# Predicting Future CSI Feedback For Highly-Mobile Massive MIMO Systems

Yu Zhang, Ahmed Alkhateeb, Pranav Madadi, Jeongho Jeon, Joonyoung Cho, Charlie
Zhang Yu Zhang and Ahmed Alkhateeb are with Arizona State University (Email:
y.zhang, alkhateeb@asu.edu), Pranav Madadi, Jeongho Jeon, Joonyoung Cho and
Charlie Zhang are with Samsung Research America.

###### Abstract

Massive multiple-input multiple-output (MIMO) system is promising in providing
unprecedentedly high data rate. To achieve its full potential, the transceiver
needs complete channel state information (CSI) to perform transmit/receive
precoding/combining. This requirement, however, is challenging in the
practical systems due to the unavoidable processing and feedback delays, which
oftentimes degrades the performance to a great extent, especially in the high
mobility scenarios. In this paper, we develop a deep learning based channel
prediction framework that proactively predicts the downlink channel state
information based on the past observed channel sequence. In its core, the
model adopts a 3-D convolutional neural network (CNN) based architecture to
efficiently learn the temporal, spatial and frequency correlations of downlink
channel samples, based on which accurate channel prediction can be performed.
Simulation results highlight the potential of the developed learning model in
extracting information and predicting future downlink channels directly from
the observed past channel sequence, which significantly improves the
performance compared to the sample-and-hold approach, and mitigates the impact
of the dynamic communication environment.

##  I Introduction

Massive multiple-input multiple-output (MIMO) technique plays a central role
in 5G and beyond for its capability of providing unprecedentedly high data
rate by leveraging the degree of freedom in the spatial domain [1]. However,
achieving such potential requires the acquisition of accurate channel state
information (CSI), which is challenging in most of the practical systems. This
is especially the case in a frequency division duplex (FDD) system where the
channel estimated in the uplink (UL) cannot be used for downlink (DL)
transmission, since channel reciprocity does not hold due to the different
carrier frequencies. In addition to the huge training overhead associated with
the massive MIMO system, another noticeable phenomenon that significantly
impacts the system performance is called _channel aging_ [2]. To put it in
simple words, channel aging models the channel variation from the time point
when it is learned to the actual time when it is used. This captures the
inherent time-varying channel characteristic and the processing delay in a
practical system. To mitigate such degradation caused by channel aging, future
channel prediction or channel tracking is required, where the overall
objective is to proactively predict what could be the actual channel state at
the time it is being used, based on available observed channels. This turns
out to be possible as wireless channel is essentially a function of its
surrounding environment and hence the channel samples are correlated over time
[3]. Therefore, in this paper, we investigate the problem of predicting future
channels by exploiting their temporal correlations with the past observed
channel samples for a downlink massive MIMO system.

Contribution: In this paper, we develop a novel 3-D convolutional neural
network (CNN) based deep learning framework for predicting the future CSI in a
massive MIMO-OFDM system. Given the huge dimension of the channel matrices in
such system, we leverage the 3-D kernels of the convolutional layer to explore
the correlations across frequency and spatial dimensions, while the temporal
correlations between channel samples are captured by using the additional “
_channel_ ” dimension of a CNN. Besides, the proposed learning model is
enhanced with a residual architecture for extracting deeper yet crucial
features that are hardly learned by shallow networks while keeping the overall
training experience efficient. The generalization capability of the model is
improved by training the network on a mixed-speed channel dataset, where the
channel sequences are collected under different mobilities. The extensive
simulation evaluations on a 3GPP channel model highlight the capability of the
proposed solution in accurately predicting future channels, which brings
significant improvement over sample-and-hold approach.

Prior work: The effects of channel aging in massive MIMO systems were first
studied in [2]. The authors of [4] proposed a time-varying channel tracking
method for massive MIMO systems based on spatial-temporal basis expansion
model. [5] utilized the delay and angular reciprocity between the uplink and
downlink frequency bands to reduce the amount of unknown parameters to be
estimated in the downlink. There are also some machine learning based channel
estimation solutions. For example, [6] introduced a neural network based
scheme for extrapolating downlink CSI from observed uplink CSI. [7] first used
neural networks to identify the channel aging pattern, then adopted an
autoregressor to forecast CSI. Our proposed solution is novel in that we
utilize a pure 3-D CNN based deep learning framework with residual
architecture to directly predict the future channel in the frequency and
spatial domain, based on the observed past channels.

Figure 1: The considered system model where a multi-antenna base station is
communicating with a multi-antenna user equipment. To perform DL data
transmission, the DL CSI should be first estimated and sent back to the base
station by the user equipment. However, due to the processing delay, the
channel varies by the time base station transmits the data.

##  II System Model

We consider a frequency division duplex (FDD) downlink single-cell system
where a base station (BS) with $N_{t}$ antennas is communicating with a single
user equipment (UE) with $N_{r}$ antennas. We assume an OFDM based system with
a total number of $K$ sub-bands. Since channel reciprocity does not hold in a
FDD system, the BS needs to rely on the channel state information (CSI)
feedback from UE to determine its downlink data transmission scheme.

Downlink Training and Uplink Feedback: In the downlink training stage, the BS
sends downlink pilots to the UE, based on which the downlink channel is
estimated. In this paper, we assume that the UE is able to estimate the
downlink channel perfectly, although channel estimation is always challenging
in the practical applications. We assume that the channel matrix at the $k$-th
sub-band is $\mathbf{H}_{k}\in\mathbb{C}^{N_{r}\times N_{t}}$. The UE will
then feed back the downlink CSI to the BS. However, as can be seen, directly
sending back $\mathbf{H}_{k}$ for all the sub-bands will incur huge feedback
overhead. Therefore, to reduce the feedback overhead, a common practice is
that the UE first calculates the desired beamforming vectors based on the
estimated downlink channel, and then feeds back those beamforming vectors to
the BS. To be more specific, the downlink beamforming vector at the $k$-th
sub-band can be obtained by using the right singular vector111Here, without
loss of generality, we assume a single layer transmission, which in practice,
is informed by rank indicator also fed back by UE. of $\mathbf{H}_{k}$, i.e.,
$\mathbf{f}_{k}=\left[\mathbf{V}_{k}\right]_{:,1}$, where
$\mathbf{H}_{k}=\mathbf{U}_{k}\boldsymbol{\Lambda}_{k}\mathbf{V}_{k}^{H}$ is
the singular value decomposition (SVD) of $\mathbf{H}_{k}$. Finally, the set
of beamforming vectors
$\mathcal{F}=\left\\{\mathbf{f}_{1},\mathbf{f}_{2},\dots,\mathbf{f}_{K}\right\\}$
will be sent to the BS.

It is worth mentioning that, in practice, such feedback overhead oftentimes is
still too expensive to afford by the system. Therefore, in the current 3GPP
release, those beamforming vectors will further be approximated by using Type
I/II codebooks [8] (which is essentially the discrete Fourier transform (DFT)
vector basis), and those coefficients will finally be sent out, as shown in
Fig. 1. As this is not the focus of this paper, we assume that the beamforming
vectors designed by the UE can be perfectly re-constructed at the BS and used
for subsequent downlink data transmission.

Downlink Data Transmission: Based on the uplink feedback, the BS then
transmits downlink data using the beamforming vectors provided by the UE.
Accordingly, the UE will use the optimal combining vectors for reception,
i.e., the left singular vector of the estimated $\mathbf{H}_{k}$, that is,
$\mathbf{w}_{k}=\left[\mathbf{U}_{k}\right]_{:,1}$. Therefore, the sum rate
across all the sub-bands is given by

$R=\sum_{k=1}^{K}\log_{2}\left(1+\frac{P_{x}}{\sigma_{n}^{2}}\left|\mathbf{w}_{k}^{H}\mathbf{H}_{k}\mathbf{f}_{k}\right|^{2}\right),$
(1)

where $P_{x}$ is the average transmit power, $\sigma_{n}^{2}$ is the noise
power, and $\mathbf{H}_{k}$ is assumed to incorporate the large-scale fading
factor.

##  III Problem Formulation

As can be seen from the above described system model, the _ideally_ achieved
system performance (1) relies on multiple factors. One of the most important
assumption is that the beamforming vector at the BS and the combining vector
at the UE indeed correspond to the channel when the BS transmits downlink
data. In practice, however, there is always a delay between the time point
when the channel is estimated and the time when actually downlink transmission
happens. In other words, by the time the BS starts transmitting data with the
beamforming vector fed back by the UE, the channel already changes. Moreover,
such delay is always in the level of milliseconds. This could be quite
problematic in high mobility scenarios where the channel changes very fast,
which degrades the system performance to a great extent. Therefore, proactive
channel prediction is necessary to mitigate such impact.

In this paper, we investigate the channel prediction problem in a downlink FDD
massive MIMO system. The objective is to maximize the throughput of future
downlink transmission, based on the knowledge of past and current channels.
Formally, the problem can be cast as

$\displaystyle\max_{\left\\{\widehat{\mathbf{f}}_{k},\widehat{\mathbf{w}}_{k}\right\\}_{k=1}^{K}}$
$\displaystyle\hskip
6.0pt\sum_{k=1}^{K}\log_{2}\left(1+\frac{P_{x}}{\sigma_{n}^{2}}\left|\widehat{\mathbf{w}}_{k}^{H}\mathbf{H}_{k}^{(t+1)}\widehat{\mathbf{f}}_{k}\right|^{2}\right),$
(2) $\displaystyle\operatorname{s.t.}~{}$
$\displaystyle\|\widehat{\mathbf{f}}_{k}\|_{2}\leq
1,~{}\|\widehat{\mathbf{w}}_{k}\|_{2}\leq 1,\forall k$ (3)
$\displaystyle\hskip 2.0pt\mathbf{H}_{k}^{(t^{\prime})}~{}\text{until time
$t^{\prime}=t$ are available},$ (4)

where $\widehat{\mathbf{f}}_{k}$ and $\widehat{\mathbf{w}}_{k}$ are the $k$-th
sub-band beamforming and combining vectors, $\mathbf{H}_{k}^{(t+1)}$ is the
downlink channel matrix at the $k$-th sub-band and time $t+1$. Therefore, the
problem is essentially to design $\widehat{\mathbf{f}}_{k}$ and
$\widehat{\mathbf{w}}_{k}$ for the transmission at time $t+1$ based on the
observed channels up to time $t$.

It is worth pointing out that there are multiple ways to proactively design
such transmission/reception schemes. In this paper, we follow the method of
first predicting the channel at time $t+1$, based on which transmission and
reception schemes can be derived based on SVD. Therefore, the core of the
problem becomes proactively predicting the channel at the transmission time,
i.e., time $t+1$, based on the previous and current estimated channels, i.e.,
up to time $t$. To put it in formal form, we want to learn a channel predictor
that takes in the past channel observations and outputs the future channel,
that is

$\widehat{\mathbb{H}}^{(t+1)}=f\left(\mathbb{H}^{(t)},\mathbb{H}^{(t-1)},\dots,\mathbb{H}^{(t-(L-1))};\boldsymbol{\Theta}\right),$
(5)

where $f$ is the desired channel predictor parameterized by
$\boldsymbol{\Theta}$ and $L$ is the number of past channel observations used
for prediction. We use $\mathbb{H}^{(t)}\in\mathbb{C}^{K\times N_{r}\times
N_{t}}$, for simplicity, to denote the 3-D downlink channel matrix in the
frequency and spatial domain at time $t$. Therefore, we have

$\left[\mathbb{H}^{(t)}\right]_{k,:,:}=\mathbf{H}_{k}^{(t)}.$ (6)

Based on the predicted channel $\widehat{\mathbb{H}}^{(t+1)}$, the beamforming
and combining vectors
$\widehat{\mathbf{f}}_{k},\widehat{\mathbf{w}}_{k},\forall k$ can be
correspondingly constructed by using SVD. Hence, the problem becomes finding
$\boldsymbol{\Theta}$ that yields accurate channel prediction, by using the
past $L$ channel observations. As the problem essentially features learning a
function, we resort to the powerful neural networks to be the channel
predictor.

Figure 2: The proposed 3-D convolutional neural network based deep learning
model for future channel prediction. The schematic shows an example of $L=3$.

##  IV Proposed Machine Learning Solution

In this section, we describe the detailed architecture of the proposed deep
learning model which is based on convolutional neural networks, and discuss
the loss function used for training the network.

###  IV-A Network Architecture

The building block of the proposed deep learning model is a 3-D convolutional
layer, where the 3-D kernel is utilized to extract the features across the
frequency and spatial dimensions (transmit and receive antennas). Different
features are then added up together to form _one of_ the input feature maps of
the next layer, where the temporal correlation can be captured. As can be seen
in Fig. 2, there are two different blocks building on top of the 3-D
convolutional layer. The first one is called convolutional block, which
basically includes extra normalization layer and activation layer. Another one
named convolutional residual block is further built on top of the
convolutional block, where a residual architecture is adopted. The introducing
of residual architecture is mainly used for extracting deeper features that
are hardly found by shallow networks while keeping the training experience
efficient. Finally, a fully-connected (FC) block is employed to reshape the
output to have the desired dimension. The detailed network architecture can be
found at Table I, where $B$ stands for the batch size, and the input dimension
of the FC block, $X$, can be calculated as follow

$X=2\times\left\lfloor\frac{N_{r}}{1}\right\rfloor\times\left\lfloor\frac{N_{t}}{2}\right\rfloor\times\left\lfloor\frac{K}{4}\right\rfloor,$
(7)

where $\lfloor\cdot\rfloor$ is the floor function. It is worth mentioning that
the main purpose of introducing the second max-pooling layer is to reduce the
number of parameters in the last FC layer, as normally, $N_{t}$ and $K$ are
relatively large.

TABLE I: Parameters of the 3-D CNN residual network Module | Parameter | Value
---|---|---
Conv Block 1 | Dimension of input & output |  $(B,2L,N_{r},N_{t},K)$ & $(B,4L,N_{r},N_{t},K)$
Kernel size & Padding & Stride |  $(3,7,5)$ & $(1,3,2)$ & (1, 1, 1)
MaxPool 1 | Kernel size & Padding & Stride |  $(3,3,3)$ & $(1,1,1)$ & (1, 1, 1)
Conv Res Block | Conv Block a | Dimension of input & output |  $(B,4L,N_{r},N_{t},K)$ & $(B,8L,N_{r},N_{t},K)$
Kernel size & Padding & Stride |  $(3,7,5)$ & $(1,3,2)$ & (1, 1, 1)
Conv Block b | Dimension of input & output |  $(B,8L,N_{r},N_{t},K)$ & $(B,16L,N_{r},N_{t},K)$
Kernel size & Padding & Stride |  $(3,7,5)$ & $(1,3,2)$ & (1, 1, 1)
Conv3D | Dimension of input & output |  $(B,16L,N_{r},N_{t},K)$ & $(B,4L,N_{r},N_{t},K)$
Kernel size & Padding & Stride |  $(3,7,5)$ & $(1,3,2)$ & (1, 1, 1)
Conv Block 2 | Dimension of input & output |  $(B,4L,N_{r},N_{t},K)$ & $(B,2,N_{r},N_{t},K)$
Kernel size & Padding & Stride |  $(3,7,7)$ & $(1,3,3)$ & (1, 1, 1)
MaxPool 2 | Kernel size & Padding & Stride |  $(1,2,4)$ & $(0,0,0)$ & $(1,2,4)$
FC Block | Dimension of input & output |  $(B,X)$ & $(B,2\times N_{r}\times N_{t}\times K)$

###  IV-B Training Loss Function

As the target of the neural network based channel predictor
$f_{\boldsymbol{\Theta}}$ is to predict the future channels, we pose our
learning problem as a regression problem conducted in a supervised learning
fashion. Furthermore, we employ mean squared error (MSE) as the training loss
function. For each predicted channel sample, the loss function is defined as

$\displaystyle\mathcal{L}$
$\displaystyle\left(\mathbb{H}^{(t+1)},\widehat{\mathbb{H}}^{(t+1)}\right)=\left\|\mathbb{H}^{(t+1)}-\widehat{\mathbb{H}}^{(t+1)}\right\|_{F}^{2},$
(8) $\displaystyle=$
$\displaystyle\left\|\mathbb{H}^{(t+1)}-f\left(\\{\mathbb{H}^{(q)}\\}_{q=t-(L-1)}^{t};\boldsymbol{\Theta}\right)\right\|_{F}^{2},$
(9)

where $\mathbb{H}^{(t+1)}$ is the ground true future channel,
$\widehat{\mathbb{H}}^{(t+1)}$ is the prediction of the network, and
$\|\cdot\|_{F}$ stands for the Frobenius norm of a matrix, which can be
straightforwardly generalized to a 3-D matrix. For the considered 3-D channel
matrix of shape $K\times N_{r}\times N_{t}$, we have

$\|\mathbb{H}\|_{F}^{2}=\sum_{k=1}^{K}\|\mathbf{H}_{k}\|_{F}^{2}.$ (10)

The objective of learning is to find a set of model parameters that minimizes
the loss averaged over the whole dataset, which can be expressed as

$\min_{\boldsymbol{\Theta}}\frac{1}{N}\sum_{n=1}^{N}\mathcal{L}\left(\mathbb{H}_{n},\widehat{\mathbb{H}}_{n}\right),$
(11)

where $N$ is the size of the dataset, the details of which will be given in
the next section, and $\boldsymbol{\Theta}$ is the model parameters to be
optimized.

##  V Experimental Setup

In this section, we describe the adopted channel model, dataset and the
training parameters used in our simulations.

###  V-A Channel Model and Dataset

In our simulations, we consider a massive MIMO-OFDM system where the base
station has $N_{t}=32$ antennas, the user equipment has $N_{r}=4$ antennas,
and there are $K=52$ resource blocks. The channel coefficients are generated
by a system level simulator from Samsung Research America, which follows 3GPP
Urban Micro (UMi) channel model with a central frequency of 2.1 GHz and
channel bandwidth 20 MHz. Both line-of-sight (LOS) and non-line-of-sight
(NLOS) scenarios are considered. We assume a periodic CSI-RS configuration
with a periodicity of 5 ms. The mobility of the UE depends on the specific
dataset.

Uni-Speed Dataset: Based on the above parameters, for each user, we generate a
long channel sequence with a total length of $Q$ channel samples under a
certain speed. It is worth noting that to ensure an efficient learning
process, the generated channel sequence is first normalized. This is found to
be very useful in the previous work [9]. In this work, we adopt the average
power normalization method. To be more specific, we first calculate the
average power of each _channel element_ as follow

$P=\frac{1}{Q\cdot
KN_{r}N_{t}}\sum_{n=1}^{Q}\left\|\mathbb{H}_{n}\right\|_{F}^{2}.$ (12)

And then, the obtained average channel element power will be used to normalize
all the channel samples

$\mathbb{H}_{n}\doteq\frac{1}{\sqrt{P}}\mathbb{H}_{n},~{}\forall
n=1,2,\dots,Q.$ (13)

Such normalized channel sequence is then truncated into segments with length
$L+1$. Therefore, we have a total number of $Q-L$ short channel sequence
segments, which forms our dataset, that is

$\mathcal{D}=\left\\{\left(\mathbb{H}^{t-(L-1)},\dots,\mathbb{H}^{(t-1)},\mathbb{H}^{(t)};\mathbb{H}^{(t+1)}\right)_{t=L}^{Q-1}\right\\},$
(14)

where, for each data sample, the first $L$ channel matrices are input of the
network and the last channel matrix is used as desired output, i.e., training
target.

Mixed-Speed Dataset: Furthermore, instead of assuming that the UE has a
constant speed, which is rarely the case in practice, we also consider a more
realistic scenario where the UE changes its speed over time. To this end, we
generate another kind of dataset that is under a range of UE speed, e.g., 30
km/h to 50 km/h. To be more specific, we first generate multiple channel
sequences under different UE speeds. Then, the channel sequence generated
under each UE speed will be truncated into short segments as before. It is
worth noting that those channel sequences will first go through a similar
normalization pre-processing step as performed in the uni-speed dataset before
being truncated. Finally, all those channel segments (under different UE
speeds) are _mixed_ together to form the ultimate mixed-speed dataset. As can
be seen, the mixed-speed dataset contains more dynamics than a uni-speed
dataset, which helps improve the generalization capability of the channel
prediction model.

###  V-B Model Training and Testing

With the network architecture and datasets described above, the 3-D CNN based
channel predictor is then trained and tested over training and testing
datasets respectively. The training is performed using PyTorch with an NVIDIA
Quadro RTX 6000 GPU. The detailed hyper-parameters of training are summarized
in Table II.

TABLE II: Hyper-parameters for model training Parameter | value
---|---
Batch size | 512
Number of epochs | 300
Optimizer | Adam
Initial learning rate | $1\times 10^{-3}$
Learning rate schedule |  $0.1$@$\left[100,200,250\right]$
Training / Testing | 70% / 30%
Figure 3: Heat map of the channels at each receive antenna for: (a) The true
channel at time $t+1$, i.e., $\mathbb{H}^{(t+1)}$, (b) the predicted channel
by the proposed deep learning model, i.e., $\widehat{\mathbb{H}}^{(t+1)}$, and
(c) the channel obtained by using sample and hold approach, i.e., channel at
the previous time step $t$, $\mathbb{H}^{(t)}$.

###  V-C Performance Evaluation Metric

We adopt two different metrics to evaluate the accuracy of the predicted
channels. One is for the predicted raw CSI, i.e., the 3-D channel matrix
$\widehat{\mathbb{H}}$ in the frequency and spatial domain. Another is for the
beamforming vector constructed based on the predicted raw channel.

Evaluating the predicted raw CSI: We use normalized MSE (NMSE) as the metric
to measure the prediction accuracy of the raw CSI, which is defined as

$\mathsf{NMSE}(\mathbb{H},\widehat{\mathbb{H}})=\frac{\|\mathbb{H}-\widehat{\mathbb{H}}\|_{F}^{2}}{\|\mathbb{H}\|_{F}^{2}}.$
(15)

It is worth mentioning that oftentimes, the decibel value of NMSE is used,
which can be converted from (15) through
$\mathsf{NMSE}_{\mathrm{dB}}(\mathbb{H},\widehat{\mathbb{H}})=10\log_{10}\mathsf{NMSE}(\mathbb{H},\widehat{\mathbb{H}})$.

Evaluating the predicted beamforming vector: As the sub-band beamforming
vectors are what will finally be fed back to the base station, they are more
closely related to the final system performance, e.g., achievable rate or
throughput. Therefore, we consider using cosine similarity as the metric to
evaluate their accuracy, which turns out to be more suitable than the NMSE.
The cosine similarity of two complex vectors $\mathbf{f}$ and
$\widehat{\mathbf{f}}$ can be defined as

$\rho(\mathbf{f},\widehat{\mathbf{f}})=\frac{|\mathbf{f}^{H}\widehat{\mathbf{f}}|}{\|\mathbf{f}\|_{2}\|\widehat{\mathbf{f}}\|_{2}}.$
(16)

As can be seen from the above definition, the cosine similarity between two
vectors are independent of their magnitudes. It essentially captures how close
they are in term of direction. Also, as both ground true beamforming vector
$\mathbf{f}$ and the beamforming vector obtained based on the predicted
channel, i.e., $\widehat{\mathbf{f}}$, are with unit norm, essentially we have
$\rho(\mathbf{f},\widehat{\mathbf{f}})=|\mathbf{f}^{H}\widehat{\mathbf{f}}|$.

###  V-D Baseline Solutions

We compare the performance of the proposed solution with the sample-and-hold
(SH) approach, and leave the comparison to other existing approaches for
future work.

Sample-and-Hold: As the name suggests, the SH approach stores the most recent
estimated channel samples and always sends them to the base station, which
will be used for DL transmission. Specifically, in our simulations, we assume
that the base station will design the DL transmission scheme based on
$\mathbb{H}^{(t)}$, while the actual downlink channel has changed to
$\mathbb{H}^{(t+1)}$. As can be seen, the SH approach essentially ignores the
inevitable transmission delay between the time point when the channel is
estimated and the time point when the actual DL transmission happens, or
assumes that period that the channel stays static is longer than the
processing delay. Therefore, the performance of SH approach is overly
dependent on how fast the channel is changing as well as the transmission
delay, i.e., the difference between $\mathbb{H}^{(t)}$ and
$\mathbb{H}^{(t+1)}$.

##  VI Simulation Results

In this section, we evaluate the performance of our proposed deep learning
based future channel prediction approach for both uni-speed and mixed-speed
datasets as described before.

###  VI-A Results for the Uni-Speed Dataset

Figure 4: The histogram of the cosine-similarity performance of the AI-based
approach and the sample-and-hold approach. Figure 5: The CDF of the cosine-
similarity performance illustrating the improvement brought by the residual
structure.

(a) 30-50 km/h

(b) 100-150 km/h

Figure 6: The CDF of the cosine-similarity performance of the proposed AI-
based approach and the sample and hold approach under the speed range of (a)
30-50 km/h and (b) 100-150 km/h.

We first consider a UE that has a constant speed of $30$ km/h. Fig. 3
illustrates the better performance of the deep learning model than sample and
hold approach by plotting the heat map of the channel at each receive antenna,
where each pixel represents the amplitude of the complex valued channel
element from each transmit antenna and at each resource block. As can be seen,
the neural network is able to capture most of the ”hot spots” of the future
channel. By contrast, due to the movement of the UE, the SH approach suffers
from the channel aging and produces significant difference with respect to the
ground true future channel.

In addition to the channel images, we further investigate the performance in
terms of the cosine similarity with the ground true layer one beamforming
vectors, which has direct influence on the achieved beamforming gain. As can
be seen from Fig. 4, the AI-based approach achieves quite excellent
performance, with most of the beamforming vectors calculated based on the
predicted channels having over $0.9$ cosine similarity with the ground true
layer one beamforming vectors. This forms direct contrast with the SH
approach, where the performance of its predicted beamforming vectors is quite
dispersive with non-negligible tails extending toward zero. And to be more
specific, the AI-based prediction achieves $25.89\%$ improvement over sample
and hold approach in terms of average cosine similarity with the true
beamforming vectors.

Furthermore, we want to understand how much of the improvement does the
residual structure bring to the system by learning the deeper patterns of the
channel. We remove the two ”Conv Res Blocks” shown in Fig. 2, leaving only the
two convolutional blocks, which is named as the 3D CNN model. As can be seen
from Fig. 5, the performance of the 3D CNN model is also quite excellent
compared to the sample and hold approach. However, by leveraging the powerful
residual structure, it allows the model to learn finer details of the channel
evolving pattern, which contributes another precious $3.6\%$ improvement.

###  VI-B Results for the Mixed-Speed Dataset

We further train and test the proposed deep learning model in two mixed-speed
datasets, that is, a mid-speed dataset of 30-50 km/h and a high-speed dataset
of 100-150 km/h. As can be seen from Fig. 6, the AI-based approach outperforms
the SH approach in both cases, with a $29\%$ improvement at the mid-speed and
a $14\%$ improvement at the high-speed. It is worth pointing out that the
considered range at the high-speed is even larger than the mid-speed, which
makes the channel prediction task even harder. Besides, we fix the number of
past observed channels used for the deep learning model to predict the future
channels in both cases. However, this parameter could be probably fine-tuned
to further enhance the performance of the AI-based approach, which is left as
future work.

##  VII Conclusions and Discussions

In this paper, we develop a deep learning based channel prediction framework
that proactively predicts the downlink channel state information based on the
past observed channel sequence. In its core, the model adopts a 3-D
convolutional neural network (CNN) based architecture to efficiently learn the
temporal, spatial and frequency correlations of downlink channel samples,
based on which accurate channel prediction can be performed. Simulation
results highlight the potential of the developed learning model in extracting
information and predicting future downlink channels directly from the observed
past channel sequence, which significantly improves the performance compared
to the sample-and-hold approach, and mitigates the impact of the dynamic
communication environment.

## References

  * [1] E. Larsson, O. Edfors, F. Tufvesson, and T. Marzetta, “Massive MIMO for next generation wireless systems,” _IEEE Communications Magazine_ , vol. 52, no. 2, pp. 186–195, Feb. 2014.
  * [2] K. T. Truong and R. W. Heath, “Effects of channel aging in massive MIMO systems,” _Journal of Communications and Networks_ , vol. 15, no. 4, pp. 338–351, Aug 2013.
  * [3] M. Alrabeiah and A. Alkhateeb, “Deep Learning for TDD and FDD Massive MIMO: Mapping Channels in Space and Frequency,” _arXiv e-prints_ , p. arXiv:1905.03761, May 2019.
  * [4] J. Zhao, H. Xie, F. Gao, W. Jia, S. Jin, and H. Lin, “Time Varying Channel Tracking With Spatial and Temporal BEM for Massive MIMO Systems,” _IEEE Transactions on Wireless Communications_ , vol. 17, no. 8, pp. 5653–5666, 2018.
  * [5] Y. Han, Q. Liu, C. Wen, M. Matthaiou, and X. Ma, “Tracking FDD Massive MIMO Downlink Channels by Exploiting Delay and Angular Reciprocity,” _IEEE Journal of Selected Topics in Signal Processing_ , vol. 13, no. 5, pp. 1062–1076, 2019.
  * [6] M. Arnold, S. Dörner, S. Cammerer, S. Yan, J. Hoydis, and S. ten Brink, “Enabling FDD Massive MIMO through Deep Learning-based Channel Prediction,” _arXiv e-prints_ , 2019.
  * [7] J. Yuan, H. Q. Ngo, and M. Matthaiou, “Machine Learning-Based Channel Estimation in Massive MIMO with Channel Aging,” in _2019 IEEE 20th International Workshop on Signal Processing Advances in Wireless Communications (SPAWC)_ , 2019, pp. 1–5.
  * [8] 3GPP, “Physical layer procedures for data (Release 15),” 2020, TS 38.214, Version 15.10.0.
  * [9] Y. Zhang and M. Alrabeiah and A. Alkhateeb, “Deep Learning for Massive MIMO With 1-Bit ADCs: When More Antennas Need Fewer Pilots,” _IEEE Wireless Communications Letters_ , vol. 9, no. 8, pp. 1273–1277, 2020.