2G

    Second Generation
3G

    3${}^{\text{rd}}$ Generation
3GPP

    3${}^{\text{rd}}$ Generation Partnership Project
4G

    4${}^{\text{th}}$ Generation
5G

    5${}^{\text{th}}$ Generation
AA

    Antenna Array
AC

    Admission Control
AD

    Attack-Decay
ADSL

    Asymmetric Digital Subscriber Line
AHW

    Alternate Hop-and-Wait
AMC

    Adaptive Modulation and Coding
AP

    Access Point
ANN

    artificial neural network
APA

    Adaptive Power Allocation
AR

    autoregressive
ARMA

    autoregressive moving average
ARIMA

    auto regressive integrated moving average
ATES

    Adaptive Throughput-based Efficiency-Satisfaction Trade-Off
AWGN

    additive white Gaussian noise
BPNN

    back propagation neural network
BB

    Branch and Bound
BD

    Block Diagonalization
BER

    bit error rate
BF

    Best Fit
BLER

    BLock Error Rate
BPC

    Binary power control
BPSK

    binary phase-shift keying
BPA

    Best pilot-to-data power ratio (PDPR) Algorithm
BRA

    Balanced Random Allocation
BS

    base station
CAP

    Combinatorial Allocation Problem
CAPEX

    Capital Expenditure
CBF

    Coordinated Beamforming
CBR

    Constant Bit Rate
CBS

    Class Based Scheduling
CC

    Congestion Control
CDF

    Cumulative Distribution Function
CDMA

    Code-Division Multiple Access
CL

    Closed Loop
CLPC

    Closed Loop Power Control
CNN

    convolutional neural network
CNR

    Channel-to-Noise Ratio
CPA

    Cellular Protection Algorithm
CPICH

    Common Pilot Channel
CoMP

    Coordinated Multi-Point
CQI

    Channel Quality Indicator
CRM

    Constrained Rate Maximization
CRN

    Cognitive Radio Network
CS

    Coordinated Scheduling
CSI

    channel state information
CSIR

    channel state information at the receiver
CSIT

    channel state information at the transmitter
CUE

    cellular user equipment
D2D

    device-to-device
DCA

    Dynamic Channel Allocation
DE

    Differential Evolution
DFT

    Discrete Fourier Transform
DIST

    Distance
DL

    downlink
DMA

    Double Moving Average
DMRS

    Demodulation Reference Signal
D2DM

    D2D Mode
DMS

    D2D Mode Selection
D-MIMO

    distributed multiple input multiple output
DPC

    Dirty Paper Coding
DRA

    Dynamic Resource Assignment
DSA

    Dynamic Spectrum Access
DSM

    Delay-based Satisfaction Maximization
ECC

    Electronic Communications Committee
EFLC

    Error Feedback Based Load Control
EI

    Efficiency Indicator
ELM

    extreme learning machine
eNB

    Evolved Node B
EPA

    Equal Power Allocation
EPC

    Evolved Packet Core
EPS

    Evolved Packet System
E-UTRAN

    Evolved Universal Terrestrial Radio Access Network
ES

    Exhaustive Search
FDD

    frequency division duplexing
FDM

    Frequency Division Multiplexing
FER

    Frame Erasure Rate
FF

    Fast Fading
FSB

    Fixed Switched Beamforming
FST

    Fixed SNR Target
FTP

    File Transfer Protocol
GA

    Genetic Algorithm
GBR

    Guaranteed Bit Rate
GLR

    Gain to Leakage Ratio
GOS

    Generated Orthogonal Sequence
GPL

    GNU General Public License
GRP

    Grouping
GRU

    gated recurrent unit
HARQ

    Hybrid Automatic Repeat Request
HMS

    Harmonic Mode Selection
HOL

    Head Of Line
HSDPA

    High-Speed Downlink Packet Access
HSPA

    High Speed Packet Access
HTTP

    HyperText Transfer Protocol
ICMP

    Internet Control Message Protocol
ICI

    Intercell Interference
ID

    Identification
IETF

    Internet Engineering Task Force
ILP

    Integer Linear Program
JRAPAP

    Joint RB Assignment and Power Allocation Problem
UID

    Unique Identification
IID

    Independent and Identically Distributed
IIR

    Infinite Impulse Response
ILP

    Integer Linear Problem
IMT

    International Mobile Telecommunications
INV

    Inverted Norm-based Grouping
IoT

    Internet of Things
IP

    Internet Protocol
IPv6

    Internet Protocol Version 6
ISD

    Inter-Site Distance
ISI

    Inter Symbol Interference
ITU

    International Telecommunication Union
JOAS

    Joint Opportunistic Assignment and Scheduling
JOS

    Joint Opportunistic Scheduling
JOELM

    jointly optimized extreme learning machine
JP

    Joint Processing
JS

    Jump-Stay
KF

    Kalman filter
KKT

    Karush-Kuhn-Tucker
L3

    Layer-3
LAC

    Link Admission Control
LA

    Link Adaptation
LC

    Load Control
LOS

    Line of Sight
LP

    Linear Programming
LS

    least squares
LSTM

    long short-term memory
LTE

    Long Term Evolution
LTE-A

    LTE-Advanced
LTE-Advanced

    Long Term Evolution Advanced
M2M

    Machine-to-Machine
MAC

    Medium Access Control
MANET

    Mobile Ad hoc Network
MC

    Modular Clock
MCS

    Modulation and Coding Scheme
MDB

    Measured Delay Based
MDI

    Minimum D2D Interference
MF

    Matched Filter
MG

    Maximum Gain
MH

    Multi-Hop
mMIMO

    massive multiple input multiple output
MIMO

    multiple input multiple output
MINLP

    Mixed Integer Nonlinear Programming
MIP

    Mixed Integer Programming
MISO

    Multiple Input Single Output
ML

    machine learning
MLP

    multilayer perceptron
MLWDF

    Modified Largest Weighted Delay First
MME

    Mobility Management Entity
MMSE

    minimum mean squared error
MOS

    Mean Opinion Score
MPF

    Multicarrier Proportional Fair
MRA

    Maximum Rate Allocation
MR

    Maximum Rate
MRC

    maximum ratio combining
MRT

    Maximum Ratio Transmission
MRUS

    Maximum Rate with User Satisfaction
MS

    mobile station
MSE

    mean squared error
MSI

    Multi-Stream Interference
MTC

    Machine-Type Communication
MTSI

    Multimedia Telephony Services over IMS
MTSM

    Modified Throughput-based Satisfaction Maximization
MU-MIMO

    multiuser multiple input multiple output
MU

    multi-user
NARX

    nonlinear autoregressive network with exogenous inputs
NAS

    Non-Access Stratum
NB

    Node B
NE

    Nash equilibrium
NCL

    Neighbor Cell List
NLP

    Nonlinear Programming
NLOS

    Non-Line of Sight
NMSE

    Normalized Mean Square Error
NN

    neural network
NORM

    Normalized Projection-based Grouping
NP

    Non-Polynomial Time
NRT

    Non-Real Time
NSPS

    National Security and Public Safety Services
O2I

    Outdoor to Indoor
OFDMA

    orthogonal frequency division multiple access
OFDM

    orthogonal frequency division multiplexing
OFPC

    Open Loop with Fractional Path Loss Compensation
O2I

    Outdoor-to-Indoor
OL

    Open Loop
OLPC

    Open-Loop Power Control
OL-PC

    Open-Loop Power Control
OPEX

    Operational Expenditure
ORB

    Orthogonal Random Beamforming
JO-PF

    Joint Opportunistic Proportional Fair
OSI

    Open Systems Interconnection
PAIR

    D2D Pair Gain-based Grouping
PAPR

    Peak-to-Average Power Ratio
P2P

    Peer-to-Peer
PC

    Power Control
PCI

    Physical Cell ID
PDF

    Probability Density Function
PDPR

    pilot-to-data power ratio
PER

    Packet Error Rate
PF

    Proportional Fair
P-GW

    Packet Data Network Gateway
PL

    Pathloss
PPR

    pilot power ratio
PRB

    physical resource block
PROJ

    Projection-based Grouping
ProSe

    Proximity Services
PS

    Packet Scheduling
PSAM

    pilot symbol assisted modulation
PSO

    Particle Swarm Optimization
PZF

    Projected Zero-Forcing
QAM

    Quadrature Amplitude Modulation
QoS

    Quality of Service
QPSK

    Quadri-Phase Shift Keying
RAISES

    Reallocation-based Assignment for Improved Spectral Efficiency and Satisfaction
RAN

    Radio Access Network
RA

    Resource Allocation
RAT

    Radio Access Technology
RATE

    Rate-based
RB

    resource block
RBG

    Resource Block Group
REF

    Reference Grouping
ReLU

    rectified linear unit
RLC

    Radio Link Control
RM

    Rate Maximization
RNC

    Radio Network Controller
RND

    Random Grouping
RNN

    recurrent neural network
RRA

    Radio Resource Allocation
RRM

    Radio Resource Management
RSCP

    Received Signal Code Power
RSRP

    Reference Signal Receive Power
RSRQ

    Reference Signal Receive Quality
RR

    Round Robin
RRC

    Radio Resource Control
RSSI

    Received Signal Strength Indicator
RT

    Real Time
RU

    Resource Unit
RUNE

    RUdimentary Network Emulator
RV

    Random Variable
SAC

    Session Admission Control
SCM

    Spatial Channel Model
SC-FDMA

    Single Carrier - Frequency Division Multiple Access
SD

    Soft Dropping
S-D

    Source-Destination
SDPC

    Soft Dropping Power Control
SDMA

    Space-Division Multiple Access
SE

    spectral efficiency
SER

    Symbol Error Rate
SES

    Simple Exponential Smoothing
S-GW

    Serving Gateway
SINR

    signal-to-interference-plus-noise ratio
SI

    Satisfaction Indicator
SIP

    Session Initiation Protocol
SISO

    single input single output
SIMO

    Single Input Multiple Output
SIR

    signal-to-interference ratio
SLNR

    Signal-to-Leakage-plus-Noise Ratio
SMA

    Simple Moving Average
SNR

    signal-to-noise ratio
SORA

    Satisfaction Oriented Resource Allocation
SORA-NRT

    Satisfaction-Oriented Resource Allocation for Non-Real Time Services
SORA-RT

    Satisfaction-Oriented Resource Allocation for Real Time Services
SPF

    Single-Carrier Proportional Fair
SRA

    Sequential Removal Algorithm
SRS

    Sounding Reference Signal
SU-MIMO

    single-user multiple input multiple output
SU

    Single-User
SVD

    Singular Value Decomposition
SVM

    support vector machine
SVR

    support vector machine for regression
TCP

    Transmission Control Protocol
TDD

    time division duplexing
TDMA

    Time Division Multiple Access
TDL

    tapped delay line
TETRA

    Terrestrial Trunked Radio
TP

    Transmit Power
TPC

    Transmit Power Control
TTI

    Transmission Time Interval
TTR

    Time-To-Rendezvous
TSM

    Throughput-based Satisfaction Maximization
TU

    Typical Urban
UE

    user equipment
UEPS

    Urgency and Efficiency-based Packet Scheduling
UL

    uplink
UMTS

    Universal Mobile Telecommunications System
URI

    Uniform Resource Identifier
URM

    Unconstrained Rate Maximization
UT

    user terminal
V2V

    vehicle-to-vehicle
V2X

    vehicle-to-everything
VR

    Virtual Resource
VoIP

    Voice over IP
WAN

    Wireless Access Network
WCDMA

    Wideband Code Division Multiple Access
WF

    Water-filling
WiMAX

    Worldwide Interoperability for Microwave Access
WINNER

    Wireless World Initiative New Radio
WLAN

    Wireless Local Area Network
WMPF

    Weighted Multicarrier Proportional Fair
WPF

    Weighted Proportional Fair
WSN

    Wireless Sensor Network
WWW

    World Wide Web
XIXO

    (Single or Multiple) Input (Single or Multiple) Output
ZF

    zero-forcing
ZMCSCG

    Zero Mean Circularly Symmetric Complex Gaussian

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# A Comparison of Neural Networks for
Wireless Channel Prediction
††thanks: The authors are with KTH Royal Institute of Technology.
Oscar Stenhammar and Gabor Fodor are also with Ericsson AB.

Oscar Stenhammar, Gabor Fodor and Carlo Fischione

###### Abstract

The performance of modern wireless communications systems depends critically
on the quality of the available channel state information (CSI) at the
transmitter and receiver. Several previous works have proposed concepts and
algorithms that help maintain high quality CSI even in the presence of high
mobility and channel aging, such as temporal prediction schemes that employ
neural networks. However, it is still unclear which neural network-based
scheme provides the best performance in terms of prediction quality, training
complexity and practical feasibility. To investigate such a question, this
paper first provides an overview of state-of-the-art neural networks
applicable to channel prediction and compares their performance in terms of
prediction quality. Next, a new comparative analysis is proposed for four
promising neural networks with different prediction horizons. The well-known
tapped delay channel model recommended by the Third Generation Partnership
Program is used for a standardized comparison among the neural networks. Based
on this comparative evaluation, the advantages and disadvantages of each
neural network are discussed and guidelines for selecting the best-suited
neural network in channel prediction applications are given.

###### Index Terms:

6G mobile communication, channel aging, channel prediction, channel state
information, deep learning, machine learning.

## Introduction

As the sixth generation (6G) of wireless communication technologies and
services emerges, higher expectations for mobile broadband services are set by
end users. Concepts of the evolving 5G and emerging 6G networks, such as
distributed multiple input multiple output systems rely critically on the
availability of up-to-date channel state information (CSI). However, obtaining
accurate CSI is non-trivial since the channel evolves over time as the
scattering environment and the position of the user equipment (UE) change. The
evolution of the channel is often referred to as channel aging and poses a
major challenge in the design of modern wireless systems. To meet this
challenge, channel prediction has emerged as a key tool to combat channel
aging [1, 2]. Most commonly, channel prediction is incorporated by exploiting
time series of past channel estimations. Using predicted CSI, it is possible
to improve the performance of wireless communication, even in the presence of
high mobility and rapidly changing channels. Accurately updated CSI allows an
adaptive transmitter to proactively tune the communication parameters, such as
the transmit power, constellation size, and coding rate to enhance the network
performance.

A general and widely used method to characterize the evolution of the wireless
channel is autoregressive (AR) models [1]. This approach is model-based
because it relies on analytical models of the dynamical evolution of the
channel. In the modeled-based approach, the wireless channel is modeled as a
linear combination of the previous realizations of the channel with some
additive process noise. Conventionally, model-based methods such as Kalman and
Wiener filtering have been used for channel prediction. Assuming that the
channel evolves according to an AR model with Gaussian noise and the second-
order statistics of the AR process are known or can be acquired, Kalman
filtering is optimal in a mean squared error (MSE) sense [3]. However, with
increasing bandwidth and number of antennas, the complexity of Kalman filters
grows relatively fast. Furthermore, the computational complexity is
proportional to the square of the amount of previous channel data used in the
model [1]. To ensure satisfactory performance, it is often necessary to use
higher-order models that employ many parameters. Thus, the inherent trade-off
between the model order and the associated computational complexity often
limits the performance of traditional model-based methods [4].

A recently proposed method to overcome channel aging is to use predictor
antennas mounted on vehicles [2]. The predictor antenna is specifically
designed for vehicles moving at high speed and is typically placed on the
exterior of the vehicle in front of the main antenna. In this way, the
predictor antenna can estimate the channel from the position that the main
antenna will reach soon. For vehicles moving at high speed, it is a valuable
suggestion. However, mounting predictor antennas on legacy vehicles may not be
viable in practice. Arguably, a more economical and viable solution is to find
a prediction scheme with satisfactory performance and to update the software
instead of installing hardware on existing connected vehicles.

In light of the above considerations, the increasing popularity and
improvements of neural networks over the last years appear as a viable
approach to wireless channel prediction. Specifically, by implementing neural
networks according to the so-called data-driven approach, no underlying model
needs to be assumed, as opposed to the model-based approach. This makes the
predicting model less sensitive to disturbances and interference since it can
learn from realistic data. In the case of predicting future channels based on
solely the previous channel estimations, the channel prediction problem
becomes a time series learning problem. Indeed, in the past years, channel
prediction has been studied extensively and numerous techniques have been
considered with the use of neural networks. For instance, channel prediction
can also be conducted using the location of the UE, which is appropriate for a
static scenario.

However, in a real-world crowded urban environment, the spatial correlation of
the channel can be very small or completely absent, due to moving objects.
Also, by depending on the location of the UE, an algorithm may become
computationally more complex. By relying solely on historical time series data
and the temporal correlation of the channel, channel prediction algorithms
become computationally more efficient and scalable among different
environments. For this reason, the present article aims to overview the most
prominent neural networks methods and to identify research gaps in channel
prediction that strictly uses historical channel measurements as input data.
The most promising neural networks for channel prediction, which has performed
well in previous studies, are compared using a dataset, with and without
noise, simulated by the common and realistic 3${}^{\text{rd}}$ Generation
Partnership Project (3GPP) tapped delay line (TDL)-A model [5]. The
performance of predicting fast-fading channels is studied over a large span of
prediction horizons.

The advantages and disadvantages of each method are discussed to ultimately
identify the most promising neural network for wireless channel prediction.
This paper is, to the authors best knowledge, the first comparison of channel
prediction methods that represent multiple different classes of neural
networks. We provide a deeper understanding of the state-of-the-art in channel
prediction to direct future research toward optimal models for real-world
implementations.

To summarize, the contributions of this paper are the following:

  * •

An overview of previous works, focusing on channel prediction that employs
data-driven machine learning (ML) methods.

  * •

An original quantitative comparison of the most promising data-driven methods
identified from previous works. The data-driven methods are also compared to
Kalman filtering.

  * •

A discussion on how to develop the state-of-the-art in channel prediction,
based on numerical evaluation arguments.

The rest of the sections in this paper are organized as follows: the
representative state-of-the-art in channel prediction using ML is overviewed;
the prediction schemes that will be compared are described and justified; the
proposed prediction schemes are numerically evaluated and compared by their
performance; the outcome of the experiments are discussed; and finally, the
results from the study we proposed in this paper are concluded.

## Overview of previous works

Model-based methods have been widely used to perform channel prediction.
However, recent advances in ML have accelerated several research areas, and
recent studies have suggested that ML has the potential to outperform
conventional channel prediction model-based methods. The ML model is a
function that maps input data to an output decision or prediction, defined by
its trainable parameters and its architecture. Training a supervised ML model
means tuning the parameters to output a satisfactory output, usually by
solving an optimization problem that minimizes a loss function. In our case,
the ML model uses historical time series of channel measurements as input and
outputs the future channel.

The performance of the model-based Kalman filter has been compared to a
multilayer perceptron (MLP) in [4]. The MLP is a basic neural network that
consists of several layers of nodes, where each node in one layer connects by
a trainable parameter to every node in the following layer. To replicate the
training process as with real channel data, [4] used noisy simulated data to
train the MLP. All other papers surveyed in our work that use simulated data
assume perfect knowledge of the channel when training the neural network. The
comparison of the Kalman filter and the MLP method, with a small advantage to
the Kalman filter, suggests a need for more advanced neural networks.

Several works have implemented more advanced structures for enhanced
prediction accuracy. A popular model in image recognition is the convolutional
neural network (CNN), which in contrast to MLP can take a matrix as input
instead of a vector. It can learn to recognize patterns in smaller sections
from an input matrix. By constructing a matrix of the size given by the time
steps and the number of antennas, a CNN is proposed in [6] to predict AR
coefficients for channel evolution. Channel prediction has also been performed
using a recurrent neural network (RNN) that utilizes the temporal correlation
in sequential data, in contrast to the CNN. A subset of frequency subcarriers
was predicted individually by an RNN in [7], followed by performing
interpolation to predict the entire frequency domain used by the antenna.
Other works have combined CNNs and RNNs to predict the channel. Both [8, 9]
have combined a CNN with a long short-term memory (LSTM) model, which is a
type of RNN. A well-known issue with RNN is that it has training convergence
issues due to vanishing or diverging gradients. LSTM alleviates these
problems. A comparison between LSTM to conventional model-based methods has
been proposed in [10], where the effect of moving at different velocities were
studied for channel prediction. In a recent paper [11], the authors modeled
the channel between a UE and base station (BS) via a reconfigurable
intelligent surface as a fast-fading channel using the LSTM, assuming
stationarity between the BS and the reconfigurable intelligent surface.

TABLE I: Contributions of the surveyed papers. Y means yes, N means no, S means simulated, M means measured. Model | Performance |  Prediction horizon |  Noisy label | Mobility |  Data generation |  Prediction procedure | Paper
---|---|---|---|---|---|---|---
LSTM |  Superior to ARIMA and SVR | 0.1-1ms | N |  Medium & High | S | Time series | [10]
Transformers | Superior to LSTM |  0.625- 3.125ms | N | Medium | S | Time series | [12]
LSTM & GRU | Superior to RNN | 1-5ms | N | High | S | Time series | [13]
RNN | Inferior to KF | 1.28ms | N |  Medium & High | S |  Time series | [7]
MLP | Inferior to KF | 40ms | Y | Low | S | Time series | [4]
LSTM & GRU | Superior to ARIMA | 1-10s | N |  Low & High | S |  Encoder- decoder | [14]
CNN/RNN | Superior to KF | - | N |  Low & Medium | S | Time series | [6]
LSTM/CNN | Superior to MLP | - | Y | None | M | Time series | [8]
LSTM/CNN |  Inferior to LSTM and CNN | - | N | Medium | S |  UL-DL & subcarriers | [9]
LSTM |  Superior to Minimum Variance Unbiased | - | N | Low | S | Time series | [11]
C-GRBFnet |  Superior to LSTM | - | N | None | S |  Spatial prediction | [15]

Aside from LSTM, gated recurrent unit (GRU) has also been proposed to improve
the sequential RNN, and is more computationally efficient than LSTM. In
channel prediction, the GRU has been tested empirically by several
researchers. The authors in [14] compared the LSTM and GRU with a proposed
prediction model exploiting an encoder-decoder scheme, with LSTM or GRU layers
at both the encoder and decoder side. Several datasets, one including 4G
measurements, revealed a slight advantage to the LSTM. In [13], an overview of
channel prediction has been made where the LSTM and GRU have been studied over
several prediction horizons. The first evaluation of a deep GRU has been
conducted, in favor of the GRU.

One additional type of neural network has been recently proposed in [12],
which adopts the transformer model to predict the channel. The transformer has
the ability to predict multiple future time steps in parallel, by learning to
identify and pay attention to critical behavior in sequential data. Another
transformer-based model has been proposed in [15] to predict the channel
impulse response, based on the location of the UE. It does not use historical
channel measurements as input like the previously discussed papers but shows
good results compared to the LSTM.

In Table I, contributions from all papers are categorized in columns and
summarized, with the topics of interest in this paper. As can be seen in the
column Prediction procedure, the channel is predicted using time series for
all papers in Table I, except for [15]. Although there are some papers using
slightly different prediction procedures, time series are the foundation to
make the prediction procedure more efficient.

One conclusion from Table I is that the majority of the papers evaluate the
channel prediction models based on simulated data. This is understandable
since it is less costly and less time-consuming to collect simulated data.
However, in a real-world implementation of channel prediction, measured data
have to be used to conduct the predictions. The channel measurement and
estimation process is unavoidably affected by noise. For this reason, noise
was introduced in the training process in [4], including the true data that is
used to update the model. The column Noisy label indicates whether the paper
considers a noisy label for training the model. If the data is generated by
measurements, the label is automatically noisy. There is only one paper that
considers noisy labels while using a simulator to generate the dataset to
train the prediction model. In our paper, we investigate its role and we show
that it can have a major effect when evaluating the performance of the
prediction methods.

The prediction horizons considered in Table I are almost exclusively
correlated with mobility. If a paper considers high mobility of the UE, the
horizon is short, and vice versa, due to difficulties of predicting the
channel over long horizons with a fluctuating channel. If the prediction
horizon exceeds the coherence time, the channel’s temporal correlation
vanishes, and it becomes impossible to infer the channel out of current or
past measurements. The prediction horizons of the papers listed in Table I are
generally short. Half of the papers do not state on what time horizon the
channel is predicted. Furthermore, no paper has included a prediction horizon
long enough for the performance to fail.

From the summary of previous works in Table I, research gaps can be found.
First, it is not obvious which neural network is the most suitable for channel
prediction. Second, although different data-driven models may each have good
results, they have not been compared to each other. In most of the existing
literature, data-driven models are compared to conventional model-based
methods. The overviewed papers generally do not perform comparisons among
data-driven models, or at best do partial comparisons. For example, [13]
compares LSTM to a deep GRU. The present paper is arguably the first to make a
comprehensive comparison among the most prominent data-driven approaches.

## Channel prediction using neural networks

To identify the most promising neural network algorithm for the purpose of
channel prediction, the most prominent algorithms found in the previous
section are further analyzed and compared. Throughout the rest of this paper,
five different regular types of neural networks are compared.

Figure 1: Distribution of simulated complex-valued channel. All samples in
every subchannel and subcarrier are included in this distribution.

(a)

(b)

Figure 2: The dynamics of the simulated channel and the predicted channel are
visualized in this figure. Apart from the true channel in both sub-figures,
one of the figures includes the channel distorted by additive Gaussian noise
with SNR of 20 dB, and the other includes the predicted channel. The duration
of this particular sample is 53 ms.

A common type of neural network is the feedforward neural network. It consists
of a set of layers, each layer with multiple neurons, and basically
constitutes the conceptual framework of all neural networks. All layers
between the input- and output layers are called hidden layers. A feedforward
neural network that has one or more hidden layers of neurons is called a MLP.
The MLP has shown convincing results in many areas and is a very general
framework compatible with many applications, since the input to a MLP must be
a one-dimensional vector. A vector that can be constructed from any type of
data. However, the ordering of the elements in the input vector is
disregarded. The result is that the potential importance of the position of
the elements in the input vector is neglected.

One type of neural network that is built to preserve spatial data is the CNN.
The idea is to identify patterns in the input matrix, making the position of
each element in the input matrix relevant compared to the MLP. CNNs have
acquired state-of-the-art status for their ability to detect patterns within
image recognition. By using previous channel samples from multiple antennas,
one can construct a matrix as a type of image as input to the CNN. The
advantages of the CNN could be utilized to find patterns across time and among
antennas.

A class of neural networks that is suitable for time series is the RNN since
it stores information from prior inputs in its internal state to influence the
current output. That makes the RNN able to benefit from sequential data better
than the MLP and CNN. However, the classical RNN has major problems with
vanishing or exploding gradients. To solve these problems, modified networks
have been suggested. One type of RNN, that has reached state-of-the-art
results in fields like speech recognition and language processing, is LSTM.
Every LSTM cell is built by three gates, the input, output, and forget gate.
As its name reveals, it has one long and one short-term memory. Since LSTM was
proposed, new algorithms with small modifications have been created. One of
those, with promising results, is GRU. The GRU has two gates and one memory,
making it computationally faster.

Another model that takes advantage of sequential data is the transfer model, a
neural network architecture that is among the recently proposed prediction
model [12]. It is most commonly used in natural language processing tasks but
can also be applicable in time-series regression. It employs a self-attention
mechanism to capture relationships among historical dynamics in a sequence.
Multiple attention heads are used to capture different dependencies and
relationships in parallel. The transformer incorporates positional encoding to
convey the timeliness of each number. With its ability to capture long-range
dependencies and parallel processing, the transformer has significant
performance in time series prediction.

To evaluate and compare the performance of the promising neural networks for
channel prediction, we consider a downlink MIMO scenario with $N_{t}$ antennas
at the BS and $N_{r}$ antennas at the UE. We model the received signal as
$\mathbf{y}[t]=\mathbf{H}[t]\mathbf{x}[t]+\mathbf{n}[t]$, where
$\mathbf{x}[t]$ is the transmitted signal, $\mathbf{y}[t]$ is the received
signal, $\mathbf{n}[t]$ is the additive noise and
$\mathbf{H}[t]\in\mathbb{R}^{N_{t}\times N_{r}}$ represents the channel. For
various adaptive wireless technologies, outdated CSI can cause heavy
performance degradation. To obtain up-to-date CSI, channel prediction is
performed. To predict the future channel $\mathbf{\hat{H}}[t+p]$ on a desired
prediction horizon $p$, we use $n$ historical measurements of the channel,
indicated as $[\mathbf{H}[t-k\cdot n],...,\mathbf{H}[t-k],\mathbf{H}[t]]$,
where $k$ determines the time interval between each sample. Since the channel
is complex-valued, real and imaginary values are separated in the input
channel matrix $\mathbf{H}[t]$.

## Experimental evaluation

To evaluate the neural networks discussed in the previous section, fast-fading
channel data is simulated using the standardized 3GPP TDL-A model [5]. The
TDL-A model is based on Rayleigh fading in a non-line-of-sight scenario and is
useful when simulating the channel for cellular systems. The BS and the UE are
assumed to have 2 antennas each, communicating at 2 GHz. The mobility of the
UE is set to 20 km/h, which gives a maximum Doppler shift of approximately 37
Hz. With 52 resource blocks, the number of subcarriers is 624. To be
consistent with the overviewed literature, only one subcarrier is considered
at a time in the input and output of the prediction. This result is a dataset
of around 26 milion data points. The distribution of the original dataset in
the complex plane is plotted in Fig. 1, showing the zero mean circularly
symmetric complex Gaussian distribution of the channel. The histogram includes
the samples from all MIMO channels and all subcarriers. The distribution
symmetry and smoothness are results of the massive dataset. From this original
dataset, the training dataset was randomly sampled to obtain 90000 samples,
and the test dataset 10000 samples.

Experiments are conducted for two versions of the dataset, one using the
original dataset and the other with the presence of noise. To imitate a
realistic scenario and represent the uncertainties from channel estimation,
the channel is distorted with Gaussian noise yielding an SNR of 20 dB. The
noise is present in the inputs and outputs of the training data. In this way,
the model is trained with realistic noisy channel data, which to the authors
knowledge, has not been studied earlier when using CNNs, LSTMs or GRUs. In the
test data however, only the inputs are distorted to evaluate the predictor
correctly. To reconstruct the predicted channel, the output vector from the
neural networks is reshaped into a vector of complex-valued channels for each
time instance. In Fig. 2(a), a small sample of the noisy-, true-, and
predicted channels are plotted.

(a)

(b)

Figure 3: The performance, measured in the MSE, evaluated for all considered
models and compared to each other, the Kalman Filter and the outdated channel.

With the Adam optimizer, the aim is to minimize the MSE over 200 epochs. The
number of historical channel measurements used to predict the future channel
is $n=5$ with a sampling time of 1 ms throughout the paper. The neural
networks are trained to optimize the accuracy for each prediction horizon. The
MLP is set to have 6 layers. For the CNN, the number of convolutional layers
is 4, each followed by a pooling layer, ending with 2 linear layers with
rectified linear unit (ReLU) activation functions just as for the MLP. The
RNNs has 3 stacked layers. The hyperparameters of the models are tuned
experimentally to obtain the best performance. For further insights of the
models, the project can be found at
Github111https://github.com/osst3224/Channel_prediction_DNN.git. The number of
hidden states is set to $150$ throughout this paper. The activation function
used is the $tanh()$, as suggested in [13]. Complexity analyses for the
different models can be found in [4, 12, 13]. An empirical evaluation of the
size of all models used in this paper is summarized in Table II. The neural
networks are trained for prediction horizons in the range of $1-40$ ms, with
the purpose of investigating longer prediction horizons, which has been
marginally done in previous works.

TABLE II: Empirical evaluation of the complexity for each prediction model. Model |  Trainable parameters |  Elapsed time per prediction
---|---|---
MLP | 1 990 402 | $27\mu s$
CNN | 4 338 | $33\mu s$
LSTM | 455 102 | $34\mu s$
GRU | 341 402 | $28\mu s$
Transformer | 463 874 | 97$\mu s$

The naive predictor, which assumes that the future channel is equal to the
outdated (current) channel is used as a benchmark. To further evaluate the
performance of neural networks in channel prediction, Kalman filtering is
implemented as well. For longer prediction horizons of several sampling times
ahead, the channel prediction of 1 ms ahead is used as historical measurements
for the next prediction. This procedure was repeated until the desired
prediction horizon is reached, just as in [4]. To fairly compare the
performance of the Kalman filter and the neural networks, the MSE of the
Kalman filter was calculated after it had reached convergence.

The MSE for the test dataset is plotted in decibels as a function of the
prediction horizon in Fig. 3(a). The performance of the RNNs are similar, with
a small advantage to the GRU. The reason they are performing very similarly is
that their architecture is closely related. With a little higher MSE, the MLP,
CNN, and transformer also have quite similar performance. The MSE grows at a
steady pace as the prediction horizon increases. As the prediction horizon
grows, the channel’s temporal correlation weakens, which naturally makes it
more difficult to predict the channel. When the prediction horizon is around
15 ms, the performance of the neural networks reaches a level of error that
stays relatively constant with a higher prediction horizon. This indicates
that the coherence time of the channel is approximately 15 ms. The rate at
which the channel ages is determined by the UE mobility in this channel model.
With a lower UE mobility in the simulation, the performance of the neural
networks would be better for longer prediction horizons.

The behavior is repeated for the performance of the neural networks trained
with the noisy dataset, visualized in Fig. 3(b). The performance of the neural
networks is kept somewhat constant for prediction horizons longer than 15 ms.
With shorter horizons, the performance of all neural networks is in the same
relative order to each other in the noisy and noise-free case. The MSE of the
test data is substantially higher when noise is introduced in the training and
test dataset. However, when noise is introduced and the prediction horizon is
short, there is no significant difference in performance between the RNNs,
MLP, CNN, and transformer.

The Kalman filter performs very well in a noise-free environment on short
prediction horizons. When the prediction horizon considered is over 5 ms, LSTM
and GRU perform better than the Kalman filter. In a noisy environment, the
Kalman filter behaves as the neural networks. But from Fig. 3, it is apparent
that the RNNs outperforms the Kalman filter over long prediction horizons.

## Discussion

From Fig. 3(a), it is evident that the recurring memory cell gives a strong
advantage in wireless channel time series prediction and constitutes a robust
performance for the GRU respectively LSTM compared to the MLP, CNN, and
transformer model, regardless of the prediction horizon. From these results,
it is concluded that GRU is the state-of-the-art in channel prediction. The
intuitive explanation for this is the GRU’s innate ability to find
correlations in sequential data. The GRU is custom-made to predict
sequentially temporal data. It has fewer parameters than the similar LSTM
network, which makes the GRU inclined to learn better and faster. However, in
form of practical feasibility in real base stations, due to constraints in
computational power and energy consumption, it might be better to consider the
MLP out of the neural networks due to its low complexity, especially during
training. The computational time and power spent to run the prediction model
is crucial for real-world implementations. Future research could further
extend the identified methods for real-world implementations such as
quantization, continual learning and one-shot learning.

For the scenario with the distorted channel, the difference in error between
the GRU and the MLP is constant around 1dB. The MLPs lower computational
complexity might make it more suitable in a real-world implementation where
computational power is limited due to time and energy constraints. As shown in
[4], the MLP has lower computational complexity than the Kalman filter.
Another fact that makes neural networks more suitable in a real-world
implementation is that the channel might not always follow a smooth pattern as
in Fig. 2. Since the MLP is data-driven, it can learn easier than the Kalman
filter in such a case.

This paper has considered the scenario of non-line-of-sight communication, for
the standardized 3GPP TDL-A channel model. The simulated dataset consists of
26 million data points and is statistically sufficient to cover the scenario
in TDL-A. Therefore, the results presented in this paper are general due to
the generality of the TDL-A model. It would be beneficial if future research
could investigate neural networks’ robustness in wireless channel prediction,
in the presence of abrupt changes in the communication environment due to the
appearance or disappearance of dominant paths. In the case of appearing or
disappearing dominant paths, the neural networks trained in this paper would
only need five consecutive samples after the abrupt change, to reinitialize
satisfactory predictions. The reason is that the input to the neural networks
is of five samples. Moreover, line-of-sight communication is more static and
fades slower than non-line-of-sight communication. Hence, neural network
models trained in this paper will perform well in scenarios containing slower
variations than those exhibited by the training dataset as well. The Kalman
filter, on the other hand, requires more than five samples to converge to
acceptable results, as seen in [4].

## Conclusions

This paper overviewed the most prominent research results from the literature
on machine learning for channel prediction. The main advantage of machine
learning is that it does not assume any underlying model, which makes it
flexible and able to learn a model from the data itself. From simulations of
the non-line-of-sight scenario of the 3GPP standardized TDL-A model, the
neural networks were trained and validated. In the scenario of a noise-free
channel, the numerical experiments of this paper established that two RNNs,
namely GRU and LSTM, achieved considerably better results for prediction
horizons up to 15 ms than the MLP, CNN, and transformer model. However, the
Kalman filter performs better than all neural networks up to the prediction
horizon of 4 ms. In the case of channel measurements with noisy data, the
difference in performance between the neural networks was not as significant.
However, from ordering the neural networks by their performance, the order was
the same in the case with and without noise. The Kalman filter performed
similarly as the MLP, CNN, and transformer model in the noisy case.

Ultimately, this overview suggests that the GRU is most suitable to perform
channel prediction and has the potential to be considered the most promising
for future real-world implementations. For future work, we plan to perform
channel prediction using more than one carrier. Also, data-efficient machine
learning models will have to be considered in future research if the
predictions have to be made in resource-constrained wireless devices.

## Acknowledgment

This work was partially supported by the Wallenberg AI, Autonomous Systems and
Software Program (WASP), funded by the Knut and Alice Wallenberg Foundation.
The work of C. Fischione was also funded by Digital Futures KTH research
center and SSF.

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## Biographies

Oscar Stenhammar [M] (oscar.stenhammar@ericsson .com) is currently pursuing
his Ph.D. at KTH Royal Institute of Technology as an industrial doctoral
student, employed by Ericsson AB and affiliated with the Wallenberg AI,
Autonomous Systems and Software Program (WASP). He received an M.S. in
Engineering Physics at Uppsala University in 2021.

Gabor Fodor [SM]<EMAIL_ADDRESS>received a Ph.D. in electrical
engineering from the Budapest University of Technology and Economics in 1998
and received a D.Sc. from the Hungarian Academy of Sciences in 2019. He is
currently a Master Researcher with Ericsson Research and an Adjunct Professor
with KTH Royal Institute of Technology, Stockholm, Sweden. He is currently
serving as an Editor for IEEE Transactions on Wireless Communications and IEEE
Wireless Communications.

Dr. Carlo Fischione [SM]<EMAIL_ADDRESS>is full Professor at KTH Royal
Institute of Technology, Network and Systems Engineering, Stockholm, Sweden.
He received a Ph.D. in Electrical and Information Engineering in 2005 from
University of L’Aquila, Italy, and has held research positions at
Massachusetts Institute of Technology, Cambridge, MA (2015); Harvard
University, Cambridge, MA (2015); and University of California at Berkeley, CA
(2004-2005 and 2007-2008). His research interests include applied
optimization, wireless Internet of Things, and machine learning. He received
the “2018 IEEE Communication Society S. O. Rice” award and the 2007 best paper
award of IEEE Transactions on Industrial Informatics.

  *[PDPR]: pilot-to-data power ratio
  *[CSI]: channel state information
  *[UE]: user equipment
  *[AR]: autoregressive
  *[MSE]: mean squared error
  *[3GPP]: 3rd Generation Partnership Project
  *[TDL]: tapped delay line
  *[ML]: machine learning
  *[MLP]: multilayer perceptron
  *[CNN]: convolutional neural network
  *[RNN]: recurrent neural network
  *[CNNs]: convolutional neural network
  *[RNNs]: recurrent neural network
  *[LSTM]: long short-term memory
  *[BS]: base station
  *[GRU]: gated recurrent unit
  *[SNR]: signal-to-noise ratio
  *[LSTMs]: long short-term memory
  *[GRUs]: gated recurrent unit
  *[ReLU]: rectified linear unit
  *[MLPs]: multilayer perceptron