# Full-Duplex Millimeter Wave MIMO Channel Estimation: A Neural Network
Approach

Mehdi Sattari, Hao Guo, Deniz Gündüz, Ashkan Panahi, and Tommy Svensson This
work has been supported by the project SEMANTIC, funded by the EU’s Horizon
2020 research and innovation programme under the Marie Skłodowska-Curie grant
agreement No 861165. Corresponding author: M. Sattari. M. Sattari, H. Guo, and
T. Svensson are with the Department of Electrical Engineering, Chalmers
University of Technology, Gothenburg, Sweden. E-mail: {mehdi.sattari, hao.guo,
<EMAIL_ADDRESS>Panahi is with the Department of Computer
Science, Chalmers University of Technology, Gothenburg, Sweden. E-mail:
<EMAIL_ADDRESS>Gunduz is with the Department of Electrical and
Electronic Engineering, Imperial College London, London, UK. E-mail:
<EMAIL_ADDRESS>
###### Abstract

Millimeter wave (mmWave) multiple-input-multi-output (MIMO) is now a reality
with great potential for further improvement. We study full-duplex
transmissions as an effective way to improve mmWave MIMO systems. Compared to
half-duplex systems, full-duplex transmissions may offer higher data rates and
lower latency. However, full-duplex transmission is hindered by self-
interference (SI) at the receive antennas, and SI channel estimation becomes a
crucial step to make the full-duplex systems feasible. In this paper, we
address the problem of channel estimation in full-duplex mmWave MIMO systems
using neural networks (NNs). Our approach involves sharing pilot resources
between user equipments (UEs) and transmit antennas at the base station (BS),
aiming to reduce the pilot overhead in full-duplex systems and to achieve a
comparable level to that of a half-duplex system. Additionally, in the case of
separate antenna configurations in a full-duplex BS, providing channel
estimates of transmit antenna (TX) arrays to the downlink UEs poses another
challenge, as the TX arrays are not capable of receiving pilot signals. To
address this, we employ an NN to map the channel from the downlink UEs to the
receive antenna (RX) arrays to the channel from the TX arrays to the downlink
UEs. We further elaborate on how NNs perform the estimation with different
architectures, (e.g., different numbers of hidden layers), the introduction of
non-linear distortion (e.g., with a 1-bit analog-to-digital converter (ADC)),
and different channel conditions (e.g., low-correlated and high-correlated
channels). Our work provides novel insights into NN-based channel estimators.

###### Index Terms:

Channel estimation, Full-duplex, mmWave MIMO, Neural networks.

##  I INTRODUCTION

Current key enabling wireless technologies, such as massive multiple-input-
multi-output (MIMO), millimeter-wave (mmWave) communication, and ultra-dense
networks have significantly improved the throughput of wireless communications
[1]. Thanks to the huge bandwidth available in the mmWave band, we can meet
the high data rate requirements for future cellular networks, and the high
path loss at these frequencies can be compensated by utilizing the beamforming
gain of MIMO antennas. Nonetheless, these technologies have been mainly
studied for half-duplex communication, and the potential of utilizing full-
duplex communication has been widely overlooked. In full-duplex transmission,
a transceiver can simultaneously transmit and receive over the same carrier
frequency, in principle doubling the spectral efficiency. Besides, the delay
associated with half-duplex transmission is not present in full-duplex
systems, thereby enabling the low latency requirements of 5G and beyond [2],
[3].

To realize full-duplex transmission, the self-interference (SI) caused by each
transmit element over the entire receive array has to be canceled out.
Consecutive cancellation procedures are usually considered in three domains:
propagation, analog, and digital [4]. In propagation domain cancellation, SI
power is suppressed before reaching the receive chain circuit. Analog domain
cancellation is performed before the analog-to-digital converter (ADC) in the
receiver analog chain. Finally, digital domain cancellation suppresses the SI
power remaining from the propagation and analog domains [5]. There are two
main antenna configurations used for full-duplex systems: separate and shared.
In the former, separate antennas are employed for transmission and reception,
yielding high isolation and relatively low interference. In contrast, in the
latter configuration, a single antenna is utilized for both functions,
offering a simpler design and lower hardware costs [6].

Due to the large number of antenna elements in massive MIMO systems, channel
estimation is a complex process with a high pilot overhead. This is the case
even in half-duplex systems, but is more pronounced in full-duplex systems due
to the critical need for SI cancellation, which requires estimation of the
large SI channel. Several studies have been conducted to address channel
estimation in full-duplex systems. In [7], a channel estimation scheme based
on least squares (LS) estimation is proposed for bidirectional communication
between a pair of transceivers. The achievable sum rate was analyzed
considering the limited dynamic range of the transmitter and receiver and
channel estimation error. In [8], an LS-based scheme was utilized for a multi-
pair two-way amplify and forward (AF) relaying system. Additionally, a low-
pilot overhead scheme was proposed, where two user equipments (UEs) employ the
same pilot sequence, while different UE pairs adopt orthogonal pilot
sequences.

In [9], an expectation-maximization algorithm is utilized to address the
channel estimation problem in a large-scale MIMO system with a full-duplex
relay. The authors explore two different scenarios: in the first scenario,
both the base station (BS) and the relay estimate their respective individual
channels, while in the second scenario, only the BS estimates the cascaded
channel between the transmitter, relay, and receiver. In [10], channel
estimation is performed in both the radio frequency (RF) and baseband domains.
In the RF domain, the authors employ the LS technique to estimate the SI
channel. Subsequently, a subspace-based algorithm is employed in the baseband
to estimate the residual SI and BS-UE channels. The study in [11] proposes a
specific frame structure for full-duplex systems, for which the achievable
rate and SI channel estimation are analyzed. Furthermore, the impact of the
calibration period on SI channel estimation is investigated in [12].

In [13], the authors have studied the problem of hybrid beamforming with low-
resolution phase shifters for full-duplex mmWave large-scale MIMO systems and
optimized the sum spectral efficiency (SE) of uplink and downlink UEs,
assuming perfect channel state information (CSI) knowledge. Two different SI
cancellation approaches, namely spatial suppression and SI subtraction, have
been compared in [14], taking into account channel estimation errors and
channel spatial correlation. The uplink and downlink achievable rates have
been obtained in [15], considering both the perfect CSI and imperfect CSI
cases in full-duplex massive MIMO systems with low-resolution ADCs. A study
has been conducted in [16] for full-duplex large-scale MIMO cellular systems,
considering both non-cooperative and cooperative scenarios. Minimum mean
squared error (MMSE) channel estimation is employed for SI and UE channels.
Analytical expressions have been derived for the ergodic achievable rate using
linear filters such as matched filter (MF) and zero-forcing (ZF). Authors in
[17] have introduced a hybrid beamforming design for mmWave full-duplex,
addressing several practical aspects such as codebook-based analog beamforming
and beam alignment.

A neural network (NN) is a specialized tool within the field of machine
learning that has demonstrated remarkable effectiveness in solving a wide
range of tasks, including computer vision, speech recognition, and autonomous
driving [18]. Thanks to the versatility of NNs, they have also found
applications in wireless communication, either in a block-structured manner or
in an end-to-end scenario [19]. Research in the wireless communication
community has explored optimizing traditional processing blocks, such as
channel estimation [20], precoding [21], signal detection [22], and CSI
feedback [23], using deep learning techniques. Moreover, in [24], an end-to-
end communication system has been represented by an autoencoder. In [25], a
communication system model is developed without prior information on the
channel, with the assistance of conditional generative adversarial networks
(GANs).

Numerous studies have focused on NN-based channel estimation. In [26],
convolutional neural networks (CNNs) are employed to estimate the mmWave
massive MIMO channel, demonstrating that CNNs can achieve superior performance
compared to MMSE by effectively exploiting spatial and frequency correlation.
In [27], the authors utilize a specifically designed CNN called “deep image
prior” to denoise the LS estimated massive MIMO channel. Their findings
indicate that deep learning can successfully estimate pilot-contaminated
massive MIMO channels. In another work, [28], fully connected neural networks
(FNNs) are utilized to estimate the massive MIMO channel from quantized
received measurements with low-resolution ADCs. The study reveals that with
sufficiently large antenna arrays, fewer pilots are required for channel
estimation. For beamspace mmWave massive MIMO systems, a learned denoising-
based approximate message passing network is used to solve the channel
estimation problem [29]. Simulation results demonstrate that deep learning-
based channel estimation outperforms compressed sensing (CS)-based schemes in
this context.

As highlighted in several papers, such as [2], [3], and [13], the channel
estimation problems in full-duplex transmission have not received extensive
research attention in the literature. The few existing works mentioned above
are primarily focused on simple channel estimation techniques, like LS
estimators, which may not ensure high-quality channel estimates in low signal-
to-noise ratios (SNRs). On the other hand, the MMSE channel estimation
requires knowledge of the channel correlation matrix and is burdened with high
computational complexity. Therefore, it is crucial to thoroughly investigate
the development of low-complexity and precise channel estimators with low
pilot overhead for full-duplex systems. Furthermore, in separate antenna
configurations, where the receive antenna (RX) chains lack the capability to
receive pilot signals, the channel estimation from transmit antenna (TX)
arrays to the downlink UEs presents an additional bottleneck in full-duplex
systems. Addressing this specific challenge is crucial to enable the downlink
transmission in such setups.

In this paper, we focus on addressing the channel estimation challenges in
full-duplex mmWave MIMO systems. Our primary objective is to propose channel
estimators that are low in complexity and require minimal pilot overhead,
while effectively estimating both SI and UE channels. To achieve this, we
explore different channel estimation techniques, including LS, MMSE, and NNs,
across various pilot dimension setups. The summary of the contributions made
in this paper is as follows:

  * •

First, we examine the NN-based channel estimator with varying numbers of
hidden layers and compare its performance to traditional LS and MMSE
techniques. We consider 0, 1, 2, and 10 convolutional hidden layers, where the
latter corresponds to the NN architecture in [26]. To delve deeper into NN-
based estimators, we introduce 1-bit ADCs at the full-duplex BS to add non-
linear distortion to the received pilot signal. Additionally, we investigate
how NNs perform under different channel spatial correlation regimes, namely,
low-correlated and high-correlated regimes.

  * •

In order to reduce the pilot overhead imposed by full-duplex transmission, we
share pilot resources between TX arrays of BS and UEs and explore different
pilot dimensions. First, we estimate the SI channel and then cancel out the SI
signal from the received pilot signal to estimate the UE channels. We discuss
how the SI power influences the channel estimates of UEs when pilot resources
are shared between TX arrays and UEs. We further analyze the effect of SI
cancellation during the pilot phase for UE channel estimation.

  * •

To acquire channel estimates from transmit arrays to downlink UEs in separate
antenna configurations, we seek a mapping from the channel of downlink UEs to
the RX arrays to the channel of TX arrays to the downlink UEs. Due to the
multi-path effect and random scattering environment, there is no simple
mathematical formulation for this mapping. Instead, we utilize an NN to
approximate it based on the universal approximation [30] theory. Accordingly,
we estimate the channels of downlink UEs at the RX arrays as well and then map
them to the channel from TX arrays to the downlink UEs using NNs. Simulation
results show that the NN can learn this mapping successfully.

  * •

Finally, we compare the computational complexity of LS, MMSE, and NN channel
estimators in terms of floating point operations (FLOPs). This comparison
indicates that the number of FLOPs increases quadratically with the number of
received antennas in MMSE estimation, whereas in LS and NN-based estimation,
it grows linearly.

The rest of this paper is organized as follows. Section II introduces the
channel models for SI and UEs, as well as pilot transmission schemes. We
examine different channel estimation approaches such as LS and MMSE, and
introduce the NN-based estimator for SI and UE channels in Section III, and
the RX-TX mapping for separate antenna configurations. In Section IV, we
provide extensive numerical simulations and compare different channel
estimators in terms of normalized mean squared error (NMSE) and computational
complexity. Section V concludes the paper.

Throughout the paper the following notations are used: Matrices are
represented using bold uppercase letters, while bold lowercase letters denote
vectors. Transpose, transpose conjugate (Hermitian), and matrix inversion
operations are denoted by the superscripts $(\cdot)^{T}$, $(\cdot)^{H}$, and
$(\cdot)^{-1}$, respectively. We use $\mathbb{E}$ to denote expectation,
$||\cdot||_{F}$ to represent the Frobenius norm, and $\otimes$ to denote
Kronecker product. The identity matrix of size $n\times n$ is represented by
$\mathbf{I}_{n}$. For a generic matrix $\mathbf{A}\in\mathbb{C}^{n\times m}$,
$[\mathbf{A}]_{ij}$ refers to the element in the $(i,j)$-th position. The
operation $vec(\mathbf{A})$ denotes a column vector obtained by stacking the
columns of $\mathbf{A}$ one below the other.

##  II System Model

###  II-A Channel Modeling

We consider a single-cell system with one full-duplex mmWave MIMO BS, serving
$K_{\text{u}}$ uplink UEs and $K_{\text{d}}$ downlink UEs as depicted in the
Fig. 1.

Figure 1: System model with a full-duplex BS serving $K_{\text{u}}$ single-
antenna UEs in the uplink and $K_{\text{d}}$ single-antenna UEs in the
downlink channel.

Both uplink and downlink UEs are half-duplex devices and the total number of
UEs in the cell is $K=K_{\text{u}}+K_{\text{d}}$. We consider separate antenna
configurations, where the BS is equipped with two uniform linear arrays to
enable full-duplex transmission. The number of transmit and receive antennas
are $N_{\text{t}}$ and $N_{\text{r}}$, respectively, and they are linearly
placed with a uniform distance. The UE channels follow a geometrical channel
model as follows

$\mathbf{h}_{k}^{(\text{u})}=\sqrt{\frac{1}{N_{\text{r}}}}\sqrt{\beta_{\text{u}}}\sum_{i=0}^{P-1}\alpha_{i}\mathbf{a}_{\text{a}}(\theta_{i}),\;\;\;k=1,2,\dots,K_{\text{u}},$
(1)
$\mathbf{h}_{k}^{(\text{d})}=\sqrt{\frac{1}{N_{\text{t}}}}\sqrt{\beta_{\text{d}}}\sum_{i=0}^{P-1}\alpha_{i}\mathbf{a}_{\text{d}}(\theta_{i}),\;\;\;k=1,2,\dots,K_{\text{d}},$
(2)

where $\mathbf{h}_{k}^{(\text{u})}$($\mathbf{h}_{k}^{(\text{d})}$) is the
uplink (downlink) channel between the $k$-th uplink (downlink) UE and receive
(transmit) antenna arrays at BS. Furthermore, $P$ represents the number of
multi-path components, $\alpha_{i}\sim\mathcal{CN}(0,1)$ are the small-scale
fading factors of individual paths, and $\beta_{\text{u}(\text{d})}$ stands
for the large-scale fading

$\beta_{\text{u}(\text{d})}=\Gamma-10\eta\text{log}(r_{\text{u}(\text{d})})+\chi,$
(3)

where $\Gamma$ denotes the average channel gain in dB at a reference distance
of 1 m and specific carrier frequency, $r_{\text{u}(\text{d})}$ is the uplink
(downlink) distance between transmitter and receiver, $\chi$ is the shadow
fading with a standard deviation of $\sigma_{\text{sf}}$, and $\eta$ is the
path loss exponent. Furthermore, $\mathbf{a}_{\text{a}}(\theta)$ and
$\mathbf{a}_{\text{d}}(\theta)$ are, respectively, the uniform linear array
(ULA) responses of the receive and transmit antennas, given by

$\mathbf{a}_{\text{a}}(\theta)=\sqrt{\dfrac{1}{N_{\text{r}}}}\left[1,\ldots,\exp\left(j2\pi\frac{d}{\lambda}(N_{\text{r}}-1)\sin\theta\right)\right]^{T},$
(4)

$\mathbf{a}_{\text{d}}(\theta)=\sqrt{\dfrac{1}{N_{\text{t}}}}\left[1,\ldots,\exp\left(j2\pi\frac{d}{\lambda}(N_{\text{t}}-1)\sin\theta\right)\right]^{T},$
(5)

where $\theta$ is the angle of arrival/departure (AoA/AoD) to/from the
receive/transmit antenna arrays at the BS, and $d$ and $\lambda$ are the
antenna spacing and wavelength, respectively. We assume AoA and AoD follow the
local scattering model with a uniform distribution
$\left[-\frac{\theta_{\text{AS}}}{2},\frac{\theta_{\text{AS}}}{2}\right]$
[31], where $\theta_{\text{AS}}$ is the angular spread (AS) of the multi-path
components.

We denote the SI channel from the transmit antenna arrays to the receive
antenna arrays as $\mathbf{H}_{\text{SI}}$, which is modeled as a Rician
fading channel due to the presence of a strong line-of-sight (LoS) path.
Hence, the SI channel consists of two components: the near-field and the far-
field terms [2]:

$\mathbf{H}_{\text{SI}}=\sqrt{\frac{\kappa}{\kappa+1}}\mathbf{H}_{\text{SI,NF}}+\sqrt{\frac{1}{\kappa+1}}\mathbf{H}_{\text{SI,FF}}.$
(6)

The near-field term $\mathbf{H}_{\text{SI,NF}}$ arises from the close-in
placement of the transmit and receive antennas, while the far-field term
$\mathbf{H}_{\text{SI,FF}}$ is a result of the reflections in the environment.
Here, $\kappa$ is the Rician factor and $\mathbf{H}_{\text{SI,FF}}$ follows a
similar geometrical channel modeling as the UE channels:

$\mathbf{H}_{\text{SI,FF}}=\sqrt{\frac{1}{N_{\text{t}}N_{\text{r}}}}\sqrt{\beta_{\text{SI}}}\sum_{i=0}^{P-1}\alpha_{i}\mathbf{a_{\text{a}}}(\theta_{i})\mathbf{a_{\text{d}}}^{T}(\theta_{i}),$
(7)

and $\beta_{\text{SI}}$ follows a similar formulation to
$\beta_{\text{u}(\text{d})}$.

The near-field term does not satisfy the far-field condition and therefore
requires a completely different channel modeling approach [32], [33]. In this
paper, we adopt the model in [6], given by

$\left[\mathbf{H_{\text{SI,NF}}}\right]_{nm}=\frac{\rho}{r_{nm}}\exp\left(-2\pi
j\frac{r_{nm}}{\lambda}\right),$ (8)

where $r_{nm}$ is the distance between the $n$-th receive antenna and the
$m$-th transmit antenna, and $\rho$ is a constant.

###  II-B Pilot transmission schemes

We consider various pilot dimensions for channel estimation in full-duplex
MIMO systems. Since uplink and downlink transmissions happen at the same time
and frequency in full-duplex systems, only uplink pilots will be transmitted
from UEs. To estimate the channels of UEs and SI, both downlink and uplink
UEs, along with transmit antennas at the BS, transmit pilot signals to the
receive antennas at the BS. Specifically, we assume that all the $K$ UEs
transmit their pilot signals to the RX arrays in the channel estimation phase,
whereas in the payload data transmission phase, a subset of $K_{\text{u}}$ UEs
will be in the uplink mode and $K_{\text{d}}$ UEs will be in the downlink
mode.

In the baseline scenario, we assume UEs and transmit antennas of the BS send
their pilot signals at different pilot resources. Due to large antenna arrays,
with this pilot transmission, the pilot overhead will be extremely high. In
the second scenario, we assume the available pilot resources will be equal to
the number of transmit antennas at the BS, and UEs will reuse the same pilot
resources utilized by the BS to estimate BS-UE channels. Compared to the first
scenario, the pilot overhead will be lower, but the received pilot signals
will be contaminated by UEs/TX arrays when estimating SI/UE channels. To
further reduce the pilot overhead, we reduce the number of pilots to the
number of UEs. In this case, besides interference, the SI channel must be
estimated with fewer pilots than the channel dimension, but with some
performance loss, we can achieve a comparable pilot overhead as half-duplex
systems. Three different pilot dimensions considered for estimating SI and UE
channels are depicted in Fig. 2.

Figure 2: Different pilot dimensions. $T$ is the frame dimension during one
coherence block and $\tau$ is the pilot dimension for estimating SI and UE
channels.

We assume that the received SI signal power remains stronger than the received
UE signal power, even after SI cancellation in the propagation domain. Hence,
our approach involves initially estimating the SI channel and subsequently
subtracting the estimated SI signal from the received pilot signal to estimate
the UE channels.

###  II-C SI channel estimation

For SI channel estimation, the received pilot signal at the RX arrays can be
written as

$\mathbf{Y_{\text{SI}}}=\sqrt{SNR_{\text{SI}}}\mathbf{H_{\text{SI}}}\mathbf{F}\mathbf{X_{\text{SI}}}+\sqrt{SNR_{\text{UE}}}\mathbf{H_{\text{UE}}}\mathbf{X_{\text{UE}}}+\mathbf{N},$
(9)

where $\mathbf{Y_{\text{SI}}}\in\mathbb{C}^{N_{\text{r}}\times\tau}$ is the
received pilot signal at receive antennas of BS during $\tau$ pilot
transmissions. $\mathbf{X_{\text{UE}}}\in\mathbb{C}^{K\times\tau}$ is the
transmitted pilot signal from the $K$ UEs,
$\mathbf{X_{\text{SI}}}\in\mathbb{C}^{\tau\times\tau}$ is a diagonal matrix
whose diagonal elements are transmitted pilot signals from the transmit
antennas at BS, $\mathbf{F}\in\mathbb{C}^{N_{\text{t}}\times\tau}$ is the
precoding matrix of transmit antennas of BS, and
$\mathbf{H_{\text{UE}}}=\left[\mathbf{h}_{1}^{(\text{u})},\mathbf{h}_{2}^{(\text{u})},\dots,\mathbf{h}_{K_{\text{u}}}^{(\text{u})},\mathbf{h}_{1}^{(\text{d})},\mathbf{h}_{2}^{(\text{d})},\dots,\mathbf{h}_{K_{\text{d}}}^{(\text{d})}\right]\in\mathbb{C}^{N_{\text{r}}\times
K}$ is the concatenated uplink and downlink channel response matrix, where the
$k$-th column corresponds to the channel between the $k$-th uplink or downlink
UE and the receive antenna at the BS. We assume all UEs have the same transmit
power $p_{\text{UE}}$, and $p_{\text{SI}}$ denotes the transmit power from
transmit antenna arrays. We define $SNR_{\text{SI}}$ and $SNR_{\text{UE}}$ as

$SNR_{\text{SI}}=\frac{p_{\text{SI}}}{\sigma_{\text{n}}^{2}},$ (10)
$SNR_{\text{UE}}=\frac{p_{\text{UE}}}{\sigma_{\text{n}}^{2}},$ (11)

where $\sigma_{\text{n}}^{2}$ is the noise variance. The matrix
$\mathbf{N}\in\mathbb{C}^{N_{\text{r}}\times\tau}$ is the additive noise
comprised of independent and identically distributed (i.i.d.) elements
following a zero-mean unit variance Gaussian distribution.

Note that when the pilot dimension is $\tau=N_{\text{t}}+K$, the SI and UE
channel estimation happens at orthogonal pilot resources. Hence, the received
pilot signal for SI channel estimation does not interfere with the pilots
transmitted by UEs, i.e.,

$\mathbf{Y_{\text{SI}}}=\sqrt{SNR_{\text{SI}}}\mathbf{H_{\text{SI}}}\mathbf{F}\mathbf{X_{\text{SI}}}+\mathbf{N}.$
(12)

During pilot transmission, we apply an orthogonal pilot codebook for the
precoded SI signals from the transmit antenna arrays and pilot signals from
UEs. Specifically, we utilize the discrete Fourier transform (DFT) pilot
codebook, given by
$\mathbf{F}\mathbf{X_{\text{SI}}}=\mathbf{W}_{N_{\text{t}}\times\tau},\mathbf{X_{\text{UE}}}=\mathbf{W}_{K\times\tau}$,
where,

$\begin{split}&\mathbf{W}_{M\times\tau}=\\\
&\frac{1}{\sqrt{M}}\begin{bmatrix}1&1&1&\cdots&1\\\
1&\omega&\omega^{2}&\cdots&\omega^{\tau-1}\\\
1&\omega^{2}&\omega^{4}&\cdots&\omega^{2(\tau-1)}\\\
1&\omega^{3}&\omega^{6}&\cdots&\omega^{3(\tau-1)}\\\
\vdots&\vdots&\vdots&\ddots&\vdots\\\
1&\omega^{M-1}&\omega^{2(M-1)}&\cdots&\omega^{(M-1)(\tau-1)}\end{bmatrix}\end{split}$
(13)

and $\omega=e^{-2\pi j/\tau}$.

We assume that SI is sufficiently suppressed in the propagation domain and the
received signal at the baseband does not exceed the dynamic range of the ADCs.
We refer to the propagation domain SI suppression factor as
$\epsilon_{\text{SI}}$. Then, we re-write the SI channel in (6) as

$\mathbf{H_{\text{SI}}}=\sqrt{\epsilon_{\text{SI}}}\sqrt{\frac{\kappa}{\kappa+1}}\mathbf{H_{\text{SI,NF}}}+\sqrt{\frac{1}{\kappa+1}}\mathbf{H_{\text{SI,FF}}}.$
(14)

Note that in the propagation domain, only the near field channel from the
transmit antenna arrays to the receive antenna arrays can be suppressed. In
the digital domain, the residual near-field and far-field SI channels can be
further suppressed after estimating the SI channel.

###  II-D UE channel estimation

To estimate the UE channels, we use the estimated SI channel and cancel out
the SI signal power from the received pilot signal. Therefore, the received
pilot signal for UE channel estimation becomes

$\begin{split}\mathbf{Y_{\text{UE}}}=&\sqrt{SNR_{\text{UE}}}\mathbf{H_{\text{UE}}}\mathbf{X_{\text{UE}}}+\sqrt{SNR_{\text{SI}}}\mathbf{E_{\text{SI}}}\mathbf{F}\mathbf{X_{\text{SI}}}+\mathbf{N},\end{split}$
(15)

where
$\mathbf{E_{\text{SI}}}\triangleq\mathbf{H_{\text{SI}}}-\mathbf{\hat{H}_{\text{SI}}}$
is the SI cancelation error and $\mathbf{\hat{H}_{\text{SI}}}$ is the
estimated SI channel.

Similarly, for the pilot dimension $\tau=N_{\text{t}}+K$, the received pilot
signal does not contain the signal from the SI channel, thus,

$\mathbf{Y_{\text{UE}}}=\sqrt{SNR_{\text{UE}}}\mathbf{H_{\text{UE}}}\mathbf{X_{\text{UE}}}+\mathbf{N}.$
(16)

Typically, the channels of nearby UEs are correlated, and for the pilot
dimension $\tau=N_{\text{t}}$ and $\tau=K$, the residual error of SI
cancellation introduces further correlation in the received pilot signals used
for UE channel estimation. We estimate the entire UE channel matrix
$\mathbf{H_{\text{UE}}}$. In this way, we take advantage of the correlation
resulting from imperfect SI cancellation and leverage the additional
information for more accurate estimation of the UE channels.

##  III LS, MMSE, NN-based Channel Estimation

In this section, we present three channel estimation methods: LS, MMSE, and
NN-based estimators for both SI and UE channels. First, we will recall the LS
and MMSE channel estimators. Subsequently, we present channel estimation using
NNs for estimating the SI and UE channels. For SI channel estimation, we
correlate the received pilot signal with the pilot matrix transmitted from
transmit antenna arrays, therefore, we define

$\mathbf{\hat{Y}_{\text{SI}}}=\mathbf{Y_{\text{SI}}}\mathbf{W}^{H}_{N_{\text{t}}\times\tau}.$
(17)

Similarly for the UE channel estimation, we have

$\mathbf{\hat{Y}_{\text{UE}}}=\mathbf{Y_{\text{UE}}}\mathbf{W}^{H}_{K\times\tau}.$
(18)

###  III-A LS channel estimator

The LS channel estimator is the most simple channel estimation technique that
does not need any prior knowledge about channel statistics and finds the
channel coefficients that minimize the mean square error (MSE) between the
estimated channel and the received pilot signal. The LS channel estimator can
be calculated as follows

$\mathbf{\hat{H}}_{q,\text{LS}}=\frac{1}{\tau\sqrt{SNR_{q}}}\mathbf{\hat{Y}}_{q},$
(19)

where $q\in\\{\text{SI},\text{UE}\\}$.

###  III-B MMSE channel estimator

MMSE is a Bayesian estimator that aims to minimize the MSE between the
estimated channel and the true channel. It offers improved performance
compared to the LS estimator by taking into account the statistical properties
of the channel and noise. To derive the MMSE channel estimator, we rewrite the
received pilot signal for SI and UE channel estimation in a vector form

$\begin{split}vec(\mathbf{Y_{\text{SI}}})=&\sqrt{SNR_{\text{SI}}}\mathbf{\tilde{X}_{\text{SI}}}vec(\mathbf{H_{\text{SI}}})+\\\
&\sqrt{SNR_{\text{UE}}}\mathbf{\tilde{X}_{\text{UE}}}vec(\mathbf{H_{\text{UE}}})+vec(\mathbf{N}),\end{split}$
(20)

$\begin{split}vec(\mathbf{Y_{\text{UE}}})=&\sqrt{SNR_{\text{UE}}}\mathbf{\tilde{X}_{\text{UE}}}vec(\mathbf{H_{\text{UE}}})+\\\
&\sqrt{SNR_{\text{SI}}}\mathbf{\tilde{X}_{\text{SI}}}vec(\mathbf{E_{\text{SI}}})+vec(\mathbf{N}),\end{split}$
(21)

where
$\mathbf{\tilde{X}_{\text{SI}}}=\mathbf{(FX_{\text{SI}})}^{T}\otimes\mathbf{I_{N_{\text{r}}}}$
and
$\mathbf{\tilde{X}_{\text{UE}}}=\mathbf{(X_{\text{UE}})}^{T}\otimes\mathbf{I_{N_{\text{r}}}}$.

The channel covariance matrix is defined as follows

$\mathbf{R}_{q}\triangleq\mathbb{E}\left[vec(\mathbf{H}_{q})vec(\mathbf{H}_{q})^{H}\right].$
(22)

The channel covariance matrix for the MIMO channel can be related to the
transmit and receive covariance matrix via the Kronecker product

$\mathbf{R}=\mathbf{R_{\text{t}}}\otimes\mathbf{\mathbf{R_{\text{r}}}},$ (23)

where $\mathbf{R_{\text{t}}}$ and $\mathbf{R_{\text{r}}}$ are the transmit and
receive covariance matrices.

The MMSE channel estimator for $\mathbf{H}_{q}$,
$\mathbf{\hat{H}}_{q,\text{MMSE}}$, minimizes the mean square error (MSE)
$\mathbb{E}\left[||\mathbf{H}_{q}-\mathbf{\hat{H}}_{q,\text{MMSE}}||^{2}\right]$.
We have

$vec(\mathbf{\hat{H}}_{q,\text{MMSE}})=\mathbf{R}_{\mathbf{H}_{q}\mathbf{Y}_{q}}\mathbf{R}_{\mathbf{Y}_{q}}^{-1}vec(\mathbf{Y}_{q}),$
(24)

where

$\mathbf{R}_{\mathbf{H}_{q}\mathbf{Y}_{q}}\triangleq\mathbb{E}\left[vec(\mathbf{H}_{q})vec(\mathbf{Y}_{q})^{H}\right],$
(25)
$\mathbf{R}_{\mathbf{Y}_{q}}\triangleq\mathbb{E}\left[vec(\mathbf{Y}_{q})vec(\mathbf{Y}_{q})^{H}\right].$
(26)

We can formulate the MMSE estimates of SI and UE channels as in equations (27)
and (28) provided at the top of the next page, respectively. In (28),
$\mathbf{R_{\text{E}}}$ represents the error covariance matrix of SI
cancellation, which can be calculated as in (29).

$vec(\mathbf{\hat{H}}_{\text{SI,MMSE}})=\sqrt{SNR_{\text{SI}}}\mathbf{R_{\text{SI}}}\mathbf{\tilde{X}_{\text{SI}}}^{H}\left(SNR_{\text{SI}}\mathbf{\tilde{X}_{\text{SI}}}\mathbf{R_{\text{SI}}}\mathbf{\tilde{X}_{\text{SI}}}^{H}+SNR_{\text{UE}}\mathbf{\tilde{X}_{\text{UE}}}\mathbf{R_{\text{UE}}}\mathbf{\tilde{X}_{\text{UE}}}^{H}+\mathbf{I}_{N_{\text{t}}N_{\text{r}}}\right)^{-1}vec(\mathbf{Y_{\text{SI}}}).$
(27)
$vec(\mathbf{\hat{H}}_{\text{UE,MMSE}})=\sqrt{SNR_{\text{UE}}}\mathbf{R_{\text{UE}}}\mathbf{\tilde{X}_{\text{UE}}}^{H}\left(SNR_{\text{UE}}\mathbf{\tilde{X}_{\text{UE}}}\mathbf{R_{\text{UE}}}\mathbf{\tilde{X}_{\text{UE}}}^{H}+SNR_{\text{SI}}\mathbf{\tilde{X}_{\text{SI}}}\mathbf{R_{\text{E}}}\mathbf{\tilde{X}_{\text{UE}}}^{H}+\mathbf{I}_{N_{\text{t}}N_{\text{r}}}\right)^{-1}vec(\mathbf{Y_{\text{UE}}}).$
(28)
$\mathbf{R_{\text{E}}}=\mathbf{R_{\text{SI}}}-SNR_{\text{SI}}\mathbf{R_{\text{SI}}}\mathbf{\tilde{X}_{\text{SI}}}^{H}\left(SNR_{\text{SI}}\mathbf{\tilde{X}_{\text{SI}}}\mathbf{R_{\text{SI}}}\mathbf{\tilde{X}_{\text{SI}}}^{H}+SNR_{\text{UE}}\mathbf{\tilde{X}_{\text{UE}}}\mathbf{R_{\text{UE}}}\mathbf{\tilde{X}_{\text{UE}}}^{H}+\mathbf{I}_{N_{\text{t}}N_{\text{r}}}\right)^{-1}\mathbf{\tilde{X}_{\text{SI}}}\mathbf{R_{\text{SI}}}.$
(29)

Note that for pilot dimension $\tau=N_{\text{t}}+K$, the MMSE estimates of the
SI and UE channels simplify to the following equations using Woodbury matrix
identity [34],

$\begin{split}&vec(\mathbf{\hat{H}_{\text{SI,MMSE}}})=\\\
&\sqrt{SNR_{\text{SI}}}\left(\mathbf{R_{\text{SI}}}^{-1}+\tau
SNR_{\text{SI}}\mathbf{I}_{N_{\text{t}}N_{\text{r}}}\right)^{-1}\mathbf{\tilde{X}_{\text{SI}}}^{H}vec(\mathbf{Y_{\text{SI}}}),\end{split}$
(30) $\begin{split}&vec(\mathbf{\hat{H}_{\text{UE,MMSE}}})=\\\
&\sqrt{SNR_{\text{UE}}}\left(\mathbf{R_{\text{SI}}}^{-1}+\tau
SNR_{\text{UE}}\mathbf{I}_{N_{\text{t}}N_{\text{r}}}\right)^{-1}\mathbf{\tilde{X}_{\text{SI}}}^{H}vec(\mathbf{Y_{\text{SI}}}).\end{split}$
(31)

###  III-C NN channel estimator

Although conventional channel estimation techniques such as MMSE can leverage
the spatial correlation present in large antenna arrays, they require prior
knowledge of the channel covariance matrix and are computationally intensive
due to matrix inversion involving a large matrix. In various fields of
research, it has been demonstrated that NNs can utilize the inherent structure
of data by solely observing a sufficient number of data points. Different
network architectures, such as FNNs, CNNs, recurrent neural networks (RNNs),
etc., can be utilized depending on the characteristics of the data. For
instance, CNNs are suitable for addressing problems with feature correlation
and local dependencies among them, while RNNs are more effective for handling
memory-based features like time series [18].

It has been shown in various studies that the wireless channel with large
antenna arrays is spatially correlated, see e.g., [35]. To exploit this
spatial correlation for channel estimation, we employ CNNs, which are suitable
for capturing the local dependency of features among data points. The input to
the CNN is the correlated pilot signal, i.e., $\mathbf{\hat{Y}}_{q}$, and the
output is the estimated channel, $\mathbf{\hat{H}}_{q,\text{NN}}$,
$q\in\\{\text{SI},\text{UE}\\}$. In convolutional layers, we apply a $3\times
3$ window size sliding through the whole input features with a unit stride
size. Different numbers of hidden layers are employed, where each layer
applies 64 convolutional kernels to extract features from the successive
windows of its input features. The effect of the number of hidden layers will
be carefully examined in Section IV.

To keep the dimensions of the output and input fixed, we utilize padding after
convolution processing. We apply rectified linear unit (ReLU) as the
activation function for the hidden layers, while for the output layer, linear
activation is used. Since tensors do not support complex operations, the input
to the CNN is converted to three-dimensional tensors, where the third
dimension stores the real and imaginary parts of the complex data samples.
Therefore, if we define $\mathbf{X_{\text{tr}}}$ and $\mathbf{Y_{\text{tr}}}$
as the input and labels of the CNN during training, we have

$\begin{split}&\mathbf{X_{\text{tr}}}\left[:,:,0\right]\triangleq\Re\left\\{\mathbf{\hat{Y}}_{q}\right\\},\\\
&\mathbf{X_{\text{tr}}}\left[:,:,1\right]\triangleq\Im\left\\{\mathbf{\hat{Y}}_{q}\right\\}.\end{split}$
(32)
$\begin{split}&\mathbf{Y_{\text{tr}}}\left[:,:,0\right]\triangleq\Re\left\\{\mathbf{H}_{q}\right\\},\\\
&\mathbf{Y_{\text{tr}}}\left[:,:,1\right]\triangleq\Im\left\\{\mathbf{H}_{q}\right\\}.\end{split}$
(33)

For training the CNN, a dataset consisting of $M_{\text{tr}}$ samples is
generated with $(\mathbf{\hat{Y}}_{q}^{(n)},\mathbf{H}_{q}^{(n)})$
representing the $n$-th sample. We employ supervised learning, where
$\mathbf{H}_{q}$ is regarded as the label during training. We apply min-max
scaling to scale the dataset in the range $(0,1)$. Such normalization is
highly recommended for training machine learning models. We consider MSE as
the loss function

$\text{MSE}=\frac{1}{M_{\text{tr}}}\sum_{i=1}^{M_{\text{tr}}}||\mathbf{X_{\text{tr}}}-\mathbf{Y_{\text{tr}}}||_{F}^{2}.$
(34)

Figure 3: Phase shift between the RX and TX arrays in the separate antenna
configuration in a full-duplex BS.

###  III-D RX-TX channel mapping

Since in separate antenna configurations, two different antenna arrays are
utilized for uplink reception and downlink transmission, uplink and downlink
channels experience different channel realizations. So far we have assumed
that RX arrays at BS receive pilot signals from both uplink and downlink UEs
and estimate all channels from both uplink and downlink UEs to the RX arrays.
For downlink transmission, the full-duplex BS needs the CSI from the TX arrays
to downlink UEs. The TX arrays are not capable of receiving and processing
pilot signals and implementing a separate receive RF chain and ADC at the TX
arrays for just pilot processing will be costly. As a result, we need to
somehow map the channel from the downlink UEs to the RX arrays to the channel
from the TX arrays to the downlink UEs. Due to the multi-path effect and
random scattering environment, it is not straightforward to derive
mathematically the relation of such mapping. More specifically, we assume that
the RX-UE channel is

$\mathbf{h_{\text{UE,RX}}}=\sqrt{\frac{1}{N_{\text{r}}}}\sqrt{\beta_{\text{u}}}\sum_{i=0}^{P-1}\alpha_{i}\mathbf{a}_{\text{a}}(\theta_{i}),$
(35)

where $a_{i}$ and $\theta_{i}$ are the amplitude and the AoA of $i$-th path,
respectively. As shown in Fig. 3, TX and RX arrays are at a close distance in
the order of a few wavelengths from each other; Therefore, the channel
amplitude for each path is essentially the same for both the RX and TX arrays,
while the antenna separation creates a delay depending on the AoA of each
path. Thus, the TX-UE channel will be

$\mathbf{h_{\text{UE,TX}}}=\sqrt{\frac{1}{N_{\text{t}}}}\sqrt{\beta_{\text{d}}}\sum_{i=0}^{P-1}\alpha_{i}\mathbf{a}_{\text{d}}({\theta_{i})}e^{-j\frac{2\pi}{\lambda}lcos\theta_{i}}.$
(36)

The mathematical relationship between $\mathbf{h_{\text{UE,TX}}}$ and
$\mathbf{h_{\text{UE,RX}}}$ is not well-defined due to the random AoA and the
effects of multi-path propagation. On the other hand, based on universal
approximation theory [30], feedforward NNs are capable of approximating any
continuous function. This theory suggests that, given enough computational
resources and data, it is possible to build NN models that can accurately
approximate any function. Therefore, universal approximation theory inspires
us to employ NNs to map the channel from the downlink UEs to the RX array to
the channel from the TX array to the downlink UEs. Furthermore, recent studies
[36] have demonstrated the existence of a spatial mapping function that can
effectively map the channel from one set of antenna arrays to another. To
accomplish this, we utilize an FNN with varying numbers of hidden layers. The
input for FNN is the channel from the downlink UEs to the RX array, while the
output is the channel from the TX array to the downlink UEs. Therefore, for
the training of FNN, we collect $M_{\text{tr}}$ samples of
$\mathbf{h_{\text{UE,RX}}}$ as the input and $\mathbf{h_{\text{UE,TX}}}$ as
the corresponding label. Again, we apply min-max normalization and MSE loss
function for training. We create the following data samples as the input and
label of FNN to work with real-valued tensors:

$\begin{split}&\mathbf{x_{\text{tr}}}[0:N_{\text{r}}]=\Re\left\\{\mathbf{h_{\text{UE,RX}}}\right\\},\\\
&\mathbf{x_{\text{tr}}}[N_{\text{r}}:2N_{\text{r}}]=\Im\left\\{\mathbf{h_{\text{UE,RX}}}\right\\}.\end{split}$
(37)
$\begin{split}&\mathbf{y_{\text{tr}}}[0:N_{\text{t}}]=\Re\left\\{\mathbf{h_{\text{UE,TX}}}\right\\},\\\
&\mathbf{y_{\text{tr}}}[N_{\text{t}}:2N_{\text{t}}]=\Im\left\\{\mathbf{h_{\text{UE,TX}}}\right\\}.\end{split}$
(38)

##  IV Simulation results

In this section, we present our simulation results for SI and UE channel
estimation. We compare channel estimators that we introduced in previous
sections, i.e., LS, MMSE, and NN for different pilot dimensions. The NMSE is
considered the performance metric to compare the different channel estimators,
and it is defined as

$\text{NMSE}\triangleq\mathbb{E}\left[\frac{||\mathbf{H_{\text{true}}}-\mathbf{H_{\text{est}}}||_{F}^{2}}{||\mathbf{H_{\text{true}}}||_{F}^{2}}\right],$
(39)

where $\mathbf{H_{\text{true}}}$ and $\mathbf{H_{\text{est}}}$ are the true
and estimated channel matrices, respectively.

We consider an operating frequency is $28$ GHz, corresponding to a wavelength
of $\lambda=10.71$ mm. We set the number of TX and RX arrays to 16, with a
distance of $10\lambda$ between them, and antenna spacing in the transmit and
the receive arrays is $d=\frac{\lambda}{2}$. The total number of downlink and
uplink UEs is set to $K=8$. The path loss parameters are based on experimental
results from [37], where the path loss constant at the reference distance is
$\Gamma=-72$ dB, the path loss exponent is $\eta=2.92$, and the shadow fading
standard deviation is $\sigma_{\text{sf}}=8.7$ dB. Unless otherwise claimed,
the number of multi-path components is $P=5$ and AS is
$\theta_{\text{AS}}=60^{\circ}$. The SI Rician factor is $\mu=40$ dB and we
assume that the propagation SI cancellation is $\epsilon_{\text{SI}}=-40$ dB.

The NN-based channel estimators are trained on Python $3.7.16$ and implemented
using the Keras libraries with a TensorFlow backend in the Jupyter Notebook
environment. We employ Adam optimizer with a batch size of $512$ to update the
network parameters. A dataset of $50,000$ samples is collected based on the
channel model and it is split into $20,000$ samples for training, $20,000$
samples for validation, and $10,000$ samples for testing. The validation data
is used to ensure that the model does not simply memorize the training data
but learns meaningful aspects of the data for effective prediction.111Upon
acceptance, we will publish our source code for reproducibility.

###  IV-A Analysing NN-based channel estimator

Before examining the different channel estimators for various pilot dimensions
discussed in Section II-B, we conducted several experiments to understand the
behavior of NN-based channel estimators under different channel conditions and
NN architectures for training. The following results are specific to SI
channel estimation with a pilot dimension of $\tau=N_{\text{t}}+K$. For the
sake of brevity, we have not included plots for other pilot dimensions and UE
channel estimation. For the MMSE channel estimator, we obtain the empirical
channel covariance matrix estimated through $M$ channel realizations, i.e.,

$\mathbf{R}_{q}=\frac{1}{M}\sum_{i=1}^{M}vec(\mathbf{H}_{q}^{(i)})vec(\mathbf{H}_{q}^{(i)})^{H},$
(40)

where throughout our simulation, we set $M=1000$.

Figure 4: NMSESI vs SNRSI with a varying number of hidden layers. Figure 5:
NMSESI vs SNRSI with a varying number of hidden layers for a full-duplex BS
with 1-bit ADCs.

Fig. 4 illustrates the NMSESI versus SNRSI with different numbers of hidden
layers. We can observe that NN-based channel estimator outperforms the LS and
MMSE estimation. Furthermore, utilizing hidden layers slightly improves the
NMSE of channel estimates. For instance, CNN with 2 hidden layers decreases
the NMSE by about 1 dB compared to CNN with no hidden layer. However,
comparing CNN with 10 hidden layers (DeepCNN) with that of 1 or 2 hidden
layers suggests that using very deep CNNs does not noticeably improve the
channel estimates (about 0.3 dB NMSE improvement), while significantly
increasing computational complexity. Furthermore, the erratic behavior of
DeepCNN at high SNRs can be rectified through fine-tuning, for example, by
using a different number of convolutional kernels, adjusting the learning
rate, etc. However, for the sake of comparison, we applied the exact same
parameters to all CNNs with different numbers of hidden layers.

In practical scenarios, various non-linear distortions in hardware components
exist, adding complexity to the problem of channel estimation. To further
analyze the impact of the number of hidden layers, we consider the
introduction of 1-bit ADCs at the BS to incorporate a non-linear distortion
effect into the received pilot signal. When employing 1-bit ADCs, the received
pilot signal becomes [28]

$\mathbf{Y_{\text{1-bit}}}=\text{sgn}(\mathbf{Y_{\text{SI}}}),$ (41)

where $\text{sgn}(\cdot)$ is the element-wise signum function.

Once again, we apply CNN with a varying number of hidden layers to address the
channel estimation problem with 1-bit ADCs. In this case, the input to the CNN
is
$\mathbf{\hat{Y}_{\text{1-bit}}}=\mathbf{Y_{\text{1-bit}}}\mathbf{W}^{H}_{N_{\text{t}}\times\tau}$,
and the output is the estimated channel. Fig. 5 presents the NMSE of LS, MMSE,
and NN-based channel estimators. As observed in the figure, the addition of
hidden layers improves the NMSE by approximately 1 dB, which is similar to the
improvement observed in the case of infinite-bit ADCs in Fig. 4.

(a) $\theta_{\text{AS}}=360^{\circ}$ (b) $\theta_{\text{AS}}=10^{\circ}$

Figure 6: NMSESI vs SNRSI for (a) low-correlated and (b) high-correlated
channels.

Then, we examine the behavior of the NN-based channel estimator under two
distinct channel correlation conditions: high-correlated and low-correlated
scenarios. The plots in Fig. 6 illustrate how the LS, MMSE, and NN-based
estimators perform under different spatial channel correlation strengths.
Larger $\theta_{\text{AS}}$ corresponds to lower spatial correlations, and
vice versa. In the high-correlated scenario, the MMSE estimator can explicitly
leverage the channel covariance matrix, resulting in a significant improvement
in estimation quality. This leads to a substantial gap between the LS and MMSE
estimations. However, in the low-correlated channel, both the LS and MMSE
estimations converge to the same NMSE at high SNRs. Comparing the NN-based
estimation with the LS and MMSE estimations in both low-correlated and high-
correlated channel conditions provides interesting insights into the behavior
of the NN-based estimator. In the low-correlated scenario, the NN-based
estimator outperforms both the MMSE and LS estimations. However, in highly
correlated channel conditions, the MMSE estimator consistently outperforms the
NN-based estimator. This observation suggests that the NN-based estimator, in
a data-driven fashion, struggles to utilize the second-order statistics of the
channel for estimation, as well as the MMSE estimator does. Furthermore, the
plots suggest that in a low-correlated scenario, increasing the number of
hidden layers does not yield any noticeable improvement. However, in a highly
correlated channel condition, the NMSE decreases with the addition of more
hidden layers to the NN architecture.

Figure 7: NMSESI vs SNRSI for different pilot dimensions, SNRUE = 0 dB. Figure
8: NMSESI vs SNRUE for different pilot dimensions, SNRSI = 0 dB.

###  IV-B SI and UE channel estimation for different pilot dimensions

Next, we examine the performance of SI channel estimation for different pilot
dimensions, as described in Section II-B. For the following simulations, we
applied a CNN with no hidden layers and set $\theta_{\text{AS}}=60^{\circ}$.

The NMSE of SI channel estimation vs SNRSI and SNRUE are provided in Figs. 7
and 8 for different pilot dimensions, respectively. The results demonstrate
that the NN-based estimator outperforms the LS and MMSE estimators across
considered pilot dimensions. Moreover, from Fig. 8, it is evident that the NN-
based approach is more resilient to interference from UEs when compared to the
MMSE and LS estimation methods. Using fewer pilot dimensions leads to a
significant reduction in the quality of channel estimates for all three
approaches. However, the reduction in pilot dimensions can allow more
resources to be allocated to transmitting payload data. The trade-off between
the performance of channel estimates and the number of pilot dimensions used
depends on the required accuracy threshold for channel estimates and the
system data rate.

We have also generated similar plots for UE channel estimation, showing the
NMSE with respect to SNRUE and SNRSI. The plots are presented in Figs. 9 and
10, respectively, with varying pilot dimensions. We employ the MMSE estimator
to cancel out the SI signal from the received pilot signal in LS, MMSE, and
NN-based channel estimators for UE channel estimation, specifically when the
pilot dimensions are $\tau=N_{\text{t}}$ and $\tau=K$. Note that we also
utilize the error covariance matrix for the MMSE estimator when estimating UE
channels as denoted in (28). From the results, it is evident that the NN-based
approach using shorter pilot dimensions like $\tau=N_{\text{t}}$ or $\tau=K$,
outperforms LS channel estimation with a pilot dimension of
$\tau=N_{\text{t}}+K$. Even the MMSE channel estimator with longer pilot
dimensions (e.g., $\tau=N_{\text{t}}+K$), is unable to outperform the NN-based
technique with shorter pilot dimensions (e.g., $\tau=N_{\text{t}}$). These
findings clearly demonstrate the superiority of NN in accurately estimating
wireless channels, particularly in scenarios with lower SNRs and higher
interference.

Figure 9: NMSEUE vs SNRUE for different pilot dimensions, SNRSI = 10 dB.
Figure 10: NMSEUE vs SNRSI for different pilot dimensions, SNRUE = 5 dB.

(a) LS  (b) MMSE  (c) NN

Figure 11: NMSEUE vs SNRSI and SNRUE for (a) LS and (b) MMSE (c) NN channel
estimators, $\tau=K$.

A higher SNRSI results in better SI channel estimation and, consequently, less
error in SI cancellation for UE channel estimation. Conversely, a higher SNRUE
increases the NMSE of SI channel estimation, leading to an increase in error
for UE channel estimation. On the other hand, a higher SNRUE leads to improved
UE channel estimates, while a higher SNRSI increases the power of interference
during UE channel estimation. To understand the joint effect of SNRUE and
SNRSI for UE channel estimation, we generated the color bar plots in Fig. 11
for LS, MMSE, and NN-based estimation considering a pilot dimension $\tau=K$.
Based on the results in this figure, we can conclude that a lower SI signal
power or higher SI cancelation will lead to better UE channel estimates.

Figure 12: UE channel estimation with/without SI cancellation, $\tau=K$, SNRSI
= 20 dB.

To further analyze the effect of SI cancelation on UE channel estimation, we
considered two cases: in the first case, we cancel out the effect of the SI
signal from the received pilot signal with the estimated SI channel, while in
the second scenario, we estimate UE channels in the presence of interference
from the SI channel. The results are shown in Fig 12. Note that for the MMSE
estimation, in the case of SI cancellation, we incoporate the error covariance
matrix of the estimated SI channel together with the covariance matrix of the
UE channels, while for the scenario without SI cancellation, we exploit the
covariance matrix of both SI and UE channels. We can observe that MMSE
estimations for these two scenarios result in the same NMSE of UE channel
estimates. By comparing the results for LS, MMSE, and NN channel estimators,
we observe that SI cancellation during the pilot transmission phase does not
provide significant improvement, while estimating the large SI channel matrix
is costly.

Figure 13: NMSE of channel mapping between RX and TX arrays vs angular spread.

To assess the performance of the NN-based channel mapping between RX and TX
arrays at the BS in the separate antenna configuration, we present the NMSE of
this mapping concerning varying angular spreads and two different numbers of
antennas in Fig. 13. We trained an FNN with a varying number of hidden layers
at angular spreads of $[10,100,190,280]$ degrees during training. From this
figure, we can conclude that an NN effectively predicts the channel from the
TX arrays to downlink UEs, given the CSI from downlink UEs to the RX arrays.
The prediction accuracy is higher in scenarios with lower angular spread,
attributed to increased spatial correlation, and vice versa. Furthermore, the
quality of prediction improves with an increased number of antennas, as
angular resolutions increase with a larger array.

###  IV-C Complexity Analysis

Finally, we compare the computational complexity of the LS, MMSE, and NN
channel estimators in terms of FLOPs. The results are summarized in Table I,
where $\zeta$ represents the side length of the convolutional window, and
$f_{l-1}$ and $f_{l}$ denote the numbers of input and output convolutional
kernel of the $l$-th layer. A bar plot is shown in Fig. 14 for SI channel
estimation and pilot dimension $\tau=N_{\text{t}}$ based on the CNN
architecture used in the simulation, i.e., $\zeta=3$, $f_{0}=f_{L}=2$, and
$f_{i}=64$ for the remaining layers ($i=1,2,\dots,L-1$). Note that we have
ignored the computational complexity of the covariance matrix calculation for
MMSE estimation, as the channel statistics do not typically change during
several coherence blocks. For the same reasons, we also ignored the
computational complexity of the matrices in the MMSE formula that depend on
spatial correlation matrices, assuming they can be pre-computed and used until
the channel statistics change substantially. According to table I, the number
of FLOPs exhibits quadratic growth with the number of received antennas in
MMSE estimation, while for LS and NN-based estimation, it is linear.
Furthermore, CNNs with no or one hidden layer require fewer FLOPs than MMSE
estimation, while increasing the number of hidden layers significantly
increases the computational complexity. In particular, adding a hidden layer
with $64$ convolutional kernels and a $3\times 3$ window size requires about
$1.5\times 10^{8}$ more FLOPs for estimating a MIMO channel with $16$
antennas.

TABLE I: Number of FLOPs for SI and UE channel estimation using LS, MMSE, and NN estimators. | LS | MMSE | CNN
---|---|---|---
SI | $N_{\text{r}}N_{\text{t}}\tau$ | $N_{\text{r}}^{2}N_{\text{t}}\tau$ | $N_{\text{r}}N_{\text{t}}\tau+N_{\text{r}}N_{\text{t}}\sum_{l=1}^{L}\zeta^{2}f_{l-1}f_{l}$
UE | $N_{\text{r}}K\tau$ | $N_{\text{r}}^{2}K\tau$ | $N_{\text{r}}K\tau+N_{\text{r}}K\sum_{l=1}^{L}\zeta^{2}f_{l-1}f_{l}$
Figure 14: Number of FLOPs for SI channel estimation of different estimators.
CNNx represents a CNN with x hidden layer(s).

##  V Conclusion

In this paper, we studied the channel estimation problem for full-duplex
mmWave MIMO systems using NNs. We found that simple NNs with no or few hidden
layers can achieve comparable NMSE to deep NNs. The conclusion is the same in
the presence of 1-bit ADC distortion, and adding more hidden layers to NNs
slightly improves the NMSE of channel estimates. The comparison between the NN
and MMSE channel estimators indicated a notable difference. In the high-
correlated regime, MMSE is the preferable channel estimator compared to LS and
NN, as it can leverage the explicit channel covariance matrix. However, in
scenarios where spatial correlation is not extremely high, our results
demonstrated that NNs outperform MMSE estimation. Furthermore, the NN-based
channel estimator performs remarkably better than MMSE when fewer pilot
resources are utilized for channel estimation. This performance allows for a
reduction in pilot overhead in full-duplex systems. We applied FNNs to
approximate RX-TX channel mapping for the separate antenna configurations in
full-duplex systems. Simulation results demonstrated that NNs effectively map
channels from downlink UEs to the receive arrays to the channel from the
transmit arrays to downlink UEs, particularly in scenarios with high
correlation and large antenna arrays. Finally, the complexity analysis
revealed that NN-based estimation with fewer than two convolutional hidden
layers requires fewer FLOPs compared to MMSE estimation.

## Acknowledgment

This work has been supported by the project SEMANTIC, funded by the EU’s
Horizon 2020 research and innovation programme under the Marie Skłodowska-
Curie grant agreement No 861165.

## References

  * [1] F. Boccardi, R. W. Heath, A. Lozano, T. L. Marzetta, and P. Popovski, “Five disruptive technology directions for 5G,” _IEEE Commun. Mag._ , vol. 52, no. 2, pp. 74–80, Feb. 2014.
  * [2] A. Sabharwal, P. Schniter, D. Guo, D. W. Bliss, S. Rangarajan, and R. Wichman, “In-band full-duplex wireless: Challenges and opportunities,” _IEEE J. Sel. Areas Commun._ , vol. 32, no. 9, pp. 1637–1652, Jun. 2014.
  * [3] I. P. Roberts, J. G. Andrews, H. B. Jain, and S. Vishwanath, “Millimeter-wave full duplex radios: New challenges and techniques,” _IEEE Wireless Commun._ , vol. 28, no. 1, pp. 36–43, Feb. 2021.
  * [4] B. Yu, C. Qian, J. Lee, S. Shao, Y. Shen, W. Pan, P. Lin, Z. Zhang, S. Kim, S. Hu, K. Lee, J. Jung, S. Choi, and C. Sun, “Realizing high power full duplex in millimeter wave system: Design, prototype and results,” _IEEE J. Sel. Areas Commun._ , vol. 41, no. 9, pp. 2893–2906, jun. 2023.
  * [5] M. Duarte, C. Dick, and A. Sabharwal, “Experiment-driven characterization of full-duplex wireless systems,” _IEEE Trans. Wirel. Commun._ , vol. 11, no. 12, pp. 4296–4307, Nov. 2012.
  * [6] Z. Xiao, P. Xia, and X.-G. Xia, “Full-duplex millimeter-wave communication,” _IEEE Wireless Commun._ , vol. 24, no. 6, pp. 136–143, Dec. 2017.
  * [7] B. P. Day, A. R. Margetts, D. W. Bliss, and P. Schniter, “Full-duplex bidirectional MIMO: Achievable rates under limited dynamic range,” _IEEE Trans. Signal Process._ , vol. 60, no. 7, pp. 3702–3713, Apr. 2012.
  * [8] Z. Zhang, Z. Chen, M. Shen, B. Xia, W. Xie, and Y. Zhao, “Performance analysis for training-based multipair two-way full-duplex relaying with massive antennas,” _IEEE Trans. Veh. Technol._ , vol. 66, no. 7, pp. 6130–6145, Dec. 2017.
  * [9] X. Xiong, X. Wang, T. Riihonen, and X. You, “Channel estimation for full-duplex relay systems with large-scale antenna arrays,” _IEEE Trans. Wirel. Commun._ , vol. 15, no. 10, pp. 6925–6938, Jul. 2016.
  * [10] A. Masmoudi and T. Le-Ngoc, “Channel estimation and self-interference cancelation in full-duplex communication systems,” _IEEE Trans. Veh. Technol._ , vol. 66, no. 1, pp. 321–334, Mar. 2017.
  * [11] D. Korpi, T. Riihonen, and M. Valkama, “Achievable rate regions and self-interference channel estimation in hybrid full-duplex/half-duplex radio links,” in _2015 49th Annual Conference on Information Sciences and Systems (CISS)_ , Baltimore, MD, USA, Apr. 2015, pp. 1–6.
  * [12] D. Korpi, L. Anttila, and M. Valkama, “Impact of received signal on self-interference channel estimation and achievable rates in in-band full-duplex transceivers,” in _2014 48th Asilomar Conference on Signals, Systems and Computers_ , Pacific Grove, CA, USA, Apr. 2014, pp. 975–982.
  * [13] J. M. B. da Silva, A. Sabharwal, G. Fodor, and C. Fischione, “1-bit phase shifters for large-antenna full-duplex mmwave communications,” _IEEE Trans. Wirel. Commun._ , vol. 19, no. 10, pp. 6916–6931, Jul. 2020.
  * [14] Y.-G. Lim, D. Hong, and C.-B. Chae, “Performance analysis of self-interference cancellation in full-duplex large-scale MIMO systems,” in _2016 IEEE Global Communications Conference (GLOBECOM)_ , Washington, DC, USA, Feb. 2016, pp. 1–6.
  * [15] Q. Ding, Y. Lian, and Y. Jing, “Performance analysis of full-duplex massive MIMO systems with low-resolution ADCs/DACs over rician fading channels,” _IEEE Trans. Veh. Technol._ , vol. 69, no. 7, pp. 7389–7403, Apr. 2020.
  * [16] J. Koh, Y.-G. Lim, C.-B. Chae, and J. Kang, “On the feasibility of full-duplex large-scale MIMO cellular systems,” _IEEE Trans. Wirel. Commun._ , vol. 17, no. 9, pp. 6231–6250, Jul. 2018.
  * [17] I. P. Roberts, J. G. Andrews, and S. Vishwanath, “Hybrid beamforming for millimeter wave full-duplex under limited receive dynamic range,” _IEEE Trans. Wirel. Commun._ , vol. 20, no. 12, pp. 7758–7772, Jun. 2021.
  * [18] I. Goodfellow, Y. Bengio, and A. Courville, _Deep Learning_. The MIT Press, Nov. 2016.
  * [19] D. Gündüz, P. de Kerret, N. D. Sidiropoulos, D. Gesbert, C. R. Murthy, and M. van der Schaar, “Machine learning in the air,” _IEEE J. Sel. Areas Commun._ , vol. 37, no. 10, pp. 2184–2199, Sep. 2019.
  * [20] M. B. Mashhadi and D. Gündüz, “Pruning the pilots: Deep learning-based pilot design and channel estimation for MIMO-OFDM systems,” _IEEE Trans. Wirel. Commun._ , vol. 20, no. 10, pp. 6315–6328, Apr. 2021.
  * [21] H. Huang, Y. Song, J. Yang, G. Gui, and F. Adachi, “Deep-learning-based millimeter-wave massive MIMO for hybrid precoding,” _IEEE Trans. Veh. Technol._ , vol. 68, no. 3, pp. 3027–3032, Jan. 2019.
  * [22] M. Khani, M. Alizadeh, J. Hoydis, and P. Fleming, “Adaptive neural signal detection for massive mimo,” _IEEE Trans. Wirel. Commun._ , vol. 19, no. 8, pp. 5635–5648, May. 2020.
  * [23] M. B. Mashhadi, Q. Yang, and D. Gündüz, “Distributed deep convolutional compression for massive MIMO CSI feedback,” _IEEE Trans. Wirel. Commun._ , vol. 20, no. 4, pp. 2621–2633, Apr. 2021.
  * [24] S. Dörner, S. Cammerer, J. Hoydis, and S. t. Brink, “Deep learning based communication over the air,” _IEEE J. Sel. Topics Signal Process._ , vol. 12, no. 1, pp. 132–143, Dec. 2018.
  * [25] H. Ye, G. Y. Li, B. F. Juang, and K. Sivanesan, “Channel agnostic end-to-end learning based communication systems with conditional GAN,” in _2018 IEEE Globecom Workshops (GC Wkshps)_ , Batam, Indonesia, Jan. 2018, pp. 1–5.
  * [26] P. Dong, H. Zhang, G. Y. Li, I. S. Gaspar, and N. NaderiAlizadeh, “Deep CNN-based channel estimation for mmwave massive MIMO systems,” _IEEE J. Sel. Topics Signal Process._ , vol. 13, no. 5, pp. 989–1000, Jul. 2019.
  * [27] E. Balevi, A. Doshi, and J. G. Andrews, “Massive MIMO channel estimation with an untrained deep neural network,” _IEEE Trans. Wirel. Commun._ , vol. 19, no. 3, pp. 2079–2090, Jan. 2020.
  * [28] Y. Zhang, M. Alrabeiah, and A. Alkhateeb, “Deep learning for massive MIMO with 1-bit ADCs: When more antennas need fewer pilots,” _IEEE Wireless Commun. Lett._ , vol. 9, no. 8, pp. 1273–1277, Aug. 2020.
  * [29] H. He, C.-K. Wen, S. Jin, and G. Y. Li, “Deep learning-based channel estimation for beamspace mmwave massive MIMO systems,” _IEEE Wireless Commun. Lett._ , vol. 7, no. 5, pp. 852–855, May. 2018.
  * [30] K. Hornik, M. Stinchcombe, and H. White, “Multilayer feedforward networks are universal approximators,” _Neural Networks_ , vol. 2, no. 5, pp. 359–366, 1989. [Online]. Available: https://www.sciencedirect.com/science/article/pii/0893608089900208
  * [31] E. Björnson, J. Hoydis, and L. Sanguinetti, “Massive MIMO networks: Spectral, energy, and hardware efficiency,” _Foundations and Trends® in Signal Processing_ , vol. 11, no. 3-4, pp. 154–655, Dec. 2017. [Online]. Available: http://dx.doi.org/10.1561/2000000093
  * [32] I. P. Roberts, A. Chopra, T. Novlan, S. Vishwanath, and J. G. Andrews, “Spatial and statistical modeling of multi-panel millimeter wave self-interference,” _IEEE J. Sel. Areas Commun._ , vol. 41, no. 9, pp. 2780–2795, Jul. 2023.
  * [33] ——, “Beamformed self-interference measurements at 28 GHz: Spatial insights and angular spread,” _IEEE Trans. Wirel. Commun._ , vol. 21, no. 11, pp. 9744–9760, Jun. 2022.
  * [34] A. Assalini, E. Dall’Anese, and S. Pupolin, “Linear MMSE MIMO channel estimation with imperfect channel covariance information,” in _2009 IEEE International Conference on Communications_ , Dresden, Germany, Aug. 2009, pp. 1–5.
  * [35] L. Sanguinetti, E. Björnson, and J. Hoydis, “Toward massive MIMO 2.0: Understanding spatial correlation, interference suppression, and pilot contamination,” _IEEE Trans. Wirel. Commun._ , vol. 68, no. 1, pp. 232–257, Oct. 2020.
  * [36] M. Alrabeiah and A. Alkhateeb, “Deep learning for TDD and FDD massive MIMO: Mapping channels in space and frequency,” in _2019 53rd Asilomar Conference on Signals, Systems, and Computers_ , Pacific Grove, CA, USA, Mar. 2019, pp. 1465–1470.
  * [37] S. Sun, T. S. Rappaport, S. Rangan, T. A. Thomas, A. Ghosh, I. Z. Kovacs, I. Rodriguez, O. Koymen, A. Partyka, and J. Jarvelainen, “Propagation path loss models for 5G urban micro- and macro-cellular scenarios,” in _2016 IEEE 83rd Vehicular Technology Conference (VTC Spring)_ , Nanjing, China, Jul. 2016, pp. 1–6.