DSRC

    dedicated short-range communications
C-ITS

    cooperative intelligent transport system
RSU

    road side unit
Bi

    bi-directional
TR

    throughput
TDL

    tapped delay line
SRNN

    Simple RNN
ITS

    Intelligent Transportation Systems
IEEE

    Institute of Electrical and Electronics Engineers
WAVE

    Wireless Access in Vehicular Environment
V2V

    vehicle-to-vehicle
V2I

    vehicle-to-infrastructure
CCH

    control channel
SCH

    service channels
STS

    short training symbols
LTS

    long training symbols
SS

    signal symbol
SoA

    state-of-the-art
DPA

    data-pilot aided
STA

    spectral temporal averaging
EL

    ensemble learning
FL

    frame length
CDP

    constructed data pilots
TRFI

    time-domain reliable test frequency domain interpolation
MMSE-VP

    minimum mean square error using virtual pilots
iCDP

    improved CDP
SBS

    symbol-by-symbol
FBF

    frame-by-frame
E-TRFI

    Enhanced TRFI
SR-CNN

    super resolution CNN
DN-CNN

    denoising CNN
RBF

    radial basis function
CNN

    convolutional neural network
TS-ChannelNet

    temporal spectral ChannelNet
WSSUS

    wide-sense stationary uncorrelated scattering
TDR

    transmission data rate
LSTM

    long short-term memory
ALS

    accurate LS
SLS

    simple LS
ChannelNet

    channel network
ADD-TT

    average decision-directed with time truncation
WI

    weighted interpolation
DD

    decision-directed
SR-ConvLSTM

    super resolution convolutional long short-term memory
RS

    reliable subcarriers
URS

    unreliable subcarriers
AE-DNN

    auto-encoder deep neural network
AE

    auto-encoder
T-DFT

    truncated discrete Fourier transform
TA-TDFT

    temporal averaging T-DFT
TA

    time averaging
PDP

    power delay profile
1G

    first generation
2G

    second generation
3G

    third generation
3GPP

    Third Generation Partnership Project
4G

    fourth generation
5G

    fifth generation
802.11

    IEEE 802.11 specifications
A/D

    analog-to-digital
ADC

    analog-to-digital
AM

    amplitude modulation
AP

    access point
AR

    augmented reality
ASIC

    application-specific integrated circuit
ASIP

    Application Specific Integrated Processors
AWGN

    additive white Gaussian noise
BCJR

    Bahl, Cocke, Jelinek and Raviv
BER

    bit error rate
BFDM

    bi-orthogonal frequency division multiplexing
BPSK

    binary phase shift keying
BS

    base stations
CA

    carrier aggregation
CAF

    cyclic autocorrelation function
Car-2-x

    car-to-car and car-to-infrastructure communication
CAZAC

    constant amplitude zero autocorrelation waveform
CB-FMT

    cyclic block filtered multitone
CCDF

    complementary cumulative density function
CDF

    cumulative density function
CDMA

    code-division multiple access
CFO

    carrier frequency offset
CIR

    channel impulse response
CM

    complex multiplication
COFDM

    coded-OFDM
CoMP

    coordinated multi point
COQAM

    cyclic OQAM
CP

    cyclic prefix
CR

    cognitive radio
CRC

    cyclic redundancy check
CRLB

    Cramér-Rao lower bound
CS

    cyclic suffix
CSI

    channel state information
CSMA

    carrier-sense multiple access
CWCU

    component-wise conditionally unbiased
D/A

    digital-to-analog
D2D

    device-to-device
DAC

    digital-to-analog
DC

    direct current
DFE

    decision feedback equalizer
DFT

    discrete Fourier transform
DL

    deep learning
DMT

    discrete multitone
DNN

    deep neural network
FNN

    feed-forward neural network
DSA

    dynamic spectrum access
DSL

    digital subscriber line
DSP

    digital signal processor
DTFT

    discrete-time Fourier transform
DVB

    digital video broadcasting
DVB-T

    terrestrial digital video broadcasting
DWMT

    discrete wavelet multi tone
DZT

    discrete Zak transform
E2E

    end-to-end
eNodeB

    evolved node b base station
E-SNR

    effective signal-to-noise ratio
EVD

    eigenvalue decomposition
FBMC

    filter bank multicarrier
FD

    frequency-domain
FDD

    frequency-division duplexing
FDE

    frequency domain equalization
FDM

    frequency division multiplex
FDMA

    frequency-division multiple access
FEC

    forward error correction
FER

    frame error rate
FFT

    fast Fourier transform
FIR

    finite impulse response
FM

    frequency modulation
FMT

    filtered multi tone
FO

    frequency offset
F-OFDM

    filtered-OFDM
FPGA

    field programmable gate array
FSC

    frequency selective channel
FS-OQAM-GFDM

    frequency-shift OQAM-GFDM
FT

    Fourier transform
FTD

    fractional time delay
FTN

    faster-than-Nyquist signaling
GFDM

    generalized frequency division multiplexing
GFDMA

    generalized frequency division multiple access
GMC-CDM

    generalized multicarrier code-division multiplexing
GNSS

    global navigation satellite system
GS

    guard symbols
GSM

    Groupe Spécial Mobile
GUI

    graphical user interface
H2H

    human-to-human
H2M

    human-to-machine
HTC

    human type communication
I

    in-phase
i.i.d.

    independent and identically distributed
IB

    in-band
IBI

    inter-block interference
IC

    interference cancellation
ICI

    inter-carrier interference
ICT

    information and communication technologies
ICV

    information coefficient vector
IDFT

    inverse discrete Fourier transform
IDMA

    interleave division multiple access
IEEE

    institute of electrical and electronics engineers
IF

    intermediate frequency
IFFT

    inverse fast Fourier transform
IoT

    Internet of Things
IOTA

    isotropic orthogonal transform algorithm
IP

    internet protocole
IP-core

    intellectual property core
ISDB-T

    terrestrial integrated services digital broadcasting
ISDN

    integrated services digital network
ISI

    inter-symbol interference
ITU

    International Telecommunication Union
IUI

    inter-user interference
LAN

    local area netwrok
LLR

    log-likelihood ratio
LMMSE

    linear minimum mean square error
LNA

    low noise amplifier
LO

    local oscillator
LOS

    line-of-sight
LP

    low-pass
LPF

    low-pass filter
LS

    least squares
LTE

    Long Term Evolution
LTE-A

    LTE-Advanced
LTIV

    linear time invariant
LTV

    linear time-variant
LUT

    lookup table
M2M

    machine-to-machine
MA

    multiple access
MAC

    multiple access control
MAP

    maximum a posteriori
MC

    multicarrier
MCA

    multicarrier access
MCM

    multicarrier modulation
MCS

    modulation coding scheme
MF

    matched filter
MF-SIC

    matched filter with successive interference cancellation
MIMO

    multiple-input, multiple-output
MISO

    multiple-input single-output
ML

    machien learning
MLD

    maximum likelihood detection
MLE

    maximum likelihood estimator
MMSE

    minimum mean squared error
MRC

    maximum ratio combining
MS

    mobile stations
MSE

    mean squared error
MSK

    Minimum-shift keying
MSSS

    mean-square signal separation
MTC

    machine type communication
MU

    multi user
MVUE

    minimum variance unbiased estimator
NEF

    noise enhancement factor
NLOS

    non-line-of-sight
NMSE

    normalized mean-squared error
NOMA

    non-orthogonal multiple access
NPR

    near-perfect reconstruction
NRZ

    non-return-to-zero
OFDM

    orthogonal frequency division multiplexing
OFDMA

    orthogonal frequency division multiple access
OOB

    out-of-band
OQAM

    offset quadrature amplitude modulation
OQPSK

    offset quadrature phase shift keying
OTFS

    orthogonal time frequency space
PA

    power amplifier
PAM

    pulse amplitude modulation
PAPR

    peak-to-average power ratio
PC-CC

    parallel concatenated convolutional code
PCP

    pseudo-circular pre/post-amble
PD

    probability of detection
pdf

    probability density function
PDF

    probability distribution function
PFA

    probability of false alarm
PHY

    physical layer
PIC

    parallel interference cancellation
PLC

    power line communication
PMF

    probability mass function
PN

    pseudo noise
ppm

    parts per million
PRB

    physical resource block
PRB

    physical resource block
PSD

    power spectral density
Q

    quadrature-phase
QAM

    quadrature amplitude modulation
QoS

    quality of service
QPSK

    quadrature phase shift keying
R/W

    read-or-write
RAM

    random-access memmory
RAN

    radio access network
RAT

    radio access technologies
RC

    raised cosine
RF

    radio frequency
rms

    root mean square
RRC

    root raised cosine
RW

    read-and-write
SC

    single-carrier
SCA

    single-carrier access
SC-FDE

    single-carrier with frequency domain equalization
SC-FDM

    single-carrier frequency division multiplexing
SC-FDMA

    single-carrier frequency division multiple access
SD

    sphere decoding
SDD

    space-division duplexing
SDMA

    space division multiple access
SDR

    software-defined radio
SDW

    software-defined waveform
SEFDM

    spectrally efficient frequency division multiplexing
SE-FDM

    spectrally efficient frequency division multiplexing
SER

    symbol error rate
SIC

    successive interference cancellation
SINR

    signal-to-interference-plus-noise ratio
SIR

    signal-to-interference ratio
SISO

    single-input, single-output
SMS

    Short Message Service
SNR

    signal-to-noise ratio
STC

    space-time coding
STFT

    short-time Fourier transform
STO

    symbol time offset
SU

    single user
SVD

    singular value decomposition
TD

    time-domain
TDD

    time-division duplexing
TDMA

    time-division multiple access
TFL

    time-frequency localization
TO

    time offset
TS-OQAM-GFDM

    time-shifted OQAM-GFDM
UE

    user equipment
UFMC

    universally filtered multicarrier
UL

    uplink
US

    uncorrelated scattering
USB

    universal serial bus
UW

    unique word
VLC

    visible light communications
VR

    virtual reality
WCP

    windowing and CP
ANN

    artificial neural network
GRU

    gated recurrent unit
RNN

    recurrent neural network
WHT

    Walsh-Hadamard transform
WiMAX

    worldwide interoperability for microwave access
WLAN

    wireless local area network
W-OFDM

    windowed-OFDM
WOLA

    windowing and overlapping
WSS

    wide-sense stationary
ZCT

    Zadoff-Chu transform
ZF

    zero-forcing
ZMCSCG

    zero-mean circularly-symmetric complex Gaussian
ZP

    zero-padding
ZT

    zero-tail

# RNN Based Channel Estimation in Doubly Selective Environments

Abdul Karim Gizzini , and Marwa Chafii  Abdul Karim Gizzini is with ETIS,
UMR8051, CY Cergy Paris Université, ENSEA, CNRS, France (e-mail:
abdulkarim.gizzini@ensea.fr). Marwa Chafii is with the Engineering Division,
New York University (NYU) Abu Dhabi, 129188, UAE and NYU WIRELESS, NYU Tandon
School of Engineering, Brooklyn, 11201, NY (e-mail: marwa.chafii@nyu.edu).

###### Abstract

Doubly-selective channel estimation represents a key element in ensuring
communication reliability in wireless systems. Due to the impact of multi-path
propagation and Doppler interference in dynamic environments, doubly-selective
channel estimation becomes challenging. Conventional symbol-by-symbol (SBS)
and frame-by-frame (FBF) channel estimation schemes encounter performance
degradation in high mobility scenarios due to the usage of limited training
pilots. Recently, deep learning (DL) has been utilized for doubly-selective
channel estimation, where long short-term memory (LSTM) and convolutional
neural network (CNN) networks are employed in the SBS and FBF, respectively.
However, their usage is not optimal, since LSTM suffers from long-term memory
problem, whereas, CNN-based estimators require high complexity. For this
purpose, we overcome these issues by proposing an optimized recurrent neural
network (RNN)-based channel estimation schemes, where gated recurrent unit
(GRU) and Bi-GRU units are used in SBS and FBF channel estimation,
respectively. The proposed estimators are based on the average correlation of
the channel in different mobility scenarios, where several performance-
complexity trade-offs are provided. Moreover, the performance of several RNN
networks is analyzed. The performance superiority of the proposed estimators
against the recently proposed DL-based SBS and FBF estimators is demonstrated
for different scenarios while recording a significant reduction in complexity.

###### Index Terms:

Wireless communications, Channel estimation, Deep learning, RNN, LSTM, GRU,
Bi-GRU.

##  I INTRODUCTION

The recent advances in beyond 5G networks enable high data rates and low
latency mobile wireless applications [1]. Wireless communications offer
mobility to different nodes within the network, however, the mobility feature
has a severe negative impact on the communication reliability [2]. In such
environment, the wireless channel is said to be doubly-selective, i.e. varies
in both time and frequency. This is due to the propagation medium, where the
transmitted signals propagate through multiple paths, each having a different
power, delay, and Doppler shift effect resulting from the motion of network
nodes. Knowing that the accuracy of the estimated channel influences the
system performance since it affects different operations at the receiver like
equalization, demodulation, and decoding. Therefore, ensuring communication
reliability using accurate channel estimation is crucial, especially in high
mobility scenarios [3].

In general, a few pilots are allocated within the transmitted frame in order
to maintain a good transmission data rate, where the state-of-the-art (SoA)
channel estimation schemes can be categorized into: (i) SBS estimators: the
channel is estimated for each received symbol separately [4, 5]. (ii) FBF
estimators: where the previous, current, and future pilots are employed in the
channel estimation for each received symbol [6]. The higher channel estimation
accuracy can be achieved by using FBF estimators, since the channel estimation
of each symbol takes advantage from the knowledge of previous, current, and
future allocated pilots within the frame. Unlike, SBS estimators, where only
the previous and current pilots are exploited in the channel estimation for
each received symbol. However, the allocated pilots are insufficient for
accurately tracking the doubly-selective channel. As a result, conventional
SBS channel estimation schemes use the demapped data subcarriers besides pilot
subcarriers to accomplish the channel estimation task. This procedure is known
as data-pilot aided (DPA) channel estimation, which is unreliable due to the
demapping errors of the data subcarriers that are also enlarged from one
symbol to another, leading to accumulated error in the channel estimation
process. Moreover, the DPA-based channel estimation schemes such as spectral
temporal averaging (STA) [4] and time-domain reliable test frequency domain
interpolation (TRFI) [5] are impractical solutions as they rely on many
assumptions such as high correlation of the channel within the received frame.
In addition, they lack robustness in highly dynamic environments. On the other
hand, several 2D interpolation methods, such as radial basis function (RBF)
[7] and average decision-directed with time truncation (ADD-TT) [8] are
employed in the FBF channel estimation. However, the performance of these
interpolation methods is limited when employed in high mobility scenarios,
since they use fixed interpolation parameters. Moreover, the well-known FBF
estimator is the conventional 2D linear minimum mean square error (LMMSE) uses
the channel and noise statistics in the estimation, thus, leading to
comparable performance to the ideal case. However, it suffers from high
complexity making it impractical in real-case scenarios. Therefore,
investigating both SBS and FBF channel estimators with a good trade-off
complexity vs. performance is a crucial need for improving the channel
estimation accuracy as well as maintaining affordable computational
complexity.

Recently, a great success of deep learning (DL) has been witnessed in several
wireless communications applications [9, 10], including localization [11, 12,
13], and channel estimation [14, 15, 16, 17], particularly when integrated
with conventional SBS and FBF estimators. This success is due to the
robustness, low-complexity, and good generalization ability of DL algorithms
making their integration into communication systems beneficial. Motivated by
these advantages, DL algorithms have been integrated into doubly-selective
channel estimators in two different manners: (i) feed-forward neural network
(FNN) and LSTM networks with different architectures and configurations are
employed on top of SBS estimators [18, 19, 20, 21, 22]. (ii) CNNs are
integrated into the FBF estimators [7, 8, 23], where the estimated channel for
the whole frame is considered as a 2D low-resolution noisy image and CNN-based
processing is applied as super-resolution and denoising techniques. These SoA
DL-based SBS and FBF still encounter a considerable performance degradation
due to the poor accuracy of the employed initial channel estimation as in [19,
20]. Moreover, they require high computational complexity due to the employed
DL architectures [7, 8, 23].

In order to achieve better performance-complexity trade-off in different
mobility scenarios according to the channel correlation, this paper sheds
light on the RNN-based channel estimation in doubly-selective environments for
both SBS and FBF channel estimation, where an optimized RNN networks
represented by a GRU and bi-directional (Bi)-GRU units are used in the
proposed SBS and FBF channel estimators, respectively. Thus, having a low-
complexity and robust channel estimation in different mobility scenarios. The
proposed GRU-based SBS estimator uses only one GRU network instead of two as
the case in the recently proposed LSTM-based estimator [21]. After that, DPA
estimation is applied using the GRU estimated channel. Finally, unlike [21]
where FNN network is used for noise elimination, in the proposed GRU-based
estimators, time averaging (TA) processing is employed as a noise alleviation
technique where the noise alleviation ratio is calculated analytically.
Moreover, motivated by the fact that Bi-RNN is designed to perform 2D
interpolation of unknown data bounded between known data [24], the proposed
Bi-GRU channel estimator is designed to overcome the limitations of the FBF
CNN-based channel estimation schemes, where an end-to-end 2D interpolation is
performed by the proposed Bi-GRU unit. In this context, the proposed Bi-GRU
channel estimator employs an adaptive frame design, where comb pilot
allocation is replaced by full pilot allocated symbols that are inserted
periodically within the transmitted frame. As a first step, the channel is
estimated at the inserted pilot symbols, after that, Bi-GRU acts as an end-to-
end 2D interpolation unit to estimate the channel at the data symbols without
the need to any initial estimation. By doing this interpolation, the proposed
Bi-GRU based estimator is able to further improve the estimation performance,
unlike the CNN-based estimators that work according to the noise mitigation
principle [25, 26] rather than doing actual interpolation. Simulation results
show the performance superiority of the proposed RNN-based channel estimation
schemes against the SoA SBS and FBF channel estimators while recording an
outstanding computational complexity reduction. To sum up, the contributions
of this paper are listed below 111We would like to mention that part of this
work related to the Bi-RNN based FBF channel estimation has been accepted for
publication in the IEEE ICC 2023 conference [27].:

  * •

Proposing low-complexity and robust RNN-based channel estimation schemes,
where an optimized GRU, and Bi-GRU units are employed to accurately estimate
the doubly-selective channel in SBS and FBF fashions, respectively.

  * •

Employing GRU unit as a pre-processing module to DPA and TA processing in SBS
channel estimation. Whereas an end-to-end 2D interpolation using Bi-GRU unit
is proposed for FBF channel estimation.

  * •

Providing a brief overview of the theoretical concept of the studied RNN
networks.

  * •

Analyzing the appropriate RNN architectures to be employed according to the
average channel correlation within the frame in different mobility scenarios,
where the advantages of using the proposed optimized GRU unit instead of
regular LSTM unit are discussed.

  * •

Showing that the proposed RNN-based channel estimators record a significant
superiority over the SoA SBS and FBF channel estimators in terms of bit error
rate (BER) and throughput for different modulation orders, mobility scenarios,
and frame lengths.

  * •

Illustrating the advantage of using the ensemble learning (EL) algorithm [28]
in the generalization of one DL model that is robust against a range of
Doppler frequencies.

  * •

Providing a detailed computational complexity analysis for the studied channel
estimators, where we show that the proposed RNN-based channel estimators
achieve substantial reduction in complexity in comparison with the SoA SBS and
FBF channel estimators.

The remainder of this paper is organized as follows: Section II presents the
system model. The SoA DL-based channel estimation schemes are thoroughly
investigated and discussed in Section III. Section IV illustrates the
framework of the proposed RNN-based channel estimation schemes, besides
providing a brief overview of the main RNN networks integrated into the
doubly-selective channel estimation. In Section V, different modulation orders
are used to present simulation results, wherein the performance of the studied
estimators is examined in terms of BER. Detailed computational complexity
analysis is provided in Section VI. Finally, Section VII concludes this study.

##  II SYSTEM MODEL

Consider a frame consisting of $I$ orthogonal frequency division multiplexing
(OFDM) symbols. The $i$-th transmitted frequency-domain OFDM symbol
$\tilde{\bm{x}}_{i}[k]$, is denoted by

$\tilde{\bm{x}}_{i}[k]=\left\\{\begin{array}[]{ll}\tilde{\bm{x}}_{{i,d}}[k],&\quad
k\in{\mathcal{K}}_{\text{d}}.\\\ \tilde{\bm{x}}_{{i,p}}[k],&\quad
k\in{\mathcal{K}}_{\text{p}}.\\\ 0,&\quad k\in{\mathcal{K}}_{\text{n}}.\\\
\end{array}\right.$ (1)

where $k$ refers to the subcarrier index, where $0\leq k\leq K-1$. Moreover,
$d$ and $p$ indices refer to the transmitted data and pilot subcarriers,
respectively. The total number of subcarriers is divided into
$K_{\text{on}}=K_{d}+K_{p}$ subcarriers in addition to $K_{n}$ null guard band
subcarriers, where $\tilde{\bm{x}}_{{i,d}}[k]$ and $\tilde{\bm{x}}_{{i,p}}[k]$
represent the modulated data symbols and the predefined pilot symbols
allocated at a set of subcarriers denoted ${\mathcal{K}}_{\text{d}}$ and
${\mathcal{K}}_{\text{p}}$, respectively. The received frequency-domain OFDM
symbol denoted as $\tilde{\bm{y}}_{{i}}[k]$ is expressed as follows

$\begin{split}\tilde{\bm{y}}_{{i}}[k]&=\tilde{\bm{h}}_{i}[k]\tilde{\bm{x}}_{i}[k]+\tilde{\bm{v}}_{i}[k],~{}k\in{\mathcal{K}}_{\text{on}}.\end{split}$
(2)

Here, $\tilde{\bm{h}}_{i}[k]\in\mathbb{C}^{K_{on}\times 1}$ refers to the
frequency response of the doubly-selective channel at the $i$-th OFDM symbol
and $k$-th subcarrier. ${\tilde{\bm{v}}}_{i}[k]$ signifies the additive white
Gaussian noise (AWGN) of variance $\sigma^{2}$. As a matrix form, (2) can be
expressed as follows

$\tilde{\bm{Y}}[k,i]=\tilde{\bm{H}}[k,i]\tilde{\bm{X}}[k,i]+\tilde{\bm{V}}[k,i],~{}k\in{\mathcal{K}}_{\text{on}},$
(3)

where $\tilde{\bm{V}}[k,i]\in\mathbb{C}^{K_{\text{on}}\times I}$ and
$\tilde{\bm{H}}\in\mathbb{C}^{K_{\text{on}}\times I}$ denote the AWGN noise
and the doubly-selective frequency response of the channel for all symbols
within the transmitted OFDM frame, respectively.

##  III SoA DL-BASED CHANNEL ESTIMATION

This section presents the recently proposed SoA DL-based SBS and FBF channel
estimation schemes, where the processing steps applied in each estimator are
presented.

###  III-A DL-BASED SBS CHANNEL ESTIMATION SCHEMES

In general, FNN and LSTM networks are employed in the SBS channel estimation,
where an optimized FNNs are integrated as a post-processing unit with
conventional SBS channel estimators as the case in the DPA-FNN [18], STA-FNN
[19], and TRFI-FNN [20]. On the other hand, LSTM networks are utilized as a
pre-processing unit in the LSTM-FNN-DPA [21], and LSTM-DPA-TA [22] channel
estimators. Both implementations are helpful in improving the accuracy of the
channel estimation. However, the LSTM-based estimation illustrates a
considerable superiority over the FNN-based estimation. In this context, and
since we are focusing on RNN-based channel estimation, this section presents
the steps applied in the LSTM-based channel estimators.

####  III-A1 LSTM-FNN-DPA

The work proposed in [21] shows that employing the LSTM processing prior to
the DPA estimation could lead to a significant improvement in the overall
performance. In this context, two cascaded LSTM and FNN networks for both
channel estimation as well as noise compensation. The LSTM-FNN-DPA estimator
employs the least squares (LS) estimated channel at the previous and current
received pilots, such that

$\hat{\tilde{\bm{h}}}_{i,p}[k]=\frac{\tilde{\bm{y}}_{i}[k]}{\tilde{\bm{x}}_{p}[k]},~{}\hat{\tilde{\bm{h}}}_{i-1,p}[k]=\frac{\tilde{\bm{y}}_{i-1}[k]}{\tilde{\bm{x}}_{p}[k]},~{}k\in{\mathcal{K}}_{\text{p}}.$
(4)

The LS estimated channel are fed as an input to both LSTM and FNN networks,
where the LSTM-FNN estimated channel is expressed as follows

$\tilde{\bm{d}}_{\text{LSTM-
FNN}_{i,d}}[k]=\mathfrak{D}\big{(}\frac{\tilde{\bm{y}}_{i,d}[k]}{\hat{\tilde{\bm{h}}}_{\text{LSTM-
FNN}_{i-1,d}}[k]}\big{)}.$ (5)
$\hat{\tilde{\bm{h}}}_{\text{DL}_{i,d}}[k]=\frac{\tilde{\bm{y}}_{i,d}[k]}{\bm{d}_{\text{LSTM}_{i,d}}[k]}.$
(6)

We note that, at the beginning of the frame ($i=1$),
$\hat{\tilde{\bm{h}}}_{i-1,p}[k]$ denotes the LS estimated channel at the
received preamble symbols, such that

$\hat{\tilde{\bm{h}}}_{\text{LS}}[k]=\frac{\sum\limits_{u=1}^{P}\tilde{\bm{y}}_{u,p}[k]}{P\tilde{\bm{x}}_{p}[k]},~{}k\in{\mathcal{K}}_{\text{on}}.$
(7)

While this estimator can outperform the FNN-based estimators, it encounters a
high complexity cost arising from the employment of two DL networks.

####  III-A2 LSTM-DPA-TA

The authors in [22] propose to use only an optimized LSTM network instead of
two as implemented in the LSTM-FNN-DPA estimator. In addition, noise
compensation is made possible by applying TA processing. This methodology only
requires the previous pilots $\hat{\tilde{\bm{h}}}_{i-1,p}[k]$ besides the
LSTM estimated channel as an input. Then, the LSTM estimated channel is
employed in the DPA estimation as follows

$\tilde{\bm{d}}_{\text{LSTM}_{i}}[k]=\mathfrak{D}\big{(}\frac{\tilde{\bm{y}}_{i}[k]}{\hat{\tilde{\bm{h}}}_{\text{LSTM}_{i-1}}[k]}\big{)},~{}\hat{\tilde{\bm{h}}}_{\text{LSTM}_{0}}[k]=\hat{\tilde{\bm{h}}}_{\text{LS}}[k],$
(8) $\hat{\tilde{\bm{h}}}_{\text{LSTM-
DPA}_{i}}[k]=\frac{\tilde{\bm{y}}_{i}[k]}{\tilde{\bm{d}}_{\text{LSTM}_{i}}[k]}.$
(9)

AWGN noise alleviation can be achieved by further applying TA processing such
that

$\hat{\bar{\bm{h}}}_{\text{DL-
TA}_{i,d}}=(1-\frac{1}{\alpha})\hat{\bar{\bm{h}}}_{\text{DL-
TA}_{i-1,d}}+\frac{1}{\alpha}\hat{\bar{\bm{h}}}_{\text{LSTM-DPA}_{i,d}}.$ (10)

Here, $\alpha$ denotes the utilized weighting coefficient. In [22], the
authors use a fixed $\alpha=2$ for simplicity. Therefore, the TA applied in
(10) reduces the AWGN noise power $\sigma^{2}$ iteratively within the received
OFDM frame according to the ratio

$\begin{split}{R}_{\text{DL-
TA}_{q}}&=\left(\frac{1}{4}\right)^{(q-1)}+\sum_{j=2}^{q}\left(\frac{1}{4}\right)^{(q-j+1)}=\frac{4^{q-1}+2}{3\times
4^{q-1}}.\end{split}$ (11)

This corresponds to the AWGN noise power ratio of the estimated channel at the
$q$-th estimated channel, where ${1<q<I+1}$ and ${{R}_{\text{DL-TA}_{1}}=1}$
denotes the AWGN noise power ratio at $\hat{\tilde{\bm{h}}}_{\text{LS}}[k]$.
From the derivation of ${R}_{\text{DL-TA}_{q}}$, it can be seen that the noise
power decreases over the received OFDM frame, i.e. the SNR increases,
resulting in an overall improved performance. The full derivation of (11) is
found in [22]. Even though the LSTM-DPA-TA improves the performance compared
to the LSTM-FNN-DPA estimator, it still suffers from high computational
complexity. Moreover, in Section IV, we show that employing LSTM unit in the
channel estimation would affect the estimation accuracy negatively. Whereas,
the proposed GRU-based channel estimation provides a better performance-
complexity trade-off.

###  III-B CNN-BASED FBF CHANNEL ESTIMATION SCHEMES

In [23], a CNN aided weighted interpolation (WI) channel estimation schemes
have been proposed. The WI-CNN estimators use adaptive frame structure
according to the mobility scenario. The idea is to avoid using comb pilot
allocation and insert $Q$ pilot OFDM symbols with different configurations
within the transmitted OFDM frame instead. In this context, the WI-CNN
estimators employ one, two, and three pilot symbols in low, high, and very
high mobility scenarios, respectively. Following the selection of the frame
structure, the WI-CNN estimators proceed as follows

  * •

Pilot symbols channel estimation: In order to estimate the channel at the
inserted pilot symbols, the basic LS denoted as _ simple LS (SLS)_ estimation
is applied using the received preambles as shown in (7), and using each
received pilot symbol such that

$\hat{\tilde{\bm{h}}}_{{\text{SLS}}_{q}}[k]=\frac{\tilde{\bm{y}}^{(p)}_{q}[k]}{\tilde{\bm{p}}[k]}=\tilde{\bm{h}}_{q}[k]+\tilde{\bm{v}}_{{q}}[k],~{}k\in{\mathcal{K}}_{\text{on}}.$
(12)

where $\tilde{\bm{v}}_{q}[k]$ represents the noise at the $q$-th received
pilot symbol, $1\leq q\leq Q$ denotes the inserted pilot symbol index, and
$\tilde{\bm{Y}}_{Q}=[\tilde{\bm{y}}^{(p)}_{1},\dots,\tilde{\bm{y}}^{(p)}_{q},\dots,\tilde{\bm{y}}^{(p)}_{Q}]\in\mathbb{C}^{K_{\text{on}}\times
Q}$. Moreover, _ accurate LS (ALS)_ can be obtained by applying the discrete
Fourier transform (DFT) interpolation of $\hat{\bm{h}}_{q,L}$ such that

$\hat{\tilde{\bm{h}}}_{{\text{ALS}}_{q}}=\bm{F}_{\text{on}}\hat{\bm{h}}_{q,L},~{}k\in{\mathcal{K}}_{\text{on}},$
(13)

with ${\hat{\bm{h}}}_{q,L}\in\mathbb{C}^{L\times 1}$ denotes the estimated
channel impulse response at the $q$-th received pilot symbol. We note that the
_ALS_ and _SLS_ are used for full pilot (FP) allocation. However, if the
number of channel taps $L$ is known, the channel estimation requires only $L$
pilots in each pilot symbol, where DFT interpolation can be applied to the
estimated channel impulse response $\hat{\bm{h}}_{q,L}$ such that

$\displaystyle\hat{\tilde{\bm{h}}}_{\text{DFT}_{q}}=\bm{F}_{\text{on}}\hat{\bm{h}}_{q,L},~{}k\in{\mathcal{K}}_{\text{on}},$
(14)

where $\bm{F}_{\text{on}}\in\mathbb{C}^{K_{\text{on}}\times L}$ denotes the
truncated DFT matrices obtained by selecting ${\mathcal{K}}_{\text{on}}$ rows,
and $L$ columns from the $K$-DFT matrix.

  * •

Data symbols channel estimation: After estimating the channel at the inserted
$Q$ pilot symbols, the WI-CNN estimator divides the received frame into
several sub-frames that are grouped as follows

$\hat{\tilde{\bm{H}}}_{q}=[\hat{\tilde{\bm{h}}}_{q-1},\hat{\tilde{\bm{h}}}_{q}],~{}q=1,\cdots
Q,$ (15)

$\hat{\tilde{\bm{h}}}_{q}$ refers to the implemented LS estimation. Then, the
estimated channel for the $i$-th received data OFDM symbol within each sub-
frame is calculated as a weighted summation of the estimated channels at the
pilot symbols, such that

$\hat{\tilde{\bm{H}}}_{{\text{WI}}_{f}}=\hat{\tilde{\bm{H}}}_{{f}}\bm{C}_{f},$
(16)

where $\hat{\tilde{\bm{H}}}_{f}\in\mathbb{C}^{K_{\text{on}}\times 2}$ denotes
LS estimated channels at the pilot symbols within the $f$-th sub-frame.
$\bm{C}_{f}\in\mathbb{R}^{2\times I_{f}}$ denotes the interpolation weights of
the $I_{f}$ OFDM data symbols within the $f$-th sub-frame. The interpolation
weights of $\bm{C}_{f}$ are calculated by minimizing the mean squared error
(MSE) between the ideal channel $\tilde{\bm{H}}_{{f}}$, and the LS estimated
channel at the OFDM pilot symbols $\hat{\tilde{\bm{H}}}_{{f}}$ as derived in
[29]. In the final step, optimized super resolution CNN (SR-CNN) is employed
on top of the WI estimators in a low mobility scenario, whereas an optimized
denoising CNN (DN-CNN) is considered in high mobility one.

The WI-CNN estimators suffer from high computational complexity. Moreover,
using noise alleviation CNNs is not sufficient to accurately estimate the
doubly-selective channel. Therefore, we propose a Bi-GRU channel estimator
that perform 2D interpolation, unlike the SR-CNN and DN-CNN networks, which
are based on noise alleviation techniques. As a result, performance
superiority of the proposed Bi-GRU channel estimator can be achieved while
recording a significant decrease of the computational complexity in comparison
to the WI-CNN estimators as illustrated in Section V and Section VI.

##  IV PROPOSED RNN-BASED CHANNEL ESTIMATION SCHEMES

In this section, RNN main concepts and extensions are first thoroughly
introduced. Then, a detailed explanation of the proposed RNN and Bi-RNN based
schemes for SBS and FBF channel estimation are presented, respectively.

###  IV-A RECURRENT NEURAL NETWORKS: REVIEW

RNN is a type of artificial neural network (ANN) designed to work with
sequential data. This sequential data can be in form of time series, text,
audio, video etc. RNN uses the previous information in the sequence to produce
the current output, where it is incorporated with memory to take information
from prior inputs to influence the current output. This mechanism is the key
essence to RNN success in sequential problems. The core concept of RNNs is to
keep/discard input data in a recurring manner. Therefore, RNN gates contain
Sigmoid activation. A Sigmoid activation regulates values in the range
$[0,1]$, which is helpful to update or forget data because any number getting
multiplied by 0 disappears and is discarded. In contrast, any number
multiplied by 1 is kept. The network can learn which data is not important,
and therefore can be forgotten or which data is important to keep. Moreover,
Tanh activation is used within the RNN architecture at some point in order to
regulate the network in the training phase by squishing values between -1 and
1, since zero values affect negatively the training accuracy.

In order to make things clearer, the RNN input is expressed in terms of the
initial estimated channel at the $i$-th OFDM symbol denoted by
$\hat{\bar{\bm{h}}}_{\rho_{i}}\in\mathbb{R}^{K_{\text{in}}\times 1}$, where
$K_{\text{in}}=2K_{\text{on}}$. Since DL networks work with real-valued
numbers, then,

$\hat{\bar{\bm{h}}}_{\rho_{i}}=\text{vec}\Big{\\{}\Re{\big{(}\hat{\tilde{\bm{h}}}_{\rho_{i}}}\big{)},\Im{\big{(}\hat{\tilde{\bm{h}}}_{\rho_{i}}}\big{)}\Big{\\}}.$
(17)

Here, $\rho$ refers to the applied initial channel estimation scheme, and
$\Re\\{.\\}$ and $\Im\\{.\\}$ denote the real and imaginary values of the
initial estimated channel
$\hat{\tilde{\bm{h}}}_{\rho_{i}}\in\mathbb{C}^{K_{\text{on}}\times 1}$,
respectively. Moreover, the RNN output is denoted by
$\hat{\bar{\bm{h}}}_{\rho\text{-RNN}_{i}}\in\mathbb{R}^{K_{\text{in}}\times
1}$. We note that FNN network treats the initial estimated channels
separately, where it produces the output
$\hat{\bar{\bm{h}}}_{\rho\text{-FNN}_{i}}$ for each input. By doing this
single input-output mapping, the FNN network is able to learn the frequency
correlation of the doubly-selective channel, besides correcting the initial
estimation error. On the contrary, RNN network treats the initial estimated
channel as a correlated sequence, where the current
$\hat{\bar{\bm{h}}}_{\rho\text{-RNN}_{i}}$ is computed using the previous RNN
estimated channel $\hat{\bar{\bm{h}}}_{\rho\text{-RNN}_{i-1}}$ and the current
initial estimated channel $\hat{\bar{\bm{h}}}_{\rho_{i}}$. This process allows
the RNN network to learn both frequency and time correlation of the doubly-
selective channel, and thus, RNN outperforms FNN in the channel estimation
task.

In general, there exist three main types of RNNs: (i) Simple RNN (SRNN), (ii)
LSTM, and (iii) GRU. The main difference between them is in how the input data
is processed by each RNN as shown in Fig. 1. For simplicity, let
$\hat{\bar{\bm{h}}}_{\rho_{i}}=\bar{\bm{x}}_{t}$ and
$\hat{\bar{\bm{h}}}_{\rho\text{-RNN}_{i}}={{{\bm{o}}}}_{t}$, where $t$ denotes
the time index.

Figure 1: Detailed architecture of the SRNN, LSTM, and GRU units.

####  IV-A1 SRNN

is useful when we need to look at recent information only to perform a present
task, where the hidden state is constantly being rewritten in each time step.
The SRNN updates the current hidden state, and its output is as follows

${\bar{\bm{h}}}_{t}=\sigma(\bm{W}_{x,t}\bar{\bm{x}}_{t}+\bm{W}_{h,t}\bar{\bm{h}}_{t-1}+\bar{\bm{b}}_{h,t}),$
(18)
${{{\bm{o}}}}_{t}=\sigma(\bm{W}_{o,t}\bar{\bm{h}}_{t}+\bar{\bm{b}}_{o,t}),$
(19)

where ${\sigma}$ denotes the Sigmoid function,
$\bm{W}_{x,t}\in\mathbb{R}^{P\times K_{in}}$,
$\bm{W}_{h,t}\in\mathbb{R}^{P\times P}$ and
$\bar{\bm{b}}_{h,t}\in\mathbb{R}^{P\times 1}$ are the weight matrices and
biases associated with the SRNN input vector
$\bar{\bm{x}}_{t}\in\mathbb{R}^{K_{in}\times 1}$ and the previous hidden state
vector $\bar{\bm{h}}_{t-1}\in\mathbb{R}^{P\times 1}$, respectively. We note
that at the first step, a hidden state will usually be seeded as a matrix of
zeros so that it can be fed into the RNN cell together with the first input in
the sequence. Training and back-propagation in RNN are similar to other forms
of ANN, where RNN needs to be trained in order to produce accurate and desired
outputs. However, when an SRNN is exposed to long sequences, it tends to lose
the information because it cannot store long sequences since it focuses only
on the latest output only. This problem is commonly referred to as vanishing
gradients [30] that occurs during the training phase, where useful gradients
cannot propagate from the output of the model back to the layers near the
input of the model. As a result, the RNN does not learn the effect of earlier
inputs and it is too difficult for RNN to preserve information over many time
steps, hence, causing the short-term memory problem. To overcome this problem,
specialized versions of RNN like LSTM and GRU are created.

####  IV-A2 LSTM

is a special kind of RNN capable of learning long-term sequences. Recall that
in SRNN, the input and hidden state from the previous time step are passed
through a simple activation layer to obtain a new state. Whereas in LSTM the
process is slightly complex, where the LSTM unit takes at each time input from
three different states defined as: (i) current input state represented by
$\bar{\bm{x}}_{t}$. (ii) short-term memory state from the previous LSTM unit
denoted by $\bar{\bm{h}}_{t-1}$, and (iii) long-term memory state from the
previous LSTM unit denoted by ${\bm{c}}_{t-1}$. These inputs are controlled by
the gates to regulate the information to be kept or discarded before passing
the updated long-term and short-term information to the next LSTM unit. We can
imagine these gates as filters that remove unwanted selected and irrelevant
information. LSTM uses mainly three gates defined as input gate, forget gate,
and output gate.

##### Forget Gate

It decides which information from long-term memory should be kept or discarded
and this is done by multiplying the incoming long-term memory by a forget
vector generated by the current input and incoming short memory. Information
from the previous hidden state and information from the current input is
passed through the sigmoid function. Values come out between 0 and 1. The
closer to 0 means to forget, and the closer to 1 means to keep, such that

${\bm{f}}_{t}=\sigma(\bm{W}_{f,t}\bar{\bm{x}}_{t}+\bm{W}^{\prime}_{f,t}\bar{\bm{h}}_{t-1}+\bar{\bm{b}}_{f,t}),$
(20)

where ${\sigma}$ denotes the Sigmoid function,
$\bm{W}_{f,t}\in\mathbb{R}^{P\times K_{in}}$,
$\bm{W}^{\prime}_{f,t}\in\mathbb{R}^{P\times P}$ and
$\bar{\bm{b}}_{f,t}\in\mathbb{R}^{P\times 1}$ are the forget gate weights and
biases at time $t$, $\bar{\bm{x}}_{t}\in\mathbb{R}^{K_{in}\times 1}$ and
$\bar{\bm{z}}_{t-1}$ represents the LSTM unit input vector of size $K_{in}$,
and the previous hidden state of size $P$, respectively.

##### Input Gate

The input gate only works with the information from the current input
$\bar{\bm{x}}_{t}$ and short-term memory $\bar{\bm{h}}_{t-1}$ from the
previous step by filtering out the information that is not useful. The input
gate proceeds as follows

${\bar{\bm{i}}_{t}}=\sigma(\bm{W}_{\bar{\bm{i}},t}\bar{\bm{x}}_{t}+\bm{W}^{\prime}_{\bar{\bm{i}},t}\bar{\bm{h}}_{t-1}+\bar{\bm{b}}_{\bar{\bm{i}},t}),$
(21)
${\tilde{{\bm{c}}}}_{t}=\text{tanh}(\bm{W}_{{\tilde{{\bm{c}}}},t}\bar{\bm{x}}_{t}+\bm{W}^{\prime}_{{\tilde{{\bm{c}}}},t}\bar{\bm{h}}_{t-1}+\bar{\bm{b}}_{{\tilde{{\bm{c}}}},t}).$
(22)

Now we should have enough information to calculate the new long-term memory
represented by the cell state. First, the cell state gets element-wise
multiplied by the forget vector. This has a possibility of dropping values in
the cell state if it gets multiplied by values near 0. Then we take the output
from the input gate and do an element-wise addition that updates the cell
state to new values that the neural network finds relevant. That gives us our
new cell state, such that

${{{\bm{c}}}}_{t}={\bm{f}}_{t}\odot{\bm{c}}_{t-1}+\bar{\bm{i}}_{t}\odot{\tilde{{\bm{c}}}}_{t}.$
(23)

Here, $\odot$ denotes the Hadamard product.

##### Output Gate

The output gate takes the current input, the previous short-term memory, and
the newly computed long-term memory (23) to produce new short-term memory
which will be passed on to the next time step. The output of the current time
step can also be drawn from this hidden state as follows

${\bar{{\bm{h}}}}_{t}={\bm{o}}_{t}\odot\text{tanh}({\bm{c}}_{t}),$ (24)
${\bm{o}}_{t}=\sigma(\bm{W}_{o,t}\bar{\bm{x}}_{t}+\bm{W}^{\prime}_{o,t}\bar{\bm{h}}_{t-1}+\bar{\bm{b}}_{o,t}).$
(25)

To summarize, the forget gate decides what is relevant to keep from prior
steps. The input gate decides what information is relevant to add from the
current step. The output gate determines what the next hidden state should be.

####  IV-A3 GRU

The GRU is the newer generation of RNN, it is based on the same concept as the
LSTM, but with optimized architecture. The GRU architecture gets rid of the
cell state, so there is no long-term memory as the case in the LSTM. Instead,
the hidden state is used only to transfer information. Moreover, GRU has only
two gates, a reset gate, and an update gate.

##### Reset gate

The reset gate is used from the model to decide how much of the past
information is needed to neglect. In short, it decides whether the previous
hidden state is important or not, such that

${\bar{\bm{r}}}_{t}=\sigma(\bm{W}_{r,t}\bar{\bm{x}}_{t}+\bm{W}^{\prime}_{r,t}\bar{\bm{h}}_{t-1}+\bar{\bm{b}}_{r,t}),$
(26)

##### Update gate

The update gate acts similarly to the forget and input gate of an LSTM. It
decides what information to throw away and what new information to add from
the current input. It is also responsible for determining the amount of
previous information that needs to pass along to the next state, such that

${\bar{\bm{z}}}_{t}=\sigma(\bm{W}_{z,t}\bar{\bm{x}}_{t}+\bm{W}^{\prime}_{z,t}\bar{\bm{h}}_{t-1}+\bar{\bm{b}}_{z,t}),$
(27)
${\tilde{{\bm{h}}}}_{t}=\text{tanh}(\bm{W}_{{\tilde{{\bm{h}}}},t}\bar{\bm{x}}_{t}+\bm{W}^{\prime}_{{\tilde{{\bm{h}}}},t}({\bar{\bm{r}}}_{t}\odot\bar{\bm{h}}_{t-1})+\bar{\bm{b}}_{{\tilde{{\bm{h}}}},t}).$
(28)

Finally, the new hidden state is calculated as follows

${{{\bar{\bm{h}}}}}_{t}=(1-{\bar{\bm{z}}}_{t})\odot{\bar{\bm{h}}}_{t-1}+\bar{\bm{z}}_{t}\odot{\tilde{{\bm{h}}}}_{t}.$
(29)

To sum up, RNN networks are incorporated with a memory to take advantage of
the prior outputs in the prediction of the current output, and thus, the
current output prediction is correlated with the previous outputs. SRNN cannot
store long sequences, since it only focuses on the latest output. On the
contrary, LSTM is capable of learning long-term sequences, and predicting the
current output is influenced by the long sequence of previous outputs.
However, LSTM is not useful in all scenarios, especially, when the successive
inputs become uncorrelated over time. Since predicting the current output will
be affected by uncorrelated previous outputs, the prediction accuracy is
negatively affected. In this context, GRU provides a trade-off; it uses a
short-term memory in predicting the current output, which improves the
prediction accuracy in comparison with LSTM. Moreover, the GRU has two gates,
whereas LSTM has three gates. Thereby, fewer training parameters and faster
execution can be achieved by using GRU instead of LSTM.

Figure 2: Correlation of the channel at the first and the last OFDM symbol
within the transmitted frame. Figure 3: Proposed RNN-based channel estimation
schemes.

###  IV-B PROPOSED SBS RNN-BASED CHANNEL ESTIMATOR

The proposed RNN-based estimation scheme sheds light on the ability of GRU in
estimating doubly-selective channels with high accuracy. As discussed in
Section IV-A, SRNN takes advantage of the previously estimated channel only
while estimating the current one. Whereas, the LSTM has long-term memory,
which means that estimating the channel at the current OFDM symbol is affected
by the older estimated channels. On the other hand, GRU provides a trade-off
between short-term memory and complexity. Therefore, in order to decide which
RNN performs better in doubly-selective channel estimation, we study the
average correlation between the channel at the first symbol and all successive
symbols within the transmitted OFDM frame, considering the frequency-time
response, such that

$\Psi_{i}=\mathrm{E}\left[\tilde{\bm{h}}_{1}\tilde{\bm{h}}^{*}_{\text{i}}\right],~{}2\leq
i\leq I.$ (30)

Here, $\Psi_{i}$ is calculated for three mobility scenarios: (i) Low mobility:
$f_{d}=250$ Hz, (ii) High mobility: $f_{d}=500$ Hz, and (iii) Very high
mobility: $f_{d}=1000$ Hz. The detailed properties of these channel models are
provided in Section V.

As shown in Fig. 2, when the mobility increases, the average correlation
$\Psi$ starts to decrease exponentially. However, as we can notice, $\Psi_{i}$
at the end of the received frame reaches around $65\%$ for low mobility
scenario, while it is around $40\%$ in high and very high mobility scenarios,
with a drastic decrease in the overall $\Psi_{i}$ curve in very high mobility
scenarios. According to the $\Psi_{i}$ values in different mobility scenarios,
we can expect that the impact of the estimated channels at earlier symbols
would affect negatively the accuracy of the estimated channel at advanced
symbols within the received OFDM frame. As a result, we can conclude that, as
the mobility increases, shorter RNN memory is required in the channel
estimation in order to guarantee the best possible performance. This is due to
the fact that, when long RNN memory is employed in a very high mobility
scenario, the older estimated channels negatively impact the channel
estimation at the current OFDM symbol because the estimated channels become
uncorrelated, i.e. the value of $\Psi$ is low. In this context, the proposed
RNN-based channel estimation scheme employs an optimized GRU unit instead of
LSTM unit in the channel estimation process due to its shorter memory. This
results in improving the accuracy of the channel estimation while recording a
significant decrease in computational complexity. Moreover, we study the
performance of the SRNN unit in order to have a complete analysis of different
RNN units.

As illustrated in Fig. 3, the RNN unit is first employed to estimate the
channel at the current data subcarriers, where it takes as an input the
previous LS estimated channels at pilot subcarriers denoted by
$\hat{\bar{\bm{h}}}_{i-1,p}\in\mathbb{R}^{2K_{p}\times 1}$, concatenated with
the previously RNN-based estimated channel at the data subcarriers
$\hat{\bar{\bm{h}}}_{{\Phi\text{-TA}}_{i-1,d}}\in\mathbb{R}^{2K_{d}\times 1}$.
Thus, the input and output sizes of the RNN unit are $2K_{on}$ and $2K_{d}$,
respectively. After that, the RNN output is fed as an input to the DPA
estimation followed by TA processing in order to further mitigate the impact
of noise. We note that our proposed estimators consider both time and
frequency selectivity, where the RNN-based pre-processing deals with the time
selectivity and the DPA estimation with the frequency selectivity. Moreover,
the RNN-based estimated channel is fed as an input to the DPA estimation
block, which further improves the DPA estimation accuracy. The proposed RNN-
based channel estimation scheme proceeds as follows

$\bar{\bm{d}}_{{\Phi}_{i,d}}[k]=\mathfrak{D}\big{(}\frac{\bar{\bm{y}}_{i,d}[k]}{\hat{\bar{\bm{h}}}_{{\Phi}_{i-1,d}}[k]}\big{)},~{}k\in{\mathcal{K}}_{\text{d}},$
(31)
$\hat{\bar{\bm{h}}}_{{\Phi\text{-DPA}}_{i,d}}[k]=\frac{\bar{\bm{y}}_{i,d}[k]}{\bar{\bm{d}}_{{\Phi}_{i,d}}[k]},$
(32)

where ${\Phi}\in\\{\text{SRNN},\text{GRU}\\}$ refers to the used RNN unit, and
$\hat{\bar{\bm{h}}}_{{\Phi}_{0,d}}=\hat{\bar{\bm{h}}}_{\text{LS}}~{}\forall~{}k\in{\mathcal{K}}_{\text{d}}$.
Finally, to alleviate the impact of the AWGN noise, TA processing is applied
to the estimated channel $\hat{\bar{\bm{h}}}_{\Phi\text{-DPA}_{i}}$ similarly
as performed in (10), such that

$\hat{\bar{\bm{h}}}_{\Phi\text{-TA}_{i,d}}=(1-\frac{1}{\alpha})\hat{\bar{\bm{h}}}_{\Phi\text{-TA}_{i-1,d}}+\frac{1}{\alpha}\hat{\bar{\bm{h}}}_{\Phi\text{-DPA}_{i,d}},$
(33)

TABLE I: Parameters of the proposed RNN-based channel estimation scheme. (SRNN units; Hidden size) | (1;48)
---|---
(GRU units; Hidden size) | (1;48)
(Bi-SRNN units; Hidden size) | (1;32)
(Bi-GRU; Hidden size) | (1;32)
Activation function | ReLU ($y=\max(0,x)$)
Number of epochs | 500
Training samples | 16000
Testing samples | 2000
Batch size | 128
Optimizer | ADAM
Loss function | MSE
Learning rate | 0.001
Training SNR | 40 dB

where $\alpha=2$ for simplicity. We note that in doubly-selective channel,
each two successive symbols are correlated regardless of the mobility
scenario, therefore, using $\alpha=2$ gives equal weights for the previous and
current estimated channel. However, However, alpha can be fine-tuned by
studying the average channel correlation between each two successive OFDM
symbols, and then assigning more accurate weights to the previous and current
estimated channels in (33).

In the proposed scheme, RNN training is performed using a high value of
signal-to-noise ratio (SNR) = 40 dB to achieve the best performance as
observed in [31]. The reason is that when the training is performed for low
noise impact, the RNN is able to better learn the channel correlation. In
addition, due to its good generalization ability, it can still perform well in
low SNR regions, where the noise is dominant. Moreover, intensive experiments
are performed using the grid search algorithm [32] in order to select the best
suitable RNN hyper-parameters in terms of both performance and complexity.
Note that the mobility conditions can be assumed known in most real case
applications. For example, in vehicular communications, the vehicle velocity
is a known parameter that can be exchanged between all vehicular network nodes
and it must be regulated according to the road conditions. In urban
environments (inside cities) the car velocity must not exceed 40 Kmphr, and
thus, the model trained on low mobility can be employed. Consequently, the RNN
training is performed for each mobility scenario separately using the same
architecture and training parameters summarized in Table I. However, when
velocity information is not available, EL algorithm can be used to combine the
weights of several trained models so that one generalized model can be
employed in all mobility scenarios as discussed in V.

###  IV-C PROPOSED FBF Bi-RNN-BASED CHANNEL ESTIMATOR

Bi-RNN networks are designed to predict unknown data that are bounded within
known data [24]. They are based on making the data flow through any RNN unit
in both directions forward (past to future), and backwards (future to past).
In regular RNN, the input flows in one direction, whereas, in Bi-RNN the input
flows in both directions to get the advantage of both past and future
information. By doing so, the Bi-RNN network will be able to predict the
unknown information in the middle based on its correlation with the known past
and future information.

Figure 4: Proposed Bi-RNN based channel estimator block diagram.

In this context, the proposed Bi-RNN channel estimator aims to utilize the
interpolation ability of Bi-RNN networks in the FBF channel estimation instead
of employing high-complexity CNN networks as it is the case in the SoA channel
estimation schemes. The proposed Bi-RNN channel estimation scheme uses Bi-GRU
unit and it inherits the adaptive frame design from the WI-CNN estimators as
shown in Fig. 4. Recall that WI-CNN channel estimation performs WI
interpolation at the data symbols, where CNN processing is applied to
alleviate the impact of noise. However, Bi-RNNs perform 2D interpolation at
the data symbols using the estimated channel at the pilot symbols without the
need for any initial channel estimation at the data symbols. Thus, the
proposed Bi-RNN channel estimator can be adapted to any existing protocols
regardless of the pilot allocation scheme. However, the employed Bi-RNN
architectures should be fine-tuned accordingly to meet the required
performance. The proposed Bi-RNN channel estimator proceeds as follows

  * •

ALS estimation at the inserted pilot symbols as performed in (13), followed by
zero insertion at all the data symbols. Thereafter, the initial estimated
channels $\hat{\tilde{\bm{H}}}_{{\rho}}\in\mathbb{C}^{K_{\text{on}}\times
I_{d}}$ are converted to the real-valued domain as performed in (17), where
$\hat{\bar{\bm{H}}}_{{in}}\in\mathbb{R}^{2K_{\text{on}}\times I_{d}}$.

  * •

Bi-RNN end-to-end interpolation, where $\hat{\bar{\bm{H}}}_{{in}}$ is fed as
an input to the optimized Bi-GRU unit. Accordingly, the Bi-GRU unit learns the
weights of the estimated channels at the OFDM data symbols. Employing the 2D
interpolation using the proposed Bi-GRU unit leads to a considerable
performance superiority in comparison with the WI-CNN estimators while
recording a significant decrease in the required computational complexity, as
shown in Section V. Also here the proposed Bi-GRU architecture is optimized
using the grid search algorithm [32] and trained using the parameters listed
in Table I. Moreover, similarly as performed in Section IV-B, the performance
of Bi-LSTM and Bi-SRNN are investigated in Section V.

##  V SIMULATION RESULTS

This section illustrates the performance evaluation of the SoA and the
proposed RNN and Bi-RNN based channel estimation schemes in terms of BER and
throughput. Vehicular communications are considered as a simulation case
study, where three mobility scenarios are defined as: (i) low mobility
($v=45~{}\text{Kmph},f_{d}=250$ Hz) (ii) High mobility
($v=100~{}\text{Kmph},f_{d}=500$ Hz) (iii) Very high mobility
($v=200~{}\text{Kmph},f_{d}=1000$ Hz). The power-delay profiles of the
employed channel models are provided in Table II. It is worth mentioning that
in order to guarantee fairness in the conducted simulations, the studied
channel estimators are trained using the same parameters shown in Table I.
Moreover, simulation parameters are based on the IEEE 802.11p standard [33],
where for the SBS channel estimation, comb-pilot allocation is employed such
that $K_{p}I$ pilots are used within the transmitted frame following the comb-
pilot allocation. Concerning the FBF channel estimation, the ChannelNet and
TS-ChannelNet estimators use $K_{p}I$ pilots per frame, whereas the WI-CNN and
the proposed Bi-RNN channel estimators employ only $K_{on}Q$ pilots per frame
following the block-pilot allocation, where $K_{on}=52$ denotes the number of
employed subcarriers within the transmitted OFDM symbol, and $Q$ is the number
of inserted pilot symbols within the transmitted frame ((i) Low mobility:
$Q=1$, (ii) High mobility: $Q=2$, (iii) Very high mobility: $Q=3$). Therefore,
the proposed Bi-RNN based channel estimator is able to outperform the recently
proposed SoA FBF channel estimators employing fewer pilots with lower
computational complexity, resulting in higher transmission data rates as
discussed in Section V-B. We also note that these simulations are implemented
using QPSK, 16QAM, and 64QAM modulation orders, the SNR range is
$[0,5,\dots,40]$ dB. In addition, the performance evaluation is performed
according to the employed modulation orders, the mobility scenarios, and
variable frame length.

TABLE II: Characteristics of the employed channel models following Jake’s Doppler spectrum. | Channel
---
model
| Channel
---
taps
| Vehicle velocity
---
[kmph]
| Doppler
---
shift [Hz]
Average path gains [dB] | Path delays [ns]
VTV-UC | 12 | 45 | 250 |  | [0, 0, -10, -10, -10, -17.8, -17.8,
---
-17.8, -21.1, -21.1, -26.3, -26.3]
| [0, 1, 100, 101, 102, 200, 201,
---
202, 300, 301, 400, 401]
VTV-SDWW | 12 | 100-200 | 500-1000 |  | [0, 0, -11.2, -11.2, -19, -21.9, -25.3,
---
-25.3, -24.4, -28, -26.1, -26.1]
| [0, 1, 100, 101, 200, 300, 400,
---
401, 500, 600, 700, 701]

(a)  BER performance employing QPSK.

(b)  BER performance employing 16QAM.

(c)  BER performance employing 64QAM.

Figure 5: BER for $I=100$, mobility from left to right: low, high, very high.

###  V-A SBS CHANNEL ESTIMATION

####  V-A1 Modulation Order

For QPSK modulation order, we can notice from Fig. 5a that FNN-based channel
estimators can implicitly learn the channel frequency correlation apart from
preventing a high demapping error arising from conventional DPA-based
estimation, where STA-FNN and TRFI-FNN outperform conventional STA and TRFI
estimators by at least $15$ dB gain in terms of SNR for BER $=10^{-3}$.
However, STA-FNN suffers from an error floor beginning from SNR $=20$ dB,
particularly in very high mobility scenarios. This is due to the STA frequency
and time averaging operations that can alleviate the impact of noise and
demapping error in low SNR regions. On the other hand, the averaging
operations are not useful in high SNR regions since the impact of noise is
low, and the STA averaging coefficients are fixed. Therefore, TRFI-FNN is used
to improve the performance at high SNRs to compensate for the STA-FNN
performance degradation in the high SNR regions. We can clearly observe that
employing RNNs as a pre-processing unit rather than a simple FNN in the
channel estimation brings a significant improvement in the overall
performance. This is because RNNs are capable of efficiently learning the time
correlations of the channel by taking the advantage of the previous output
apart from the current input in order to estimate the current output. Even
though the recently proposed LSTM-based estimators are able to outperform the
FNN-based estimator, but using LSTM in the channel estimation is not the best
option, due to LSTM long-term memory problem. In contrast, we can notice that
the proposed GRU-DPA-TA estimator is able to outperform the LSTM-DPA-TA
estimator by around $6$ dB gain in terms of SNR for BER $=10^{-5}$,
especially, in very high mobility scenario. This is due to the fact that LSTM
employs long-term memory, thus, the current estimated channel is affected by
older estimated ones. This process harms the performance as the mobility
increases, and the channel at successive received OFDM symbols becomes
uncorrelated. Whereas, the GRU uses shorter memory than LSTM, Thus, leading to
the superiority of the proposed GRU-DPA-TA estimator in comparison with the
LSTM-DPA-TA estimator. However, we can notice that in low mobility scenario,
both LSTM-DPA-TA and GRU-DPA-TA estimators achieve almost similar performance.
This is because of the negligible impact of Doppler interference in low
mobility scenario, thus, the channels at successive symbols within the
received OFDM frame are highly correlated. So, considering long or short
memory while estimating the current channel will not lead to considerable
performance degradation. Concerning 16QAM and 64QAM modulation orders, the
proposed GRU-DPA-TA estimator outperforms the LSTM-DPA-TA estimator by more
than $5$ dB and $7$ dB gains in terms of SNR for BER $=10^{-4}$ and BER
$=10^{-3}$, respectively, in very high mobility scenarios, as illustrated in
Fig. 5b and Fig. 5c. However, it can be noticed that FNN-based channel
estimators suffer from severe performance degradation when 64QAM modulation is
employed. This is because of the remarkable accumulated DPA demapping error
that cannot be eliminated by simple FNN architectures. A nice observation can
be noticed from Fig. 5 where employing SRNN in the channel estimation performs
similarly to the LSTM-FNN-DPA estimator in all mobility scenarios. This
reveals that using SRNN combined with TA processing records similar
performance as LSTM combined with FNN. In other words, the performance
degradation caused by the LSTM long-term memory is compensated by the FNN
network in the LSTM-FNN-DPA estimator. However, SRNN unit can be used instead
to eliminate the LSTM long-term memory problem as well as mitigating the noise
by simple TA processing as the case in the SRNN-DPA-TA estimator.

Fig. 6 illustrates the throughput of the studied SBS channel estimators
employing QPSK modulation. It can be seen that the proposed RNN-based channel
estimators perform higher throughput than conventional and FNN-based channel
estimators, especially in low SNR regions. This is due to the accurate channel
prediction.

Figure 6: Throughput employing QPSK, mobility from left to right: low, high,
very high.
Figure 7: BER employing very high mobility and QPSK, frame length from left to
right: I = 10, I = 50, I = 100.

(a) Robustness against Doppler frequencies, SNR = 40 dB.

(b) Robustness against mobility, testing scenario: High mobility.

Figure 8: Robustness analysis of the proposed GRU-DPA-TA channel estimator
employing QPSK modulation.

####  V-A2 Mobility

The impact of mobility can be observed in Fig. 5. The performance behavior is
influenced by the following factors: (i) channel estimation error, (ii) time
diversity due to increased Doppler spread, since the Doppler spread and the
time diversity gain are proportional, i.e. more time diversity gain can be
obtained in very high mobility scenarios, and (iii) frame length. As shown in
Fig. 5, where the frame length is fixed ($I=100$), the performance of all the
studied channel estimation schemes degrades with the increase of mobility.
This is because the channel estimation error increases with the increase of
Doppler frequency. Moreover, we can notice that the conventional STA and TRFI
channel estimators suffer from severe performance degradation in the very high
mobility scenario, since the impact of the AWGN noise and DPA demapping error
is much more dominant than the time diversity gain. On the contrary, the time
diversity gain is dominant in DL-based channel estimators, since DL networks
are capable of reducing the channel estimation error resulting from the AWGN
noise and the DPA demapping error, leading to a performance improvement in
very high mobility scenarios. Note that the net time diversity gain is also
related to the employed frame length, since increasing the frame length
increases the time diversity gain. This is clearly illustrated in Fig. 7,
where QPSK modulation order with very high mobility is utilized. As we can
notice, the performance of the proposed RNN-based channel estimators improves
when a longer frame length is employed. It is worth mentioning that, the
proposed GRU-DPA-TA and LSTM-DPA-TA estimators perform similarly when $I=10$,
since in shorter frames the impact of long and short-term memory cannot be
clearly illustrated. On the contrary, when using longer frames, i.e, $I=50$
and $I=100$, we can notice the superiority of using the GRU-based estimator
instead of the LSTM-based one.

In order to further illustrate the importance of channel tracking, Fig. 7
shows the performance of the proposed GRU-DPA-TA channel estimator when the
outdated estimated channel is used. In this context, the received OFDM symbols
are equalized by the DL-based estimated channel at the beginning of the frame.
As shown in Fig. 7, equalizing by the outdated estimated channel significantly
degrades the performance even when shorter frames are employed. Therefore,
this shows the importance of applying channel tracking to guarantee good
performance in different mobility scenarios.

Fig. 8 illustrates the robustness of the proposed GRU-DPA-TA channel estimator
against the change in Doppler frequency, where QPSK modulation is employed.
Fig. 8b shows the performance of the proposed channel estimator when trained
on one Doppler frequency and tested on the entire range of Doppler
frequencies. In this context, the entire range of Doppler frequencies is
divided into 3 ranges: (i) Low mobility (0 Hz - 300 Hz), (ii) High mobility
(300 - 600 Hz), and (iii) Very high mobility (600 Hz - 1000 Hz). As we can
notice, training on one Doppler frequency and testing on the same one gives
the best performance. However, training on the highest Doppler frequency
within each range shows a satisfactory performance when tested on different
Doppler frequencies within the considered range. This can be explained by the
fact that when the model is trained on the worst conditions, i.e., high
Doppler frequency, it can perform well when tested on better conditions, i.e.,
low Doppler frequency. We note that the SNR = 40 dB is considered to show the
impact of Doppler interference. Therefore, the performance of the trained
models can be generalized, where we can train 3 models only. In addition, we
can notice that training on lower Doppler frequencies, (for example, $f_{d}$ =
250 Hz) and testing on higher Doppler frequencies lead to a severe performance
degradation which is expected as shown in Fig. 8a, since the model is trained
in the absence of Doppler interference. It is worth mentioning that, further
model generalization can be achieved by using the concept of ensemble learning
(EL) in case the velocity range is not known, where the weights of several
trained models can be averaged in order to produce one generalized model. We
note that, in Fig. 8a, the EL results are obtained by averaging the weights of
the trained models on 700 Hz, 800 Hz, and 900 Hz. This combination can be
optimized and fine-tuned according to the requirements of real-time
applications.

(a)  BER employing QPSK modulation.

(b)  BER employing 16QAM modulation.

Figure 9: BER for $I=100$, mobility from left to right: low, high, very high.

###  V-B FBF CHANNEL ESTIMATION

In this section, performance evaluations of the CNN-based estimators,
conventional 2D LMMSE estimator as well as the proposed Bi-RNN based channel
estimator are discussed using the same criteria as Section V-A. We note that
we only consider the ALS-WI-CNN among the WI-CNN estimators since it has the
best performance.

####  V-B1 Modulation Order

Fig. 9a and Fig. 9b depict the BER performance employing QPSK and 16QAM
modulation orders, respectively. The performance of channel network
(ChannelNet) and temporal spectral ChannelNet (TS-ChannelNet) accounts of the
predefined fixed parameters in the applied interpolation scheme, where the RBF
interpolation function and the ADD-TT frequency and time averaging parameters
need to be updated in a real-time manner. On the contrary, in the ALS-WI-CNN
estimator there are no fixed parameters, and the time correlation between the
previous and the future pilot symbols is considered in the WI interpolation
operation. These aspects lead to the performance superiority of the ALS-WI-CNN
compared to the ChannelNet and TS-ChannelNet estimators. Although CNN
processing is applied in the ChannelNet, TS-ChannelNet, and ALS-WI-CNN
estimators, they suffer from a considerable performance degradation that is
dominant in very high mobility scenario. This show that the CNN processing is
not able to effectively alleviate the impact of Doppler interference,
especially in very high mobility scenarios, where the proposed Bi-RNN based
channel estimation scheme outperforms the WI-ALS-CNN estimator by at least $5$
dB and $12$ dB gain in terms of SNR for a BER = $10^{-5}$ employing QPSK and
16QAM modulations, respectively. We note that the robustness of the proposed
Bi-RNN based channel estimator against high mobility is mainly due to the
accuracy of the end-to-end 2D interpolation implemented by the utilized Bi-GRU
unit. Moreover, we can see that employing Bi-LSTM performs similarly to the
ALS-Bi-GRU estimator, this is due to the used frame structure, where the
variation of the doubly-selective channel within each sub-frame is low.
However, it can be noticed that employing CNN performs better than the Bi-
SRCNN unit in low and high mobility scenarios, while using Bi-SRCNN unit leads
to around $2$ dB gain in terms of SNR for a BER = $10^{-4}$ in comparison with
the ALS-WI-CNN estimator in very high mobility scenario as shown in Fig. 9b.
As a result, we can conclude that employing Bi-GRU unit instead of CNN network
leads to more accurate channel estimation with lower complexity. Finally, we
note that the performance of the 2D-LMMSE estimator is comparable to the
performance of the ideal channel but it requires huge complexity as we discuss
in the next section, which is impractical in a real scenario. Moreover, the
proposed Bi-RNN based estimator records almost close performance as the
2D-LMMSE estimator. Therefore, the proposed Bi-RNN based channel estimator is
an alternative to the 2D-LMMSE estimator where it provides a good performance-
complexity trade-off.

####  V-B2 Mobility

The impact of mobility can be clearly observed in Fig. 9b, where the
performance of the ChannelNet and TS-ChannelNet channel estimation schemes
degrades as the mobility increases, and the impact of the time diversity gain
is not dominant due to the high estimation error of the 2D RBF and ADD-TT
interpolation techniques employed in the ChannelNet and TS-ChannelNet
estimators, respectively. In contrast, the time diversity gain is dominant in
the ALS-WI-CNN and the proposed Bi-RNN based channel estimator, since the
initial ALS and WI estimations are accurate, thus, the SR-CNN and DN-CNN
networks are capable of overcoming the Doppler interference. However, using
the ALS estimation at the pilot symbols followed by Bi-GRU unit for 2D
interpolation at the data symbols reveal considerable robustness against
mobility. This is due to the ability of the optimized Bi-GRU unit to
significantly alleviating the impact of Doppler interference, where it can be
noticed that the proposed Bi-RNN estimator is able to outperform the ALS-WI-
CNN estimators in different mobility scenarios. As a result, the proposed Bi-
RNN based channel estimator provides a good performance-complexity trade-off
between the CNN-based estimators and 2D-LMMSE estimator.

$10^{4}$$10^{5}$LSTM-FNN-DPALSTM-DPA-TAGRU-DPA-TADPA-FNNSRNN-DPA-TATRFI-
FNNSTA-FNNReal-Valued
OperationsMultiplications/DivisionsSummations/Subtractions

Figure 10: Computational complexity of the studied DL-based SBS channel
estimators.

##  VI COMPUTATIONAL COMPLEXITY ANALYSIS

$10^{6}$$10^{7}$$10^{8}$$10^{9}$$10^{10}$$10^{11}$$10^{12}$$10^{13}$2D-LMMSEChannelNetTS-
ChannelNetALS-WI-DNCNNALS-WI-SRCNNALS-BiLSTMALS-BiGRUALS-BiSRNNReal-Valued
OperationsMultiplications/DivisionsSummations/Subtractions

Figure 11: Computational complexity of the studied DL-based FBF channel
estimators.

This section provides a detailed computational complexity analysis of the
studied channel estimation schemes. The computational complexity analysis is
performed in accordance with the number of real-valued
multiplications/divisions and summations/subtractions necessary to estimate
the channel for one received OFDM frame [34].

###  VI-A SBS CHANNEL ESTIMATION

The computational complexity of the SRNN lies in the calculation of
${\bar{\bm{h}}}_{t}$ in (18), where $PK_{in}+P^{2}$ multiplications and
$PK_{in}+P^{2}$ summations are required. The computation of ${{{\bm{o}}}}_{t}$
requires $P^{2}$ multiplications and $P^{2}$ summations. Therefore, SRNN
processing requires $PK_{in}+2P^{2}$ multiplications and $PK_{in}+2P^{2}$
summations. Similarly to SRNN, each LSTM gate requires $PK_{in}+P^{2}$
multiplications and $PK_{in}+P^{2}$ summations. Therefore, the total
computations performed by the LSTM unit can be expressed as $4PK_{in}+4P^{2}$
multiplications and $4PK_{in}+4P^{2}$ summations, in addition to $3P$
multiplications and $P$ summations required by (23) and (24). The GRU unit
employs fewer computations compared to the LSTM unit, where it requires
$3PK_{in}+3P^{2}+3P$ multiplications and $3PK_{in}+3P^{2}+2P$ summations. Fig.
10 shows the required multiplications/divisions and summations/subtractions
required by various examined SBS channel estimators. A detailed derivation of
the required operations is provided in [34].

The proposed optimized GRU unit is configured with $P=k_{d}$ and
$K_{in}=2K_{on}$. Therefore, it requires $6K_{on}K_{d}+3K^{2}_{d}+3K_{d}$
multiplications and $6K_{on}K_{d}+3K^{2}_{d}+2K_{d}$ summations. After that,
the proposed GRU-DPA-TA estimator applies the DPA estimation followed by the
TA processing that requires $18K_{on}$ real-valued multiplications/divisions
and $8K_{on}$ summations/subtractions. Hence, the GRU-DPA-TA estimator
requires $6K_{on}K_{d}+3K^{2}_{d}+3K_{d}+18K_{on}$ real-valued
multiplications/divisions and $6K_{on}K_{d}+3K^{2}_{d}+2K_{d}+8K_{on}$
summations/subtractions. Therefore, the proposed GRU-DPA-TA estimator is able
to decrease the required complexity by $48.22\%$ compared to the LSTM-DPA-TA
estimator. In other words, the proposed GRU-DPA-TA estimator is $2x$ less
complex than the LSTM-DPA-TA estimator, at the same time, it achieves a
significant performance gain as discussed in Section V. Employing SRNN instead
of GRU unit requires $K_{d}K_{in}+K^{2}_{d}+4K_{d}$ multiplications/divisions
and $K_{in}+4K_{d}-2$ summations/subtraction. Therefore it achieves $76.54\%$
and $54.69\%$ computational complexity decrease in the required real-valued
operations in comparison to the LSTM-DPA-TA and GRU-DPA-TA estimators,
respectively. Therefore, using SRNN is $4$x and $2$x less complex than using
LSTM and GRU units. However, the RNN-DPA-TA estimator performs similar to the
LSTM-FNN-DPA estimator as shown in Section V. Finally, we note that a trade-
off between the desired performance and the accepted complexity should be
taken into account, in order to optimize the use of the RNN-based channel
estimators.

###  VI-B FBF CHANNEL ESTIMATION

The computational complexity of any Bi-RNN unit is twice the required
complexity for the regular RNN unit since both forward and backward data flows
are applied. The proposed Bi-GRU estimator is optimized where $P=32$.
Moreover, we use $Q=4$, i.e. assuming very high mobility scenario, in order to
have a fair comparison with the ALS-WI-CNN estimator, and
$K_{in}=2K_{\text{on}}I^{\prime}$, where $I^{\prime}=I+Q$. The ALS channel
estimation at the inserted pilot symbols requires
$4K^{2}_{\text{on}}Q+2K_{\text{on}}Q+2K_{\text{on}}$ multiplications/divisions
and $5K^{2}_{\text{on}}Q$ summations/subtractions. Therefore, the overall
computational complexity of the proposed Bi-GRU channel estimation scheme can
be expressed by $16K^{2}_{\text{on}}+39946K_{\text{on}}+6336$
multiplication/divisions and $20K^{2}_{\text{on}}+39936K_{\text{on}}+6272$
summations/subtractions. We note that employing Bi-LSTM instead of the GRU
unit increases the computational complexity by around $26.29\%$ where
$16K^{2}_{\text{on}}+53258K_{\text{on}}+8384$ multiplications/divisions and
$20K^{2}_{\text{on}}+53248K_{\text{on}}+8256$ summations/subtractions are
needed without any gain in the BER performance as discussed in V. Moreover,
using Bi-SRNN requires $16K^{2}_{\text{on}}+13322K_{\text{on}}+4096$
multiplications/divisions and $20K^{2}_{\text{on}}+13312K_{\text{on}}+4096$
summations/subtractions. Therefore, the overall computational complexity is
decreased by $73.63\%$ and $64.22\%$ in comparison to the ALS-BiLSTM and ALS-
BiGRU estimators, respectively. However, Bi-SRNN unit suffers from limited
performance due to its simple architecture.

Fig. 11 illustrates the computational complexities of the studied CNN-based
FBF channel estimators. We can notice that the conventional 2D LMMSE estimator
records the highest computational complexity [35], making it impractical in
real-time scenarios. Moreover, the ChannelNet, TS-ChannelNet, and the WI-CNN
estimators did not provide a good complexity vs. performance trade-off. In
contrast, the complexity is significantly decreased by the proposed ALS-BiGRU
channel estimator which is $10$x and $115$x less complex than the ALS-WI-SRCNN
and the ALS-WI-DNCNN estimators, respectively. Moreover, the proposed ALS-
BiGRU channel estimator is $10^{6}$x less complex than the conventional 2D
LMMSE channel estimator. Therefore, we can conclude that employing the
proposed optimized Bi-GRU network instead of CNN networks in the channel
estimation is more feasible and at the same time it offers better performance.
Thus making it a good alternative to the 2D LMMSE channel estimator.

##  VII CONCLUSION AND FUTURE PERSPECTIVES

In this paper, RNN-based channel estimation in doubly-selective environments
has been investigated. The recently proposed DL-based SBS and FBF channel
estimators have been presented and their limitations have been discussed. In
order to overcome these limitations, we have proposed optimized RNN-based and
Bi-RNN estimators for SBS and FBF channel estimation respectively. Moreover,
the performance of several RNNs architectures including SRNN, LSTM, and GRU
has been thoroughly analyzed based on the channel correlation within the
received frame. Moreover, we show that the proposed GRU and Bi-GRU units
result in a better performance-complexity trade-off in different mobility
scenarios. Simulation results have shown the performance superiority of the
proposed channel estimators over the recently proposed DL-based SBS and FBF
estimators while recording a significant reduction in computational
complexity. As a future perspective, we will investigate the ability to extend
the proposed RNN and Bi-RNN based channel estimators for MIMO and mmWave
communications taking into consideration the constraints of each scenario. In
order to further improve the online performance of the proposed channel
estimators and their generalization capabilities, advanced DL algorithms such
as transfer and meta-learning can be investigated. In addition, more advanced
architectures such as the transforms can be tested for channel estimation,
considering the trade-off analysis between complexity and performance.
Moreover, working on interpretable and explainable theoretical DL models is a
crucial future step that would ensure the reliability and transparency of
employing DL networks in the domain of wireless communications, especially,
channel estimation, where the intuitions behind our proposed work can be
further validated.

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  *[OFDM]: orthogonal frequency division multiplexing
  *[CP]: cyclic prefix
  *[SBS]: symbol-by-symbol
  *[FBF]: frame-by-frame
  *[LSTM]: long short-term memory
  *[CNN]: convolutional neural network
  *[RNN]: recurrent neural network
  *[GRU]: gated recurrent unit
  *[SoA]: state-of-the-art
  *[DPA]: data-pilot aided
  *[STA]: spectral temporal averaging
  *[TRFI]: time-domain reliable test frequency domain interpolation
  *[RBF]: radial basis function
  *[ADD-TT]: average decision-directed with time truncation
  *[LMMSE]: linear minimum mean square error
  *[DL]: deep learning
  *[FNN]: feed-forward neural network
  *[CNNs]: convolutional neural network
  *[Bi]: bi-directional
  *[TA]: time averaging
  *[BER]: bit error rate
  *[EL]: ensemble learning
  *[AWGN]: additive white Gaussian noise
  *[FNNs]: feed-forward neural network
  *[LS]: least squares
  *[WI]: weighted interpolation
  *[SLS]: simple LS
  *[ALS]: accurate LS
  *[DFT]: discrete Fourier transform
  *[MSE]: mean squared error
  *[SR-CNN]: super resolution CNN
  *[DN-CNN]: denoising CNN
  *[ANN]: artificial neural network
  *[RNNs]: recurrent neural network
  *[SRNN]: Simple RNN
  *[SNR]: signal-to-noise ratio
  *[ChannelNet]: channel network
  *[TS-ChannelNet]: temporal spectral ChannelNet