}}_{h}^{1/2}(N,T,q_{h})}}_{(B)}.$

By Theorem 4.1, we have $(A)\rightarrow_{d}\mathcal{N}(0,1)$. We have
$\bar{V}_{h}(N,T,q_{h})\geq||q_{h}||^{2}||v_{h}^{\star}||^{2}\sigma_{\varepsilon}^{2}\geq
T_{1}N_{1}\sigma_{\varepsilon}^{2}$. Therefore $(B)=o_{p}(1)$ under Equation
(7). The result follows from Slutsky theorem.